PREDICTION OF SONIC BOOM SIGNATURE USING EULER-FULL POTENTIAL CFD WITH GRID ADAPTATION AND SHOCK FITTING*

O. A. Kandil 1 and Z. Yang 2, Old Dominion University, Norfolk, VA 23529and

P. J. Bobbitt 3 Eagle Aeronautics, Inc., Hampton, VA 23666

Abstract

A computational fluid dynamics (CFD) methodology has been formulated and a prototype computer code has been developed to propagate Euler-equations-based, near-field supersonic pressure calculations to the ground through a real atmosphere. This near-field supersonic flow is propagated to the ground using the full-potential equation. The scheme is called the Euler-full potential (EFP) scheme. Once shocks are captured, a grid adaptation scheme, based on the density gradient, is implemented to obtain a crisper shock. This is followed by a shock-fitting scheme that is based on the Rankine-Hugoniot conservation equations. Computational results for uniform (isothermal) and non-uniform (standard) atmospheric conditions are presented. For computational validation, the far-field results of the Euler equations for a two-dimensional biconvex airfoil are compared with the results of the far-field potential equation. The results on the ground are in excellent agreement with each other. The three-dimensional capability of the code is illustrated by calculations of on and aft track ground pressure distributions for a three-dimensional delta wing cruising at 52,000 feet altitude and at a Mach number of 2.01.

SymbolsC wing or airfoil chordH distance below source locationM∞ Mach numberp static pressureq∞ dynamic pressurer radial distance from aircraftS wing areaU, V,W longitudinal, lateral, and vertical velocity X longitudinal or streamwise coordinate _____________________________________*Work performed under Lockheed-Martin contract Number EM1424560E to Eagle Aeronautics, Inc.1 Professor & Eminent Scholar, Associate Fellow AIAA.2 Graduate Research Assistant, Member AIAA3 Director of Aerodynamics, Fellow AIAA Copyright 2002 by O. Kandil. Published by the AIAA, Inc., with permission

Y lateral distanceZ vertical distance CD drag coefficient = drag/ q∞SCL lift coefficient = lift/ q∞SCP = ∞∞− qpp /)(

∆p ∞− ppρ Ambient density

φ Velocity potentialAbbreviationsRMS root mean squareCFD computational fluid dynamicsEFP Euler-Full PotentialGASF grid adaptation and shock fitting

Subscripts∞ freestreamx, y, z derivative relative to these coordinates

ζηξ ,, derivative relative to these coordinates

Introduction

A number of methods and computer codes are available for predicting the evolution of supersonic aircraft’s pressure signature as it propagates to the ground through a “real” stratified atmosphere 1- 8.The most utilized sonic boom propagation codes are based on linear acoustic ray tracing and the Blokhintsev invariant. If, in addition, equivalent axisymmetric-body, F-function methodology is utilized, then the code developed by Hayes (ref. l) is employed. Where a pressure distribution is available, then the Thomas code (ref. 6) is the preferred approach. This method is also based on the acoustic ray tracing methodology but employs several waveform parameters to describe the evolution of the pressure wave. It is this latter code that is normally used in combination with a near-field Euler CFD calculation to propagate sonic boom signatures to the ground.

8th AIAA/CEAS Aeroacoustics Conference & Exhibit<br> <font color="green">Fire17-19 June 2002, Breckenridge, Colorado

AIAA 2002-2542

Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

One of the problems in matching (or connecting) the Thomas propagation code to an Euler near-field solution is that the Thomas solution in the near-field

has a r

1 pressure dependency (r is radial distance).

Near-field Euler solutions generally do not reach this "asymptotic" state at the outer boundary of the calculated flow field where the match is made. Consequently, there is an error introduced. One improvement is provided by the introduction of a mid-field pressure propagation routine that more exactly matches the Euler solution rate of radial

decay as well as the r

1decay at the other

boundary where the Thomas waveform code is implemented.

