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

REFRACTION CORRECTIONS FOR THE KIRCHHOFF METHOD

Anthony R. Pilon*Department of Aerospace Engineering, Penn State University, University Park, PA 16801

Anastasios S. LyrintzisfSchool of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907

AbstractThe Kirchhoff method of computational aeroa-

coustics has been modified to account for the refrac-tion caused by non-uniform mean flows in jet noisecalculations. The Kirchhoff method allows radiatingsound to be evaluated based on quantities on an ar-bitrary control surface, if the linear wave equationis assumed valid in the exterior of the surface. Thecontrol surface is assumed to enclose all nonlinearflow effects and noise sources. The solution on thecontrol surface is evaluated using a computationalfluid dynamics (CFD) code. The corrections pre-sented here are based on simplified geometric acous-tics principles. An axisymmetric, parallel shear flowis assumed for the jet mean flow which causes the re-fraction. The corrections are shown to qualitativelyapproximate the "zone of silence" near the jet axisobserved in jet acoustics experiments.

IntroductionNoise generated by supersonic and subsonic jets

is important for both civil and military aircraft. Thesuccess of the High-Speed Civil Transport (HSCT)and other aircraft depend on a substantial reduc-tion of the radiated jet exhaust noise. Because ofthis, it is necessary to have accurate jet noise pre-diction methods, so that future aircraft designs canbe assessed. One attractive prediction technique isthe Kirchhoff method. The Kirchhoff method con-sists of the calculation of the nonlinear near-filed,usually numerically. The far-field acoustics are thendetermined through Kirchhoff's integral formulationevaluated on a control surface surrounding the non-linear field.

Kirchhoff's integral equation,1 as a tool for nu-merical acoustic prediction, has become popularrecently.2 Methods based on this integral relationare attractive because they utilize surface integrals,not the volume integrals found in acoustic analogymethods, over a source region to determine far-fieldacoustics. Additionally, Kirchhoff methods do notsuffer the dissipation and dispersion errors foundwhen the mid-field and far-field sound is directlycalculated with an algorithm similar to those used

* Research Associate, Member AIAA.f Associate Professor, Associate Fellow AIAA.Copyright ©1997 by A.R. Pilon and A.S. Lyrintzis. Published by the American Institute of Aeronauticsand Astronautics, Inc. with permission.

in computational fluid dynamics studies.The Kirchhoff method has been used successfully

in the prediction of jet noise by several researchersrecently.3"8 Shih, et. al.7 showed that the Kirch-hoff method can predict results nearly identical tothose obtained with a direct calculation method,with a substantial savings in CPU time. However,there are some difficulties involved with using theKirchhoff and related methods for jet aeroacousticproblems. For an accurate prediction, the Kirchhoffcontrol surface must completely enclose the aerody-namic source region. This is often difficult or impos-sible to accomplish with the source regions found injet acoustics problems. The validity of predictions isalso dependent on the control surface being placedin a region where the linear wave equation is valid.Difficulties meeting these criteria frequently arise injet acoustics studies. Additionally, the existence of asteady mean flow outside the Kirchhoff surface willcause refraction of the propagating sound. Failureto account for this refraction will also lead to errorswhen the observer location is near the jet axis.

This paper outlines the development of correc-tions to the Kirchhoff method to account for thedifficulties caused by mean flow refraction. The cor-rections are based on geometric acoustics principles,with the steady mean flow approximated as an ax-isymmetric parallel shear flow. Sample calculationsare presented which show the corrections to predicta "zone of silence" in qualitative agreement with ex-perimental observations.

The Kirchhoff Integral FormulaIn this section, we present the Kirchhoff integral

formula for computational aeroacoustics. The for-mula has been derived and presented extensively bythe authors and others recently,9"12 so only a briefreview will be given here.

