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Appendix
A.1.0 Aerothermal
A.1.1 Introduction
We discuss in detail all of the required aerodynamic and aerothermal information that is needed
in order to launch a 200g, 1kg, and 5kg payload into low earth orbit. We cover everything from
the research and development costs to detailed numeric values for all of the aerodynamic
coefficients, aerodynamic loads, and aerothermal heating.
Aerodynamic terms, such as coefficients of drag, lift, pressure, and moment (among others) are
needed in order to predict the trajectory, devise the external structure, and confirm the control
system. Generally aerodynamic characteristics are determined via wind tunnel testing; however,
due to the nature of the coursework, making a model and performing wind tunnel tests is not
viable. Therefore, all of our coefficients are devised numerically through extensive codes and
analysis.
The aerodynamic coefficients are predicted to the best of our ability through a multitude of
engineering methods. One such method, Linear Perturbation Theory, is implemented in order to
determine a majority of the aerodynamic coefficients. Programs such as Gambit, FLUENT,
MATLAB, and EXCEL are also employed to aid in determining the aerodynamic coefficients.
We will discuss our methods in detail, in the sections to follow.
Author: Brian Budzinski
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A.1.2 Design Methods
A.1.2.1 Research and Development
The key component for research and development from an aerodynamic standpoint is wind
tunnel testing. Wind tunnel testing is essential in many aeronautical design processes. Wind
tunnel testing gives us data on a scaled down model, which we can then relate to our current
design. This data includes: drag, lift, moment, dynamic and static stability, surface pressure
distributions, flow visualization, wind effects, and heat transfer properties.
Due to the nature of the coursework, we are not using a wind tunnel for our design process.
However, if a full design and build process were to be done, a wind tunnel would be necessary.
The wind tunnel needs to be applicable for subsonic, transonic, supersonic, and possibly
hypersonic regimes. We also need the wind tunnel to allow for changes in temperature, pressure,
and density. This is because the launch vehicle will be traveling through the atmosphere where
these values will vary, thus affecting the design parameters.
We also need to take into account the scaling effects, flow blockage, presence of the model in the
test section, and wall boundary layers. To simulate the real conditions, we must keep the
dimensionless parameters constant when building our scaled down model (Reynolds, Mach, and
Prandtl). Flow blockage occurs in wind tunnels of limited size when testing relatively large
models. The blockage is defined as the ratio of the frontal area of the model to the area of the
test section. Ideally blockage ratios of less than 5% are necessary for aeronautical testing. The
presence of the model in the test section blocks the incoming flow and has the effect of
increasing the pressure on the tunnel walls.
The size of our scaled down model depends on the wind tunnel we use, and there are a variety of
candidates available for us to use. The tunnel needs to take into account the blockage we
mentioned earlier. Therefore, we investigated three different locations, which we chose based on
the upper limits of the free stream velocity they could achieve. However, each location also has
limits on the size of the model that can be used.
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The three locations are the NASA Glenn Research Center (GRC), located in Cleveland, Ohio,
NASA Langley Research Center in Hampton, Virginia, and Purdue University in West
Lafayette, Indiana. Both Purdue and GRC are able to reach a maximum test section Mach
number of 6.0. Glenn Research Center also provides ten discrete flow velocities between Mach
1.3 and 6.0 for their 1’x1’ Supersonic Wind Tunnel (SWT). Glenn also has four additional and
distinct wind tunnels located at the same facility. Table A.1.2.1.1 gives a comparison of the
three different tunnels available, showing parameters which would be important to future testing.
Table A.1.2.1.1: Hypersonic Wind Tunnel Comparison
Test Section
Maximum Mach
Simulated Altitude
Reynolds Number
Dynamic Pressure
Temperature
Maximum Area
Estimated cost
Footnotes:
Purdue’s
“Quiet” Mach 6
6.0
-3e6 – 20e6
616-1862
418
2116
10
NASA
GRC SWT
6.0
3.35 – 35.05
0.4e6 – 16.5e6
3.83 – 83.8
288.9 - 611
929
34
NASA
Langley SWT
5.0
-5e4 – 20e6
0.191 - 167
297 – 366.48
645
--
Units
-km
1/m
kPa
K
cm2
$/hr
Purdue’s “Quiet” Mach 6 wind tunnel is the most feasible for testing a scaled down model of our
particular launch vehicle. It offers the cheapest running rate, the largest allowable model size,
and also proximity since it is located near the main campus in West Lafayette.
Authors: Jason Darby and Brian Budzinski
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A.1.2.2 Sizing Function
The purpose of this part of the project is to come up with a method for determining the shape of
the launch vehicle. The first method we use is to linearly scale the vehicle by payload mass. To
accomplish this, we use the dimensions of two rockets for data points to make the sizing
functions; the Vanguard rocket and the Purdue Hybrid Launch Vehicle. This method, however, is
ineffective at sizing the vehicle because it yields unrealistic overall lengths for small payload
masses. We choose to abandon the linear scaling in favor of sizing the vehicle based on the
volume of propellant in each stage.
The method of sizing the vehicle based on fuel volumes yielded realistic lengths for every
vehicle. The size was more realistic because it was based off of how much propellant each stage
needs instead of a scaling factor based off of the payload mass. However, we had to manually
optimize the length and diameter of the vehicle to obtain the final vehicle dimensions. Since this
proved time consuming, we discontinued use of the Excel version due to a similar method
employed in a large optimization code (MAT code).
To begin the initial sizing of the vehicle, a sizing function was needed. We decided to size the
vehicle by linearly scaling the Vanguard rocket based on payload mass. The linear relationship
was calculated using Vanguard payload mass data along with stage length and diameter data
found from an online source for historical rockets.1 For a second set of data points, the payload
mass, stage length, and stage diameter data from the Purdue Hybrid Launch Vehicle were used.2
This data was then entered into Excel and a linear relation between length and diameter was
found with respect to payload mass for each stage. An example of how the sizing functions were
calculated is shown in Fig. A.1.2.2.1 below.
Author: Chris Strauss
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Project Bellerophon
2nd Stage Length vs Payload Mass
6
y = 0.5702x
R2 = 0.9506
5
Length (m)
4
3
2
1
0
0
2
4
6
8
10
Payload mass (kg)
Fig. A.1.2.2.1: Sizing function regression plot for vehicle second stage.
(Chris Strauss)
Figure A.1.2.2.1 shows the regression plot for the length of the second stage of the launch
vehicle along with the sizing function associated with the stage. This was created by entering the
data for second stage length of Vanguard and the Purdue Hybrid Launch Vehicle versus the
payload mass of each. A linear regression line between the points was then plotted and the
equation of the line was used as the sizing function for the stage length where x is the payload
mass.
We used a similar method on each stage length and diameter until a complete set of dimensions
was calculated for each launch vehicle, and each different payload mass. The results of this
scaling are shown in Table A.1.2.2.1 below for the overall length of the rocket. The results by
stage are shown in Tables A.1.2.2.2 through A.1.2.2.4.
Table A.1.2.2.1 Initial Scaling Overall Length vs. Payload
Mass
Variable
Payload Mass 1
Payload Mass 2
Payload Mass 3
Overall Length 1
Overall Length 2
Overall Length 3
Value
0.20
1.00
5.00
0.51
2.56
12.78
Footnotes: 1,2,3 for lengths refer to masses 1,2,3
Author: Chris Strauss
Units
kg
kg
kg
m
m
m
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Project Bellerophon
Table A.1.2.2.2 Initial Scaling Stage 1 Dimensions vs. Payload
Mass
Payload (kg)
0.20
1.00
5.00
Length (m)
Diameter (m)
0.25
1.26
6.32
0.70
0.74
0.94
Footnotes:
Table A.1.2.2.3 Initial Scaling Stage 2 Dimensions vs. Payload
Mass
Payload (kg)
0.20
1.00
5.00
Length (m)
Diameter (m)
0.11
0.57
2.85
0.39
0.43
0.63
Footnotes:
Table A.1.2.2.4 Initial Scaling Stage 3 Dimensions vs. Payload
Mass
Payload (kg)
0.20
1.00
5.00
Length (m)
Diameter (m)
0.06
0.28
1.38
0.12
0.15
0.33
Footnotes:
From the data presented, we found that a linear scaling method was not a good method to use. As
evidence for this, we looked at the overall length for the 200 gram payload mass and found that it
was unreasonably small at 0.51 meters. The only reasonable dimensions calculated using this
method were those for the 5 kilogram payload where the overall length was nearly half that of
the Vanguard rocket. This is reasonable for a launch vehicle size considering the smaller payload
that will be carried. A new method for sizing the launch vehicle needed to be devised to provide
more accurate results reflecting the actual size of the launch vehicle and payload.
After the linear scaling method was proven to be very inaccurate, we based our next attempt at
sizing the vehicle based on fuel volume. This method relied on finding the amount of fuel burned
for each stage and the densities of the fuel being burned. This information was provided by the
propulsion group. Below are tables showing the results of sizing the vehicle based on fuel
volume.
