5.0 Aerothermodynamics
Giles Goetz
5.1 Nomenclature
K = Proportionality constant used in the modified Newtonian theory
K = 2, from Reference 2
S = Arbitrary Reference Area set to be the projected area of the cylinder with radius R.
S = R2
  Arbitrary reference length, set to be the length of the cylinder, L along the x-axis.
 = Angle of attack, the angle between the vehicle and the velocity vector
 = Fin angle, the angle between the horizontal of the fin and the horizontal of the vehicle
body,
L = Length of the cylinder
R = Radius of the cylinder and the hemisphere
Le = Length of the leading edge of the fin.
Re = Radius of the leading edge of the fins.
 = Sweep angle of the fin
Lf = Length of the fin, dependent on the length of the leading edge and the sweep angle
Wf = Width of the fin, dependent on the length of the leading edge and the sweep angle
Xcg = Distance along the x-axis from the reference point to the center of gravity with
positive direction being forward.
Zcg = Distance along the z-axis from the reference point to the center of gravity.
5-1
All the coefficient equations are non-dimensionalized by the reference area and
reference length in the case of the moment coefficients.
5.2 Vehicle Parameters and Design
The vehicle used for the case studies is blunt with a low L/D ratio. A
diagram of the vehicle can be seen below in Figure 5.1. The coordinate system for the
vehicle is labeled below in Figure 5.1 and has the variables x, y and z for the three
dimensions. The positive directions are labeled with the bold arrows.
x
y
Coordinate
Reference
Point
z
z
x
y
Figure 5.1: Diagram of Vehicle with Coordinate System
The different vehicle parameters can be seen in Figures 5.2, 5.3 and 5.4. The nose
is a hemisphere with radius R, and is attached to a cylindrical body of length L with same
radius R. A set of control fins are attached perpendicular to the body. The control fins
are of length Lf and width Wf. The control fins have a leading of length Le, radius Re and
a sweep angle . The different vehicle parameters are described in detail later.
5-2
5.3 Methods
5.3.1 Aerodynamics
The aerodynamics of the vehicle was modeled using a combination of methods.
The different methods were hypersonic Newtonian theory, hypersonic skin friction,
supersonic skin friction, supersonic linearized flow, wave drag and viscous interaction
effects. The results from the various methods were used to form a model of the vehicle to
get Cl, Cd, and Cm for a given velocity, angle of attack, and a set of vehicle parameters.
The different parameters used can be found in Table 5.1 in section 5.3.
Hypersonic Newtonian Theory
Newtonian Theory uses the pressure distribution of air flowing over a body to
calculate forces and moments at hypersonic speeds. The equations used for the
Newtonian theory came from Reference 3. The method used was to take simple shapes
and combine them to get an overall model of the vehicle. Clark and Trimmer’s paper
contained base shapes along with their Cl, Cd and Cm calculations. By using the reference
points and equations for each shape, it was possible to combine the shapes to create an
aerodynamic model of the vehicle.
Hemisphere
R
Reference
Point
Figure 5.2: Diagram of a Hemisphere with Reference Point
5-3
The following equations are for the Cn,Hemi and Ca,Hemi for a hemisphere of radius
R with a reference point at the center of curvature. The moment coefficients for a
hemisphere are zero about the center of curvature. The equations can be found in
Reference 3, pp 28-29.

 KR
C n , Hemi   sin  (1 cos  )
4
 S

 KR
C a , Hemi   (1 cos  ) 2 
8
 S
2
(Ref 3, Equation 120)
2
(Ref 3, Equation 121)
Reference
Point
Cylinder
L
Radius R
Figure 5.3: Diagram of Cylinder with Reference Point
The following equations are for the Cn,Cylinder, Ca,Cylinder and Cm,Cylinder for a
cylinder of radius R with a reference point at the end. The equations can be found in
Reference 3, pp 34.
5-4
C n ,Cy ln der
(Ref 3, Equation 155)
4
 KLR
  sin 2  
3
 S
C a ,Cy ln der  0
C m ,Cy ln der  C N
(Ref 3, Equation 156)
L
2
Fins
(Ref 3, Equation 157)

1/3 Lf
Reference
Point
1/3 W f
Wf
Le
Center of Pressure
Lf
Figure 5.4: Diagram of Fins and Reference Point
The Newtonian calculations treat the leading edges of both fins as a single unit for
analysis. The equations for the leading edges are from Reference 3, pp 12-13.