Farassat and Myers9 have derived a time domainKirchhoff formula valid for moving and deformingsurfaces. Assume the linear, homogeneous waveequation,

= 0 (1)

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is valid for some acoustic variable </>, and soundspeed a0, in the entire region outside of a closed andbounded smooth surface, 5. Then (j> for all points inthe exterior of 5, with spatial and temporal coordi-nates (x, t) is defined in terms of surface quantities(with coordinates ( y , r ) ) by

, t) = / r ( l - A / r ) TdS

r 2 ( l - M r ) J r

(2)dS

where f = x — y, r$ = ri/r, and n, is the unit vectornormal to 5. E\ and E? are defined by

E, = (Ml - 1) - + MnMt • V20 - ̂^ '

(cos0- Mn)

Here Mn is the local Mach number (of the surfacemotion) normal to the surface, MT is the Mach num-ber in the radiation direction, Mj is the Mach num-ber vector tangent to the surface, and Va is thesurface gradient operator. Also, (a dot indicates asource time derivative, with y kept fixed)

nr =

M —

Mr =

Mn =

<j> = a0Mn(d<t>/dn) + a0Mt

The subscript r in (2) indicates evaluation of theintegrand at the retarded (emission) time, which isthe root of

g=rr(r)•t+-^- =0 (3)

In the study of static jets, it is possible to write(2), in a simple form valid for stationary surfaces.The Kirchhoff formula is then

(4)f \$\•ys^f dS

where cos 9 — f • n. The the use of a Fourier trans-formation, equation (4) can be expressed in the fre-quency domain as

= / eiur/a° -(Js r\

--—cosr a0

a~dn

(5)

J

where (/> is the Fourier transform of 0, and w is thecyclic frequency. An equivalent to (5), valid for sur-faces and observers in rectilinear motion was pre-sented by Pilon.12

At this point some discussion should be made re-garding the choice of acoustic variable (j>. Most of-ten, the disturbance pressure p' = p - pa is usedfor (j>. Mitchell, et. al.4 however, advocate the useof (j> = V • u where u is the fluid velocity. They feelthat this is a logical choice, because this quantity ap-proaches zero as r increases. The authors feel thatthe best choice for <f> lies in the density perturba-tion (4> = a%p'). The authors have shown10"12 thatwhen this variable is used the Kirchhoff formula canbe reduced to a porous surface form of the FfowcsWilliams-Hawkings13 formula for noise generated bymoving surfaces, plus an additional volume integralof quadrupoles. Thus, nonlinear sound sources atthe Kirchhoff surface can be partially accounted forthrough the use of this variable.

Refraction EffectsThe Kirchhoff formulas presented in (4) and (5)

can efficiently and accurately predict aerodynami-cally generated noise, so long as the Kirchhoff sur-face surrounds the entire source region. In jet noisepredictions, however, it is usually impossible, withcurrent numerical methods, to determine the entirenear-field source region. This is due to time andmemory limitations imposed by the computer archi-tecture, as well as dispersion and dissipation con-straints. Thus, a significant nonlinear source region,as well as a steady mean flow, will exist outside ofthe Kirchhoff surface. The jet flow field and Kirch-hoff surface for a circular jet are depicted in figure1.

JetPlume

^Kirchhoff/ Surface

Figure 1. Jet flow and Kirchhoff surface.

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The large extent of the source region describedabove can be seen in numerical data made availableto the authors. The axisymmetric Large Scale Sim-ulation CFD code of Mankbadi, et. al.,L4~15 basedon the 2-4 MacCormack method of Gottlieb andTurkel,16 was used to simulate an excited, Mach 2.1,unheated (T0 — 294A"), round jet of Reynolds Num-ber Re = 70000. The jet exit variables were per-turbed at a single axisymmetric mode at a Strouhalnumber of St = 0.20. The amplitude of the per-turbation was 2% of the mean. (See equation 22 inreference 14.) The flow data was converted to thefrequency domain at all spatial points using a FastFourier Transform algorithm.17 Figures 2 and 3 showthe axial variation of the L/ighthill stress tensor com-ponents, SR(Tn) and 9(Tn) on the jet axis from 5to 70 jet radii (the extent of the available data), forthe first and second Fourier wave modes. The fre-quencies of these modes correspond to St = 0.20 andSt = 0.40. (3? and 9 denote the real and imaginaryparts respectively). All variables are normalized byjet nozzle conditions. Higher order wave modes showsimilar results. It is evident that the disturbanceamplitude is quite large at the end of the compu-tational domain. Thus, some approximation of thesources in the region downstream of the Kirchhoffsurface is necessary. The authors have proposed asuitable approximation in earlier works.10"12 In thecurrent work, the emphasis will be on the refractioncaused by the steady mean flow, so any nonlinearsources outside the surface will be ignored here. Inthe future, the nonlinear source approximations willbe included along with the refraction corrections de-rived here.

