rarefied gas dynamics
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Dynamics of Rarefied Gas DynamicsTRANSCRIPT
Ann. Rev. Fluid Mech. 1989.21: 387-417 Copyright © /989 by Annual Reviews Inc. All rights reserved
RAREFIED GAS DYNAMICS
E. P. Muntz
Department of Aerospace Engineering, University of Southern California, University Park, Los Angeles, California 90089- 1 1 91
INTRODUCTION
Rarefied gas dynamics is a diverse field, encompassing, for example, highaltitude hypersonic flow fields, the reflective and reactive characteristics of gases interacting with solid and l iquid surfaces, energy-transfer phenomena in molecular collisions, aerosol dynamics, cluster formation and topology, flows induced by evaporation and condensation, upper-atmospheric dynamics, and the attainment of milli-Kelvin temperatures by flow cooling techniques. Other subj ects in the field include vacuum-pump performance, spacecraft contamination, a variety of interactions due to thruster plumes, spacecraft charging, and gas and isotope separations. Underlying all of these subjects is the central theme of the field of rarefied gas dynamics: the study of gas flow phenomena in which the discrete molecular nature of the gas cannot be safely ignored. The field has a rich heritage of analysis applied to the s tudy of flows where concep ts and techniques related to nonequilibrium statistical mechanics are important. An equally respected tradition is the development and application of instrumentation techniques that can be used to study the details of molecular motion in gas flows, as well as to study flow-generated populations of internal energy states and the characteristics of gases after surface encounters.
Rarefied gas dynamics is founded on the pioneering work in the kinetic theory of gases begun in the latter half of the nineteenth century, which has continued to the present. What today we call rarefied gas dynamics was discussed as early as 1 934 by Zahm (1 934), but research began in earnest shortly after World War II (Tsien 1 946). Because of growing interest in space and flight at extreme altitude, attention was drawn to a flow regime that had not been seriously considered as a concern of either aerodynamics or gas dynamics. In this flow regime, the molecular mean free path in the gas, .A., becomes significant compared with either a charac-
387 0066-4189/89/0 1 1 5 -0387$02.00
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teristic distance for important flow-property changes to take place or the size of the flow field, or more simply the size of the object creating the flow field. From about 1 950 to the late 1 960s there was active research, supported by several governments; these efforts were principally in the United States and the USSR, but major contributions were also made in France, Japan, Canada, Australia, Italy, the Federal Republic of Germany, The Netherlands, India, Poland, Y ugoslavia, the United Kingdom, and elsewhere. During this period a biannual series of International Symposia on Rarefied Gas Dynamics was initiated under the auspices of an informal advisory committee. The proceedings of these meetings have been published, beginning in 1 960; they form a unique record of the field (see Table I).
As it turned out, rarefied gas dynamics initially was somewhat ahead of its time. When all was said and done there were no early, critical aerospaceengineering problems associated with the regime. In the 1 950s and 1 960s reentries were ballistic, and thus vehicles quickly transited the uncertainty of the rarefied transitional flow regime at altitudes around 80-1 00 km. Besides, important things like heat transfer and aerodynamic forces were small in the rarefied regime compared with their effects at lower altitudes. In this time period, continuous or quasi-continuous flight in the 80-1 50 km altitude range was not a major concern. At higher altitudes satel lites did not have sufficient mechanical or electronic life for the low-level effects of the ambient atmosphere to be of great significance. As a result , from the late 1 960s to the early 1 980s governmental interest in the field, particularly in the United States, was at a low level . However, work continued in many countries, most notably in the USSR, Australia, Japan, and Europe. The International Symposia on Rarefied Gas Dynamics also continued because of intrinsic academic interest and because the field impacts many areas outside of space and high-altitude flight. Despite difficulties, significant scientific progress was made during this period.
We now come to what is the principal subject of this review: the likely consequences of a modern resurgence of governmental interest in rarefied gas dynamics and the direction work in this field can be expected to take in the coming decade. While there may be intense academic stimulation in a field, a fact of modern research life is that governments are usually required to energize major research endeavors. In the early and mid 1 980s several events have conspired to once again trigger governmental interest in work related to rarefied gas dynamics.
During the flights of Space Shuttle's STS-3 and STS-4 a low-light-level television camera on the orbiters recorded a gaseous glow above the windward tail surfaces of the vehicle (Banks et al. 1 983, Mende et al. 1 983). In the same period it was noticed that thermal insulation blankets returned
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Table 1 List o f the published proceedings o f the International Symposia o n Rarefied Gas Dynamics
RGD 1. Rarefied Gas Dynamics. 1960. Proceedings oJthe 1st International Symposium,
ed. F. M. Devienne. Paris: Pergamon RGD 2. Rarefied Gas Dynamics. 1 961. Proceedings oj the 2nd International Symposium,
ed. L. Talbot. New York: Academic RGD 3. Rarefied Gas Dynamics. 1963. Proceedings oJthe 3rd International Symposium,
Vols. 1 ,2, ed. J. A. Laurmann. New York: Academic RGD 4. Rarefied Gas Dynamics. 1965. Proceedings oj the 4th International Symposium,
Vols. 1,2, ed. 1. H. deLeeuw. New York: Academic RGD 5. Rarefied Gas Dynamics. 1967. Proceedin.qs oJ the 5th International Symposium,
Vols. 1, 2, ed. C. L. Bmndin. London: Academic RGD 6. Rarefied Gas Dynamics. 1969. Proceedings oj the 6th International Symposium,
Vols. 1 ,2, ed. L. Trilling, H. Y. Wachman. New York: Academic RGD 7. Rarefied Gas Dynamics. 1 971. Proceedings oJthe 7th International Symposium,
Vols. 1,2, cd. D. Dini, C. Ccrcignani, S. Nocilla. Pisa: Editrice Tecnico Scientifica RGD 8. Rarefied Gas Dynamics. 1974. Proceedings oJ the 8th International Symposium,
Vols. 1 ,2, ed. K. Karamcheti. New York: Academic RGD 9. Rarefied Gas Dynamics. 1974. Proceedings oj the 9th International Symposium,
Vols. 1,2, ed. M. Becker, M. Fiebig. Porz-Wahn, Germ: DFVLR-Press RGD 10. Rarefied Gas Dynamics. 1977. Proceedings oJthe 10th International Symposium,
Parts I and 2 of Vol. 51 of Progress in Astronautics and Aeronautics, ed. L. Potter. New York: AIAA
RGD 1 1. Rarefied Gas Dynamics. 1979. Proceedings oj the 11 th International Symposium,
Vols. 1 ,2, ed. R. Campargue. Paris: Commissariat a l'Energie Atomique RGD 12. Rarefied Gas Dynamics. 1 981. Proceedings oJ the 12th International Symposium,
Parts 1 and 2 of Vol. 74 of Progress in Astronautics and Aeronautics, ed. S. Fisher.
New York: AIAA
RGD 13. Rarefied Gas Dynamics. 1985. Proceedings oj the 13th International Symposium,
Vols. 1,2, ed. O. M. Belotserkovskii, M. N. Kogan, C. S. Kutateladze, A. K. Rebrov. New York: Plenum
RGD 14. Rarefied Gas Dynamics. 1 984. Proceedings oJ the 14th International Symposium,
Vols. 1,2, ed. H. Oguchi. Tokyo: University of Tokyo Press RGD 1 5. Rarefied Gas Dynamics. 1986. Proceedings oj the 15th International Symposium,
Vols. 1,2, ed. V. Bofti, C. Cercignani. Stuttgart: B. G. Teubner RGD 1 6. Rarefied Gas Dynamics. 1989. Proceedings oj the 16th International Symposium,
Progress in Astronautics and Aeronautics, ed. E. P. Muntz, D. Weaver, D. Campbell.
