Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.

AIAA Meeting Papers on Disc, January 1996A9618738, NCC2-374, AIAA Paper 96-0783

Spatial simulations of a confined supersonic shear layer at two density ratios

Oshin PeroomianCalifornia Univ., Los Angeles

Robert E. KellyCalifornia Univ., Los Angeles

Sukumar ChakravarthyCalifornia Univ., Los Angeles

AIAA 34th Aerospace Sciences Meeting and Exhibit, Reno, NV Jan 15-18, 1996

Spatial 2D simulations of a confined shear layer are carried out for two density ratios using a second-order TVDscheme. The density ratios are chosen based on the linear stability results of Peroomian and Kelly (1995). For both setsof parameters, only the acoustic instability modes exist. Solutions of the steady compressible boundary layer equationswhich are taken to be functions of the cross-stream direction only are used to initialize the computational box. For thefirst case, formation of strong shock-expansion structures is observed within the flowfield. The acceleration androtation of the lower part of these structures due to the generation of vorticity near the walls increase the magnitude ofthe pressure change near the lower wall, causing a blowing effect from the wall which improves the entrainmentprocess. For the higher density ratio, the generation of Kelvin-Helmholtz type structures in the subsonic portion andcompression-expansion waves in the supersonic portion of the shear layer were observed. The fundamental mode wasobserved to saturate quickly and then, after a long plateau, was observed to grow again. (Author)

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AIAA-96-0783

Spatial Simulations of a Confined SupersonicShear Layer at Two Density Ratios

Oshin Peroomian, Robert E. Kelly, and Sukumar ChakravarthyUniversity of California Los Angeles

Abstract

Spatial 2-D simulations of a confined shear layer arecarried out for two density ratios using a second order TVDscheme. The density ratios are chosen based on the linearstability results of Peroomian and Kelly[l]. For both sets ofparameters, only the acoustic instability modes exist.Solutions of the steady compressible boundary layer equationswhich are taken to be functions of the cross-stream directiononly are used to initialize the computational box. For the firstcase, formation of strong shock-expansion structures areobserved within the flowfield. The acceleration and rotationof the lower part of these structures due to the generation ofvorticity near the walls increase the magnitude of thepressure change near the lower wall, causing a blowing effectfrom the wall which improves the entrainment process. Forthe higher density ratio, some interesting phenomena such asgeneration of Kelvin-Helmholtz type structures in thesubsonic portion and compression-expansion waves in thesupersonic portion of the shear layer were observed. Also, thefundamental mode was observed to saturate quickly and then,after a long plateau, was observed to grow again.

Introduction

hi recent years, studies of the instability ofcompressible shear layers have intensified due to applicationsto future generation aircraft propulsion systems (e.g.SCRAMJETS). Linear analysis has been used extensively byresearchers to gain insight into the initial dynamics of theshear layer. Jackson and Grosch[2], as an example,investigated the spatial instability modes in a free shear layerand for the supersonic case, using the hyperbolic tangentapproximation for the basic profiles, found two modes. One ofthe modes was categorized as a 'fast mode' whose phasespeed is supersonic with respect to the slow stream<^ = £/* - c / f l " < 1 and A/ - £ / - c / a >1), andthe second was categorized as a 'slow mode' whose phasespeed is supersonic with respect to the fast stream (Mci > Iand Kid < 1 ). The growth rates of these modes were found tobe nearly 1/5 of those found for the Kelvin-Helmholtz (K-H)mode in an incompressible shear layer and were also muchsmaller than the K-H instability growth rates found for low tomid-subsonic shear layers. This drop in growth rates as theMach number of the shear layer increased was shownexperimentally in the work by Papamoschou and Roshko[3,4].They found that the growth rate curves could be

parameterized fairly well with the use of a Mach numberrelative to the phase speed of the instability wave. This wascalled the convective Mach number. Its formulae in the twostreams are given above and, by assuming a similar ratio ofspecific heats and isentropic flow in the two streams, a singleconvective Mach number can be written as:

Me =a? + a\

The assumption of isentropic flow does not hold forhighly compressible shear layers where shocks can form. Asshown by Papamoschou[5], M el & A/c 2 for these cases;however, he also showed that the above convective Machnumber is a good parameter for characterizing the first ordereffects of compressibility.

Tarn and Hu[6] and Greenough et al. [7] conductedspatial and temporal linear analyses for a bounded shearlayer. Greenough et. al used a hyperbolic tangent profilebased on the cross-stream coordinate and only considered theconstant density case. Multiple modes were found when theconvective Mach number was supersonic. For the ratio of wallheight to vorticity thickness used, these modes are differentfrom the radiating vorticity modes found in the unboundedshear layer and were referred to as supersonic wall modes byGreenough and acoustic modes by Mack[8]. Tarn and Huconducted a thorough spatial analysis of these modes usingboth a hyperbolic tangent profile (based on the cross-streamcoordinate and using the Busemann-Crocco relations) as wellas a vortex sheet model. For the vortex sheet model, theyfound two families of instability modes and labeled themClass A and Class B modes. They also found two families ofneutral modes that they labeled Class C and D acousticneutral modes. For the finite shear layer case, only theunstable solutions were shown because the contour ofintegration was not deflected in their analysis to avoid thecritical layer (Private Communication). Tarn and Hu foundthat for a relatively thick shear layer the most unstable modeis the first Class A 2-D mode. Mack[8] discussed in detail thenature of these acoustic modes as they pertain to boundarylayers, wakes, jets and confined shear layers. The underlyingcondition which needs to exist in order for acoustic modes tobe observed is a bounded region of supersonic flow relative tothe phase speed of the instability wave. This means thateither two sonic lines (Mc =1.0), two walls or a combinationof the two must exist in the flow.

Peroomian and Kelly[l] investigated the effects ofdensity gradients on the acoustic modes in a confinedsupersonic shear layer by conducting a spatial analysis of the

steady compressible boundary layer profiles. For theparameters matching those of Tarn and Hu[6] (density ratiopl I>, = 1.398) they found three generalized inflection pointsfor the basic profiles which for the confined case lead to anew class of modes related to the minimum of ~pDU andwhich possess the largest growth rates. As the density ratiowas increased from this point, two of the inflection pointsdisappeared and the basic profiles were changed Thesechanges in the profiles lead to a linear resonance between theacoustic modes resulting in an emergence of a mode with amuch larger unstable bandwidth and multiple growth ratepeaks which are much higher than those observed for themodes at the lower density ratio, hi this paper, thesignificance of these linear results upon- the nonlineardevelopment of the unstable shear layer will be examined byspatial 2-D simulations using a TVD scheme.