4
 KLe Re
C n , Lead   sin  cos  cos   1  sin 2  cos 2  
S
3

C a , Lead

 sin 2 
cos 2  cos 2  
 4 cos  

 2
3

2
2

 cos  cos  1  sin  cos 

C m , Lead  C n , Lead
Le sin 
2

 KLe Re

S


(Ref 3, Equation 38)
(Ref 3, Equation 42)
(Ref 3, Equation 43)
5-5
It is also necessary to include the effect of the flat plate part of the fins. The
equations for coefficients of the flat plate come from Reference 1, pp 51. At the center of
pressure, the Cm,Plate is equal to zero, and flat plates do not have any axial force. Since the
center of pressure is not the reference point of the fins, the Cm,Plate of the flat plate part of
the fins needs to be transferred to the reference point of the fins. The distances from the
center of pressure to reference point are used to calculate Cm,Plate at the reference point
with Equation 161, from Reference 3, pp 35. The CN,Plate value for the flat plate part is
also multiplied by two to take into account two fins. The two in front of the equation for
CN,Plate in Anderson’s book, is the K factor, and not the two representing the two fins.

C n , Plate  2 sin 2 
 KLSW
f
(Ref 1, Equation 3.9)
f
(Ref 2, pp 51)
C a , Plate  0
C m , Plate  C N , Plate
(Ref 3, Equation 161)
1
Lf
3
Final Composite Configuration
In order to combine everything, need to first make sure everything is in the right
frame. Since the fins can move relative to the body, a coordinate transformation must be
made. The fins are related to the coordinate system of the vehicle body with the angle .
 is the pitch angle between the fins and the x-axis of the body.  = 0 when the fins are
parallel to the x-axis of the vehicle body and the z-axis points straight up through the fins,
and  = 90 when the fins are parallel to the z-axis and the x-axis points through the
bottom of the fins. Using the transformation equations from Reference 1, pp 17,
Equations 1.1 and 1.2, the coefficients for the parts of the fins were added and applied to
a coordinate transfer to the body frame, shown in the following equations. Also note that
since the two plate’s moment coefficients were about their centers of pressure, the
5-6
coefficients of moment had to be moved to the reference point of the body by use of
Reference 3, Equation 161, pp 35.
C n , Fin  C n, Lead  C n , Plate * cos   C a , Lead  C a , Plate * sin 
C a , Fin  C n , Lead  C n , Plate * sin   C a , Lead  C a , Plate * cos 
Lf 

C m, Fin  C m, Lead  2 C m, PlateC N , Plate 
3 

(Fin Equation 1)
(Fin Equation 2)
(Fin Equation 3)
Now that all the parts are in the same frame, it is possible to just add the normal
and axial coefficients. The moment coefficients for the hemisphere and fins need to be
translated to the reference axis of the whole vehicle using Equation 161, Reference 3, pp
35. These final composite equations can be seen below.
C n ,newt  C n , Hemi C n , Fin C n ,Cylinder
C a ,newt  C a , Hemi C a , Fin C a ,Cylinder
Lf
L
L
1  cos    C a, fin f sin 
C m,newt  C m, Hem i C n , Hemi  C m, Fin C n , Fin

3
3
Finally, need to shift the center of the moment coefficient to the center of gravity
of the vehicle. Again, this is done using Reference 3, Equation 161.
C m,cg  C m,newt  C n ,newt
Xcg
Zcg
 C a ,newt