Washington: AIAA. In press
from orbit on the STS orbiters had been severely eroded (Whitaker 1 983, Peters et al. 1 983). In the early 1 980s i t also became evident that modern satellite computer systems frequently required re-booting due to upsets as a result of discharges traced to differential charging on satellite surfaces. The upsets were most frequent as the vehicles passed through the midnight to dawn quadrant of their orbits (Garrett 1 980, Fennell et al. 1 983) . All of these phenomena indicated that in addition to the effects of the
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magnetosphere's radiation belts, there are interactions between local space environments and satellite systems of practical, measurable significance. As extended satellite lifetimes are realized, the contamination of optical components and thermal control surfaces becomes increasingly important. Major sources of contamination are the flow fields of controland station-keeping thrusters. Optical contamination as a result of the ambient high-speed atmosphere interacting with exhaust gases has also been recognized as a potentially serious space-system problem. These occurrences represent phenomena that have been studied within the field of rarefied gas dynamics. Generally, increasingly frequent visits to space, as well as the intention of several countries to man permanent low-Earthorbit space stations and to establish permanent lunar or other bases, leads one to expect numerous rarefied-gas-dynamic problems associated with activities such as the mining and processing of native resources in space. Even the old idea of space scoops (Berner & Camac 1 961 ) has recently been rejuvenated in both its original form (Ramohalli et al. 1 987) and a variant (Muntz & Orme 1 987) .
Several nations have announced their intent to pursue the goal of transatmospheric flight with hypersonic winged vehicles. A representative flight corridor for such vehicles is shown in Figure 1 . While these airplanes tend to fly at rather low altitudes during acceleration in order to operate airbreathing engines, they also necessarily have quite sharp leading edges.
ItO 100
90
80
E = 70
� 60 � � 50 ..J <[
40
30
20
10
NASP RE-ENTRY
NASP CRUISE
AOTV-AFE '--I i I I I t /
NASP SPACE LAUNCH
8s � 10% STANDOFF RN- fern
o 2 3 4 5 6 7 8 9 10 11
SPEED (km/sec.) Figure 1 Flight corridor for NASP and aerobraking vehicles, with the ratio of the maximum slope shock thickness to shock standoff distance superimposed for leading-edge radii RN•
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RAREFIED GAS DYNAMICS 39 1
Chapman et al. (1987) have pointed out that at the sharp leading edges of the vehicle, where heat-transfer problems are severe, a significant portion of the trans atmospheric flight corridor will produce shock-wave thicknesses that are at least 1 0% of the shock standoff distance. To illustrate this point, lines indicating where the shbck-wave thickness has reached 1 0% of the standoff distance for 1 cm and 1 0 cm leading-edge radii are superimposed on the flight corridor in Figure 1 . For aero assisted orbital transfer trajectories [Aeroassist Flight Experiment (A FE)], the shock thickness is of the same order as the shock standoff distance even at a typical minimum altitude of 75 km (Bird 1 986). Thick shock waves are important because significant radiation and chemistry can take place within the shock wave (Bird 1985, 1986, 1987, Moss 1987) . Thus an old rarefiedgas-dynamics problem- shock-wave structure-appears with the additional complications of chemistry and radiation within the shock structure. Transatmospheric and aeroassisted vehicles also bring up, through attempts to extend continuum approaches to higher altitudes, velocity-slip and temperature-jump boundary conditions associated with the kinetic or Knudsen layer at the interface between the gas and the vehicle's surface. The hypersonic context of many of these effects has been surveyed by Cheng (1 988).
At any particular time, researchers in rarefied gas dynamics seem to be involved in one or more areas that are of great i nterest to other scientific fields. During the 1970s, gas-dynamic isotope separation techniques and the development of flow cooling to aid in the spectroscopic study of complicated molecules were two examples of such involvement. For the mid-1 980s and into the 1 990s, the unique properties of clusters created by gas-dynamic sources provide possibilities for stimulating interactions with investigators studying surface depositions of cluster complexes, cluster structures, and cluster chemistry. Related to this is the study of the dynamics of aerosol clouds, where the particles have very small diameters and may be continually condensing and evaporating, as well as chemically active. The presence of such clouds both in nature and in industrial applications has attracted the rarefied-gas-dynamics research community to address these problems.
In common with the rest of fluid mechanics, rarefied gas dynamics applies to a variety of technological devices in disparate scientific areas while at the same time having its own intrinsic attraction. In this article I review the status and some likely future directions for research in rarefied gas dynamics, keeping in mind the weighting factors introduced by governmental involvement. The particular subjects have been selected for two reasons: first, to represent the status or scientific position of the field in what are considered to be important areas, and second, to raise possibilities for interesting and useful future research.
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AVAILABLE LITERATURE
Rarefied gas dynamics' literature, as well as its basis in the kinetic theory of gases, is well organized. The chronological l isting of books, monographs, and reviews presented here in Table 2 are for the convenience of readers who would like to obtain either background or in-depth knowledge of the field. A quick catch-up can be obtained from the reviews of
Table 2 Background reading list (in chronological order) for rarefied gas dynamics
Knudsen, M. 1934. The Kinetic Theory o/Gases. London: Methuen & Co. Loeb, L. B. 1 934. Kinetic Theory 0/ Gases. New York/Toronto/London: McGraw-Hill
Kennard, E. H. 1 938. Kinetic Theory o/Gases. New York: McGraw-Hill Jeans, J. 1 940. An Introduction to the Kinetic Theory o/Gases. London: Cambridge
Univ. Press Dushman, S. 1944. Scientific Foundations of Vacuum Technique. New York: Wiley Chapman, S., Cowling, T. G. 1 952. The Mathematical Theory 0/ Non-Uniform Gases.
Cambridge: Cambridge Univ. Press
Jeans, J. 1954. The Dynamical Theory o/Gases. New York: Dover. 2nd ed. Hirschfelder, J. 0., Curtis, C. F., Bird, R. B. 1954. Molecular Theory o/Gases and
Liquids. New York: Wiley Patterson, G. N. 1956. Molecular Flow o/Gases. New York: Wiley
Grad, H . 1958. Principles of the kinetic theory of gases. In Encyclopedia 0/ Physics, ed. G. Flugge, XII: 205-93. Bcrlin/G6ttingen/Heidclberg: Springer-Verlag
Present, R. D. 1958. Kinetic Theory o/Gases. New York: McGraw-Hill Schaaf, S. A., Chambre, P. L. 1 958. Flow of rarefied gases. In High Speed Aerodynamics
and Jet PropulSion, ed. H. W. Emmons, Vol. 3, Sect . H. Princeton, NJ: Princeton Univ. Press
Schaaf, S. A. 1958. Mechanics of rarefied gases. In Encyclopedia of Physics, ed. S. Flugge. VllI/2: 59 1-624. Berlin/G6ttingen/Heidelberg: Springer-Verlag
Vincenti, W. G., Kruger, C. H. 1 965 . Introduction to Physical Gas Dynamics. New York: Wiley
Shidlovskii, V. P. 1965. Introduction to the Dynamics 0/ Rarefied Gas. Moscow: Nauka (Trans\., 1967, ed. J. A. Laurmann. New York: Elsevier)
Cercignani, C. 1 969. Mathematical Methods.in Kinetic Theory. New York: Plenum Kogan, M. N. 1969. Rarefied Gas Dynamics. Moscow: Nauka (Transl., 1969,
ed. L. Trilling. New York: Plenum) Sherman, F. S. 1969. The transition from continuum to molecular flow. Ann. Rev. Fluid
M echo 1: 317-40 Patterson, G. N. 1 97 1 . Introduction to the Kinetic Theory o/Gas Flows. Toronto:
Univ. Toronto Press Kogan, M. N. 1 973. Molecular gas dynamics. Ann. Rev. Fluid Mech. 5: 383-404 Cercignani, C. 1 975. Theory 0/ Application o/the Boltzmann Equation. New York:
Elsevier Bird, G. A. 1 976. Molecular Gas Dynamics. Oxford: Clarendon Bird, G. A. 1978. Monte Carlo simulation of gas flows. Ann. Rev. Fluid Mech. 1 0:
ll-31
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Sherman (1969), Kogan (1973) , and Bird (1978). The books by Bird (1976), Cercignani (1975, 1988), and Vincenti & Kruger (1965) are also recommended. Since 1960, a detailed and comprehensive record of activity in the field is available from the published proceedings of the International Symposia on Rarefied Gas Dynamics (RGD 1-16 in Table 1). These proceedings are of high quality and represent refereed and invited articles that form a unique, concentrated reference source. As a reasonable rule of thumb, most work in the field since 1960 has appeared in some form in these proceedings. One unfortunate aspect is that there is no cumulative subject index for the RGD Proceedings. This problem has been recognized, and the International Advisory Committee will attempt to provide such a data base sometime after the upcoming RGD symposium (RGD 16, July 1988, Pasadena, California).
