AIAA 95-0833 A Data-Parallel TVD Method for Sonic Boom Calculations

A.R. Pilon University of Minnesota, Minneapolis MN A.S. Lyrintzis Purdue University, West Lafayette IN

33rd Aerospace Sciences Meeting and Exhibit

January 9-1 2,1995 / Reno NV For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 370 L'Enfant Promenade, SW, Washington DC 20024

A DATA PARALLEL TVD METHOD FOR SONIC BOOM CALCULATIONS

Anthony R. Pilon * Department of Aeraspace Engineering and ?dechanics

University of Minnesota, Minneapolis MN 55455

Anastasios S. Lyrintzis t School of Aeronautics and Astronautics

Purdue University, West Lafayette IN 47907

Abstract Sonic boom predictions are shown for the near-

and mid-field, and comparisons are made with ex- perimental data. The computations are performed on the Thinking Machines' CM-5 massively parallel su- percomputer to utilize it's large available memory and high floating point performance. A second order ac- curate total variation diminishing scheme is used to solve the Enler equations in the computations. Addi- tionally, a recently developed implicit method, based on the LU-SGS algorithm, is used to speed the conver- gence and accuracy of the steady state computations. The method is shown to work well on near- and mid- field sonic boom predictions for several test cases.

*

Introduction The projected use of the high speed civil transport

(HSCT) has drawn attention to the problems asso- ciated with the noise due to sonic booms.' An ac- curate and efficient sonic boom prediction methodol- ogy is needed for the assessment of various proposed HSCT configurations. One such method which uti- lizes the power of massively parallel supercomputers is presented here.

A review of current sonic boom prediction meth- ods is given by Plotkin.' Many of these methods are based on the modified linear analysis of Whitham3 and Wz~lkden .~ Other methods have been developed based on a modified method of characteristics which approx- imately account for the effects of three dimensional flows. Experimental and analytical studies have shown that these methods lose effectiveness as free stream Mach number approaches 3 . In high Mach number regions there are strong shock waves with significant entropy generation. The linear based methods neglect these production terms.5 A modified method of char- acteristics has been developed which can account for these hut it requires nonlinear near-field ini- tial data which is difficult to obtain computationally

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'd or experimentally.

* Graduate Research Assistant, AIAA Student Member t Associate Professor, AIAA Member

Copyright 0 1 9 9 5 by A S . Lyrintzis. Published by the American Institute of Aeronautics and Astro- nautics, Inc. All rights reserved.

Several authors have developed prediction methods based on near-field solutions of the Euler or Navier- Stokes equation^.^^' &lost of these methods involve marching in one spatial dimension with an implicit Euler scheme or a parabolized Navier-Stokes (PNS) algorithm. These solutions are used in conduction with Whitham's F-function to evaluate the far field pressure signature. The existing prediction methods are unattractive for several reasons. First, the spatial marching is often difficult, and additional accuracy in the marching direction greatly increases the computa- tional time. Also, they do not employ methods de- signed to resolve shock waves effectively. Finally, they do not utilize the power available with massively par- allel supercomputers.

Accurate shock resolution is needed in the CFD methods to determine the far field pressure signa- ture effectively. Since the prediction methods are dependent on shock resolution it is desirable to use a computational method that is designed to capture shock waves. The total variation diminishing (TVD) schemes developed by Harten,$ Yee," and others are well suited for this purpose. In this study a second order accurate TVD scheme is used to determine the pressure field about several models. These solutions are then compared with experiment.

Also important in accurate prediction is the dis- cretization of the flow domain. I t would be beneficial to have as large a number of computational cells as possible, while performing the calculations as qnickly as possible. For this reason the computations are per- formed on a massively parallel supercomputer with a very large available memory and high peak floating point performance.

