Adiabatic Waves in Liquid-Vapor Systems

International Union of Theoretical and Applied Mechanics

G. E. A. Meier' P. A. Thompson (Eds.)

Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Gbttingen, 28.8. - 1. 9.1989

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong

Prof. Dr. Gerd E. A. Meier

Max-Plank-Institut fUr Stromungsforschung BunsenstraBe 10 H2 0-3400 Gottingen

Prof. Dr. Philip A. Thompson

Department of Mechanical Engineering Rensselaer Polytechnic Institute Troy, New York 12181 USA

ISBN-13:978-3-642-83589-6 001: 10.1007/978-3-642-83587-2

e-ISBN-13 :978-3-642-83587-2

Library of Congress Cataloging-in-Publication Data Adiabatic waves in liquid-vapor systems: IUTAM Symposium Gottingen, 28.8.-1.9.1989 G. E. A. Meier, P. A. Thompson, eds. (IUTAM symposium Gottingen, 1989) At head of title: International Union ofTheoretical and Applied Mechanics. Proceedings of the I UTAM Symposium on Adiabatic Waves in Liquid-vapor Systems.

ISBN-13:978-3-642-83589-6 (U.S.) 1. Hydrodynamics--Congresses. 2. Gas dynamics--Congresses. 3. Cavitation--Congresses. 4. Evaporation--Congresses. 5. Condensation--Congresses. I. Meier, Gerd E. A. (Gerd Emil Alexander) . II. Thompson, Philip A. 1II.Internationai Union ofTheoretical and Applied Mechanics. IV.IUTAM Symposium on Adiabatic Waves in Liquid-vapor Systems (1989 : Gottingen, Germany) V. Series. VI. Series: I UTAM Symposium. QC150.A35 1990 536'.44--dc20 90-9486

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© Springer-Verlag, Berlin Heidelberg 1990 Softcover reprint of the hardcover 1st edititon 1990

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In Memoriam Yakob Borisovich Zel'dovich

The Symposium on Adiabatic Waves in Liquid-Vapor Systems is dedicated to the memory of

Academician Ya. Zel'dovich, who died on December 2, 1987. His contributions to the field

of Physico Chemical Hydrodynamics have been numerous and fruitful. He had the intention

to participate in the Symposium. This was expressed in the letter reproduced below, which

was written in the summer of 1987. The decision to dedicate the Symposium to Zel'dovich

carries the consent and approval of all of the participants .

Prof. Dr. Philip A. Thompson & Dr. Gerd E.A. Meier IUTAhl .. Sympo~ium MPI fur Stromungsforschung Bunsenstrabe 10 H2 D-3400 Gottingen, FRG

Dear Professor Thompson,

Academician Ya.B.Zeldovich Institute of Chemical Physics of the USSR Academy of Sciences Moscow USSR

I congratulate you with finding an interesting topic for

an international conference.

As for me, I would be glad to take part and deliver a lec­

ture, tentatively on the history of shock wave structure theory,

with emphasis on new phase formation in general and in shocks.

I vlOUld propose also the Borissov and Uakorjakov brothers

from Uovossibirsk and also l'rolov, Dremin, Kuznotzov and Ji1ortov

from Iiloscow. Detailed description of their Works, their mailing

addresses and, perhaps other candidates will be sent to you

later, by separate cover.

Your truly

academician Zel 0 ich Ya..B.

General Information

International Scientific Committee

D.G. Crighton (UK) W. Fiszdon (Poland) V.K. Kedrinskii (USSR) A. Kluwick (Austria) G.E.A. Meier (FRG), Co-Chairman

Executive Committee

T.A. Kowalewski, Symposium Secretary G.E.A. Meier, Co-Chairman P.A. Thompson, Co-Chairman

S. Morioka (Japan) W. Schiehlen (FRG) B. Sturtevant (USA) P.A. Thompson (USA), Co-Chairman L. van Wijngaarden (Netherlands), ex officio

Local Organizing Committee

R. Breyhan E. Gossieaux T.A. Kowalewski U. Mader G.E.A. Meier D. Porsiel K. Resner C. Schwabe

Supporting Organizations and Companies

International Union of Theoretical and Applied Mechanics (IUTAM) U.S. Department of the Navy, Office of Naval Research European Office, London (UK) U.S. Army Research, Development & Standardization Group, London (UK) Deutsche Babcock Werke AG, Oberhausen (FRG) Krupp Atlas Elektronik GmbH, Bremen (FRG) Kreissparkasse Giittingen (FRG) Phywe Systeme GmbH, Giittingen (FRG)

Preface

The planning for the IUTAM Symposium on Adiabatic Waves in Liquid-Vapor Systems

began in May of 1986 in G5ttingen. The Symposium was held in August of 1989 in the

Max-Planck-Institut fUr Str5mungsforschung.

The invitations to participants suggested that the written papers concern Fast Adiabatic Phase

Changes in Fluids and Related Phenomena. Particular topics suggested were: Liquefaction

shockwaves and Shock splitting; Evaporation waves; Condensation in Laval nozzles and

turbines; Stability in multiphase shocks; Non-equilibrium and near-critical phenomena;

Nucleation in dynamic systems; Structure of transition layers; Acoustic phenomena in two­

phase systems and Cavitation waves. All of these topics should have been treated with

emphasis on physical results, new phenomena and theoretical models. Participants from

fourteen nations took part in the Symposium and presented papers which were within the

range of suggested topics.

The organization and execution of the Symposium was performed by the Max-Planck-Institut

fUr Str5mungsforschung in G5ttingen. In particular, the meeting has been promoted under

the leadership of Professor Dr. E.-A. MUller, who has for many years given his support for

international exchanges in science. The detailed work of organization up to and during the

Symposium was in large part due to Dr. T. Kowalewski, who served as Symposium Secretary.

Many other persons at the Max-Planck-Institut fUr Str5mungsforschung contributed to the

success of the Symposium. In particular, we would like to thank Professor Dr. H.-G. Wagner

and the members of the local committee including Emmanuel Gossieaux, Katja Resner, Dirk

Porsiel and Irmhild Meier. Both Renate Breyhan and Ursula Mader secretaries in the office

of Dr. G.E.A. Meier were extremely helpful.

Among those supporting this Symposium from the American side at Rensselaer Polytechnic

Institute, Esther Rendano, Hollis McEvilly, Jennifer Tolson, John Gulen, Hung-Jai Cho,

Susan McCahan, Professor Joseph Shepherd and Nancy Thompson performed valuable

services.

In this Symposium, the Scientific Committee formed a working group which ultimately made

the success of the Symposium possible. We are especially grateful for the many efforts of

these scientists, whose names are listed above. Professor Dr. L. van Wijngaarden formally

represented the IUT AM Headquarters. He was especially helpful in sorting out the many

papers which we received and making suggestions for the conduct of the Symposium. We are

grateful to him for this very useful assistance.

We are also grateful to the Deutsche Forschungsanstalt fUr Luft- und Raumfahrt (DLR,

Gottingen) for opening their facilities for a visit of the Symposium participants. The

cooperation of the Springer Verlag has been noteworthy in producing the Proceedings of the

Symposium in a timely manner. Very helpful financial support for the Symposium was

provided by the above listed institutions.

This scientific Symposium had also a lighter side. There was an Aus/lug to Duderstadt in the

Eichsfeld with inspection of the many half-timbered houses in the old city. There was also

an evening of music in the Symposium Hall, provided by a string quartet made up of

students from the University in Gottingen - the Georgia Quartet.

In recent years, a variety of rapid phase-change phenomena, often with surprising properties,

have been discovered. These phase changes are typically driven by pressure changes, rather

than heat transfer (an exception is the vapor explosion, sometimes called explosive boiling).

At the same time, new developments have modified our understanding of old fields such as

cavitation and boiling. We are dealing with those developments of the last fifteen years

which have not yet, until now, been the subject of an international meeting. The interest is

in both the physical phenomena themselves and their applications. This volume contains the

papers accepted for publication. As care has been taken to invite the competent active

workers in the field, we hope those papers contain the latest development in research on the

adiabatic waves in two-phase, liquid-vapor systems. We have been impressed with the quality

of the contributions and wish to thank the authors. Even so we wish to thank the participants

for their lively discussions during the Symposium. In total , thirty eight papers were prepared

for publication. Their usefulness can be judged from this book.

Gottingen, October 1989

d{vlJ Gerd E.A. Meier Phi lip A. Thompson

List of Participants

Prof. Dr.-Ing. J. Ballmann Lehr- und Forschungsgebiet Mechanik der Rheinisch-Westfalischen Technischen Hochschule Aachen Templergraben 64 5100 Aachen, FRG

Prof. Dr.-Ing. A. E. Beylich Technische Hochschule Aachen StoBwellenlabor Templergraben 55 5100 Aachen, FRG

Prof. Dr. K. Bier Institut fur Technische Thermodynamik und Kaltetechnik der Universitat Karlsruhe Postfach 6380 7500 Karlsruhe 1, FRG

Dr. Zbigniew Bilicki Institute for Fluid Machinery Polish Academy of Sciences ul. Gen. Fiszer 14 Gdansk Poland

Dipl.-Phys. Jurgen Bode Max-Planck-Institut fur Stromungsforschung BunsenstraBe 10 3400 Gottingen, FRG

Prof. Dr. A.A. Borisov Institute of Thermophysics of the Siberian Branch of the USSR Academy of Sciences 630090 Novosibirsk USSR

Dr. J.A. Boure Centre d'Etudes Nucleaires de Grenoble SETh/LEF, 85 x Avenue des Martyres 38041 Grenoble Cedex France

Dr. G. Brusdeylins Max-Planck-Institut fur Stromungsforschung BunsenstraBe 10 3400 Gottingen FRG

Ir. J. Buist university of Twente Laboratorium voor Warmteen stromingsleer Gebouw WB, Postbus 217 7500 AE Enschede The Netherlands

Dr. Humberto Chaves Max-Planck-Institut fur Stromungsforschung BunsenstraBe 10 3400 Gottingen FRG

Dr. N.N. Chernobayev Lavrentyev Inst. of Hydrodynamics Siberian Branch of the USSR Academy of Sciences 630090 Novosibirsk USSR

Dr. Can F. Delale Bogazici University Department of Mathematics P.K.2 Bebek 80815 Istanbul Turkey

Dr. Georg Dettleff Deutsche Forschungsanstalt fur Luft- und Raumfahrt e.V. DLR SM-ES BunsenstraBe 10 3400 Gottingen FRG

Prof. Dr. Nguyen van Diep Institute of Mechanics National Center for Scientific Researches Hanoi vietnam

Dr. A. Dinkelacker Max-Planck-Institut fur Stromungsforschung BunsenstraBe 10 3400 Gottingen, FRG

Dr. ir. M.E.H. van Dongen Eindhoven university of Technology Physics Dept. P.O. Box 513 5600 MB Eindhoven The Netherlands

Dr. phil. F. Ehrler Institut fur Technische Thermodynamik und Kaltetechnik der Universitat Karlsruhe Postfach 6380 7500 Karlsruhe 1, FRG

Prof. Dr. G.R. Fowles Washington state university Department of Physics Pullman, WA 99164 USA

Prof. Dr. David L. Frost McGill University Mechanical Engineering Dept. 817 Sherbrooke st. W. Montreal Quebec H3A-2K6 Canada

Prof. Dr. S. Fujikawa Inst. of High Speed Mech. Tohoku University Katahira 2-1-1 Sendai 980 Japan

Prof. Dr. Henri Gouin Dept. of Mathematics and Mechanics University of Aix-Marseille Faculty of Science and Technology Av. Escadrille Normandie­Niemen 13397 Marseille CEDEX 13 France

XI

Prof. Dr. rer. nat. H. Gronig StoBwellenlabor R W T H Templergraben 55 5100 Aachen FRG

Mr. A. Guha Whittle Laboratory Madingley Road Cambridge CB3 OHE England

Mr. S.C. Gulen Department of Mech. Eng. 12-1 Georgian Terr. Rensselaer Polytechnic Inst. Troy, NY 12180 USA

Dr. Harumi Hattori Department of Mathematics University of West Virginia Morgantown, WV 26506 USA

Dipl.-Phys. M. Herrmann Max-Planck-Institut fur Stromungsforschung BunsenstraBe 10 3400 Gottingen FRG

Dipl.-Phys. W.J. Hiller Max-Planck-Institut fur Stromungsforschung BunsenstraBe 10 3400 Gottingen FRG

Dr. Hiroyuki Hirahara saitama University Faculty of Engineering Dept. of Mechanical Engineering Shimo-okubo 255 Urawa 338 Japan

XII

Prof. Dr. V.K. Kedrinskii Member of IUTAM Scientific committee Lavrentiev Institute of Hydrodynamics Siberian Branch of the USSR Academy of Sciences Lavrentiev Ave. 15 630090 Novosibirsk USSR

Prof. Dr. Joseph Kestin Division of Engineering Brown University Providence, RI 02912 USA

Prof. Dr. A. Kluwick Member of IUTAM Scientific committee Techn. Universitat Wien Karlsplatz 13 1040 Wien Austria

Prof. Dr. Yasunori Kobayashi Inst. of Eng. Mech. University of Tsukuba 1-1-1 Tennodai Tsukuba, Ibaraki 305 Japan

Dr. Tomasz A. Kowalewski Max-Planck-Institut fur Stromungsforschung BunsenstraBe 10 3400 Gottingen FRG

Prof. Dr. Heinz Lang Max-Planck-Institut fUr Stromungsforschung BunsenstraBe 10 3400 Gottingen FRG

Prof. Dr. Kazuo Maeno Department of Industrial Mechanical Engineering Muroran Inst. of Technology 27-1 Mizumoto Muroran, Hokkaido 050 Japan

Dr. M. Maerefat Dept. of Mechanical Eng. Faculty of Engineering Kyoto University Kyoto Japan

Priv.-Doz. Dr. G.E.A. Meier Member of IUTAM Scientific committee Max-Planck-Institut fur Stromungsforschung BunsenstaBe 10 3400 Gottingen, FRG

Prof. Dr. K.A. M0rch Laboratory of Applied Physics Bldg. 307 Technical University of Denmark DK-2800 Lyngby Denmark

Prof. Dr. shigeki Morioka Member of IUTAM Scientific committee Dept. of Aeronautical Engineering Kyoto University Kyoto 606 Japan

Prof. Dr. E.-A. Muller Max-Planck-Institut fur Stromungsforschung BunsenstraBe 10 3400 Gottingen, FRG

Prof. Dr. T. Niimi Nagoya University Nagoya Japan

Prof. Dr. F. Obermeier Max-Planck-Institut fur Stromungsforschung BunsenstraBe 10 3400 Gottingen, FRG

Prof. Yoshimoto Onishi Dept. of Applied Mathematics and Physics Faculty of Engineering Tottori University Tottori 680 Japan

Dr. Bettina Paikert Universitat Essen Fachbereich 12 Maschinentechnik Schiitzenbahn 70 4300 Essen 1 FRG

Dr.-Ing. Franz Peters Universitat Essen Fachbereich 12 Maschinentechnik Schiitzenbahn 70 4300 Essen 1 FRG

Dr. K. Piech6r Warsaw university Dept. of Mathematics PKiN p. IX 00-901 Warszawa Poland

Prof. Dr. B.G. Pokusaev Institute of Thermophysics of the Siberian Branch of the USSR Academy of Sciences 630090 Novosibirsk USSR

Dr. N.A. Pribaturin Institute of Thermophysics of the Siberian Branch of the USSR Academy of Sciences 630090 Novosibirsk USSR

Priv.-Doz. Dr.-Ing. G. H. Schnerr Institut f. stremungslehre und Stremungsmaschinen Universitat Karlsruhe (TH) KaiserstraBe 12 7500 Karlsruhe, FRG

Priv.-Doz. Dr.-Ing. habil. Stefan U. Scheffel Universitat Kaiserslautern Mechanische Verfahrenstechnik und Stremungsmechanik Erwin-Schredinger-straBe 6750 Kaiserslautern FRG

XIII

Prof. Dr. J. Shepherd Department of Mech. Eng. Rensselaer Polytechnic Inst. Troy, NY 12180-3590 USA

Prof. V.P. Skripov Institute of Thermal Physics Ural Branch of the USSR Academy of Sciences pervomaiskaya Str. 91 GSP-169 Sverdlovsk,620219 USSR

Dr. Marshall Slemrod University of Wisconsin­Madison center for the Mathematical Sciences 610 Walnut Street Madison, WI 53705 USA

Ir. H.J. Smolders Eindhoven university of Technology Physics Dept. P.O. Box 513 5600 MB Eindhoven The Netherlands

Prof. Yoshio Sone Kyoto University Dept. of Aeronautical Eng. Honmachi, Yoshida sakyo-ku Kyoto 606 Japan

Dipl.-Phys. H. Stern Deutsche Babcock Werke AG Duisburger Str. 375 4200 Oberhaus en FRG

Prof. Dr. B. Sturtevant Member of IUTAM Scientific committee California Institute of Technology CALTECH 301-46 Pasadena, CA 91125 USA

XIV

Dr. hab. A. Szumowski Inst. of Aircraft Eng. and Appl. Mechanic Tech. University Warsaw ul. Nowowiejska 24 00-665 Warszawa Poland

Prof. Dr. Kazuyoshi Takayama Katahira 2-1-1 Institute of High Speed Mechanics Tohoku University Sendai, 980 Japan

Dr. V.M. Teshukov Lavrentyev Inst. of Hydrodynamics Siberian Branch of the USSR Academy of Sciences 630090 Novosibirsk USSR

Mr. Thomas Teske Max-Planck-Institut fur stromungsforschung BunsenstraBe10 3400 Gottingen FRG

Prof. Dr. Philip A. Thompson Member of IUTAM Scientific Committee Rensselaer Polytechnic Inst. Dept. of Mech. Eng. & Mech. Troy, N.Y. 12181 USA

Dr. Stephen C. Traugott Fluid Dynamics Program Room IIIOB National Science Foundation 1800 G. Street N.W. Washington DC 20550 USA

Dr. G.B. Truong Institute of Mechanics National Center for Scientific Researches Hanoi Vietnam

Prof. Dr. Leen van Wijn­gaarden Member of IUTAM Scientific committee Techn. Hogeschool Twente Postbus 217 7500 AE Enschede The Netherlands

Dr. J.B. Young Whittle Laboratory Madingley Road cambridge CB3 OHE England

Dr. Erwin Zauner Asea Brown-Boveri AG Funktionsbereich Forschung 5405 Baden-Dattwil Schweiz

Contents

I. Liquefaction shock waves and evaporation waves

Chairmen: V.P. Skripov, G.R. Fowles

Evaporation wave model for superheated liquids J.E. Shepherd, S. McCahan, J. Cho (Troy)

Evaporation waves in fluids of high molar specific heat H. Chaves, T. Kurschat, G.E.A. Meier (GOttingen)

An experimental study of evaporation waves in a superheated liquid L. G. Hill, B. Sturtevant (Pasadena)

Waves in reactive bubbly liquids A.E. Beylich, A. Giilhan (Aachen)

Chairmen: J. Shepherd, V.K. Kedrinskii

Determination of condensation parameter of vapours by using a shock-tube S. Fujikawa, M. Maerefat (Sendai/Kyoto)

Vapor condensation behind a shock wave propagating through vapor-liquid two-phase media Y. Kobayashi (Tsukuba)

Shock wave propagation in low temperature fluids and phase change phenomena K. Maeno (Muroran)

On an inviscid approach to phase transition problem H. Hattori (Morgantown)

Interaction of underwater shock wave with air bubbles K. Takayama, A. Abe (Sendai). K. Tanaka (Tsukuba)

II. Condensation in flow. boiling

Chairmen: B. Sturtevant, A.A. Borisov

Phase changes of a large-heat-capacity fluid in transcritical expansion flows E. Zauner, G.E.A. Meier (GOttingen)

Experimental investigation and computer analysis of spontaneous condensation in stationary nozzle flow of COZ-air mixtures K. Bier, F. Ehrler, M. Niekrawletz (Karlsruhe)

Spontaneous condensation in stationary nozzle flow of carbon dioxide in a wide range of density K. Bier, F. Ehrler, G. Theis (Karlsruhe)

An asymptotic predictive method for gas dynamics with nonequilibrium condensation C.F. Delale (Istanbul)

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13

25

39

49

59

69

79

91

103

113

129

143

XVI

Chairmen: K. Takayama, D.L. Frost

Stationary and moving normal shock waves in wet steam A. Guha, J.B. Young (Cambridge)

Numerical investigation of nitrogen condensation in 2-D transonic flows in cryogenic wind tunnels G. H. Schnerr, U. Dohrmann (Karlsruhe)

Explosive boiling: some experimental situations V.P.Skripov, O.A. Isaev (Sverdlovsk)

On the similarity character of an unsteady rarefaction wave in a gas-vapour mixture with condensation H.J. Smolders, E.M.J. Niessen, M.E.H. van Dongen (Eindhoven)

Chairmen: K. Bier, Y. Sone

Properties of kinematic waves in two-phase pipe flows J.A. Boure (Grenoble)

Growth of n-propanol droplets in argon studied by means of a shock tube expansion-compression process F. Peters, B. Paikert (Essen)

A discrete kinetic model resembling retrograde gases K. Piechor (Warszawa)

III. Non-equilibrium in dynamic systems. critical phenomena

Chairmen: K.A. March, M.E.H. van Dongen

159

171

181

197

207

217

227

Internal gravitational waves near thermodynamic critical point 239 A.A. Borisov, Al.A. Borisov, V.E. Nakoryakov (Novosibirsk)

Effect of thermodynamic disequilibrium on critical liquid-vapor flow conditions 247 Z. Bilicki, J. Kestin (Gdansk/Providence)

Wave propagation in flowing bubbly liquid 261 S. Morioka (Kyoto)

Stability of shock waves and general equations of state 271 V.M. Teshukov (Novosibirsk)

Rarefaction and liquefaction shock waves in regular and retrograde fluids 281 with near-critical end states S.C. Gulen, P.A. Thompson, H.J. Cho (Troy)

IV. Cavitation waves and evaporation waves

Chairmen: J. Kestin, S. Morioka

Strong evaporation from a plane condensed phase Y. Sone, H. Sugimoto (Kyoto)

293

Film boiling phenomena in liquid-vapour interfaces H. Gouin, H.H. Fliche (Marseille)

On the macroscopic boundary conditions at the interface for a vapour-gas mixture Y. Onishi (Tottori)

Remarks on the traveling wave theory of dynamic phase transitions M. Slemrod (Madison)

V. Acoustic phenomena in two-phase systems. cavitation

Chairmen: L. van Wijngaarden, A. Kluwick

Cavitation behind tension waves J. Bode, G.E.A. Meier, M. Rein (Gottingen)

Acoustics of travelling bubble cavitation J. Buist (Twente)

Modeling of shock-wave loading of liquid volumes N.N. Chernobaev (Novosibirsk)

Liquid-vapour phase change and sound attenuation H. Lang (Gottingen)

Nonstationary wave processes in boiling media V.E. Nakoryakov, B.G. Pokusaev, N.A. Pribaturin, S.I. Lezhnin, E.S. Vasserman (Novosibirsk)

VI. Vapor explosions

Chairmen: E. Zauner, H. Gouin

Explosion hydrodynamics: experiment and models V.K. Kedrinskii (Novosibirsk)

Vapor detonations in superheated fluids G.R. Fowles (Washington)

Propagation of a vapor explosion in a confined geometry D.L Frost, G., Ciccarelli, C. Zarafonitis (Montreal)

The development of cavity clusters in tensile stress fields K.A. M0rch (Lyngby)

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305

315

325

341

351

361

371

381

395

407

417

427

Liquefaction Shock Waves and Evaporation Waves

Evaporation Wave Model for Superheated Liquids J. E. SHEPHERD, S. McCAHAN, and Junhee CHO

Department of Mechanical Engineering Rensselaer Polytechnic Institute Troy, NY 12180-3590 USA

Summary

Experiments with rapid decompression of superheated liquids and droplets exploding near the superheat limit reveal the existence of steady evaporation waves. An idealized model for steady evaporation waves has been analyzed. A evaporation wave is treated as a jump or discontinuity between metastable liquid and an equilibrium vapor or liquid-vapor mixture.

Numerical solutions of the jump conditions have been obtained using Starling's equation of state to represent the thermodynamics of equilibrium and metastable states of hydrocarbon fluids. For simple fluids (small specific heat), only solutions with two-phase downstream states exist. Single-phase downstream states (complete evaporation waves) are predicted for complex fluids with a specific heat comparable to or greater than octane, given a sufficiently superheated initial state. Possible wave velocities range between zero and a maximum value determined by a Chapman-Jouguet condition.

This wave model is combined with a simple similarity description of liquid and vapor motion to predict the rates of steady spherical bubble growth in superheated liquids. The Chapman­Jouguet hypothesis is used to fix the evaporation rate and the results are compared with ob­servations in bubble column experiments.

Introduction

Metastable liquids can spontaneously and rapidly change phase under adiabatic condi­tions. Metastable liquid states (shown in Fig. 1) can be experimentally reached from equilibrium states through processes of adiabatic depressurization and/or isobaric heat­ing. The region of metastable liquid states is bounded by the equilibrium or saturation boundary on one side and the superheat limit on the other. The superheat limit is the temperature at which phase change is initiated by homogeneous nucleation; this tem­perature lies close to but below the absolute limit of thermodynamic stability, (Bp/ BV)T = 0, known as the spinodal.

Superheated liquids and associated rapid phase changes appear in a variety of phys­ical phenomena. A simple laboratory experiment is to heat a small droplet of fluid by immersing it in a column of denser fluid which is heated at the top and cooled at the bottom, a "bubble column". When the droplet temperature approaches the superheat limit, the droplet rapidly vaporizes (in less than 100 fJS for a 1 mm diameter hydrocar­bon droplet) with an audible pop. Detailed investigations (Shepherd and Sturtevant [1], Frost and Sturtevant [2]), have shown that under certain conditions (low ambient pressure, large superheat) the vaporization has an explosive character. In explosive

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium G6ttingen, Gennany, 1989· © Springer-Verlag Berlin Heidelberg 1990

4

boiling, the liquid is converted to a liquid-vapor mixture by a traveling wave which sweeps through the liquid at a velocity of 5-15 m/s. While this wave moves with a con­stant mean velocity, the surface is rough and unsteady, indicating violent instability and fragmentation processes occurring at the interface between liquid and liquid-vapor mixture.

300

r critical point

200 spinodal

metastable liquid region

100 saturation boundary

Figure 1. Pressure-volume diagram of a fluid showing the region of liquid metastable states bounded by the saturation line a and the spinodal (Op/OV)T = o.

Evaporation waves are also observed in other laboratory experiments such as rapid depressurization of liquid-filled containers. Experiments, reviewed by Hill and Sturte­vant [3] in this symposium, reveal that following the expansion wave created by depres­surization, a much slower (0.5-25 m/s) evaporation wave follows. The wavefront moves at a constant mean velocity, but is highly disturbed and produces fine liquid fragments in the downstream flow.

The examples quoted above involve very low-speed waves but higher velocity evap­oration waves may also exist. Molten metal-water interactions, often known as steam explosions, can result in quite violent interactions involving transient evaporation waves with velocities of up to 500 m/s. While the evidence for such high-speed waves is lim­ited, numerous industrial accidents and large-scale experiments (see the references in Frost et al. [4] of this symposium) have demonstrated the destructive nature of these events.

These experimental observations of high and low speed waves have naturally led to the suggestion that evaporation waves are analogous to combustion waves and both subsonic (deflagration) and supersonic (detonation) waves may exist in superheated liquids. Indeed, as we discuss below, this analogy is exact in that two such solution branches do exist for steady waves in superheated liquids. This analogy has further suggested the special role of solutions with sonic (relative to the wave) flow downstream;

5

these are the Chapman-Jouguet (CJ) solutions of combustion. Frost [5] has estimated sound speeds in the liquid-vapor mixtures downstream of the evaporation wave observed in bubble growth and finds that the CJ condition appears to be satisfied. Chaves [6] investigated the same hypothesis for evaporation waves created in depressurization experiments and also obtained good agreement with the CJ hypothesis. It is not possible to obtain a CJ solution for a high-speed wave in a pure liquid (see the discussion in Fowles [7], this symposium) but wave speeds between 300 and 500 m/s have been predicted for supersonic waves which satisfy the CJ condition in liquid metal/liquid water/vapor water mixtures (Board and Hall [8]). Note that the water downstream of the wave in these "thermal detonations" is in a single fluid, supercritical state.

Analysis of Planar Steady Waves

The experimental and theoretical evidence reviewed above indicates that there are at least two classes of evaporation waves. In a variety of situations, these waves appear to be steady in an average, macroscopic sense, although violently unstable at a microscopic level. This suggests that a steady wave analysis, identical to that used for waves in inviscid reacting flow, can be used to explore the admissible solutions for steady evaporation waves in superheated liquids.

Consider a control volume surrounding a portion of a steady evaporation wave (Fig. 2); superheated liquid (labeled state 1) flows in from the left, vapor or liquid­vapor mixture (labeled state 2) flows out to the right. We suppose that the control volume can be chosen so that even if the wave is curved, it can locally be considered planar and any possible flow divergence will be neglected. In this frame of reference, the wave is at rest and the fluid moves into the wave with velocity WI, density Ph pressure PI, enthalpy hI and entropy 81 and leaves with corresponding quantities at state 2. We assume that there is no storage within the control volume, the flow is effectively inviscid, and the control volume has been chosen so that energy transport by heat conduction is negligible.

(Liquid) (Vapor-Liquid or Vapor)

Evaporation Wave

Figure 2. Control volume surrounding an steady evaporation wave. Upstream states (1) are superheated liquid; downstream states (2) can be either equilibrium liquid-vapor mixtures, pure liquid, or pure vapor.

6

Under the restrictions outlined above, upstream (1) and downstream (2) states are related by the jump conditions across an interface:

PtWt = P2W 2 (1)

PI + PtW~ = P2 + P2W~ (2) w2 w2

ht + ---1. = h2 + ---1. (3) 2 2 82 ~ 8t (4)

which are identical to the conditions used to describe inviscid gasdynamic discontinu­ities. These relations are supplemented by the equations of state p(p, T), h(p, T); we have used Starling's equations for hydrocarbons and Kennan and Keyes equations for water. Equations and parameter values were taken from Reynolds [9] compilation. So­lutions for downstream states are obtained as a function of a single parameter such as wave speed or downstream pressure once the upstream state is specified. A variety of upstream states have been explored, these states are usually metastable liquid near the superheat limit. For a specified initial temperature and pressure, the initial enthalpy, density, and entropy are computed by extrapolating the equations into the metastable region. Inspection of property values estimated in this fashion show that reasonable results, i.e., smooth extrapolation of equilibrium properties, no singularities near the spinodal, are obtained.

The downstream fluid is considered to be an equilibrium state, which could be either liquid, a liquid-vapor mixture, or pure vapor depending on the solution branch and fluid type. For single-phase downstream states, the jump equations (1-3) must be solved numerically by an iterative procedure. This is most conveniently performed by combining equations 1-3 to obtain the Rankine-Hugoniot equation,

(5)

which yields the locus of possible downstream states as a pressure-volume (P2, V2) curve known as the Hugoniot or shock adiabat. The adiabat can be obtained by direct computation without iteration when the downstream state is a mixture state, since the final pressure and temperature must lie on saturation curve u. A complete adiabat curve for water with an initial state on the spinodal at Po = 2.89 MPa, To = 600 K is shown in Fig. 3.

The adiabat shown in Fig. 3 has a strong resemblance to the adiabats predicted for chemical explosives [10]. The similarities include the displacement of the locus of downstream states from the upstream state; the division of the adiabat in two branches; and the existence of a Chapman-Jouguet point on the lower solution branch. These features are generic to adiabats of systems making a transition from a metastable to an equilibrium state and involving a release of energy, either chemical or thermal. An important distinction is the existance of phase boundaries in the superheated liquid case, these can result in slope discontinuities or kinks in the adiabat. Such features can give rise to wave instabilities or shock splitting [11] when constructing simple wave solutions to initial value problems.

25.0

20.0

~ 15.0

6 A..

10.0

5.0

0.0 .001

I '"

i """"

. - . - metastable liquid shock adiabat

"'" ,~~~~~hYSiCal

I

• initial state

..........

.Q1

---'-'----~SOniCbranCh C.J. point-=------

.1 v (m3/kg)

Figure 3. Complete adiabat solution for water with a superheated liquid initial state starting on the spinodal at Po = 2.89 MPa, To = 600 K. The dashed branch marked nonphysical represents solutions to the adiabat which have imaginary steady wave velocities. The dot-dashed line from the initial state up to the saturation curve is the isentrope in the metastable liquid.

7

The lower (1'2 ::; PI) branch represents subsonic waves, for a given wave velocity o ::; WI ::; W max there are usually two solutions for the downstream state. Using combustion terminology, this is known as the de:f1agration branch of the adiabat. There is only one solution at the Cbapman-Jouguet point, where the wave has the maximum velocity and the downstream flow is sonic relative to the wave. In the case shown in Fig. 3, the CJ deflagration speed is 9.8 m/s. For solutions above the CJ point downstream flow is subsonic; below the CJ point, supersonic. The defl.agration branch terminates at a point where the entropy inequality, Eqn. 4, is violated. This occurs when the downstream pressure becomes sufficiently small.

The upper (P2 ~ PI) branch represents supersonic waves. As shown in Fig. 3, a kink occurs in the adiabat where it crosses the phase boundary, this results in two solutions for the downstream state for given wave velocity 00 ~ WI ~ Wmin. Using combustion terminology, this is known as the detonation branch of the adiabat. The solution passing through the kink has the minimum wave velocity (520 m/s) but unlike chemical explosives, this is not a CJ solution since the downstream flow is not sonic. States on the small portion of the adiabat accessible within the saturation region have supersonic downstream flow as discussed by Fowles [7]. No CJ point exists further up the adiabat from the kink since these states are simply a continuation of the shock adiabat in the metastable liquid.

8

Complete Evaporation Waves

For the example of superheated water, the subsonic branch of the adiabat terminates inside the mixture region so only liquid-vapor mixture downstream states are possible. Is it possible to obtain downstream states which are pure vapor, i.e., a complete evap­oration wave? The possibility of such waves is suggested by the existence of the inverse process, the complete liquefaction shock [12]; that is, shocks from a pure vapor to a pure liquid state. This process can only occur in a fluid of high molar specific heat, a retrograde fluid in which the saturation boundary on the vapor side has positive slope in T-s coordinates.

This is also true for the complete evaporation wave. A simple criterion for the molar specific heat needed for complete evaporation can be obtained by considering the limit of Eqns. 1-3 as WI approaches zero. If complete evaporation occurs for this case, then the entire subsonic portion of the adiabat will correspond to complete evaporation. This will be the case if the Jakob number is greater than unity:

Ja == Cp,I(TI - T,,) > 1 flh Jg - (6)

Using the simple rules of thumb for the superheat limit TI '" 0.9Te, the normal boiling point T" '" 0.6Te, and the latent heat flh Jg '" lORTe, this implies that Cp(T" )IR > 33 for complete evaporation of a liquid at the superheat limit and one atmosphere initial pressure. The simplest molecule that meets this criterion is octane, CSH1S'

2.5

2.0

1.5 '2 0...

6 0...

1.0

0.93 •

0.5 0.917 •

0.9108 •

0.0 .001 .01 .1

v (m3/kg)

Figure 4. Portion of subsonic adiabats for n-octane with upstream states on se­lected spinodal points and subsonic downstream states. Initial states are labeled by the reduced temperature Tr = T ITc. Downstream states are mixtures for initial points of Tr = 0.95 and 0.93; the adi­abat crosses the saturation curve for Tr = 0.917; the adiabat lies completely in the vapor region for Tr = 0.9108.

9

Numerical solution of the jump conditions for octane [13] verifies this simple es­timate. The subsonic-subsonic portion of the adiabats are shown in Fig. 4 for four upstream states on the spinodal. As the upstream pressure is decreased, the adia­bat moves toward and across the saturation boundary. At an initial pressure of one atmosphere (T = 0.91 Tc), the subsonic adiabat lies completely in the vapor state. Computations [13] with lower hydrocarbons in the alkane series, butane and pentane, show only mixture downstream states.

Bubble Growth Model

Subsonic evaporation waves can be combined with a simple similarity solution to the radial continuity and momentum equations to obtain [13] an idealized model for rapid bubble growth in superheated liquids. This model is based on the experimental observa­tions [1,2,5] that the bubble radial velocity and evaporative mass flux are approximately constant for the explosive boiling mode of evaporation near the superheat limit.

~ 0...

6 0...

16.0 -U

12.0

R

Compressed 8.0 Liquid

4.0 L-------'_-----''------1_---'_---'_---'-_---'-_---'

0.0 2.0 4.0

rlR 6.0 8.0

Pressure Wave

Undisturbed Fluid

Figure 5. Schematic of the postulated similarity flowfield for the steady growth of a liquid-vapor mixture bubble within a superheated liquid. Radial variation of pressure is shown for a bubble radial velocity of 147.1 mls in water superheated to the spinodal point of 600 K and 2.89 MPa. The bubble velocity corresponds to an evaporation wave velocity of 72.85 mis, slightly above the CJ velocity of 67.7 mls but below the maximum velocity of 78.5 m/s.

In our idealized model, shown in Fig. 5, a single spherical bubble of radius R(t) = Rt grows in a superheated liquid of infinite extent. A steadily-moving evaporation wave separates the liquid from a stagnant liquid-vapor mixture within the bubble. As shown in Fig. 5, the pressure within the bubble is much higher than the ambient pressure at large distances from the bubble but lower than liquid pressure just outside the bubble due to the pressure decrease across the evaporation wave. The pressure drops off rapidly with increasing radial distance from the bubble due to the divergence of the streamlines, i. e., the 1/r2 dependence ofthe fluid velocity. At a distance r I R = cl R ~ 10, where c is the metastable liquid sound speed, the pressure wave terminates in a weak shock front,

10

beyond which the liquid is undisturbed. This shock can be located and its' strength determined by the nonlinearization technique of Whitham [14].

The entire flow field outside the bubble has been constructed as a similarity solution, in which the similarity parameter is r It. Our solution is a simple extension of Taylor's original similarity solution [15] for the flow outside of a sphere (or nonevaporating bubble) expanding at a constant radial velocity. Taylor showed that for small radial Mach numbers, the acoustic approximation to the full Euler equations yielded a good approximation to the exact solution obtained by numerical methods. In the present' study, an alternative approximation of an almost inCompressible liquid (used in most bubble growth analyses) has been used. We have compared our approximate solutions (in the limit of vanishing evaporation mass flux) with Taylor's exact results and find good agreement for low (less than 0.2) interface Mach numbers.

Using the incompressible flow approximation for the radial motion outside the bub­ble, the velocity u can be derived from a potential function rjJ:

(7)

where the relative velocity U in the liquid just outside the bubble can be computed from the Gallilean transformation to the evaporation wave coordinates

(8)

in terms of the evaporation wave normal velocity Wl. Integration of the inviscid mo­mentum equation yields:

(9)

assuming isentropic flow in the liquid. These additional equations plus the jump conditions, Eqns. 1-4, are solved numeri­

cally for a given equation of state model and specified conditions for the liquid far from the bubble. A one-parameter family of solutions is obtained for a range of radial veloc­ities, 0 :5 R :5 Rm .. .,. The maximum velocity is limited by the entropy inequality, Eqn. 4. Solutions for the liquid at the bubble surface and the liquid-vapor mixture within the bubble are shown in Fig. 6 for the particular case of water at the spinodal, T = 600 K. Inertia of the liquid surrounding the bubble results in the pressure at the bubble surface increasing with increasing bubble velocity. As shown, the pressure increases almost (see inset) to the saturation pressure at the maximum velocity point B. For this example, the maximum bubble radial velocity is 150 m/s. For a special upstream state just below state B in Fig. 6, the downstream state for the bubble solution is the CJ point on the corresponding adiabat.

In order to compare the model predictions with experiment, a choice of bubble growth velocity must be made. As mentioned earlier, experimental evidence suggests that nature selects the velocity which yields the CJ state. No rigorous theoretical justi­fication of this choice is available and it must be considered a plausible but speculative hypothesis at this time.

Experiments with butane droplets [1] yielded estimates of 7 m/s for the equivalent evaporation wave velocity and an estimated density of 200 kg/m3 for the density of

11

the liquid-vapor mixture inside the bubble. Computations [13] using the measured superheat limit as the ambient condition, yields a CJ evaporation wave velocity of 19 m/s and a bubble density of 140 kg/m3 • The corresponding bubble radial velocity is predicted to be 56 m/s, much higher than the measured radial velocity of about 15 m/s. However, in the experiments only a fraction of the bubble surface was evaporating and it makes more sense to compare equivalent evaporation wave velocities rather than bubble radial velocities.

25.0

20.0 p[2t] 15.0

'2 v Q.,

6 B' Q.,

10.0

5.0

A

0.0 .001 .01 .1

v (m3/kg)

Figure 6. Solution of bubble growth model for water superheated to 600 K at 2.89 MPa (on spinodal). Curve AB is the locus of liquid states just upstream of the bubble (evaporation wave). CurveA'B' is the locus of states for the liquid-vapor mixture within the bubble. CJ indicates the Chapman-Jouguet point. The inset shows the distinction between the maximum velocity point B and the saturation curve (j.

This comparison, while not definitive, is certainly encouraging and suggests further exploration of the CJ hypothesis for bubble growth is worthwhile. Our hypothesis is an alternative to the ad hoc assumption of Nguyen et al. [16], who postulate a relationship between interfacial liquid velocity and the degree of superheat. Their assumption results in the prediction of downstream states and wave velocities that are significantly different from the present analysis. Application of the present model to other types of evaporation waves, such as the Hill and Sturtevant experiment [3], is in progress.

Acknowledgement

This research was supported by Lawrence Livermore National Laboratories.

12

References

1. Shepherd, J. E.; Sturtevant, B.: Rapid Evaporation at the Superheat Limit. J. Fluid Mech. 121 (1982) 379-402.

2. Frost, D.; Sturtevant, B.: Effects of ambient pressure on the instability of a liquid boiling explosively at the superheat limit. J. Heat Trans. 108 (1986) 418-424.

3. Hill, 1. G.; Sturtevant, B.: An experimental study of evaporation waves in a· superheated liquid. Presentation at the IUTAM Symposium on Adiabatic Waves in Liquid-Vapor Systems, Gottingen, 1989.

4. Frost, D. 1.; Ciccarelli, G.; Zarafonitis, C.: Propagation of a vapor explosion in a confined geometry. Presentation at the IUTAM Symposium on Adiabatic Waves in Liquid-Vapor Systems, Gottingen, 1989.

5. Frost, D. L.: Dynamics of explosive boiling of a droplet. Phys. Fluids 31 (1989), 2554-2561.

6. Chaves, H.: Phaseniibergiinge and Wellen bei der Entspannung von Fluiden hoher spezifischer Wiirme. Dissertation, Georg-August-Universitiit zu Gottingen 1983.

7. Fowles, G. R.: Vapor detonations in superheated liquids. Presentation at the IUTAM Symposium on Adiabatic Waves in Liquid-Vapor Systems, Gottingen, 1989.

8. S. J. Board, R. W. Hall and R. S. Hall "Detonations of Fuel Coolant Explosions" Nature 254 (1975) 319-321.

9. Reynolds, W. C.: Thermodynamic Properties in S1. Stanford University, 1979.

10. Thompson, P. A.: Compressible-Fluid Dynamics. McGraw-Hill (1972) 347-358.

11. Thompson, P. A.; Chaves, H.; Meier, G. E. A.; Kim, y'-G.; Speckmann, H.-D.: Wave splitting in a fluid of large heat capacity. J. Fluid Mech. 185 (1987) 385-414.

12. Detleff, G.; Thompson, P. A.; Meier, G.E.A.; Speckmann, H.-D.: An experimen­tal study of liquefaction shock waves. J. Fluid Mech. 95 (1979) 279-304.

13. Cho, Junhee: Evaporation waves in superheated liquids. MS Thesis, RPI, Troy, NY 1988.

14. Whitham, G. B.: "The propagation of spherical blast," Proc. Royal Soc. London 203 (1950) 571.

15. Taylor, G. 1.: "The air wave surrounding an expanding sphere," Proc. Royal Soc. London 186 (1946) 273.

16. Nguyen, V. T.; Furzeland, R. M.; Ijpelaar, M. J. M.: "Rapid Evaporation at the Superheat Limit," Int. J. Heat Mass Transfer 31 (1988) 1687-1700.

Evaporation Waves in Fluids of High Molar Specific Heat H. Chaves, T. Kurschat, G.E.A. Meier

Max-Planck-Institut fur stromungsforschung Bunsenstr. 10, D-3400 Gottingen, FRG

Summary In a fluid of high molar specific heat, i. e. composed of molecules with a large number of atoms, a complete phase transition can occur adiabatically because at high temperatures the energy stored in the large number of internal degrees of freedom of the molecules exceeds by a great margin the energy involved in the phase transition. Jet experiments were carried out to obverve the case of evaporation from an initially saturated state. The phenomena observed for the superheated liquid jets are partial and complete evaporation discontinuities with sonic outflow into an underexpanded supersonic jet and a new type of cylindrical recompression shock wave. A comparison of the experimental results is done with the theory for deflagration waves.

Introduction

Past experiments performed to observe the fast adiabatic

evaporation of fluids of a high molar specific heat [1], [2]

showed that it is indeed possible to completely evaporate

these sUbstances adiabatically, but not in a one dimensional

duct. The large change in specific volume inherent to the

phase transition and the very low sound speed of a two-phase

mixture cause choking and thus limit the mass and volume flow

rate through a one-dimensional duct. The value of the

vOlumetric flow rate is not high enough to permit complete

evaporation. The idea to study the evaporation of superheated

liquids in a free jet is a consequence of this argument,

because in jet flow there are no solid boundaries to limit the

expansion.

Experimental set-up

Figures 1 and 2 show schematically the experimental set-up.

Figure 1 gives a global picture of the apparatus. Most of the

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Giittingen, Germany, 1989 © Springer-Verlag Berlin Heidelberg 1990

14

testf luid

. ~/ massflow

.-----.-1 --,----,-- j 0 ~_1*:J===::;---- pressure reductor

pressure supp ly ---

bellows --------L-J_ -._---'---' N2 -- bottled nitrogen

va lve V1

/ vacuu m pump

fast action electromagnetic va lve

V2 _ f/\ t-- c>'Cl------+ - -------0 initial pressure

- V3

initial temperature

endpressure

1------- regulated heating nozzle - ------- foil

.l~I--+------ t hermocouple

vacuum chamber ~window

Figure 1: Experimental set-up

equipment is needed to assure the necessary initial conditions

for the experiment, Le. initial temperature To and pressure

Po of the liquid, back pressure Pe in the expansion chamber of

1 m3 in volume. These are also the parameters which were

varied in the course of the measurements. The initial

temperature could be varied between room temperature and 1800

C, the in ita I pressure between atmospheric and 2MPa, and the

expansion chamber pressure between atmospheric and 10Pa.

15

Measuring the displacement of the bellows the volume flow rate

could be measured. The bellows also inhibited the contact of

the pressurizing nitrogen with the liquid and thus avoiding

the dissolution of gas in the liquid.

Figure 2: Optical set-up of a differential interferometer

Figure 2 shows the differential interferometer used to make

video pictures of the jet flow. By extracting the Wollaston

prisms and the polarization filters from the set-up a

shadowgraph of the jet could be made. Both techniques used

together permit a clear picture of the two-phase flow. The

typical exposure times were around 10 microseconds.

Since the vapour volume production rate of the evaporation is

large a continuous experiment would mean that a

correspondingly high capacity pump would be needed to keep the

back pressure constant. Therefore the experiments were carried

out during a short opening time of an electrically actuated

injector valve, typically 10-40 ms. The flow is stationary

after a period of approximately 2ms. This was checked by

making pictures at different delay times after opening of the

injector. Even the lowest back pressure used (200 Pal did not

change significantly after twenty single injections because of

the large volume of the expansion chamber.

The nozzle itself is a convergent nozzle of 0.35 mm exit

diameter, a length to diameter ratio of 5.7 and a convergence

half-angle of 100 • These contructive details were done to

insure a quick and unperturbed expansion of the liquid from a

subcooled to a superheated state.

16

a) H = 0.5 b) H = 2.4

T o = 20 · C, Po = 2 bar, Pc = 500 mbar T o = 20·C, Po = 2 bar, P" = 100 mbar

c) H = 6.8 d) H = 370 T o = 20 · C, Po = 2 bar, P" = 35 mbar T o = 100 · C, Po = 15 bar, Pc = 10 mbar

e) H = 600 f) Po/Pe = 1600 T o = 120 · C, Po = 15 bar, Pc = 10 mbar T o = 175·C, Po = 15 bar, Pc = 10 mbar

Fig. 3 Jets of different appearance depending on the superheat of the liquid. H is the ratio of the vapour presssure corresponding to the initial temperature to the external pressure. H is only a relative measure of the superheat of the liquid, because the pressure within the supersonic region is lower than the external pressure.

Experimental results

The resul ts obtained range from the

thermodynamically stable subcooled liquid

mechanically

jet through

17

and

the

superheated liquid jet to the underexpanded supersonic gas jet

depending on the initial temperature and end pressure. If the

back pressure is higher than the vapour pressure of the liquid

jet then the jet is mechanicaly and thermodinamicaly stable

(Figure 3a). Because of the low compressibility of the liquid

the temperature decrease for the expansion of the liquid

within the nozzle is small. Therefore as long as the fluid

exiting from the nozzle is pure liquid, its temperature is

close to the reservoir temperature. wi th increasing ini tial

temperature or decreasing back pressure the vapour pressure of

the liquid jet can exceed the back pressure, then the jet is

superheated. with increasing superheat the jet first breaks up

at singular points where heterogeneous nucleation nuclei grow

to form large bubbles (Figure 3b). This phenomenon has been

studied already in the literature (e. g. Fuchs and Legge [3]).

At somewhat larger superheat levels the number of nucleation

nuclei increases so much that no singular bubble can be

discerned (Figure 3c). The jet breaks up after an intact

length given by the mean lifetime of the superheated liquid

state (see Skripov [4]) and the exit velocity. These jets are

usually refered to as atomizing jets although the atomizing

mechanism here is a thermodynamic one and not the mechanical

Kelvin-Helmoltz instability of the shear layer assumed

normally in the literature [5]. The jets are still subsonic,

but with decreasing back pressure and increasing initial

temperature the evaporation rate at the jet surface becomes so

high that the sonic limit is reached. The jet structure

changes radically (Figure 3d). The jet composed of a two-phase

fluid is surrounded by a supersonic flow region terminated by

a barrel shock, a quasi-cylindrical shock and a Mach disc. At

the intersection between two of these shocks triple points

(also called Mach reflection) are present. Both the barrel

shock and the Mach disc are well known in the dynamics of

underexpanded supersonic jets. The cylindrical shock though is

new and gives the decisive clue to the phenomena occuring in

the jet. At high superheat levels on a shadowgraph of the jet

18

it can be seen that the liquid is evaporating completely

within a very small region at the jet surface (Figure 3e). The

dimensions of this region are small in comparison to the

length scales of the global flow. The evaporation region can

be treated then as a discontinuity in the flow field. The

geometry of this region is a truncated quasi-cylindrical cone.

This fact together with the cylindrical shock and the fact

that the supersonic region is flattened in comparison to the

typical supersonic region of an underexpanded supersonic gas

jet (Figure 3f) leads to the conclusion that the flow is

mainly a cylindrical expansion from the surface of the liquid.

This type of supersonic fl~w is consequently terminated by a

more or less cylindrical shock surface. Because of the

similarity of supersonic flow the distance of a cylinder shock

normalized with the nozzle diameter is only a function of the

pressure ratio. In figure 4 the measured normalized distances

of the cylinder shock are drawn versus the pressure ratio. On

a double logarithmic plot a slope equals to one of this curve

corresponds to cylinder source flow, if the slope is equal to

60

m = 0.9

100 500 1000 4000

Fig. 4 Normalized distance of the cylinder shock

19

0.5 spherical flow would be present. Since the liquid jet is

not infinitely long the total flow field is a result of

cylindrical, of the spherical expansion flow resulting from

the tip of the liquid jet and of the effect of the wall. The

assumption of radial expansion from the liquid surface is

therefore reasonable.

When the initial pressure is lower than the vapour pressure at

the initial temperature only vapour flows through the nozzle

and one obtains a well - known underexpanded supersonic gas jet

(Figure 3f).

Model of the flow

Based on the cylindrical geometry (Figure 5) of the

evaporation discontinuity a model was formulated to describe

the changes of state of the fluid, shown in the pressure

volume plane in Figure 6. The liquid is expanded from the

initial state 0 isentropically in the nozzle to a state 1 at

the exit of the nozzle. The evaporation occurs within a

discontinuity or deflagration from state 1 to state PC. The

stability of this wave is given by the evaporation rate of the

liquid and by the fact that the outflow is sonic at a point

close or at the discontinuity itself. There is no crosssection

which can limit the mass flow rate in the given jet geometry

PO

nozzle

~""k quasi-cylindrical exp~

I I r -------evaporation -),(1 r discontinuity

P1

superheated liquid core

Fig. 5 Geometry of the evaporating jet

20

a,s

0,0

metastable V liquid isentrope

\ \ \ \

P1--t- .

\ - . /.

saturation boundary

- .

equilibrium R-H-Adiabat for state P1

...... PE R- H-Adiabat for state PS _./Rayleigh line

\ Rayleigh line -. ___ . Isentrope

\ \

2 10

4 10

\ 10 v/vc

Fig. 6 Changes of state in the pressure volume plane

and undoubtedly the flow is supersonic after the

discontinuity. A stability anal isis following Landau-Lifshitz

[6] shows that the mass flow rate for a deflagration with

subsonic inflow velocity and sonic or subsonic outflow

velocity has to be determined by a rate of reaction or in this

case by the evaporation rate. The state which the liquid

attains on exit from the nozzle is therefore given by the

evaporation rate, which in turn is a gas kinetic process at

the liquid surface . Unfortunately it is not easily possible to

measure the pressure within a nozzle of such small dimensions

wi thout disturbing the flow. Such a measurement would

determine the state 1. The mass flow rate is insensitive to

small changes of the pressure PI' because PI is much smaller

than Po and therefore also the square root of pressure

difference determining the mass flow rate. In order to obtain

a pressure value for the state 1 the observed similarity of

supersonic flow (from state C to state S in Figure 5) was

used. This closes the system of equations and was used to

determine iteratively the state 1. Assuming a cylindrical

21

source flow of the diameter of the nozzle with an estimated

state I, the cylindrical shock has to match the given end

pressure Pe at a given distance from the nozzle, so that by

succesive aproximations the state 1 was found.

x

0.9

0 .7

0.5

0.3 after evap . discontinuity after supersonic expansion final state

0.1~~~~~~~~~~~~~~~~~~~~~~~~~

20 40 60 80 100 120 140 160 180 initial temperature To rOC]

Fig. 7 Evaporated mass fraction

Figure 7 shows the mass fraction of evaporated liquid at the

different states Xc, XS and Xe of the flow field as a function

of the intial temperature. At an initial temperature of 60 0 C

the model predicts for the first time complete evaporation

(Xs=l) at the state upstream of the shock after the supersonic

expansion, and recondensation behind the shock (Xe less than

one), this can be seen on figure 3d. At a temperature of 1200

C complete evaporation at the liquid surface is predicted,

compare with figure 3e.

Globaly the qualitative behaviour of the flow is described by

the model. At high initial temperatures though the measured

mass flow rates are significantly lower than those predicted

by the model, (Figure 8). From the literature it is known that

22

70 11..._--&.._-...... -- __

--~~~-~-~-c--=-.---_______________ _

9U 60 -I -[m~2s] 50 1 -

1 -

40 1 •

1

30 ------- - - incompr . calc. 1

model calc. • exp. results

-To 0.9 Te

•

20 0 2 0 40 60 80 100 120 140 160 180

initial temperature To [DC]

Fig. 8 Measured and calculated mass flow rates

nucleation at temperatures greater than 0.9 times the critical

temperature is extremely fast and of an explosive character .

This causes evaporation to occur within the nozzle. The fluid

emanating from the nozzle is not pure superheated liquid but a

two-phase mixture fluid. The model is not strictly valid any

more, although the basic behaviour is still described.

Conclusions

-Complete evaporation of a fluid with high molar heat by

adiabatic expansion was verified by injecting a superheated

liquid jet into a low pressure chamber.

-At a high level of superheat the liquid evaporates completely

within a stable and stationary evaporation discontinuity at

the liquid surface. This expansion discontinuity is a defla­

gration with sonic outflow velocity (Chapman-Jouget condi­

tion). The evaporation rate at the surface is given by gas

23

kinetic processes and not by the flow field. This could permit

a study of the microscopic evaporation processes.

-At lower values of superheat the evaporation is not complete

within the deflagration, but the remaining liquid evaporates

completely in the supersonic expansion flow.

-The supersonic flow is terminated by a barrel shock, a cylin­

drical shock and a curved Mach disc with triple points at

their intersections (Mach reflexion). The cylindrical shock

results from the evaporation flow of the liquid being almost

perpendicular to the jet axis.

-Based on the similarity of supersonic cylindrical source flow

a model was formulated for the distance of the cylinder shock

from the nozzle. This value allows a calculation of the

thermodynamic states of before and after the evaporation

discontinuity. The measured mass flow rates agree well with

the calculated values.

-Above initial temperatures of 0.9 times the critical tempera-

ture the liquid

and choking is

is most likely nucleating within the nozzle

limiting the mass flow rates. This is not

allowed for in the model and thus explains the differences in

the measured and calculated mass flow rates.

References

1. Kurschat, T; Vollstandige adiabatische Verdampfung stark uberhitzter Flussigketsfreistrahlen, Ph. D Thesis, Mathematisch-Naturwissenschaftlichen Fachbereiche der Georg-August-Universitat zu Gottingen 1989, in print

2. Chaves, H.; Kowalewski, T.A.; Kurschat, T.; Meier, G.E.A.; Muller E.A.; Similarity in the behaviour of initially saturated or subcooled liquid jets discharging through a nozzle. Chemical Physics 126 (1988) 137-143

3. Fuchs, H.; Legge, H.; Flow of a water jet into vaccum. Acta Astronautica 6 (1979) 1213-1226

4. Skripov, V.P.; Metastable liquids. New York, Toronto: Wiley & Sons 1974

5. Reitz R.D.; Atomization and other breakup regimes of a liquid jet, Ph. D. Thesis , Department of Mechanical and Aerospace Engineering, Princeton University, 1978.

6. Landau, L.D.; Lifschitz E.M.; Lehrbuch der theoretischen Physik Band IV, Kapi tel IX und XIV. Berl in: Akademie verlag 1966

An Experimental Study of Evaporation Waves in a Superheated Liquid L. G. HILL and B. STURTEVANT

Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125

Summary

The dynamical behavior governing the propagation of evaporation waves in chlorinated fluorocarbons is studied in a constant-diameter vertical glass test cell which exhausts into a large, low-pressure reservoir. Care is taken to suppress heterogeneous nucleation within the liquid column. The test liquid is initially in equilibrium with its own vapor, sealed by a foil diaphragm. Upon diaphragm rupture, a series of expansion waves depressurizes the liquid to approximately the reservoir pressure, during which nucleation and subsequent rapid vaporization begin at the free surface. After an approximately 10 InS long start-up transient, a quasi-steady process develops during which the wavefront propagates into the stagnant liquid column at constant average velocity, generating a nonunifonn high-speed two-phase flow. The leading edge of the wavefront consists of smooth and rough bubbles with maximum diameters of order 1 mm and characteristic lifetimes of order 1 ms. High speed movies show that the nucleation rate is both spatially nonunifonn and temporally nonsteady, which leads to significant unsteadiness in the propagation of the wave. Fragmentation of the liquid into fine droplets occurs primarily as the result of the violent break-up of the leading-edge bubbles coincident with explosive bursts of aerosol, which occur in the region extending about 1 em downstream of the leading edge bubble layer. These two processes appear to be mutually interactive. Three distinct modes of flow initiation are observed depending on the liquid superheat. Moreover, a self-initiation threshold is observed, below which waves do not occur. We observe that waves can propagate at slightly lower superheats if they are started artificially. However, an absolute threshold for wave propagation exists which is related to the nonsteady processes alluded to above.

1. Introduction

It is well known that highly superheated liquids can release their stored thennal energy through rapid, and often violent, evaporation. The mechanisms by which rapid evaporation proceeds are

complex; they include interaction between evaporation (which gives rise to forces on the liquid) and

the resultant liquid motion (specifically, fragmentation which enhances evaporation). Consequently,

the range and nature of such interactions are not well understood. In some cases, rapid vaporization

proceeds via a wavefront propagating through the superheated liquid and producing a high-speed

two-phase flow downstream. This phenomenon offers a unique well-controlled environment for the

study of the mechanisms of rapid evaporation.

Evaporation waves were first proposed by Bennett et al. [1] to explain anomalies in exploding

wire experiments. The first direct observation of an evaporation wave known to the authors was

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Gottingen, Germany, 1989 © Springer-Verlag Berlin Heidelberg 1990

26

made by Friz [2]. In water that was rapidly depressurized, Friz observed many bubbles throughout the entire length of the liquid column "immediately" after the arrival of the initial (acoustic) expansion wave. An "acceleration front" several centimeters thick then propagated into the

expanding bubbly liquid. Thompson et at. [3] observed similar behavior in a highly retrograde fluorocarbon (C6 F14) when the initial temperature was sufficiently close to the critical temperature.

In such cases the initial acoustic wave was observed to cause homogeneous nucleation in the liquid

column; a relatively slow evaporation wave then followed.

Evaporation waves have also been observed without any nucleation occurring in the upstream liquid. In particular, Grolmes and Fauske [4] conducted rapid depressurization experiments using a

variety of liquids (refrigerants, alcohol, and water). They observed waves in all the liquids when the superheat exceeded a threshold value which depended on the liquid and the container. Similar

behavior was observed on a smaller scale by Shepherd and Sturtevant [5], and by Frost and

Sturtevant [6]. In those experiments, a bubble column was used to heat millimeter-scale

hydrocarbon droplets to the superheat limit; whereupon, a single vapor bubble nucleated near the

droplet boundary. Evaporation then propagated across the drop. For lower superheats the evaporating surface remained smooth and the evaporation was relatively slow. For higher superheats

the surface appeared rough and generated a jet of aerosol, and the vaporization was explosive.

Only the studies of Sturtevant et at. have addressed the specific physical mechanisms by which liquid fragmentation occurs. The purpose of the present study is to explore this issue further (also in

waves without upstream nucleation) using a rapid depressurization facility. The experimental methods are outlined in the following section, and the results for the quasi-steady and start-up regimes are presented in sections 3 and 4, respectively. Results concerning an absolute threshold for

wave propagation are presented in section 5. Finally, our experimental measurements are summarized in section 6.

2. Experimental

The experimental facility and instrumentation are shown in Fig. 1. The flow facility consists of a

vertical 2.5 cm diameter, 15 cm long Pyrex glass test cell mounted beneath a low-pressure reservoir of volume 0.27 m3. The test cell is sealed by an aluminum foil diaphragm, which is ruptured by

pneumatically driven knife blades. The test cell is surrounded by a liquid-filled jacket with flat

windows. This serves two purposes: to eliminate the cylindrical lens effect of the test liquid (by chOOSing a jacket liquid with a refractive index similar to that of the test liquid) and to provide

temperature control (by circulating the jacket liquid through a heat exchanger).

The test cell base and exit pressures are measured with piezoelectric transducers. The exit

transducer is thermally insulated from the cold evaporating flow by a plug of neoprene rubber

cemented to the transduccr face. Nucleation from the base transducer is suppressed using a coating

of gelatin, or alternatively, a liquid layer of Refrigerant 113. The evaporation wave dynamics are

photographed using a 16 mm high speed motion picture camera at framing rates up to 6000 fps and,

for higher temporal and spatial resolution, 35 mm still photographs. The illumination for the still

photographs is by spark flash of 1 lls duration. For the motion pictures four different views are

studied: side, bottom, top 45° oblique, and bottom 45° oblique. For the still photographs two

views-side and bottom-are taken simultaneously.

27

I---~;;;;;;~== --Nitrogen

Low Pressure Reservoir -----.1 Pneumatic Cylinder

Test Cell With Liquid Jacket

Bose Pressure Transducer Thermal Insulotion /

Fig. 1. Schematic drawing of the evaporation wave facility.

The test liquids are Refrigerants 12 (C02F2, R12) and 114 (C2C12F4, Rl14). R12 is a normal

liquid; Rl14 is retrograde. At 20°C the densities of saturated liquid R12 and Rl14 are 1329 kg/m3

and 1472 kg/m3, respectively (Reynolds, [7]). Particulate impurities are removed by distilling the

test liquid directly within the test cell. The nominal test temperature is 20°C in all cases. The low­

pressure reservoir is tilled with air at pressures from 0 to 1 bar.

3. Quasi-Steady Propagation

After an initiation phase of order 10 ms duration, a quasi-steady condition is reached in which,

on time scales large compared to 1 ms, the average speed of the evaporation wave V w and the

average base pressure Pbase and exit pressure Pexit reach nearly constant values. However on time

scales of order 1 ms or less, the mechanisms by which two-phase flow downstream of the wave is

generated are nonuniform and nonsteady, so the wave speed and base pressure are nonsteady. Fig. 2

shows side and bottom views of an example which we take as the representative one, namely R12

exhausting into an evacuated reservoir. The quasi-steady behavior for other conditions is

qualitatively similar. The two views are simultaneous still photographs taken 25 ms after diaphragm

burst. The free surface was initially at the 8 cm mark. The side view is back lit, so the upstream

liquid region at the bottom appears light and the highly light-scattering two-phase flow above it

appears dark. The wave propagates at a speed of 0.63 mis, and the two-phase flow travels upward at

35 m/s.

The side view of Fig. 2 suggests that the flow can be divided into several regions. The upstream

liquid region is uniformly superheated, essentially stagnant liquid. The wavefront is of order 1 em

thickness within which liquid fragmentation and flow acceleration take place. The wavefront is

composed of two subregions. The formation of bubbles at the leading edge bubble layer, readily

apparent in Fig. 2, is the first step in the fragmentation process. Further fragmentation occurs during

the violent, nonsteady disintegration of the leading edge bubbles into explosively expanding and

translating clouds of aerosol, which we call 'bursts'. The fragmentation/acceleration region, which

28

Developed two-phase flow region

Side View

__ .... -Fragmentation/acceleration region

----l.eaclingedge bubble layer

----Upstream liquid region

Bottom View ------ Leading edge bubble layer

Fig. 2. Side and bottom views of quasi-steady evaporation wave. Liquid: R12; T1iq = 20oe; Pres = 0; Large ruler divisions: cms; t = 25 ms from diaphragm burst; Initial liquid height 8.0 cms; Run #: SPR4

extends about 1 cm downstream of the leading edge bubble layer, is evidently the ensemble of

individual bursts of various strengths. Downstream of the fragmentation/acceleration region the two-phase flow travels at a nearly constant speed, indicating that the evaporation rate is

comparatively small. Accordingly, we call this region the developedjlow region.

During the quasi-steady phase (about 100 ms) the base and exit pressures are nearly constant

(Fig. 3). Indeed, the pressure in these experimentS is much more constant than in many so-called

blowdown experiments, in which nucleation and bubble growth typically occur in the bulk of the

liquid (see, e.g., Winters and Merte [8]). The difference between Pbase and Pexil is the 'wave amplitude', in this case 0.53 bar. In the steady state the upstream pressure is approximately maintained by the thrust of rapid vaporization. The fact that the exit pressure is greater than the

reservoir pressure (namely, 0) indicates that the flow is choked. The instantaneous pressure

fluctuates with frequency components between 0 and about 2 kHz; the strongest peaks are below 250

Hz. The rms pressure fluctuations are approximately 3% of the mean pressure for both Pbase and

Pexil'

29

7 . 0

6 . 0

L 5.0 IU .0 4.0 Q) 3 . 0 L :J Ul 2.0 Ul Q) Base Pressure L 1.0 0...

0 Exit Pressure

-1. 0 -1 0 1 2 3 4 5 6 7 8 9 10

Time (msec)

Fig. 3. Base and exit pressure traces. Liquid: R12; T/iq = 20°C; Pres = 0 bar; Run # : MPR 25.

3.1 Wavefront Region

The leading edge of the wavefront is primarily composed of smooth bubbles with diameters

ranging from less than 100 j.lII1 (approximately the smallest bubble that can be distinguished in the still photographs) to a few mm. Features on the leading edge interface that are smaller than about

100 11m (down 'to the resolution of the film, about 10 j.lII1) appear rough. The characteristic features

of the bottom view of Fig. 2 occur uniformly over the cross-sectional area, suggesting that the walls

do not play a causative role in the propagation of evaporation waves. Even on the scale of the

diameter of the test cell the wavefront is convoluted and often oblique, indicating that the stabilizing

effects of gravity and surface tension are relatively weak. As a consequence, the leading edge

bubble layer often appears thicker than it really is. However, when it is nearly planar and its plane

is parallel to the line of sight, as in Fig. 2, it is apparent that the leading edge bubble layer is only

about one bubble thick.

The upper (downstream) portions of the leading edge bubbles are obscured by the

fragmentation/acceleration region, so we cannot determine by direct observation whether they are

capped by a liquid film (characteristic of non-evaporating bubbles at a free surface) for some or all

of their observed lifetime, or whether they are open. Fig. 4a compares the growth rate of 7 such

bubbles to classical thermal bubble growth theory (e.g., Prosperetti & Plesset [9]). Samples are

selected from bubbles that can be tracked from nucleation, and whose growth is not hidden or

visibly impaired by neighboring bubbles. The measured growth rates fall below the classical value

(solid line), which is not surprising since evaporation occurs primarily from the upstream side. The

growth rate for a hypothetical capped bubble in which only half the surface is evaporating is also

plotted (dashed line), and the actual growth rate is found to lie about midway between this and the

classical value.

Fig. 4b shows a lifetime histogram for 100 samples taken from bottom view motion pictures.

The most probable lifetime tlife is about 1 ms, but lifetimes up to about 5 times this value were

30

E 1 . 2 20 E

1 . 0 100 Samples

(J1 . 8 ::J

' M

;., 15 0 c

'0 .6 co L

QJ . 4 rl

QJ ::J 10 0-QJ L

LL 5 .0 .2 .0 ::J 0 m o

0 . 4 . 8 1.2 1 . 6 o 1 . 0 2 . 0 3 . 0 4 . 0 5.0 SQRT (T ime. ms) Bubble lifetime (msec)

a) b)

Fig. 4. Growth rates (a) and lifetime distribution (b) of selected bubble samples. Liquid: R12, Pres =

O. Run #: MPR 25

observed. The average nucleation rate can be estimated from the ratio of bubble number density per

unit area (obtained from bottom view still photographs) to the characteristic bubble lifetime given by

Fig 4b. Since the spatial resolution of the.film is finite, the estimate yields a lower bound. For R12

exhausting into vacuum the nucleation rate per unit area of the leading edge is estimated to be at least 106 s-lcm-2.

The evaporation occurring at the wavefront draws heat from the upstream liquid. Since

convection in the upstream liquid is small, heat transfer in it occurs primarily by diffusion, and the

time-averaged thermal boundary layer thickness is

(1)

where 1C is the thermal diffusivity of the liquid and Vw is the average wave speed (see Frost [10]).

For R12 exhausting into vacuum 8s == 0.1 1J.1ll, which is much smaller than any scales visible in our photographs. Because of unsteadiness and spatial nonuniformities, this expression does not predict

the instantaneous thermal boundary layer at a point, but suggests that on the average only small

surface perturbations are necessary to bring highly superheated liquid to the surface. The nonsteady

thermal boundary layer thickness for an individual bubble is approximately given by [9]

8 =0.35~~ lIS 11fa PI

(2)

where 11 is a (constant) correction factor which reconciles the observed growth rate of individual

bubbles in Fig. 4a with classical theory, R(t) = bubble radius, fa == cp[Tliq -Tsat(P)]IL is the

Jakob number and Pv/pi is the ratio of vapor and saturated liquid densities. For a typical bubble

diameter of 1 mm, and using 11 = 0.75 from Fig. 4a, 8ns == 3 Ilm.

An expression for the wavefront velocity can be obtained using the information in Fig. 4, by

modeling its propagation as a sequence of growing thermal bubbles, each new bubble growing for

the time tlife on the upstream surface of the former:

31

[ ] 112 []1I2 12 PI K Vw = - 11-Ja - .

7t Pv tli!e (3)

Using tlife = 1 ms and 11 = 0.75, Eq. (3) yields Vw = 0.85 mIs, compared to the measured value of

0.63 m/s. This agreement is reasonable considering that the model does not account for the apparent

statistical nature of the wavefront processes. In particular, it suggests that the limited number of

sample bubbles may represent the majority.

The fragmentation/acceleration region is a dark, fuzzy region approximately I em thick within

which the leading edge bubbles are fragmented into droplets. Significant unsteadiness can be seen

within this region in the high speed motion pictures, which takes the form of explosively expanding,

and often translating, clouds of aerosol, or 'bursts.' Bursts are detectable because they are regions of

relatively high droplet concentration which appear darker than their surroundings. Bursts occur just

downstream of the leading edge region; they are usually several mm in extent and their expansion

and translation rates are of the same order as the developed two-phase flow downstream. Visual

estimates of bursting frequency (for all identifiable bursts) compare favorably with the frequency of

the strongest pressure oscillations in the liquid: a few hundred Hz. The fragmentation/acceleration

region appears quite dark in the motion pictures (less so in still photos). This, combined with the

fact that the view is a superposition of many high-speed events, obscures the details in this region.

3.2 Two-Phase Flow Region

Downstream of the fragmentation/acceleration region the developed flow is a high-density

aerosol (density about 25 kg/m3 for Rl2 exhausting into vacuum). On the whole, this region is

significantly more transparent than the fragmentation/acceleration region. Clearing of the flow

occurs in the fragmentation/acceleration region and for several centimeters downstream. The aerosol

is spatially nonuniform and contains dark streak-like structures reminiscent of those observed by

Anilkumar [11]. These structures travel at approximately constant speed, 35 mls for Rl2 exhausting

into vacuum. Since the streaks emerge directly from the fragmentation/acceleration region, it is

likely that they are material from bursts. Even when the wavefront is substantially inclined, the flow

leaves normal to it and is turned in the axial direction by the wall upon which it impinges. In this

case, the smaller flow structures merge to form larger continuous streamers near this wall, while the

flow near the opposite wall is primarily vapor. When the wavefront is curved, the structures tend to

concentrate where the curvature is concave· up, forming streamers, and spread out where it is convex

up. A thin, rough liquid layer of order 100 J.1Ill thickness begins climbing the wall when the wave

starts, and proceeds at a constant velocity of a few rn/s, which is much slower than the flow through

the center.

Although the streaks do not accelerate noticeably in the developed flow region (suggesting that

the evaporation rate is slow), the fact that the flow visually clears in the first few em downstream

indicates that the two-phase mixture may not be in eqUilibrium. Still photographs show many liquid

droplets to be in the 10-100 Jlm diameter range. For our test cell and two-phase flow speeds,

thermal equilibrium is not expected to be achieved by the time the droplets exit the container unless

the characteristic droplet size is less than about 1 J.1Ill, the diameter for which the thermal boundary

layer reaches the center of the drop when the drop reaches the exit

32

Thompson et at. [3] have suggested on the basis of their evaporation wave experiments that the Chapman-Jouget (Ch-J) condition (sonic outflow in the reference frame of the wavefront) which

applies to classical detonation also applies to evaporation waves. In our experiments, a choked exit

is essentially the same as the Ch-J condition, since the wave motion relative to the exit is negligible

compared to the two-phase flow velocity. We observe waves both with choked and unchoked flows

(see Table 1); hence, subsonic outflow is also possible, and the Ch-J condition is not necessary for

the propagation of evaporation waves.

4. Start-Up

4.i Depressurization/rom initial Condition

As in the previous section, we use R12 exhausting into vacuum as the primary example. All

quoted times are given with respect to the sharp initial drop in the exit pressure unless otherwise

noted. Prior to diaphragm rupture the test cell is typically half-filled with liquid in equilibrium with

its overlying vapor. Diaphragm rupture, which takes less than 1 ms, generates an expansion wave in

the vapor. During the start-up transient the pressure in the test chamber falls to a value close to the

reservoir pressure, and then, after rapid vaporization initiates, both the base and exit pressure rise to

the quasi-steady levels (Fig. 3). The source of the compressive precursor and the subsequent

oscillations superimposed on the decreasing signal recorded by the base transducer in Fig. 3 are not

understood. An analogous negative precursor preceding shock waves in liquid Rll was noted by

Tepper [l3], who suggested interaction with the tube walls as one explanation. Motion pictures for

R12 exhausting into vacuum show that rapid vaporization initiates at timesyarying from 1 to 3 ms

i.e., when the pressure is very close to the reservoir pressure (Fig. 3). The small increase of pressure

to the steady-state value at about t = 4 ms in Fig. 3 is observed even in the absence of rapid

vaporization.

4.2 Start-Up Modes

In contrast to the steady-state behavior, there are significant qualitative differences in start-up

behavior depending on the test liquid and reservoir pressure. Three different modes of flashing

initiation are observed, which, for the range of our experimental conditions, can be ordered

according to the liquid superheat based on the reservoir pressure (i.e., the Jakob number). Representative examples of the three modes, taken from high speed motion picture frames, are

presented in Fig. 5.

At the highest superheats (Mode 1), individual nucleation sites rapidly initiate at random spots

on the liquid free-surface and at the glass/liquid contact line. Flashing spreads to the remaining

surface in less than one frame of the motion pictures (150 J.l.s). At intennediate superheats (Mode 2),

nucleation begins only at many sites on the glass/liquid contact line. Flashing then spreads radially

inward toward the center, for the example of Fig. 5 at 10 mis. At low superheats (Mode 3),

nucleation begins at one or more sites on the glass/liquid contact line and propagates across the

surface, for this example at 1 mis.

33

Mode # 1 2 3 Liquid R12 R1l4 R12

Pres (bar) <0.01 <0.01 0.97 (Near threshold)

0.86* 0.77* 0.35

Run #

Time (ms)

2.90; 1.15; 20.8

3.05; 1.55; 26.0

3.20; 1.95; 31.2

3.35; 2.35; 36.4

Fig. 5. Representative examples of the three observed modes of flashing initiation. Tliq = 20 °e. View from below at 45°. Time relative to diaphragm burst. Mode 1 and 2 examples: first picture corresponds to nucleation onset, in Mode 3 example to wave transition.

4.3 Self-Initiation Threshold

The condition for which nucleation at the lowest superheats ceases (which evidently depends on

factors such as surface roughness and cleanliness) approximately corresponds to a threshold below

which waves will not self-initiate. fa for both liquids at this condition is about 0.35. Just above

this threshold (as in the Mode 3 example of Fig. 5) the rapid spread of flashing across the surface

does not immediately follow the onset of nucleation. Rather, bubbles grow and break in the

34

immediate vicinity of the initial site until a cluster composed of several bubbles is formed. At some

time (17.8 ms after the appearance of the first nucleation site for the example of Fig. 5) the bubble cluster bursts, driving a cloud of aerosol rapidly upward and triggering the spread of rapid flashing.

Below the self-initiation threshold the free-surface remains quiescent. The mass transfer rate,

limited by convective heat transfer from the warmer liquid below, is about three orders of magnitude

less than just above threshold. Our observations in this regime are quite similar to those of Peterson et al. [13].

4.4 Transition to Quasi-Steady Propagation

The development of quasi-steady propagation is similar for each of the three modes of initiation,

with the exception of Mode 3 near the self-initiation threshold. Generally, the initial wavefront

scales are smaller (compare Mode 1 start-up, Fig. 5, with Fig 2) and the initial wave speed is

significantly higher than the corresponding quasi-steady values. For example, the initial wave speed

for R12 exhausting into vacuum is about 2 mis, compared to 0.63 mls for quasi-steady propagation.

Bursting on smaller temporal and spatial scales than for quasi-steady propagation can be seen in the

bottom view during start-up. On the other hand, near the self-start threshold the initial wave

velocity is less than in the quasi-steady state and, as seen in the Mode 3 example of Fig. 5, it already

has the appearance of the quasi-steady wavefront.

5. Absolute threshold

Conditions near the self-start threshold produce waves that appear quite vigorous once initiated.

Therefore, experiments were conducted to determine if waves could propagate under conditions below the self-start threshold if started artificially, a procedure we call 'jump starting'. R12 is

placed above the denser, less volatile Rl14 under conditions for which R12 self-starts but R114 does

not. We find that Rl14 can be jump started over a modest range of reservoir pressures: the self-start

threshold for R114 at 20°C is about 1/3 bar, and waves can be successfully jump started for reservoir

pressures up to 1/2 bar. The latter pressure is an absolute threshold which reflects the wave's ability

to maintain rather than to start itself. At reservoir pressures between the two thresholds, waves can

also be initiated by other forms of external perturbation. In fact, most of the R114 runs exhausting

into 1/3 bar were initiated by a small bit of falling diaphragm striking the surface.

The wave amplitude approaches zero as the absolute threshold is approached, but the wave speed

does not. Wave breakdown at the absolute threshold is illustrated in Fig. 6. Rl14 in the test cell

below the 6.5 em mark is jump-started by R12 exhausting into 1/2 bar reservoir pressure. While the

wave propagates from 6.5 cm to 4.5 cm bursting is sporadic, and at the 4.5 cm mark it stops.

Immediately following the last burst, wave propagation ceases. Nucleation still occurs, but at a

greatly reduced rate, and those bubbles which do nucleate grow large and form a froth. Slow

bubbling occurs for several hundred ms; but, since cold liquid is no longer carried out of the

container, the bulk of the liquid cools. Eventually all frothing ceases, at which point the liquid level

has receded little from the time of the final burst. The dramatic decrease in nucleation rate and the

increase in bubble size following the final burst indicate that nucleation and bursting are interactive

processes.

35

t=O SOms lOOms lS0ms

Fig. 6. Sequence of photographs illustrating behavior of wave breakdown at the absolute threshold. Time measured relative to the start of the final burst. Liquid: Rl14, Tliq = 20°C, Pres = 0.5 bar. Run#: MPR72

Table 1. Summary of Experimental Measurements

Property Symbol Unit Refrigerant 12 Refrigerant 114

Initial Test Cell Pressure Po bar 5.67 1.82

Reservoir Pressure Pres bar 0 1/2 1 0 1/3 1/2 Motion Picture Run Number MPR - 25 26 31 55 57 72

Results for Quasi-Steady Propagation

Exit Condition: ChokediUnchoked

- - c u u c u u

Average Base Pressure Pbase bar 0.78 0.95 1.01 0.37 0.48 0.50

(liquid)

Average Exit Pressure Pexil bar 0.27 0.50 1.01 0.15 0.33 0.50

(2-phase flow)

Average Wave Speed Vw mls 0.63 0.57 0.36 0.32 0.27 0.21

(from motion pictures)

Average 2-Phase Flow Speed V 2$ mls 35 35 5 25 15 5

(from motion pictures)a

Average 2-Phase Flow Speed (from pressure traces)b V2$ mls 31 - - 26 - -

Average mass flux p/vw m kg s 1 m 2 837 758 478 471 397 309

Average mixture density m/V2$ - kg/m3 25 20 95 20 25 60

Jakob Numberc Jabase 0.37 0.36 0.35 0.30 0.27 0.26

(based on base pressure) -

Results for Start-Up Phase

Observed Start-Up Mode - - 1 3d 3 2 3d _e

Number (see Fig. 6)

Jakob Numberc Jares 0.8(/ 0.46 0.35 0.771 0.34 0.26

(based on reservoir pressure) -

a : Visual speed of structures away from wall. Accuracy is believed to be about +1- 10%. b: Obtainable in choked cases only. c: cp = specific heat [14]; L = latent heat [7] d: Near self-initiation threshold. e : Near absolute threshold. I: Tsal evaluated at 0.01 bar.

36

6. Summary of Experimental Results

Measurements for the six experimental conditions are summarized in Table 1. For both of the

test liquids only the lowest pressure cases (Pres = 0) are choked. In these cases the exit speed can be inferred from the time of flight of the evaporation wave reflected from the test cell bottom. In

general, for a given liquid the wave speed, two-phase flow speed, mass flux, two-phase flow mixture

density and wave amplitude increase with Jakob number, while the base and exit pressure

individually decrease.

The lowest superheat cases for each liquid have in common negligible wave amplitudes, along

with dramatically smaller V 2<1>' and somewhat smaller V w. P2cj> is significantly larger than for the

higher superheats, indicating a corresponding decrease in evaporation rate. Since the corresponding

Jakob number (and hence the energy available for evaporation) decreases only slightly, it can be

inferred that evaporation wave breakdown has more to do with a decrease of fragmentation

efficiency (which is supported by the visual observations discussed in section 5) than with any

energetic considerations. Since Rl2 exhausting into 1 bar (near the self-start threshold) behaves

qualitatively like R114 exhausting into 1/2 bar (near the absolute threshold), the two thresholds may

be quite close together for R12.

Acknowledgments

This work was supported by the National Science Foundation under research grant #EAR-8512724. L. G. H. was supported in part by a fellowship from Caltech's Program in Advanced Technologies, sponsored by Aerojet General, General Motors, and TRW, and by a fellowship from the ARCS foundation.