There exists no comprehensive recent review of the experimental techn iques of rarefied gas dynamics. Schaaf (1958) lis ts in his references (nos. 209-262) work on a number of experimental rarefied-gas-dynamics techniques. Sherman & Talbot (1960) present the status of comparisons between theory and measurement at that time. Sherman (1963) discusses early experimental techniques. The electron-beam fluorescence technique, which has provided many types of measurements in rarefied gas flows, is reviewed by Muntz (1969, 198Ia,b) and by Biitefisch & Vennemann (1974). To date, there has been no extensive application of laser techniques to the study of rarefied flows, although this subject is to be reviewed by Lewis in the proceedings of the upcoming RGD 16 (Lewis 1989). Hagena (1984) has reviewed cluster formation in expansions, and Fenn (1985) has discussed the future (and the route to get there) for molecular beams. Thomas (1985) and Hurlbut (1985) have examined surface accommodation coefficient measurements. Measuremen ts of nonequilibrium processes in free-jet expansion flows have been reviewed by Rebrov (1977, 1985) .
THEORETICAL RAREFIED GAS DYNAMICS
There are many surveys of theoretical techniques in rarefied gas dynamics. A review of Cercignani (1975,1988), Kogan (1967,1973), Sherman (1969), Shidlovskii (1965), Grad (1958), and Schaaf (1958) provide a comprehensive coverage. Works dealing specifically with discrete velocity models and Monte Carlo approaches arc brought up in more detail later. In this section I briefly look at theoretical techniques in order to provide a basis for later discussions.
The fundamental equation of the kinetic theory of gas flows, if only two-body collisions are important, is the Boltzmann equation:
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al +� a] = J(J,I) at } axj, Kn
(j= 1,2,3). (1)
Following Kogan's (1967, 1973) notation, Xj and 0 are nondimensional coordinates and speeds, respectively, of a molecule in thej-direction based on a characteristic length L and a characteristic speed [2(k/m)T]1/2. The local static temperature is T, k is Boltzmann's constant, and m is the molecular mass. The nonlinear collision integral J is complicated; it describes the net effect of populating and depopulating collisions on the distribution function. The distribution function 1 is the nondimensional molecular number density (based on a reference number density no) in a six-dimensional phase space formed from three velocity and three space coordinates. The Knudsen number Kn is the ratio of a molecular mean free path to the characteristic flow dimension. By multiplying the Boltzmann equation in turn by molecular mass, momentum, and energy and inte
grating over all possible molecular velocities in each case, five conservation equations can be found for the transport of mass, momentum, and cncrgy.
As the average distance a molecule travels between collisions (its mean free path) becomes large compared with a characteristic flow dimension, the Knudsen number approaches infinity and the role of gas-phase collisions becomes unimportant. This limit is the free-molecule flow regime, where only gas-surface collisions between the gas and an obstacle or conduit determine the flow-field properties. In this regime the flow field can be determined even for complicated geometries by using computational techniques, which are limited only by the accuracy to which the surfacereflection characteristics are known.
For Kn --+ 0, gas-phase collisions become important and the flow approaches the continuum regime of conventional gas dynamics. In the limit of small Kn, the Boltzmann equation can be solved by expanding the distribution function as a series in Kn:
J= pO)+KnJ(I)+Kn2j<2)+ . . . . (2)
Substituting for J in Equation (I) and equating terms of equal order results in a set of integral equations (the Hilbert-Chapman-Enskog approach; Kogan 1973 ). Once the form of the distribution function is assumed, all of the hydrodynamic quantities including the stress terms and heat-flux vector can be written in terms of the distribution function, which can thus be specified in terms of flow quantities and their derivatives (cr. Kogan 1 973) .
The first term on the right-hand side of Equation (2) is the Maxwell or equilibrium distribution function, which is closely approximated if the flow is isentropic, and is given by
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� and u represent the nondimensional speeds of a molecule and of the flow, respectively. The first term in Equation (2) corresponds to the Euler equations. The second term corresponds to the Navier-Stokes equations, and the third to the Burnett equations. Later we wil l have occasion to use the form of Equation (2) that is consistent with the Navier-Stokes equations, namely
(4)
In Equation (4) the molecular speeds Cj and Ck are "peculiar" speeds, or speeds relative to the local flow speed, where Cj = 0-Sj' Sj = u) [2 (k/m)T] 1/2, and T is the local static temperature. The speed ratio Sj is the j-component of local flow velocity Uj' made nondimensional using the most probable molecular peculiar speed [2(k/m)Tf/2.
The distribution function of Equation (4) i s a local Maxwellian (i . e. isentropic flow) if the macroscopic gradient terms AO In T/oxj and AoS)oxk approach zero. Note that these terms are really the fractional change in temperature or change in speed ratio per molecular mean free path A. As is described by Holtz et al . (1971), limits on the size of these terms before the general form of Equation (4) becomes invalid can be estimated from comparison to experimental measurements of the distribution function.
For flows with large Kn, the term Kn-I can be used as a small parameter to expand the distribution function and substitute into Equation (l); a solution is subsequently obtained by equating terms of the same order from the left- and right-hand sides. This is the nearly free-molecule flow limit, where the distribution function is disturbed slightly from its collisionless value (see summary in Sherman 1969).
Much of the initial impetus for studies in rarefied gas dynamics was to be able to describe flows for arbitrary Kn, with particular emphasis on the transition regime between the continuum and free-molecule limits. There have been a variety of approaches developed to achieve this purpose. In addition to solutions based on expansions of the distribution function in terms of a small parameter (the Knudsen number for small Kn, or Kn-I
for large Kn near the free-molecule limit) or variants of this general approach (see Kogan 1967, Cercignani 1975), Equation (1) can be solved
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by linearizing the collision integral for flows where the average speed and temperature exhibit l ittle variation. This linearization has been extensively exploited by Cercignani (1975) to obtain solutions in the transitional-flow regime.
Another method for simplifying the collision integral can be found in models or intuitive forms. One popular model is the B GK equation (Bhatnagar et al. 1954), which is given by
aJ - aJ _ NN J(O) J ;1- + �i;;- - nv( - )/Kn. ut uXj
(5)
Here v is a dimensionless molecular-collision frequency that does not depend on molecular velocity but may depend.on static temperature. This model equation can be solved for many situations where a solution of the Boltzmann equation is not practical. An example is the shock-wave solution of the B GK equation obtained by Liepmann et al. (1962) that was used to study thc characteristics of strong shock wavcs.