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Project Bellerophon
Table A.1.2.2.5 5 kg Payload Launch Vehicle Dimensions for
Various Fuel Combinations
Fuels
LOX/HTPB
H2O2/RP-1
AP/HTPB/Al
Length (m)
5.04 (stage 1)
7.63 (stage 2)
1.88 (stage 3)
9.25 (stage 1)
5.57 (stage 2)
2.33 (stage 3)
16.41 (stage 1)
9.36 (stage 2)
2.63 (stage 3)
Diameter (m)
3.00 (stage 1)
4.00 (stage 2)
1.00 (stage 3)
6.00 (stage 1)
4.00 (stage 2)
0.75 (stage 3)
6.00 (stage 1)
4.00 (stage 2)
0.75 (stage 3)
Footnotes:
Table A.1.2.2.6 1 kg Payload Launch Vehicle Dimensions for
Various Fuel Combinations
Fuels
LOX/HTPB
H2O2/RP-1
AP/HTPB/Al
Length (m)
10.35 (stage 1)
6.28 (stage 2)
2.75 (stage 3)
10.96 (stage 1)
4.58 (stage 2)
1.92 (stage 3)
13.51 (stage 1)
4.93 (stage 2)
2.17 (stage 3)
Diameter (m)
6.00 (stage 1)
4.00 (stage 2)
0.75 (stage 3)
5.00 (stage 1)
4.00 (stage 2)
0.75 (stage 3)
6.00 (stage 1)
5.00 (stage 2)
0.75 (stage 3)
Footnotes:
Table A.1.2.2.7 0.2 kg Payload Launch Vehicle Dimensions
for Various Fuel Combinations
Fuels
LOX/HTPB
H2O2/RP-1
AP/HTPB/Al
Length (m)
14.26 (stage 1)
6.01 (stage 2)
2.64 (stage 3)
10.48 (stage 1)
4.38 (stage 2)
1.83 (stage 3)
12.93 (stage 1)
4.72 (stage 2)
2.08 (stage 3)
Diameter (m)
5.00 (stage 1)
4.00 (stage 2)
0.75 (stage 3)
5.00 (stage 1)
4.00 (stage 2)
0.75 (stage 3)
6.00 (stage 1)
5.00 (stage 2)
0.75 (stage 3)
Footnotes:
From the data shown in Tables A.1.2.2.5 through A.1.2.2.7, it can be seen that the vehicle sizes
are all comparable to each other when similar diameters are used. This implies that a single
Author: Chris Strauss
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Project Bellerophon
launch vehicle could be used for all three payloads. This conclusion is based on very minimal
optimization of each stage diameter, however. This data also shows that the method of sizing the
vehicle based on fuel volume provides us with better results than linearly scaling the vehicle
based on payload mass. Since the vehicle has realistic lengths, this method could be used for a
more in depth sizing analysis once a particular fuel combination is chosen for each stage.
This exact method for determining the size of the vehicle’s stages is not used as the final sizing
method; however, since an automatic size optimization routine is included into the MAT code.
The MAT code is then used for all sizing problems through the end of the design process.
References
1
Wade, M., “Vanguard”, 1997-2007. [http://www.astronautix.com/lvs/vanguard.htm]
2
Tsohas, J., “AAE 450 Spacecraft Design Spring 2008: Guest Lecture Space Launch Vehicle Design”, 2008
Author: Chris Strauss
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Project Bellerophon
A.1.2.3 Aerodynamic Coefficients
A.1.2.3.1 Drag Coefficient
The coefficient of drag CD is one of the most important aspects of launch vehicle aerodynamics.
This small, non-dimensional number impacts many features of the overall launch vehicle design.
A few examples include the amount of thrust needed for an appropriate thrust to weight ratio, the
overall ΔV required to reach orbit, and the ability to control the launch vehicle. The CD is
essentially a means of representing the impact a launch vehicle’s shape will have on the amount
of drag incurred as the launch vehicle speeds through the atmosphere. The manner in which the
CD achieves this impact can be seen through Eq. (A.1.2.3.1.1)
D  C D qA
where D is the total drag, q is the dynamic pressure, and A is the area.
(A.1.2.3.1.1)
One of the first goals of the aerothermodynamics group was to further understand the impact of
launch vehicle geometry, Mach number, and angle of attack on the CD. Doing so would allow us
to put some preliminary limits on certain aspects of the launch vehicle design, such as diameter,
and maximum tolerable angle of attack.
The CD is highly dependent upon Mach number. In the subsonic regime CD is relatively low. In
the transonic regime it raises to its highest value, and in the supersonic regime it reduces back to
CD
a lower value. An example of this trend is shown in Fig. A.1.2.3.1.1.
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
Mach number
Fig. A.1.2.3.1.1: Impact of Mach number on CD for V2 rocket.1
(Jayme Zott)
Author: Jayme Zott
7
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Project Bellerophon
Not only is the CD defined by the speed of the launch vehicle, it is also defined by aspects of the
geometry such as diameter, number of fins, and length. Referencing data from the Vanguard and
other historically successful launch vehicles, we realized that as the rocket diameter increased, so
did the CD. To get an idea of exactly how much the CD increased with respect to diameter, we
referenced established model rocket programs. Table A.1.2.3.1.1 shows outputs from the
Aerolab3 model rocket program using Vanguard geometry with varying diameter.
Table A.1.2.3.1.1 The impact of Diameter on CD
Base Diameter
1.00
1.14*
1.25
1.50
1.75
2.00
2.25
2.00
Units
m
m
m
m
m
m
m
m
Max CD
0.37
0.42
0.47
0.70
1.10
1.50
2.10
2.70
Footnotes: * Vanguard base diameter
(Jayme Zott)
From this information we were able to determine that our final launch vehicle diameter should
not exceed 2.00 meters in length. Doing so would lead to undesirable CD values in the transonic
regime.
The angle of attack also has a noticeable impact on the CD. In order to deduce the magnitude of
this impact, we referenced historical data from various launch vehicles.1,2 Using this historical
data, we created general trends for the subsonic, transonic, and supersonic regimes shown in Eqs.
(A.1.2.3.1.2), (A.1.2.3.1.3), and (A.1.2.3.1.4) respectively.
𝐶𝐷 = 0.2083𝑀3 + 0.0445𝑀2 − 0.1494𝑀 + 0.18 + 𝐶𝐷0 + 𝑎10−1 (0.0034𝑀2
− 0.0003𝑀 + 0.4283)
𝐶𝐷 = 0.5293𝑀−1.2374 + 𝐶𝐷0 + 𝑎10−1 (0.0034𝑀2 − 0.0003𝑀 + 0.4283)
(A.1.2.3.1.3)
(A.1.2.3.1.4)
is the initial coefficient of drag, and M is the Mach
𝐶𝐷 = 0.355𝑀−0.5162 + 𝐶𝐷0 + 𝑎10−1 (0.0034𝑀2 − 0.0003𝑀 + 0.4283)
where CD is the coefficient of drag, CD0
(A.1.2.3.1.2)
number, and α is the angle of attack.
Author: Jayme Zott
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Project Bellerophon
By extrapolating these empirical results we were able to show the impact of a wide variety of
angles of attack on CD.
Cd vs. Mach Number for Various AOA
1.1
Angle of Attack (deg)
1
0.9
9 deg
8 deg
Cd
0.8
7 deg
6 deg
0.7
5 deg
4 deg
0.6
3 deg
0.5
2 deg
1 deg
0.4
0 deg
0
1
2
3
Mach number
4
5
6
Fig. A.1.2.3.1.2: Impact of Angle of Attack on CD.3
(Jayme Zott)
Knowing historical trends for the impact of Mach number, launch vehicle geometry, and angle of
attack on the CD is of great use in preliminary analysis. When the team began work on creating a
final design configuration, it was necessary to solve for the CD in a much more refined manner.
In order to take into consideration all elements of the launch vehicle geometry, angle of attack,
and Mach number for the final design analysis, linear perturbation theory was used.
Linear perturbation theory is the method in which the pressure over the top and bottom surfaces
of the launch vehicle is integrated to solve for axial and normal force coefficients acting on the
launch vehicle. From these axial and normal force coefficients, we are then able to use Eq.
(A.1.2.3.1.5) to solve for the CD.
𝐶𝐷 = 𝐶𝑁 sin 𝛼 + 𝐶𝐴 cos 𝛼
(A.1.2.3.1.5)
where CD is the coefficient of drag, CN is the normal force coefficient, CA is the axial force
coefficient, and α is the angle of attack. An explanation of how linear perturbation theory is
implemented can be found in the following sections on aerodynamic forces, A.1.2.3.2-A.1.2.3.7.
Author: Jayme Zott
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Project Bellerophon
References
1
Sutton, George P., and Oscar Biblarz. Rocket Propulsion Elements. New York: John Wiley & Sons, Inc., 2001.
The Martin Company, “The Vanguard Satellite Launching Vehicle”, Engineering Report No. 11022, April 1960.
3
Toft, Hans Olaf. “Aerolab” Software. [http://users.cybercity.dk/~dko7904/software.htm. accessed 1/30/08].
2
Author: Jayme Zott
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Project Bellerophon
A.1.2.3.2 Design Considerations
Aerodynamic forces quickly emerge as a major component of the Project Bellerophon design
process. Center of pressure, normal and axial forces, pitching moments, bending moments and
shear stresses all require analysis. In a more detailed design process (perhaps one with an
operating budget), our analysis would include extensive wind tunnel testing. As a class however,
our hands are tied to theoretical models and numerical solutions. If we were to build this launch
vehicle (LV), wind tunnel testing would be absolutely necessary to ensure its operability. The
following sections describe in detail the process by which we predict all of the aerodynamic
forces on our launch vehicle.
Our aerodynamic forces code was the main platform from which we offered solutions for the
other members of the design team. D&C, Structures, and Trajectory were all affected by its
results. The seed from which the code grew was the search for a valid center of pressure (CP).
We surmised early on in the design process that the CP would be required by the dynamics and
controls group. The CP is the point along the LV body where the various forces acting on the
body act as one force (and by extension, one moment). Our initial research on this subject
yielded several interesting processes by which we could calculate a reasonable CP location.
The simplest method of determining the center of pressure is one very familiar to the model
rocket builders of the world. For these cases, maintaining a CP behind the center of gravity (CG)
is necessary for static stability. In subsonic conditions, a conservative estimate for the LV’s CP is
located at the center of its lateral area.1 For an amateur rocketeer, a common way of using this
information is to make a thin cardboard cutout in the shape of their rocket, and suspend the
cutout across a sharp edge, like a ruler. Since cardboard is of uniform density, and is assumed to
be of negligible thickness, the point about which the cutout is stable is the rocket’s CP.
As a method of doing this computationally, we set up a computer code to determine the projected
area of each launch vehicle section, using a triangle for the nose, a rectangle for the cylindrical
stages, and a trapezoid for the “shoulder” or skirt sections. The code then summed these sections
from the nose to determine the overall area. Halving this value, we designed the code to step
from the nose until it reached the half area, and then determine the fraction of the current section
Author: Alex Woods
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Project Bellerophon
that was on either side of the center point. This resultant point was the CP by the lateral area
method.