References

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2. Friz, G.; Coolant ejection studies with analogy cxperiments, Proc. Conf. on Safety of Fast Reactors, ANL-7l20, p. 890 (1965)

3. Thompson, P.A.; Chaves, H; Meier, G.E.A.; Kim, y'-G.; and Speckmann, H.-D.; Wave splitting in a fluid oflarge heat capacity, J. Fluid Mech., Vol. 185, pp. 385-414 (1987)

4. Grolmes M.A.; and Fauske, H.K.; Axial propagation of free surface boiling into superheated liquids in vertical tubes, Proc. Fifth Intl. Heat Transfer Conf., Vol. 4, pp. 30-34 (1974)

5. Shepherd, J.E.; and Sturtevant, B.; Rapid evaporation at the superheat limit, J. Fluid Mechanics, Vol. 121, pp. 379-402 (1982)

6. Frost, D.; and Sturtevant, B.; Effects of ambient pressure on the instability of a liquid boiling explosively at the superheat limit, J. Heat Trans., Vol. 108, pp. 418-424 (1986)

7. Reynolds, W.C.; Thermodynamic properties in S.l., Dept. of Mech. Eng., Stanford Univ. (1979)

8. Winters, W.S.; and Merte, H.; Experiments and nonequilibrium analysis of pipe blowdown, Nuc. Sci. Eng., Vol. 69, pp. 411-429 (1979)

9. Prosperetti, A.; and Plesset, P.; Vapour-bubble growth in a superheated liquid, J. Fluid Mech., Vol. 85, Part 2, pp. 349-368 (1978)

37

10. Frost, D.; Effects of ambient pressure on the instability of a liquid boiling explosively at the superheat limit, Ph.D. Thesis, Caltech, p. 95 (1985)

11. Anilkumar, A.V.; Experimental studies of high-speed dense dusty gases, Ph.D. Thesis, Caltech (1989)

12. Tepper, W.; Experimental investigation of the propagation of shock waves in bubbly liquid­vapour-mixtures, Proc. 14th 1m. Symp. on Shock Tubes and Waves, edited by R.D. Archer and B.E. Milton (1983)

13. Peterson, R.J.; Grewal, S.S.; and EI-Wakil, M.M.; Investigations of liquid flashing and evaporation due to sudden depressurization, Int. Jour. Heat Mass Transfer, Vol. 27, No.2, pp. 301-309 (1984)

14. DuPont Corp., Freon fluorocarbons: properties and applications, Publication G-l

Waves in Reactive Bubbly Liquids

ALFRED E. BEYLICH and ALI GtiLHAN

StoBwellenlabor, Technische Hochschule Aachen

0-5100 Aachen, Federal Republic of Germany

Summary Shock tube experiments on wave propagation in a two-phase mixture, consisting of glycerine as matrix and bubbles with an Ar+(2H2+02) mixture as disperse phase, indicate that, above a certain threshold pressure of the initial shock wave, reaction starts inside the bubble, causing the separation of a precursor with a sol iton 1 ike structure. A theoretical model is developed in which the bubble dynamics including bubble interaction and the combustion process are considered. Numerical results, using the full nonlinear governing equations, show good agreement with the experimentally obtained profile.

1.Introduction

In connection with the handling of reactive liquids containung en­

trained bubbles or inert liquids with reactive bubbles the basic

questions are of interest, up to which pressure thresholds explo­

sions can be excluded, what are the mechanisms of initiation of

detonation waves, and under which conditions stable waves are propa­

gated. Previous experiments [1] on bubble chains have shown that

bubble - shock wave interaction leads to an explosion of a reactive

gas mixture in the bubble; the pressure pulses due to the explosion

in turn initiate further explosions: a chain reaction is started.

Further studies in nonreactive liquids with bubbles containing reac­

tive gas mixtures, H20 with (C 2H2+2.502),[2], and glycerine with

(noble gas+(2H2+02»,[3], as well as fuels with 02 - bubbles [4]

clearly indicate that, beyond a minimum pressure step Ap of the

incoming shock wave, a precursor wave is produced. For 70%Ar+

(2H2+02) one finds APmin= 8.5 bar, and the limit goes up to 13.5 bar

for 30%Ar [3]. The Mach number Mo of this precursor is independent

of Ap and in good approximation also independent of the void frac­

tion <!I o and the bubble radius Ro; one finds Mo 0/. 5.0 - 5.5 which is

about twice the value of the following shock wave. From the study of

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Gottingen, Germany, 1989 © Springer-Verlag Berlin Heidelberg 1990

40

wave phenomena in inert bubbly systems [51 one may obtain the follo­

wing picture of the start-up of the precursor: When in the first

peak of the oscillating structure of the shock profile conditions

are reached which allow initiation of the reaction, this peak will

be strongly amplified and accelerated in upstream direction. It

separates like a single soliton wave using up all chemical energy.

The present work is an attempt to model theoretically the phenomenon

of ~his precursor-soliton-detonation wave; it is an extension of

previous work on waves of moderate strengths [61. With respect to

the experiments that have been performed [71 it is important to note

that quantitative information an wave structure was obtained for the

first time by using methods of superposition of profiles.

2.Experimental Results

The two-phase mixture consisting of diluted (85%) glycerine and of

bubbles with radii of Ro= 1.4 and 1.8 cm, respectively, containing a

gas mixture of 70%Ar+30%(2H2 +0 2 ), was investigated in a vertical

shock tube [2,3,81. Pressure gauges, equally spaced along the two­

phase part of the shock tube, were used to record the pressure

profiles which were stored in transient recorders. To reduce scat­

ter, profiles from repeated shots were superimposed.

Some initial investigations on shock wave - single bubble interac­

tion were performed using optical techniques. When the gas in the

bubble is compressed due to the passage of the shock wave in the

Fig.1. Reactive gas bubbles with emitting pressure waves. Left: He, right: Ar, with 30%(2H 2 +0 2 ). Llp/po CJ. 8.2, Ro = 1.86 rom, Llt = 70 vs.

liquid, at a certain point, the reaction starts creating a wave

inside the bubble which is reflected several times at the bubble

wall emitting spherical waves into the 1 iquid. These waves can be

41

seen in Fig.1 and, knowing the bubble size, allow to get an estimate

of the speed of propagation inside the bubble. In general, lumines­

cence can be observed even if there are no chemical reactions,

however, when the reaction takes place, the light emission is in­

creased by an order of magnitude; the width of the light pu 1 se is

typically 1-2 vs. When the pressure step of the initial shock wave propagating into

the two - phase mixture is gradually increased, a detonation wave is

initiated above a certain threshold. For the 30%(28 2 +°2) mixture

this lower limit is at ~p ) 8 bar. This detonation wave separates

600 ,us / DiY

600 ,us / DiY

600 ,us / DiY

600 ,us / DiY Fig.2. Pressure records. Uper gauge (#1) is in the gas part of the shock tube; #2 at 285 mm, #3 at 815 mm, and #4 at 1345 mm below phase boundary. 30%(282 +0 2 )+Ar, Ro = 1.4 mm, Ql o = .35%.

from the initiating shock wave and moves at a considerably higher

speed. Using the Mach number related to the speed of sound in the

mixture, c o 2="tPo/(QfQlo), (with "t being the ratio of specific heats,

42

Po the initial pressure, Pf the liquid density, and ~o the initial

void fraction), we find typically Mo= 5.0-5.5 • This is approximate­

ly twice the speed of the shock wave. The detonation wave is a very

stable system and it has a soliton-like structure. In Fig.2, a

typical system of pressure profiles gives an impression of the

evolution of the waves and the separation process. Unfortunately,

due to the enormous scatter in the pressure, quantitative informa­

tion on the wave profile cannot be obtained from a single pressure

recording. Therefore, up to 50 profiles were superimposed in order

to achieve a certain degree of smoothness in the profiles.

3.Theoretical Model

The development of the model for the wave phenomena is based on a

hierarchy of length scales [6]: On the microscale this is the bubble

radius Ro' on the mezoscale it is the cell radius Ao' and on the

macroscale we deal with a characteristic wave length Lo. We make the

assumption that the bubble distribution is monodisperse, that bub­

bles keep their spherical shape and do not fragment. The void frac­

tion ~ is then related to the bubble number density n and the radius

R by

(1)

or ~ n R 3 - = -f;-J ; ~o = (Ro fAo )3, ~o no 0

(2)

where index 0 refers to the initial state. It can be shown [6] that

( 3)

For the one-dimensional unsteady problem the averaged equations of

conservation of number, of mass, and the momentum equation are (with

index g for gas and f for liquid)

a a -<n> + -<nu > 0, (4) at ax g

a a -<P> + -<pu> 0, (5) at ax

a a a -<pu> + _<pu2 > - -<p> (6) at ax ax

with

<0> = ~<Og> + (l-~)<Of> ~ (l-~)Ofo'

<ou> = ~<OgUg> + (1-~)<Ofuf> ~ (1-~)Ofo<u>.

since

43

(7)

(8)

(9)

We shall nondimensional ize (4-6) by using the reference time to 2= 2 ". ... ,..

Ro Ofo/(3lo Po) and the lenght Ao' thus. t = t/to' x = x/Ao' u = ~oAo/to' P = p/po. Furthermore. we introduce a new variable

We obtain from (5)

and from (6)

~ = - ~P>x + £uxx + O(~o) • loco

where an eddy viscosity has been introduced in the form

with

and

a _a _ -(e-<u» • ax ax

Co 2 = 1/(3~J/3 being the speed of sound. Combining (11) and (12) yields

(10)

(11)

(12)

.. 2 Co ..

ntt = - --<P>xx + entxx + O(~o)· (13) lo

For a steady wave moving at constant speed V we perform a Galilean

transformation ~ = x + vt. and find

o. (14)

where Mo = V/co is the Mach number.

The connection between the gas state and the liquid state in the

cell is provided by an averaged form of the Rayleigh - Plesset

equation [6] which contains also bubble interaction terms

44

(15)

Using (3,10), we finally obtain the governing equation

(16)

The state inside the Bubble

Inspection of the experimental results guides the modeling of the

state inside the bubble. We consider several steps, see Fig.3: From

* Po to Pg the compression is adiabatic, then follows an isochoric

combustion leading to the state Pg + ,Tg +. On the new adiabate 3, a

further compression to the maximum pressure Pgmaxmay take place and,

finally, the products will be expanded to Po' Since at 1)", the gas

temperature will still be high, there is little condensation up to

this point, especially due to the buffering effect of the inert gas

component. The process will be finished by a slow isobaric cooling

and condensation to the end point 1)e' In this first approach, ef­

fects like radiation, heat conduction, and compressibility are neg­

lected.

P

19 Pg

p~ g

hr ___ pmax g

1}min 1}" -1}

Fig.3. state changes inside the bubble. Gas pressure p~ as function of void fraction 1)=~/~o' (1) adiabatic compression, r~) isochoric combustion, (3) adiabatic expansion, (4) isobaric cooling and conde­nsation.

Using a stoichiometric mixture of H2 and 02 and with xl as mole

fraction of the inert gas component, we have

+ Tg

(1-x1)6EO + [(1-X1)cv2+(1-x1)cv3/2+x1cv1] (T-To )

(1-xl)cv 4 + x1cv1 (17)

45

+ * Pg = t.P+Pg (18 )

where t.Eo is the internal energy of the reaction process, cvi are

the mean specific heats (l=inert gas, 2=H2 , 3=02' 4=H20), Ci is the

mass fraction and mi molecule mass. The gas pressure in (16) can now

be replaced by

Pg = a/Tlno, (19)

with

a 1, e = 1 for adiabate 1, (20)

and

a (21)

Using (14,16,19) and with the Galilean transformation, a fundamental

equation 1n Tl can be formulated

(22)

with

a 1, e = 1, q = 1 before reaction,

- +/~ *~1 I I Pg Pg , e = ~3 ~o' q = ~o ~m after reaction,

where

(23)

and

Tl m = a1/ll. (24)

For small amplitudes (22) can be expanded in ~=Tl-1 , and one obtains

a Korteweg-deVries equation which can be integrated to yield a

relation between the Mach number and the maximum amplitude (Pmax> l +1 2 M = 1 + __ o __ [(p > -11 + O(~ ).

o 6~ max o

(25)

However, this type of expansion cannot be used for the present case

where large amplitudes are to be expected; as a consequence, (22)

cannot be integrated, and the relation corresponding to (25) has to

be obtained by numerical techniques.

To start the numerical integration, we have to study the behaviour

near the equilibrium points. We use the direction field Tl~ = 6 ,6~.

From linearization at the equilibrium points, characteristic roots

are found (viscosity terms are neglected here)

46

1/(~~V) = ±[(1+«o~o1/3)~~7/3(M62 _ q1/3)]1/2,

thus for ~ -+ -~

~~ = 1, q = 1; -+ (M62 - 1) > 0, -+ node.

For ~ -+ +~

~~ > 1, q < 1; -+ (M 2 0

_ q1/3) > 0, -+ node.

4.Results and Discussion

(26)

Equation (22) was integrated numerically in the 8-~ field after

transformation to polar coordinates. From previous numerical studies

[3], the point of ignition for a single bubble was obtained; in the * case of an Ar concentration of 70% we used T ITo = 4.0 • One has to

raise the question why a limiting Mach number of about 5 exists. If

one increases the Mach number Mo when integrating (22), one obtains

a decreasing minimum of ~ which again corresponds to a maximum in

8

o o

, , "

'~

.05 .1 -----Tlmin

Fig.4. Mach number Mo as function of minimum void fraction ~;n for different values of internal energy ~o. ~o =.4%, Ro=1.4 nun, 'il1~ Ar.

the pressure <p> according to (14). In Fig.4, Mo is plotted versus

~min for 25, 50, and 100% 8Eo • For Mo > 7, the maximum pressure

inside the bubble will be more than 1000 bar; it is reasonable to

expect that real gas effects, compressibility of the liquid, losses

due to acoustic and electromagnetic radiation, and heat losses will

lead to a finite Mach number. To show the influence of energy losses

~Eo was varied. The behaviour in the 8-1j1 phase diagram is shown in

Fig.5 for Mo= 5.0. Combustion takes place at IjI*=~*-l, and the curve

ends at the new equilibrium point 8~=0, IjI~= 81/~1 - 1. The pressure

<p>-l as function of time is plotted in Fig.6; comparison with the

experimental profile, which had been obtained from superposition of

50 single profiles, shows good agreement with respect to amplitude

47

and half-width. But even for 50 profiles the scatter is still consi­

derable and can be only reduced by further averaging •

. 5

THETA

- 1 - .5 .5

-.25

- .5

60

50

20

10

PSI

.05

Fig.5. 8-W phase diagram for Mo=5.0,

~0=.4%, Ro =1.4 rom, 50% ~Eo (with ~Eo=

2.406.106 Joule). 70% Ar + 30%(2H2+02)

in glycerine. Ww = .496.

. 1 . 15 .2 .25 .3 __ U(msl

Fig.6. Sol iton-detonation wave for Mo = 5.0, ~ = .4%, Ro = 1.4 rom, 70%Ar + 30%(2H2+02). Full curve: numerical resul~s, data points from superposition of ~O profiles [3].

48

References

1. Hasegawa,T. and Fujiwara,T.: Detonation in oxyhydrogen bubbled

liquid. 19th Symp. on Combustion. The Combustion Inst.1982, 675.

2. Sychev.A.I: Detonation waves in a liquid-gas bubble system.

Fizika Goreniya i Vzryva 21 (1985)103-110.

3. GUlhan,A.: StoBwellen in Fltissigkeiten mit inerten und reaktiven Blasen. Dissertation, Technische Hochschule Aachen, 1989.

4. Pinaev,A.V. and sychev,A.I.: Structure and properties of detona­

tion in a liquid-gas bubble system. Fizika Goreniya i Vzryva 22

(1986) 109-118.

5. Beylich.A.E.: Shock tube studies in bubbly liquids. In: Shock

Tubes & Waves. Proc. 16th Int. Symp. on Shock Tubes & Waves,

H.Gronig(ed), VCH Verlagsges.mbH, Weinheim, FRG (1988)101-115.

6. Beylich,A.E. and Gtilhan,A.: On the structure of nonlinear waves

in liquids with gas bubbles. To be published.

7. Gtilhan,A. and Beylich,A.E.: Detonation wave phenomena in bubbled

liquid. Proc. 17th Int. Symp. on Shock Tubes & Waves, July 1989,

Bethlehem, USA, to be published.

8. Kusnetsov,V.V., Nakoryakov,V.E., Pokusaev,B.G., and Shreiber,

loR.: Propagation of perturbations in a gas-l iquid mixture. J.

Fluid Mech. 85 (1978) 85-96.

Determination of Condensation Parameter of Va pours by Using a Shock-Tube

ABSTRACT

S.Fujikawa and M.Maerefat*

Insti tute of Fluid Science Tohoku University, Sendai 980, Japan

*Department of Mechanical Engineering Kyoto University, Kyoto 606, Japan

The condensation parameter (c.p.) and the condensation coefficient (c.c., sticking probability) of vapours consisting of polyatomic mole­cules are determined by using a shock-tube. The c.p. and the c.c. for three kinds of vapour are deduced from the comparison between experiment and gasdynamical theory with molecular gasdynamical boundary conditions. It is found that the experimental values of c.p. and c.c. of these vapours are significantly less than those for the complete cap­ture of molecules on their condensed surfaces. The transition state theory suggests that these small values are due to that the rotational degrees of freedom of molecules in the liquid state are hindered in com­parison with those of molecules in the gas state.

I. INTRODUCTION

Vapour condensation is a fundamental and important problem of

thermo-fluiddynamics in a sense that it must be treated from true

molecular aspects such as collisions between molecules at the

vapour-liquid interface but internal motions of the molecules. It also

involves a problem of the physics of liquid state. Many important

contributions to condensation problem have been made in the field of

molecular gasdynamics (for example. Pao ell. Sone and Onishi [2l.

Sone and Onishi [3l. Labuntsov and Kryukov [4l. Fiszdon [5l. etc.).

These works can be characterized as taking account of collisions

between evaporating molecules and condensing molecules by making use

of the great variety of solution methods of the kinetic equations.

Of particular interest for the present paper are the condensation

parameter (c.p.) and the condensation coefficient (c.c.) of vapours

consisting of polya tomic molecules; The meaning of c. p. and c. c. will

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems

50

be given in the next section. Numerous experiments on c.c. have been performed, however, they seem to have failed to determine reliable

values of it except a few examples. The reason may be attributed to the lack of reasonable understandings of condensation phenomena and

to the lack of microscopic condensation theories on the levels of atoms

and molecules. Recently, Fujikawa and Maerefat [6] have treated; in a purely theoretical way, the c.p. and the c.c. by using the

transition state theory on the basis of statistical mechanics of gases and liquids.

In the present paper, the c.p. and the c.c. of methanol, water, and carbon tetrachloride vapour will be obtained by using a shock-tube

(Maerefat et al. [7]) and be explained reasonably by the transition state theory.

Vapour Gasdynamical region

condensationl 1 Evaporation

------------------------------Vapour-liquid Knudsen

Ts .Ps interf~ce\ layer ,/ S

----------- - - ---:-_-___ -_-_-.Liquid film _-~:_

Fig.1 The state produced by the reflection of the shock wave.

II. NON-EQUILIBRIUM STATE PRODUCED BY SHOCK WAVE REFLECTION

A. Molecular gasdynamical boundary conditions

Figure 1 shows a typical state produced by the reflection of a

shock wave near the endwall of a shock-tube. Just at the instant when the shock wave reflects at the endwall, fluiddynamical quantities such

as the pressure, the temperature, and the density of the vapour increase in a stepwise fashion from an initially low state to a high one. The vapour begins to condense in the form of a liquid film on

51

the endwall surface, and at the same time an unsteady thermal boun­

dary layer develops into the vapour region. The film grows according

to the time lapse. Near the vapour-liquid interface a characteristic

region is formed, the length scale of which is of orders of the mean

free path of molecules and which is called the Knudsen layer. The

vapour region may be divided into two regions: a Knudsen layer and

an outer gasdynamical region. A fairly detailed analysis of vapour

behaviour in the Knudsen layer has been carried out from the viewpoint

of molecular gasdynamics. For example, Sone and Onishi [3] have

obtained the following expressions for the extrapolated molecular flux

j and extrapolated vapour temperature T at the vapour-liquid inter­

face (S-S in Fig.l) as follows:

j (10)

T (lb)

where j denotes the net rate of evaporation and condensation, the

number of molecules transfered per unit time and area at the

interface, n the number density of vapour molecules at the interface,

Ts the liquid temperature at the interface, p the vapour pressure at

the interface, k the Boltzmann constant, c (~(8kT/7CJfl)1/2) in which III

denotes the mass of a molecule, and the ac and at denote the conden­

sation parameter and the temperature parameter, respectively. The

subscripts 00 and s denote quantities relating to the vapour far from

the interface and quantities relating to the saturated vapour (ps,ns)

or quantities estimated at the surface temparature (T.) of the liquid.

In the case where vapour molecules approaching the liquid surface

are completely captured by the surface and the molecules emitted from

the surface have a Maxwellian distribution corresponding to the sat­

urated vapour, then, ac~O.46904 and at~O.44675 (Sone and Onishi [3]).

B. Condensation parameter and condensation coefficient

According to the transition state theory (Fujikawa and Maerefat

[6]), the net rate of evaporation and condensation is given as

follows:

52

j I -

_(.9..!.) .~.n .pm-ps q~ S 4 s pm (2)

where q~ denotes the rotational partition function of a molecule in

the liquid state, q~ the rotational partition function of a molecule in the gas state. The interference of evaporation and condensation

fluxes and the influence of the temperature jump at the interface on

the flux j are not considered in Eq.(2). Let us compare Eq.(2) with

the following Labuntsov-Kryukov's equation [4] which has been expressed on the same level of approximation as that of Eq.(2):

j _ _ a . ~ . n . P- Ps 1-0.399' a 4 s pm (3)

where a denotes the condensation coefficient which represents the sticking probability of vapour molecules on the liquid surface. When

the pressure is uniform throughout the va pour, 1. e., when P= pm, Eq. (3)

just corresponds to Eq. (2) except the factors a/(1-0.399· a) and (qVq~)s. Furthermore, for

0.399' a «1 (4)

Eq.(3) is approximately reduced to the so-called Hertz-Knudsen equa­tion in which the interference of evaporation and condensation fluxes

is not taken into account. Under the condition (4), Eq.(3) becomes identical with Eq.(2), in consequence, we obtain

(5)

On the same level of the above approximations, Eq.(la) can be reduced

to the same functional form as that of Eq. (2). Then we obtain

Gc '" (6)

The factor (q~/q~)s is given on the basis of statistical mechanics

of gases and liquids as follows (Fujikawa and Maerefat [6]):

I ( Sk)

q'¢ S

53

(7)

where V9 denotes the molar volume of the saturated vapour, V' the molar

volume of the liquid, ~ the molar evaporation entropy, and R the

molar gas constant.

-;;; 200 c

180 '" '" II>

160 c -'" u

.r: 140

E 120

-u 100 :::J

80 cr

~ 60 0

c 40 0

~

0 20 <-0

0 > 0

Temperature- 290 , 1 K

Pressure- 5,813 kPa

Mach Na, - 1.350

/ /

/

I

i /uc~0.05

Complete capture ;/

(a ~O. 46904) ''y' c /'

/ /

/ ,/

/

2

/

/ /

/

/

4

/ /

6 ( T ime (usJJ I12

8

III. EXPERIMENTAL APPARATUS

Fig.2 The growth liquid film on shock-tube end wall the reflected shock

10 methanol vapour.

of the the

behind wave:

Details of the experimental apparatus which has been used in

the present study are given in a previous paper (Maerefat et

al. [7J). The reader can refer to this paper.

IV. EXPERIMENTAL RESULTS AND DISCUSSION

Figure 2 shows the variation in time of the thickness of the liquid

film for methanol vapour. The experimental condition is indicated

in the figure. The abscissa is, up to 100 ~s, the square root of the

time. The origin of the time is taken to be just the instant of the

54

initiation of condensation. The experimental result (circles) is com­

pared with theoretical ones (solid lines) obtained by Fujikawa et al. 's

theory [8] using molecular gasdynamical boundary conditions (la)

and (lb) for three different values of ac = 0.03, 0.04, and 0.05. The

initial thickness of the liquid film is taken to be 13 nm (Maerefat et

al. [7]). The temperature parameter at is taken to be 0.44675, the

theoretical value [3]. This figure shows that good conformity between

the experiment and the theory is given for the value of ac = 0.04, which

is of the magnitude of one tenth of the value, 0.46904, for the complete

capture of the vapour molecules on the liquid surface [3]. This value

corresponds to 0:=0.14. The theoretical values of c.p. and c.c.,

evaluated by Eqs.(5), (6), and (7), are respectively 0.08 and 0.28 and

they are in rather good agreement with the above experimental values

if the assumptions and approximations made in the transition state

theory are considered. Equations (5) and (6) suggest that these small

values of c.p. and c.c. are due to that the rotational degrees of

freedom of molecules in the liquid state are hindered in comparison

with those of molecules in the gas state.

- 80 E C /-

Temperature- 298.8 K I " 70 Pressu r e- 1. 9T3kPo I "' " / c Mach No . ... 1 .355 /

.>0: 60 a =0.14 u c L

, ~ /

E 50 /

/ <!)

/

"U 40 Complete capture /

(a =0.46904) / / :J CT 30 c \/ .-

/ 0.08 ~

/ 0 20 / c / 0

10 /

~ / 0 /

'- /. 0

0 > 0 2 4 6 8 10

( 1 i me ( ~ s ) ) 1/1

Fig.3 The growth of the liquid film on the shock-tube endwall behind the reflected shock wave: water vapour.

55

- 400 E c

Temperature- 298.3 K It> Pressure- 8.053 kPa '" ~ach Na.-1 . 330 '" c ~ 300 L

E

~ 200 "0

:J cr

~

0 100 c 0

.... 0

L

0 0 > 0 2 4 6 8 10

( 1 i me ( us I 1112

Fig.4 The growth of the liquid film on the shock- tube endwall behind the reflected shock wave : carbontetrachloride vapour .

Figures 3 and 4 show the variations in time of the thickness of

liquid film for water and carbontetrachloride vapour. The experi­

mental conditions are indicated in the figures. The initial thickness

of the f ilm is taken to be 11 nm for water vapour and 16 nm for car­

bontetrachloride vapour. The temperature parameter O t is taken to

be 0.44675 for the both cases. These figures show that good conformity

between the experiments and the theory is given for Oc = 0.10 ( 0: = 0.35)

in the case of water vapour and for Oc = 0.20 ( 0: = 0.71 ) in the case

of carbontetrachloride vapour. Theoretical values of c.p. and c.c.,

evaluated by Eqs.(5), (6), and (7), are 0.14 and 0.50 respectively for

water vapour and are 0.20 and 0.71 for carbontetrachloride vapour.

The value of c . c. of water vapour is in rather good agreement with

the Hatamiya and Tanaka's value [9]. The experimental and theoret­

i cal values of c.p. a nd c.c. of water vapour and carbontetrachloride

vapour are significantly less than those for the complete capture of

molecules on the condensed phases. The reason is the same as for

methanol vapour.

Figure 5 shows the temperature distributions in the vapour, the

liquid film, and the shock-tube endwall: (a) methanol ~apour, (b)

56

1. 00

0.90

1.0

Shock-tube endwall

8 5

1.0

L iqui d f i 1m

9 1

t - I

e

t-IO

Vapour

t r- 0 . 008 x r - 2. 522 Tr -340 . 1

usee urn K (a)

0.80+----,-----,----+-------+-------,-------.-------.------. 15

1.00

0 .9 0

10 5

1.0

Shock-tube endwa I I

8 5

o o

0.0

1.0

L i qu i d f i 1m

9 1

t -IO

t·1

e

2. 5

Vapour

tr-0.0 19 x r -9 .205 Tr -36 4 .4

usee urn K

5.0

(b)

0 . 80+----,-----,----+-------+-------,-------.-------.------. 15

1.00

0 .95

10 5

1. 0

Shock -tu be endwall

8 5

o o

0.0

1.0

L i qu i d f i 1m

9 1

t· 1

e

2.5

Vapo u r

t r -0.028 xr-3 .875 Tr-332.0

~ sec

urn

5 . 0

K (c)

0.90+----,-----,----+-------+-------,-------.-------.------. 15 10 5 o

o 0 .0

1

1.0 2.0

Fig.5 The temperature distributions in the vapour (0), the liquid film (8,), and the shock-tube endwall (85 ): (a) methanol vapour, (b) water vapour, and (c) carbontetrachloride vapour.

57

water vapour, and (c) carbontetrachloride vapour. The space coordi­

nate is normalized by the reference length Xr (indicated in the figures) for the gasdynamical region, by the instantaneous thickness of the film for the film region, and by the reference length

(Dsr)1/2-0.7575/lm (Ds -5.74xlO-7 m2/s, ,=l/ls) for the endwall region. The

temperatures are normalized by the vapour temperature Tr in the reflected shock region, obtained by Rankine-Hugoniot relations, and

the time by the reference time t r. It is found that the temperature

Jump at the interface is very small for the three cases. This suggests that the estimation of c.p. and c.c. at an equilibrium state is

reasonable. which has been made on the derivation of Eqs.(5), (6). and (7) (Fujikawa and Maerefat [6]).

V. CONCLUSION

The shock-tube has been applied to determine the c.p. and the c.c. of methanol, water, and carbontetrachloride vapour. The values

of c.p. and c.c. are 0.04 and 0.14, respectively. for methanol vapour, 0.10 and 0.35 for water vapour, and 0.20 and 0.71 for carbontetrach­loride vapour. which are all significantly less than the values for

the complete capture of the molecules. These values have been

explained by the transition state theory on the basis of statistical mechanics of gases and liquids. The theory has clearly suggested that the c.p. and the c.c. are relating to the rotational degrees of

freedom of molecules in both gas and liquid state.

ACKNOWLEDGEMENTS

The authors would like to express their gratitude and apprecia­

tion to Professor T. Akamatsu of Kyoto University for his encourage­

ment throughout this work and also to their former colleagues, M.

Okuda. T. Goto, and T. Mizutani for their significant contributions to this work. The research was supported by the Grant in Aid for

Scientific Research of the Ministry of Education. Science and Culture

of Japan.

58

References

[ 1] Pao, Y. P • : Applica tion of Kinetic Theory to the Problem of Evaporation and Condensation. Phys.Fluids 14(1971) 306-312.

[2] Sone, Y. and Onishi, Y. : Kinetic Theory of Evaporation and

Condensation. J.Phys.Soc.Japan 35(1973) 1773-1776. [3] Sone, Y. and Onishi, Y. : Kinetic Theory of Evaporation and

Condensation Hydrodynamic Equation and Slip Boundary

Condition. J.Phys.Soc.Japan 44(1978) 1981-1994. [4] Labuntsov,D.A. and Kryukov,A.P. Analysis of Intensive

Evaporation and Condensation. Intl.J. Heat Mass Transfer

22(1979) 989-1001. [5] Fiszdon, W. Remarks on the Boundary Conditions at the

Liquid-Vapor Interface for the Continuum Vapor Region.

Ann.Nuc.Energy 7(1980) 227-234. [6] Fujikawa,S. and Maerefat,M. : A Study of Molecular Mechanism

of Vapour Condensation. Trans.JSME(1989), (in press). [7] Maerefat,M.; Fujikawa,S.; Akamatsu,T.; Goto,T. and Mizutani,T.:

An Experimental Study of Non-equilibrium Vapour Condensation in

a Shock-tube. Exp.in Fluids(1989), (in press). [8] Fujikawa,S.; Okuda,M.; Akamatsu,T. and Goto,T.

Non-equilibrium Vapour Condensation on a Shock-tube Endwall

behind a Reflected Shock Wave. J.Fluid Mech.183(1987) 293-324. [9] Hatamiya,S. and Tanaka,H. : A Study on the Mechanism of

Dropwise Condensation. - 2nd Report. Condensation Coefficient of Water at Low Pressures - Trans.JSME 52(1986) 2214-2221.

Vapor Condensation Behind a Shock Wave Propagating Through Vapor-liquid Two-Phase Media

Y. Kobayashi Institute of Engineering Mechanics University of Tsukuba Tsukuba, Ibaraki 305, Japan

Abstract

Vapor condensation behind a shock wave propagating through two­phase media with excessively large void fraction (vapor to liquid volume ratio) was investigated experimentally by using shock tube facility. A flow field of such media revealed itself largely different from that of pure gases in the process of attaining thermal equilibrium condition. In the former shock intensities P2/ P1 realized are much weaker, and larger heat energy are trans­ferred to the tube wall in the form of condensates. A series of investigations including schlieren photographs with the aid of high-speed drum camera illustrate a sequential change of generat­ion, growth and evaporation of fluid condensates in the flow field. These imply that the phase change phenomenon caused by thermo-fluid dynamic behavior plays an important role in such flow field immediately after the shock. The relaxation time of the vapor flow behind a shock wave is in orders of magnitude longer than that predicted by one-dimensional analyses based on kinetic theory.

Introduction

A shock wave propagation in a two-phase medium has been

studied extensively due to its importance both in academic and

engineering fields. The publicized works relating to this phenom­

enon therefore are diverse, encompassing from basic studies of

shock wave structure or relaxation mechanism to its application

techniques in industrial problems. Theoretical and/or analytical

efforts of the shock wave structure or relaxation zone behind a

shock wave in the two-phase medium preceded by G. Rudinger and 1 -5

F.E. Marble have been continuously developed last couple of

decades, and many analytical predictions describing the flow

field are proposed. The idea of the flow field which these theo­

ries draw behind a shock wave is characterized as such that it

has a longer relaxation zone with phase change (evaporation or

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Giittingen, Germany, 1989 © Springer-Verlag Berlin Heidelberg 1990

60

condensation) involved depending on its thermal condition. On the

other hand, a relatively fewer experimental results which will

evaluate these works have been available. In recent investigatio-6-8

ns specific natures of the vapor are introduced both in the 6,7

analyses and experiments. For example, P.A. Thompson et. al

categorize vapors into two classes of "retrograde" and "regular"

in accordance with possible condensation by shock compression or

by liquefaction shock waves. Discussion of the phase change 9

phenomena in relation to film detonation may also give a new

approach of interpreting the phenomena, and should be developed

further. With all current discussions associated with phase

change in such vapor flows, there still remain a lot to be resol-

ved in this essentially nonequilibrium phenomenon. investigated

associated with phase change in such a flow field.

In the present study a normal shock wave was injected into

the flow field of a single-component two-phase medium in its

thermal equilibrium condition, and detailed measurements of the

flow field behind a shock wave were conducted to measure vapor

condensation and heat transfer characteristics associated with

phase change of the medium.

Experimental Method

A conventional shock tube facility was used in the experime­

nt. It has a 5 m-long low pressure chamber with rectangular

cross sectional area of 20 mm-by 30 mm, the one end of which is

attached to a high pressure tubular chamber of 1 m-long with 30

mm in diameter and the other end to a dump tank. An aluminum

diaphragm of different thickness is set at the flange section

connecting two chambers, and is ruptured by a needle driven by

pneumatic means. A pair of optical windows of 30 mm by 45 mm for

flow field visualization are opened in the low pressure chamber

at 4 meters down-stream from diaphragm position. Shock wave

structure was examined by pressure transducers, thin film gage,

tungsten hot-wire temperature gage, laser beam attenuation and a

series of high speed schlieren photographs. A schematic lay-out

of the experimental apparatus together with main instrumentations

is illustrated in Fig. 1.

@/"'--.. @ ~ <:( ,~

® \.1000. \.

1 Light Source 2 Optical Window 3 Concave Mirror

I \ I I " lib I

4 High Speed Camera 5 Vacuum Pump

5000

6 Liquid Injector 7 Dump Tank 8 Diaphragm 9 Rupture Needle

10 Auxiliary Tank

(mml

11 Driver Gas Bottle 12 DC Amplifier 13 DigitalOsciloscope 14 Osci loscope 15 Wave Memory 16 Micro Computer

Fig. 1 Schematic layout of experimental apparatus

Nitrogen gas was used as a driver in the pressure range from

61

0.5 MPa to 5 MPa. To realize a single component two-phase therm­

al equilibrium condition in the low pressure chamber, a certain

amount of pure liquid was filled after evacuating the chamber.

Such common liquids as distilled water, ethanol, benzene, acetone

and refrigerant-11 (R-11 I, were examined, but in the this paper

discussions are focused mostly on the results of R-11 and benzene

because their shock wave structures are more distinctive due to

their relatively high saturated vapor pressure at room temperatu­

re. It should be noted that these are both "regular" fluids 6

according to Thompson's category. Shock wave data for the dry

air in various conditions of initial pressure P were obtained as 1

a reference in evaluating those of liquid-vapor media.

Typical shock Mach numbers Ms realized were in the range of

1.5 < Ms < 3.2 depending on the medium used in the experiment,

where Ms was evaluated by a measured shock speed and vapor prope­

rties of the medium. Void fraction of the medium realized in the

low pressure chamber was around 99.5 %, and quality (vapor to

total fluid mass ratio) was 0.056 for benzene and_ O.3 5 for R-11.

62

A saturated vapor condition (i.e. quality equals unity ) was also

e xamined for R-11.

Results and Discussion

a. Intensity of shock waves

Profiles of pressure distributions observed in the vapor are

largely different from those of pure gases at their shock front

region. Theoretical study of shock induced dissipation in the 10-13

boundary layer of pure gases can well explain the transport

phenomena taken place in the shock tube. In reference to t hese

ga seous theories, shock strength P /p of the vapor is much smal-2 1

ler than that of gas, indicating a weak compressibility effe cts

of a shock wave compared with those for pure gas. In addition,

the time duration of uniform high temperature and/or pressure

thermodynamic equilibrium condition behind a shock wave becomes

much shorter, as typically shown in Fig. 2(a) and (b) for

air and R-11 under the same initial pressure P. Two solid curv-1

es in the figure represent corresponding predictions with Rankin-

e-Hugoniot gasdynamic theory and that with dissipation effect 10

accounted (develope d by Mirels et. al ) for pure gases. This

figure also indicates that dissipation of R-11 is much larger

than that of the air which could be explained only when phase

change of the medium is incorporated in the former.

AIR \ PI = 101 KPo R -11 , 15 15

U) U) M i re~

E \ E Theory , R - H Relation

<1.>1 0 <1.> 10

~ E E f= • f= c: c: ,

~ ..g .2 •• 5 - 5 0

\~ R-H Rel ati on

~~-~ ~ ... => .......... ~ =>

Cl Cl

Mirel s' Theory -- • • 00 f

I t I .. 00 I I I

1.5 2.0 1.5 2.0 Shock Mach Num ber, Ms Shock Mach Number,

(a) (b) Fig. 2 Duration time of high temperature region for

dry air and R-11 vapor

,. ...

Ms

63

b. Heat transfer to the wall

Figure 3 shows typical trend of sequential temperature chan­

ges after a shock passage through three different media measured

by the film gage on the tube wall flush with its surface. The

origin of the abscissa corresponds to the time when the shock

front arrives. An excessively large and abrupt change of temper­

ature in vapors of benzene and R-ll compared with that of air is

apparent. A gradual temperature rise in the air shows a typical

trend of heat dissipation from heated gas to the relatively cold

wall surface by conduction, while those of vapors suggest a late­

nt heat associated with condensation of the vapor onto the wall

surface (heterogeneous condensation). Figure 4 shows comparison

between pressure change of R-ll, PrS, measured by pressure

transducer and the one, Ps(T), converted from the data taken by

film gauge under the assumption that thermal equilibrium is main­

tained in the v icinity of the wall where boundary layer is devel­

oped. Agreement of the two curves are quite good until 1000 ps

after a shock passage in this case. Same results were confirmed

under the different shock intensities and also for the case of

benzene, although time duration of the agreement between two

curves is different. These facts in turn imply the validity of

the assumption that thermal equilibrium is realized in the vicin­

ity (i.e. in the boundary layer) of the shock-tube wall. In

addition, a series of schlieren photographs indicates that

60

o 0

Fig.3

600

.., ~ .., tl: 200

AIR

R - 11 Ms=1.94 1.0

PrS

.~

---- ----__ Ps IT ) 0 .5 ~ .......... ~

............ co

U---~====k===----~-=~O 1000 2000 1000 2000

Time After Shock Passage (JL s)

Typical temperature Fig.4 profiles of three media

Time Afte r Shock Passage ( JLs)

Typical output profiles of three sensors

64

=> 0.08 Q.. -.J '-

£ 0.06 o

0:: ~ Q)

'"Vi 0.04 c:: o ~

'5 ~ 0.02

. - R -11 &-B ENZEN E

•• "r Mirel s' Theory

• ~ # BENZENE ...... R-l 1

°o~r--~~==~~~~--~----~ 1.5 2.0 2 .5 3 .0 3.5

Shock Mach Number , Ms

Fig.5 Heat transfer rates from vapors to the shock tube wall measured by thin film gage

deviation of the two CUrlres.,. a ,f-t.e-F- 1 OO()'ps ' r n Fig. 4 almost coinc­

ide s with the arrival of a contact surface.

Non-dimensionalized heat flux in the boundary layer calcula­

ted from temperature data similar to those of Fig. 3 is shown in

Fig. 5 with respect to shock Mach numbers. The value of q repre­

sents a heat flux calculated from temperature data (see Fig. 3),

and (LPU) in the figure represents a potential latent energy

carried b y the vapor flow per unit area and time in the region

be hind a shock front, where L, p ,U are latent heat, density and

axial velocity of the vapor. The solid curves in the figure

indica t e a prediction of aerodynamic heating predicted by Mirels

theory in the boundary layer behind a shock wave provided that

the va por would behave as if it were a perfect gas. This figure

suggests again a possibility of he terogeneous vapor condensation.

c. Heat transfe r in the main flow

Typical results of temperature and ' pressure distributions of

R-ll in the main flow field measured by hot-wire temperature

gauge and pre ssure transducer, PrS, are shown in Fig. 6. Symbols

of HandLon the temperature curve stand respectively for the

sensor aligned with parallel and perpendicular to the flow direc­

tion, and the arrow points a tifue wh~n - t~e contact surface arriv­

es. It should be noted that temperature data are converted to

600

0 400 0...

:::.:::

~ t -'

:::J

'" '" <1> ..... 200 0...

o 0

• .".,r ••••••••

. ~ .. -L !

.~ ...... "-.• ' .. .........

II ... -----_ .. _----' ...... ,

PrS

R -11 Ms = 1.65

......... ~ .. "

... ----.... :...;;::.~ '.\

1000 2000 3000 Time After Shock Passage (fL s)

u o

60 ~ ~ ~ <1> Cl.

E <1> f-

40 "0

~ ~ :::J

-0 20 c.n

o

Fig.6 Typical temperature and pressure distribution in the main flow field

65

their corresponding saturated vapor pressures under the assumpti­

on that thermal equilibrium condition holds in the flow field.

The most distinctive feature obtained from the figure is that

temperatures measured by the cross-field (perpendicular) sensorJL

are much higher than those by the parallel # sensor. This indic­

ates that -L sensor detects much higher vapor temperature which

far from thermal equilibrium. It should be noted that no such

discrepancy of the output is observed in the measurement of air.

A qualitative interpretation of these results is that the

hot-wire is heated by collision of supersaturated vapor molecules

and clusters which will condense and discharge condensation heat

energy to the wire surface. Although we have not directly ident­

ify a formation of clusters or liquid droplets in the vapor flow

yet, we have another clue to imply their formation in the flow

field measurement by laser beam attenuation method. A typical

signal output from photodiode detector of R-11 vapor is shown in

Fig. 4 (as symbolized by "LB"). The intens i ty of transmitted

laser beam I/I starts decreasing at the incidence of shock wave o

due probably to scattering loss by condensates generated in the

vapor flow, and after 140 microseconds of constant intensity,

66

it rapidly decreases to zero, and recovers in about a few

milliseconds. The elapsed time from shock wave arrival to zero

signal output in the photodiode detector corresponds to a black­

out phenomenon in the measurement of schlieren photograph (refer

to section d.). Effect of relaxation for thermal equilibrium due

to internal energy of a vapor should be negligibly small because

temperature level of the medium is low enough in this experiment.

A vapor in the compressed region behind a shock front would beha­

ve so as to adjust itself to the initial thermal equilibrium

condition in pressure and/or temperature by converting its energy

given by shock compression into phase change of the vapor. The

relaxation time that characterizes these thermo-fluid dynamic

behavior of the vapors behind a shock wave is much larger than

the one of thermodynamic properties of the vapor.

d. Flow field visualization

A typical sequential change of series of the flow field

behind a shock wave for R-11 at shock Mach number Ms=2.5 is illu­

strated in six sheets of schlieren photographs, as shown in Fig.

7. The time of t=O us in the figure is arbitrarily referenced,

and elapsed time after this moment are shown. The revealed flow

field in the figure is rather complicated, and its shock front is

somewhat analogous to that of exothermic detonation wave. Those

facts that the shock front is not clear enough (which is not due

to poor focusing of optical system), that an uncertain spots

moves on the shock front, and that stripe patterns are observed

to grow at the foot of the shock on the wall until it crosses the

whole flow field are also characteristic features of the flow

field. After a few hundreds of microseconds, these stripe

patterns are completely disturbed to block the optical source

light to pass across the flow field (as named "black-out"). It

is believed these stripe patterns represent a trace of

condensates generated by phase change of the medium, but it has

not been identified directly yet. It should also be noticed that

these stripe patterns were similarly observed in the flow field

in the saturated vapor medium (i.e. quality equals unity), which

implies liquid portion contained in the shock tube chamber is not

necessarily a main cause of black-out phenomenon.

67

t=O ps t=100 ps

t=33 ~s t=133 ~s

t=67 ~s t=167 ps

Fig.7 A series of schlieren Photographs of R-ll flow field

Conclusion

A series of shock tube experiments on vapor-liquid two-phase

media were conducted to examine characteristic features of the

system. Followings is the provisional conclusion obtained in

this study:

The experimental results of relatively weak intensity of a

shock front in the two-phase medium, complicated stripe patterns

and their sequential change, and heat transfer rates both in the

boundary layer and main flow field are all implying vapor

condensation in the flow field for a short period of time of at

most several hundreds of microseconds.

68

References

1. Rudinger, G.: Some Properties of Shock Relaxation in Gas Flows Carrying Small Particles, P.Fluid, Vol.7, No.5 (1964), 658-663

2. Marble,F.E.: Some Gasdynamics Problems in the Flow of Condensing Vapors, Astro.Acta, Vol.14 (1969), 585-613

3. Young-Ping Pao: Application of Kinetic Theory to the Problem of Evaporation and condensation, P.Fluids, Vol.14, No.2, (1971),306-312

4. Panton,R. and Oppenheim,A.K.: Shock Relaxation in a Particle­Gas Mixture with Mass Transfer between Phases, AIAA J., Vol.6, No.11, (1968), 2071-2077

5. Lu,H.Y. and Chiu,H.H.: Dynamics of Gases Containing Evaporable Liquid Droplets under a Normal shock, AIAA J., (1966), 1008-1011

6. Thompson, P.A. and Sullivan,D.A.: On the possibility of complete condensation shock waves in retrograde fluids, J.Fluid Mech. Vol.95, (1975), 639-649

7. Dettleff,G., Thompson,P.A., Meier,E.A. and Speckmann,H.: An experimental study of liquefaction shock wave, J.Fluid.Mech. Vol.95 (1979), 279-304

8. Thompson, P.A., Kim, Y.-G., Meier, G.E.: Shock-Tube Studies with Insident Liquefaction Shocks, Proc. 14th Shock Tubes and Waves, (1984), 413-420

9. Borison,A.A., Gel'fand,B.E., Sherpanov,S.M. and Timofeev,E.I.: Mechanism for Mixture Formation Behind a Shock Sliding Over a Fluid Surface, Combustion and Explosion Shock Waves, Vol.17, No.5, (1980), 558-563

10. Mirels,H. and Mullen,J.F.: Small Perturbation Theory for Shock Tube Attenuation and Nonuniformity, P.Fluids, Vol.7, No.8 (1964), 1208-1218

11. Roshko,A.: On Flow Duration in Low-Pressure Shock Tubes, P. Fluids, Vol. 3, No.3, (1960), 835-842

12. Ardonceau,P.L.: The Structure of Turbulence in a Supersonic Shock-Wave/Boundary-Layer Interaction, AIAA J •• Vol.22, No.9, (1984), 1254-1262

13. Dillon Jr,R.E. and Nagamatsu,H.T.: Heat Transfer and Transition Mechanism on a Shock-Tube Wall, AIAA J., Vol.22, No.11,(1984),1524-1528

Shock Wave Propagation in LowTemperature Fluids and Phase Change Phenomena

K. Maeno

Department of Industrial Mechanical Engineering Muroran Institute of Technology Muroran, Hokkaido, 050 Japan

Summary

In this paper shock wave experiments in low temperature gases are studied by using of a diaphragmless and cryogenic shock tube combined with cooling by liquid nitrogen. Together with the normal test gases of NZ or 02' the re­frigerant R-12 is used as test gas. Shock wave propagat10n ana the flow field behind incident and reflected wave are studied in comarison with the results of normal temperature range. Nonequilibrium and very fast condensa­tion at the cold tube wall and the surface of 2-dimensional wedge is ob­served by flow visualization technique. For the flow field around the 2-dimensional wedge behind the propagated shock wave, simple estimation of condensing characteristics for the oblique shock relations including the condensation effect is conducted, and the results are discussed. Estimated condensing velocity normal to the inclined wedge surface ranges from 30m/s to SOmis, which is about one tenth of the incident velocity to the wedge. Vapor bubble dynamcis in cold R-12 liquid which is compressed by the shock reflection are also presented. Interaction between the existing bubbles and also between the bubbles and emitted pressure waves are observed.

1. Introduction

It is known that many kinds of fluids are utilized and investigated in

the research of mechanical and aerospace engineering, physical chemistry,

cryogenics, and in the industrial plants, and moreover, most of these gases

have their points of phase change in cold temperature region to cryogenic

level. If an adiabatic wave propagates in such fluid of near phase change

point, the phase change may be initiated by the wave and many interesting

phenomena can be observed behind the wave.

As for the condensation of gases, typical experiments have done by using

an expansion fan. For instance, Spiegel and Wittig [1] reported binary nu-

cleation in the gas/vapor (water, propanol) by shock tube of heated temper­

atures. The homogeneous nucleation in the fluids is also investigated mainly

by the supersonic expansion into the vacuum (Wilcox and Bauer [2], Froger

and Rosengard [3],etc.). On the other hand, the sho~k wave propagation (com-

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium G6ttingen, Germany, 1989 © SpringerNerlag Berlin Heidelberg 1990

70

pression) and phase changes in retrograde fluids have been systematically

investigated by Thompson et al. [4,5]. Furthermore, the condensation phenom­

ena for R-ll vapor behind the incident shock wave have been studied by

Carruthers et al. [6] from the aspects of condensability criteria including

the vibrational relaxation effect, and also by Outa et al.[7] for non­

equilibrium condensation in wall boundary layer. The condensation of normal

vapors on the tube end wall behind a reflected shock wave has been reported

by Fujikawa et al.[8].

These investigations have been restricted within the range of normal tem­

perature or high temperature vapors except the expansion experiments into

vacuum. The condensation of R-12 behind the shock waves in low temperature

conditions has reported by authors [9] for the initial visualization. In the

present paper the shock propagation and condensation experiments by dia­

phragmless, cold shock tube for R-12 vapor are studied in comparison with

normal test gas data, and R-12 flow field around 2-dimensional wedge behind

the propagating shock wave with condensation is also investigated, together

with the shock compressed vapor bubble dynamics in cold R-12 liquid.

2. Experimental Setup

The experimental apparatus and instrumentation are shown in Fig.l. This

diaphragmless shock tube uses the main and auxiliary nylon pistons which are

controlled by a solenoid valve in evacuation chamber behind the driver cham­

ber. The details of diaphragmless shock tube have been already reported by

authors [9,10]. The merit of this tube is its freedom from pollutions of

broken diaphragm or freezing influx of air and water vapor. To maintain the

low temperature conditions in test section, the low pressure tube of 19.4mm

inner diameter has 90° bend portion after 2.3m horizontal tube, and it has

1.2m vertical portion including 20x20mm2 square test section which is cooled

by liquid nitrogen.

The pressure variation by incident and reflected shock waves are moni­

tored by piezoelectric transducer (PCB-lllA24 or 102AllB), and shock veloc­

ities are calculated with from pressure histories by microcomputer. Initial

vapor temperature in the test section is measured by CA thermocouple. Fur­

thermore, flow visualization are conducted by shadowgraphs with YAG laser

(second harmonics) and by consecutive shadowgraphs with high speed image

converter (Imacon-790). The temperature of test section ranges down to 150K

and satisfactory operation of the shock tube can be obtained.

3. Results and Discussion

Fig. 1 Experimental Apparatus

71

As the example of measured results, Fig.2 shows the relations between

incident shock Mach number Ml and driver/driven pressure ratio P41 for test

gases N2 and O2 , One-dimensional and idealized calculation curves for normal

and cold temperature ranges are also plotted. Due to the decrease of sound

velocity at low temperature test gas, calculated curves for cold region

1

/

/ C{j.~ " 6l> 9;..<If' •

Q, • ~ 'l'. • o

• ~ • •

Calculated - - Tl =290K

- --- Tr =200K

Measured O T 1=189 . 7-209.8(K)

~ Tl=232.7 - 248 . 4(K)

• Tl =294.3(K)

o ~------~--------~--~ 1 100 200

'P41

(a) Driver gas; N2-Test gas; N2

2

1

o Oe °~ -

~. . o ", • o

•• ----- T l = 200K

Calculated ------ T l = 290K

Measured

o ;T l = 187 . 2 - 208.7 (K)

. ;T l = 295 (K)

200

(b) Driver gas; N2- Test gas; O2

Fig. 2 Incident shock Mach number and initial pressure ratio

72

indicate the higher distributions than those for room temperature data.

Measured data show the similar trend to calculated curves. As discussed by

Liepmann and Laguna [11], stronger shock wave can be realized in colder test

gases. The results for test gas R-12, on the other hand, present the differ­

ent trends as shown in Fig.3. In this figure the measured data for room tem­

perature are distributed over slightly higher range than the calculation,

while the data for cold temperature are distributed in rather lower rang e ,

This lower distribution can be interpreted as the results from condensation

of R-12 at the cold wall behind the incident shock wave. These phenomena

will be discussed in the following section.

9 .----------------------,

8

7

6

5

4

3

2

Calculated

-- Tl =290 [K]

--- - Tl =200 [K]

0 00

• ... • 0

100

'P41

.. o o

Measured

P 4 (kPa)

306-317 352-358 400-422

348-46 3

200

~.-- -

Tl (K)

293 293 293

182 -235

300

Fig. 3 Relation between in­cident Mach number Mt and initial pressure rat10 P41

Driver gas;He - Test gas;R-12

3.1 R-12 Condensation behind Incident and Reflected Waves

As regards the shock propagation and condensation in low temperature re­

frigerant R-12, Fig.4-a presents the pulsed YAG laser shadowgraph of reflec­

ted shock wave from liquid free surface [10]. In this figure the flow field

near the reflected shock shows very complicated features. We can see the

reflected shock wave, and in front of the wave there exist an interaction

zone of R-12 condensed boundary layer with pressure waves and precursor

condensation layer in this zone. Behind the reflecte d wave there are also

condensing region above the liquid free surface. The rapid pressure decay is

observed, which is regarded as the results from non- equilibrium condensation

at the cold wall behind the compressive shock wave.

Boundary l ayer­pressure wave i nter act i on

~ Pr ecur so r condensation

~layer

0« I Ref l ected shock wave

~ Condensing zon e

< I R- 1 2 liqu id free surface

Driver gas;He- Test gas;R-12 Pl=40kPa, T1=230K, P4=425kPa, M1=1.7

Fig . 4-a Shadowgraph of shock reflection

73

400 Cri teal point

T [K]

300 v

200

o 1.0 2.0

Specific entropy S [kJ/kg·K]

Fig.4-b Temperature-entropy diagram for refrigerant R-12

In order to explain this phenomenon, we consider a T-s diagram as shown

in Fig.4-b. From this figure it can be said that R-12 is not the retrograde

substance [4,5] but normal (or intermediate) vapor, and it can be hardly

liquefied by merely isentropic compression. Then the nonequilibrium conden­

sation at the cold tube wall seems to be the main origin of the phenomenon.

The temperature of the gas adjacent to the cold wall cannot follow Rankine­

Hugoniot temperature jump and rapid nonequilibrium condensation is initi­

ated at the wall.

3.2 Condensation by Supersonic 2-Dimensional Wedge Flow

If the 2-dimensional wedge is settled in our cryogenic shock tube test

section, the supersonic flow behind the incident shock wave produces oblique

shock waves from the apex of the wedge. Behind these oblique shock waves the

possibility of wall condensation of R-12 can produced in accordance with

initia l conditions. Figure 5 shows the shadowgraphs of oblique shock waves

around the 2-dimensional wedge in R-12 supersonic flow behind incident shock

wave taken by Imacon-790. The right hand wave from the apex of wedge is Mach

wave, and the left hand side wave is oblique shock wave. In these results

for cold but dry R-1 2 vapor, condensation effect can be hardly observed, and

the measured angles of oblique shock and Mach wave are almost constant

before the reflected shock wave arrives.

74

In case of colder and wet vapor, the Z-dimensional wedge can produce the

compressed and supersonic condensing flow at the inclined surface as shown

in Fig. 6. We can see immediately after the attack of incident shock wave a

very rapid condensation zone that propagates from the apex of wedge at

series No.3,4,5,6 and the left hand side of the wedge becomes strongly

Driver gas; Pl= 6.7kPa, v Z= 370m/s,

He, T =

1 a = Z

19.

P4= 509k~a (room ~emperature)~ Test gas; R:1Z ZZOK, Ml - 3.3, PZ- 79kPa (PZm- 137kPa), TZ- 419K 184m/s, MZ= Z.Ol, laps/frames

Fig. 5 Shadowgraphs of oblique shock wave by Imacon-790

Driver gas; He, P4= 53Zk~a (room ~emperatu£e)~ Pl= 6.7kPa, Tl = 195K, Ml - 3.9, PZ-l06kPa (PZm­v Z= 415m/s, a Z= 189m/s, MZ= Z.ZO, laps/frames

Test gas; R-1Z 136kPa), TZ= 44ZK

Fig. 6 Shadowgraphs of condensing oblique shock wave by Imacon-790

75

darkened, while the Mach wave side indicates no significant condensation. As

shown in the strong condensing photographs, the angle of oblique shock wave

to incident flow direction decreases firstly by condensation, and then

increases gradually (but within about lOOps). This is somewhat peculiar

result combined with the results in previous sections. In usual condensation

of vapor at cold wall the phenomenon can be initiated from the nucleation of

liquid droplet in supersaturated vapor in the temperature boundary layer

adjacent to wall surface. The latent heat is released mainly into the vapor

in the layer, and droplet temperature becomes slightly higher than cold wall

temperature, while the temperature of vapor boundary becomes much higher to

eliminate the super-saturated conditions. According to this explanation, the

temperature of the flow behind the oblique shock wave should be raised by

the latent heat release from nucleation, Mach number should be decreased,

and the oblique shock angle should be increased. Also in the previous sec­

tion the pressure behind incident shock wave should be increased, and

possibly the incident shock Mach number should be augmented. The reverse,

however, is the case in our measurements.

These discrepancies will be carefully checked and discussed in the future

study. But up to now for the results in Fig. 6, an explanation can be set­

tled. At the very beginning of condensation in the supersonic flow on cold

wall as shown in our shadowgraphs, there exists a direct condensation zone

to the cold wall surface before the development of boundary layer. In short,

the kinetic region can exist on the wall surface, and the latent heat is

absorbed into the cold surface at very initial stage. Actually the oblique

angle becomes increased in later series of photographs.

Keeping this explanation in mind, an estimation for condensing parameters

can be performed. We take the minimum angle of oblique shock wave after the

onset of strong condensation. If the oblique shock wave is affected by con­

densing mass transfer (no heat addition), the shock angle will be decreased

as shown in Fig. 7. Then from the geometrical relation the velocities normal

and tangential to the oblique shock are obtained. By using the usual shock

wave relations, we can estimate condensing velocity component normal to the

2-D wedge surface as follows:

VzcosS; tan(S} -~ - VzcosSl tan(Sl -6)

sin(S; -6)tan(Sl -6) + cos(Sl -6)

76

Obliq ue s hock wave without cond ensation

Oblique s hock wave with considering co nd e nsation

------~-----

2-D Wedge

Fig. 7 Oblique shock wave relation with condensation effect (oblique shock angle is assumed to cbange by condensing mass transfer)

80r---------------~

60 r-Vl '- 8@'6 E

40 t- 0

?; go c

> 20 r-

o '----____ i-'--___ --'

1 2

Driver gas; N2 Tl = l8S-2l7K

Fig. 8 Estimated normal ve­locity to wedge surface by condensation on incident flow Mach number M2

3

In Fig. 8 the calculated normal condensing velocity component is indicated.

Though the estimated data are too scattered to discuss some clear trends,

the positive condensing velocity and mass flux can be obtained from our

assumption. The condensing velocity normal to the cold wedge surface ranges

from 30m/s to SOmis, which is about one tenth of the incident supersonic

flow velocity to the wedge.

Driver gas; He, ~ p4=48SkPa (room temperature). Test gas; R-12 Pl= 19.1kPa, Tl~ 210K(a = 873m/s), Ml~ 2.5, laps/frames

Fig. 9 Shadowgraph of co~lapse and rebounds of cavitation vapor bubbles in shock compressed R-12 liquid by Imacon-790 (five bubbles can be seen)

77

3.3 Vapor Bubble Collapse in Liquid R-12

In our test section cooled enough by liquid nitrogen below the phase

change point of R-12, condensed liquid phase accumulates at the tube end.

With controlling the initial pressure on such liquid-vapor equilibrium,

vapor bubbles are produced in R-12 liquid. If the shock wave reflects from

the liquid free surface, existing vapor bubbles starts to collapse or re­

bound to produce stronger pressure waves into the liquid phase. Tepper [12]

reported the similar experiment in normal temperature, and low temperature

results were reported by the authors [10]. In Fig. 9 the flow visualization

results for compressed liquid including several vapor bubbles by shock re­

flection are indicated. The bubbles are seen in the right side of the

figures, which make strong pressure emissions into R-12 liquid from the

upper side of the bubbles in turn. The strong interaction between the lower

side bubbles and the emitted pressure waves are observed.

4. Conclusion

Experimental investigation has been conducted for shock waves and phase

change in low temperature R-12 or other test gases by means of diaphragmless

cryogenic shock tube. The parameters for shock propagation and condensation

phenomena behind shock waves in low temperature R-12 have been investigated

with the aid of flow visualization. Nonequilibrium and very fast condensa­

tion at the cold tube wall and the surface of 2-dimensional wedge is

observed. For the flow field around the wedge, simple estimation of condens­

ing characteristics for oblique shock relations is conducted. Estimated

condensing velocity normal to the wedge surface ranges from 30m/s to SOmis

at the very beginning of condensation initiation. For the vapor bubble

dynamics in cold R-12, interaction between the bubbles and emitted pressure

waves are observed.

References

1. Spiegel, G.H. and Wittig, S.L.K.; Binary nucleation and condensation: Theoretical analysis of the relaxation process and its experimental ver­ification in a shock tube, Shock Tubes and Waves (H. Gronig, ed.), VCH, pp.297-303 (1988).

2. Wilcox,Jr.,C.F. and Bauer S.H.; Stochastic simulation of condensation in supersonic expansion (Ar), Shock Waves and Shock Tubes (D. Bershader and R.Hanson, eds.), Stanford Univ.Press, pp.783-792 (1986).

78

3. Froger, A. and Rosengard, A.; Rayleigh diffusion measurements and calcu­lations for SF condensation in a supersonic nozzle expansion, Rarefied Gas Dynamics,I~ (H.Oguchi, ed.), Univ.of Tokyo Press, pp.759-766 (1984).

4. Thompson, P.A. and Sullivan, D.A.; On the possibility of complete con­densation shock waves in retrograde fluids, J. Fluid Mech., Vol.70, pp. 639-649 (1975).

5. Thompson, P. A., Carofano, G.C., and Kim, Y.G.; Shock waves changes in a large-heat-capacity fluid emerging from a tube, Mech., Vol.166, pp.57-92 (1986).

and phase J. Fluid

6. Carruthers, C., Francoeur, K., and Teitelbaum, H.; Studies of condensa­tion kinetics behind incident shock waves, Shock Tubes and Waves (H. Gronig, ed.), VCH, pp.327-333 (1988).

7. Outa,E., Matsuda, E. ,and Tajima, K.; Shock tube study on non-equilibrium wall condensation of a super-heated freon vapour flow, Shock Tubes and Waves (H.Gronig,ed.),pp.289-295 (1988).

8. Fujikawa,S., Okuda,M., Akamatsu,T., and Goto, T.; Non-equilibrium vapour condensation on a shock-tube end wall behind a reflected shock wave, J. Fluid Mech., Vol.183, pp.293-324 (1987).

9. Maeno, K. and Orikasa, S.; Study on shock waves in low temperature gas by means of a non-diaphragm shock tube, Shock Waves and Shock Tubes (D. Bershader and R. Hanson, eds.), Stanford Univ. Press, pp.563-569 (1986).

10. Maeno, K., Shizukuda, Y., and Hanaoka, Y.; Experiment of vapor bubble collapse in low temperature R-12 under shock compression, Shock Tubes and Waves (H.GrHnig,ed.),pp.273-279 (1988).

11. Liepmann, H.W. and Laguna, G.A.; Nonlinear interactions in the fluid me­chanics of helium II, Ann. Rev. Fluid Mech., Vol.16, pp.139-177 (1984).

12. Tepper, W.; Experimental investigation of the propagation of shock waves in bubbly liquid-vapour-mixtures, Shock Tubes and Waves (R.D. Archer and B.E. Milton, eds.), New South Wales Univ. Press, pp.397-404 (1983).

On an Inviscid Approach to Phase Transition Problem

Harumi Hattori

Department of Mathematics West Virginia University Morgantown. WV 26506

*This paper was partly supported by the Army Grant DAAL03-B9-G-00BB.

I. INTRODUCTION

An interesting aspect of van der Waals' type fluids is that the

systems describing the motion of the fluids are of hyperbolic-elliptic

mixed type. when the dissipative and/or capillary terms are neglected and

the systems are inviscid. As in the hyperbolic systems of conservation

laws. the weak solutions are not unique and to pick up a physically

relevant solution various admissibility criteria have been proposed. There

are mainly two distinct approaches to this admissibility problem. One

approach is to consider the viscous or/and capillary effects of the fluids.

Slemrod [1). [2) has proposed the viscosity-capillarity criterion in which

not only the viscous but also the capillary effects of the fluids are taken

into account in order to select the admissible solution. Pego [3). on the

other hand. has considered the viscous effect only and has shown the

stability of coexistent phases in a viscoelastic bar. In the above

approaches the higher order effects are considered. Another approach is to

consider the inviscid systems. Shearer [4) considered the Riemann problem

for (2.1) assuming all the stationary phase boundaries are admissible. He

also considered the double phase boundary problem of (2.1) in [5). Hattori

[6). [7). [B) has shown that the application of the entropy rate

admissibility criterion to the above fluids is useful. Note that a

comparison of the various admissibility criteria is given in Slemrod [9).

In this note we summarize the applications of the entropy rate

admissibility criterion to the above fluids. discussed in [6). [7). [B).

and extend the results in [6). [7). This criterion. which has been

proposed by Dafermos [10). (11) originally for hyperbolic systems of

conservation laws. roughly says that the entropy decays (physically

increases) with the highest rate for the admissible solution. As mentioned

above. the systems we discuss are of hyperbolic-elliptic mixed type. We

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Gottingen, Germany, 1989 © Springer-Verlag Berlin Heidelberg 1990

80

see here an extension of this criterion to nonhyperbolic systems. In

Section 2 we describe the van der Waals' fluids and the Riemann problem.

In Section 3 we discuss the results in [6], [7], namely the single phase

boundary problem, and extend the results. The result in [8], namely the

double phase boundary problem will be discussed in Section 4.

II. VAN DER WAALS FLUID

We consider the following systems

(2.1)

(2.2)

where u, v, e and p are velocity, specific volume, internal energy, and

pressure, respectively. Systems (2.1) and (2.2) describes the one

dimensional motion of isothermal and nonisothermal flow, respectively. In

a typical van der Waals fluid the pressure is given by RB a

p= v-b - VZ·

The relation between pressure and specific volume for various values of

B(temperature) is given in Fig. 2.1.

81

p

Pm -----

---r-------+b--L--L----~~--~---------> V

Fig.

For a fixed B, where ( sa) . B (Be - 27bR ,the l.nterval (b,a) where

Pv(v,B) < 0, is called the a-phase (liquid phase) and the interval [P,ro)

where again Pv (v,B) < 0, is called the p-phase (vapor phase). The

interval (a,p), where Pv (v,B) > 0, is called the spinodal region and

assumed to be unstable. Also, in the isothermal case it should be noted

that the horizontal line p = Pm for which areas A and B are equal is

called the Maxwell line and in the steady state case two phases coexist

when the pressure is at Pm This construction of steady state solution

with two coexistent phase is called the Maxwell line construction and the

derivation is discussed in Fermi [12). Also, in the steady state case the

states (b,am) and (Pm'ro) are stable, [Il!m,a) and [P,Pm) are metastable

and am and Pm are neutrally stable. If the state is metastable, a small

disturbance may change the phase drastically. The discussion in this note

will be focused around the case where B < Be' so that we may observe phase

boundaries. The phase boundary in the inviscid case is defined to be a

discontinuity across which the Rankine-Hugoniot condition is satisfied and

the phase change takes place.

In the next two sections we discuss the Riemann problem where the

initial data is given by

(U,v) - {

x < ° (2.4)

x> ° for (2.1) and

82

x < 0

(u,v,e) (2.5)

x> 0

are constants. In section 3

we discuss the case where Vo and v 1 are in the different phases. This

problem is called the single phase boundary problem. In this case for a

technical reason we further require that Uo and u1 are the same. In

Section 4 we discuss the case where Vo and vl are in the same phase.

This problem is called the double phase boundary problem. These problems

are introduced in James [13].

III. SINGLE PHASE BOUNDARY PROBLEM

If we assume that we observe only one phase boundary when Vo and v 1

are specified in the different phases, it is plausible to employ the one­

parameter (denoted as ~) family of solutions of Fig. 3.1 in the isothermal

case and Fig. 3.2 in the nonisothermal case when the initial data are

appropriate. In these figures all the intermediate states (vL,vM,vR, etc.)

are constants. The abbreviations F.W., B.W., C.D., and P.B. stand for

forward wave, backward wave, contact discontinuity, and phase boundary,

respectively. The forward and backward waves consist of a shock or a

rarefaction wave. In the isothermal case this one-parameter family of

solutions was discussed first time in [13]. We take ~ to be vL - Vo in

the isothermal case and uL - Uo in the nonisothermal case. Then the decay

of entropy (the entropy rate) is given by

Wi(O ~ L jump

discontinuities

in the isothermal case and

2:

1 - 2

jump discontinuities

(3.2)

in the nonisothermal case. The jump discontinuities mean shocks and phase

boundaries, and a is the speed of a jump discontinuity given by

t

phase boundary (P.B.)

Figure 3.1a, E = 0

t

P.B.

backward (B. W. ) wave

Figure 3. Ib, f> 0

t

P.B.

Figure 3.lC f < 0

x

forward (F. W.) wave

x

x

83

84

t

P.B.

--------------~----~~--~~~--------~ x Figure 3.2a, ~ - 0

contact discontinuity (C.D. )

t

P.B.

Figure 3. 2b, € > 0

t

P.B. C.D.

(uM' v M' eM)

(uR' v R' e R) F.W.

Figure

x

x

85

a = ± (3.3)

The subscript + and denote the state on the right and on the left of

a jump discontinuity. Here, v+ and v_ stand for v o, v L, vM, vR, or

v 1 and others have the same meaning. After checking the signs of ~'(~)

and ~"(O, we obtain the following theorem. The proof is given in [6] and

[7].

Theorem 3.1. Assume that there exists one-parameter family of solutions as

in Fig. 3.1 and Fig. 3.2. Then, we have the following conclusions.

(i) In the isothermal case, if Po = Pl = Pm and Uo = u 1, the

stationary phase boundary (the case where ~ = 0) is admissible in the

sense that it locally minimizes the entropy rate, namely,

~~(O) o and ~~(O) > O. On the other hand, if Po = Pl ¢ Pm and

Uo = u 1, the stationary phase boundary does not minimize the entropy rate.

Particularly, if Po = Pl > «) Pm' ~~(O) < (» O.

(ii) In the nonisothermal case, if So = sl' Uo = ul and Po = P1'

then the stationary phase boundary is admissible, and if So ¢ sl' Uo = u l

and Po = P1' then the stationary phase boundary is not admissible. In

the same manner as the isothermal case, if So > «) Sl' then

~~(O) < (» O.

The above results agree with the classical results. Namely, (i)

agrees with the Maxwell construction of steady state solutions with

coexistent phases and (ii) agrees with the fact that for any transformation

occurring in an isolated system the entropy of the final state can never be

less than that of the initial state. Therefore, in order to support the

feasibility of the assumption of existence of one-parameter family of

solution (Fig. 3.1 and 3.2), it is desirable to have nontrivial solutions

which minimize locally the entropy rate among the solutions in Fig. 3.1 and

3.2. For this purpose, considering that the entropy rate ~ also depends

on u o, u l ' and etc., we express ~ as follows

Then, using the implicit function theorem, we obtain the following

86

Theorem 3.2. Assume ~ is twice continuously differentiable in the

isothermal case, if Uo and u1 are sufficiently close and Po and P1

are sufficiently close to Pm' then there exists a nontrivial solution of

Fig. 3.1 type which minimizes locally the entropy rate. In the

nonisothermal case suppose ~~(~=O; Uo - uC' u1 - uC' po(vo,eo) = PC'

P1(v1,e1) - PC' so(vo,eo) - sC' sl(v1,e1) = sc) = 0 and ~~(O;···) > 0,

where uC' Pc and Sc are constants. Then, if Uo and u1 are

sufficiently close to uC' Po and P1 are sufficiently close to PC' and

So and sl are sufficiently close to sC' then there exists a nontrivial

solution of Fig. 3.2 type which minimizes locally the entropy rate.

Proof. Note that

~~(O; ... ) > O.

Therefore, the implicit function theorem and the continuity of ~" imply

that there exists an interval of ~i - ~i(uO u1,PO,P1) for which

~i(~l) = 0 and ~i(~l) > 0, and ~i(uc' uC' Pm' Pm) =0. The nonisothermal

case is also proved in the similar manner.

Remark 3.3. It is shown that for the van der Waals fluid it is possible to

choose vo, v1, eo, and e1 to satisfy the above conditions in the

nonisothermal case. Here Vo and v1 are specified in the different

phases.

IV. DOUBLE PHASE BOUNDARY PROBLEM

In this section we discuss the double phase boundary problem in the

isothermal case and see what the entropy rate admissibility criterion

indicates. First, Shearer [5] has shown that if Vo and V1 in (2.4) are

specified in the same phase, there are cases where there exist two

solutions with the same initial data, both satisfying the viscosity

capillarity admissibility criterion. One solution lies in the same phase

and consequently there is no phase boundary. Another solution consists of

two phases and two phase boundaries. So, it is interesting to see if the

entropy rate admissibility criterion distinguished these solutions.

backward rarefaction wave (B.R.W

F.R.W. curve

u

B.R.W. curve

Figure 4.1

B.P.B.

Figure 4.2

--r-------------~v Figure 4.3

F.P.B.

87

forward rarefaction (F.R.W) wave

F.R.W.

88

small disturbance will change the phase drastically. Therefore, it is

instructive to see how the entropy rate admissibility criterion explain the

meaning of the metastability.

Shear's solutions [5) are the following. One solution consists of the

backward and forward rarefaction waves and all the constant states are in

the same phase; see Fig. 4.1. Another solution consists of the backward

and forward rarefaction waves and two phase boundaries which are symmetric;

see Fig. 4.2. If we draw these solutions in the state space, we obtain

Fig. 4.3 and see the relation between these two solutions. If we assume

that the two phase boundaries are symmetric when the both backward and

forward waves are the rarefaction waves and apply the entropy rate

admissibility criterion, we obtain

(4.1)

for the entropy rate. Here, up and vM is a function of vL •

Checking the sign of ~(vL) in (4.1), we obtain the following theorem.

The proof of the theorem is given in [7).

Theorem 4.1. If vM ~ Qm' the solution with the single phase (Fig. 4.2) has

the lower entropy rate than the solution with two phases (Fig. 4.3). On

the other hand if vM > Qm' the solution with two phases has the lower

entropy rate than the solution with the single phase.

References 1. Slemrod, M., Admissibility criteria for propagating phase boundaries

in a van der Waals fluid, Arch. Rat. Mech. Anal. 81 (1983), 301-315.

2. Slemrod, M., Dynamic phase transitions in a van der Walls fluid, J. Diff. Eqns. 52(1984), 1-23.

3. Pego, R., Phase transitions: Stability and admissibility in one dimensional nonlinear viscoelasticity, Arch. Rat. Mech. Anal. 97 (1987), 353-394.

4. Shearer, M., The Riemann problem for a class of conservation laws of mixed type, J. Diff. Eqns. 46(1982), 426-443.

5. Shearer, M., Nonuniqueness of admissible solutions of Riemann initial value problems for a system of conservation laws of mixed type, Arch. Rat. Mech. Anal. 93 (1986), 45-59.

6. Hattori, H., The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion - Isothermal case, Arch. Rat. Mech. Anal. 92 (1986), 247-263.

7. Hattori, H., The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion - Nonisothermal case, J. Diff. Eqns., 65 (1986), 158-174.

8. Hattori, H., The entropy rate admissibility criterion and the double phase boundary problem, Contemporary Mathematics Vol. 60 (1987), 51-65.

89

9. Slemrod, M., Admissibility criteria for phase boundaries. Non-linear hyperbolic problems (St. Etienne, 1986), 196-171. Lecture Notes in Math., 1270, Springer, Berlin-New York, 1987.

10. Dafermos, C. M., The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Diff. Eqns. 14 (1973), 202-212.

11. Dafermos, C. M., The entropy rate admissibility criterion in thermoelasticity, Atti Accad. Naz. Lincei Rend. cl. Sci. Fis. Mat. Natur. (8), (1974), 113-119.

12. Fermi, E., Thermodynamics, Dover, New York, 1956.

13. James, R. D., The propagation of phase boundaries in elastic bars, Arch. Rat. Mech. Anal. 73 (1980), 125-158.

Interaction of Underwater Shock Wave with Air Bubbles

K. Takayama and A. Abe

Shock Wave Research Center, Institute of Fluid Science, Tohoku University, Sendai, JAPAN

and K. Tanaka

National Chemical Laboratory, Tsukuba, JAPAN

Summary

An experimental and numerical study was made of the interaction of underwater shock waves with air bubbles which may appear in human tissue damage occurring in the extracorporeal shock wave lithotripsy. In order to visualize the shock-bubble interaction, double exposure holographic interferometry was applied to a 1.7 mm diameter air bubble exposed to underwater shock waves which were initiated by a 10 mg silver azide (AgN3 ) pellet. By counting the interference fringes, the formation of the high pressure spot and the resulting rebound shock and also microjets were quantitatively clarified. The numerical result agreed qualitatively with the experimental result.

Introduction

In the extracorporeal shock wave lithotripsy [1], immediately

after the focused high pressure disappear~cavitation phenomena

occur near the focus region. It is reported that even

though the greatest majority of these bubbles disappea~ bubbles

still remain for one tenth of,second from the shock focusing

[2]. Therefore, if the shock wave exposures are done repeatedly

the shock-bubble interaction occurs in the real lithotripsy.

Recent investigation revealed that the cavitation bubble is

responsible for damage"", the solid wall on which the bubble was

located. Therefore, it is of importance to understand the

cavitation damage the kidney or gallbaldder tissue.

It is also known that the ellipsoidal reflector is occasionally

filled with very small air bubbles which introduced by

circulating the water in the ellipsoidal reflector, the focused

pressure could never be enhanced to the expected level. The

shock wave energy absorbed by the tiny bubbles stimulates their

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Gottingen, Germany, 1989

92

motion and finally the shock waves are significantly

attenuated. Therefore, it is important to know how much energy

is absorbed into bubbles and how much is released later in the

form of rebound shock waves or etc.

The goal of the present investigation is to observe

quantitatively the shock-bubble interaction in water by pulsed

laser holographic interferometry. A 1.7 mm diameter air bubble

was placed 20 mm apart from a 10 mg silver azide (AgN3 ) pellet

in water. When the pellet was detonated, the spherical

underwater shock wave interacted with the air bubble. The

interaction initiated the bubble collapse and was followed by

the generation of the rebound shock and microjet [3].

Consequently by counting the spatial fringe distribution of

infinite fringe holographic interferograms, it is revealed

that the pressure at the side where the shock wave at first

impinged was enhanced to the peak pressure which, in a few

~sec, turns into a rebound shock and then re-expansion of the

bubble associated with the micrajet. The numerical simulation

was also conducted to interpret the shock bubble interaction

and its results agree qualitatively with the experimental

result.

Experiments

1. Holographic interferometry

Figure 1 shows experimental setup and optical system [4]. The

present optical system consisted of a collimating mirror of 300

mm in diameter and 3 m in focal length, and auxiliary mirror

and lenses. The light source was a holographic ruby laser,

Apollo Laser Inc., 22HD, 2 J/pulse, 25 nsec pulse width. Double

exposure holographic interferometry was used. By this technique

the phenomena occurring in an inhomogeneous medium such as water

is quantitatively observable.

In the double exposure holographic interferometry, two

exposures are required. The first exposure was done beforet~e

phenomenon and the second exposure was made simultaneously with

the phenomenon. After reconstruction of holograms, the fringe

93

intensities on the hologram are related as follows,

I cc 1 + cos AlP (1)

where ~¢is the phase change between the two exposures, and is

related to the integral of refractive index n along the light

path length L as

2'11' AlP = TIn dL (2)

where A = 694.3 DID is the wave length of the light source. In

the two-dimensional case, since n is constant along the

light path, each fringe corresponds to an isopycnic. The basic

arrangement of the present optical system is similar to the

shadowgraph. A beam splitter is used to split the source beam

into reference and object beams. Films used here were 100 mm x

125 mm Agfa 10E75 Holo Test sheet films and 100 mm x 125 mm

Neopan SS films for reconstruction. An argon-ion laser was used

for reconstruction of the holograms.

2. Experimental setup

Experiments were conducted in a water filled chamber of 0.5 m x

0.5 m x 0.5 m having observation windows made of 400 mm

diameter plastic glass. In order to obtain underwater shock

waves, 10 mg silver azide (AgN3 ) pellets were ignited. The

pellets have a cylindrical shape of 1.5 mm in diameter and 1.5

mm in height and density of 4 g/cm3 and were supplied by

Chugoku Kayaku Co. Ltd. The pellet was pasted on the polished

tip of fiber-glass of 0.6 mm in diameter. When a Q-switched

ruby laser beam of a few hundred mJ was transmitted through the

fiber - glass, the pellet was ignited with "time delay of less

than 0.1 ~sec [5].

Both two-dimensional and three-dimensional experiments were

conducted. In two-dimensional experiment, a 1.5 mm diameter

cylindrical air bubble was placed between two 3 mm thick

acrylic plates which were 1.5 mm apart. Thus a

cylindrical air bubble was supported between these plates. The

94

center of the bubble was 20 mm apart from the explosive. When

the pellet is ignited, the gap of the two plates is so narrow

that a spherical transmitting shock turns quickly into

cylindrical shape with propagatio~. In the acrylic plates

precursing longitudinal and shear waves exist. Although they

propagate faster than the shock wave in water, they are too

weak to disturb the shock-bubble interaction. In the three­

dimensional experiment, a 1.7 mm air bubble was produced by a

syringe from the bottom of the water chamber. When the air

bubble

position

ignited.

bars.

moved toward the water surface and arrived at a

20 mm from the explosive center, the pellet was

The shock overpressure in this distance is about 330

Results and discussion

1. Two-dimensional case

In the two-dimensional case, sequential interferograms are

shown in Fig.2. When the shock collided with the bubble, the

bubble started deforming. The side of the bubble on which the

shock at first impinged was flattened. Less than 1 ~sec later,

this flattened face was deformed concave toward the center of

the bubble. This is a very ear1y stage of formation of the

microjet. Figure 2(a) taken 26 ~sec from the ignition is a

single exposure hologram which is equivalent to the

shadowgraph. On the left hand side of Fig.2(a), many small

circles are observable. They are reflected expansion waves

occurring when the shock wave impinge very small air bubbles

initially attached tome plastic glass plates. Figure 2(b) is a

double exposure holographic interferogram. The fringes are

correspond to isopycnics. It is known that when the bubble

converges to a minimum volume, the pressure at the side where

the shock first impinged is a maximum. In other words, at

the final stage of bubble collapse, the maximum pressure is

induced and acts as a weak point explosion source. The point

explosion finally drives the rebound shock wave.~ ~rectlY

counting the fringes, we are able to evaluate the pressure

distribution around the collapstng bUbble.

In water, the refractive index is related to the density by the

Lorenz-Lorentz formula,

1

P = const.

In the two-dimensional case, the integral in

straightforward. Therefore, in putting L = 1.5 mm,

95

(3)

Eq. (2) is

the density

increment corresponding to one fringe shift is given by

-3 1.266 x 10 /fringe (4)

where ~ is the water density at the standard condition. If the

Tait equation is used as the equation of state of water[61, the

pressure increment /'>, p corresponding to one fringe shift in

Eq. (4) is given by,

~~ = 27.7/fringe (5)

Fringes concentrate on the side of the cylindrical bubble where

the shock first impinged is equivalent to 135 bars. This high

pressure drives the microjet which finally penetrates the

other side of the bubble. In less than 1 ~sec, the peak

pressure profile turns into the rebound shock. In Fig.2(c),

single exposure hologram, the rebound shock can be seen. This

trend is more clearly seen in Fig.2(d), double exposure

holographic interferometry as concentric fringe distributions.

It is noted that the center of concentric circular fringes

indicates the narrow region where the pressure has been a

maximum.

2. Three-dimensional case

In general three-dimensional cases, the integral of the

refraction index n along the light path in Eq.(2) is not as

easy as intwo-dimensional cases, since the light path L changes.

Nevertheless, in the axisymmetric flow field the isopycnics

can be determined numerically by using the Abel transformation

[71. By introducing concentric sectors in which density is

constant, Eq.(2) is approximated, using a finite difference

approach, in the form of a system of equations of the linear

combination of the unknown density corresponding to each

96

section. When the measured fringe number and distribution are

given to the system of equations, then the system is solved and

the density distribution is determined. Once the isopycnics and

the isobars are evaluated.

It was examined in the preparatory experiment whether or not

the collimated object beam can be distorted through the

phenomenon. Passing through the density gradient across the

shock, the collimated beam may be deflected from the original

direction. It was indeed found that a very small deflection of

the parallel beam existed only at the shock front, but

it was negligible.

Figure 3 shows sequential interferograms. It is clearly seen in

Fig.3(a) tha~ when the spherical shock wave collides with the

bubble, the primary reflected wave is an expansion wave.

Figure 3(a) and (b) were taken at 18.4 ~sec and 20.2 ~sec from

the ignition, respectively. The bubble converged and drastical

fringe concentrations could be clearly seen on the side where

the shock first impinged the bubble. By using the

aforementioned data processing,the local peak pressure was 70

bars in Fig.3(b) . As soon as the collapsing bubble has its

minimum volume, the bubble starts re-expanding and the high

pressure turns into a rebound shock wave as found in the two-

dimensional case.

Figure 3(b) corresponds to the moment very close to the

rebound shock initiation. In Fig.3(c) taken at 22.1 ~sec after

the ignition, the peak pressure turns into the rebound shock

and the bubble is re-expanding. It is clear that the rebound

shock wave has its center at the point where the pressure was a

maximum. The penetration of the microjet to the other side of

the bubble is delayed at least a few ~sec after the

generation of the rebound shock. Figure 3(d) was taken at about

6 ~sec from Fig.3(c).