10 15 20 25 30 35 40 45 50 55 60 65 70

Figure 2. Centerline axial variations of TH atSt = 0.20.

Even if the unsteady sound sources outside of theKirchhoff surface can be ignored, there is still a sub-stantial steady mean flow in the region near the jet

axis, downstream of the Kirchhoff surface. Figure 4shows the decay of averaged axial velocity along thejet axis. At the downstream end of the Kirchhoffsurface the mean axial velocity is still over 98% ofthe jet exit value. The linear wave equation (1) isnot valid for acoustic propagation through the re-gion near the jet axis, downstream of the Kirchhoffsurface. Thus, some means of approximating the ef-fects of this steady flow are required if an acousticprediction is desired for observer points lying nearthe jet axis.

5 10 15 20 25 30 35 40 45 50 55 60 65 70x/R,

Figure 3. Centerline axial variations of TH atSt = 0.40.

Flow Approximation and Effects

A suitable approximation to the downstream flowis necessary, in order to determine the refraction ef-fects. In the past, several researchers have used anaxisymmetric parallel shear flow model to determinesound produced by point acoustic sources within cir-cular jets.18"21 This approach is adopted here aswell. A real jet has non-zero radial velocity, but therefracting effect of this component is minimal, cancan safely be ignored. The numerical simulationsused to determine the near-field source terms onthe Kirchhoff surface are axisymmetric in nature, sothe lack of azimuthal variation in the parallel shearflow approximation will not have an effect here. Thevalue of the axial velocity to be used in the shear flowapproximation can be taken directly from the near-field numerical simulation, at the downstream endof the Kirchhoff surface, as an average of the timedependent axial velocity at each radial grid point.

The refraction problem now consists of a collec-tion of point acoustic sources (the integrands of (5)or (4)) acting at radial location R scaled by dif-ferential area AS = RARAtp, (where <£> is the az-imuthal angle), and the parallel shear flow with U

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determined at each R. If the acoustic wavelength,A = 2na0/ui, is assumed to be small compared tothe shear layer thickness 5, then geometric acousticsprinciples hold.

1.0050

I.0025

1.0000

0.9975

=̂ 0.9950D

0.9925

0.9900

0.9875

0.98500 10 20 30 40x/R,

60 70

Figure 4. Decay of averaged centerline axial velocity.

If the steady velocity at the downstream end ofthe Kirchhoff surface is denoted Us, the sound emis-sion angle with respect to the jet axis $s, and theemission angle in the stagnant, ambient air is de-noted i?0, then the axial acoustic phase speeds arepreserved by the stratified flow, i. e.20

a0

COS7?C(6)

Here it is assumed that the speed of sound at thesource is equivalent to that in the ambient air. Thisequation can be rearranged to show that there is acritical angle, $c defined by

= COS-11

(7)

If the the observer angle $0 is greater than T?C thanno sound emitted at the source on the KirchhofF sur-face can reach the observer. This criterion is easilyadded to the stationary surface Kirchhoff program.(Note that Ms is the Mach number of the mean shearflow, and not the KirchhofF surface, which is assumedstationary.)

An additional correction is necessary to accu-rately account for the mean flow refraction. Impos-ing the local "zone of silence" condition describedabove can allow a surface source at a relatively largeradial location to radiate sound into and throughthe shear flow. This is because the local "zone of si-lence" decreases in size with the radial location of the

source, due to the decrease in source Mach number.The simple correction is to set the source strengthto zero if the observation point is located closer tothe jet axis than the source point on the Kirchhoffsurface,

. _ _J ~

f0

R0 > RsR0 < R3

(8)

It should be noted that the azimuthal variationbetween the source and observer points has beenignored in the analysis presented here. The az-imuthal variation should have some effect, but itis most likely secondary to those effects describedabove. (Though the near-field CFD calculations areaxisymmetric, the Kirchhoff surface is a full three di-mensional cylinder, so discrepancies between sourceand observer azimuthal location can exist.) Also,the geometric acoustics approximation is only validfor J/A > 1. It is assumed here that the down-stream end of the cylindrical Kirchhoff surface is lo-cated far enough downstream of the jet potentialcore that the shear layer thickness is large comparedwith the acoustic wavelength. Regardless, Morfeyand Szewczyk20 have shown that jet mixing noisecan be effectively modeled with geometric acousticsprinciples even when 6/X < 1.