Reference 11 describes a modified propagation code based on the multipole linear scheme of George (Ref. 12). While this is clearly a more accurate method to propagate the pressure signature than connecting the ray tracing code directly to the CFD solution, there is still a concern. If the linear multipole solution is matched to the CFD solution where there are still significant nonlinear effects, then one would expect some error to creep in as the multipole solution is propagated to the far-field. Clearly, the further one can take the CFD solution (larger radial distance) into the mid-field, the moreaccurate will be the multipole solution. It will also be easier to match to the CFD pressures with the multipole solution. While the multipole solution can be matched to the CFD pressures, no comparisons were given in Ref. 11 of the radial pressure gradients, which are intrinsic to the multipole method, with those at the CFD boundary. In addition, results were only given for a simple triangular wing. Since the publication of Ref. 11 the multipole method has been used for a wide range work and gradually thought to be an improvement over the direct coupling of near and far-field solutions.

The precision with which a complex near field solution with the nonlinear effects and radial gradients can be matched and propagated can still be improved. Use of a higher order methodology in both the near field and far field is feasible and desirable. Ideally, this "higher order" solution would be matched to the near field solution point for point including radial gradients and shock jumps. The study described herein provides such a solution.

In this paper, a computational fluid dynamics (CFD) method and a solver have been developed to accurately compute the propagation of an aircraft's sonic boom pressure wave through a real atmosphere. The near field flow is predicted byusing the Euler equations while the far-field flow is predicted by using the full-potential equation. This scheme is called the Euler full-potential (EFP) scheme. Once shocks are captured, a grid adaptation scheme, based on the density gradient, is implemented to obtain a crisper shock. This is followed by a shock-fitting scheme that is based on a searching algorithm and the Rankine-Hugoniot conservation equations of mass, momentum and energy equations. The repetitive solution cycles of the grid-adaptation and shock-fitting (GASF) schemes minimize the root mean square of relative errors percentage in the mass, momentum and energy equations across the shock.

The methodology of EFP and GASF has been applied to a biconvex airfoil with a 5 percent thickness ratio and to a 60 degree Delta wing with a 5 percent thickness ratio thickness ratio biconvex airfoil section. The wing has a chord length of 50 feet and is at 52,000-foot altitude, a Mach number of

2.01 and an angle of attack of 0°.

Near Field Calculation

As noted in reference 13, "If one calculates the near field of an aircraft using a CFD Euler code, the shock jumps are consistent with the Rankine-Hugoniot equations. With this "constraint" there are shocks of different shape and strength followed by pressure recoveries of different strengths (rates of recovery). Normally CFD calculations are made with grids that are stretched radially as one moves to the outer boundaries, since the flow gradients become smaller and far-field boundary conditions can be imposed. Outer boundaries are typically placed one to two body lengths from the centerline depending on the Mach number and shape. Solution schemes are generally of the shock capturing variety. All of this combines for most solution schemes, to smear the shocks and reduce their maximum pressure jumps from their proper value at the outer boundary." Reference 13 further stated "That shock-fitting schemes are more appropriate and have been available for many years. However, they are not normally used or needed where the primary interest is the pressures on the configuration itself. Whether a procedure can be formulated to recover an

"accurate'' finite shock jump at the near field boundary from a smeared one is not clear. Lacking a "shock fitted" Euler (or Navier-Stokes) code it is desirable to utilize a grid that conforms as close as possible to the shocks and to have as high a concentration of grid points in the vicinity of the shocks as can be accommodated. The latter is desirable since shock jumps are usually resolved in three or four grid points. The use of a grid adaptation scheme in an Euler code with a shock capturing solution algorithm is one way of further improving resolution. The combination of shock capturing and shock fitting is the preferred or ideal approach."

In the present study an early version of NASA Langley's CFL3D code was modified to include both grid adaptation as well as shock fitting. Shock fitting yields the most accurate flow solution available. It provides a procedure for explicitly computing the jump conditions across shocks. It locates the shocks and treats them as boundaries between regions where the solution is “regular.” Moretti developed a method known as floating shock fitting where shocks "float" during the iterative calculation. A number of papers treating the Euler equations with shock fitting have been published over the years and the technique is fairly well established. Shock fitting has also been applied to various versions of the potential equations for both transonic and supersonic flows (Refs. 14 and 15).