There were a few problems associated with this method of finding the Center of Pressure. First,
the model essentially assumed an angle of attack of 90º, which is only present as a launch stand
condition for most LVs. As a result of this initial assumption, the CP location is very
conservative for the purposes of small LVs, and more importantly, not necessarily indicative of
an actual value. Instead it gives a maximum limit for the CG if static stability is required. For the
purposes of the design project, this stipulation was unnecessary, since with the use of gimbaling
or LITVC, static stability is unnecessary, or even undesirable.
With this in mind, our efforts turned toward the Barrowman Method. This is a method of
analytically determining the CP by using the LV’s geometry. The advantages of this method are
several: it gives a much less “conservative” location for the CP, it is relatively simple to
calculate, and it is relatively accurate for the conditions it is designed for. The method involves
dividing the LV into several portions (nose, cylinders, shoulders/boat tails, and fins), and
determining the surface area and volume of each section. These geometric components are
directly related to the coefficients of normal force and pitching moment (CN and Cm respectively)
as follows:
8
 S ( L)  S (0)
d2
8
CM 
 L * S ( L)  V 
d3
CN 
 A.1.2.3.2.1
 A.1.2.3.2.2 
where L is the length of the section, S is the cross-sectional area of the section at the given
location, d is a reference length equal to the diameter of the base of the nosecone, and V is the
volume of the section.
If we take XCP to be the location of the center of pressure, then
C 
X CP  d *  M 
 CN 
 A.1.2.3.2.3
where CM is computed from the tip of the nose cone. By computing XCP for each section, it is
possible to determine the overall location as so:
Author: Alex Woods
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Project Bellerophon
X CP ,Overall 
CN , Nose * X CP , Nose  CN ,Shoulder * X CP ,Shoulder  ...
C
 A.1.2.3.2.4 
N
For a more detailed treatment of the Barrowman method, please refer to “The Theoretical
Prediction of Center of Pressure”.1
Using Vanguard geometry, we designed a computer program to calculate XCP using the
Barrowman method. This gave us a result of 25.9% of the body length from the tip of the
nosecone. This was where we ran into Barrowman’s limitations. The report states a series of
rather strict assumptions, including that the angle of attack is approximately zero, and that the
flow is both steady state and subsonic. The Vanguard report published a wind tunnel center of
pressure graph that starts at the beginning of the transonic regime, but it suggests that the
subsonic CP was around 40%. Furthermore, the Vanguard report includes data for the CP vs.
angle of attack, which deviates quite significantly from the initial zero case. The Barrowman
Method is unable to account for this change and thus we renewed our search for a serviceable
aerodynamic model.
We then considered a third model, Newtonian Theory. The advantages of this model are apparent
simplicity, and a high degree of accuracy even at very high angles of attack. The main
disadvantage is that the method is only valid at very high Mach numbers; starting at about Mach
5 – hypersonic speeds. It was certainly a valuable tool if we found our LV would reach those
speeds, but lacked the range of Mach numbers we would need for the overall design.
For the purposes of Project Bellerophon, our aerodynamic model required several characteristics.
First, we needed it to function over a large range of Mach numbers; i.e. we needed subsonic,
supersonic, and perhaps even hypersonic values. Since transonic flows are poorly understood
even by state of the art computational models, we were forced to “fudge” these values in the final
design, keeping some eye to historic launch vehicles – realizing that wind tunnel testing would
be required for proper analysis here. We also needed our model to accurately predict CP’s across
a range of angles of attack. A study of the Vanguard report revealed that our angle of attack
range would probably not progress past six degrees, but we wanted to be prepared for as much as
fifteen. 4
Author: Alex Woods
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Project Bellerophon
We searched through aerodynamic texts and consulted knowledgeable professors, and finally
came upon the most serviceable method for our needs. Linear perturbation theory allowed us to
compute all of the major aerodynamic forces in a relatively simple fashion. By computing
coefficients of pressure over the surfaces of the launch vehicle, perturbation theory gave us the
building blocks of normal forces, axial forces, shear stresses, bending moments, and the ever
elusive center of pressure.
It is important when using linear perturbation to study the theory’s underlying assumptions. The
first is that the flow is steady and isentropic. The second is that the airflow is irrotational. Third,
that the flow is inviscid. The theory goes on to assume that the changes in the vehicle geometry
(the perturbations) are small, and that the vehicle is at a small angle of attack. The resulting
equation is as follows


2  2 
(1  M 2 ) 2  2  0
x
y
 A.1.2.3.2.5 

where  is the velocity potential, x and y are Cartesian axis directions, and M is mach number.2
The above equation may be written as well in cylindrical coordinates as follows
( M 2  1)
 2  2 1 


0
x 2 r 2 r r
 A.1.2.3.2.6 
where φ is the perturbation velocity potential, r is the radial direction and x is the axial direction.3
Using Eq. (A1.2.3.2.6), the term that governs Cp for slender, axially symmetric bodies is
Cp  
2
2
2  1     1   
 2 

 
 
u0 x u0  r   r   
 A.1.2.3.2.7 
where θ (please note that theta is used for more than one quantity in this report) is the angular
direction in the cylindrical coordinate system and u0 is the free stream velocity.3
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The problem that the aerothermodynamics group ran into was the difficulty in applying the
velocity potential φ – itself a differential equation – to a “real world” problem. As a result, the
decision was made to use a method more correct for airfoils than for launch vehicles, for which a
simpler equation was available. The reason the equations are not the same is that the shape of an
airfoil allows certain cross flow velocities to be neglected.3 Due to the small size of our LV, we
decided that the simplification was reasonable. Also, because we continue to integrate around the
longitudinal axis of the LV, our theory retains some of the accuracy that would otherwise be lost.
Discussion with knowledgeable professors and accuracy of our final numbers has served as
justification of our choice. 4, 5, 6
We make use of an equation from Anderson to calculate our data:
Cp 
Cp 
2
M 2 1
C p ,0
(supersonic)
 A.1.2.3.2.8 
(subsonic)
 A.1.2.3.2.9 
1 M 2
where θ here is the geometric angle of the launch vehicle geometry with respect to the free
stream velocity, and Cp,0 is the incompressible pressure coefficient, for which we also used 2θ.
We
implement
Eqs.
(A.1.2.3.2.8)
and
(A.1.2.3.2.9)
respectively,
using
the
code
CP_Linear_two.m, which runs off the master call_aerodynamics.m. We use the lengths and
diameters of each vehicle section to find the angle θ of the geometry. We then take angle of
attack (α) into account by adding α to θ for the lower surface of the LV and subtracting α from θ
for the upper surface geometry. We compute Cp at many different points along the launch vehicle
surface at regular intervals, creating a pressure distribution vector.
The pressure coefficients form a distribution along the launch vehicle body as shown below.
Please note that the geometry used for the figures in this section is not final. Please refer to the
detailed design of the report for numbers related to our final designs:
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Fig. Section A.1.2.3.2.1: Pressure distribution over the length of a 3 stage launch vehicle at Mach 4.5 and
0° angle of attack
(Alex Woods)
We can see here that geometry, Mach number, and angle of attack are the primary variables that
affect pressure distribution. As geometry changes, the shape of the Cp spikes change, with
higher, thinner spikes coming with shorter, higher angle changes in geometry. The overall
magnitude of the distribution changes with Mach number and the difference between the upper
and lower surfaces grows with angle of attack.
A.1.2.3.3 Normal Force Coefficient
Once CP_Linear_two.m forms pressure distributions, we can integrate those distributions to
derive the aerodynamic forces acting along the LV body. The first of these is the normal force
coefficient, CN. We can integrate using the equation:
L
2
1
CN    r  dz  C p cos  d
S0
0
 A.1.2.3.3.1
where S is a reference area in square meters, r is the radius of the LV at a given point in meters,
L is the overall vehicle length in meters, and θ is the angle of the LV, in radians, with respect to
the windward point. For the purposes of Project Bellerophon, we use the base of the first stage as
the reference area.
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Within CP_linear_two.m, we make this equation work by first integrating numerically around
the LV body for the lower and upper surface, resulting in an “average” C p for each. This is then
integrated along the axial direction by subtracting the upper surface Cp from that of the lower
surface, giving a resultant pressure difference, and multiplying by the radius and the step size
(taken to be 0.1 meters in the analysis). Summing and dividing by the reference area, we
compute CN for the launch vehicle. The behavior of the resulting coefficient may be seen in the
figures to follow.
Fig. A.1.2.3.3.1, Normal coefficient vs. angle of attack for a 3 stage launch vehicle
(Alex Woods)
Fig. A.1.2.3.3.2, Normal force coefficient vs. vehicle length at 6º aoa and M = 3.5 for a 3 stage launch vehicle
(Alex Woods)
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Fig. A.1.2.3.3.3, Normal Force Coefficient vs. Mach number for a 3 stage launch vehicle at 0º angle of attack
(Alex Woods)
We can see from Fig. A.1.2.3.3.1 that normal forces increase with angle of attack. Also we see
from Fig. A.1.2.3.3.2 that CN is distributed over the length of the launch vehicle in a fashion
similar to that of the Cp distribution. Of note is that for zero angle of attack, the output of CN
from CP_Linear_two.m is non-zero, when theoretically it should be zero. This is caused by a
flaw in the code that could not be resolved before the conclusion of this project. The slope of the
CN vs α curve should be steeper than is represented as well. Furthermore, theory predicts a linear
relationship between CN and α, but in the real world this relationship is non-linear, with CN
increasing at a greater rate than predicted. This non-linearity begins around 6º angle of attack,
and becomes too great to neglect at least as early as 14º. Finally, bear in mind that the values
within the transonic region of the graph are of place-holder value only; they are not based on any
valid theory.