The flow fields in this study are steady. Because of this, an implicit temporal differencing method is employed. Such a method greatly reduces the time to compnte a steady solution. One implicit method, specifically designed For use on massively parallel com- pnters, is utilized here. The implicit method does not

2 AIAA 95-OR3

effect spatial differencing, so it. works well with thc T V D scheme.

In this paper the benefits of using a TVD scheme in conjunction with the implicit method on massively parallel snpercomprsters will be shown. It is possible to obtain accurate flow results at a small computational cost.

Total Variation Diminishing Scheme The limitations of linear theory in sonic booin pre-

diction are well For this reason sev- eral researchers have nsed computational solutions of the Euler or iiavier-Stokes equations in near- and mid-field sonic boom This research has pointed out the nccd for a high order accurntc, shock capturing method to solve the flow in the near- and mid-fields of an aircraft. One family of such methods ate the total variation diminishing (TVD) schemes of Barten,9 Yee," and others. One of these methods, the Harten-Yee modified-flux scheme is outlined below.

Consider the two dimensional Euler equations

U , F , and G are the vector of conserved quantities, and x and y fluxes. Thus,

Where p is the density, u and the 1: and y veloci- ties, p the pressnre, and E total energy. A first, order, explicit, finite volume formulation is then

Where X , j i s the volume of cell i , j. S is the interfacc area, and At is the time step.

A second order accurate scheme is proclnccd with modifications to the fluxes. F and are now

Horc: R is the right eigenvector mntrix of the (lux ,jato- bian evaluated at the cell interlace. R,oe's symmclric averaging" is used to determine t,hc flow properties a t the cell interface.

Conventional high order acciirnti: schemes requirc Llie addition of linear dissipatioii; or .'nrtificial viscos- ity" t,o prevent instabilities whcn calculating flow ilis- continuities. In the TVD scheme tlic necessary dissipa- tion is added only where it is necdcd, near discontinn- ities. This dissipation enters the calcnlations through the vector a. Terms in @ have the effect of limiting the influence of certain cells in the computational stencil. These terms are the "limiter" fnnctions. ID i s defined

W*

!

where

Here the index I is 1 - 4 for tmo-dimensional perfect gas flow and 1 - 5 for three-dimensional flow, corre- sponding to the components of the vector of conserved quantities. Also, A' are the eigenvalnes of the flns in- cobian. (I and 'W' are defined by

When (I' = 0, 7 = 0. Also

In ( F ) the first definition of u produces second order accuracy in time as well as spacc. While this woolil seem to be an unnecessary calcnlation when using an implicit method, it does improve convergence proper- ties. Thus, the first definition is nscd Ihroughout this study. In (3) the term 61 is a entropy correction pa- rameter. This prevents entropy violations in steady and nearly steady shocks. Values of 0.10 to 0.125 are rised in this study, as recomnmided by Yee.'"

The terms y' in equation (2) have not yet h e m ad- dressed. These are the "limiter" functions. Near i l is-

continuities, the limiter functions reduce or eliminate llre effects of points in the computational stencil which fall on the opposite side of a disconl.inuity. bIany forms

L/

A I/l A 95- 08.13 3

of limiter function are availahle. The easiest and most robust of these is

minmod(z, w) = sgn(z) . max(0, min(lz) , ysgn(a))}

Many other limiter fnnctions are available," hut they all serve the same prirpose, and most are not as stable as (i), so this is the only limiter used in this stndy. It should he noted here that while the limiter functions preserve computational stability, they also cause the scheme to lose second order accuracy in the region of discontinuities. Thus, TVD schemes are not globally high order accurate.

