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Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

AIAA-97-1629-CP

Numerical Study of Supersonic Jet and Instability Wave

San-Yih Lin and Yu-Fen ChenProfessor Master

Institute of Aeronautics and AstronauticsNational Cheng Rung University70101, Tainan, Taiwan, R. O. C.

Abstract

Numerical investigations of the unsteady flowfield and the associated instability wave due tothe non-ideal expansion supersonic jet were per-formed numerically. The axisymmetric Euler equa-tions were solved by a proposed finite volumemethod. The numerical method is based on athird-order upwind finite-volume scheme in spaceand a second-order explicit Runge-Kutta schemein time. Details of the flow structures, such asshock cell, shear layer, and the corresponding in-stability waves with different jet pressure ratioswere investigated. The predicted time-averagedpressure field at the jet centerline was comparedwith the experimental and other numerical data.By using the spectrum analysis and the correla-tion method, three families of instability waves,Kelvin-Helmholtz, supersonic and subsonic in-stability waves, proposed by Tarn are analyzed.Two families of instability waves, Kelvin-Helmholtzand subsonic, can be identified in this study.

1. Introduction

The unsteady flow field and the associatedsound radiation of non-ideal expansion are im-portant in both the engineering applications andthe fundamental aerodynamic and aeroacousticresearch. For a high-speed civil transport plain,how to reduce the jet exhaust noise is one ofmain subjects on jet noise problems. Under non-ideal expansion jet, the shock cell structure wasgenerated and the instability of shear layer wasdeveloped. A lot of experimental and theoret-ical works have studied the mechanisms of jetnoise.1""5 Generally, there are three main mech-anisms: (1) turbulent noise, (2) the shock cell

"Copyright © 1997 by San-Yih Lin and Yu-FenChen. Published by the American Institute ofAeronautics and Astronautics, Inc. with permis-sion."

and shear layer interaction, and (3) the high-amplitude discrete tone due to the fluctuationpressure feedback mechanism. Earlier, Oertel6

experimentally observed three sets of waves inhigh-speed jets. Tarn and Hu7 investigated thesethree families of instability waves analytically andcomputational and classify them as the Kelvin-Helmholtz, the supersonic and the subsonic in-stability waves. Among of them, the Kelvin-Helmholtz instability was well known in the jetnoise research. In this paper, we numerical inves-tigations were conducted to analyze these threewaves. Under certain flow conditions, we are ableto identify two families, Kelvin-Helmholtz andsubsonic instability waves.

In numerical point of view, direct numeri-cal simulation based on the compressible Navier-Stokes equations is a way to analyze the unsteadyflow field and the associated instability waves.However, due to the resolution requirement ofhigh-Reynolds number flows, the direct numeri-cal simulation is impractical in the modern com-puter capacity. Proper turbulent model simula-tion and large-eddy simulation as tools are pro-posed. On the other hand, in a supersonic jetflow, the viscous effects are not very importantin the near field except in the shear layer. Forthe aeroacoustic computations, one need use veryhigh accurate schemes with lower dissipation anddispersion.8'9 In this paper, the two-dimensionalaxisymmetric Euler equations was solved by aproposed finite volume method, MOC scheme.10

2. Computational Methods

Flows of axisymmetric, compressible, invis-cid, and non heat conducting fluid can be de-scribed in conservation form by the Euler equa-tions:

Wt -y

Gy + -S = 0 (i)352

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where the parameters a» , bi , Q.11 Let £» represents thewidth of the ith cell, then

F =pu

pu2 +ppuv

•p ) .

S =

si __

pvpuv

pvpuv

pv2 +pv(pe + p}.

di =-

•p) .here p, p, u, v, and e are the pressure, density, x-and y-directional velocity components, and thetotal energy per unit mass, respectively. Thepressure p is given by the equation of state fora perfect gas:

12'

where 7 (= 1.4 for air) is the ratio of specificheats.