Governing Equations and the Discretization

The equations used to model a compressible two-speciesnon-reacting mixing layer will be the Euler equations whichcan be written in a non-dimensional conservation form:

or

where

dt

dt dx dy

+ V • F = 0 F = fi + gj

Ppv.pv

puui+puv

\. puC, )

pvpuv

pv* +p

The Euler equations are solved using a finite volumeformulation. The equations are integrated over a control areaA,

ff .***

and by using Leibniz's and Gauss's Rules one can obtain,

where /J = « f + « j is the outward unit normal to the controlsurface, jj - -xn, - yn is the movement of the controlsurface, and - ^ J r f 's ^e avera8e °f me

conservation variable over the control surface. For the presentpurposes, let us assume a stationary grid; however, the codewritten for this analysis can handle moving grids. Also, it isassumed that the discretization of the domain is done by theuse of quadrilaterals with nodes ABCD (i.e. structured grids).

Based on these assumptions, the equations can now bewritten as the following:

1

TVD Formulation

The TVD scheme used in this analysis is a highresolution second order TVD scheme based on the limiting ofthe characteristic variables and it is presented inChakravarthyflO]. The equations are solved in conservationform. The scheme uses the cell averages, q~, and the left andright eigenvectors of the Jacobian matrix, t1 and f. FollowingChakravarthy[10] let,

x+l/2

Since the aim of this analysis is to investigate the initialnonlinear evolution of the shear layer and not the processes oftransition and turbulence, the viscous terms will not beincluded in the simulations. It is well known that as Reincreases the mesh size required for capturing the smallKolmogoroff scales arising in turbulence decreases; thereforelarger and larger grids are necessary to resolve both the smalland large scales of the flow. However, for the initialinstability, Reynolds number effects seem to be of secondaryimportance (Ragab and Wu[9]), and so it is hoped that certainaspects of the actual nonlinear evolution will be modeled bythe in viscid approximation.

where

2)andm=jatkandN = (nx,«,). Wherenxandny are the cellface outward pointing normals (unit normals of the cell facesmultiplied by the cell face length). Note that the parameter a'corresponds to the jump across each characteristic and is usedin Riemann solvers such as Roe's Riemann solver. Thisparameter is now used to construct the slopes in a secondorder TVD scheme in order to come up with a betterapproximation than the cell averages for the left and rightstate at each integration node. First a slope limiting process isused in the following way:

minmod [a'mtlll,ba'm.in]

minmod [a'm.in,ba'mt.ln]

The parameter b is known as a compression parameterand is chosen to be 2, and the parameter ^ is an accuracyparameter which determines the truncation error of thescheme and is chosen to be 1/3. This choice of <j>corresponds to third order overall spatial accuracy of the basicunlimited form of the scheme on uniform girds. The minmodoperator is defined as:

minmod [*,y] = sign(x)max[0,min{|x|,ysign(x)}]

Now, within a computational cell, the left value at theface m+1/2 and the right value at the face m-1/2 are given as:

where

Once all the left and right states are computed at the cellfaces, a Roe's solver is used to come up with the value at eachface center. The TVD spatial operator described above is thenused along with a second order Runge-Kutta scheme tocalculate and advance the solution.

Numerical Simulations

The two cases considered for numerical simulations aredone so based on the analysis of Peroomian and Kelly[l]which will be referred to hereforth as P&K.

Case I

p2 = 1.398 A/, =1.836 C/2 = 0.276

M,=4 .500 (He) M1 = 1.600 (JV 2 )

Case I

p2 = 3.000 Mc = 1.836 C/2 = 0.276

M, = 3.876 (He) M2 = 2.019 (N2)

Problem Formulation (Case I)

TVD schemes have been used by previous researchers(Sigalla et al. [11,12] and Lu and Wu[13]) to conductnumerical simulations of the acoustic modes. The first casesimulated is that of the test case parameters. Previousresearchers, namely, Lu and Wu and Gathmann et al. [14]have conducted 2-D and 3-D simulations respectively withthese parameters. There are some differences between thecharacteristics of the modes and profiles used in their analysisand the ones used in the current case. Since Lu and Wuconducted 2-D simulations, the current case will be comparedto their conditions and results. Lu and Wu used theapproximate hyperbolic tangent profiles from Tarn and Hu[6]as their basic flow profiles for which the maximum growthrate was found to be 0.426 at o=L8 corresponding to thefirst Class A 2-D mode called the Aoi mode. They prescribedthree types of disturbances: 1) fundamental forcing , 2)fundamental and sub-harmonic, and 3) broadband. In eachcase they found that the fundamental mode was selected andgrew in the downstream direction. The forcing in theiranalysis was applied only to the pressure using only the realpart of the eigenfunctions, i.e.p' = pr(y)cos(cot + <j>)

There are several differences in the current simulations.First, the basic flow profiles used to initialize thecomputational box are those from P&K, i.e. the solution ofthe steady compressible boundary layer equations. This wasdone by adding subroutines to the code which calculated theprofiles at the integration nodes of each computational cell,thus enabling a cell average of the initial profiles to becalculated P&K showed that these profiles are different fromthe hyperbolic tangent profiles used in previous analysis.They have three generalized inflection points which in turngive rise to a new class of modes. The maximum growth rate,-ki - 0.395, of this mode happens to be very close to that ofthe Aoi mode in Tarn and Hu[6] for the test case parameters.Another difference, which arises when these profiles areused, concerns the magnitudes of the disturbanceeigenfunctions. When the hyperbolic tangent profiles areused, the streamwise velocity possesses the maximummagnitude among all the eigenfunction; however, this is notthe case when the boundary layer profiles are used. Here theperturbation density eigenfunction has the maximummagnitude, and it is nearly four times that of the streamwisevelocity. However, the shape of the eigenfunctions from thetwo sets of profiles are somewhat similar.

It is interesting to see how these differences in profiles,eigenfunctions and modes can impact the nonlinear dynamicsof the shear layer. The linear dynamics of the shear layer are,of course, affected as well; however, the mode with themaximum growth rate, probably by coincidence, happens tohave a similar growth rate, frequency and wavenumber tothose found by Tarn and Hu for the AOI mode. We say bycoincidence because changes in the parameters such as thedensity ratio change the linear dynamics of the shear layerconsiderably, and the use of hyperbolic tangent profiles doesnot give results anywhere close to those found by P&K.