5-7
Once the coefficients are known, Cl and Cd are calculated using Equations 1.1 and
1.2, Reference 1, pp 17.
Cl ,newt  C n,newt cos   C a ,newt sin 
C d ,newt  C n,newt sin   C a ,newt cos 
Hypersonic Skin Friction
Since Newtonian theory only takes into account the shape of the vehicle, other
techniques are needed to further analyze the effects of the flow around the vehicle. The
first such method is hypersonic skin friction. Using Reference 4, the process was coded
as follows.
First the Mach number is determined from the velocity. This indicates if the flow
is hypersonic or supersonic. The subsonic case will not be analyzed for the Mars entry
vehicle. Next, the local edge Mach number is calculated by using the crude relation of
Me = Mcos(). Now the location of transition on the surface of the vehicle is found by
calculating the Reynolds number at that location by using Equation 72, Reference 4, pp
35. Finally the location of the transition is found by using Equation 71, Reference 4. In
Equation 71, ’ is the reference viscosity, found using the Sutherland law, ’ is the
reference density and can be found from the reference temperature, which is calculated
from the wall temperature and free stream Mach number and temperature. This is an
important step because the value for the hypersonic skin friction is dependent on whether
the flow is turbulent or laminar.
Now the Cd,hyper can be found using Equation 86 on page 37 of Reference 4. In
Equation 86, Re’l is Reynolds number at the end of the vehicle using reference density
and viscosity, Re’x,t is the Reynolds number at transition using reference density and
5-8
viscosity, Rex,t is the Reynolds number at transition using free stream density and
viscosity, xt is the point on the body where transition occurs, T’ is the reference
temperature, T is the free stream temperature and l is the length of the vehicle which is
computed from the sum of the arc length of the hemisphere, and the length of the
cylinder.
The process is then duplicated for the fins, substituting in the different lengths,
and instead of using , the code uses the angle  +  to represent the angle of attack of
the fins. Then the two hypersonic skin friction values are added together to generate a
total hypersonic skin friction value.
Supersonic Skin Friction
Since the vehicle will not always remain at hypersonic speeds, calculations for
skin friction at supersonic speeds must be made. Again, the Reynolds number and
position of transition are found using Equations 71 and 72 from Reference 4. Now the
incompressible skin friction is found using Equation 88 from Reference 4, on page 38,
and then factoring in compressibility by using Equation 89. Finally Cd,super is calculated
by using Equation 91 on page 39 of Reference 4. Instead of multiple pieces, the CF for
just the vehicle is used, with the Sref being the projected area of the cylinder, and the Sfuselage
being the surface area of the vehicle. The supersonic skin friction is also calculated
for the fins using the surface area of the fins, and the same reference area for the body.
The two values are then added to get a total supersonic skin friction value.
Supersonic Linearized Theory
Since the hypersonic equations do represent the vehicle well at lower supersonic
speeds, supersonic linearized theory was used to calculate the coefficient of life for the
vehicle. Using Equations 12.23, and 12.24 from Reference 1, pp 576, to calculate the
coefficient of life and drag for the vehicle under supersonic conditions based on the angle
of attack and Mach number.
5-9
Wave Drag
Also under the supersonic method is wave drag due to boundary layer thickness.
The equations used are 92, 93 and 94 from Reference 4, with the sweep angle being the
90 - angle of attack and considering the whole vehicle as a single wing.
Viscous Interaction Effects
Finally there are the viscous interaction effects on the vehicle. The viscous
interaction occurs when the boundary layer becomes thick due to the low densities at high
altitudes. The effect is found by first calculation the viscous interaction coefficient, VI.
The equation for VI can be found on page 41 in the methods hand out. Next VI is used to
calculate the ratio of actual L/D and L/Dinviscid, using Equation 95 from Reference 4, on
page 41. Finally, the CD,actual is found by using Equation 96 in the methods handout and