The time variation of the pressure profiles evaluated from the

interferograms corresponding to Fig.3(a)-(c) are summarized in

Fig.4. The ordinate designates the pressure in bar and the

97

abscissa is the distance normalized by the interval between the

explosive and the center of the bubble. The pressure profiles

of reflected expansion wave, formation and the propagation of

the rebound shock wave are clearly seen.

3. Numerical Simulation

The finite difference numerical simulation was carried out by

solving the Euler equations by the Lagrangian approach. The

DANE code was used for computation. The air bubble radius was

1.0 mm and the shock overpressure was 1 kbar. Computational

grids were, in the axisymmetric Cartesian coordinates, 150 x

300 and one grid size was 0.025 mm. Figure 5 shows the

sequential isobars. It is clearly seen that the peak pressure

appear on the side where the shock first impinged the bubble,

and the bubble deformation starts which indicates the microjet

initiation. However, the rebound shock is so weak that, if

compared with the incident shock, the wave front could not be

resolved in this numerical scheme.

It is found that through the complicated wave interaction, the

incident shock energy was in part absorbed by reflected

expansion wave, rebound shock wave and then the bubble

deformation, and the incident shock wave is significantly

atten~ated. The present study has been done under an idealized

condition. Therefore, more works have to be done in the future

with combining various parameters such as shock overpressures

and the size and number of bubbles.

Conclusions

Results obtained are summarized as follows

(1) When air bubbles are exposed to shock waves in water, the

peak pressure appears on the side of the bubble where the shock

impinged the bubble. This peak pressure turns into a rebound

shock.

(2) The microjet formation process proceeds while the

bubble contracts and re-expands. Its driving force is the

abovementioned peak pressure. The microjet penetrates the other

side of the bubble in a few ~sec after re-expansion of the

bubble.

98

(3) The similar trend exists between two-dimensional and

three-dimensional shock-bubble interactions.

(4) The finite difference numerical simulation agreed

qualitatively with the interferograms.

References

1. ChaussY,Ch. et.al.:"Extracorporeal Shock Wave Lithotripsy". KARGER,1982. 2. Kuwahara. M, et.al.: "Acoustics Cavitation Bubbles in the Kidney Induced by Focused Shock Waves for the Extracorporeal Shock Wave Lithotripsy(ESWL)",Proc. 17th ISSW&T (to appear 1990) . 3. Shima, A., Takayama, K. and Tomi ta, Y.: "Mechani sm of impact pressure generation from spark-generated bubble collapse near a wall". AIAA J., Vol.21, 1983, pp. 55-59. 4. Takayama, K.: "Application of holographic interferometry to shock wave research". SPIE Proc. Vol. 398 1983, pp. 174-180. 5. Takayama, K., Esashi, H. and Sanada, N.: "Propagation and focusing of spherical shock waves produced by underwater microexplosions". Proc. 14th Int.Symp. on Shock Tubes & Waves, 1984, pp. 553-562. 6. Glass, I. I. and Henckroth, L. E.: "Low energy spherical underwater explosion". UTIAS Rep. 96, 1964. 7. Abe, A and Takayama, K. : "Shock Wave Diffraction from the Open End of a Shock Tube". Proc. 17th ISSW&T (to appear 1990).

1. Holographic ruby laser 2. Ruby laser for ignition 3. Laser power supply 4. Laser power supply 5. Fiber glass 6. Silver azide pellet 7. Air bubble 8. Test chamber 9. Syringe

10. He-Ne laser for monitoring the bubble motion

11. Photo sensor 12. Delay 13. Beam splitter 14. Paraboloidal mirror 15. Plane mirror 16. Object beam 17. Reference beam 18. f = 75 mm lens 19. f = 200 mm lens 20. Film holder

Fig.1 Optical arrangement and experimental setup.

99

(c) ld) Fig.2 Shock-bubble interaction in the two-dimensional case.

(a) (b)

(c) (d) Fig.3 Shock-bubble interaction in the three-dimensional case.

! :::1

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10 l

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~~

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~~~

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0.8

1

.6

a 0

.4

0.8

b

ub

ble

sh

ock

X

fr

on

t C

b)

(a)

(b)

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.2

1.6

b

ub

ble

sh

ock

fr

on

t

10 3

-;:;

10 2

'" .0 ""'

10 l

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(el

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a 0

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T

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b

ub

ble

(e)

Fig

.4 P

ressu

re p

rofi

les

ev

alu

ate

d

from

F

ig.3

(a).

(b)

an

d

(e).

2.5

ra

diu

s (m

m)

Fig

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Condensation in Flow, Boiling

Phase Changes of a Large-Heat-Capacity Fluid in Transcritical Expansion Flows

E. Zauner* and G.E.A. Meier

Max-Planck-Institut flir Stromungsforschung Bunsenstrasse 10, D-3400 Gottingen, Federal Republic of Germany * Present adress: Asea Brown Boveri AG, Corporate Research

CH-5405 Baden-Dattwil, Switzerland

Summary Substances of large heat capacity are able to undergo complete adiabatic liquid-vapour phase changes. The soundspeed discontinuities at the phase boundaries give rise to various real flow effects. The investigation of Laval nozzle flows assuming phase equilibrium shows discontinues choking at the liquid phase boundary. At the vapour phase boundary up to three shocks can occur simultaneously in the nozzle: one expansion shock and two compression shocks. In the experiment the initial conditions in the blowdown reservoir are chosen such that the expansion adiabats intersect the two-phase region close to the critical point. The phase boundary reached first during the expansion appears as a pronounced nucleation front. At the second phase boundary the influence of the distribution of liquid and vapour on the transfer processes becomes apparent. During the expansion process on subcritical adiabats the remaining liquid forms relatively large droplets. The transition to the pure gas flow spreads over a large spatial region. On supercritical adiabats, however, the two­phase state is reached coming from the gas phase. The re-evaporation of the small condensation nuclei is completed in a distinct evaporation front.

1. Introduction

Phase changes under isothermal conditions and those in flowing fluids show a fundamental

difference. In the first case the latent heat of evaporation has to be transferred between

the system and the environment. Many flow processes, however, are adiabatic and some

of them are almost reversible. In such adiabatic flows the latent heat must be provided

from the internal energy of the fluid. The ratio of internal energy and latent heat which

depends mostly on the molar specific heat of the substance characterizes the extent of

phase change attainable in adiabatic processes. Obviously, adiabatic phase changes can

take place much faster than isothermal phase changes.

The temperature-entropy diagram in fig. 1 clearly shows that fluids of low molar specific

heat, e.g. water, can only partly be evaporated or condensed by reversible adiabatic

processes. Fluids of high molar specific heat, e.g. perfluOIo-n-hexane (C6F14), however,

make complete adiabatic phase changes feasible. Physically, the difference between

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid·Vapor Systems IUTAM Symposium Giittingen, Germany, 1989 © Springer-Verlag Berlin Heidelberg 1990

104

substances of low and high specific heat is the number of internal degrees of freedom

which characterizes the ability of the molecule to store energy. Furthermore, for

substances of high molar specific heat the onset of vibrational excitation takes place at

lower temperatures. Molecules of higher complexity such as hydrocarbons and fluorinated

hydrocarbons with more than four C-atoms show the behaviour presented in fig. 1. The

fact, that most fossile fuels which are mixtures of many organic components have this

property underlines the great technical importance.

TITe

1.0

0.9

0.8

0.7 -15 -10 -5

, \

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

o 5

(S-Se) I R

Fig. 1. Liquid-vapour phase boundary in the temperature-entropy diagram for a substance of low and high specific heat. Water: cv=3.5, PPI (C6F14): cv=39.3.

Surprising wave phenomena like liquefaction shock waves, evaporation waves and wave

splitting have recently been investigated in these fluids (Thompson [1], Meier and

Thompson [2], Chaves et al. [3], Thompson et aI. [4], [5]). Moreover, these studies show

that under certain conditions some simple rules of gasdynamics are inverted by increasing

the specific heat. The possibility of rarefaction shocks is an example (Thompson and

Lambrakis [6]).

An interesting phenomenon is the behaviour of fluids near the critical point. Approaching

the critical state the time required to reach equilibrium increases with an inverse power of

the difference between actual and critical temperature. In the experiments of Borisov et al.

[7] for instance it took about 20 hours. As a consequence of large scale critical fluctuations

the propagation of sound is severely impeded. Soundspeed measurements in the kHz and

105

MHz range show a pronounced minimum at the critical point (Tielsch and Tanneberger [8],

Garland and Williams [9]). Critical phenomena in unsteady flows where the transit time

through the critical region is not sufficient to establish equilibrium have been investigated

by Thompson et al. [10]. They have found the soundspeed minimum and the disappearance

of phases at a pressure about 25% above the critical value.

In this study we present theoretical considerations for Laval nozzle flows of a large-heat­

capacity fluid at pressures and temperatures including the critical region. The

corresponding experiments show phase changes close to the critical point during

blowdown from supercritical stagnation conditions.

2.Laval nozzle flows of real fluids

We consider single phase and two phase flows of a pure substance through a Laval

nozzle. We presume that mechanical, thermal and chemical equilibrium between the

phases can be maintained. The flow will be treated as steady and one-dimensional. To

verify the flow model we apply a cubic equation of state with linear temperature

dependence (Abbott [11]). Internal energy and specific heat, respectively, take into

account vibrational excitation of the molecules (Chaves [12]). As a parameter

characterizing the behaviour of fluids in case of adiabatic phase changes we define the

nondimensionalized specific heat of the perfect gas at the critical temperature:

(1)

For cv;::.11 a part of the dew line in the temperature-entropy diagram shows a positive

slope (ds/dT>O) as is typical for large-heat-capacity fluids.

As an important quantity in nozzle flows fig. 2 presents the mass flux density along a slightly subcritical isentrope (s<sc) for different stagnation pressures. The substance,

C6F14 (cv=39.3), has also been used in the experiment. It should be called in mind that for

a perfect gas of constant heat capacity ratio a universal curve could be obtained

independent of stagnation pressure and entropy. This universality gets lost in case of real

fluids. At fairly low stagnation pressures we still find a behaviour similar to perfect gases.

At the phase boundaries, however, the mass flux density curves have nondifferentiable

points which are caused by the jump of the sound speed. This also means that the Mach

number is discontinous. The sound speed of the mixture is always lower than that of the

single phase fluid (fig. 3) since both bubbles in a liquid as well as droplets in a gas

increase the compressibility. If the mass flux density maximum is located at the liquid

106

p u / ( Pc / V c ) 1 12

0. 8 ,-----r---.,..--..,....,---""T"T"""-----.---,

VAPOUR

0.6

0.4

0.2

0. 0 L-_-L ___ L-__ LLL-~-L~~_L~

0.0 0.5 1.0 1.5

P / Pc

Fig. 2. Mass flux density for a large heat-capacity-fluid (C6F14) along a slightly subcritical isentrope «s-sc)/R=-O.OS) starting at different stagnation pressures.

C / ( Pc V c ) 1 /2 1. 5 ~--':"'-.--""::"'-:~r-----'-----'

1.0

0.5 LIQUID VAPOUR SIDE I SIDE /

~ ~

~

---0.0

0.10 0.05 0.00 0.05 0.10

(Tc-T)/Tc

Fig. 3. Sound speed for a large heat-capacity-fluid (C6F14) along the phase boundary (-- single phase fluid, ------ two-phase mixture).

107

phase boundary the transition from subsonic to supersonic flow takes place without

passing a sonic state. This behaviour is called discontinous choking (Collins [13]). The

wellknown picture representing solutions for different exit pressures in Laval nozzles is

changed according to fig. 4 .

P/p.o

.LLt/////)//// ///L I

777777?7/;V777l /)T?T I

I

x

single phase M<l

M>l mixture

Fig. 4. Solutions for Laval nozzle flow in case of discontinuous choking at the liquid phase boundary.

From the gasdynamics point of view the mass flux density minimum at the vapour phase

boundary is very interesting. In general a minimum occurs if a sonic state is located in a

region with negative fundamental derivative (Thompson [14]):

r = - .Y.( a2p J 1(~)2

2 av2 av S S (2)

i.e. along isentropes with negative curvature in the pressure-volume plane. For r>o (i.e.

perfect gases: r=(1C+ 1)/2) shock formation is observed in compression waves, while for

r <0 expansion waves are steepening. The convex kink of the isentropes at the phase

boundary leads to a behaviour similar to that in a region of negative curvature. The

consequences for the nozzle flow are as follows. Obviously the highest mass flux density

is reached at the nozzle throat. A local minimum, however, cannot occur in a simple

convergent-divergent nozzle. It has to be 'bridged' by a discontinuity which is an expansion

shock. Depending on the shock position in the nozzle (Le. stagnation pressure) either

upstream or downstream conditions are sonic. At the throat even double sonic shocks are

possible. It can be shown that these expansion shocks satisfy the second law (Thompson

and Lambrakis [6]) and that their position in the nozzle is stable. For certain values of

stagnation pressure and exit pressure up to three shocks can occur in the nozzle: one

108

expansion shock and two compression shocks. The latter are due to shock splitting at the

phase boundary during re-compression. The situation is schematically shown in fig. 5.

P/Po

M>l M<l

M>l

mixture

adiabat

x

single phase

Fig. 5. Solutions for Laval nozzle flow with multiple shocks at the vapour phase boundary.

3. Blowdown experiments from supercritical stagnation conditions

The experiments were performed in a V-shaped expansion tube shown in fig. 6. The

volume below the nozzle inserts forms the reservoir for the substance. The test section is

115mm long with a cross section of 20x15mm2 at the bottom and an opening angle of 11°.

The throat of the arc nozzle has a cross section of 20x4.5mm2. Front and back walls made

of glass allow optical observation of the flow. Pressure transducers (Kistler 603 B) or fast

acting iron-constantan thermocouples can be mounted in the sidewalls.

High speed photography ('" 6kHz) which uses parallel light from an LED-stroboscope is

applied to visualize the flow in the nozzle. The test substance used in our experiments is PPI (QjF14) having a critical temperature of Tc=451K (177.9°C) and a critical pressure of

PC=18.7bar. The initial conditions are chosen such that the state in the nozzle and in the

reservoir pass the critical region during blowdown both left and right from the critical point.

Corresponding computations are based on a quasi-steady approximation of the flow.

Fig. 7 shows a typical blowdown transient from an initial state of 1.90Pc on a supercritical

isentrope (s-sc>O). At the beginning sonic conditions at the throat are well above the

critical pressure. According to the high stagnation soundspeed the pressure drops initially

very fast. It is reduced by the subsequent enormous soundspeed decrease (the minimal

vacuum vesse l

Fig. 6. Axial cut through the expansion tube .

L a

.D

109

50r---.----.----.----r----,---~ 120

~

40 CJJ "-60 ...s

30 thr oat velocity stognot. soundspeed 0

u

0... 20 __ .&_

10

o

P3

o 5 10 15 20 25

t ems]

Fig. 7. Blowdown transient of PPI along a super­critical isentrope «s-sc)/R=0.06): PO=34.8bar, TO=192°C.

value predicted in the single phase region is 30m/s) which raises the sonic pressure ratio

to 0.85. At about 2.5ms the throat pressure reaches the phase boundary. It is arrested at

this value for that time in which the throat velocity decreases from the value of the single

phase soundspeed to that of the mixture soundspeed. The unusual plateau in the throat

pressure gives rise to an acoustic wave propagating upstream (fig. 8, 3.5ms). About Ims

later the condensation front follows. Because of the large number of droplets which does

not change significantly during the condensation and re-evaporation process along a

supercritical isentrope also the second phase boundary appears fairly pronounced (7ms).

But it cannot be decided from these observations if an expansion shock is embedded in the

front as predicted from the simplified theory.

Nonequilibrium during the evaporation process is very sensitive to the initial conditions.

On subcritical isentropes (s-sc<O) the vapour bubbles in the liquid are subject to a

selection mechanism which prefers the larger bubbles (Meier [15]). This leads to a strong

reduction of the bubble number during the growth process. Consequently, also the number

of droplets formed during the evaporation process is reduced. The distribution of the liquid

in a small number of large droplets, however, results in a reduction of the surface between

the phases and considerable vaporization delay. The frames in fig. 9 indicate the

-2m

s

1.4

6m

s

2m

s

2.S

4m

s

Fig

. 8.

H

igh

spee

d ph

otog

raph

s o

f P

Pl-

blow

dow

n al

ong

a su

perc

riti

cal

isen

trop

e «s

-sC

>/R

=O.1

3):

PO=3

3.3b

ar,

To=

192

°C.

Tim

e t=

O c

oinc

ides

with

the

ons

et o

f fl

ow i

n th

e re

serv

oir.

The

fl

ow d

irec

tion

is

upw

ards

.

3.S

ms

Fig

. 9.

H

igh

spee

d ph

otog

raph

s of

PP

l-bl

owdo

wn

alon

g a

subc

riti

cal

isen

trop

e «s

-sc)

/R=

-O.3

0):

PO=3

0.7b

ar,

TO

=187

°C.

.... o

Sm

s

7m

s

13

ms

13

ms

1

6m

s

18

ms

111

widespread evaporation zone on a subcritical isentrope. Curvature effects in the nozzle

even lead to separation of vapour and droplets.

Finally we discuss frames from the beginning of the blowdown in fig. 8. They show fast

growing wedge-shaped perturbations in the shear layers at the windows. The

perturbations are convected at a velocity of about 15m/s. It is interesting to note that the

perturbations propagate at different velocities in the diverging and the converging part of

the nozzle, respectively, which can be seen from the wedge angles. With increasing time a

transition to an irregular fine-grained structure occurs. An estimate for the Reynolds

number in the shear layer can be obtained by idealizing the flow as being instantaneously

accelerated to a constant velocity. The shear layer thickness where 99% of the outer flow

velocity has been reached is according to Schlichting [16}: o=4(vt)1/2. Applying a velocity

of U=15m/s which is due to the initial expansion wave and estimating the kinematic

viscosity at the critical point from a correlation given by Reid et al. [17], vc=5.7xlO-8m2/s,

gives a Reynolds number of Re=2.5xl02[s-l/2]tl/2. After 0.15ms for instance Re would

have reached a value of 3000 which is approximately the transition value for a boundary

layer at a plane wall. This supports the interpretation of the peculiar structure as turbulent

spots. Of course, a. more careful investigation is necessary which takes account for the

considerable changes of fluid properties in the critical region.

4. Conclusions

Real gas effects in steady Laval nozzle flows of fluids undergoing adiabatic phase changes

lead to discontinuous choking and in case of large-heat-capacity fluids to multiple shocks

including expansion shocks.

Experimental investigations of transcritical expansion flows show a pronounced sound

speed minimum close to the critical point. Phase changes develop significantly different on

subcritical and supercritical isentropes, respectively, although the initial conditions are

fairly close to each other.

Acknowled~ement

One of the authors (E. Zauner) gratefully acknowledges the hospitality of the Max-Planck­

Institut ftir Stromungsforschung, Gottingen, and the possibility to perform this work.

References

1. Thompson, P.A.: On the possibility of complete condensation shock waves in retrograde fluids. J. Fluid Mech. 70 (1975), 639-649.

112

2. Meier, G.E.A. and Thompson, P.A.: Real gas dynamics of fluids with high specific heat. Lecture Notes in Physics: Flow of Real Fluids (G.E.A. Meier and F. Obermeier, eds.) vol. 235,103-114, Springer 1985.

3. Chaves, H.; Lang, H., Meier; G.E.A. and Speckmann, H.-D.: Adiabatic phase transitions and wavesplitting in fluids of high specific heat. Lecture Notes in Physics: Flow of Real Fluids (G.E.A. Meier and F. Obermeier, eds.) vol. 235, 115-124, Springer 1985.

4. Thompson, P.A.; Carofano, G.C. and Kim, Y.-G.: Shock waves and phase changes in a· large-heat-capacity fluid emerging from a tube. J. Fluid Mech. 166 (1986),57-92.

5. Thompson, P.A.; Chaves, H., Meier; G.E.A., Kim, Y.-G. and Speckmann, H.-D,: Wave splitting in a fluid oflarge heat capacity. J. Fluid Mech. 185 (1987),385-414.

6. Thompson, P.A. and Lambrakis, K.C.: Negative shock waves. J. Fluid Mech. 60 (1973), 187-208.

7. Borisov, A.A.; Borisov, Al.A.; Kutateladze, S.S. and Kakoryakov, V.E.: Rarefaction shock waves near the critical liquid vapour point. J. Fluid Mech. 126 (1983), 59-73.

8. Tielsch, H. and Tanneberger, H.: Ultraschallausbreitung in Kohlensaure in der Nlihe des kritischen Punktes. Zeitschrift fUr Physik 137 (1954),256-264.

9. Garland, C.W. and Williams, R.D.: Low-frequency sound velocity near the critical point of Xenon. Phys. Rev. A 10 (1974), 1328-1332.

10. Thompson, P.A.; Kim, Y.-G. and Chan, Y.: Nonequilibrium, near-critical states in shock-tube experiments. Shock Tubes and Waves (H. Gronig, ed.), 343-349, VCH Publ. 1987.

11. Abbott, M.M.: Cubic equations of state. AIChE J. 19 (1973),596-601.

12. Chaves, H.: Phaseniibergange und Wellen bei der Entspannung von Fluiden hoher spezifischer Warme. Mitteilg. Max-Planck-Institut fUr Stromungsforschung, Gottingen, Nr. 77 (1984).

13. Collins, R.L.: Choked expansion of subcooled water and the I.H.E. flow model. Trans. ASME, J. Heat Transfer 100 (1978), 275-280.

14. Thompson, P.A.: A fundamental derivative in gasdynarnics. Ph.Fluids 14 (1971), 1843-1849.

15. Meier, G.E.A.: Zur Realgasdynamik der Fluide hoher spezifischer Warme. Habilitationsschrift, Universitat Gottingen, 1987.

16. Schlichting, H.: Boundary Layer Theory. McGraw Hill 1979.

17. Reid, R.C.; Prausnitz, J.M. and Sherwood, K.T.: The Properties of Gases and Liquids. McGraw Hill 1977.

Experimental Investigation and Computer Analysis of Sontaneous Condensation in Stationary Nozzle Flow of CO2-Air Mixtures

K. Bier, F. Ehrler and M. Niekrawietz

Institut fUr Technische Thermodynamik und Kaltetechnik, Universitat Karlsruhe (TH)

Summary

The influence of an admixture of air on the onset and the progress of spontaneous condensation of carbon dioxide in a stationary supersonic nozzle flow has been studied for mixtures with CO?-mole fractions ranging from 5 to 75 per cent. For mole fractions of 75 and 50 per cent, the supersaturation of the carbon dioxide at the Wilson point is scarcely influenced by the addition of air. In the range of lower CO 2-concentrations, the condensation of carbon dioxide is promoted by the presence of the non-condensing component, as was to be expected from various results reported in the literature.

The thermodynamic state in the Wilson point and the progress of condensation in the flow were calculated with a model of the condensation process based on the classical nucleation model and a formulation of droplet growth proposed by Gyarmathy. It is remarkable that the results of the calculation, obtained with a uniform choice of the free parameters of this condensa­tion model, agree well with the experimental results, both for pure CO 2 and for C0 2/air-mixtures.

Introduction

For various technical flow processes with spontaneous condensa­tion of a supersaturated vapour it is desirable to calculate the' state, where condensation begins, the Wilson point, and the influence of the condensation on the downstream flow field. Important parameters of such calculations are the formulation of the nucleation rate and of the further growth of stable nuclei. Theoretical models for calculating the nucleation rate by computer simulation are available, if at all, only for gases of simple molecular structure. Moreover, calculations of this kind appear to be rather complicated if integrated into a flow

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium G6ttingen, Germany, 1989 © Springer-Verlag Berlin Heidelberg 1990

114

calculation programme. Therefore, the classical formulation of the nucleation rate seems more suitable for this purpose. This model as well as the formulation of droplet growth contain parameters that cannot be prescribed from theoretical conside­rations; they have to be determined by comparison with experi­mental results.

The verification of a calculation mod91 for describing a con­densing flow seemed to be more secure if applied to mixtures of a vapour and a non-condensing inert gas because of the in­fluence of the gas molecules on the kinetics of nucleation and droplet growth. Since we had already investigated spontaneous condensation in a stationary supersonic nozzle flow of pure CO 2 in a wide range of density t1J, we carried out a similar syste­matic investigation of CO 2-air mixtures. The results of these measurements and of the calculation of the condensing flow will be reported in this paper.

Experimental

The measurements were performed with the flow system described in the preceding paper for low pressure-expansions of CO 2 [lJ. The CO 2-air mixtures were expanded to the atmosphere. Addi­tional effort was necessary to dry the components and to con­trol the dosage of CO 2 and air to the mixing chamber in order to maintain a stationary flow of a mixture with constant composition. Strong desiccation of the compressed air with a molecular sieve yielded a dew point of lower than -83°C; the residual moisture content was less than 0.4 ppm. A correspon­ding treatment of the CO 2 vapour reduced the moisture content at least below 5 ppm. The relative uncertainty of the mole fraction of CO 2 in the mixtures was 19ssthan 2 per cent.

A supersonic nozzle with rectangular cross-section, denoted as nozzle B2 in Fig.2 of the preceding paper, was employed [lJ. The expansion rate at the majority of Wilson points was between

115

8'10 4 s-l and about 15'104 s-l. The stagnation pressure varied

between 7 bar and 55 bar, the stagnation temperature between -43°C and +40°C. A block diagram of the flow system is shown in Fig.l. For further experimental details cf. reference [2.1.

As a typical example, pressure profiles for a CO 2-air mixture with 75 mole per cent CO 2 are given in Fig.2. The upper diagram shows an expansion without condensation, employed to determine the effective nozzle profile, whereas the lower diagrams indi­cate the deviation of the static pressure downstream of the Wil son poi nt.

For the evaluation of these measurements the thermodynamic

properties of the CO 2-air mixtures were described with the equation of state by E. Bender, applying the mixing and combi­nation rules proposed by Sievers and Schulz 13J. As has already been described for pure gases, the temperature TW at the Wilson point was calculated with this equation of state for the pres­

sure PW at the Wilson point and the stagnation entropy IlJ.

In order to calculate the isothermal supersaturation, the dew line of the respective mixture hat to be known 1). For a given temperature and the corresponding vapour pressure of pure CO 2 the molar density of saturated CO 2 vapour, gOco (T), was calcu­lated with the equation of state. From this val~e the molar total density of a saturated mixture with the CO 2-mole fraction

yco follows as 2

g'M (T)

1) Although the temperature Tw was lower than the temperature of the triple point of pure CO (216 K) for all the Wilson points determined in this inve~tigation, the supersaturation was related to the properties at the vapour-liquid equi­librium of the respective mixture. This seemed to be justi­fied by Ostwald's phase rule according to which the forma­tion of liquid droplets is to be expected at first. Perhaps, this assumption may not be valid any more for expansions with very low Wilson points (TW~150 K).

116

r-- -~~~ l

: calibratianr L _____ .J

to atmosphere.

bypass

r------, ~CQl ibrotion :

~ __ _.--~ L ____ __ ~

Fig.l. Block diagram of the flow system for the investigation of CO 2-air mixtures

