0

TON THE ORIGIN AND EVOLUTION OF THREE DIMENSIONAL

EFFECTS IN THE MIXING LAYER

Final Technical Report

• by

by LEVEV.Javier JIMENEZ

Rodrigo MARTINEZ-VAL

Manuel REBOLLO

Laboratory of Fluid Mechanics, School of Aeronautics

Universidad Polit~cnica de Madrid

Madrid - 3 (SPAIN) D TI 0December, 1979 OR 0 b •81 3

EUROPEAN RESEARCH OFFICE

United States Army

London England

GRANT NUMBER DA-ERO 78-G-079

E'xcmo. Sr. D. Jose Luis Ramos Figueras

Rector Magnifico de la Universidad Politdcniea

Cen Bermddez, 10, Madrid - 3 (SPAIN)

ApprOvcd ;or Public Roleiese- distribution unlimited.~--ULL-

9'nn

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On the Origin and Evolution of Three Dimensional 'inal echnical geport.Effects In the Mixing Layer. 1. "Jl 7- 9 Jul 79"

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Javierjlimenez /_ .; D•R-78_G_079!( Rodri golMarti nez-Val

Manuel IRebol loPERFOR ING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENI. PROJECT, TASK

Labor'atory of Fluid Mechanics, School of AREA_6Vi.•RK UNIT NUMBERS I

Aeronautics 6.11. 02A1 T1611 (2BH57-p6Universidad Politecnica de Madrid, Spain

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Instability of 2-D TurbulenceMixing Layers and Coherent StructureStreaksEjections

2•. ABSTRACT (Coeefrui - re wer ed. d ft rn..oaiy mrd Idr•Vit by block nu.b.r)

There seems to be at least two different phenomena connected with the three-dimensional breakdown of the plane mixing layer. First, at least in somecases, there is a macroscopic instability of the large structure whichproduces a spanwise system of longitudinal jets whose amplitude builds upvery quickly and then remains approximately constant while their separationadjust itself to a value that is somewhat higher that the vorticitythickness of the layer. This phenomenon may be due to a collectiveinstability of the vortex cores of the same type as the on• found in vortex

DID 1473 EDITION OF I NOV65 IS OBSOLETE .... A S SI.SECURITY CLASSIFICATION OF THIS PAGE (When V.I. Enrat-d)

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20. Contd.

rings.

The second phenomenon is the development of a 3-D inertial subra, 3e in thespectrum of the small scales of the velocity fluctuations. In t,,e only castin which we have documented both phenomena the second one does not developfully until the macroscopic instability has not been growing for some timeand has actually arrived to what seems to be its final configuration.

Moreover there is some indication that there are two qualitatively differentways for the establishment of the 3-D inertial ranges, examplified here bythe water and air experiments. In the first one the transition is abruptand there is an initial part in which the inertial range has a slopecharacteristic of the 2-D enstrophy cascade. In the second one thetransition is more gradual and there is no evidence of a 2-D regime. It isnot clear at present which is the condition responsible for the differencesbetween the two cases.

The work presented in this report should be seeý as a preliminary explorationof the effects connected with the 3-D transition. Much more research is neede(dis all cases.

Ur•CLASS I F I ED

ýLCURHVTY C> A sF ICAT ;f7.. Or I W~S V A, Cv 4 o

LAD]ON THlE ORIGIN AND EVOLUTION OF THREE DIMENSIONAL

EFFECTS IN THE MIXING LAYER

Fin~lTechnical Report

by

Javier JIMENEZ

Rodrigo MARTINEZ-VAL

Manue] REBOLLO

Laboratory of Fluid Mechanics, School of Aeronautics

Universidad Polit6cnica de Madrid

Madrid - 3 (SPAIN) Accezion "'or"--

Di'IC ~~December. 1979

EUROPEAN RESEARCH OFFICE

United States Army --

London England JJet ,

GRANT NUMBER DA-ERO 79-G-079

Fxcmo. Sr. D. Josu Lujs Ramos Ficueras

Rector Maunifico de ]a Universidad Politicnica

Cen llerIyirndez, 10, Madrid - 3 (SPAIN)

A1,Ij.i ov('d it)r IIul]J c P t] s,; d ,i It 7biti on un I ji teed.

ii

ABSTRACT

The transition to three-dimensionality in a plane mixing layer isstudied through exploration of the flow field and analysis of the powerspectrum of longitudinal velocity fluctaations.

Two different phenomena are identified. First the appearance of 3-Dsmall scale fluctuations is marked by the transition of the slope of theinertial subrange towards the value of -5/3. Second we find in one casethat the large scale itself shows an instability, with wavelengths of theorder of. the vorticity thickness of the layer, and that seems to be expla-ined by a collective "corrugation" ot the vortex cores.This second pheno-mena is studied in some detail..

' ' ! II I I

iii •

TABLE OF CONTENTS

Part Title Page *

AbsL'ract

Table of contents iii

1. Introduction i

2. Flow Facilities and Equipment 3

3. Measurement Techniques 5

4. Results 8

5. Spectral Results 10

6. Spanwise Flow Structure 14

7. Conclusions 22

Appendix A

Stability characteristics of the splitter

plate wake. 24

References 26

Figures 31

1. INTRODUCTION

It is generally agreed that three-dimensionality is anessential property of fully developed turbulence. Even if2-D turbulence is possible it is only under three-dimension-al conditions that a fully developed energy cascade isestablished and leads to the production of enough smallscale motion to account for the rapid mixing of concentra-tion and momentum gradients that is characteristic ofturbulent flows.

The plane mixing layer is a well studied flow in whichthe initial conditions are two-dimensional and the flouremains two-dimensional for a long time after separation. Infact there is evidence that the large scales of the flow maystay two-dimensional essentially forever [1,21. Since it isobvious that the small scale motion in the mixing layershould, and indeed does, eventually become three-dimension-al, this flow is a good test case to study the instabilitiesthat are respunsible for the breakdown of two-dimensionalmotion.

This transition has been studied previously by variousinvestigators from several points of view. Bradshaw [31measured the evolution of the turbulence level in the mixinglayer and its approach to its equilibrium value. Perhapssignificantly this approach was found to depend on whetherthe initial boundary layer was turbulent or laminar andthus, possibly, on the amount of initial three- dimensional-ity in the flow. The evolution of the turbulence level hasbeen measured more recently by other investigators fordifferent initial conditions [4,51 but it is still not clearwhether the establishment of three-dimensionality coincideswith the attainment of the equilibrium value of the fluctua-tion intensity.

Mora recently Konrad [61 and Breidenthal [7 1 haveidentified a transition in the amount of molecular mixingpresent in a two-species shear layer. Since enhanced mixingmust come from an increase of small scale structure thistransition is probably associated with the onset ofthree-dimensinality [81.

We have used still another property, the evolution ofthe slope of the inertial subrange of the longitudinalvelocity power spectrum towards its 3-D value of -5/3, as anindicator of three-dimensionality. The transition isclearly visible in the three cases that we have studied andis discussed in section 5.

Another interesting and related problem is the possibleappearance of three-dimensionality in the large structures,as opposed tu the small scales. Chandrsuaa et. al. r 9 1reported helical distorsion of the vortices in a planemixing layer and similar effects have been described in the

2

initial region of circular Jets [10]. Kourad has reportedthe appearance of longitudinal sLreaks in his concentrationpictures and Breidenthal has associated these streaks withsinusoidal corrugations of the vortex cores. In one of ourflows, the only osae checked for detailed spanwise structure,we have found permanent longitudinal jets that may corres-pond to the same effect and seem to be associated to,although they precede, the transition to small scale three-

dimensionality. These results are described in section 6.Some of our measurements were carried out with hot film

anemometers in water. Since that technique is not too welldeveloped a significant amount of time was spent checkingand improving the accuracy of the method, and the resultsare briefly discussed in section 3.

Due to the time limitations involved in this study manyof the experiments presented are exploratory end many of theconclusions tentative. Further work is being carried out toconfirm and extend them.

3

2. FLOW FACILITIES AND EQUIPMENT

The experiments have been performed in two independentfacilities: a water tunnel and 2-D air half-Jot.