by using the Cd calculated from Newtonian Theory. Cl is assumed to be unaffected by
any viscous interaction effects for this method.
Combined Effects
In order to have a smooth transition from hypersonic to supersonic flow, at Mach
5, it is assumed the flow starts to be come supersonic, and becomes totally supersonic
flow at Mach 4. In between Mach 5 and 4, a weighted average based on the Mach
number is used to combine the two different flight conditions. Finally, the viscous
interaction effect is added in for all three cases.
5-10
5.3.2 Aerothermodynamics
The next step was to calculate the heating values of the Mars entry vehicle. There
were three different types of heating points considered, stagnation points, leading edge
points and flat plate thermal points. The stagnation points were found first, then using
the location of the stagnation points to calculate the heating values at the thermal points,
then using both the stagnation and flat plate heating values to calculate the leading edge
heating values. Currently there are two stagnation points, one on the front of the body at
45 from the x-axis and at the intersection of the fins and the body. The reason the 45
point is the stagnation point is because most of the time the vehicle is flying at that angle
of attack and that would cause the stagnation point for the body to appear there. There
are four flat plate thermal points along the body and one on the fin, and there are two
leading edge points on the fin, the first having an arc length of rlead, and the second
being on the tip of the fin. Figure 5.5 shows the locations of each of the thermal points
on the vehicle.
Stagnation Points
The stagnation point heating values were found using Equation 47 Reference 4
and Equation 1 from Reference 5. Equation 47 is the stagnation heat-transfer rate to the
vehicle were V is the free stream velocity,  is the free stream density, rn is the nose
radius of the vehicle or the radius of the leading edge of the fins depending on which
stagnation point, and Cpw is the specific heat of the wall, and Tw is the wall temperature at
the stagnation point. Equation 1 is the radiative heating at the stagnation point were C is
a constant depending on the atmosphere, rn is the radius of the nose of the vehicle or the
radius of the leading edge of the fins depending on which stagnation point,  is the free
stream density, a and b are constants depending on the atmosphere, and f(V) is a function
depending on velocity. A curve fit was created from the values found in Table 1 from
Reference 5. In the code the velocities used to in the f(V) calculation are limited to 60009000 m/s. If the velocity is larger then 9000, it is set to 9000, if it is smaller then 6000, it
is set to 6000. The reason for doing this is because the peak radiative heating values
5-11
occur within that velocity span so there is no need to calculate beyond those velocities.
Once the two sets of values are calculated, they are added together to form a total value
for the one stagnation point at the front of the vehicle, and for one at the stagnation point
on the fins.
Flat Plate Thermal Points
Once the stagnation points have been found, the thermal points were found by
using the flat-plate heat-transfer equations. The basic equation used to find the heating
value at any given point is Equation 52 from Reference 4. The  value is the free stream
density and the V is the free stream velocity. The C is the heating parameter. C will vary
depending on the conditions of the flow and the angle of attack of the vehicle. First the
angle of attack is checked against the equation Msin  found on page 30 in Reference 4.
If the value for Msin  is greater than 1, then the flow has large angles of attack and
Tauber method is used. If the value for Msin  is less then or equal to 1, then White’s
method for small angles of attack is used.
For the two different methods, the location of transition is needed. Reusing
Equations 71 and 72 from Reference 4 to get the transition location. For the large angle
of attack with laminar flow, C is found from Equation 53 on page 30 of Reference 4. The
equation has the variables , the angle of attack for this case, gw, the ratio of wall
enthalpy to total enthalpy, found by equation 54, and x is the distance from the stagnation
point. For the turbulent case, there are two different C values, one for a free stream
velocity greater then 3962 m/s, Equation 56, and one for less then or equal to 3962 m/s,
