METHODS FOR OPTIMIZATION OF A LAUNCH VEHICLE FOR PRESSURE
FLUCTUATION LEVELS AND AXIAL FORCE
Except where reference is made to the work of others, the work described in this thesis is
my own or was done in collaboration with my advisory committee. This thesis does not
include proprietary or classified information.
________________________
Scott Walter Thomas
Certificate of Approval:
________________________
Brian Thurow
Assistant Professor
Aerospace Engineering
________________________
Roy J. Hartfield, Chair
Professor
Aerospace Engineering
________________________
Robert Gross
Associate Professor
Aerospace Engineering
________________________
George T. Flowers
Interim Dean
Graduate School
METHODS FOR OPTIMIZATION OF A LAUNCH VEHICLE FOR PRESSURE
FLUCTUATION LEVELS AND AXIAL FORCE
Scott Walter Thomas
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
August 9, 2008
iii
METHODS FOR OPTIMIZATION OF A LAUNCH VEHICLE FOR PRESSURE
FLUCTUATION LEVELS AND AXIAL FORCE
Scott Walter Thomas
Permission is granted to Auburn University to make copies of this thesis at its discretion,
upon request of the individuals or institutions and at their expense. The author reserves
all publication rights.
________________________
Signature of Author
________________________
Date of Graduation
iv
VITA
Scott Walter Thomas was born on February 24, 1982, in Huntsville, Alabama to
James Thomas and Violet Rigdon. He grew up in a small town south of Huntsville called
Lacey?s Spring and went to high school at A. P. Brewer High School. Upon graduating
high school he started college at Wallace State Community College in August 2000 then
transferred to Calhoun Community College the next semester in January 2001. He later
transferred to Auburn University in August 2002 where he pursued an Aerospace
Engineering degree. As an undergraduate student he became a member of several honor
societies and also worked alternating semesters as a co-op engineer with Aerotron
AirPower, Inc. in LaGrange, GA through the Auburn University Co-op program. Scott
graduated Magna Cum Laude with a Bachelor of Aerospace Engineering degree in May
2006. He went on to attend graduate school at Auburn University pursuing a Master?s of
Science in Aerospace Engineering.
v
THESIS ABSTRACT
METHODS FOR OPTIMIZATION OF A LAUNCH VEHICLE FOR PRESSURE
FLUCTUATION LEVELS AND AXIAL FORCE
Scott Walter Thomas
Master of Science, August 9, 2008
(B.A.E., Auburn University, 2006)
83 Typed Pages
Directed by Roy J. Hartfield, Jr.
A computational fluid dynamics (CFD) code has been combined with a Genetic
Algorithm (GA) to perform a shape optimization study on a two dimensional
axisymmetric model of a typical launch vehicle. The objective of this study was to
demonstrate a methodology for reducing pressure fluctuations and the axial force
coefficient for a launch vehicle throughout a typical ascent trajectory. Due to the high
computational expense and difficulty of generating an adequate mesh autonomously, few
CFD driven GA optimizations have been conducted. Some of the complexity of this
process was alleviated by using a simple two dimensional axisymmetric geometry to
model the vehicle.
The optimization process involved the GA selecting a set of geometric parameters
that define the shape of the vehicle. A grid generator created a mesh based on these
vi
parameters and a CFD solver calculated the flow parameters. The grid generator is a
FORTRAN routine written for this particular geometric shape. The FORTRAN code
created a mesh file dependent only on the geometric variables chosen by the GA. The
pressure fluctuation level and axial force coefficient are calculated by the flow
parameters that are obtained from the CFD solution.
A pressure fluctuation level minimization study and axial force minimization
study were conducted separately using the same CFD model. The results of each
optimization study were compared to a baseline geometry having a very similar shape to
the Ares I Crew Launch Vehicle. The results of the pressure fluctuation study yielded a
reduction in the average RMS pressure fluctuation level throughout the ascent trajectory.