Water Tunnel

The experimental apparatus is baoically a blow-downtunnal in which a plane mixing layer is established betweentwo streams of water coming from two large independentreservoirs (figure 1). Both tanks are pressurized with airat 1.5 kg/cm2 to keep the change in velocity, due to thedrop in water level throughout the shot, below a maximum of.5%.

The two streams go through a conventional sequence ofcontrol valves, flow-meters, settling chamber, straightnersand screens and a 4:1 contraction before they are broughtinto contact to form a plane mixing layer at the exit ot two7.5 x 15 cm nozzles in the test section, from where they arediscarded through a siphon.

The test section is 15 x 15 cm across and 100 cm ling.In the lateral walls there aro three pairs of registerswhich, together with a sliding probe holder, allow manualpositioning of the probe at any downstream coordinate up to4U cm from the edge of the splitter plate. Cross-streamprobe motion is achieved by means of a traversing gear whichincorporates a stepping motor, with a .2 mm step, and iscontrolled by a HP 2100S minicomputer which automatizes thewater experiments.

Al~houh L'. ';ppazatus can provide a flow with tempera-tures as well as velocity differences, the experimentsreported in this work have been conducted in a homogeneousshear layer.

A steady flow is established in less than a second atuominal velocities of 19 and 37 cm/s. Under these ronditionsthe momentum thickness of both bounda,'y layers, at Lited P,,of the splitter plate, are .7 and .5 mm respectively. Awaiting time of about 15 a is needed before the waterinitially in the settling chamber is swept completely andthe temperature is uniform enough to allow the use of hotfilm sensors. With this in mind, maximum useful running timeis about 50 a.

We have run performance tests on the apparatus and thefree stream turbulence level was found to be about .2%. Thefree stream velocities were found repeatable for 100 differ-ent shots and constant throughout a given one to within .5%.The shear across the central part of the free stream highvelocity side was less than .2% and of the order of 1% inthe low velocity side.

Even if the tanks are thermally insulated it is unnvoid-

4I

ible to have some temperature difference between them. Thisdifference is monitored by means of thermistors and can bekept below .1 "C which represents a velocity error of about.2%.

Air Half Jet

The air flow facility was used during the last part ofthe experiments. A blower-tunnel was utilized to supply atwo-dimensional mixing layer in air. A blowing fan poweredby a 3.7 Ky, 2870 rpm motor supplied the air flow through adiffuser to a plenum chamber. The plenum chamber con.ajuedfoam. screens and a deep honeycomb in order to reduce theturbulence level at the nozzle exit. The air was exhaustedthrough a rectangular nozsle of 7x13 cm (figure 2.). Thecontraction ratio was 10:1.

The jet wAs allowed to mix on one of its boundaries withthe surrounding quiescent air, the c.her three being solidto prevent the formation of mixing layers.

A traversing gear provides hree degrees of freedom witha precision of .05 mm in the x-y plane and .2 mm in thespanvise direction.

The maximum velocity could be as high as 50 m/s althougpiall the experiments in air have been performed at 25 or 16,"m/s. A more detailed description of this facility is giv, •in section 6.

Anemometry and Data Acquisicion System

Velocity measurements were carried out in both mix ,layers with a DISA 55D01 anemometer in constant tempera'u.emode without linearization. The output of the anemometer' wasfed into a 10 bit HP5610A analog-digital convetter an~d thedigitized data were stored by the HP 2100S minicomputer inmagnetic tapa. Data rate was zoverned by tn independentprogrammable crystal clock which provided a 8.. ',Ie frequencystandard. The tape thus produced was then processed off-linein a large IBM 370/158 computer.

In the water tunnel facility the ex ,-: •ri•i•nts werecontrolled by the IP 2100S that -pened ,Pc' closed thevalves, moved thp probe and monitored sJ .',i data astank temperature, flowmeter output and prc-e poa-.txon.

5

3. MEASUREMENT TECHNIQUES

Ho. Film Anemometrj in Water

The velocity measurements in water were carried outusing a hot film quartz coated single sensor DISA 55A81wedge probe. Probe overheat wan kept at .05, which gave usenough sensitivity while minimizing the risk of bubbleformation and deposition of dirt on the probe surface F11".Even at this low overheat, and even if we filtered our vaterthrough a 3 um filter to minimize the number of nucleationpoints, bubbles were present (see figure 3) and about 10% ofthe runs had to be discarded because of this problem.

Surface contamination produced a day to day drift in thethermal characteristics of the fil, which caused a change inthe calibration curve. To study this effect we calibratedour probe in a small, low speed recirculating tunnel withvery low turbulence level, in which the velocity wasmeasured simultaneously with the probe and with a calibratedflowmeter placed in the recirculating duct. The upper partof figure 4 shows different calibrations, carried out ondifferent days and showing c~learly the drift. To correctthis effect we tested several heat transmission models basedon the expression

Nu - Nu - dNun (3.1C

where Nu is the Nusselt number as actually measured by theprobe. Nu,. i- the corrected Nusselt number that would bemeasured by an ideal uncontaminated film and 8 is a parame-ter related to the ratio of thermal conductances of thecoating and the fluid that must be found empirically foreach calibration. A detailed account of this method can befound in V121. There we find that the beat collapse of thedata happens for n-2 (see figure 5) in agreement with themodel proposed by Morrow and Kline in ý13. Applying thiscorrection to the calibration data, all points fall close tothe "ideal" curve that is shown in the lower part of figure4. The standard deviation with respect to chis curvedecreases from 8% before correction to L6Z.

The application of this correction model to experimentalsituatiers presents additional problems, since at least thereal velocity at one point has to be known to compute .Diaphragm flowmeters were introduced in the sLpply lines tomonitor the volumetric flows but, since the rest sectionwalls were fixed and we expected some pressure gradient todevelop on the streamwice direction, these flows gave U9only the velocity at the exit of the nozzles. In facr thepressure gradient had to be computed as part of the probe

calibration procedure.

i

6

Let us assume that one data point is measured in eachstream giving Nueselts NuI and Nu. and that the velocitiesat the exit of the nozzles are known to be UpF and Ua . Thevelocities in the sections under consideration can berelated to the pressure through Bnrnouilli'e equation whichafter linearization gives,

UiIE iE 3.2)

If the calibration formula fo, the probe is

U - f(Nu ) - f(Nu - 3Nu ), (3.3)C

linearizing again and substituting in (3.2) we get from thetwo streams a system of linear equations permitting us tocompute 6 and the pressure as functions of the measuredNusselt numbers and velocities at the nozzle. Figure 6 showsthe pressure gradient computed in this way in five differentco-ordinates along the test section, confirming the consis-tency of the above procedure and the applicability of thecorrect ion method.

Taking into account the scatter and errore associatedwith the calibration of the probe, the flovasters and themeasurements of water temperature and pressure gradient weeatimate the accuracy of our velocity measurements to beapproximatoly 3%.

Hot Wire Measurements in Air

In the mixing layer in air classical hot wire anamometrytechniques were applied. Velocity 2'luctuations were measuredwith a DISA 55F31 hot wire probe, "- um in diameter and 1.25mm in length. An overheat ratio of 0.8 was used in allcases.

Data processing

Two different types of data were taken during the watertunnel experiments: a) profile measurements at four down-stream sections and b) spectral measurements at fixedpositions within the shear layer.

Ten Co twelve points were taken in Pach traverse. Thedata collection period per point was 3 seconds which corres-ponds to approximately 10 large structures passing bv at themost downstream section, x-37 cm.

The spectral measurements were taken leaving the probestanding still at fixed positions within the layer and, inorder to resolve very low frequencies, the data collectionperiod at these points was the whole duration of the run,i.e. 40 to 50 seconds.

7

In the air eyperiments the measurements taken wereessentially the same, although the exploration was extendedrO th"ee dimens'ons, and tae data c~lleccion peeiod was

changed to corresPond to the different frequencies involved.Mean and rms velocities ii water were computed by using

the calibration curve on the digital data, and taking theaverage and st&ndar deviat*.on -if the resulting velocityrecords. Our previous knowledge of the free stream veloci-ties-, the pressure gradient and the hoc film drift was usedin his reduction.