Equation 55. Once C is found, the heating rate at that point can be found.
For the small angle of attack case, there is also a laminar and turbulent case. The
laminar case requires the calculation of the reference temperature from the free stream
temperature, wall temperature and the free stream Mach number using equation 57.
Next, use the reference temperature to calculate C* from equation 58. Then use Equation
59 to get the adiabatic wall temperature, and Equation 60 to get the skin-friction
5-12
coefficient at the thermal point. In Equation 59, Pr is a constant equal to approximately
0.72. And in equation 60, Rexe is the Reynolds number calculated at the thermal point,
using free stream conditions. Finally Equation 61 is used to get the heat-transfer
coefficient from the skin-friction coefficient, which is then used with Equation 62 to get
the heat transfer to the wall. For the turbulent case for a small angle of attack, the skinfriction coefficient is calculated using Equation 63, and the adiabatic wall temperature is
calculated from Equation 64. The rest of the steps are the same as the laminar condition.
Leading Edge
Since the fins have the two leading edge points, a special heating equation is
needed to calculate the heat flux, Equation 65 from Reference 4. The equation uses the
sweep angle , and the heating values from the stagnation point and the flat plate heating
value at that leading edge point to create a heating value for the leading edge.
Where the hemisphere
meets the cylinder
End of cylinder
Midpoint of
fin
Leading
edge on
tip of fin
Stagnation
Point
10 from
Stagnation
Point
Midpoint of
Cylinder
Stagnation Point and
leading edge calculation
point
Figure 5.5: Location of thermal points on the vehicle
5-13
5.3.3 Stability
Stability of the vehicle is important; in order for the vehicle to be stable the
moment coefficient must be zero and slope of the moment coefficient vs angle of attack
has to be negative. This means that is the system is perturbed from its flight path; it has
the ability to move itself back to its flight path. The stability of the vehicle is based on
the location of the center of gravity, and the control effectiveness of the fins. Using the
code developed for the Newtonian theory, along with another Fortran program,
aeroprop.f, the stability of the vehicle was analyzed. The program allowed the user to
move the cg of the vehicle, while adjusting the angle of the fins, through a series of angle
of attacks. The program then stored the Cl, Cd and Cmcg of the vehicle at those different
points. By plotting the Cm vs. angle of attack, it is possible to see the stability of a
vehicle. In order for a vehicle to be stable at a given angle of attack, the Cm slope must
be negative, as well as equal to zero. Later on the trajectory code was altered so the user
could input cg locations and the code would find what angles of attack the vehicle could
fly at and still be stable.
5.3.4 Aerothermodynamics Code
Several different programs were created and used for the aerothermodyanmic
analysis of the project. Aerodat.f and heatflux.f contain the two sets of calculations for
aerodynamics and heat-transfer calculations respectively. Aeroprop.f, aerotest.f, and
altvmap.f were used to analyze aerodat.f’s code under different conditions. Aeroprop.f
would set the cg at different locations and generate Cl, Cd and Cmcg values for a range of
alpha and beta angles. These values could then be used to find where the vehicle was
stable. Aerotest.f would do almost the same thing, but it would calculate the Cl and Cd
values generated when the vehicle was trimmed at a given angle of attack. Altvmap.f
was used to generate actual Cl and Cd values for a range of actual flight conditions.
These values could then be used to show the best ranges for the vehicle to fly in.
Aerotest.f and altvmap.f both used another program aerotrim.f to generate the trimmed
values for the vehicle. Aerotrim.f would call aerodat.f and locate where the vehicle was
5-14
stable. The trajectory code also used aerotrim.f along with the aerodat.f and heatflux.f
code.
5.4 Trade Studies
Using the different sets of code, the following trade studies were done. For the
aerodynamic case there were two different vehicles looked at, as well as two different
sets of fins. The numbers for the different configurations can be found in Table 5.1.
