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

AIAA Meeting Papers on Disc, May 1996A9630823, NAG1-1605, AIAA Paper 96-1709

An improved Kirchhoff method for jet aeroacoustics

Anthony R. PilonMinnesota Univ., Minneapolis

Anastasios S. LyrintzisPurdue Univ., West Lafayette, IN

AIAA and CEAS, Aeroacoustics Conference, 2nd, State College, PA, May 6-8, 1996

A modified Kirchhoff approach is presented for the use in computational aeroacoustics. The method is derived, usinggeneralized function theory, in a similar fashion to the traditional Kirchhoff and Ffowcs Williams-Hawkings methodswhich are currently employed in aeroacoustic predictions. Test calculations with favorable results are shown forsupersonic jet noise. The modified method should also be applicable to other types of aeroacoustic problems. (Author)

Page 1

96-1709

A96-30823

AN IMPROVED KIRCHHOFF METHOD FOR JET AEROACOUSTICSAnthony R. Pilon*

Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455

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

AbstractA modified Kirchhoff approach is presented for

the use in computational aeroacoustics. The methodis derived, using generalized function theory, ina similar fashion to the traditional Kirchhoff andFfowcs Williams-Hawkings methods which are cur-rently employed in aeroacoustic predictions. Testcalculations with favorable results are shown for su-personic jet noise. The modified method should alsobe applicable to other types of aeroacoustic prob-lems.

IntroductionRecent interest in the development of a high speed

civil transport (HSCT) aircraft has increased theneed for accurate aeroacoustic predictions. Predic-tion and reduction of the noise due to supersonicjets will be essential in the development of such anaircraft. This paper outlines a new strategy for theprediction of supersonic jet noise with a modifiedKirchhoff approach.

Recently, several different types of predictionmethods have been utilized in computational aeroa-coustics. Some of the methods can be categorizedas direct calculation methods. In these methods,the mid-field or far-field sound is calculated directlyby extending a computational fluid dynamics (CFD)algorithm from the source region to the observer re-gion. These methods are disadvantageous becauseof the large amount of computational resources re-quired to extend accurately the calculations to thefar-field. Additionally, dispersion and dissipationerrors in the CFD algorithm can render the predic-tion useless. Acoustic analogy methods, based onLighthill's equation, can also be employed. In thesemethods the governing equations are re-arranged toform an inhomogeneous wave equation. A volumeintegral over the sound source region determinesthe far-field sound. This volume integral requiresa great deal of computational resources, so a less in-tensive aiternatrve^desirabie: One alternative is touse Kirchhoff's integral solution of the linear waveequation.

Traditional Kirchhoff methods use flow data (usu-ally obtained from a CFD calculation) on a con-trol surface in a linear region of a flow to predictmid-field and far-field acoustic signatures. Lyrintzisgives a review of Kirchhoff methods and their ap-plications in reference 1. Several researchers haveattempted to utilize these Kirchhoff methods foruse in jet noise predictions.2'3 These attempts havemet limited success however, because the traditionalKirchhoff method is not suitable for this type of aero-dynamically generated noise.

In jet noise and other aeroacoustics problems, thecontrol surface cannot be placed in a linear region.This is due to the large extent of the non-constantflow domain in the axial direction. Placing the con-trol surface in a nonlinear region results in the gen-eration of spurious waves and a loss of accuracy inthe acoustic predictions.

In this paper some modifications to the Kirch-hoff method are developed in order to deal withthese difficulties. The proposed modified methodaccounts for the nonlinearities by placing additionalacoustic sources on the prediction surface, and byapproximating the solution to Lighthill's equationin the remaining non-constant flow region. Themodified Kirchhoff method is derived for use in jetaeroacoustics problems. However, it is believed thatthis approach can be utilized to improve predictionsin other aeroacoustic problems where the Kirchhoffmethod has been employed.

The Traditional Kirchhoff IntegralThe traditional Kirchhoff integral formulation4

for a stationary control surface can be written as

(1)P(X>t) = ̂ ls[^-0!Hdn

_ + dSr dn r2 dn \ ret

Here p' is the acoustic pressure fluctuation, and 5is the surface area of a control surface, with unitnormal n, which encloses all acoustic sources. Sub-script ret denotes evaluation at the retarded time,

* Graduate Research Assistant, AIAA Student Member.Current Address: School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907\ kssatiajfc PTOfesam, M.AA Swriov Membev.Copyright ©1996 by A.R. Pilon and A.S. Lyrintzis. Published by the American Institute of Aeronautics and Astronautics,Inc. with permission.

tret = t - r/a0, where r is the distance between thesource and observer points, (r = x - y) and a0 isthe ambient sound speed, (dr/dn = cosip where ^is the angle between the emission vector, F and n.)Farassat and Myers5 give a modern derivation of (1)using generalized function theory.