The spectra, bcth ir air and vater, were also computedin this way. The mean velocity wa's reinoved for each run, andan a ceraged FFT technique [14], with a Blackmann-Harris datawindow [15] , wa,_ "•ed to ,btain the pow-r spectra. Finallyall the spectra ccrrespoding to a given station were aver-age•. together. Sxne the le-igth of a long run is 40 to 50 sLid 4 to 6 runs were made on every single point the minimumnumber of large structures presented -in each averagedspe.ctrum is close ýo 1000, altiough in some cases it is muchlar'er.

Mean quartitie, in air 479re averaged analogically fromth� •inemozaeter output au4 la:er reduced to velocity andtuxbulence levelpi.

4. RESULTS

Flow description: water

The top part of figu rq 7 shows nondimenoional profilesof mean velocity which fito fairly well to an error functioncurve. In spite of the slight favourable pressure gradientmentioned above the growth is linear and similarity isachieved. The vorticity thickness of the layer was measuredand the spreading rate found to be .0793, with a virtualorigin located 26 mm upstream of the splitter plate edge.This spreading rate is perhaps a little high but within therange of values given in previous studies [16], [17]. Thedividing streamline is n--.0065 where the nondimensionalmean velocity is (U-U 2 )/(Ui-U2)-S.58.

The bottom part of the figure presents the rms value oflongitudinal velocity fluctuations at two sections. In themost upstream one some influence of the initial conditionscan still be detected and the general turbulence level ishigher than in the other one. The Reynolds numbers for bothsections, xAU/v, are 34000 and 63000 which are below thelimits given in [3] and [10] as needed :o attain similarityof the fluctuation profiles. In fact figure 9 shows themaximum value of fluctuations measured as a function ofdownstream coordinate and it is clear that it has not yetattained its equilibrium value. This asymptotic value can beestimated to be between .18 and .19 which is again a littlehigh but reasonable. The facts that both the growth rate andthe turbulence level are slightly high might nor be inde-pendent E18].

Some time lapse pictures of the shear layer were takenby injecting dye at the spiitter plate and these pictureswere used to measure the visual spreading rate. The result-ing value 6 i(x-xo)i.125 is consistent with that given in[191, even if it is not clear whether dye visualizationshould give the same spreading rates as the shadowgraphs.

Flow description: air

A study of the two-dimensional air half jet was under-taken with the splitter plate of type I of figure 2 , whichis essentially a thin aluminium sheet and a jet velocity of25 m/s. The mixing layer thus formed has been analyzedthrough mean and fluctuation velocity profiles and spectrataken at several positions.

The top of figure 8 presents nondimensional profiles ofmean velocity, which fit fairly well to the expected errorfunction curve. From these profiles we see that similarityhas been attained at least since x-80 turm. The vorticitythickness of the layer was measured and the spreading rate

• m i I i -I I I •I

9

was found to be .195, with a virtual origin located 9 mmupstream of the splitter plate edge. This spreading rate ishalf way between those reported by Wygnansky and Fiedler[20] and Liepmannand Laufer [21). The dividing streamline isfl0-r0

3 and the corresponding non-dimensional mean velocityis (U-U 2 )/(U1-U2)'.57. This result is the sama as the onefound in [213.

The bottom part of figure 8 shows the rms value oflongitudinal velocity fluctuations at three sections. Themost upstream profile indicates that the turbuleat intensityhas not achieved its equilibrium value, and this isconfirmed in figure 14. The asymptotic value of the maxinumu'/AU is .17 which again agrees fairly well with thosereported in [20] and [21].

A third experiment using the same facility and a veloci-.ty of 16.4 m/s is described in section 6.

• I I I I I II I I I I I I-T --I I II

10

5. SPECTRAL RESULTS

In order to study the development and interaction of thelarge coherent structures and their possible relation to theorigin and evolution of three-dimensionality we have carriedout a detailed spectral analysis of the velocity fluctua-tions in our three mixing layers: the two-stream waterfacility and the air half-Jet at two different velocities.

All spectra were taken at the mid-span plane. Those inwater were measured either at n-y/(x-xo)-O or n-.043. Thelast •osition permitted a reliable detection of the largestructure peaks, not yet swamped by the strong low-frequencynoise that seems to be associated to the center of themixing layer, while still showing a high-frequency inertialsubrange characteristic of fully turbulent flow. A typicalevolution of the power spectra across the mixing layer isshown in figure 10.

For the high-Apeed air jet (25m/a) all spectra weretaken at n-0 while those for the low speed one (16.4 m/s)fall in two series, none of them along a similarity ray. Oneseries was measured at U1/IU-.5 and was used to study theinertial subranges while the other, at U/U0,-.08 and there-fore in the outer edge of the layer, was used for thedetection of the large structures. Since this work isrelated to mixing layer transition and most of the data arefar from the asymptotic similarity range, this laterapprcech waa found more satisfactory than keying themeasurements to geometric variables that are relevant mostlyfor the downstream, well developed, part of the layer.

The frequency behaviour of the spectra can be dividedbroadly into three ranges: low frequencies, large structuresand inertial range. Only the last two are studied separatelybelow since the low frequency one, though interesting [221,does not seem to be influenced by three-dimensionality.

The spectrum of the large eddies

The part of the spectrum corresponding to the largestructures is the most interesting one but also the mostcomplex as can be seen in figure 11, which shows the down-stream evolution of the water spectra.

One of the most noticeable aspects is the existance ofpeaks which can be followed across several stations with avery definite constant freqency, while the low frequencyones gain importance as we move downstream. The evolution ofthe position of these peaks with downstream coordinate isgiven in the bottom part of figures 12-a,b and c, for thethree layers. For the purpose of these figures a peak wasincluded if its maximum power was at least 10% above thespectral continuum at that frequency.

11

An important phenomenon is that the ratio betweensuccessive spectral lines is mostly two, although differentfactors, like 1.5, are also present. These values agree withthose given by Liernan and Jimenez [23) and Jimenez et al.[22 ] and can be explained by means of the amalgamationprocess. The doubling is easy to understand through thecoallescence of two equal eddies. The factor 1.5 is alsoexplained assuming the amalgamation between structures oftwo successive generations, which implies that some eddiesof the last one have escaped without pairing and hencecoexist and eventually amalgamate with vortices whose scalesare twice as large.

The same variables are 8hown nondimensionalized infigure 13. The frequencies are made non-dimensional with theconvection velocity, •f-U 1+U2 /2, and with'a characteristiclenght which is supposed to be proportional to the wavei-eugth of the initial instability of the layer. For the airjets this length is the momentum thickness, 62 , of theboundary layer [3, 19]; for the water layer we have used anequivalent thickness, related to the most unstable wavelength of the wake profile of the splitter plate (seeAppendix A). The longitudinal coordinate is made non-dimen-sional with the length U612/AU, proportional to the lengthneeded for a spatially amplified disturbance to develop[241.

In these dimensionless variables the frequencies of mostsoectral peaks agree for the three mixing layers, especiallynear the splitter plate, indicating that instability theorydominates the flow for at least a few pairings. Thosespectral peaks which appear only in one of the layers canusually be traced to forcing frequencies, e.g. blowerfrequency, which are not intrinsic to the flow. Moreover, inthe downstream region in which some degree of self-preserva-tion has been achieved, all layers follow a unique frequencylaw which can be written as

X = .3 AU x/U (5.1)

and agrees very well with the data reported by other inves-tigators 125,26].

Some coupling between the coherent structures and theinstabilitie3 of the initial boundary layer in the airexperiments is apparent in that frequency peaks that willlater be dominant in the mixing layer are visible veryearly in the downstrem development (fig. 12-b,c). Theboundary layers in these cases are marginally unstable(U ai/v- 650 and 520), and some of the frequencies inducedeiýher by the blower fan or possibly by coupling withdownstream structures are amplified and seem to survive inthe mixing layer. Other frequencies which are far from the

12 |

instability region in the boundary layer do not survive andare lost quickly, even if their absolute amplitude is larger(see also figure 16 and discussion in section 6). In thewater mixing layer the boundary layers at both sides of theplate are stable (U6,/v

13

fluctuations begin to decrease to their asymptotic level,confirming that the two phenomaena are not unrelated.