Massively Parallel Supercomputer Used Before discussion of the implicit algorithm it is nec-

essary to discuss the Thinking Machines' C W 5 mas- sively parallel supercomputer used in the calculations. This computer, located a t the Army High Performance Computing Research Center, i s able to compute in sin- gle instruction, multipledata (SIMD) mode, or in mul- tiple instruction, multiple data (MIMD) mode. It is composed of 896 32 Mflop processors, arranged in a fat tree architecture. Each processor has four vector units, which leads to a maximum of 128 M o p per processor. Thus, the 512 processor partition (the largest avail- able) has a peak theoretical floating point performance of 65.5 GRop. Although only fractions of this value are obtained for practical problems, the CM-5 does present the researcher with a large amount of avail- able memory (16 Gbyte) and computational speed. It is necessary to keep inter-processor communication to a minimum with this architecture, as there is a large latency and the communication bandwidth is only 5 Mbyte/sec.

The high theoretical operating speed and large available memory make the CkI-5 an appealing plat- form for CFD problems. However, it is difficult to im- plement efficiently an implicit method on a massively parallel computer. This is because true implicit meth- ods involve the solution of very large, sparse matrices. Solution of these matrices will usually leave many pro- cessors idle and thus greatly diminish performance. It is possible to decrease the idle time by decomposing the computational domain into sub-domains, and then solving the implicit prohlem in each sub-domain. This reqrrires a MIMD algorithm. hdditionally, there is a great deal of necessary inter-processor communication which also hurts performance.

An easier solution to the communication difficulty can be found in the SIMD mode. It is possible to

4

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develop a data-parallel implicit, metlrod which can op- erate in this mode. The method uses only nearest- neighbor communications, which greatly rednces [,he overall comrnnnlcation time of the code. The method can be programmed in a high-level Iangnage like POR-

T R A N Q O . This allows the programmer to iitilizc opti- mized communication commands like CSHIFT. There is one drawhack to the data-parallel approach. To obtain optimal performance a structured grid must he used to maintain the nearest-neighhor communica- tions. The data-parallel implicit algorithm is ontlined next.

I ,..~..i

Data-Parallel Lower-UDper R.elaxat ion M e t h o d As mentioned above, true implicit methods re-

quire the solution of a very large, sparse malrix. To eliminate the inter-processor communication inherent in such a solution an approximate implicit method has been developed for data-parallel supercompntcrs. This new method, developed by Candler et al," is based on the Lower-Upper Symmetric Gauss-Seidel (LUSGS) method of Yoon and J a m e ~ o n . ' ~ In the LUSGS method, the flux jacobians are approximated to eliminate costly matrix inversions.

Consider a fully implicit representation of the Euler equations

D-n+l - u" aF"+l ap+' +- + - = O At az aY

The implicit flux vectors are now linearized by

Gn+' 2 G" + (E) ( U"+' U") = C" + R"6O-n

These flux vectors are then split with a modified Steger-Warming approachI4

F = F+ + F- A = A+ +A.

Thus, the implicit finite volume formulation is

Tlie jacobians appearing on the left side of (S) can now he approximated. These approximations are

Wit11 this approacli, tlic required dat;L for each sub- itrmtion have been calculated in the previous s u b iteration. This allows for efficient operat,ion as only

A + = ? ( A + ~ ~ I ) ~ - = , ( . ~ t - ~ , , r ) nearcst-neighbor commnnication i s rcmiired. iiddi- 0 I I 1 1 -

tionally, the implicit flus vectors can he calculatcd and then communicated. This e1iminati:s tlie nerd to coin- municate the entire flux jacobian.

where p,, is the spectral radius of A , /u/ + a , and (1 is the local speed of sound. Then the differences hctween jacobians become, for example

A+ - A- = pa I = (1.1 + a ) I

Moving the remaining approximated jacobians to the right side produces

1 + B;i,;- Sj,j + 6Ut j -1 - Ei,; + f Si,j+ + 6Ui':j+1

where A*, = v p ~ . The solution to (9) i s straight- forward. On a scalar or vector computer (9) is solved using two sweeps. The first from the lower right to tlie upper left solving for 6M" by ignoring tlie negat,ive flux jacobians. Then 6U is found by sweeping from upper right to lower left with 6M*, ignoring the positive flux jacobians.

