Aertothermodynamics
Giles Goetz
Vehicle Diagram
The vehicle used for the sample case is blunt with a low L/D ratio.
A diagram of the vehicle can be seen below in Figure (1). The coordinate system
for the vehicle is labeled below in Figure (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 1: Diagram of Vehicle with Coordinate System
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 W f. The
control fins have a leading of length Le, radius Re and a sweep angle . The
different vehicle parameters are described later on in detail with other figures of
the vehicle. Currently the vehicle dimensions are as follows, radius of 6.5
meters, cylinder length of 10 meters, leading edge length of the fins is 3 meters
with a 45-degree sweep, and a radius of 0.05 meters.
Aerodynamics
The aerodynamics of the vehicle was modeled using a combination of
different aerodynamic methods. The different methods were Hypersonic
Newtonian Theory, Hypersonic Skin Friction, Supersonic Skin Friction, 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 characteristics.
Hypersonic Newtonian Theory
Newtonian Theory uses the pressure distribution on a body to calculate
forces and moments at hypersonic speeds. The equations used for the
Newtonian Theory came from Clark and Trimmer, “Equations and Charts for the
Evaluation of the Hypersonic Aerodynamic Characteristics of Lifting
Configurations by the Newtonian Theory.” The method used was to take simple
shapes and combine them to get an overall model of the vehicle. Clark and
Trimmer’s packet contained base shapes along with their Cl, Cd and Cm
calculations. By using the reference points and equations of each shape, it was
possible to combine them into the vehicle used for the sample case. The
nomenclature is as follows.
K = Proportionality constant used in the modified Newtonian theory
K=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
 = 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
Zcg = Distance along the z-axis from the reference point to the center of gravity
All the coefficient equations are non-dimensionalized by the arbitrary
reference area and arbitrary reference length in the case of the moment
coefficients.
Hemisphere
R
Reference
Point
Figure 2: Diagram of a Hemisphere with Reference Point
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 C & T pages 28 and 29.

 KR
C n , Hemi   sin  (1 cos  )
4
 S

 KR
C a , Hemi   (1 cos  ) 2 
8
 S
2
(C &T Equation 120)
2
(C & T Equation 121)
Reference
Point
Cylinder
L
Radius R
Figure (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 C & T page 34.
4
 KLR
C n ,Cy ln der   sin 2  
3
 S
(C & T Equation 155)
C a ,Cy ln der  0
(C & T Equation 156)
C m ,Cy ln der  C N
L
2
(C & T Equation 157)
Fins

1/3 Lf
Reference
Point
1/3 W f
Wf
Le
Center of Pressure
Lf
Figure (4): Diagram of Fins and Reference Point
The Newtonian calculations treat both leading edges of the two fins as a
single unit for analysis. First the equations for the leading edges from C & T
pages 12 and 13.


4
 KLe Re
C n , Lead   sin  cos  cos   1  sin 2  cos 2  
3
 S
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

(C & T Equation 38)
(C & T Equation 42)
(C & T Equation 43)
Also need to include the effect of the flat plate part of the fins. The
equations for coefficients of the flat plate come from Anderson,”Hypersonic and
High Temperature Gas Dynamics,” page 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.
Using the distances from the center of pressure to reference point to calculate
Cm,Plate at the reference point with C & T’s equation 161 on page 35. The C N,Plate
value for the flat plate part is also multiplied by two to take into account for there
being 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
f
C a , Plate  0
C m , Plate  C N , Plate
(Anderson Equation 3.9)
(Anderson HHTGD pg 51)
1
Lf
3
(C & T Equation 161)
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. Using the transformation equations from
Anderson, “Fundamentals of Aerodynamics,” page 17, equations 1.1 and 1.2, the
coefficients for the parts of the fins were combined and transformed 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 coefficients of
moment had to be moved to the reference point of the fins by use of C & T’s
Equation 161 on page 35.
Cn , Fin  Cn , Lead  Cn , Plate * cos   Ca , Lead  Ca , Plate * sin 
(Fin Equation 1)
Ca , Fin  Cn , Lead  Cn , Plate * sin   Ca , Lead  Ca , Plate * cos 
(Fin Equation 2)
Lf 


Cm , Fin  Cm , Fin  2 Cm , PlateC N , Plate
3 

(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 moved by use of C & T’s Equation 161 on page 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    Ca, 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 C & T’s Equation 161.
C m,cg  C m,newt  C n ,newt
Xcg
Zcg
 C a ,newt


