1

SONIC BOOM

MINIMIZATION OF

AIRFOILS THROUGH

COMPUTATIONAL FLUID

DYNAMICS AND

COMPUTATIONAL ACOUSTICS

Michael P. Creaven*

Virginia Tech, Blacksburg, Va, 24060

Advisor: Christopher J. Roy†

Virginia Tech, Blacksburg, Va, 24060

Abstract

This project analyzes 2-D, inviscid, steady

supersonic flow over different airfoil designs at

Mach 2.2 while at 60,000ft. The airfoils tested

have sharp leading and trailing edges. The shapes

range from diamond to convex to a combination of

the two. A hybridization of Computational Fluid

Dynamics (CFD) and Computational Acoustic

simulations are used to obtain values for the lift

coefficient, drag coefficient, and maximum

overpressure. The trends obtained from this very

specialized case show that flat bottomed airfoils

generate the smallest overpressures, and the

highest lift to drag ratios. The reason for this is

that the thinner shapes create smaller disturbances

in the flow and thus generate smaller shock waves,

which in turn reduce the drag, and the

overpressure. This study does not take into

consideration structural issues, viscosity, differing

angles of attack, or 3-D effects.

* Undergraduate Student, Aerospace and Ocean

Engineering, 125 Lee Hall, mcreaven@vt.edu † Associate Professor, Aerospace and Ocean Engineering, 330 Randolph Hall, cjroy@vt.edu

Nomenclature

𝐿 = chord length

𝑥 = axial coordinate

𝑦𝑢 = maximum distance from the

centerline to the upper surface of the

airfoil with respect to L

𝑦𝑙 = maximum distance from the

centerline to the lower surface of the

airfoil

𝑥𝑢 = location of 𝑦𝑢 from the leading

edge

𝑥𝑙 = location of 𝑦𝑙 from the leading

edge

𝛿 = shape function of the airfoil

𝑃 = free stream pressure

𝛿𝑃 = over pressure

𝑀 = Mach number

𝑇 = temperature

I. Background

The main challenges facing commercial

supersonic flight are the increased drag due to

shock waves and the resulting sonic boom.

Both of these issues need to be addressed and

overcome before supersonic commercial flight

is considered a viable option.

Current Federal Aviation Administration

regulations prohibit commercial aircraft from

reaching supersonic speeds over the United

States. The reason for this ban is the loud

“boom” that is generated by aircraft flying

close to or faster than the speed of sound. As

an aircraft flies through the atmosphere it

creates pressure waves in the air which travel

at the speed of sound and propagate away

from the aircraft. As the speed of the aircraft

increases the distance between waves becomes

smaller. At supersonic speeds the plane is

traveling faster than the pressure waves thus

the pressure waves compress and form a thin

shock wave. The sudden change from high

pressure in front of the shock wave to low

pressure behind the shock wave is what

generates a sonic “boom.”1 The pressure over

2

the ambient pressure is referred to as the

overpressure.

This loud boom can be intense enough to

damage weak structures, as well as cause

significant disturbances in human and animal

populations. It is predicted that federal

regulations will set the maximum allowable

overpressure to 0.4 psf. The first country to

produce an economically viable aircraft that

meets federal regulations will be purchased by

airlines from countries around the world.2

The idea of minimizing a sonic boom has

been around since the 1955. In fact even the

idea of boomless supersonic flight has been

mentioned. There has been much research

done in the area recently, most notably the

shaped nose cone configuration of the F-5E3,

and the Gulf Stream “Quiet Spike.”4 Both of

these were in conjunction with NASA Dryden,

and demonstrated that the sonic boom could

be shaped such that the intensity was

significantly decreased. Experimental tests

and demonstrations such as these are

extremely expensive, and require a great deal

of preparation.3

The use of CFD is advantageous in many

ways, but primarily due to its lower cost

compared to experimental tests. Another

benefit of CFD analysis is that it eliminates

flow field disturbances as seen in wind tunnel

testing, were pressure waves generated by the

tunnel walls create an unrealistic and

unfavorable test environment. The two main

challenges in applying CFD to the sonic boom

problem are the need for accurate prediction

of the shock wave structure in the near-field

region and the prevention of numerical (i.e.

non-physical) dissipation of the sonic boom

pressure wave in the far field. These

challenges are addressed with a combination

of careful numerical error estimation and

comparison with existing experimental sonic

boom data.