The average RMS fluctuating pressure level was reduced by approximately 17.5%
compared to the baseline geometry; however the optimized geometry would not be
favorable as a practical design for a launch vehicle shape. While the resulting optimized
geometry for the pressure fluctuation study is not an ideal design, the methodology for
reducing pressure fluctuations using a GA combined with CFD is shown. The axial force
minimization study yielded a reduction in the axial force coefficient of approximately
56%. The resulting shape from the axial force minimized solution was found to resemble
that of a blunted ogive, as expected.
vii
ACKNOWLEDGEMENTS
The author would like to thank Dr. Roy Hartfield for his guidance and patience
with this thesis as well as Ravi Duggirala and Josh Doyle for their assistance regarding
technical matters involving the Linux cluster and CFD solver. The author also wishes to
thank the author of the IMPROVE 3.1 Genetic Algorithm, Dr. Murray Anderson. The
author would also like to thank his friends, family, and fianc? for their ongoing support
and motivation as he worked to complete this thesis.
viii
Style manual or journal used:
The American Institute of Aeronautics and Astronautics Journal
Computer software used:
Improve 3.1 Genetic Algorithm, Fluent, Tecplot 360, Force 2.0 Fortran Compiler,
Microsoft Excel, Microsoft Word
ix
TABLE OF CONTENTS
LIST OF FIGURES ............................................................................................................. x
LIST OF TABLES ........................................................................................................... xiii
NOMENCLATURE ......................................................................................................... xiv
1 INTRODUCTION ....................................................................................................... 1
2 LAUNCH VEHICLE MODEL .................................................................................... 4
2.1 MODEL GEOMETRY ......................................................................................... 4
2.2 FLIGHT CONDITIONS ...................................................................................... 6
2.3 CFD MODEL ....................................................................................................... 8
2.4 MESHING THE MODEL .................................................................................... 9
2.5 GRID REFINEMENT STUDY .......................................................................... 12
2.6 PRESSURE FLUCTUATION MODEL ............................................................ 19
3 LAUNCH VEHICLE OPTIMIZATION ................................................................... 22
3.1 CODE STRUCTURE ......................................................................................... 22
3.2 PRESSURE FLUCTUATION MINIMIZATION STUDY ............................... 26
3.2.1 CONVERGENCE CRITERIA ............................................................ 26
3.2.2 PRESSURE FLUCTUATION MINIMIZATION RESULTS............. 29
3.3 AXIAL FORCE MINIMIZATION STUDY ...................................................... 44
3.3.1 CONVERGENCE CRITERIA ............................................................ 44
3.3.2 AXIAL FORCE MINIMIZATION RESULTS ................................... 47
4 CONCLUSIONS AND RECOMMENDATIONS .................................................... 61
REFERENCES .................................................................................................................. 65
APPENDIX A: GA Input File ........................................................................................... 69
x
LIST OF FIGURES
Figure 1: 3D Representation of Launch Vehicle Geometry ................................................ 4
Figure 2: Launch Vehicle Geometry with Design Variables .............................................. 6
Figure 3: Altitude and Dynamic Pressure as a Function of Mach Number ........................ 7
Figure 4: Example of Launch Vehicle Mesh .................................................................... 10
Figure 5: Close-up of Grid Near Conic Sections .............................................................. 11
Figure 6: Comparison of Course and Fine Mesh .............................................................. 13
Figure 7: Axial Force Convergence for Considered Meshes in Refinement Study .......... 14
Figure 8: Axial Force Coefficient Throughout Ascent for Course and Fine Mesh ........... 16
Figure 9: Comparison of Computation Times at Each Flight Condition .......................... 17
Figure 10: Pressure Distribution Plot for Course Grid ...................................................... 18
Figure 11: Pressure Distribution Plot for Fine Grid .......................................................... 18
Figure 12: Location of Pressure Fluctuation Level Calculation ....................................... 19
Figure 13: Effect of Flight Speed on Velocity Profiles ..................................................... 20
Figure 14: Flow Diagram for a Typical Genetic Algorithm ............................................. 23
Figure 15: Flow Diagram for GA Running Multiple Members Simultaneously .............. 24
Figure 16: Residuals and Local Pressure Convergence for the Course Mesh and Mach
0.85 Flight Condition ...................................................................................... 28
Figure 17: Fluctuating Pressure Level throughout Ascent for Baseline and Optimized
Geometries ...................................................................................................... 31
Figure 18: Maximum Average and Minimum Pressure Fluctuation Level Evolution ...... 32
xi
Figure 19: Minimum Fluctuating Pressure Level Evolution ............................................. 33