Sample Validation CalculationAs an initial test of the refraction corrections for

the Kirchhoff method, a simple acoustic monopolewas placed inside a cylindrical Kirchhoff control sur-face. The monopole was located at (x, R) = (5 A, 0).The cylindrical Kirchhoff surface had dimensions(Lk,Rk) = (10 A, 1.5 A). The surface was discretizedwith 130, 40 and 90 quadrature points, in the ax-ial radial and azimuthal directions respectively. Thevalue of the Kirchhoff integrands was determined an-alytically on the Kirchhoff surface at each quadra-ture point. For comparison, the radiated sound fieldwas first calculated in the absence of a mean shearflow. Then, a parallel shear flow was imposed onthe region downstream of the Kirchhoff surface. Theflow velocity was governed by

Us(R)=Msa0exp[-(R2/b2)]

where M3 = 1.4, and b = 0.153 A. This is not a re-alistic scenario, as the shear flow is created at theend of the Kirchhoff surface, and no refractive ef-fects are included in the analytical determination ofthe Kirchhoff integrands, but it serves the purposeof demonstrating the nature of the proposed correc-tions.

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Figure 5 shows instantaneous contours of </> cal-culated with the Kirchhoff method for the casedescribed above. The contours shown above thecenterline are those obtained with no refractingflow field, while those shown below the centerlinewere obtained with with refraction corrections de-scribed above. The effect of the parallel shearflow is seen in the region downstream of the Kirch-hoff surface the propagating sound waves are bentaway from the centerline by the imposed shearflow. This causes a "zone of silence" near thecenterline. Note that a null sound field is calcu-lated inside the Kirchhoff surface in both cases.This is a result of the outgoing radiation condi-tion imposed by the Green's function solution of thewave equation used in the derivation of the Kirch-hoff method.9 It serves as a validation of the nu-merical implementation of the Kirchhoff algorithm.

15.0

Figure 5. Instantaneous contours of <f>. R > 0: Noshear flow. R < 0: Shear flow imposed at KirchhofTsurface.

Jet Noise CalculationThe axisymmetric near-field jet CFD calculations

discussed above were used to determine the inte-grands in the KirchhofT integral formula, and alsoto predict the parallel shear flow downstream ofthe Kirchhoff surface. The surface was chosen tomatch lines in the mesh used for the CFD calcu-lations, so that Lfc = 64.67Rj. (The surface ex-tended axially from x = 5Rj to x — 69.67/?_,-.)The cylindrical Kirchhoff surface had a radius ofRk = 8.56.Rj. These values were deemed to bethe best choices among the available data, based onmesh spacing and the linearity of disturbances near

the surface. However, it is evident from figure 2that the disturbance amplitudes are still quite largeat the downstream ends of the surfaces, so a muchlonger Kirchhoff surface would be preferable. How-ever, the extensive CFD calculations, needed to de-termine the acoustic quantities on a longer surfacewould lead to prohibitive CPU times. There were389 axial, 167 radial, and 90 azimuthal quadraturepoints on the Kirchhoff surface. The radial mesh wasexponentially stretched about R = Rj. Mid-pointquadrature22 was used in the determination of theintegral solutions in the frequency domain formula-tion of the Kirchhoff method. The mesh enrichmentprocedure of Meadows and Atkins23 is a means ofimproving accuracy in Kirchhoff predictions whenthere is an insufficient number of quadrature pointson the Kirchhoff surface. This procedure was notnecessary in these calculations, but may be requiredin the future. The time step used in the CFD calcu-lations corresponds to 1/64 of the period of the exci^tation frequency. Because of constraints imposed bythe number of temporal points per period requiredfor an accurate prediction, only the first four Fouriermodes are used in the Kirchhoff predictions.