Another approach that has been used in specialized situations is the combination of two or more Maxwell distributions. Perhaps the most famous of these is the Mott-Smith assumption that the distribution function in a shock wave is the weighted combination of the upstream and downstream Maxwell distributions (Mott-Smith 1951, Muckenfuss 1962). This rather simple assumption actually works very well for the prediction of shock-wave thickness over a wide range of shock Mach numbers (Alsmeyer 1 976).
COMPUTATIONAL RAREFIED GAS DYNAMICS
The dominant predictive tool in rarefied gas dynamics for the past decade at least has been the direct simulation Monte Carlo (DSMC) technique. This approach, which was introduced in 1963 and 1 964 by Graeme Bird (Bird 1963, 1965), has been developed, nurtured, and brought to an impressive level of ptoductive capability by Bird and others in the intervening years. The work reported by Moss ( 1 986) using the DSMC method to predict species concentrations and radiation in very energetic flow fields indicates thc power of the technique. The basic technique is described by Bird in h is book (Bird 1976); its use in chemicall y reacting flows has been presented by Koura ( 1 973) and Bird (1976, 1979). Recently, the technique has been extended to radiating flows (Bird 1 987). A short review of the DSM C technique, as well as the Hicks-Y en-N ordsieck method (N ordsieck & Hicks 1967, Yen 197 1 ) and the molecular-dynamics method, has been
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given by Bird (1978, 1 989; see also Yen 1984). The Hicks-Yen-Nordsieck (HYN) approach is a Monte Carlo method, but in this case the collision integral is solved by a Monte CarIo sampling technique; the remainder of the Boltzmann equation is solved using standard finite-difference methods. Appropriate implementations of the DSMC technique have been shown by Bird ( 1 976) and by Nanbu in a series of papers (Nanbu 1986) to be in principle an exact solution of the Boltzmann equation, although as Bird ( 1 989) argues, it is not entirely clear that such a connection is necessary.
In the DSMC method a large number of simulated molecules are followed simultaneously (Bird 1 976, 1989). Coll isions are handled on a probabilistic basis using the molecules found in a small geometric cell after each computational time step. The computation is started from some initial condition and followed step by step in time; steady flow is the condition that is reached at large times. The computational cell network is in physical space, and the time steps can be directly related to physical times. A most important feature of thc technique is that it can be applied so that the computation time is proportional to the first power of the number of simulated molecules.
The DSMC technique is a pure form of computational fluid dynamics. In principle it can contain all of the physics needed for any problem without the necessity of non equilibrium thermodynam ic assumptions that are required in nonequilibrium continuum-flow calculations. In practice the technique is computationally intensive compared with its continuum counterparts. However, calculations that overlap continuum calculations (with added slip effects) for STS orbiter flow fields have been accomplished (Moss 1986). The success of the technique depends on computational performance, so it is easy to anticipate further advances in its use. A number of research centers are now looking at efficiently matching DSMC and continuum techniques in a computational hybrid approach (Cheng & Wong 1 988). As indicated by Bird (1985), hypersonic flow fields can show widely different degrees of rarefaction at different locations, which makes a hybrid approach attractive for such flow fields. On the same subject, it was noted by Yen ( 1 984) that the HYN approach offers the possibility of not having to match d ifferent numerical techniques for solving mixed continuum and rarefied-flow problems.
Since in many situations where extreme nonequil ibrium effects exist, such as in shock transition zones around sharp hypersonic leading edges, the DSMC technique is the only realistic method for obtaining solutions, its validity is important. In some sense its validation may well serve as a prototype for many other computational fluid-dynamic techniques. The usual situation, and the one that applies here, is that for flow fields in which one really wants to use the computational technique, it i s very
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difficult to check it directly. The status of DSMC experimental validation is reviewed in a following section.
The success of the DSMC technique, as well as a continued desire to extend calculations to lower Knudsen numbers, has revived interest in discrete-velocity gas models. In these models, gases are allowed to have molecular velocities with only a l imited set of possible values. It is surprising how successful the very simplest velocity model of a gas, frequently presented in introductory chapters of kinetic-theory texts, can be in predicting approximate transport properties and the equation of state. This simplest model permits molecular motion only parallel and antiparallel to the coordinate axes of a Cartesian system. Following the suggestions of Krook (1955) and Gross (1960), Broadwell (1964a,b) has extended versions of the simple model to solve Couette and Rayleigh flows and shock-wave structure. Broadwell's use of a very few discrete velocities permitted him to solve analytically both the infinite-Mach-number shock-wave transition problem and translational Couette flow for low Mach number and arbitrary Knudsen number. Hamel & Wachman (1965) have used what they call the "discrete ordinate" technique for a study of Couette flow.
For discrete models, v Boltzmann equations can be written for v discrete velocities:
(7)
The rate of change of the number of molecules N(v) with velocity (v) on the left-hand side of Equation (7) is just equal to the difference between gain (G) and loss (L) terms on the right-hand side. There is clearly a set ofv differential equations that can be written where the "collision integral" (G - L) will be very simple if v is a small number (Broadwell 1 964b) . The surprising qualitative and even quantitative accuracy of the discretevelocity models is related to the question expressed recently by Hasslacher (1987): "How detailed must micromechanics be to generate the qualitative behavior predicted by the Navier-Stokes equations?" A related question is, is there a use for discrete models to help accelerate DSMC-type calculations? If there is, a h ybrid DSMC-discrete-velocity approach would permit calculations to smaller Knudsen numbers, which is a worthwhile objective.
The discrete-velocity model of Broadwell retains the notion of continuous space and time. Other models go beyond the discrete-velocity assumption to discrete physical space and time. As reviewed by Hasslacher (1987), the particles exist at nodes in a lattice; in each time step a particle will move in one of the possible discrete directions a d istance of exactly
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RAREFIED GAS DYNAMICS 399
one node. Simple conservation collision rules are implemented at each node to determine the discrete particle velocities for the next step. In all cases the particle speeds are constant. Thus at the beginning of a time step only the particle velocities in the nearest neighbor nodes need be known in order to completely determine the results of the time step. Simple, massively parallel computations are possible. With appropriate choices of lattice configuration and collision rules, cellular spaces, or cellular automata as these models are called, have been observed to evolve at least approximately the dynamics of nonlinear physical systems. The connection between the lattice gas and real gases (as represented by the Navier-Stokes equations) can be formed using analogs of the kinetic-theory techniques developed for obtaining transport coefficients from the Boltzmann equation (Frisch et al. 1986). Results of lattice-gas calculations have provided remarkable, apparently qualitatively reasonable continuum flow field visualization (Hasslacher 1987). The quantitative accuracy of lattice-gas simulati on of continuum-gas-dynamics phenomena remains to be established, but the application or adaptation of these new directions to massively parallel rarefied-flow computations are of great interest. The singlespeed lattice gas has no way to have a gas temperature independent of gas velocity. To remedy this, a multispeed lattice gas is required and has been proposed by d'Humieres & Lallemand (1 986). Calculations based on this model will be reported at RGD 16 (Nadiga et al. 1989). It is clear that the computational efficiency of at least the simplest collision-rule automaton can be extremely attractive. The inclusion of more complicated rules to accommodate real gas effects (rotation, vibration, chemistry, etc.) is another matter; however, there is no question that the automata offer intriguing possibilities and certainly will be the subject of much future research.
EXPERIMENTAL RAREFIED GAS DYNAMICS
The history of experiments in the field of rarefied gas dynamics involves two rather different mainstreams that have, however, interacted in mutually beneficial ways. Special effort has been expended on developing techniques for producing and measuring rarefied flow fields. Determination of the details of the molecular motion and the populations of internal energy states has been a particular emphasis and has led to the development of several new techniques. In parallel, much effort has gone into the development of supersonic free-jet molecular-beam sources. This work was originally done for surface-interaction studies but has now been extensively used in chemical kinetics and a wide range of related investigations. The development of sources in the period up to about 1975 leaned heavily
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on flow-field instrumentation techniques that could observe energy-state populations and molecular motion in various gas species. In return, the simple free-jet flows were an ideal test-bed for fl ow-field instrumentation techniques. The two parallel experimental developments are a remarkable success story. Spectacular progress has been made in molecular beams and flow cooling for both chemical kinetics and surface studies. Detailed flowfield measurements were made a number of years ago and have been used very recently for comparison with DSMC calculations.