A.1.2.3.4 Moment Coefficient
Directly related to the coefficient of normal force is the pitching moment coefficient, CM. We
chose the pitching moment to be the moment about the nose, caused by the normal force acting
at the center of pressure. This quantity is determined theoretically as such:
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Project Bellerophon
L
CM 
2
1
(r )( z )dz  C P cos 
SL 0
0
A.1.2.3.4.1
where z is the distance of the current point from the tip of the nose cone.
By this method, each point along the distribution vector generates a separate moment, and the
magnitude of that moment tends to increase as we move along the body of the launch vehicle. By
summing the vector (integrating in theory) we calculate the overall scalar value of C M. Since CM
is directly related to CN, the changes of CM with angle of attack and Mach number are very
similar in behavior, as can be seen in the figures to follow.
Fig. A.1.2.3.4.1, Variation of pitching moment coefficient with Mach number at 0º aoa
(Alex Woods)
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Project Bellerophon
Fig. A.1.2.3.4.2, Variation of pitching moment coefficient with angle of attack at Mach 3
(Alex Woods)
We can see here that once again, CM has a linear progression with angle of attack and a nice
curve with changing Mach number. There are several characteristics that we must note about
these plots. First, the slope of the plot in Fig. A.1.2.3.4.2 is slightly steeper than that in the
normal load. This is expected, as it produces a changing CP with changing angle of attack. Also,
in reality the plot in Fig. A.1.2.3.4.2 would have some non-linearity, but in a less pronounced
fashion than what one would find in the normal coefficient.3 Finally, we note once again that this
data is not reasonable for the transonic or hypersonic regions.
A.1.2.3.5 Center of Pressure
Since we calculate both normal and moment coefficients, we can produce a reasonable location
for center of pressure using Eq. (A.1.2.3.2.3). For the purposes of this analysis, we found it more
useful to use the following modification, which outputs the CP location as a fraction of the body
length from the tip of the nosecone:
C 
X CP   M 
 CN 
 A.1.2.3.5.1
This equation was used directly to produce a CP location that changes with angle of attack, as the
Vanguard report suggests it should.2 A visualization of this variance can be found in the
following figure.
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Fig. A.1.2.3.5.1, XCP vs. angle of attack for a 3 stage vehicle at Mach 3
(Alex Woods)
Figure A.1.2.3.5.1 shows that the center of pressure will move aft along the LV body as angle of
attack changes, which is what we expect for a launch vehicle.3 We have some issues with the
validity of the results however. We found that the CP values being output by the code tend to
begin lower and higher than real world data, by as much as 30% of the actual value. The change
of CP from minimum to maximum also occurs faster than the Vanguard data would suggest. 2
Finally, the location of the CP does not vary with Mach number in our results. While this is
consistent with linear theory, it does not agree with information found in the Vanguard report.
Vanguard has wind tunnel data showing an aft CP in the subsonic region, and a spike even
farther aft in the transonic region.
In the subsonic region this difference can probably be attributed to viscous effects. Since the
location of the CP is determined by an integral of forces acting along the vehicle surface, it
seems reasonable that if viscous effects were included, they would heighten the effect of long
cylindrical stages present on the launch vehicle. This would be particularly true if flow
separation occurred on the aft surfaces, which is also something not modeled by the aerodynamic
codes. We note the same characteristics in the transonic region, with the addition of possible
shocks as flow accelerates over the vehicle surfaces.
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A.1.2.3.6 Axial Force Coefficient
We find axial force along the launch vehicle is the prime component of drag for low angles of
attack. As such, deriving an axial force coefficient (CA) based upon vehicle geometry is a top
priority for the design team. Once again we turn to linear perturbation theory for a solution.
Using the pressure distribution as described in A.1.2.3.2, we integrate with respect to body
thickness as shown below:
L
2
1
C A   (2 r )dy  C p cos  d
S0
0
(A.1.2.3.6.1)
where dy denotes that we are integrating with respect to thickness, lengthwise along the LV.
Figure A.1.2.3.6.1 provides a visual example.
Fig. A.1.2.3.6.1, Axial force acting along the launch vehicle body
(Alex Woods)
CP_Linear_two.m calculates the CA value in a similar fashion to CN, with the major exception
being that we left out the integration around the launch vehicle (leaving the analysis in two
dimensions). We did this in order to more accurately fit our results to historical data, which was
larger than we were predicting. These differences may have been in part due to viscous or
separation effects along the vehicle body.
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Project Bellerophon
Fig. A.1.2.3.6.2, Variation of axial force coefficient vs. angle of attack for a 3 stage LV
(Alex Woods)
Fig. A.1.2.3.6.3, Variation of CA with Mach number for a 3 stage LV
(Alex Woods)
The model axial force coefficient does not change with angle of attack. This is consistent with
historical and experimental data.3 These results should be reasonable up to at least 10º angle of
attack, and the non-linearity experienced afterwards is not very significant.
We find that the axial force coefficient for a range of Mach numbers is fairly accurate when
compared to historical data. The exception to this is that the Vanguard results have a decrease in
drag for the middle of the subsonic region, while our model predicts a small increase. Recall as
well that the transonic results from our model are not to be trusted.
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Project Bellerophon
The axial force coefficient can be used to find a simple drag value by using the equation:
CD  CN sin( )  CA cos( )
 A.1.2.3.6.2 
where CD is the drag coefficient for the LV. For 0º angle of attack the drag coefficient is equal to
the axial force coefficient. Eq. (A.1.2.3.6.2) predicts an increase in drag as angle of attack
increases, as we expect. Once our model progresses past approximately 14º it no longer predicts
an accurate drag, because sizable flow separations will occur on the leeward side of the LV.
A.1.2.3.7 Shear Coefficient
We find that the derivation of normal forces and pitching moments allows us to derive some of
the forces working within the launch vehicle. Shear stresses and bending moments are important
considerations for the structures personnel to factor in to their analysis. To provide a solution, the
aerothermodynamics group developed a code called CP_Structures.m. This code analyzes the
lowest “connection point” on the LV at any given time. We define this as the point where the
skirt meets the lowest stage; for our final models this was always the top of the first stage.
We first derived the theory behind the shear stress on the LV, and ran the method by our
structures contacts to promote accuracy. We defined shear stress as the force of one stage acting
on another in a horizontal fashion.
Fig. A.1.2.3.7.1, Normal coefficient along a LV surface
(Alex Woods)
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Project Bellerophon
The shear stress is the differential between the normal forces acting on the LV on either side of
the shearing point. This means that if the sections of the launch vehicle are causing different
amounts of aerodynamic force, the differences between those sections is going to manifest as
shear forces within the vehicle structure. Or more bluntly,
x
L
0
x 1
Shear   CN   CN
 A.1.2.3.7.1
where x is the shear point. If this value emerges negative it means that the forces acting on the
lower portion of the vehicle (the first stage) are greater than those on the rest of the vehicle.
CP_Structures.m is designed to ignore shoulder sections such that the shear output is for each
stage along the vehicle. The maximum loads experienced are at the junction between the first
stage and the upper stages.
A.1.2.3.8 Bending Moment
The bending moment is slightly more complicated to compute than the shear stresses. In
principle it once again uses the normal force. Since moment equals force multiplied by distance,
and each section of the vehicle geometry has a local center of pressure, we can say that the
normal force acting over a section of the LV will cause a moment acting about a point from the
local CP. This can be visualized in Fig. A.1.2.3.8.1 below.
Fig. A.1.2.3.8.1, Bending moments caused by normal forces acting at local centers of pressure
(Alex Woods)
If we take a point within the vehicle geometry to sum the moments about, we have an opposing
moment pair causing the structure to fold in on itself. If we take nose up to be a positive pitching
moment, the value for the first moment will be:
Cbend ,1  CN ,1 *  X CP,1
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(A.1.2.3.8.1)
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Project Bellerophon
where Cbend,1 is the portion of the bending moment caused by the upper stages, CN,1 is the normal
force coefficient acting on the same section and XCP,1 is the local center of pressure. XCP,1 is
negative because we take the nose tip to base direction to be positive. We then take a moment
about the point caused by forces acting on the other side of the launch vehicle. Please note that
XCP,2 will be positive because it is on the opposite side of the summing point:
Cbend ,2  CN ,2 * X CP ,2
(A.1.2.3.8.2)
where all the variables are identical to those in Eq. (A.1.2.3.8.1), but for the opposite side. These
two moments can then be summed to create the overall bending moment:
Cbending  Cbend ,1  Cbend ,2
 A.1.2.3.8.3
where Cbending is the bending moment. Since CN,2 is causing a nose down moment, this will be
subtracted from the first moment, causing us to sum moments, just like what common sense
would dictate. Because of the way we defined the unit vectors, the moment being output by
CP_Structure.m is negative, but the magnitude is the same as if it were a positive moment, and
just as important to the design process.
References
1
Barrowman, James and Barrowman, Judith, "The Theoretical Prediction of the Center of Pressure" A NARAM 8,
August 18, 1966. www.ApogeeLVs.com
2
Anderson, John D., Fundamentals of Aerodynamics, Mcgraw-Hill Higher Education, 2001
3
Ashley, Holt, Engineering Analysis of Flight Vehicles, Dover Publications Inc., New York, 1974, pp. 303-312
4
Klawans, B. and Baughards, J. "The Vanguard Satellite Launching Vehicle - an engineering summary" Report No.
11022, April 1960
5
Steven Collicott, Ph.D., In personal communication regarding linearized perturbation theory, 2:00-2:30 at his
office in Armstrong Hall, Purdue University on Feb. 6 th 2008.
6
Marc Williams, Ph.D, Personal communication regarding pressure distribution and derivation of aerodynamic
forces, 2:30-3:30 at his office in Armstrong Hall, Purdue University on Feb. 19 2008
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A.1.2.4 Lift and Lifting Bodies
Though lifting bodies are not implemented on the final design, they are still researched in order
to determine a cost effective means of launch. Lifting bodies, such as a wing, are beneficial for
an aircraft launch. We discuss in detail the aerodynamic coefficients which include lift, drag, and
moment that are created with the addition of lifting bodies.