The effect of the refraction corrections on thisjet noise prediction is shown in figure 6. The fig-ure shows instantaneous contours of a2p'/p0 on aplane passing through the jet axis, calculated usingthe numerical data described above. The contoursshown above the jet axis are those obtained whenthe mean flow refraction effects were ignored. Thecontours shown below the jet axis were calculatedin an identical fashion, except that the effects ofmean flow refraction were included in sound gener-ated at the downstream end of the Kirchhoff surface.Both calculations capture the Mach wave radiationin the region R > 10 Rj identically. The steadymean flow has little effect on the radiation in thisregion. Downstream of the Kirchhoff surface, thesound waves appear to propagate away sphericallyfrom an equivalent source located near x w 30 Rj.In the prediction without refraction correction, thesound waves have large amplitude near the end ofthe Kirchhoff surface, and propagate as through auniform stationary medium. The corrections, how-ever, reduce the amplitude in the region near the jetaxis, and adjust the phase of each disturbance. Thecorrected sound waves propagate away from the axisat a modest angle. This creates a "zone of silence"near the axis, similar in nature to those observed ex-perimentally. The zone of silence is also evident infigure 7, which shows sound pressure level contoursin the near and mid acoustic fields. The reductionin amplitude near the jet axis caused by the refrac-

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tion corrections is again evident. Also noteworthyis the prediction of sound inside the Kirchhoff sur-face. As discussed earlier, a null acoustic field shouldbe calculated inside the surface. The sound fieldinside the surface shown here is a result of severalfactors. Among these factors are numerical round-off errors, and the interpolation routine used by thegraphics program. Also, the upstream end of theKirchhoff surface was left open in the predictions.

Refraction Corrections ''(//.-, , , i , , , , i ,

10 20 30 40 50 60 70 80 90 100x/R,

Figure 6. Instantaneous contours of a%p'/p0.R > 0: No refraction corrections. R < 0: Refractioncorrections imposed.

0 10 20 30 40 50 60 70 80 90 100x/R,

Figure 7. Sound Pressure Level contours(Re: 2 x 10~5 Pa). R > 0: No refraction correc-tions. R < 0: Refraction corrections imposed.

In the past, researchers utilizing Kirchhoff meth-ods to predict jet noise have ignored sound gener-ated at and outside of the downstream end of theKirchhoff surface.3-0 If the observer lies in an areain which a majority of the sound is predicted bythe constant radius portion of the Kirchhoff surface,then this omission may not pose a problem. How-ever, the authors have shown that "open surface"Kirchhoff methods are not acceptable for jet acous-tics predictions when the observer is in the regiondownstream of the Kirchhoff surface. The refractioncorrections presented here can aid in the accurateprediction of sound in this region. While these cor-rections are crude, and most likely overly simplified,they do represent a first step towards efficient, ac-curate determination of acoustic propagation at andnear a jet axis. Further development of the correc-tions is required. Emphasis should be focused oninclusion of azimuthal and amplitude variations inthe refraction effects.

Conclusions

This paper has presented results of the initial de-velopment of refraction corrections for use with theKirchhoff method. The corrections were developedin an effort to enhance the prediction capabilities ofthe Kirchhoff method in jet noise studies. In thepast, the Kirchhoff method has proven to be an effi-cient and accurate tool for the prediction of jet noise,and other aeroacoustic phenomena. It is hoped that,with further development, the simple corrections de-rived here will allow for accurate prediction of the"zone of silence" and downstream jet acoustics ra-diation fields. These capabilities are currently notavailable with the Kirchhoff method. The correc-tions are based on geometric acoustics principles andparallel shear flow assumptions. In the future, theseassumptions will be relaxed, to improve the accu-racy of the Kirchhoff predictions. The correctionswill also be employed in conjunction with Kirchhoffmethods for acoustics in two dimensions, Kirchhoffmethods for moving surfaces, and approximationsto the volume integral found when nonlinear sourcesexist outside of the Kirchhoff surface.

Acknowledgments

This work was sponsored by NASA Langley Re-search Center under research grant no. NAG 1-1660.Dr. Kristine Meadows served as the technical moni-tor. The authors also wish to thank Drs. Steve Shihand Reda Mankbadi of NASA Lewis Research Cen-ter for performing the jet CFD calculations used inthis research.

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This research was supported in part by grantnumber ASC950019P from the Pittsburgh Super-computing Center, sponsored by the National Sci-ence Foundation (NSF).

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