For sufficient accuracy in the far-field calcula-tions, high order quadrature should be used to solvethe surface integrals in (1). The predicted surfacequantities (p', dp/dn, dp/dt) should also be veryaccurate. This can be achieved through the use ofa very fine mesh in the CFD calculations. However,memory and time constraints often make this im-practical. Meadows and Atkins6 have shown thatit is possible to obtain highly accurate Kirchhoffpredictions from relatively coarse-grid CFD solu-tions. Through an interpolation process, more spa-tial points are added to the Kirchhoff quadraturecalculations without additional effort in the CFDprocess. This has the effect of refining the CFDmesh with almost no additional cost. They referto this process as "enrichment". High order quadra-ture, temporal interpolation, and enrichment are im-portant for accurate far-field noise predictions withthe Kirchhoff method, especially if the CFD grid res-olution is somewhat coarse. Fourth order compositequadrature, third order temporal interpolation, andenrichment are used in this study.

Equation (1) works well for aeroacoustic predic-tions when the control surface is placed in a region ofthe flow field where the linear wave equation is valid.In jet acoustics problems, the logical choice of con-trol surface is a cylinder surrounding the flow. Thisis shown in figure 1. However, since it is currentlynot practical to extend CFD calculations axially toa linear region, equation (1) will not be valid at thedownstream end of the cylindrical control surface.Thus, it is necessary to include additional terms inthe derivation of the Kirchhoff integral formulation.

Freund, et. al.7 have used a standard Kirchhoffformulation for use in jet noise predictions. Theyused a frequency-domain formulation of the Kirch-hoff formula, and a cylindrical control surface withno end surfaces. They showed acceptable resultswhen the emission angle with respect to the jet axisis large. Their criterion was based on whether theemission path from a virtual source to the observercrossed the portion of the Kirchhoff surface in a linerregion. If this is the case, they determined the tradi-tional Kirchhoff method provided adequate predic-tions. Near the jet axis, they corrected for the errorsassociated with the omission of the cylinder end sur-face by using an asymptotic correction to the Hankel

functions that arise from the reduced Green's func-tion in the traditional Kirchhoff formula.

The research presented here is aimed at account-ing for the nonlinearities outside of the Kirchhoffsurface in a different fashion. Instead of derivingan integral solution to the linear homogeneous waveequation, and obtaining the standard Kirchhoff for-mula, an integral solution to Lighthill's equation issought. This solution leads to the modified Kirch-hoff method.

(x,t) -Jet Volumeplume Integral

n

—v,Figure 1. Kirchhoff surface and volume integral region.(Not to scale)

The Modified Kirchhoff Formula

In this section LighthuTs equation8 is derived,and then used in the derivation of the modifiedKirchhoff formula. Several coordinate systems willbe employed throughout the derivations. The sourcecoordinates and time are denoted (y, r) or (2/1,7")-When cylindrical coordinates are employed, thesource coordinates will be (xs,rs,0) for the axial,radial, and azimuthal directions. The observer co-ordinates and time are denoted (x, t) or (xj,i). Incylindrical coordinates, the observer coordinates are(x, R) for the axial and radial directions. In polarcoordinates, the observer field will be expressed with(r, 9), where 6 is the angle F makes with the jet axis.The acoustic field is considered axisymmetric.

The conservation relations for mass and momen-tum are

(3)

Here, it is assumed that there are no additionalsources of mass, or externally applied forces in theflow region, p and p are the fluid density and pres-sure, Ui are the cartesian components of the velocityvector, <7jj is the viscous stress tensor, and <5jj isthe Kronecker delta. Taking the inner product of MJ

with (2) and adding to (3) yields

ddt - - (pUiUj + pdij - aij) = 0 (4)

Taking the divergence of (4), subtracting the timederivative of (2), and adding a-2d2p'/dt2 - 82p/dt2

to both sides yields

ay ay> Oj.9 O *?& rj/^ r7TD Ul UJjia2 , 1 d2 (5)

Viscous effects are generally quite small, and canusually be ignored. Non-isentropic effects are im-portant in cases where the jet is heated, or con-tains shock waves, and should generally be included.However, as a first approximation, they are omittedhere, so that equation (5) can be written as

(6)

where T,j = pUiUj is the approximated Lighthillstress tensor.