0.6

05 Y C02 = 0.75 A eff Po= 9.98 bar

A * 0.1,. To= 265 .2 K 2.0

L isentrop ic Pa 0.3

1 5

0.2

0 .1 1.0

0.6 -,-OS Po=998 ba r

To =2494 K

P 04 isentropic

Po 0 .3

0.2 " " " 0.1

~~~.~~ o 1 2 3 I, 5 mm 6

f l ow coordinat e x

Fig.2. Typical pressure profiles of two expansions of a CO 2-air mixture without and with spontaneous condensation

117

Finally, the saturation pressure of the mixture is calculated with the equation of state for the mixture. In the following, the supersaturation of a mixture is defined with respect to the partial density of CO 2

= YC02 . 9M (T.p)

g"c02(T)

Experimental results for CO 2-air mixtures

(1)

Fig.3 shows two series of expansions for mixtures with 50 and 10 mole per cent CO 2, respectively. The pressure profiles were measured at constant stagnation pressures and different stagna­tion temperatures. They look very similar to those for pure CO 2 [1]. The Wilson point is shifted upstream with decreasing tem­perature.

The results for an equimolar C02~air mixture, obtained with several series of expansions, are presented in Fig.4. The pressure-temperature diagram in the upper part shows the expe­rimental Wilson pOints and the dew line, which is extrapolated below the triple point 1). For comparison, the frost line is plotted, too. The lower diagram shows the same Wilson points in a plot of the partial density of CO 2 versus temperature. Here, the full curve represents the Wilson line for pure CO 2, obtai~

ned in the preceding investigation [lJ. Obviously, spontaneous condensation of CO 2 in the equimolar CO 2-air mixture begins nearly at the same partial density as with pure CO 2 ,

Fig.5 gives a survey of the results for mixtures with different CO 2-mole fractions. The supersaturation at the Wilson point according to equation (1), is compared with the experimental Wilson line for pure CO 2 , A significant influence of the ad­mixture of air on the onset of CO 2-condensation is observed

118

P Po

010 Yco ,=0,10

Po = 28.76 bar D.06

3.5 ':'0 4.5 5.0 5.5

flow coordinate x

l

6.0 mm 65

Fig.3. Pressure profiles for a series of expansions af CO -air mixtures with 50 and 10 mole per cent of CO 2 , respecti~ely

16 ,---~--,----,---,--~----~,-,

bar

12 CO 2 / a i r

P 8

o

05

mol

dm3

0.3 g e02

0.2

0.1

Yeo/ OSO

Wilson -line dew pOint line, /' 1

extra pol /' / ),.// --1

_/ .... '\ I 1_._ -----==--;-.:.::.-d'--; frost lineJ

,- ,- 'I ------,----1-- "

Wilson-line pure C02

a

o dew point line" ,/ / extrapol . // /

/ / ".-,,// ,,/ --- .....-"

----_---- frost line o '--===-="--'-~=ct=-::-_-_-'-_'

140 150 160 170 180 190 200 K 210

T

Fig.4. Wilson points of an equimolar CO? - air mixture in plots of total pressure and of partial CO/-density versus tempera ­ture. The dew line of the mi xture is extrapolated below the triple paint of CO 2

15

10 3_ OJ

(J)

c 5 0

..... 0 ~

:::J -0 Ul ~

Q) 0.. :::J VI

5

150

tI

T

CO 2 ' air Yco z = 075

o <?

9

210 130

0>0 00

o())

150 170 K 190

T

Fig_5. Supersaturation at the Wilson point for CO 2-air mix­tures of different composition. The full curve is the Wilson 1 i ne for pure CO 2 [lJ

20

15

Sg.w

10

0.5 0.6

TlTc 0.7

1.0

119

Fig.6. Correlation of the experimental results for the super­saturation of CO?-air mixtures at the Wilson point as a func­tion of the normalized temperature and of the mole fraction of CO 2

120

only for the mixtures with 25 and 10 mole per cent CO 2 . For these mixtures the supersaturation at the Wilson point is markedly reduced by the presence of the inert component, espe­cially at lower temperatures. This tendency, which can also be derived from literature data [4J, was confirmed by additional measurements with a mixture of 5 mole per cent CO 2; these measurements were, however, limited to a small range of low Wi 1 son temperatures [2]. For CO 2-mol e fracti ons of 50 per cent and more there is practically no influence of the admixture of air on the onset of the spontaneous condensation of the CO 2 component.

These results have been summarized by an empirical correlation expressing the supersaturation at the Wilson point as a func­tion of the normalized temperature and of the CO 2-mole fraction (Fig.6). This correlation, given explicitly in reference [2], represents the experimental supersaturation values of about 80 Wilson points with a mean deviation of 4 per cent.

Calculation of the condensing flow

The first complete calculation for the onset and the progress of spontaneous condensation was described by Oswatitsch for a one-dimensional expansion of a vapour with ideal gas properties [5J. By means of a computer such calculations can now be per­formed even for vapours at high density with a complex equation of state to describe the real gas behaviour. In addition to the equations for the conservation of mass, momentum and energy and to the equation of state, a further equation for the increase of the mole fraction of condensate

Y cond = (2 )

has to be introduced (n molar flow rate of condensed CO 2 and

of the mixture, respectively). Following the concept of

Oswatitsch one obtains x

d Ycond = f dmC02' r

. J (x') . A (x') dx' dx'

x+

121

(3 )

In this equation MCO is the molar mass of CO 2 , J(x') is the nucleation rate and 2 A(x') the cross section of the flow at co­

ordinate x'; (dm CO /dx') is the local increase of the mass of 2' r

a spherical C02~droplet which has originated by homogeneous nucleation at x' and which has grown to radius r at the flow coordinate x. x+ is the flow coordinate where the expansion

intersects the saturation curve.

In the equation for the nucleation rate J the classical expression was supplemented by the non-equilibrium factor Z, introduced by Zeldovich, and by the so-called non-isothermal

f act 0 r e (c f. [21):

with 6G* work of formation of a critical cluster

PCO particle density of single CO 2 molecules C 2 mean molecular velocity of CO 2 molecules

0* surface of a critical cluster

condensation coefficient

In this expression, the condensation or "sticking" coefficient

a c is a first parameter that is not prescribed by theoretical considerations; it is determined by comparison with the experi­

mental pressure profiles.

The nucleation rate is sensitively dependent on the assumption

concerning the dependence of the surface tension of small drop­lets on droplet radius r. We applied the relation proposed by

To 1 man [6J

122

a (T,r I = Goo (T I 1+2·6/r

(5 )

The parameter 0 of this relation was also determined by compa­rison with the measurements.

A third parameter which was fitted to the experimental results

is the thermal accommodation coefficient a A in the formulation

of droplet growth as proposed by Gyarmathy [7J.

The complete set of equations for the one-dimensional super­

sonic flow was solved numerically, using the Runge-Kutta-Verner

procedure. Up to 70 classes of droplets originating within

successive intervals of the flow coordinate were treated. The

cal c u 1 at ion pro c e d u rei s des c rib e din de t ail i n ref ere n c e [2J.

In a first step, the free parameters of the condensation model

(ac , 0 and a A) we red e t e r min e d by com par i n g cal c u 1 ate dan d mea­sured pressure profiles. As an example, results for an expan­sion of the equimolar mixture are shown in Fig.7. In the left

part, the condensation coefficient a c was varied with constant values of the parameters a A and o. In the right diagram, the Tolman parameter 0 of the surface tension is varied, while a c and a A are kept constant. As a result of a great number of such comparisons it followed that the agreement between measured and

calculated pressure profiles was the best, on the whole, if the following set of parameters

O. 1 nm

is chosen 2).

a c O. 1 1 . (6 )

2) According to Tolman, the value of the parameter 0 should be between 25 and 60 per cent of the average molecular distance in the liquid phase. For CO 2 this distance amounts to about 0.4 nm; hence, the value 0= 0.1 nm is in agreement with Tolman's proposal.

P Po

0.1.

03

0.2

o

CO2 / a i r

Yeo 2 = 0.50

Po = 16.73 ba r

To = 243 .3 K

isentrop ic , /' without condens .

f low coordinat e x mm 5 0

--<,

CO2 / a i r

Yeo 2 = 0.50

Po = 16 .73 bar

To = 2/,3.3 K

B=O.32Bnm

isentropic, \ without condens . \ ,

\ experimental

6 :: O.Ot. nm

2 1. mm 5 fl ow coo rd inat e x

Fig.7. Calculated pressure profiles for an expansion of an equimola r CO 2- air mi xture (dashed curves) in comparison with exper i mental results

123

Some details of the calcu l ation of the condensing flow concer­

ning the expansion shown in Fig . 7 are demonstrated in Fig.8 and 9. The nucleation rate and the supersaturation pass a max i mum at the flow coordinate x = 2.3 mm . In that range, the mole

fraction of the condensate begins to increase significantly.

Accordingly, the maximum of the supersaturation is taken as c r iterion for the calculated Wilson po i nt. The upper diagram of Fig.9 shows the increase of the droplet radius for three clas­ses of droplets starting at different flow coordinates slightly

upstream of the Wilson point. Since t he supersaturation increa­

ses in that range, the starting size of these droplets, i.e. the critical radius of stable nuclei, decreases slightly and

passes a flat mi ni mum at the Wilson pOint. The radius of the droplets increases rapidly from this initial size to:=:::30 nm and

then grows more slowly to a final size which is practically

reached at the flow coordinate x:=:::3 mm. This corresponds fairly well with the pressu r e profile in Fig . 7.

124

1020'~--__ ----~---.----r---, 1

cm3 s

J 1010

nucleation rate

10°L--LJ-__ ~-L~L-__ -L __ ~ Fig.8. Calculated nucleation rate, supersaturation and condensate mole fraction as

6

5

4

3

2

o

, : Wilson-point

calculated

2 3 4 flow coordinate x

---

008

0.06

Ycond

0.02

a function of the flow coor­dinate for the expansion shown in Fig.7

---'-----/"

./

.-- ....... ·/~criticol radius r*

Wilson-point:

10-1 L ____ --L ______ -L--1' ____ -L-____ ---1 ______ -.l

o 2 3 4 mm 5 flow coordinate x

Fig.9. Growth of droplets generated at different cross-sec­tions upstream of the Wilson point (upper diagram) and size distributions of droplets for different cross-sections of the expansion shown in Fig.7 (lower diagram)

125

The lower diagram of Fig.9 shows the size distribution of all droplets passing a certain cross section; it was calculated from the results for droplet growth together with the corres­ponding nucleation rates. In the range where the two-phase equilibrium is nearly attained, (between x::::::3 mm and x::::::4 mm) the calculation yields a most probable droplet radius of about 40 nm and a maximum radius of about 70 nm.

With this calculation programme and the uniform set of para­meters (equ.(6» the theoretical Wilson points were calculated for all our experiments with pure CO 2 and with CO 2-air mixtu­res. In Fig.10 the calculated Wilson points of pure CO 2 are compared with the experimental Wilson line obtained in the previous investigation [1]. The agreement is very good in a wide range of temperature, from about 160 K up to 300 K. Only for expansions with very low temperatures of the Wilson point, - between 140 K and 160 K -, the deviations between the calcu­lated and the measured values of PW increase up to about 10 per cent. Corresponding results for the investigated CO 2-air mix­tures are shown in Fig.11. Again, the calculated Wilson points agree well with the experimental Wilson lines, even if the relative deviations in PW are somewhat larger than for pure CO 2, especially at temperatures below 150 K. However, in this temperature range, about 70 K below the triple point of CO 2, the assumption that the condensate will originate only in the liquid phase, may not be valid any morel). Some experiments with water vapour, reported in the literature, indicate that solid particles may be formed if spontaneous condensation takes place at temperatures that far below the triple point.

Besides of this uncertainty concerning the interpretation of the experiments with very low Wilson temperatures, the compari­son between calculated and experimental results indicates that a rather reliable description of a supersaturated condensing flow has been achieved in the present investigation.

126

80

bar

50

1.0

P

20

CO 2

W i lson- points calculated

vapo r pressure curve

Fig.10. Wilson points of pure CO?' calculated with the des­cribed condensation model, in comparison with the experimental Wilson line

15 bar CO 2/a ir 12

p 8

1.

0

15 bar Yeo 2

= 0.25

12

P 8 Wilson lines,

ex perimental

1.

11.0 150 160 170 180 190 200 K 21 0

T

Fig.ll. Wilson points, calculated for CO 2-air mixtures of different composition, in comparison with the respective ex­perimental Wilson lines

127

Financial support of this investigation by Deutsche Forschungs­gemeinschaft, Bonn - Bad Godesberg, is gratefully acknowleged.

References

1. Bier, K.; Ehrler, F.; Theis, G.: Spontaneous condensation in stationary nozzle flow of carbon dioxide in a wide range of density. IUTAM Symposium: Adiabatic Waves in Liquid-Vapour Systems, Gottingen, 1989.

2. Niekrawietz, M.: Experimentelle Untersuchungen und Modell­rechnungen zur spontanen Kondensation in DUsenstromungen Ubersattigter Kohlendioxid/Luft-Gemische. Dissertation, Universitat Karlsruhe (TH), 1989.

3. Sievers, U.; Schulz, S.: Berechnung thermodynamischer Eigen­schaften fluider Gemische mit einer Zustandsgleichung unter Verwendung neuer Mischungs- und Kombinationsregeln. Forsch. Ing.-Wes. 48 (1982) 143-153.

4. Barschdorff, D.: Carrier gas effects on homogeneous nuclea­tion of water vapour in a shock tube. Physics of Fluids 18 (1975) 529-535.

5. Oswatitsch, K.: Kondensationserscheinungen in Uberschall­dUsen. ZAMM 22 (1942) 1-14.

6. Tolman, R.C.: The superficial density of matter at a liquid-vapour boundary. J. Chem. Phys. 17 (1949) 118-127.

7. Gyarmathy, G.: The spherical droplet in gaseous carrier streams: Review and Synthesis. In: Multiphase Science and Technology, 1, Hemisphere Publ. Corp., Washington & London, 1980.

Spontaneous Condensation in Stationary Nozzle Flow of Carbon Dioxide in a Wide Range of Density

K. Bier, F. Ehrler and G. Theis

Institut fUr Technische Thermodynamik und Kaltetechnik, Universitat Karlsruhe (TH)

Summary

The onset and the progress of spontaneous condensation in a supersaturated, supersonic nozzle flow of carbon dioxide has been investigated by precise measurement of the static pressure along the nozzle axis. The supersaturation at the Wilson point, where condensation begins, has been determined for pressures from below the triple point up to about 90 per cent of the cri­tical pressure. The Wilson lines of carbon dioxide, obtained from series of expansions performed with two nozzles of diffe­rent divergence angle, are compared with results for two refri­gerants, CF Cl and CHF Cl, and for water vapour. For compa­rable flow ~eo~etry the 2Wilson lines of carbon dioxide and of the refrigerants are nearly identical in a normalized pressure­temperature diagram. For water vapour the supersaturation at . the Wilson point is lower than for the non-associating sub­stances.

Introduction

The phenomenon of spontaneous condensation in a supersaturated vapour is of interest in different fields of natural science and in various technical applications, e.g. wind tunnels and steam turbines. Expecially with respect to Organic Rankine Cycles for heat recovery there exists some interest in the con­ditions for the onset of spontaneous condensation in the so­called Wilson point and in the influence of the released heat of condensation on the flow field for different working fluids. From this point of view, we started systematic investigations on spontaneous condensation of pure vapours in stationary su­personic nozzle flow with the intention to cover a wide range of thermodynamic state for substances of different molecular structure. In this paper we present experimental results for carbon dioxide and compare them in a first step with results

G. E. A. Meier· P. A. Thompson (Erl!I.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Goltingen, Germany, 1989 © Springer-Verlag Berlin Heidelberg 1990

130

Table 1. Thermodynamic and molecular data of the investigated substances.

H2O CO2 CHF2Cl (R22)

CF2Cl 2 (RI2)

~lolar moss g/mol 18,02 44,01 86,47 120,92 Dipole moment Debye L87 ° 1,42 0,51 Crit. temperature K 647,25 304,2 369,25 385,15 Cr i t. pressure bor 220,94 73,84 49,90 41,41 Cr i t. dens i ty k9/dm3 0,317 0,466 0,513 0,555 Satur. temperoture K 373,15+) 216,6') 232,2+) 243,35+)

Condens. entholphy kJ/kg 2255,50+) 352,20 ) 233,7+) 166,5 +)

0) p = 5,18 bor (triple point); +) p = 1 otm

for the refrigerants R12 (CF 2C1 2 ) and R22 (CHF 2C1). In a second

step the comparison is extended to water vapour, i.e. to a sub­stance associating by hydrogen bonds. Thermodynamic and mole­

cular data of these test fluids are compiled in Table 1.

Experimental

A schematic diagram of the apparatus installed to achieve a

stationary flow of vapour is shown in Fig.1; it had been used previously for the investigation of the refrigerant R22 [11. The vapour is expanded in a supersonic nozzle from the stng­nation chamber into the condenser which is cooled by evapo­rating ammonia. The condensate is collected in the reservoir

and flows, slightly subcoo1ed, to the feed pump which delivers the fluid at a supercritica1 pressure and with a very even flow

rate into the flow heater. Here, heat is supplied in eight

electric heater elements, each one having a separate tempera­ture control. The superheated vapour is throttled to the stag­

nation pressure Po in a controlled throttle valve and returns

through a thermostatted section to the stagnation chamber. The

mass flow rate in the nozzle, depending on substance and expe­

rimental conditions, is limited by the available refrigerating capacity which amounts to about 30 kW at a condensation tem­

perature of -30°C.

3

specific ent halpy h

Fig.l. Schematic diagram of the flow system; the process is demonstrated in the inserted pr essure - enthalpy diagram.

type A type 8

A1 B1 B2

t hroa t 1. .0mm 0.0. H =0.75 mm 0.67 mm 2.0mm i .O. B=2.S 1 mm 2 51 mm

mouth 6.1 mm 0.0. H= 1.82 mm 2.71. mm 2.0mm i o. B = 2.S1 mm 2. 5 1 mm

distance 21. .l.mm 21. .1. mm 9. 5 mm throat-mouth

Fig.2. The two nozzle types employed in this investigation.

131

132

With CO 2, the stagnation temperature was varied between -37°C and +142°C, the stagnation pressure between 4 bar and 130 bar. At stagnation pressures of 20 bar or less, a closed-cycle ope­ration was not possible because the vapour pressure at the lowest attainable temperature of the condenser was too high (14.3 bar at -30°C). Under these conditions the vapour was ex­panded to the atmosphere as shown in the upper ri~hthand of Fig.l. Because of the over-expansion in the nozzle Wilson points could be detected at static pressures as low as 0~2 bar.

The addition of condensation enthalpy to the supersonic flow causes an increase in static pressure, in temperature and in vapour density and a decrease in the flow velocity, compared to the non-condensing isentropic flow. Since deviations of the static pressure can be determined reliably, the onset and the progress of condensation were observed from precise measurement of the static pressure along the nozzle axis.

Due to the given limits in mass flow rate and to the relative high vapour density in the high-pressure experiments, nozzles of rather small cross section had to be employed. Fig.2 illu­strates the two types of Laval nozzles applied in this investi­gation. Type A is of rotational symmetry with a central pres­sure probe; the flow is annular. Type B has a rectangular cross section; the static pressure probe is integrated into one of the non-profiled side walls. The characteristic dimensions of these nozzles are given in Fig.2. More details of the experi­mental arrangement and of the measuring technique are described elsewhere [1,2J.

Results for carbon dioxide

Fig.3 shows typical pressure profiles of CO 2-expansions ob­tained with the rectangular nozzles B1 and B2 at equal stagna­tion conditions. The expansion rate

. p = 1 dp

p'df

0.8 : p= 2. 10 4 5 - 1

0.6 po = 45bar To=300 .12K

P 0.4 c

P o . ~ rQJOv "§ :0

0.2 0 <f) C i sentropic

·0 a.

0 z a

P -Pi s ~~LZ l P is

0 CO 2

-4 0 4 8 mm 12

133

nozz le 8 2 : P= 8 .101. 5 . 1

c ·0 a.

z

IT -4 0 4

Po= 45 ba r To :3000 5 K

~o , l 8 mm 12

flow coordinate x

Fig.3. Typical press~re profiles of CO?-expansions with diffe ­rent expansion rates P. The lower diagrams show the relative deviation of the pressure in the condensing flow from that of a ~on-condensing isentropic flow.

is about four times greater for nozzle B2 than for nozzle Bl;

the values of P given in Fig . 3 are average values for the

supersonic part of the expansions. Both expansions intersect the saturation line in the subsonic part of the nozzle.

To locate the onset of condensation, the measu r ed pressure

profiles are compared with reference profiles calculated for one-dimensional non-condensing isentropic flow. For these

calculations the equation of state by E. Bender [3J, which satisf i es the Maxwell criterion , is extrapolated beyond the saturation line into the metastable super s aturated vapour

state. Boundary layer and other dissipative effects are con­sidered by determining an effective nozzle profile from an expansion starting from the same stagnation pressure at a highe r stagna t ion temp erature and extending completely in the

superheated regi on [ 1 ,2]. The Wi 1 son poi nt is defi ned as that

metastable vapour state , where the relative difference between

the measured and the calculated pressure profile increases sig ­

nificantly (cf. lower part of Fig.3). The temperature at the

Wilson pOint, TW' is calculated with the equation of state from

134

the measured pressure at this point, PW' and from the stagna­

tion entropy So{To'po)'

Fig.4 shows a comparison of two series of expansions obtained

with the nozzles Bl and B2 at equal stagnation pressure and

varied stagnation temperature. The expansions with To~ 325 K show no condensation effect within the nozzle; they were used for calibration of the effective nozzle profile. With decrea­

sing stagnation temperature the Wilson-point moves upstream. Downstream of the Wilson-point the supersaturation is quickly reduced and the expansion obviously tends to a two-phase flow

in which equilibrium between vapour and droplets is nearly attained. For equal values of the stagnation temperature, the Wilson points obtained in the faster expansion (nozzle B2) are

found at a somewhat lower pressure ratio p/po' i.e. at a higher supersaturation than with nozzle Bl.

In Fig.5 the experimental Wilson points of CO 2 , obtained with

both nozzles, are plotted in a pressure-temperature diagram and

in a temperature-entropy diagram, respectively. Whereas the Wilson lines of both nozzles are nearly coincident in the pres­

sure-temperature diagram, a distinct difference is found in the entropy diagram, and the influence of the expansion rate on the

supersaturation at the onset of condensation is clearly percep­t i b 1 e .

Comparison between carbon dioxide and the refrigerants

Experiments with the refrigerants R12 and R22 were performed

with the annular nozzle Al having a similar profile as nozzle Bl. The expansion rates were comparable with those of the CO 2-

expansions in nozzle Bl. The results for R12 are shown in Fig.6; Wilson points could be determined only in the range between 50 and 90 per cent of the critical pressure. Above the

upper limit, about 6 K below the critical temperature, the

Wilson line cannot be distinguished from the saturation line. This is reasonable because of the rapidly decreasing values of

135

O.B ~---,------,---.......,

0.6

L Po

0.4

0.2

a -5 a

Po= 45 bar

nozzle 81

31060K/

325.76 K

5 mm 10 flow coo rd inate x

Fig.4. Pressure profiles for two series of CO - expansions starting at equ~l stagnation pressure and at different stagnation temperatures.

bar,------.-----,------,----.-, 40 ,---,----r----.--~

p

70

60

50

40

30

20

10

e nozzle B 1 '" nozzle B 2

I /

. / SpinO dol cur ve " ;' '"

" . e i~ I '"

./ ,"e

/ '" / ~

./ '"~ '" ./ €je saturation

// ~ // ~

'"~ '"~

°C

20

o

T

-20

-40

-60

'" \ \ \

"' i i \ i i

spinoda l .--l cu r ve I

i i i i

saturated vapour

- - - -- I -W~L~~N- ~n-e -

i nozz l e B 2

i " triple point .

0L-----L-----L-----L---~ -80 L-----~1-----L-----L----~ -80 - 50 -20 10 'C 40 1.0 1.2

T

1.4

5

1.8

Fig.5. Wilson lines of CO , obtained with the rectangular nozzles Bl and B2, in a pr~ssure - temperature and in a tempera ­ture - entropy diagram. For the evaluation of the lowest Wilson pOints the properties of the vapour - liquid equilibrium were extrapolated beyond the t r iple point.

136

condensation enthalpy and surface tension when approaching the critical point. On the other hand, the lower limit of the Wilson line of R12 was surprising insofar, as the strength of the pressure effect, depending on the amount of the released condensation enthalpy, normally increases with decreasing temperature.

In order to understand why no spontaneous condensation of R12 could be observed at lower pressures, the results of Fig.6 were compared with those of previous measurements with refrigerant R22 [11. In the upper part of Fig.7 the experimental Wilson points of both refrigerants are presented in a common diagram of normalized pressure plpc versus normalized temperature T/T c . Both Wilson lines can be represented, within the experimental uncertainty, by the same function. When this common Wilson line is transformed to an extended temperature-entropy diagram of R12, the dashed line in Fig.8 is obtained. This line passes a flat entropy maximum just below the lowest measured Wilson

point. Therefore, expansions with so> smax' which intersect the saturation line at lower pressures, do not reach the Wilson line, i.e. the vapour is not supersaturated to a degree neces­sary for spontaneous condensation in expansions with comparable expansion rates. At least, this is valid if the expansion does not extend to a very low temperature l )

In Fig.7 the common Wilson line of the refrigerants R12 and R22 is also compared with the Wilson points of CO 2. In this norma­lized presentation the Wilson points of CO 2 agree very well with the Wilson line of the refrigerants. The relative diffe-

1) From the same diagram (Fig.8) it is evident, why spontaneous condensation was observed d~wn to very low pressures with the refrigerant R22. On the other hand, the problems of spontaneous condensation will be practically irrelevant for turbine flows of such retrograde fluids as e.g. Rl14 (C2F4Cl~ If it is assumed that the same normalized Wilson line wTl r be valid for this fluid, it follows from Fig.8 that sponta­neous condensation would be possible only in a very small range below the critical pOint. In this range, however, the influence of condensation on the flow field is negligible [1].

45 ,-----,---_,~--_,----_,

bar R12

p

40

35 ;:

spinodal cu rve , l /

30 /

/ /

/

i

25 // "- satura lio n

////

15 L---~----~----~----~

120',---------,.------,.------,-------

' c

110 ...... ,

100

T

90

80

70

\ \ \ i

spi nodal - j cu rve j

i i i i i I i I WILSON

i i

____ saturated va pour

60~~~----~--~~~-L~

60 75 90

T 105 ·C 120 100 103 106

S

1.12

137

Fi g .6. p,T- and T,s-diagram of r~frigerant R12 with Wilson lines determined with nozzle Al (P ~ 7500 1/5).

1.0

0.8

0. 6

£... Pc

0 .1.

0 . 2

0

-- spinodal curve - Wi lson line - saturati on

R22 /' /

I /' ...... ...... . / . / -...... ,-

0.7 0.8

TlTe

) / .

0.9

1.0

0.8

0.6

~ Pc

0.1.

0.2

1.0

Fig.7 . Experimental­ly determined Wilson points of R12, R22 and CO in a no r mali ­zed pr~ssure - tempera­turediagram. The Wilson points of R12 and R22 are interpo ­lated by a common Wilson line.

138

0,95

0,90

lO,85

Tc 0,80

0,75

0,70 /

/ /

/ /

R 114

/ /

/

/ /

/

0, 65 O°C-=±cJl ___ -L ___ ~ __ __._J ___ ~--------'

0,95 1,00 1,05 1,10 1,15 1,25 5

Fig.S. Comparison of the vapour pressure curves and of the common Wilson line of R12 and R22 in a common temperature­entropy diagram; O--<>Wilson line, experim., ---Wilson line, extrapol., -- saturation curve

rence in the normalized pressure, PW/pc' is less than 4 per cent (lower diagram). Considering the difference in the mole­

cular structure of CO 2 and of the investigated refrigerants, it may be assumed that the same Wilson-line, formulated in norma­lized pressure and temperature, will also be valid for other

substances not yet investigated. Of course, a reservation has to be made concerning substances with an essentially different

type of intermolecular interaction, expecially substances asso­

ciating by hydrogen bonds. As an example of this group of sub­

stances water vapour will be discussed in the next section.

Comparison with water vapour

Numerous experimental investigations on spontaneous condensa­

tion have been performed with water vapour, either pure or in mixtures with inert carrier gases. Thereby, different experi­

mental techniques and evaluation methods have been employed by different authors. In order to guarantee a really consistent

139

comparison of our results for carbon dioxide and the refrige­rants with those for water vapou~ a limited number of conden­sation experiments with pure water vapour has been carried out applying precisely the same measuring technique and evaluation method as with the other fluids [4]. These experiments were performed in a nearby heating and power station, because supercritical heat addition to water was not possible.in the flow system in our laboratory. Due to the conditions of the steam supply, the expansions were carried out in a medium pressure range, yielding Wilson points between 0.6 and 6.8 bar. Three Laval nozzles of rotational symmetry were employed; one of them had exactly the same geometry as nozzle Al of Fig.2.

The Wilson points obtained with this n6zz1e match very well with comparable literature data. Therefore, it seemed to be justified to include further literature data for water vapour into the intended comparison. After an accurate re-evaluation of the relevant literature data for water vapour, using Pol­lak's equation of state ~] and the same criterion for defining the Wilson point, we obtained an extended average Wilson line which is representative for the relative high expansion rates realized in supersonic nozzle flow ~].

This average Wilson line for water vapour is shown in the right diagram of Fig.9 as center line of the hatched zone, which re­presents the scatter of the Wilson points for all evaluated nozzle experiments 2). For comparison, this normalized pres­

sure-temperature diagram also contains the Wilson lines of CO 2 for the nozzles Bl and B2 as well as the saturation curves and the spinodal curves of both fluids, indicating the range of the metastable, supersaturated vapour state. Obviously, the norma­lized Wilson lines of water vapour and of CO 2 differ markedly, contrary to the result found for CO 2 and the refrigerants. This agrees with the well-known fact that the normalized presenta-

2) The width of this Wilson zone can be explained mainly by differences in the expansion rate of the different experi­ments.

140

P Pc

TlTc 1000;::.5~ __ =;0.-=-6_---=T-_OT·8_'-711.0

C02 extrapol . //

/ /

/

",/ /

/ /

10-3 L-L----L-_--'-_ _ '-----_-':-_~. 2.0 1.8 1.6 11.

- Tc /T

C02 } saturat ion curves H20

O~ O~ 1.0

T fTc

Fig.9. Comparison of the vapour pressure curves of different substances (left) and of the Wilson lines of CO and of water vapour (right) in a normalized pressure - temperafure diagram.

0.20

<l-

~ 015 f-<l

'0 0 U

.D

~ 005

'" L

0

f-u 15

t: V1

<l-

<l-10

c 0

"2 ;:)

0 5 ~ '" <l-OJ Ul 0

01,

spinodal cu rves H2O'. \ . CO 2

"\ Y \ \

\ \ \

\

H2O

spinodal curves

H20,-\1 CO2

( I. II

\ \. \\

0.6

TITe

CO 2

=8 10" S-1 \ =210" S-1

CO 2

.8 · 10"S-1

=2 10"S-1

0.8 1.0

Fig.10. Supersaturation and relative isobaric sub­cooling at the Wilson lines of CO and of water vapour; the h&tched zone indicates the scatter of the re-eva­luated literature data for H20.

tion of the vapour pressure curve of water also deviates di­

stinctly from those of other, non-associating substances (cf.

left diagram in Fig.9).

1~

The quantitative difference in the spontaneous condensation of

CO 2 as an example for a non-associating fluid and of water vapour is shown more clearly in Fig.10, where the supersatura­

tion and the relative subcooling,i .e. the isobaric subcooling

~T related to the respective saturation temperature, are plot­

ted as a function of the normalized temperature. Clearly, the onset of spontaneous condensation requires a lower supersatura­

tion or a lower relative subcooling for water vapour than for CO 2 , the difference increasing with decreasing temperature. Apparently, the strong interaction by hydrogen bonds promotes

nucleation and thus the process of spontaneous condensation of

water vapour.

This investigation was supported by Deutsche Forschungsgemein­schaft, Bonn - Bad Godesberg.

References

1. Bier, K.; Ehrler, F.; Kissau, G.; Lippig, V.; Schorsch, R.: Homogene Spontankondensation in expandierenden Dampfstrahlen des Kaltemittels R22 bei hohen normierten DrUcken. Forsch. Ing.-Wes. 43 (1977) 165-175.

2. Theis, G.: Spontankondensation in Ubersattigten Dampfstro­mungen von Kohlendioxid und von Difluordichlormethan. Dissertation, Universitat Karlsruhe (TH), 1985.

3. Bender, E.: The Calculation of Phase Equilibria from a Thermal Equation of State Applied to the Pure Fluids Argon, Nitrogen, Oxygen and their Mixtures. C.F. MUller-Verlag, Karlsruhe, 1973.

4. Hechler, C.: Untersuchungen zur spontanen Kondensation in Ubersattigten Stromungen von Wasserdampf und Entwicklung eines Streulichtverfahrens zur Bestimmung der TropfengroBe und -konzentration. Dissertation, Universitat Karlsruhe (TH), 1988.

5. Pollak, R.: Die thermodynamischen Eigenschaften von Wasser. Dissertation, Ruhr-Universitat Bochum, 1974.

An Asymptotic Predictive Method for Gas Dynamics with Nonequilibrium Condensation C.F.DELALE

Department of Mathematics Bogazici University, Istanbul

Summary An asymptotic method for one-dimensional nozzle flows with nonequilibrium condensation that reveals the structure of possible condensation zones is presented. The streamtube formulation for expansion flows on walls with nonequilibrium condensation employed with the asymptotic method is discussed for both smooth flows and flows with an embedded, frozen, oblique shock wave. In particular the location of the oblique shock wave is predicted by employing Barschdorff's shock fitting technique. The extension of the method for shock tubes is also summarized.

Introduction

This paper is a brief discussion of some gas dynamic problems

(one-dimensional nozzle flows, steady two-dimensional supersonic

expansion flows, etc.) with nonequilibrium homogeneous condensation

using the asymptotic approach [1]-[3]. This approach, in constrast

to direct numerical calculations based on various theories of

condensation [4]-[7], offers analytic structure built in by

homogeneous nucleation and droplet growth. It is also predictive

since it explicitly determines the onset of condensation and

location of the characteristic condensation zones. In addition

the condensing substance need not be specified and the nucleation

and droplet growth rate laws may be left almost as arbitrary in

the general analysis. Detailed numerical calculations based on

this approach are in progress and will be communicated later.

Nozzle Flows

We consider the supersonic expansion of a condensible vapor with

a carrier inert gas through the diverging section of a Laval

* This paper is dedicated to the memory of Professor Joseph H. Clarke.

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Gatlingen, Germany, 1989 © Springer-Verlag Berlin Heidelberg 1990

144

nozzle. With a suitable normalization carried out in [2], the

mixture flow and state equations for one-dimensional nozzle

flows take the form

p u A = u c ( 1 )

du dp p u (2) dx dx

2 2 C T + u - Lg Cpo + u po

2 2 c

(3)

and

p ( 1 -1

g) pT - H . (4)

In Eqs. (1)-(4) x is the scaled axial coordinate measured from

the throat of the nozzle, A is the normalized nozzle area, p, p,

T and u are respectively the normalized pressure, density,

temperature and flow velocity of the mixture, g is the weighted

condensate mass fraction, Cpo is the normalized mixture specific

heat, L is the normalized latent heat of vaporization and H a

suitable normalization constant. The subscript c refers to

saturation conditions.

The nonequilibrium rate equation is constructed from a suitably

normalized local nucleation rate

J -1

L: (p, T , g) exp [- K B (p , T , g) 1

and a local droplet growth rate law with a power law radius

dependence

dr

dx A r-n n(p,T,g)

1

where r is the normalized radius of the droplets, B is the

normalized activation energy,L: is a normalized functional factor

for nucleation, n is a normalized functional factor for droplet

growth, K is the nucleation parameter assumed to be small and

145

Al is the droplet growth parameter assumed to be large (for

details see [3]). If the critical radii of condensation nuclei

are neglected, the resulting condensation rate equation takes

the form

g(x) A 3 fX M (x, 0 L: (t;)A(t;) exp[-K- 1 B (t;) Jdt; x c

where we have defined

M(x,t;) 2

(5)

(6 )

with A = [A 1 (1+n)J 1/(1+n) and m = 1/(1+n). In particular the

case m=1 (n=O) corresponds to flows with large Knudsen number

where the droplet growth law becomes independent of the droplet

radius.

The condensation rate equation (5) contains two disparate

parameters K and A. K « 1 signifies a small nucleation rate and

A » 1 signifies a small droplet growth time. In the asymptotic

method we thereby study the double-limit process K + 0 and

A + 00. This limit has a physical counterpart in observations and

experiments.

We let Xc be the scaled distance to the saturation point and x k be the scaled distance to the onset point, a point in the onset

zone that represents the beginning of the collapse of the

supersaturated vapor state (Fig.1). In Eq. (5) the double limit

K + 0 and A + 00 is ordered such that a geometrically defined

onset zone appears for finite xk-xc . The condensate-free flow at Xc

is frozen at g=O (isentropic flow) and all the flow variables

at saturation are to be calculated from the frozen solution.

However, this solution can not persist throughout the interval

Xc ~ x ~ xk . For this reason we introduce nearly frozen flows.

For such flows it is more convenient to use the momentum theorem

pA + p U Z A 1 + U Z + fA pdA c 1

(7 )

instead of Eq. (2). In Eq. (7) the major contribution to the

integral on the right hand side near Xc arises from the frozen

solution so that

146

A A J pdA '" J Pf dA 1 1

(8)

where the subscript f denotes frozen solution. Furthermore since

L(T) is a slowly varying function of T, T can be downgraded by

the frozen solution to yield

(9)

In the nearly frozen approximation given by Eqs. (8) and (9), the

algebraic set of Eqs. (1)-(4) with Eq. (2) replaced by Eq. (7) can

be solved to yield the local nearly frozen solution [2]

u(x) u (g(x) ,x) , p(x) = p (g(x),x), etc. (10)

in the entire invertal Xc ~ x ~ x k . For nearly frozen flows we

also have

B(x) B (g (x) ,x) , z (x) Z (g (x) I x), etc.

so that the condensation rate equation

x exp {_K- 1 B [g(I;), I;]} dl; (11)

with

M(x,!;) x - 3m J rl[g(n),n]dn) (12 ) I;

becomes a nonlinear Volterra integral equation for g(x). In the

limit K + 0 Eq. (11) is dominated by the exponential factor

containing the activation function B shown in Fig.1. From the

behavior of B it follows that six physically distinct zones which

are in one-to-one correspondence with the characteristic condensa­

tion zones can be distinguished. These zones are the initial

growth of condensate (IGZ), further growth (FGZ) , rapid growth

(RZ), onset (OZ), nucleation with growth (NZ) and droplet growth

zone (DGZ).

B

J 1\ I I

I :~ I I\~

I I I I I

1:' o

u~ :0 I I I I

I G Z

147

B, frozen

x

FIG 1 Typical variation of the normalized activation function Band the six characteristic condensation zones occuring in nozzle flows.

148

within the finite IGZ, B almost equals Bf . This is the only zone

where B is known a priori. For the finite FGZ, B becomes

numerically distinct from Bf by definition. In this zone dB/dx=0(1

on measure K as K + O. In this and all subsequent zones d 2B/dx2

is numerically compared to unity. The rapid growth zone (RGZ)

begins where dB/ dx diminishes to O(K 1/2) as K + 0 and ends a,s

x + x£' the turning point of B. In this zone the magnitude of g

grows rapidly from exponentially small values to 0(K1/ 2 ) on

measure K. The onset zone (oz) ends at x=x£ and extends upstream

to define a convenient zone for the collapse of the supersaturated

vapor state. As x + x£' the exponential fUnction in Eq. (11) exhibi

a marked peak. Thus g(x) will exhibit of finite slope change and

thus a very large value of d 2g/dx 2 , but with very little change

in g. These three features occur over a vanishingly thin zone and

define geometrically the unique onset zone containing the onset

point xk ' the beginning of the collapse of the supersaturated

vapor state. Our result for x k is

as K + 0 •

Indeed OZ and RGZ overlap such that the thickness of OZ,

00z 0(K1/ 2 ) and (orgz/ooz) = 0(10) on measure unity. The

nucleation zone with growth begins at x=x£ and extends downstream 1/2 ' a distance 0nz = O(K ) on measure K. Moreover NZ and RGZ+OZ

have some similar, symmetric, overlapping features in the

asymptotic sense. Finally, in the droplet growth zone (DGZ) g

grows to 0(1) on measure K and equilibrium is approached over

thickness 0d 0(K1 / 3 ). The magnitudes in each zone is gz summarized in Table I [2].

The nearly frozen solution for the dependent flow variables in

the'zones IGZ, FGZ, RGZ and OZ given by Eq. (10) require the

solution of the rate equation (11) for g(x). For evaluating the

integral in Eq. (11) as K + 0 we consider Laplace's method.

According to this method the only important contribution to the

integral arises from the immediate vicinity of the point at

which B assumes its least value. With respect to B in the zones

discussed we identify (i) end point minima for the zones IGZ,

FGZ and RGZ and (ii) a turning point for the zone oz. The

corresponding asymptotic solution for g(x) in these zones togetheI

149

with the onset criterion for xk are carried out in detail in

[2]. For NZ and DGZ, the dependent flow variables can now be

expanded in appropriate perturbation series about the frozen

solution so that consisting matching with the upstream zones is

obtained. Details can be found in [2]. It follows that NZ has

thickness 0(K1/ 2 ) and DGZ has thickness 0(K1/ 3 ) on measure K. In

the droplet growth zone if we introduce the scaled coordinate

X of 0(1) on measure K as in [2] and neglect terms of O(K 1/ 2 )

and 0(K1/~), we obtain the following structured-shock flow

relations:

p(X) u(X) (13)

(14)

1 2 Cpo Tf(xk ) + 2 uf(xk ) (15)

p(X) [1-H- 1 g(X)] T(X) , (16)

and

X do (17) * n (0)

where n*(o)= n~1 n(X) (the subscript k refers to onset conditions

and all barred variables denote variables with magnitude 0(1) on

measure K). Eqs. (13)-(17) are the rectilinear, one-dimensional

flow equations of a normal shock wave structured or resisted by

droplet growth (similar equations were derived in [1] somehow

differently). The solution of the structured shock relations are

given in [2] with the downstream limit of solutions corresponding

to saturated states in local thermodynamic equilibrium. The droplet

growth zone discussed above as a structured shock solution is

accompanied by the release of a remarkable amount of heat of

condensation. If the initial specific humidity of the mixture

exceeds a certain limit, then the amount of heat released may

become very large and it may outweigh the influence of increasing

cross section moving the flow Mach number toward unity. When this

occurs, the flow is said to be thermally choked and heat addition

in excess of thisamount is called supercritical. In this case the

150

TliliLE I.. Orders of magnitude, on measure l~ as K __ 0 t of key quanti ties in the si>:, successive charac­

teristic zones of cOnUensation along the· nozzle fl0l4.

'J

E!l. ax

;0. dx2

IGZ fGZ RGZ OZ uz IJi;2

0(1) 0(1) O(Kl12 ) O(Kl12 ) O(Kl12 ) 0(Kl/3 )

expo small expo small from ey.p. small O(Kl12 ) ~o O(Kll2) O(Kl12 ) 00)

expo small .xp. small from expo small 0(;;-113) ~o 00) oell 00)

O(K-In ) O(K- l12 ) 0(1;-2/3)

9" -------------

o o x

FIG. 2 Typical droplet growth shock structure for subcritical (I) and supercritical (II) flows.

151

solution of Eqs. (13)-(17) of the droplet growth shock structure

has to be modified by the inclusion of a frozen normal shock

* wave [2] . Furthermore the location of the shock X in the

scaled coordinate of the droplet growth zone is determined by

a shock fitting technique originated by Barschdorff [8]. Fig.2

shows typical droplet growth shock structure for subcritical and

supercritical flows.

Two-Dimensional Supersonic Flows We now extend the asymptotic analysis of the previous section

to solve two-dimensional supersonic expansion flows with

nonequilibrium condensation. In this case it is convenient to use

the streamtube formalism (e.g.see [9]-[11]) with natural coordinates

as shown in Fig.3. With the normalization carried out in [2],

the flow, state and rate equations take the form

puA = u along a streamtube, (18 ) c

au ap (19 ) pu as as

2 ~ 2 a6 u (20 ) P = pu

R an as s

C T + 1 2 - Lg C + 1 2 along a streamtube, u u po 2 po 2 c

(21)

-1 (22) p pT (l-H g) ,

and

-ex>

along a streamtube (23)

where

M(s,~) (24 )

152

In the above equations s is the scaled streamwise coordinate, n

the scaled normal coordinate, A=~n/~nc with ~n denoting the

normalized separation between streamlines constituting a

streamtube, Rs is the normalized radius of curvature of the

streamlines,8 is the direction of flow velocity, and the rest of

the variables are the same as those of the previous section.

Moreover, for use in nearly frozen flows the momentum theorem

2 2 A (p+pu )A = 1 + u + J pdA along a streamtube (25) c 1

can be used instead of Eq. (19). Equations (18), (21)-(25) are

identical with the governing equations for nozzle flows if the

streamwise coordinate s is replaced by the axial coordinate x

and if A(s) is thought to act as the given nozzle cross­

sectional area A(x) .

The construction of the network of streamtubes will logically

start on the prescribed curved solid boundary along which the

flow expands. The curved surface then acts as the lower streamline

of the first streamtube and the construction is completed if the

upper streamline can be drawn for an arbitrarily initial

separation. The network of streamtubes can then be generated

successively. For this reason we consider an initial supersonic

flow field designated by points 0, 0', etc. and separated by

arbitrarily small normal distances ~n(O), ~n1 (0), etc. as shown

in Fig.3. We assume that the lower streamline passing through ° is known. We further assume that upstream of point (s,n), point 1

in Fig. 3, is known ( s=O corresponds to the initial supersonic

field). To extend the construction we draw a normal line of length

~n A(s+~s) from point 3 to arrive at point 4 where c

(26 )

In order to complete the construction of the upper streamline

from point 2 to pOint 4, we evaluate 84 from

~s (27)

~n (s) +~n1 (s)

Moch

FIG 3

FIG 4

sl r eom l ines

5 ,

I I 6 s +ttos6 n

'" t n 8, I eto n 6s 6n +-6 s

3 d s

so lid wall

Natural coordinates and construction of network of streamtubes in expansion flows.

7

Configuration of streamtubes in supercritical expansion flows with an embedded oblique shock wave.

153

154

obtained by introducing the second dynamical consideration needed

for two-dimensional flows, i.e. Eq. (20). The locus of the direc­

tions characterized by ~2 and 64 extend the upper streamline from

point 2 to point 4, hence extending the construction of the

streamtube from s (point 1) to s+~s (point 3) along the given

lower streamline. The extension of the network of the next

stremtubes from s (points 2,5, etc.) to s+~s (points 4,6, etc .. )

follow in the same manner. The flow variables at s+~s (points 3,

4,6, etc) can then be evaluated by substituting for A from

Eq. (26) along each streamtube into the nozzle flow equations of

the previous section and by following the procedure given for

nozzle flows. In this way the characteristic condensation zones

along each streamtube can be constructed.

When supercritical flow conditions are reached in the droplet

growth zone along each streamtube, the inclusion of an embedded,

frozen, oblique shock wave in the droplet growth zone becomes

necessary to redirect a flow sufficiently diverted by large

enough heat addition. Fig.4 shows a typical construction with an

embedded, frozen, oblique shock wave. If we let point 1 denote

the point of initial formation of the oblique shock, then the

formalism and construction developed for smooth flows hold only

upstream of point 1 and the equations of oblique shock waves for

condensing flows developed in [3] then yield the flow properties

downstream of the shock at 1b provided that the angle of inclina-

* tion ~ and the frozen condensate mass fraction g at point 1 are

known. The equation for ~ is determined so as to satisfy the * conservation of mass along each streamtube and the value of g

is determined by Barschdorff's shock fitting technique [8].

Details are to be found in [2] and [3].

Further Applications & Conclusions

The asymptotic method discussed above can also be extended to

solve steady axisymmetric supersonic flow with condensation and

unsteady one-dimensional constant area flow with condensation

(e.g.condensation in shock tubes). In the case of steady

axisymmetric flow, one has to first modify the equations of

two-dimensional supersonic flows ([9]-[11]) and then proceed in

the same manner. The asymptotic method can be incorporated in

H

I I I I I I I I I

f

I I

~----\-----------~---- t

I I I I

I

x

x

FIG 5: Wave diagram for nonequilibrium condensation i n shock tubes.

155

156

the solution of condensation phenomena in shock tubes ([12]-[13]),

In Fig.5.the diaphragm located at x=O is instantly removed at

t=O+ ~nd the contact surface moves off along path 00. The lines

OH and OT correspond respectively to the rarefaction wave head

and virtual frozen rarefaction wave tail and OS represents the

wave at which saturation is reached. The curve EN corresponds

to the onset of condensation and in the region between the lin~

OS and the curve OEN the flow is nearly frozen. The actual f-rozen

compression waves emerge from 0, coalesce along the curve EK and

form the frozen, adiabatic, gas dynamic shock wave KC (for

supercritical flow) embedded in the droplet growth zone of the

shaded condensation region. The broken line EA is approached

asymptotically. That part of the asymptotic flow in the shaded

condensation region is shown in Fig.5.

The above discussed gas dynamic problems treated qualitatively

by the asymptotic method with detailed analytic structures need

to be substantiated by. actual numerical computations and compared

with experimental results and numerical calculations based on

the method of characteristics. Detailed numerical computations

are being carried out and will be communicated later.

References

1. Blythe, P.A., Shih,C.J.:Condensation shocks in nozzle flows.

J.Fluid Mech-. 76 (1976) 593-621.

2. Clarke, J.H., Delale,C.F. : Nozzle flows with nonequilibrium

condensation. Phys.Fluids 29(5) (1986) 1398-1414; Supercritical

shocks in nozzle flows with nonequilibrium condensation. Phys.

Fluids 29(5) (1986) 1414-1418; Expansion flows on walls with

nonequilibrium condensation. Quart. Appl. Math. Vol.XLVI, No.1

(1988) 121-143.

3. Delale,C.F. :Two-dimensional supersonic expansion flows on

walls with nonequilibrium condensation. Ph.D.Thesis, Brown

University (1983).

4. Wegener,P.P., Mack,L.M. : Condensation in Supersonic and

Hypersonic Wind Tunnels. Adv. in Appl. Mech. Vol. 5, Academic

Press, New York (1958).

157

5. Wegener,P.P. : Gas dynamics of expansion flows with condensation

and homogeneous nucleation of water vapor. in Nonequilibrium

Flows, Ch.4, Part 1,Ed.P.P.Wegener, Marcel Dekker, New York (1979).

6. Gyarmathy,G. : Theorie de la Condensation en Cours de Detente

dans les Turbines a Vapeur. Rev.Frencaise de Mecanique 57(1976)

35-48; "Condensation in Flowing steam" in Moore, M.J. and

Sieverding, C.H. (eds.) Two-Phase Steam Flow in Turbines and

Separators, Hemisphere Publ.Corp., Washington and London (1976).

7. Stever,H.G. : Condensation Phenomena in High Speed Flows.

in Fundamentals of Gas Dynamics, Vol.3, Princeton Series in High

Speed Aerodynamics and Jet Propulsion, H.W.Emmons (ed.), Princeton

University Press, Princeton (1958).

8. Barschdorff,D. :Verlauf der Zustanagrossen und gasdynamische

Zusammenhange bei derspontanen Kondensation reinen Wasserdampfes

in Lavaldtisen. Forsch. Ing. Wes. 37(1971) 146-156.

9. Liepmann,H.W., Roshko,A. :Elements of Gas Dynamics, Wiley,

New York (1957).

10.Vincenti,W.G., Kruger, C.H.Jr. :Introduction to Physical Gas

Dynamics, Wiley, New York (1965).

11.Thompson,P.A. :Compressible Fluid Dynamics, McGraw-Hill,

New York (1972).

12.Sislian,J.P. :Condensation of water vapor with or without a

carrier gas in a shock tube.UTIAS Report No.201, Toronto (1975).

13. Sislian,J.P.,Glass,I.I., AlAAJ. 14 (1976) 1731.

Stationary and Moving Normal Shock Waves in Wet Steam

A. GUHA and J.B. YOUNG

Whittle Laboratory, University of Cambridge,

Madingley Road, Cambridge, CB3 ODY, England.

SUMMARY

Steam of low wetness fraction «20%) is a dispersive medium and two limiting sound speeds can be identified corresponding to complete equilibrium and fully frozen flow respectively. A stationary normal shock wave in wet steam can exhibit either a fully dispersed or a partially dispersed structure. A theoretical analysis of the structure of such shocks demonstrates the role of the three different time scales associated with the relaxation processes occurring within the wave. In contrast, the internal structure of a moving shock wave can change with time. An unsteady time-marching computational method has been developed and has been applied to solve the classical piston and cylinder problem for wet steam. The results have a direct bearing on the periodically unsteady flow of a condensing medium in a converging-diverging nozzle and the non-equilibrium two-phase flow in low-pressure wet­steam turbines.

INTRODUCTION

In engineering, the study of vapour-droplet wet steam flows of high quality is most important in the context of the flow in the final stages of large, low-pressure, steam turbines used for electrical power generation. Within these turbines, condensation occurs spontaneously in the form of vast numbers of submicron liquid droplets shortly after the steam crosses the saturation line. The interaction of the droplet cloud with the vapour results in a very significant loss in turbine efficiency known as the wetness loss. Despite intensive theoretical and experimental work over a period of seventy years, the source of the wetness loss is still not clearly understood. Recently, engineers have tended to the view that the vapour-droplet interaction may be responsible for changes in the flow dynamics which are, in turn, responsible for generating the loss. If this is so, improved understanding of the two-phase flow is likely to lead to real improvements in turbine design.

The flow in the final stages of large steam turbines is transonic with maximum Mach numbers of about 2 and shock waves are always present. The flow may not always be steady relative to the blades and periodically oscillating waves of the type observed in nozzles by Barschdorff [1] and Skillings [2] may be the norm rather than the exception. The three­dimensional geometry of turbine blading is complex and most attempts to calculate nucleating and wet steam flows have adopted numerical rather than analytical methods. Because of this,

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Gattingen, Germany, 1989 © Springer-Verlag Berlin Heidelberg 1990

160

some basic phenomena have never been investigated in isolation. with the result that the underlying physics is still not properly understood. One such area is the structure of shock waves in wet steam.

Theoretical treatments of shock waves in vapour-droplet flows are rare in the literature. Partly dispersed shock waves are discussed by Marble [3] (who made some incorrect deductions concerning the magnitude of the relaxation times). Konorski [4] and Bakhtar and Yousif [5]. Fully dispersed waves dominated by just one relaxation process were treated by Petr [6]. but few details of his analysis appear in the paper. No experimental measurement of shock wave structure is available. although the work by Barschdorff [1]. and Schnerr [7] shows interesting shock formation patterns and has stimulated the present work.

Although the results presented below are confined to flows of pure water-steam mixtures. the same phenomena also occur in other vapour-droplet flows with or without inert carrier gases present and are therefore of interest in connection with other scientific and engineering applications.

BASIC EQUATIONS

We consider the one-dimensional. frictionless flow of wet steam in a duct of constant cross-sectional area. The vapour is the continuous phase and. for simplicity. the droplet population is assumed to be monodispersed with droplets of radius r • density Pf and

mass m = 41tr3pd 3. (Methods of dealing with a polydispersed droplet cloud are discussed

in [8]). If there are n droplets per unit mass of mixture, the wetness fraction is given by y=nm.

The analysis is restricted to low wetness fractions (y < 0.2) and pressures such that the droplets occupy negligible volume. It is assumed that the droplets are sufficiently large (r > 0.02 Jlm) that changes in vapour pressure due to surface curvature are negligible.

In deriving the two-phase flow conservation equations, a number of minor assumptions and approximations are introduced which will not be discussed here. For a full discussion of these and other details, the interested reader is referred to [8].

For one-dimensional unsteady flow, the conservation of droplets is represented by

1... (npg) + 1... (npgUf) = 0 ( 1) at l-y ax l-y

The mass continuity, momentum and energy equations for the two-phase mixture are,

;t (pg + m~~~) + tx (pg Ug + mni=/f) = 0 (2)

a ( mnpg Uf) a ( 2 mnpg Uf2) at pg Ug + l-y + ax p + pgUg + l-y 0 (3)

;t [P~hg + ¥) + 7!yg (hf + Uf)] -~ (4)

161

where p is the pressure, P the density, h the specific enthalpy and U the velocity.In

equations (3) - (6), subscript 9 denotes the vapour phase and subscript f refers to the liquid properties, neither of which are necessarily at saturation conditions.

The vapour phase is assumed to behave as a perfect gas with constant specific heat

capacity. Thus p = Pg RTg (5)

where Tg is the vapour temperature and R is the specific gas constant. For greater accuracy, more realistic equations of state can be introduced, but these complicate the development and do not provide further physical insight.

The thermodynamic equilibrium state is usually specified by the saturation temperature Ts rather than the pressure. The two are related by the Clausius-Clapeyron equation which, for low pressure, is

dTs = 0.Ts) 2£ Ts \hfg P

( 6)

where h fg is the specific enthalpy of evaporation.The equation set (1) - (5) is incomplete and must be supplemented by three equations representing the interphase transport of mass, momentum and energy. The interphase transfer mechanisms are quantified in terms of relaxation times which represent the rates at which the two-phase system reverts to equilibrium following a disturbance. As non-equilibrium variables, we choose Llu = u g - Uf to represent velocity relaxation, LlT f = T s - T f to represent droplet temperature relaxation

and LlT = Ts - Tg to represent vapour thermal relaxation. (Note that LlT representsthe negative of the vapour superheat). It is shown in detail in [8] that the interphase transfer equations can then be written,

dUf _ Llu

dtf - 'tv

dTf _ LlTf dtf - 'tD

dm (l-y) Cpg LlT y Cpf LlTf (hg - hf) n dtf +

'tT 'tD

(7)

(8)

( 9)

where d/dtf = a/at + Uf a/ax is the substantive derivative following the droplets

and Cpg and Cpf are the vapour and liquid isobaric specific heat capacities.. The three relaxation times, 'tV! 'tD and 'tT are given by , [8],

2r2 Pf 'tv = --- [<\>(Re) + 4.5 Kn]

9 Ilg

't = 0.Ts)2 (rPf Cpf) ~ D \hfg 6R P

(l-y) c pg Pf r2 'tT = (1 + 4.5 Kn/Prg)

3Ag y

(10)

(11)

(12)

where <\> (Re) is an empirical correction for large slip Reynolds numbers

(Re = 2pg rILlul/llg) givenby

<\>(Re) = [1 + 0.15 Re O. 687 ]-1 (l3)

162

and A.g and Jlg are the vapour thennal conductivity and dynamic viscosity respectively. Prg

is the vapour Prandtl number and Kn = 19/2r is the droplet Knudsen number, 19 being the molecular mean free path of the vapour.

Equations (10) - (12) are by no means definitive. They represent expressions in fairly common use and are supposedly valid for all droplet Knudsen numbers from the continuum to the free molecule regime. The method of analysis is not dependent on the fonns of (10) - (12),

however, and other, possibly more

,.,. ~ bl)

g

-3.0 p = 0.5 bar y = 0.1

-4.0

-5.0

~.-.. -6.0 .g ~ c'-' -7.0

·15 i -8.0 ~

-9.0 +--.--.----r--r--c.-.-.--.--. 0.1 0.5 1.0

Droplet radius ij.un)

Fig.} Different relaxation times for mo­-nodispersed water droplets in pure steam.

suitable, expressions could easily be incorporated if desired.

For a given droplet radius and wetness fraction, the three relaxation times are of quite different magnitudes. Droplet temperature relaxation is extremely rapid and is about one order of magnitude faster than velocity slip relaxation. Dividing (10) by (12), it is seen that 'tv / 'tT - y. Fig. 1 shows

curves of 'tD, 'tv (for <!>(Re) = 1) and

'tT as functions of droplet radius for

steam at 0.5 bar pressure and wetness fraction 0.1.

NUMERICAL SOLUTION OF THE EQUA nONS

Despite the complexity of the governing equations, it is possible to make considerable progress using analytical methods, at least for the comparatively simple case of stationary waves in one-dimensional, steady flow. This approach is described in detail in [8]. For accurate quantitative results, however, and especially for the analysis of waves in unsteady flow, it is necessary to adopt a numerical method of solution which we now describe.

For the investigation of stationary waves in steady flow, all partial derivatives with respect to time are removed from equations (1) - (4) and (7) - (9) . With the help of equations (1) and (5), the continuity, momentum and energy equations (2) - (4) may then be recast as a set of three simultaneous equations for dUg/dx , dTg/dx and dp/dx. Equations (1) and (7) - (9) furnish expressions for dut/dx , dn/dx , dTt/dx and dm/dx. The resulting set of seven simultaneous first order differential equations can then be integrated numerically using a fourth order Runge-Kutta procedure.

A computational procedure that marches forward in space must necessarily start from an initial condition that represents a deviation from equilibrium. For a pardy dispersed shock wave, the difference in the vapour and liquid phase flow variables just downstream of the frozen shock discontinuity constitute the required initial departure from equilibrium. For a fully dispersed shock wave an initial, arbitrary perturbation of the flow must be specified. Step-by-

163

step integration of the conservation equations then automatically generates the wave profile. Providing the initial perturbation is sufficiently small, the numerical results closely approach the exact solution. Thus, if two calculations are perfonned for the same upstream flow conditions but with different initial small perturbations, it is possible to superpose the results by a relative shift in the x-direction.

For calculating moving shock waves, the unsteady fomls of the basic equations are retained and solved by a mixed Eulerian-Lagrangian time-marching technique. For this purpose, the continuity, momentum and energy equations (2) - (4) are solved by the Denton, finite volume, timemarching method [9]. The interphase transfer processes, however, are more easily describable in a Lagrangian framework. Equations (7) - (9) are therefore integrated along the droplet path lines and are coupled to the unsteady Euler solver by introducing their effects as source tenns in the continuity, momentum and energy equations. For simplicity, droplet temperature relaxation is neglected in the unsteady flow analysis and the droplet

temperature is assumed to equal the local saturation temperature at all times.

SPEEDS OF SOUND

We now introduce four reference velocities corresponding to the speed of sound in two­phase flow under different thermal and mechanical constraints. The full frozen and full equilibrium sound speeds (denoted here by af and ae3 respectively) are well known. The lWO

intennediate speeds correspond, (a) to the case of equilibrium droplet temperature relaxation, but frozen momentum and heat transfer (ael), and, (b) to the case of equilibrium droplet temperature and velocity slip relaxation but frozen heat transfer (ae 2).

The derivation of the sound speeds is straightforward [8]. The results are,

af2 = )'RTg (14 )

2 )'RTg a e1 (15)

1 + (y/ (1-y) ) (YCpf/R) (RTs/hfg) 2

2 (l-y) )'RTg a e2 = ( 16)

1 + (y/ (1-y) ) (ycpdR) (RTs/hfg) 2

2 (l-y) )'RTg a e3

y[l - (RTs/hfg) (2-cTs /hfg) 1 (17)

where C = cpg + YCpf/(l-y).

The relationship of the four reference velocities to each other is of great importance. As an example, for steam at 0.35 bar pressure and 0.1 wetness fraction,

af : ael : ae2 : a e 3 =1 : 0.997 : 0.945 : 0.878

STATIONARY WAVES IN STEADY FLOW

We now consider the structure of stationary, finite amplitude waves in one-dimensional steady flow. Far upstream of the wave the flow is assumed to be in thennodynamic and velocity equilibrium with a specified pressure, wetness fraction and droplet radius. Far downstream of the wave a new equilibrium condition is re-established. The far upstream

164

condition is denoted by subscript o.

By applying the conservation equations in integral form between the upstream and downstream eqUilibrium states, it follows that the upstream flow velocity ugo must exceed

the full equilibrium speed of sound ae30 in order for a solution to exist other than the trivial

Gase when all flow properties remain constant. It is also found that a continuous transition of fluid properties can occur between the upstream and downstream states, providing the upstream

flow velocity is less than the fully frozen speed of sound afo. For this range of upstream

flow velocities (ae30 < Ugo < afo), the steepening effect of the non-linear terms in the equations of motion is just balanced by the dispersive effect of the relaxation processes. Such

waves are described as fully dispersed.

For Ugo > afo, the conservation equations do not admit a continuous solution and a very steep fronted shock wave forms in which the flow is dominated by the effects of viscosity

and thermal conductivity. This type of wave is normally modelled by a discontinuity in vapour

flow properties followed by a continuous relaxation zone where the flow relaxes to its final equilibrium state. Such waves are described as partly dispersed.

From a qualitative point of view, we have described nothing other than the well-known

types of stationary wave to be found in all relaxing gas flows. An important result of our work, however, is that fully dispersed waves in vapour droplet flows can naturally be

subdivided into three further categories depending on the upstream vapour phase velocity. We

thusdefme

Type I waves corresponding to ae 30 < Ugo < a e20

Type II waves corresponding to ae20 < Ugo < aelo Type ill waves corresponding to aelo < Ugo < afo

Type I waves are dominated by vapour thermal relaxation, Type II waves by both velocity and thermal relaxation and Type ill waves by all three relaxation processes.

In Type I fully dispersed waves, the gradients of the flow properties are comparatively

mild and an excellent approximation is to assume that the droplet temperatures and velocity slip instantaneously relax to their equilibrium values. Thus, in equations (9) and (10) we let 'tv ~

o and 'to -+ 0 I the ratios ~u/'tv and ~Td'to remaining finite such that (in steady flow)

~u dUg ~Tf dTs - -+ Ug dx and -- -+ Ug dx . The structure of the wave is governed by 'tv 'to the vapour thermal relaxation process and the reference velocity a e2 takes on the characteristic

of a frozen speed of sound. This implies that, were we to maintain the constraints on droplet

temperature and velocity slip for upstream velocities in excess of ae2, it would be necessary

to insert a discontinuity in the flow to obtain a solution of the conservation equations. An approximate, but acceptably accurate, analytical solution of Type I waves is possible

[8] and yields much physical insight into the general wave structure. Here we concentrate on

numerical solutions, Fig. 2 displaying curves ofug i P, Tg , ~T and y as functions of x for a typical example. Note that the origin, x = 0, has been arbitrarily set at the equilibrium

sonic condition, Ug = ae30'

1.14

I 12 ,..... 1.10

If 1.08 -a 1.00

§ I O.

102

&: 100

o D8

O.DO

~ mlO

~ 357 S

3570

~ 3511 S

i 3511.0

ass s 5 asso I-< 3S. S

354 0 ,..... ~ I.'

I-< 1.2

1 I 0

bO 08

c:: 00 .~

!'.l o.

1: 02

~ 00 CI)

000 ,....

D.g4 .s <II o D2 -e ODO

088

~ 0811

'g 08'

43 082 > 080

0.101 >- Po = O.S bar c:: 0.100

Yo = 0.1 0 .~

o oeD fo = O.S 11m ~

.):; a,oge

'" '" 0) o OD7

~ ~ = LOS ~ o oeo 1le30

... ~ -100 .. so so 100 150

Distance,x (mm)

Fig. 2 Numerical solution for a

165

Fig.2 shows that velocity changes in the wave are approximately centred on x = 0 (as follows from Prandtl's relation

Ul U2 = a; for weak shock waves) but

that the wave is asymmetrical in the x­direction about this point. Indeed, as the upstream velocity ugo approaches the upper limit ae20 the front part of the wave becomes steeper and departures from velocity slip equilibrium become increasingly important although they never attain sufficient magnitude to significantly change the wave profile even near the loading tip. The maximum vapour superheat, -~ T, is attained where Ug = ae3 and the vapour temperature T g overshoots the far downstream equilibrium value. The conditions for the latter behaviour are discussed in [8].

When ae20 < Ugo < aelo it is necessary to relax the constraint of equilibrium velocity slip in order to obtain a continuous solution to the conservation equations. Type II waves are therefore. dominated by velocity relaxation in the steep forward part of the wave and by thermal relaxation in the long tail. A transition section connects the two regions where the effects of velocity slip, governed by the short time constant 'tv, are decaying, and

where the effects of heat transfer, governed by the much longer time constant 'tT , are increasing in importance. Maximum velocity slip occurs near Ug = ae2 and maximum superheat near Ug = a e 3' The reference velocity ael takes on the characteristic of a frozen speed of sound.

Fig.3 is a numerical solution showing the structure of a Type II wave. The forward

part of the wave is, indeed, very steep and very small integration steps are required to resolve this region accurately.

Type I fully dispersed stationary wave in steady flow

166

Type III waves correspond to upstream velocities in the range aelo < Ugo < afo

• 25

"6 1.20 ~

......... I 15

.e.

~ 1..0

£ 105

'00 D.gS

302

~ 3D •.

~ 300.

i 35g.

[ 358. 357.

e 350

~ 355

354.

..... 5.

~ ~

3.

·r 2.

~ l-i:

fr (I)

1.00

... I 2

;.. ..... 1.0 -58

0 .... 0.8 -* ~ Ja 0.0

os ~- o • =~ 02 (I) .....,

00

Po = 0.5 bar Yo = 0.1 To = 0.5 p.m

.!!&2.. = 1.1 1Ie30

-40. -20 0 20. 040. DO SO 100

Distance ,x (mm)

Fig. 3 Numerical solution for a Type II fully dispersed

stationary wave in steady flow

and continuous velocity profiles can only be obtained if the constraint on the droplet temperature is removed. The difference between ael and af is always very small and Type III waves are of little practical importance. We shall discuss them no further here.

When ugo > afo, it is necessary to insert a discontinuity in the flow in order to obtain a solution to the inviscid conservation equations. Across the discontinuity, all interphase transfer processes are effectively frozen and hence no changes occur in droplet radius, temperature and velocity. The vapour phase conditions donwstream of the discontinuity are easily calculated by a standard Rankine-Hugoniot analysis and these provide the initial conditions for the numerical integration procedure.

Fig.4 shows the structure of a typical partly dispersed shock wave. For strong shocks such as this (u go / a f 0 = 1. 5)

very large departures from equilibrium occur just downstream of the discontinuity. In the relaxation' zone the droplet temperature rises very rapidly to the saturation value on a time scale 'tD and this is followed by velocity and

thermal relaxation on time scales 'tv and

'tT respectively .

Two interesting phenomena can be observed in FigA which are directly attributable to velocity slip. Firstly, the vapour temperature initially increases after the discontinuity before decreasing to the downstream equilibrium value. This is due directly to the effect of frictional heating caused by velocity slip. Secondly, the wetness fraction also increases before

droplet evaporation begins to dominate further downstream. This is a result of an increase in droplet concentration due to the higher velocity droplets overtaking the lower velocity vapour.

o . Q

O. B

0.7

o.e

Vapour phase temp. (T g / T go.stgn)

Pressure far downstream : Pressure

\ (P / Po.stgnG- _ - =.-...,""== =-=-=--=-=-=-__ I

'- I

" \'- Droplet velocity (Uf I 3[0) , , " ,

Droplet temp.

"~::::::~ _ Vapour phase velocity (ug / afo) -:;; ..

Droplet temp. relaxn ......................... __ -----Inertial reln.-l

Thermal relaxn.- Velocity far downstream

.2

. 0

o

0 . 6

0 . 4

0 . 2

0 . 0

Distance (mm)

--- .... _---

Wetness fraction (y /Yo)

Droplet conc. (N / 3No)

( the dOllcl1 curve will be followed ir there were no

vc locilY slip 1

0 . 0 0 . 5 1.0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 .0

Distance (mm)

167

Fig. 4 Numerical solution for a partly dispersed shock wave ill sleady flow

SHOCK WAVES IN UNSTEADY FLOW

The physical significance of the various wave profiles discussed above can be appreciated more readily by considering their development under unsteady flow conditions. As a typical example, we now discuss wave generation in one-dimensional flow by an instantaneously accelerated piston in a frictionless pipe initially containing stationary wet steam.

Unsteady flows with shock waves are difficult to compute even in single-phase flow. To demonstrate the accuracy of the numerical scheme, we therefore present results for the piston-in-a-pipe problem using a perfect gas rather than wet steam. Fig.S shows the theoretical (t-x) characteristics diagram which consists, quite simply, of a shock wave of

168

2:! ::I

f 'B .~ 'iI c: .S! '" ~ ~ 0 Z

I.

01

01

0.7

0.1

0."

0.'

0.1

0.1

0.1

0.0

If rressure profiles at equal time intervals

AI. Al2 II ~ ~

AXI = AX2 = AXn ~

O.

60

so

10

o

~ ~ a ~ ~ ~. H ~

Distance (mm)

ZhOCkP81h

5 10 IS 20 2S

Distance (mm)

Fig. 5 Shock propagation in a pipe : single-phase flow of a perfect gas

constant strength propagating at constant velocity into the stationary gas. The gas velocity behind the wave is constant and equal to the speed of the piston. The numerical solution for the pressure distribution is also shown in Fig.5 and it is evident that the computed shock profile and wave velocity are extremely accurate and remain remarkably constant as the wave propagates along the pipe.

Fig.6 shows the rather different behaviour when the wave propagates into wet steam of pressure 0.35 bar, wetness fraction 0.1 and

. droplet radius 0.1 mm. The (t-x) diagram was constructed from the results of the unsteady time-marching calculation. At the instant of initiation, all the interphase transfer processes are frozen and the shock velocity corresponds to the propagation velocity into a single­phase vapour at the same temperature. Behind the shock, the mixture relaxes to equilibrium along the particle

pathlines. The droplet temperature relaxes first on the very short timescale 'to and is followed

by the velocity slip and vapour temperature on timescales 'tv and 'tT respectively. Changes along the particle paths are propagated upstream and downstream along the left and right running Mach lines (based on the frozen speed of sound). The right running Mach lines overtake the shock wave, weakening it and causing it to slow down. The shock path therefore curves in the (t-x) diagram until it reaches a constant equilibrium speed. When this occurs, the dispersive effects of the relaxation processes are just balanced by the steepening effects of the non-linear terms and the wave structure is identical to that of the stationary waves in steady flow described earlier. Whether the final equilibrium structure is partly or fully dispersed depends on the piston velocity.

The variation of the wave pressure profile with time is also shown in Fig.6 and the deceleration and weakening of the wave front are clearly visible. The behaviour of the superheat, vapour temperature, droplet radius and wetness fraction is shown by the curves in Fig.7 which are self-explanatory. As with stationary partly dispersed shock waves,the increase in wetness fraction downstream of the frozen shock wave is due to the effects of velocity slip.

t a

0 9

a 8

e 0 7 ::l

Vl Vl ., ....

0 . 6 0.

"0 ., .!:l 0 . 5 OJ c: 0 0 . 4 'in c: ., E 0 . 3 '5 <.: 0 a 2 Z

O . t

0 . 0

60

SO

40

O.

The shock wave in weI RUm constantly Jags behind , hal in the dry .... pour. (or the same COnsWlt piston veloc ity.

5 . to. t5 . 2 0 . 25 .

I wet steam I Distance (m m)

.. '

\ , , i , , , , , , , , , , ,

pressure profiles at same time step

---- dry vapour

-- wet steam

30 . 35 . 40 .

~ 1.05 'g

169

ypsrream conditjnns •

pressure : 0 .35 bar

temp. : 345.9 K

radius : 0.1 ~m

wc:mCS$ : 0.1

piston velocity •

267.85 rrJs

~ g

30 ., .§

"0 1.00 ;>

..>0: u

0.95 0 ..c •• ••• · · · ·[:~:~k path in

.... 20

10

o o 5

•••• •• dry vapour

10 IS

Distance (mm)

Mach line droplet path

20 25

Fig. 6 Shock propagation in a pipe

CONCLUSIONS

Vl

"0 0.90 ., .!::l OJ 0.85

E 0 0.80 Z

0 to 20 30 40

Time (~scc)

Two-phase flow of wet steam

In this paper we have described all the major features of stationary and moving, partly and fully dispersed, normal shock waves in wet steam. Numerical calculation schemes have been developed which accurately predict the phenomena in one-dimensional steady and unsteady flow. The results of the analysis have application in the interpretation of oscillating condensation shock patterns in nozzles and turbine cascades and work is in progress to extend the applicability of the numerical techniques to deal with these, more complicated, flows.

ACKNOWLEDGMENTS

During the course of this work, which was carried out at the Whittle Laboratory, University of Cambridge, A. Guha was supported by a Nehru Scholarship.

170

Q' 440 e.-co 420 !-<

CI)

3 400 ~ S. 380 E ~

360 <> ~ .c 340 Q,

~ 8. 320

'" > 300

60

Q' 50 e.-!-< 1 40

bO C . ., 30 i1

i 20 :s

CIl

10

o

gradual reduction in the strength of frozen shock

..... \ . '.

o

.. luration temp. colTclponding 10 pressure at Ihe piston end

gradual reduction in the

.:~:ng~~. of fraU/Ck

-..... -.....

10

-"'''-

20

Distance (mm)

frozen ,hock

~

prof Lie. al equ~ time intervals

fraun shock

~

~

30 40

1.00'

~ 0 .95 ... 0 ....

-=- 0.90 VI :s :a C! 0.g5 ., ~ 0.80

Q 0.75

0.70

10

~

~ 9

c 8 0

';:J u

.£l 7 VI VI <> 5

6 <> ~

4

0 10

plOfilcs al equ:.I1 time intervals

fro zen shock

profiles at equal time inlervals

20

Distance (mm)

30

Fig. 7 Variation of flow parameters during shock propagation in wet steam

REFERENCES

40

1. Barschdorff, D., "Droplet formation, influence of shock waves and in stationary flow patterns by condensation phenomena at supersonic speeds", Proc. 3rd Int. Conf. on "Rain Erosion and Associate Phenomena", R.A.E., Famborough,1970, pp.691-705.

2. SkilJings,S.A. ct. aI., "A study of supercritical heat addition as a potential loss mecha­nism in condensing steam turbines", IMechE Conf.,Cambridge,U.K., 1987 ,pp 125-134

3. Marble, F.E., "Some gas dynamic problems in the flow of condensing vapours", Astronautica Acta, Vo1.l4, 1969, pp.585-614.

4. Konorski, A., "Shock waves in wet steam flow", PIMP (Trans. Inst. Fluid Flow Machinery, Poland),1 Vol.57, 1971, pp. 101-109

5. Bakhtar, F., and Yousif, F.H., "Behaviour of wet steam after disruption by a shock wave", Instn. Chern. Engnrs., Symp. on "Multi-phase Flow Systems", Univ. of Strathclyde, 1974, paper G3.

6 . Petr, V., "Non-linear wave phenomena in wet steam", Eighth conf. on "Steam Turbines of Large Output", Plsen, 1979, pp.248-265.

7 . Schnerr, G., "20 transonic flow with energy supply by homogeneous condensation ", Experiments in Fluids,Vol.7, 1989, pp. 145-156.

8. Young, J.B., and Guha, A., "Normal shock wave structure in two-phase vapour­droplet flows", submitted for publication to J. Fluid Mechanics.

9. Denton, J. D.," An improved time-marching method for turbomachinery flow calculation", Trans. ASME, J. Eng. for Power, Vol. 105, 1983, pp 514-524.

Numerical Investigation of Nitrogen Condensation in 2-D Transonic Flows in Cryogenic Wind Tunnels G. H. Schnerr and U. Dohrmann

Institut fUr Stromungslehre und Stromungsmaschinen Universitat (TH) Karlsruhe, F. R. of Germany

Summary

This paper discusses 2-D transonic flows of nitrogen, relevant to cryogenic wind tunnels, at the most critical initial values of high pressure and low temperature and with cooling rates (-dT/dt) of about 0.02 - 0.05 0 C/~ which are representative of cryogenic transonic flows. For the first time inviscid steady 2-D flows of nitrogen with nonequilibrium phase transition are investigated numerically using the Euler equation coupled with the classical nucleation theory. The main droplet growth rates are calculated by a macroscopic law and the surface averaged droplet radius. Real gas effects are not yet included. The theoretical values of adiabatic supercooling become nearly constant 15 - 16 K, and without exception, the onset Mach numbers are below 1.3. Due to the lower rates of heat addition in nitrogen flows near the Wilson point only smooth compressions are observed for the lowest condensation onset Mach numbers, too. In detail the 2-D propagation along subsonic heating fronts and the effects of different cooling rates are demonstrated in appropriate nozzle flows. Heat addition by homogeneous condensa­tion in the flow around an inclined NACA-0012 airfoil affects the form of the supersonic regions essentially. Pressure drag and lift coefficients decrease simultaneously of about 19% and 37%, respectively.

Introduction

Reliable aerodynamic tests in transonic flows require agreement with free flight conditions, i. e. the Mach

number and the Reynolds number. This applies essentially to the topical transonic problems such as shock

wave boundary layer interaction with and without control and laminar wing design. In experiments such high

Reynolds numbers of about 40 . 106 can only be achieved by cooling of the test gas (N2) close to the vapor

pressure curve which causes a decrease of the viSCOSity and increases the density. Simultaneous

pressurizing of the circuit of the wind tunnel to about 5 bar allows a model size reduction by a factor 25.

Hence, free flight Reynolds numbers can be achieved in test sections having a cross section of 2 x 2.4 m2,

e. g. the new European Transonic Wind Tunnel (ETW). Reynolds numbers up to 60 ·106 are expected for

the new cryogenic Ludwieg tube of the DLR (Gottingen) in a cross-sectional area of the test section of only

0.4 x 0.35 m2and initial values for pressure and temperature of 1 0 bar and 1 00 K, respectively. By changing

to such low temperatures and high pressures condensation can occur after crossing the saturation

boundary which causes experimental deviations from the results in adiabatic flow e.g. for the static

pressure, drag and lift coefficients. Since a possible supersaturation will allow an additional extension of

the operation range the nonequilibrium phase transition by homogeneous condensation of nitrogen has

been investigated in many papers (see Ref. [1]). The condensation problem with respect to airfoil tests in

cryogenic wind tunnels has been reviewed by Hall [2]. [3]. Free jet experiments with ETW relevant

stagnation conditions and the so called stream tube duplication method are presented by Koppenwallner

and Dankert [4] and DQker [5]. The first theoretical validation of this method by equilibrium condensation

G. E. A. Meier· P. A. Thompson (Ed,.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Gatlingen, Germany, 1989 © Springer-Verlag Berlin Heidelberg 1990

172

calculations were provided by Wagner [6]. Heterogeneous condensation from foreign nuclei or small droplets existing in the approching flow is also of interest but not treated in the present work. With the assumption of an ideal fluid in this paper the homogeneous condensation in transonic flows of cryogenic N2 has been investigated by a numerical study taking into account the cooling rates of typical airfoil wing sections at the most important operating state at low temperatures and high pressures. Emphasis is given to the quantitative analysis of condensation effects in steady two-dimensional flows in Laval nozzles and around airfoils and the comparison with similar water vapor condensation phenomena in transonic flows of moist air.

Theory

The steady two-dimensional diabatic flow is described by the equations for mass, momentum and energy in conservation form (Schnerr and Dohrmann [7], Dohrmann [8]). Real gas effects are not yet included and inviscid fluids are assumed. Here the classical nucleation theory of Volmer [9] is used which gives a good qualitative representation of the behavior of condenSing N2 in the supersaturated state (Wegener [1'0]). Oswatitsch [11] introduced this theory into the calculation of flow processes, a summary of all basic relationships for compressible flows with heat addition is given by Zierep [12]. To compute the nucleation rate J per unit time and volume, we take

~ p2 (16 ( m )2( a )3) J = - am ·312 -y- exp - _ 1& -- -'-1& ' Pc 3 pcln(s) kTy

(1)

Here m is the mass per molecule, pyand Pc are the density of vapor and condensate, respectively, s is the supersaturation ratio, k the Boltzmann constant and Ty the temperature of the vapor. To limit the un­certainty in the surface tension a, the dominating quantity of the nucleation rate, presently only tempera­tures above the triple point (T tr = 63.2 K) are investigated. In Fig.1 -left one isentrope is calculated, starting form typical values for the initial pressure (POl) and temperature (T01 )' togehter with Wilson points measurements from new unpublished results from the Jiaotong University Xi'an (Guo Youyi, Ji Guanghua, Wang Qun et. al. [13]). The numerical result shows good agreement at the onset, and the further development of the nonequilibrium heat addition. For the vapor pressure curve we use the formula from Wagner [6], the subscripts .. s • and • 00· indicate saturation state and plane liquid surface, respectively.

( 701.7) ps~ = 8705· 105 exp t -r:- [N/m2J (2)

To compute the density of liquid condensate Pc' we take a linearly decreasing relationship (see Ref. [6]). The surface tension a JT) of a plane liquid surface (T > T tr) is given by the formula of Eotvos Eq. (3) (see Ref. [14]). Here Mv is the molecular weight and T crit the critical temperature (126.2 K) of N2•

(3)

10'

P ...

T.P.

70 80 90 T [Kl

100

2.0

1.6

1.2

a 10' [N/ ml

0.8

0.4

0.0

110 30

173

. _._- - -- -- -

, a.(T)

_ . -DOker

- - Tolman

- Wegener, Mack , ,

T~

40 50 60 70 80 90 100 110 T [Kl

Fig.1. Vapor pressure curve and surface tension of N2, - theory, • experiment from Ref. [13]

The validity of this relationship is expected to be limited to droplet radii not to small. In principle the surface tension is a decreasing function of the radius r for small clusters. Hence, the result of Eq. (3) are corrected by Tolmans formula (see Ref. [14]) with /) = 10·'0 m, the intermolecular distance in the liquid, see

Fig. 1 - right. If the droplet radius is small compared with the mean free path A., the molecular growth is described by the Hertz Knudsen formula

dr - = -dt Pc

Pv - ps.r

y21tRT' v v

(4)

Here t is the time and Rv the specific gas constant of the vapor. For r > A. macroscopic thermodynamic

processes of heat removal by conduction limit the growth rate

dr

dt _ k _ (T -T), k=2.067 ' 10.3 Tv L Pc r s,r v 1 + ~

T v

[J/(msK)] (5)

where k is the heat conductivity (see Ref. [6]) and L the latent heat. Analogous to the numerical calculations for flows of moist air the surface averaged droplet radius f of Hill [15] is introduced. Our numerical solver

is a further development of the explicit adiabatic finite volume method of Eberle [16].

Comparison of homogeneous condensation in N2 and moist air

On the basis of numerical and experimental results of homogeneous condensation processes in transonic

flows of moist air we estimate the corresponding effects in flows of cryogenic N2• The specific values of the

heats cp and gas constants R are quite similar but the latent heat of N2 is more than one order less. Eq. (6) gives by a one-dimensional approximation the relationship of the Mach number M in a nozzle to the

174

amount of the normalized heat addition q I cp T 01 and the nozzle geometry A(x)

1 dM

M d~ y*

Y-1 1 + -- M2

2 (6)

Here y is the specific heat ratio and y* is half the height in the nozzle throat (Refs. [11] and [12]). Eq. (6)

turns out that in geometrical similar nozzles the identical Mach number distribution is to be expected if the

normalized heat addition q I cp T01 = f(x I y*) keeps unchanged. By a one-dimensional flow approximation,

too, the transonic onset of heat addition by homogeneous condensation is controlled by the initial conditions

at zero velocity (P01' T 01) and the cooling rate (-dT/dt)* at the throat being proportional to T01 312 (Wegener

and Mack [14])

( dT J * - 2312 y- 1 lirR T 312 - dt - (y+ 1)2 ~ Y*Fr 01

(7)

Forthe present comparison two sets of initial conditions and the parameters y*, R* (R* = radius of curvature

of the wall in the throat) of geometrical similar nozzles are correlated that way to get approximately constant

values of the cooling rate and the condensation onset Mach number Me' In spite of the smaller latent heat

of nitrogen q I cp T01 rises quickly to the same order dueto the higher vapor pressure (Fig. 2). The weaker onset in N2 with smaller gradients of the heat addition results only in a smooth compression (X-shock,

Fig. 3 -left); normal shocks, typical for such low onset Mach numbers in flows of moist air (Fig. 3 - right),

are not to be expected in flows of cryogenic N2. The plotted frozen Mach number contours Mf = const. represent the ratio of the local flow velocity to the frozen velocity of sound. In moist air with a large

amount of noncondensing carrier gas, i.e. a low vapor pressure, molecular droplet growth rate (f < A)

develops, contrary to f > A in N2. Comparing with atmospheric supply this pressure increase and temperature decrease improves the Reynolds number by a factor 20 at which the effect of cooling prevails

against the pressurizing.

0.14,--------------,

0.12

0.10

q 0.08

Cp T010.06

0.04

0.02

0.4 0.8 1.2 1.6 x/y*

moist air

2 2.4

Fig.2. Normalized heat addition in geometrical similar nozzles - flows of cryogenic N2 and moist air

LIM, = 0.02

~o = 84.3 % TOI = 92.1 K POI

I I

log\OJ I + I (-£!)

·C = 0.110-[m-3.s-l ] I dt ~s

35

30

25

20

15

10

5

0.6 I g/ gmax --------

0.10 0.5 diabat lc

0.08 0.4

0.06 0.3

0.04 0.2

0.02 0.1

0.00 0.0 - 1 0 2 3

~::: l~~~O / r

-7.0 J-;- , ~: . : if--t-----------~. - 8.5 I

I -9.0 I

I _9 . 5+-~f___,____,~,_~x,.:_:.[c:..:.m:.,:]

- 1 o 2 3 4

nitrogen

LIM, = 0.02

~o X

TOI

logloJ [m-3. s-l ] 35 0.6

g/gmox 30 1.0 0.5

25 0.8 0.4

"20 0.6 0 .3

15 0.4 0 .2

10 0.2 0. 1

5 0 .0 0 .0

4

o

I I I dT ' ·C : (-at) = 0159 ~s

I

9/9 ....

x [cm]

o 5 10 15

5 10 15

moist air

175

Fig.3. Geometrical similar hyperbolic nozzles - left: y* = 15.75 mm, R* = 52.5 mm, Me = 1.167, X-shock - right: y* = 60 mm, R* = 200 mm. Me = 1.150, supercritical heat addition, normal shock

Cooling rates for transonic airfoil tests

Here two different airfoils are investigated. the circular arc CA-0.1 (thickness ratio of 10 %) and the NACA-0012. Due tothe approximately constant cooling rate at local Mach numbers close to one the circular arc airfoil allows some simplifications. The value at Mach number one. denoted by the star. represents the time scale at the entire transonic airfoil section. whereas the cooling rate at the NACA-0012 changes considerably. Condensation onset Mach numbers very close to one are correlated with higher cooling rates and vice versa (Table 1). Usually tt is assumed that the temperature gradients in adiabatic flow and with heat addition are identical up to the condensation onset for free stream Mach numbers M~ :5 1, too. However, the numerical results of diabatic flows show a more or less pronounced precompression

176

NACA - 0012 CA - 0.1

To1 [K] c[mm] M c (- ~~l [~] To1 [K] c[mm] (_~T)*[~]

91 .1 100 1.000 0.100 91.1 80 0.046

91 .1 100 1.128 0.041 91.1 200 0.019

91.1 100 1.204 0.025

91 .1 100 1.238 0.011 Table 1. Airfoils - cooling rates (ex = 00 , theory)

upstream of the onset, depending on the onset location and the amount of heat addition. The wind tunnel

experiments of Wegener [17] and DOker [5] with nozzles were performed at cooling rates of about

0.1 0 C/~ (condensation onset). Following Wegeners argument a factor of two of this variable is not too

important. On the other hand for mean values of transonic onset Mach numbers ~ 1.25 at the NACA-0012,

the cooling rate becomes about one order less, in particular for model cord length typically greater than

100 mm. To demonstrate the effects of different cooling rates the size of one circular arc nozzle

(y* = 30 mm, R* = 200 mm, Fig. 4 -left) was enlarged by a factor 2.5 (Fig. 4 - right). This decreases the

cooling rate inversely proportional. The time scale covered by this two nozzles is in accordance with the

relevant cooling rates of the airfoil models collected in Table 1. Except the typical decrease of the onset

Mach number from 1.074 to 1.054 all further results agree nearly completely, e. g. the plot of the iso-Mach

lines. The slightly reduced maximum value of the nucleation rate obviously is compensated by the lower

time scale in the droplet growth phase. To conclude, for determination of the onset Mach number the

y ' = 30 mm . R = 200 mm

dT ' °C <- aT) = 0.042 ~s

M, = 1 --"'i'""-".£L.L~?la.w p/ Po,

log,oJ 0.7 [m-J. s-']

35 0.6 g / g"",.

30 0.10 0.5

25 0 .08 0. 4

20 0 .06 0. 3

15 0.04 0.2

10 0.02 0.1

5 0.00 0.0 0 4 8

g/g"",.

diabat ic

adiabat ic x [cm]

12

<= 75 mm R = 500 mm

<_QI) • = 0.017 °C -dt ~s

M,= -"'f"'-<.'-'L.<~0'&~

log, oJ [m-J.s-' ]

35

p/ Po, 0.7

0.6 g/g...,.

30 0.10 0.5

25 0.08 0.4

20 0.06 0.3

15 0.04 0.2

10 0.02 0.1

5 0.00 0.0 0 10 20

Fig.4. Geometrical similar circular arc nozzles - To. = 91.1 K, po. = 3.604 bar, 11>0 = 91.7 %

30

177

accurate cooling rates should be investigated, especially in transonic flows. In cryogenic flows of N2 subcritical heat addition prevails at the lowest onset Mach numbers, too. This differs completely from

homogeneous condensation processes in moist air where at comparable low onset Mach numbers

supercritical heat addition produces normal shocks and unsteady phenomena (Schmidt [18], Zierep and

Lin [19], Frank [20]).

T" (Kl 105.5 95.5 90.7

1.8 4.251 3.412

1.7 POl (barl

1.5

1.5

M,1.4

1.3

1.2 T, = 88.5 K ,n ad = 15.1 K

1.1 (_!!I) aC

= 0.110 -dt "S

1.0 20 30 40 50 50 70 80 90 100

<Ilo (%l

Fig.5. Onset Mach number Me for different initial values of P01' TOI and constant cooling rate (-dT/dt)

On the basis of one Wilson point further onset Mach numbers easily can be computed for different supply

conditions if the initial values of pressure and temperature are varied along an isentrope (see Fig. 1 -left) .

For a constant cooling rate (see Eq. (7)), the saturation temperature T. and the adiabatic supercooling

iH ad = T. - Te will be constant, too. Fig. 5 shows the increase ofthe onset Mach number Me by decreasing

the relative humidity <1>0 in the supply (increase of pressure and temperature). Simultaneously the Reynolds

number increases by a factor 2.5.

60

LIp, = 2.7 <Ilo = 91.7 %

50 T,,= 91.1 K POI = 3.604 bar

40 0.5

LIp 9 0.4 9 [mbarl 30 (g/kgl

0.3 20

0.2

10 0.1

0 0.0 f---------j

9c = 0. 1 g/kg 10 mm

Fig.S. Circular arc nozzle, y' = SOmm, R* = 100 mm - adiabatic precompression (subsonic heating fronts)

178

As a rule it is assumed that condensation effects in supersonic flow cause an increase of static pressure compared to the isentropic value at the same position. However, in Fig. 6 compression already is observed

in the subsonic flow, whereas the condensate mass fraction glgmax' i. e. the heat addition, starts in supersonic flow. This precompression of about 50 mbar is caused by subsonic heating fronts which typically develop in well curved nozzles at supersonic Mach numbers close to one (Schnerr [21], [22]) . In principle this characteristic two-dimensional propagation of heat addition already has been discussed by Bartlma [23] and Bratos and Meier [24]. Consequently in this flows static pressure or density measurements are not appropriate for Wilson point detection. Hence, an accurate theoretical onset condition has to be defined by a minimum value of condensate or heat addition, in our computation by a condensate mass fraction of

0.1 glkg, equ ivalent to a mean pressure increase of 2.7 mbar which corresponds to the repeatability of wind

Mf.~ = 0.8. a = 1.25'. c = 100 mm -1.0 TOI = 91.1 K. POI = 3.604 bar

-0.8

-0.6

- 0.2

0.0

-1.0

- 0.8

- 0.6

* CpOl ~ -0. 4

-0.2

0.0

-- - $0 = 0 %

- <1>0 = 91 .7 %

Pressure side

0.2 0.4 0.6 0.8 1 x/c

... --- -, /'", .......... - \

// \ I * t -- Cp02

I

: I

-- - --- I-

ACo = - 19.48 %

ACl = -36.69 %

Suction side

0.2 0.4

I

: I I

:

0.6 0.8 1 x/c

g, =0.1

Fig.7. Condensing flow around a NACA-0012 - static pressure distribution - right, from top: Mach number contours adiabatic (Mf ;:>: 1, ~Mf = 0.015), diabatic, condensate mass fraction contours (~gc = 2 g/kg)

179

tunnel measurements. In condensation processes of moist air similar effects has been established

(Refs. [8] and [22]).

Airfoil NACA-Q012

In condensing flows around inclined airfoils development and reaction of the nonequilibrium heat addition to the static pressure distribution is to be expected different along the suction and the pressure side. This is shown for the NACA-0012 (chord length 100 mm) at a free stream Mach number 0.8 and with an angle

of attack a = 1.250 • In adiabatic flow already two local supersonic areas exist (Fig. 7) having a maximum Mach numberof 1.373 and 1.088 on the airfoil surface ahead olthe shocks. Initial pressure and temperature are identical with that of the nozzle flows shown in Fig. 4. The relevant cooling rates at the suction and pressure side are reproduced in both nozzles, too, and in the limits of those shown in Table 1. For the condensation onset Mach number we get 1.110 at the suction side and 1.080 at the pressure side. Due to the compression the shock waves move forward to the leading edge. Obviously the aerodynamic properties in this diabatic flow are quantitatively decided by the effects arising at the suction side. Drag and lift coefficients decrease simunaneously about 19% and 37%, respectively. In tendency a pressure drag decrease is expected if the shock at the suction side is located near the trailing edge, e.g. at a high free stream Mach number or a high angle of attack (Schnerr and Dohrmann [25]). Otherwise the pressure decrease at the rear airfoil section leads to a drag increase. The contours of constant condensate mass fraction demonstrate another interesting detail. Whereas along the suction side of the airfoil the heat addition starts far upstream of the shock and heat is removed from the rear airfoil section due to evaporation, along the lower side the main heat addition develops outside of the supersonic area in subsonic flow. For freestream conditions, just a little bit supersaturated, wake flow measurements, e.g. for the drag, will become incorrect. In that case a distinct distribution of condensate mass fraction remains in the wake and may evaporate in the flow around rakes of pitot probes. I n the numerical results for the same airfoil and freestream Mach number at an angle of attack of a = 0.37· and with a simplified assumption of an equilibrium condensation process (Ref. [6]) the static pressure distributions of adiabatic and diabatic flows differ much less and as distinguished from the present results the form of the supersonic region is still not affected essentially. Hence, an equilibrium condensation process undervalues the effects of ho­mogeneous condensation, especially near the onset. On the other hand, the known sensitivity of any result calculated by the classical theory to the surface tension needs some further experiments to confirm the estimate introduced in our calculations.

Conclusions

In conclusion we find for condensation onset Mach numbers below 1.2 significant effects in transonic flows around airfoils on the form of the supersonic regions, shock positions, pressure drag and lift coefficients for initial states close to the vapor pressure curve. A comparison with homogeneous condensation in moist air shows weaker compression near the onset; for N2 flow supercritical heat addition is not observed and unsteady phenomena are not to be expected. Starting from the same initial values and with the same

cooling rates the Wilson points of Ref. [13] are good reproduced. Due to the actual estimate olthe surface tension including the Tolman correction and the lag of further experimental datas at this high pressures