The Euler solver, MOC scheme, solves theaxisymmetric, unsteady, Euler equations. Thisscheme, in strong conservation form, is based ona third-order upwind finite-volume scheme anda second-order explicit Runge-Kutta scheme. Abrief description of the implementation of theMOC scheme is demonstrated here for one di-mensional flow. For more details, please see Linand Chin.10 Consider the flux F at a surface i —-|,where i is the index of grid in x-direction. First,the state of the flow at each interface is describedby two vectors of conserved variables on eitherside, WL and WR, as follows:

[(a - 2 • AS • 60 • Ai+i W

Cj

(2)

where « is a constant, — ! < « < ! , and the non-uniformity of cell sizes is taken into account using

li4 + 4+i4 + 4-i

For the uniform grid, a» = ^,6^ = |,Cj = 1, thescheme is exact Osher-Chakravarthy scheme.12

Furthermore, to prevent the numerical wig-gle around the shock regions, we use the follow-ing limiters on WR and WL to stabilize the MOCscheme:

(4)where m is the minmod function.

The flux at the interface is then obtained byRoe's approximate Riemann solver13:

lF;_i = -[F(WL)+F(WR)-R\A.\L(WR-WL)],

(5)where L and R are the left and right eigenmatri-ces of the Jacobian matrix A at the Roe averagedvalues of WR and WL, and A is the correspond-ing diagonal matrix of eignvalues. After spatialdiscretization, a second-order three-stage Runge-Kutta scheme14 is used for the time integration.Overall, the MOC scheme is formally third-orderaccurate in space and second-order accurate intime.

3. Numerical Tests

The test case chosen represents a supersonicjet by an exit Mach, Me, 2 at two pressures ra-tios. One is an under-expanded jet with pressureratio, Ss-, 1.4454 and the other one is an over-expanded jet with pressure ratio 0.839. Wherepe is the jet exit pressure and pa is the ambi-ent pressure. A schematic of the test geome-try is shown in Fig. 1. According to the di-mensional scale of Seiner and Norum's experi-mental data15 and Zhang and Edwards's com-putational results9, the jet exit diameter, D, is

353American Institute of Aeronaustics and Astronaustics


5 cm. The streamwise size of the flow domainis 31 D, and transverse size is 10 D. Since theflow is axisymmetric, only half of the flow do-main is used as the computation domain. A to-tal of 620 x 140 uniform grid cells with Ax of0.05 and Ay of 0.0375 are used. The boundaryconditions are following. At the exit of the noz-zle, Bl, p = l,u = l,v = 0, c = -jfi-. Alongthe solid wall, B2, the slip boundary conditionis used. The characteristic boundary conditionapplies on the far field boundaries, B3 and B4.Symmetric flow conditions are applied to the jetcenterline, B5. At the downstream of the jet,B6, the conservative variables are extrapolated.The nondimensionlized initial data are given by,at the jet exit, pe = l,ue = l,i>e = 0,Me = 1.The rest of initial data are computed by the jetpressure ration. With long computation times, aself-sustained oscillatory state is obtained. Theoscillatory jet is demonstrated by the instanta-neous density and pressure contours.

For the under-expanded jet, 2s. = 1.4454,Figures 2 and 3 show instantaneous density andpressure contours at t = 100, 106, and 112, re-spectively. At the near jet exit, x < 15, theshock cell structure is very clear. At the endof the shock cell structure, the vortices becomestronger and dominated the fluid flow oscillation.Figures 4 and 5 show the averaged pressure fieldand Mach number on the centerline of the jet.the results were compared well with Seiner andNorum's experimental data15 and Zhang and Ed-war ds's numerical data9. In Figure 5, there has aphase shift between Seiner and Norum's experi-mental data and ours. Comparing to the pressurefiled, we thought this may be some mistakes inthe Seiner and Norum's experimental data.