Lifting bodies create additional nose up pitching moments that would allow for the launch
vehicle to pitch from an initial horizontal configuration, which is assumed to be angle of attack
zero degrees, to a final vertical configuration, which is assumed to be an angle of attack of 90
degrees. This extra nose up pitching moment is needed if an aircraft launch configuration is
considered.
To help us better visualize this configuration, refer to Fig. A.1.2.4.0 below.
Fig. A.1.2.4.0 Launch Vehicle with a Delta Wing Configuration
(Kyle Donahue)
A.1.2.4.1 Drag and Drag Coefficient
Though the pitching moment is a known benefit of the wing, induced drag is not. Induced drag
is defined as a drag force which occurs whenever a lifting body or a finite wing generates lift. If
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Project Bellerophon
all other parameters are held constant, the induced drag will increase with increasing angle of
attack. Let us look deeper into this subject.
The induced drag is calculated using
D
1
   V 2  S C D
2
(A.1.2.4.1.1)
where D is the induced drag, ρ is the air density, V is the true airspeed, S is the reference area,
and CD is the coefficient of drag.1
It was previously noted that induced drag increases with increasing angle of attack. But this is
not apparent from Eq. (A.1.2.4.1.1). Therefore, in order to see this relation we must further
dissect Eq. (A.1.2.4.1.1). The variable that changes with angle of attack is the coefficient of
drag. This is shown using
C D  C N  sin   C A  cos 
(A.1.2.4.1.2)
where CD is the coefficient of drag, CN is the normal force coefficient, α is the angle of attack,
and CA is the axial force coefficient.1
The normal force and the axial force coefficients can then be computed for a lifting body. The
derivation of the coefficients follow three basic steps: first we must determine the geometric
shape of the body, next we must integrate the theoretical pressure coefficients over the body and
evaluate the basic force coefficients, and finally we must determine the appropriate moment
coefficients from the vehicle center of mass. All of the extensive integrations necessary to derive
the aerodynamic force coefficients are omitted and only the results are presented.
For this analysis, we assume an aircraft launch, being that an aircraft launch is the only launch
configuration that requires a wing. In order to determine the normal and axial force coefficients
we make several assumptions. We implement the Newtonian Model; this assumption is made
because the launch vehicle is traveling at supersonic and hypersonic speeds throughout most of
the trajectory. We assume turbulent flow; once again this is a valid assumption due to the high
velocities. Finally a delta wing configuration is employed.
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A.1.2.4.2 Normal and Axial Force Coefficients
With the assumptions stated, we can now determine the axial and normal force coefficients. In
order to determine the total axial and normal force coefficients we must divide the wing surface
up into two separate parts, the leading edge and the lower surface. The leading edge and the
lower surface are chosen because they are the two portions of the wing that are exposed to the
relative wind given an angle of attack. The normal and axial force coefficients from the leading
edge are found using
 4  RLE  l LE 
CN  
  k LE  sin  cos  e  cos   cos  
3 S


(A.1.2.4.2.1a)
 4  RLE  l LE  k LE
2
CA  
 cos cos  e  cos   cos  

3 S

 2
(A.1.2.4.2.1b)
where CN is the normal force coefficient, RLE is the radius of the leading edge, lLE is the length of
the leading edge, S is the reference area, kLE is the correction factor for the leading edge, Λ is the
wing sweep, Λe is the effective wing sweep, α is the angle of attack, and CA is the axial force
coefficient.1
Next we must look at the lower surface of the wing. The normal and axial force coefficients
from the lower surface can be found using
S 
C N  k LS   LS   sin 2 
 S 
(A.1.2.4.2.2a)
2.2
 S  0.45 cos   4.65V 10,000sin   cos 
CA  G   w 
V  c   0.2
 S 
(A.1.2.4.2.2b)
(Laminar Flow)
2.25
1.5
 S  0.048 sin 4.5   0.70V 10,000 cos   sin 
CA  G   w 
V  c   0.2
 S 
(A.1.2.4.2.2c)
(Turbulent Flow)
where kLS is the lower surface correction factor, SLS is the lower surface area, S is the reference
area, α is the angle of attack, Sw is the wing area, V∞ is the relative velocity, c is the chord length,
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μ∞ is the relative air viscosity, G 
2 1  m1 n 

 , n = 0.5 laminar, n = 0.8 turbulent, and m
n(1  n)  1  m 2 
is the planform taper ratio.1
Once we find the normal and axial force coefficients for the leading edge and the lower surface,
the total normal and axial force coefficients are determined by summing the two.1
C N  C N , LE  C N , LS
(A.1.2.4.2.3a)
C A  C A, LE  C A, LS
(A.1.2.4.2.3b)
Now that the axial and normal coefficients are known, they can be substituted back into Eq.
(A.1.2.4.1.2) to solve for the coefficient of drag. Prior to doing that though, let us first look at the
behavior of the normal and axial force coefficients against angle of attack. Logically the normal
force should be the greatest when the launch vehicle is at a high angle of attack. Therefore, as the
angle of attack is increased, the normal force should also increase. This can be shown through
Fig. A.1.2.4.2.1.
Fig. A.1.2.4.2.1 Normal Force Coefficient vs. Angle of Attack
(Brian Budzinski)
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On the other hand, the axial force should be the greatest when flying directly into the relative
wind, or at a zero degree angle of attack. As the angle of attack is increased, the axial force
should decrease. This can be shown through Fig. A.1.2.4.2.2.
Fig. A.1.2.4.2.2 Axial Force Coefficient vs. Angle of Attack
(Brian Budzinski)
Now we are ready to further discuss the performance of the drag coefficient versus angle of
attack. Understandably, the drag coefficient increases with increasing angle of attack. This
behavior can be seen through Fig. A.1.2.4.2.3 below. The addition of the wing will generate a
drag coefficient of approximately 1.1 at a 90 degree angle of attack, as shown by Fig.
A.1.2.4.2.3.
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Fig. A.1.2.4.2.3 Drag Coefficient vs. Angle of Attack
(Brian Budzinski)
A similar process can be used in order to determine the drag imparted through the addition of
fins. Equation (A.1.2.4.1.1) and Eq. (A.1.2.4.1.2) still apply; however, the axial and normal force
coefficients will be different. In order to determine the normal and axial force coefficients, we
must look at Eq. (A.1.2.4.2.4) below. If we assume a pair of fins,
CN  
8  RF  l F  k LE
cos 2  F   sin  F
3 S
CA  2  kF 
(A.1.2.4.2.4a)
SF
8  RF  l F  k LE
 3 cos 2  
cos 2  F   cos  F (A.1.2.4.2.4b)
S
3 S


where CN is the normal force coefficient, RF is the radius of the fin(s) leading edge, lF is the
length of the fin(s), kLE is the correction factor for the leading edge, S is the reference area, Λ F is
the sweep of the fin(s), α is the angle of attack, CA is the axial force coefficient, SF is the fin area,
and λ is the correction for the sweep angle.1
To help us better visualize this configuration, refer to Fig. A.1.2.4.2.4 on the following page.
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Fig. A.1.2.4.2.4 Launch Vehicle with a Pair of Fins
(Kyle Donahue)
Similar to the wing, once we know the axial and normal force coefficients for the fins, those
values can be inserted into Eq. (A.1.2.4.1.2) in order to determine the generated drag. If a delta
wing and a pair of fins are added to the launch vehicle, the individual axial and normal force
coefficients are summed to determine the total axial and normal force coefficient, much like Eq.
(A.1.2.4.2.3). For a pair of fins and a delta wing configuration the total axial and normal force
coefficient is calculated as shown through Eq. (A.1.2.4.2.5) below.1
C N  C N , LE  C N , LS  C N , F
(A.1.2.4.2.5a)
C A  C A, LE  C A, LS  C A, F
(A.1.2.4.2.5b)
These values can then be inserted into Eq. (A.1.2.4.1.2) in order to determine the total induced
drag generated by this configuration.
A.1.2.4.3 Moment and Moment Coefficient
Now that the drag and drag coefficient have been thoroughly covered, let us discuss in further
detail the pitching moment that is incurred. As aforementioned, the addition of a wing will
increase the nose up pitching moment, thus allowing the launch vehicle to pitch into a vertical
configuration. Let us discuss this phenomenon in more detail. In order to determine the pitching
moment by the addition of a wing, we once again must divide the wing up into two separate
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Project Bellerophon
sections: the leading edge and the lower surface. The pitching moment coefficient for the
leading edge is calculated by means of
C m  C N , LE
x LE
c
 C A, LE
z LE
c
(A.1.2.4.3.1)
where Cm is the moment coefficient, CN is the normal force coefficient about the leading edge,
xLE is the axial distance from the leading edge to the center of mass, c is the chord, CA is the axial
force coefficient about the leading edge, and zLE is the radial distance from the leading edge to
the center of mass.1
Similarly we find the moment coefficient about the lower surface
C m  C N , LS
x LS
c
 C A, LS
z LS
c
(A.1.2.4.3.2)
where Cm is the moment coefficient, CN is the normal force coefficient about the lower surface,
xLS is the axial distance from the lower surface to the center of mass, c is the chord, CA is the
axial force coefficient about the lower surface, and zLS is the radial distance from the lower
surface to the center of mass.1
Comparable to the total normal and axial force coefficients, the total moment coefficient is found
by summing the leading edge term and the lower surface term. As one may assume, the moment
coefficient will increase with increasing angle of attack. This is because the upward pitching
exposes more of the lower wing surface to the relative wind, increasing the force applied. This
increase in moment coefficient versus angle of attack can be seen through Fig. A.1.2.4.3.1 on the
following page.
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Fig. A.1.2.4.3.1 Moment Coefficient vs. Angle of Attack
(Brian Budzinski)
In order to calculate the moment coefficient for the addition of a pair of fins, the mathematics
become a little more involved. We now can calculate the moment coefficient for a pair of fins
using Eq. (A.1.2.4.3.3).