Instrumentation Techniques for Flow Fields The requirements for understanding rarefied flows stimulated the development of special instrumentation techniques, which were developed to sense the rotational, vibrational, and translational energy-state population distributions at points in rarefied flow fields.
The first flow-field instrument to indicate the directional details of the molecular motion in a gas flow was the free-molecule pressure probe developed by G. N. Patterson and his students (Patterson 1 956, Enkenhus et al. 1960). This device is a cylindrical probe with a small orifice in its side
wall. It is used in flows where the mean free path is much greater than the probe diameter. By placing the probe's axis in arbitrary directions and rotating the probe about its longitudinal axis, the plane of the probe's orifice can in principle be oriented in any direction. Pressure measurements as a function of orifice orientation provide information about the molecular motion as a function of direction. There have been many variants of this probe, including one in which the probe was used to feed a mass
spectrometer for the study of mixture flows (Koppenwallner 1986). The long time response of very small probes attached to finite gauge volumes and the difficulty of probe manufacture have l im ited the use of freemolecule pressure probes to flows that are at lower dens ity than is convenient for many laboratory fl ow studies.
A second-generation flow-field probe is the electron-beam fluorescence technique. A collimated beam of, say, 20-kV electrons is projected through a rarefied flow. The electrons stimulate fluorescence from the flow gases, which can then be used to measure under various particular circumstances the translational, rotational, and vibrational population distributions of the gases in the flow. First introduced by Schumacher & Gadamer (1 958) for density measurements, the technique was later developed for translational, rotational, and vibrational population distributions by Muntz and many other investigators (Muntz 1969, 1981a,b). Spatial resolution is obtained by observing the fluorescence at posit ions along the beam's trajectory and maintaining a beam diameter of around 1 mm or so. A wide
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RAREFIED GAS DYNAMICS 401
variety of studies have been reported using electron-beam fluorescence measurements (Muntz 1 969, 1981a,b, Butefisch & Vennemann 1 974) .
A lmost all of the measurements that have recently been developed for optically accessible molecules using laser-induced fluorescence had been made 20 years ago using electron-beam fluorescence. Laser-induced fluorescence can operate at higher densities using saturation optical pulses and is more sensitive on a per-molecule basis. However, seeding the flow with optically accessible atoms or molecules i s frequently necessary. In rarefied gas flow it is not convenient to use seed molecules for use with laserinduced fluorescence, since they may not follow the flow that is being traced owing to mass differences; additionally, their internal energy-state population distributions will not reflect the flow molecules in a wellunderstood way. An alternative would be to study pure flows of, say, iodine or sodium vapor at relatively high pressures. This method presents its own set of problems.
Flow-field measurements over the next decade will likely continue in the tradition of electron-beam fluorescence, but with a difference. As discussed by Muntz et al. (1987) there are real advantages to be obtained by combining electron-beam- or X-ray-induced excitation with laser-induced fluorescence. The electron or X-ray beam acts to seed optically accessible species along its trajectory. A crossing laser beam can then probe these species. If great care is taken to follow the excitation kinetics, it is believed that a flow-measurement technique can be developed that will exhibit the best features o f both electron-beam and laser-induced fluorescence. I f optically accessible molecules are being studied, laser-induced fluorescence offers a number of advantages. It will certainly be applied on i ts own in future rarefied-gas-dynamics studies.
Gas-Dynamic Molecular Beams and Free Jets
The suggestion by Kantrowitz & Grey (1 95 1) that intense molecular beams be formed using supersonic or hypersonic flows, from which a high-fluence beam is skimmed, has resulted in a large body of research in the area of free-jet expansions and molecular beams. A convenient source for hypersonic gas was found to be the free jet (Ashkenas & Sherman 1965, French 1 966). Research on both the characteristics of free-jet flows and on the formation of intense molecular beams has been prolific. Freezing of the translational random motion in the direction of the expanding flow (the parallel "temperature") was identified by Anderson & Fenn (1965) using molecular-beam sampling techniques and time-of-flight analysis of the parallel molecular velocity d istribution in the molecular beams produced by the sampling process. The freezing was subsequently predicted
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by Hamel & Willis (1966) and Edwards & Cheng (1966). The transition region from equil ibrium to frozen translational temperature in jet expansions was observed initially by Muntz (1967) and later more comprehensively by Cattolica et al. (1974). The internal-state relaxation in jets was initially studied by Marrone (1967) and M iller & Andres (1967). There have been a large number of subsequent publications on this subject, and these are perhaps best summarized by Rebrov (1977, 1985) and by Fenn ( 1 985).
Because of their convenience, free jets are the method of choice for generating gas-dynamic molecular beams. Flow-field measurement techniques, such as the electron-beam fluorescence probe and the molecular beams obtained by skimming free-jet flows, have been used to study jet flow properties (Anderson & Fenn 1965). After considerable research by many investigators, the art of flow sampling has been refined to such an extent that very l ittle interference is introduced by the sampling process (cf. Bossel 1 971) . Because of this capability, extremely low temperatures can be measured in the molecular beams sampled from the jet flows.
Free-jet expansions have been used extensively as refrigerators. Fenn (1985), in a wonderful review of his long affair with jets, discusses several uses of jet cooling. By taking advantage of large increases in coll ision cross sections at very low temperatures, groups led by Campargue (Campargue & Lebehot 1974, Campargue et al. 1981) and by Toennies (Miller et al. 1974) have managed to avoid translational freezing and to generate extremely low temperature (milli-Kelvin) molecular beams. The low temperatures permit surface-reflection studies that can sense single-phonon interactions of, say, helium w ith a surface (Brusdeylins et al. 1981). These extremely monoenergetic beams have also been used in inelastic gasscattering experiments (Faubel et al. 1980).
Cooling with the jet gas used as a bath makes it possible to depopulate the majority of the rotational and vibrational levels of complicated molecules. Indeed, owing to their low density in the bath, the molecules can be cooled well below their normal condensation temperatures. This effect has spawned a remarkable quantity of recent papers on laser-inducedfluorescence molecular spectroscopy, as described by Fenn (1985).
The ubiquitous free-jet expansion into a h igh vacuum has been used extensively for generating controlled agglomerations of molecules in the condensed phase. The ability to generate molecular groups ranging from dimers to aerosol-size particles allows technologically important phenomena to be studied (Fenn 1985). Cluster formation in jets has been reviewed by Hagena (1984) and by Rebrov (1985) as part of a review of nonequilibrium processes. The subject is wide ranging and interesting but cannot be discussed here in detail owing to space l im itations. The many scientific studies
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made possible by a convenient source of clusters indicate that this subject will continue to attract attention in the future.
ASSESSING COMPUTATIONAL RAREFIED GAS DYNAMICS
Because of its central importance in predicting rarefied flows, the validation of the DSMC technique by experiments is discussed in this section. The status of the validation is important for simple gases, for chemicall y reacting gases, and for gases with internal degrees of freedom. The DSMC technique, or some variant, can in principle account for all of the correct physics during a collision. However, to do so implies an extreme computational overhead. The question raised earlier about how much microknowledge has to be included to arrive at a macroscopically satisfactory result needs to be answered; however, note that there is no one answer, since the level at which macroscopic satisfaction is achieved will depend on the application. Comparing DSMC results with experiments can achieve this goal. As an illustration of the possibilities, two flow fields are examined here in order to present the status of comparisons between DSMC results, other predictions, and measurements. Additionally, it is important to validate the DSMC method because of the typical computational-fluid-dynamics role it has assumed as a surrogate for experiment.