Eventually an optimized design was found, and the values for it are also located in Table
5.1.
Vehicle Configurations
Cylinder Cylinder
Length
Radius
(m)
(m)
One
10
6.5
Two
10
4.5
Fin Configurations
One
*
Two
*
Leading
Edge
Length
(m)
*
*
*
*
Leading
Edge
Radius
(m)
*
*
Sweep
Angle
(deg)
*
*
4
3
0.05
0.05
45
45
4
0.05
45
Optimized Design
10
4.5
Table 5.1: Different Vehicle and Fin Configurations Used
First the stability of various cg locations was analyzed by viewing the Cmcg vs
AoA plots. Figures 5.6 and 5.7 show two example cases and Figure 5.8 shows the Cmcg
vs AoA plot for the optimized design. All three cases use the same fin configuration two.
5-15
Xcg at 6.5, Cmcg vs AoA for Vehicle Configuration One
0.15
0.1
Cmcg
0.05
0
0
10
20
30
40
50
60
70
80
90
-0.05
Beta = -90
Beta = -80
Beta = -70
Beta = -60
Beta = -50
Beta = -40
Beta = -30
Beta = -20
Beta = -10
Beta = 0
Beta = 10
Beta = 20
Beta = 30
Beta = 40
Beta = 50
Beta = 60
Beta = 70
Beta = 80
Beta = 90
-0.1
AoA (deg)
Figure 5.6: Cmcg vs AoA plot for Xcg at 6.5 of Vehicle Configuration One
Xcg at 6.5, Cmcg vs AoA for Vehicle Configuration Two
0.2
0.15
0.1
0.05
Cmcg
0
0
10
20
30
40
50
-0.05
-0.1
-0.15
60
70
80
90
Beta = -90
Beta = -80
Beta = -70
Beta = -60
Beta = -50
Beta = -40
Beta = -30
Beta = -20
Beta = -10
Beta = 0
Beta = 10
Beta = 20
Beta = 30
Beta = 40
Beta = 50
Beta = 60
Beta = 70
Beta = 80
Beta = 90
-0.2
-0.25
AoA (deg)
Figure 5.7: Cmcg vs AoA plot for Xcg at 6.5 of Vehicle Configuration Two
5-16
Xcg at 6.554755, Zcg at 0.133519, Cmcg vs AoA, Optimized Design
0.2
0.15
0.1
0.05
0
Cmcg
0
10
20
30
40
50
60
70
80
90
-0.05
-0.1
-0.15
-0.2
Beta = -90
Beta = -80
Beta = -70
Beta = -60
Beta = -50
Beta = -40
Beta = -30
Beta = -20
Beta = -10
Beta = 0
Beta = 10
Beta = 20
Beta = 30
Beta = 40
Beta = 50
Beta = 60
Beta = 70
Beta = 80
Beta = 90
-0.25
-0.3
AoA (deg)
Figure 5.8: Cmcg vs AoA plot for Xcg at 6.5 of Optimized Design
Figure 5.7 shows that the larger vehicle has a much smaller range of angles the
vehicle can fly at. Because the vehicle is longer, at the same given cg location, the longer
vehicle will only trim at higher angles of attack. Also, the second configuration has a
much broader range of angles of attack the vehicle can fly at. The second vehicle was
able to generate more lift because it has a more slender shape when compared to the other
vehicle configuration.
Another stability issue was the size of the fins. The size of the fins
determined how large the range of angle of attack the vehicle could fly in. Figure 5.9
shows the different fin configuration for the same cg location as Figure 5.7.
5-17
Xcg at 6.5, Cmcg vs AoA for Vehicle Configuration Two with Smaller Fins
0.2
0.15
0.1
0.05
0
Cmcg
0
10
20
30
40
50
60
-0.05
-0.1
-0.15
70
80
90
Beta = -90
Beta = -80
Beta = -70
Beta = -60
Beta = -50
Beta = -40
Beta = -30
Beta = -20
Beta = -10
Beta = 0
Beta = 10
Beta = 20
Beta = 30
Beta = 40
Beta = 50
Beta = 60
Beta = 70
Beta = 80
Beta = 90
-0.2
-0.25
AoA (deg)
Figure 5.9: Cmcg vs AoA plot for Xcg at 6.5 of Vehicle Configuration Two and
Smaller Fins
By increasing the length of the leading edge of the fin by 1 meter, the fin surface
area went from 4.5 meters to 8.0 meters, almost double what the original value was. This
increased our angle of attack range from 15 degrees to 25 degrees, an increase of 10
degrees. This allowed the vehicle to trim over a wider range of angle of attack as well as
the vehicle being able to trim at higher angles of attack.
Next looked at what basic Cl and Cd values were generated at the different cg
locations to determine how effective the vehicle was and what would be a good flight
path angle for the vehicle. Figures 5.10 and 5.11 show two example cases and Figure
5.12 shows the optimized design for the vehicle
5-18
Trimed Cl vs AoA for Different Xcg Locations for Vehicle Configuration One
0.5
0.4
Trimmed Cl
0.3
Xcg 6.5
Xcg 7.0
Xcg 7.5
Xcg 8.0
Xcg 8.5
0.2
0.1
0
-0.1
-0.2
0
10
20
30
40
50
60
70
80
90
AoA
Figure 5.10: Trimmed Cl Values for Different Cg Locations for Vehicle
Configuration One
5-19
Trimmed Cl vs AoA for Different Xcg Locations for Vehicle Configuration Two
0.8
0.6
Trimmed Cl
0.4
Xcg 6.0
Xcg 6.5
Xcg 7.0
Xcg 7.5
Xcg 8.0
Xcg 8.5
0.2
0
-0.2
-0.4
0
10
20
30
40
50
60
70
80
90
100
AoA
Figure 5.11: Trimmed Cl Values for Different Cg Locations for Vehicle
Configuration Two
Trimmed Cl vs AoA for Optimized Design Xcg = 6.554755, Zcg = 0.133519
0.7
0.6
Trimmed Cl