The high frequency end of the inertial range is markedby a corner in the spectrum beyond which the slope becomesmuch steeper. The position of the corner is given in thebottom parts of figurez 12ab,c and is very near in all casesto. the inverse of the sensor length, so that the new rangeis probably explaineAd by the limitations of the sensingequipment.

Finally a word must be said concerning the dimensionlessvariables used. Considering that only three independentexperiments are available it is not really possible todecide in moat cases which is the appropriate normalizationand the variables have been chosen with some particulartheoretical model in mind. The growth of the large struc-tures is the best understood process and conacquently thenon-dimensionalization works best in that case (figure 13).The cause of the three-dimensionality is less clear and wehave done little else than trying several normalizingfactors. Using at attcissa the ratio x/62 gave slightlybetter results than the one used here, but was difficult tojustify theoretically for the tv,-stream case. The obviousparameter ,AU Sw/v, increased the differences between thethree experiments to almost an order of magnitude.

i4

6. SPANWISE FLOW STRUCTURE

An important question is whether the transition tothree-dimensionality is itself two-dimensional. An obvioitspossibility, for oxample, is that perturbations are injectedfrom the end plates that limit the shear layer in span. andthis mechanism has been shown to be important in some casesof transition in the two-dimensional wake behind a cylinder[281. If this be so in our case it should be expected thatthe three-dimensionality will appear further upstream as weapproach the end plates and, to test this hypothesis, wehave explored the flow in a shear layer along the z coordi-nate as well as in the x-y plane. The somewhat surprisingresults are presented in this section.

Experimental arrangeuent.

The measuremento were carried out in the air half jetadjusted to a nominal velocity of 16.4 a/s. Two slightlydifferent splitter plate configurations were used which areshown in outline in figure 2. Cronologically, configurationI was used first and later taken appart and substituted byconfiguration Ii. The boundary layer in both cases waslaminar with a momentun thickness of about .2 mm at the exitsection, corresponding to an effective linght of flat plateof 90 mm in case I, and slightly less in case IH. Thisthickness was both computed from the area distribution ofthe exit nozzle and measured directly by exploration of theflow with a hot wire just downstream of the plate edge. Whe"severe spanwise nonuniformities in the flow were discoveredin shear layer I the two-dimensionality of the edge waschecked and found to have irregularities with an amplitudeof nearly .5 mm and a characteristic wavelength of 50 mm.Also a slight wobble (.2 mm) was found in the y position ofthe probe as it was traversed in the z direction. As aconsequence, and even if it was doubtful whether such smoothirregularities could cause the observed effects, the split--ter plate was substituted by a carefully machined squareprism (II) whose two-dimensionality is thought to be betterthan .1 mm in the 130 mm span. The traversing mechanism wasalso rebuilt to minimize the wobble and any remainingdeviations were calibrated against the edge of the splitterplate. This calibration procedure was repeated several timesand the resulting positional accuracy of the probe withrespect to the plate edge was estimated to be about .05mm(rms). The x and y movement of the probe was done by apair of crossed lathe beds whose positional resolution isalso .25 mm, which probably accounts for much of the uncer-tainty quoted above. The data reported in this section referto layer II unless explicity stated otherwise.

ISI

A general sketch of the nozzli geometry, together withthe coordinate system used and the the approximate positionof the mixing region is also shown in figure 2.

All measurements were taken with a single hot wire 1.25mm long and 5 pm in diameter, whose holder, with a diameterof 3 mu, was held at an angle of 45 degrees out of the planeof the flow to minimize interference. The wire itself wasaligned parallel to the splitter plate edge.

The spanvise uniformity of the free stream was checkedand found to be better than .5%. The residual deviationswere smooth and showed no sign of oscillatory behaviour. Thefree stream was 2% faster near the splitter plate than atthe center of the jet in a way consistent with the expectedflow at the exit of a concentration. The level of longitudi-nal velocity perturbations ,in the stream was comparativelyhigh (.8Z) and concentrated in a fAw well defined frequen-cies, mainly harmonical of the rotation frequency of theblower fan (48 Hz), and as a consequence, the boundary layercontained a rather high level of perturbations (4Z). Powerspectrr for rho free stream and boundary layer longitudinalvelocities are given in figure 16 and show a clear corres-pondance between the important peaks in both cases. Thefrequencies of these peaks are also plotted on the Toll-mien-Lin stability diagram for the instability limit fortwo-dimensional perturbations F291 and those peaks at theapproximate frequencies show signs of alight amplification.Although no measurements were taken of the mechanicalvibrations of the splitter plate it was clear y feeling

manually the side walls of the tunnel that pait of theperturbations to the boundary layer were transmitted thatway, and by taking a series of steps to change the rigidityand mass of the tunnel walls the level of perturbations wasreduced from about 10% to the final 4%.

The boundary layer, measured just downstream of thesplitter plate, was not very two-dimensional. Four profilesare given in figure 17 in which the transverse coordinateshave been normalized with the local measured momentumthickness. The average velocitie are seen to collapsesatisfactorily and to follow qui.. closely the Blasiusprofile for the laminar layer, but the turbulence levelsshow a large scatter. Figure IS shows the spanwise variationof the momeatum thickness and the maximum turbulence levelacross half of the total span. The momentum thickness showsvariations of almost 20% and the non-uniformity of theturbulence level is even higher. It is not clear which isthe cause of this nonuniformity although it is presumablyconnected either with small geometric irregularities in thenozzle walls or w:..h changes in the amplitude of the mechan-ical vibrations. The fact that the layer is just transition-al probably amplifies the ^ffect of any such perturbations.

16

The measuremients were done over a period of several days andenough points were re-checked to make sure that the effectwas repeatable and that the position of the nonuniformitiesremained constant with time, but no detailed comparison wasdone between the two splitter plate configurations and it isnot known which was the effect of rebuilding the plate onthe boundary layer.

Results

Most of the results presented in this section refer toaverage properties integrated analogically over periods ofabout one minute and reduced to velocity without making anyallowance for effects due to hot wire nonlinearity. Anexception is the data used in power spectra, which weredigitized directly and treated in the way described earlierin the report. Each spectrum derives from about 400 s ofmeasurements.

Using splitter plate I, we took data at two differentspanwise locations, mid-span (z-0) and z-35 mm. Averagelongitudinal velocities and turbulence levels correspondingto the mid-span position are shown in figure 19 where thetransverse coordinate is made nondimensional with the localcomputed vorticity thicknasa and normalized to the localcentral streamline, defined as the point where U-.5 U.. Itis seen that the profiles become self similar beyond x-60mm, and that this is true even for the turbulence levels,although with a larger scatter.

A less reassuring view is presented in figure 20 wherethe growth of the vorticity thickness and the evolution ofthe maximum turbulence levels are given for three cases, thetwo spanwise locations in layer I and the mid-span positionof layer II. It is specially interesting to note that thegrowth of the thickness of layer I is not the same, even atthe most downstream coordinate, for the two snanwise loca-tions, and that the data for the other configuration do notagree either with any of the two curves, even if the threeasymptotic growth rates are within the range of values foundby other investigators.

As a consequence of this result the nozzle configurationwas rebuilt and a detailed spanwise exploration of layer IIwas undertaken. Figure 21 shows the turbulence levelsmeasured on the plane y--,082(x-3.45 mm), and it is clearthat, also in this case, there are soanwise nonuniformitiesamountit.g to nearly 15% of the average turbulence level andthat these nonuniformities organize themselves broadly inlong longitudinal streaks.

The next three figures show maps of average velocitiesand turbulence levels in three roughly orthogonal sectionsof the shear layer. Figure 22 is a conventional x-y map

17

taken at mid-span and shows t.- expeCte' velocity distribu-tions. In particular the peak ehat appears near x-30 mm inthe turbulence level seems to be charanteristio of flowswith a laminar initial boundary layer r31.