additional experiments are necessary in this area of initial states in undisturbed quasi 1-0 nozzle flows, e. g. with supersonic heating fronts. In condensing transonic airfoil flows the diabatic pressure gradients should be taken into account being lower or higher at the suction and pressure side, respectively, at most

of about 50%. The following aspects are of interest to further computations of condensing airfoil flows: Onset conditions below the triple point, pressure drag and lift coefficient variation, depending on the angle of attack and investigation of different airfoil geometries.

180

Acknowledgement

The authors are particularly grateful to Professor P. P. Wegener of the Yale University and to Professor Guo Youyi and hers colleagues oftheJiaotong University Xi'an (China) to make available new unpublished results of cryogenic nitrogen flows.

References

[1) Proceedings of the First International Symposium on Cryogenic Wind Tunnels. Department of Aeronautics and Astronautics at the University of Southampton, England (1979) [2) Hall, A.M.: Onset of Condensation Effects as Detected by Total Pressure Probes in the Langley 0.3-Meter Transonic Cryogenic Tunnel. NASA Technical Memorandum 80072 (1979) [3) Hall, A.M.: Onset of condensation effects with an NACA 0012-64 airfoil tested in the Langley 0.3-metertransonic cryogenic tunnel. NASA Technical Paper 1385 (1979) [4) Koppenwallner, G., Dankert, C.: The homogeneous nitrogen condensation in expansion flows with ETW relevant stagnation conditions. In: Proc. First Int. Symp. on Cryogenic Wind Tunnels [1) (1979) [5) DOker, M.: Nitrogen condensation in stream tubes duplicating the airfoil flow In a cryogenic wind tunnel. DFVLR­AVA, Bericht No. IB 222-83 A 08 (1983) [6) Wagner, B.: Estimation of Simulation Errors and Investigations of Operating Range Extensions forthe European Transonic W1ndlunnel ETW. BMFT-FB, Bericht No. W 82-003 (1982) [7) Schnerr, G., Dohrmann, U.: Theoretical and experimental investigation of 2-Ddiabatictransonic and supersonic flow fields. Proc.IUTAM Symp. Transsonicum III, ed: Zierep, J., Oertel, H., 125-135, Springer (1989) [8) Dohrmann, U.: Ein numerisches Verfahren zur Berechnung transsonischer StrOmungen mit Energiezufuhr durch homogene Kondensatlon. Dissertation UniversitAt (TH) Karlsruhe (1989), in press. [9) Volmer, M.: Klnetik der Phasenblldung. Steinkopff, Leipzig, (1939) [10) Wegener, P.P.: Nucleation of nitrogen: Experiment and theory. Journ. Phys. Chemistry, Vol. 91, 2481 (1987) [11) Oswatitsch, K.: Die Nebelbildung in WindkanAlen und ihr EinfluB auf Modellversuche. Jahrbuch derdeutschen LuftfahrHorschung, Vol. 1, 692-703 (1941) [12) Zierep, J.: Theory of flows in compressible media with heat addition. AGARDograph No. 191 (1974) [13) Guo Youyl, JI Guanghua, Wang Qun et. aI., Jlatong University Xi'an, private communication (1989) and [10) [14) Wegener, P.P., Mack, L.M.: Condensation in supersonic and hypersonic wind tunnels. Advances in Appl. Mechanics, Vol. 5, ad: Dryden/~rmAn, 307-447 (1958) [15) Hill, Ph. G.: Condensation of water vapour during supersonic expansion in nozzles. J. Fluid Mech., Vol. 25,593-620 (1966) [16) Eberle, A.: A new flux extrapolation scheme solving the Euler equations for arbitrary 3-D geometry and speed. MBB/LKE 1221S1PUBl140 (1984) [17) Wegener, P.P.: Study of experiments on condensation of nitrogen by homogeneous nucleation at states modelling those of the National Transonic Facility. Final Report to the NASA Langley Research Canteron Grant NSG-1612, prtvate communication (1980) [18) Schmidt, B.: Beobachtungen Oberdas Verhalten derdurch Wasserdampfkondensation ausgelOsten StOrungen in einer Oberschall-WindkanaldOse. Dissertation UniversitAt (TH) Karlsruhe (1962) [19) Zierep, J., Lin, S.: Eln Ahnlichkeitsgesetz far instationAre KondensationsvorgAnge In der Laval-DOse. Forschung im Ingenieurwesen 34, 97-99 (1968) (20) Frank, W.: Condensation Phenomena in Supersonic Nozzles. Acta Mechanica 54,135-156 (1985) [21) Schnerr, G.: Homogene Kondensation in stationAren transsonischen StrOmungen durch LavaldOsen und um Profile. Habilitationsschrlft UniversltAt (TH) Karlsruhe, FakultAt far Maschinenbau (1986) [22) Schnerr, G.: 2-D transonic flow with energy supply by homogeneous condensation: Onset condition and 2-D structure of steady Laval nozzle flow. Experiments in Fluids, Vol. 7,145-156 (1989) [23) BartlmA, F.: Ebene OberschalistrOmungen mit Relaxation. In: Appl. Mech., ad.: H. GOrtler, 1050-1060, Springer (1966) [24) Bratos, M., Meier, G. E. A.: Two-dImensional, two-phase flows in a Laval nozzle with nonequilibrium phase transition. Archives of Mechanics 28,5-6,1025-1037, Warszawa (1976) (25) Schnerr, G.H., Dohrmann, U.: Transonic Flow Around Airfoils with Relaxation and Energy Supply by Homogeneous Condensation. AIM paper 89-1834, Fluid Dynamics and Lasers Conf., Buffalo, N. Y. (1989)

Explosive Boiling: Some Experimental Situations

V. p. SKRIPOV, o. A. ISAEV

Institute o~ Thermal Physics Ural Branch o~ the USSR Academy of Soiences Sverdlovsk

Summary

A model of explosive boiling making use of the idea of inten­sive homogeneous nucleation allows us not only to give a qua­litative explanation to effects observed in two-phase nonequi­librium flows. but also to make trustworthy quantitative eva­luations (for example, of critical flow rates through short channels) which cannot be obtained with the aid of traditional. schemes of heterogeneous media mechenic6. The field of appli­cability of the model is outlined quite definitely. This model is a useful addition to all other models of fluid mechanics.

1. Introduction

Boiling assumes an explosive character when the power of heat release increases sharply or a volume of hot liquid is rapidly depressurized. The structure of a two-phase flow depends on the development of instabilities of different nature in it. It is di~ficult to choose an appropriate model and ensure a sa­tisfactory accuracy of calculations for different apace-and­-time scales of flow of an extensively boiling liquid. In such a situation the study of separate fragments of a complicated problem making use of private models is of great importance. High-gradient flows are usually nonequilibrium because of the local incompleteness of the phase transition. It is connected with the insufficient density of the number of heterogeneous boiling centres.

Fig. 1 shows the variation of the specific mass flow rate in n-pentane flowing through a short channel into the atmosphere [11. The initial liquid state corresponds to the saturation line. The upper curve has been plotted by the experimental

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems JUTAM Symposium Gatlingen, Germany, 1989 © Springer-Verlag Berlin Heidelberg 1990

182

%, Kt/Cm'.'

o p,MPa.

Pig. 1. Specific mass flow rate of boiling n-pentane (from the saturation line) flowing through a short channel into the atmosphere. The dashed line shows calculation by an equilib­rium model.

values of 9 ' the lower one shows calculation by an equilibrium model for a critical regime. A considerable excess of the ob­served flow rates over those calculated is caused by the fact that the jet remains in a one-phase state in spite of a high liquid superheat at the entrance into the channel. A similar pattern has been observed for water [2] and carbon dioxide [3]. The discrepancy under discussion increases with increasing initial temperature (pressure), reaches its maximum at T;Tc~ ~ 0.9 and vanishes in approaching the thermodynamic critical

point ( Tc , r c. ).

~he limit thermodynamic nonequilibrium corresponds to the at­tainment of liquid superheats at which intensive spontaneous boiling is observed on nucleus bubbles of fluctuation nature. The physical definiteness of this boundary is conditioned by a very sharp dependence of the nucleation rate J ( T , P ) on the Gibbs number G- = W,/kg T , where W* is the work of formation of a critical. nucleus 14, 5], ksiS the Boltzmann constant. By the homogeneous nucleation theory the value of J is calculated making use of thermodynamic parameters. Thus, for water at at­mospheric pressure ( ~ = 373 K) and T = 573 K we have J ~

183

~10-3 8-1m-3, and at T .. 583 K the value of J increases by 23 decimal orders. It is important that the homogeneous nucle­ation theory is in good agreement with the experimental data [5, 6] obtained on small samples at high superheats.

The intensity of a flow of fluctuation vapor nuclei increases practically from zero to a very large value in a narrow tem­perature and pressure range. On aT, p constitutional diagram we have a belt that contracts at the thermodynamiQ critical pOint. This allows us to introduce the notion of shock or ex­plosive boiling as a limiting regime under a rapid change of the liquid state with a deep penetration into the metastable region [5-7]. It is supposed that the action of ready and heterogeneouB vaporization centres gives a relatively weak and "slurred over" contribution to the boiling process. The higher is the concentration of Buch centres in the system, the higher must be the rate of the state change to ensure ex­plosive boiling. The term "limiting regime" is used in the sense that the evaluation of the number of arising vapor bub­bles may be made by the homogeneous nucleation theory.

At our Institute various manifestations of the explosive (shock) boiling of liquids in high-gradient flows have been studied for many years. Intensive homogeneous nucleation re­sults in a strong hydrodynamic response of the system. The observed sign of this response (the liquid flow rate through a short channel, the f9rm of a free jet, the reactive force, the speed of motion of the boundary of a two-phase cloud) de­pends on the organization of the experiment. Every realized situation (see the Table) is characterized by correspondence between some obvious peculiarity of the process and the con­ditions of explosive boiling. If in an adiabatic process To'

Po are the initial parameters, ~>O is the terminal pressure, explosive boiling sets in at To~O.9Tc. We also have Po>~(;;)'

where p. is the pressure on the saturation line. s

Let us discuss in a nutshell the manifestation of explosive boiling in the situations presented in the Table.

184

Ifable

Situation Substance Characteristic property

L;i.Qw.i,d ~-pentane [1,8} Flow -flo. water 12], Channel critical of hot rate nitrogellr choking ( To ~ 0/3 Tc ) liquid oxygen 19

through a short

[101 channel Jet form ~-pentane Complete jet opening

Jet [11 ]

Sharp decrease of R. reaction in-pentane depending on tem)erature

R, Tv (pressure Po . Hot liquid scattering

t12] Increase in the rate of

in envelope failure lFreon-113 scattering of a vapor cloud

Pipe uncorking Water, carbon dioxide i13J

Splitting of a relief (boiling) wave

2. Liquid Flow Rate

Experiments were carried out with a set-up of short-time action (the time of outflow from 5 to 20 s) with a cylindrical chan­nel (the inner diameter d .. 0.5 DIm, the relative length tid • • 1.4) with a sharp entrance edge. Channels of somewhat dif­ferent dimensions were also used. The liquid outflow into the atmosphere was accompanied with a sharp decrease in the flow rate at Po" const when the initial temperature was about or exceeded 0.9~. Earlier choking does not occur because of a short time of the liquid stay in the channel ( ~10 mcs in our experiments). But intensive volume boiling on fluctuation centres ensures a vaporization rate necessary for the choking of the flowc Fig. 2 gives the results of experiments with seve­ral liquids, the values of 9h have been calculated by formula (1). A solid line has been used to approximate data for water [2J with channels of a considerably larger diameter.

It is important to establish if the decrease of the flow rate ~ observed on short channels is connected with the critical

flow of a two-phase medium. A direct method of solving this

185

'i/i" t.OO

4 I>

0. 0- 1 0- 2

(),50 1>- 3

Fig. 2. Relative flow rate in different liquids flowing out t{).rough a short channel: 1) n-pentane ( Pol Pc = 0.11; ci = 0.5 mm; f,,/cL.,. 0.8 and 1.4) l141; 2) n-hexane ( I2/Pe- = 0.86; d,= 0.5 mm; e/ d= 1.4) /151; 3) dibromotetrafluoroethane (Po/Pc = 0.90; d.. 0.) mm; Uri. 1.4 and 2.6) [15J; 4) water (p.,/Pc= 0.81; rim ).5-4.5 mm; .t/d-.,. 0.1-).0) [2J

problem presupposes setting up experiments with a variable backpreasure. Such experiments have been carried out with n­-pentane [8], 'fc.,. 469.1 K, Pc. = ).)1 lIPa. In Fig. ) one can see a phase diagram for n-pentane, oircles show the initial states of liquid in a big volume. They correspond to three se­ries of experiments for different initial pressures (2.6; 3.6;

p.MPa. 5 0 1

r<4 o 1

I o I

I 1

I

o 0 01 000 0/ 0 0

5-1 6,/ o 0 9 000/0 0 0

II I III C

00'1

Fig. J. Phase diagram of n-pentane: 1) saturation line, C) critical point, 2) liquid spinodal, J) line J( T , P ) .,. .,. 1020 s-1m-J. The rest of the symbols are in the text.

186

4.9 MFa). A dashed line 4 shows the adiabat that separates the region of "cold" liquid I from region lIt where below the saturation line 1 there is a mode~tely superheated state (without any appreciable vaporization in the short nozzle). Adiabat 5 reaches the region of intensive spontaneous nucle­ation at atmospheric pressure. For line 3 we have assumed the condition J= 1020 s-1m-3 t 2 is the spinodal determined by the condition (oploV) = 0; 6 is the adiabat that passes

T through the thermodynamic critical point. Rapid adiabatic ex-pansion from region III results in explosive boiling. Fig. 4a refers to the outflow of cold liquid. A step is conditioned by the tra~ition from a flow with a cavity (at small ap) to a regime with a separation of the jet from the wall along the whole length of the channel. The calculation of the flow rate by a hydraulic formula

(1)

where fa is the liquid density and)A is the discharge coef­ficient, is in good agreement with the experimental data.

b

Fig. 4. Dependence of the specific mass flow rate of n-pentane on the pressure differential AI' = R, ,.. f1 at p .. = 3.6 JlPa for two initial temperatures: -,;,. 293 K, To/To. = 0.624(a) and r., = 451 K, To /Tc = 0.913(b). Dots show the experiment, solid

lines - calculation by formula (1), and a dashed line - calcu­lation by (3)

187

(On the right-hand side of (1) for a flow with a cavity there appears an additional faotor ~ =v~l~' ~CJ). Fig. 4b shows the onset of a critical regime in the case of hot liquid (shown wi th an arrow). An inorease in the pressure differential A P was oreated by decreasing backpressure ~ • Adiabata plotted for this kind of flows are in the belt between linea 3 and 2,

see Fig. 3.

The choking of a channel shown in Fig. 4b means that the velo­city of propagation of small perturbations ~ near the outlet section decreases to the value of the axial flow velocity U • (In our experiments U~102 m/s). The decrease of 1Af by appro­ximately an order of magnitude is due to the fact that the me­dium compressibility increases when small vapor bubbles appear in it.

For short channels we suggest identifying the choking pressure (effecti ve critical backpresaure) with the pressure p;... ( T ),

at which the homogeneous nucleation rate has a sufficiently high value. For definiteness we assume

(2)

Here T = To -8T, 8T is a correction allowing for the effect connected with the adiabatic liquid expansion. In such a model the critical discharge is determined by a simple expression

0) ~he difference from formula (1) is that instead of the ambient pressure ~ we use the pressure found from condition (2). The validity of (3) has been corroborated by our experiments. The critical flow rate shown in Fig. 4b with a dashed line has been obtained by formula (3) •

.3. The Jet Form

The specific character of explosive boiling is reflected in the form of the jet beyond the outlet section at a short chan­nel ~ol. The same channel as in the experiments on the mea­surement of the flow rate has been used. Fig. 5 gives a series

188

a

b

-d

e

f

g

~ig. 5. Photographs ofn-pentane jets at a pressure in the chamber po = 2.6 'MPa for a number of temperatures To: 293 K(a}; 354(b); 369(0); 385(d); 419(e); 433(f); 451(g)

ot photographs of n-pentane jets flowing out into the atmo­sphere (Pc./Pe = O.76) with a successive increase of temperature 't/Te from 0.62 to 0.96. Photographs with an exposure of about 1 mCB have been obtained. A cold and weakly superheated jet is characterized by a base expansion angle eX that is close

189

to zero. With increasing liquid superheat the jet swells, pass­es into a dispersed state, the angle ex, increases up to 90-1000 •

Then an abrupt complete expansion of the jet is observed at To/ Tc ::::t 0.92, its trace is lost in the photograph. The ex­

pansion angle is confined by the design of the outlet washer. At a higher initial temperature a gas dynamic jet with a rela­tively dense core is formed. It is characteristic that upon achievement of the conditions of explosive boiling the tran­sition to a complete expansion of a jet proceeds by a leap and not gradually.

4. The Jet Reaction

Liquid flowing out into the atmosphere creates a reactive force R that acts on the chamber. A freely suspended apparatus devi­ates to the side opposite to the flow, then it is considered that K~ O. For a weakly diverging jet

R. = c,.u )

where ~ is the mass flow rate, U is the velocity in the sec­tion where the pressure in the jet is equal to atmospheric pressure. If there is a radial velocity component, and the chamber is flowed over by accompanying flows, the jet reaction may be smaller than that calculated by formula (4). For con­ditions of explosive liquid boiling we expect a considerable decrease of R. due to a complete expansion of the jet and. de­creasing d.ischarge G- = all ,J2 is the channel sectional area. i! 0 0

Experiments have been carried out with n-pentane [11]. The va­lue of the reactive force in a stationary flow regime was de­termined by the established deviation of the chamber from the rest position in the gravitational field. The same channel as in the experiments described above was used. Fig. 6 gives the measured valuea of R as a function of the initial temperature at a given pressure ~ = 2.6 MPa. ~he fall-through of the re­active force and even the change of its sign look impressive. The region of the fall-through correspond.s to the values of To, Po at which a complete expansion of the jet is observed.

190

R,N

0,8 0 0

0 0

0.6 ~ f I

0,4 I I I

0,2 I I I I I I

0 I I

~

Fig. 6. Temperature dependence of the reactive force in n­-pentane flowing out through a short channel into the atmo­sphere at Po = 2.6 MFa

R,N 1,0

0.8 0 - 1

0.6

0.4

0,2

0 - 2

o ~,2L-~--~---=~~ o i 3 .1p.MPa

Fig. 7. Reaction of a jet of n-pentane: 1) cold liquid, To = • 293 K; 2) in flowing out from the saturation line

Fig. 7 shows how the force R changes wi th increasing pressure in the chamber for cold liquid ( To= 293 K) -1, and in a flow

from the saturation state - 2, Po = Ps ( 1;,>, Ap= Po - Pa.tl't!' A fall­-through in the form of a wide step for case 2 connected with sharp boiling of a liquid contrasts especially with the linear growth of R in flowing out cold pentane.

191

An "anomalous" decrease of the reactive force with a corres­ponding choice of the interval of the initial parameters To, P is caused by a fundamental reason. But the fact of changing o

the Sign of R.. beara the imprint of the geometrieal parameters of the apparatuB, depends on its size. A decayed jet causes a flow along the front wall of the chamber, which results in decreasing static pressure ~t and appearance of other accom­panying motions near the chamber. For a complete reaction one may write

R- = f U h ~ Jil -t- J Ap cASl (5) (..D.1 ) (122 )

where 'Un is the projection of the velocity on the direction of the jet axis, b. p = Pst - Pcxbn' fl1 is the surface of the he­mi~phere that limits the jet at the exit section, .122 is the total Burface of the chamber. At Ilf< 0 the second term in (5) may prove to be predominant, then R<.o. If the chamber were located in a vacuum, there would be no negative reaction.

5. Scattering of a hot liquid in the process of the envelope de~truction

Experimen t~ were carried out with freon-113 ( Tc = 481.2 K, Pc = = 3.41 MPa) [12]. The Gubstance wae in a sealed glass spheri­c.!>;1 ampoule (]) ~ 2.5 cm). The mean denei ty of filling was

, 0

c1o~e to the cri tical one (fc). The ampoule was placed in a chamber made of ~heet duralumino The chamber acted as an air thermo~tat and was supplied with sight glasses. The temperature of the experiment~ was in the interval (0.8-1.0)Tc • A striker wa~ used to destroy the envelope. In photographing use was made of eo light pulse with a duration of '" 1 mcs and a controlled delay id with respect to the beginning of the deatruction. Fig. 8 givea photographs of explosions for To/Tc = 0.92 at three valuea of td: 50, 100 and 200 mes.

In proceossing the results of photographing we introduced into consideration the value of the relative volume of an explosive cloud 'f = (D FDa)3 - 1, where ])0 is the ampoule diameter, D ia the diameter of an explosive cloud, i.e. the distance be­tween the extreme boundaries of a vapor-liquid mixture that

192

Fig. 8. Explosive cloud of freon-113 for a reduced initial temperature TolTe = 0.92 at three values of the delay time (from left to right): 50, 100 and 200 mea

2,0

1,0

Fig. 9. Dependence of the rate of increase of the volume of an explosive boiling on the initial temperature of the liquid

are the most remote from eaeh other. Fig. 9 shows the tempe­rature dependence of the value of ~ whieh charac~erizea the rate of increase of the vapor cloud volume. Here ~ has been evaluated by two points at t:.cl = 50 mes and Crt = 100 mes. It i6 evident that at Te/Tc;;:O 0.9 the rate of the volume growth increases sharply.

6. Pipe uncorking The pressure field inside a pipe during its depressurization haa been investigated in experiments with water and carbon

193

dioxide. The inner diameter of the pipe was 2.5 mm, the length was 3 m. One end face was plugged. The uncorking was realized by breaking the membrane at the free end. The initial liquid temperature To before every separate act of depressurization was maintained constant along the pipe. The value of To in the experiments was from 0.51 to 0.91 of 1 . The pressure on the outside was equal to atmospheric pressure.

Experiments showed that the character of the pressure decrease at TaiTe- below and above 0.9 was different. Let us dwell on the initial stage of the outflow, where the plugged end of the pipe does not have any effect yet.

The rarefaction front propagates along the pipe at a speed clOHe to the sound speed in a liquid. The pressure falls to a certain value of P .... ·.( Pr (T). At To ITc. <. 0.9 we have R . ~

<nth S 1->"11

~0.5 P, • At ToITe >- 0.9 the value of P ~' .. exceeds 0.5 D ,p, '" S ... t,. ri: 1'>1-1'",-

~P.., i.e. the maximum. liquid superheat in a rarefaction wave *" ia close to a etate of intensive homogeneous nucleation. A si-

milar result is observed in Ref. [11]. The first front is fol­lowed by the second relief wave directed inside the pipe (see Fig. 10), which is absent at TaiTe <: 0.9 [13, 181. The rate of the preaaure decrease in the second wave is two or three orders lower than in the first stage of the pressure drop. The velo­city of propagation of the second front along the pipe is se­veral tens of metres per second.

" the first rare­'action \,lave

the second rare­faction \.Jove

2~ __ ----,-___ -,--_ o 10 20 t,ms

Fig. 10. Time dependence of the pressure in the pipe during an ejection of boiling carbon dioxide (tit = 0.96) at a dis-tance of 12 cm from the outlet section C

194

The discovered correlation of P.... with R must be connected ,,,'h If-with the process of vaporization inside the pipe. Vaporization on the rarefaction front at a pressure close to p~ leads to a splitting of the front of a rapid pressure drop into two.

References

1. Reshetnikov A.V., Isaev O.A., Skripov V.P. Kriticheskiye rashody v~kipayuahchey zhidkoaty i kondensiruyuehchegosia gaza pri neravnove~nom regime istecheniya (Critical flow rates of boiling liquid and condensing gas under a none qui­librium regime of outflow) II Teplofizika Visokih Tempera­tur. - 1988. - V826, N38 - P.544~548.

2. Kh.le~tkin D.A., Kor~hunov A.C., Kanishchev V.P. Opredele­niye ra~hodov vodi visokih parametrov pri istechenii v at­mospheru cherez tailindriche~kiye kanaly (Determination of water flow rates of high parameters in the outflow into the atmosphere through cylindrical channels) II Izvestiya Aka­demii Nauk SSSR, Energetika i transport. - 1978. - N5. -- P.126-135.

3. Hesson G. t Peck R. Flow of two-phase carbon dioxide through orifice~ II AIChE J. - 1958. - V.4, N2. - P.207.

4. Volmer M. Kinetik der Phasenbildungt Dresden - Leipzig, 1939.

5. Skripov V.P. Metastabil'naya zhidkost. - M.: Nauka. - 1972. - 312 p .. (Skripov V.P. Metastable Liquidf;\l, Halsted Press, New York, 1974).

6. Skripov V.P., Sinitayn E.N., Pavlov P.A., Ermakov G.V., Mu­ratov G.N., Bulanov N.V., Baidakov V.G. Teplofizicheskiye $voistva zhidkostey v metastabil'nom aostoyanii. - M.: Atomizdat. -1980. - 208 p. (Skripov V.P., Siniteyn E.N., Pavlov P.A., Ermakov G.V., Muratov G.N., Bulanov N.V., Bai­dakov V.G. Thermophysical Properties of Liquids in the Me­tastable (Superheated) State, Gordon and Breach Science Publishers, New York, London, Paris, 1988).

7. Pavlov P.A. Dinamika vskipaniya sil'no peregretih zhidkoa­tey (Boiling dynamics of highly superheated liquids). -- Sverdlovsk. 1988. - 244 p.

8. Isaev O.A., Reshetnikov A.V., Skripov V.P. Izucheniye kri­ticheskogo zapiraniya stataionarnih potokov vskipayushchey zhidkosti (Study of the critical choking of stationary flows of boiling liquids) II Izvestiya Akademii Nauk SSSR, Ener­getika i transport. - 1988. - N6. - P.114-121.

9. Baidakov V.G., Maltsev S.A., Pozharskaya G.I., Skripov V.P. Vzrivnoye vakipaniye zhidkih azota i kisloroda pri iateche­nii cherez korotkiye naaadki (Explosive boiling of liquid nitrogen and oxygen in the outflow through ahort nozzles) II Teplofizika Visokih Temperatur. - 1983. - V21, N5. -

P.959-964.

195

100 Ieaev O.A., Nevolin M.V., Utkin S.A. Formi raspada svobod­noy struyi vskipayushchey zhidkosti (Forms of decay of a free jet of boiling liquid) II Termodinamika metastabil' nih eistem. - Sverdlovek, 1989. - P.J3-39.

11. Ieaev O.A., Nevolin M.V., Skripov V.P., Utkin S.A. Reak­tsiya strui vskipayushchey zhidkosty (Reaction of a boiling liquid jet) II Teplofizika Visokih Temperatur. - 1988. -

V.26, N5. - P.1028-1030.

12. Ieaev O.A., Nevolin M.V., Skripov V.P., Utkin S.A. Razlet goriachey zhidkosti pri razrushenii obolochki (Scattering of a hot liquid after the ~hel1 destruction) II Zhurnal prikladnoy mehaniki i tehnicheskoy fiziki. Novosibirsk, 1988. - N4. - P.72-75.

13. Ieaev O.A., Pavlov P.A. Vskipaniye ~hidko~ti v bol'~hom obiome pri biatrom sbrose davleniya (Liquid boili.ng in a large volume with a rapid preasure release) II Teplof.i­zika Vi~okih Temperatur. - 1980. - V.18, N4. - P.812-818.

14. Shuravenko N.A., Iseev O.A., Skripov V.P. Vzrivnoye vski­paniye peregretoy zhidkosti pri techenii ohere:?: korotkiye nasadki (Explosive boiling of a superheated liquid in the eu tflOVI through :ahort nozzle;;;) I I Teplofi2ika Vi!'liokih Tem­perature - 1975. - V.13, N4. - P..896-898.

15. Skripov V.P., Shuravenko N.A., 18~ev O.A. Zapiraniye pato­ka v korotkih kanalah pri udarnom viikipanii 2ihidko!!liti (Flow chok.ing in short nozzples in shock boiling of liqu.ida) II Teplofizika Visakih Temperatur. - 1978. - V.16, N3. -

P .. 563-568.

16. Slov B.N. Istecheniye zhidkOGti chere£; naaadkl v eredi s protivodavleniem (Liquid outflow through nozzle~ into me­dia with backpreaaure) I I M.: Ma",hinol;; troyeniye.. 1968. - 140 p.

17. Lienhard J.R., Alamgir Md., Trela M. Early response of hot water to sudden release from high pressure II Tranaactions of the ASME, Journal of heat transfer. - 1978. - V.100, N3. -- P.473-479.

18. Edwards A.R., O'Brien T.P. Studies of phenomena connected with the depressurization of water reactors II British Nuclear Ener. Soc. - 1970. - V.4, N2. - P.125-135.

On the Similarity Character of an Unsteady Rarefaction Wave in a Gas-Vapour Mixture with Condensation H.J. Smolders, E.M.J. Niessen, M.E.H. van Dongen.

Dept. of Physics, Eindhoven University of Technology, P.O. Box 513, NL-5600 MB Eindhoven, The Netherlands.

Abstract The self-similar solution of an unsteady rarefaction wave in a gas-vapour mixture with condensation is investigated. If the onset of condensation occurs at the saturation point, the rarefaction wave is divided into two zones, separated by a uniform region. If condensation is delayed until a fixed critical saturation ratio Xc > 1 is reached, a condensation discontinuity of the expansion type is part of the solution. Numerical simulation, using a simple relaxation model, indicates that time has to proceed over more then two decades of characteristic times of condensation before the self-similar solution can be recognized. Experimental results on heterogeneous nucleation and condensation caused by an unsteady rarefaction wave in a mixture of water vapour, nitrogen gas and chromium-oxide nuclei are presented. The results are fairly well described by the numerical relaxation model. No plateau formation could be observed.

Introduction

The unsteady expansion of a gas-vapour mixture in e.g. the expansion section of a

Ludwieg tube or the high pressure section of a shock tube, caused by the instantaneous

opening of a membrane, is usually characterized by strong non--equilibrium processes with

a relaxation time 7. When t / 7 -+ CD the solution tends to a self-similar solution. If

condensation begins at a saturation ratio of unity, the rarefaction wave is split into two

zones, separated by a uniform region. This is due to the discontinuous character of the

equilibrium speed of sound at the onset of condensation. If condensation is delayed until a

fixed critical supersaturation is reached, a condensation discontinuity is expected to be

part of the solution. The condensation discontinuity has been subject of many theoretical

and experimental investigations [1,2,3,4]. Much work has been published about

non--equilibrium condensation in an unsteady rarefaction wave [5,6,7,8]. In the present

paper the asymptotic behaviour of such a system is investigated. The allowed types of

condensation discontinuities are discussed and a numerical simulation of the approach to

the asymptotic solution is presented. Finally some experimental results are compared with

theory.

The condensation discontinuity

We start with the assumption that in an adiabatically expanding mixture of vapour and

gas, condensation will occur at a given critical value Xc of the saturation ratio X defined

198

as:

(1)

where Pv is vapour pressure and Pve is equilibrium vapour pressure. The latter is given for

a perfect gas by the Clausius-Clapeyron equation:

(2)

R denotes the specific gas-constant, cp and Cv the specific heat at constant pressure and

constant volume, respectively, and cl the specific heat of the liquid phase. Further, gas,

vapour and liquid are indicated by subscripts g, v and l, respectively. The latent heat L

varies with temperature according to:

(3)

The critical value Xc at which the onset of condensation actually will take place depends

on the mechanism of nucleation that controls the onset. From this point of initial

condensation on, the mixture relaxes towards a new equilibrium by the exchange of heat

and mass between droplets and gas.

If the characteristic time of relaxation tends to zero, the transition occurs instantaneously

in a sharply defined front. We denote the velocity of the front by U and the velocity of the

mixture by u. The state of the mixture is characterized by pressure p, temperature T and

liquid mass fraction fp. The gas and the vapour are considered perfect so they obey

Dalton's law. For simplicity we shall from now on refer to the supersaturated vapour state

and to the state of liquid-vapour equilibrium as state 1 and state 2, respectively. It is clear

there exists no liquid phase in state 1, fpl = 0, and the vapour mass fraction fVl is equal to

the total mass fraction of the condensable component of the mixture. The vapour pressure

Pv2 of state 2 has to satisfy the Clausius-Clapeyron equation (2). The laws of conservation

of mass, momentum and energy are, applied to the condensation discontinuity:

(4)

(5)

(6)

199

where the specific enthalpy h of the mixture can be written as:

(7)

The last term in equation (7) represents the heat released due to condensation of vapour.

Introducing E = fgRg + fvRv' cp = fgcpg + fvcpv + fpc l , Cv = cp - E and ;Y = cp/cv' the frozen and equilibrium speeds of sound are:

(8)

and:

(9)

respectively. The Mach numbers for state 1 and state 2 can now be defined as:

(10)

A numerical evaluation ofthe condensation discontinuity is performed for several values of

the critical saturation ratio Xc. State 1 is related isentropically to a fixed reference state O.

In Fig. 1. the Rankine--Hugoniot curves and the Ma2-Ma1 relations are shown for a

mixture of water vapour and nitrogen gas. Only those parts of the curves are shown that

correspond to entropy increase, to real massflux and to positive droplet mass fraction

downstream the discontinuity. The Chapman-Jouguet points, defined by Ma2 = 1,

separate the curves in four different regions:

strong compression

weak compression

weak expansion

strong expansion

: Mal> 1, Ma2 < 1,

: Mal> 1, Ma2 > 1,

: Mal < 1, Ma2 < 1,

: Mal < 1, Ma2 > 1.

Following the arguments of Landau and Lifshitz [9] strong compression and strong

expansion discontinuities are excluded, because systems with strong compression

discontinuities are overdetermined and strong expansion discontinuities are internally

unstable [10]. So the only allowed condensation discontinuities in gas dynamic flow are

weak compression and weak expansion discontinuities with the Chapman-Jouguet

solutions as limiting cases.

200

1.2

~ LO ----~,:----0.8 I X=2~

I c A I

I

M'~2 lfi2 LO --\\~-t-- ----

10 2 1.05 I

1.4

0.8

1.4

Fig. 1. The condensation discontinuity for a mixture of water vapour and nitrogen gas; A. Rankine-Hugoniot curve; B. Mach number relation; Xc: 1.05, 2 and 10; state 1 is related isentropically to reference state 0: Po = 1 bar, To = 295 K, fvo = 0.0136. -+-: Chapman-Jouguet point; ~: zero droplet mass fraction downstream the discontinuity; -0: zero entropy increase.

The self-similar solution

In analyzing the self-similar solution of an unsteady rarefaction wave in a gas-vapour

mixture with condensation, we will first discuss the situation where the onset of

condensation occurs at a saturation ratio of unity. Then, the change in the state of the

mixture is continuous. The solution is fully isentropic and follows from the characteristic

form of the Euler equations, which for a left running wave is:

u+F=O (11)

-t~ F- " PoP a

(12)

x/t=u-a (13)

This set of equations is completed with the isentropic condition. Due to a jump in the

speed of sound at the onset point, equation (13) results in a solution as depicted in fig. 2A.

Two rarefaction fans are separated by a region of constant critical state. The tail

characteristic of the leading fan travels at x/t = Uc - afc' while the first characteristic of

the second fan travels at a speed Uc - aec'

If condensation starts at a fixed critical saturation ratio Xc > 1, the change of the

supersaturated vapour state to the state of liquid-vapour equilibrium can only be

201

discontinuous. The position of the condensation discontinuity in the self-similar solution

is completely determined by the value of Xc. As stated in the preceding section the

discontinuity is either a weak compression, a weak expansion or a Chapman-Jouguet

discontinuity of either two types. A weak compression discontinuity and a

Chapman-Jouguet compression discontinuity are not allowed, because Mal> 1, which

means that the discontinuity should travel at supersonic speed. This cannot be matched

with the prescribed value of Xc. The Chapman-Jouguet expansion discontinuity is a

possible solution. In this case Mal < 1, Ma2 = 1. This means that the discontinuity lags

behind the leading rarefaction fan, thus creating a uniform region. The second rarefaction

fan remains connected to the discontinuity. Finally we consider the weak expansion

discontinuity. Again Mal < 1, but now M~ < 1 , which implies that every left running

wave will overtake and change the discontinuity. A self-similar solution then exists when

the state behind the discontinuity is uniform. Which type of discontinuity occurs depends

on the depth of the expansion. Some typical examples of possible self-similar solutions

with condensation discontinuities are shown in Fig. 2B.

p

a

a b

A B

x x/t

Fig. 2. Self-similar rarefaction waves. A. x-t diagram of self-similar isentropic rarefaction wave; a: head of wave; b: head of plateau; c: tail of plateau; d: tail of wave; e: piston. B. Typical examples of pressure profiles for self-similar rarefaction wave; a: with Chapman-Jouquet discontinuity; b: with weak expansion discontinuity.

The self-similar solution is an asymptotic solution that develops when tiT -till, with T as

the characteristic time of the condensation process. This development is studied

numerically for a mixture of nitrogen gas and water vapour. At t = 0, a piston is

accelerated instantaneously to a constant velocity, causing a one-<llmensional unsteady

rarefaction wave in a mixture of gas and vapour. The wave is travelling to the left and the

wave front runs with x/aot = -1. A simple relaxation model is assumed:

-= f -f ve v (14)

dt T

202

1.0

p/po

0.8

0.6 A

1.0 1-------,,,-----.,----;;----1

0.8

0.6

c

-1.0 -D.8 -D6 -D.4 x/aot

Fig. 3. Numerical simulation of unsteady rarefaction wave in mixture of nitrogen gas and water vapour; A. pressure, B. temperature, C. vapour mass fraction, D. saturation ratio; curves a: frozen, f: equilibrium; b, c, d, e correspond to t/T = 10,20,40,80, respectively; Po = 1 bar, To = 295 K, fvo = 0.0136, ao = 350 m/s; Xc = 2; piston velocity is 125 m/s.

where T is taken constant and d/dt denotes the material derivative. The equilibrium

vapour mass fraction fve is taken equal to:

(15)

* Here p is the local density of the mixture and T is the temperature the mixture would

attain in a state of equilibrium with the local value of density and internal energy. For the

calculation, the Random Choice Method is used, adapted by an operator splitting

technique to account for relaxation [11,12]. The condensation discontinuity is not a priori

included in the algorithm. The numerical results are given in Fig. 3. The formation of a

203

uniform region of critical state is already observed in the early stages of the process. The

condensation occurs at the rear end of the plateau. Due to relaxation of the mixture the

saturation ratio evolves to unity. With increasing time the wave following the plateau

steepens to form a discontinuity. It appears that time has to proceed over more then two

decades of characteristic times before the self-similar solution can be recognized.

Experiments

The unsteady rarefaction wave in a gas-vapour mixture is studied experimentally. The

experimental set-up used, is described in detail elsewhere [13]. It consists of a tube of 12.8

m length and 0.1 x 0.1 m2 cross-section, initially separated from a large vacuum vessel by

a polyester membrane. All the experiments are performed with a mixture of water vapour and nitrogen gas at an initial pressure of 1 bar in the test section. Condensation nuclei,

chromium-oxide particles with a size of the order of 10 nm, are added to the mixture to

stimulate heterogeneous condensation. Bursting the membrane causes a rarefaction wave

to travel into the test section. The depth of the expansion is controlled by an orifice

between test section and vacuum vesseL The mixture expands adiabatically and the

degree of saturation will increase, resulting in dropwise condensation of water vapour. At

the observation point at 6.6 m from the orifice measurements are performed of pressure

with a piezo-electric transducer and of gas density with a Mach Zehnder interferometer

[14]. In addition, a three-wavelengths light extinction set-up is used to determine modal

droplet radius, relative width of the droplet size distribution function, droplet number

density and liquid mass density [8].

A typical experiment is shown in Fig. 4. Pressure, temperature, vapour mass fraction and

saturation ratio are compared with numerical calculation. The characteristic time T,

required for numerical evaluation, is obtained by a fit of the experimental vapour mass

fraction signal, resulting in T = 5 ms. The chosen values of critical saturation ratio and

piston velocity are rather arbitrary. The experimental saturation ratio is calculated from

pressure, temperature and vapour mass fraction. When no liquid mass can be detected, the

vapour mass fraction is set to the initial value. Experiment and numerical simulation

agree fairly well. The first part of the expansion of the gas-vapour mixture is isentropic

and accounts for an increase of the saturation ratio. Condensation on the heterogeneous

nuclei starts at a value of the measured saturation ratio of about three. After the onset of

condensation a rise in temperature is observed due to the release of latent heat. The

saturation ratio tends to unity as time increases. The plateau formed in the numerical

solution is not observed in the experimental signal. Obviously, the experimental condition

is far from self-similarity, and the expansion process is still in its early stage, where

relaxation is dominant. The simple numerical model does not describe accurately the

204

details of the relaxation process, so that a more refined relaxation model has to be

implemented.

1.0 1.0

p/po T/To

0.9 0.96

0.8

0.92

1.0 3.0

fv/fvo b );

0.9 a 2.0

a

0.8 b

C 1.0 D

0 10 20 30 0 10 20 30 t (ms) t (ms)

Fig. 4. Unsteady rarefaction wave in mixture of nitrogen gas and water vapour. Typical experiment compared with numerical calculation; A. pressure, B. temperature, C. vapour mass fraction, D. saturation ratio; curves a: experimental b: numerical; Po = 1.00 bar, To = 295.7 K, fvo = 0.0139; numerical: T = 5 ms, Xc = 2.3, piston velocity is 75.6 m/s.

Discussion

It is interesting to note that the ultimate self-similar solution discussed before has got

very little attention in literature. Zel'dovich and Raizer [15] discuss the interesting and

related problem of the reflection of a strong shock wave from the free boundary of a solid.

Due to the peculiar thermodynamic properties of the material, both an expansion shock

wave and a plateau appear. In the final asymptotic self-similar solution the expansion

shock wave and the plateau develop into the same type of asymptotic solution as given

here. Heybey and Reed [4] discuss the possibility of the formation of a weak compression

condensation discontinuity that moves supersonically with respect to the gas and

overtakes the rear part of the rarefaction wave. In their view, the discontinuity becomes

weaker and finally becomes sonic. Yet, it is not well explainable from the theory given

before how such a compression discontinuity can ever become sonic. A detailed model of

homogeneous nucleation and growth of droplets in an unsteady rarefaction wave is

developed by Sislian [6]. Numerical evaluation of the model is performed for the initial

205

tage of the expansion process. The effect of condensation is observable as a condensation

~axation zone followed by a compression shock wave. However, the time interval is too

mited to analyse the asymptotic behaviour. It is a challenging task to analyse how the

omplicated flow pattern of Sislian will develop into the asymptotic solution predicted in

he present study.

leferences Oswatitsch, K.: Kondensationserscheinungen in Uberschalldiisen. ZAMM 22 (1942) 1-14.

:. Wegener, P.P.: Water vapor condensation process in supersonic nozzles. J. Appl. Phys. 25 (1954) 1485-1491.

:. Belenky, S.Z.: Condensational jumps. Comptes Rendus (Doklady) de l'Academie des Sciences de l'URSS 48 (1945) 165-167.

l. Heybey, W.H.\ Reed, S.G. Jr.: Weak detonations aild condensation shocks. J. Appl. Phys. 26 (1955) 969-974.

,. Wegener, P.P.j Lundquist, G.: Condensation of water vapor in the shock tube below 1500K. J. Appl. Phys. 22 (1951) 233.

;. Sislian, J.P.: Condensation of water vapour with or without a carrier gas in a shock tube. Institute for aerospace studies, univ. of Toronto, report no. 201 (1975). Glass, I.l.j Kalra, S.P.j Sislian, J.P.: Condensation of water vapor in rarefaction waves. III. Experimental results. AIAA J. 15 (1977) 686-693.

I. Smolders, H.J.j Willems, J.F.H.j de Lange, H.C.j van Dongen, M.E.H.: Wave induced growth and evaporation of droplets in a vapour-gas mixture. Proc. 17th Int. Symp. Shock Waves and Shock Tubes, Ed. Kim,Y.W.j LeHigh Univ. (1989).

). Landau, L.D.j Lifshitz, E.M.: Fluid Mechanics, Volume 6 of Course of Theoretical Physics, Pergamon Press (1959) 327-329, 496-498.

to. Hayes, W.D. in Fundamentals of Gasdynamics. Ed. Emmons, H.W., Princeton, New Jersey: Princeton University Press (1958) 442-448.

t1. Chorin, A.J.: Random choice solution of hyperbolic systems. J. Compo Phys. 22 (1976) 517-533.

L2. Sod, G.A.: A numerical study of a converging cylindrical shock. J. Fluid Mech. 83 (1977) 785-794.

L3. Goossens, H.W.J.j Cleijne, J.W.j Smolders, H.J.j van Dongen, M.E.H.: Shock wave induced evaporation of water droplets in a gas-droplet mixture. Exp. Fluids 6 (1988) 561-568.

14. Goossens, H.W.J.j van Dongen, M.E.H.: A quantitative laser-interferometric measurement of gas density in a gas-particle mixture. Exp. Fluids 5, (1987) 189-192.

15. Zel'dovich, Ya.B.j Raizer, Yu.P.: Physics of shock waves and high-temperature hydrodynamic phenomena. Volume II. Academic Press (1967) 757-762.

Properties of Kinematic Waves in Two-Phase Pipe Flows

J.A. BOURE

Commissariat a l'Energie Atomique, Centre d'Etudes Nucleaires de Grenoble, Service d'Etudes Thermohydrauliques, 85 X F3804l, Grenoble Cedex, France

Abstract

In two-phase pipe flows, the slip velocity or,equivalently, the drift flux is related to the void fraction. This leads to the occurrence of kinematic 'waves which convey void fraction signals. Kinematic waves have been investigated experimentally in air-water two-phase flows by inducing small void fraction disturbances at the inlet of vertical ducts, the average void fraction varying from 0.01 (bubbly flows) to 0.41 (slug flows). The temporal fluctuations of the void fraction are detected in regularly spaced cross sections by non intrusive impedance probes. The statistical processing of the data reveals the existence of two kinematic modes and provides their velocities and damping or amplification coefficients. The results are presented and discussed. One of the modes, practically absent at low void fractions, is predominant at larger void fractions. As the void fraction increases it changes from damped to amplified and controls the bubble slug transition

1. INTRODUCTION

Having more degrees of freedom that single phase flows, two-phase flows are

subject to more complicated wave phenomena. Some of these phenomena

correspond more or less to those occurring in single phase flows (e.g.

pressure waves) but others are essentially kinematic waves.

Kinematic waves result from the existence of a functional dependency

between some "flux" and the corresponding "density", otherwise related

through a balance equation (Whitham, 1974, p. 27). In two-phase flows, the

foregoing flux and density are respectively the drift flux (volumetric flow

rate of one phase with respect to the mixture) and the void fraction

(volumetric concentration of the same phase in the mixture). Kinematic

waves convey information on the flow structure and the associated variables

such as the void fraction and the drift flux. In particular, they control

flow pattern changes and density wave oscillations.

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Gottingen, Germany, 1989 © Springer-Verlag Berlin Heidelberg 1990

208

The present paper reports an experimental investigation on the properties

of kinematic waves of small amplitudes in bubbly flows up to the

bubble-slug transition. Except for the pioneering works by Wallis (1961)

and Zuber (1961), publications were scarce until 1980. Since then, a few

laboratories worked on the subject. Most investigations are based on

cross-correlating the signals of two void fraction probes (impedance or

~-attenuation probes). The results reported here were obtained by.

processing the simultaneous signals of a significant number (5 to 9) of

successive impedance probes, a particularity which turns out to be quite

important since it enables detecting the existence of two kinematic modes.

2. EXPERIMENTAL CONDITIONS

MODES

DATA PROCESSING EXISTENCE OF TWO KINEMATIC

The conditions of the experiments made in Grenoble are summarized in

table 1. The results presented hereafter are primarily based on the data of

Tournaire (1987a,b) because they are recent, accurate (in particular, a

good accuracy on velocity differences implies low mixture velocities) and

encompass large void fraction and frequency ranges. The other results are

compatible with Tournaire's data.

For each run (characterized by an average void fraction and a disturbance

frequency) and for each of the 8 pairs of successive probes 1 to 9, the

data processing provides (Tournaire, 1987a,b)

- the apparent velocity Ce of the void fraction signal, deduced from the

peak of the cross correlation function. Tournaire noted that two

neighboring peaks were clearly distinguishable in some occurrences.

- the apparent velocity Cs of the void fraction signal, deduced from the

difference of phase between successive probes.

- the coherence function ~ (0 < ~ < 1) which provides an evaluation of the

quality of the correlation between the signals of successive probes.

the apparent gain factor H, ratio between the signal amplitudes at

successive probes.

The results may be grouped in two categories :

1. The apparent velocities Ce and Cs coincide and this velocity and the

apparent gain factor H do not change significantly over the test section

length or at least its downstream half. (To account for the experimental

errors, differences are ignored if Ce , Cs and H are within ± 2.3 cm/s

and ± 13 % respectively of their average values C and H). The values of

TA

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(upw

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are

2 x

2 cm

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cm

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m

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dia

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3.8

em

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cm

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cm2

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(up

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210

the coherence function ~ remain larger than 0.67. The corresponding

data may be straightforwardly interpreted in terms of a single wave

whose velocity is C and whose spatial amplification coefficient is :

Ln H d

d being the distance between successive probes (here 0.2 m)

2. In some part of the test section, the data exhibit some of (in most

cases all) the following characteristics :

There are significant differences between

significantly from one pair of probes

Cc and

to the

CS ' and Cs varies

next. The cross

correlation function may present two neighboring peaks.

- H varies significantly form one pair of probes to the next, and the

the product of Hmin by Hmax is of the order of 1.

- Coherence "accidents" (very low isolated values) may occur.

As it can be shown by computing the apparent velocity and gain resulting

from the superimposition of two waves (Boure, 1988), this is exactly

what is to be expected when two modes are coexisting, in which case Cs

and H lose their significances.

The existence of two modes is confirmed by plotting (fig. 1) the data for

the wave velocity and the drift flux as functions of the void fraction (the

data is tabulated in Boure, 1988). According to the theory (Boure, 1988)

the significant wave velocity is supposed to be the velocity C-W relative

to the mixture center of volume with the definitions :

WG and WL being the average velocities of the two phases. On fig. 1, the

points are strikingly distributed into two families which correspond to the

two modes. One mode, referred to hereafter as mode 3 (the subscripts 1 and

2 being reserved for pressure waves), is predominant at small void

fractions (~< 0.25), while the other (mode 4) is predominant at larger

void fractions (~> 0.30). The two modes probably coexist at least for

0.20 < ~ < 0.40.

c - W (em/s)

30~------+-------,-------~-------+--~

~~------~--------~--------~------~--~

10~------T---~~1--------r-------+----

a OL-------~------~------~------L-~

5 (em/s)

5~------~------~------~~~--~~

a O~------~------~--------~------~--~

.1 .2

c- W= 5~

Fig. 1 Wave velocities (top) and drift flux (bottom). Data (segments and crosses) and conjoint correlations (solid lines).

211

The existence of two modes appears also on the plot of the amplification

coefficient as a function of the frequency or of the wavelength. This plot

is not reproduced here because only orders of magnitude can be derived from

the available data for mode 4.

It is pointed out that, as shown by fig. 1, the domains of predominance of

the two modes correspond to different relationships between the drift flux

& and the void fraction~, i.e. to different flow regimes. It is suggested

that mode 3 may result from individual interactions between otherwise

independent bubbles, while mode 4 involves interactions between individual

bubbles and swarms of bubbles (in active interactions themselves).

212

3. KINEMATIC WAVE VELOCITIES

Whether kinematic waves are dispersive or not cannot be ascertained because

beyond a moderate frequency (say 3 Hz for mode 3 and 5 Hz for mode 4 in the

experimental conditions of Tournaire, 1987a,b) the waves are strongly

damped and cannot be studied. For frequencies below these limits, no

significant frequency effect on the wave velocity can be detected.

The velocity of mode 3 is, except for the smallest void fraction (~ ~ 0.05)

comprised between WL and WG, and even between Wand WG . The difference C-W

decreases as ~ increases. The low void fraction "anomaly" was also observed

by Saiz-Jabardo & Boure (1989). Fig. 1 reveals that the anomaly does not

rest with C but with & (i.e. with WG) which is much smaller than an

interpolation would let expect. This could be due to a distribution effect

since at low void fractions the bubbles remain close to the wall, a low

velocity region. Anomalies attributed to distribution effects were also

observed by Micaelli (1982).

The velocity of mode 4 is larger than WG and does not significantly vary

with ~. It increases slightly (a few percent) along the test section,

indicating development effects. Tournaire (1987a) and Saiz-Jabardo & Boure

(1989) noted that it is of the same order of magnitude as the predicted or

observed gas plug velocity in slug flow.

The velocity of either mode can be correlated conjointly with the

corresponding fully-developed drift 0FD' in terms of W and ~, by

correlations of the forms :

Such a relationship between C and 0FD appears in the simplest available

theory, in which a single kinematic mode is predicted. Surprisingly, it

applies to both modes.

For mode 3, such conjoint correlations are (0, C and W in mjs)

0FD = 0.22~(l-~)[l - 1.25 ~(l-~)l ± 0.0023 for 0.07 < ~ < 0.26

C - W = 0.22(1-2~)[1 - 2.5 ~(l-~)l ± 0.01 for ~ < 0.31

213

The corresponding limit for the velocity difference WG - WL when ~ tends

towards zero is 0.22 m/s. There is no significant effect of W, at least in

the explored velocity range 0.26 < W < 0.41. It may be worthwhile to note

that the above correlations fit also the data, obtained with a different

geometry by Mercadier (1981).

For mode 4, conjoint correlations are, for 0.25 < ~ < 0.41

&FD - 0.22~ - 0.028 ± 0.005

C - W - 0,22 ± 0.021

Again, there is no apparent effect of W, but the explored range

(0.37 < W < 0.52) is fairly narrow.

The two correlations for &FD lead to the same value &FD = 0.033 for

~ - 0.28. This may be adopted as the practical limit between the domains of

predominance of mode 3 and of mode 4.

4. KINEMATIC WAVE DAMPING OR AMPLIFICATION FLOW PATTERN TRANSITION

The damping or amplification of a wave may be described either by its

spatial amplification coefficient k j (frame of reference fixed to the pipe)

or by its time amplification coefficient (~ - kjC) in a frame of reference

moving with the wave. The dependency of ~ on the overall mixture motion (W)

is likely to be weak, which is not the case for k j •

The time amplification coefficient ~ of mode 3 is plotted in fig. 2 as a

function of the square of the wave number kr' as suggested by a simple

theory. It is always negative (damping). The wave number, or the wavelength

A - 2 n/kr , is related to the wave frequency f (through C = Af) but, unlike

f, it is an intrinsic (frame independent) wave parameter. ~ does not depend

significantly on~. A simple correlation of the data is (~ in s·l ; A in m)

The data cover the

1.3 ~ - - ----± 0.09

1 + 50 A2

ranges ~ < 0.31,

0.5 < f < 2.6 Hz), 0.26 < W < 0.41 m/s.

0.17 < A < 0.82 m (i.e.

214

o

-.1

-.2

-.3

-.4

-.5

-.6

10-3 k~ (m-2 )

+ IX = 0.047 x IX = 0.075 A IX = 0.123 0 IX = 0.191 0 IX = 0.255 0 IX = 0.306

Fig. 2 Time amplification coefficient of mode 3. Data and correlation

1.5

In most runs involving mode 4, mode 3 remains sufficiently present to

prevent the quantitative determination of ~. In all likelihood however :

- for ~ = 0.26, ~ = 0.31 and ~ - 0.37, some wavelengths at least (roughly

0.2 < A < 0.4 m) are neither significantly damped, nor significantly

amplified (~~ 0).

- for

[~max

~ = 0.41, the same

~ 0.33 s-1, which means

after 3 s].

wavelengths are slightly amplified

that the amplitude is multiplied by e

When mode 4 is amplified, it is logical to conjecture that its instability

results in a flow pattern transition. Visual observation shows indeed

(Tournaire, 1987a, Saiz-Jabardo & Boure, 1989) that the instability of mode

4 is generally accompanied with the occurrence of gas plugs in the

downstream part of the test section. As already noted, the velocity of

mode 4 is of the same order of magnitude as the gas plug velocity in slug

flow.

215

5. CONCLUSIONS

The experimental knowledge reached to date on kinematic waves in two-phase

flows has been summarized. The existence of more than one kinematic mode

has been brought out naturally, thanks to the use of several pairs of

probes distributed along the test section and to a double determination of

the apparent signal velocity.

The phenomena occurring in a bubbly flow may be summarized as follows :

1. At low void fractions (typically ~ < 0.2) mode 3 (individual interactions

between bubbles) is predominant. Its velocity is comprised between Wand

WG . It is always damped, the damping decreasing when the wavelength

increases.

2. At intermediate void fractions (typically 0.25 < ~ < 0.37) mode 3 and

mode 4 (interactions between swarms and individual bubbles) coexist. The

velocity of mode 4 is larger than WG . Both modes are damped, as a rule.

However, for mode 4, wavelengths around 0.2 - 0.4 m are neither

significantly damped nor significantly amplified.

3. At larger void fractions (typically ~ > 0.40) mode 3 is still present

with the same properties, but mode 4 is slightly amplified, at least for

a certain range of wavelengths. It results in the development of swarms

of closely spaced bubbles which swallow up the slower mode 3 waves. When

some critical packing is obtained, the swarms coalesce into gas plugs

and a slug flow pattern results. The larger the void fraction or the

inlet disturbance, the fastest the process.

The properties of kinematic w~ves may be accounted for by two-phase flow

models, provided adequate closure laws are used.

216

REFERENCES

Boure, J.A., 1988, Properties of kinematic waves in two-phase pipe flows; consequences on the modeling strategy, European Two-Phase Flow Group Meeting, Brussels.

Mercadier, Y., 1981, Contribution A l'etude des propagations de perturba­tions de taux de vide dans les ecoulements diphasiques eau-air A bulles. These de Docteur-Ingenieur, Universite Scientifique et Medicale et Institut National Poly technique de Grenoble. .

Micaelli, J.C., 1982, Propagation d'ondes dans les ecoulements diphasiques A bulles A deux constituants. Etude theorique et experimentale, These de Doctorat es Science, Universite Scientifique et Medicale et Institut National Poly technique de Grenoble.

Saiz-Jabardo, J.M. & Boure, J.A., 1989, Experiments on void fraction waves, Int. J. Multiphase Flow, 15, 483-493.

Tournaire, A., 1987a, Detection et etude des ondes de taux de vide en ecou­lement diphasique A bulles jusqu'A la transition bulles-bouchons, These de Docteur-Ingenieur, Universite Scientifique et Medicale et Institut National Poly technique de Grenoble.

Tournaire, A., 1987b, Compilation des resultats experimentaux obtenus sur la boucle Oscar, Internal report, SETh/LEF, Centre d'Etudes Nucleaires de Grenoble.

Wallis, G.B., 1961, Two-phase flow phenomena associated with the boiling of liquids, PhD Thesis, Cambridge University.

Whitham, G.B., 1974, Linear and nonlinear waves, J. Wiley & sons.

Zuber, N., 1961, Steady state, transient response, operating limits and continuity waves in two-phase flow systems, General Electric 61 GL 215.

Growth of n-Propanol Droplets in Argon Studied by Means of Shock Tube Expansion­Compression Process F.Peters and B.Paikert Stromungslehre, FB 12 Universit&t Essen 4300 Essen FRG

Summary Growth of monodisperse alcohol droplets suspended in a mixture of supersaturated vapor and argon is studied. The droplets originate from a special expansion-compression shock tube process by homogeneous nucleation of vapor molecules. The droplet radius is measured between 100 and 2200 nm as function of time by Mie-light scattering. The results are compared with Gyarmathy's growth model. A growth coefficient is found to represent the growth data by a single curve. The extrapolated curve suggests the often quoted condensation coefficient to be close to unity.

Introduction

Dropwise condensation of a vapor diluted in a carrier gas

occurs whenever the vapor becomes supersaturated due to a

change of state of the gas/vapor mixture. By supersaturation we

define the actual vapor pressure over the equilibrium vapor

pressure (over a flat surface) at the same temperature. The

change of state may be an adiabatic expansion as in cloud

chambers, supersonic nozzles and shock tubes

process as in diffusion cloud chambers [1].

or a diffusion

The dropwise condensation can be looked at as a two stage

process. In the first stage nuclei of vapor molecules are

formed. The critical size of the nuclei is a few molecular

diameter. Critical refers to the fact that for thermodynamic

reasons concerning the free energy only nuclei above critical

size are able to grow.

In the second stage nuclei grow into droplets. The driving

potential for growth is the non-equilibrium between vapor and

the new phase measured in terms of supersaturation. Therefore,

growth ceases when conversion of vapor into liquid has reduced

supersaturation to unity. The final state is a cloud of

droplets suspended in the vapor/gas atmosphere.

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Gatlingen, Germany, 1989 © Springer-Verlag Berlin Heidelberg 1990

218

Our objective with this paper is to contribute to the problem

of growth. We will present experimental data on alcohol (n­

propanol) droplets condensing in argon. Applied to these data

the validity of mass and heat transfer laws with Gyarmathy's

[2] correction for the molecular range will be shown. Finally,

an important conclusion on the condensation coefficient can be

drawn. In condensation studies this coefficient is a frequently

employed parameter of uncertain magnitude.

The paper is

method and

confined

results

background work.

Growth law

to essentials

referring the

on theory, experimental

reader to respecti ve

We consider a suspension of spherical droplets. The droplets

are at rest relative to the ambient vapor/gas mixture. The

droplet concentration is sufficiently small to rule out droplet

interactions. According to the droplet concentration a

spherical volume is assigned to every droplet as vapor

reservoir. The temperature T and the partial pressures pv

(vapor) and pg (gas) at the outer edge of the volume represent

the thermodynamic state. The droplet temperature Td is taken to

be constant throughout the droplet however may vary during the

growth process. The partial pressure of the vapor at the

surface Pd is set equal to the equilibrium vapor pressure over

a spherical surface as given by the Kelvin equation

2 0 exp [ ] r 91 Rv Td

(1)

Here 0 is the surface tension and 91 is the density of the

liquid. Rv denotes the gas constant of the vapor. Pd is not

much different from the equilibrium vapor pressure p. (Td) over

a flat surface for droplets with radius greater than 10 nm. The

difference between the vapor pressure pv at the outer edge of

the volume and the pressure at the surface of the droplet Pd is

the driving force for diffusion of vapor molecules towards the

droplets surface. The mass flux M at which vapor mass is

transported to the spherical droplet surface 4 n r 2 is readily

obtained from the diffusion law [3]

, M 4 n r2 D

r

pv - Pd (Td ) Rv T

219

(2)

D is the diffusion constant of the vapor in the gas. The mass

flux increases the droplet radius as

dr dt

. M

4 n r2 91 (3)

Obviously Eqs. (1-3) cannot be integrated readily for r(t)

because Td is not known. An addi tional equation is required.

Neglecting the work of formation of the droplets surface and

keeping the droplet temperature Td unchanged for a short time

interval of computation (quasi-stationary treatment) means that

the latent heat of condensation M L has to be conducted away

from the droplet by the heat flux

M L.

The heat flux Q is obtained in analogy to the diffusion law

4 n r 2 D'/r) (Td - T)

where A is the heat conductivity of the vapor/gas mixture.

From Eqs. (2-5) we now have the two implicit equations

dr D pv - Pd (Td) --dt r 91 Rv T

dr A Td - T dt r 91 L

(4)

(5)

(6)

(7)

which have to be integrated for r(t) together with Eq. (1). With

the liquid, vapor and gas properties 0, 91, D, p-, A and Rv

known this is easily achieved on a personal computer. Vapor

depletion and increase of ambient temperature T by the released

heat can be taken into account when the droplet number concen­

tration is given, i.e. when the volume per droplet is known.

Obviously these equations reflect the view of heat and mass

transfer theory in continuous media between two spherical

shells. They are applicable for droplets much bigger (at least

5 times) than the mean free path between collisions of

220

molecules. (In our experiments the mean free path ranges below

100 nm). When the droplets become smaller than the mean free

path continuous transport breaks down at the inner shell, the

droplets surface. Molecular transport of mass and heat must now

be considered. The reader is referred to the comprehensive work

of Gyarmathy [2] who has worked out correction functions for

the right sides of Eqs. (6 ,7) of the form (1 + C l/r) - 1 .1

denotes the mean free path and C includes properties on

molecular and continuous transport. Practically C turns out to

be close to a constant number for each of Eqs. (6 and 7).

With these terms Eqs.(6,7) are applicable over the entire size

range from nucleus to droplet size.

We will use these equations to compare our results with

starting integration for r(t) at critical size.

Condensation coefficient

In condensation problems dealing with very small droplets (with

radius far below the mean free path) Gyarmathy's equations

converge with the much simpler Hertz-Knudsen model involving

the condensation coefficient ~. This model is widely used

although the condensation coefficient is not known. Mostly it

is taken to be unity, however values as low as 0.01 are

perpetuated [2]. Some of the uncertainty may be due to

considerable confusion in the literature regarding its

definition. Here we deal with it because our results render

valuable information on its magnitude.

For the Hertz-Knudsen model and the definition of the

condensation coefficient we follow Hill [4]. The bombardment of

small droplets by vapor molecules is described by the

impingement rate

pv ~ = m (2 n Rv T)1/2

(8)

Here m denotes the molecular mass. The impingement rate gives

the number of impinging molecules per unit area and time. If

all these molecules would be incorporated into the droplet the

greatest possible growth would result at the constant rate

221

dr / d t = m ~ / 91 (9)

The real growth rate is smaller. A part of the hitting

molecules is rejected. Only the fraction a ~ is incorporated. A

continuous flux of molecules is leaving the droplet. This is

equal to the fraction of incoming molecules when the droplet is

in equilibrium with its vapor

(10)

The growth rate then becomes

dr/dt (11)

An experiment devised to evaluate a on the basis of this

equation would have to measure dr/dt and to know ~ and ~d. Pd

in ~d is taken from Eq. (1), however for Td the equation of

molecular heat transfer has to be solved [4]. Although in the

literature a is often regarded a constant this

neccessarily the case. The difficulty to carry out

experiment seems to be the main reason for the

consistent and reliable data.

Experimental

is not

such an

lack of

Our experimental method employed to measure ret} above 100 nm

has been described in detail previously [5]. The basic idea of

the experiment is an expansion/recompression process of a

special shock tube to subject a vapor to a very short super­

sa tura tion pUlse. Nuclei are born during the pulse and grow

beyond the pulse in a remaining supersaturated atmosphere.

We restrict ourselves to outline the wave diagram (Fig.l) and

the optical detection system.

The shock tube comprises a short driver section (1m) and a long

driven section (8m) featuring a constriction close to the

diaphragm. Prior to the experiment the driver section is filled

with a vapor/gas mixture (here n-propanol/argon). The driven

section contains pure gas. The pressure ratio is roughly 2:1.

Everything is at room temperature. The experiment starts with

rupture of the diaphragm. An expansion wave guided by head and

222

tail moves towards the end wall of the driver section from

where it is reflected. A shock wave runs down the driven tube.

The constriction produces a small reflected shock that follows

the tail.

The following process developes close to the end wall. Upon

arrival of the head, gas and vapor pressure start to drop. The

temperature follows according to isentropic expansion

condi tions. At some point the vapor reaches saturation and

shortly afterwards a desired supersaturated state is attained

where nucleation takes place. The nucleation period is

deliberatly terminated by the small shock. This procedure

confines nucleation to a very short period (0.2-0.4 ms) in

contrast to the subsequent growth period which extends to 45

ms. All droplets have nearly the same history and thus grow

monodisperse.

The growth of the monodisperse droplets is observed close to

the end wall by a 90 0 Mie-light scattering method. The system

consists essentially of a laser and a photomultiplier (PM)

collecting light of droplets growing within the laser beam. The

ratio of scattered over laser intensity 1/10 is an oscillatory

function of radius at fixed optical parameters (Fig. 2). The

peaks of the function, appearing in the experimental PM-signal,

are sui table to identify the droplet radius as function of

time. Fig.3 shows an example with the peaks marked after

identification in Fig.2.

Results

Growth of monodisperse alcohol droplets (n-propanol) in argon

was investigated. The initial temperature of everything before

the experiment was around 298 K. The initial partial pressure

of n-propanol ranged from 1-10 Torr at fixed total pressure of

760 Torr. At nucleation supersaturations between 4 and 6 were

selected accompanied by nucleation temperatures from 240 to 265

K. The involved droplet concentrations (corresponding to a rate

of nuclei formation of 108 cm- 3 s- 1 ; see [5]) is roughly 10~cm-3.

Fig.4 presents the growth results in terms of radius r plotted

versus time t. The origin of the coordinate system coincides

with the nucleation period. The symbols show at which times the

223

peaks of the light scattering curves are found. Each chain of

symbols along a solid line represents one experimental run. The

runs are distinguished by the nucleation temperature and the

initial partial pressure of n-propanol which drops from top to

bottom curve. The solid lines are computed from Gyarmathy's

equations (Eqs.6,7 plus correction terms incorporating

depletion and heating of the ambient vapor/gas mixture). The

equations are integrated starting at the critical radius. An

excellent agreement can be stated.

Since Gyarmathy's growth law represents the experimental data

very well it may conveniently serve to express dr/dt as

function of rand ini tial condi tions . dr / d t may then be put

into the Hertz-Knudsen model (Eq.ll) solving for a. For small

droplets in the molecular range a keeps its meaning as conden­

sation coefficient. For bigger droplets in the domain of

continuous heat and mass transfer a takes the meaning of a

growth coefficient measuring the real growth rate with a hypo­

thetical maximum. The plot of a in Fig.5 reveals the remarkable

result that all experimental data collapse onto a single curve

a (r). Therefore, a normalized representation of growth for a

certain substance appears to be possible. A theoretical

substantiation of this finding has not yet come to hand. As to

the magnitude of the condensation coefficient a we conclude

from the extrapolation of the a curve to small droplets that a

is greater than 0.5 and most likely approaches unity.

Conclusions

An expansion/compression shock tube process in combination with

a Mie-light scattering system has delivered growth rates of n­

propanol droplets suspended in supersaturated vapor and argon.

Excellent agreement with Gyarmathy's growth model has been

found. Relating the experimental growth rate with the hypo­

thetical rate of the Hertz-Knudsen model results into a single

curve a(r) which we call the growth coefficient. In extension

to very small droplets a(r) recovers the meaning of the often

quoted condensation coefficient. Its value exceeds 0.5 and

seems to approach unity.

224

References 1. Kotake, S.; Glass,I.I. : Flows with nucleation and

condensation. Prog. Aerospace. Sc. 19 (1981) 129 - 196

2. Gyarmathy,G.: The spherical droplet in gaseous carrier streams: review and synthesis. Multiphase Science and Technology Vol.1 edited by Hewitt,G.F. et al. (1982) Hemisphere Publishing Corporation New York

3. Eckert,E.R.G.; Drake,R.M.Jr.:Analysis of heat and mass transfer. (1972) McGraw-Hill Book Company

4. Hill,P.G.:Condensation of water vapour during superso­nic expansion in nozzles. (1965)J.Fluid Mech.25,593-620

5. Peters,F.; Paikert,B.: Nucleation and growth rates of homogeneously condensing water vapor in argon from shock tube experiments. Exp. Fluids 7 (1989) in press

a ~ u 2

I

Xw

D 'I driver section

Photomult~lier laser

) I

diaphragm

shock \ --

:\ !ii'] constriction

x

Fig.1: wave diagram of the shock tube experiment with pressure

history at the end wall (observation station)

1000 I (~~2 -= - xT(r)

10 2TTd

n- propanol

100

T (r)

10

n :1.385 . ~=514.2 nm

0.1

laser (1 0 )

0.Q1 o 200 400 600 800 1000 1200 1400 1600 r(nm) 2000

Fig.2: relative scattered intensity according to Mie-theory

initiol stote-

75 Torr I

n - propanol I orgon

...... pressure signal

'-nucleation

Hie sJ,jnol

0.1 V I ""

o 2 4

®

6 8 10 t (ms) 14

Fig.3: simultaneous recording of pressure and light signal

of a growth experiment

225

226

n- propanol Jargon 2400

10

IX \ \ \

0.8 \

0.7

0.6

0.5

0.4

0.3

0.2

0.1

\ \ \ \

4 8 12

\ \

\ ___ extrapolation \

\ ,

16

Fig.4: droplet radius

growing versus time,

experiment (0) and

theory (-)

20 24 t (m s ) 32

/

Fi g .5: experimental growth coeffi­

cient versus radius with

extrapolation to condensation

coe ffici e nt

IX = dr/clt (e xperiment) m((3-fJd )/Sl

experimental range

0.0 '--_,--.l-,--_,--_~~~~-~-~-~-~-~---'

o 100 200 300 400 500 600 700 800 900 1000 r(nm) 1200

A Discrete Kinetic Model Resembling Retrograde Gases

K. PIECH6R

University of Warsaw Institute of Applied Mathematics and Mechanics 00-901 Warszawa, PKiN, p.IX, POLAND

Summary A discrete kinetic system modelling some properties of retro­grade fluids is proposed. Plane shock waves corresponding to the model Euler, Navier-Stokes and kinetic approximations are studied. It turns out that in some cases the number densit,y must decrease in order to obtain a stable shock wave. The shock structure and its thickness in the kinetic approximation are determined and are consistent those of Cramer and Kluwick [1i.

1. Introduction Retrograde fluids, as defined by Meier and Thompson [~ , are fluids for which the specific heat capacity multiplied by temperature is greater than the latent heat of evaporation. This propert,y allows a complete phase transition from the liquid to the gas state and vice versa only by pressure changes without heat exchange with external sources.

Flows of retrograde fluids exhibit many spectacular properties, and their analysis from the molecular point of view may be of some interest. This is, however, ver.y difficult because retrograde fluids are complex organic substances whose molecules have very many internal degrees of freedom.

However, being conviced that one can better understand even complicated phenomena by means of simple mathematical models we propose a model of kinetic character which exhibits some properties typical for retrograde fluids. Since no simple model can be used to describe properly all features of such media, we will concentrate on one aspect of the

228

problem, namely the shock wave problem in the gaseous phase of a retrograde fluid.

The proposed model belongs to the class of so called discrete velocity models of the Boltzmann equation [ 2J ' L 3J which are deduced from the Boltzmann equation b.y diseretizing the velocity domain. The discrete velocity models were successfully applied to gasdynamics, in particular to the shock wave problem in single monatomic gases (t~ - 15J ) and in their mixtures C[ eg - [9} ).

The model, which we propose, is a modification of Broadwell's celebrated model [4J , L5]. We modify it so as to obtain a simplified van der Waals pressure formula.

2. Presentation of the model

We consider the following discrete kinetic models

;) N1 'd N1 1 n-+ c -rr-::: -£,Q,

'4 N2 'C) N2 1 ~-c~=-Z'Q,

1 :::-Q t

(1 )

where (x, t) (i R >< R+ ' the space and time. The scalar functions N1(x,t), N2(x,t). N3(x,t) represent the probabi­lity densities of particles moving in the positive x-direction, the negative x-direction, and perpendicularly to it, respectively. All particles move with the same speed c. The parameter £.):. 0 represents the mean free path and is treated as small. Finally, Q ::: Q(N) is the collisional term which must be constructed.

The number density n and the mean velocity u are defined as usual by ([ 21)

(2)

and

respectively. The transport equations corresponding to model (1) are following

? n ? ( ) ~ + ~ nu = 0

Eqs. (4), (5) correspond to the Euler approximation it Ni = Mi(n,u), i = 1,2,3. where Mi(n,u) are components of the equilibrium distribution, which is also undetermined so far. Hence for Ni = Mi there follows

1 2( ) 2 4 c M1 + M2 = nu + p (6)

where p is the pressure, which is assumed to be given by the simplified version of the van dar Waals equation

p = RnT - an2 (7)

tor an infinitely compressible gas. In (7) R is the gas constant T is the temperature, and a is a positive constant characterizing the coupling of molecules.

Since the molecules move with the same speed a, the temperature T is equal to

c2 _ u2 T = 2R (8)

229

Inserting (7), (8) into Eq. (6) we obtain a linear algebraic equation tor three unknowns M1t M2, M3 • This equation along with (2) and (3) constitutes a system at three algebraic equations tor the three unknowns. The solution is the equilibrium state

230

M1 n(1 + ~)2 2a n2 = -~ c c

M2 n(1 _ ~)2 2a n2 (9) = --::-2" c c

M3 = n(1 u2 2a n2 - -::-2") +-::-2" • c c

Thus we have co~structed an equilibrium distribution with an unknown collisional term. Of course Q must be such that

Q(M) = 0 (10)

Assuming that Q is a polynomial of N = (N1 ,N2 ,N3) of the third degree we obtain C1~

Q = (N + ~ n2 )(N + ~ n2) - (N3 - ~ n2)2 (11) 1 cl!:: 2 cl!:: cl!::

Hence, our model is described by equations (1) with Q given by (11).

Setting a = 0 in (11) we obtain a version of the one­-dimensional Broadwell model ([2] - [51 and references therein).

An analogue of the Boltzmann H-theorem can be proved for model (1), (11). The H-function reaches its minimum if and only if the densities Ni (i = 1,2,3 ) are the equilibrium densities (9). Therefore the state (M1 ,M2 ,M3) can be called Maxwellian.

Applying the Chapman-Enskog procedure as in [2J we obtain the model Euler equations

~ n + ~~ (nu) = 0 ,

? ::J 2 ~ (nu) + ~x (nu + p) = 0 ,

(12)

with

1 (2 2 2 p = ~ n c - u ) - an

(cf. Eqs. (4) - (8)), as well as the model Navier-Stokes

231

equations [10] • In what follows we assume that (n.u) belong to the

following subset U of the (n,u)-plane

/u/< c, o<,_an<~in[(c _ lui )2,

~(c2 - u2~J .

u=(n,u): n)O,

(14)

This is necessary to have M1• M2 • M3 and p positive as well as to satisfy inequalities (17) given in the next section.

3. Shocks in the Euler approximation

The Euler equations (12), (13) can be shown [10] to be strictly hyperbolic with the characteristic speeds A + given by

1 \/ 2 2 I ~ t (ntu) = ~(n ± 2c - u - 8an ) t

and the corresponding right eigenvectors

rt(ntu) = (n,-A;)

If (n, u) E U then we have [1Ql

-c < ~ _ < 0 < A+ < c

~_ <u <A+

After some calculations we obtain 2

~ c2 _ 6an -+ = r + • grad ~ + = :;: . . - - ~ 2c2 _ u2 - 8an I

(16)

(17)

(18)

where grad = (adn • o~u) • It follows from (18) that the characteristics are neither genuinely nonlinear nor linearly degenerate in the sense of Lax [121. Following Liu [13] we define '( ± curves as follows

232

r:!: = ( (n,u) eu: r:!:(n,u) = 0 J. It can be shown (cf. [101) that the vectors r+(n,u) are transversal to ~:!: ' respectively. Hence, we can make use of the theory of hyperbolic systems of conservation laws develop­ed by Liu in [1~. We define the R-H curves s(no'uo) through a given state (no,uo)eU as follows t1~

S(no'uo) = i (n,u) 6 U: v(n-no) = nu-nouo and

( ) 1 (2 2 1 (2 2 ) () nu-nouo = ~n c +u -2an) - ~no c +uo-2ano

(20)

Let x = x(t) be the equation of a line of discontinuity. Then across it the Rankine-Hugoniot jump conditions hold. With the notion of the R-H curve they are written as

where (nl,ul ) and (nr,ur ) are the values of the flow on the left and on the right of the line of discontinuity, respectively.

Either directly [1<11 or from the general considerations [1~ t 0~ it follows that the R-H curve S(no'uo) consists of two smooth curves such that

(22)

respectively. Liu in [1~ introduced the following entropy condition:

a discontinuity (nl'u1;nr,ur ) is an admissible disconti­nuity if (nr,ur ) 6' S+(nl'u1 ) (orS_(ul'ul ) and if it satisfies the following entropy condition

v (nl'ul;nr,ur ) ~ c;-(nl'ul;n,u) (E)

for any (n,u)6 S+(nl'ul ) (S_(npul ), respectively)

233

between (nl,ul ) and (Dr'ur ) •

Applying the entropy condition (E) we obtain (at least for weak shock waves): i) for (nl,ul)E:s+(nr,ur )

if r+(nr,Ur ) > ° then nl>nr and ul <ur (23) if r+(nr,ur ) (0 then n1 <nr and ul)ur

ii) for (Dr'ur ) ~S_(nl'ul)

if r _(nl'ul )<0 then Dr)nl and Ur<Ul (24) if r _(nl'ul» 0 then nr(Ul and ul)Ur •

The first results in (23) and (24) can be interpreted as statements that the shock wave is a compression wave, whereas the second statements in (23), (24), resemble retrograde gases, since the density is smaller.behind the shock than that before it. We call such waves expansion waves.

4. Shock waves with dissipation

The Boltzmann shoek profile is a density vector N(,y) , y 6 R, such that

N(x,t) = N(x-st) (25)

is a solution of the model kinetic equations (1), (11), with the limiting values

H(y = - (0) = _1, N(y = + (0) = Mr , (26)

where )(1 ,Mr are the equilibrium densities (9) with hydrodynamical moments (n1,ul > and (Dr'ur ), respectively. We assume that both (nl,u1) and (nr,ur ) are elements of the set U.

We make the sUbstitution

Ni = ~~Mi + Mi) + G(y)(Mi - Mi~ ,i=1,2,3 (2?)

where G(y) is an unknown function such that

234

G(y = - (0) = 1 • G(y = +00) = -1 (28)

It can be shown L1~ that G(y) satisfies the equation

where the constant B is given by

2 2 ( c -s -2a nl +2nr ) B = 1 - 2 -----=---=- (0)

Equation (29) was obtained earlier by Cramer and Kluwick [1~ • who considered weak. shocks in the true Navier-Stokes equations, This time, Eq. (29) is a rigorous consequence of the model. The parameter B has to satisfy the condition I B I ~ 1 in order for the function G to satisfy the boundary conditions (28). We have from (30)

B > 1 if and only if one of the following relations takes place i) either

s2"/ c2 - 2a(nl +2nr ) and nl ) nr

or

11) s2 < c2 - 2a(nl +2nr ) and nl ( nr (32)

The equation B=1 defines the transition line

Similar results can be obtained if B ~ -1. Qualitatively, these results agree with those of Section 3, but this time equation (33) of the transition line is different. In can be shown however (see [1Q]) that the present results agree with those for the Euler equations in the ease of weak shock waves.

For solutions of Eq. (29) and their graphs for various

values of B see the cited paper by Cramer and Kluwick [1~. We define the shock thickness by:

From this definition and Eq. (29) it follows i) if B = 0(1), then

L "'J __ 1:......~

(nl -nr )2

this confirms the Cramer-Kluwick' s conj ecture • 1.1.) if B -? c;>o, corresponding to a ~ c:>o

2 C s 3a( n 1+ nr ) ) L ~ - (1 +

,nl-nr \ c2 _ s2

reminding the results of [15l •

235

A similar analysis can be performed for the shocks in the model Navier-Stokes equation 11q. It is noticeable that the value of the parameter B remains the same, but the Navier­-Stokes profile coincides with the Boltzmann shock profile for weak shocks only.

References

1. Meier G.E.A., Thompson P.A. in Lecture Notes in Pbysics, vol. 235 (1985), 103-114. Springer-Verlag

2. Gatignol R., Theorie Cinetique des Gaz a. Repartition Discrete de Vitesses, Lecture Notes in Pbysics, vol.36, (1975), Springer-Verlag

3. Platkowski T., Illner R., SIAM Review, vol.30 (1988), 213-255

4. Broadwell J.E., Phys. Fluids, vol.7 (1964), 124}-1247

5. Caflish R.E., Comm. Pure Appl. Math. vol.32 (1979), 521-554

236

6. Monaco R., Acta Mech. vol.55 (1985), 239-251

7. Monaco R., Proc. 15th RGP Symp., Grado; ed. V.Boffi, C.Cercignani, Teubner, 1986, 245-254

8. Platkowski T., Mech. Res. Comm. vol.14 (1987), 347-354

9. Platkowski T., in Discrete Kinetic Theory, Lattice Gas Pynamics and Foundations of Hydrodynamics, ed. R.Monaco, World Scientific, 1989. 248-255

10. Piechor K., To be published in Archive of Mechanics

11. Thompson P.A., Lambrakis K.C., Journal Fluid Mech. vol.60 (1973), 187-208

12. Lax P.D., Comm. Pure Appl. Math. vol.10 (1957), 537-566

13. Liu T.-P., Memoirs of AMS, No 240, 1981

14. Cramer M.S., Kluwick A., Journal Fluid Mech. vol.142 (1984), 9-37

15. Chaves H., Hermann E., Meier G.E.A., Kowalczyk P., Walenta Z.A., To be published.

The work was supported by the Pol. Gov. Program CPBP 02.01.

Non-equilibrium in Dynamic Systems, Critical Phenomena

Internal Gravitational Waves Near Thermodynamic Critical Point

A. A. BORISOV, AL. A. BORISOV, V. E. NAKORYAKOV Institute of Thermophysics, Siberian Branch of the USSR Acade~ of Sciences, Novosibirsk - 90, 6}0090, USSR

Summary The problems of internal wave propagation in a medium whose thermodynamic parameters are near the critical "liquid - va­pourtl point are investigated experimentally and theoretical­ly.

Apparatus

For substance Freon - 1} (OOlF}) under study the critical pressure Pc = }.968 MPa, density .5'c. = 580 kg/m} and tempe­rature Tc = }02.02 K. Under such conditions the wave proces­ses can be observed only in a closed channel, in our case -in a tube. Tube 3 m long with inner dia. 0.057 m was placed in a thermostat which was a dosed hydrodynamic circuit. The basic units of the apparatus are described in detail in the paper by Borisov et ale /1/. Tensometric pressure transducers measured static pressure. The tube was filled with Freon-13· after its pressurization to residual pressure 1 Fa. The sub­stance amount and mean filling density were determined by the weighting method. Wave producer generated waves at the interface. The positive ("humpstl) and negative ("valleystl) solitary waves were generated. The velocity of mixed liquid was always less than the wave velocity.' Three optical windows located along the tube were used to observe the substance state and to record solitary waves. Two couples of windows are located on the side surface of the tube and one - at the tube end. Observations were carried out using a shadow device based on laser, collimator, light dividing prisms and Fouca­ult's knives.

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Gatlingen, Germany, 1989

240

Transition from one isotherm to another and thermostatic cont­rol were performed with a step of 1 grade/hour in the inter­val (297.15 - 300.15) K and with a step of 0.1 grade/hour in the interval (300.15 - 302.15) K. The convection disappearan­ce and pressure constancy served as a criterion of establish­ing equilibrium in a vapour - liquid system.

Motion - picture cameras with the given photography fre­quency were switched on simultaneously with starting the wave structure and velocity were determined using the obtained ci­negrams.

2) Transversal internal waves generated by oscillating source (Fig. 1) have been obtained for the first time near the critical "liquid - vapour" point in which two mutually intersecting planes forming dihedron angle are phase sllrtaces.

Fig. 1

exists between them.

RESULTS

1) Fig. 2 shows the expe­rimental values for the velocity of solitary wave propagating along "liquid­vapour" interface. The range of temperatute vari­ation is 1.5-10-3 < 1.8 10-2• The density 9 of filling the working vo­lume 9 is equal to 9c • The analysis of dependences for the wave propagation veloci ty V on temperatu­re 1: allows to assume

( 1 )

where ?JO and J... are the constants. The data processing in logarithmic coordinates is shown in Fig. 3. It is seen that atT~ 3.10-3 the regime of wave propagation changes that is the result of washing out the interface. Thus at 1:' > 3.10-3

the superscript .d.. equals 0.19, at T < 3 10-3 - 0.48. Let us

I,m/s o ° 3 .6.x~ , .

0,25

0,2

x o~

.6.-x~ o

- " .6. )(0

--.6. X o Ole

.6.~

o x

0,75 ~--------~--------~------~

2,8 't·/O;) 16

3,2 ..

3,7 -

3,0

2,9 -

2,8 .

9,5

Fig. 2

IX "'-" If '" = 0, 23 ."x Vo =79,8Sm/s

~x , x" x

I "'=0,79 " . ~=52,2Sm/S ~

.~

'.

• m d=O,48

.1;; =340Sm/S

241

consider the rirst case T> 3.10-3 • For two homo­geneous layers of densities

9L and .91l. and heights h.J. and h/l. in a plane

channel the expression for phase velocity ~ of small perturbations exists, Lan­dau, Lif shi t s /2/:

'2f =[g kih;.(.5l. - .9t}/ (9dL.~ +92.ht )Jf2. (2)

The formula is valid in linear approximation of ideal incompressible li­quid as well as in the ca­se of long waves. Let us apply this expression to our case. Taking into ac­count the curve existence relation as well as the fact that at critical

filling density the layer heights are equal he = h~ and 9L -+ 9i- c. 2. 90 after substitung in (2) we obtain

f,!~ 1]3/2 '?r = (g/LlJ/2.) l% -T/% . (3)

According to the data of the paper by Shavandrin, Li/3/, j) =3.7, .J3 =0.34.

-4 -5

In the second case 1_______ we sugge sted the model

-5 til.".

Fig. 3 which describes qualitati­vely the transition to the second propagation regime.

Compressibility increases near the critical point. As a re­sult, the density gradient along the tank height arises.The

242

displacement of liquid field from equilibrium hydrostatic le­vel, for example, to denser layer, gives rise to the appearan­ce of buoyancy force which is directed to return this mode in­to the initial equilibrium state.

Let us consider a liquid layer near the critical point. The liquid ( gas ) parameters are so close to the critical ones and in a certain section along the height are exactly equal to them, that one should take into account compressibility un­der the effect of proper weight. For unperturbed state we have conditions of hydrostatic equilibrium.

110 = a , v Po = .90 (a)-9 ( 4 ) from which we find unperturbed density distribution in the la­er height. Since Po is a function of 90 one should add the equ­ation connecting Po and 90 to Eq. (4), which can be obtained from asymptotic state equati~ of the form, Landau,Lifshits /2/

p-~t= :f:1~1~(l+ett;1~i~) t =1'-% J 2 =-n. - ne. ,n, is the particles number density.

At the critical isotherm we obtain i). [

('aP/89)t = B! fi.9 - ( 5 ) Here fi is the exponent of the critical isotherm. P:= p- ~

;> = 5' - .9c. • 81. is the constant. Solving Eqs (4) and (5) jointly we shall find the density distribution in the dimension-

less form [ ] VO' .9-9c./9c= 9o/!t8t·gl~-l/!I·8/t-1 (6)

Let perturbations of small amplitude are imposed for a liquid

.91=Sl~5'o(Z}, 11'= 'If-?fo, ?J(?J, u, LtJ), ('lro=O) , p'=p_~ The system of conservation describing the internal wave reduced

to one equation, where S is entropy, ~ is the medium velocity in a laboratory system of coordinates X , t .

In W{l!)-coordinates with due account of density (6)

takes the form.

d~dl.2.-yblt-l:cl.dlV/dr+JL-9/rSll-lcl·w=a (?)

with the boundary conditions W(O)=lJJ(-I-I)={) The general solution of Eq. (7) can be written in the form

243

5 -t '( [ . fllf .I1;!.]1) h 5 + 1) W=l1li-kI Y ' ~%!' 2.1[~ll-i~ 'r QZ-ZJ 2/5 +

( r fiT II" Jil 0-+1) +Q ~ b-f / l2-y!f !l- l=c\i'2 1 r0l-l~r R% ( 8 )

The fracti~nal functions in (8) are even. If solution (8)

satisfies the boundary conditions, the equation for eigenvalu­es appears I9r ~ ~ vrlzo - ~c.1 = ~n- ( 9 ) where } i"l.. are the roots of the Bessel function of indices

± (6-+ 0/5 • From (9) we Shall obtain the ~xpression for phase velocity of internal waves of n-mode

VI?- -:. {/5; '4/0' . 9' k/<2- (10 ) From formula (10) one can estimate the maximum velocity of the wave. Substituting in (10) the least positive root of the Bes­sel function of in1ex (fJ+I)/li , ;1.which has the value(litl)/(j', we shall obtain V{ = R..9 h 5/( 0+ f) • Note that the value of phase velocity of the first mode is essentially less than the velocity of gravitational wave on interface surface of analogo­us layer of heavy liquid with height h and gas. Using formula (9) one can obtain the wave velocity at the existence curve near the critical pOint. For this, we shall sUbstitute i- ~c

according (8) via the density difference 9 - S'~

Ii! - ~cl =1t1.9/9cI 5 (6 - f) /;, 8[ ]]c/90 9 but !J.9 =.9 - .9c via the tem:Qerature difference Ll r the existence curve 4.9/9c =1' I ~jr/7'cl13 Here !J.9=9I,2.-9c; i)/T'=7i,2.- n where .9~/2. are densities at vapour and liquid phase boundaries, respectively.

v:= 4.(b-I}~.8l[.D·IL\1'/7'cl} +(¥Z)Y'~ (5+ f)/5];S:' 5~ j=>c.

At the critical isotherm~T= 0 the phase velocity of in­ternal waves remain finite and proporti'onal to the picnocline thickness. For L\ T f:. 0 ath-O the power dependence is of the form Vn.."'-'IAT/'7'c\J3OfoZ

According to known data: S' = 3.37 and for j3 = 0.34 we . \/ rrO'S"¥-shall obta~n V~ ~ ~ • Comparing with our experimental data

on velocity for .9 < gef V ~ TO' 4 we obtain a qualitative agreement.

244

Thus it has been shown that the density distribution near the critical point determines the velocity of internal waves. This distribution depends on the critical exponents of the exis­tence curve and critical isotherm.

CONCLUSION

The created experimental setups allowed to investigate for the first time the dynamics of internal gravitational wa­ves near thermodynamical critical point.

The velocity of internal waves propagating in pinocline, when approaching the critical point, decreases according to power law 'if= v;, 'l:ol with the power exponent cJ.. less than un.i ty. Depending on the vicinity to the critical point two re­gimes of internal wave propagation exist, i.e. far from the critical point both over the two-layer liquid and near the cri­tical point - as in substance with continuously varying densi­ty profile due to aritical compressibility.

In the immediate proximity to the critical point ?: <. 3.10-3 the internal waves were found in liquid and vapour

phases. The presence of such waves is due to stratification ge­nerated in the near-critical region. The results are compared with experimental data obtained in linearly stratified salt solutions.

The model is developed and the equation describing the propagation of linear waves in a medium near the thermodynamic cri tical "liquid - vapour" point is obtained.

The revealed internal waves are a new instrument for the investigation of-phase transitions and critical substance state. A new method of modelling the media with density stra­tification in height is proposed, in which the density varia­tion with height is adjusted by the closeness of the substance state to the critical point.

245

References

1. Borisov A.A., Borisov Al.A.,Kutateladze S.S., Nakoryakov V.E. Rarefaction Shock wave near the critical liquid-vapour pOint. - J. Fluid Mech. 126, 1982.

2. Landau L.D., LifShits E.M. Hydrodynamics, M., Nauka, 1986. 3. Shavandrin A.M., Li S.A. Experimental study of temperature

density parameters at Freon - 13 saturation line. Inzh.-fiz. Zh., 37, N 5, ppa 830 - 834, 1979.

Effect of Thermodynamic Disequilibrium on Critical Liquid-Vapor Flow Conditions

Z. Bilicki and J. Kestin

Brown University, Providence, RI, USA and Polish Academy of Sciences, Gdansk, Poland

Abstract

In this lecture we characterize the effect of absence of unconstrained thermodynamic equilibrium and onset of a metastable state on the adiabatic flow of a mixture of liquid and its vapor through a convergent-divergent nozzle. We study steady-state flows and emphasize the relations that are present when the flow is choked. In such cases, there exists a cross-section in which the flow is critical and in which the adiabatic wave of small amplitude is stationary. More precisely, the relaxation process which results from the lack of equilibrium causes the system to be dispersive. In such circumstances, the critical velocity is equal to the frozen speed of sound, ar corresponding to w -+ co.

The relaxation process displaces the critical cross-section quite far downstream from the throat and places it in the divergent portion of the channel. We present the topological portrait of solutions in a suitably defined state-velocity space and discuss the potential appearance of normal and dispersed shock waves.

In extreme cases, the singular point (usually a saddle) which enables the flow to become supercritical is displaced so far that it is located outside the exit. Then, the flow velocity is everywhere subcritical (w < a f) even though it may exceed the equilibrium speed of sound (w ~ ae> beyond a certain cross-section, and in spite of the presence of a throat.

1. Introduction

In several papers [1,2,3], the present authors and their collaborators have made a

systematic study of adiabatic one-dimensional flows of single substances present

in the stream in a liquid and a gaseous phase, with emphasis on critical flow as a

cause of choking. The study fell naturally under two headings. First, we

explored the general qualitative features of such flows as they are described by a

wide class of mathematical models [I]. Given that flows of this kind invariably

start from rest with a liquid which expands into a metastable state, we have,

secondly [2,3], undertaken an analytic study of the effect of the realxation

mechanism which drives the local state to one of unconstrained equilibrium.

Clearly such flows are dispersive and the attainment of a critical velocity at

248

choking is governed by the physical properties of waves. i.e. disturbances. which

propagate through the system.

In the present lecture we give an account of both sets of results. For the sake

of brevity and clarity. the results are stated with a minimum of proofs. because

the audience can find them in the original references.

2. General features

The preponderant majority of c~ntemporary mathematical models which are used

to analyze a wide variety of steady. adiabatic. one-dimensional two-phase flows.

with the single space coordinate z. can be reduced to the standard form

dOj B .. ( 0) - = c .• ( o.z)

IJ dz (i.j = 1.2 •...• n) (1)

Here 0 denotes a state-velocity vector of the dependent variables of the system of

n ordinary quasi-linear differential equations which constitute the model. For

example. in the elementary (rather obsolete) homogeneous equilibrium model the

components of vector 0 = {h.P.w} are conveniently chosen as enthalpy h. pressure

p. and veloctiy w [4]. In two-fluid models the number of equations. n. may

reach. or even exceed. six.

To study choking. it is sufficient to concentrate on the steady-state form (1); in a

parallel study of waves. the model will acquire an additional. time-dependent term

and will become

(i.j = 1.2 •...• n) . (1 a)

The time-dependent term plays a role when dispersion is analyzed. The

important features of form (I). which lead to a number of very general

statements. consist in the fact that the matrix Bjj is generated by the

conservation laws. Consequently. it contains only elements of a to the

exclusion of z. The vector c j contains both a and z; its elements are

generated by the closure conditions. and the presumption here is that they are

not expressed in terms of the derivatives da/dz.

the form of the prescribed channel shape A(z).

The coordinate z appears in

When resolved by Cramer's rule, equation (1) assumes the equivalent form

Here

Ni (a,z)

8(a)

249

(2)

(2a)

and the matrices N arise from Bij by successively replacing a column by the

column vector c.

Application of the center manifold theorem [l] to system (1) leads to a statement