The basic flow profiles used in this analysis, unlike thehyperbolic tangent profiles, are asymmetric with eachvariable having a different y location for its extrema in slopeand curvature. Also these extrema. say for ~p(y), are muchlarger in magnitude, especially for the curvature, than those inthe hyperbolic tangent approximations. Therefore, in order toresolve these high gradients in the>> direction, 171 grid points(170 cells) were used in that direction with a specialclustering of cells between -O. l iy^O. l and near thewalls. The clustering in the middle of the shear layer isasymmetric in order to resolve the basic flow profiles as wellas the eigenfunctions. The minimum Aj> = 0.002 -In the xdirection, for reason of comparison, the length of thecomputational box is taken to be 16 as was the case in Lu andWu. Based on the finding of Sigalla et al. [1 1,12] that 32 gridpoints per wavelength are needed in this type of second orderTVD scheme in order for the wave to grow at the correctgrowth rate, a uniform grid of 241 points (240 cells) was usedin the x direction corresponding to Ax = 0. 0 (T ( kr = 1 . 75 ) .A CFL of 0.5 was chosen to insure the stability of the schemeas well as to correctly model the time evolution of the shearlayer. After a time-periodic state had been reached, aconstant time step of A/ = 0. 00 1 was used in order to beable to run FFTs in the post processing of the data.

The boundary conditions used for this simulation aresimple because both the inflow and outflow are entirelysupersonic. Based on the theory of hyperbolic equations, allthe quantities are therefore specified at the inflow. At theoutflow, the values obtained from the scheme at the boundaryare enforced. At the top and bottom of the computationaldomain, non-permeable wall boundary conditions areemployed. At f=0, the entire computational box is initializedwith the boundary layer profiles, q(y), obtained by P&K. Atthe inflow, the following condition is prescribed:

q(x = 0,y,t) =

Based on Lu and Wu's observations on the behavior ofthe supersonic shear layer for different types of forcing, only afundamental forcing is applied to the system. A forcingamplitude of 0.05 is chosen which coincides with theamplitude in Lu and Wu; however, as stated previously, themaximum eigenfunction magnitude for the current profiles isfor the density eigenfunction as opposed to the streamwisevelocity for the hyperbolic tangent profiles. The equations areintegrated in time using a TVD spatial discretization until atime-periodic state is reached at which point the datacollection process begins.

Results (Test Case)

In order to monitor the evolution of the shear layerdynamics, the centerline pressure at six streamwise locations(x=2.0, 4.0, 6.0, 9.0, 11.0, and 14.0) were plotted againsttime much like Lu and Wu[14]. This achieves two goals. Itallows us to visually see when the shear layer reaches a time-

periodic state at each spatial location, and it indicates theresponse of the shear layer at different spatial locations. Thepressure histories are shown in Figure 1. The farthest spatiallocation from the inflow where statistics are collected is x =14.0, and from the figure, it can be seen that the flow reachesa time periodic state at about t«44 for this spatial location.The pressure histories from Lu and Wu[14] are similar to theones found in the current simulation. In order to gain insightinto the frequency response of the shear layer, FFTs were runon the pressure histories between 45.23 ^ t ^ 56.00 whichcorresponds to three periods of the fundamental frequency.The magnitude of the FFTs at the six spatial locations areshown in Figure 2. The results lead to the conclusion that atleast near the centerline the shear layer dynamics areessentially linear or only weakly nonlinear up to x = 9. Thisis a little longer than Lu and Wu's simulations and canprobably be attributed to the fact that a different type offorcing (all of the eigenfunctions) was used in this analysis.Beyond this point, the shear layer dynamics becomes moreand more nonlinear giving rise to successive harmonics of thefundamental mode. At x=14.0, four harmonics of thefundamental have been generated which are two more thanfound by Lu and Wu. Also from the magnitude of the FFTs,one can conclude that, following the initial growth of thefundamental mode, the growth of successive harmonics isobserved and, as these harmonics are generated, thefundamental mode saturates ( x « 11). Beyond this point, thefundamental begins to decay and energy is transferred into theharmonics causing further growth of the harmonics. Thissuggests that a secondary instability of the finite amplitudestate might exist. Lu and Wu relate this growth to the otheracoustic modes such as AO?, however, the initial generation ofthe harmonics is a natural nonlinear response of the shearlayer and need not be related to any linear modes. Also, thisapparent secondary instability occurs at a finite amplitudestage where the mean profiles are different from those at theinflow and so the results of the initial linear analysis mightnot apply. In any case, as alluded to by Lu and Wu, in thistype of instability growth of the harmonics (as opposed tosubharmonics hi K-H instability) are seen in the nonlinearresponse at least in the middle of the shear layer.

hi order to see whether a secondary instability is actuallytaking place in the strongly nonlinear region of the shear layerevolution, FFTs were run on the pressure histories aty =-0.49, -0.235,0.235 and 0.49 . The x locationscorresponded to the ones used for the centerline pressurehistories. These chosen y locations roughly correspond to thelocations where the pressure eigenfunction has its localmaxima and minima, i.e. y = ±0.49 are near maximay = ±0.235 are near minima (see Figure 2.7c in P&K). Also,y = ±0.49 correspond to regions near the walls where strongshock formation is observed (see Figure 6) and largerharmonic content is expected in this region. The pressureeigenfunction also has a local maxima near the centerline.The FFTs for the four different locations are given in Figure 3a,b,c and d. From these figures one can conclude that thesaturation and decay of the fundamental is confined to aregion near the center of the shear layer. Of course a betterapproach to this analysis would be to run FFTs of the integral

of the average pressure subtracted by the instantaneouspressure, i.e.

H 12j [p(x,y) - p(x,y,

-H 12

p(x,y) = — J p(x,y,t

1

This will be considered for later simulations. Anyhow, itseems that if there is a secondary instability at work, it is atleast for x w 14 confined to the center of the shear layer. Thecurrent results are inconclusive regarding this point; however,one should keep in mind that at the center of the shear layerthree generalized inflection points exist, one of which (middleone) corresponds to the critical layer of the instability wave inthe current analysis. As the fundamental equilibrates thecritical layer corresponding to the region near this middlegeneralized inflection point becomes important. If there is asecondary instability at work, it might initially be confined tothe critical layer, assuming that both waves have almost equalphase speeds (non-dispersive). This seems to be the case hereat least in the upper portion of the shear layer. Of course thelinear modes for this case are highly dispersive but the growthobserved corresponds to a harmonic of the fundamental. Thisharmonic also has nearly twice the wavenumber of thefundamental which will lead to a non-dispersive combination.Anyhow, a simulation with a longer computational box havingmany probes in the highly nonlinear region will go a long wayin answering the questions about the secondary instability.