The other two figures are spanwise sections and areconstructed in a slightly different way. For each x-yposition a representative value of velocity and turbulencelevel is defined by averaging over the spanwise coordinate,z, and the figures show the difference between actualmeasured properties and those locally constructed averages;they represent the effect of spanwise nonuniformitiesindependently of the general background levels of thevelocity and turbulence fields.

The x-z section in figure 23 shows clearly the stream-vise turbulence streaks, The corrdsponding average velocitymap ahows that these streaks are associated to long stream-wise jets of higher than normal velocity. The jets appear toattain their full strength near x-40 mm and are involvedlater in a series of interactions that keep increasing theirsepatation progressively, although discontinuously, withoutappreciably decreasing their intensity.

The transversal structure of these streaks is shown inthe y-z section in figure 24 where the jets are shown tohave transversal as wall as longitudinal coherence. Perhapsthe most signtficant feature ia Lhis figure is that thevelocity field is roughly simmetrical about the centralstreamline, while the turbulence level is approximatelyantysimmetrical. This is in contradiction with the explana-tion of the nonuniformities as streamwise vortices of thetype postulated by Benney C30] ani known to be important inthe transition of the laminar boundary layer, and agreesmore with a model involving streamwise jets, or cross-strea:ivortices. This will be discussed later in more detail.

In understanding these jets an obviaus possibility isthat they are somehow inherited from some irregularity inthe initial conditions. We saw before that the free streamin the neighbourhood of the splitter plate was examinatedand shown to be uniform within the accuracy of the hot wireequipment, but that the spanwise nonuniformities found inthe boundary layer just after separation were importantenough to be considered as likely candidates to be the causeof the stronger features found further downstream.

Before this matter is examined n word must be saidregarding the accuracy of the measurements. It was notedabove that the position of the probe was known only towithin .05 mm and, since near the splitter plate there arestrong transverse velocity gradients, this uncertaintyinduces errors in the measurement that can be important inthe upstream part of the layer. These errors are largeenough that data on velocity variations upstream of x-15 mm•

are not really signliicant and could be consistent with auniform average velocity while turbulence data, being lessaffected by strong gradients, are qualitatively, althoughnot quantitatively, reliable all the way to the splitterplate.

Having in mind this warning, the x-z map in figure 23suggesta that there is no direct continuity between thefeatures in boundary layer and those downstream. It. particu-lar the region between x-5 and 20 mm seems to be one ofreorganization of the apanwise structure, vhile the factthat the separation of the jets changes downstream and thatthere is a rapid increase in their intensity near x-40 amsuggests that some kind of local instability of the mixinglayer is responsible for them.

If we think of the large coherent eddies of the mixinglayer as representative of the dominant longitudinal mode oflayer instability, the spanwise nonuniformities can beconsidered as a secondary wave superimposed on that mainone, and by comparing the power spectra of the flow atvarious z positions it slould be possible to study theproperties of that vave.

For a particular=,_-y position we define a referenceturbulence level 77- -u- and povrer spectrum P by averagingover the span. At a givan z the devigticn from these v&luuacan be ascribed to the secondary- spanwise wave and, to afirst approximation, we could expect that the deviation inthe spectra are proportional to those in 02. If this is truethe normalized excess power spectrum

P(f,Z) -P(f) Tp(fz) (6.1)

P(f) c:(z) - o

should be independent of z and represent the frequencystructure of the secondary wave. Figure 25 shows an estimateof this quantity for three different x-y points, obtained byevaluating equatizn (6.1) at several spanwise positions andaveraging over z,

pýM = -P(f, )- (6. 2)

Since the integral of the power spectrum should equal 0.it is easy to see that p should, in the average, be equal to1, with those frequencies where ý>I representing the spec-tral peaks associated with the secondary wave. The mainneaks evident in the spectra in figure 25 are represented inthe lower part of the figuire superimposed on the x-frequencydiagram that gives the evolution of the large structure inthe layer (see also figure 12-c). If the jets were just

• . , , , i----• - r I I I I I I I I III

19

"fossil" effects deriving from activity in the boundarylayer one would expect their spectral behaviour to remainconstant downstream, with peaks associated either to thespanwisa instabilities, of the transitional boundary layer[31] or with the forcing frequency from the blower fan.Instead. the dominant frequency shifts downstream andremains locked approximately with the frequencies associatedto the large eddies in the mixing region. This gives furthersupport to the view that the streaks are caused by &nintrinsic local instability.

A final piece of evidence, although fragmcntary anddifficult to interpret, is important enough that it shouldbe presented here. Before the apparatus for layer I wastaken appart and the exit nozzle rebuilt, some exploratorymeasurements were taken of the spanwise distribution ofvelocity and turbulence. -Since it wag expected that thereworking of the splitter plate, with its complete change inthe position and nature of all random irregularities, wouldchange also the nonuniformities in the flow, no measurementswere taken for layer II at the same points as in I.Nevertheless the measurements taken in I 4re represented infigure 26 together with their closest equivalent in II.While the mest upstream data do not show any similarity inboth cases, due may be as much to the inaccuracy of themasoramenta i. Liar station as to anything else, thesituation is different in the downstream sections where thesimilaritits are high enough that it is difficult to escapethe conclusion that the two flows have basically the samespanwise stcucture.

Since there is a very low probability that the spanwisestructure of the initial boundary layer is even roughlysimilar in both cases, the similarity of the final flowseems to imply that the initial conditions are not importantfor the development of spanwise nonuniformities. Whetherthis persistance of the structure after the reworking of thesplitter plate is due to some intrinsic property of thelayer or to an irregularity of the free stream, not theboundary layer, small enough to be undetectable by ourequipment but still important for the flow ia not clear andmust await further investigation.

Theoretical considerations

We have shown that under some conditions a mixing regionthat should be expected to be two-dimensional contains adefinite three-dimensional structure consistine of hglh

turbulence streaks associated to high velocity longitudinaljets. The position of these streaks remains fairly constantfrom one day to another and even shows some degree ofpermanence after severe %hanges in the initial conditions.

20

Similar three-dimensional effects have been observed inother flown, notably in the transitional boundary layer [31]And in the wake behind a two-dimensional body r281. In themixing layer Konrad [61 observed longitudinal streaks in theconcentration interface while Breidenthal r7 ] repeated t heobservation under different conditions and shoved that thestreaks arise f.rom "corrugations" in the vortex cotes (,f thelares structure.

Bennsey 30) studied theoretica.lly the nonlinear .,,e:meat of the instability of a laminar mix.in,•t ) .i SUdconcluded that a three-dimensional vav, can itnraruer withthe dominant three-dimensional mode to prvcduce :or'nanentthree dimensional variations of average velo•t~ly inociatedto longitudinal vortices aligned with the x-axis. Th-t workwas aimed at explaining the transition in Lhe ; 41inarboundary layer, where longitudinal vortices arcs kn-'- toapperr, but unfortunately does not explain the transversalstructure of the ;treaks observed in our case.

In fact a longitudinal vortex transfers momentum acerossthe mixing layer producing an average antysimmetric distri-bution of velocity exces5, in contradiction with the resultspresented in figure 24.

The structure shown in that figure suggest that thevortices responsible for the streaks are aligned in thetransverse, y, direction, while the spectral results showthat these vorticta are aouihow L11 phase with rho largestructure. When the two results are put tqgether we arriveat a picture in which the vortex cores of the primarytwo-dimensional instability are deformed or corrugated inaccordance with Breidenthal's pictures, and the corrugationhas at least a component in the y-z plane. At cthe pointswhere the corrugation is concave towards the high velocityaide the velocity is higher while the center of the core,and therefore the region of higher turbulence, i displacedtowards the low speed aide of the layer (see figure 27).This picture is in qualitative agreement with the data infigure 24 but, if these features are to be visible in theaverage, all the cores have to deform "in step".

Saftman r321 has suggested that the instability respon-sible for the corrugation of the cores is the same oneappearing in vortex rings. Briefly, the criterion forinstability is that a core deforms in such a way that itaself-induced rotational speed ic zero [331, in which casethe strain field induced by other cores in the layer is ableto pull it appart and feed the growth of the deformation.This growth starts at an angle of 45 degrees to the flowdirection and has therefore the transverse component Impliedby our observations.