of very great generality. With its aid it is possible to characterize the topology

of the ensemble of all solutions of system (1). This is done in a phase space n

of n+l dimensions consisting of the union a U z. The key to the understanding

of choking is in the identification of the singular points of system (1) whose

coordinates in phase space n are solutions a*,z* of the simultaneous equations

8(a) o (3a)

o (3b)

where Ni is anyone element of N. The pattern of the trajectories in phase

space n consists of non-intersecting curves at all points for which 8 'I- O. The

solutions cease to be single-valued and can become intertwined at singular

points a*,z*. In turn, the pattern around these is given uniquely with

reference to a plane II of two dimensions which is tangent to all solutions at

the singular point that pass through it. This plane is located with the aid

of two eigenvectors ~,7) of the linearized matrix eij of ciBij-1, regardless of the

complexity residing in the number n of independent equations.

The properties of plane II admit the possibility of only three patterns: saddle

points, nodal points and spirals, but only saddle points are relevant to nozzle

flows without appreciable body forces. Figure I recalls the topology of a saddle

point in the P,z projection and makes the unexpected statement: the pattern of

250

solutions for the most complex system of n equations, possessing the features

studied here, is identical with that familiar from gas dynamics.

~----------------~--------r-----~~

I I I I I

, o

, , ....

L . __ .

~=O

---1P{3

z* z z=L

THROAT

Figure I. Portrait of solutions induced by a saddle point S. I - Continuous solutions, subcritical flows; 2 - Subcritical branch passing through saddle point; 2 I - Subcritical branch emerging from saddle point; 2" - Supercritical branch emerging from saddle point; 3 - Solution with turning point; F 1 - States unattainable by continuous solutions; F2 - States unattainable from state R

It is clear that eqn. (3a) determines the critical velocity w· in terms of the

thermodynamic components of o.

this is the classical spced of sound

When three conservation equations are used,

(4)

Equation (3b) then locates the critical cross-section which is also the choking

cross-section.

251

The physical conclusions that follow, are:

1. The necessary condition for choking is (3a), or (4) where applicable.

2. The necessary and sufficient conditions for choking are (3a) and (3b).

3. A transition from subcritical flow with l;. > 0 to supercritical flow with

l;. < 0 through l;. = 0 can occur only in the presence of a singular point,

in practice a saddle point.

These principles were successfully embodied in our re-interpretation of the Moby

Dick experiments [2].

3. Relaxation

The simplest, qualitative study of relaxation is based on a set of three

conservation laws supplemented by a single evolution equation. The general

form of such a mathematical model of four equations still conforms to eqn. (la).

All relaxation mechanisms are modeled on a single one with relaxation time 9.

The simplest version of such an evolution equation is that used, e.g., by E. G.

Bauer et al. [5] for the analysis of two-phase flows (and others for other

applications).

In this model the state-velocity vector a incorporates a composition parameter, in

our case [3] the dryness fraction x, so that

a = (P,h,x,w)T

In contrast with the equilibrium model in which x

equilibrium value in

x = it - (it - xo) exp(--tj9) ,

(5)

x(P,h), where x is the

(6)

with t denoting time, the composition x has been assigned the role of an

independent variable.

form

The corresponding, simple evolution equation has the

Dx

Dt

x-x

9 (7)

which is consistent with the principles of linear irreversible thermodynamics

[6].

252

By the use of standard methods [3], the model equation, quoted in the Appendix,

leads to the familiar dispersion relation

(8)

The physical consequence is that the full model introduces two speeds of

sound, the

equilibrium speed of sound

which is identical with a in eqn. (4), and the

frozen speed of sound

it being important to recall that

as a requirement of negative damping.

a 2 f = (8P/8p). '

(8a)

x'" x; (8b)

(9)

A further application of standard methods, as shown by J. D. Ramshaw & J. A.

Trapp [8], would prove that the equations of the characteristics of the system

(apart from dz/dt = w) are

dz/dt = W:l: af (10)

The physical consequence of this result is that in steady-state choking, when

the slowest characteristic must be stationary in the critical cross-section (J. A.

Trapp & V. H. Ransom [9]), the flow velocity in it is the frozen speed of

sound

(11)

and not, as might superficially be expected, the equilibrium speed of sound,

253

4. Portrait of solutions in w,z projection in presence of relaxation

We have by now determined that the portrait of solutions in the presence of

relaxation is governed by the steady-state set of equations

dp dw w-+p-

dz dz

AI -"A Pw (12a)

dw 1 dP CT w-+-- -- (12b)

dz pdz pA

dh 1 dP CT ---- -- (12c) dz pdz pA

dx x-x - --- (12d) dz w9

which fits into the general form (1).

The key quantities for the understanding of the physical fel\tures of the flow are

consequences of the two equations (3a,b); they specialize to

o (13a)

Cfw"D1 A I w2 wDs ------ + --(x-X)

2A A 9p o . (l3b)

Here N; = N w has been chosen for future convenience in preference to N p' N h,

or Nx which, as we know, are guaranteed to vanish simultaneously when A and

N w have been made to vanish. Explicit forms of the thermodynamic

derivatives Dl' D2, Ds are quoted in the Appendix.

The necessary condition for choking (13a) leads to

w* = llr = [(ap/ap).,x y/2 (x ~ X) (14)

254

Compared with a non-dispersive model, the propagation velocity a f has

increased by an order of magnitude, i.e., compared with that stipulated in eqn.

(4). This is so over most of the composition range but in the range of small

values of the dryness fraction, x, i.e. close to the onset of boiling the

difference reaches several orders of magnitude. This is due to the fact that

atx) is continuous across the two-phase boundary in a thermodynamic diagram

whereas ae jumps discontinuously from a low value in the two-phase region to

a very high value in the single-phase liquid.

The location of the expected saddle point results from an insertion of (14) into

(l3b), or explicitly

C(z*)f w*3D1 A '(z*)w* D3 --------- + --(x-X)

2A(z*) A(z*) 9p o (15)

Figure 2. Portrait of solutions in w,x projection. 2, 2', 2" - trajectories which pass through S,a solution of eqns. (13a,b); A - area of continuous solutions, sensitive to back-pressure with mass-flow rates increasing from in = 0 at wa = 0 to m* at we; B - area of turning solutions; C - area of inaccessible states; D - area of enhanced entropy production

The general theory expounded in Ref. [1] allows us to sketch the w,z projection of

the portrait of solutions shown in Fig. 2. This starts at z = -zo -- a hypothetical

255

stagnation point R -- where w = O. This portrait possesses the familiar

features known from elementary gas dynamics. Curves 2-2'-2" represent the

physically meaningful solutions of eqns. (12 ) which start with w = 0; they all

pass through the singular point S at which they are double-valued. Curve 2

represents subcritical flow which can pass through critical and turn supercritical

only by way of the saddle point. Depending on the back-pressure, the stream

chooses to proceed along 2' or 2", once it reached the choked cross-section at z*.

Curve 2 I II does not pass through w = 0, and is, therefore, of no importance here.

Area A contains the continuous solutions which start with w = 0 at z = - Zo and

are sensitive to the back pressure. The mass-flow rate ill increases monotonically

from 0 at wa reaching its choked, critical value ill* at wS. Solutions in area B

turn on themselves and are devoid of physical significance in the present context.

Solutions in area C cannot be reached under any circumstances. Finally, area D

represents states which cannot be reached from S along a continuous solution; they

can only be attained across an area of enhanced entropy production or a

discontinuity in the form of a normal shock which we discuss in the next Section.

It is remarkable that the addition of the dispersive equation (12d) moves the

location of the critical cross-section by a substantial distance downstream from

the throat at A I (Zt)=O. This is clear from eqn. (15) in which the last term, due

to relaxation, is always positive, because D3 < 0 and (x-x) < 0 in expansion. In

consequence, the solution z = z* is displaced into a region of a positive A I (z) and

one, moreover, whose numerical value is enhanced by Dix-x)/ep.

5. Shock waves

The classical studies of chemically reacting, dissociating or ionizing flows. [10],

have revealed the existence of partly dispersed and fully dispersed shock waves.

Their appearance can be clearly traced to the operation of one or several

relaxation processes in the fluid. The "smearing out" of the tail end of a normal

shock wave is fully described by the mathematical model (12a-d).

When the flow reaches choked conditions at a saddle point z = z* and becomes

supercritical along curve 2 continuing along 2" in Figure 1, the setting of a

back-pressure Ps < P a < P a will sometimes force the flow to produce a normal

shock wave followed by a dispersed tail, as shown in Figure 3. We disregard

here the possibility of the appearance of a system of oblique, "criss-cross" shock

waves in the divergent section and the adjustment by compression from Ps to

256

Figure 3. Partly dispersed shock wave. 3 - normal branch; 4 - dispersed tail

R~~~------------.--'~-----r----------~

.... __ --1 ........... .... .... --1 - --.,.,.,

o z z=L z*

Figure 4. Case when saddle point is outside nozzle

P a > P/3 through a system of external oblique shock waves. This is the case of

a partly dispersed shock wave.

An interesting situation arises when the flow reaches a velocity ae ~ w <ar at the

throat. Contrary to what some authors maintain, this set of circumstances does

257

not cause the flow to be choked at or near the throat. The flow simply continues

to expand subsonically; it mayor may not turn supersonic within the confines of

the nozzle depending on the location of the singular point S. It can be

confidently asserted that no choking conditions will set in unless eqns. (13a,b) lead

to a solution z* which lies in the divergent section upstream of the exit at z = L.

The tendency of shifting such a cross-section far into the divergent portion of the

nozzle, inherent in the relaxation effect, may sometimes present experimenters

with a situation which is difficult to interpret. Such a shift causes the span

between P ex and Pa to narrow considerably. In particular, when z* ~ L, all

solutions incorporate a fully dispersed shock wave and present the pattern shown

in Figure 4. Here the flows along trajectories I are subsonic (w < af) even

though at some points w ~ a •.

exit without causing choking.

Then, any back-pressure can be set at the nozzle

Such a set of conditions was discovered by the

authors in the Moby Dick series [2].

Acknow ledgmen ts

We wish to thank Professors G. E. A. Meier of Gottingen, FRG, and P. A.

Thompson of Troy, NY, for their invitation to participate in the IUT AM

Symposium ADIABATIC WAVES IN LIQUID VAPOR SYSTEMS and for their

generous offer of financial assistance with the travel expenses.

Appendix

The homogeneous relaxation model

Dp 8w -+p-Dt 8z

Dw I 8P --+---Dt P 8z

Dh I DP ------Dt pDt

Dx --Dt

with

AI -A Pw

CT ---pA

CTW

pA

x-x ----e

(AI)

(A2)

(A3)

(A4)

258

A(z) -stress

D

Dt

channel

at wall,

a a +w-at az

shape, A'

p(P,h,x)

(A5)

dA/dz, C(z) - circumference, T(p,w) - shearing

local density, it(P,h) dryness fraction at

unconstrained equilibrium, 8(P,h,x,w) - relaxation time.

D1 [:;] (>0) , (A6)

h,x

D2 [:~] «0) , (A7)

P,x

Ds [~] «0) (A8)

P,h

References

1.

2.

Bilicki, Z., Dafermos, C., Kestin, J., Majda, G., & Zeng, D.-L. 1987.

Trajectories and singular points in steady-state models of two-phase flows.

Int. J. Multiphase Flow 11, 511-533.

Bilicki, Z., Kestin, J. & Pratt, M M. 1988. A re-interpretation of the

results of the Moby Dick experiments in terms of the nonequilibrium

model. Fundamentals of Gas-Liquid Flows, ed. by E. E. Michaelides and

M P. Sharma. ASME Winter Annual Meeting, pp. 11-18.

3. Bilicki, Z. & Kestin, J. "Physical Aspects of the Relaxation Model in

Two-Phase Flow," submitted to Proceedings of the Royal Society, London.

4. Bilicki, Z., & Kestin, J. 1983. "Two-phase flow in a vertical pipe and

the phenomenon of choking: homogeneous diffusion model-I.

Homogeneous flow models," Int. J. Multiphase Flow -.2., 269-288.

5. Bauer, E. G., Houdayer, G. R. & Sureau, H. M. 1976. A nonequilibrium

axial flow model and application to loss-of -coolant accident analysis: The

CL YSTERE system code, Proceedings of CSNI Meeting on Transient

Two-Phase Flow, Toronto, Canada.

259

6. Meixner, J. & Reik, H. G. 1959. Thermodynamik der Irreversiblen

Prozesse, contribution to Handbuch der Physik, Vol. III/2, ed. by S.

Flugge, Springer, 413-523.

7. Whitham, G. B. 1974. Linear and Nonlinear Waves, John Wiley and Sons.

8. Ramshaw, J. D. & Trapp, J. A. 1978. Characteristics, stability, and

short-wavelength phenomena in two-phase flow equation systems. Nuclear

Science and Engineering. 66, 93-102.

9. Trapp, J. A. & Ransom, V. H. 1982. A choked flow calculation criterion

for nonhomogeneous, nonequilibrium, two-phase flow. Int. J. Multiphase

Flow~, 669-681.

10. Vincenti, W. G. & Kruger, C. H. 1965. Introduction to Physical Gas

Dynamics. John Wiley and Sons.

Wave Propagation in Flowing Bubbly Liquid

S. Morioka

Department of Aeronautical Engineering Kyoto Uni versi ty , Japan

Summary Propagation characteristics of waves out of a source placed in uniform bubbly liquid flow are discussed on the basis of the dispersion relation for a dilute bubbly liquid model. It is shown that the pressure fluctuation associated with the voidage wave disappears with vanishing velocity slip and that the cut­off frequency of the pressure wave decreases with increasing flow velocity and with decreasing angle between the wave vec­tor and the flow. Corresponding initial-boundary value prob­lem is solved, and it is shown that most disturbance of high frequency is cut off and there is a little propagation wave. Pressure fluctuation characteristics of bubbly liquid flow observed in a converging-diverging nozzle are interpreted by the above theoretical results.

Introduction

The purpose of this paper is in pointing out that there are

remarkable features in generation and propagation of the waves

out of a disturbance source relative to flowing bubbly liq­

uid.

Particularly, we intend to show that there are some mechanisms

for disappearance of high frequency pressure fluctuation, apart

from the effects of viscosity and heat conduction.

Pressure Fluctuation of Bubbly Liguid Flow in Converging-

Diverging Nozzle

First we show the characteristic pressure fluctuation ob­

served in the bubbly liquid flow in a converging-diverging nozzle.

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Gottingen, Germany, 1989

262

Figure 1 shows the schematic diagram of a water-nitrogen two­

phase flow blowdown facility. The facility consists of the

upstream tank, the test section and the downstream dump tank.

The detail of the test section is shown in Fig. 2. The nitro­

gen-gas is supplied by a bomb, and it passes through a pres­

sure regulator and a flowmeter and then blows out through a sin~

tered metal ring on the head of the injector.

The experiment is performed as follows: First an amount of water (8.7 x 10-3m3 ) is put into the upstream tank. The pressure in

the upstream tank is opened to the atmosphere (0.1 MPa). The

pressure in the downstream dump tank is reduced by using a vac­

uum pump. The mixing gas flow rate out of the injector is set

by the adjusting valve. Then, the shutter valve is opened and

the water in the upstream tank is blown down toward the down­

stream dump tank, mixing with gas out of the injector and flow­

ing through the test section. The blowdown is finished in a

few second, and the measurement is performed during this pe­

riod, by using semiconductor pressure transducers (averaged

pressure, intensity of fluctuation, power spectrum density

function (PSDF», a double resistivity probe (velocity and chord­

length of bubbles, void fraction) and a laser Doppler anemome­

ter (flow velocity of water).

The PSDF of the pressure fluctuation at four measuring points

(a, b, c, d; see Fig. 2.) are shown in Fig. 3, for the mixing

gas flow rate 2.12 x 10-"m3/s at 0.1 MPa and the pressure in the

downstream dump tank 0.074 MPa (left column) and 0.034 MPa (right

column). At the throat, the PSDF is considerably intensified

and extended to high frequency region. However, at the down­

stream of the throat, the intensity of PSDF is appreciably reduced

and the high frequency spectrum disappears. These features

in the pressure fluctuation seem to be closely connected with

the wave characteristics in bubbly liquid flow.

Dispersion Relation

In order to predict the wave characteristics in two-phase flow

theoretically, the modeling and formulation are necessary at

the first place. Here we consider a dilute gas-bubbles and liquid

263

mixture model, and we adopt Wijngaarden's formulation [1,2,3]

as a typical one. We consider the case of no velocity slip in

the main flow, and hence the argument about so-called complex

characteristics is not required directly. Further we neglect

the effects of viscosity, gravity, surface-tension and evapo­

ration-condensation. Constant temperature and isothermal

variation of state are assumed.

We consider a plane monochromatic wave propagating to the di­

rection of angle (J with the uniform flow. Then, the disper­

sion relation is obtained in the form

o ( 1 )

where OJ-ku cos (J ,

c 2 p a(1-3a)pL'

where OJ is the angular frequency, k the wavenumber, u the veloc­

ity of uniform flow, c the maximum propagation speed of pres­

sure wave, OJB the angular frequency of natural volume oscil­

lation of a bubble. The solution of the dispersion relation

[eq. (1)] is shown in Fig. 4, for real OJ and m=O.S [see eq. (4)].

For the voidage wave [~~=O], we find that

( 2 )

h .6 1\ .6. .6 L\ "" £:> h d f were p, PG , PG , R, uil vii 0: denote t e amplitu es of waves or

averaged pressure, gas-phase pressure, gas-density, bubble

radius, liquid-phase velocity, gas-phase velocity and void frac­

tion, respectively. The above result means that the pressure

fluctuation associated with the voidage wave should disappear

as the velocity slip in the main flow vanishes, when the voidage

wave is degenerated to a convection wave.

For the pressure wave, the dispersion relation [OJ~(c2k2+OJi)­

OJic2k2=O] is the second order with respect to OJ. This shows the

264

existence of two pressure wave modes. However, the propaga­

tion velocity relative to the flow changes from the finite maximum

value c at long-wave limit to zero at short-wave limit. There­

fore, if the main flow velocity is less than the maximum propa­

gation velocity, the flow field is subsonic for long-wave but

supersonic for short-wave. This fact is reflected on the dis­

persion relation, which is the fourth order with respect to

k. Physically this means that an imposed pressure disturbance

is divided into four propagating wave modes.

However, two of k's become complex conjugate values for fre­

quencies above we. The critical frequency we is given by

( 3 )

Where m= ~ cosO. ( 4 )

The value of m decreases as the flow velocity decreases and the

angle between the wave vector and the flow increases. In view

of the functional form in eq. (3), we is appreciably small com­

pared with wB for m<l. The large imaginary part suggests the

strong damping as if they are cut off.

Initial-Boundary Value Problem

The above dispersion relation exhibits the existence of cu­

rious waves, such that the group velocity is positive but the

phase velocity is negative and the amplitude increases in the

direction of the phase velocity. Thus, it is difficult to pre­

dict what waves are generated by imposing disturbance and how

they propagate subsequently. Then, we return to the original

differential equation and we consider the initial-boundary value

problem, where a disturbance with a certain frequency is im­

posed at a location in the uniform flow and from an initial instant

and to a specified direction.

Then, the differential equation is given by

( 5 )

265

corresponding to the dispersion relation [eq. (1)]. Here the

nondimensional time .,;=t;wsucos8/ c and the nondimensional space

coordinate ~=ZOJs/c have been used. The initial condition and

the boundary condition are taken as follows,

op P=(JT"=O

P = 0

p = l-cos(0J7:),

at .,;=0,

as ~-±oo,

op = 0 o~

at ~= O.

( 6 )

( 7 )

The solution can be found conveniently by using Laplace trans­

form technique with respect to time, but the solution takes

somewhat complicated form, since the inversion integral in­

cludes the branch points corresponding to the critical fre­

quency. The detail of the solution has been described by Toma

& Morioka [4]. Here we say only that the present result, gen­

eralized for arbitrary propagation direction, can be obtained

by replacing Min [4] to m = ucos8/c.

Now we ask for the initial amplitudes of the four wave modes

for high frequency wave, on the basis of the above analytical

solution. Then, we find out that the amplitudes of the prop­

agating wave modes are small in the order of m/2 compared with

the initial amplitude of the cutoff mode. This means that most

energy of the imposed pressure disturbance is distributed in

the cutoff mode for high frequencies above we.

Discussion

The PSDF, intensified and extended to higher frequency region,

at the throat observed in the previous experiment suggests that

a kind of instability occurs in the throat section. This in­

stability seems to be essential in two-phase flow, because the

fluctuation is of larger order of magnitude than in single phase

flow of water at the same pressure condition. The dependence

of the instability on the main flow condition is quite consis­

tent with the growth rate of unstable voidage wave on the ba­

sis of Wijngaarden's classical model. That is

266

I v I 1/2 Wi = U -1 (2a) uk, ( 8 )

where Wi is the temporal growth rate, v/u the velocity slip ra­

tio, a the void fraction and u the flow velocity of liquid.

On the other hand, the reduction of the fluctuation intensity,

particularly the disappearance of high frequency power spec­

trum density observed at the diverging nozzle section can be

well interpreted by the present theoretical results. In the

above experiment, the flow velocity is less than the maximum

propagation velocity of pressure wave throughout the nozzle.

Hence, the velocity slip is rapidly reduced in the diverging

nozzle section, because of the viscous drag on the bubbles and

the positive pressure gradient. Accordingly, the pressure

fluctuation associated with the voidage wave should disappear.

The flow velocity is also reduced in the diverging nozzle sec­

tion, and then almost pressure fluctuation with high frequency

will be cut off as the pressure wave is reflected and gener­

ated on the nozzle wall and in the adjacent boundary layer.

The PSDF's in Fig. 3 also exhibit a tendency that the critical

frequency is reduced and the propagating modes are rising up,

as the flow velocity is increased.

The probe measurement shows that the chord-length of bubbles

has become smaller at the downstream of the throat. This fact

suggests that the energy of absorbed waves has been converted

into the surface-tension energy by the breakup of bubbles with

increase in the bubbles-liquid interface area.

There are arguments that an instability will be essential in

bubbly liquid flow, but the divergence at short wavelength is

the consequence of the neglect of some dissipation process that

occurs when short wave passes through disperse media, and the

growth rate of the unstable wave is relatively small if the

wavelength shorter than the interbubble distance is neglected

(Klebanov et al. [6], Drew [7]). However, the physical mecha-

267

nism of such a dissipation process has not been shown explic­

it l y. The above dissipation mechanism may be one of answers

to their problem.

9 Fig . 1. Schemaic diagram of experimental facility .

3 10

2

( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) ( 6 ) ( 7 ) ( 8 ) ( 9 ) ( 10)

Fig.2. Detail of thest section. Cross-section of converging­diverging nozzle is 40 x 15 mm2

at entrance, 10 xIS mm2 at throat and 15 x 15 mm2 at exi t .

upstream tank downstream dump tank gas injector test section shutter valve vacuum pump pump flowmeter pressure regulator nitrogen bomb

1Il

1Il 1Il

268

o 4x 10' .---~-----.----,-----. Q)

</)

~~ 2x 10'

o (j)

CL

I x 10' FREQUENCY

point a

2 x 10' [Hz)

o 4xlO'nr---,-----.----~--~ Q)

</)

~~ 2x 10'

o o I x 10'

FREQUENCY point b

2x 10' [H z )

"; ::::: iM:. : : I o I xlO' 2xlO'

FREQUE NCY [H z )

point c o [I x I 0' rlr---,---_,__----,_--__, Q)

~~ 2x 10'

o (j)

CL o o I x 10'

FREQ UENCY

point d

2x 10' [Hz)

4x 10' r----,----_,__----~--_.

2x 10'

o ~.A o I x 10'

FREQUENCY

point a

2 x 10' [H z )

[I xl 0' m---~----,-_,__----,---_,

2x 10'

o o I x 10

3

FREQ UENCY point b

2x 10' [H z )

4x 10'r---~-----,------~--~

2x 10'

o A ;w. N>-

O I x 10' FREQUENCY

point c

2x 10' [H z)

4x 10' .---~-----.----,-----,

2x 10'

I x 10' FR EQUENCY

point d

2 x 10' [H z )

Fig. 3. PSDF of pressure fluctuation at four measuring points for gas flow rate = 2.12 x 10-4rn3 /s, pressure in usptream tank = 0.1 MPa, pressure in downstream dump tank = 0.074 MPa (left column) and 0.034 MPa (right column).

-2 -1

I I I I I I I I I , ,

Kr' ,'1 , , , , I I

I I I

o

269

1 2 3 4

Fig. 4. Numerical solution of dispersion relation [eq.(l)] for m = 0.5 < 1. Imaginary part of complex conjugate roots is plot­ted only in the first quadrant.

References

1. van Wijngaarden, L.: On the equation of motion for mixture of liquid and gas bubbles. J. Fluid Mech. 33 (1968) 465-474.

2. van Wi jngaarden , L.: One-dimensional flow of liquid con­taining small gas bubbles. Ann. Rev. Fluid Mech. 4 (1972) 369-396.

3. Biesheuvel, A. and van Wijngaarden, L.: Two-phase flow equa­tions for a dilute dispersion of gas bubbles in liquid. J. Fluid Mech. 148 (1984) 301-318.

4. Toma, T. and Morioka, S.: Acoustic waves forced in flowing bubbly liquid. J. Phys. Soc. Jpn. 55 (1986) 512-520.

5. Toma, T., Yoshino, K. and Morioka, S.: Fluctuation charac­teristics of bubbly liquid flow in converging-diverging nozzle, Fluid Dynamic Research 2 (1988) 217-228.

6. Klebanov, L.A., Kroshilin, A.E., Nigmatulin, B.I., Nigmatu­lin, R.I.: On the hyperbolicity, stability and correctness of the Cauchy problem for the equations of two-speed motion of two-phase media. PMM 46 (1982) 66-74.

7. Drew, D.A.: Mathematical modeling of two-phase flow. Ann. Rev. Fluid Mech. 15 (1984) 261-291.

Stability of Shock Waves and General Equations of8tate V.M. TESHUKOV

Lavrentyev Institute of Hydrodynamics Siberian Division of the USSR Academy of Sciences Novosibirsk 630090, USSR

Introduction

The experimental study of liquefaction shocks in large-heat ca­

pacity fluids has revealed the shock front instability (P.A.Thom­

pson, G.C. Carofano, Y.G. Kim, 1986). With growth of the Mach

number the following types of irregular shock front behaviour

were observed: appearance of disturbances in the vicinity of the

shock tube symmetry axis, regular waves at the shock front, chao­

tic behaviour and fracture of the shock front.

A number of papers are devoted to the theory of shock stability.

The majority of them consider the problem of shock wave propaga­

tion in space without boundaries. In present paper we consider

three-dimensional flow disturbances downstream of the shock pro­

pagating in a cylindrical tube. The possible types of transver­

sal waves at the shock front are found. We study the properties

of equations of state which guarantee the fulfilment of shock

stability conditions and obtain the classification of equations

of state. Stability of shock waves in two-phase flows is discus­

sed.

Shock front disturbances

Consider the stationary gas flow in a cylindrical tube: X~ + 2 2 + X2 ':::; ro ,-00< X,3< +00 ( X1 ' X2 , X,3 - Cartesian coordinates

in space). Let the velocity vectors ui=(O,O,Wi ) ,pressure

Pi ,entropy 8i (i=1,2) be constant within the domains X,3< ° and X,3 > ° on the both sides of stationary shock front X,3=O . At

X,3=O this quantities satisfy the Rankine-Hugoniot conditions.

Consider the small disturbances of this flow. Downstream of the

shock we define

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Gottingen, Germany,1989

272

S = 82 + 82S

whereU = (U,V,W) ,p ,S are the non-dimensional small

disturbances, Jl2 is the undisturbed density at X3> 0. Introduqe

the non-dimensional independent variables

x=r~1x1' y=r~1x2' z=r~1x3' t=a2r~1t'

(a2 is the sound speed of the main flow, t' is time). The li­

nearized gas dynamics equations are to be satisfied at Z > ° : (1)

At t ~ 0 ,r=1 ( X =r cos 'f, y = r sin If' ) the following ini­

tial and boundary conditions

U(O,x) = Uo(x),

S(O,x) = So{x), U(t,oos I() ,sintp ,z), n = 0

(2)

have to be satisfied ( n =(cos~ sin

the prescribed initial disturbances).

defined by the equation z=F(x,y,t) Rankine-Hugoniot conditions:

If ,0), Uo, pO, SO are

Let the disturbed front be

. At z=O the linearized

U - M F = 0, X

w + p p = 0,

v - ~F = ° y

S-,tiP=O

are satisfied. HereM = w2a21, ~ = M(R - 1), R = U1 V'21 ( 11 =.P -1 - specific volume), .I' = (1 - ~ ) (2WI)-'

(3 )

273

2 2 1 .. f V = M R(1+.6)«1-M )(1-'&»-, A= (P-P1)(V'1-U2)( czr/~P)H ( ( aV" I () P)H is the derivative along the Hugoniot curve at point

tJ= '/J' 2 ,P=P2 ), ~ =( j>2a~(1-(aplds)lY (dtrlap)H)(s2(cp/a s)tr )-1

As a consequence of (1) - (3), we obtain the following problem

for pressure disturbances:

(4)

( z ~ 0, t~ 0, O~ r~ 1 )

p(O,x,y,z) = pO(x,y,z), Pt(O,x,y,z) = p1(x,y,z)

with the boundary condition at z=O

and the condition

dP/an(t,oosi/, sin I{? , z) = 0 (6 )

on the tube wall. The solution of this problem and one of con­

ditions (3) allow to determine shock front disturbances. We shall

construct this solution as a series 2 00 00

P = .(-.-, ') L Pn1i Ct,z)Qnli (x,y) (7) J.=1 W n=O

where Qnli are the eigenfunctions of V2. ( \72. - Laplace ope­

rator in variables x, y ) :

\J 2Qnli = -( A nl)2Qn1i , (8)

a Qnlil a nCoos If, sin If) = 0

The functions Qnli could be taken in the form

Qn11 = JnC) nlr)oos nlP , Qn12 = JnC;\ nlr)sin nIP

and eigenvalues are defined by the conditions: J~C~nl) = 0

I n is the Bessel function;Aoo=O ). For determining the coef­

ficients Pn11 we have to solve the problem of the form (4), (5)

where Vtpnli will be substituted by _(>.n1)2Pn1i . The initial da-

274

ta for Pnli series (7).

may be obtained after expansion of pO , p1 into

After the Laplace transformation in time we integrate equation

(4) and

lInli ( ).., z) = J e- ,\ t Pnli (t,z)dt o

is found in explicit form with the help of condition (5) and

(9 )

condition of vanishing of the disturbances at infinity. (pO, p1

are assumed to vanish at large values of z and in~the vicinity

of z=O ). We represent F in form (7) and define Fnli analo­

gously to (9). We obtain with the help of (3):

where

00

Bnli (,\ ) = J e-k2( ¢ nli'( t\ ,t )dt o

(10)

A behaviour of Fnli(t) at large values of t is determined by

peculiari ties of~nli()") . With new complex variable ¢ ,

). = "nl(1_lVl2)1/22-1(¢ _?-1) (n2+l2 :1: 0)

f ( >') is transformed to the form

f ( ,\) = ¢ -2 ,\ ~l N( c;- ) (11 )

N( ~ ) = (1+ j-f) ~ 4 + 2,1(.,(2)1 -1) ~ 2 - (1-.,M)

Notice that a half-plane Re A > 0 corresponds to the domain

I?-I> 1 ,I arg?1 < 'J{/2 • According to 121, the roots of equation

N( ~ )=0 are located in the circle I? I -< 1 , if

-1 < A < (1_lVl2_RlVl2 )(1_M2+RlVl2)-1 (12)

275

If'

(13)

then the equation N( ~ )=0 has two pure imaginary roots ~;j=~1zi

in the domain I ~ I> 1 and two roots in the domain ,~ 1< 1 . When

f::. < -1, A > 1+2M (14)

the equation N( ~ )=0 has two complex roots in the domain J~ 1>1 • The inverse Laplace trans~ormation determines Fn1i(t)

2 Fnli (t) = L res j 0/ nli (t, ~) +

j:::1

..,.. (2 $'1)-1 J 0/ nli (t,~ )d~ 1~/;;:1

III nli (t, ~ ) = A nl2- 1 (1_M2 ) 1/2Fnli (? »)(

"'(1+?-2) exp(t "n12- 1 (1-M2 )1/2(<,; _?-1)

(15)

where residues are calculated at the points ~ : I~j/> 1 ,N(~;j)=O At large t the integral in (15) decays as t-J / 2 • So, in the ca­

se o~ ~ul~ilment o~ (11) the ~ront disturbances vanish at t ~OO.

The ~irst term o~ (15) determine a behaviour o~ disturbances i~

(13) or (14) are valid. In the domain o~ parameters (14) we ob­

tain instability. In the domain (13) the non-decaying transver­

sal waves appear at the shock ~ront. For the large values o~ t

!l.

F(t,x,y) = L.. L. (Aruisinol nli) I n (,, nlr ) ;.::1 h,t

(16)

The coe~~icients ~i ,mn1i are univalently determined by (15). Each term in (16) represents a transversal wave propagating with

ib:langular Velocity W nl =.Anl (2n)-1 (1_M2) 1/2( J1 + 'l-1 )a.2r;1 When n=O ,we have the aXi"symmetrical disturbances. These types

o~ disturbances were realized in experiments [11 ~or small Mach

276

numbers. The appearance of these disturbances may be due to vis­

cosity effects near the tube walls. The appearance of non-axisym­

metric disturbances for larger Mach numbers may be due to small

deviations of downstream state from gas-liquid mixture equilib­

rium. Within the domain of parameters (14) the disturbances are

also determined by formulae (16) with the complex ~ • The most

rapid growth is inheI;'ent in terms with the large values of>' nl ' and we obtain chaotic behaviour of the disturbances.

Classification of equations of state

The following question arises: what properties of equations of

state provide the fulfilment of inequalities (12) or (13) at any

shock front? Let

z = P 11 (RT)-1 (17)

'i = -( a lnp/ oln V') s

G - Grueneisen parameter, Z - compressibility factor, R -

gas constant, T - temperature, 6 - internal energy). We shall

assume the following assumption for equations of state:

G - 21 $ 0 (18)

Inequality (18) provides boundness of (aif/ap)H (otherwise,

quantity (O~/~P)H is very large at some points and conditions

(12) and (13) are violated). We assume that equations of state

p=g( 1/ ,s) , E = c. (zr,p) satisfy assumptions:

g>O, glr'" 0, gv-zr> 0, gs>-O, c> 0,

011' = t T( V',T» 0

(called in literature as Bethe, Weyl conditions 141).

(19)

Statement 1. Condition (12) is fulfilled at any shock wave if

and only if the compressibility factor Z does not increase on

isentropes with growth of specific volume «~Z/'atr) s:!f. 0 ) •

Inequality (12) may be reduced to the following form:

277

(20)

(inequali ty M < 1 for the downstream state was taken into ac­

count, index "1" designates the upstream state). Inequality (20)

is satisfied for any P1> 0 only in the case when

(21)

The equivalent formulations of (21) have the form:

(a) G+1- i ~ 0, (22)

For ideal gas equations of state z= const~and we obtain equa­

lity in (22). This means that small deviation of equations of

state from ideal ones can produce non-decaying oscillations at

the shock front in certain ranges of upstream and downstream pa­

rameters. To determine the upstream state in this case , we have

to construct an inverse adiabat from the point lJ, P and

find v.t ,P1 on this curve, wi th

0< P1 < p(G+1- ¥ )(G+1 )-1 (23)

This is an example of above-mentioned small "disturbance" of the

ideal gas equation of state:

where r:J. ,,ft ,M , S are positive constants, ~>.1 ,6«1

The shock adiabat of this fluid is a monotonic curve, but oscil­

lations of the shock front are possible. Assume additional con­

dition

lim e ( 1r ,p) = 0 p~ 0

(24)

Statement 2. Inequalities (12) or (13) are fulfilled at any

shock front if and only if equations of state satisfy the con-

di tion a is the sound speed):

G (25)

This condition appears after determining the extremum with res­

pect to parameters V;, P1 in (13) [51. Assumptions (19) do

278

not provide the fulfilment of (25). This is an example of equa­

tion of state satisfying (19). but violating (25) in some do­

mains:

s ::; 0 (ft .eonst,~ =const )

s ~ 0

dt> 1 .}> O/2)1~3Hence. conditions (19) do not provide stabi­

lity of the shock waves.

Inequalities (22). (25) classify the equations of state in con­

nection with the shock stability conditions. Notice that (22).

(25) may be used not only in the case of class (19) equations of

state.

Two-phase flows

We shall consider an equilibrium liquid-vapor model. In the do­

main p ~ Pc • V'1 (p) ~ tr S V'2(P) the equations of state are found

from conditions of equilibrium of liquid-vapor mixture. For

~~V2(P). p~ Pc vapor equations of state are prescribed. and

for tr~V'1 (p) • P ~ Po liquid equations of state are given. The

coordinates of the critical point are the following: P=Pc

tr=u,(pc )=lJ2(pc )=V-c • Let condition; (18) and

Pv-~ 0, PT> 0, Clf = C T > 0

in the domains of single phase states be fulfilled (the symbols

PIT • PT • e T denote the partial derivatives of functions

of V. T ; gif. gs -designation of derivatives ofp=g(V',s) ).

The saturation line is determined by equation p=pQ (T) with

p~(T»O (derivative with respect to T ). In this case in­

equality (20) may be obtained from (12) at admissible shocks

wi th the downstream number M ~ 1 • For the rarefaction shocks

(appearing in the case of changing the sign r . r is the fun­

damental derivative 111) (20) is satisfied. This means that the

disturbances of the negative shocks decay with growth of time.

279

The oscillations of the compression shock front (or its instabi­

lity) may occur only in the case of the downstream states ~, T

satisfying the inequality (inverse to (21))

Since in the vicinity of the critical point c'V- is large (some

known equations of state do not provide this property) and

Ptr -> 0 , inequality (26) is valid in the neighbourhood of the

critical point for single-phase states. Inequality (26) is also

valid for single-phase states with P+~~> 0 and sufficiently

large values of C If far from the critical point.

For equilibrium liquid-vapor mixture states we define an analo­

gous quanitity

"" () ) II f -1 I 'I-' m if,T = (1J -1.T i TPG'" (pc;) + V'(p-TPcr ) +

I I I '-1 + pT(si- ti,: Po- ) (Po- )

where Si (T) ,if i (T) are the values of S ,11 on the boun-

daries of mixture region. The maximum value of?m(V',T) at p =

=const is attained on the boundaries. The values of~m(ifi(T),T)

on the boundaries are

cp m(1{,T) = T2pptr p; (ctr g ur 1 «dSi /dp)2+ V iT(dsi/dp)­

- PT(TPTPV.)-1 cP ( if i,T»

(quanti ties p ,PT ,g If ' c1f /P(rr,T) are calculated for single­

phase state). InequalitY<l'm(U'i,T) > 0 is fulfilled for the sta-

tes lfi(T) ,T on the boundary of the mixture region in two

cases: if the inequality

<p (Vi,T)~ -l!~P1Y P(4TPT)-1 (28)

holds, or if (28) fails, but the inequality

280

is valid. This consideration shows that the ranges of values of

p ,where inequality ~>O holds, expands after transition in­

to the mixture region from the liquid side (where ds1/dp > 0 ), or from the vapor side in the vicinity of states V', P wi th

retrograde behaviour (ds2/dp > 0). Inequality (29) is also ful­

filled in the neighbourhood of the critical pOint,where dsi/dp is large. The admissible compression shock waves with downstream

states in the domain 9' (tr,T) > 0 (fm(tr,T) > 0) and amp litudep-P1 exceeding a certain value (see (23» are characterized by irregu­

lar shock front behaviour (non-decaying oscillations or instabi­

lity). We can conclude that the desired inequalities are usual­

ly fulfilled in some neighbourhood of the critical point. For

large-heat capacity fluids, the corresponding domains e~pand up

to the non-critical ones.

References

1. Thompson, P.A.; Carofano, G.C.; Kim, Y.G.: Large-heat capaci­ty fluid emerging from a tube. J. Fluid Mech. 166 (1986) 57-92.

2. Jordansky, S.V.: On the stability of plane stationary shock wave. PMM 21 (1957) 465-472.

3. Kontorovich, V.M.: Concerning the stability of shock waves. Sov. Phys.: Tech. Phys. 6 (1957) 1179-1181.

4. Rojdestvensky, B.L.;. Yananko, N.N.: Systems of quasilinear equations and applications to gas dynamics. Moscow, Nauka, 1978.

5. Teshukov, V.M.: On the conditions of shock wave stability. Dinamika sploshnoi sredy, izd. Inst. Gidrodinamiki SO AN SSSR, 75 (1986) 134-147.

Rarefaction and Liquefaction Shock Waves in Regular and Retrograde Fluids with Near-Critical End States

S. C. Gulen, P. A. Thompson, H. J. Cho Department of Mechanical Engineering Rensselaer Polytechnic Institute Troy, New York U. S. A.

Abstract

Near-critical states are obtained behind reflected liquefaction shock and rarefaction waves in shock tube experi~~nts. :rhermodyn~c prop~rties C., c, and r (nonlinearity coefficient) are determined using a near-critical, linear parametric equation of state, and the possibility of rarefaction shocks are investigated based on these results. The power law behavior of C. as predicted by the parametric EOS is incorporated into a BWR type classical EOS using a switching function. Hence a single EOS, valid over an extended temperature range, can be used in shock calculations with reasonably accurate predictions in the critical region.

1. Near-Critical Flows in Shock Tubes

In shock tube experiments, states near the liquid-vapor thermodynamic critical point

can be reached in three distinctive ways: (i) Behind a direct shock wave in a retrograde

fluid; (ii) behind a reflected shock wave in a retrograde fluid; and (iii) behind a reflected

rarefaction wave in a regular or retrograde fluid. The idealized space-time wave diagrams

for three cases are shown in Figure 1. The first two cases involve the liquefaction shocks

and are possible only when substances with large molar heat capacities, i. e. substances

with many molecular degrees of freedom are used. The vapors of such substances tend

to condense on adiabatic compression, as distinct from the vapors of fluids of lower heat

capacity, which tend to condense on adiabatic expansion. In Figure 2 T - s diagrams of

substances with increasing characteristic heat capacities are shown.

In the third case, beginning from a supercritical state on the critical isentrope, a reversible

expansion wave of suitable amplitude will produce a stationary near-critical state by re­

flection from the endwall. This process has the advantage that the nominally uniform,

near-critical state may be long in duration. An interesting feature is that the (ideal) termi­

nation reflected wave travels at the critical soundspeed. As the critical point approached,

the isochoric heat capacity increases asymptotically according to the power law

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Giittingen, Germany, 1989

282

Cv '" (T - Tc)-a, where a '" 0.09. This implies that the sound speed will approach zero

and that the nonlinearity coefficient will change sign from positive to negative sufficiently

near the critical point. Thus, the conventional rules for wave steepening will reverse,

with the attendant possibility that the rarefaction wave will steepen to form a rarefaction

shock, which are discussed in detail in the following section.

2. Rarefaction and Liquefaction Shocks

A finite wave may be thought of as a succession of infinitesimal pressure pulses. When

the velocity and temperature gradients are not too steep, viscous and heat conduction

effects are negligible. Thus, the wave is isentropic, and each elementary part of the wave

travels at the local speed of sound with respect to the fluid in which it is propagating.

The change in the propagation velocity of a part of the wave with respect to pressure, in

a fixed reference frame is given as,

dVw v2(cJ2Pjov2)s V -=-- =-T dP 2c (8Pj8v)s c2

where c is the soundspeed. r is referred to as fundamental derivative or nonlinearity coef­

ficient by various authors. It necessarily has the sign of (82 P j 8v2 )., because (8P j 8v)s < 0

from the requirement for thermodynamical stability. The sign of r is therefore geomet­

rically associated with the curvature of an isentrope in the P, v plane, that is, whether

the isentrope is concave upwards or concave downwards. The following identities can be

obtained for r by thermodynamic manipulation [1]:

r = (8(PC)) = ~ {8(P + pc2)} = (~)

c 8P 2 8P 1 + pc 8P s s s

In addition, a formula given by Bethe is most useful for finding r from conventional

thermodynamic information:

= (82 Pj8v2) _ 3.1:.. (&P) &2p + 3 y2 (&P)2 (82Pj8T2) T Cv &T v &v&T c~ &T v v

+.1:.. &P 1 _.1:.. !!!2u. ( ) 3 { () } c~ &T v Cv &T v

If dVwj dP (i. e. r) is positive, the higher-pressure parts of the wave overtake the lower­

pressure parts, and a compression wave steepens as it progresses whereas a rarefaction

wave becomes flatter, and the gradients of velocity, density, etc. become smaller. Thus the

wave remains isentropic. However, if dVwjdP is negative, a compression wave becomes

283

less steep and a rarefaction wave steepens into a rarefaction shock. In order to determine

whether or not rarefaction shocks can in fact exist in real fluids, particularly in the

vapor phase, it is first of all necessary to find regions where r < o. Using arbitrary

equations of state, for retrograde fluids with sufficiently large heat capacities r becomes

negative. Thompson and Lambrakis [1] computed the minimum values of r along the

critical isotherm from various state equations. Their results are shown in Figure 3. Except

the van der Waals equation, the smallest value of C~ for which a negative r exists is around

50. Furthermore, the results of different equations are far from being close.

Thus, the theoretical prediction of the rarefaction shocks in the superheated vapor region

primarily depends upon the choice of the equation of state, and upon the molecular

complexity of the particular substance.

On the other hand, prediction of the liquefaction shocks solely depends upon the magni­

tude of the characteristic heat capacity of the substance (retrogradicity), and the equation

of state used in computations does not have a crucial effect on the qualitative behavior.

For sufficiently high values of the characteristic heat capacity, compression of the vapor

will lead to condensation, because the work of compression will produce a correspondingly

small increase in temperature. Retrograde substances show the typical tendency for the

saturated vapor boundary (j to lean over to the right such that (ds / dT)" takes on positive

values with increasing heat capacity. For smaller values of Cv, compression of the vapor

will lead to such a large temperature rise that no condensation is possible.

In the next section near-critical behavior of soundspeed and r will be examined using a

parametric equation of state based on the power laws.

3. Importance of Equation of State in Shock Calculations

It is a known fact that the classical equations of state which are analytical at the critical

point are not adequate enough to describe the anomalous thermodynamic behavior in the

critical region. They are not able to yield the horizontal course of the critical isotherm and

the flat top of the coexistence curve in that region. Furthermore, they fail to predict the

asymptotic divergence of the isochoric specific heat as given by the power laws. Therefore,

an effort has been made by several researchers to develop special equations of state for

the critical region.

Expanding the reduced chemical potential of a classical equation in powers of

b..p* = (p - Pc)/ pc and /::;'T* = (T - Tc)/Tc around the critical point in the form of a

284

Taylor series, in first approximation, Levelt-Sengers et. al. obtained the following scaling

law equation [2):

* * 1 * 15- 1 (D ( tlT*)) tlJ1 = tlp tlp 1 + (B 1 tlp* 1)1/,6

The linear parametric equation of state proposed by Schofield [3), is maybe the most

versatile model based on the scaling law. The equation of state is in terms of parametric

variables rand 0, where r is essentially the radial distance to the critical point, and 0

describes an angular position in the p, T plane (Figure 4). All anomalies represented by

the power laws are incorporated in the r-dependence, whereas the O-dependence is strictly

analytic. The transformation proposed by Schofield is:

(p - Pc)/ Pc = r,6 M(O) = r,6kO

tlJ1* = (J1(p, T) - J1(Pc, T)) = r,65 H( &) = r,65 aO(l _ 92)

Pc/Pc

where J1(Pc, T) is the chemical potential on the critical isochore at temperature T. The

functions T(O), M(O), and H(O), are the simplest choices compatible with the scaling

requirements. Thus 0 = 0 represents the critical isochore, 0 = ±l the two sides of the

coexistence curve, and 0 = ±b-1 the two parts of the critical isotherm to the left and

right of the critical point. The name linear refers .to the linear dependence of the function

M(O) on O. Using the thermodynamic relation fL = (8A/8p)r we can find the reduced

Helmholtz free energy per unit volume as

A* = : = A~(T*) + p* J1*(P:, T*) + r2-" f(8) c

where the integration constant A~(T*) and J1*(p~, T*) are assumed to be analytic functions

of T* near critical point in the form of a Taylor series. Other thermodynamic functions

can be found by direct integration of the Helmholtz free energy.

The linear parametric model is valid in a limited region around the critical point, located

within approximately 40% of the critical density and 1 % of the critical temperature.

Outside this region it fails severely. Thus in order to carry out the necessary shock

calculations we need a classical far-field equation of state.

A new equation of state in the acentric factor system developed by Schreiber and Pitzer

[4) is used in this work. It is an extended Benedict-Webb-Rubin (eBWR) type equation

285

which, in addition to critical temperature and pressure, employs the acentric factor w as

a third parameter. The complete equation is

Z = 1 + (Cl + * +"* + #)PT + (cs + ¥,: +"* + ~exp(-p;))p; + (eg + ~ + W)p~

+(~ + W- )p; + (fFt + '¥t )p; + (W + (W + W- )exp( -p~))p~ Tr Tr Tr Tr Tr Tr Tr

where

In determining the coefficients of the equation, extensive use was made of many near­

critical data such that the equation has a good fit in the critical region, and yields the flat

dome of the coexistence curve along with the horizontal course of the critical isotherm

quite accurately. Nevertheless, as e>.-pected, it fails to predict the asymptotic divergence

of the isochoric heat capacity approaching the critical point. Furthermore, significant

deviations from the fit appear in the dense fluid region for reduced densities higher than

1.5-2. A comparison of the two equations can be seen in Figure 5. For reduced densities

less than unity, both equations yield similar results.

The main problem is to find the thermodynamic properties in a region intermediate be­

tween the critical region in which the scaling laws apply (described by Schofield's linear

model), and regions further from critical, where classical behavior prevails (described by

the eBWR equation). Chapela and Rowlinson [5] solved this problem by combining two

representations of pressure, analytic and scaled, using a switching function f in the form

P = f(r)PA + (1- f(r))Ps

where r is the distance from the critical point as defined in the linear model. The switching

function f is a smooth function of r. It is equal to zero at the critical point, and approaches

unity as r becomes large. All derivatives of f(r) vanish at r = 0 to preserve the order of

critical singularities. Chapela and Rowlinson used the following function

where k2 > kl and n2 > nl. Although the pressure is represented rather well by this

method, derivative properties such as heat capacity show spurious behavior unless the

two equations are nearly identical near the critical point. In any event, this method is

very inconvenient for practical use.

286

In this work a restricted use of the switching function method is proposed for practical

application to the shock tube experiment calculations. The eBWR equation of state

proposed by Schreiber and Pitzer represents the pressure in the critical region rather

accurately. The equation's major deficiency is in predicting the power law behavior of

the isochoric heat capacity and the isothermal compressibility near the critical point.

This, in turn, leads to incorrect numerical results for soundspeed and r. A switching

function similar to that of Chapela and Rowlinson is used to combine analytical and

scaled representations of Cv :

Thus, more realistic estimates are obtained for soundspeed and fundamental derivative to

facilitate more accurate design of rarefaction experiments.

4. Reflection and Soundspeed Calculations

In Figure 6 the (critical) rarefaction isentrope for SF6 is shown in the P - v plane. The

state numbers correspond to those shown in Figure lc. The isentrope is calculated using

the eBWR equation of state. The reflection condition is given by the following relations:

1 (8P) 1/2 c du = ±- - dp = ±-dp

p 8p s p

integrating we find

r2 :::'dp = r1 :::'dp J1 p Jo p

where c is the soundspeed and u is the fluid velocity. Using this integral condition, the

initial state shown on the isentrope, which produces the critical state after the reflection

of a rarefaction wave, can be found. Below the critical point the isentrope continues in

the mixture region.

The question whether a rarefaction shock develops or not, necessitates the calculation

of the nonlinearity coefficient r. From the definition of r given in section 2, it is to be

expected that, sufficiently near the critical point, the logarithmic divergence in Cv will

force a sign change in r due to rapidly diminishing soundspeed. The change in sounds peed

on the critical isotherm, as predicted by the linear parametric EOS, is given in Figure 7.

It can be seen that the soundspeed decreases rapidly while approaching the critical point

287

from sub critical and supercritical states. Hence, the nonlinearity coefficient takes negative

values near the ?ritical point at subcritical pressures as shown in Figure 8.

The soundspeed and the nonlinearity coefficient on the critical isotherm, calculated using

the eBWR EOS with a modified Cv is shown in Figures 9 and 10. There is a clear

improvement in the results compared to the case when Cv is not corrected to agree with

the power law behavior. Also shown in Figure 11 is the soundspeed on the rarefaction

(critical) isentrope. It is obvious that there is no possibility of a rarefaction shock to

develop in the single-phase region due to decreasing soundspeed with decreasing pressure

(r > 0). However, below the critical pressure, in the mixture region, r changes from

positive to negative so that the rarefaction wave can steepen into a rarefaction shock.

The incident and reflected, inverse adiabats for a liquefaction shock can be calculated

using the Rankine-Hugoniot equation together with the shock reflection condition

[Plr[vlr = [Pldvli. The results for PP1 are shown in Figure 12. The reflected adiabat

crosses the saturated vapor boundary and approaches the critical point in the two - phase

region. Sufficiently near the critical point, the modified Cv can be used to calculate the

soundspeed on the adiabat in a straightforward manner.

5. Conclusions

Near-critical states can be obtained behind the reflected compression shock waves and

rarefaction waves. For the existence of a rarefaction shock, the nonlinearity coefficient r must be negative. Using a parametric equation of state, it has been shown that this is only

possible when expansion on the critical isentrope is extended into the two-phase region.

However, it must be noted that the development of the rarefaction shock is not due to

the phase change. It is a direct result of the sharp decrease in the sound speed due to the

logarithmic divergence of the isochoric specific heat near the critical point. Furthermore,

a rarefaction shock is possible in the near-critical single-phase region, although not on the

critical isentrope.

On the other hand, in the superheated vapor region far from the critical point, rarefaction

shocks may exist for fluids with very high characteristic specific heats (high molecular

complexity). The eBWR EOS used in this work did not predict their occurrence for the

test fluid P PI (Cv = 40.45).

It has also been shown that correcting the behavior of the Cv in the critical region com­

bining classical and parametric expressions using a switching function, greatly improved

288

the performance of a classical BWR type EOS which otherwise fails in this region. Thus,

reasonably accurate predictions of c, Cv , r, etc. can be obtained using a single EOS, valid

over an extensive P, T range, without resorting to rigorous mathematical formulations.

References

[1] P. A. Thompson, K. C. Lambrakis 1973 J. Fluid Mech., vol. 60, part 1, pp. 187-208. [2] J. M. H. Levelt-Sengers, S. C. Greer, J. V. Sengers 1976 J. Phys. Chern. Ref Data, Vol. 5, No.1, pp. 1-51. [3] P. Schofield 1969 Phys. Rev. Leiters, vol. 22, No. 12, pp. 606-608. [4] D. R. Schreiber, K. S. Pitzer 1988 Int. J. Thermophys., vol. 9, No.6, pp. 965-974. [5] G. A. Chapela, J. S. Rowlinson 1973 J. Chern. Soc., Faraday Trans. 1, vol. 70, pp. 584-593.

/ (0 )

/ /

x

/ /

// I

/

,<,

x x

Figure 1. Idealized space-time wave diagrams for the creation of a near - critical state in a shock tube: (a) Direct shock compression in a retrograde fluid; (b) Shock compression with reflection in a retrograde fluid; (c) Isentropic expansion with reflection in a regular or retrograde fluid.

Cy =4.03

t j 0.5

(a) (b) (c) (d) s( R

Figure 2. Temperature-entropy diagrams of substances with increasing characteristic heat capacities 6.

1-0

a'o

O-S 1. a, ·b-' a'b- ' Abbott

T 0 0

0

Clausius

- 05 0 2-phase

Redlich & Kwong

- 1-0 L......_-'. __ -'--"-....JL.._....L. __ .L..:>._-'. __ -'

o 20 40 60 100 120 140

Figure 3. The minimum value of r along the critical Figure 4. Schematic representation of the p - T plan isotherm calculated from various equations of state. near the critical point in terms of the parametric

variables r, (J.

,. , p,

Figure 5

Figure 7

o·

2(cr)

289

RAREFAcn~~.ISENTROPE

v,

Figure 6

f1JND. DER. ~~l'l C.';ll- ISOTHERlof

Figure 8

290

SOUNDSPEED O~rfRIT . ISOTHERM

....................................•.

~9 'iW ' ( /

~ .... ---...=--...

0.99 P,

Figure 9 Figure 10

SOUNDSPEED ON ~fJTICAL ISENTROPE INCIDENT &: REF'LECn:p INVERSE ADlABATS

2 (er)

:::1 U(Ift,,)

.awl _/.,...d, cy

v, v,

Figure 11 Figure 12

Cavitation Waves and Evaporation Waves

Strong Evaporation from a Plane Condensed Phase Yoshio Sone and Hiroshi Sugimoto

Department of Aeronautical Engineering,

Kyoto University, Kyoto 606, Japan

Abstract

The formation and propagation of disturbance in an initially

uniform gas bounded by its plane condensed phase in nonequilibrium

with the gas are investigated in detail on the basis of the Boltzmann­

Krook-Welander equation when evaporation takes place from the condensed phase. From the long time solution, the steady behavior of

the gas (the transition region from the condensed phase to the uniform state at infinity, the relations among the variables at infinity and

on the condensed phase) is clarified. Further the effect of different

boundary conditions at the condensed phase on the steady evaporation is discussed.

I. Introduction

Consider a semi-infinite expanse of an initially uniform gas

bounded by its plane condensed phase. Depending on the conditions of

the gas and the condensed phase, condensation or evaporation will take

place on the condensed phase; the disturbance induced by their

interaction will propagate in the gas; and after a long time a steady

condensation or evaporation flow will be established. The senior

author (Y. S.) considered the problem on the basis of kinetic theory

in Ref. 1 when condensation takes place. In Ref. 1 the behavior of

the gas is analyzed numerically by a finite difference method for a

large number of initial situations, from which the transient behavior

to a final steady state 1s classified and the steady behavior,

especially the relation satisfied among the parameters at infinity and

on the condensed phase in a condensation flow, is clarified.

In this paper we consider the problem when evaporation takes

place from the condensed phase and investigate the time development of

the disturbance, especially the propagation and decay of the

discontinuity of the velocity distribution function, and the steady

behavior of the evaporation from a plane condensed phase. The

relations among the variables at infinity and on the condensed phase in the steady evaporation serve as the boundary condition for the

294

macroscopic gas dynamic equations on the interface of a gas and its condensed phase. Thus, we also investigate the effect of different

microscopic boundary conditions on the steady evaporation.

Some of the results in this paper are announced in letters2 ,3 in

Japanese.

H. Problem and Assumption

Consider a semi-infinite expanse of a rarefied gas bounded by its

plane condensed phase with a uniform and constant surface temperature

Tw. Let the gas occupy xl > 0, where xi is the space rectangular

coordinate system. At time t = 0, the gas is in a uniform equilibrium

state with pressure Pw' temperature Tw ' and velocity (uw ' 0, 0), which is not in equilibrium with the condensed phase. We investigate the

time development of the disturbance produ~ed by the interaction of the

gas with the condensed phase on the basis of kinetic theory and

clarify the behavior of the steady evaporation in the half space from

the long time behavior of the transient solution.

We analyze the problem under the following assumptions:

i) The behavior of the gas is described by the Boltzmann-Krook­

Welander (BKW) equation4 .

ii) In Sec. ill we consider the problem under the conventional boundary

condition on the condensed phase. That is, the gas molecules leaving

the condensed phase constitute the corresponding part of the

Maxwellian distribution pertaining to the saturated gas at rest and

with temperature of the condensed phase. In Sec. N we discuss the

effect of different boundary conditions at the condensed phase on the

steady solution. The explicit form of the condition to be considered

is given there.

The BOltzmann-Krook-Welander equation in the present one­

dimensional case is written in the form:

ef + 1;1

ef AcolP(fe - fl, et eXl

(1;1 _ u )2 + I;~ + 1;2

fe e exp[- 1 3], (2nRT)3/2 2RT

p Ifdl;ldl;2dl;3' ul * II; l fdl;ldl;2 dl;3'

T 1 I 2 1;2 2 p = RpT, 3Rp [(1;1 - ul ) + 2 + 1;3 1fdl;ldl;2dl;3'

where f(t, Xl' I;i) is the velocity distribution function, I;i is the

molecular velocity, p is the density of the gas, u i = (ul ' 0, 0) is

its velocity, T is its temperature, p is its pressure, R is the gas

constant per unit mass, Acol is a constant (AcolP is the collision

(1)

(2)

(3)

295

frequency of the gas molecules), and the domain of integration is the

whole space of ~i' The conventional boundary condition on the condensed phase is

f 2pw exp(- ~i n3/2(2RTw)5/2 2RTw) , for ~1 > 0, at xl = 0, (4)

where Pw is the saturation gas pressure at temperature Tw' In the

present paper, however, the relation between Pw and Tw (the Clausius­Clapeyron relation) is never used, and thus they can be chosen freely

in the results. The boundary condition at infinity and the initial

condition are

f

[for ~1 < 0, as xl ~ -J and [for all ~i' at t = OJ. (5)

According to Ref. 5, we can eliminate the two molecular velocity

components ~2 and ~3 from our system (1) - (5). That is, multiplying (I), (4), and (5) by 1 or ~; + ~~ and integrating the results over the

whole ~2-~3 space, we obtain the system (cf. Ref. 1) for the reduced distribution functions g and h:

g (2RTw)3/2( 2Pw)-1 Jfd~2d~3' (6)

h = (2RTw)I/2( 2Pw)-1 J(~; + ~~)fd~2d~3'

The system arranged in nondimensional variables contains three parameters M_ (= u (5RT /3)-1/2), p /p , and T /T , for various sets

....... (I) (X) <lO W 00 w of which we analyze the problem numerically.

In describing our results, the following characteristic length !w

and time tw are introduced: !w = (8nRTw/n)I/2(AcOIPw/(RTw»-1 (the

mean free path of the equilibrium state at rest with pressure Pw and temperature Tw) and tw i w(2RTw)-1/2.

m. Behavior of Gas

A. Transient behavior

At t = 0, there is a discontinuity between the boundary condition

at xl = 0 and the initial condition. This discontinuity propagates

into the gas, decaying owing to molecular collisions. Since the

standard finite difference method cannot describe this discontinuity

accurately, we rewrite the system in the characteristic variables

(t, xI-~lt, ~1) and integrate this system until the discontinuities of g and h decay sufficiently. After that, we analyzed the original

system by a finite difference method. 6

The transient behavior for M .. = 0, p .. /pw I is

296

0.4

0.3

0.2 ill

(2RTw)'/2

0.0 t/ tw= 2 4 6 8 10

-0.1 0.7

0.6 t/ tw= 2 4 6 8 10

p 0.5

Pw 0.4

0.3

0.2 1.3

1.2

T 1.1

Tw 1.0

0.9 t/ tw= 2 4 6 8 10

0.80 5 10 15 xl/lw

Fig. 1. Transient behavior 1a

(M", = 0, p",/pw = 1/4, T",/Tw = 1).

0.4

0.3

0.2 ill

(2RTw)V2

20

0.0 \ \ I

-0.1 t/ tw= 50 100 150

0.7

0.6

p 0.5

Pw 0.4

0.3

0.2 1.3

1.2

T 1.1

Tw 1.0

o .9

0.8 0 50 100 150 200 xl/lw

Fig. 2 . Transient behavior 1b

(M", = 0, p",/pw = 1/4. T",/Tw = 1).

250

shown in Figs. 1 - 3. Fig. 1 shows the development of disturbance at

initial stage, and Fig. 2 shows the separation process of the

disturbance into a shock wave, a contact layer, and a Knuds en layer,

accompanying the development of uniform regions between these wave or

layers. Fig . 3 shows the decay of the discontinuity of the reduced

velocity distribution function g, where the discontinuity is invisible

at t = 50tw'

Figs. 4 and 5 show the case with Moo = 0.75, poo/pw = 1, and Too/Tw = 1. Since the gas is initially receding from the condensed

phase, a rarefaction region develops near the condensed phase, from

0.6 0.7 x, / lw=O.O

0.1 0.6

0.4 0.2 0.4 0.5

g 0.8 u , 1.2

0.2 1.6 (2RTw)'/2

0.3 t/tw = 2 4 6 8 10

0.0 0.2 0.6 1.2

x,/lw=O.O t/ t w= 2 4 6 0.1

1.0 0.5 0.4 1.0

g 2 . 0 6.0 L 0.8 4.0 8.0 Pw 0.2 / 10 0.6

0.0 0.4 0.6 1.1

x, / lw=O.O t/ t w= 2 4 6 8 10 0.1

1.0 0.5 0.4 1.0

g 5.0 T 0.9 10 Tw 0.2 20

0.8

0.0_2 0.7 0 5 10 15 xl/lw

Fig. 3. Transient behavior 1c Fig. 4. Transient behavior 2a (Mw = 0, pw/Pw ~ 114, Tw/Tw = 1). (Mm = 0.75, P"'/Pw = 1, Tm/Tw = 1).

which an expansion wave, a contact layer, and a Knudsen layer are separated and uniform regions develop between the wave or layers.

297

20

The long time behavior of the case Mm = 1 . 0924, Pm/pw = 0.1850,

and Tm/Tw = 0 .6152 in Figs. 6 and 7 is an example to demonstrate that

a supersonic region in the gas, if any, moves away up to infinity. In the figures, c = (5RT/3)1/2 is the local sound speed, and XC is the

point where (u1 - C)/(2RTw)1/2 takes a given value and thus a

function of (u l - C)/(2RTw)1/2 and t/tw' In Fig. 6 the supersonic

region is shifting toward infinity. In Fig. 7 the shifting velocity dXc/dt of the point XC of a given (u1 - C)/(2RTw)l/2 versus u1 - C is

plotted for various large t, where the curve t ~ m is estimated by

298

0.7 1.2 ,---------,----,---,------,

0.6 1.1

0.5 u l

~1 0 C • 20000 15000

t/ tw=5000 10000 (2RTw)'/2

0.3 t/ tw= 50 100 150

0.2 1.2

t/ tw= 50 100 150

1.0

L 0.8 Pw

0.6

0.4 1.1

1.0

~ 0.9 Tw

50 100 150 200 250 xl/lw

Fi g. 5. Trans i ent b ehavior 2b

(M~ = 0.75. p~/pw = 1. T~/Tw = 1).

Table I. p~/pw and T~/Tw vs M~in the steady evaporation.

0.9

500 1000 1500 2000 xl/ lw

Fig . 6. Shift of the supersoni c

region (M~ = 1.0924. p~/pw = 0.1B50.

T~lTw = 0.6152) .

0.06,--,--.--.--'--'-n~ t/tw=2000

0.04 ~ggg , dXY d t 8000 '

10000 , (2RTw)1/2

O. OOF::;;;;;;;;;~~:::¥

20000 : t~oo

-0.02L--L--~~--~~--~ --{). 06 --{). 04 --{). 02 O. 00 O. 02 O. 04 O. 06

(UI-C) / (2RTw) 1/2

Fig. 7. dXc/dt vs u1 - c in

the case of Fig. 6.

~ T M oo ~

Pw TW

0.0000 1. 0000 1. 0000 0.04999 0.908 3 0.9798 O. 1000 0.8267 O. 9599 0.2000 0. 6891 0.9212 0.2999 0. 5790 0.8836 0.3999 0. 4901 0.8 470 0.5001 0. 4177 0.8113 0.59 97 O. 3587 O. 7766 0. 6997 0.3100 O. 7425 O. 8000 0. 2695 O. 7088 0.8 994 0.2359 0. 6760 O. 9897 0.2102 0. 6467

299

extrapolating the approximate behavior Bt-a of d2Xc /dt2 arouna t = 20000tw up to t = 00 The dXc/dt approaches u l - c in the

supersonic region and 0 in the subsonic region. This suggests that an expansion wave in the classical gas dynamics is propagating toward

the condensed phase relative to the gas motion. Since any supersonic XC point moves away from the condensed phase with a speed larger than u1 - c, the supersonic region finally disappears from the flow field

to infinity.

B. Steady behavior

Making use of the preceding transient behavior, we construct a

large number of steady solutions, from which the behavior of steady evaporation is clarified. That is, we pursue the time development

until a uniform state ahead of the Knudsen layer develops enough and

confirm that g and h there correspond to those of the Maxwellian; if

necessary, we introduce the following cut and patch process several

times: replace the contact layer etc. ahead of the nearly uniform

region by a uniform state and pursue the time development. Since we

are interested only in a steady solution, we don't have to follow the

time development accurately, and the time consuming characteristic coordinate method is unnecessary. Instead, in order to avoid to miss

any possible stable steady solution, we try various nonuniform

initial conditions.

From a large number of steady solutions, we find that poo/pw and

Too/Tw are determined by Moo' where Pm' Too' and Moo are the data at infinity in the steady solution (but not those of the initial data).

They are tabulated in Table I. No steady solution exists for Moo > 1.

Three examples of the transition region or Knudsen layer (Moo = 0.1, 0.5001, and 0.9897) are shown in Fig. 8, where x denotes the value at

~~o~wO~r :~~00~oow~e~~9::7[~h~p:~::~~;:~;w~~~~~:le::e~:: :::: ;~:~tpath of the equilibrium state at rest with pressure p and temperature T .

In contrast to the case of condensationl ,6, the ~hickness of the 00

transition region depends much on Moo'

When Moo ~ I, convergence to its steady state is very slow. In

order to confirm the convergence, we examine the decay of the shifting

velocity dXM/dt of the point XM with a given local Mach number M

(= U1 (5RT/3)-1/2). If the slope of the curve l~glO(dXM/dt)(2RTw)-1/2 versus loglO(t/tw) is less than -1, the point X converges to a finite

point from the condensed phase. Fig. 9 shows the curves for various M

in a case, where the flow converges to the steady state with

Moo 0.9897. From the decay curve of dXM/dt, we can also estimate the

error of the steady profiles. This method is also applied in Ref. 6.

300

o • 090.---.----.-----r----,---,

0.089

0.088

0.087

o • 08f1L----l.-L-----L-----L--.J o . 87.---.---,-----,------r---,

0.8 ~ Pw

0.8267

0.83

o . 82L-..l--'-----'---'---l o . 966.---.---.---,-,----.------,

T Tw

0.9599

0.960

0.45

0.40

0.45

0.40 0.88

0.86

0.82

0.80

0.70

V '0:7266 --------

0.4112 0.60

0.50 U l

(2RTw)V2 0.40

0.30 0.60

~ 0.50 Pw

0.40

0.4177 0.30 0.2102

~ / --------0.20 0.90

T T Tw 0.80 Tw

0.8113 0.70 0.6467

---.1---------

O. 9580~--!:-1---!2:;---:!o-3 ---':-4---'..5 0.800 xl/lw

2 50 100 150 200 x/lw

(a) M", = 0.1. (b) M ... = 0.5001. (c) M", = 0.9897.

Fig. 8. Transition region in the steady evaporation. Here. x indicates the value at Xl = o.

-4 M=0.989

0.98

1 dX;:Yd t o~O(2RTw)1/2

-8

0.96

0.9

0.8

0.6 -10L-~~~L-~~~

345 loglo(t/tw)

Fig. 9. Convergence test to the steady profile with M.., = 0.9897.

1.0

@

0.5

o .0'==========::c...-t--=-L2:::l -1.5 0.0 1.5

U "'/ (2RT w) 1/2 Fig. 10. .Classification of the asymptotic behavior (cf. See. me).

301

C. Asymptotic behavior

From about 70 examples of the numerical solution as shown in Sec. mAo we induce that the long time behavior of our system (1) - (5) is classified in the following cases.

Q): Knudsen layer + contact layer + shock wave. @: Knudsen layer + contact layer + expansion wave R. @: Knudsen layer + expansion wave L + contact layer + shock wave.

@): Knudsen layer + expansion wave L + contact layer + expansion wave R.

where each' element is arranged in order of the position from the condensed phase. The shock wave and the expansion wave Rare propagating away from the condensed phase relative to the gas motion.

The expansion wave L is propagating toward the condensed phase relative to the gas motion. and its front is at sonic condition. The contact layer corresponds to the contact discontinuity in the classical gas dynamics.

After each element is separated enough from each other. the Knudsen layer. practically steady. corresponds to the steady solution

in Sec. mB. and the uniform regions on both sides of shock wave or expansion wave or contact layer are confirmed numerically to be related by the classical gas dynamic relation of shock wave or simple wave or contact discontinuity. This is reasonable from the asymptotic

theory7. With the aid of these relations we can derive. by a simple

calculation. which asymptotic behavior Q) . @ . @. or @)occurs from a given initial condition. The map of the asymptotic behavior on the

initial data u~-p~ plane for T~ = Tw is given in Fig. 10. where the circled number except ® corresponds to the number of the preceding classification. In the region ® condensation takes place. which is duscussed in detail in Refs. 1 and 6. As the shock or expansion (R) wave vanishes with the approach of the initial condition to the

boundary of ® and ® o~ ® and ® regions. a weak pulse propagating with attenuation as Fig. 4(e) of Ref. 1 dominates in the microstructure.

w. Effect of Boundary Condition

So far we discussed the problem under the conventional boundary

condition on the condensed phase. where the velocity distribution

function of the molecules leaving the condensed phase is independent of the velocity distribution of the molecules incident on the

condensed phase and its shape is the half of a stationary Maxwellian.

Now we investigate the effect of different boundary conditions at the condensed phase on the steady evaporation.

First we consider the effect of the incident molecules. The generalized boundary condition suggested by Wortberg's experiment8 on

302

-O.O,--,---.---,--~--~

- 0.2

- 0.4 K(MJ

- 0.6

Fig. 11.

1.0 --

0.8

p= 0.6

~ 0.4

0.2

0.0

0.2 0 .4 0.6 0 .8 1. 0 M=

N1=-0.5 0.0 0.5

/

0.2

ac=0 .2 0.4 0 .6 0.8 1.0

O.O~--L---L---L-__ L-~ 0.0 0 .2 0.4 0.6 0.8 1.0

M=

Fig. 12. P IpG vs M_ [Eq. (9)] . = w -

0.5

T= 0.8

T S N1=-0.5 w 0.7

0.6 1.0 1.5

0.5 0.0 0 .2 0.4 0 .6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

M= M= Fig . 13. p=/p! vs M= [cf. Eq . (11a) ] . Fig. 14 . T=/T! vs M= [ c f . Eq. (11b) ] .

a solid surf ace is gi ven as follows . The veloc i ty distribution of the

leaving molecules is given by the sum of two t e rms: ~c times of the distribution of the conventional boundary condition (Sec. H) and

~c (1 - ~c) times of the diffuse reflection distr i bution, where

(0 < ~c ~ 1) is a constant called condensation factor . That i s, the

distribution of the leaving molecules is given by replacing pw

boundary condi tion (4) by the following Sw'

in the

conventiona l

1/2 J - (1 - ~ )(2nRT ) ~lf(t, 0, ~ . )d~ld~2d~3· c w 1; <0 1

1

(7)

Comparing the conventional condition (4) and the gene rali zed

condition, we find a simple relation between the solutions of the two

types of the boundary condition . Let f (Xl' ~ i ' pG, TG) be the ~ w w G

sol ution unde r the generalized bounda ry condit i on wi th p = p and T = TG, and let f(x1 , ~., pc, TC ) be the solution underWthe w

w w 1 W W C c conventional boundary condition (4) with pw = pw and Tw = Tw' where

p~, Tm' and M~ are assumed to be common. Then the following r e lations hold between the two solutions:

303

(8)

(9 )

where

K(Moo ) = (10n/3)1/2(p~/P~)(Too/T~)-1/2Moo - 1 . (10)

The p Ipc and T lTc, the pressur e and temperature ratios for the Q) w 0) w

conventional boundary condition, are the functions of Moo given in Table I with p and T replaced by pC and TC respectively. The K(M..,)

w w w G w versus Moo is plotted in Fig. 11, and p~/pw versus M~ for various a c in Fig. 12 .

Next we consider the effect of the shape of the velocity distribution of the leaving molecules on the stea dy evaporation. Let

the velocity distribution of the molecules leaving the condensed pha se be the corresponding part of the Maxwellian ¥ with pressure p, temperature ~ , and velocity (u, 0, 0), and put

p! 2J ~i¥d~ld~2d~3' (lla) ~1>0

T! = 3~ I ~if'd~ld~2d~3(J f'd~ld~2d~3)-1. (llb) ~1>0 ~1>0

Taking ~ [= u(5R~/3)-1/2] as the shape factor, we consider the cases

~ = -0.5, 0.5, 1.0, and 1.5 [~ = 0 corresponds to the conventional boundary condition (4)]. For each~, we look f or the possible steady

solutions as in Sec. IDB and find the relations among p..,/P!, T~/T!, and M_ that allow a steady solution. The p Ips and T ITs are the

....... (OW CDW

functions of Moo plotted in Figs. 13 and 14. When ~ < 1, the steady

solution exists for Moo < 1; when M > I, the steady solution exists for ~2 ~2 1/2

M~ < ML, where ML = [(M + 3)/(5~1 - 1)] (the standing shock wave

relation between upstream and downstream Mach numbers M and ML), besides an obvious isolated solution, Maxwellian with pressure p, temperature ~, and gas velocity (u, 0, 0), for p = p, T =~, and

~ ~

u~ = u (thus M.., > 1). Finally, the distributions of the leaving molecules considered here are compared in Fig. 15.

1 . 2 .--,---,--,--,

0.8 Nt=-o . 5

0.0 o . 5 1.0 1.5

0 . 00 1 [1 / (2RT~)l/2

2

304

References

1. Y.Sone, K.Aoki, and I.Yamashita, in. Rarefied Gas Dynamics, edited by V.Boffi and C.Cercignani (Teubner, Stuttgart, 1986), Vol. II, p.323.

2. Y.Sone and H.Sugimoto, J.Vacuum Soc. Jpn. 31, 420 (1988). 3. H.Sugimoto and Y.Sone, J.Vacuum Soc. Jpn. 32, 214 (1989). 4. C.Cercignani, Theory and Application of the Boltzmann Equation

(Scottish Academic, Edinburgh, 1975) p. 95. 5. C.K.Chu, Phys. Fluids 8, 12 (1965). 6. K.Aoki, Y.Sone, and T.Yamada, Phys. Fluids A (to be submitted). 7. H.Grad, Phys. Fluids 6, 147 (1963). 8. R.Mager, G.Adomeit, and G.Wortberg, in Rarefied Gas Dynamics, edited by

D.P.Weaver, E.P.Muntz, and D.H.Campbell (AlAA, New York, 1989) (to be published).

Film Boiling Phenomena in Liquid-Vapour Interfaces H. GOUIN and H. H. FLICHE

Departement de MatMmatiques et Mecanique, Faculte des Sciences et Techniques, Case 322 F-13397 Marseille Cedex 13 - France Centre de Physique TMorique, CNRS , Case 907, Luminy F-13288 Marseille Cedex 9 - France

SummaiY

The equations of motion for a fluid whose internal energy is a function only of density, entropy and their spatial derivatives are able to explain motions through non-isothermal liquid-vapour interfaces. Molecular and statistical models which take the local state of molecules into consideration lead to such an internal energy of the fluid. An interpretation of film boiling phenomena (a liquid is on a very hot plate and sustained by a vapour layer) is obtained as follows: a liquid bulk:, a very thin interface, a film of vapour when the density p decreases a little bit until p reaches the value for which temperature is equal to that of the plate. That leads to the evaluation of the temperature from which the fllm boiling occurs: Leidenfrost's temperature. In the case of Van der Waals fluids, a numerical analysis is given. These results give a better understanding of annoyance for nuclear engineering and metallurgy.

Introduction

Water splashed on a hot metallic flat plate, spreads out, comes to the boil and quickly

evaporates. It is not the same when metal is red hot: water remains cool, breaks up into many

drops that roll, bound and are thrown on all sides.

J.G. Leidenfrost first experimented the process in 1756: a little water poured on a red hot

spoon, does not damp the spoon and takes the same shape as mercury. A strong motion of

vapour laying between liquid and metal supports liquid masses and causes fast vibrating motion

in the liquid bulk:. This is the film boiling phenomenon [1]. The phenomenon was the subject of

numerous studies during the nineteenth century [2].

The interest in the Leidenfrost phenomenon rises from an explosive behaviour associated with

non-equilibrium of pressures in dynamic change of phases. That affects all the industrial sectors

where hot temperatures are used (for example in metal industry: quenching of metals [3]). This

generates the unsteadiness of distillation plants in petroleum industry and many acCidents in

nuclear engineering (TchernobyI1986) [4].

The phenomenon is experimentally well-known. Effects can be foreseen in some definite

conditions. Nevertheless, no satisfying theory has been elaborated [5,6].

In recent papers [7,8], P. Casal and H. Gouin built up a model of thermo capillary fluids that

allows to study flow through non-isothermal liquid-vapour interfaces. Such a model cannot be

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IVTAM Symposium Gottingen, Germany, 1989 © Springer-Verlag Berlin Heidelberg 1990

306

expected to solve all the problems coming from the film boiling, but it can simply explain the

phenomenon and interprets some experimental results.

EQuation of motion for thermocapillmy fluids

Recall the main results [7].

For a fluid whose internal energy E is a function not only of density p and specific entropy!),

but also of gradients of p and s, we can imagine non-isothermal equilibrium in zones with large

density gradients. We call such a medium a thermocapillary fluid. This extends Yan der Waals

and Cahn and Hillard's models [9,10].

Internal specific energy is written:

E = f(p, s, grad p, grad s) (1)

The equation of motion is:

(2)

Here, r denotes the acceleration vector, n is the extraneous force potential, crv is the viscosity

stress tensor and cr is the generalization of the stress tensor:

0{ = - (p - P div cI» B{ - cI>j P,i - \}Ij S,i

where p = p2 E,p , cI>i = P E'N ' \}Ii = P E,S,i •

(The theory yields two new vectors cI> and \}I ).

The conservation of energy yields:

aetat + div[(e - cr - crv)Y] - div U + div q - r = p antat (3)

where e = p ( 1/2 y2 + E + n), Y is the velocity field, q is the heat flux vector, r is the heat

dp ds source term and U = dt cI> + dt \}I •

In the case of an inviscid fluid with infinite thermal conductivity, the equations of motion and

energy yield:

(4)

where eo is a constant along the motion. This is the case we study.

e =E,s.

e is the partial derivative of the internal energy with respect to the entropy; eo represents the

temperature in the liquid bulk [8].

Equ. (4) points out that a thermocapillary fluid may be in non-isothermal equilibrium depending

on the temperature e. In the simplest case, the internal energy is given by:

E = a(p, s) + Q/2p (5)

307

with Q = C (grad p)2 + 2 D grad p. grad s

C and J) are constant and a(p, s) is the internal energy of a compressible fluid whose a partial

derivative gives the pressure:

P= p2 u· p.

The quadratic form is not complete and fits with a statistical model [11]. In this case, the

following values are inserted in the former formula:

p=P-Ql2

<P=Cgradp+Dgrads

'¥=Dgradp

Flow. density and temperature distributions through flat interfaces

System (2)-(4) and boundary conditions are insufficient to determine a motion of

thermocapillary fluids. A state equation as Van der Waals'

is necessary to get a complete set.

P = P R eo _ a p2 1 - bp

(6)

A system of units in which the various coefficients a, b, Rand C are equal to 1 is convenient

for calculations. These units, named internal capillary units (or I.e. units), have been studied

in [7].

For example, in the case of water and considering Vander Waals equation, it follows:

Length:

Mass:

Time:

L = 5.10-10 m

M = 1.5 . 10-22 g

T=7. 10-13 s

(7)

With respect to such a scale, an interface cannot be considered as a surface of discontinuity.

This justifies the use of equations of thermocapillary fluids.

With such units, Vander Waals equation is:

P = p eo _ p2 1 - P

A liquid-vapour transition lets itself be crossed by mass and the equation of motion allows to

study this problem.

Assume that terms associated with entropy gradients (coefficient D in the equation of motion)

are negligible in comparison with those corresponding to density gradients. Then, a simplified

308

equation of motion which does not take into account terms with gradients of entropy is

obtained. The extraneous force potential being neglected,

pr=-gradP +pgrad~p (8)

The velocity of the fluid is assumed to be normal to the flat interface (z denotes the normal

direction) and equ. (8) is expressed as a function of flow Q (Q = P IIVII ) and of density p. With

an unit of time such as in equ. (7), the flow is considered as permanent and the acceleration -

with a term of lip - modifies the differential equation explaining density.

Equ. (8) yields: Q2 1 •

(P+ - -pp"+ - p.2) = 0 P 2

(The derivation noted' is expressed with respect to z).

Two integrations yield the equation of motion in the form:

with

P .2= 2 G(p)

G(p) = - 80 P Ln (lIp-I) - pL Q2 + A + B P 2p

where A and B are two constants.

(9)

(10)

At a given flow Q, equ. (10) allows to compute the distribution of density through the interface.

Equ. (4) can be written

(11)

Equ. (11) gives the distribution of temperature through the capillary layer; 80 denotes the

temperature in the liquid bulk.

Motion of a thermocapillaty fluid in contact with a flat wall

Far from the wall, the fluid is in liquid state. Density p and temperature e are uniform. The

temperature of the plate is 8p. The geometry of the fluid is invariant by translation parallel to the

wall.

The condition at the wall is

C (~)p = ~ (12)

( or P'p = ~ ); ~ denotes a wall characteristic and n denotes the normal direction.

Equ. (12) is given by P. Casal [12]. Demonstrated in case of equilibrium, D = 0, this condition

can easily be extended to motions of thermocapillary fluids. Equ. (10) implies that p' keeps a

constant sign (positive). The wall is said neutral if ~ is null ( this case is studied) and the

fluid density at the wall is a root of equation:

G(p) =0

(Effects of the wall are evaluated by the values (positive or zero) of ~).

Relations

G(pp) =0

G(PI)=O

(a)

(c)

D G'(pp) = Pp (9p - 90 )

G'(PI) = 0

are the boundary conditions of the thennocapillary fluid.

(b)

(d)

309

(13)

(14)

(Relation (14b) is deduced from equ. (11). Relations (14 c,d) correspond to boundary

conditions in a bulk [7]).

For a vapour layer in contact with a wall, the fluid density is below than PI. A complete

interface is obtained in a particular case: equ. (13) has an even order multiple root. In this case,

the first simple root of equ. (13) below than PI is pp. It follows that:

G'(pp) > 0 and 9p - 90 > O.

The density Pv is the one of the vapour bulk; numerically studied in the next paragraph, this

extends results in [7].

Let us denote f(p) = 90 P Ln ( l/p - 1) + p2

In general case, system (14) becomes

- E/PI + A + BPI= f(PI),

- E/pp+ A + Bpp= f(pp)'

g(p) = f(p) - P 9 0 + p2, E = Q2/2. 1 - P

E/PI + BPI= g(PI),

2 9p-90 E/pp + Bpp= g(pp) + Pp D

If 9 0 , PI and Pp are given, this leads to a system with four unknowns A, B, E, 9 p that

determines the motion of the thennocapillary fluid.

So,

Complete interface of a thennocapillary fluid in contact with a flat wall

This case is directly studied [7]. Repartition of density is obtained by taking into account

boundary conditions expressing that G(p) and G'(p) are zero in liquid and vapour bulks. Note

that G(p) is infinite for p = O. This behaviour is different from the static case (Q= 0).

Let us denote X = Pv' Y = PI, L = In (1/X- 1), M = In( 1/Y - 1), E = Q2/2 ,

XI =X/(1-X), Y I =Y/(I-Y)

310

The four boundary conditions G(X), G'(X), G(Y), G'(Y) null can be written:

- eo x L + A + B X - E/X - :xl = 0 - eo Y M + A + B Y - ElY - y2 = 0

and that leads to:

eo = (X_Y)3/F(X,Y) with F(X,Y) = 2 X Y (L-M) + (X-Y) (X1+Y 1)

XY E =""2 {eo /[(I-X)(I-Y)] -(X+Y)}, A = - (X2 +XY + y2) + eo (X+Y-XY)/[(I-X)(I-Y)]

2B = 3 (X+Y) - eo {l/[(1-X)(1-Y)] - 2(XL-YM)j(X-Y)}

which allows to calculate the temperature, the flow and constants A and B according to X = Pv and Y = PI' Moreover, it is necessary that G(p) > 0 on the open set ]X,Y[.

Since lim (p ~ 0) G(p) = - 00 and lim (p ~ 1) G(p) = + 00 , equ. (13) has two double

roots X and Y.

There exists a third root Z between 0 and 1, that must be smaller than X. This corresponds to

the value Pp of the density at the wall. The associated temperature ep is given by equ. (11).

On fig. 1, the solution of our equations matches exactly the film boiling phenomenon: a liquid

bulk, a very thin interface, then a film of vapour where P decreases a little bit until preaches

the value for which temperature is equal to that of the plate. It must be larger than the

maximum temperature in the interface. That leads to the evaluation of the temperature from

which the film boiling occurs (Leidenfrost's temperature). If the temperature on the plate was

smaller, the solution would have stopped in the interface, the fluid would have damped the

surface and would have come to the boil immediately. On the contrary, in the case of film

boiling, the thickness of the vapour film has an order of size more important than the one of

interface.

In this particular case, we quantitatively interpret some results by using Van der Waals'

equation. These results match with the experiments and are represented by the following

graphes.

For all figures, horizontal dashes correspond to points for which the Leindenfrost temperature

is upper than the wall temperature. In the contrary vertical dashes correspond with the film

boiling area. Borders (a,b), (a,c) and (b,c) are respectively associated with the maximum flow,

the null flow and Pv= PI' The critical point of the fluid is denoted by c ( Q = 0, Pv = PI ' eo = ec ). A dynamical critical point is denoted by b; the temperature eo and the flow are

maximum [7] ( eo= 1.068 ec' Pp = Pv = PI ).

311

G

O~T-~~--------------------~~~---L- p

p

z

.......................... 1 L~~b~ ! In1erface

........................................................................................................ : ........... .......... .

Vapour film

p Hot wall.

Fig. 1. Case of film boiling with complete interface (I.C. units)

312

0.9

I a

o.

Fig. 2. Domain of pairs (Pv'p\) . (I.C. units).

5000 .

o.

,._,'-tl" I I I

" . . ' , -, . _ \ I ,

,". '.

e. o. dynamical pressure

0.7

b

I

350. e.1lnOspheres

Fig_ 3. Case of water. Domain of values for dynamical pressure and flow

5000.

o ... '1 S " ...

o. a -30'C

_0 _ 0 0 000

liquid bulk 1emperature

b

o I I ," I

420'C

Fig. 4. Case of water. Domain of values: temperature in the liquid bulk, flow.

350.

o. a -30'C

,0

0'

11 I I

liq uid. bulk ~mperature

313

b

420'C

Fig. 5. Case of water. Domain of values: temperature in the liquid bulk, dynamical pressure.

314

Fig.3: Bands of points are associated with temperature zones of 10°C range in the liquid bulk

(bands are 10°C apart).

For dynamical pressure upper than the one of the critical point c (220 atm. in the case of

water), a null flow is impossible.

- A temperature in the liquid bulk corresponds to a pair pressure, flow.

- For a given flow, the temperature 80 is an increasing function of the dynamical pressure.

- For a given dynamical pressure, the temperature 80 is an decreasing function of the flow.

FigA: Bands are associated with dynamical pressure zones of 10 atm. range.

For a given temperature 80 , the flow is an increasing function of the dynamical pressure. For

reaching areas without film boiling, the dynamical pressure has to increase ( area close to the

limit curve (a,b».

Fig. 5: Bands are associated with flow zones of 250g cm-2s-1 range.

- A null flow ( curve (a,c» is possible only for a liquid bulk temperature lower than the critical

temperature ( i.e. 374°C).

- For a given temperature 80 , the flow is maximum if the dynamical pressure is maximum.

These results are in agreement with Boutigny's experiments [2].

References

1. Leidenfrost, J.G.: De aquae communis nonnullis qualitatibus tractatus. Duisburg (1756); translated by Wares, c., in Int. J. Heat Mass Transfer 15 (1966) 1153-1166.

2. Boutigny, M.: Sur les phenomenes que presentent les corps projetes sur des surfaces chaudes. Annales de Chimie et Physique 3, IX (1843) 350-370 and 3, XI (1844) 16-39.

3. Flament, G.; Moreaux, F.; Beck, G.: Film boiling instability at high temperature on a vertical cylinder quenched in a subcooled liquid. Int. J. Heat Mass Transfer 22 (1979) 1059-1067.

4. Delhaye, 1M.; Giot, M.; Riethmuller, M.L. (eds). Thermohydraulics of two-phase systems for industrial design and nuclear engineering. New York: Mac Graw Hill 1981.

5. Hahne, E; Grigull, U. (eds). Heat transfer in boiling. Washington: Hemisphere PubL Corp. 1977.

6. Joerd, S.; Van Stralen, S.; Cole, R. (eds). Boiling phenomena. Washington: Hemisphere Pub!. Corp. 1979.

7. Casal, P.; Gouin, H.: Non-isothermal liquid-vapour interfaces. Journal Th. and App!. Mech.7 (1988) 689-718.

8. Casal, P.; Gouin, H.: A representation of liquid-vapour interfaces by using fluids of second grade. Annales de Physique, Supp!. 3,13 (1988) 3-12.

9. Van der Waals, ID.: Theorie thermodynamique de la capillarite dans l'hypothese d'une variation continue de la densite. Archives Neerlandaises 28 (1894-1895) 121-209.

10. Cahn, lE.; Hilliard, J.E.: Free energy of a non-uniform system. Journal of Ch. Physics 31 (1959) 688-699.

11. Gouin, H.: A molecular interpretation of therrnocapillary fluids. Comptes Rendus Acad. Sci. Paris 306, II (1988) 755-759.

12. Casal, P.: La theorie du second gradient et la capillarite. Comptes Rendus Acad. Sci. Paris 274, A (1972) 1571-1574.

On the Macroscopic Boundary Conditions at the Interface for a Vapour-gas Mixture Yoshimoto ONISHI

Department of Applied Mathematics and Physics Faculty of Engineering Tottori University, Tottori 680, Japan

Summary Motions of a binary gas mixture of a vapour and an inert gas in a general flow

domain are investigated on the basis of kinetic theory in order to derive a set of macro­scopic equations and the appropriate boundary conditions at the interface between the condensed phase and the gas phase. The analysis has been carried out based on the Boltzmann equation of BGK type subject to the diffuse-reflection condition for the distribution functions at the interface under the following situations: (i) the Knudsen number of the system defined by the ratio of the molecular mean free path of a vapour to a characteristic length of the system is small compared with unity; (ii) the deviation of the system from a certain stationary equilibrium state is small but is of the order of the Knudsen number. In this case, the Reynolds number of the system is of order unity, because, in general, it is proportional to the ratio of the Mach number to the Knudsen number, where the Mach number is a measure of the magnitude of the deviation. The derived system of macroscopic equations and boundary conditions makes possible at the level of ordinary fluid dynamics the treatment of various flow problems of a binary gas mixture involving evaporation and condensation processes which require kinetic theory consideration, giving adequate description of the behaviour of a mixture and its com­ponent gases when the Reynolds number is finite. Although this macroscopic system is meant only for steady problems, the boundary conditions derived here may serve to provide appropriate conditions at the interface even for unsteady problems.

1. Introduction

Various flow problems involving evaporation and condensation phenomena are quite

common in ordinary circumstances and have aroused an interest of scientists not only

in the field of fluid dynamics but also of kinetic theory. The reason for this is that

the ordinary continuum-based fluid dynamics cannot'describe qualitatively correctly

the process of evaporation and condensation occurring at the interface even in the

continuum limit because of the existence of a nonequilibrium region, the thickness of

which is of the order of the molecular mean free path, in the close vicinity of the

interface between the condensed phase and the gas phase. Such a nonequilibrium region

is called the Knudsen layer, in which collisions between molecules are not so frequent

that the momentum and energy exchanges between the molecules leaving the interface

and coming from the outside of this region are not sufficient. Therefore, flow problems G. E. A. Meier' P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Gottingen, Germany, 1989

© Springer-Verlag Berlin Heidelberg 1990

316

of this kind must be based on kinetic theory. Essentially the same situation, although

more complex and difficult, arises for a binary gas mixture of a vapour and an inert

gas. The present study is concerned with the appropriate description of the motions

of a binary mixture of this kind at the continuum or fluid dynamic level, deriving a

qualitatively correct set of macroscopic equations and the boundary conditions at the

interface from the analysis of kinetic equation. With this set becoming available, one

does not have to start from kinetic theory, thus being able to avoid laborious works ill handling the kinetic equation directly. For a pure-vapour case (absence of inert gas),

the corresponding set of macroscopic equations and the boundary conditions together

with the Knudsen-layer structure has already been presented for two different ranges

of the Reynolds number: one given in [1] for the case of negligibly small values of the

Reynolds number and the other in [2] for the case of its finite values, each of which gives

an asymptotic solution to the kinetic equation for small Knudsen numbers. The present

study will give an extended version of [2] to -the case for a binary gas mixture, which is

valid, at present, only at the continuum level. For the purpose of obtaining the above

mentioned macroscopic system, it is necessary to carry out the analysis for the steady

behaviour of a binary mixture of a vapour (say, gas A) and an inert gas (say, gas B) in

a general flow domain. Here the Boltzmann equation of BGK type [3] is adopted as the

governing equation and the half-range Maxwellian associated with diffuse reflection as

the boundary condition for the distribution function at the interface. Furthermore, the

present analysis will be carried out under the following assumptions: (i) the amount of

at least one of the component gases is enough so that not only the molecular mean free

path of the dense component gas but also that of the other component is small compared

with the characteristic length L of the system. Hence, the mean free path of the vapour

IA will be adopted, without loss of generality, for the definition of the Knudsen number

Kn of the system, i.e., Kn = IA / L, and Kn « 1; and (ii) the deviation of the system

E from a reference stationary equilibrium state is small (E « 1), but its magnitude is

of the order of Kn (E "" Kn). The problem, then, becomes nonlinear (weakly nonlinear,

properly speaking), and the kinetic equation and its boundary condition cannot be

linearized. Since the Mach number of the system is a measure of the degree of the

deviation of the system and is proportional to the ·Knudsen number times the Reynolds

number, the Reynolds number in the present flow system becomes finite.

Recently, the one-dimensional evaporation and condensation problem between the

two plane interfaces has been studied in [4] and [5] under the same assumptions as made

above; the former paper for the case of moderate values of the concentration of the inert

gas and the latter for the case of its small values. Unfortunately, these two cases cannot

be combined into one single form covering the whole range of the concentration of the

inert gas, even if the problem is of weakly nonlinear nature. The reason for this is that, as

was pointed out in [4,5], when the concentration of inert gas becomes small in systems

317

where weakly nonlinear or slightly strong processes of evaporation and condensation

are taking place, molecules of the inert gas are likely to be driven toward the surface

of condensation and accumulated there by frequent collisions with those of the flowing

vapour, thus macroscopically forming the large gradients of the partial pressure and

number density near the surface, other fluid dynamic quantities being kept within the

range of weak nonlinearity. This is the manifestation of the full nonlinear behaviour of

inert gas in a mixture associated with weakly nonlinear processes of evaporation and

condensation with small concentration of inert gas. It is this nonlinear behaviour that

forces one to treat problems with small concentration of inert gas separately from the

ones with the moderate concentration. Therefore, the present study also deals with

these two cases separately. In the first case labeled by Case I, the concentration of

inert gas is of moderate values and, hence, the space variations of all the fluid dynamic

quantities always remain within the range of weak nonlinearity; in the other labeled

by Case II, the concentration is small (the ratio'" Kn or less), and the full nonlinear

behaviour of some of the fluid dynamic quantities may appear and, hence, the variations

should properly be taken into account in the analysis.

By the singular perturbation method, we have derived, as an asymptotic solution

to the Boltzmann equation for small Knudsen numbers, the macroscopic equations gov­

erning the fluid dynamic quantities and the appropriate boundary conditions for them

at the interface together with the Knudsen-layer strncture up to the order of approx­

imation sufficient for the description of the behaviour of a gas mixture at the fluid

dynamic level. The equations obtained are of incompressible Navier-Stokes type; the

macroscopic boundary conditions are no slip and no jump conditions for Case I, whereas

they are jump conditions for Case II and comprise a term which is expressed by the

normal component of the vapour flow velocity times a constant related to the jump

coefficient. The Knudsen-layer structure, which is not given explicitly here, is of the

nature of the correction to the solution of the macroscopic equations and has the same