Adaptive Grid

In order to have an accurate shock jump calculation, using a shock-fitting technique, the shock jump should first be resolved as accurately as possible using the basic solution algorithm. A two-step adaptive- grid scheme has been employed, along with the pseudo-time shock-capturing solution methodology of the basic code to solve for the flow about a simple slab-sided delta wing. Its effect on the shock jumps, lift, drag, and solution convergence has been examined. Figures 1 and 2 show pressure distributions on the centerline and near the tip of a slab-sided delta wing at a Mach number of 2.01 and an angle of attack of 5 degrees. Results are given for the original grid and after the first and second grid

adaptations. Figure 3 shows color contour plots of the pressure field for the same 3-grid configurations. As in Figs. 1 and 2, there is a significant difference between the original grid and the first grid adaptation and not much difference between the first and second. Of equal importance is the effect of grid adaptation on convergence. Figure 4 shows that once the grid is adapted, lift and drag converge after only 30 or 40 time steps; several hundred were needed for the fixed grid.

Shock Fitting

The three-dimensional shock fitting is based on the pseudo-time captured shock with adaptive grid. First, the captured shock is located using a searching algorithm. Once the shock is located, shock-fitting equations are implemented. Shock-fitting equations are based on the Rankine-Hugoniot conservation equations of mass, momentum and energy. The equations are used to fit the three-dimensional shock surface. Next, the grid is adapted around the fitted shock surface and the flow is recomputed. This cycle is repeated and the errors in mass, momentum and energy are monitored until they reach certain minima.

Figure 5, for the same slab-sided delta wing of the earlier figures, shows a comparison of CL and CD histories for the original grid case, the adaptive grid-only case, and the shock-fitting along with the adaptive grid. It is observed that the latter technique requires about 25 iteration cycles to obtain the shock. The adaptive grid requires about 45 cycles and the original grid requires about 65 cycles. Figure 6 shows comparisons of the relative errors in mass, momentum and energy across the projected shock surface. The figures show the relative errors as viewed by an observer looking upstream at the shock surface. It is clear that shock fitting coupled with an adaptive grid technique produces the minimum relative errors. The root-mean square of the relative errors in mass, momentum, and energy for adaptive grid-only and shock fitting with grid adaptation for 1 of 5 cycles of iteration is given in the following table.

Mass. Momentum EnergyAdaptive 4.86E-002 3.00E-002 7.93E-003Fitting (1 iteration) 2.65E-003 1.54E-003 6.49E-004Fitting (5 iterations) 1.24E-003 1.13E-003 3.94E-004

Figure 7 shows a comparison of the root-mean square of the relative errors in mass, momentum, and energy or the adaptive grid-only and the shock fitting with grid adaptation for several iterative cycles. Figure 8 shows a comparison of shock thickness as obtained by grid adaptation only and shock fitting with grid adaptation.

Full Potential Equation Solution for Flow Field Propagation

The full potential equation is used to propagate, to the ground, the near field signature calculated using the Euler equations. Since the Euler equations solution does not exactly satisfy the potential equations, a methodology is formulated to determine the potential and its derivatives at the interface between the Euler and potential "domains." The conservative form of the potential equation is solved using a space marching, upwind scheme (see Refs. 15-19). This scheme is "augmented" by a sub-block technique, which facilitates the treatment of the varying speed of sound.

Details of the Solver

Governing Equation

The 3D full-potential equation written in conservation-law form is given by 15

0)()()( zzyyxx =φρ+φρ+φρ (1)

[ ] 11

2z

2y

2x

22

1 1(M1 −γφ−φ−φ−+=ρ ∞−γ

In computational domain, it can be written as

0J

W

J

V

J

U =

ρ+

ρ+

ρζηξ

(2)

where

φφφ

=

ζ

η

ξ

333231

232221

131211

aaa

aaa

aaa

W

V

U

(3)

2z

2y

2x33

zzyyxx3223

2z

2y

2x22

zzyyxx3113

zzyyxx2112

2z

2y

2x11

a

aa

a

aa

aa

a

ζ+ζ+ζ=

ζη+ζη+ζη==

η+η+η=

ζξ+ζξ+ζξ==

ηξ+ηξ+ηξ==

ξ+ξ+ξ=

(4)