The lowest mode for which this happens is a corrugationof the core in which both radial and spanwiae vorticity

21

distributions are changed. The most unstable wavelengthdepends on the vorticity distribution assumed initiallywithin the cores but it falls between 2 and 2.5. times thevortex radius C34, 35], and, if we estimate this radius asabout half the vorticity thickness of the layer we arrive at

A 1 to 1.25 SW (6.3)

While this is only a rough estimate, the wavelengthsmtasured in the layer and shown in figure 28 are indeed ofthis order of magnitude, specially in the downstream part ofthe layer where the streak system is welt developed. Thus,this mechanism might be able to explain the instability of asingle vortex core, but the question of whether it ispossible to lock all the vortices in the layer into acoherent instability that will produce the observed jets isstill open and needs further investigation.

22

7. CONCLUSIONS

There seems to be at least two different phenomenaconnected with the three-dimensional breakdown of the planemixing layer. First, at least in some cases, there is amacroscopic instability of the large structure which produc-es a spanwise system of longitudinal jets whose amplitudebuilds up very quickly and then remains approximatelyconstant while their separation adjust itself to a valuethat is somewhat higher that the vorticity thickness of thelayer. This phenomenon may be due to a collective instabili-ty of the.vortex cores of the same type as the one found invortex rings.

The second phenomenon is the development of a 3-Dinertial subrange in the spectrum of the small scales of thevelocity fluctuations. In the only case in which we havedocumented both phenomena the second one does not developfully until the macroscopic instability has not been growingfor some time and has actually arrived to what seems to beits final configuration.

Moreover there is some indication that there are twoqualitatively different ways for the establishment of the3-D inertial ranges, examplified here by the water and airexperiments. In the first one the transition is abrupt andthere is an initial part in which the inertial range has aslope characteristic of the 2-D enstrophy cascade. In thesecond one the transition is more gradual and there is noevidence of a 2-D regime. It is not clear, at present, whichis the condition responsible for the differences between thetwo cases.

An intet--ing coincidence is that in the three casesstudied here, the transition to three-dimensionality in thesmall scales happens near the second or third vortex amalga-mation (see figures 12-a,b and z), despite differences ofone order of magnitude between the Reynolds numbersinvolved. It may be that the amalgamations act as triggerfor the transition, but the small number of cases makes itimpossible again to derive definitive conclusions.

We believe the work presented in this report should beseen as a preliminary exploration of the effects connectedwith the 3-D transition. Much more research is needed in allcases. In particular it is important to check whether thespanwise macroscopic breakdown can be reproduced in differ-ent circumstances and how it depends on initial conditionsand Reynolds number. An interesting question, in view of theimportance of 3-D structure for a vigorous mixing, iswhether this instability can be enhanced and used to shortenthe distance needed for transition in practical flows.

Secondly the influence of vortex amalgamations ontransition should be checked, Again more experiments with

23

different initial conditions are needed for this, but thetools developed in this study using the power spectra ofvelocity fluctuations should be enough to detect both theaverage position of the pairings and the position of three-dimensional breakdoun.

We want to acknowledge the suppott of part of thisinvestigatkon by the Spanish Coi •sioa Asesora de Investiga-cion as well as the help of the UAM-IBM Scientific Center inMadrid which gave us access to its computer center for thedigital processing of the data in this report.

I

I24

APPENDIX A

Stability charactaristics of the splitter plate wake.

The wavelength of the initial instability in a mixing layeris strongly influenced by the wake of the splitter plate inthose cases in which both free stream velocities are differ-ent from zero. The downstream distance needed for viscosityto fill the dip in the profile produced by the initialboundry layers is of the order of the equivalent flat platelength of those layers while the amplification distance forthe unstable disturbances is proportional to the layerthickness and is usually much smaller. Moreover, for thecommonly found Reynolds numbers the wake flow is unstablefrom the moment it leaves the splitter plate and the nonli-near regime of the instability is achieved before anyappreciable evolution of the profile has taken place.

It seems likely therefore that the wavelength of theinitial instability is fixed by the initial wake profile andwe present here an analysis of a simple straight lineapproximation to this profile. Approximation of this type

have been shown to be useful in other cases [36, 371 andgive a quantitative basis for comparison between differentprofiles. In particular the variation of wavelengths betweendifferent velocity ratios should be given approximatly bythis model.

The profile we use is shown in figure 29. If we intro-duce a perturbation stream function

T(x, y, t) = P(x,y) + ý(y) exp(ica (x-ct)), (A.1)

and assume infinite Reynolds number and parallel flow, :heperturbation amplitude satisfies the equation [38 1,

(U - c) (0" - a2,) - U"D =0. (A. 2)

In the segments in which U is linear the solution of (A.2)in given by

Dexp(±+ ay), (A.3)

while at the corners, where U" is infinite, the streamfunction remains continuous but the derivatives satisfy ajump condition

¢+' - _= (u -U ')/(U-c)_ (A.4)

When we add the conditions that ,P remains bounded at infini-ty and assuming that the real part of a is positive, thesolution of (A.1) can be divided in the segments shown in

25

figure 29. The jump conditions give relations between thevarious coefficients, and the final result is a cubiceigenvalue equation to be satisfied by c.

In the case of the mixing layer the equivalent flatplate length for both boundary layers is usually the samc,and the thickness are related by

h 2/h, = (U,/U 2 ) 1/ 2 (A.5)

The solutions to the eigenvalue equation with this conditionis given in figure 30, where the stability boundary is givenin terms of the vave number and the velocity ratio, togetherwith isoline. of the spatial amplification rate ahlci-/cr.The properties of the most amplified wave are also given infigure 31 as functions of the velocity ratio.

For our water mixing layer U2 /U 1w.51 and the mostunstable wave number is 1.25 times that of the half jetwith the same boundary layer thickness at the high speedside, Therefore the vavelength and the "equivalent" initialthickness can be taken as .8 times the real high speed sidevalue, which is the thickness used in section 5.

Some of the ideas presented in this appendix wheresuggested by Prof. A. Lifan.

26

REFERENCES

1. Dimotakis P.E. and Brown G.L. (1978). The mixing layerat high Reynolds numbers. J. Fluid Mech., 78, pp.535-560.

2. Wygnanski I., Oster D., Fiedler H. and Dziomba B.(1979). On the perseverance of quasy-two-diuensionaleddy-structure in a turbulent mixing layer. J. FluidMech., 93, pp. 325-335.

3. Bradshav F. (1966). The effect of initial conditions onthe development of the free shear layer. J. FluidMech., 26, pp. 225-236.

4. Hussain A.K.M.F. and Zedan M.F. (1978). Effects of theinitial condiLion on the axisymmetric free shear layer:Effects of the initial momentum thickness. Phys.Fluids, 21, pp. 1100-1112.

5. Crow S.C. and Champagne 0.H. (1971). Orderly structurein jet turbulence. J. Fluid Mech., 48, pp. 547-591.

6. Konrad J.1_. (1977). An experimental investigation ofmixing in two-dimensional turbulent shear flown withapplications to diffusion- limited chemical reactions,Ph, D. Thesis. California Institute of Technology.

7. Breidenthal R.E. (1978). A chemically reacting, turbu-lent shear layer. Ph. D. Thesis, California Institute ofTechnology.

8. Bernal L.P.. Breidenthal R.E., Brown G.L., Konrad J.H.and Roshko A. (1979). Proc. 2nd Symp. on Turbulent shearFlows, London, pp. 8.1 - 8.7.

9. Chandrsuda C., Mehta R.D., Weir A.D. and Bradshaw P(1978) Effect of free-stream turbulence on large struc-ture in turbulent mixing layers. J. Fluid Mech., 85.,pp. 693-704.

10. Yule A.J. (1978). Large scale structure in the mixinglayer of a round jet. J. Fluid Mech., 89. pp. 413-432.

11. Rasmussen C.G. (1967). The air bubble problem in waterflow hot film anemometry. DISA Information, 5. pp.21-22.

27

12. Jimenez J., Martinez-Val R. and Rebollo M. (1979). Hotfilm sensors calibration drift in water. Submitted to J.Physics E.