Once the coefficients are known, Cl and Cd are calculated using the same
Anderson transfer matrix to get the equations below. Since no other methods
used affects the lift coefficient, Cl is set equal to the Cl,newt value.
Cl  C n,newt cos   C a ,newt sin 
C d  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 the methods
handout, the process was coded this way.
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, local edge Mach number is calculated by using the
crud 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 on page 35 of the methods hand out. Finally the location of the
transition is found by using Equation 71. In Equation 71, ’ is the reference
viscosity, found using 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 the
methods handout. 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 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.
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 the methods handout. Now the incompressible skin friction is found using
Equation 88 from the methods handout on page 38. Then factoring in
compressibility by using Equation 89. Finally Cd,super is calculated by using
Equation 91 on page 39 of the methods handout. 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.
Also under the supersonic method is wave drag due to boundary layer
thickness. The equations used are 92, 93 and 94 from the methods handout,
with the sweep angle being the 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 large due to the
low densities at high altitudes the vehicle is flying at. The effect is found by first
calculation the viscous interaction correlation, 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 on page 41 of the methods
handout does this calculation. 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.
Aerothermodynamics
The next step was to calculate the aerothermodynamic properties of the
Mars entry vehicle. This was done by first finding the stagnation point on the
vehicle, and then using that point to calculate the heating values on the rest of
the vehicle.
Stagnation Point
The stagnation point heating was found using equation 47 from the
methods handout. In the equation, V is the free stream velocity,  is the free
stream density, rn is the nose radius of the vehicle, and Cpw is the specific heat of
the wall, and Tw is the wall temperature. Once the stagnation point is calculated,
it can be used to calculate the heating values at the other thermal points as well
as to calculate the change in temperature at the stagnation point. Equation 50
can be used to find the change in temperature at any given point on the vehicle,
by using the heating rate calculated at that point, along with the wall temperature,
wall density, specific heat of the wall material, the thickness of the wall, and the
emissivivity of the surface.
Thermal Points
Once the stagnation point has been found, the other points are found by
using the flat-plate heat-transfer equations. Currently there are seven thermal
points on the vehicle. The basic equation used to find the heating value at any
given point is equation 52 from the methods handout. 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 the methods handout. If the value for Msin  is
greater then 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. Again
the location is calculated using Equations 71 and 72 from the methods handout.
For the large angle of attack with laminar flow, C is found from equation 53 on
page 30 of the methods handout. 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 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 from 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.
Since the fins have a leading edge, a special heating equation is needed
to calculate the heat flux at that thermal point, Equation 65 in the methods
handout. Currently this code has not been installed with the rest of the heat flux
code and will be added later on.
Stability
The stability of the vehicle is based off 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 program, 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 be equal to zero at that angle. Several plots were made of
the sample case for different cg locations along the x-axis. Figure (5) shows the
location at 7 meters from the reference point, and Figure (6) shows the cg at 8.5
meters from the reference point, the current location of the cg for the sample
case.
xcg 7.0
0.06
0.05
0.04
0.03
Beta = -60
Beta = -45
Beta = -30
Beta = -15
Beta = 0
Beta = 15
Beta = 30
Beta = 45
cmcg
0.02
0.01
0
0
5
10
15
20
25
30
-0.01
-0.02
-0.03
AoA deg
Figure (5): Cm vs AoA Plot, for cg at 7.0 meters
35
xcg 8.5
0.015
0.01
0.005
0
0
5
10
15
20
25
30
cmcg
-0.005
-0.01
-0.015
35
Beta = -60
Beta = -45
Beta = -30
Beta = -15
Beta = 0
Beta = 15
Beta = 30
Beta = 45
Beta = 60
-0.02
-0.025
-0.03
AoA deg
Figure (6): Cm vs AoA Plot, for cg at 8.5 meters
From the 7.0 xcg plot, it can easily be seen that none of the beta angle
conditions would satisfy the stability requirements. The 8.5 xcg plot shows that
around 13 to 15 degrees of angle of attack, that the vehicle is stable, provided
the beta angle is between –30 and 0 degrees. If the cg cannot be moved for the
vehicle, altering the size and sweep of the fins is possible in order to provide a
more stable vehicle. This will be one process of the trade studies for the design
of the Mars landing vehicle.
Aero Code
Currently the code used for aerodynamic and aerothermodynamic
analysis for the vehicle is contained in aerodat.f and heatflux.f. Aerodat.f
contains the Newtonian, skin friction and viscous interaction code. Only the
Newtonian code is active because the other parts of the code have not been
tested thoroughly. The heatflux code contains the stagnation code, as well as
the code to generate the heatflux at the other thermal points. The heatflux code
will compile, but there is a runtime error that causes the code to crash, still trying
to track down the source of it. The heatflux code for the fins has not been added
yet. For trade studies, any of the vehicle parameters can easily be adjusted, as
well as the placement for the thermal points. The aeroprop code that calculated
the stability of the vehicle can be used to generate data for plots of C m vs AoA for
any configuration.