Solution of (Q) on a massively parallel computer rc- quires a modification to the LUSCS algorithm. Us- ing sweeps on this type of computer is ineficicnt be- cause only a small fraction of the processors would be busy during the evaluation of the off-diagonal terms. The data-parallel lower-upper relaxation (DPLUR) method" modifies the LUSGS algorithm in the follow- ing fashion. Instead of sweeps the equations are solved with a series of sub-iterations. These sub-iter a t ' ions allow the computations to be performed in an almost perfect data-parallel fashion. First, the right side of (8) i s pre-conditioned to obtain 6U(").

(9)

.,,

Then the k,,, sub-iterations are made using

Mesh AdaDtation Consider a t' ions Tlie T V D scheme, parallel computing power, and

implicit method combine to allow lor accurate and ef- ficient computations. It is desirable, however; to i l l -

crease accuracy by placing more ct:lls near important areas of the Row, (eg shock waves) and less in relatively iinchanging areas. This non-uniform distribution can be obtained with an a-priori knowledge of the flow so- lution, but such knowledge is nsually not available. A more applicable method is the solution adaptive mesh generation algorithm of Benson and McRae". In this method, new locations are determined for the cell vertices based on weighting functions. T h e weight- ing functions are determined from the strength of flow gradients. The flow gradients come from an interme- diate flow solution.

This mesh adaptation algorithm is preferable over others because it i s simple to employ, is easily adapt- able to parallel computations, and preserves tlie struc- tured mesh required with the implicit algorithm. How- ever, this portion of the method i s still under devel- opment. Efforts are focused on reduction of inter- processor communication. No solutions using this por- tion of the method are included here, hut the process may become necessary in the future if increased accu- racy i s desired.

d

Results The data-parallel T V D method has been imple-

mented to solve several simple two and three dimen- sional flows for which wind-tunnel da ta is available. Comparisons with this data will determine the valid- ity of the method. Performance and accuracy issues

for k = 1, k,,,, arc also addressed in a qualitative rashion with this preliminary data. 6U$) = { I + A;I + A;I}:l{AUi,j 1 7

.I

Flow Calculations. At *"@-c & " W l ) +G(A:i-+, js i -+, j . - l , j - A?i++,jSi++,j ,+I , ; Supersonic flows over three sirnple geometries have

heen calculated with the TVD method. These geome- tries were also tested in wind-tunnels in references I F and 17. The first of these geometries, model 7 in ref- erence 17, is a cone-cylinder. Tlie Mach number for t,liis case was 2.96. The second geometry, model 'D'

S. . ,&U(!-') - B" S. 6 ( " k L ) ) } , B?i,j-; % , I - - - ~ *.I-1 - i , j + + % , I + + ' i , I t l

then

6f,rtT1 = 611(!*""") i ,I

( L O )

W

in reference 16, is a simple parabolic body of revolu- tion. T h e third geometry is that of a thin symmetric wing, model 'C ' in reference 16. The second and third cases were run at a Mach number of 2.01. In an at- tempt to more closely approximate the wind-tunnel conditions, stings were approximated on the first two models. I%owever, extensive information on the shape and size of the stings was not available, so the repre- sentation is only an approximate one.

Pressure signatures from the flows arc compared with those measured in experiments. Figure 1 shows the nomenclature used in subsequent plots. EIere, ,8 is tlie Mach angle, G. The pressure is measured on a stationary boundary layer bypass plate, while the model is moved in the flow field.

t Model

\

Year- and Mid-ficlil prcssure contours for the flow are plotted in figures 4. Thc bow shock and cxpansion region can he seen to be sharply resolved he the TVD method. In t h mid-field the shock waves can Re seen t,o 1 x 5 propagating a t the Mach an&.

\ I Figure 2a . Near-field symmetry planes of the 32 x 256 x 12 computational m e s h

Boundary Layer Bypass Plate

Figure 1. Measurement nomenclature.

