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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, [email protected] † Associate Professor, Aerospace and Ocean Engineering, 330 Randolph Hall, [email protected]
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