ζζζηηηξξξ

=∂ζηξ∂=

zyx

zyx

zyx

)z,y,x(

),,(J (5)

Space Marching Scheme

a. Treatment ofξ

ρJ

U

Upwind scheme is used. Consider the direction ξ to be the marching direction. Since ρU is a function of φ and the unknown (ρU)i+1 , the latter can be expanded as

( )

+ζ∂φ∆∂

−+

η∂φ∆∂

−+ξ∂

φ∆∂

−

ρ≈ρ +

i

i213

i212

i

2

2

11

i1i

U)(

a

UWa

)(

a

UVa

)(

a

Ua

U (6)

where i1i φ−φ=φ∆ + . Using upwind differencing

})()({1)(

i1i −ξ∆=ξ∂∂

+

this approximation produces the positive artificial viscosity

ξ∆φ

−ρ

ξξξ2

211

2

2U

U

aa1

Ja if 112

2

aa

U >

b. Treatment ofη

ρJ

V

The 2nd order central differencing is used here. A second-order finite-difference approximation is given by

21

21 jj J

Vˆ

J

Vˆ

J

V

−+η

ρ−

ρ=

ρ(7)

The artificial density is given by

)()1(ˆ m21jm2j21

jj 21

21 +−+++ ρ+ρν+ρν−=ρ (8)

where

>−−>−

=

<>

=01

00

0,1

0,0

2

2

222

2

2

2

22

22

aV

V

aaaV

afor

afor

V

Vm νThe treatment of the term of

ζ

ρJ

W is similar to

the one described above.

Implicit Algorithm

A fully implicit model will be

RJ

a

J

aA

J

a

J

aA

=∆

∂∂

∂∂

+

∆∂∂

+∂∂

∆+

∂∂

∂∂

+

∆∂∂

+∂∂

∆+

φηρ

ηβξρ

ηβηξβ

ζρ

ζβζρ

ζβζξβ22212

33313

ˆ1ˆ11

111

(9)

where

21

2133

21222

2

111

)(,

,,

ξβρ

ρρ

∆=

−=

−=

−=

A

a

UWa

JA

a

UVa

JA

a

Ua

JA

i

i

i

i

i

i

(10)

Ref. 15 gives the detail of equations and solution

method.

Interface Conditions Between Euler Solution and Full Potential Solution

In the sonic boom computation, the near field is calculated using an Euler solver (CFL3D) and the far field is calculated using the developed full potential solver. At the interface, the velocity components (u, v, w) of the Euler solution are transformed into a velocity potential that is used for the initial condition of the full potential solver. The figures and the steps below give the detailed procedure:

Step A. Cut planes at x2X,xX,Xx 000 ∆∆ ++= from 3D Euler solution and obtain the velocity (u, v, w) on the cutting planes by interpolation.

Step B. Generate a new grid for the full potential solver. The new grid is totally independent of the grid used in 3D Euler solution. But the first three planes of the new grid are at the same locations of the cutting planes of the Euler grid

x2X,xX,Xx 000 ∆+∆+= , respectively. Then

the velocity is interpolated from the cutting planes to the first three planes of the new grid.

Step C. For the first three planes in the new grid, calculate the velocity potential φ from velocity (u, v, w). This is accomplished by rewriting the full potential equation in terms of ζη φφ , as unknowns

on the left hand side and the remaining terms as source terms and the equation is solved using an ADI scheme. Once φ is obtained on the third plane, backward differencing is used to obtain φ on the second and first planes. The details of the procedure are given below: Rewrite full potential equation, Eq.(2)

( )

( ) RaaJ

aaJ

=

+∂∂

+

+∂∂

ζη

ζη

φφρζ

φφρη

3332

2322

(11)

where

( )

( ) ( )

∂∂−

∂∂−

++∂∂−=

ξξ

ζηξ

φρζφρ

η

φφφρξ

3121

131211

aJ

aJ

aaaJ

R

(12)

and R can be calculated using ζηξ φφφ ,, , which can

be derived from u, v, w by

=

φφφ

ζηξ

ζηξ

ζηξ

ζ

η

ξ

w

v

u

zzz

yyy

xxx

(13)

For the third plane in new grid, Eq. (11) can be discretized using a 2nd order central differencing scheme. For the 2nd and 1st planes in new grid, φ can be obtained by

ξ∆φ−φ=φ ξ+1ii (14)

Because only φ at 3rd plane and φζ at 2nd plane will be used in full potential solver, the method above should produce the correct initial condition for full potential space-marching scheme.