The first model tested was . the cone-cylinder. T h e reference length of the conical forebody wns 2 inches, while the cylindrical afterbody approximated thc wind-tunnel sting. Figures 2 show two views of the computational mesh about the model in the near- field.

Pressures were calculated at five body lengths nor- mal to the body and compared with those found experimentally. The results are shown in figure 3 . Two different compiitational grids were used in the calciilations. It is evident from the figure that the 128 x 236 x 12 grid provides suitahle accuracy, while t,hc 3'2 x 2.56 x 12 grid is not fine enorigh to xesolvr the flow accurately. Deviation from the expcrimentally measwed pressures downstream is most, likely due i o the inaccuracies involved in niorleling the shape and size of tlic wind-tunnel sting.

1

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Figure 2b. 256 x 12 compritational mesh

Near-field cross-flow planes of the 32 X

Thc DP-TVD method W R S also used to calculate the flow about a parabolic hody of revolution, model 'D' in reference 16. Computational grids of varying sizes W E ~ P iised t,o determiiie t,he iieitr- and micl-fidil pressure signatures. Pressure contours iii the sy inme t r y planes are shown for Llacli 2.01 Now in figuces .5 A 125 x 256 x 12 mesh was used in these coinputa- tions. The approximation to the windMunnel sting is evident in the figiires. The sting was approximated as one body length (2 inches) behind the hody nnd 0.25

F A 111 A 95- O R 3 3

inches in diameter. This is most, likely not an effective approximation. Placing more cells in the region of the sting with the solution adaptive mesh algorithm would help in resolving the the Row more effectively.

A " ' " 1 0040

(X-PRIIL

Figure 3. Experimental and computed pressnre sig- natures for the cone-cylinder body at 5 cone lengths normal to the body.

Pressure signatures were extracted from the flow field at four locations normal to the body axis. These signatures were then compared to the experimental da ta of reference 16. This data is presented in figures 6. It is evident that the data-padlei T V D scheme adequately resolves the shock structure in the near- field for a relatively small amount of cells in the flow direction. 32 to 64 cells seems to adequately resolve the bow shock and expansion region. However, in the mid-field, a finer mesh is required. The best signature resolution came with the use of a 256 x 256 x 12 mesh. Even with this fine mesh there is still difficulty resolv- ing the flow in the region of the wind-tunnel sting, but

cylinder a t M = 2.96. 125 x 266 x 12 mesh. cylinder a t M = 2.96. 128 x 256 x 12 1nts11.

this appears to he due to the approximate nat,ure of the sting definition in the calculations.

The last flow tested was that of a thin, symmet- This case

was used to test the DP-TVD method for fully tlirce dimensional Row. Due to the dimculty in mesh gener- ation, no approximation to the wind-tunnel st,ing was made for this flow case. Figures 7 show near- and mid-field pressure contours on the vertical symmetry planes. Excellent shock definition is again shown for both cases.

Pressure signatures are again compared with exper- iment for this case. Figure S shows the experimen- tal and computed pressure signatures in the mid-field. The solution, calculated on a 125 x 2.56 x 22 mesh. adequately predicts the amplitude of the pressure dis- turbance, but does not capture the shape as well. This is most likely due to the lack of an approximation to the wind-tunnel sting. Viscous effects may also effect the signature.

These three sets of calculations have shown the DP- T V D scheme to adequately predict the near- and mid- field pressure signatures of vehicles moving at super- sonic speeds. Judicious mesh sizing and generation are essential for accurate results. A solution adaptive mesh portion of the method will improve accuracy, but is still in development.