Before looking at specific flow fields it is useful to remember that molecular interactions are generally modeled by point centers of repulsion in computational rarefied gas dynamics. In this model the intermolecular force during a collision is assumed to be inversely proportional to a power 1] of the distance between the molecules. Based on the Chapman-Enskog theory described earlier, the coefficient of viscosity can be related to the intermolecular force law by {t = K\Tw, where w = (1]+3)/2(1]-1 ) and K\ is a constant given by kinetic theory and the characteristics of the molecule. The mean free path of a molecule can be calculated from its coefficient of viscosity and is thus directly related to w or 1]. In the DSMC technique a collision cross section is required. As discussed by Bird ( 1 976), an effective hard-sphere momentum-transfer cross section that varies with the relative kinetic energy of the collision partners can be written in terms of w or 1]. Calculations made for different theories or models can be compared by using the same intermolecular force law to provide either viscosity coefficients or collision cross sections as necessary.
Normal Shock Waves There is now a considerable body of data on the density profiles of normal shock waves in monatomic gases for a wide range of shock Mach numbers.
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Most of the available data are reviewed by Alsmeyer (1976), who also presents important new data with very little relative scatter. Other data for very low shock Mach numbers have been presented by Garen et al. ( 1 977). These two studies were done in shock tubes; one used observations of the attenuation of an electron beam to measure the density profiles, the other a laser interferometric technique. Argon and xenon were the gases used in the studies. We first consider Figure 2, where various quantities are defined that relate to a shock-wave density profile. It is generally convenient to use the nondimensional density fi as a position variable in the shock and to nondimensionalize distances by the free-stream mean free path. Using only Alsmeyer's ( 1 976) data for maximum-slope shock thickness, which really lies in the middle of a rather wide scatter band from many other investigations based on a variety of techniques, the comparison shown in Figure 3 (due to Fiscko & Chapman 1 988) between experiment and prediction with 11 = 10 (w = 0.72) is consistent with the temperature sensitivity of measured argon viscosities at temperatures above 3 00 K (appropriate for shock-tube experiments). The trend for the DSMC technique to predict a thinner shock at low Mach numbers was also noted by Alsmeyer ( 1 976). He ascribed this tendency to the rather significant attractive well of argon (which of course is not modeled by a simple inverse-power repulsion) playing a more important role in the low-Machnumber shocks. Sturtevant & Steinhilper ( 1 974) performed DSMC calculations incorporating various attractive-well depths but at higher Mach numbers, finding smaller thicknesses in terms of the upstream mean free path for decreased attractive-well depths.
It is interesting to note that Alsmeyer's data are fit quite well by the M ott-Smith model as developed by Muckenfuss ( 1 962). The fit is best at the lower Mach numbers. A similar satisfactory fit was found recently by Velikodnyi et al. ( 1 985), who applied Struminskii's ( 1 980) method of
>- n2 I-en z w o a: w CD � Z
FLOW -
Q=� A2
SHOCK WAVE NUMBER DENSITY PROFILE
o L-______ -L __ L- ______________ � o
Figure 2 Characteristic parameters that can be derived from a shock-wave density profile. The maximum-slope shock thickness is.5" and AI is the upstream mean free path.
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0.3 A- DSMC
b b
/---. ......... � / CURVE THOU:::':' " ' :..:.. ;- - -
ALSMEYER'S DATA. ARGON , T1 " 300 K
MOTT - SMITH, w = 0.72
o 1!-L--±-3.....l.----;!5,------lL...,7!;--L--±-9.....l.�11 SHOCK MACH NUMBER
RAREFIED GAS DYNAMICS 405
Figure 3 Comparison of experiment and predictions for argon normal shock waves (after Chapman et al. 1987).
systems of identical particles to the shock-structure problem. The Burnett equations, which were once thought to have no solution above M � 1.9, were integrated in 1 976 to M = 4 by Simon & Foch ( 1 977) using modern numerical techniques for the case of a normal shock with w = 1 . Very recently, Fiscko & Chapman ( 1988) have integrated the Burnett equations (less one small but numerically troublesome term) to M ::::: 35. The shockthickness predictions by F iscko & Chapman are in quite good agreement with DSMC calculations for the case of OJ = 1 .
A convenient single-parameter measure of the shape of a shock profile is the quotient Q defined in Figure 2 (Schmidt 1 969). The plot in Figure 4 shows Q as a function of shock Mach number as calculated by S imon & Foch ( 1977) with OJ = 1 for the Navier-Stokes and Burnett equations. Superimposed are the results of Oaren et al. (1977). The data seem to support the Burnett results above M � 2.0 (however, remember that OJ = 1
f.20
1.15 GAREN .f 01 DATA. xENON T,= 300K
BURNETT--7" ....... -w . f ,, _ or
o // / ' , _ . ' . ' 4.05 // ,
If' ,/ , I ,/ 1.00 / '
___ ALSMEYER
I " ,.,.-DATA. ARGON I Y T ' 300K J '/ 4
I :
2 3 4 5 SHOCK MACH NUMBER
Figure 4 Comparisons of the quotient Q (see Figure 2) for Navier-Stokes and Burnett solutions to normal shock waves with two sets of experiments.
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in the predictions). Between M = 1.0 and 2 .0 the data do not really favor either version. I am not aware of Q-values having been presented for lowMach-number DSMC calculations, although accurate values could surely be obtained with present techniques and computers (cf. Bird 1989).
So far we have looked only at density, which i s a very simple moment of the molecular-velocity distribution function in a shock wave. It is of interest to see what level of complexity in the description of collisions is required for the DSMC or other techniques to be able to predict the distribution function in a shock wave. M easurements are available from electron-beam fluorescence studies done a number of years ago in helium at M = 1.59 and M = 20 (Muntz & Harnett 1969; E. P. Muntz, unpublished work, 1968) and in argon at M = 7 (Holtz & Muntz 1984). For lowMach-number shock waves, a convenient single-parameter assessment of the shape of the molecular-velocity distribution function is its full width at half-maximum, i .e. its half-width (Holtz et a1. 1971). Data for the halfwidths of the distribution for molecular motion in the direction of the flow (parallel) and perpendicular to it (perpendicular) are shown in Figure 5 as a function of non dimensional density rise fi in the M = 1.59 helium shock. For comparison, plots are also shown of the Navier-Stokes half-widths [using Equation (4) for the distribution function], M ott-Smith half-widths, and DSMC half-widths, all with w = 0.647. Note that in this case, the DSMC and HYN techniques do a good j ob of matching the data, whereas the Chapman-Enskog distribution is quite poor. The M ott-Smith dis-
1 .2 r-r-'-.,...,�'-,..,.-'-"""--,,,,,,,,,,.-r-,-,......,--',-,,,,,-,
1.1
<;; 1.0 :i 0.9 .... e 0.8 ;;
..:. 0.7 ...J � 0.6 (f) 0.5 (f)
':J 0.4 z � 0.3 z � 0.2 a 0.1
o
W = W - W1
� .or:..
�_
w2 - w, y .....
.
W : half width
{
.Y,.:.<
--
-
-
-
4-
�
;r;
1 - Upstream t/.:,.-'- ;?p 2 - Downstreom k/ ./ . . ;.:'
I ....... ;tjr / ..... /.:
/ ..... f ,/.�� Experiment
/ /. ,::/' Muntz B Harnett
/ .... / /;/' <} .L d istribution ;.r . . ;§ { /1 distribution I .. ):/ .. ' f .