Interface Conditions for the Sub-Blocks

The atmospheric conditions are varying continuously with the altitude. To simulate the real flow, a sub-block technique has been developed to account for this variation. In each sub-block, flow is assumed as uniform flow and the free-stream conditions are determined from the altitude conditions at the

midpoint altitude of this sub-block. To transfer the shock from one block to other block, the pressure jump introduced by shock has been kept the same across the interface. This means that

*B

*B

*A

*A PPPP ∞∞ −=− (15)

where A and B refer to the sub-blocks on the two sides of interface. Consider

[ ] 12z

2y

2x

22

1 1(M1P

P −γγφ−φ−φ−+= ∞

−γ

∞ (16)

Equation (15) can be rewritten as

StepA:Cut planes from Euler solution and interpolate velocity (u,v,w) to slices

Step B: Interpolate velocity (u,v,w) from the slice plane to new grid

Step C: Transfer velocity (u,v,w) to velocity potential φ

( )[ ]

( )[ ]

−−−

=

−−−

−∞−

∞

−∞−

∞

11~

1

11~

1

1222

1*

1222

1*

γγγ

γγγ

BBB

AAA

VMP

VMP

(17)

where *AP ∞ and *

BP ∞ are the dimensional free-stream

pressure at sub-blocks A and B, respectively; 2AV

~ and

2BV

~ are total velocities at sub-blocks A and B. If

2AV

~ in sub-block A is known, 2

BV~

can be calculated by using Eq. (17).

Two types of interfaces are used: one is the interface in marching direction (Interface Type I) and the other is the interface in the perpendicular direction (Interface Type II).

Interface Type I

For Interface Type I, the flow is supersonic and the direction of flow is perpendicular to the interface. Iteration is not needed here. Assume the sub-block A is the upwind block and the sub-block B is the downstream block. To satisfy Eq. (15), the total

velocity 2BV

~, which is calculated from velocity

potential Bφ , must satisfy Eq. (17), where Bφ is

derived from Eq. (17) and Aφ is known.

There are three planes in this interface. First, assume

AB φ=φ at the 3rd plane. Considering Eq. (3), we have

)(aWW

)(aVV

)(aUU

AB31AB

AB21AB

AB11AB

ξξ

ξξ

ξξ

φ−φ=−φ−φ=−φ−φ=−

(18)

at plane No.3. Then, by using

ζηξ φ+φ+φ= WVUV~ 2 ,

we can find

)()(

~~

3121

22

ξξζξξη

ξξφφφφφφ

φφABAB

AABBAB

aa

UUVV

−+−+−=−

(19)

Let ξξξ φ−φ=φ∆ AB , Eq. (19) becomes

0)V~

V~

(U2a 2B

2AA

211 =−+φ∆+φ∆ ξξ (20)

Solving for ξφ∆ and considering 2B3BB φ−φ=φ ξand 2A3AA φ−φ=φ ξ , we can get

ξφ∆−φ=φ 2A2B (21)

where 2 and 3 represent planes No. 2 and No.3, respectively. Similarly, 1Bφ can be obtained.

Interface Type II

For Interface Type II, the flow perpendicular to interface is subsonic flow and iteration is needed. This means that after sub-block A is done, velocity potential is re-created based on sub-block A and Eq. (17). These values are used as boundary conditions for sub-block B. After sub-block B is done, using the same procedure, this process is iterated until the error is less than a prescribed tolerance.

The procedure of re-creating velocity potential is:

First, Eq. (17) is applied to get total velocity 2BV

~ at

the interface. Second, the velocity component directions are assumed fixed. Thus, the velocity components are

2A

2B

ABV~V~

uu = , 2A

2B

ABV~V~

vv = , 2A

2B

ABV~V~

ww =

Then doing similarly to what have been done in Eq. (11)– (14), the velocity potential at interface is obtained.