II. Introduction

This study uses commercially available

computational software to approximate the

propagation of pressure waves from an airfoil

traveling at an assumed cruise of a supersonic

transport of Mach number 2.2, at 60,000ft, and

at a 0o angle of attack. The purpose of this

study is to analyze different supersonic airfoil

configurations and their effects on lift, drag,

and overpressure (𝛿𝑃). Airfoils were analyzed

because they are the most essential aspect to

an aircraft and have received much less

attention in sonic boom studies than fuselages.

Two shapes were analyzed on the upper

surface, convex and diamond and these same

two shapes were analyzed on the lower

surface. A total of four different surface

combinations were possible and thus 4

different airfoil shapes. The thicknesses (yu,

yl) and the thicknesses location (xu, xl) were

varied between each configuration. A mesh

was created for each of these configurations,

and a grid study was performed to ensure that

the created grids were adequate resolved. The

grids were run through a CFD simulation that

returned the L/D ratio and the pressure profile

at the end of the near-field, which in turn was

input into an acoustics code which returned

the sonic boom pressure footprint on the

ground. The results show that thinner airfoils

with a flat lower surface have the highest L/D

ratios and lowest peak overpressures.

III. Airfoil Configurations

Four different airfoil configurations were

used. These configurations can be broken up

into four surfaces, a round upper surface, a

diamond upper surface, a round lower surface,

and a diamond lower surface. The

combination of these four surfaces results in

the four different configurations or series: the

1 series has a round upper and lower surface,

the 2 series has a diamond upper and lower

surface, the 3 series has a round upper surface

and a diamond lower surface, and the 4 series

3

has a diamond upper surface and a round

lower surface. The different configurations

can be seen in Figure 1. Each configuration is

a function of the maximum thickness on the

upper surface (yu), the maximum thickness on

the lower surfaces (yl), and the location of

these thicknesses xu, and xl respectively.

Figure 2 shows an airfoil described by these

parameters.

The parameters have been grouped into a

single number for convenience. An example is

-308065025. The format of the number is as

follows: series number (3), yu (0.08c), yl (-

0.06c), xu (0.5c), xl (0.25c). If yl is negative

the negative sign is placed in front of the

entire number.

The upper surface of the 1 and 3 series

airfoils are described by the piece-wise

equation (1).

𝑦 𝑥 =

𝑦𝑢 cos

𝑥 +𝐿2− 𝑥𝑢 𝜋

2𝑥𝑢 , 𝑥 ≤ 𝑥𝑢

𝑦𝑢 cos 𝑥 +

𝐿2− 𝑥𝑢 𝜋

2 𝐿 − 𝑥𝑢 , 𝑥 > 𝑥𝑢

(Eq 1)

The lower surface of the 1 and 4 series

airfoils are described by equation (2).

𝑦 𝑥 =

𝑦𝑙 cos 𝑥 +

𝐿2− 𝑥𝑙 𝜋

2𝑥𝑙 , 𝑥 ≤ 𝑥𝑙

𝑦𝑙 cos 𝑥 +

𝐿2− 𝑥𝑙 𝜋

2 𝐿 − 𝑥𝑙 , 𝑥 > 𝑥𝑙

(Eq 2)

The upper surface of the 2 and 4 series

airfoils are described by equation (3).

