CFD JET TO FREESTREAM VELOCITY RATIO COMPUTATIONS FOR A JET IN A.pdf

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AM 93 4860 CP

JET TO FREESTRE M VELOCITY R TIO COMPUT TIONS FOR JET IN

C R O S S F L O W

Richard J. Margason , NASA Ames Research Center, Moffett Field, CA

Jin Tso , California Polytechnic State University, San Luis Obispo, CASummary

The flowfield induced by a single, subsonic jetexhausting perpendicularly from a flat plate into asubsonic crossflow has been numericallyinvestigated. Time-averaged solutions were obtainedusing the thin-layer Navier-Stokes equations and twooverlapping grids. Test cases were chosen to matchavailable experimental data where the jet Machnumber was 0.78 and the freestream Mach numberwas varied to represent effective velocity ratios Rfrom 4 to 12. Comparisons of the pressures inducedon the flat plate are presented. The results show thatthe best agreement is obtained for R of 4 and 6, Rof 8 is more difficult to resolve, and the R of 12solution was not satisfactory. It is anticipated thatthe resolution could be improved by using eithersmaller time steps or finer grid spacing as Rincreases.

Nomenclature

speed of soundconstants used in empirical equationfor the vortex curve (ref. 8)boundary condition designations, seetable I1 for descriptionsgrid factor in Roberts' algebraic gridgeneration modelpressure coefficient based on freestreamdynamic pressurejet diametercomputational time steptotal energy per unit volumeinviscid flux vectors of Navier-StokesformulationJacobianresidual of the right hand side of finitedifference formulationMach number

Aeronautical Engineer. Associate Fellow AIAA

Associate Professor, Senior Member AIAA

Copyright Q 1993 by the American Institute of Aeronautics andAstronau tics Inc. No copyright is asserted in the United States underTitle 17, U.S. Cod e. Th e U.S. Governmen t has a royalty-free licenseto exercise al l r ights under the copyright claimed herein forGovernmental purposes. All other r ights are reserved by thecopyright owner.

pressure

viscous flux vectors of Navier-StokesformulationPrandtl numbervector of conservative variables ofNavier-Stokes formulationreciprocal of effective velocity ratiocylindrical coordinatescenter of radial clustering, r/D 0.5

radial coordinate indexReynolds number, p , V d / btimeCartesian velocity componentscontravariant velocity components

effective velocity ratio,( ~ j ) ' / ~ ( v a j )Cartesian coordinates non-dimensionalized by Dincrement between adjacent radialcoordinatesratio of specific heatscoefficient of thermal conductivitycoefficient of viscositygeneralized coordinatesdensitygeneralized time variable gridstretching parameterangle between Z axis and the normal tothe vortex curve

radial index, 1 for the smallest jet gridspacing which is located next tor/D 0.5maximumminimumcomponents in Cartesian directionsfreestream

Introduction

The jet in a crossflow (JICF) has been the subject ofnumerous experimental and computationalinvestigations. Earlier this year the first authorpresented a review1 of this research over the last fiftyyears. That paper includes a bibliography ofreferences. The reader is referred to that paper for a morecomplete description of previous research; the presentpaper will cite only those papers which are directlyrelated to this numerical investigation.


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A common geometry for JICF investigations has been asingle, subsonic jet exhausting perpendicularly from aflat plate into a subsonic crossflow (figure 1). Theavailable experimental results for this simple geometryprovide a relatively well defined qualitative andquantitative description of the JICF and its associatedflowfield. The results of a companion experimentalinvestigation2 in which the present authors participatedare also being presented at the present meeting. As

shown schematically on figure 1 there are three vortexsystems which characterize this flow. The jet efflux isdeflected by the freestream and rolls into a contrarotatingvortex pair which dominates the flow. When the locusof maximum vorticity is projected to the symmetryplane the resultant line is called the vortex path. Thereis also a locus of maximum velocities in the symmetryplane which is called the jet centerline path. Theupstream recirculation region generates a horseshoevortex system which is analogous to the vortex at thewing-body juncture of an airplane. The vortical flow inthe w ake-region has been defined experimentally to besimilar to vortex shedding from solid bluff bodies. Arecent investigation3 using a rectangular jet has shownthat the upstream recirculation is unsteady and periodic.The frequency of this periodic motion appeared to be thesame s that produced by the vortex shedding in the nearwake. The Strouhal number based on the vortexshedding frequency varied uniquely with the velocityratio.