The centerline pressure at / = 56.0 is plotted in Figure4. Again, it can be seen that the behavior of the shear layer isnearly linear up to x « 9, i.e. the sinusoidal disturbancegrows exponentially. In order to obtain the growth rateobserved in the shear layer, the first 5 pressure peaks in thissinusoidal response are taken and plotted in a Figure 5 whichdepicts \n(p -p)vs.x. A least squares fit is used to fit astraight line through these five points whose slope should bethe growth rate of the shear layer due to the form of thegrowth in the linear dynamics of the shear layer. The slope ofthe line is 0.389 which is within 2% of the growth rate foundin P&K for the Coi mode at the maximum growth rate( - k j = 0.395). Also note that the line fit is nearly perfectwith all five points lying in a straight line.

So far, the data which has been described points to thevalidity of the linear analysis and gives us some insight intothe nonlinear dynamics of the shear layer evolution. Lu andWu observed a strange wave phenomenon on both sides of theshear layer which were compression and expansion waves(shocks and expansions) resulting from the growth of thedisturbances in the shear layer. A similar type of result wasfound by Sigalla et al. [11,12] in relation to the mode whichwas supersonic with respect to both streams. (They also saw asimilar type of phenomenon in modes which were supersonic

with respect to only one stream; however, as expected, thesecompression and expansion waves were only on one side ofthe layer). Based on the results of these researchers, a similartype of phenomenon is expected in this analysis since theinitial disturbance given here also corresponds to an acousticinstability wave which is supersonic with respect to bothstreams, and the form of the disturbances applied are verysmall amplitude compression and expansion Mach waves.Pressure contours in the channel for t.= 56.0 are given inFigure 6. The same type of compression and expansion wavesare present within the flow, and they become stronger as theyare converted downstream. Eventually the compression wavesform shocks at about x » 10. The speed of these structures isvery close to that found in linear analysis, i.e. Uc = 0.6.Again, as expected from the results of the linear analysis, itcan be seen from the pressure contours that in the convectiveframe the velocity seen by the structures is in the flowdirection in the upper portion of the shear layer and oppositeto the flow direction in the lower region. To see this moreclearly, the pressure along the upper and lower walls areplotted in Figure 7a and b. Along the upper wall the flow firstgoes through a shock at the peak labeled as A and then goesthrough an expansion. On the lower wall, however, thereverse happens. Since the relative flow here is in thenegative x direction, the shock and expansion are reversednear peak B. The relative flow directions are given on Figure6. Another interesting phenomenon, which can be seen bothin Figure 6 and Figures 7a, b, is the change between thedistance of the peaks located in the lower portion of the shearlayer. Interpreting these peaks as corresponding to flowstructures, the distances between the last three structures inthe lower part of the shear layer seem to be increasing as thestructures are converted downstream. Meanwhile, thedistances between the structures in the upper portion of theshear layer remain relatively constant. This implies that theconvection speed of the lower part of the structure (in thelower part of the shear layer) increases after x « 12.5 in thenonlinear region, while the upper and middle portions of thestructures which extend throughout the channel still travelwith roughly the phase speed from the linear analysis, i.e.only a slight deceleration is observed for the upper portions asopposed to a large acceleration of the lower portions of thestructures. To see this more clearly, the pressure contours at6 different times in one period ( T * 3.59) of the fundamentalfrequency is given in Figure 8. From these figures one can seethe evolution of the structure A The change in speed of thelower part of the structure along with a nearly constant speedof the upper portion imparts a rotation on the structure whichis clearly seen in Figure 8. The developing structure isinitially at angles roughly corresponding to those of the Machangles of the convective Mach numbers in the two streamswhich for this case are very close to one another. However, asthe structure is converted into the highly nonlinear region andthe speed of the lower part increases, it is rotated so that itbecomes nearly perpendicular. An explanation of thisphenomenon is now given. As the shocks develop near thewall, positive vorticity (same direction as the rotationobserved and opposite the basic flow vorticity) is generated inthe regions near the upper and lower walls. This can be seen

in the vorticity contours given later (Figure 9d). As a rotationis imparted on the structures, the bottom portion acceleratescausing the Mach number (Figure 9b) at the lower part of thestructure to increase leading to a large variation of pressurenear the bottom wall, i.e. A/> = pmix - pmfa becomes verylarge at the lower wall. At the same time the upper portion ofthe structure only decelerates slightly and a relatively smallerchange in Ap is observed at the upper wall. The relativelylarge A/> at the lower wall seen in the color contours ofFigure 8 has implications on the flowfield which will bediscussed later.

This rotation is observed in the region of the flow wherethe "apparent" secondary instability of the flow is beginning.In the strongly nonlinear region of the flow, near thecenterline of the shear layer, the growth of the first harmonicalong with the decay of the fundamental is observed. If thistrend is observed for x>\6 (in a simulation using a largercomputation box) for the integrated quantities, it would meanthat the finite amplitude state of the initial instability is itselfunstable to a wave which has nearly twice the frequency ofthe fundamental and that this secondary instability issomehow related to the rotation of the structures. Asecondary instability is of course present in low subsonicshear layers where the subharmonic begins to grow during thesaturation of the fundamental and gives rise to the vortexpairing phenomenon seen in such shear layers. Of course,nonlinear processes can give rise to harmonics in many typesof instabilities; however, it is the rapid growth of theseharmonics while the fundamental declines that is intriguing inthe current case.

The density, Mach number, species concentration, andvorticity contours at / = 56.0 are given in Figures 9a, b, cand d. The density and Mach number contours again show theevolution of the disturbances into strong shocks andexpansions. It is interesting to note the formation of A -shocks, especially on the lower wall which indicate thereflection of the initial shock from the wall. Also, note that inthe Mach number contours, the light colored area near thelower wall towards the end of the computational domain showa rise in the Mach number of the flow where the structuresare accelerating and rotating. The species concentrationcontours show a totally different type of mixing between thetwo species than that occurring in K-H type of instability.Figure 10 shows the species concentration contours of asubsonic spatial simulation from Sigalla et al. in which thefamiliar doubling of the shear layer thickness after eachvortex pairing and role-up is seen. In the current analysis, itseems that the shear layer begins to deform in a sinusoidalfashion and then, under the effect of these shock-expansionstructures, starts expanding near the region where the rotationof the structures is observed. One should note that the speciesconcentration profile, Ua (y~), and disturbance eigenfunction,

ca ( y ) , used here are different from Lu and Wu. In fact, thiseigenfunction has the second largest magnitude. The vorticitycontours given in Figure 9d show a somewhat similarbehavior to those seen in the species concentration contourswith dashed lines representing positive vorticity which isconcentrated near the walls. The vorticity initially present in