𝑦 𝑥 =

𝑦𝑢𝑥𝑢

𝑥 , 𝑥 ≤ 𝑥𝑢

−𝑦𝑢(𝐿 − 𝑥)

𝐿 − 𝑥𝑢 , 𝑥 > 𝑥𝑢

(Eq 3)

The lower surface of the 1 and 4 series

airfoils are described by equation (4).

𝑦 𝑥 =

𝑦𝑙𝑥𝑙

𝑥 , 𝑥 ≤ 𝑥𝑙

−𝑦𝑙(𝐿 − 𝑥)

𝐿 − 𝑥𝑙 , 𝑥 > 𝑥𝑙

(Eq 4)

The airfoils used in this study have yu

values that range from 0.02c to 0.08c, yl

values that range from -0.04c to 0.04c. The xu

and xl values range from 0.25c to 0.75c.

IV. Setup

The airfoils that were designed and tested

are designed for a supersonic transport similar

to the Concorde. The flight altitude is at

60,000ft, and the cruise Mach number is 2.2,

and the angle of attack is 0o. In this project it

is assumed that the wing is inside the Mach

cone of the aircraft, and does not experience

free stream conditions. The conditions inside

of the Mach cone were estimated by solving a

conical flow problem for the nose of the

Concorde. The results were a Mach number of

2.15, pressure of 7757.6Pa, and a temperature

of 220.8K. The simulations also assumed that

the flow was inviscid, and the fluid was an

ideal gas.

V. Grid Generation

There were a total of 60 different airfoil

designs. Each airfoil was imported into

Gridgen, a commercial grid generation

program. Two 321x129 node blocks were

generated around the airfoil, one along the

upper surface and the other along the lower

surface, thus the final mesh for each

configuration was a 321x257 node grid. The

grid dimensions are 5 chord lengths above and

below the airfoil, half a chord length in front

of the airfoil and 11 chord lengths behind the

airfoil. Figure 3 and 4 show one of the grids

that was generated, and Figure 5 shows a

schematic of the distribution of the nodes

along the boundaries of the lower block.

The dimensions are based on the height

which extends 5 chord lengths above and

below the airfoil. This is the distance were

diffraction, or interaction between shock

waves and expansion waves becomes

negligible.5 The length was then selected to

make sure that the shocks were captured

within the 5 chord height domain.

4

VI. Grid Study

The grid is a simple rectangle for the

reason that other shaped grids would not

iteratively converge sufficiently. Grids that

were directly lined up with the shock and

expansion waves were tested, however these

shaped grids only converged 3.5 orders of

magnitude. It was determined that this lack of

convergence was due to the cells being

skewed, which was due to the steep angle of

the domain.

Before the majority of the simulations

were run a grid study was performed to ensure

that the generated grids were adequate. It was

performed on the top block of the 202005050

(2 series airfoil, with an xu value of 0.5c, a yu

value of 0.02cand a flat lower surface) airfoil

grid. The simulation conditions were set to the

Mach cone conditions, and the simulation was

run until the scaled residuals converged 13

orders of magnitude. A refinement factor of 2

was selected, thus a coarse mesh of 161x65, a

medium mesh of 321x129, and a fine mesh of

641x257 were tested. Because the simulations

were assumed to be inviscid, it was possible to

calculate an exact solution. Figure 6 shows the

regions were the exact solution was compared

to the simulated result, and Figures 7 and 8

show the percent error between the exact and

simulated results, H = 1 corresponds to the

fine mesh, and H = 4 corresponds to the

coarse mesh. The study shows that the

medium mesh is accurate enough, and that the

lift and drag coefficients stop oscillating after

the residuals have been resolved 5 orders of

magnitude.

VII. Computational Fluid Dynamics

The flow calculations were performed

using Fluent, a finite volume solver. In this

case Fluent was set to use an implicit method,

second order upwind method, and a density

based solver. The fluid was defined as an ideal

gas with a molecular weight of 28.966

kg/kmol, and a specific heat capacity of

1006.43 J/kg K. The airfoil within the grid

was assigned a “wall” boundary condition,

and the edge of the grid was assigned a

“pressure far-field” boundary condition. The

boundaries as well as the domain were

initialized with the interior Mach cone values

of M=2.15, P=7757.6Pa, and T=220.8.