.. )Y" . . . . ......... ... Mott - Smith II
fi .. . ' ... - . , . Moll -Smith .L I ,:/' - - - Chopmo n - Enskog
I ; :t. - -- - .. HYN, w : 0.5 .. 'I' - - '-- HYN , w : 1.0
-- DSMC II , w :O.647 - - - - - - - - DSMC .L w :0.647
-0.1 L..J......L.J.......l.-'--...L....L..l.-'-l.....t.....l....-'-'---'-....L-<-...l.-,-..L....Jc....J o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1.0
R EDUCED NUMBER DENSITY, � Figure 5 Half-widths of distribution functions for an M = 1 .59 normal shock wave.
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RAREFIED GAS DYNAMICS 407
tribution is nearly as successful as the DSMC results. Further results of a similar nature, including comparisons to M = 7 argon data, will be presented soon (Erwin & Pham Van Diep 1 989). As an example, the MottSmith parallel distribution function for an M = 20 helium shock is compared with experimental results (E. P. Muntz, unpublished work, 1 968) at fi = 0.62 in Figure 6. Note that while the Mott-Smith shock thickness is in reasonable agreement with experiment (Alsmeyer 1 976), the distribution function is not, exhibiting a large population of scattered molecules unaccounted for in the Mott-Smith model. The distributions in Figure 6 have each been normalized by their maximum values in order to compare the shapes of the two functions.
The preceding detailed comments are presented here as an example of one of the opportunities that exist for establishing true-bench-mark validations of computational rarefied gas dynamics. Similar studies of mixture shocks and shock waves in gases with internal degrees of freedom in order to study the details of inelastic collisions would be extremely useful. In both of these cases, measurements beyond those already available may be necessary. Space limitations do not permit further discussion on these subjects here, although both arc extremely important for the complete validation of computational rarefied gas dynamics.
Flow-Field Studies and Knudsen Layers
The prediction of rarefied flow fields about simple shapes with detailed comparisons to experiments remains to be completed. The issue here is essentially how the kinetic layer near the interface between gas and solid (the Knudsen layer) is treated. Kogan ( 1 973, 1 985) and Cercignani ( 1 978,
1 .0
0.9 z � 0.8 3 0.7 ::::> g, 0.6 a.
0.5 w � 0.4 :5 0.3 w 0:: 0.2
Mp = 20
;; = 0.62
x EXPERIMENT. Fp
-- PREDICTION. MOTT-SMITH
0.1
O �����WL�����LL����LLLL�� 2.0 1.5 1.0 0.5 0 -0.5
(Veil X W- 5 ( cm/s )
Figure 6 Comparison of the shapes of parallel distribution functions at a point in an
M = 20 helium shock wave.
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1 979) have addressed the problem from a theoretical point of view. Cheng ( 1 988) describes the near-continuum rarefaction problem as shock slip and surface slip. Shock slip is the interaction between a thick shock wave and a fully viscous layer between the shock and the body. Surface slip refers to the jump conditions that occur in the kinetic layer (a thickness of order one mean free path) at the interface between gas and surface.
Crawford & Vogenitz ( 1 974) presented detailed DSMC flow-field calculations for a cylinder. These results included perpendicular and parallel temperature predictions. Davis & Harvey ( 1 979) have studied a highMach-number nitrogen flow over a flat plate (actually a large-diameter hollow cylinder aligned with the flow) and a plate with a forward facing step. The latter authors compared experiments with DSMC calculations by using several energy-transfer models to account for rotational energy transfer in collisions. Crusciel & Pool ( 1 986) have studied the Knudsen layer and other flow-field details for flows over a sphere and a sharp cone (Crusciel & Pool 1 987). In the latter case the flow-field density measurements of Hickman ( 1 967) were compared with the predictions. Measurements by Krylov et al. ( 1 985) of surface stress using a floating element balance are available for comparison to predictions. Molodtsov & Ryabov ( 1 985) compare measurements of flow around a sphere with DSMC and Navier-Stokes calculations. The Knudsen layer has been probed by Gottesdiener ( 1 984) with a free-molecule wire probe; this experiment is one of the few to resolve profiles in the Knudsen layer. Earlier velocity-profile measurements in a Knudsen layer have been presented by Reynolds et al. ( 1 974). Electron-beam fluorescence density measurements have been compared by Allegre et al . ( 1 986) with Navier-Stokes solutions with slip boundary conditions for a NACA 001 2 airfoil in an M = 2 flow. Velocitydistribution function measurements in the flow field over a flat plate made by Becker et al. ( 1 974) have been compared with DSMC predictions by Hermina ( 1 987) and Hurlbut (1 987).
It is clearly time for a detailed review of flow-field comparisons between experiments and predictions, as well as of the related issue of the Knudsennumber limits of continuum-flow approaches that include slip. Space limitations do not permit such a review here.
RADIATING, REACTING, NONEQUILIBRIUM COMPUTATIONAL RAREFIED GAS DYNAMICS
As noted in the introduction there is a growing need to be able to predict highly nonequilibrium, radiating and reacting flow fields. A major triumph of the past 5 to 10 years has been the development of the DSMC technique to provide this capability. The DSMC, or any similar particle treatment
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of a gas where classes of particles are followed (in this case only in a statistical sense), intrinsically is able to provide a representative reflection of nature and can be correct in highly nonequil ibrium situations. This of course is only true if all important elastic and inelastic collision cross sections are known as a function of energy. However, the possibil ity is there, whereas in continuum treatments reliance must be made on the assumptions of nonequilibrium thermodynamics. In the past decade, Graeme B ird and his associates at the Langley Research Center have been able to develop in "an engineering context" (Bird 1 985) the DSMC techn ique so as to provide predictions for h ighly nonequilibrium real air flow fields including radiation (Moss et al. 1 988). The "engineering context" refers to the numerous necessary approximations that must be made in describing the collision processes and energy transfers during collis ions. The computational resources and physical information needed to avoid these approximations are lacking at this time. The exciting point is that the engineering calculations are possible and provide reasonable agreement with the very few experimental results that are available (Bird 1 987). It is also clear that the agreement may be fortuitous, since there are a very large number of interacting approximations and not very well known collision cross sections involved.
The possib il ity of making flow-field predictions for multidimensional, reacting and radiating flows (also of course viscous) with the DSMC technique impl ies a clear direction for future work. Significant effort in rarefied gas dynamics will be aimed at developing inelastic coll ision models and establishing their validity in situations that are less complicated than a nonequilibrium air chemistry flow field. As an example, the standard way of treating inelastic collisions with transfer of rotational energy in the DSMC technique is based on the model of Borgnakke & Larsen ( 1 975). In the model a certain fraction of collisions are considered inelastic; for these collisions, new translational and rotational energies are sampled from the distribution of these quantities that would occur in an equilibrium gas with a spcci fic cnergy that is the same as the specific energy available in the collisions. This of course is not very realistic, but if the fraction of inelastic collisions is chosen so that the rotational relaxation rate in the gas is matched, it seems to work quite satisfactorily in an "engineering context." Vibration and electron excitation have been handled by analogous techniques. Experiments can be done quite easily to investigate models of, say, rotational energy transfer [for instance, rotational-level population distributions in a shock wave are already available (Robben & Talbot 1 966)]. It is to be expected that different inelastic collision models (e.g. Anderson et al. 1 986) will appear, and that experimental studies in simple situations will be undertaken to validate their predictions.
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In summary, because of the DSMC technique's potential importance in many scientific and technological fields (e.g. astrophysics, radiation transfer, exhaust gas radiation) the details of i ts application to reacting, radiating flow fields should be studied and improved where appropriate in order to construct a sound foundation for the technique's predictive capabilities. I t is likely that more detailed knowledge can be included in the future owing to increasing computational performance. The necessary research will be an active area in rarefied gas dynamics in the next decade (for an example, see Kunc 1 989).