the shear layer develops into elongated vortices which werealso seen by Lu and Wu. Based on the magnitude of thevorticity in the last vortex in the core of the shear layer, it isclear that a generation of vorticity has occurred due to thedevelopment and convection of the structures. Most of thisgeneration occurs hi the last 1/5 of the computational domainwhere the rotation of the structures is observed. Again, ageneration of vorticity occurs near the, wall again due to theformation and convection of shocks within the flow, hi orderto look at the vorticity dynamics during the evolution of theshear layer, each term in the vorticity equation will beanalyzed at different spatial locations. This approach wasadopted by Soetrisno et al. [15], Lele[16], and Lu andWu[14]. The inviscid 2-D vorticity equation hi a convectiveframe can be written as:

cto dco d® dco \ ch dt

Vpx Vp

—-where Uc is the convective speed. Of course this convectiveframe is valid up to x * 12.5 where the structure begins torotate and Uc is no longer constant. But up to that point, theequation is valid and can give us insight into the vorticitydynamics. The first two terms on the right hand side governthe convection of vorticity with respect to the convectiveframe, the third term is a vorticity generation term due tocompression and expansion (generation and destructiondepend on the sign of the initial basic flow vorticity), and thelast term is the baroclinic torque, hi 2-D flows the last twoterms are the only terms which can generate or destroyvorticity within the flowfield If the last two terms arenegligible, say in a constant density low subsonic shear layer,then the only vorticity dynamics which would be observed is arearrangement of the vorticity due to the convective terms.Figures 11 a-e show the variation of the terms hi equation(13) within the channel at five different spatial location. Atx = 6.0 the dynamics of the shear layer are nearly linear andthe maximum vorticity near that station is almost the same asthe maximum vorticity, 0)«-8.5 , in the initial basicprofiles. The baroclinic term and the expansion term actopposite to the convection term and combined have a smallermagnitude than the convection term. Virtually no region in yexists where the convective terms and the generation termshave the same sign; therefore, only a rearranging of vorticitytakes place hi the convective frame. At x = 9.0, which isnear the core of first vortex, the terms have a similar structureto those found at x = 6.0; however, the convection term ismuch more dominant. At x = 10.75, the magnitude of thebaroclinic torque becomes very small compared to the othertwo terms. This is due to the alignment of the gradients of pand p as the instability evolves in the nonlinear region. Thiswas also seen by Lu and Wu. Also, there now exist regions iny where the expansion term and the convection terms have thesame sign which can lead to a generation of vorticity. hi factthe maximum vorticity near this region is ca » -10 which

means that a generation of vorticity has taken place. This is incontrast to subsonic shear layers. For those shear layers,Lele[15] showed that the expansion term opposed theconvection term causing the suppression of the 'cats-eye'vortex development seen in the development of the K-Hinstability. That is to say that the magnitude of the vorticity atthe vortex centers was lower than a nearly incompressibleshear layer since the convection of the vorticity towards thevortex cores was suppressed by the expansion term. It is clearthat the role of the expansion term due to the action of theshock-expansion structures is very different from thesuppression effect seen in the subsonic shear layer.

The region where the expansion and convection termshave the same sign increase further into the nonlinear regionwhich is evident in the figures provided for x = 11.1 andx = 12.5. More and more vorticity is generated as thestructures move downstream and the deflection of the shearlayer along with the convection of the vorticity becomesgreater and greater. Beyond this point the structures begin torotate due to the generation of positive vorticity near the wallsand away form the walls more negative vorticity is generateddue to the expansion term. At the core of the last vortex in thecomputational domain the maximum value of the vorticity isnow co » -18 as previously shown on Figure 9d. Theinstantaneous streamlines and vector plot of the velocity aregiven in Figures 12 a and b. The effect of the deflection of theshear layer, the elongated vortices and the positive vorticitynear the walls is more prevalent in the lower portion of thechannel. The magnitude of the normal velocity is similar inthe upper and lower part of the shear layer as can beexpected from the shape of the eigenfunction given in Figure13. However, the average streamwise velocity in the lowerpart of the shear layer is approximately 1/4 that of the upperportion (C/2 = 0.276); therefore, the effect of the normalvelocity is more prevalent in the lower part of the shear layerthan the upper part as seen from the vectors in Figure 12a.The consequence of the large Ap due to the strongacceleration of the lower portion of the structures is evident inthe instantaneous streamlines shown in Figure 12b. A regionof very high pressure exists below and to the left of regionswith much lower pressures causing a upward "blowing" typeof flow away from the lower wall. This region is enclosed bythe dashed curve given in Figure 12b.

From the results presented above, it is clear that theevolution of the shear layer is governed by the growth of thefundamental and generation of successive harmonics of thefundamental. The given results also reinforce those given bySigalla et al. and Lu and Wu. It is clear that for thesupersonic convective Mach numbers considered in theseanalyses, a different type of instability dynamics is observedfor the acoustic modes in comparison to the subsonic modesand further investigations are necessary in order tounderstand them thoroughly.

Assuming that the shear layer response is truly at thefundamental frequency corresponding to the maximum growthrate of the Co/ mode even when a broadband disturbance isused, then using a fundamental forcing model, investigationsof the highly nonlinear region should be the next step. Asseen from the results given in this chapter, in the highly

nonlinear region, the structures begin to rotate and at leastnear the centerline the fundamental disturbance begins todecay. However, the computational box is just long enough tosee the structure become perpendicular to the walls before itexists. A simulation with a larger computational box shouldbe conducted to examine in more detail the dynamics andmixing of the shear layer in the strongly nonlinear region. It isobvious from the species concentration contours that thespreading of the shear layer is confined to a region near theend of the computational box and a larger box with manyprobes in the nonlinear region is needed to elucidated theconsequences of the phenomena described above and toanswer the questions about secondary instabilities.

Problem Formulation (Case II)

There are several differences between the characteristicsof the most unstable mode for these parameters than those forthe test case. As shown by P&K, the most unstable mode forthis density ratio is the Boi mode whose growth rate curveshows multiple peaks of which two have virtually the samegrowth rate. The frequency range over which this moderemains unstable is much larger than that of the Cot mode inthe test case. In fact, the first maximum growth rate peakoccurs at a> = 5.775 which is three times larger than that ofthe test case. Also, as the density ratio is increased, we knowfrom the linear analysis that the phase speed of the instabilitytends towards the slower stream velocity. In effect, at thesehigh frequencies, the wavenumbers of the waves becomemuch larger than those seen for the test case, i.e. kr «16.18for the first maximum growth rate peak. The shapes of theeigenfunctions are also very different from those in the testcase. The eigenfunctions for the most unstable waves showlarge spikes near the 'critical layer1 (although a singularitydoes not exist for the unstable case), and the magnitude of thedensity eigenfunction compared to the others is now muchlarger. Each of these characteristics of the Boi mode canpresent problems for the numerical simulations. The highwavenumber means that the wavelength of the instabilitywave becomes very small which leads to finer and finer gridsin the x direction. For kr «16.18, a Ar = 0.0125 is neededwhich is less than 1/5 that of the test case. This can result in alot of grid points if a simulation between 0 £ x ^ 16 isconducted. The large gradients associated with theeigenfunctions as well as the basic profiles for this densityratio along with the highly oscillatory behavior of theeigenfunctions on the supersonic side lead to a much finergrid in the y-direction then in the test case. Several testswere done, and a Aymin = 0.001 was deemed adequate. Thisresulted in 248 grid points in the y direction, and thereforenearly 1/2 the time step in the test case. Not only were theremore points in x and y but the number of time steps needed toreach a certain time was doubled. This coupled with the factthat the instability wave at the first maximum growth ratepeak travels at Cph » 0.36 which is 60% of the speed for thetest case, it was decided that a much shorter computational