C m  2  k F 
z F , LE
 x F , LE

SF 3
 z  8  RF  l F  k LE
 cos 2   F  
cos 2  F     
sin  F 
cos  F 
S
3 S
c
 c 
 c

(A.1.2.4.3.3)


where most of the variables were defined by Eq. (A.1.2.4.2.4) above, and xF and zF are the axial
and radial distances from the fin leading edge to the launch vehicle center of mass respectively.1
Once the moment coefficients have been calculated, we determine the pitching moment using
Eq. (A.1.2.4.3.4).
M  Cm  q  S  c
(A.1.2.4.3.4)
where M is the moment, Cm is the moment coefficient, q is the dynamic pressure, S is the
reference area, and c is the chord length.
Though it may be difficult to tell from the previous equations, through the addition of a wing, the
nose up pitching moment is increased. Seeing as the wing is mounted on the first stage of the
Author: Brian Budzinski
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Project Bellerophon
rocket, it is aft of the aerodynamic center. Since the moment caused through the addition of the
wing is aft of the aerodynamic center, the launch vehicle pitches upward.
A.1.2.4.4 Lift Coefficient
Lift is yet another important aerodynamic characteristic that should be reviewed. Any structure
or body can generate lift once an angle of attack is encountered. Moreover, the addition of a
wing, referred to previously as a lifting body, will create lift due to reaction forces. The lift force
is the equal and opposite force created from an object, such as an airfoil, turning the relative fluid
flow perpendicular to its original direction. Therefore, the lift coefficient, much like the drag
coefficient, is calculated using the axial and normal forces as shown in Eq. (A.1.2.4.4.1).
C L  C N  cos   C A  sin 
(A.1.2.4.4.1)
where CL is the lift coefficient, CN is the normal force coefficient, α is the angle of attack, and CA
is the axial force coefficient.1
As expected, the lift coefficient increases with increasing angle of attack. We can see this
through Fig. A.1.2.4.4.1.
Fig. A.1.2.4.4.1 Lift Coefficient vs. Angle of Attack
(Brian Budzinski)
Author: Brian Budzinski
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Project Bellerophon
At approximately 53 degrees angle of attack, the wing reaches the maximum lift. Once the angle
of attack is pushed beyond the maximum, the lift begins to decrease dramatically. Though an
angle of attack of 53 degrees may seem excessive for a traditional configuration, for a hypersonic
vehicle with a delta wing design, this is commonplace. To help better understand this
phenomenon, let us briefly discuss how a delta wing generates lift. A delta wing uses vortices to
generate lift rather than straight air flow. Since straight flow is disrupted by high angles of attack,
a traditional wing becomes dysfunctional at high angles. However, with a delta wing
configuration, high angles of attack increase vortices, thus increasing the lift.2
Additionally, the relationship between lift and drag is shown in Fig. A.1.2.4.4.2.
Fig. A.1.2.4.4.2 Drag Coefficient vs. Lift Coefficient
(Brian Budzinski)
A.1.2.4.5 Shear Coefficient
The final aerodynamic force that we discuss is shear force. A shear force occurs when shear
stress is encountered. Shear stress is defined as the stress that acts parallel or tangential to the
face of a material as opposed to normal stress which acts in a perpendicular manner. Though the
details of shear stress are not thoroughly covered in this section, particularly because shear is a
structural problem, the results from the addition of a wing and/or fins are covered.
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For the simplicity of an aerodynamic viewpoint, the shear force imparted on the launch vehicle
through the addition of a wing is considered equal to the normal force acting on the wing itself.
This concept is more easily seen through Fig. A.1.2.4.5.1 below.
Fig. A.1.2.4.5.1 Shear Imparted on the Launch Vehicle by the Wing
(Brian Budzinski)
Therefore, as the angle of attack of the wing increases, the normal force also increases. This
increase in normal force thus increases the shear induced on the launch vehicle. The maximum
shear coefficient is found to be approximately 1.1 which can be shown through Fig. A.1.2.4.5.2
below.
Fig. A.1.2.4.5.2 Shear Coefficient vs. Angle of Attack
(Brian Budzinski)
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The analysis of the shear induced on the launch vehicle from the addition of fins follows suit.
We find the shear force imparted on the launch vehicle through the addition of fins by assuming
that it is equal to the normal force acting on the fin itself. Once again, this can be more easily
shown through Fig. A.1.2.4.5.3 below.
Fig. A.1.2.4.5.3 Shear Imparted on the Launch Vehicle by the Fins
(Brian Budzinski)
A more in depth analysis is required in order to determine the cost effectiveness of fins. We
neglect to go into great detail of this matter. The addition of fins would require less stabilization
control from D&C. However, the method for stabilization control that we implement does not
require the addition of fins.
In summary, the use of a wing and/or fins is very beneficial if an aircraft launch configuration is
to be considered. The additional nose up pitching moment is advantageous if the launch vehicle
is launched from a horizontal configuration. Furthermore, fins are a favorable method for
stabilizing the rocket as they eliminate the need for a costly thrust vectoring method.
References
1
Hankey, Wilbur L., Re-Entry Aerodynamics, AIAA, Washington D.C., 1988,
pp. 70-73
Rhode, M.N., Engelund, W.C., and Mendenhall, M.R., “Experimental Aerodynamic Characteristics of the Pegasus
Air-Launched Booster and Comparisons with Predicted and Flight Results”, AIAA Paper 95-1830, June 1995.
2
Author: Brian Budzinski
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Project Bellerophon
A.1.2.5 Computational Fluid Dynamics
As computer technology has greatly advanced, it has become an industry standard to use
Computational Fluid Dynamics, CFD, as a preliminary form of aerothermodynamic analysis. A
cheaper alternative to wind tunnel testing, CFD allows engineers to obtain accurate solutions to a
variety of aerothermodynamic concerns. Because most aerodynamic theory falls apart in the
transonic regime, it is hard to get accurate results using basic equations and analytical solutions.
It is much more accurate to create a mock up of the launch vehicle and place it in a wind tunnel
to retrieve physical results.
Creating a mock up of the launch vehicle becomes a very time consuming and costly task
however, when the design begins to advance. As the design progresses, the launch vehicle
geometry begins to change; since most aerothermodynamic loads are based on geometry, they
are constantly changing as well. Every time the geometry of the launch vehicle changes, a new
launch vehicle mock up needs to be built, and more wind tunnel tests need to occur. The
alternative to these costly wind tunnel tests is CFD.
A CFD analysis can output the same type of information as a wind tunnel test in a timelier, more
cost effective manner. Instead of paying for new launch vehicle mock ups to be created with
each change in geometry, changes can simply be made in a computer aided design, CAD,
software program such as CATIA, ProEngineer, or SolidWorks. CFD can then be completed for
each phase of the design, and costs associated with wind tunnel testing become obsolete.
Completing a CFD analysis on a launch vehicle can be broken down into a four step process:
1. Create a model of the launch vehicle in a CAD software program.
2. Import the launch vehicle geometry into a meshing program, such as Gambit or
StarCCM+, and mesh the geometry.
3. Import the meshed geometry into a CFD program such as Fluent or Stardesign, set design
parameters and environmental conditions, and run the program.
4. Post-process the output and analyze the results.
Author: Jayme Zott
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Project Bellerophon
The results can then be used to determine whether or not the aerothermodynmic loads exceed
tolerable values. If they do, a new design will need to account for these loads, and if not, more
analysis can be done on other components of the launch vehicle design.
What makes CFD nearly as accurate as wind tunnel testing are the numerical methods imbedded
internally within the CFD program. By meshing the CAD model first, the launch vehicle is
broken down into small pieces. When placed into the CFD program, solutions to Navier-Stokes
equations are integrated across each of these small pieces, and summed in order to solve for a
multitude of aerodynamic loads. Outputs can range from pressure, temperature, and velocity
distributions to coefficient of drag, coefficient of pressure, and moment coefficient acting on the
launch vehicle.
CFD is an incredibly advantageous tool because it allows for geometry changes as well as
environmental changes to be taken into consideration. By specifying the appropriate boundary
conditions one can change the speed and angle of attack of the launch vehicle, account for
changes in temperature, density, and pressure of the surrounding atmosphere, and even include
viscous effects and shock waves.
Due to the cost, time, and inaccessibility of a wind tunnel, we decided to use CFD as a means of
determining aerothermodynamic loads at designated intervals throughout the launch. In order to
exploit Fluent’s symmetry capability we created a model of half of the 1 kg launch vehicle using
CATIA. Splitting the launch vehicle in half reduces the complexity along with the amount of
time to needed to solve the problem. We then saved this model as a “.igs” file, and imported into
GAMBIT.
Once in GAMBIT, the model was nearly ready to be meshed. In order to account for the fact that
air flows around the launch vehicle and not through it, the area surrounding the launch vehicle
model needed to be meshed, rather than the launch vehicle itself. To do this, we created a large
rectangular prism surrounding the launch vehicle. The launch vehicle geometry was then
subtracted from this rectangular prism leaving only the area surrounding the launch vehicle to be
meshed.
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Project Bellerophon
To begin, we meshed the edges of the rectangular prism with a spacing of 0.8. Next, the longest
symmetry plane edges of the launch vehicle were meshed with a spacing of 0.13, and the
smallest symmetry plane edges of the launch vehicle were meshed with a spacing of 0.05. Using
the edge mesh sizes as guides, we meshed the faces of the launch vehicle and the faces of the
rectangular prism next. We created both of these face meshes using a triangular mapping pattern.
Finally, the volume surrounding the launch vehicle was meshed using a tetrahedral hex-core
pattern. The results of the mesh can be seen in Fig. A.4.1.2.5.1 below.
Fig A.4.1.2.5.1: Mesh of 1Kg launch vehicle in GAMBIT
(Jayme Zott, Chris Strauss, Brian Budzinski)
After meshing was complete, we broke up the launch vehicle into zones. We designated the face
of the rectangular prism in front of the launch vehicle as a pressure inlet, and the face behind the
launch vehicle as a pressure outlet. We designated the face aligned with the symmetry plane of
the launch vehicle as symmetry, and the remaining faces as walls.