Many numerical analyses have attempted to predict theflow field induced by a jet exiting perpendicular to acrossflow from a flat plate. These analyzes include twopapers4.5 by the present authors. Good agreement hasgenerally been achieved between computations andexperiments for the jet trajectory, the contrarotatingvortex pair, the horseshoe vortex, and the overallrecirculation region downstream of the jet. However,due to the turbulent and separated flow on the flat plateand near the jet circumference, difficulties remain inpredicting the surface pressure distribution and fine flowstructures like the horseshoe vortex in the boundarylayer of the plate. This has prompted the presentinvestigation whose goal is to evaluate computationalresults for the influence of effective velocity ratio, R, onthe pressures induced on the flat plate as well s on theoverall flow field.

The emphasis on accurate surface pressure prediction ismotivated by the relationship of JICF to the lifting jetsof Short Takeoff Vertical Landing (STOVL) aircraft.The jets typically induce a reduction in lift and anincrease in nose-up pitching moment during thetransition between jet-borne and wing-borne flight. Onthe AV8A Harrier V I S TO L ~ he lift losses range from afew percent of thrust in hover to s high s 30 percent ofthrust at the velocity typically required for transition toor from conventional flight at altitudes away from the

ground. The corresponding induced nose-up pitchingmoments range from zero near hover to as high as avalue equal to the thrust times the effective et diameternear transition.

Governing Eauations

The flowfield induced by a JICF is characterized by largepressure variations, a pair of diffuse contrarotating

vortices, a separated upstream region containing ahorseshoe vortex, tomado-like vortices on the lee side ofthe jet and a three-dimensional, turbulent wakecontaining both separated and reversed flow. Thus, thethree-dimensional Reynolds-averaged Navier-Stokesequations are required to adequately simulate thisflowfield. These equations can be written inconservation, non-dimensional form.

In order to develop a numerical tool that may capture thepertinent flow features using a reasonable amount ofcomputer resources, simplifications to the full three-dimensional Navier-Stokes equations must be made.

First, since the Reynolds number of the flow is high,the thin-layer Navier-Stokes equations are used in thenumerical simulation. Similar to the method used inboundary layer theory, the thin-layer approximation usesorder of magnitude arguments to drop all viscous termsthat contain derivatives that are parallel to the bodysurface. However, unlike boundary layer theory, thethin-layer approximation retains all three momentumequations and no limitation is put on the pressure field.Thus, the thin-layer equations permit the calculation ofseparated and reversed flows s well s flows with largenormal pressure gradients . Further, they are valid inboth inviscid and viscous flow regimes. Previouslaminar, steady flow computational studies have capturedthe global features of the JICF using grid sizes less than200,000 for the full domain. For computationalefficiency, this simulation also assumes steady, laterallysymmetric flow.

The three-dimensional Navier-Stokes equations withthin-layer viscous terms retained in all three coordinatedirections are cast in generalized coordinates and writtenin conservation law form s follows:

where Q is the vector of conservative variables, E F andG are the inviscid flux vectors and the viscous fluxvectors are S, R and P. The generalized coordinates 5q and T are functions of the Cartesian coordinates x.y z and time, t. The conservative variables and the fluxvectors are given by


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The viscous flux vectors in the q direction, R, and inthe 5 direction, P, may be obtained by substituting qand 5 for 5 in the expression for S.

The conservative variables are based on density, p, thethree cartesian velocity components, u v and w, and thetotal energy per unit volume, e; these variables arenondimensionalized by p,, a nd paw2 respectivelywhere a denotes the freestream speed of sound. The

pressure, p, is defined as

In these equations, U, V and W are the contravariantvelocity components and J is the Jacobian of thecoordinate transformation which is calculated using themetric terms, ex, q Lt, and so on. The viscosity is

expressed by p he Prandtl number is expressed by Pr,the coefficient of thermal conductivity is given by K andthe ratio of specific heats is denoted by y.

Numerical Methods

The present investigation used the multi-grid, overset-mesh flow solver designated OVERFLOW^ Withinthis solver an implicit, three-factor, diagonalized, centraldifference scheme, also known as the diagonalized Beam-Warming scheme8-lo was selected to solve a thin-layerformulation of the three-dimensional, compressibleNavier-Stokes equations. In the computations, thin-

layer terms were applied in all three coordinatedirections. This solution scheme which is alsocontained in the A R C ~ D ~ode is first order accurate intime and second order accurate in space. Second andfourth order artificial dissipation terms are used. Aspatially varying time step was employed to accelerateconvergence to a steady state solution. For most of thesolutions, the time step was started at 0.01 for 50iterations, increased to 0.1 for the next 1950 iterations,increased to 0.5 up to 4000 total iterations, thenincreased to 1.0 for more than 4000 iterations.