In the future, other additions will be made to the current method so that it may calculate viscons and possible chemical reaction effects. These effects domi- nate hypersonic re-entry flows, and will be needed in flow predictions of a future aerospace plane or hyper- sonic transport. This met,hod should work well for the calculation of other supersonic and trimsonic flows as well.

ric wing; also a t a Mach number of 2.01. W'

4

Figure 4s. Near-field pressure contours for thi: cone- Figure 4b. Mid-field pressure contours for the cone

W

Figure 5a. Near-field pressure contours for the Figure 5b. Mid-field pressure contours for the parabolic projectile a t A4 = 2.01. 128 x 256 x 12 mesh. parabolic projectile at A4 = 2.01. 125 x 256 x 12 mesh.

6 4 x 126 x 12Gnd 128 x 2 5 6 x 12 Grid 256x256x 12Gnd

....__ 02

0 1

,, . e . . . . .\ Aplp 00 ~ . .

' . .\,

02 -

64~12Ex12Gnd 126x256 x 12 Grid 2561256 x 12 Grid

..' ... , - \. j . , \\. I . .

* ,

-0 2 1 0 2

(X-pWL

Figure 6a. Experimental and computed pressure sig- natures at H/L = 1.0. for the parabolic projectile.

Figure 6b. Experimental and computed pressure sig- natures at H/L = 2.0 for the parabolic projectile.

010 , , , , , , , , , , Experiment 6 4 x 126x 12Gnd l 2 6 x 2 5 6 x 12 Grd

~ 256xZ56x12Grid

......

.

005 -

-0.10 0 2 'd V-DW

Figure 6c. Experimental and computed pressure natures a t H/J, = 4.0 for the parabolic projectile.

0.06 , . I

-0.06 " " " " ' I " " ' 0 2 - 1 , -2

V -PW

Figure 6d. Experimental and computed pressure sig- natures at H/L = S.0 for the parabolic projectile.

Figire 7a. Near-Field Pressure contours for thc thin wing. 128 x 256 x 22 mesh at 12.I = 2.01.

IX-PRY

Figure 8. Experimental and computed pressure signa- tures a t A/L = 8.0 for the thin wing.

Performance Issues. T h e DP-TVD method has been shown to ade-

quately predict the pressure signatures of supcrsonic Rows, but the method must have excellent performance qualities for it to replace existing prediction methods. Some of the performance issues involved are discussed here in qualitative fashion.

Foremost of the performance issues involved is code size. T h e size of the code will determine the amount of cells that can be used in the computational domain. Since more cells leads to increased accnracy it is desir- able to use as many as allowed hy the availablc nreni- ory. Currently, a 256 x 256 x 12 mesh uscs approx- iinatcly 750 Mhyte, and fits easily on the CAI-5’s 64 processor partition. Thus, a mesh with over 6 million cells, such as a 512 x 512 x 24 mesh, could bc cm-

Figure 7b. Mid-Field Pressure C:ontours for the thin wing. 128 x 256 x 12 mesh at )\I = 2.01.

ployed on the 512 processor partition. This is a con- servative estimate. With a dedicated partition even larger codes can be run. T h e available memory allows for considerably more cells than can currently he used with traditional prediction methods. Use of the solu- tion adaptive mesh algorithm will use more memory, but will increase accuracy and shock capturing effec-

High floating point performance is also required of the new prediction method. Computational simula- tions are considerably more cost effective than wind- tunnel measurements, but an effort must bc made to reduce the amount of time required for an adequat,e prediction. Thus, computational efficiency is essential. Thc program comput,ed on a 128 x 256 x 12 mesh at, approximately 1.7 Gflop on the C$l-5’s 64 processor partition. Similar values were obtained with similar meshes on 32 processor. If lhe problcm size is increased accordingly, the floating point performancc scales with an increase in the number of processors utilized. Thus, a performance of approximatcly 14.0 Gfiop would he expected on the 512 processor partition.

Codes of the size presented here usually convergc to machine zero in several thousand iterations. This tnlies about 30 to 45 minutes oil a 32 or 64 processor partition, depending on the load level. Total CPU time is usnally much less than this, 011 thc order of 10 to 20 minutes. This is far faster lhan most codes used i n current prediction methods.

