Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

Large-Scale Simulation of Supersonic Jet Noise

S. H. Shih* and D. R. Hixon*Institute for Computational Mechanics in Propulsion

NASA Lewis Research Center, MS 5-11Cleveland, Ohio 44135

RedaR. Mankbadi**Institute for Computational Mechanics in Propulsion, also

Mechanical Power Engineering DepartmentCairo University, Cairo, Egypt

L. A. Povinelli***NASA Lewis Research Center, MS 5-3

Cleveland, Ohio 44135

Abstract

Direct computation of the unsteady characteristicsof supersonic jet flow and its radiated noise is performed.The computation consists two parts: nonlinear soundsource generation in the near field and linear acoustic ra-diation in the far field. The numerical method for thenear field, based on the large-eddy simulation technique,split the fluid motion into three kinds of motion: a time-averaged motion; a periodic, large-scale wavelike struc-ture; and a background, fine-scale random turbulence.The nonlinear disturbance equations are derived andsolved to obtain the unsteady fluctuations in the nearfield. The Kirchhoff s method is used to obtain far fieldnoise with the information provided by the solution ofnonlinear disturbance equations. Qualitative agreementwith the measured sound directivity of a heated super-sonic jet is obtained.

1. Introduction

The existence of large-scale coherent structures inthe initial region of a round jet has been well establishedby several investigations. By using the flow visualizationand by measuring the fluctuating velocity, Crow andChampagne [1] were able to detect a travelling wave sys-tem the amplitude of which reached a maximum and thendecayed gradually downstream. The importance of thelarge-scale structures in the mixing of the free shearflows and their role in supersonic jet noise has been in-vestigated by many researchers [2-6]. For supersonic jetsand especially high temperature jets, the large-scale

structures propagate downstream at supersonic Machnumber relative to ambient air resulting in intensivenoise radiation in the form of Mach wave emission. ThisMach wave radiation predominates over the noise fromfine-scale turbulence and makes the large-scale struc-tures the dominant noise source of supersonic jets.

By realizing the role of large-scale structures hi su-personic jet noise, Morris and Tarn [5] and Tarn and Bur-ton [6] developed the instability wave models for jetmixing noise. More recent development hi jet noise sim-ulations have focused on utilizing high performancecomputers. Several first-principle approaches for jetnoise prediction have recently been developed. Directnumerical simulations (DNS) can simultaneously cap-ture both the flow fluctuations representing the soundsource and the associated acoustic field. The approach issuitable for simple flows at low-Reynolds numbers [7].However, for the technologically important, high Rey-nolds number flows, the current computer capabilitydoes not allow resolving all the scales involved.

In the large eddy simulations (LES) approach for jetnoise prediction [8-12], the large scale structures whichare efficient hi generating noise are numerically re-solved. The effects of the unresolved scales on the re-solved ones are accounted for via subgrid scalemodeling. The computed field encompasses both theflow fluctuations and the associated sound emission.High-order algorithm must be used to minimize the dis-persion and dissipation errors to capture the wavelike na-ture of the disturbances. The results given hi Mankbadiet. al. [10] showed that clean, direct prediction of boththe sound source and the associated field can be ob-

* Senior Reserach Associate, Member AIAA** Professor, Associate Fellow AIAA*** Chief Scientist, Turbomachinery and Propulsion Systems Division, Fellow AIAAThis paper is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

1

Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

tained. However, three-dimensional simultaneous pre-diction of both the flow and the far-field noise is CPUintensive. To reduce the computational time, a zonal ap-proach for noise prediction was developed [13]. In thezonal approach, LES is used to obtain the inner, nonlin-ear source field separately from the outer, linear, soundpropagation field. Other techniques were developed toextend the near field solution to the far-field [14,15].While the zonal approach considerably reduced the CPUtime, three-dimensional LES is still expensive.