OVERFLOW uses the Chimera grid embeddingt e ~ h n i ~ u e l ~ , ~ ~ .ince the JICF has bilateral symmetryfor steady flow, the present investigation used twooverlapping grids (fig. 2) on a symmetrical half plane.A Cartesian grid with 79 x3 3~ 66 oints in the X, Y andZ directions respectively was used for the surroundingfreestream and boundary layer flow on the flat plate.Then a cylindrical grid with 51x 33 ~6 6 oints in the R,0 and Z directions respectively was embedded for the jetplume. This grid increased in diameter along the Z

direction and deflected along the jet vortex curve whichwas given by Fearn and ~ e s t o n s l ~ mpiricaldescription

z=aRbxcThe empirical parameters are a 0.35 15, b 1.122, andc 0.4293

Both the rectangular plate grid and the cylindrical gridwere designed for turbulent flow and were clustered inthe expected high shear regions near the plate and nearthe edge of the jet plume. Approximately 30 grid pointswere clustered vertically within the plate boundary layer.Adjacent to the plate, the finest spacing is 6*10-5.Preliminary computations of laminar and turbulent flowover the flat plate (jet off) were made using this griddistribution. The turbulent results obtained using eitherthe Baldwin-Lomax model15 or the Baldwin-Barthmodel l were in close agreement with experimentaldata17.

The jet plume grid was generated algebraically with caretaken to ensure near-orthogonality everywhere. Gridpoints were clustered at the shear region between the jetand external flow using an algebraic model developed by~ o b e r t s

sinh[q i, B ]

sinh(z,~)

where


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region immediately downstream of the jet exit. Theseresult^^ ^ show that while the turbulence models

improve the fidelity of the solutions relative toexperimental data, there are other factors (such asboundary conditions and jet grid radial distribution)which need to be resolved before an adequate solution forV/STOL or STOVL aircraft application is obtained.

The present numerical investigation was conducted to

assess the abilty of the code to evaluate a range of Rvalues which include the following: 4, 6, 8, and 12.The Baldwin-Barth turbulence model16 was used. Sincethe highest R value generates a jet with much greaterpenetration in the Z direction, the maximum coordinateof the rectangular grid was increased from 20 to 40 jetdiameters. This was done to reduce any effect that mightbe induced if the jet efflux came too close to themaximum Z grid boundary. The effect of these twovalues of the maximum z coordinate was compared forthe R 6 case in figure 5. While there is little effect at0 azimuth, a noticible difference is observed at 90azimuth. For zmax 20, the computed cp is below the

experimental data. For zmax 40, the computed cp isabove the experimental data by a simlar increment. Asmaller difference is noted at 180' azimuth. In figure 5(d), it is shown that the path of maximum local Machnumber are similar in both cases and both computedresults penetrate the freestream less than theexperimental data19.

R 8: The jet grid for the previous R 6 calculationswas deflected along the vortex curve for R 6 (figure2(a)) to concentrate the grid in the region of the jet flow.In an attempt to assess the importance of the griddeflection for the R 8 case, two jet grid trajectorieswere compared for results at 10,000 iterations using jetgrid 2. These results are presented in figure where thejet Mach number was 0.78 and the freestream Machnumber was 0.0975 for both cases. In one case the jetgrid was deflected along the R 6 vortex curve and inthe other case the jet grid was deflected along the R 8vortex curve. The data show very little difference in theinduced pressures. This indicates that the cp results arenot sensitive to moderate changes in the jet grid. As aconsequence, it appears that moderate inaccuracies in thejet grid location are not critical.

second set of computations was done with the jet griddeflected along the R 8 vortex curve (figure 2(d)) to

determine if changing the progression of time stepsimproved the computation of steady state results. Whenall of the previous computations were done the time stepwas started at 0.01 for fi s t 50 iterations, 0.1 for thenext 1950 iterations, 0.5 for the next 4000 iterations,and increased up to 1.0 for all the rest of the iterations.

computational time was then computed by taking thecumulative sum of the product of number of iterationsand the time step. In figure 7 these results are presented

for 12,000 iterations which represent a computationaltime of 8195.5. This result is identified by the averagetime step DT which is 8195.5 divided by 12,000 or0.68. A second progression of time steps was also triedto assess the effect of a slower rate. In this case, thetime step was 0.01 for the first 2000 iterations, 0.5 forthe next 12,000 iterations, and 1.0 for the last 2000iterations. In figure 7 these results are presented for16,000 iterations which represent a computational time

of 8020. This result is identified by the average timestep D T which is 0.50. The results show that thesmaller average time step provides results in much betteragreement with the experimental data. The residualspresented in figures 7(d) and 7(e) are similar.