Once meshing was complete and the launch vehicle had been broken up into zones, we exported
the mesh into Fluent. Table A.4.1.2.5.1 describes the settings and boundary conditions we chose
within Fluent.
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Project Bellerophon
Table A.4.1.2.5 Summary of Fluent settings and boundary conditions for 1 Kg launch
vehicle at 350 m/s.
Setting/Boundary Condition
Solver
Pressure based
GG node based
Implicit
Steady
Energy
On
Viscosity
Inviscid
Materials
Ideal Gas
Operating Conditions
Pressure Inlet Boundary Conditions
Total Pressure
Supersonic
Pressure Outlet Boundary Conditions
Outlet Pressure
Solution Controls
Pressure/Velocity
Pressure Model
Pressure Accuracy
Courant Number
Relaxation factor
Value
-----------0.00 [atm]
1.30 [atm]
0.65 [atm]
0.65 [atm]
Coupled
Standard
2nd order upwind
5.0
0.5
We based our choices for the settings and boundary conditions shown in table A.4.1.2.5.1 on
Fluent tutorials1, Fluent webinars2, conversations with graduate students and professors3, and
trial and error. The solver was chosen to be pressure based because pressure based is most
accurate for supersonic flows. The energy equation was turned on as a requirement for
incompressible flow. The viscosity was chosen to be inviscid, because viscous forces are
negligible at zero angle of attack. The boundary conditions for the pressure inlet and outlet were
chosen based on the desired launch vehicle velocity. The remaining settings and boundary
conditions were based more on trial and error than anything else. Overall, we attempted many
different solution possibilities, from adapting the gradient of the grid to account for the formation
of shock waves, to testing out a density based solver, to reducing the Courant number all the way
to 0.01. There were many different options tested, and while our output seems intuitively
reasonable, it is hard to say whether or not the settings displayed in table A.4.1.2.5.1 are the best
for analyzing the supersonic flow of air around our launch vehicle.
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Project Bellerophon
The results for the pressure distribution of the 1kg launch vehicle traveling 350 m/s at zero angle
of attack can be seen in Fig. A.4.1.2.5.2. The scale on the left displays a color schematic
representing the range of pressures distributed across the launch vehicle. The lowest pressure,
colored blue, begins at 0.37 atm, and the greatest pressure, colored red, stops at 1.56 atm. The
pressure is highest at the locations where a sharp edge occurs, and lowest in the areas
immediately after them. Based on our initial boundary conditions, and the high probability that
the flow is separating near the base of each skirt, these results seem reasonably accurate.
Figure A.4.2.1.5.2 Pressure distribution of 1 Kg launch vehicle
(Jayme Zott)
The results for the velocity distribution of air surrounding the 1kg launch vehicle traveling 350
m/s at zero angle of attack can be seen in Fig. A.4.1.2.5.3. The scale on the left begins in blue at
5.89 m/s, and ends in red at 411 m/s.
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Project Bellerophon
Figure A.4.2.1.5.3 Velocity distribution of air surrounding 1 Kg launch vehicle traveling 350 m/s
(Jayme Zott)
The velocity is greatest at the locations where the skirts end, and lowest slightly after that
location. Shocks are most likely forming at the base of the skits where a significant change in the
launch vehicle geometry occurs. These probable shock locations correlate well with the velocity
distribution, and the velocity magnitudes correlate well with our initial boundary conditions. We
therefore assume that the results are reasonably accurate for use in our aerodynamic analysis.
Since the bottom line aerodynamic analysis for the launch vehicle design was completed using
call_aerodynamics.m, we used CFD as a sanity check for the linear perturbation theory output.
With both of these methods working together, we were able to get a solid idea of the type of
aerodynamic loading the launch vehicle was likely to experience throughout its flight.
References:
1
Ansys Fluent “Fluent 6.3 Tutorial Guides”, Fluent Inc. 2006.
2
Fluent: Fluid Flow Modeling Webinars, Compressible Flows - Solving Compressible Flow Problems; June 25,
2005. [http://www.fluent.com/elearning/resources/webinars/webinars_cfd.htm. accessed 1/29/08].
3
Charles Merkle PhD, Reilly Professor of Engineering, Personal communication, Mechanical
Engineering Building, Purdue University, 1/29/08.
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Project Bellerophon
A.1.2.6 CMARC
We now detail our process to determine the aerodynamic coefficients using the computational
fluid dynamics package CMARC. This process is used in place of Fluent due to Fluent’s
exceptionally long run time; also the results obtained in CMARC are reasonably close to those in
calculated in Fluent.
We found that while running a full three dimensional CFD simulation to obtain aerodynamic
coefficients, took a long time to reach a converged solution. This led to our decision to come up
with an alternative method to find these coefficients.
We decided to use a panel method solver called CMARC and its post-processing program
POSTMARC for this analysis. The reason we chose this program is because of the time it takes
to run a full three dimensional viscous case.
Fluent, the program we were using prior to CMARC, took several hours to run only a small
fraction of one case, while CMARC can run a full three dimensional viscous case in
approximately five minutes. There is a slight difference between the results obtained from
CMARC and Fluent. This is because Fluent is a full Navier-Stokes solver whereas CMARC is a
panel method solver where the accuracy of the solution is based on the number of panels in the
model. The results, however, are close enough to the Fluent solution to be useful.
To begin our CMARC model, we enlisted the help of a doctoral student, Liaquat Iqbal, who has
had much experience working with CMARC and POSTMARC. We used his method of creating
CMARC input files in Excel to create our model geometry. A sample of the input prompt is
shown in Figure A.1.2.5.1 below.
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Project Bellerophon
Figure A.1.2.5.1: Sample CMARC Parameter Input
(Chris Strauss)
From Figure A.1.2.5.1, we can see that different design parameters of the launch vehicle such as
stage length and diameter, nose cone length, skirt length, and (if a wing is present for an aircraft
launch case) wing parameters can be easily changed to analyze the current configuration.
The launch vehicle, for this case, had wings because this method was originally used to analyze
aircraft launch configuration. The launch vehicle would have a wing attached to the first stage
enabling it to pitch into a vertical trajectory. After the first stage burned out, the wing would be
discarded along with the first stage. This method, however, is flexible enough so that nonwinged rockets can also be analyzed. We accomplished this by setting the wing span to zero.
After the parameters are entered, the Excel sheet is saved in a format that is readable in CMARC.
The input file is then run in CMARC which creates an output file for use in POSTMARC. Once
this output file is entered into POSTMARC, the pressure distribution and aerodynamic
coefficients are found using the program’s aerodynamic coefficient calculation routine. The
pressure distribution on a winged aircraft launched vehicle can be seen below in Figure
A.1.2.5.2.
Author: Chris Strauss
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Project Bellerophon
Figure A.1.2.5.2: Pressure distribution on winged air drop rocket at a 0 deg. angle of attack
produced in POSTMARC
(Chris Strauss)
As seen in Figure A.1.2.5.2, the pressure is at a maximum on the nose cone and the leading edge
of the wings. This is as expected and thereby supports the accuracy of using CMARC and
POSTMARC for the calculation of the aerodynamic coefficients. The model’s flexibility is
shown in Figure A.1.2.5.3 below.
Figure A.1.2.5.3: Pressure distribution on wingless rocket at 0 deg angle of attack
(Chris Strauss)
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Project Bellerophon
Figure A.1.2.5.3 shows a modification to the original model. This model uses the same input
sheet as the winged model except in this case the wingspan was set to zero to allow a wingless
rocket to be analyzed. Again, the figure shows that the pressure distributions are as expected
with the highest pressure on the nose cone and lower pressures along the rest of the rocket body.
This again shows that the model is reasonable for aerodynamic analysis of the vehicle.
While this model appears useful when the preliminary cases are run, a major flaw is present. This
is not a flaw in the model, but rather with the limitations of the CMARC/POSTMARC package.
We find that CMARC calculations are only valid up to Mach 0.9. This effectively ends the use of
CMARC as a primary CFD tool because the launch vehicle quickly achieves supersonic
velocities after being launched. Had these supersonic cases been run in CMARC, erroneous
results would have been obtained and jeopardized the integrity of the project.
Author: Chris Strauss
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Project Bellerophon
A.1.2.7 Ascent Aeroheating Analysis
Due to the nature of the coursework and limited time constraints, a thorough ascent aeroheating
analysis was “black boxed”. A subsequent analysis would be necessary for a final design;
however, we do not believe that glossing over this subject was detrimental to our design.
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A.1.3 Closing Comments
The aerothermodynamics group survives trial and tribulation to bring you, the reader, reasonable
aerodynamic data. This data includes aerodynamic coefficients and their corresponding forces.
For example: drag, lift, moment, normal and axial, and shear. We also find the location of the
center of pressure and aerodynamic heating.
If the project is revisited, aerothermodynamics has several recommendations for the design. For
future improvements, the launch vehicle geometry should be a prime driver. The large variance
in stage to stage diameter produces excess drag that can be minimized with a more slender and
uniform configuration. Additionally, the drag versus launch vehicle diameter should be taken
into consideration when determining the nominal launch vehicle geometry.
The consideration of drag was not taken into account when first determining a cost effective
means of launch. Therefore, with more time and more tools, it is beneficial to further study the
effects of drag with launch vehicle diameter.
We find the aerodynamic coefficients to the best of our ability, though several complications
arise in the transonic and hypersonic regimes. The models we implement are not applicable for
transonic and hypersonic flows. Further study must be done in order to determine the exact flow
effects in those regimes. The best method to determine the flow effects in the transonic regime is
wind tunnel testing. As aforementioned, due to the nature of our coursework, making a model
and performing wind tunnel tests is not viable. However, the predictions that we make for the
transonic regions are unsound, they err on the high side. Being that our vehicle is making it into
an acceptable orbit indicates that the aerodynamics experienced in reality would cause improved
performance.
As for the flow in the hypersonic region, the Newtonian method should be employed. Due to the
launch configuration, the launch vehicle is far out of the atmosphere before reaching hypersonic
speeds. This ensures that our model remains valid for the implemented configuration. On the
other hand, if a different configuration is to be used, rather than the model that we implement,
our model would need to be reformatted in order to take into account the hypersonic regime.