Linearized Euler equations (LEE) can be used as ap-proximate, fast approach for jet noise prediction. In theLEE approach, the starting point is the compressible,time-dependent Navier-Stokes equations. The flow andacoustic parameters are split into mean flow and fluid/acoustic fluctuations. The mean flow is assumed to begiven by other means, e.g. solution of Reynolds-aver-aged Navier-Stokes equations, experiments, or analyticalsolution. The governing equations are then linearizedaround the mean flow to obtain the linearized Eulerequations upon neglecting the viscous effects. In manysituations the linearized equations adequately describedthe flow fluctuations and the associated acoustic field. InMankbadi et al. [16] the predicted sound source and as-sociated sound field based on LEE were found in closeagreement with the experimental observations of Troutand McLaughlin [4] for a supersonic jet at Mach number2.1. Since LEE does not require much computer CPUtime, it is quite useful for studying various concepts andfor performing parametric studies. Similar approach hasalso been employed by Viswanathan and Sankar [17] forM=2 heated and unheated axisymmetric jets.

In Morris et al. [18,19] each flow variable is splitinto a mean flow, a resolvable large-scale fluctuation,and an unresolved small-scale fluctuation. LES equa-tions are derived and solved for the fluctuations while ac-counting for the effect of the unresolved ones on theresolved ones. The mean flow is assumed to be given bysome others means. The approach is similar to the LEEapproach in that the mean flow is assumed to be given,and, as such it might be expected to be faster than LES.The nonlinear effects are accounted for in the distur-bance equations. The mean flow however is consideredindependent and is not influenced by the calculated fluc-tuations. The nonlinear disturbance equations are thensolved for both the flow fluctuations and their radiatednoise for a supersonic jet

The approach presented in this paper follows Morriset al. [ 18] and the zonal approach of Shin et al. [ 15]. TheLES is performed for the jet near field fluctuations alone.The mean flow is initially assumed to be given. The in-stantaneous flow variables are decomposed into time-av-eraged mean values, time-dependent large-scale andsmall-scale fluctuations. The large-scale structures arecomputed directly from the nonlinear disturbance equa-

tions, while the small-scale fluctuations are modeled us-ing Smagorinsky's subgrid scale model. The solution ofthe near field fluctuations is used to obtain far field noiseradiation via the Kirchhoff s method.

2. Governing Equations

2.1 Nonlinear Disturbance EquationsThe flow field of a supersonic jet is governed by the

compressible Navier-Stokes equations. There are twosteps involved in the derivation of the nonlinear distur-bance equations for the large-scale fluctuations. Firstfollowing the LES approach, the instantaneous variablescan be decomposed into the resolved, large-scale part fand the unresolved, small-scale part f' by applying aspatial filter to each flow variables.

f = (1)

f = Jf(X,t)G(X-Y)dYD

where G is the filter function. Then time averaging is ap-plied to the large-scale part f to split it into mean valuef0 and a perturbation f.

f = f0 + f (2)Consequently, the flow variables are split into mean val-ue, large-scale and small-scale fluctuations. Upon substi-tuting this splitting in the full Navier-Stokes equations,and neglecting the viscous terms for present free jet sim-ulation, the nonlinear disturbance equations hi cylindri-cal coordinates can be written as

(3)

(4)

1 3 f ' ' "Ni^-|H + H + H = Sr3<j>^ n )where' TQ = [p, u + p'u', v + pV, w + pV, e]

T= [p1, (pu)' + p'u', (pv)' + pV, (pw)' + p'w', (pe)']Assume U, V and W are the mean axial, radial and azi-muthal velocities. The velocities are normalized by thejet exit centerline velocity Ue, time by R/Ue, density bythe mean exit centerline value, and pressure by the exitdynamic pressure. The linear convective fluxes for thelarge-scale perturbations F', G', and H' are given as

F =p' + 2uU-pU2

f iV+vU-pUVwu-puw

(u-pU)E

(5)

Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

G =

H =

Q V + v U - p U V

p' + 2vV-pV2

vW + wV-pVW_(P '+e)V + (v-pV)EJ

wuW + wU-pUWvW + wV-pVW

,2(7)

p' + 2wW-pW>' + 8)W + (w-pW)EJ

while the nonlinear contribution of the large-scale per-turbations Fn', Gn', and H,,' are given as

Fn =

Gn =

p'u'u'(2u-p0u' + p'u')

u'v + p'v'U + p'uVu'w + p'w'U + p'u'w'

(p* + e)u' J

pVu'v + p'v'U + p'uV

(9)v'w + p'w'V + p'v'w'

(p' + e)v'

p'w'u'w + p'w'U + p'u'w'v'w + p'w'V + p'v'w'w'(2w-p0w' + p'w')

(10)

(p1 + e)w'The effect of small-scale perturbation on the large-scalestructure needs to be modeled. These terms F', G" andH" can be written as

F = ~Txr

q _UT -Vt -4x xx xr

(1 1)

G =

0-Txr

-Trr (12)

H =-T,

(13)

where the subgrid-scale stresses are defined as

(8) and the subgrid-scale heat flux is expressed as(14)

where h is the enthalpy. The source term S consists of themean flow convective fluxes S0 and the contributionfrom the use of cylindrical coordinates Sc.

00

r-pw2p.' + 2wW-pW +w'(2w-pow' + p'w')-T(j.

- (vW + wV - p VW + v'w + p'w'V + p'v'w') + T,

0The fluctuating pressure is related to the fluctuating en-ergy as

p' = (Y-l)f"e-(uU

1 ,(2fl _ p „. + ,u.} + y,(2v _ , + V)2 v Ko * > v Ko v '

w'(2w - p w' + p'vw') (16)

Note that in the present formulation, the Favre averagedquantities of the large-scale perturbations are used. TheFavre averaging is defined as

f = | (17)

The unresolved stresses TJ: and heat fluxes qj appearingin equations (14-15) need to be modeled.

2.2 Subgrid-Scale ModelingThe effect of unresolved scales on the resolved ones

is accounted for through the use of Smagorinsky's sub-grid scale model [20]. The subgrid-scale turbulencestresses are represented as follows:

Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

2pvJS::-k. <18>

where kg is the kinetic energy of the residual turbulenceand is neglected with respect to the thermodynamic pres-sure. The strain rate of the resolved scale is given by

(19)

VR is the effective viscosity of the unresolved field,

and Af is the filter width given by

Af = <VrThe heat flux in equation (15) is modeled as

(20)

(21)

(22)

where Prt is the subgrid-scale turbulent Prandtl number,which can be taken as 0.7. Smagorinsky's constant Cs inequation (20) is 0.1 as was used in previous study [10].

3. KirchhofTs Method

In the outer region surrounding the jet the acousticwave propagation is governed by the linearized Eulerequations. Shih et al. [IS] did a comprehensive compar-ison of various acoustic methodologies for far field noiseprediction. The Kirchhoff-type methods were found tobe as accurate as the linearized Euler solutions, whilemore efficient and much less computer CPU time wererequired. Hence, the Kirchhoff s method is adopted here-in. A cylindrical Kirchhoff s surface S is assumed to en-close all the nonlinear effects and sound sources. In theouter region surrounding the jet where the mean flow ve-locity is zero or constant, the linearized Euler equationsreduces to the convective wave equation

<23>where U^ and c^ are the velocity and speed of soundof the ambient air. In the classical Kirchhoff s theory, aGreen's function approach is used to derive a represen-tation for the solution of the convective wave equationin terms of the surface pressure and its derivatives. Thepressure field can be expressed using surface integral

23n r_9n

_Lf47tJ

OL-p

J_§E&.