This result is consistent with unpublished resultsobtained by the authors which used a minimum CFLnumber of 5 for a R 6 case. The computations failedto converge. This case was equivalent to a much fasteraverage time step than 0.68. It is concluded that sincethe JICF problem has two velocities whose ratio is R,there are two time scales which must be resolved. This

resolution should become more difficult as R increases.It is anticipated that the resolution should be improvedby using either smaller time steps or finer grid spacing.

R 4: The effect of the number of iterations for the R4 case (Mj=0.78, L = O 195) is presented in figure 8.

Along the 0 and 90 azimuths the cp converges to agreewith the experimental data. As expected along the 180azimuth less satisfactory results are obtained. In thiscase the diference in time scales between the freestreamand jet velocities is reduced and good results areobserved.

R 12: The effect of the number of iterations for the R12 case (Mjz0.78, M,=0.65) is presented in figure 9.

Along all three azimuths the agreement of the computedresults with the experimental data are poor even though16,000 iterations were completed. The residual (fig. 10)does not show any unusual behaviour. These resultsindicate that the jet grid which was adequate for effectivevelocity ratios from 4 to 8 may be too coarse for R12. Additionally this case might improve if a loweraverage time step were used.

Concluding Remarks

The emphasis in the present paper was to show theeffect of variation of the effective jet to freestreamvelocity ratio R on the computation of the pressuredistribution on the flat plate. This was evaluated usingnumerical parameter variations in a series ofcalculations. The solutions obtained from theOVEFUXOW code provided the best correlation with allof the experimental cp data at R values of 4 and 6. Lesssatisfactory results were obtained for R of 8.Unsatisfactory results were obtained for R of 12.


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It is concluded that since the JICF problem has twovelocities whose ratio is R, there are two time scaleswhich must be resolved. This resolution is moredifficult as R increases. It is anticipated that theresolution could be improved by using either smallertime steps or finer grid spacing as R increases.

References

Ma rgason, R.J., Fifty Years of Jet in Cross FlowResearch, presented at the72nd AGARD FluidDynamics Panel Meeting and Symposium onCom putational and Exp erimental Assessment ofJets in Cross Flow, April 1993.

Dennis, R.F., Tso, J., and Margason, R.J.,Induced Surface Pressure Distribution of a

Subson ic Jet in a Cross Flow, AIAA Paper No.93-4861, Dec. 1993.

Krothap alli, A., Lourenco, L., and Buchlin, J., Onthe Separated Flow Upstream of a Jet in a CrossFlow, AIAA Paper No. 89-0571, Jan. 1989.

Chiu, S., Roth,K. R., Margason, R. J., and Tso,J., A Numerical Investigation of a Subsonic Jet ina Crossflow, AIAA Paper No. 93-0870, Jan. 1993.

Chiu, S.H., Roth, K.R., Margason, R.J., and Tso,J., A Numerical Investigation of a Subsonic Jet ina Crossflow, presented at the 72nd AGARD FluidDynamics Panel Meeting and Symposium onCom putational and Exp erimen tal Assessment ofJets in Cross Flow, April 1993.

Margason, R. J., Vogler, R. D., and Winston, M.M., Wind Tunnel Investigation at Low Spee ds of

the K estrel (XV-6A) Vectored-Thrust V/STOLAirplane, NASA TN D-6826, July 1972.

Renze, K. J., Buning, P. G., and Rajagopalan, R.G., A Com parative Study of Turbulence Modelsfor Overset Grids, AIAA Paper No. 92-0437, Jan.1992.

Pulliam, T. H. and Chaussee, D. S., A DiagonalForm of an Imp licit Appro ximate FactorizationAlgorithm, J. Comp. Physics Vol. 39, No. 2,Feb. 1981, pp. 347-363.

Beam, R. M. and War ming, R. F., An ImplicitFactored Scheme for the Com pressible Navier-Stokes E quations, I J. Vol. 16, No. 4, A pr.1978, pp. 393-402.

Pulliam, T. H. and Steger, J. L., Recent Improve-ments in Efficiency, Accuracy, and Convergence forImplicit Appro ximate Factorization A lgorithms,AIAA Paper No. 85-0360, Jan. 1985.

11. Pulliam, T. H. and Steger, J. L. Implicit Finite-Difference Simulations of Three-DimensionalCompressible Flow, I J. Vol. 18, no. 2,Feb. 19 80, pp. 159- 167.

12. Benek , J. A.; Buning , P. G.; and S teger, J.L., A3-D Chim era Grid Embedding Technique,AIAAPaper No. 85-1523, July 1985.