The CFL number is a parameter used to

define the stability criteria for time marching

processes. In this project a CFL number

between 2 and 6 was used was used for each

case. The case was run until the residuals

converged at least 6 orders of magnitude, and

took about 15 minutes per case. Figure 9

shows the convergence of one of the grids.

The lift and drag coefficients of the airfoil

were recorded, and the 1-D pressure

distribution along the bottom of the grid was

saved and used as the input for the acoustic

code.

VIII. Geometric Acoustics

An acoustic wave propagation code was

used to propagate the near-field pressure

disturbances through the far-field to the

ground. The solution to the near field problem

is used as an input for the wave propagation

code which solves for the sonic boom

footprint on the ground. The wave propagation

program used is PCBoom4. It is based on the

original Thomas code and uses geometrical

acoustics and ray tracing to propagate waves.6

The program is initialized with a height of

60,000ft, and the atmospheric distribution of a

standard day. The model length was set to

60.5ft which is the mean aerodynamic chord

of the Concorde. The trajectory is a straight

line as if cruising. The results from the

program give the footprint of the sonic boom,

and the maximum overpressure can then be

recorded. A footprint of the -308045075

airfoil can be seen in Figure 10.

5

IX. Results

60 airfoils were tested at a simulated

cruising condition at an altitude of 60,000ft

and a Mach number of 2.2 (values from the

interior of the Mach cone were used). CFD

was used to compute the near-field solution

and computational acoustics was used to

compute the far-field solution. The

computational near-field flow solution shows

that there is an attached oblique shock wave at

leading edge then a expansion waves at the

points of maximum thickness(xu, xl), and then

another shock wave at the trailing edge. This

is the general solution trend of all of the cases.

Figure 11a and 12a show the Mach

number contours and the pressure contours of

a single solution in the extreme near field.

These figures show that the Mach number

drops and the pressure increases, through the

first shock wave. Then the Mach number

increases and the pressure decreases through

the expansion wave. The Mach number then

drops, and the pressure increases through the

trailing edge shock wave.

Figures 11b and 12b show the complete

near field. These figures show that past 0.5

chord lengths away. The shock and expansion

waves begin to interact. By 5 chord lengths

away the interactions become negligible, and

it is assumed there is no more diffraction.

The pressure distribution from the edge of

the grid is extracted and run through

PCBoom4. Figure 10 shows a sonic boom

footprint of an airfoil. The footprint of this

airfoil is representative of the other footprints,

in that the behavior is similar, the only

differences are the peak 𝛿𝑃 and the time

interval. The peak 𝛿𝑃, is defined as the

maximum overpressure, in a signature.

Effect of Upper Thickness

The effect of the upper surface on the L/D

ratio and the maximum peak overpressure on

the ground was evaluated. 16 airfoils were

tested varying the shape, thickness, and

location of thickness of the upper surface

while the lower surface was maintained flat

and shapeless. The results show that thinner

airfoils with an xu value of 0.5c have higher

L/D ratios, and lower peak 𝛿𝑃 as can be seen

in Figure 13. The results also show that a

diamond upper surface produces a higher L/D

ratio than a convex upper surface, and that the

upper surface shape is independent of the

maximum overpressure.

Effects of Lower Thickness

The shape of the upper surface was varied

between diamond and convex, however the yu

and xu values were fixed at 0.08c and 0.5c

respectively. The bottom of the airfoil was

varied between shape, thickness (yl), and

thickness location (xl).

Figure 14 shows the L/D and peak 𝛿𝑃

values as functions of the lower surface

thickness (yl) and location of maximum

thickness (xl) for a 1 series airfoil. The

behavior shown in this figure is very similar to

the behavior of the 2, 3, and 4 series airfoils.