P P P

c r_M

dS

dS (24)

where subscript p denotes evaluation at the Prandtl-Glauert transformation

and Qp is the outward normal to the surface S. The dis-tance between the observer and the surface point inPrandtl-Glauert coordinates are

2 2 I 211/2J (y.y.) +^(Z_Z')ZJ (25)

t 2\l/2= I I - M l

-[<.- x')

•JAll the values are calculated at the retarded time T,

= t-- (26)

Equation (24) describes the sound pressure at a point(x,y,z) in terms of the information prescribed on theKirchhoff s surface. The large eddy simulation using thenonlinear disturbance equations as described in section2 can provide the pressure and its normal and timederivatives.

4. Numerical Method and Boundary Conditions

The nonlinear disturbance equations for the largescale simulation in the near field of the jet are solved nu-merically using the 2-4 MacCormack scheme by Gottli-eb and Turkel [21]. The scheme is fourth-order accuratein space and second-order accurate in time. The compu-tational domain for the nonlinear sound generation re-gion as shown in Fig. 1 extends axially from x/R=5 to x/R=70, using 196 equally spaced points. The computa-tional grid was begun 2.5 diameters downstream of thejet exit due to the numerical problem associated with thesteep mean-flow gradients near the nozzle. In the radialdirection, the grid extends from the centerline to r/R=10,with a total of 262 points. The grid is uniform from thecenterline to r/R=2, with a spacing of Ar/R=0.02. At thispoint, the grid is stretched geometrically by a factor of1.01 until the outer radius boundary. In the azimuthal di-rection, 37 points is used with a uniform spacing of 10degree.

Boundary condition is an important issue in thecomputation of jet noise. Proper boundary treatment

Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

should allow waves to pass through the boundary with-out generating reflecting waves. The present work em-ploys the same boundary treatments as in references [10-13] for the nonlinear sound generation region. On theouter radial boundary, an acoustic radiation treatment isspecified. On the jet outflow boundary, the asymptoticoutflow boundary condition of Tarn and Webb [22] isused. On the centerline, an averaging procedure is used[11]-

At the inflow boundary, the acoustic radiationboundary condition is used for r/R>4. For r/R<4, the in-flow disturbance is specified in a random fashion bothtemporally and spatially, as described by Morris et. al.[18] in order to simulate the natural jet exit conditions asclosely as possible. The spatial distributions are chosento be a combination of the axisymmetric and the first he-lical modes. The axis of symmetry of the helical modesis chosen at random at each time step. The temporal vari-ation of the disturbances are chosen to be the sum of thebase frequencies with random phase shifting between thebase frequencies. The Strouhal numbers for the base fre-quencies are chosen to be 0.05,0.1,0.2 and 0.4 with in-dividual random phase additions. The radial distributionof the disturbances is chosen to be a Gaussian distribu-tion centered around the nozzle lip line with a half widthequal to 5% of the jet radius.

To apply the nonlinear disturbance equations, themean flow field of the jet must be specified. In Seiner et.al. [23], the jet flow is expressed by means of a half-Gaussian profile described by the parameters, half-widthof the jet b(x), potential core radius h(x), and local axialcenterline velocity Uc(x). There are three regions in thejet exhaust The first two are the potential core and thetransitional region, where the mean velocity profile is de-scribed as:U = Uc(x)

U =

r<h(x)^

^p) j r>h(x)(27)

In the potential core, Uc(x) is equal to one, while in thetransitional region, the centerline velocity is decreasing.At the end of the transitional region, the parameter h(x)becomes zero, and the fully developed region is

*\U = Uc(x)exp(Un(2)(-^) j (28)

The parameters h(x) and Uc(x) are related to b(x)through the conservation of axial momentum. Using theboundary layer assumption, the pressure across the jet istaken as constant. The density in the jet is related to themean velocity using Crocco's relation:

T t jJu j(29)

Once the axial velocity and density are known, the con-tinuity equation for the mean flow can be integrated toobtain the mean radial velocity distribution.