13. T rame l, R. W. and Suhs, N. E., PEGSU S 4.0User's Manual, AEDC TR -9 1-8, June 1991.

14. Feam, R. L. and Weston , R. P., Velocity Field ofa Round Jet in a Cross Flow for Various JetInjection Angles and Velocity Ratios, NASA TP-1506, Oct. 1979.

15. Baldw in, B. S. and Lomax, H., Thin Layer Ap-proximation and Algebraic Model for Separated Tur-bulent Flows, AIAA Paper No.78-257, Jan. 1978.

16. Baldw in, B. S. and Barth,T J., A On e EquationTurbulence Transport Model for High ReynoldsNumber Wall-Bounded Flows, NASA TM 102847,Aug. 1990.

17. W hite, F. M., Viscous Fluid Flow, McG raw-HillPublishing Co., 1974.

18. R oberts, G. O., Computational Meshes for Boun-dary Layer Problems, Proceedings of the SecondInternational Conference on N umerical M ethods inFluid Dynamics, Vol. 8, pp. 171-177, Aug. 1971.

19. Feam. R. L. and We ston. R. P.. Induced Pressure~i str ibu tio n f a Jet in a'crossflow, NASATN D-7916, July 1975.

Table I. Jet grid radial spacing parameters.

GRID

Note: One radial point is added to Grid 3 atr/D 0.1 1818 to obtain Grid 4.


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orseshoe vortex3 Wake vortex str t

Fig. 1. Sketch of the three vortex systems associ-ated with the jet in crossflow: (1) rolled upjet vortex pair, 2) horseshoe vortex; 3)weak wake vortex street.

Fig. 2. Cor

b) R = 4 and zma/D = 40.

~tinued.

(a) R = 6 and z m D = 20.

Fig. 2 Chimera grid structure showing therectangular grid for freestream and thecylindrical grid for the jet.

(c) R = 6 and z m D = 40.

Fig. 2. Continued.


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Turbulence model Grid

2 Baldwin-Lomaxexp. ref. 19)

0

1

2 Turbulence model grid

Paldwin-Barth 2

3 Baldwin-Lomax 4exp. ref. 19)

4

2 Baldwin-Lomaxexp. (ref. 19)

CP

0

(c) 180 azimuth4 6

Fig 4 Examples of previous solutions fromreferences 4 and 5 for R = 6 and 10 000iterations

z max)

exp. ref. 19)

F a) 0 azimuth

4 0exp. ref. 19)

4 0exp (ref. 19)

C

0

(c) 180 azimuth2

r / D

Fig 5. Effect of variation of the maximum z valueof the rectangular grid for R = 6 et grid 2and 10 000iterations


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- -40

exp. ref. 14)

d) jet maximum V path

Fig 5 Concluded

Jet Grid for R

2exp. (ref. 19)

(a) 0 azimuth' ' ~ ' ' ~ ' ' ~ ~ ' ~ ' ~

5 4r / D S 2

1

Jet grid for R

- 4i e x p . (ref. 19)1 (b) 90 azimuth I

Fig 6 Effect of jet grid geometric trajectory for

Jet Grid for R

exp. (ref. 19)

0

(c) 180 azimuth

Fig 6 Concluded

average DT

0 . 5 0

0 . 6 8

exp. (ref. 19)

(a) 0 azimuth

~ ' ~ ~ ' m ~ ~ ' ~4r / ~ - ~

1

0

1

a v e r a g e DT

C -0 s o

P 3 0 . 6 8e x p . (ref. 19)

(b) 90 azimuth

Fig 7 Effect of time step sizes during iterationof the OVER FLOW solutionR = 8 flow conditions


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4 ,average DT- 0 . 5 0

2 - - 0 . 6 8exp. ref . 19)C

P

0

c) 180 azimuth

l tera t ions x 1 0-

2

exp. ref. 191

CP

a) 0 azimuth

- K d) Residual foravg. DT 0.50

4L

n o r m

x 1 08'

00 4 0 0 0 8 0 0 0

i t e r a t i o n s

6e) Resid ual for

avg. DT 0.68

i t e r a t i o n s

Fig 7 Concluded

lterations x 1 0

I 0.... 1 2Fearn

4l te ra t ions x 1 0 -

2

Fig . Effect o f number of iterations on the inducedpressures for R = 4 and jet grid 2


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4lterations x 1 O1 0 I

10

8

L

n o r m4

x l o

2

00 4000 8000 12 000

i t e r a t i o n s

lterations x 10-

1exp. ref. 2)

11 Iterations x o ~

Fig 10 Residual for R = 12 and grid 2

Fig 9 Effect of number of iterations on the lnd uc dpressures for R = 12 and jet grid 2

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