The results from all four series show that the

maximum L/D is achieved when the lower

surface is flat, and consequently the minimum

overpressure also occurs when the lower

surface is flat. However if the lower surface is

not flat the results suggest that the location of

thickness (xl) should be at 0.5c because at 0.5c

the L/D is maximized and the 𝛿𝑃 is minimized

compared to the other xl locations.

Figure 15 shows the L/D and 𝛿𝑃 values as

functions of the airfoil shape, and thickness

(yl), while the thickness location (xl) is kept

constant at 0.5c. When the lower surface is

flat the L/D is maximum, and the peak 𝛿𝑃 is

minimum. The general trends in this figure

show that a diamond upper surface (2 and 4

series) produces a higher L/D ratio, and that a

convex lower surface (1 and 4 series) produce

a lower peak 𝛿𝑃.

6

Physics

The above results suggest a general theory

that a thinner airfoil produces a higher L/D

and a lower peak 𝛿𝑃. This makes sense

because a thinner airfoil would generate

weaker shocks. In supersonic flight the

majority of drag is due to wave drag,7 thus

weaker shock waves translate to less drag and

higher L/D ratios. Stronger shock waves also

create a larger overpressure, and when

propagated to the ground create a larger peak

overpressure. Therefore a thinner airfoil is

generates higher L/D ratios and lower peak 𝛿𝑃

because it generates weaker shock waves.

X. Future work

This was a very narrow and specialized

project. The leading and trailing edges of the

airfoil were sharp. This allowed the shocks in

the near field solution to be perfectly attached.

The shocks may have also been perfectly

attached since the solution was calculated

assuming inviscid flow. For a supersonic case

inviscid flow is not a poor assumption,

however it is still an assumption and thus may

have affected the results.

The structure of the airfoil was not

considered, the optimum airfoil was a

diamond shaped airfoil with a 0.02c maximum

thickness(202005050). The L/D ratio of this

airfoil at the specified conditions is 2.1, and

the peak overpressure is 0.001psf. This shape

has not been structurally analyzed, however it

visually appears too thin to be a realistic

airfoil/option along the entire span of the

wing.

Another aspect that was not considered

was the effect of angle of attack. These cases

were run at a cruise condition were it was

assumed that the aircraft would be at 0o angle

of attack.

In future work these limitations will be

addressed. The leading and trailing edges will

be rounded to represent an actual airfoil.

Structural analysis will be preformed to

evaluate how realistic an airfoil design is. The

flow calculations will take viscous effects into

consideration. The numeric results will be

verified with wind tunnel tests.

XI. Conclusion

This project analyzed different supersonic

airfoils using computational methods. The

airfoils were tested at what is an assumed

cruise for a supersonic transport (altitude =

60,000ft, M =2.2). The near-field is calculated

using CFD, and the far field is calculated

using a geometric acoustics code. The L/D

ratio is taken from the near field solution, and

the peak overpressure is taken from the far-

field acoustics solution. The results show a

general trend that thinner airfoils produce

weaker shocks which produce larger L/D

ratios and smaller peak overpressures. More

specifically the results show that a diamond

shaped upper surface, with a flat lower surface

produces the maximum L/D and minimum

peak 𝛿𝑃. The convex lower surface produces

the maximum L/D and minimum peak 𝛿𝑃

after the flat lower surface configuration.

These results appear to be correct for this

limited case. Angles of attack other than zero

were not tested, structural analysis was not

performed to ensure that configurations were

realistic, the airfoils had unrealistic sharp

edges, and the flow was assumed inviscid.

These limitations will be considered in future

work, and under these new conditions the

conclusions may change.

7

XII. Figures

Figure 1. Different Airfoil Series

Figure 2. Airfoil Schematic

Figure 3. The 321x257 node grid that was used to run the simulation.