Equation (5) is now used to derive the modifiedKirchhoff formula. First, the control surface is de-fined by f ( x ) = 0 such that the surface unit out-ward normal, n = V/. The perturbation densityand pressure, and all other flow variables are math-ematically set equal to zero inside the surface via

/(£) < 0C =

Where £ is p', p', or Ui. Then, generalized functiontheory9 is used to write equation (5) so that it isvalid over all space and time. An overbar denotes ageneralized derivative.

(7)dt2 ax\

Expanding the generalized derivatives in (7) will pro-duce the modified Kirchhoff integral formulation.The expanded derivatives for a stationary surfaceare

dt2 ~ dt2

Here 6(f) is the Dirac delta function. These defini-tions become more complicated when the surface ismoving or deforming.9 Using these definitions, (7)becomes

(8)

,'

H(f), the Heaviside, function, is included to en-sure that the last term is only calculated outsideof the control surface (x > Lk or R > Rk infigure 1). Equation (8) is the Ffowcs Williams-Hawkings equation10 for a stationary, porous controlsurface. A similar derivation, for a moving surface,is in reference 11. Equation (8) is valid through allspace and time, and can be solved with the free-space Green's function,

4?rr where g = t - T + -

This leads to

(9)

/Integrating and converting the divergence operatorsto time derivatives (see reference 5, equation 15)gives the modified Kirchhoff formulation

l l dr

j__aa0 dr^ ~ ^ _- —•a-riTiri +

retdS(10)

HI ret

where r is the unit radiation vector, r = r/\f\. Tothe authors' knowledge, equation (10) is a new for-mulation, and is presented here for the first time.

The modifications to the traditional Kirchhoffmethod consist of three additional sources on theKirchhoff surface, and a volume integral. It is easyto show that this equation reduces to the traditionalKirchhoff integral if the control surface is placed ina fully linear region. That is, the additional surfacesource terms and volume integral are zero if thereis no flow through the control surface or outside ofit. This equation can be extended to handle a con-stant, non-zero freestream velocity. In this case, aconvective form of Lighthill's equation is used in thederivation.

The modified Kirchhoff formula, as presented inequation (10), represents an improvement over tra-ditional Kirchhoff calculations, in that sound gener-ated by flow through the control surface is accountedfor. However, the volume integral that arises fromthe non-linear region outside of the control surfacepresents a challenge. A major motivation for theuse of Kirchhoff methods is the lack of volume in-tegrations, which reduces necessary calculations byan order of magnitude. CFD calculations are notlikely to be performed in the non-zero flow regionoutside the control surface, because of memory limi-tations. (If a larger computation domain is used, theKirchhoff surface should be expanded.) However, ig-noring this region is also not acceptable, when usingtraditional acoustic analogy methods,12 or Kirchhoffmethods. Thus, the source region inside the volumeintegral should be approximated. This approxima-tion is outlined in the next section.

Volume Integral Approximation

With any prediction method, calculation of theentire source region in a jet flow is difficult. If CFDcalculations are performed far enough downstreamto capture the entire region, memory constraints willlimit the amount of grid points that can be used toresolve acoustic scales, and accuracy will be sacri-ficed. But, if grid points are spaced to resolve smallscales, calculations cannot extend through the en-tire source region. Here, the demonstrated wave-likestructure of a supersonic jet is used to determine thesource region downstream of the calculation domain.

Mankhadif-et,_aL12 and Mitchell, et. a!.13 sh&wedthat supersonic jets (both excited and natural) haveorderly wave-like structures. These structures existmany diameters downstream of the jet nozzle, ex-tending beyond the end of the calculation domain.The orderly nature of the of the structures allows

them to be predicted downstream of the end of thecomputational domain.

A simulation of a Mach 1.5, axisymmetric, per-fectly expanded, heated jet (Tjet = 2T0) of ReynoldsNumber Re — 1.27 x 107 was performed with the2-4 MacCormack method of Gottlieb and Turkel.14

The effect of small scale turbulence on the larger,noise producing eddies was modeled with a largeeddy simulation (LES) algorithm. The jet exit ve-locity was excited at a Strouhal number, St = 0.125and fifteen harmonics. (St — fD/U where / is thenatural frequency of excitation, D is the jet diame-ter, and U is the jet exit velocity.) The CFD sim-ulation was performed on two domains. The firstused a 300 x 172 grid which extended 50 jet radiiin the axial direction and 5.04 jet radii in the radialdirection. The second use a 450 x 172 grid, over adomain of 75 Rj x 5.04H,. 64 temporal points wereused per period of the excitation frequency. Detailsof the calculation method can be found in reference12.