Results for 2D Biconvex Airfoil

Propagation calculations were first carried out for two-dimensional flow. They were done to establish the grid adaptation and shock-fitting techniques, developed for the Euler equations, for the potential equation formulation. The geometry that was used is a biconvex airfoil with t/c of 5 percent with a 50 feet chord. The match between the Euler and potential equation solutions for the potential propagation was made at 3 chord lengths (150 feet) below the wing. The angle-of-attack was zero degrees and the Mach number is 2.01. The calculation was started at an altitude of approximately 52,000 feet. Both Euler and potential-equation propagations were done to "check" the accuracy of the potential equation results. The Euler calculation is, of course, very time consuming but doable in two dimensions. No such check would be practical for the 3D propagation. Finally, propagation calculations were made for both uniform (isothermal) and non-uniform (standard) atmospheres.

Figure 9 shows the results for a uniform (isothermal) atmosphere. It illustrates the grid, density contours and pressure distribution for both the Euler and potential calculations side by side. The results are for 9.8 miles below the airfoil (-52,000 ft.). The initial pressure jump at this distance for the Euler calculation is 1.908 psf and that for the potential calculation 1.898 psf. Computation time for the Euler code on a 1.4 GHz PC was approximately 5 days; and for the potential equation, approximately one half hour. Note that no ground reflection factor has been used on the pressure distribution at 9.8 miles. Also, note that the pressures are plotted against vertical distance; consequently, the duration of the pressure signal on the ground cannot be obtained directly from these plots.

Results for the non-uniform (standard) atmosphere are given in Figure 10. The results are for 9.8 miles below the airfoil. The Figure shows that the initial pressure jump for the Euler formulation was 7.142 psf while that for the potential was 7.230 psf. Computation times were only slightly larger than those for the uniform atmosphere.

Results for Delta Wing

The delta wing used in the present calculations had a t/c of 5 percent and a root chord of 50 feet. The angle-of-attack was zero degrees. The Mach number

was 2.01 and the starting altitude was 52,000 feet (9.8 mi). The match between the Euler and potential solutions was made at 1.8 chord lengths (90 feet) below the wing. As with the two-dimensional configuration, both uniform (isothermal) and non-uniform (standard) atmosphere results were obtained and the pressures are plotted against vertical distance. Figure 11 gives plots of the mesh, pressure contours and centerline (on-track) pressure distributions for both of these atmospheres side by side for two distances. Figure 11a is for a distance of 5000 feet (0.9 mi) below the wing; Figure 11b is for 52,000 feet (9.8 mi) below the wing. A plot of the variation of pressure jump with altitude is given in Fig. 12. It is similar in character to the results obtained by wave propagation methods (see Fig. 4 of Ref. 20).

Figure 13 gives the grid and pressure distributions along the ground at the symmetry plane (without reflection factor) for the uniform (isothermal) and non-uniform (standard) atmospheres. It is approximately 4.2 chord lengths long (210 ft.) which converts to 108 milliseconds in duration; consistent with experimental data for an aircraft of this size. Off-track pressure distributions (without reflection factor) are given in figures 14 and 15 for the uniform (isothermal) and non-uniform (standard) atmospheres, respectively. Plots are given for one, two, three, four and five hundred chord lengths (50 feet) off the plane of symmetry. Initial ∆p values are tabled at the end of each figure.

Concluding Remarks

The methodology developed in this study represents a giant step forward in our ability to predict the sonic boom signatures of supersonic configurations. No longer will linear mid-field codes be required to link up with a nonlinear near field code on one side and linear far field code on the other that assumes a cosine-like lateral variation of pressure. The basis has been established to treat complex configurations that attempt to modify the off-track pressure distributions, with much greater precision. Configurations that intuitively ought to have lower booms can now be analyzed with a confidence not heretofore possible.

Acknowledgement

The authors wish to thank the Lockheed-Martin Skunkworks for their support of this research. Work performed under Lockheed-Martin contract numberEM1424560E to Eagle Aeronautics, Inc.