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
50
60
70
AoA (deg)
Figure 5.12: Trimmed Cl Values for Optimized Design
5-20
80
90
From the first two figures, it shows that for vehicle configuration one has a max
Cl of 0.4 and for the second vehicle configuration has a max Cl of 0.6. Also there are
small fluctuations in the Cl values at the lower angles of attack. This is because the fin
angle changed from approximately 60 degrees to –30 degrees, in order for the vehicle to
trim at those angles of attack and cg locations. Because the second configuration had
more lift, it was able to perform better inside the atmosphere, and so it became our
optimized design. Figures 5.13 and 5.14 shows the trimmed Cd vs AoA plots for the two
different vehicle configurations and Figure 5.15 shows the trimmed Cd vs AoA plot for
the final configuration.
Trimmed Cd vs AoA for Different Xcg Locations for Vehicle Configuration One
2.5
2
1.5
Cd
Xcg = 6.5
Xcg = 7.0
Xcg = 7.5
Xcg = 8.0
Xcg = 8.5
1
0.5
0
0
10
20
30
40
50
60
70
80
90
AoA (deg)
Figure 5.13: Trimmed Cd Values for Different Cg Locations for Vehicle
Configuration One
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Trimmed Cd vs AoA for Different Xcg Locations for Vehicle Configuration Two
3
2.5
Trimmed Cl
2
Xcg 6.0
Xcg 6.5
Xcg 7.0
Xcg 7.5
Xcg 8.0
Xcg 8.5
1.5
1
0.5
0
0
10
20
30
40
50
60
70
80
90
100
AoA
Figure 5.14: Trimmed Cd Values for Different Cg Locations for Vehicle
Configuration Two
Trimmed Cd vs AoA for Optimized Design Xcg = 6.554755, Zcg = 0.133519
2.5
2
Trimmed Cl
1.5
1
0.5
0
0
10
20
30
40
50
60
70
AoA (deg)
Figure 5.15: Trimmed Cd Values for Optimized Design
5-22
80
90
From the figures, the max Cd for the first vehicle configuration is almost 2, while
for the second vehicle configuration the max Cd was 2.5. This meant that for the second
vehicle configuration, the vehicle would be able to slow down faster inside the
atmosphere, so the vehicle would not have to make as many passes in order to land.
Finally data was generated for the actual Cl and Cd in-flight conditions for the
optimized vehicle. Figure 5.16 shows the values for Cl and Cd for the hypersonic flight
of the vehicle, and Figure 5.17 shows the changes of Cl and Cd values, as the vehicle goes
from supersonic to hypersonic flow.
Hypersonic Cl and Cd vs Mach Number for Optimized Design
1.20
1.00
Cl, Cd
0.80
Hypersonic Cl
Hypersonic Cd
0.60
0.40
0.20
0.00
0
5
10
15
20
25
30
35
Mach Number
Figure 5.16: Hypersonic Cl and Cd Values for the Optimized Design
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Cl and Cd vs Mach Number, Transition from Supersonic to Hypersonic Flow for Optimzied
Design
7.00
6.00
5.00
Cl,Cd
4.00
Cl
Cd
3.00
2.00
1.00
0.00
2.50
3.00
3.50
4.00
4.50
5.00
5.50
6.00
Mach Number
Figure 5.17: Transition from Supersonic to Hypersonic Flow for Optimized Design
Figure 5.16 shows that for the hypersonic flow, Cl remains constant, because it is
only dependent on Newtonian theory, and Cd increases but only slightly due to the
addition of the hypersonic skin friction to the Newtonian theory. Figure 5.17 shows that
the vehicle has much more drag in the supersonic flow then in the hypersonic flow. This
means the vehicle will slow down even faster as it approaches Mach 3, were the
parachutes are deployed.
Overall the optimized design was a low L/D vehicle with large amounts of drag
that allowed it decelerate quickly, but at the same time had enough lift and control
surface to be able to land and stay with in the necessary g-loading and weight
requirements.
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5.5 References
1. Anderson, J. D.: Fundamentals of Aerodynamics 2nd Edition, McGraw-Hill, New
York, 1991.
2. Anderson, J. D.: Hypersonic and High-Temperature Gas Dynamics, McGrawHill, New York, 1989. Reprinted by AIAA Publications, Fall 2000.
3. Clark, E.L. and Trimmer, L. L.: Equations and Charts for the Evaluation of the
Hypersonic Aerodynamic Characteristics of Lifting Configurations by the
Newtonian Theory, Arnold Engineering Development Center, Arnold Air Force
Station, Tennessee, March 1964.
4. Schneider, Steven: Methods for Analysis of Preliminary Spacecraft Designs,
Purdue University, Indiana, 2001.
5. Tauber, M. E. and Sutton, K., “Stagnation-Point Radiative Heating Relations for
Earth and Mars Entries,” Journal of Spacecraft and Rockets, Vol. 28, Jan.-Feb,
1991, pp. 40-42.
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