To demonstrate the wave-like structure of the jetflow, a Fast Fourier Transform15 is performed on thetemporal variable at all spatial points,

~ f °C(*;w)= /

J —0

where £ is again any flow variable. Figures 2 and 3show the axial variation of SR(ptlu) and ^(puu) onthe jet centerline from 0 to 75 jet radii, for the firstand second Fourier wave modes, which correspondto St = 0.125 and Si = 0.250, (ift and 0 denotethe real and imaginary parts respectively), ti andv now represent the axial and radial components ofvelocity, and all variables are normalized by jet exitconditions. Higher order wave modes show similarresults. It is evident that the amplitude of the dis-turbances is still quite large, and still growing, atthe end of the computational domain. Thus, a pre-diction of these modes in the region downstream ofthe computational domain is needed.

Mitchell, et. al.13 give an approximation to thewave modes in the region where they are decayingfar downstream of the jet nozzle. Their method isalso applied here. However, in the calculations here,

--most of the calculated waves-have-not yet startedto decay at the end of the computational domain.This will introduce errors in the acoustic predictions.But, the calculations performed here should sufficeto demonstrate the applicability of the method. Inreference 13, the Lighthill stress tensor components,

in Fourier space, are calculated via gives

(11)

where K is a complex wave number with positiveimaginary part. .4^ and K are determined fromthe flow calculations at and near ( x , r a ) = ( L k , r 3 ) .The positive imaginary part of K ensures that thewave amplitudes are decaying with x downstreamof x0, which is taken as Ljt- If the waves arenot actually decaying at x = L* spurious waveswill be generated. Thus, in future works it willbe necessary to calculate the jet flow downstreamto a point where equation (11) gives a more ac-curate representation of 7\,. Aij and K are as-sumed to be constant with x. This is a reason-able assumption for lower frequencies, but Mitchell,et. al.13 have shown it to break down as ui gets large.

5.04.03.02.01.00.0

-1.0-2.0-3.0-4.0-5.00 15 30 x/R 45 60

Figure 2. Centerline axial variations at St =(first mode). ——— $t(puu); - - - Q(pmi).

75

0.125

The limit of frequency resolution attainable withKirchhofF methods is approximately /max = (8A*)"1

or /max = a0/(8Ax), where At is the temporaldiscretization, and Ax is the spatial discretization.For the temporal discretization used here, /mai isapproximately equal to the frequency of the sixthwave mode. Because of this, and the variability ofA^ and K at high frequencies, only the first six wavemodes are employed in this analysis.

If the generation of unwanted waves can be ig-nored, equation (11) can be used to calculate a so-lution to the volume integral in (10). The generalsolution to Lighthill's equation is

where subscript ret again denotes evaluation at theretarded time. Taking the Fourier transform of (12)

Pi = f eiulr/a* d2l}

Jv 4.7rr dytdyj (13)

Substituting equation (11) into (13), making a far-field approximation, and expanding the axial inte-gral with a Taylor series allows equation (13) to bewritten as (see details in reference 13)

UJ2PI = -—530 j=o

where the Green's function is

(rja0K — ux)

(14)

d</> (15)

and

= (x - L fc)2 + R2 + r\ - 2Rr, cos((j>)

cos(0) = (x-

B\ = sin(29)puv0 = cos2(e)puu= sin2(d)pvv

where puu is An, PUV is AIZ, and pvv is A-M. Equa-tions (14) and (15) can now be used to calculate thevolume integral in the Modified Kirchhoff formula-tion. An inverse Fourier transform is then used toconvert the volume integral predictions to the timedomain. The computational domain for the modi-fied formulation is shown in figure 1. The form of(14) and (15) is important, because this calculationamounts to that of a surface integral on the end ofthe cylindrical control surface. The reduction froma volume integration to a surface integration repre-sents an order of magnitude savings in computationtime.

Kirchhoff Surface Sizing

The modifications to the traditional Kirchhoffmethod will improve predictions when a portionof the control surface is placed in a nonlinear re-gion. However, the flow predictions must be accu-rate at the surface for valid far-field acoustic cal-culations. The CFD algorithm employed to de-termine the near-field flow variables and deriva-tives is an important part of the calculation pro-cess. Inaccuracies in the near-field variables andderivatives (in particular dp/dn) are translated intoincorrect acoustic predictions with the Kirchhoffmethod. These inaccuracies often occur from thetreatment of non-reflecting boundary conditions and

grid stretching ia the CFD algorithm. Properplacement of the control surface to partially alle-viate these problems is discussed in this section.