13. Morrow T.B. and Kline S.J. (1974). The performance ofhot vire and hot film anemometers in water. Flow: itsmeasurement and control, (R.E. Wendt Ed.), InstrumentSociety of America. Pittsburg, Penn.

14. Welch P.D. (1967). The use of FFT for the estimation ofpower spectra. IEEE Transactions Audio Electroacustics,15, pp. 70-73.

15. Harris F.J. (1978). On the use of windows for harmonicanalysis with DFT. Proc. IEEE, 66, pp. 51-83.

16. Mills R.D. (1968). Numerical and experimental investiga-tions of the shear layer between two parallel streams.J. Fluid Mech., 33, pp. 591-616

17. Miles J.B. and Shih J. (1968). Similarity parameter fortwo-stream turbulent jet mixing region, AIAA Journal, 6.pp. 1429-1430

18. Pui N.K. and Gartshore I.S. (1979). Measurements of thegrowth rate and structure in plane turbulent mixinglayers. J. Fluid Mech., 91. pp. 111-130.

19. Roshko A. (1976). Structure of turbulent shear flows: anew look, AIAA Paper 76-78.

20. Wygnanski I. and Fiedler H.E. (1970). The two dimension-al mixing region. J. Fluid Mech., 41, pp. 327-362.

21. Liepmann H.W. and Laufer, J. (1947). Investigation offree turbulent mixing. NACA Tech. Note 1257.

22. Jimenez J., Martinez-Val R. and Rebollo M (1979). Thespectrum of large scale structures in a mixing layer.Proc. 2nd Symp. on Turbulent Shear Flows. London, pp.8.7-8.12.

23. Hernan M.A. and Jimenez J. (1979). The use of digitalimage aaalysis in optical flow measurements. Proc. 2ndSymp. on Turbulent Shear Flows. London, op. 7.7-7.14.

24. Jimenez J. (1980). On the visual growth of a turbulentmixing layer. J. Fluid Mech., 96, pp. 445-458.

28

25. Brown G.L. and Roshko A. (1974). On density effects andlarge structure in turbulent mixing layers. J. FluidMech., 64, pp. 775-816.

26. Winant C.D. and Browand F.K. (1974). Vortex pairing: themechanism of turbulent mixing layer growth at moderateReynolds numbers. J. Fluid Mech., 63, pp. 237-255.

27. Batchelor G.K. (1969). Computations of the energyspectrum in homogeneous 2-D turbulence. !P.!j* FluidsSu221. II, pp. 233-239.

28. Garrard J.R. (1966). The three-dimensional structure ofthe wake of a circular cylinder, J. Fluid Mech., 25, pp143-164.

29. Shen S.F. (1954). Calculated amplified oscillations inplane Poiseuille and Blasius flows. J. Aero. Sci., 21,pp. 62-64.

30. Benney D.J. (1961). A non-linear theory for oscillationsin a parallel flow. J. Fluid Mech., 10, pp. 209-236.

31. Klebanoff P.S., Tidstron K.D. and Sargent L.M. (1962).The three-dimensional nature of boundary layer instabil- -Iity. J. Fluid Mech., 12, op. i-34.

32. Saffman P.G. and Baker G.R. (1979). Vortex interactions,Ann. Rev. Fluid Mech., 11, pp. 95-122.

33. Windnall S.E., Bliss D.B. and Tsai C.-Y. (1974). Theinstability of short waves in a vortex ring. 3. FluidMech., 66, pp. 35-47.

34. Tsai C.-Y. and Widnall S.E. (1976). The instability ofshort waves in a straight vortex filament in a weakexternally imposed strain field. J.Fluid Mech., 73, pp.721-733.

35. Saffman P.G. (1978). The number of waves on unstablevortex rings. J. Fluid Mech., 84, pp. 625-639.

36. Esch R.E. (1957). The instability of a shear layerbetween two parallel streams. J. Fluid Mech., 3, pp.289-303.

37. Michalke A. and Schade H. (1963). Zur Stabilitat vonfreien Grinzschichten. I-,. Archiv., 33, pp. 1-23.

29

38. Lin C.C. (1955). The theory of hydrodinamic stability.Cambridge Univ. Press. Cambridge. England.

'I

I

30

31

PRESSURIZATION

S~VALVES

FLOWME TERS Ii •ON -OFF

VALVESFLOW

STRAIGHTENERS

TEST SECTION

STO SINK

Fig. I.- Schematic view of water installation.

32

10 10m 1 0

/z

//X XI

I ~I: UG160

civ 2.-(:eoMe.trv of te'st se~ction in iir hafio. Prwn oil~itternlates* we~re use~-d which are represented )n rop (oord inaito.ývstem shown is that used in ;ocltion 0 of text.

33

I

Fig. 3.- Effect of an air hubble in the voltape response c.fthe hot film sensor.

.35- o

Nu %X

0

a OX A Nuc

625- 0 ~OXA A-3

.24 o cA 0

ODAA

.2* .•A 25

-. 20

4 6 8 e

FI i{. 4.- FoLtr different hot film ,ca1ibra ions, showiiii' drilt(top) mýtd "" ile" curve oh r i ed throt,,h ,forrecrt ion(3.1) with n 2.

34

Xx

xx Xi x X X X

X X

xx !

0 1 2 3

Fig. 5.- Variation of rms error after correction of calibration

drift as a function of the exponent in equation (1.1).

-. 202AP o

§-. !0 o

0

I °

0 20 x (cm) 40

Fig. o.- Pressure grradient in tt,.t section of water tunnelcomputod as part oi- the drift correction procedurefor the hot film.

35

I I A I

0. 0

0 oxoXOX

0 . I I.1 .05 0 -. 05 -.1

y/(x-xo)

U4-U, C ••2M xXXx x x.2 o x=37 x

Fi. 7- Avrg Xoiy tp n lctainpoie

XX l e XX 0 0OaxSX X o

| I - xX o00 XSX O0 Oo X

SX 0 X

X 0 , X

.1.05 0 -. 05 -.I

y/ (x-X.)

Fig. 7.- Average velocity (top) and tluctuat-ion profilesfor the water mixing laver; U2/U,=.5].

36

I III

o x=40mm n,0OxAOAxOA

x x=80 ax

U A x=120U®

A

0 Ix I I I-.3 -.-. 0 .1 .2 .3

yl(x-x*)

.20 1 001 o

o x=40mm 0xX=80 : •x.15 Xxo

Sx =120

.10 x°x

0X

.05 - OA X °xA AO x 0 ,

o ,, I I I I.3 -2 0 .1 .2 .3

y/(x -xo)

Fig. 8.- Average velocity (top) and fluctuation profiles for the airhalf-jet (U= 25 m/s).

11

37

x(cm) I.30- 10 20 30 40

! I I

.301

.25 a

UmooAU

00

.2 X X x 0 0

xx x x x x x x

.15- x

.10 I , I I0 50 100x(mm)

Fig. 9.- Variation of turbulence level with downstream coordinates.

0 , water layer, top abcissae; X , air (25 m/s) bottom.

38

0,- o' -5/3 !

10

tog P

10

I I I I.5 1I (z 5 10

f (Hz)

Fig. 10.- Evolution of power spectra accross mixing layer inwater. U2 /LTI = .51; x= 250 mm.

39

10

I__ __ __ __ _ __ __ _5 10 50f (Hz)

Fig. 11.- Pcwer spectra variation with downstream coordinate.Water layer at n = .043.

40

-;I I I I

01 OOD 0 0

"-20

n 00 0

-3 o 0

-4-I

-5 I ! I

0 10 20 x 30 40

100,

f(Hz)

00

o o10\

0 "•-- o-o----o-- .. _0 0 00

0

0 10 20 x(cm) 30 40

Fip. 12a.- Por leeend see nage 43.