5. Results

The nonlinear disturbance code is first validatedwith an axisymmetric single frequency disturbance at theinflow. The mean flow profile is specified according toTroutt and McLaughlin's experimental measurement[4]. The disturbance at the nozzle exit is specified ac-cording to the linear instability theory. They follow theform

(30)where CD is the input frequency, 3>;(r) is the eigenvectorsassociated with the eigenvalue a, and e is the initial ex-citation level. The linear instability equations are solvedto obtain the eigenvectors <E>j(r) based on the input fre-quency eo. where co is set to 0.2n corresponding to theStrouhal number (fD/Ue) of 0.2. An initial investigationof the nonlinear effect has been made by using variousexcitation level e.

Figure 2 shows the perturbation pressure amplitudedistribution along the streamwise direction in the jetshear layer for e=10'4, 10'3' and 5xlO"3. The results arescaled to e=10"3 case for comparison purpose. One cansee that at eslO"4, 10"3 the development of pressure dis-turbance are essentially linear until x/R=2S. When thedisturbance grows, higher modes are generated becauseof the nonlinear effect. At high excitation level e=5xlO"3

the disturbance growth is different from the low excita-tion level cases due to the strong nonlinear interactions.The computed pressure spectra at certain axial locationsin the shear layer confirmed that higher modes are negli-gibly small for the low excitation level cases.

The full three dimensional computation is carriedout for Mach 2 heated supersonic jets issued from a con-vergent-divergent nozzle to ambient stationary air. Theexperimental work was done by Seiner el. al. [23]. Thecase of jet total temperature 755 °K is tested. The ratio ofjet total temperature to ambient static temperature is2.7 1 8. The computation was begun with the initial distur-bance specified at the inflow boundary, as described insection 4, and carried out until the initial transient purgedout of the computational domain. The unsteady fluctua-

Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

tion data are collected from the nondimensional timet=120tot=160.

Figures 3a-3c show the instantaneous distribution ofaxial velocity, pressure and density at t=134 at <(>=0 andjc planes. These are the sum of the mean and fluctuationquantities. The wave-like structure of the flow field is ev-ident in this figure. The contours exhibit some out ofphase behavior above and below the centerline due to thehelical nature of the flow. The mean flow spreads furtherdownstream of the jet, and the large scale structure be-gins to break down into smaller vortices. Figures 4a-4cshow the instantaneous distributions of axial momentum,density and total energy disturbances. The data haveshown that the linear fluctuation terms are the major con-tributions, though the nonlinear terms are of comparablemagnitudes at some locations.

Figure 5 shows the predicted root-mean-squarepressure fluctuation on the jet centerline and on the noz-zle lip-line as a function of downstream distance. Thepressure fluctuation on the centerline rise sharply nearthe end of potential core which is about x/R=20. Thepressure fluctuation at the nozzle lip-line peaks a littlefurther downstream. Both show comparable magnitudesafter downstream of two potential core lengths. Figures6a-6b show the axial distributions of the root meansquare velocity components on the centerline and nozzlelip-line. The axial velocity fluctuation is the largest com-ponent with the other two components of nearly equalmagnitudes. A rise of fluctuations after the potential coreis shown in this figure. The peak values of the velocityfluctuations on the centerline are greater than those onthe lip-line until two potential core lengths downstream.After that, the velocity fluctuations are approximately ofequal magnitudes on both centerline and lip-line.

The time history of the pressure fluctuation on theouter radius boundary (x/R=10) is obtained from the nearfield, nonlinear disturbance calculation. The Kirchhoffsmethod is then applied with the obtained pressure fluctu-ation and its normal derivative on the cylindrical surface.Figure 7 compares the computed directivity of soundpressure level to the experimental results of Seiner et al.[23]. The data was gathered at itfJR=&0, 0<xob/R<124,and 45<rob/R<80, x^/R=l24. The observation angle ex-tends from 80 degree to 160 degree relative to inlet axis.The experimental data are at four different frequencies,while the computed result is the overall sound pressurelevel. The computed peak noise radiation angle is consis-tent with the measurements, though slightly highersound pressure level is predicted near 90-110 degree.