Figure 4. View of the grid around the airfoil (Flow is to the left)

Figure 5. Schematic of the node distribution along the boundaries of the lower face

Figure 6. Schematic of the Tested Regions around the airfoil

Figure 7. Percent Error in the Pressure

1 2 3 4410

-4

10-3

10-2

H

Pe

rce

nt E

rro

r

Pressure

P2

P3

P4

8

Figure 8. Percent Error in Mach number

Figure 9. Iterative convergence

Figure 10. Sonic Boom Footprint of -308045075 airfoil

Figure 11a. Mach Contours around -408025075 airfoil

Figure 11b. Mach contours around -408025075 airfoil

Figure 12a. Pressure contours around -408025075 airfoil

1 2 3 4

10-4

10-3

10-2

10-1

H

Pe

rce

nt E

rro

rMach Number

M2

M3

M4

9

Figure 12b. Pressure contours around -408025075 airfoil

Figure 13. L/D ratios and δP values, for 1 and 2 series airfoils with varying upper surface thicknesses (yu) and thickness locations (xu). The lower surface is kept at yu = 0, and xu = 0.5c.

Figure 14. L/D ratios and δP values, for a 1 series airfoil with varying lower thicknesses (yl) and thickness locations (xl). The upper surface is kept at yu = 0.08c, and xu = 0.5c.

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

0 5 10

δP

and

L/D

Upper Thickness (percent chord)

Effects on δP and L/D for DifferentUpper Surfaces

L/D T Location 0.25c (1 series)

L/D T Location 0.5c (1 series)

L/D T Location 0.25c (2 series)

L/D T Location 0.5c (2 series)

δP T Location 0.25c (1 series)

δP T Location 0.5c (1 series)

δP T Location 0.25c (2 series)

δP T Location 0.5c (2 series)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

-5 0 5

δP

and

L/D

Lower Thickness (percent chord)

Effects on δP and L/D for different Lower Surface Thicknesses

L/D T Location at 0.25c

L/D T Location at 0.5c

L/D T Location at 0.75c

δP T Location at 0.25c

δP T Location at 0.5c

δP T Location at 0.75c

10

Figure 15. L/D ratios and δP values, for different series airfoil with varying lower thicknesses (yl). The upper surface is kept at yu = 0.08c, xu = 0.5c, the lower surface thickness is kept at xl=0.5c.

XIII. References 1 John D. Anderson. Modern Compressible Flow

3rd

edition. New York NY, 2003 2 National Research Council. “Commercial

SUPERSONIC Technology The Way Ahead”.

Washington D.C. 2001 3 Joseph W. Pawlowski, David H. Graham,

Charles H. Boccadoro, Peter G. Coen, Domenic J.

Maglieri “Origins and Overview of the Shaped

Sonic Boom Demonstration Program”, AIAA

paper 2005-5, January 2005.

4 Donald C. Howe, Kenrick A. Waithe, Edward A.

Haering. Jr. “Quiet SpikeTM

Near Field Flight Test Pressure Measurements with Computational Fluid

Dynamics Comparisons”, AIAA paper 2008-128,

January 2008. 5 Laflin, K.R., Klausmeyer, S.M., Chaffin M., “A

Hybrid Computational Fluid Dynamics Procedure

for Sonic Boom Prediction”. AIAA 2006-3168 6 Plotkin, K.J., and Grandi, F., “Computer Models

for Sonic Boom Analysis: PCBoom4, CABoom,

BooMap, CORBoom”. Wyle Report WR 02-11,

June 2002. 7 Bertin J.J., Cummings R.M. Aerodynamics For

Engineers 5th edition. Pearson Prentice Hall,

Upper Saddle River NJ 2009.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

-5 0 5

δP

and

L/D

Lower Thickness (percent chord)

Effects on δP and L/D for Different Configurations on the Lower

Surface

L/D 1 series

L/D 2 series

L/D 3 series

L/D 4 series

δP 1 series

δP 2 series

δP 3 series

δP 4 series