For the over-expanded jet, ^ = 0.839, Fig-ures 6 and 7 shows instantaneous density andpressure contours at t = 100, 106, and 112, re-spectively. At the near jet exit, x < 8 , the shockcell structure is very clear. At the end of theshock cell structure, the vortices become strongerand dominate the fluid flow oscillation. Figures 8and 9 show the averaged pressure field and Machnumber on the centerline of the jet.

4. Results and Discussions

The unsteady flow field and the associatedinstability wave due to the non-ideal expansion

supersonic jet were investigated in this section.To analyze the instability waves, the sound pres-sure level (SPL), power spectrum density (PSD),and correlation method are used.

4.1 Sound Pressure LevelThe SPL is defined by

SPL(dB) =Pref

where pre/ = 2 x 10~5. The root-mean-squarevalue of pressure, prms is obtained from t = 100to 124.

Figures lOa and b show the distributions ofthe sound pressure level, SPL, with respect tothe under-expanded and the over-expanded su-personic jets. It is clear to see that, at the nearfield, x < 7, the values of SPL decrease alongabout 135° to the jet flow direction and at themiddle field, 7 < x < 12, they decrease alongabout the 90° direction.

4.2 Power Spectrum Density

Again from t = 100 to 124, six position,(x, y) = (10D, OD), (10D, 0.5D), (20D, OD), (20D,0.5D), (30D, OD), and (SOD, 0.5D), are chosen tocompute their PSD. To compare with the Zhangand Edwards's results9, the dimensional frequency,/, is computed by

t

For the under-expanded case, Figure 11 showsPSDs at six position. At x = 10D and 20D, theirmain frequency is 2972Hz. At x = SOD, the mainfrequency is 1486Hz. For the over-expanded case,Figure 12 shows PSDs at six position. At x =10D the main frequency is 3219Hz. At x = 20D20D, the main frequency is 1238Hz, and at x =SOD, the main frequency is 990Hz. For both oftwo cases, one can see that the PSDs have a peakat low frequency. This peak is then following bya drop in the PSD value. This phenomena issimilar to Norum and Seiner's experimental dataand Zhang and Edwards's computational results.As Norum and Seiner pointed out that this typeof spectrum is associated with broad band shockrelated noise radiation.



4.3 Correlation Method

Earlier, Oertel6 experimentally observed threesets of waves in high-speed jets. Tarn and Hu7 in-vestigated these three families of instability wavesanalytically and computational and classify themas: (a) the Kelvin-Helmholtz instability wave,the interaction angle between its wave front andthe direction of jet is about 50°, (b) the super-sonic instability waves, its propagation speed isslower than Kelvin-Helmholtz wave and the in-teraction angle between its wave front and thedirection of jet is about 90°, (c) the subsonic in-stability wave, it only exits in the interior of thejet and its propagation speed is subsonic. Toanalyze the character and propagation speed ofthe instability waves, the auto correlation, RI,I,and cross correlation, #1,2 are computed at fourposition, ( x , y ) = (2.975D, 1.44585D), (6.975D,1.44585D), (11.975D, 1.44585D), and (16.975D,1.44585D). Figures 13 shows the diagram of thebasic correction point and its related directionfor the cross correlation. The definitions of RI,Iand Ritz are given by

1 rRI,I(T}=-=1 Jo

For the under-expanded case, Figure 14 showsthe correlation at (x,y) = (6.975D, 1.44585D).One can see that the maximum propagation speedis in the 6th direction. Figure 15 shows the cor-relation at (x,y) = (11.975D, 1.44585D), and itsmaximum propagation speed is also the 6th di-rection. On the jet centerline, Figure 16 showsthe correlation at x = 11.975D. Table 1 show thelocal velocity and the wave speed. One can seethat the wave speed is close to the value of u — c.

For the over-expanded case, Figure 17 showsthe correlation at (x,y) = (6.975D, 1.44585D).One can see that the maximum propagation speedis in the 6th direction. Again for the correlationat (x,y) = (11.975D, 1.44585D), its maximumpropagation speed is also the 6th direction. Onthe jet centerline, Figure 18 shows the correlationat x = 6.975D. Table 2 show the local velocityand the wave speed. One can see that the wavespeed is close to the value of u — c.