-5.0,

Figure 3. Centerline axial variations at St —(second mode). ——— 3?(puu); - - - S(puu).

Whenever possible, the Kirchhoff surface shouldbe placed in a linear region of the flow field. There-fore, the cylindrical Kirchhoff surface used in thisstudy should have a radius Rk so that flow distur-bances at the surface are governed by the linear waveequation. Axially, the flow is nonlinear throughoutthe domain of interest, so the cylinder end surfaceshould be placed at an axial location L* where theCFD data is reliable.

Proper placement of the control surface can betested by calculating the sound speed at the surface(a = \/7P/P) • If ^e surface is in a linear region,the sound speed should be nearly constant. Thesecalculations can also give insight on the reliabilityof the data from the CFD program. Figure 4 showsthe sound speed at r, = 3.90H, at several instancesin time, determined with the CFD code discussedabove. The disturbances are smoothly varying, andhave magnitude less than one percent of the am-bient sound speed, to approximately 45 — 50 jetradii axially. In this region the disturbances growand become chaotic. This may be due to nonlin-ear interaction between wave modes, as discussedby Mankbadi, et. al.,12 or to the boundary condi-tions employed in the CFD solver. To test the effectsof surface size, two separate Kirchhoff surfaces areused-ia-this study. - The first has a length of 44,147jet radii, and a radius of 3.90.R,-. The second hasa length of 71.00.Rj, and a radius of 3.90/?j. Thisradius appears to be an acceptable location betweenthe nonlinear flow region, and the area where gridstretching reduces the accuracy of the flow solver.

0.0100.0080.0060.0040.0020.000

-0.002-0.004-0.006-0.008-0.010-0.012-0.014-0.016-0.018'0 15 30

x/R.45 60 75

Figure 4. Aa/a0 vs. axial location, at several instancesin time, at R = 3.90Hj

Results

The CFD code described above was used to calcu-late acoustic sources on the Kirchhoff surfaces. 133and 81 points were used in the axial and radial direc-tions on the shorter surface, while 213 and 81 pointswere used on the longer one. Enrichment was usedin the axial direction, so that a total of 265 and 425axial points were used. 33 temporal points were usedper period of excitation. (Because of the already finemesh, enrichment had little effect on the calculationsin this study.) Figure 5 shows a comparison of theacoustic pressure signatures predicted by the modi-fied Kirchhoff method using both surfaces, and theCFD program at (x,R) = (33.24^,5.04^). Thesignals match reasonably well. Differences in the sig-nals are most likely caused by boundary conditionerrors in the CFD algorithm, and the inaccuraciescreated by the assumption that all wave amplitudesare decaying at the end of the control surface.

The effects of the additional surface sourcesand volume integral in the modified Kirchhoff for-mulation were also investigated. Figure 6 showseach surface source components' contribution tothe overall Kirchhoff integral sound prediction at(r,(9) = (150J?j,46°). The total signal is largelydetermined by that from the traditional Kirchhoffintegral, while the additional terms act as a correc-tion to this solution.

The modified Kirchhoff formula should properlycapture the directionality of the noise radiated bythe jet flow. Figure 7-sh9w&-tbe--variation-with-9-of-the RMS averaged values of the traditional Kirch-hoff integral, the additional surface sources, and theapproximated volume integral at a constant observerradius of r = l50Rj. The overall signal is seen tobe dominated by the traditional Kirchhoff integral.

The effects of the additional Lighthill surface sourcesremain nearly constant with 6, while the volume in-tegral contribution shows a decrease with 0.

-°-%0 0.2 0.4 0.6time/period

Figure 5. Comparison of acoustic signatures at(x,R) = (33.24^,5.04^). ———CFD; - - -Longsurface; — — Short surface.