form as the boundary conditions with the constant replaced by a rapidly decreasing

function of the distance from the interface. The constant and function are universal in

the sense that they are totally independent of the geometry of specific problems. It will

be seen in the present weakly nonlinear case that the vapour mass transfer is governed

by the diffusional ability of the vapour through the inert gas, being of small values of

the order of E Kn or E2 for Case I, whereas, for Case II, the mass transfer is governed

by kinetic effects, being of the order of E.

2. Fundamental Kinetic Equations and the Boundary Conditions

The Boltzmann equation of B-G-K type [3] for a binary gas mixture may be written

for steady state as

ei ~~ = NSK, .. (F: - f") + NtK,st(F:t - f"), (2.1)

318

(2.2)

F" = N"(21rR T")-3/2 exp{ (e, - U;')2} e " 2R.T" '

(2.3)

F"t - F'(N' - N" U~ - u~t T' - T8t) e-e - "~-I' - , (2.4)

Ur = U:8 = p.,U;' + p.tU:, (2.5)

(2.6)

where 1', = ms/(m. + md and d{ stands for d6d6d6. s appearing as superscript

and subscript refers to A or Band t refers to B or A. X, is the position vector in

the rectangular coordinate system; e, the molecular velocity vector; f", NS, U;', T8

and P' (= N 8 kT') are, respectively, the distribution function, the number density, the

mean flow velocity vector, the temperature and the partial pressure of S component

gas. m, and R, are the molecular mass of gas S and its gas constant per unit mass,

respectively; k is the Boltzmann constant. F: and F:t are the local Maxwellian distri­

butions for molecules of gas S characterized by the macroscopic quantities (N', U;" T')

and (N', U;"', T·t ), respectively. N' "' .. and Nt "'.t represent the number of collisions

per unit time of a molecule of gas S due to its collision with the other molecules of

its own component and with the molecules of another component, respectively, thus

NS "'ss + Nt "'st being the average collision frequency for a molecule of gas S irrespective

of its collision partners. The collision frequencies are related to the transport coefficients

of the mixture and its component gases [3] and, hence, "' .. and"',t (= "'to), which are

assumed to be constant and of the same order of magnitude, can be evaluated from

these coefficients at some equilibrium reference state (see Eq.(3.12)).

The kinetic boundary condition for f" on the interface is given by the following

half-range Maxwellian distribution

f " - fB - FB (N S - N" U· - U T' - rp ) - W = e - W, ,- ,w, -.LW, for e,n, > 0, (2.7)

where n, is the unit normal vector to the interface pointed to the gas phase. Tw and

UiW are the temperature and the velocity of the interface, UiWn, being here assumed

to be zero (i.e., U,wn, = 0). Nw represents the number density of the molecules of

gas S leaving the interface after the interaction with it. N~ is taken to be equal to the

319

saturated vapour number density corresponding to Tw, which is given uniquely by the

Clapeyron-Clausius relation [6J as

and A A { hL (To )} P =P. exp --- --1 w 0 RATw Tw '

(2.8)

where pa, and pc} are the saturated vapour pressures corresponding to Tw and To,

respectively. hL is the latent heat of vapourization per unit mass of gas A. N~, on the

other hand, is to be determined, as a part of the solution, by the condition of no net

mass flow of gas B across the interface, i.e.,

or B 27r 1 B ~ Nw = - (2 R T, )1/2 ~ind d~, (2.9)

7r B W (ini<O

where the condition (2.7) for fB (~ini > 0) has been used in the last expression above.

At this stage, the transport equations for gas S are presented, which are derived by

multiplying Eq.(2.1) by 1, ~i, e and integrating the resultant expressions with respect

to the whole molecular velocity space, i.e.,

(2.10)

These equations play an important role in the actual analysis that follows, although the

explicit mention is not made there.

Now, before going into the analysis, we take a stationary equilibrium state of the

system as the reference state for the present study and introduce some reference quan­

tities of fluid dynamic interest to which a suffix 0 is attached: namely, To as the refer­

ence temperature of the mixture and its component gases; N&, PJ (= N&kTo) and Po

(= msN~) as the reference number density, partial pressure and mass density of S com­

ponent gas, respectively; No = Nc} + Nt! and Po = p~ +pg as the reference number and

mass densities of the mixture, respectively. With the choice of these reference quantities,

it may be noted from assumption (ii) in §1 that max(ITw - Tol/To, IPa, - pc} 1/ Pc})

should be at most of the order of E. If there is some representative velocity Uo imposed

on the system (e.g., uniform streaming velocity at infinity), Uo/(2RsTo)1/2 should also

be at most of the order of E. In what follows, we proceed to the analysis based on the

assumptions made in the previous section for the following two cases: Case I where

Nt! /Nc} rv 0(1) and Case II where Nt! /Nc} rv O(Kn). The present analysis essentially

follows [4J for Case I and [5J for Case II.

3. Macroscopic Equations and the Boundary Conditions

As is seen from [4,5], the present problem is of singular nature, and it is the ap­

pearance of a small Knudsen number that makes the problem singular. This fact will be

320

elucidated by displaying the kinetic equation (2.1) in nondimensional form. The solu­

tion, therefore, may be sought in the form as was done in [4,5]: the moderately varying

part of the solution, called the fluid dynamic part (suffix H is attached) plus the rapidly

changing part, called the Knudsen-layer correction part (suffix K attached); the former

varies over a distance of the order of the characteristic length L of the system and the

latter varies over a distance of the order of the molecular mean free path and becomes

appreciable within a thin layer called the Knudsen layer near the interface, vanishing

completely outside the layer. Thus we write the desired solutions for the distribution

function and the fluid dynamic quantities of each gas as

(3.1)

(3.2)

with (3.3)

where aB stands for any fluid dynamic quantity of S component gas defined by f" as in Eq.(2.2). 'fJ is a nondimensional stretched coordinate along the normal to the

interface, being of the order of unity in the Knudsen layer within which Xi - X iW '"

O(lA) where [A is taken here as the mean free path of gas A evaluated at the reference

state; Xiw is a position vector to the interface and a function of (1 and (2 which are

non dimensional curvilinear coordinates on the surface 'fJ = const. The fluid dynamic part

of the distribution function iiI with the fluid dynamic quantities aH associated with it

of course satisfies Eqs.(2.1)-(2.6) and (2.10) just as f" does. In the actual analysis, for

the quantities whose variations are expected to remain safely within the magnitude of

the order of i, the nondimensional perturbed quantities from the reference state were

introduced for both Case I and Case II, whereas for some quantities of gas B which may

exhibit nonlinear behaviour'in Case II, simple nondimensionalization was introduced

with respect to the reference state. Each part of these non dimensional quantities was

expanded in terms of the Knudsen number Kn at the reference state in the following

form

(3.4a)

for any of the perturbed quantities of gas S in Case I and Case II, and

Q~ = Q~o + Kn Q~l + ... , (3.4b)

for the quantities of gas B which are expected to make large variations of the order of

unity in Case II. The expansions in Eq.(3.4a) start from O(Kn) to make ',OHm and ',OKm

321

(m = 1,2, ... ) be of order unity, because 'PH and 'PR- are of the order of E and hence of

Kn. However, the expansions in Eq.(3.4b) must start from 0(1) because QZ and Qj{. are

considered to be of the magnitude of order unity. These expansions were substituted into

the nondimensional kinetic equation, and the analysis proceeded from the lowest order in

the Knudsen number. In general, at each level of approximation, the analysis of the fluid

dynamic part gives macroscopic equations, and that of the Knudsen-layer correction

part a set of boundary conditions for the macroscopic equations at the interface and

the Knudsen-layer corrections near the interface (see [4,5]). Here the present analysis

has actually been carried out up to the order of approximation sufficient for the present

purpose, giving a set of macroscopic equations and the boundary conditions at the

interface which should be used for the appropriate formulation at the fluid dynamic

level of flow problems involving evaporation and condensation processes.

In the following, this set of macroscopic equations and the boundary conditions will

be listed in dimensional form (with the suffix H attached to the quantities omitted).

The governing macroscopic equations are given as

/JU/ = 0 /JXi '

A fJU/ 1 /Jp T/M fJ 2ut U· --=---+---

J fJXj Po fJXi Po fJX]'

TA -To

To

and the remaining quantities are obtained from the following relations

fJpB fJpA NB_Nt pB -pt TB -To

fJXi fJXi , NB p'B To 0 0

Ui = UiB = uri, T=TB =TA, P = p _ (pA + pB),

for Case I, and

fJpB fJpA

/JXi = - fJXi '

Ui =UiA ,

pB

N B = kTo'

T=TB =TA,

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

(3.11)

322

for Case II. Ui , T and P are the mean mass flow velocity, the temperature and the total

pressure of the mixture, respectively, and P represents a part of the total pressure which

changes with the velocity field, being at most of the order of Po IUd 2 or P /(Pt + pf) '"

0«('2). It may be noted that the expressions for N A in Eq.(3.9) and N B in Eq.(3.10)

can be used in place of N A = pA /(kTA) and N B = pB /(kTB), respectively, at the

present continuum level of approximation. Also in the expression for N B in Eq.(3.11),

To may be adopted instead ofTB, because (L/NB)(fJNB /fJXi) and (L/pB)(fJpB /fJXi)

are of order unity, whereas (L/TB)(fJTB /fJXi ) is of order of (' and hence the difference

between TB and To does not cause any appreciable changes in N B at the present level

of approximation in Case II. (5/2)kNo in (3.7) corresponds to the heat capacity per

unit volume of the mixture. "1M, AM and DAB are, respectively, the viscosity, thermal

conductivity and diffusion coefficient of the mixture at the reference state. For the

present BGK type of kinetic equation, these coefficients may be given by (see [3])

, kTo "I =-,

x;" \' 5R , A = '2 ,"I, (3.12)

where "18 and A' are, respectively, the viscosity and thermal conductivity of S component

gas also given by the BGK model equation (see [3,7]). Po, No, "1M and AM appearing in

the macroscopic equations (3.6) and (3.7) may be replaced, respectively, by p~, Nt, TJA

and AA in Case II. Incidentally, it is noted that the mean free path I' for the molecules of

S component gas at the reference state is defined by /' = (8R.To/7r)1/2 /(N~x; •• +N6x;.t). With the definition for the mean free path, the diffusion coefficient DAB at the reference

state is expressed in terms of the Knudsen number Kn as

(3.13)

where au = (N! /Nt)a21 and a21 = "'AB/"'AA. Except for an extreme case in which

the molecular mass of gas B is very small compared to that of gas A, DAB is of the order

of Kn and, hence, of the order of (' here regardless of the values of the concentration

ratio N! /Nt.

Now, the boundary conditions for the macroscopic equations (3.5)-(3.8) at the

interface (Xi = Xiw) are given as follows:

U/ = UiW, pA = Pa. (3.14)

323

for Case I, and

B A aNB N Ui ni = DAB ni aX; ,

TA -Tw * ufn; To = d4 (2RA ToP/2'

(3.15)

with C: = -2.132039, d: = -0.446749 for Case II, where ti is a unit tangential vector

on the interface. The constants C: and d: are universal in the sense that they are

independent of the geometry of specific problems and are identical to those given in [5].

In Case I, it is seen that no slip and no jump conditions (adhesive conditions) hold at

the interface, and the normal component of the vapour velocity relative to that of the

interface or the evaporation or condensation rate is zero. This means that the process

of evaporation and condensation does not occur at the continuum level but it may occur

at the higher level of approximation in the Knudsen number. Consequently, the vapour

mass transfer is small in this case as will be shown later. In Case II, on the other hand,

the jumps in the partial pressure and temperature of the vapour occur at the interface,

which are related to the normal component of the vapour velocity and are of exactly

the same form as in the pure-vapour case (see [1,2]) except that the normal component

of the vapour velocity in this case is related to the normal gradient of the logarithmic

number density of the inert gas at the interface. It is this temperature jump that is also

responsible for the existence of the negative temperature gradient in a gas mixture (see

[5,8,9]), the phenomenon of which was first noted in [10] and later discussed in [1] for

the pure-vapour case.

Now, the formulas for the vapour mass flow per unit time and unit area from the

interface m will be given below:

. DAB No apA m = -----n·--

RATO Nil • aXi ' (3.16)

for Case I, and

(3.17)

for Case II. It is pointed out that the expression (3.16) is obtained from the analysis at

the level of approximation one step higher than the continuum level, which necessarily

gives a comparatively small vapour mass flow. This is also seen from the estimation of

the order of magnitude: since DAB I[(2RATo)1/2 L] rv O(Kn) from Eq.(3.13) and also

(LI Pt)ni(apA laX;) rv O(E), it follows that m/[p~(2RATo)1/2] rv O(E Kn) rv O(E2).

The expression (3.17), on the other hand, gives the vapour mass flow, which is of the

324

same order of magnitude as the deviation of the system from the reference state as is

seen from Eq.(3.15), i.e., m/[p~(2RATo)1/2) rv O(E) rv O(Kn), because (pA - pa,)/ pl,

(TA - Tw)/To rv O(E) and (L/NB)n;(oNB /OXi) rv 0(1). It is noted that, as the

ratio N! /Nt is forcibly increased up to 0(1), m/[pc;(2RATO)1/2) given from Eq.(3.17)

goes down to 0(E2), which reduces to that given from Eq.(3.16), as far as the order of

magnitude is concerned.

Finally, the present set of macroscopic equations and the boundary conditions are,

strictly speaking, applicable only to steady problems. The extension of the macroscopic

equations to unsteady cases may be straightforward but whether or not the boundary

conditions are applicable to these cases is open to discussion. However, it is expected

that the form of these conditions may not change appreciably over a macroscopic time

scale which is far greater than a microscopic time scale of the order of the time between

molecular collisions and, therefore, it may be applied with sufficient accuracy to various

unsteady problems at the continuum leveL

References

1. Sone, Y.; Onishi, Y.: Kinetic theory of evaporation and condensation - Hydrody­namic equation and slip boundary condition -. J. Phys. Soc. Japan 44 (1978) 1981-1994.

2. Onishi, Y.; Sone, Y.: Kinetic theory of slightly strong evaporation and condensation - Hydrodynamic equation and slip boundary condition for finite Reynolds number-. J. Phys. Soc. Japan 47 (1979) 1676-1685.

3. Hamel, B.B.: Kinetic model for binary gas mixtures. The Physics of Fluids 8 (1965) 418-425.

4. Onishi, Y.: Nonlinear analysis for evaporation and condensation of a vapor-gas mixture between the two plane condensed phases - Concentration of inert gas rv

0(1) - . Muntz, E.P. (ed.) In Rarefied Gas Dynamics (Progress in Astronaut. and Aeronaut. VoL 117). New York: AIAA 1989 (to be published).

5. Onishi, Y.: Nonlinear analysis for evaporation and condensation of a vapor-gas mixture between the two plane condensed phases Part II: Concentration of inert gas rv O(Kn) . Muntz, E.P. (ed.) In Rarefied Gas Dynamics (Progress in Astronaut. and Aeronaut. VoL 117). New York: AIAA 1989 (to be published).

6. Landau, L.D.; Lifshitz, E.M.: Statistical Physics. New York: Pergamon Press 1969. 7. Vincenti, W.G.; Kruger, C.H. Jr.: Introduction to Physical Gas Dynamics. New

York: John Wiley 1965. 8. Onishi, Y.: A two-surface problem of evaporation and condensation in a vapor-gas

mixture. Oguchi, H. (ed.) In Rarefied Gas Dynamics, VoL II. Tokyo: University of Tokyo Press 1984, pp. 875-884.

9. Onishi, Y.: The spherical-droplet problem of evaporation and condensation in a vapour-gas mixture. J. Fluid Mech. 163 (1986) 171-194.

10. Pao, Y.P.: Application of kinetic theory to the problem of evaporation and conden­sation. The Physics of Fluids 14 (1971) 306-312.

Remarks on the Traveling Wave Theory of Dynamic Phase Transitions M. SLEMROD*

Center for Mathematical Sciences University of Wisconsin Madison, Wisconsin 53706

O. Introduction

The purpose of this paper is to review the traveling wave theory of phase transitions

especially as this theory relates to the dynamics of van der "\Vaals like fluids. The paper

is divided into three sections after this one. The first section derives the balance laws

of mass, momentum, and energy for the one dimensional motion of a viscous, heat

conducting fluid possessing a Korteweg-van der Waals contribution to the stress [28],

[47]. Also derived are the ordinary differential equations governing the motion of

traveling waves. The second section describes some solutions of the traveling wave

boundary value problem and their relevance to the phenomena of shock splitting. The

third section discusses possible relevance of the traveling wave theory to experiment.

1. The Equations of Motion

We consider the one dimensional motion of fluid possessing a free energy of the form

52 (BV)2 f(V, e) = fo(V, e) + 2Bx . (1)

Here V is the specific volume, e the absolute temperature, x the Lagrangian mass

variable, and 5 > 0 is a small parameter. The graph of fo as a function of V for fixed

e will vary smoothly from a single well potential for e > ecrit to a double well potential

for e < ecrit. The value ecrit is called the critical temperature. The term ~(~)2 is

the specific interfacial energy. A discussion of such free energy formulations may be

found in [2-10, 12, 14-18, 22-24, 31-32, 46].

In the absence of viscous forces the variation of (1) with respect to V over an appro­

priate spatial domain delivers the stress T = ~(V, e) _.:;2 a;x;" Inclusion of a viscous

* Sponsored in part by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Contract/Grant No. AFOSR-87-0315.

The United States Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright herein.

Also supported in part by a Meyerhoff Visiting Professorship, Weizmann Institute of Science.

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Gatlingen, Genmany, 1989 © Springer-Verlag Berlin Heidelberg 1990

326

stress delivers the natural modification of (2) given by Korteweg's theory of capillarity

[28], i.e.,

(2)

where u = u(x, t) denotes the velocity of the fluid, p. > 0 is the viscosity, and the

pressure p is defined by -~.

The one dimensional balance laws of mass and linear momentum are easily written

down: oV ou

(mass balance) , at = ox

(3)

ou oT (linear momentum balance) . =

ot ox (4)

The equation for balance of energy is more subtle. While a thorough examination

of the energy equation appears in Dunn & Serrin [12] it is the conceptually simple

approach of Felderhof [13] we recall here. Let e(V, B) denote the internal energy.

Felderhof's postulate is that the internal energy is influenced only by the component

of internal stress 7" = -p(V, 8) + p.~, i.e., the balance of energy is given by

(5)

where h is the heat flux.

Unlike equations (3) and (4), equation (5) is not in divergence form. To alleviate this

difficulty we consider the specific total energy E = 3f + e(V, B) + ~ (-%¥-) 2 made up

the specific kinetic, internal, and interfacial energy. Now compute the time rate of change of E: 8E = u 8u + 8e +c:28V 82V = u 8T + T 8u +c:282V 8u + 8h +c:28V 82u 7ft 7ft at ax 8im ax & 8:1)2 & & ax 8:1)2 where we have used the relation T = 7" - c:2 ~:l{. We easily see that the balance of

energy can be written as

oE = ~(uT) + c:2~ (ou OV) + oh . ot ox ox ox ox ox

(6)

The term c:2~ ~ represents the "interstitial working" [12].

For simplicity we constitute h by Fourier's law: h = I\:M; where I\: > 0 is the (assumed

constant) thermal conductivity. Then we may collect the balance laws and write them

as

(mass) ,

ou 0 { ou 202V} -=- -p(V,8)+p.--c:­ot ox ox ox2 (linear momentum) ,

oE 0 { ou 202V 2 (OU OV) 00} - = - u( -p + p.- - c: -- + c: - - + 1\:-ot ox ox ox~ ox ox ox

(7)

(energy) .

327

The traveling wave theory of phase transitions attempts to determine when two ho­

mogeneous phases (V, u, B), (V,;a=, B) may be joined by a traveling wave solution

V = Vex - ct) u = u(x - ct) B = B(x - ct) (8)

of equations (7). Here c represents the speed of the traveling wave. For small values of

c:, p" K, these waves represent the structured shock waves joining the liquid and vapor

phases to themselves or each other.

Substitution of (8) into (7) yields the following system of ordinary differential equations

dV du -cd{ = df. '

du d { du 2 d2 V } -c df. = df. -p(V, B) + p, df. - c: df.2 ' (9)

dE d {( du 2d2V) (du dV) dB} -cdt = df. u -p + f-L df. - E df.2 + E d~ d{ + K, d~

We impose the boundary conditions that (V, u, B)(~) connects the homogeneous phases

(V,u,B) and (V,IT,e), i.e.,

lim (V,u,B) = (V,u,e), lim (V,u,B) = (V,IT,e) . ~->-oo ~->oo

(10)

We can integrate (9) from -CXl to f. to find

- c(V - V) = u - U ,

( ) ( ll) du 2 d2V - -- c u - U = -p V, u + p, df. - e de2 + p(V, B) ,

( u2 e2 (dV)2 -2 ) - c 2" + e(V, B) + 2 de - ~ - e(V, B)

(11)

( ( du 2d2V) - - 2 (du dV) dB = u -p V, B) + p, de - e de2 + up(V, B) + e de d[ + K, de .

After some manipulations we find that (11) can be rew,ritten as the first order system

dV d[=v,

2 dV (- --E d[ = -c V - V) - (p(V,B) - p(V,B» - f-LCV, (12)

de _ - c3 _ E2cv2 K, df. = -c( e(V, B) - e(V, B» + 2(V - V)2 + -2- - cp(V, B)(V - V) ,

328

We notice of course that the consistency conditions between two equilibria (V, 'fl, e) and (V, iiI, 8) are given by the classical Rankine-Hugoniot jump conditions:

0= c[V][u]

0= -c2 [V] - [P]

3 0= -c[e] + c2 [V]2 -cp(V,e)[V]

for the "inviscid" (10 = {t = I>, = 0) conservation laws (7).

System (12) can be rewritten slightly differently if we define new parameters

for {; > 0 small. Then (12) takes the form

dV d[=v, 105 ~~ = -c2(V - V) - (p(V, B) - p(V, e)) - {tacv ,

dB - - c3 - 2 €5 Cv2 - - -I>,a d~ = -c(e(V,O) - e(V,O)) + 2(V - V) + -2- - cp(V,O)(V - V) ,

where for convenience we have written Va as v.

(13)

(14)

We can now define the viscosity-capillarity admissibility criterion for shocks. We say

a shock solution of (7) (with {t = I>, = 10 = 0) propagating with speed c satisfying the

Rankine-Hugoniot jump conditions (13) of the form

u = 'fl, V = V,B = e, ex < ct), u = iiI, V = V, 0 = e, (x > ct) (15)

is admissible according to the viscosity-capillarity criterion if there is a continuously

differentiable solution of the boundary value problem (10), (14). Notice in this case

the structured solution (u, V, 0)(0 approaches to solution (15) as the shock width

8 -> 0+.

2. Solvability and Qualitative Behavior of the Structure Equations

The issue of solvability of (10), (14), for double well potentials is a rather complicated

problem in the theory of ordinary differential equations. The most up to date results

on the full system (14) are contained in the thesis of M. Grinfeld [19], his papers

[20, 21], and the recent paper of K. Mischiakow [29]. Earlier results were given in

the papers of Slemrod [38-40], Hagan and Slemrod [26], Hagan and Serrin [25], and

Shearer [34-36]. Related approaches for general Riemann problems are contained in

[35], [37]. Within the class of equilibrium problems, the fundamental role of gradient

329

theories was discussed by Aifantis and Serrin [2, 3], Berdichevskii and Truskinovskii

[4], and Carr, Gurtin and Slemrod [5]. Related discussions of gradient theories appear

in the papers of Gouin [14-18], Casal [6], and Casal and Gouin [7-11].

In this paper we consider some special problems for which the qualitative behavior of

(10) and (14) is easily ascertained.

First consider the case when we are looking for an admissible stagnant phase boundary

c = O. In this (10) and (14) reduce to

2d2V - --eo dE2 = -p(V, 8) + p(V, 8) (16)

where 8 = 8. Multiplication of (16) by ~ and integration from -00 to 00 show

1: (-p(V, 8) + p(V, 8») ~~ dE = 0 (17)

where we have made use of the fact that ~ --+ 0 as lEI --+ 00. A change of variable of

integration shows

J;(p(V,lJ) - p(V,lJ))dV = 0 (18)

which is the Maxwell equal area rule i.e., the only admissible stagnant phase boundary

is given by

U=Ul, V=V, 8=lJ, (x<O), U = Ul, V = V, 8 = 8, (x > 0) (19)

where V, V satisfy the Maxwell rule (18). Notice the presence of the capillarity

coefficient e5 > 0 was crucial here, for if eo = 0 we would lose the ability to have

smooth structured solutions connecting two homogeneous phases.

As this was an equilibrium result, viscosity played no role. To consider the role of

viscosity take the simple case when (14c) is viewed as balanced by a heat source term

so as to keep 8 = 8 for all E i.e. isothermal motion. In this case (14) reduces to

2d2V 2 - - --eO dE2 = -c (V - V) - (p(V, 8) - p(V,8» - 110cv ,

with boundary conditions

lim Vee) = V , e--oo lim Vee) = V . e-+oo

(20)

(21)

Now we observe the consequences of setting the viscosity 110 to zero. For if 110 = 0,

equation (20) is exactly the same as equation (16) with p replaced by p(V, 8) + c2V so

that the necessary condition for a connecting trajectory (IS) becomes

(22)

330

where

c2 = _ (P(V,8) - p(V, 8») . V-V

(23)

Graphically this means that the signed area between the graph of p(V, 8) between V

and V and the chord with slope _c2 joining V and V sums to zero. While the result

is pleasing to the eye, it should be rejected on physical grounds: our dynamic shock

will do no work. In fact for regions where :v < 0 and g~~ > 0 (say in an ideal gas) the admissibility criteria says there will be no admissible shocks. Of course the error

made is omitting viscosity. Viscosity will introduce dissipation and yield admissibility

of the usual compressive shocks in regions where ~ < o. This is discussed in Slemrod

[38].

Reiterating the above remarks, (i) a criterion omitting capillarity will rule out equilib­

rium smooth structured phase transitions, and (ii) a criterion omitting viscosity will

rule out classical isothermal compressive shock waves in regions of "normal" behavior,

i.e., 3V < 0, g~~ > o. It thus seems reasonable to retain all the singular terms in

(14), i.e., fi.0 > 0, 1\;0 > 0, EO > 0 at least for modeling materials where (i) and (ii)

are undesirable features. For materials which are viscously dominated however, e.g.

viscoelastic fluids and solids exhibiting phase transitions, a reasonable model may be

one with E = o. This issue has been admirably pursued by R. Pego [30]; (see also

[33]).

p

Figure 1

To see when a dynamic phase transition for the isothermal case is possible, return to

(20), (21). Set V = VI fixed in the liquid phase. Let VI be above the left Maxwell

point m as shown in Figure 1. As we vary 0 :5 c < 00, the intersection of the chord

-c2 (V - VI) with the graph of p picks out the equilibria of (20) of which there may

331

number as few as one (VI large, c = 0) and as many as four. For fixed c > 0 we denote

the equilibria as V2(C) for the equilibrium lying in the spinodal region -3il > 0, V3(C)

for the first equilibrium lying on the stable (vapor) branch of p(V,8), V4(C) for the

second equilibrium lying on the stable (vapor) branch of p(V, 8). Of course for different

values of c and Vb not all of these equilibria will exist.

An easy calculation shows for c = 0 trajectories emanating from V = VI, v = 0 in the

V -v phase plane will overshoot all the possible equilibria and no connection is possible

(consistent with the Maxwell rule (18)). Not quite so obvious is the following: If there

is a if on the stable (vapor) phase branch of p(V, 8) so that equal areas can be obtained

between the graph of p(V, 8) and the chord joining (Vl, p(V}, 8)) and (if, p(if, 8)) with

slope c > 0: -2 (pcY,8) - P(VI, 8)) c = - A ,

V-Vl (24)

then there is a speed 0 < C*(VI) < c and a vapor p. 'ase equilibrium of (20), V3(C*) for

which the boundary value (20) with lime-+-oo VCe)'= Vl, lime-+o") Vee) = V3(C*) has a solution. Of course we must have c*2 = - (P(V~-~Vl». Notice V and hence V3(C*)

will always exist provided VI is not too large. Large Vl states preclude connections to

intermediate states V3 and only admit connections to the far vapor state V4. A proof

of this statement may be found in [26].

We thus see for a not too large homogeneous liquid phase Vl there is a speed c* and

intermediate vapor phase V3( c*) to which a comi.ection may be made by a traveling

wave. In fact V3 is in the metastable vapor branch of p(V,8) where V < M and

-3il < 0 and c* and V3 are unique (given VI). These facts have been shown by M.

Shearer [34, 36].

In this case in the limit as the shock width 6 -+ 0+ we have constructed an admissible

condensation wave with metastable vapor in front of the wave and stable liquid behind

the wave. The speed c* satisfies 0 < c* < (-pl(VI))~' 0 < c* < (-pl(V3(C*))~ and

hence is subsonic with respect to the states in front and behind the propagating phase

boundary.

Notice that according to Figure 1 there is a fourth possible equilibrium to consider,

i.e., V4. The existence of V4 brings us to the topic of shock splitting. As we remarked

earlier for fixed VI trajectories emanating from V ::;:: VI, v = 0 in the V - v phase

plane will overshoot V = V3(C) for speed 0 < c < C*(VI). In fact they have no choice

but to flow to V = V4(C), v = 0 as e -+ 00. So for all values V4 > V4(C*) (see Figure

1) the liquification wave

V = Vb U = uI, x < ct, V = V4(C), U = ul + c(V4(c) - VI), x> ct

is an admissible phase boundary propagating with speed c > 0: c2 = _ (p(V~=\{Vl)). What happens as we reduce V4 less than or equal to the triple point val~e V4(C*)?

332

The answer is simple and follows from our earlier discussion. For c = c* we know the

orbit emanating from V = Vb v = 0 must go to V3(C*) and cannot reach V4(C*). The

connection to V4( c*) is obtained by a second orbit connecting V3( c*) to V4( c*). Thus at

the triple point value the connection is made with two trajectories possessing the same

speed c*. For V3( c*) < V4 < V4( c*) we continue this two trajectory construction. The

first trajectory connects Vl to V3(C*) with speed c*, the second trajectory connects

V3(C*) to V4(C) with speed Cshock > 0 satisfying

c2 -shock -

cshock > C*(Vl) > 0 .

Hence decreasing V4less than the triple point value V4(C*) causes the originalliquifica­

tion wave to split into a liquification wave moving with speed c* and a faster forerunner

vapor phase ,hock propagating with speed Cshock. The liquification wave is subsonic

with respect to the states in front and behind the wave; the forerunner vapor shock is

supersonic with respect to the state in front of wave and subsonic with respect to the

state behind the wave. The values c* and V3(C*) are invariant with respect to V4 and

depend only on Vl. A graphical description of the split wave is given in Figure 2.

x

Figure 2

where U4 - U3 = -Cshock(V4 - V3(C*)), U3 - Ul

considered given.

As to evaporization waves an analogous discussion can be given except there is no

possibility of shock splitting. This is due to the asymmetry of the van der Waals

isotherms and hence the lack of existence of a state playing the role of V4 above.

333

Finally we note that isothermal tangential C - J constructions going from stable

liquid (vapor) states to the top (bottom) of the vapor (liquid) loop are not possible in

this theory.

Further discussion of these ideas and rigorous proofs may be found in the paper of

Hagan and Slemrod [26].

3. Connection with Experiment

The traveling wave theory of phase transitions delivers the local shock structure of van

der Waals like materials. It is a rigorous mathematical theory based on a postulated

constitutive relation. Whether real materials exhibiting phase transitions can be ac­

curately modeled by double well potentials possessing spinodal regions and gradient

terms is a topic of some controversy - though the actual concept is a fundamental

starting point in the theory of phase transitions and is usually termed the Landau­

Ginsburg theory (see for example [1]).

With respect to the topic of this conference, the work of P.A. Thompson and his

collaborators on liquid-vapor interfaces are most pertinent: [11], [41-45], and the thesis

of Thompson's Ph.D. student Y.-G. Kim [27]. (Of course dynamic phase transitions

and shock splitting are known to occur in other contexts [48].)

The above referenced experiments (adiabatic) occur in flow geometries much more

complicated than the simple (isothermal) planar Riemann (shock structure) problems

discussed above. Furthermore, the interesting results on shock splitting given in [27],

[41-44] all involved piston and not Riemann initial value problems. Hence a direct com­

parison with the local shock structure theory given here may be difficult. Nonetheless

for completeness we record some obvious consistencies and discrepancies between the

experimental results and traveling wave theory.

Consistency:

(i) For the classical diaphragm rupturing (Riemann problem) of the type de­

scribed in the traveling wave theory the experiment of [11] did provide com­

plete liquification waves: vapor ---+ liquid.

(ii) The waves and splitting of vapor ---+ liquid waves described by [27, 41-44]

possess the same ordering of speeds (sonic, supersonic) as those described

here. That the experimentally observed vapor intermediate state after shock

splitting is metastable points out the advantage of using a non-convex (as

opposed to a convexified) free energy contribution fo.

With respect to the fact that traveling wave theory given above was isothermal while

the experiments are not we recall that the papers [40], [21] show that the same qualita-

334

tive results hold for the full system (14) is restricted to fluids with large heat capacity

such as those used by Thompson et al.

Discrepancy: A major discrepancy between the traveling wave theory and the shock

splitting results of [41-44], [27] is the fact that phase transition in these experiments

goes from vapor --t a liquid, vapor mixture. Several reasons could be suggested for

this discrepancy. Of course there is an obvious one, i.e., the van der Waals equation

of state may be a poor model of the material under consideration. On the other hand

one must in fairness recall

(i) the above experiments are piston driven and

(ii) the traveling wave theory delivers only local shock structure. A traveling wave

theory will not deliver the global solution of a complicated problem and it may

be that geometry of the experiment is a central reason for the vapor --t liquid,

vapor mixture transition.

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Acoustic Phenomena in Two-Phase Systems, Cavitation

Caviation behind Tension Waves J. Bode, G.E.A. Meier and M. Rein Max-Planck-Institut fur Stromungsforschung Bunsenstr. 10 D-3400 Gotti ngen

Summary

The abrupt deceleration of a flow and the suitable reflection of the resulting pressure wave is the most convenient method for the generation of tension waves and for the exam­ination of cavitation behind those waves. In this paper two facilities using this method are described, allowing the examination of the propagation of tension waves and the resulting cavitation bubble clusters. In a tube the interaction between the cavitation nuclei and the reflected wave trains causes periodically generated bubble fields. Various physical effects are taking place in the bubble cluster at the same time, but it was possible to observe some aspects seperately. The dynamic instability of a bubble during its collapse permits to change the nuclei density in the liquid. Pressure oscillations inside the bubble cluster are explained by a comparison of numerical and experimental results. A bubble selection mechanism in the cluster, which was examined in the experiments, is caused by differences in the surface tension between bubbles of different radii. Pressure oscillations and bubble selection can be described in a first order approximation by an approach of Meier (1987), who proposed to consider each bubble of a cluster to be the centre of a surface tension driven flow. For a detailed description of cavitation behind tension waves these effects have to be taken into account.

1. Introduction

Liquids undergoing tensions In stressing waves have been frequently investigated. A

main goal of these investigations has been the determination of the threshold for cavitation

inception and its dependence on various parameters. In this paper special attention will

be paid to the details of the breaking dynamics of the liquid. Such dynamics are strongly

influenced by cavitation nuclei which are almost always present in ordinary liquids.

<surface Low pressure section /""* ... U' .... ' Surface

~ Vacuum

Expansion 1 . Expansion Piston

-------------- Li quid Diaphragm Compression High pressure section Liquid 2.Expanslon --/ " 1. Expansion 2.Expansion Liquid

I \ Explosion

Liquid Compression

0 1. Expansion

\ / 2.Expanslon Shock Open end

Fig. 1: a) Underwater explosion, b) Shock tube, c) Impulse slug, d) Closing valve.

Tensions can be dynamically applied to liquids by several methods, most of them being

due to the reflection principle (Trevena 1984). Based on this principle, a strong compression

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems lUTAM Symposium Gattingen, Germany, 1989 © Springer-Verlag Berlin Heidelberg 1990

342

or expansion wave is reflected at a boundary to a low or high impedance regime and thereby converted into a tension wave. Examples for the generation of tension waves (Fig. la-d)

are underwater explosions (Kedrinskii 1978), shock tubes (Szumowski 1989), the impact of an impulse slug by a piston (Brown 1967; Couzens and Trevena 1974) or fast closing valves

in a piping system (Joukowski 1898; Kojima, Shinada and Shindo 1984). The shock tube method has also been used by Hill and Sturtevant (1989) to generate evaporation waves in

superheated liquids.

In this paper a method similar to the one of a fast closing valve is used by modifying an approach of Favreau (1984). It is based on the same mechanism which causes waterhammer

oscillations in a hydraulic piping system (Streeter and Wylie 1974).

On impact of a liquid column in a pipe by stopping its motion, a waterhammer wave is formed which propagates upstream towards the open end. A strong pressure rise occurs over the whole length of the pipe. The subsequent reflections of this pulse generate the

desired tension wave. In this manner, large tensions can be produced within the entire pipe.

The presence of minute cavitation nuclei in the liquid leads to a sudden rupture of the liquid

and to a violent growth of these nuclei. Connected with the cavitation a pressure rise occurs behind the tension wave. This pressure rise can only be due to the development of the cavitation zone since no other causes such as a finite width of a tension pulse are present in this case. This highlights the advantage of the special experimental arrangement used

here. Actually, the pressure rise is not monotonical, but pressure oscillations around a time average p = 0 occur. These pressure oscillations were observed by Rein and Meier (1990a) using numerical simulations and they were also reproduced in experiments by Bode, Meier

and Rein (1990). The influence of various parameters on these oscillations was investigated

numerically and experimentally. In the experiments the nuclei density was changed using a

mechanism for nuclei production by bubble collapse (Kedrinskii 1987). It turns out that the number density, rather than the bubble size plays the dominant role in the formation of the

pressure oscillations.

In the case of high nuclei densities Bode, Laake and Meier (1987) observed a bubble se­

lection mechanism. The different sizes of bubbles in a cavitation cluster cause an interaction of the bubbles by pressure and velocity waves propagating between the single bubbles, so that the smaller bubbles tend to collapse earlier than the larger ones because of differences

in the surface tension ( Meier 1987). All these effects occur simultaneously, therefore an

accurate interpretation of the results is difficult.

2. Generation of a Tension Wave

The kinetic energy accumulated in a stationary flow can be transformed into sound energy by stopping the flow abruptly. This may be achieved, for example, with fast closing

valves, carried along with the flow, until they reach their stop. The inertia of the liquid

compresses the liquid column near the valve. The kinetic energy of the liquid is transformed

into sound energy in the tube and into vibrational energy of the apparatus. In the liquid

a pressure wave of amplitude ap is generated. In a first order approximation for the one­

dimensional case the amplitude of the compression wave is obtained by the product of the

impedance pa of the liquid and the differential change au of the velocity in the fluid column

343

where p is the density of the liquid and a is the velocity of sound:

.C!t..p = pa.C!t..u. (1)

In the experiments considered later, water is used as the liquid. Since water has a relatively

high impedance, a compression wave of high amplitude is generated . The abruptly generated

pressure wave propagates through the tube with the velocity of sound, provided the influence

of the elastic tu be walls ca n be neglected.

Here, it is desired to first generate a waterhammer wave which is subsequently trans­

formed into a strong tension wave by using the reflection principle . A nescessary condition

for generating waterhammer waves is that the liquid is stopped in a time intervall .C!t..t which

is shorter than the time ts the pressure wave needs to propagate through the tube twice.

This leads to the following inequality for the length L of the tube :

2L ~t < ts = a

(2)

Satisfying equation (2) guarantees that no interference between the generated and reflected

wave will occur. In the special case that the test liquid is accelerated in a cylinder and

stopped abruptly ( Fig . Id ), the length of the pressure pulse equals the length of the liquid

column. If such a wave is used for the generation of a tension wave, the wave pattern of

the main wave fronts running back and forth in the tube, becomes quiet simple.

--- Water

- WdlCI lank

Fig. 2 : Cavitation tube.

I=;;:;;:f~:c.j c __ Magnelic holding

--- Measul-ing vessel

-- Guicies

./Guide roil

_ I~cce ivi ng vessel

-Shock cy linci er

---Damping maleria l '7£1

Fig. 3 : Falling tube.

In Fig. 2 a sketch of the experimental apparatus, called cavitation tube, in which most

of the experiments are performed is shown. In a cylinder filled with the test liquid, a piston

and the liquid column are accelerated by a pressure difference between the upper and lower

side of the piston . Here, a difference of one atmosphere is applied. When the piston reaches

its stop, the liquid column is abruptly stopped and a pressure rise over the whole tube occurs .

The subsequent reflections of the pressure wave at the open and the closed end of the tube

generate the desired tension wave.

Fig. 3 elucidates another method for generating tension waves. The falling tube shown

in this figure is an improved development of the cavitation tube, allowing easy prediction

344

of the amplitude and the wavelength of the pressure wave. The test liquid contained in a

plexiglas tube has an open and a closed end . When the plexiglas tube is dropped on a shock

cyli nder of the same mass and geometry, an elastic shock takes place and the tube and thus

the liquid column, is abruptly stopped . The pressure wave which is generated in the liquid

is again converted into a tension wave via the reflection principle .

1""'--~-""'---10"-----210-x [ em 1

Fig. 4 : Wave pattern.

Closed end Open end

.6..t = 940J.l,S

.6..t = 850J.l,S I t

.6..t = 690J.l,S

.6..t = 510J.l,S

.6..t = 430J.l,s

.6..t = 290J.l,s

.6..t = 140J.l,S

.6..t = 80J.l,s

x -Fig . 5 : Developement of the cavitation zone .

In the case of the cavitation tube and the falling tube the wave patterns of the main

wave fronts (Fig. 4) and the pressure signals (Fig. 7) are similar. Fig . 4 shows the wave

pattern diagram. First, the compression wave propagates through the tube to the open end

and thereby increases the pressure of the und isturbed region (j) to a high level (region

()). At the open end it is reflected as an expansion wave, which propagates back in the

direction of the closed end . Behind the trailing edge of the expansion wave the pressure

again decreases to the ambient pressure (region @) . When the expansion wave reaches the

piston, it is reflected as a tension wave. Behind this tension wave the pressure drops far

below the vapour pressure (region @) and pressu re osci ll ations occur (region @). However,

the pressure does not reach the minimal value Pmin = 1 -1.6..pl which might be expected in

the case of the reflection of a tension wave in a pure liquid . Here, a bubble field is generated

preventing that small a pressure. Some photographs , similar to those obtained by Hansson,

Kedrinskii and Mcerch (1982), which are taken at different times corroborate this kind of

wave propagation (Fig . 5) . At the open end of the tube the tension wave is reflected with

a phase change as a compression wave and forces the bubbles to collapse . The reflection of

the wave gives rise to a new pressure pulse (® Fig . 7a) . An interfering wave is radiated from

the piston, which also forces the bubbles to collapse. It is produced by waves propagating

345

in the piston and the plexiglas tube . All waves are damped by friction at the walls and by internal viscosity.

The breaking of real liquids behind tension waves gives rise to dynamic cavitation clusters (Fig. 6). The main physical aspect of these dynamic cavitation clusters is the stiff coupling between the single bubbles by the surrounding liquid (Meier 1987) . . Meier proposed to consider each bubble to be the centre of an outward or inward moving flow. The difference of the surface tension pressure which depends on the bubble size, is the driving force for this flow . The interaction among the bubbles influences the entire dynamics of the cavitation~

the pressure oscillations, the bubble selection and the bubble collapse inside the cluster. The experimentally observed effects are handled seperately and are compared with a numerical simulation.

3.1 Bubble Collapse and Nuclei Production

In cavitation physics it is always a problem to take the nuclei density n into proper account. This quantity which is of greatest importance for the interaction between the bubbles (Rein and Meier 199Gb), is related to the distance Al between the bubbles and to the void fraction {3 and the radius R of the bubbles by :

(3)

The characteristic time for the interaction between the bubbles depends on the reciprocal distance 1/ Al multiplied by the sound speed a of the liquid . Only for very small bubble distances the linear dependence is no longer valid.

a.) t = 0.7 msec b.) t = 0.9 msec c.) t = 1.0 msec d.) t = 1.5 msec

Fig. 6a-d : Collapse and generation of secondary nuclei ( water, ambient temperature)

In experiments it is very difficult to influence the number density of the nuclei in the liquid . For the case of ultrasonic cavitation Kedrinskii (1987) proposed a multiplication mechanism of cavitation nuclei by bubble collapse which is also used in the experiments performed here. It is based on the radial symmetry of a cavitation bubble being unstable during its collapse (e.g. Plesset and Prosperetti 1977). The contraction of the bubble causes a dynamic instability of the bubble wall. During the collapse separate fragments are formed at the wall which are considered to become new cavitation centres. High speed pictures

346

(Fig. 6) obtained by a Cranz - Schardin technique support the hypothesis of Kedrinskii . In

Fig . 6a a spherical cavity can be seen . The collapse of such a cavity inside a bubble field

is not influenced by a geometric effect of a solid wall or an incident shock wave, it is only

stimulated by a small pressure disturbance. Fig. 6b shows the bubble surface oscillation

(Hentschel and Lauterborn 1984). The oscillation becomes nonlinear with a large amplitude

(Fig. 6c) . Small bubbles are separated from the bubble by the surface tension which cuts

them off. A large number of bubbles is produced (Fig . 6d). By a fast repetition of the

experiments an increase of the nuclei density in the liquid is possible .

In the cavitation tube there exist no bubble oscillations due to ultrasound (Crum 1984)

but the experiments can be repeated with a maximum frequency of 0 .3 Hz. This repetition

creates new nuclei of similar size and of comparable gas content . The number of experiments

done is thus a measure for the nuclei density. This number is called repetition number n,'ep

here . In the experiments the nuclei density is changed via the parameter n,'ep. This

happens as follows. First the liqu id rests for at least ten minutes. Then the experiment

is repeated n ,'ep - times. The mesurements are carried out during the last (i .e . n,'ep­

th ) repetition only. In th is manner the nuclei density which is present right before an

experimental measurement is started, is changed by varying n ,'ep'

3 .2 Pressure inside the Cavitation Cluster

Inside the bubble field pressure oscillations are present which depend on the nuclei

density. These oscillations were predicted by Rein and Meier (1988) and are confirmed by

experimental evidence . A numerica l model of the cavitation in the tube permits a qualitative

comparison between experiment and numerical simulation.

1,---~----------------~ CV p

(bar)

5

3

-1

-3+------------,..,.-,........< o 2 3 t(ms) 4

1.---------------------------. p (bar)

5

3

-1

~~----~------~--------~~ o 2 3 t(ms ) 4

Fig. 7a-b : Dependence of the pressure signa l on the repetition number n,'ep'

All pressure measurements are made at that position where the bubble field exists for

347

the longest time. This place is 3 cm below the piston. A typical pressure measurement for

the case of nrep = 2, is shown in Fig. 7a. It exhibits all the patterns described in Fig. 4.

When the piston arrives at its stop a pressure pulse is generated, increasing the pressure

from CDto @ At the open end of the tube the compression wave is reflected as an expansion

wave thereby reducing the pressure to the ambient pressure 0). After arriving at the piston

the expansion wave is reflected without phase reversal as a tension wave. Behind the tension

wave pressure oscillations occur Q. These oscillations have a constant frequency and an

exponentially damped amplitude @.

Behind the tension wave the pressure decreases far below the vapour pressure and the

microbubbles are forced to grow by several orders of magnitude. The growing volume of

the bubbles causes a flow velocity which is directed out of the tube and a fast pressure rise.

With rising pressure the stability of the bubbles changes. The bubble growth is decelerated

and finally the bubbles begin to shrink. At the same time new tensions are generated in

the liquid due to the velocity being directed out of the tube. These tensions cause a new

bubble growth and a new cycle of the oscillations starts. - The reflections of the initial

pressure wave which is running back and forth between the open end of the tube and the

cavitation cluster (d. Fig. 4), decomposite the bubble field. By this process the direction

of the velocity is reversed. As soon as the bubble field is decomposed, again an impact of

the liquid column takes place at the closed end of the tube and a second pressure pulse is

generated (Fig 7a: @). In the experiments the nuclei density is not known quantitatively but can be varied qual­

itatively by changing the repetition number. The nuclei density affects both, the amplitude

and the frequency of the oscillations. This can well be seen by comparing Fig. 7a and Fig.

7b where the density was increased in the case of Fig. 7b by performing the measurement at

a higher repetition number. In the case of the higher nuclei density (Fig. 7b) the pressure

oscillations disappear. This is in agreement with numerical simulations (Rein and Meier

199Gb) which have shown that high nuclei densities cause pressure oscillations of very small

amplitude and high frequency, such that the oscillations virtually disappear. Using a model

describing the interaction of two single bubbles Meier (1987) derived a relation between the

frequency v, the amplitude pose and the interbubble distance Ai:

v= (4)

The experimental results are well reproduced by this formula. It can be checked by taking

the values of PO"e and Ai, and then calculating the frequency v. Keeping in mind that

Pose = posc(n) , a qualitative good correspondence can be obtained.

In the following the experimental and numerical results are compared. In all cases the

amplitude Pini of the initial pressure pulse is Pil1i = 5bar. The parameter changed is

the nuclei density. In the numerical calculation it is prescribed via the initial conditions

and remains constant during the calculation. In the experiment it is varied qualitatively by

changing the repetition number. In Fig. 8 a plot of the oscillation amplitude Po .• e versus the

oscillation frequency v is shown. The parameter of the curves is the nuclei density which

increases with the frequency. For high frequencies the amplitude approaches the value

pose = obar in the theoretical as well as in the experimental case. For low frequencies

348

both curves approach the value p ose = 4bar which is the value that is obtained when no breaking of the liquid occurs.

]lose/Po 5~--------------______________ ~

11;1/.; = 5ba:r

n u merica I ca Icu lation

25 50 75 v[kHz]

Fig. 8 : Comparison of experimentally and numerically observed pressure oscillations.

Although the theoretical model is based on some idealizing assumptions such as all bub­bles are assumed to have the same initial radius , a qualtitative agreement of the experimental and the numerical curve in Fig. 8 is obvious.

3.3 Bubble selection in a cluster.

In section 3.2 it became apparent that for similar behaviou r of cavitation behind tension waves a main question is whether the number density of the nucleis remains constant . By performing experiments in water using high repetition numbers n 7•ep > > 20 it was possible to observe bubble dynamics for a high nuclei density. The stability of a cavitation bubble in a cluster depends on different factors . For example, bubbles generated out of nuclei with a high gas content are more stable than those with a low gas content. The disappearance of a bubble thus means that the bubble collapses to a size comparable to the init ial nuclei size. These dynamic effects are very difficu lt to deal with analytically, however static considerations give a qualitative explanation for differences in stabil ity.

In Fig. 9 the pressure in the liquid is plotted against the corresponding eq uil ibrium radius of a cavitation bubble. The dashed curve represents a gas bubble and the solid cu rve a vapour bubble. Cavitation bubbles in a cluster are very sensitive to differences in the radius . In the case of a constant pressure, all bubbles with a radius smaller than the equil ibrium radius are unstable and collapse. For gas bubbles this only holds as long as its equilibrium rad ius is greater than the radius rk (d. Fig . 9) . This mechanism, in addition to stabil ity differences due to a different amount of gas/vapour contained in the bubbles,

349

results in a bubble selection . The selection of the more stable bubbles takes place by a

pressure driven flow of liquid between the bubbles, which causes the collapse of the less

stable bubbles.

In a first order approximation the characteristic time of such a selection process is given

by the pressure difference ap and the flow velocity u . Using Bernoulli 's equation the

following formula has been derived for the characteristic time of the selection process by

Meier (1987):

al ~R2 t=-=al --u 4<TaR'

(5)

where al represents the bubble distance, <T the surface tension, p the density of the liquid

and ~i the relativ radius difference . For example, assuming Ai = l.o,R = O.Olcm and

al = O.lcm yields a typical selection time of t = O.6ms.

o~~--~~--------~ r

Fig. 9 : Radial bubble motion in equilibrium.

(after Meier 1987)

Fig . 10 : Dynamic nuclei density

variation .

Image processing of 600 high speed photographs yields a detailed history of the develop­

ment of the nuclei density in time (Fig. 10, see also Fig. 5). Immediately after the passage

of the tension wave a huge number of bubbles is expanded. After the first expansion there

is a rapid decrease in the nuclei density nand n reaches a minimum. The time needed for

this nuclei selection is about O.5ms (Fig. 10) which is nearly the time predicted by equation

(5) for a typical example. Before the decomposition of the bubble field starts a rebound

of the number density takes place (Bode, Laake and Meier 1987) . The nuclei expansion

affects the increase of the pressure, the stability of the bubbles is shifted, and a selection

process takes place. Under the assumption of a uniform gas content of the bubbles the less

stable, namely the smaller bubbles collapse and the larger nuclei reach their maximum size.

In a tension generated bubble field the nuclei density thus changes nonlinearly and cannot

be assumed to be constant .

350

4. Conclusions

Several aspects of cavitation behind strong tension waves have been investigated. In experiments tension waves were produced in a tube using the reflection principle. The nuclei density which is of greatest importance for the cavitation event, has been changed by

taking advantage of the dynamic instability of the bubble surface during a collapse. Behind a tension wave a cluster of cavitation bubbles is formed and the tensions are virtually annihilated. Inside the bubble cluster pressure oscillations have been observed. It turned out that the frequency and amplitude of these oscillations depend very much on the nuclei

density. This dependence is in a good qualitative agreement with results obtained from a numerical simulation of this event. Further, a relation, obtained earlier by Meier (1987) which holds between the frequency and amplitude of the oscillation, and the interbubble

distance (which depends on the nuclei density) was verified by the experimental results. An investigation of the time history of the bubble density in a cluster revealed that there exists a selection process among the bubbles. Only some of the bubbles increase in size in the

long run. The characteristic time of this selection process is in excellent agreement with a value obtained using a model of Meier (1987) for the interaction of two individual bubbles.

5. References

Bode, J., Laake, A. and Meier, G.E.A. (1987) Daga' 87

Bode, J., Meier, G.E.A. and Rein, M. (1990) ZAMM (accepted for publication) Brown, S.J. (1967) M. S. Thesis, Pennsylvania State University

Couzens, R.C.F. and Trevena, D.H. (1974) J. Phys. D: Appl. Phys. 7,2277-2287

Crum, L.A. (1984) Ultrasonics 22, 5, 215-223 Favreau, C. (1984) Revue Phys. Appl. 19,951-961

Hansson, J., Kedrinskii, V.K. and Mrerch, K.A. (1982) J. Phys. D.: Appl. Phys. 15, 1725

- 1734

Hentschel, W. and Lauterborn, W. (1984) DAGA' 84, 469-472 Hill, L.G. and Sturtevant, B. (1989) (Proc.: IUTAM Symposium Adiabatic Waves in Liquid - Vapour System) Joukowski, N.E. (1898) Ac. Imp. des Sc. St Petersburg, Vol. 9 Nr. 5

Kedrinskii, V.K. (1987) Proc. ICA 12, J4-8 Kedrinskii, V.K. (1978) J. Appl. Mech.and Tech. Phys. Vol. 19

Kojima, E., Shinada, M. and Shindo K. (1984) Bull. of JSME, Vol. 27, No. 233, 2421 -

2429

Meier, G.E.A. (1987) MPI fur Stromungsforschung Ber. (7/1987) Gottingen

Plesset, M.S. and Prosperetti, A. (1977) Ann. Rev. Fluid Mech., 9,145 - 185

Rein, M. and Meier, G.E.A. (1988) In: Furuya, O. (ed.) Cavitation and Multiphase Flow

Forum - 1988. New York, pp.40 - 44.

Rein, M. and Meier, G.E.A. (1990a) Acustica (publication scheduled for Vol. 10, part 4)

Rein, M. and Meier, G.E.A. (1990b) to be published

Szumowski, A. (1989) private communication

Streeter, V.L. and Wylie, E.B. (1974) Ann. Rev. Fluid Mech. 6,57-73

Trevena, D.H. (1984) J. Phys. D.: Appl. Phys. 17,2139-2164

Acoustics of Travelling Bubble Cavitation

J. Buist

Department of Mechanical Engineering University Twente, The Netherlands

Summary. Experiments on cavitation prove that the noise level is highly dependent on the type of cavitation. Usually it is expected that bubble cavitation is much less severe than cloud cavitation. To verify this expectation, noise measurements on a bubble stream over a hydrofoil were done at Marin (Wageningen ,The Netherlands). These experiments show that there is not much difference irrespective of the fact whether the bubbly flow is clustering or not. In this paper the noise level produced by a macroscopically steady bubbly layer on a hydrofoil is predicted analytically, based on the stochastic properties of the fluctuating quantities in the layer. This analysis uses the technique of Fourier-Stieltjes transformation as used by a.M. Phillips. In this way, the sound spectrum outside the bubble layer can be correlated to the covariance of the fluctuating quantities, as gasfraction, velocity etc. ,assuming for instance that the bubble layer is stochastically stationairy and almost homogeneous. Based on this theory an estimate can be made of the relative importance of cluster formation, in dealing with the sound production of bubble layers.

I nt roduct ion.

Cavitation often occurs in hydrodynamical equipment. Our research is

concerned about cavitation near ship propellors. Here cavitat ion causes

erosion, lift oscillations and sound. The purpose of our research is

eventually to find scaling rules for the sound emission of propellor

models to full scale noise. Several types of cavitation can be

distinguished, see for example Knapp, Daily and Hammitt 1]. The

occurence of those types depends on, for example, the cavitation number,

free stream velocity and the angle of incidence. Here a division as

scetched in figure 1 will be given.

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Giittingen, Germany, 1989 © Springer-Verlag Berlin Heidelberg 1990

352

Fig. 1. Several types of cavitation.

These types are:- Bubble cavitation.

- Sheet cavitation.

- Sheet-cloud cavitation.

- Vortex cavitation.

It is expected from experimental research that sheet-cloud cavitation

causes most of the sound production by a cavitating ship propellor.

Therefore Omta [ 2J analysed the sound production of a spherical bubble

cloud analitycally. At MARIN the results of Omta where verified in a

water tunnel [ 3J. In these experiments the sound radiation of a bubble

stream, injected into the fluid near a two dimensional hydrofoil, was

measured. These bubbles where injected both intermitted and continouosly,

in order to distinguish a cloudy bubble stream and a continuous bubble

stream. The remarkable result of this research was that no difference

between the two, above mentioned, situations was observed.

In that connection we address in this paper the question of the intensity

of sound emitted by a stochastically stationary bubble stream.

Stochastic Model.

To model the above mentioned situation, an arbitrary hydrofoil will be

expanded in a Fourier-sine series. For one Fourier mode the problem can

be sketched as in figure 2 . Suppose a wavy-wall, l)l(X), with a thin

bubbly layer on it, whereby l)l(X) is defined in the complex notation as:

l)l(x) = c exp(ikx) and ck « 1. Outside the bubbly layer, in the far

- - --o 0

o g o 0 00

x

o

T) (x l 1

Fig. 2 The stochastic bubbly layer

353

field, an uniform flow U~ of pure water exists. The interface between the

layer and the outer flow is represented by 1)2(x, t). It is assumed that

the bubbly layer is stochastically stationary, which means that all

quantities, averaged in a sample volume, are independent on time. The

sound radiation into the outer flow is caused by stochastic fluctuations.

These fluctuations can either be in gasfraction, bubble radius or

pressure. The stationary solution of this problem can be derived by using

the two-phase flow equations of Biesheuvel and Van Wijngaarden [ 4] and

some proper assumptions on the behavior of the bubbly layer, see Acosta,

Brennen and Agostino [ 5] and Buist [ 6].

The propagation of sound in the outer flow will be described by using the

two dimensional wave equation. And the sound intensity outside the bubbly

layer, defined as,

1 = py. (1)

will be related to the covariance of one of the fluctuating ' quantities in

the mixture, using proper boundary condition on the interface 1)2(x,t).

Fourier-Stieltles Transform.

In this paper the sound intensity will be defined in a stochastic way

using the technique of O.M. Phillips [ 7] . Consider an arbitrary varying

function in both space and time, f(x,t). The joint probability function

P(f1 ,f2, · · ,fn ) can be defined such, that

354

represents the probability, that f(x,t) at point (X1,t1), .. (xn,tn) should

all lie within assigned limits f1,f1+df1 etc. The mean part of the random

function is defined in such way, that

f r

'" J

f P(f ) df r r r

-'" equals zero for all r. The second moment of the probability density

P(f1,f2 ) is defined as

Z(x,r,t,'r) JJ"'f1f 2 P(f1,f2 ) df1df2

-'"

f(x,t)f(x+r,t+T)

in which the points 1,2 are taken as (x,t),(x+r,t+T). The wave spectrum

is the Fourier transform of the covariance Z(r,T).

lI(K,W) -iwT -iKr e e drdT

the integration being over the horizontal r-plane and over all time

intervals, If the random function can be considered as a stationary

random function of time and homogeneous in space, then it can be

represented as a Fourier-Stieltjes integral

f(x,t) II dF(w,K)

W,K

iwt iKX e e ( 2)

where the integration is over all wave-number frequency space. This way

the spectrum can be defined in terms of this Fourier-Stieltjes components

. , , dF(K,w)dF (K ,w ) o if K $ K , W $ W ( 3)

lI(K,w)dKdw K,W = K ,w

where the asterisk denotes a complex conjugate. In the following all

covariances, for example in ~2' a etc., will be defined in this way.

Definition of the Sound Spectrum,

The radiation of sound in the far field can be described by the two

355

dimensional wave equation. For convenience this equation will be

transformed to a frame moving with the free stream velocity Uoo '

+ o. ( 4)

The sound intensity perpendicular to the the wavy wall can be written in

terms of the potential ~ with use of Bernoulli's equation as,

a~ a~ I(r,y,.) = PI --(X,y,t)--(X+r,y,t+.).

at ay

The frequency spectrum can be defined in terms of the Fourier-Stieltjes

components as, see [ 6]

foo /2 2 2 ' • I(y,w) = -iw VK -w Ic dV(K,y,w)dV (K,y,W) dK ( 5)

-00

After Fourier-Stieltjes transformation the general solution for the

different Fourier components dV(K,y,w) of~, is

/2 2 2' /2 2 2' dV(K,y,w) = C(K,w)e-y VK -w Ic + D(K,w)eY VK -w Ic

As boundary condition at infinity in the subsonic area, K2 > w2 /c2 , can

be taken that the solution must be finite. In the supersonic region,

K2 < w2/c 2 , the condition states that no energy can propagate towards the

wall. Both conditions imply that D(K,W) equals zero. As boundary

condition at the interface, ~2(X,t), will be taken

a~1 ay y=H

Resul t ing in:

(w + U K)3

I(w, K) 00 ---;.=======; IT (K, w+U K).

/ 2 2 2' TI 00

w - K Ic

( 6)

Here IT is the covariance of the fluctuations in the interface, defined ~

in the fixed frame. If it is assumed that Uoo «c, then in the far field,

where the components from the subsonic region vanish, the terms with UooK

356

can be ignored and as integration interval in equation C 5) can be taken

{-wlc, wlc}.

The Rayleigh-Plesset equation.

The covariance of the interface can be related to the other quantities in

the mixture by using the dynamic boundary condition on the interface.

This condition gives for the Fourier-Stieltjes component of the

fluctuation in the pressure of the mixture,

P CK,y=H,w) = -iwp dVCK,y=H,w). m 1

In the next part of the paper the covariance in gasfract ion wi 11 be

discussed. The gasfraction is defined as,

a(x,y,t) = n(x,y,t)4n/3 R(x,y,t)3 ( 7)

where n(x,y,t) means the bubble number density and R is the radius of the

bubbles. In this report only monopole sound will be considered, so only

fluctuations in the bubble radius will be taken into account. Further the

mass of one bubble, ap, remains constant, where p represents the 9 9

density of the gas in the bubbles. We shall assume the isothermal

relation,

P Ip = <P >I<p > + second order terms. 9 9 9 9

The relative motion between the phases can be ignored. If the fluctuation

in gasfraction is defined as

a(x,y,t) = <a(x,y» + a'Cx,y,t),

then the relation between the pressure, P , and the gasfraction is m

P = <P > __ P R2 _ <a> {[ a ]1/3 m gal 0 <a>

d2

[a ]1/3 3 [d [a ]1/3]2} dt2 <a> -; dt <a> . .

( 8)

Linearisation of this equation about <a> and using the definition of the

spectrum in the Fourier components gives,

n (IC,W) 11

[ 2]2 W 222 1 - _ W Ic -IC n (IC W)

2 4 a' W W

b

357

( 9)

where wb is the bubble resonance frequency for a single bubble at

isothermal conditions in an unbounded liquid,

The covariance in the gasfraction.

The covariance in the gasfractlon is defined as,

a' (x, y, t )a' (x', y' ,t')

In this equation the gasfraction in point (x,y,t) is correlated to the

gasfraction in a point (x', y' ,t'). A condition for appl ication of the

Fourier-Stieltjes transform was, that the wave field must be homogeneous.

This means that all probability-densities are invariant under the

addition of a constant vector to all space points. Strictly speaking this

condition is not fulfilled for a wavy wall. But when the length scale on

which the mean quantities change, is small in respect with the variation

of Z in the separation variable r, then the wave field can be assumed to

be almost homogeneous. A second condition on the transformation yields

the stationarity of the wave field, so the second moment function will be

only dependent on the separation variable T. This condition is satisfied

when the mean quantities are independent in time. In this way the

frequency, wave-number spectrum can be written as,

Q)

n (K,W) = IT R (T,r.) e e drdT. I 2 -iWT -ilCr a s d

-Q)

Where IT is the standard deviation of the gasfraction a, or s

and Rd(T,r) is the correlation coefficient defined as,

" 2 Rd(T,r) = <a (x,y,t)a (x+r,y+r,t+T»/lTs .

The correlation coefficient must satisfy the following two necessary

358

requirements:

Rd(O,O) = 1

R d (-t"±IXI, r±lXI) -> 0

This way the correlation coefficient takes the form of

for a gaussian process, where r m

is the correlation length and liT the m

cut-off frequency, which will be determined later. Substituting this

covariance in a into the frequency spectrum gives:

4>(w) 2 <P >2 [ w2] 2 rW/Cj 2 2 2' 2 2 2 2 _---=m=__ 1 _ (lr T W - Ie C e -W T m e -Ie r m die cp <a> 2 w2 s m m W

1 b 0

(10)

Figure 3a gives a sound spectrum of a cavitating ship propel lor based on

consideration of bubble oscillation, see van Wijngaarden [ 81.

120

I()Q

\"

~r -------------.Iv

10 10' ,

10

~ _2

~ ~

Fig 3a Usua.l expected spectrum

dB/ /J,lz ~

~ol~ .lb

, I I I

''LoO 101 102 10' 104

Frequency

3b Spectrum from equation (10)

In the low frequency region the spectrum behaves like w4 , due to volume

oscillation and in the high frequency region a behaviour like l/w2 •

Figure 3b gives a spectrum from equation (10) which shows a dependency

proportional to w.

Estimates of the correlation scales.

To make an estimate of the different correlation scales and also the

359

standard deviation, something must be known about the way two-phase flows

have been modelled . Each quantity characterising the flow will be

averaged. Therefore Biesheuvel and van Wijngaarden make a distinction in

different length-scales. The meso-scale 1, see figure 4, is defined as

the scale on wich the averages takes place.

Fig 4 A sample of the bubble mixture (meso-scale , 1)

Due to the requirement of homogeneity the correlation length must be no

longer than this mesoscale. The mesoscale will be a few bubble distances

a, which is proportional to the n- 1/ 3 and can be determined with the

definition of 0:.

So a good estimate of r will be,

r m

= 10 R / <0:>1/3 o

m

The break-off frequency can be estimated by demanding that the pressure

changes must be slow enough for the bubble oscillation to react on it . A

suitable time-scale therefore is the bubble distance devided by the sound

speed in the mixture ,

m R / «0:>1/3 ) o c m •

where c is the sound speed of a bubble mixture in the absence of m

dispersive effects [ 81,

Finally last the standard deviation will be estimated by assuming that it

360

is half of the mean gasfraction. The sound spectrum gives for large

enough I" and T a maximum of m m

I max

For <P> = 105 p&, <R>

approximately 90db.

Discussion.

w .sw b

O.OOlm and <ex> 0.1 this gives a sound level of

The sound level given in the previous section can be regarded as an upper

limit of stochastic bubble sound. In respect to noise levels from

cavitating ship propellers of aproximately 150 dB and flow noise of

40 dB, it can be concluded that this noise production I ies somewhere in

between the two forementioned ones.

Acknowledgment.

This research is supported by the foundation S. T. W. under contract TTN 77.1308 and suported by the "Maritiem Research Instituut Nederland" within the scope of a cooperative project on scaling cavitation noise.

References.

1. Knapp, R.T.; Daily J.W.; Hammitt F.G.: Cavitation. McGraw-Hill Book Company 1970

2. Omta, 0.: Oscillations of a cloud of bubbles of small and not so small amplitude. J. Acoust. Soc. Am. 82 (1987) 1018-1033.

3. van del" Kooij, J.: Geluidsproduktie door luchtbellen in water bij passage langs een vleugelprofiel. MARIN Rapport No. 45978-1-VT (1985) In Dutch.

4. Bi esheuve I , A.; van Wijngaarden, L.: Two phase flow equations for a dilute dispersion of gas bubbles in liquids. J. Fluid. Mech. 148, (1984) 301-318.

5. Agostino, L.; Brennen, C.E.; Acosta A.J.: Linearised dynamics of two dimensional bubble and cavitating flows over slender surfaces. J. Fluid. Mech. 192 (1988) 485-511.

6. Buist, J.; Sound production of a stochastically stationary bubbly layer. Forthcoming PH. D. thesis.

7. Phillips, O.M.: The dynamics of the upper ocean, Cambrigde University Press 1966

8. Wijngaarden, L.; Omta, R.; Hydrodynamics of cavitation nOise, THD Schriftenreihe Wissenschaft und Technic 28 (1986) 219-226

Modeling of Shock-Wave Loading of Liquid Volumes

N.N.CHERNOBAEV

Lavrentyev Institute of HydredYflamicS Siberian Division of the USSR Academy of Science Novosibirsk, 630090, USSR

Summary

Cavitation evolution dynamics in cylindrical liquid volumes under the axial loading by an exploding wire is studied experi­mentally and theoretically. The method of dynamic head regist­ration is used to study the structure of two phase flows for­med and evaluate characteristic time of cavitation liquid frac­ture. As a result of numerical simulation of the experiments, which was performed in a single-velocity two-phase model appro­ximation, the energy transformation mechanism is determined at shock interaction with a free real liquid surface. A two-phase model is suggested to describe the irreversible development of a cavitation zone formed as a result of the mentioned interac­tion. The model is based on practically instantaneous tensile­-stress relaxation in a centered rarefaction wave and further inertial evolution of the process.

Introduction

The problem of dynamic strength of a liquid was considered in

a number of works, where pulse methods [1-6] were used. The am­

plitude of maximum tensile stresses achieved in liquid depends,

as it shown in [2,3], on the parameters of rarefaction wave

(RW) and on the parameters of initial gas-containing of liquid.

The fast growth of cavitation nuclei in RW leading to the re­

laxation of tensile stresses in liquid in a time of order of

10-6 s [2-4]. Further two ways of the cavitation process deve­

lopment are possible: (i) the bubble damped oscillations occur

and (ii) the irreversible development of cavitation zone take

place [5,6], which leads to the formation of foaming structure

and the liquid fracture into discret particles. The main prin­

ciple structure peculiarities of process in the second case re­

main vague. The structure of flows which forms an axial explo-

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Gotlingen, Germany, 1989 © Springer-Verlag Berlin Heidelberg 1990

362

sive loading of liquid cylinders followed by the irreversible

development of cavitation zone is studied experimentally and

theoretically in this paper.

Experiment and results

The experiments were carried out according to the technique sche­

matically shown in Fig. 1. The shock wave (SW) was initiated by

Fig. 1. The scheme of experimental setup.

exploding wire-1 placed along axis cylindric liquid volume-2,

limited by thin wetted paper shell-3. The liquid cylinder

was fixed rigidly on faces by the acrylic plastic plates-4,

through which the optic vizualization by high-speed photoregis­

ter-5, was carried out. The experiments were carried out using

settled distilled water. The liquid cylinder radius ro =2cm,

and its length is 3 cm. In this case the loading is estimated

to be quasi- one-dimensional. It is shown in [5] that the tran­

sition to irreversible development of cavitation zone is obser­

ved at fixed duration of positive phase of the pressure in SW

(near 2 ro ), when the SW amplitude increases. The picture of

the process, corresponding to the given transition (the energy

in capacitor bank Eo = 145 J, the energy release time 1: =

= 4ps) is shown in Fig. 2. The corresponding oscillogram of

the pressure changing in SW - P = P (YO, t) registered by the

pressure piezo gauge in "unlimited" plane channel with the width

3 cm, filled by water, is shown in Fig. 3. It is seen from the

photography analysis in Fig. 2 that the cavitation qevelops

over the whole liquid volume after SW interaction with its

free surface. Then the visible cavitation disappears near the

internal boundary r, . The velocity of external boundary r2 being defined according the photography, within the initial

interval of time I:l t near 100 ps is constant. It is equal

363

Fig. 2. The process photograph.

P,atm

400

200

o 20

Fig. 3. Pressure in SW (p=p(ro,t) in unlimited liquid. 1 is the expe­riment; 2 is the numerical calculation.

to the doubled mass velocity beyond the front of initial SW

within the limits of measurement errors.

For the purpose of the fine structure investigation of two-pha­

se flow forming at the explosion, the method based on the coinc·i­

dence of optical visualization and the registration of dynamic

head pressure is suggested. The pressure was measured by pin

pressure gauge with the body tube diameter of 4 mm and the pie­zoelement diameter of 1 mm, is mounted along the normal to the

flow (see Fig. 2), and is equidistant from the lateral bounda­

ries of liquid cylinder. The pressure P(t) = J3 ~ (t) U 2 (t) was registered, where fo is the coefficent constant for the

gauge of given fonn g is the mixture densi ty , u. is the

mixture mass velocity. While placing the gauge at different dis­

tance ~ from the initial free surface and registrating pet)

364

the main stages o~ the process development in time may be ~ol­

lowed qualitatively.

The results o~ measurements are shown in Fig. 4. The registered

signal consists o~ two pulses 1, 2 when 5 = 0 in Fig. 4. The

P,atm

3~---~--""'~

5,cm

/

500

/

t,~s / \000

/

Fig. 4. The pressure registration results o~ dynamic head o~ gas-liquid ~low

signal structure is clear ~rom photography analysis in Fig. 2:

~irstly the cavitation ~low, corresponding to the ~irst pulse

leaks in on the gauge at point ro ( ~ = 0), and then the internal

liquid layer corresponding to the second pulse links in at the

same point. Thus, the presence o~ the second pressure pulse is

explained by the density jump in the flow. The dynamics o~ the

dispersion process o~ gas-liquid ring is such that the time in­

terval II t betweet; pulse ~ronts increases wi th ~ increasing

(see Fig.4): llt = 100 ps ( 5' = 0), At = 150 ps ( 6' = 0.5 cm), At = 180 )ls ( 6" = 1 cm). It shows that the ve­

locity o~ the ~low external part is higher than that o~ the in­

ternal part, i.e. the cavitation zone development occurs. In

this case the mixture density over the cavitation zone decreases a nd

in time that leads to the decreasing o~ registered pressure of

dynamic head over the given region. In the time moment t* z 150+ 200 Jls ( li = 1 cm) the mixture density over cavitation

zone decreases so that the pressure registrated in the first pulse

approaches to zero. The given e~fect is connected with the li­

quid cavitation destruction. The internal layer of liquid cor­

responding to the second pulse in Fig. 4 is destructed due to

365

the instability development at its boundaries which is-shown

in Fig. 2 in the view of "broadening" of the boundary r, in

time. This fact is confirmed by the absence oL a sharp jump in

pressure on the second pulse profile when the internal bounda­

ry of liquid ring crosses the gauge. The destruction by this

mechanism occurs slower than by cavi.tation: the second pulse

of pressure decreases in amplitude and changes its. form sharp­

ly at 700 + 800 .fs ( 15' = 3 cm).

Thus, it is shown that the analysis of fine structure of pres­

sure profile of dynamic head allows to registrate the characte­

ristic peculiarities of the flow and destruction times of li­

quid volume at the pulse loading. This method allows also to

registrate the dynamics of reversible development of cavitation

zone [6].

Computer Simulation and Results

The computer simulation of the experiments in the frame work of

two-phase single-velocity model [7] has been carried out in or­

der to clear up the mechanism leading to the formation and ir­

reversible development of the cavitation zone. The one-dimensio­

nal problem was solved in the cylindrical coordinate system (r, e ,Z ). The impulsive energy release occurs in time moment t=O

along the axis Z . When t > 0 three areas are singled out in

the problem: .Q. l are the explosive products (0 < r < rl ), .0.2

is the two-phase gas-liquid medium ( rl < r < r2. ), n 3 is

the air (r2 < r ), where If is the boundary coordinate of the

explosive bubble, r2 is the coordinate of free surface. The

flow over area (12 was calculated. The calculations were car­

ried out in mass lagrangian variables using the introducing of

artificial viscous pressure [10]. The system of equations [7]

has the form:

VOr acp+ q.) ~ a~ (1 )

m P = B[ (Vo/V) - i] + Pc, I

366

where r, ~ are the Euler and Lagrangian coordinates, U is

the mass velocity; Vo I V are the initial and present speci~ic

volumes o~ liquid, V is the speci~ic volume o~ gas-liquid mix­

ture; Vs 'N are the volume and number o~ cavitation nuc­

lei per uni t o~ liquid mass respectively; Ro,R are the initial and

present radii o~ bubble; Po ,p are the initial and present

pressures in liquid; t is the time; q. is the term o~ arti~i­

cia1 viscous pressure which is matched so that the spreading

of SW front occurs in some cells of the network. In RW, where

aU/a~ > 0, q = 0 was assumed in the calculations. In the

Tait equation of the state B = 3045 atm, m = 7.15 for water. At

the stage of SW propagation over the liquid with cavitation

nuclei the bubble radius was assumed to change according to the

adiabatic dependence:

i R = Ro (pi Po) - ~ (2)

where ~ is the gas isentropic exponent. The pressure nonequi-

1ibrium in different phases was considered over the flow area

beyond the RW ~rom the moment the tensile stresses appear

in liquid ( P < Po ). The bubble dynamics was determined in

the given area ~rom the Rayleigh equation solution:

where 6 is the coe~~i.cient o~ surface tension, Y is the

coefficient of kinematic viscosity of liquid. Eqautions (1)-(3)

form the close system for the calculation of two-phase flow

which may be used under the condition R« e« L ,where f is the d·istance between the bubbles, L is the averaging scale.

The solution o~ system (1)-(3) was determined over the area !22

with the initial conditions at t = 0: U(r) = 0, P (r) -= Po • Vcr) = Va , r(~) = ~ . The bubb.le distribution in size was believed to be monodisperse with size Ro = 1.S.10-4 cm [8].

The magnitude N = 105 l/g is constant [9J. (5 = 72 erg/cm, 2 V = 0.001 cm Is, ~ = 4/3 were considered in the calcula-

tions for water. The motion velocity of the internal boundary

367

was determined experimentally by the method of slit photoscan­

ning. According to discrete quality of the points measured ex­

perimentally ~(tL) the splain interpolation was carried out

with the following use of restored function as a boundary con­

di tion of the domain Sl2. . It was believed that PCr;J::-R, when

t > 0, where Po = 1 atm. The equation (3) was solved after

appearance of the tensile stress P < Po in liquid at the mo-

ment when t::; to with the ini tial condi tions

where U was determined from (1) when t:: to' The

calculation of the process is limited by time of RW approach to

explosive bubble as far as the boundary rJ looses its stabili­

ty later on.

The calculation of integral energies was carried out in the

problem per unit length along the axis Z: E i is the kinetic

energy, E 2 is the elastic energy, E 3 is the energy connected

with the cavitation bubbles and containing gas internal energy

of the bubbles, kinetic energy of their radial pulsations, free

surface energy.

12 - 2

E{ = j ~u 211Tdr

II (4)

The flow calculation was carried out by finite-differential

scheme, sugge.sted in [10] and the experiment was modulated

on liquid cylinder loading ro = 2 cm ( Eo = 145 J, 'l: = 4 Jls) •

The comparison of change of pressure measured experimentally in

SW with calculated one was carried out. The close values of

two curves in Fig. 2b allow to speak about the computer simula­

tion of loading process.

The dynamics of wave process over domain n 2 is shown in Fig.

5a, b. The RW (2,3) in Fig. 5a spreads in liquid after SW for­

mation (1) and its interaction with free surface. Due to the

cavitation development a relaxation of tensile stresses occurs

368

P,atm

400

200

II "L-.o..--...... --.... ro

o 0.5 r/ro Fig. 5a. The pressure change

over the domain.o2: 1 - t= 12 ps; 2 -t = 18 ps; 3 - t = = 24}J.s

5.0

30

IO~O------~~----~+-~~ 0.5 I. r/IO

Fig. 5b. The velocity dis­tribution over the domain 0. 2 i-t = 12 ps; 2 - t = 18 ps; 3 - t 24 ps

in RW as far as the compressibility of two-phase medium is de­

termined in this case by gas component compressibility. The am­

plitude of fine-scale pulsations of pressure in RW, connected

with inertia joint to the bubbles of liquid mass, is much less

than the pressure in initial SW. In this case the velocity pro­

file shown in Fig. 5b is formed in gas-liquid fluid. The dist­

ribution of mass velocity beyond RW is seen to be stationary.

The velocity stability follows from the equation of pulse con­

servation (1), as far as ap/a~ > 0, over the area under con­

sideration, U(t) ~ const (~) • The velocity of free surface

coincides with the experimentally measured one 67~3 m/s.

The time-dependent balance of integral energy of liquid volume,

calculated according to 4), is shown in Fig. 6, where t i

8 E, J/cm

4 £2 I

EI I I t,J.ls

0 10 t. 20 til.

Fig. 6. The dynamics of integral energy change in gas-liquid ring

369

is the moment of SW emergence on free surface; t2 is the mo­

ment of RW approach to an explosive bubble. The value of ener­

gy E 3 connected with cavitation bubbles is much less than E 1

For comparison E,(t 2) = 6.5 J/cm, E3 (t 2 ) = 6.10-3 J/cm.

Thus, the elastic energy of a liquid in SW converts into kine­

tic in RW when tl < t < t2. .If in the initial SW ap/ar> 0,

div 11 > 0 over the flow domain beyond the RW. The latter cQn­

dition due to velocity quasistationarity corresponds to the

inertial tension of two-phase medium.

Taking into account the given pecularities one may suggest a

two phase model for description of irreversible development of

cavitation zone which occurs experimentally. The SW spreading

over real liquid is calculated as an approximation of ideal

compressed liquid due to the low initial concentration of ga­

seQUS phase. When in the RW the pressure of cavitation thres­

hold Pc [1] is achieved its instantaneous .relaxation occurs

up to the magnitude P'II" (the pressure of saturation vapor).

After that the medium is considered to have no strength, and

the change of mean density of porous medium occurs in accordan­

ce with the equation of continuity-of system (l}.The s>ystem of

equations for flow calculation beyond the RW in mass Lagrangian

coordinates has the appearance:

ar = at u, au =0 at

av at

Vo ~n

'0 (r"u) n_ p a~ ,'--...,. (5 )

where V=Vo+VS ,VB is the porous volume per mass unit, n = 0,1,2 for plane, cylindrical and spherical symmetry of the

flow. The given model has no limits in volume concentration

of gaseous phase, which is necessary to be introduced from

the physical considerations. The bubbles are supposed to be

sph~rical and they are distributed uniformly in liquid •. Two li­

mit cases for bubble packing may be distinguished: (i) the

cubic one of the least density and (ii) the tetrahedral one of

the most density. As far as the real arrangement of bubbles

occupies the intermediate place between the given limit cases,

their junction occurs at the concentration range 0.52 ~ Vg / (Vo+Vg) ~ 0.74. The medium converts into foaming state when

370

the given critical concentrations are achieved and the medium

decomposition into discrete particles is possible.

The calculation results according to the suggested model when

~ = - 10 atm are denoted by the points in Fig. 5a, b, 6.

The good agreement of the calculations with the results obtain­

ed in the model [7] shows that the principle peculiarities of

flow are considered by the suggested model.

In conclusion the author acknowledge useful discussion of

the work with ProfessorV.K. Kedrinskii.

References

1. Driels M.R. Estimation of the dynamic cavitation tension of water by a shock tube method. J.Sound and Vibration, 98 (1985) 365-377.

2. Erlich D.C.; Wooten D.C. Dynamic tensile failure of glucero­leo J.Appl.Phys. V.42 13 (1971) 5495-5500.

3. Kedrinskii V.K. Dynamics of cavitation zone at underwater explosion near free surface. PMTF, 51 (1975).

4. Carlson G.A., Henry K.W. Technique for studying tension fai­lure in application to glycerol. J.Appl.Phys. V.42 5(1973) 2201-2206.

5. Stebnovskii S.V., Chernobaev N.N. Energetic threshold of pulse failure of liquid volume. PMTF 1 (1986) 57-61.

6. Chernobaev N.N. The peculiarities of pulse failure of liquids with different physical properties. Dinamika sploshnoi sredy izd.lnst.Gidrodinamiki SO AN SSSR. 84 (1988) 135-143.

7. Iordanskii S.V. On the equations of motion for liquids con­taining gas bubbles. PMTF, 3 (1960) 102-110.

8. Besov A.S., Kedrinskii V.K., Pal'chikov E.I. The study of initial stage of cavitation using diffraction optic techni­que. Pis'ma v ZhTF, V.10 4 (1984) 240-244.

9. Kedrinskii V.K. Peculiarities of bubble spectrum behavior in cavitation zone and its effects on wave field parameters. Proceed of Ultrasonic-85 Conference London 225-230.

10. Wilkins M.L. Calculation of eleastic plastic flows. Compu­tational methods in hydrodynamics. Moscow, Izd. Mir, 1967.

Liquid-Vapour Phase Change and Sound Attenuation H . Lang

Max- Planck-Institut fUr Stromungsforschung D-3400 Gottingen FRG

Summary

Starting from the entropy produced at the liquid surface the transport matrix for the heat and mass transfer is derived. Onsager symmetry and the role of the evaporation coefficient, the condensation coefficient and the energy ac­commodation coefficient are discussed. Based on the acoustic equations, Millikan's formula for the drag of droplets and the transport matrix for heat and mass transfer between droplets and their surrounding vapour, a numerical solution for the sound attenuation is given and the influences of droplet size and the evaporation coefficient are discussed. A simplified relationship for the sound damping is suggested which reveals 3 typical relaxation times characterizing the dependence of sound at­tenuation on the sound frequency.

The reflectance of a liqUid surface to sound incident from a saturated va­pour depends on the evaporation coefficient, which offers a possibility for the direct measurement of this coefficient. The dependence of the reflectance on the nature of the substance, the evaporation coefficient and the sound frequency is discussed and demonstrated for PPl (n-C6F14).

Entropy production. transport of mass and heat at a liquid surface

First let us discuss some basic quantities necessary to calculate the transfer

of mass and heat at the liquid-vapour phase change.

T'

Fig.1 Scheme of the transport fluxes at the liqUid surface where Jm + -

is the total mass flux ( Jm = J - J ), Ju is the total internal energy flux

(Ju = J~ - J ~), and J~ and J~ are the total entropy fluxes in the vapour

and the liquid phase; liquid phase (,), vapour phase (" ).

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Gotlingen, Germany, 1989 © Springer-Verlag Berlin Heidelberg 1990

372

In order to understand the meaning of the different transport coefficients

arising in the liquid vapour phase change we consider a liquid surface at temperature T"with an adjacent vapour phase at temperature T' (Fig.1). The

vapour pressure is assumed to be so low that gas collisions can be neglected

(Knudsen gas). The entropy flux Js can be expressed in terms of the flux of

internal energy Ju and the mass flux Jm by the following equation

J=_1J-tLJ sTu T m'

with the chemical potential tL and the absolute temperature T.

The entropy production rate dsirr/dt at the surface is given by

dSirr = J" _ j' dt s s

The following bilinear relationship can be found from Eqs. (1) and (2)

dSirr dt

J (1 - 1 ) _ J (tL" _ tL') u T" T' m T" T' .

(1) .

( 2)

(3 )

Further conclusions can be drawn from the entropy prod uction (3) applying

the principles of linear irreversible thermodynamics [1] - [3]. By assuming a

near equilibrium situation, an ideal gas state in the vapour and by taking in­

to account that the pressure dependence on the chemical potential in the liquid phase is negligible, Eq.(3) can be recast into the following form:

(4)

with the molar gas constant R , the fluxes Jm and Jq and the .. forces .. sm

and St. The modified heat flux Jq is defined by

Q. RT

(5)

where Q is the heat flux from the liquid surface to the vapour, and T is a

mean value of the temperature and the forces are given by

S = m ps( T') - p(T")

ps (T') S t = T'- T"

T' ( 6)

where p is the vapour pressure due to the surface temperature T' and p is s

the pressure of the vapour due to the temperature T:'

Because we restrict ourselves to the near- equilibrium region we can write

the following phenomenological equations:

Jm ocmm sm + ocqm St (7)

Jq = ocmq S m + oc St qq

with the .. transport coefficients .. oc ij

373

According to Onsager's reciprocity theorem [1] , the matrix of the phenome­

nological equations (7) is symmetric and we have

(8)

Moreover, by the second law of thermodynamics, the entropy production is

always positive for irreversible phenomena and zero for reversible phenomena.

If the Eqs. (7) are introduced into the expression (4) for· dsirr/dt one ob­

tains a quadratic form which has to be non-negative. Hence, all principal mi­

nors of the quadratic form must be non-negative and we find

cx > 0 , mm

(9)

Although the relationships (8) and(9) are generally valid and supply impor­

tant information about the transport coefficients cx •• their relation to the kine-IJ

tic quantities determining the molecular mass and energy transfer must be

known. Among these kinetic quantities the most important ones are the eva­

poration coefficient Cl and the thermal accommodation coefficient ct: e.

The evaporation coefficient Cl is the ratio of the evaporating particle flux Je to the particle flux Jo of a Maxwellian distribution function in thermal equi­librium with the surface:

(to)

where ns is the particle density, ps is the vapour pressure and v s is the mean thermal velocity due to the surface temperatur T', respectively.

Since the fraction of molecules that condense may differ from the fraction

evaporating some authors distinguish between the condensation coefficient Clc and the evaporation coefficient Cl , where

Cl = lL c r

( 11)

with the condensing particle flux Jc and the incident particle flux J:-But,the equality of Clc and Cl is only certain valid in the thermal equilibrium

case, when J = J and J- = J . The evaporation coefficient Cl is only dependent ceo

on the nature of the liquid surface and its temperature, but the condensation

coefficient Clc also depends upon the incident distribution function of \:he

molecules [4] - [6] .

Usually, the thermal accommodation coefficient 1s defined as the ratio of the

actual mean-energy change of molecules colliding with the wall to the mean­

energy change if the molecules came into equilibrium with the surface [7].

Knudsen's definition is valid for an exchange of energy at a solid surface

where the particle flux of molecules impinging to the surface is equal to the

particle flux of molecules emerging the surface ( J = 0). m

374

However, if mass exchange takes place, this definition is only valid for mo­

lecules which , after falling on the surface, are not condensed and are re­

flected or re-emitted from it. Obviously their accommodation coefficient cx e

cannot be determined directly by bulk measurements.

In the case of a liquid surface, where mass exchange by evaporation and con­

densation takes place, we found it convenient to define a modified thermal

accommodation coefficient &'e [4, 5,6] as

&' e

where the .. reduced energy flux" (]) is defined by

(])= J-u(T')J, u

( 12)

(13)

with the energy flux J , the corresponding mass flux J and the energy of u

transport u which is a function of the temperature. The flux (])o refers to

thermal eqUilibrium and vanishes because

(])o = J~,o - u(T')J+ with J:,o = u(T')J+

The energy of transport u is written in molar quantities as

u = 2 R T' + c. (T') T' , mt

with the molar internal specific heat cint. Furthermore, (])- is the reduced

energy flux of the molecules impinging to the surface and (]) + is the reduced

energy flux of the molecules emerging from the surface. Substitution of Eq.<t3) into Eq.(12) yields

J~ - J: + u J m

J~ - (,0 + U J m

Hence, it can be immediately seen that, in the case of no mass exchange

Om= 0), &'e takes the form of Knudsen's accommodation coefficient cxe' The

modified accommodation coefficient ae acts as substitute for cxe in the more

general case of both mass and heat exchange.

The question arises as to which kind of .. accommodation coefficients .. can be expected ? To answer this question we took the starting point for our

research as the surface scattering laws for monatomic gases [4,5] and for

polyatomic gases [6]. Although so far our understanding of the microscopic

gas-liquid surface interaction is very incomplete, useful conclusions can be

drawn by considering microscopic reversibility and the preservation of thermal

equilibrium together with the scattering law. As a consequence of the mi­

croscopic reversibility the symmetry of the transport matrix (7) follows.

Furthermore, relations between the transport coefficients Q}j and the reflec­tion probability of the molecules at the surface have been found . In this way further accommodation coefficients have been represented , especially

partial thermal accommodation coefficient OCe,trans' OCe,rot and oce,vib for the

different internal degrees of freedom of a polyatomic gas [6].

375

With regard to our restricted knowledge of the interaction of gas molecules

with a liquid surface we use Maxwell's suggestion [7] dividing the non-con­

densed part of the incident particle flux into a speculary reflected and a

diffusively reflected fraction. Experimental results [8.9] indicate that the

evaporation coefficient of pure liquids is nearly unity ,Le. the reflection

probability is very small. Consequently an approximative treatment of the

reflection problem using Maxwell's method turns out to be sufficient.

To show our results in an appropriate form we transform the flux 1q into 1<p

and the force sm into ~ ;

~= S -LE m 2 t (14)

We obtain the following transport matrix for the mass and heat exchange in

of a liqUid surface of temperature T' with a Knudsen gas of temperature T"

the linear region where 1<1"- T')I T'I «1

o 1 ~

°

with the dimensionless internal specific heat

important relation [4 -6] :

c~ = c. / R and with the mt mt

At a soli d surface we have 0 = ° and the well known result

1q = CJ(e10(2 + Ci~t)\ ' (with 1m = 0) ,

follows from Eqs.(4)-(161.

(15)

(16)

(17)

Sound attenuation of very small droplets; Sound reflectance at a liqUid surface.

Absorption of sound in air-water fogs of submicron droplets (droplet size

comparable with the mean free path) has been investigated by some authors

[10], [11]; in this case the air molecules cause the heat transfer through the

gas and mass is transported through the gas phase by diffusion. However in

the case of a pure substance , the mass flow in the gas is a gasdynamical

process. The basic equations for the multiphase flow of a condensing vapour

have been published by several authors [12] . To be concise, let us start di­

rectly with the acoustic version derived by linearizing the basic equations

[12], taking into account the fact that the volume part of the droplets is generaI1y negligible compared with the volume part of the vapour. Continuity Equations:

~v + pO auv a t v ax

~ + pO ~p = _ no M, dt P ax

08a)

where n is the droplet number density per unit volume of the droplet-vapour

system, M is the mass flow from the droplet to the vapour; pv is t~e mass

of the vapour and R the mass of droplets where both are defined per unit

volume of the dropl~t-vapour system. The velocities are given by Uv and up'

The index ° refers to values in the sound-free system.

376