As Tarn and Hu mentioned that the interac-tion angle between the wave front of the Kelvin-Helmholtz wave and the jet direction is about50°. In our analysis, the 6th direction is 135° tothe jet direction. Comparing with the SPLs, Fig-ure 10, the Kelvin-Helmholtz wave is captured inour computations. According our analysis, in theinterior of jet, the subsonic wave is also capturedin our computations. As far as the supersonicwave, it is not clear in our computational results.

5. Conclusion

The finite volume scheme, MOC, is appliedto study the unsteady flow field and the asso-ciated instability wave due to the non-ideal ex-pansion supersonic jet. Details of the flow struc-tures, such as shock cell, shear layer, and thecorresponding instability waves with different jetpressure ratios were investigated. The predictedtime-averaged pressure field at the jet centerlinewas compared with the experimental and othernumerical data. By using the spectrum analysisand the correlation method, two families of in-stability waves, Kelvin-Helmholtz and subsonic,can be identify in this study.

ACKNOMLED CEMENT

This work is partially supported by the Na-tional Science Council of R.O.C. We would liketo thank Institute of Aeronautics and Astronau-tics, National Cheng Kung University, for facilitysupport.

References:

1. Tarn, K. W., "On the Noise of a NearlyIdeally Expanded Supersonic Jet," J. FluidMech., Vol. 51, Part 1, 1972, pp. 69-95.

2. Seiner, J. M. and Norum, T. D., "Experi-ments of Shock Associated Noise on Super-sonic Jets," AIAA Paper 79-1526, July 1979.

3. Norum, T. D. and Seiner, J. M., " Broad-band Shock Noise from Supersonic Jets," AIAAPaper 80-0983, June 1980.

4. Troutt, T. R. and Mclaughlin, D. K., "Ex-periments on the FLow and Acoustic Prop-erties of a Moderate-Reynolds-Number Su-personic Jets," J. Fluid Mech., Vol. 116,1982, pp. 123-156.

5. Tarn, K. W., "A Multiple Scales Model ofthe Shock-Cell Structure of Imperfectly Ex-



panded Supersonic Jets," J. Fluid Mech., Vol.153, 1985, pp. 123-149.

6. Oertel, EL, "Mach Wave Radiation of HotSupersonic Jets," In Mechanics of Sound Gen-eration in Flows, ed. E. A. Muller, Springer,1979, pp. 275-281.

7. Tarn, K. W. and Hu, F. Q., "On the ThreeFamilies of Instability Waves of High-SpeedJets," Journal of Fluid Mechanics, Vol. 201,1989, pp. 447-483.

8. Tarn, C. K. W. and Webb, J. C., "Dispersion-Relation-Preserving Finite Difference Schemesfor Computational Acoustics," Journal of Com-putational Physics, Vol. 107, 1993, pp. 262-281.

9. Zhang, X. and Edwards, John A., "A Com-putational Analysis of Supersonic Jet andInstability Wave Interaction." AIAA Paper94-2194, June 1994.

10. Lin, S. Y. and Chin, Y. S., "Comparison ofHigher Resolution Euler Schemes for Aeroa-coustic Computations," AIAA Journal, Vol.33, No. 2, 1995, pp. 237-245.

11. Lu, P. J., and Yeh, D. Y., "Transonic Flut-ter Suppression Using Active Acoustic Exci-tations," AIAA Paper 93-3285, July 1993.

12. Osher, S. and Chakravarthy, S. R., "HighResolution Schemes and the Entopy Condi-tion," NASA-CR-172218, Sep. 1983.

13. Roe, P. L., "Approximate Riemann Solvers,Parameter Vectors and Difference Schemes,"Journal of Computational Physics, Vol. 43,No. 2, 1981, pp. 357-372.