BRIEF DISCUSSIONS OF SOME INTERESTING SUBJECTS
Evaporation- and Condensation-Induced Flows
In 1970 Bochkaryov et al. ( 1 970) reported on the study of radially expanding flows from cylindrical and spherical sources constructed from perforated sheets. A similar, although plane, source was used by Y trehus ( 1977) to simulate an evaporating surface. Considerable attention has been paid to the evaporation-condensation problem because of its technological importance and i ts intrinsic interest as a problem in rarefied gas dynamics. A Knudsen layer is at the interface between the surface and the vapor for the case of evaporation or condensation. Condensation or evaporation can be described as strong or weak (Kogan 1985). Strong conditions are identified with sufficient vapor mass flow such that, to leading order in the approximation at the outer edge of the Knudsen layer, the Navier-Stokes terms can be neglected; a drifting Maxwellill;n and a finite flow velocity outside of the Knudsen layer are appropriate. If, at the edge of the Knudsen layer, Navier-Stokes terms are present in the approximation, then the flow is considered weak. While the evaporation and condensation Knudsen layers are symmetric and seem theoretically well understood for weak evaporation and condensation, this is not the case for strong condensation layers. Hatakeyama and Oguchi have identified a subsonic-supersonic bifurcation (Hatakeyama & Oguchi 1979, Oguchi 1 9 8 1 ) that involves conditions in the flow outside of the Knudsen layer. Kogan ( 1 985), in his RGD 1 3 review of transport processes in Knudsen layers, outlines the theoretical status as of 1 982. Also at the same symposium (RGD 1 3), Schilder et al. ( 1985) reported on the first known measurements in an evaporative Knudsen layer, which in this case was produced by intensive evaporation of iodine. They used laser-induced fluorescence to measure fluorescence emission-line profiles from which temperatures parallel and perpendicular to the evaporating flow were determined. Further theoretical studies of the subsonic-supersonic transition in strong condensation have
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been presented by Hatakeyama (1 984), who expands on the properties of the subsonic and supersonic branches. Ytrehus & Aukrust (1 984) also discuss flow breakdown of condensation Knudsen layers. Unsteady strong condensation has been analyzed by Sone et al. ( 1986) with emphasis on the establishment of a steady-state flow.
Two situations of practical interest for space applications-the evaporation of spherical and cylindrical sources into a high vacuum-have been studied by Edwards & Collins (1 969) and by Knight (1 976). Based on the work of Fuchs & Legge (1 979), Muntz et al. (1 984), Muntz & Orme (1 987), Anders & Frohn (1 984), and Faubel (1 989), it has become clear how to investigate evaporating liquid surfaces in a vacuum. It is expected that a number of studies concerning the characteristics of liquid surface evaporation in vacuum will appear over the next few years. Detailed studies of the associated vapor flow fields are possible, with some initial work already reported (Takens et ai. 1984, Faubel 1989). Surface Interactions at Medium Energies
The study of surface interactions at low energies (up to say I eV) has long been of interest to the rarefied-gas-dynamics community. Unfortunately, there is insufficient space in the present review to address this area. New emphasis, however, has been given to surface interactions at intermediate energies (say 1 to 20 eV) as a result of the observations of glow and material erosion in low Earth orbit, as mentioned in the Introduction. Even prior to this most recent interest, Knuth and his associates had studied momentum transfer for satellite surfaces (Knuth 1 980, Liu et al. 1 979). An interesting aside is the work of Steinheil et al. (1 977), which demonstrated an impressive reduction in tangential accommodation coefficient for specially prepared surfaces, with at least one surface being stable to exposure to atmospheric-pressure gases. The advent of new surface preparation techniques and designer-type materials, as well as of the new surface measurement techniques such as scanning tunneling microscopy, indicates that the time may be about right for significant advances in the control of surface accommodation coefficients. Uses for such control are not entirely clear, since the drag of practically shaped space vehicles is not very sensitive to the tangential accommodation coefficient. Nevertheless, a very low coefficient could significantly affect the drag of stabilized slender shapes. It is also worth noting that an overall reduction in momentum accommodation will reduce the density of the cloud of reflected gases that concentrate ahead of the windward surfaces of space vehicles.
The status of intermediate-energy surface interaction has been reviewed by Hurlbut (1 985, 1 989) . Studies of surface-interaction measurements from space flights are reported by Karr et al. (1 986) and Gregory & Peters (1 986) .
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The production of medium-energy oxygen atoms in ground facilities has been reported by Cross et al. (1 986) and by Caledonia (1 989). A useful review of low-Earth-orbit atmospheric effects is given by Green et al. ( 1 985).
Non-Navier-Stokes Gas Dynamics
Kogan and his colleagues (cf. Kogan 1 986) have investigated continuumlimit flows that have Burnett terms contributing effects of the same order as the Navier-Stokes terms. When a uniformly heated body is immersed in a gas at a temperature different from the gas temperature, such as when a hot body is immersed in a significantly cooler gas, stress terms appear in the Burnett approximation for so-called slow, nonisothermal flows. Kogan indicates that these terms can lead to surprising results. If a hot sphere is in the Stokes-flow regime, then for large enough heating (say 1 0 times the gas temperature) the drag o n the sphere can become negative. The energy for the work done comes from maintaining the sphere's temperature (Kogan 1 986). Burnett term effects related to the ones just described have been observed experimentally for one specific case in elegant work by Hermans et al. ( 1 977, 1 979). In these experiments, Burnett viscomagnetic heat-flux predictions have been found to agree well with experiments.
Based on the Burnett terms' importance even in low-Mach-number shock waves, as well as the recent successful integration of the Burnett equations at higher Mach numbers and their significant contributions to non-Navier-Stokes continuum flows, it seems that the earlier perception that the Burnett approximation was not very useful may be mistaken. Investigations involving the Burnett equations appear likely to continue.
CONCLUSIONS
Several research subjects within the broad field of rarefied gas dynamics have been discussed here, chosen on the basis of what I consider to be their importance as research directions in the next decade. In this effort I have undoubtedly slighted some, but I hope only a limited number, of my professional colleagues; I trust they will forgive the sins of my omissions. Briefly, I expect the most fundamental issues in rarefied gas dynamics during the next decade to be the following:
I . Assessing the accuracy of computational rarefied-gas-dynamics techniques so that they can be used as surrogates for experiments, including investigations of the level of knowledge of the microkinetics necessary for appropriate descriptions of gas flows with active internal degrees of freedom and chemistry.
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2. Development of hybrid flow-field computational techniques that match continuum (Navier-Stokes and Burnett computations) with particle computations (DSMC, cellular automata) and that can apply to highly nonequilibrium, radiating, chemically active flows.
3. Validation of computational rarefied gas dynamics by means of experimental studies of Knudsen layers in simple flows with a detailed following of internal energy-state populations, and gas-mixture flows.
4. A continuing interest in the theoretical analysis of rarefied-gas-dynamics problems, which is essential for understanding and developing computational methods.
5. Investigation of designer surfaces for achieving stable, low momentum and thermal accommodation coefficients.
6. Studies of the interactions at 5-20 eV between atmospheric species and both surfaces and gases.
7. Evaporation and condensation experiments and theory relating to events close to surfaces.
8 . Studies relating to the formation, properties, and use of clusters. 9. The analysis of evaporating and condensing particle and aerosol clouds
in both Earth and space environments.
ACKNOWLEDGMENTS
I thank Gail Dwinell and Carolyn Gautier for much assistance with the preparation of this article. In part, the time spent on this review was supported by a Hypersonic Training and Research grant from NASA, AFOSR, and ONR (NASA NAGW 1 06 1 ). Finally, my greatest obligation is to my colleagues from around the world, with whom I have had many exciting discussions mixed with an eclectic assortment of beverages over the past 25 (maybe a few more) years.
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