box was to be used. This decision was substantiated by thefact that the growth rates for this mode were nearly doublethat of the test case. A computational box of 8 x 1corresponding to 641 x 248 grid points was used for this case.

As a final note, one should keep in mind that the viscouseffects are ignored both in the linear analysis and in thenumerical simulations. In the linear analysis, the order of theviscous terms is ( *3 / Re) for high wavenumbers. Now, theratio between the two wavenumbers as stated above is about5 which means that the Re at which the viscous effects can beignored for this density ratio is about 225 times larger thanthat of the test case. Of course this is only a order ofmagnitude estimate and more analysis needs to be done inorder to find the exact effect of viscosity especially at thehigher frequencies and wavenumbers. Again, this analysis isdeferred to a later time.

Results (Case II)

First, a simulation was run with a perturbation on onlythe pressure which included seven frequencies (o>i=3.040,0)2=4.000, co3=4.905 co4=5.775, <o5=6.623, 0)6=7.548 and0)7=8.283) corresponding to seven of the peaks seen in Figure2.14b. A randomly selected phase for each frequency waschosen so that the initial perturbation did not show any auto-correlation in time and a very small amplitude was chosen forthe perturbations. The computational box of 3.5 x 1corresponding to a 281 x 248 grid points was used. The goalof this simulation was to see if a certain frequency wasselected during the evolution of the shear layer as was thecase for the test case parameters in Lu and Wu's analysis.FFTs of the centerline pressure history at x = 3.1 are shown inFigure 14. The shear layer response seems broadbandedespecially for the frequencies below a> « 6. There might notbe enough resolution to correctly model the higherfrequencies and wavelengths. Based on looking at FFTs fromprevious spatial locations, the frequency at the peak with thehighest growth rate co = 5.775 seems to become moredominant as the wave moves downstream. It is well knownthat broadband forced shear layer simulations requirecomputational boxes which are much longer than thoserequired for a shear layer forced at a specific frequency. Inorder to gain some insight into the nonlinear dynamics of thismode, it was decided to run a simulation with fundamentalforcing only. First the results of this simulation will be givenand then a discussion will follow on how to proceed withfuture work to improve upon the results found.

The computational box was chosen to be 8 x 1 and theforcing amplitude of 7% was prescribed at the inflow. TheCPU time per time step for this simulation was nearly fourtimes that of the test case. Figure 15 shows the centerlinepressure histories at six different x«locatrons <AT= 1.0, 2:0,3.0, 4.5, 6.0 and 7.0). The shear layer seems to behavelinearly up to x » 3. The two "wiggles" seen in the figuresfor x = 6.0 and x = 7.0 at t » 16.0 and t » 19.0 respectivelycorrespond to the time when the instability wave (Cfh =

0.356) first reaches these locations. The behavior before thattime is due to the « and u+c disturbances (eigenfunctions ofthe Jacobians of the Euler equations) reaching these spatiallocations. FFTs were run on the pressure histories between27.0 £ f 5 31.532 which corresponds to four periods of thefundamental frequency. The results are shown in Figure 16.Again, like the test case, generation of harmonics as opposedto subharmonics is observed. For the^ first three x-locations,the frequency response is at the fundamental frequency. Butat later stations, the response becomes nonlinear andharmonics of the fundamental frequency are generated. Thegrowth of the first harmonic seems to differ from that seen inthe test case. The harmonic initially begins to grow atx » 4.5 and then saturates and starts decaying (see fourthand fifth plots in Figure 16). However, the decay of theharmonic stops further downstream and a growth is againobserved. Based on the centerline pressure FFTs, acomparison can be made between the saturation amplitudesbetween the test case and the higher density ratio case. Forthe test case, the saturation magnitude of the FFT is roughlytwo orders of magnitude higher than that of the initialdisturbance as opposed to the single order of magnitudeincrease seen in the current case. From the FFTs shown inSigalla et al. for the subsonic shear layer forced by abroadband disturbance, there is also a two orders ofmagnitude increase in the FFT magnitudes near the saturationregion.

The centerline pressure at t = 31.5 is given in Figure 17.Between 0.0 <. x £ 3.0, as will be shown, an exponentialgrowth of the fundamental mode occurs. Then between3.0 £ x £ 5.0 the fundamental begins to saturate and nearlylevels off. Once this occurs, as seen in the FFTs and alsoseen in the current figure, harmonics to the fundamentalbegin to grow. But from the instantaneous centerlinepressure, it seems that between 7.0 <. x £ 8.0, a rapidgrowth of a wave with a wavelength very close to that of thefundamental begins. The behavior between 6.0 ̂ x ^ 7.0 isnot due to boundary effects since the flow exiting thecomputational domain is supersonic ( Mmin * 1-7 ) and theregion in question is nearly 80 to 160 grid points away fromthe boundary. Unfortunately the length of the computationalbox is not long enough to capture the strong nonlinear region.However, it seems from the current case that for this densityratio, the imposed initial disturbance should have beenbroadbanded. Only then can one say without any uncertaintyhow the shear layer will respond. Based on the currentfigures, it seems mat although the linear growth rate of thefrequency chosen was almost double that of the test case, themode at this frequency seems to saturate quickly. Anexplanation for this can be found in the relatively largedifferences in magnitude between the eigenfunctions. For thisfrequency, the eigenfunctions were given in Figures 2.15a, b,c, d and e in P&K. The spike in the density eigenfunction isvery large in magnitude compared to _the_other_ eigenfunctionand even compared with the rest of its profile. This can leadto a quick saturation of the mode, especially near this spikeregion where due to the larger increase of the initially largeramplitudes, nonlinear effects set in much earlier than otherlocations. This also implies that the linear analysis is only

pertinent for a very small location after the inflow andnonlinear terms at least near the spike region cannot beignored for long.