6. Conclusions

Direct computation of the unsteady characteristics

of supersonic jet flow and its radiated noise is performed.This zonal approach consists two parts of computations:nonlinear sound source generation in the near field andlinear acoustic radiation in the far field. The nonlineardisturbance equations are derived and solved to obtainthe unsteady fluctuations in the near field The Kirch-hoff s method is used to obtain far field noise with the in-formation provided by the solution of nonlineardisturbance equations. Qualitative agreement with themeasured sound directivity is obtained. It is demonstrat-ed that the approach is suitable for full three dimensionalsimulation of large scale perturbations and their radiatednoise in supersonic jets.

Acknowledgement

This work was carried out under grant NCC3-531from the NASA Lewis Research Center. Dr. L. A. Pov-inelli was the Technical Monitor.

References

1. Crow, S. C. and Champagne, F. H., "Orderly Struc-ture hi Jet Turbulence", J. of Fluid Mechanics, Vol.48, pp. 547,1971.

2. Mankbadi, R. R., "Dynamics and Control of Coher-ent Structure in Turbulent Jets", Applied MechanicsReview, Vol. 45, No. 6, pp. 219-248,1992.

3. Tarn, C. K. W., "Supersonic Jet Noise", Annual Re-view in Fluid Mechanics, Vol. 27, pp. 17-43,1995.

4. Troutt, T. R. and McLaughlin, D. K., "Experimentson the Flow and Acoustic Properties of a ModerateReynolds Number Supersonic Jet," Journal of FluidMechanics, Vol. 116, pp. 123-156,1982.

5 Morris, P. J. and Tarn, C. K. W., "On the Radiationof Sound by Instability waves of a CompressibleAxisymmetric jet", Mechanics of Sound Generationin Flows, pp. 55-61, Springer-Verlag, Berlin, 1979.

6. Tarn, C.K.W. and Burton, D.E., "Sound Generatedby Instability Waves of Supersonic Flows, Part 2:Axisymmetric Jets," Journal of Fluid Mechanics,Vol. 138,1984, PP. 273-295,1984.

7. Lele, S. K., "Computational Aeroacoustics: A Re-view", AIAA Paper 97-0018, Reno, NV, January 6-9,1997.

8. Mankbadi, R. R., Hayder, M. E. and Povinelli, L. A.,"Structure of Supersonic Jet Flow and Its RadiatedSound," AIAA Journal, Vol. 31,1994, pp. 897-906.

9. Mankbadi, R. R., 'Transition, Turbulence, andNoise," Kluwer Academic, Boston, 1994.

10. Mankbadi, R. R., Shih, S. H, and Hixon, D. R.,"Large-Scale Simulations of Flow and AcousticField in a Supersonic Jet," AIAA Paper No. 95-0680, Reno, Nevada, January, 1995.

Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

"Three-Dimensional Structure in a Supersonic Jet:Behavior Near the Centerline", AIAA Paper No. 95-0681, Reno, Nevada, January 1995.

12. Shih, S. H., Hixon, D. R., and Mankbadi, R. R. andPovinelli, L. A., "Noise Generation Due to VortexPairing," Proceedings of the ASME, Int. Conf. onFluid Dynamics and Propulsion, pp. 349-359, De-cember 29-31,1996, Cairo, Egypt

13. Shih, S. H., Hixon, D. R., and Mankbadi, R. R.,"Zonal Approach for Prediction of Jet Noise," J. ofPropulsion and Power, Vol. 13, No.6, Nov.-Dec.,1997, pp. 745-752.

14. Lyrintzis, A. and Mankbadi, R. R., "On the Predic-tion of the Far-Reld Jet Noise Using KirchhoffMethod," AIAA Journal, Vol. 34, 1996, pp. 413-416.