14. Jameson, A. "Multigrid Algorithms for Com-pressible Flow Calculations," Multigrid MethodII, edited by W. Hackbusch and V. Trot-tenberg, Lecture Notes in Mathematics, Vol.1228, Berlin, Springer-Verlag, 1985, pp. 166-201.

15. Seiner, J. M. and Norum, T. D., "Aerody-namic Aspects of Shock Containing Jet Plumes,"AIAA Paper 80-0965, 1980.

B3

Bl

B4

B2

B5 10D B6

ID SOD

Fig 1. Schematic of the computational domain ofthe jet flow.

a "* 0 of'fl'i

Fig 2. The instantaneous density contours at t =100, 106, and 112 for the under-expandedjet.



t=100 t-100

t=106 t=l06

t=H2

Fig 3. The instantaneous pressure contours at t = Fig 6- The instantaneous density contours at t =100, 106, and 112 for the under-expanded 100, 106, and 112 for the over-expanded jet.jet.

X/D

Fig 4. The averaged pressure field on the center line Fig 8. The averaged pressure field on the centerlineof the jet for the under-expanded jet. of the jet for the under-expanded jet.

I'

X/D X/D

Fig 5. The Mach number on the centerline of the Fig 9. The Mach number on the centerline of thejet for the under-expanded jet. jet for the under-expanded jet.



t-H2

0OOOOeO I 000064 2 OOOOC* 3 OOOGE* 4 000X4

OOOOOEO 1000064 IOOOOE4 30000C4 40000E4

Fig 7. The instantaneous pressure contours at t —100, 106, and 112 for the over-expanded jet.

1 OOOOE4 1OOOOE4 J OOOOE4 4 000084

1000W4 ZOOOOC4 30(

10000C4 10000C4 10000C4

Fig 11. The PSDs for the under-expanded jet at sixposition: (x,y) = (a) (10, 0), (b) (10, 0.5),(c) (20, 0), (d) (20, 0.5), (e) (30, 0), and (f)(30, 0.5).

Fig 10. The SPL contours for (a) under-expandedand (b) over-expanded jets.



Fig 12. The PSDs for the over-expanded jet at the .,above six position,

r

* 5 2 / 2 3 *

Fig 13. The diagram of the basic correlation point Fig 14. The correlation at (x,y) = (6.975, 1.44585)and its related direction for the cross correc- for the under-expanded jet.tion at four position, (x, y) = (2.975, 1.44585),(6.975, 1.44585), (11.975, 1.44585), and (16.975,1.44585).



i.Hf7S>.f.1<4M8O

«.119750,1.1 44SMO(g) Ai-02457

Fig 15. The correlation at (x,y) = (11.975, 1.44585)for the under-expanded jet. pig 17 Tae correlation at (x,y) = (11.975, 1.44585)

for the over-expanded jet.



|«t»119Ti0.r-C.3M«Q

Fig 16. The correlation on the jet centerline at x =11.975 for the under-expanded jet.

Fig 18. The correlation on the jet centerline at x =6.975 for the over-expanded jet.

Table 1. For the under-expanded jet, (a) The localvelocity at the basic points: x = 11.975 andy = (1) 0.01785,(2) 0.16065,(3) 0.30345, (b)the propagation speed of wave at the basicpoints 1, 2, and 3.

Table 2. For the over-expanded jet, (a) The local ve-locity at the basic points: x = 6.975 and y= (1) 0.01785, (2) 0.16065, (3) 0.30345, (b)the propagation speed of wave at the basicpoints 1, 2, and 3.

(a)

123

u0.99110.99781.0178

c0.50380.50060.4926

u + c1.49491.49881.5104

u-c0.48720.49710.5251

(b)(a)

fiS 10.4247

20.4678

30.4571

123

u0.90720.91990.9457

c0.53410.52940.5202

u + c1.44131.44931.4659

u-c0.37310.39050.4254

(b)/-^^B 1

0.3532

0.38463

0.4597


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