0.002

0.001

0.000P'/P0

-0.001

-0.002

-0.00^0.2 0.8 1.00.4 0.6

time/periodFigure 6. Source components of the total Modi-fied Kirchhoff prediction at (r,Q) = (150^,46°).—•— Total integral prediction; ——— TraditionalKirchhoff integral; — — Lighthill surface sources;- - — Volume integral.

above the x axis (jet centerline) is the predictionobtained from the modified Kirchhoff method, whilethe portion below the centerline is that obtainedfrom the standard Kirchhoff integral.. Figure 8 shows that the shorter control surfaceis not long enough to adequately capture the soundgeneration which occurs at and near the downstreamend of the surface. The approximation to the vol-ume integral should be able to predict some of theneglected sound sources, but the assumption of de-caying amplitude at x = Lk has hampered that pro-cess. Thus, the predictions from the modified andtraditional Kirchhoff methods are very similar, ex-cept near the end of the control surface. Acous-tic disturbances are also apparent inside the controlsurface, where they are assumed to be zero. Thesedisturbances are caused by nonlinearities on the con-trol surface.

Figure 8 also shows a lack of refraction causedby the mean jet flow downstream of the control sur-face. The effects of this mean flow were ignored inthe volume integral approximation, and the Kirch-hoff integral assumes zero mean flow. So, the spheri-cal wave pattern in the figure should be expected. Inthe future, the modified Kirchhoff formulation willaccount for the non-zero mean flow by using a re-fraction correction.

Figure 9 shows the same conditions as figure 8, ex-cept that the longer control surface was used in thecalculations. The discussion above applies here, ex-cept that the acoustic waves calculated at and nearthe jet axis are not as large in magnitude. It is notclear which of figures 8 and 9 is a more accuraterepresentation of the actual jet acoustic field.

0.005

0.004

0.003

0.002

0.001

The modifications to the Kirchhoff integral are p'/pprimarily intended to improve mid-field and far-field acoustic predictions. Several calculations wereperformed to determine the applicability of themethod for jet acoustics predictions, and to iden-tify areas that need further attention. First, themodified Kirchhoff method was used to predict _the_ .g.QQG,axisymmetric sound field of the supersonic jet dis-cussed above at one instant in time. Figure 8 showsthe results obtained using the short control surface,while figure 9 shows the results obtained with thelonger control surface. The portion of each figure

0 10 20 30 40 500 - deg.

60 70 80 90

Figure 7. Angular variation of the components of theModified Kirchhoff formulation. —•— Total integralprediction; ———— Traditional Kirchhoff integral;— — Lighthill surface sources; — - — Volume integral.

The modified and traditional Kirchhoff methodswere also used to calculate sound pressure levels inthe acoustic field of the jet. Contours of constantSPL (dB re 2 x KT5 Pa) are shown in figure 10.The longer control surface was used in these calcula-tions. The lobed nature of the contours shows strongacoustic radiation in a direction approximately 40°to the jet axis. This is in qualitative agreement withother predictions and experimental results. Furtherwork on the CFD calculations is required before adetailed comparison can be made. However, theseresults are encouraging.

R/R, o

150

Figure 8. Instantaneous pressure contours predicted bythe modified (top) and traditional (bottom) Kirchhoffmethods. The control surface is shaded.

Conclusions and Future DirectionsThe traditional KirchhofF integral formulation is

a powerful tool in aeroacoustic analysis. A mod-ified KirchhofF method was derived here in an at-tempt to make the method more applicable to super-sonic jet noise calculations. The modified Kirchhoffapproach accounts for noise generation at and out-side of the control surface. Encouraging results havebeen shown for axisymmetric supersonic jet noisepredictions. The method should also be applica-ble to other types of aeroacoustics problems whereKirchhofF methods are employed.

Several improvements in the modified KirchhofFmethod are planned for the near future. First, themestr and-calculation domain for the CFD calcula-tions will be changed to increase the accuracy of thenormal derivative calculations, and to extend the so-lution to a region when all wave modes are damped.The Lighthill surface source terms will be expanded

to include the effects of a convective freestream, andnon-isentropic effects. A refraction correction willbe included in the modifications. Calculations ofthree dimensional, imperfectly expanded jet will beperformed. The results will be compared with ex-perimental data.

R/R, o

150

Figure 9. Instantaneous pressure contours predicted bythe modified (top) and traditional (bottom) KirchhofFmethods. The control surface is shaded.

R/R, 0

25 30 35 40 45 50 55 60 65 70 75 80 85

x/R,

Figure 10. Sound pressure level contours (dB) predictedby the modified (top) and traditional (bottom) KirchhofFmethods. The control surface is shaded.

Acknowledgments

This work is sponsored by NASA Langley Re-search Center under research grant no. NAG 1-1605.

Kristine Meadows is the technical monitor. The au-thors appreciate her advice during the course of thiswork.

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

The Authors also wish to thank Steve Shih andReda Mankbadi of NASA Lewis research center forperforming the CFD calculations used in this re-search.

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