It should he noted here that tlic soliltion adaptive mesh portion of the program will dccrease performance considerably. This is due to thc extensive amount of

tiveness. u,

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A I A A 9.5-0833 9

inter-processor communication that is required with the algorithm. In the future efforts will he concen- trnted on reducing this communication as much as

30, No. 3, May-June 1993, pp. 309-014. 6 Sic-ari, M., and Darden, C . , “An Euler Code Predic-

tion ofNear to Mid-Field sonic B~~~ pressure sig. e/ possible. natures,” A I A A Paper No. 90-4000, October 1990.

Concluding Remarks A new data-parallel total variation diminishing

method for sonic boom calculations was introduced. This new method was shown to produce good results efficiently for several sonic boom test cases. This method has potential to he used as a primary tool in the evaluation of new HSCT configurations. Vis- cous and chemical reaction terms can he added to the method to account for those effects when calculating the higher Mach number flows associated with research in hypersonic flight.

The DP-TVD method also has the ability to be utilized for the calculation of other types of supersonic and transonic flows.

Acknowledgments Support for this research was provided by the Army

Research Office contract number DAAL03-0038 with the University of Minnesota Army High Performance Computing Research Center (AHPCRC) and the DoD Shared Resource Center a t the AHPCRC.

This work was also supported by allocation grants from the Minnesota Supercomputer Institute.

The authors wish to thank Dr. Graham Candler and Michael Wright for their assistance in developing the implicit portion of the code.

w

References

Darden, C.M., and Shepherd, K.P., “Assessment and Design of Low Boom Configurations for Super- sonic Transport Aircraft,” DGLR/AIAA Paper No. 92-02-063, May 1992. Plotkin, K.J . , “Review of Sonic Boom Theory,” A I A A Paper N o . 89-1105, April l9S9. Whitham, G.B., “The Flow Pattern of a Supersonic Projectile,” Commzln. Pure 63 Appl. Malii., Vol. V , No. 3, Aug. 1952, pp. 301-348. Walkden, F . , “The shock Pattern of a Wing Body Comhination, Far From the Flight Path,” Aeronaut. Q., Vol. IX, P t . 2 , May 1958, pp. 164-194. Darden, C.M., “Limitations of Linear Theory for Sonic Boom Calculations,” Journai of Aircrafl, Vol.

Cheung, S.H., Edwards, T .A. , and Lawrence, S.L., “Application of CFD to Sonic Boom Near and Mid Flom-Field Prediction,” Journal of Aircraft, Vol. 29, No. 5, Sept.-Oct. 1992, pp. 920-926. Siclari, M., and Darden, C., “CFD Predictions of the Near-Field Sonic Boom Environment of Two Low Boom IISCT Configurations,” A I A A Paper No. 91-1631, April 1991. Harten, A , , “On a Class of High Resolution Total- Variation-Stable Finite Difference Schemes,” SIAM J . Num. Anal., Vol. 21, pp. 1-23, 1984.

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l5 Benson, R.A. and McRae, D.S., “A Solution Adap- tive Mesh Algorithm for Dynamic/Static Refine- ment of Two and Three Dimensional Grids,” Pro- ceedings: Numerical Grid Generation in CFD and f i e l a l e d Fields, Barcelona, Spain, .June 3-7 1991, North Holland, pp 185-200.

l6 Carlson, H.W., “An Investigation of Some Aspects of the Sonic Boom by means of Wind Tunnel Mea- surements of Pressures ahout Several Bodies a t a Mach Number of 2.01,” NASA TND-161, Dec. 1959.

l7 S h o u t , B.L., Mack, R.J . , and Dollyliigh, S.M. , “A Wind Tunnel Investigation of Sonic-Boom Pressure Distrihutions of Bodies of Revolution a t Mach 2.96, 3.83, and 4.F3,” iVASA TND 6195, April 1971.