U

1=

41

-rI I I I-5/3

0 0 0 0 0 0 0

-2 0n 0

0

-3

-4

-50 20 40 60 80 100 120X(mni)

A

1

10.000 , I I ,

ft-,- -A

f (Hx) - 011

00060

1.0t- 000oo0 0 b-o---o-o

00000 o o o o o o 0-0,)•

0

100 0o0 0 o \---o

0p0 I I I I I I0 20 40 60 80 100 120

"x(mm)

"ilj. 12b.- Pot le•,end see pare 43.

42

•._ T I 1 I 1I T- 1 "

0

0 0

103

* 0

0 02

10- L I

0 50 I (mm) 100

Figz) 0l0- 0 0 l "p "0-.,

Figl. l~c.- F'rlege•nd s•ee page 4+3.

43

Fig. 12 a, b, c.- Evolution of the frequency of the spectral peakscorresponding to large scales (o), and the frequencyof the "nose" marking the end of the inertial range(A). Also slope of the inertial range. a) Water layer;b) Air (25 m/s); c) Air (16.4 m/s).

44

100

1~~~~ 6xAx

0 X x

10-4£ A X AxXX X0 0 \x x x,

0 \"00

0 0 0-0 '\

10x x x x x xxx A

OO I

0~ x

x \ A

Ax

if-Ox x 4AX x

xx x

10 100 1000U Ux

Fig. 13.- Dimensionless representation of large scale frequencies versus

downstream coordinate. 0, water; X Air (25 m/s); A , Air(16.4 m/s).

45

.3

0 0

AU 0

XAA x X0A A X C

A A a x X AX x x

xA

A

.1-1

- 5/3& o A x Ax X X

-2 a x40 A x

n xo X

00 A- o

A

-40 500 1000 1500

AU xU 62

Fig. 14.- Dimensionless evolution of turbulence level (top) and -lopeof inertial subrange. 0, water; X . Air (25 m/s); A Air(16.4 m/s).

46

I\

P

100 1000 f(Hte) 10.000

Fig. 15.- Definition of the inertial and probe-limited ranges in thepower spectrum, showing the "nose" separating them.

47

7--

,41o-1 -

p

10,j

10-S

t II

PV

! Il ;- 1~ I 1•1

10 100 f H)1000

-.3

goo v1120v x103

600 k I xl .

600 1.2f(Hz)

400 V

IV

200 111 0

10L_ L___0 1000 2000

UO 61/v

Fig. 16.- Power spectra of free qtro:im (---) and boundary laver -- ) measured

at x- I mim, IT/U. .5. O-iýni, icint neiks -ire represented on Tollmien' ;stab i I itvy diavram for the lamitimr hobind.irv laver. laver 11. z =O.

48

u0.8V

0.4

oo

0 0x x

"34 - 0 x0

Um - x

.02 0xO A 7 x

O_< loaLna,10 y/6 2 1 10

Fig. 17.- Nondimensional profiles of boundary laver at severa.lspanvise loeations. 0 z -,-35 mm; X , z--25;V1 ,z-15; & , z--5. Solid line is Blasiu. profile.L.aver IT, x- I mm.

.04 -

.02

S- I

-4.0 -20 z(mm) 0

i'i,.1.- ;panwise v,'irid.tion o max imuzm ttu•ltil-nc-c- l |vel I-)..ItCd morientiim thickneý- f ..... ,r 'Toind~irv L over.T.a,'er IT, x - I mm.

!I

49

.75 .

CD.50

.2S5

0 0

-' 0

.2s

0X

L__05~ -- 0 V16-10 (Y Y¶%

Fig. 19.- Nundiniensional. Ion.it-udinal ve-locliv and trubulencelevel prof ileg. For the air !aver. 0 , x=40 wma;X, x 60, A ,Y.=80: V x ý 100. L.)ver T, z ý0;

50

30

20

15Crnm)

S.•'.

10

0 so Xlmm) 100

.20

U I t - -

.16 Or/L Ii'12 I

'II

0 50 lc

Fig. 20.,, Vorticity thickness and maximum turbulence levels for

three I tvers in air U..- 16.4 m/s in all cases,i- , z --0; T, z= 35 mm;-..-. IT, z= O.

- 51

UL1

-6

-4

2I

IsI-30 -20 -i0 0 10 20 30

rig. 21.- Turbulence level is Furyetji, of streaimwise and Piqnwi-3e (Z)coordinates. r~vure corresp~onds to lo,,we- part of fieure 23ana it5 nosition in ýhe Inver is marked in ficvirp 22

-- - - - -- - -

52

30 i ý ...... .... Y - itI20 U0,

10

y(Mrp)

0 .8

-20 1 .1

-30 I0 20 0 6080 10010X (mm)12

30y -Z

U, .02

-10

.06

320 L J50~

Fig 2.- Trnsersl X(M ) 0700 120Pig 2 .-. rnfg versation c-y) maps of mixing la yer at center of span.Conigra~ 0~IT, o=.

53

40

22

20 -J---------------- - -- ---

20 --- - - - -

0 -- - -2-

r22

-202

-20

2 -

0 20 40 60 80 100 120x (tym)

40 g

-4 L2 c=.i =-

20

-20

.4

0 20 40 60 60 100 120x(MM)

Fig. 223.- Longitudinal (x-z) maps of velocity and turbulence level devia-tions from spanwise uniformity. Configuration II; position ofmap defined in figure 22.

54

Si. -- I\ ' \ - .

- I -' / I I-~

"1"" .-- -, 4' ,I , , \ .

.,. ,,, Il ..

I 1 I I I 11 1 I I f I I1

SI II -

C),-4

a:l

ctut4-4

f 4 -4 -'

> or-U

___________o 0I-E C-)~1~

L T \ g , '-., 44.4r "

CI ,l \;, A Is ;

N I'.\II ."""\I li,I I I Il\I I,

, ,( , tI 111 { Vii i'" ' ' " i-_'l-H- I

IIl II !r'l l

I •I '' ' IiI '\ iil l )/,II ?• I I I ' I \ I

I \ - -* \- I

'- 4)I. IN V -'--[ @ • , S

l ] i '

_ i 1 L __L... i_•--. i I.----• .I I iI

3X4

3

10 100 ICH) 1000 10000

10X 1x04 UCO XIV Jos

104

102

10L

Fie~. 25.- For lei~cnd see next paRe.

56

Fig. 26.- Normalized deviation of power spectra from spanwiseuniformity; p defined in text. Each curve is averageof several z positions. x- 1, y- .05 mm: z=-40,-30,-15, 10, 20, 35 mm; x-20, y--. 9 mm: z=-35, -25,-15, 5, 10, 25, 40 mm; x=60, y=-6 mm: z--40, -35,-25. -20, -10, -5, 5, 15, 25nm.. Spectral peaks areoverlaid on frequency diagram for layer, (see figure12c) to show relation to dominant peaks in the spectrum.

57N IN

44 II ,II II

0I oj I

0 0 0

-,4 U

C a)

• • II

E ." ,'w- E *

k,,

- X

II..

0 g - II I - g I

w a)C 5,.'4

5'D - a4- Iu

I / vi

I / ii ~ -' U7

W--

58

'S'

Fig. 21.- The corrugations of the vortex cores cause displacementof the turbulence level and variations in mean velociLty.

30 1 1 , ,

00

101

00 40 x(mm) 80 120

Fitg. 28.- Spanwise waveleng.th of jet svstem in air laver. 0wavelength; sol.idi line is vorticitv thickness.

I

59

-=a, eay

-• a,• e 01

11 - U2 b2 ec " + c2e -d"

""v 4- -a

h2 hi

Fig. 29.- Straight line approKimation for the wake profile ofthe splitter nlate. Expressions are the perturbationamplitudes for the stream function.

60

II I / : I!2LlSI I /

I. ,/ .K Cr0 2

2I i-

U1 I I /

Ui,

I INu2/ /

/0ji / 1

4

0 1 2 3

Fig. 30.- Spatial amplification rates and stability boundary for splitterplate wakc -when equivalent plAL- lenvth is the same at both sides.

1.5

0.

5 -

0 .5 U//u

Fig. 31.- Properties oF most qnatiri1lIv impplifieo( wave ibove.

c r /I - - h " / r