15. Shih, S. H., Hixon, D. R., Mankbadi, R. R., Pilon, A,and Lyrintzis, A., "Evaluation of Far-Field Jet NoisePrediction Methods," AIAA 97-0282, Reno, NV,Jan. 6-10,1997.

16. Mankbadi, R. R., Hixon, D. R., Shih, S. H., and Pov-inelli, L. A., "Use of Linearized Euler Equations forSupersonic Jet Noise Prediction," AIAA Journal,Vol. 36, No. 2, Feb. 1998, pp. 140-147.

17. Viswanathan, K., and Sankar, L. N., "Toward theDirect Calculation of Noise: Fluid/Acoustics Cou-pled Simulation", AIAA Journal, Vol. 33, No. 12,Dec. 1995, pp. 2271-2279.

18. Morris, P. J., Wang, Q., Long, L. N., and Lockard,D. P., "Numerical Predictions of High Speed JetNoise", AIAA 97-1598, 3rd AIAA/CEAS Aeroa-coustics Conference, Atlanta, GA, May 12-14,1997.

19. Morris, P. J., Long, L. N., Bangalore, A., and Wang,Q, "A Parallel Three-Dimensional ComputationalAeroacoustics Method Using Nonlinear Distur-bance Equations", J. of Computational Physics, inpress.

20. Smagorinsky, J., "General Circulation Experimentswith the Primitive Equations, I. The Basic Experi-ment," Monthly Weather Review, Vol. 91, pp. 99-164,1963.

21. Gottlieb, D. and Turkel, E., "Dissipative Two-FourMethods for Time-Dependent Problems," Mathe-matics of Computation, Vol. 30, No. 136, pp. 703-723,1976.

22. Tarn, C. K. W. and Webb, J. C., "Dispersion-Rela-tion-Preserving Finite Difference Schemes forComputational Acoustics," Journal of Computation-al Physics, Vol. 107, pp. 262-281,1993.

23 Seiner, J. M., Ponton, M. K., Jansen, B. J., and La-gen, N. T., "The Effects of Temperature on Super-sonic Jet Noise Emission", Proceedings of the 14thAeroacoustics Conference, DGLR/AIAA 92-02-046,1992.

Far field: Kirchhoff s Method Observer (x,r,<|>,t)

Acoustic Radiation

Acoustic RadiationI \o

Thompson Inflow (hydrodynamic regime)

Centerline condition

Near field: Nonlinear Disturbance Equations

Acoustic Radiation

Tarn and Webb out-flow

Fig. 1 Computational Domain

Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

0.0015

0.00103

"o.

VJI 0.0005

0.00000 20 40 60 80

x/RFig. 2 Axial distribution of pressure perturbation amplitude at r/R=l, axisymmetric case.

Fig. 3a Instantaneous distribution of axial velocity at t=134.

Fig. 3b Instantaneous distribution of density at t=134.

Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

Fig. 3c Instantaneous distribution of pressure at t=134.

V / / ^ VFig. 4a Instantaneous distribution of axial momentum disturbance at t=134.

Fig. 4b Instantaneous distribution of density disturbance at t=134.

V th /Fig. 4c Instantaneous distribution of total energy disturbance at t=134.

Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

0.010

0.008

0.006

0.004

0.002

0.0001 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0

x/R

0.20

0.16

0.12

0.08

0.04

0.001 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0

X/R

Fig. 5 Root mean square values of pressure perturbations on Fig. 6b Root mean square values of velocity perturbationsjet centerline and nozzle lip-line. on nozzle lip-line.

0.20

0.16

0.12

~>

0.08

0.04

O.X 0 10 20 30 40 50x/R

60 70 80

120

110

100

90 •

8080 100 120 140angle to inlet axis, Deg.

160 180

Fig. 6a Root mean square values of velocity perturbations onjet centerline.

Fig. 7 Far field noise directivity.

10