References

1. Hays, W.D., Haefeli, R. C.; and Kulsrud, H. E., “Sonic Boom Propagation in a stratified Atmosphere, With Computer Program,” NASA CR 1299, Aero. Res. Associates. of Princeton, Inc., April 1969.

2. Hayes, Wallace D.; and Runyan, Harry L., Jr., “Sonic Boom Propagation Through a Stratified Atmosphere,” Journal of Acoustical. Soc. of America, Vol 51, No. 2 (part 3), pp. 695-701, 1972.

3. Randall, D. C., “Sonic Bang Intensities in a Stratified, Still Atmosphere,” J. Sound Vibrations, No. 8, pp.196-214, 1968. Also RAE Tech. Report. No. 66002, 1966.

4. Friedman, Manfred P., “A Description of a Computer Program for the Study of Atmospheric Effects on Sonic Booms,” NASA CR-157, MIT, Feb. 1965.

5. Friedman, M. P.; Kane, E. J.; and Sigalla, A., “Effects of Atmosphere and Aircraft Motion on the Location and Intensity of Sonic Boom,” AIAA Jour., Vol. 1, p. 13-27, June 1963.

6. Thomas, Charles L., “ Extrapolation of Sonic Boom Pressure Signatures by the Waveform Parameter Method,” NASA TN D-6832, June 1972.

7. Chambers, James P.; Cleveland, Robin, O.; Bass,Henry E.; Blackstock, David T. and Hamilton, Mark F., “ Comparison of Computer Codes for the Propagation of Sonic Booms Through Realistic Atmospheres Utilizing Actual Acoustic Signatures,” NASA Conference Publication 3335,pp. 151-175, July 1996.

8. Plotkin, K, J.; and Cantril, J. M., “ Prediction of Sonic Boom at a Focus,” Wyle Laboratories Res. Staff Report. WR 75-7, Oct. 1975.

9. Cliff, Susan E.: On the Design and Analysis of Low Sonic Boom Configurations. NASA Conf. Pub. 10133, Ames Research Center, May 12-14, 1993.

10. Siclari, M. J.; and Fouladi, Karoran: A CFD Study of Component Configuration Effects on the Sonic Boom of Several High-Speed Civil Transport

Concepts. NASA Conf. Pub. 10133, Ames Research Center, May 12-14, 1993.

11. Page, Juliet A.; and Plotkin, Kenneth J., “ An Efficient Method for Incorporating Computational Fluid Dynamics Into Sonic Boom Prediction,” AIAA paper 91-3275, Baltimore, MD, September 23-25 1991.

12. George, A. R, “ Reduction of Sonic Boom by Azimuthal Redistribution of Overpressure,” AIAA Paper No. 68-159, 1968.

13. Maglieri, Domenic J.; and Bobbitt, Percy J., “History of Sonic Boom Technology, Including Minimization,” Eagle Aeronautics, Inc. under Lockheed Martin Contract EM1424560E, Nov. 2001.

14. Hafez, M. M.; and Murman, E. M., “ A Shock-Fitting Algorithm for the Full Potential Equation,” AIAA Paper 77-632, 1977.

15. Shankar, Vijaya, “ Conservative Full Potential, Implicit Marching Scheme for Supersonic Flows,” AIAA Jour., Vol. 20, No. 11, pp. 1508 - 1514, Nov. 1982.

16. Shankar, Vijaya and Osher, Stanley, “An Efficient, Full-Potential Implicit Method Based on Characteristics for Supersonic Flows,” AIAA Jour., Vol. 21, No. 9, pp. 1262-1270, Sept. 1983.

17. Holst, T. L and Ballhaus, W. F., “ Fast, Conservative Schemes for the Full-Potential Equation Applied to Transonic Flows,” AIAA Jour., Vol. 17, No. 2, pp. 145-152, Feb. 1979.

18. Holst, T. L., “Fast Conservative Algorithm for Solving the Transonic Full-Potential Equation,” AIAA Jour., Vol. 18, No. 12, pp. 1431-1439, Dec. 1980.