Momentum Equations:

+ ~ = no F ox po ~=-n F

p ot 0

where F is the drag force of a single droplet on the surrounding gas. Energy Equations:

po ~ v ot

po ~ p ot

n Q + op o p at

=- n Q -n M H o p 0 v'

08 b)

08c)

where hv and hp are the specific enthalpies of the vapour and liquid, respec­tively, Q is the rate of heat transfer from a droplet to the gas, Hv is the latent hlat of vaporization Hv and p is the gas pressure . We assume that viscosity,heat conduction and internal relaxation cause no noticable sound at­tenuation in the sound frequency region under consideration . Like the trans­port properties in the gas, the transport processes between droplets and va­pour involve relaxation and we express the mass exchange M and the heat exchange Qp of a droplet by a "relaxation matrix" [13] as

M m {_1_~ _1_ ~} p t P t Tv mm v mq

Qp ( 19)

m { 1 ~p 1 ~T} , c p,v Tp p tqm T tqq Tv

where

~P= p (Tp) - Pv , ~T= T - T p v

where mp is the droplet mass, cpv the specific heat of vapour at constant pressure, Tp the droplet temperature, Tv the vapour temperature and p(Tp)

is the vapour pressure due to the droplet temperatur . The relaxation times due to the different exchange processes are given by : t mm , t mq (mass ex­change) and tqm ,t (heat exchange>. The relaxation times of the matrix (19)

can be found by c~~parision with the corresponding transport matrix [14]

and for our problem we applied the transport matrix given in [1S] where Maxwell's reflection law has been used. By the Onsager principle (8), the re-

laxation times t and t are related by mq qm

t ..I....=......! t (specific heat ratio y). ~ y ~ (~

The relaxation time for the momentum exchange td is defined by and calcu­lated by use of [16]

F = --1L.(u - u) td P

( 21)

Plane wave solutions are sought by substituting solutions for each dependent

variable of the form <I> = <Dexp{j(Kx-wt)},

with the circular acoustic frequency w . The complex wave number K is rela­ted to the energy attenuation coefficient 0: by the relation 0: = 2 1m ( K) .

377

Performing this substitution into Eqs.(18) leads to a relationship for K

where tqq w tqq , ao the sound velocity and C is the ratio of the mass of the

the droplets to the mass of the droplet-vapour system in the undisturbed state,

2 01 = WT B N Q [WT L N Q + NLE ( NQ - ST L)] ,

02= WT y N Q [(NQ -2) NLE + NQ1,

03 = B QT (WT e y + y - WT ST L ) ,

04= y QT, Ds= WT N~ NLE , D6 = B QT WT L 2,

...:mm., tqq

WT Y - 1 QT =

~, tqq

(y -1)

L ~ R· Tp ,

y, ST = y + 1

The energy attenuation coefficient ex can be obtained from relationship (22)

by complex computer calculations. If C « 1 we found the following useful

approximation [131:

~=C{ N 2'71:

where T mm

,(

(l

D3 - D6 tmm + td LE Dl (C D )2 ", 2 'r2

NLE~ + t + d Dl

mm

+ D2 - Os tag } OS (~JT+ t 2

qq

Wtmm Td w td and A is the acoustic wavelength .

100

10- '

10-2

10- 3

10- "

10-" 1 0 0 10' 1 0 2 1 0 3 10" 10" 106 107

Freq uency (Hz)

Fig . 2 Sound attenuation a.A versus frequency as functions of droplet radius rp; solid line. exact so­lution (22). dashed lin .. , approximation (23) System= water-vapour/droplets at the boiling point C=O.OOl; Kn=O.063 for rp=lJ.£m Knudsen number Kn=mean free path/droplet radil.lS

( 23)

378

10°,-------------------------------------------.

10 -'

0= 0 .0

10-·

10' 102 1 O~ 10· 1 0' Fre qu e n cy (Hz)

Fig .3 Sound attenuation a.A ver""sus frequency as o function of the evaporation coefficient a Systerr> water-vapour/droplets at the bailing paint C=O,001; rp=1~m; Kn=0.063

Equations (22) and (23) are shown graphically in Fig,2 for diffel'ent values of

the droplet radius for the watel'-vapour/droplet system at the boili ng point.

We note the appearance of three attenuation maxima [13 J, The mass exchan­

ge accounts essentially for the first maximum , which is shifted to very small

values of frequency , A similal' situation has already been found fOl' sound

attenuation in atmospheric fog [11 J, The second maximum is due to the vis­

cous contl'ibution to the sound atte nuation and the third term represents a

contribution due to simultaneous mass and heat transfer. The numerical solu­

tion of Eq ,(22) and the appI'oximation (23) are found to be in very good

agl'eement, Fig ,3 shows that the dependence of the dimensionless sound atte­

nuation c£. A on the evaporation coefficient is stl'onger at high frequencies f

( f '" 1 Mhz), The attenuation reaches the value obtained for non-volatile

particles in the limit of vanishing evaporation coefficient. Up until now , we

assumed in our calcultation that c£e = 1 and that 0 is changed , It should be

mentioned that a similar situation exists when 0 = 1 and the thermal accom­

modation coefficient c£e changes, Measurements of sound absorption in vola­

tile dl'oplets supply intel'esting information on the sizes and mass fl'action of

the dl'oplets, Howevel' , uncel'tainties about the droplet size distribution and

the cort'ect sul'face temperature of the droplets make measurements of the

evaporation coefficien t 0 di fficul t, Another way to obtain information about the evapol'ation coefficient 0 are

measurements of the reflectance I' of sound waves at a plane liquid surface,

The reflectance ( sometimes also called reflection coefficient) is the ratio of

the time-avel'aged energy flux in the reflected and incident sound waves , Due

to the irreversible processes of heat and mass transfel' at the sUl'face the

sound wave loses energy and I' deviates from unity, Furthermore, the surface

tempel'atlu'e is known because changes of the surface temperature can be ne­

glected in the acoustic experiment. This method has ah'eady been demonstl'a­

ted by observing standing-wave I'esonances in a tube, The tube is closed by a

sound generator on top and by boiling water below El71 A detailed in­

vestigation of the influence of evaporation and condensation on sound reflec-

379

tion has been given in [18] . While the vapour is generally not in thermal

equilibrium with the liquid surface there is a Knudsen layer at the surface and boundary conditions include slip and jump conditions. We obtained , as

an good approximation, the following expression for the specific acoustic ad­

mittance Z at the surface:

L + pp ( 1 - i)· (w T )1/2

( 24)

This approximation is valid provided that the internal relaxation effects in the vapour have no mal'ked influence on the sound damping. Based on an

elementary consideration (Maxwell's method) we found for Lpp

_ ( 11: \ 1/2 { 2 - 0 I Lpp - ~ --0- + "2 'I-I} Yj+1 , (25)

where 'I = cp,l/ CV.I and the subscripts are 1 for vapour and 2 for liquid. A special case of the formula (25) for the jump coefficient, Lpp ' has already been represented in [18] for perfect accommodation ( C(e = 1). The relaxation time '[ in expression (24) becomes [18]:

'[ = 'I A2 92 cP,2 R T3

292 M q4 1

( 26)

with A" is the heat conductivity 9" the mass density and c 2 the specific ~ ~ p,

heat at constant pl'essure of the liquid , M the molecular mass and q the

heat of vaporization per unit mass. For example. if we consider PPI

( n- C 6 F12) at 25° C and the_~alues 'I = 1.03 and C(e = 1. We find from equa-tion (26) that '[ = 0.27 x 10 sec.

The reflectance r can be expressed in terms of specific admittance as [18]:

I' = 1 ( 1 - Z)/ ( 1 + Z) 12 . (27)

The values of the reflectance r for PPl, obtained from Eqs .(24) - (27), are

shown in Fig.4.

III u c .3 u

" ;;::

" Ir

1.0"'----____________ -.

o.g (,rr-O.OOl

0.8

0.7

0.6

0.5

0.4

0.3

0.2 c.JT=l

0.1

O.O.T--r--.--r-'--.--'r--'--r-~~ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Evaporation Coefficient a

Fig.4. Dependence of the reflectance r on frequency and the evaporation coefficient a for PPl (n-C6F14) in the saturation state at 25" c.

3~O

In the limits " ~ 0 ( non-volatile substance ) and w t ~ 0 the reflectance approaches unity. For very small frequencies the vapour-liquid system is able to adjust itself to momentary thermal equilibrium with no irreversible energy losses; thus , r is equal unity. A marked sensitivity of the reflectance r to the evaporation coefficient " is found for w t > 0.1 , i.e. for sound frequen­cies higher than 600 kHz for PPl at 25° C. No noticable influence of the thermal .accommodation coefficient on r was found by changing O(e and kee­

ping " constant, " = 1. Experiments show [19] that the adjustment of translational and internal de-grees of freedom requires relatively few collisions in the case of polyatomics like n- C6 H14 . Hence, we can conclude that the relaxation time of PP1- at the saturation state at 25°C must be very small ( t R:I 3. 10-9 sec ). Thus, internal relaxation can be neglected in the sound frequency range under con­sideration and the validity of Eq.(24) is justified. It follows from Eqs.(16) and (25) that the reflectance is independent of the thermal accommodation coef­ficient O(e for " = 1 . Hence, there is only a weak influence of O(e on r near perfect evaporation (" R:I 1 )

References 1. de Groot, S.R.; Mazur, P. (eds.l , Non-Equilibrium Thermodynamics,

North- Holland ,Amsterdam (1962) 2. Bornhorst, W. J.; Hatsopoulos, G.N.; J.AppI.Mech. 34, 840 (1967) 3. Waldmann, L.; RUbsamen, R.; Z.Naturforschung 27a, 1025 (1972) 4. Lang, H.; Kuscer, I.; Proceedings of the 9th Int. Symposium on Rarefied

Gas dynamics, DFVLR Press, Porz Wahn, Vo1.2, F.t2-t (1974) 5. Lang, H.; J .Chem.Phys. 62, 858 (1975) 6. Lang, H.; Proceedings of the 12th Int. Symposium on Rarefied Gas dy­

namics, American Institute of Aeronautics and Astronautics 74, 346 (1981) 7. Present, R.D.; Kinetic Theory of Gases, McGraw-Hill 1958 8. Cammenga, H.K.; Klinge, H.; Rudolph.; Fortschr. Kolloide u.Polymere 55,

118 (1971) 9. Cammenga, H.K.; Current Topics in Material Science, North-Holland ,Am-

sterdam, Vol. 5, 335 (1980) 10. Jaeschke, M.; Hiller, W.J.; Meier, G.E.A.; J. Sound Vib. 43, 467 (1975) 11. Cole, J.E.; Dobbins, R.A.; J.Atmos. Sci. 27, 426 (1970) 12. Jackson, R.; Davidson, B.J.; Int. J. Multiphase Flow 9, 491 (1983) 13. Lang, H.; Z .angew.Math. Mech.69, T631 (1989) 14. Lang, H.; Bulgarian Academy of sciences, Theoretical and Applied Mecha-

. nics XIX , No 3, 46 (1988) 15. Lang, H.; Phys.Fluids 26, 2109 (1983) 16. Millikan, R.A.; Phys.Rev. 22, 1 (1923) 17. Maurer, 0.; Bestimmung des Kondensationskoeffizienten durch Messung

der Schallreflexion an der GrenzfUiche Dampf-FlUssigkeit, Diss.Giessen, unpublished, (1957)

18. Robnik, M.; Kuscer I.; Lang, H.; Int. J. Heat Mass Transfer 22, 461 (979) 19. Lambert,J.D.; Vibrational and rotational relaxation in gases, Clarendon

Press, Oxford (1977)

NonstationaryWave Processes in Boilding Media

V. E. NAKORYAKOV, B. G. POKUSAEV, N. A. PRIBATURIN, s. I. LEZHNIN, E. s. V AS S ERMAN

Inatitute of Thermophysics, Siberian Branch of the USSR Academy of Sciences, 630090 Novosibirsk, USSR

Summary Boiling medium is a typical example of heterogeneous multiphase medium with various internal structure. Modelling the wave pro­cesses in such medium will be difficult due to the complex na­ture of mass and energy transfer at a "vapour-liquid " inter­face. However the basic effects governing the formation and motion of compression waves in such medium can be obtained considering this medium as a superposition of the character­istic flow regimes and analysing the wave processes in these regimes. This lecture considers the models of compression wave propagation at different structures of two-phase flow and out­lines the general mechanisms of wave formation for various structures, the scope of these mechanisms and, finally, the basic peculiarities of the propagation of strong compression waves.

Pressure Perturbations of Moderate Intensity

Previously in our theoretioal and experimental works [1, ~ it has been shown that the behaviour of pressure waves of small (APo /Po < 1) intensity, their structure, attenuation in a boiling bubble medium are well described by a model equation

where ,.co·tlto,}=.xlto ; P"'=APIA!'O,to is the length of initial perturbation, APo is the amplitude of ini­tial perturbation. Similarity criteria (5 , M , W define the contribution into distortion of wave profile of disperse, nonlinear effects and interphase heat transfer process. Their

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Giittingen, Germany, 1989 © Springer-Verlag Berlin Heidelberg 1990

382

expressions for bubble regime are given in the upper line of

the Table.

Table

Co M 6' W

00 (TP f2 1'+1 LlPO !Ia ta t ~ c8 °0 JG a 1 0) 513. 00 fif/1~,{;) (sMlftlf-'fd» R Q

2{" Po 2 .%7; R/M r fa 00 0 0

00 0 C'p T 00

0 0 . __ 1 _ 00

,flo, 'I' °0 r>

~~ jp '12 T+1 f:JPa a 12 to

f 1/3 a

(r, 'I/; (!~ v;;J ~al 0 ) c2 Pt 2r Po

24M -J7i1;t2i.fJ~ .". fa e

. CPt T r

~-9I ~. tfa Tef' +! LlPO (-P1 6M'ib J: f ( a t ~ 1/2 C 2'p

• (; I- (1- '1'0) J', 1 0 2. 1

~J1 2Te, Po fa 1-~ "2 .JLCa])2cp2 r jJ2

112 eo CPt T ~~(jp (n)

D(1- YO) r

~l ~ Tf. f12 r+-f tlPo t G Ro ~ :o~ Y1<jJ(f~'?)~ 2r Po

'/vf. + " == 'f'. 6'f(l!- ~)~

-:- 8 -00 t~- y;s) 2 fJ'" 0 00

0 134 (t- 'P)2 t'a 00

'I' = 'IS . 'f' 8

It has been shown that the variants of shock perturbation evo­lution in a vapour-liquid mixture of bubble structure can be shown graphically as a map of regimes [2J • The value of 6~ (5 Jt (6"* ~ 9 for experimental conditions ) divides the plot into two regimes. If the generation of sign-alternating pulses is possible in the lower region, in the upper one it is principally impossible. With the criterion values VV ~ 0.1, heat and maSS transfer on interphase boundary is the determin­ing mechanism of the wave behaviour in a boiling medium. In this case the pressure pulses quickly attenuate and the lead­ing fronts of initially shock waves flatten. Inertial effects

are restrained.

383

As the value of ~ decreases, the role of nonlinear and iner­tial effects becomes more pronounced, which result in the em­ergence of pulsations in propagating perturbation and the pre­servation of shock wave profile at certain distances. The va­lue of ~* ' where nonlinear effects may emerge, is defined to a considerable extent by thermophysical properties of va­pour-liquid mixture [3J and can vary from 0.01 m to 100 m (thus for boiling water and LlPO #o~ o. 3; x* ~ 0.012 m atpo=0.1 MFa and X* = 14 m at Po = S.5 MFa). The analysis shows that prac­tically with W < 0.1 at distances tV 2 m inertial and nonlin­ear effects are determining ones in pressure wave formation and evolution. Experimental data [4+7J agree well with the properties of equa­tion (1) discussed above. Thus, for example, the evolution of shock waves observed in the experiments [4J is different at different parameters of boiling mixture and initial perturba­

tion (Fig. 1). For the case "a" we have M = 0.1S,.x*~O.17 m, respectively, and the shock wave actually degenerates at such distances. For the case lib" M = 0.47, X*~2 m. In the experi­ments the nonlinearity is pronounced up to distances ~ 4 m. As water, which boils at pressure close to the atmospheric one, has the small values of :r * and the large ones of W , it becomes evident why no dependence of initial shock wave velo­city on its amplitude was found in the work [5J • On the cont­rary, shock waves can be generated in water boiling at high pressures that was also observed in the experiments [6, 7J Consider the evolution of pressure perturbations in a two-phase media of another structure, first of all, slug and stratified ones. In the work [SJ an assumption is made based on theoretic­al analysis that the above generalization of wave phenomena may be extended to these two-phase media as well. Really, within quasi-continuum model (slug structure) 'and two-continuum model (stratified structure), the wave process can be described again in the frame of Eq. (1) with the similarity criteria given in the Table. In contrast to bubble regime, here the wave disper­sion is attributed only to geometric inhomogeneity of two-phase mixture. The estimates of the values of e> , VV ,A1 for real values of geometric parameters of pipelines, two-phase mixture

384

p,MPa a

0.8 X=D.2Jm :J. 5Jm

0.8

0 . 7

0.8

0.6

t,s

g p,MPa x=3.5Jm

0.1 t,s Fig.1. Hydraulic shock in bubble vapour-liquid flow. Experi­ment [ 4 J • Freon-113 Po = 0.62 MFa, a-'I'o = 0.0)2, 6 - 0.008

~ ] 20m,y ~

=25mm L'2o = 62mm ~=0!8 .;r=O. 78m

6=2.9

=25mm C'2/'65m "8=018 x=O.78m

6=27

Fig.2. Types of pressure perturbations in two-phase mixture of slug structure (water-air). - experiment, --- calculation by (1)

4P, CJ MPa

0.06

O.OJ

0

20 40 60 t,ms Fig.3. Shock wave in two-phase mixture of slug structure. Wave in: 1 - slug region, 2 - liquid plug. J) = 25 mID, 1'20 = 60 mID,

'1'0= 0.18

LJp

20 40 50 80 t,ms Fig.4. Comparison of experiment [10J with calculation (---) by (1). D = 5 mID, e = 0.1 m, '1'0 = 0.2, tlp = 96 KPa

385

-and thermophysical properties of the applied coolants, show that inertial effects contribute most (even in comparison with bubble regime) into the wave process at slug structure of two­phase flow. At stratified regime their role is insignificant and the waves emerging here should be characterized by a strong manifestation of nonlinearity. This conclusion is confirmed by the experiments conducted with a two-phase media of slug structure (case W = 0 [9J, Fig. 2). As in a bubble regime, there exists such value of (5 C::. 6'* ( de­fined by initial perturbation shape) below which a wave packet is formed (Fig. 2a). At (5 = 7.4 (here ~ ~ 6) a solution is formed in the medium and at (5 > (5 '* the initial perturbation evolves in the form of nonlinear wave with oscillating back front. Consider the propagation of shock wave. As is seen from Fig 3, the wave profile has an oscillating structure in the re­gion of gas slug. Since the neighbour slugs oscillate in phase opposition under shock load, pressure oscillations in the liquid plug should be restrained. Really the wave here turns out to be monotonous and its amplitude coincides practically with pressure in the incident wave (Fig. 3). The applicability of Eq. (1) to the wave analysis is confirmed by experimental results [10J • Fig.4 shows the comparison of these experiments conducted in a capil­lary horizontal tube ( 5 mm diam.) with air-water medium of slug structure, with the data of calculation by (1). The diversity of the available structures of two-phase flow requires to Single out the basically different "canonical" re­gimes, such as bubble (waves running in a liquid scatter on bubbles), stratified (waves run simultaneously in liquid and gas phase), slug (waves propagate by turns in liquid and gas phase). While investigating the wave dynamics in real complex structures, any regime can be presented as a superposition of canoni~al regimes in a first approximation. With intensive dy­namic effect ~Po/Po'> 0,5 on the slug structure of the flow, a slug may, for example, change into a cloud of small bubbles. One can consider the destructed slug structure in which liquid phase and bubble mixture with voiumetric void fraction Y'6 al­

ternate. The real void fraction of the mixture Y'= ~'~6

386

Slug and bubble dispersions of the waves coexist in such medi­um. Using relations for the accustics of complex media, after simple but clumsy calculations, one can obtain the appropriate evolution equation, the parameters and criteria of which are

given in the Table. Fig. 5 presents experimental and theoretic­al results on evolution of slight shock wave in a destroyed slug structure. Representing any regime as a superposition of "canonical" structures, one can investigate the propagation of slightly intensive signals in other real structures of two-phase flow: annular bubble (economizer boiling), crest (horizontal slug flow), slug bubble (slug structure at which liquid contains bubbles) etc.

Peculiarities of Intensive Pressure Wave Propagation

The above mechanisms are valid only for low-frequency waves with the amplitude less than APo /Po < 1. As the wave amplitude increases, its evolution deviates from the already known laws, and, moreover, effects related to the change in the structural characteristics of two-phase mixture begin to manifest them­selves. Fig. 6 shows the typical examples concerning the pro­pagation of pressure waves of different intensity in a boiling

steam-water medium. The synchronous record of void fraction behaviour in a wave is also given here. The wave of relatively small intensity (na") acquires an oscillating structure at the initial stages of evolution. Void fraction (condensation front) also behaves monotonously. The evolution of stronger wave differs appreciably - it is seen that its amplitude has in­creased by several times. The void fraction at transition

through the wave front decreases from the value of ~ to 0 • Thus vapour phase practically disappears bBhind the leading front of such wave. Fig. 7 shows the shadow motion picture of vapour bubbles in a shock wave which has been obtained using a powerful pulse laser stroboscope. The interval between the frames is 90 ms, and the exposure time of a frame is 10 ns. Such high time resolution allowed to register the secondary shock waves occurring at the moment of vapour bubble collapse.

387

LIp, _,Poo qpo8o f °0800 <f' =S$''d MPa

rr/-~?I . f· ~ I ~ 0.07 R ._ ---.----./'

x =0.65m ~x=O 0

4 12 20 28 t,ms

Fig~5. Wave in periodic bubble structure (water-air). 1) = 25 mm, R = 1.5 mm, e = 0.41 m, 'Ps~0.15, 'l' z O.1

LlP, MPa

Llp,MPa a 0.2 0.8

~ 0 0 /--

1 ~t,ms 1 t,ms

0 <fJ/</O 0 <p/~

Fig.6. Pressure waves and void fraction variation. cident wave. Po = 0.1 MPa, .x = 0.4 m, a-Ro= 2.8 mm, 0.018; g - 0.7 mm, 0.002

i+ 15mm->f

~ ¥o

t(m.CS) o 90 180 270 360 450

Fig.7. Vapour bubbles collapse with shock wave radiation. Freon-113, Po = 0.1 MPa, 'PazO.12, IlPo'" 0.1 MPa

- in­'1'0 =

388

They are seen as circumferences and arcs at late frames. This motion picture allows to compare the dynamics of bubbles dif­fering considerably in size. Large bubbles collapse monotono­usly and generate a powerful system of concentric waves. The experiments have shown that it is possible to choose the parameters of medium and initial perturbation so that a short. solitary pressure pulse of large amplitude will be generated (Fig. 8). The possible mechanism of this process can be presen­ted as follows: secondary waves which are generated by the bub­bles collapsing behind the wave front, have the velocity ex­ceeding that of the principal wave (due to the decrease of ~ ) and reach its front. In this case a powerful solitary wave borns as a result of interference. It may travel considerable distances without any substantial change of its shape (Fig. 8) and one can assume that such a wave can be long-living. Its amplitude can attain that of reflected wave and exceed the in­cident one by an order of magnitude. We shall present some quantitative relations for stationary shock wave with complete condensation. Continuity condition and momentum conservation law define the expressions for the wave velocity a::fiPo!R(t-'4)tf/;JI/a1. and the liquid velocity behind its front V= f2lPo~tq (1-~21 Va. The necessary condition of stability a>co gives lower limit for the incident wave intensity..:jpo/po>r. The threshold of shock pulse genera­tion, which has been determined experimentally, corresponds approximately to this value. The reflected wave propagates ~

through pure liquid and its amplitude ..:!3Pr="p, V01 ~£AAPo"b/~ (for ~ «1 ). The maximum pressure in a direct wave which is calculated on the basis of collapsing bubble cell model is gi ven by the expression [ 11 J

2 II n C ~ )12 J-' 1 0

P, 1-~

,/ 2 -4 _ ~ For 1'::>). 'f'o).':>I!Pt C, (~0.4·10 for water),Pma.:r--0A~~) C1 •

Really, in the experiments the amplitude of the formed wave ap­proaches the pressure in the reflected wave.

389

0,8 x=o 0,4m >-- -_._. o

2 4 6 t,m3

Fig.S. High-pressure pulse generation in vapour-water media. Po = 0.1 j\'[Pa, 'Fa = 0.002, Ro = 0.6 mm. r -wave ref­lected from the bottom of the shock tube

a-water 6 a jjjJ.,. 2.2 4.4 t,mS e - Freon-!! 0

Po -2 arrivot Of' 40

-4 ref'/. wove 20 .11,.00 2 -6 - VI·tO .c:.H,mm Po

<1 8 /2

Fig.9. Motion of liquid column behind the wave (a) and ampli­tude of reflected wave (b). -- calculation

t- 25mm-----l tJp,MPa

t(rns) a 20 40

tfO =.JOmm teo =fOOmm

t,ms

Fig.10. Destruction of slug under shock wave (a) and pressure pulse generation (b)

390

As it follows from Fig. 9, where variation of the mixture lev­el is shown when shock perturbation passes through it, the li­quid velocity behind the front is constant that corresponds to the above considerations. The amplitude of reflected wave (Fig. 9b) behaves also in accordance with formula for (Fig. 9b). We face the effect of pulse increase in the direct wave ampli­tude also in a two-phase system of slug structure (Fig. 10). Here the pressure wave generated at oscillation of one slug under the effect of shock wave is shown. The emerging pulse is related to the slug destruction. The destruction threshold ap­peared to be small and it amounts the value of 0.02 MPa and doesn't depend on the slug diameter, gas density in the slug and liquid viscosity. As is seen from Fig. 10, destruction oc­curs due to the formation of cumulative jet, at the outflow of which into the liquid plug, a hydraulic shock occurs. This ac­counts for the emergence of pulse in Fig. 10. Its amplitude de­pends on the velocity of cumulative jet which, in its turn, is defined by the amplitude and time of the increase of initial pressure perturbation, slug dimensions and liquid plug. There are good reasons to assume that in a real slug structure (with the bubbles in the plug) of vapour-liquid flow the effect of such hydraulic shocks may result in the generation of power­ful pressure pulses in a liquid plug, the nature of which has been considered above.

References 1. Nako ryako v , V.E.; Shreiber, I.R.: Model of perturbation pro­

pagation in vapour-liquid mixture. TVT 17 (1979) 798-803.

2. Nako ryako v , V.E.; Pokusaev, B.G.; Shreiber, I.R.; Pribatu­rin, N.A.: The wave dynamics of a vapour-liquid medium. Int. J. Multiphase Flow 14 (1988) 655-677.

3. Nako ryako v , V.E.; Pokusaev, B.G.; Pribaturin, N.A.; Shrei­ber, I.R.: Shock waves in boiling liquids. Int. Com. Heat Mass Trans. 11 (1984) 55-62.

4. Akagama, K.; Fujii, T.: Development of research on water hammer phenomena in two phase flow. ASME-JSME Thermal Engi­neering Joint Conference. Honolulu, 1987. P. 333-349.

391

5. Grolmes, M.A.; Fauske, N.K.: Comparison of propagation cha­racteristics of compression and rarefaction pressure pulses in two-phase, one-component bubble flow. Trans. Amer. Nucl. Soc. 11 (1968) 683.

6. Sarkisov, A.A.; Popov, N.A.; Lukyanov, A.A.: Boiling flow dynamics at shock perturbations. Heat Transfer, Temperature Regime and Hydrodynamics at Vapour Generation. Leningrad: Nauka 1981. P. 39-48. .

7. Neshchimenko, Yu.P.; Suvorov, L.Ya.: Weak shock waves in a boiling water and gas-liquid suspensions. J.Atomic Energy 591/6553. P. 11.

8. Lezhnin, S.l.; Pribaturin, N.A.: Nonstationary pressure waves for different flow regimes of vapour-liquid medium. Izv. SO AN SSSR, sere tekn. nauk 2 (1983) 20-26.

9. Nakoryakov, V.E.; Pokusaev, B.G.; Pribaturin, N.A.; Lezhni~, S.I.: Pressure Ware Dynamics in slug regime of gaS-liquid flow. Int. Seminar "Transient Phenomena in Multiphase Flow". Dubrovnik, Yugoslavia, 1987.

10.Matsui, G.; Sugihara, M.; Arimoto, S.: Propagation Charac­teristics of Pressure wave through Gas-Liquid Plug Systems. Bull. JSME 22 (1979) 1562-1569.

11.Nigmatulin, R.I.: Dynamics of multiphase media. Moscow: Nauka 1987.

Vapor Explosions

Explosion Hydrodynamics: Experiment and Models

V. K. KEDRINSKII

Lavrentyev Institute of Hydrodynamics Siberian Division of the USSR Academy of Sciences Novosibirsk 630090, USSR

Summary

Explosion hydrodynamics is related to the investigation of a wide range of unsteady processes developing under pulse load­ing of liquids, such as wave processes, bubble cavitation and high-rate cumulative jet flows. These phenomena are possible to be analysed in detail only by the combined investigations, both experimental and theoretical, with developing appropriate physical and mathematical models. This approach will be demon­strated below on the examples of some principal results and will refer mainly to the so-called surface effects and the problem of wave field parameters control.

Spiral Charges

The problem of shock wave generation by underwater detonation

of explosive charges has been studied in detail by many authors

[lJ for several tens of years. As a rule, these investigations

deal with one-dimensional and axisymmetrical statements. In

these case, the influence on the parameters of shock waves, ge­

nerated in the medium, is restricted by physical or scale fac­

tors. For example, the explosion of a linear cord charge initi­

ates a cylindrical shock wave in a liquid. Its characteristic - 0.244

time, determined by the relation e = G co/R* = 8.4(r/R*)

[2J for a stqndard charge, R*=0.15 cm in radius, at a distance

r= 20 m is only 85 ps and is independent of the charge length

L if the condition r «( L is valid.

The structure and parameters of the wave field charge in prin­

ciple if the detonation front propagates not along the straight,

as for a linear charge, but circumferentially rotates. This

system was realized on the basis of a cord charge having the

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM SYmposium Giittingen, GermanY,1989

396

form a plane or spatial spiral {3]. The characteristic highspeed

photo~raphs of the underwater explosion of the spatial (a) and

plane (b) spiral charges generating a succession of shock waves

are shown in Fig. 1, the interval between frames being 8 ps and

from 6 to 16 ps, respectively. The acoustic signal shown here in

the form of the wave packet at a distance of 20 m on the axis of ·

the plane spiral, with a 1 m outer diameter and 10 cm step, has

a duration of 5 ms (time scale is 400 ps) that is 20 to 30 times

as high as the above-mentioned values of e , the cord length L

a

b

c

t

r~~~' ~~~~j

~ Fig. 1. Generation of the sequence of shock waves at underwater explosion of a spatial (a) and plane (b) spiral. An acoustic signal is at a distance of 20 m, the time scale is 400 ys.

and detonation velocity D being the factors responsible for this

value. If the detonation velocity component along the axis of

the spatial spiral is equal to the shock wave velocity in the

liquid, the wave packet is transformed into one long wave having

the amplitude modulated in accordance with the frequency of the

detonation front rotation. An exact calculation of wave field

for such charges is difficult to be made. The estimates show,

however, that the acoustic source model describes the wave struc-

ture rather well [3J : j3

p = ( 9oa / 4 'J1) 50 S(t-'C )/f d"( (1)

-J 2 2 2 i where f = z + r + a - 2ar cos.,( , Z, r,"( are the cylind-

rical coordinates, 1.: (f/co-oI..a/D) is the delay time, a is the

ring radius, S = 11 R2 is the cross section area of a toroidal ca­

vity. The source power may be estimated by calculating the dyna­

mics of the cylindrical cavity containing the detonation pro­

ducts [4J

397

• .. :3 -2 - 1 - • • R (1 - R/ c ) R + "4 R (1 - R/ 3c) = "2 (1 + R/ c ) w + W R (1 - R/ c ) / c

Here c and Ware the local velocity of sound and enthalpy,

determined by the pressure of detonation products peR) according

to Shvedov at al. data [lJ.

Cumulation at Underwater Explosion

Fig. 2. A vertical spout with the jet at its top at underwater explosion of a 10 kg charge at a depth of 1 m.

Underwater explosion at small depth result in the formation of

vertical directed ejections, or spouts, on a free surface. Their

jet character (Fig. 2) was revealed and analysed within the fra­

mework of a plane statement, when a cylindrical charge was placed

normally to tw'o parallel planes, an optically transparent part of

which being above a free surface. The frames of a high-speed pho-f ..

Fig. 3. Plane cavitating spalls at underwater explosion.

tography (Fig. 3) display the formation of cavitating spalls, as

a result of interaction between a strong shock wave and a free

surface. As a result of their fracture, there develops a splash

structure of a white dome on a surface. A further flow develop­

ment is characterized by the formation of vertical jets of pur~

liquid under the dome (Fig. 4) [5J. The analysis of the experi­

mental results shows that the problem of potential liquid flow

398

Fig. 4. Formation of a vertical jet under the splash dome.

with two unknown boundaries is to be formulated in oder to deter­

mine the jet flow formation mechanism :

Q (t)

S (t) R (t)

A 'fl = 0, If-.O at r -- 00

p Pa' '-Pt - (v'-P)2/2 = 0 (2)

p Po [S ( t) / Sol -0 , lP t - (V <f) 2/2 = P - 1

At t = 0 ~ (0) is the horizontal surface, "f = 0 on ~ (0),

R(O) is the circumference, H = 4R(0) is the submergence depth of

the charge having the radius R(O), Po = 4.103 MPa is the initial

pressure in the detonation products. The calculated results show

Q(t) R(t}

Fig. 5. The jet flow develop­ment upon expension of cavity near the initial plane free surface.

(Fig. 5) that the jet flow mechanism is associated with both the

rarefaction zone formation near the axis, which develops at the

stage of explosion cavity boundary retardation by atmospheric

pressure, and the flow cumulation in this direction on the back­

ground of continuous expansion of the cavity with detonation pro­

ducts [5J.

The next figure illustrates a hydrodynamical model of the forma­

tion blast mechanism of vertical spout: to create the same condi-

399

tions for jet development a solid sphere (for example) has to be

forced abruptly up outside (it was initially submerged into a li­

quid on a small depth) and then also fast to be decelerated [61. It is seen that on the sphere surface there occurs vertical jet :

t •

air

Fig. 6. A hydrodynamical model of a blast spout formation: deve­lopment of water jet upon pulse motion of a solid sphere.

Bubble Cavitation and Rarefaction Wave Structure

The cavitation development process is the formation of a cloud

of vapour-gas bubbles (bubbly cluster) under the action of a ra­

refaction phase in the liquid containing microinhomogeneities,

such as microbubbles of free gas, solid particles or combinations

of them. Under an intense development of cavitati9n the liquid

physically transforms into a two-phase state changing substanti­

ally the structure and parameters of the applied field. In order

to construct a mathematical model of this process, one should

know, first of all, the state of real liquid and the mechanism

of the bubbly cluster formation.

The notion of the real liquid state in cavitation problems is

related to the existance of cavitation nuclei in medium and the

conditions of their stabilization. A minimum size of nuclei is

likely to be determined on the basis of a statistic model of a

"fluctuating holes", which gives a few angstrems [7]. Some inter­

mediate size (1.5 Fm) is detected from the Harvey .model, which

is the most real one from the viewpoint of stabilization .[8] •

This model presupposes the existance of hydrofobous particles

having gas nuclei in crevices. Their model admits the existence

of even empty nuclei: from condition Pg = Po - 2 Giro at Pg=O

2 ~/po' In an upper part of we can get the nucleus radius

t he spectrum, approximately, a

r =: o

5 pm size is estimated as a re-

400

sult of nuclei stabilization in natural convective flows taking

into account the lifting and Stokes forces, on the basis of the

determining parameters, such as temperature T, viscosity y

density 90 ' gravity acceleration g:

ro ~ ( v 2 kT/ ~o g2)1/7

The second important parameter of real liquid state is the nuclei

number densit; (N, cm-3 ) or the volumetric concentration of nu­

clei (ko ). The experimental data [9-11J for the settled tap wa­

ter provide the following values: from 1 bubble, ~6 ym in size,

to 100 ones, ,.. 3 )lm in size, may be contained in 1 cm3 of water

at room temperature, the volumetric concentration being 10-8 _ -12 - 10 • The size close to 1.5 pm was detected in [12J on the

basis of the light scattering method and the analysis of scatte­

ring indicatriss dynamics in shock wave. These nuclei was proved

to be ones of a free gas.

Up to the recent time the notion about the mechanism of cluster

development was based on the theory of an avalanche-like multi­

plication of cavitation nuclei (from~l cm-3 , a single original

nucleus, to _106 cm-3 ) as a result of the instability of the ca­

vitation bubble form and its fragmentation on a separate parts

in an ultrasonic field [13J. However, the distribution character

of fragments and its rate suggest that an observed effect of clu­

ster development is interpreted incorrectly. Besides,. there

exists a great number of experimental data which record the deve­

lopment of dense bubbly clusters in the field of single rarefac­

tion pulses [3, 5J. After all, the estimates of scattered light

intensity and density of diffraction spot number in the volume

of the laser beam illuminating the liquid determine the number

of microinhomogeneities of any origin in the distilled water by

value of 105 - 106 cm-3 . This result corresponds to the bubble

density in the finally formed cluster [13J. Due to the above­

mentioned factors, the necessity arose for re-considering the me­

chanism of nuclei multiplication because of instability.

The approach, which is new in principle, is the following [14,15J:

1. A real liquids contain the spectrum of cavitation nuclei, from

401

10-7 to 10-3 cm in size, with constant value of N (""-106 cm-3 ).

The notion of a visible size of a cavitation bubble is introduced,

i.e. such one which is detected within the framework of the me­

thod used.

2. The seeming multiplication of cavitation bubbles takes place

in relatively weak ultrasonic fields and represents a subsequent

zone saturation by bubbles achieving a visible size for different

periods of time depending on the initial size of a nucleus.

3. When the rarefaction phase intensity is high, the whole spect­

rum of nuclei may achieve a detectable size simultaneously, the­

rewith saturating the zone by bubbles up to a maximum density.

This model of cavitation development allows explaining the above­

mentioned experimental facts and essentially simplifies the prob­

lem of constracting a mathematical model of the process. In [16J

the author proposed a real liquid to be considered as a two-phase

medium, in spite of the infinitesimal parameters of its inhomoge­

neity, and to describe the cavitation processes with the use of

the system of Iordansky equations for bubbly medium [17J. In the

case of incompressible liquid component, under some additional

assumptions, this system is reduced to a rather simple mathemati­

cal model [16J

A S = \

k tt -(~ k 1 / 6 )2 (~oko)-lS + k~ /6k (3)

where k=(R/Ro )3, 01.. = (3k /R2)1/2, S = P - p k- t , and the space o 0 1/6 0

coordinate has the scale factor ~ k . This model was success-

fully used for analysing wave processes in cavitating liquids,

and recently, after some small modification, it was also used to

describe the phenomena of bubble detonation [18J.

Cavitation development at underwater explosion near a free sur­

face was investigated numerically within the framework of the mo­

del (3) in an axisymmetrical statement ~,16J. The initial parame­

ters of the incident of rarefaction wave ~ere determined by the

method of superposition of an imaginary source. The model began

to work only in the rarefaction phase. The calculation (Fig. 7)

shows that in the zone of developing cavitation the value of k

402

increases by 7-8 orders (a) and the maximum negative pressure (b)

admitted by a real liquid significantly depends on the increase

time (curves 1,2,3 correspond to its values of 0,0 . 1 and 1 ys)

of tensile stresses in the front of the incident wave: when the

front steepness is 1 ps, the maximum amplitude drops by 2 orders.

K R1Ro 1)7

105 102

105

10 4

Kif 10'

102

a b

1o-J "'10-2 " .,0-1' , :; " ''''10 ' "102 o

-10

-20 ~L-~ __ 1--L ______________ __

t ,fit;

Fig. 7. Explosion of 1 g charge at a depth of 5.3cm. Initial parameters of in-

C homogeneities: R = 0.5 Fm, k = 10--11 • Dynam~cs of bu­bBle in cavitation zone(a) and profile of rarefaction wave(b) for different ste­epness of front:O,O.l,l FS. Comparison of the experi-

d mental frame(c) and the calculated visible zone of bubble cavitation (t=64ys after reflection of s.w.).

The cavitation effects near the bottom of a vertical tube, filled

with a liquid, abdruptly accelerated vertically down due to the

impact were analysed in [19J. This experiment may be considered

as some model of a dynamic fracture of a liquid sample at a ten­

sion. The very important peculiarity of this statement is that

system (3) in this case allows an analytical solution to be ob­

tained for the relaxation time of the tensile stress, if the low

aCt) (acceleration) is known. Actually, the solution of the first

equation of system (3),

p = A exp ( - 7 ) + B exp (~ ) (4 )

403

where 1/2 ~ =oLy z, y = R/Ro ' z is the vertical coordinate,

taking into account the boundary condition dp/dz =- ~o a(t)

when z=O and a restriction of the solution at infinity (B=O),

takes the form

1/2 P = - ~ 0 I a ( t) I exp ( - ~ ) / oL y ( 5)

For this form of p(t) the second equation of system (3), re­

written with respect to the dimensionless radius y (k=y3)

admits the solution

(1 5 c(t) /cJ. R2 )2/5 y = + 2" t S 0

where c(t)=J J aq ) d ~ dS In particular, 0 0

stant, the relaxation time (the maximum amplitude

factor of e) is determined by the expression

~ = 14.3 (k R2 / a 2 )1/4 000

(6 )

if a=a is con-0

decreases by a

(7)

8 -2 11 For example, for a o= 1.643 10 cm s , k o= 10- , Ro= 0.1 ~m,

corresponding to an incident wave amplitude of -30 MPa, ? is

6.3 10-9s. This value is rather close to the calculated one, con­

sidered in the above-mentioned problem.

Previously it was mentioned that the rarefaction wave front steep­

ness was the major factor responsible for achieving maximum nega­

tive pressures, admitted by cavitating liquid. The obtained solu­

tion makes it possible to determine their maxima:

p = _ 0.77 0 (a2 R3 / k t )2/5 max JO 0 0 0 *

(8 )

where t* is the rarefaction wave front steepness. For the above­

mentioned values of Ro ' ko' a o ' calculated for a -30 MPa inci­

dent wave, we obtain Pmax=-1.82 MPa at t*=l ps and Pmax=

= -4.58 MPa at t*=O.l ys, that fully corresponds to the data on

underwater explosion (Fig. 7 a).

Among another statements, one should be ,emphasized, which con.­

cerns a hydrodynb.mical shock tube. Propagation of one-dimensional

rarefaction wave in a real liquid filling in the high pressure

chamber was investigated numerically within the framework of prob­

lem of discontinuity decay. A low pressure chamber contains a gas,

A fine wave structure was analysed within a full model taking in­

to account the compressibility of the liquid component and the

interphase heat exchange.

404

A characteristic wave profile for one of the time instants is

displayed in Fig. 8. It fully confirms an adequacy of transfor­

mation mechanisms of shock waves in bubbly media and rarefaction

waves in real liquids and their structures with an accuracy of

1,5 PlPo

fO

a5

0

-14 -10 -5 -2

Fig. 8. Two-wave structure of rarefaction wave in real liquid: separation of wave into osci-lla­ting precursor1 and main distur­bance 2.

inversion of processes: the shock wave consumes its energy to

the bubble collapse, the rarefaction wave does it to the cavita­

tion zone development. As seen from the Figure, the rarefaction

wave is stratified into the oscillating precursor and the basic

disturbance, which propagate with sonic velocity characteristic

for pure liquid and equilibrium one, respectively [20J.

References.

1. Korobeinikov, V.P.; Khristoforov, B.D. Underwater explosion. In: Itogi nauki i tekhniki. Gidromekhanika. 9(1976).

2. Kedrinskii, V.K. On the parameters of weak cylindrical shock wave at a great distance from a charge. In: Dinamica splosh­nykh sred. Novosibirsk: IGIL (1972), v.10.

3. Kedrinskii, V.K. Peculiarities of shock wave structure at underwater explosion of spiral charges. PMTF(1980), 5.

4. Kedrinskii, V.K.; Kuzavov, V.T. Dynamics of cylindrical cavi­ty in compressible liquid. PMTF(1977), 4.

5. Kedrinskii, V.K. Surface effects at underwater explosion (review). PMTF(1978), 4.

6. Kedrinskii, V.K. Explosion hydrodynamics (review). Pf.1TF (1987),4.

7. Frenkel, Ya.I. Kinetic theory of liquid. Izd. AN SSSR, 1945.

8. Harvey, E.N. et al. J. Cell. Comp.Physiol.,24,1(1944).

9. Strasberg, M. In: Cavitation in Hydrodynamics. London: NPL 6(1956) ,1-l3.

10. Gavrilov, R.L. In: Powerful ultrasonic fields. Moscow:Nauka (1970), part 1V.

405

11. Hammit, F.G. et al. In: Proceedings of Cavitation Conference Edinburgh, 3-5 Sept. ,1974, 341-354.

12. Besov, A.S. et al. In: Optical Methods in Dynamics of Fluids and Solids. Libice: IUTAM Symposium,1984, 129-135.

13. Sirotyuk, M.G. In: Powerful ultrasonic fields. Moscow:Nauka (1969), part V.

14. Kedrinskii, V.K. On multiplication mechanism of cavit~tion nuclei. Proceedings of 12th ICA, Canada, Toronto, 1986.

15. Kedrinskii, V.K. Peculiarities of bubble spectrum behaviour in cavitation zone and its effect on wave field parameters. Proceedings of International "Ultrasonic-85", London, UK, 2-4 july, 1985, 225-230.

16. Kedrinskii, V.K. Cavitation zone dynamics at underwater exp­losion near free surface. PMTF(1975),5.

17. Iordansky, S.V. On equation of motion for the liquids contai­ning gas bubbles. PMTF(1960),3.

18. Kedrinskii, V.K.; Mader, Ch. Accidential detonation in bubble liquids. Proceedings of 16th ISSTW, Aachen, FRG,26-31 July, 1987.

19. Hansson, I. et al. On the dynamics of cavity cluster. Journal Physics D: Applied Physics,15(1982),1725-173~.

20. Kedrinskii, V.K.; Plaksin, S.I. Rarefaction wave structure in cavitating liquid. Proceedings of Xl th ISNA, "Novosibirsk, USSR, 24-28 August, 1987, 51-55.

Vapor Detonations in Superheated Fluids

O. RICHARD FOWLES

Department of Physics Washington State University Pullman, Washington

Summary

The theory of detonation is applied to the liquid-vapor phase transition in superheated fluids. It is shown that such detonations are always weak detonations, characterized by supersonic flow of the shocked region. The detonation state is therefore determined by the transport properties-the viscosity and reaction rate-rather than by the boundary conditions. A numerical example is presented using the Van der Waals equation of state with parameters appropriate for water superheated approximately 100 degrees at a pressure of 5 bars. Detonation pressures of the order of 100 bars and explosion energies of the order of 105 J/Kg are predicted for this example.

I. Introduction

Vapor explosions are observed in a variety of circumstances when metastable, superheated

fluids suddenly transform to vapor. Rapid transformations of this kind occur on much smaller

time scales than those that obtain in ordinary boiling, and in consequence they can be very

destructive when they occur in large volumes, as in underwater volcanoes, nuclear reactors, natural

gas spills on water, or in the metals and paper industries. l Numerous investigations of these

phenomena have focussed on the initiation process at the "superheat limit," identified

approximately with the spinodal point of the isothermal equation of state, at which homogeneous

nucleation takes place.2 However, few studies have been made of the mechanism of propagation

of the transformation once it is initiated. In this paper we describe a model for the propagation of

the phase change that is fundamentally similar to the theory of detonation of chemical explosives.3

The principal difference is that, in the case of vapor explosions, the energy to drive the detonation

wave derives from the excess energy of superheat rather than from stored chemical energy. The

details of the initiation process are not essential to the steady state theory; it is only required that the

detonation shock produce an environment in which nucleation, whether homogeneous or

inhomogeneous, is triggered. This might occur, for example, by shock heating sufficient to cause

homogeneous nucleation or, more likely, by adiabatic compression heating of small gas bubbles

that in the ambient state are too small to grow. The large local temperatures produced could

provide nucleation sites at the shock front that may then grow rapidly. Such bubble compression

is an important initiation mechanism for fluid explosives such as nitromethane or nitroglycerine.4

G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid·Vapor Systems IUTAM Symposium Gottingcn, Germany, 1989 © Springer-Verlag Berlin Heidelberg 1990

408

Most chemical detonations are thought to be Chapman Jouguet detonations or, if supported

by the boundary conditions, to be strong detonations. Weak detonations are not usually

considered attainable because of the large reaction rates required, although there is some evidence

that they exist in special circumstances.S Vapor detonations, by contrast, are essentially all of the

weak variety because of the nature of the equation of state, independently of the reaction rate.

II. Theory

a. Overview

The theory is perhaps most easily described by means of an example. Figure 1 is a

pressure-volume plot of Van der Waals' equation showing the binodal and spinodal curves in

addition to several isotherms. The numbers are appropriate for water and are shown for later

reference. We consider a situation in which the initial state of the fluid is in the metastable,

~

co () III 111 Q.

ID 0 ... ~

ai ... ~ III III GI ... Q.

30

20

10

T 647K

640K

560K

500K

300K

A'J I

I I

I

0; 0.. 'b 15.0 ~

a:-ID 10.0 :; ., ., CD

5

A ~ 5.0

o.o-+f--,,----,-, --,..-, --" -----" 2.9 3.0 3.1 3.2 3.3 3.4

Specific Volume, V, (1 a-3m 3/Kg)

o.a~----+L-L---.-L--.--.-.-----,,-----.---,--.--,

2 4 6 8 10 20 40 60 80 100

Specific Volume, V, (1 0-3m3/Kg)

Fig. 1. Van der Waals' equation of state for water. The saturation (binodal) curve and the spinodal curve are shown as dashed lines. A metastable initial state in the fluid is indicated by point A, superheated with respect to the boiling point at that pressure, indicated as point A'. A steady detonation shock produces state B on the Hugoniot curve; subsequent relaxation to the initial pressure at point C takes place along the isentrope labeled S.

409

superheated state indicated as point "A." The boiling point at this pressure is denoted as point

"A'." If a small (or large) pressure pulse in the metastable medium causes the reaction to proceed

then a steady shock can form that takes the material to state "B" on the equilibrium Hugoniot

curve centered on state A. Note that this curve has a very sharp break in slope at its point of

intersection with the binodal curve and that it is bounded in a narrow pressure range by that point

and by the volume of the initial state. Subsequent expansion takes place along an adiabatic curve,

labeled "S" in the figure, to the pressure of the rear boundary, indicated here as the ambient

pressure of the initial state. At that point there has been partial reaction and the internal energy is

reduced, corresponding to the work done on the surroundings during the adiabatic expansion.

In conventional detonation theory the Chapman-Jouguet (C-J) state is the point of tangency

of the Hugoniot curve with the Rayleigh line joining the end states. In the case of vapor

detonations this point of tangency is seen to be the single point corresponding to the intersection of

the Hugoniot curve with the binodal curve; it is only realizable if no reaction whatever occurs in the

shock transition layer. Even a small degree of reaction is sufficient to cause the detonation state to

lie at pressures below the C-J point. If the detonation is supported by an applied pressure above

the C-J point there is a range of such pressures for which an overdriven detonation bifurcates into

two shock fronts traveling at different velocities. At still higher applied pressures a single shock

becomes stable again; in this case the phase reaction proceeds in both directions in the shock layer

to produce no net reaction.

A close-up view of the details of the Hugoniot curve is shown in the inset of Fig. 1. The

line labeled "R" is the Rayleigh line joining the initial and final states of the detonation shock.

The detonation state, B, lies on the Hugoniot curve as shown. Its location is determined by the

reaction rate and by the transport coefficients that apply to the shock transition layer. The flow

behind the shock is supersonic; consequently, once a steady shock is established the boundary

conditions have no influence on its behavior. The determination of the detonation state can be

understood by considering the behavior of the integral curve in the specific volume-reaction

coordinate plane, or phase plane.

b. Phase Plane

The integral curve joining the two equilibrium end states of the shock transition is defined

to be the locus on the thermodynamic surface of the states through which the material passes

during the transition. It is distinguished from the Rayleigh line in that the Rayleigh line includes a

viscous stress contribution. This integral curve exhibits critical behavior at the end states where the

gradients of the dependent variables vanish. The critical points are of either nodal or saddle

character depending on the circumstances of the flow. The detonation state, referred to as the

"eigenvalue" state, is determined by the intersection of the separatrix of a saddle point singularity

410

with the Hugoniot curve.5 The corresponding flow is supersonic with respect to the shock, as the

following argument shows.6

The flow equations in material coordinates and in one dimension can be written:

p + pUx = 0

pu -ax = 0

e-aV=O

(1)

(2)

(3)

where X is the material, or Lagrangian, coordinate, equal to the spatial coordinate in the initial

state; r is the density; sigma the normal stress component in the direction of the flow; u the particle

velocity; and e the energy density. Heat flow is ignored in this formulation.

To these equations are added the equation of state and constitutive relations that specify the

material. These are written:

P = pep, e, l)

a=1t-P

l = - (klRT)A

(4)

(5)

(6)

(7)

Equation 4 is the equation of state including the reaction coordinate, "1", as an independent

variable; the material is thus assumed to be in mechanical and thermal equilibrium, but not

necessarily in phase equilibrium. Equation 5 expresses the stress as the sum of the equilibrium

pressure and the viscous stress, which is assumed proportional to the velocity gradient [Eq. (6)].

The rate equation [Eq. (7)] assumes that the rate of change of the reaction coordinate is

proportional to the chemical affmity, A, defined as the difference in the electrochemical potentials

of the two phases at the same pressure and temperature. The exact forms of the rate equations,

Eqs. (6) and (7), are not essential to the argument concerning the nature of the critical points

provided only that they are valid in the limit as the shock end states are approached. They may

thus be considered as first order expansions about equilibrium of more general nonlinear rate

equations. Whatever rate equations are assumed they must reduce to the forms shown in order to

conform to the thermodynamic requirement that entropy production be second order as equilibrium

is approached.

In the case of steady flow the flow equations are integrated to read:

pv == j = constant

. .2 JV - cr == J a = constant

_ kA Ix - - jRTV

Vx = (~j)[l(v-a)+p]

411

(8)

(9)

(10)

(11)

(12)

where v = u - D is the steady velocity in a coordinate frame whose velocity with respect to the

original laboratory frame is D and x is now measured with respect to the shock front.

The latter two equations form an autonomous pair of differential equations that exhibit

critical behavior; the nature of the critical points is found by standard procedure. Forming the ratio

ofEqs. (11) and (2) gives

dV/dl = _ ~T[l(v;)+p] The numerator and denominator both approach zero as equilibrium is approached.

Expanding each of these in series gives, to fIrst order:

where

dV/dl = aAV + ~Al yAV+MI

~ = - (l) [V(dP(dl) ] ~k v,s

1 (aAJ y = RT avj l,s

~ = iT(~~) v,s

(13)

(14)

(15)

(16)

(17)

(18)

412

The critical point is of either nodal or saddle character depending on the sign of the

determinant, q = no - ~y which, after some reduction can be written as,

V (aA) (av) [ .2 (av) ] q = - f.l.k: RT Tl V,s ap A,s 1 + J ap A,s ;

>0 node

<0 saddle (19)

In this form it is easy to see that nodal character corresponds to subsonic flow and saddle.

character to supersonic flow. Thus, the coefficient is always positive to insure thermodynamic

stability. Moreover, the Mach number of the flow in stationary coordinates, MI = Iv/eJ, can be

shown to be given by 7

~l = -/<oV/()P) A,S

The sound speed, "c", is the sound speed in chemical equilibrium and is

(20)

(21)

It can be further shown that Mi > 1 whenever the Rayleigh line is steeper than the

Hugoniot curve, as is clearly the case shown in Fig. 2, and is true in general for weak detonations.

The conclusion is that weak detonations correspond to supersonic flow whereas strong

detonations correspond to subsonic flow. In the latter case of course the boundary conditions

determine the shock state.

Numerical Example

Experiments have not yet confirmed this model. However, computer simulations have

been done that illustrate the behavior. These use a version of the SALE code developed at Los

Alamos for two-dimensional, time-dependent compressible flow in either Eulerian or Lagrangian

coordinates.8 It incorporates artificial viscosity to accommodate shock fronts. The equation of

state used is the Van der Waals equation as generalized by Callen to include the internal energy.9

In this form it is a complete equation of state and can be expressed by the pair of equations,

P = RT _ arl V - b

e cRT- aN

(22)

(23)

413

The corresponding fundamental relation is

(24)

In addition to this equilibrium equation of state we assume a reaction rate equation of the

formlO

= k [ 1 - exp(A/Rnj (25)

It is seen to conform to the requirements of irreversible thermodynamics, Eq. (7), when expanded

to fIrst order about an equilibrium state.

The equation of state for the mixed phase, of the form ofEq. (4), is determined by

invoking the lever rule for total energy and volume in addition to Eqs. (22) and (23). Thus,

denoting the pure phases by subscripts 1 and 2, equality of pressure and temperature requires,

(26)

and

(27)

These are solved simultaneously with,

(28)

and

(29)

where I, e, and V are given. Pressure and temperature follow immediately from resulting values

of eI, VI (or e2, V2) by means of Eqs. (22) and (23).

The parameters assumed for the calculation and the results of the numerical solution are

listed in Table 1. The parameters are appropriate for water superheated by about lOOK at a

pressure of 5 bars. The initial state is indicated on Figs. 1 and 2 as point A and the shocked state

is shown as point B. The fInal equilibrium state upon relaxation to the initial pressure is shown as

point C.

414

Table 1

Values for Numerical Example

Equation of State Parameters

critical pressure, critical temperature, critical specific volume, viscosity coefficient, reaction rate coefficient,

Flow Parameters

Parameter

I e(x 105 J/kg) vex 1O-3mT/kg) P(x 106 Pa) T (K) T(K) ignition

Initial State, A

1.0 2.650 3.175 5.0 560 565

Energy of explosion = 8.2 x 104 J/kg Power of explosion - 1010 watts/m2

Shock speed - 103 mls Rarefaction speed - 2 x 102 m/s

Shocked State, B

0.991 2.657 3.10

13.82 573

2.21 x 107 Pa 647K 5.067 x 10-3 m3/kg -102 1()6 s-1

Relaxed State, C

0.66 1.84

13.29 5.0 466

Ignition of the reaction is assumed to occur at a critical macroscopic temperature. This assumption

provides a simple phenomenological criterion for purposes of displaying the wave propagation

behavior, although it may seriously oversimplify the physics of initiation in detail.

Figure 2 shows the wave profiles when a constant pressure is applied to the rear boundary

that is somewhat less than the eigenvalue pressure. The ignition temperature is slightly above the

initial ambient temperature. The various profiles are the pressure, temperature, internal energy,

and reaction coordinate, defined as the mass fraction of the liquid phase. The state parameters are

seen to quickly attain equilibrium values on the Hugoniot curve. In Fig. 1 this state is labeled

"B." The constant shocked state expands as time progresses, in accordance with the supersonic

condition. Following the constant state is a rarefaction wave that adjusts to the boundary

conditions. Most of the reaction occurs in the rarefaction wave; the extent of the reaction in the

shock transition layer is very small. Similar results have been obtained for methanell and for

freon12 using modifications of the Van der Waals equation.

415

A ~38

B 16

.;;--14 ",E 3.7 , , , ,

~ 3. 6 to 12 , I

n. , >< .. I - 3 .5 0 10 I , :> ,

I Ii 3.4 ~

, , E :::I

, ~ 3 .3 I/) ,

I/) , 0 ., > 3 .2 I c: · 4 .g \ , , ·u 3 . 1 , " '---"--------"----- - ---- -, .,

/J; 3.0

C 1.06

0 2.68

Iii 2.66 ..... 1.04

:.: (/ .......... ---------- Ii " ;;> 2.6 4 ... 1.02

0 C - 2.62 :;:; 1.00

~ 0

t' ar 2.60 0 0 .9 8

I U ,;; I C 0 .96 D> 2.58 I 0

Gi I

~ I 0 .94 c 2 .56

/ to I W ., I

a: 0 .92 , ,

---=~-_...../.(--- - --------,

2 .54

0 .90 2.52 0

0 6 8 1 0 12 14 16 18 2 0 2 2 2 4 8 10 12 14 1 6 1 8 20 22 24

Distance x (1 0-3m)

Fig. 2. Wave profiles from a numerical simulation. A: pressure, B: specific volume, C: internal energy, D: reaction coordinate.

Conclusion

The model proposed provides a straightforward explanation for the steady propagation of

vapor explosions that does not depend on detailed knowledge of initiation mechanisms. Peak

pressures and explosion energies can be estimated or bounded and appear to be comparable to

those observed or inferred. Experiments to verify the model are clearly desirable; some attempts

have been made to measure shock wave speeds and pressures but these have not yet been carried to

sufficient completion to draw definite conclusions. 12 The experimental results of Chaves do not

verify these predictions, but may be that steady state was never reached in his experiments. 13

References

1. Witte, L.C.; Cox, J.E.; Bouvier, 1.E.: The Vapor Explosion. 1. Metals 22 (1970) 39.

2. Shepherd, 1.E.; Sturtevant, B.: Rapid Evaporation at the Superheat Limit. J. Fluid Mechanics 121 (1982) 379.

416

3. Fowles, G.R.: Vapor Phase Explosions: Elementary Detonations? Science 204 April 1979 168.

4. Campbell, A.W.; Davis, W.C.; Travis, J.R.: Shock Initiation of Liquid Explosives. Phys. Fluids 4 (1961) 498.

5. Fickett, W.; Davis, W.: Detonation. Berkeley, CA: University of California Press 1979.

6. Rabie, R.L.; Fowles, G.R.; and Fickett, W.: The Polymorphic Detonation. Phys. Fluids .22(3) (1979) 422.

7. Fowles, G.R.: Stimulated and Spontaneous Emission of Acoustic Waves from Shock Fronts. Phys. Fluids 24(2) (1981) 220.

8. Amsden, A.A.; Ruppel, H.M.; Hirt, C.W.: SALE: A Simplified ALE Computer Program for Fluid Flow at All Speeds. Los Alamos Report LA-8095, Los Alamos, N.M.

9. Callen, HB.: Thermodynamics and an Introduction to Thermostatistics, 2nd Ed., p. 74 ff. New York: Wiley and Sons.

10. Reichl, L.E.: A Modem Course in Statistical Physics, p. 627. Austin: University of Texas Press 1980.

11. Hixson, R.S.: Vapor Phase Detonations in Light Hydrocarbons. Ph.D. Dissertation, Washington State University 1980.

12. Flock, R.A.: Vapor Explosions in a Superheated Liquid. Ph.D. Dissertation, Washington State University 1986. Also Flock, R.A.; Fowles, G.R.: Explosive Phase Transitions in Superheated Freon, Shock Waves in Condensed Matter, 1983, p. 273. Asay, l.R.; Graham, R.A.; Straub, G. K. (eds.). Elsevier Science Publishers B.V. 1984.

13. Thompson, P.A.; Chaves, H.; Meier, G.E.A.; Kim, Y.: Speckmann, H.D.: Wave splitting in a fluid of large heat capacity. l. Fluid Mech. 185 (1987) 385.

Propagation of a Vapor Explosion in a Confined Geometry

D. L. Frost, G. Ciccarelli, and C. Zarafonitis

Mechanical Engineering Department McGill University, Montreal, Canada

Abstract

The propagation of a vapor explosion within a narrow channel has been investigated experimentally with high-speed photography and fast-response pressure instrumentation for two different liquid pairs: (i) a liquid refrigerant and hot oil, and (ii) molten tin and water. In each case, the explosive interaction propagates as a front with a typical velocity in the range of 40-80 mis. The pressure rise associated with the front has a peak overpressure in the range of 2-9 bar. The low propagation velocity and relatively long pressure rise times (~1 ms) indicate that the propagation is not coupled to a leading shock wave. With liquid refrigerant drops immersed in oil, spatial propagation of the interaction is due to sequential explosion of the drops. The pressure rise and associated flow from a local explosion collapses the vapor film surrounding a nearby drop, leading to explosion propagation. The conversion ratio of thermal to mechanical energy was estimated to be about 5% in the refrigerant/glycol system. A similar propagation mechanism is observed for molten tin drops immersed in water. In a stratified molten tin/water system, the explosive interaction propagates along the surface of the tin, producing a wedge-shaped wake region that lofts the overlying water and the majority of the tin.

1 Introduction

A violent vapor explosion can result when a cold volatile liquid is suddenly brought into contact

with a hot liquid. The explosion is due to the rapid vaporization of the cold liquid from heat

transfer from the hot liquid. Such explosions are referred to as vapor, steam, physical, or

thermal explosions; rapid-phase-transitions (RPTs) (typically when referring to explosions

involving cryogenic liquids); and molten fuel-coolant interactions (FCIs) when applied to

nuclear reactor accidents. Accidental vapor explosions are frequent occurrences in the

metallurgical, pulp and paper, and cryogenic industries. General reviews of the various aspects

of vapor explosions can be found in Reid [1] and Corradini et al. [2].

The general requirements for a vapor explosion to occur following mixing of a hot and cold

liquid are well established: (1) direct liquid-liquid contact is required, and (2) the temperature of

the hot liquid must exceed a threshold value. For vapor explosions involving a cryogenic liquid

and a hot liquid, strong evidence exists showing that this threshold temperature is equal to the

superheat limit temperature of the cold liquid [1]. If the temperature of the hot liquid is

G, E, A, Meier' p, A, Thompson (Eds,) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Gettingen. Germany. 1989

© Springer-Verlag Berlin Heidelberg 1990

418

considerably higher than this threshold value (e.g., for molten metal-water interactions), then

stable film boiling between the two liquids occurs initially and some external disturbance is

necessary to initiate a vapor explosion. In a large-scale vapor explosion, an additional

requirement is that a propagation phase must occur in which a local interaction escalates into a

coherent large-scale explosion.

Very little is known regarding the propagation mechanisms of vapor explosions. Propagation

refers to the progressive spatial development of the explosion process from a localized region

where an interaction is first initiated. to the rest of the coarse mixture of the hot and cold liquids.

The propagation limits, i.e., the limiting conditions required to support the self-sustained

propagation of a vapor explosion, have yet to be determined. Analogous to a combustion

wave, it may be possible for different propagation regimes or modes to exist for the same

liquid-liquid mixture, depending on the initial and boundary conditions (e.g., thermodynamic

conditions, mixture geometry, trigger amplitude, degree of confinement). A fast propagation

mode has been proposed (Board et al. [3]) in which the release of thermal energy is coupled to a

leading shock wave, in analogy with a chemical detonation. In this propagation mode, shock­

induced fragmentation of the dispersed molten fragments leads to rapid heat transfer to the

coolant. If the timescale for energy release is sufficiently short, the pressure generated within

the reaction zone can reinforce the strength of the propagating front. Although numerical

calculations of thermal detonations (e.g., [4]) have been carried out for molten metal-water

systems, the results have not been validated experimentally. To improve the numerical models,

results from well-characterized experiments are required to clarify the physical mechanisms

involved. Even for the case of shock interaction with a single molten drop, the timescales and

rates associated with the processes that occur during the vapor explosion (e.g., heat transfer,

vaporization, fragmentation) are poorly understood.

The propagation of a vapor explosion through a coarse mixture of hot and cold liquids has been

observed in medium-scale tests [5] as well as in laboratory-scale experiments (e.g., [6])

although the details of the propagation mechanisms have not been resolved. Small-scale

experiments have shown that vapor explosions can also propagate in systems in which the hot

and cold fluids form an initially unmixed stratified configuration [7]-[9]. The objective of the

present study is to i~vestigate the physical mechanisms involved in the propagation of a vapor

explosiol1 in laboratory-scale experiments in both coarse mixture and stratified geometries with

the use of high-speed photography and fast-response pressure instrumentation. Results will

first be given illustrating the propagation of a RPT in a liquid refrigerant-oil system. These

results will then be compared with the propagation of a vapor explosion in molten metal-water

systems confined within a narrow channel.

419

2 Apparatus

RPTs are generated by spilling ofrefrigerant-114 (C2Cl2F4) into ethylene glycol. The choice of

the volatile liquid is governed largely by convenience: R-114 has a boiling temperature and

superheat limit temperature of 4°C and about 102°C, respectively. The apparatus for the R-114-

ethylene glycol tests consists of an aluminum channel (length 35 cm, height 9.5 cm, width 1.9

or 5.4 cm) with lexan windows. Four piezoelectric pressure transducers (PCB 113A26, 10

m V /psi) are flush-mounted on the base of the channel with a spacing of 10.2 cm. The R-114 is

contained in a plexiglas tube with a narrow slot spanning the length of the tube that is mounted

above the channel. Rotation of the cylinder causes the refrigerant to discharge into the glycol,

forming a coarse mixture of liquid R -114 drops with sizes on the order of 1 cm surrounded by

refrigerant vapor. RPTs are triggered by a shockwave generated by discharging a HV capacitor

(0.05 J.!f charged to 20 kV) across a spark gap at one end of the channel in the glycol. The spark

discharge is triggered when the R-114 intercepts a low-power laser beam within the channel,

causing a photodiode detector circuit to generate a triggering pulse. A standard video camera

(30 frames/s) and a high-speed Hycam 16 mm camera (5000 frames/s) are used to visualize the

development of the RPT.

A similar apparatus is used for the molten tin-water experiments and has been described earlier

[10]-[12]. The channel (length 40 cm, height 15 cm, width 1.27 cm) is placed within a larger

glass tank filled with water. Interactions are triggered by discharging a high-voltage capacitor

(4.0 J.!f charged to 4 kV) through a thin wire mounted within the channel at one end. Four

pressure transducers are mounted in the back window, 10.2 cm apart and 2.2 cm above the

channel base.

3 Results and Discussion

3.1 R-114/ethylene glycol RPTs. A series of trials was carried out to investigate the

propagation of a RPT through a coarse mixture of R -114 and glycol. The characteristic features

of a propagating interaction are illustrated in Fig. 1 which contains a series of single frames

(250 J.!s between frames) taken from the high-speed film record of one of the trials. In this case

80 ml of saturated R-l14 were spilled into glycol at 130°C within a channel of width 1.9 cm.

The interaction was initiated 4 cm to the left of the field of view about 0.5 ms before the first

frame shown in Fig. 1. The local interaction causes adjacent drops to explode, and the RPT

propagates through the field of view at a speed of about 70 mls. During the expansion phase,

the refrigerant vapor regions grow and coalesce and the majority of the glycol in the channel is

accelerated vertically and ejected from the channel. Fig. 2 shows the pressure that is recorded

420

Fig. 1 Propagation of RPT through coarse mixture of saturated drops of R-114 immersed in ethylene glycol at 130·C (250 ~ between frames, background grid has a length scale of 0.64 cm)

along the channel base for a trial with similar conditions. The propagation velocity calculated

from the pressure pulses corresponds to the visual propagation speed estimated from the high­

speed films. An expansion of the early part of the signals is shown on the right in Fig. 2,

illustrating the noise generated by the spark discharge and the propagation of the triggering

o 2 4 6 8 10 12 14 16 Time (ms)

0.0 0.5 1.0 1.5 Time (ms)

Fig. 2 Overpressure generated along channel base during propagation ofRPT in R-114-g1ycol mixture (transducer spacing: 10.2 cm). Pressure traces are shown on the right with an expanded timescale illustrating noise from spark discharge and propagation of triggering shock wave.

421

2.0

shock wave which moves at the acoustic speed in the glycol, or about 1800 mls. The impulse

of the triggering pressure wave is negligible compared to that associated with the RPT. In 15

trials in which 80 m1 of R-114 were used, the propagation velocity varied between 55 and 80

mls with peak overpressures ranging from 3 to 8 bar (with pressure risetimes varying from

several hundred microseconds to a millisecond). Although some of the variation in the

propagation speeds obtained is due to variations in the initial geometry of the mixture, there was

no clear correlation between propagation velocity and pressure generated or channel width.

The long risetimes and slow propagation speeds, relative to the sound speed in the glycol,

indicate that the propagation of the RPT is not coupled to a leading shock wave. The pressure

generated by the RPT is due to the flow of the relatively incompressible glycol due to the

expansion of the high-pressure vapor generated. From the high-speed films it is evident that

after the RPT is initiated, it propagates as a sequential series of local explosions of the coarsely

mixed fragments of liquid R-114. The speed of propagation is governed by the timescales

associated with several important processes: film collapse, mixing and heat transfer following

liquid-liquid contact, and vaporization of the volatile liquid, although the details of these

processes are too complicated to be able to predict a priori the propagation speed for a given

coarse mixture.

422

A second series of trials was carried out to study the effect of R -114 volume on the yield of the

explosion. In each trial the initial glycol temperature, volume, and layer depth were 138'C, 1

liter, and 5 cm, respectively, and the channel width was 5.4 cm. The results of the trials are

shown in Table 1 below. Self-sustained propagation was observed for each trial except for a R-

114 volume of 25 ml. The average propagation velocity was determined from the time required

for the pressure pulse to move from the first to fourth transducers. The efficiency of the

conversion of thermal to mechanical energy may be estimated by considering the kinetic energy

of the glycol that is ejected from the channel during the interaction. Although the glycol velocity

is not measured directly, if the flow of the glycol is assumed to be one-dimensional, then the

glycol velocity can be related to the pressure field at the channel base by integrating Newton's

first law for the glycol to obtain an expression for the velocity, i.e., V = J Fdt/m = A J Pdt/m =

A//m, where A = surface area of glycol, m is the accelerated glycol mass, and / is the pressure

impulse !pdt. In RPTs involving cryogenic liquids, it is conventional to define the conversion

ratio as the ratio of the kinetic energy of the hot fluid to the thermal energy stored in the volatile

liquid, assuming that the cold liquid has been heated to the superheat limit temperature. Using

this definition and the above expression for the velocity gives the following expression for the

conversion ratio:

CR _ KEg/yeo/ _ (Alp/2m - [mCp(Ts/- Tsat)JR-114 - [mCp(Ts/- Tsat)JR-114

(1)

where the subscripts sl and sat refer to superheat limit and saturation conditions, respectively.

The results are shown in Table 1 below. The energy yield rises roughly linearly with the

volume of the volatile liquid. For the trials in which self-sustained propagation of the explosion

occurred, the conversion ratio was on the order of 5%. This is a conservative value because the

energy imparted to the channel walls is neglected and also the thermal energy available in the R-

114 is calculated using the initial R-114 volume, whereas the actual volume may be somewhat

less due to evaporation of the R-114 prior to the RPT.

R-1l4 vol Pmax,avg Vprop,avg KEglycol KE/volume Conv.ratio

(ml) (bar) (m/s) (J) (J/I) (%)

80 5.6 77 595.8 7.45 5.5

50 3.4 75 310.7 6.21 4.6

40 2.7 80 187.2 4.68 3.5

30 2.9 55 198.4 6.61 4.9

25 0.8 prop. failed 35 1.40 1.0

Table 1

423

3.2 Molten tin-water steam explosions. Experiments have also been carried out to

study the propagation of a vapor explosion through a mixture of molten tin and water and are

described more fully in earlier publications [10]-[11]. The propagation of an explosive

interaction through multiple drops of tin immersed in water and confined within a narrow

channel is illustrated in Fig. 3. In this trial 20 5 g tin drops at 700·C were dropped into water at

58·C. The exploding wire trigger (visible at the left of the first frame) initiates a local interaction

and the explosion propagates to the right at a speed of about 40 mls. Peak overpressures of up

to 5 bar are recorded within the channel.

Fig. 3 Propagation of a vapor explosion through 20 5 g drops of molten tin immersed in water and confined within a channel of width 1.27 cm (1.2 ms between frames).

If the tin is allowed to settle to the bottom of the channel a propagating interaction may still be

initiated. A series of trials was carried out in which 400 g of tin at 700·C were poured into

water at 70·C and allowed to form a stratified layer. Interactions were initiated with an

exploding wire trigger and self-sustained propagation of the interaction occurred consistently

with a water height of 12.7 cm. Fig. 4 illustrates the characteristic features of the interaction.

424

Fig. 4 Propagation of an explosive interaction in a stratified molten tin-water system (400 J..lS between frames) . Diameter of transducer plug mounted on back of channel (visible at left) is 2.5 em.

The interaction propagates along the surface of the tin layer with an average speed of about 40

mls. The wake region behind the leading edge of the explosion zone has an overall shape of a

wedge (fonning a wedge angle of about 10°) and the water above the tin layer is accelerated

vertically by the high pressure within the reaction region. Fig. 5 shows the characteristic

pressure pulses recorded within the water (2.2 cm above the channel base) during an

interaction. The long risetime (-1 ms) of the pressure pulses and slow propagation speed of the

interaction (-40 mls), relative to the acoustic speed in water (1500 mls), indicate that the

explosion is not coupled to a leading shock wave in the water. The pressure rise in the water is

due to the hydrodynamic flow generated by the high-pressure produced within the interaction

region. The stagnation pressure associated with a propagation speed of 40 mls is 8 bar, the

same order as the peak pressures recorded. The general shape of the pressure profile in the

425

water can be reproduced with a s imple potential flow model in which the interaction zone is

replaced with a moving wedge that displaces the water above it [12].

Fig. 6 shows a plot of the average peak overpressure (averaged over the four transducers) as a

function of the average propagation velocity for trials in which a self-sustained propagation was

observed. Although the peak overpressure generated varies over a wide range (2 to 9 bar), the

propagation velocity has a smaller variation (35 to 47 mls). It was found that the degree of

inertial confinement provided by the water layer above the tin controls whether a self-sustained

propagation is possible. For a water height of 5.1 cm, in 7 trials propagation never occurred

whereas for a water height of 12.7 cm, self-sustained propagation occurred in each of the 25

trials. The kinetic energy of the water and tin lofted during the interaction was estimated to be

about 25 1. Since the initial sensible enthalpy of the molten tin is large (-40 kJ), the conversion

ratio based on the available thermal energy of the tin is small (-0.06%).

o 2 4 6 8 10 12 14 16 Time (ms)

Fig. 5 Overpressure recorded during propagation in stratified tin-water mixture

4 Conclusions

12

ro1O e ~ 8 :::::l en

6 en ~ 0.. ~ 4 ~ Q)

a... 2

0 0

Fig.

10 20 30 40 50 60 Propagation ve locity (m/s)

6 Peak overpressure vs. propagation velocity for stratified experiments

The propagation ofa vapor explosion has been investigated in laboratory-scale experiments

using two different systems: (i) a liquid refrigerant immersed in hot oil, and (ii) molten tin

immersed in water. Despite the differences in geometry, fluid properties and thermodynamic

conditions, vapor explosions were observed to propagate in both systems with a speed in the

range of 40-80 mls with peak overpressures ranging from 2-9 bar. The slow risetime (-1 ms)

of the pressure pulses and slow propagation speed, relative to the ambient liquid sound speed,

show that the propagation is not coupled to a leading shock wave. In the refrigerant-oil system,

426

spatial propagation of the interaction is due to sequential explosion of the refrigerant drops. The

pressure rise and the associated flow of the surrounding fluid due to violent boiling of one drop

causes the film surrounding an adjacent drop to collapse, sustaining the propagation. In this

system, the conversion ratio of thermal to mechanical energy based on the maximum superheat

of the refrigerant available was estimated to be about 5%. Propagating interactions are also

observed in a stratified system of molten tin and water. The interaction propagates along the

surface of the tin, producing a wedge-shaped explosion zone, and lofting the overlying water

and the majority of the tin. The degree of inertial confinement provided by the water layer

above the tin controls whether a self-sustained propagation is possible. Only a small fraction of

the total thermal energy of the tin is converted to mechanical energy during the interaction.

Acknowledgements

The authors would like to thank Prof. J. H. S. Lee for many useful discussions. Financial support was provided by the Natural Sciences and Engineering Research Council of Canada, Shell Thornton Research Centre and Sandia National Laboratories.

References

l. Reid, R. c.: Rapid phase transitions from liquid to vapor. Adv. Chern. Eng. 12 (1983) 105-208.

2. Corradini, M. L.; Kim, B. J.; Oh, M. K.: Vapor explosions in light water reactors: a review of theory and modelling. Progress in Nuclear Energy 1 (1988) 1-117.

3. Board, S. J.; Hall, R. W.; Hall, R. S.: Detonation of fuel coolant explosions. Nature 254 (1975) 319-325.

4. Fletcher, D. F.; Thyagaraja, A.: Multiphase detonation modelling using the CULDESAC code. Presented at 12th ICDERS, Univ. Michigan (1989).

5. Mitchell, D. E.; Corradini, M. L.; Tarbell, W. W.: Intermediate-scale steam explosion phenomena: experiments and analysis. NUREG/CR-2145 SAND81-0124 Sandia National Laboratories 1981.

6. Baines, M.: Preliminary measurements of steam explosion work yields in a constrained system. Inst. Chern. Eng. Sym. Series 86 (1984) 97-108.

7. Board, S. J.; Hall, R. W.: Propagation of thermal explosions: I-Tin/water experiments. CEGB RD/B/N2850 1974.

8. Anderson, R.; Armstrong, D.; Cho, D.; Kras, A.: Experimental and analytical study of vapor explosions in stratified geometries. ANS Proc. 1988 National Heat Transfer Conference 3 (1988) 236-243.

9. Bang, K. H.; Corradini, M. L.: Stratified vapor explosion experiments. ANS Proc. 1988 National Heat Transfer Conference 3 (1988) 228-235.

10. Frost, D. L.; Ciccarelli, G.: Dynamics of explosive interactions between multiple drops of tin and water. Progress in Astronautics and Aeronautics 114. AIAA Press (1988) 451-473.

11. Frost, D.; Ciccarelli, G.: Propagation of explosive boiling in molten tin-water mixtures. Proc. of 1988 ASME/AIChE National Heat Transfer Conference 2 HTD-96 (1988) 539-548.

12. Ciccarelli, G; Frost, D. L.; Zarafonitis, C.: Dynamics of explosive interactions between molten tin and water in stratified geometry. Presented at 12th ICDERS, Univ. Michigan (1989).

The Development of Cavity Clusters in Tensile Stress Fields K.A. Morch

Laboratory of Applied Physics Technical University of Denmark DK-2800 Lyngby, Denmark

Summary

The development of a cavity cluster from a distribution of supercritical cavitation nuclei at their exposure to tensile stress is discussed. An approach to this problem was presented by Hansson et al. [1], and is the basis of further analysis and comparison of planar and spherical cavity cluster development. The stress penetration into the cluster depends primarily on the inter---cavity distance and on the cluster form. In interplay with the cavity dynamics it determines an acoustic impedance of the cluster boundary which approaches zero during cavity growth, and so the tensile stress at the boundary resulting from the incident and the reflected waves becomes small which indicates that not only this pressure but also the equilibrium pressures of the cavities are important for the cluster development.

Introduction

In a flow of liquid around a solid body cavity clusters may develop by growth of cavi­

ties from nuclei which are convected with the flow into a region of tensile stress. Such

cavities develop continuously or intermittently at a certain location at the body surface

and they are convected with the flow typically into the wake where they accumulate

into clusters whose development depends on the pressure field and which themself may

influence this field and possibly the conditions of cavity form~tion at the site of incep­

tion, so that a cyclic cavitation process is set up. Such a cavitation cycle is quite com­

plicated, and it is feasible to study the basic character of the different phases of cluster

development by considering liquid systems in which flow problems do not add to the

complexity, i.e. systems in which cavitation is generated by simple waves such as the

planar wave set up by the motion of a piston, or a spherical radially symmetrical

standing wave in a resonating spherical bottle. The boundary conditions at the inter­

face between single-phase liquid and cavity cluster change during the cavitation cycle

which roughly can be divided into a cavity cluster nucleation phase, a cavity growth

phase, and a cluster collapse phase. The latter has been the subject of several papers

[2]-[5] and the growth of the cavities in a planar cluster was considered in [1], while

acoustically generated cluster nucleation was treated recently in [6]. The existence of a

distribution of supercritical cavitation nuclei is the fundamental basis for the formation G. E. A. Meier· P. A. Thompson (Eds.) Adiabatic Waves in Liquid-Vapor Systems IUTAM Symposium Gottingen. Germany, 1989 © Springer-Verlag Berlin Heidelberg 1990

428

of a cavity cluster, but it is in general difficult to give details about the nuclei in a real

liquid. If gaseous (subcritical) nuclei are present in the undisturbed liquid a tensile

stress wave of sufficient strength causes growth of these nuclei and the pressure drops

until critical conditions are reached. Beyond this point the wave is not propagated in

the cavitating medium but it is reflected at its boundary and the pressure and velocity

distributions inside the zone of cavitation are governed by the cavity dynamics. In the

present paper the cavity development in planar clusters is considered in continuation.of

[1] and the growth of a spherical cluster exposed to a focused radial wave field is dealt

with. One of the objectives is to determine during cavity growth the reflection condi­

tions for the wave that generates the cluster, another to analyse the cluster structure.

The Theoretical Model

The planar cavity cluster

We consider a one-dimensional field of liquid in which a stress wave has caused the

cavitation nuclei to grow beyond critical size, thus forming a column of microcavities of

radius ao at coordinates x ~ O. The cavities are assumed to be uniformly distributed,

forming an hcp-Iattice with the c-axis in the x-direction and with an inter-cavity

distance f!. As a low frequency sound speed does not exist in the cavitating medium

stress propagation beyond the interface between the two-phase and the single-phase

medium at x = 0 is connected to the liquid phase and governed by the cavity dynamics

which is described by the Rayleigh-Plesset equation

(1)

where p is the local pressure, Peq is the equilibrium pressure of a cavity of radius a and

Po is the density of the liquid.

For the two-phase medium the equation of mass conservation

(2)

reduces to

(2a)

as p aul ax > > u Bpi ax, and with the assumed cavity structure the void fraction is

429

(3)

which gives the density of the medium P = Po (1-,8) and with (2a)

(4)

The equation of motion

(5)

can be reduced to

(5a)

Using P ~ Po differentiation of (5a) with respect to x gives with (4)

2 2 8 p + 16v'27r !Y2 (~ a 8 a + ~ (8a)2) - 0 axx 3 £3 4 "0[2" 2 at - (6)

Combining (1) and (6) we find

2 2 ~ _ 16v'27r a (_p + ~ 8 a P ) = 0 axx 3 1JP eq 4"0[2"0 (7)

This equation essentially corresponds to the one found in [1], where it should be noted

that the exponential pressure change from the cluster boundary (eq. (7) in [1]) is based

on an approximation implicating that the change of cavity size is small, which it is not,

neither in the numerical examples given nor in real experiments. The development in

time and space of a planar cavity cluster from a distribution of micro-cavities at expo­

sure to a tensile stress at the cluster boundary must be determined by simultaneous

numerical solution of the two differential equations (1) and (7). The front of the pres­

sure disturbance imposed at the cluster boundary propagates into the cavitating me­

dium at the speed of sound in the liquid co, but quickly it is attenuated as energy is

used to set up the radial motion of the liquid around each cavity and to establish the

superposed translational motion of the liquid, connected to the growth of the cavities.

430

The velocity of the liquid at the cluster boundary x = 0 connected to a pressure distur­

bance (p )x=o is found from (4)

cot

(u)x=o = - J~ a2 ~ dx (8) o

The boundary condition (P)x=o can be interpreted as the result of the reflection of an

incident acoustic wave at the cluster boundary which exhibits a nonlinear acoustic

impedance connected to the cavity dynamics [6]. The reflection coefficient is

k = ~ = (P-PoCou) Zlpi P+Pocou

(9) x=o

in which .6.Pi and .6.Pr are the pressures of the incident and reflected waves, respective­

ly.

Likewise, a piston at the position x = 0 moving with the velocity (u)x=o would set up

the pressure (p )x=o at the boundary of the ca vi ty cluster.

The spherical cavity cluster

The formation of a cluster of supercritical cavities within a confined spherical region

around the focal point of a focused acoustic field is considered in [6]. It is shown that

micro-<:avities are produced throughout a region of radius R by an acoustic field of

sufficient strength, but at larger radii cavities do not grow. At exposure to a tensile

stress the cluster boundary is considered to constitute a compliant interface at which a

tensile stress wave is reflected with a phase shift 11" and on this basis the development of

the cluster mean void fraction is determined. However, it remains to be discussed how

cavity growth is established throughout the cluster, as planar theory gives stress decay

within a very short range from the cluster boundary.

In the analysis of the spherical cavity cluster development a radial coordinate system

with origin at the focal point of the acoustic field is used, and the cluster radius is R. Equation (1) is independent of the cluster type but the mass conservation equation is

§.£ 1 {} at + ~ (Ji (r2pv) = 0 (10)

in which v is the radial velocity of a fluid element. It can be reduced to

while the equation of motion remains

Using the same procedure as in the planar case we obtain

and with (1)

2 Q..Q + ~ Up + 0 (4/lJra2 oa) = 0 &2 r dr P at -r.r at

431

(lOa)

(11)

(12)

(13)

which for large r-values reduces to the planar case, but for small r the second term

causes deep penetration of the pressure disturbance at the cluster boundary into the

cluster.

Likewise, from (lOa) the velocity at the cluster boundary

R

(v)r=R = J (~~- ;v) dr (14)

r=O

Numerical results

In the numerical calculations it is assumed that the cluster develops from micro--cavi­

ties of initial radius ao = 10 pm, and that the imposed pressure (tensile stress) causing

cavity growth I pi> > I Peql , so that void dilatation due to different equilibrium si.zes

of the cavities during their growth is negligible. It is chosen to apply a pressure distur­

bance at the cluster boundary

(P)x=o = (P)r=R = - Pm sin wt t ~ 0 (15)

432

From (7) and (13) it is apparent that f is an important parameter for the cluster deve­

lopment. The initial cavity size ao may be of importance directly as well as through Peg

if p is small.

The planar cavity cluster

The numerical results obtained from simultaneous integration of (1) and (7) are shown

in figs 1-3 for a planar cavity cluster at Pm = 20.103 Pa with f = w/27r = 20 kHz when

f = 0.3 mm. It is apparent from fig. 1 that initially the tensile stress in the cluster

drops exponentially with the distance from the cluster boundary [1], but after 4 JtS the

pressure profile steepens near the boundary due to the significant growth of the cavities

here. The tensile stress decays beyond the first few cavity "layers" where growth of the

cavities primarily occurs. The profile of increase of cavity radius vs. position is found to

be essentially exponential, fig. 2. The pressure assumed from (15) is set up if an inci­

dent acoustic wave with a pressure ~Pi vs. time t as shown in fig. 3 reaches the cluster

boundary. It is noticed that very quickly the reflection coefficient approaches -1,

showing that the boundary becomes an essentially compliant interface.

A piston at the cluster boundary could produce the planar cluster described, if moved

with the velocity (u)x=o also shown in fig. 3. The associated acceleration is high for a

mechanical system and a frequency f = 2 kHz might be more reasonable. For such a

case the tensile stress penetration into the region with microcavities is weaker, but

after 30 JtS the cavity growth is approximately as after 16 JtS at 20 kHz.

If the stress amplitude is reduced to Pm = 2 kPa with f = 2 kHz a cavity growth as

above is obtained after about 70 JtS, and then Ux=o = - 0.49 mls with k = - 0.996. The

pressure at the cluster boundary penetrates slightly deeper than above, but still only

the outermost cavity "layers" are appreciably affected and the cavity growth profile is

essentially the same in all cases, only the time of development changes.

A change of the initial cavity size to ao = 5 /.Lm is also of minor importance, except that

it causes a steeper initial growth of the cavities.

As mentioned above the primary parameter for the cluster development is the inter-ca­

vity distance f. A change of £from 0.3 mm to 3 mm (at f = 2 kHz, Pm = 2 kPa, ao = 10

/.Lm) gives the pressure penetration shown in fig. 4 and an associated increase of the

cavity radius vs. position given in fig. 5. It appears that the effects of the imposed

E 0.

"-0. 1

1~

0.001+-0 ---~---.-----0.5 x/mm

Figure 1. The pressure p vs. position x in a planar cavity cluster in which ao= 10 f.1m, e = 0.3 mm when f = 20 kHz, Pm = 20 kPa. The pressure at the clus­ter boundary (P)x=o = -Pm sin (27f ft).

tillS

433

10

0.1

4 Ils 0.01+-----r'---~-­

o 05 x/mm

Figure 2. The increase of the cavity radius (a-ao) vs. position x as con­nected to the pressure distributions shown in figure 1.

~ 0 t--===-----TS ___ ....:1rO ___ ..:.c1S"-E "-o

11 X

:3 ",,'-05

ru Q..

L "­a: <I -1

Figure 3. The pressure of the incident wave flPi vs. time t at the boundary of a planar cluster causing the pressure and cavity growth distributions shown in figures 1-2, the associated wave reflection coefficient k and the velocity (u)x=o of the fluid elements at x = o.

434

0: 001 "-CL I

O'O01+----~--~--~ o 10 20 30

x/mm

Figure 4. The pressure p vs. position x in a planar cavity cluster in which ao = 10 j.tm, e = 3 mm when f = 2 kHz, Pm = 2 kPa and (p h=o = -Pm sin(21l" ft)

r/mm

0~1 ___ ~ ___ ~? ___ ~3

~---------------_.£J:lS R , -----I

Figure 6. The pressure p vs. radius r in a spherical cavity cluster in which ao = 10 j.tm, e = 1 mm when f = 20 kHz, Pm = 20 kPa and (P)r=R= -Pm sin(27r ft)

o d "-

o d I

d

10

Figure 5. The increase of cavity radius (a-ao) vs. position x as con­nected to the pressure distributions shown in figure 4.

101

! R1!? __ --...--1

1[_~==-_~S __ -~ dO I ~

I

a o,j 4~ 1--------

I _,_"'------~ om r--------------- I 023

r/mm

Figure 7. The increase of cavity radius (a-ao) vs. position r as con­nected to the pressure distributions shown in figure 6.

435

pressure (p )x~o penetrates about 30 times deeper into the cavitation region at an increa­

se of £ by a factor 10.

The spherical cavity cluster

Spherical cavity clusters are typically generated by focused acoustic waves, and from

experiments R = 3 mm, £ = 0.3 mm are realistic values at f = 20 kHz. Assuming that

ao = 10 11m and Pm = 20 kPa we find from (1) and (13) that when Ipi » I Peg; I the

effects of sphericity are of minor importance as the stress penetration is increased by

only 10 % compared with the planar case.

However, changing the intercavity distance to f = 1 mm causes the tensile stress and

the cavity growth to penetrate to the cluster centre as shown in figs. 6 and 7, respecti­

vely.

Discussion

The development of a planar cavity cluster from a uniform distribution of micro--cavi­

ties is connected to the imposed stress field which decays from the cluster boundary, at

first essentially exponentially [1], but as the cavities grow the decay becomes stronger

because the cavity radius influences the second term in (7). However, the change of the

cavity radius vs. position in the cluster turns out to remain almost exponential over a

long period of time. The penetration of the stress into the cluster depends primarily on

the inter--cavity distance f and increases significantly with £, but still the decay con­

stant is of the order of one to a few inter-cavity distances only. During cavity growth

the reflection coefficient for tensile stress waves is found to approach -1 very quickly,

and thus the cluster is essentially comparable to a compliant wall with an acoustic

impedance approaching zero. Consequently, the stresses at the cluster boundary be­

come small during cavity growth.

The calculations indicate that realistic values of the tensile stress occuring at the

cluster boundary are of the order of the critical stress for normal cavitation nuclei, and

it means that in (7) it may be a very crude approximation to consider I pi> > I Pegl.

In addition differences of initial cavity radius with position may influence the cluster

development.

Considering a spherical cluster it appears that when I pi> > I Peg I the effect of spheri­

city is small at cluster radii and inter--cavity distances as expected from experiments,

436

but moderate increase of e causes stress penetration and cavity growth throughout the

cluster. Thus the sphericity term in (13) is close to being significant. The cavities are

calculated to develop strongest at the cluster boundary, but from experiments it ap­

pears that those at the center (the focal point) develop strongest [7]. This indicates

that though the cavities reach critical size almost simultaneously [6] those at the center

relax first to supercritical conditions, and so, Peq decreases (the tensile stress increases)

in radial direction ·which causes growth of the central cavities at the expense of the

outer ones. In (13) Peg causes reduction of the third term, thus strengthening the

influence of the second term. It suggests that Peq is an important parameter for ex­

plaining the spherical cluster development. A major problem in this context is that

very little is known about the initial size distribution of cavitation nuclei that can be

used to make reliable calculations of its effect on the cluster development.

References

1.

2.

3.

4.

5.

6.

7.

Hansson, 1.; Kedrinskii, V.; Morch, K.A.: On the dynamics of cavity clusters. J. Phys. D: Appl. Phys., 15 (1982) 1725-1734.

v. Wijngaarden, L.: On the collective collapse of a large number of gas bubbles in water. Proc. VIth Int. Congr. Applied Mech. Munich 1964 (H. G6rtler ed.) 854-861, Springer Verlag.

Morch, K.A.: On the collapse of a cavity clusters in flow cavitation. Springer Ser. in Electrophys., 1: (1980) 91-100.

Hansson, 1.; Morch, K.A.: The dynamics of cavity clusters in ultrasonic (vibrato­ry) cavitation erosion. J. Appl. Phys., 51 (1980) 4651-4658 and 52 (1981) 1136.

Chahine, G.: Pressure field generated by the collective collapse of cavitation bubbles. Proc. IAHR Symp. Operating Problems of Pump Stations and Power Plants. Amsterdam 1982. Paper 2, 1-l4.

Morch, K.A.: On cavity cluster formation in a focused acoustic field. J. Fluid Mech. (1989) 201, 57-76.

Ellis, A.T.: Techniques for Pressure Pulse Measurements and High-Speed Pho­tography in Ultrasonic Cavitation. Proc. N.P.L. Symposium on Cavitation in Hydrodynamics 1955. Paper 8, 1-32, Her Majesty's Stationery Office, London.