As alluded to earlier, this type of spike seen in thedifferent eigenfunctions ( p and ca ) correspond to the locationof the critical layer (Ufa) - cr = 0), although one does notexist, strictly speaking, for the unstable case since\cr I c i \ > \ . This value of \cr Ict\ is too big for the "criticallayer" to be solely responsible for the large spikes seen in theprofiles, and more will be said regarding the terms that leadto these spikes in a later paragraph. Many researchers haveshown (e.g. Goldstein[17]) that the nonlinear dynamics ofshear layers and boundary layers can initially be contained inthe critical layer and then spread into the entire flowfield.Figures 18a-d, Figures 19a-d and Figures 20a-d show theinstantaneous pressure, density, species concentration andnormal velocity at y * -0.047, y » 0.0 and y « 0.04,respectively. The first location corresponds to the spike region(critical layer), and the latter two are in the region where theeigenfunctions have oscillatory behavior. From thecomparison of these figures, it is clear that the nonlinearbehavior (harmonic generation and deviation from a sinusoidat a single wavelength) at y « -0.047 starts at a downstreamlocation much closer to the inflow than the other two ylocations. All the twelve figures show an exponential growthof the disturbances between 0.0 £ x £ 2.8, but beyond thatpoint the dynamics at the three locations are very different.

The pressure is the only variable which does not showthis phenomenon. In fact, the figures for the pressure at thedifferent y locations seem very similar. In order to explainthis deviation in behavior between the pressure and the otherflow quantities, the relationship between the disturbanceeigenfunctions were analyzed. These relationships were givenin P&K. The spike seen in p and ca can be explained bythe combination of several terms. First, note that oncesubstitutions are done on these equations, the denominators ofu,ca and/5 are like (U -c)~2 as opposed to (U -c) forv . Also, the denominators of expressions for p and ca arenot multiplied by p. Second, after substituting for Dv in theexpression for p in terms such as pDU,DpsndD2p showup which have large values near the generalized inflectionpoint which correspond to the location where U(yc) = cr. Thecombination of these terms (smaller denominator and largernumerator) give rise to the large spikes in p and ca.However, it is not clear from these relations why the pressuredoes not show the nonlinear behavior near the spike region asdo the other variables. It might be due to the fact that amongall the eigenfunctions the pressure has the lowest magnitudeand so a smooth variation even in the spike region. Evaluationof the nonlinear terms might be necessary in order to shedsome light on this question. Anyhow, the nonlinearphenomenon seen in the critical layer spreads throughout theshear layer further downstream near x » 6 as seen in thefigures fory = 0 andy = 0.04. Based on all this it seems thatin order to gain better insight into the shear layer dynamics at

this density ratio, a broadbanded disturbance and a muchlonger computational box is necessary.

The first ten peaks from Figure 17 were plotted in Figure21 in a ln(.p - p) vs. x plot. The line shown on the figure isa least squares fitted line through the first seven points (up tox » 3). The slope of the line is 0.684 which is within 7% ofthe growth rate (0.74) found by P&K for this frequency. Itseems that unless a higher order method is used, more gridpoints are necessary in the y direction 'to get more accurategrowth rates. Reducing Ax to 0.012 will also help.

As mentioned before, the length of the computationalbox was not long enough to capture most of the nonlineardynamics of the shear layer, however the emergence of someinteresting phenomena can be seen even in a computationalbox of this size. The pressure contours at t=31.5 are given inFigure 22. In the upper portion of the shear layer, convertedMach waves can be seea The magnitudes of these waves(compression and expansion) increase as they are converteddownstream. In the lower region of the shear layer, thepressure disturbance is of a different form. It is more likewhat is seen in subsonic simulations. Even more interestingare the contours for density and species concentration whichare given in figures 23a and b. For both contours, in the layerwhere the spike in the density and species concentrationeigenfunction takes place, it seems that small structures aredeveloping very much like in a Kelvin-Helmholtz instability(this is typical of the nonlinear critical layer). Of course inthis portion of the shear layer the convective Mach number issubsonic (A/c < 1 for the region including and below thegeneralized inflection point) and such a development at leastinitially can be seen as consistent. In the upper portion of theshear layer toward the end of the computational box, thefamiliar deflection of the shear layer seen for the test case isbecoming visible. Note that the differences seen in thedensity and species concentration contours outside the shearlayer are due to the fart that ca « 0 outside the shear layerwhere DCa » 0. Clearly there are two types of phenomenaworking in the evolution of the shear layer, a Kelvin-Helmholtz type and an acoustic type. However, it is not clearwhether both or only one will be seen if a broadbandedsimulation in a large enough computational box to capture thenonlinear region were to be conducted.

It is clear from the results presented above that thesimulation of the higher density ratio case based on thesebasic profiles and eigenfunction is not a simple task. Thesimulation whose results were shown took roughly 7 weeks tocomplete on a IBM 9000 machine (using an average of 8 CPUHours per day). In order to conduct simulations withbroadbanded disturbances, a computational box of roughly 24x 1 is needed. This coupled with the fart that more gridpoints might be necessary in the y-direction will multiply thetime required by a factor of roughly 5. In order to overcomethis obstacle the code must be vectorized before any furthercalculations are done. The vertorization of the code has beenachieved. Current tests show a nearly fourfold decrease inCPU time per time step.

Two simulations will be conducted using this vectorizedcode. The first will involve an extended computational boxfor the current case, and the second will be a broadbanded

simulation. These two simulations should shed some light onthe type of mixing enhancement (if any) that the higherdensity ratios might offer. It seems currently that although themaximum growth rate of the mode simulated is roughly twicethat of the test case, the shape and relative differences in theeigenfunction magnitudes cause it to saturate very quickly.Thus, at least in the weakly nonlinear simulation, the resultsshown above indicate no clear enhancement of mixing.However, the emergence of the structures in the spike layer ofthe eigenfunctions seems to indicate that something might bevery different in the strongly nonlinear region.

Acknowledgments

This work was supported by NASA DrydenResearch Center under Grant NCC2-374.

References

[1] Peroomian, O. and Kelly, R.E. (1995) "Effect of densitygradients in confined supersonic shear layers", Accepted forpublication in Physics of Fluids A.

[2] Jackson, T.L. and Grosch, C.E. (1989) "Inviscid spatialstability of a compressible mixing layer", J. Fluid Mech., 208,609-637.

[3] Papamoschou, D. and Roshko, A (1986) "Observations ofsupersonic free shear layer", AIAA Paper No. 86-0126.

[4] Papamoschou, D. and Roshko, A (1988) "Thecompressible turbulent shear layer, an experimental study", J.Fluid Mech., 197,453-477.

[5] Papamoschou, D. (1989) "A structure of the compressibleturbulent shear layer", AIAA Paper No. 89-0126.