19. Hirsch, Charles, “Numerical Computation of Internal and External Flows,” John Wiley and Sonic, Ltd., Vol. 2, 1990.

20. Thomas, Charles L, “Extrapolation of Wind-Tunnel Sonic Boom Signatures Without Use of a Whitham F-Function,” NASA SP 255, pp. 205-217, 1970

Figure 1. Pressure distribution for slab-sided delta wing along grid line J = 1

determined with and without grid adaptation.

Figure 2. Pressure distribution for slab-sided delta wing along grid line J = 7 determined with and without grid adaptation.

Original Grid

Adaptation 1

Adaptation 2

Figure 3. Pressure contour plots for slab-sided delta wing with and without grid

adaptation.

Original Grid

Adaptive Grid 1

Adaptive Grid 2

Figure 4. Variation of CL and CD with time step for slab-sided delta wing with and without grid adaptation.

Original Grid

Adaptive Grid

Fitting and Adaptive Grid

Figure 5. Comparison of lift and drag coefficient history for slab-sided delta wing with

and without grid adaptation and shock fitting.

Original Grid Adaptive Grid Fitting and Adaptive Grid

Figure 6. Comparison of relative errors in mass, momentum and energy across shock surface For a slab-sided delta wing with and without grid adaptation and shock fitting.

RM

S(

Mas

sE

rror

)

0

0.01

0.02

0.03

0.04

0.05

0.06

Mass Error Comparison between Adaptation and Fitting

adaptation

Fitting

I II III IV V

RM

S(

Mom

entu

mE

rror

)

0

0.01

0.02

0.03

Momentum Error Comparison between Adaptation and Fitting

adaptation

Fitting

I II III IV V

RM

S(

Ene

rgy

Err

or)

0

0.002

0.004

0.006

0.008

0.01

Energy Error Comparison between Adaptation and Fitting

adaptation

Fitting

I II III IV V

Figure 7. Comparison of root-mean square, RMS, of mass, momentum and

energy relative errors

Shock Thickness Comparison Between Adaptation and Fitting

Adaptation

Fitting, 3rd Iteration

Fitting, 5th Iteration

Figure 8. Comparison of shock thickness between adaptive grid only and shock fitting with grid adaptation.

On ground, H = 9.8 mi below the source location

Figure 9.Two-dimensional far field computation using full potential code and comparison with Euler Code for uniform (isothermal) atmosphere at α = 0, M = 2.01, initial altitude 52,000 ft.

On ground, H = 9.8 mi below the source location

Figure 10. Two-dimensional far field computation using full potential code and comparison with Euler Code for non-uniform (standard) atmosphere at α = 0, M = 2.01, initial altitude 52,000 ft.

(a) H = 0.9 mi below the source locationUniform (isothermal) Flow ∆∆∆∆P = 0.90 psf Non-uniform (standard) Flow ∆∆∆∆P = 0.9709 psf

Figure 11. Comparison of uniform (isothermal) and non-uniform (standard) atmosphere sonic boom pressure for a 60-degree delta wing with 5 percent biconvex profile at α = 0, M∞ = 2.01, initial altitude of 52,000 ft

(b) H = 9.8 mi below the source location: Near Ground with No Reflection FactorUniform (isothermal) Flow ∆∆∆∆P = 0.150 psf Non-uniform (standard) Flow ∆∆∆∆P = 0.3285 psf

Figure 11. Concluded.

. Figure 12. Variation of pressure jump with altitude for initial altitude of 52,000 ft., α = 0, M∞ = 2.01, uniform (isothermal) and non-uniform (standard) atmospheres.

Uniform (isothermal) Atmosphere Non-uniform (standard) Atmosphere

Figure 13. Ground pressure distribution and grid blocks for a 60-degree delta wing with 5 percent biconvex profile at α = 0, M = 2.01, initial altitude 52,000 ft. No reflection factor.

Y 0 100 200 300 400 500∆P (psf) 0.15 0.1485 0.1454 0.1411 0.1346 0.1275

Figure 14. Pressure distribution on ground for uniform (isothermal) atmosphere

Y 0 100 200 300 400 500∆P (psf) 0.3446 0.3424 0.3363 0.3271 0.3151 0.3017

Figure 15. Pressure distribution on ground for non-uniform (standard) atmosphere