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All in all, the applied model is satisfactory for our launch configuration and all of the
aerodynamic coefficients are reasonable for our specific design.
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A.1.4 User’s Guides for Aerothermal Codes
User’s Guide for call_aerodynamics.m
Written by Jayme Zott
Revision 2.0 – 3/18/08
Description:
The purpose of the code ‘call_aerodynamics.m’ is to output all aerodynamic loads pertinent to
the launch vehicle design. These outputs include Mach number, coefficient of drag (both
historically based and dimensionally based), normal coefficient, pitching moment coefficient, the
center of pressure location, shear forces acting on the launch vehicle, and moments acting on the
launch vehicle. All of these aerodynamic loads are output with respect to Mach number, and
angle of attack (in essence, time).
Files necessary to run call_aerodynamics.m:
Filename
Author
Solve_cd_int.m
(Jayme Zott)
CP_Structure.m
(Alex Woods)
CP_Volume_two.m
(Alex Woods)
CP_overall_int.m
(Jayme Zott)
Atmosphere4.m
(historical code)
CP_Dimensions.m
(Jayme Zott)
CP_Linear_two.m
(Alex Woods, Jayme Zott)
Assumptions:
There are quite a few assumptions made within the call_aerodynamics.m code. Assumptions can
be broken down for each output.
Mach number – comes directly from trajectory’s output. Any assumptions trajectory
makes in their code can also be regarded as assumptions for the aero code.
Cd (historical) - The coefficient of drag with respect to Mach number and angle of attack
for a multitude of historically successful launch vehicles was analyzed and compressed
into a single equation for each region of flight, subsonic, transonic, and supersonic. These
equations became the basis for the historical coefficient of drag code.
Normal – the normal force acting on the launch vehicle was found by assuming linear
supersonic flow. The launch vehicle was broken up into sections so the coefficient of
normal force could be calculated for each section. Once each section had been found,
their values were added together to output the final coefficient of normal force, which is
then multiplied by the dynamic pressure.
Pitching moment – the pitching moment acting on the launch vehicle was found in the
same manner as the normal force stated above. Linear supersonic flow was also assumed.
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Xcp – the location of center of pressure is measured in meters from the tip of the nose.
The same assumptions used in solving CN and CM are also assumed in the case of solving
Xcp as the location of the center of pressure is simply the ratio of the pitching moment
coefficient to the normal coefficient.
Shear – The shear force acting on the launch vehicle is found by integrating the normal
force over a specified length of the launch vehicle.
Bending Moment – the moment is computed by measuring the center of pressure for a
specified length of the launch vehicle and then multiplying the distance between that
center and the stage by the integrated normal force.
Axial – The axial force acting on the launch vehicle is found by integrating the pressure
distribution around the launch vehicle and multiplying it by the radius of the launch
vehicle at each measured location. These values are then added up in much the same
manner as the CM and CN were.
CD (dimensional) – Calculated using the axial force coefficient, normal force coefficient,
and angle of attack. Assumptions used in finding those values hold here as well.
Important Notes:
In the final design, trajectory’s code uses Cd (historical), and the angle of attack is considered to
be 0 degrees at all times.
Input Section:
The call line of the function is:
[M, Cd, Normal, Pitching_Moment, Xcp, Shear_Force, Bending_Moment, Axial, CD] =
call_aerodynamics(V,r,a,D)
All of the variables that are passed into the function are described below:
Variable Name
Description
V
Velocity – can be input in the form of an array, or as a
single variable [UNITLESS]
a
Angle of Attack – measure of the angle between the free
stream velocity and the centerline of the launch vehicle.
Can be input in the form of an array, or as a single
variable. It must, however, have the same length as M.
It would be wise to check: length(a) = length(M)?
[DEGREES]
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Project Bellerophon
r
Radius – measure of distance from surface of the Earth to
the rocket. This value cannot exceed 290,000 ft, as this is
considered the end of the atmosphere. It also happens to
be the location at which atmosphere4.m falls apart.
Rocket Dimensions – Must be input as a matrix
comprised in the following manner:
D = [N,L,D,T]
N - the location of the geometry in order starting from the
nose of the launch vehicle
- example: nose = 1, shoulder = 2, stage 2 = 3,…
L - length of specified portion of launch vehicle
[METERS]
-example: nose = 3, shoulder = 4, stage 2 = 6,…
D
D - diameter of specified portion of launch vehicle
[METERS]
-example: nose = 1, shoulder = 2, stage 2 = 2,…
T - Type of geometry used in the vehicle
1 specifies conical nose
(example: nose cone)
2 specifies cylinder
(example: all stages)
3 specifies Shoulder or Boattail (example: skirt)
This is an example of what the Vanguard launch vehicle's
"D" matrix would look like:
D = [N, L D T]
D = [ 1,86.4,.84, 1
2,159-86.43,.84,2;
3,358-159,.84,2;
4,57.638,1.14,3;
5,448.19,1.14,2; ]
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Project Bellerophon
Output Section:
Variable Name
Description
M
Mach number [UNITLESS]
Cd
Coefficient of drag (historical)[UNITLESS]
Normal
Normal Force [N]
Pitching Moment
Pitching moment [N-METERS]
Xcp
Location of Center of Pressure – measured from tip of
nose cone [METERS]
Shear
Shear force – shear force acting on each stage [N]
Bending Moment
Moment – moment force acting on each stage [NMETERS]
Axial
Axial Force – axial force acting on the launch vehicle [N]
CD
Coefficient of Drag (dimensional) [UNITLESS]
Sample Output:
The variables named in the output section will print to the screen. Outputs occur at intervals of 1
second. Nothing other information will be output. Suggestion: write all output variables to an
excel file using the xlswrite(‘filename’, variable) command in MATLAB.
Author: Jayme Zott
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Project Bellerophon
User’s Guide for wing_fin.m
Written by Brian Budzinski
Revision 2.0 – 3/29/08
Description:
The purpose of the code ‘wing_fin.m’ is to output all aerodynamic coefficients and loads needed
for the addition of a wing and/or fins for an aircraft launch. These outputs include lift coefficient,
drag coefficient, normal coefficient, axial coefficient, pitching moment coefficient, and shear
coefficient. All of these aerodynamic coefficients and loads are output with respect to Mach
number, and angle of attack.
Assumptions:
Several assumptions are made within the ‘wing_fin.m’ function. The assumptions can be further
explained as follows:
Mach number – The Mach number is assumed to range from Mach 1.0 to Mach 15.0.
This is assumed because for an aircraft launch the initial Mach would be near 1.0 and the
wing would remain attached up until near Mach 15.0.
Angle of Attack – The angle of attack is assumed to range from an initial horizontal
configuration (AoA = 0°) to a final vertical configuration (AoA = 90°).
Normal – The normal force was found assuming the Newtonian Method. Though the
Newtonian Method is geared towards hypersonic flow, the launch vehicle will quickly
enter the hypersonic regime with an aircraft launch. Therefore, this remains a reasonable
assumption.
Axial – Similarly, the axial force was found assuming the Newtonian Method.
Shear – The shear force acting on the launch vehicle from the wing and/or fin is assumed
to be equivalent to the normal force acting on the wing and/or fin itself.
Pitching Moment – The pitching moment is found by calculating the normal and axial
forces and multiplying them by their corresponding distances from the center of mass.
Input Section:
All of the variables that are passed into the code are described below:
Variable Name
Description
h
Altitude – measure of the height above ground level, and is initially
chosen to begin at 12,000 m for an aircraft launch. [m]
rho
Air Density – an array initially chosen from an altitude of 12,000 m to
a final altitude of 205,000 m. [kg/m3]
T
Air Temperature – an array initially chosen from an altitude of 12,000
m to a final altitude of 205,000 m. [K]
Author: Brian Budzinski
205
Project Bellerophon
Variable Name
Mach
Description
Mach Number – an array initially chosen to range from Mach 1.0 to
Mach 15.0. [Non dimensional]
alpha
Angle of Attack – measure of the angle between the free stream
velocity and the centerline of the launch vehicle. Initially chosen to
range from 0 to 90 degrees for an aircraft launch. [Degrees]
taper_ratio
Taper Ratio – measure of the taper ratio of the wing attached to the
rocket. [Non dimensional]
c
sweep
lambda_eff
S, S_w, S_LS
Chord – measure of the chord length of the wing. [m]
Wing Sweep – the angle at which the wings are swept backwards.
[Radians]
Effective Sweep – the effective sweep angle [Radians]
Reference Wing Area [m2]
S_fin
Reference Fin Area [m2]
length_LE
Leading Edge Length [m]
length_fin
Fin Length [m]
Rn
RLE
Rf
Xn
Zn
XLE
ZLE
XLS
ZLS
Xfin
Nose Radius – Radius of the nose of the launch vehicle. [m]
Leading Edge Radius – Radius of the leading edge of the wing. [m]
Fin Radius – Radius of the leading edge of the fin(s). [m]
Axial distance from nose to center of mass. [m]
Radial distance from nose to center of mass. [m]
Axial distance from wing leading edge to center of mass. [m]
Radial distance from wing leading edge to center of mass. [m]
Axial distance from wing lower surface to center of mass. [m]
Radial distance from wing lower surface to center of mass. [m]
Axial distance from fin to center of mass. [m]
Author: Brian Budzinski
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Project Bellerophon
Zfin
Radial distance from fin to center of mass. [m]
Output Section:
All of the variables that are passed out of the code are described below:
Variable Name
Description
Normal Force Coefficient [Non dimensional]
Cn
Ca
CL
Axial Force Coefficient [Non dimensional]
Lift Coefficient [Non dimensional]
CD
Drag Coefficient [Non dimensional]
CM
Pitching Moment Coefficient [Non dimensional]
q
Shear Force Coefficient [Non dimensional]
Sample Output:
The variables listed in the output section will be plotted as functions of angle of attack.
Author: Brian Budzinski