[6] Tarn, C.K.W. and Hu, F.Q. (1989) "The instability andacoustic wave modes of supersonic mixing layers inside arectangular channel", J. Fluid Mech., 203, 51-76.

[7] Greenough, J., Riley, J., Soetrisno, M. and Eberhardt, D.(1989) "The effect of walls on a compressible mixing layer",AIAA Paper No. 89-0372.

[8] Mack, L. M. (1990) "On the inviscid acoustic-modeinstability of supersonic shear flows—Part 1: Two-Dimensional waves", Theoret. Comput. Fluid Dynamics, 2,97-123.

[9] Ragab, SA and Wu, J.L. (1989) "Linear instabilities intwo- dimensional Compressible mixing layers", Phys. FluidsA, 1, 957-966.

[10] Chakravarthy, S. (1986) "The Versatility and Reliabilityof Euler Solvers Based on High-Accuracy TVDFormulations", AIAA Paper No. 86-0243.

[11] Sigalla, L, Eberhardt, D., Greenough, J., Riley, J. andSoetrisno, M. (1990) "Numerical simulation of confined,spatially-developing mixing layers: comparison to thetemporal shear layer", AIAA Paper No. 90-1462.

[12] Sigalla, L, Eberhardt, D., Greenough J. and Riley, J.(1991) "Numerical simulation of confined, spatiallydeveloping shear layers: Nonlinear interaction betweeninstability modes", AIAA Paper No. 91-1643.

[13] Lu, P.J. and Wu, K.C. (1991b) "Numerical investigationon the structure of a confined supersonic mixing layer", Phys.Fluids A, 3, 3063-3079.

[14] Gathmann, R.J., Si-Ameur, M. and Mathey, F. (1993)"Numerical simulations of three-dimensional naturaltransition in the compressible confined shear layer", Phys.Fluids A 5,2946-2968.

[15] Soetrisno, M. Eberhardt, D., Riley, JJ and McMurtry, P.(1988) "A Study of Inviscid, Supersonic Mixing Layers usinga Second-Order TVD Scheme", AIAA Paper No. 88-3676.

[16] Lele, S.K. (1989) "Direct numerical simulation ofcompressible free shear flows", AIAA Paper No. 89-0374.

[17] Goldstein, M.E. (1994) "The role of critical layers in thenonlinear stage of boundary layer transition", Submitted forpublication.

10

Fig. 1 Centerline pressure histories at different xlocations for the test case parameters.

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14

0.00y

Fig. 11 c Vorticity equation terms as a function of y for x= 10.75.

•OJO

Fig. 1 Id Vorticity equation terms as a function of y for x

0.50

0.00 E

-0.500.0

Fig. 1 le Vorticity equation terms as a function of y for x= 12.5.

4.0 8.0 12.0

Fig. 12a Instantaneous streamlines at time = 56.0.

16.0

0.50 M. ...... = -

0.25

0.00

-0.25

-0.50

: ! ! : ! : : : : : : : : : h H ! M

0.0 4.0 8.0 12.0 16.0

Fig, 12b Vector plot of velocity at time = 56.0.

15

M, . 4.500 (H() U, • 0.271M,.t.600(N,) p,-1395M..1836 5,. 0100

-0.5 -0.4 -04 -0.2 -0.1 0.0 O.I 0.2 0.3 0.4 0.5'

Fig. 13 Normal velocity eigenfunction at the maximumgrowth rate of the Coi mode for test caseparameters.

I

1.0 J.O la 4.0 (A 4.0 70 10 1.0 10.0Frequency.«

Fig. 14 FFT magnitude of centerline pressure histories.

0.0405 •, (TTTo]0.0400-1

0.0395 '

Fig. 15 Centerline pressure histories at different xlocations for pi = 3.0.

(jt.26-4

"-S.06-S

O.OEO

-

-

-

0 1Fit

9.06-4

4.0E-4

l.OE-4

0.060

M-

„'

J.St-4f 7T*

I7TR1 2-«64

2.1E-<

1.46-4

7.06-5

... . . ,

"

•

.

..im ••«•<

EH34.06-4

206-4

'

•

• .»TT1

E35

*§- '1 M

,1 .U£. X ' "° ' « » » " - . .0 20 30

• n. 1^E-S

EHB«.OE-<

-MM 4.0E-4

-J^-—— 0^60

«.oe-4 .-•<"»• -irn

EZ3I.OE-4

106-4M. II H

-U ————— 0060

• •H M

EH3

..i;m

• -n *, 1 ,.

0 10 20 X " 0 10 M 30

Fig. 16 FFT magnitudes of centerline pressure historiesbetween 27.0 < t < 31.532.

Fig. 17 Instantaneous centerline pressure as a functionofx.

16

ft U 14 74 t>

Fig. 18a Instantaneous pressure at y=-0.047 and time =31.5.

Fig. 19a Instantaneous pressure at y=0.0 and time = 31.5.

M >4 M 1-0 *4

Fig. 18b Instantaneous density at y=-0.047 and time =31.5.

a «.n

M »4 14 II '0 i« «• '»

Fig. 19b Instantaneous density at y=0.0 and time -31.5.

M 14 M II <t U M T« II

Fig. 18c Instantaneous species concentration at y=-0 047and time = 31.5.

Fig. 19c Instantaneous species concentration at y=0.0 andtime = 31.5.

,,-n

Fig. 18d Instantaneous normal velocity at y=-0 047 andtime = 31.5.

I* U M 4»

Fig. 1 9d Instantaneous normal velocity at y=0.04 andtime = 31.5.

17

t» )« « tC M 7* II

Fig. 20a Instantaneous pressure at y=0.04 and time:

31.5.

UN

ttn

—ir II >• I*

Fig. 20b Instantaneous density at y=0.04 and time = 31.5.

«J I* >• U «« >• '» " •»

Fig. 20c Instantaneous species concentration at y=0.04and time = 31.5.

U tj U 1« <• U II IJ M

Fig. 20d Instantaneous normal velocity at y=0.04 andtime = 31.5.

1.0 U

Fig. 21 First ten pressure peaks vs. x.

18

E3 c C4472

I j 14201

0.50 i

0.25

0.00

-0.25

-0.50

Fig. 22 Pressure contours at time = 31.5.

050

025

-0.25

•0.50

o o O O O } } . ^ Do o o o\o 0 C 0^ C

o 00D000000

0.0 1.0 2.0 3.0 40 50 60 7.0 80

Fig. 23a Density contours at time = 31.5.

0.50

025

i 0.00

•025

•05000 10 2.0 3.0 4.0 5.0 60 70 80

Fig. 23b Species concentration contours at time = 31.5.

19