Figure 20: Pressure Fluctuation Level Study Geometry Comparison............................... 34
Figure 21: Pressure Fluctuation Study Variable Distribution for Rc1 .............................. 36
Figure 22: Pressure Fluctuation Study Variable Distribution for Rc2 .............................. 37
Figure 23: Pressure Fluctuation Study Variable Distribution for LcTot ........................... 38
Figure 24: Pressure Fluctuation Study Variable Distribution for Lc1 .............................. 39
Figure 25: Pressure Fluctuation Study Variable Distribution for Lc3 .............................. 40
Figure 26: Pressure Fluctuation Study Pressure Distribution Plot for Baseline and
Optimized Geometries for Mach 0.85 Flight Condition ................................. 41
Figure 27: Pressure Fluctuation Study Dynamic Pressure Distribution Plot for Baseline
and Optimized Geometries for Mach 1.50 Flight Condition .......................... 42
Figure 28: Pressure Fluctuation Study Velocity Distribution Plot Comparison at Mach
0.85 Flight Condition ...................................................................................... 43
Figure 29: Velocity Distribution Close-Up for Baseline and Optimized Geometries for
Pressure Fluctuation Study at Mach 0.85 ........................................................ 44
Figure 30: Residuals and Axial Force Coefficient Convergence for the Course Mesh and
Mach 0.85 Flight Condition ............................................................................ 46
Figure 31: Axial Force Coefficient throughout Ascent for Baseline and Optimized
Geometries ...................................................................................................... 48
Figure 32: Maximum Average and Minimum Axial Force Coefficient Evolution .......... 49
Figure 33: Minimum Axial Force Coefficient Evolution ................................................. 50
Figure 34: Best Performer and Baseline Geometry Comparison ...................................... 52
Figure 35: Axial Force Study Variable Distribution for Rc1 ............................................ 53
xii
Figure 36: Axial Force Study Variable Distribution for Rc2 ............................................ 54
Figure 37: Axial Force Study Variable Distribution for LcTot ........................................ 55
Figure 38: Axial Force Study Variable Distribution for Lc1 ............................................ 56
Figure 39: Axial Force Study Variable Distribution for Lc3 ............................................ 57
Figure 40: Axial Force Study Pressure Distribution for Baseline and Optimized
Geometries at Mach 1.50 Flight Condition ..................................................... 59
xiii
LIST OF TABLES
Table 1: Geometric Parameters and Dimensions of Baseline Launch Vehicle ................... 6
Table 2: Flight Conditions Considered During the Ascent Trajectory ............................... 8
Table 3: Mesh Sizes Considered for Grid Refinement Study ........................................... 13
Table 4: Prescribed Design Space for Both Optimization Studies.................................... 26
Table 5: Optimized Parameters and Design Space ........................................................... 51
xiv
NOMENCLATURE
CFD - Computational Fluid Dynamics
GA - Genetic Algorithm
Rc1 - Radius of 1st Conic Section
Rc2 - Radius of 2nd Conic Section
LcTot - Total Length of Conic Sections
Lc1 - Length of 1st Conic Section
Lc2 - Length of 2nd Conic Section
RANS - Reynolds Averaged Navier Stokes
LES - Large Eddy Simulation
k - Turbulent Kinetic Energy
? - Turbulent Dissipation Rate
RNG - Renormalization Group
i - Grid Direction Parallel to Wall
j - Grid Direction Perpendicular to Wall
y+ - Normalized Turbulence Length
1
1 INTRODUCTION
Pressure fluctuation levels have long been a concern among launch vehicle and
aircraft designers.1-4 These pressure fluctuations are the result of turbulent flow passing
over the surface of the vehicle and are often referred to as aerodynamic noise. The
pressure fluctuations can be severe for certain high Reynolds number, high dynamic
pressure flows, and the fluctuations can be translated to the structure of the vehicle as
vibration which is of great significance concerning structural integrity. The resulting
vibration can lead to fatigue and potentially severe damage to the vehicle or sensitive
equipment during flight. Pressure fluctuations are predominately important during the
ascent phase of the launch vehicle and are particularly important during the transonic
flight regime and near the maximum dynamic pressure point in the flight. The most
severe cases occur when transonic conditions overlay with maximum dynamic pressure
conditions. An additional interest during the ascent phase of launch vehicles and for
missiles is axial force reduction. This thesis presents a methodology for accomplishing
both pressure fluctuation reduction and axial force reduction.
Adjoint methods5 and other gradient based approaches6 have been demonstrated to
be effective for well-behaved aerodynamic shape optimization applications. For less
well-behaved problems and for problems with discrete design variables or problems with
discontinuous objective functions, population based techniques offer a more versatile and
2
robust approach to optimization, especially if multiple design goals are considered. In
particular, binary encoded Genetic Algorithms have been shown to be effective and
robust for a range of complex aerodynamic design optimization applications.7-40 For
robustness in dealing with a range of potential objective functions involving the
prediction of pressure fluctuation levels, the Genetic Algorithm based approach was
chosen for this effort.
Applications of Genetic Algorithms (GA?s) in the aerospace industry include
design of wings, airfoils and propellers,7-17 missiles and rockets,18-25 structures,26 flight
and orbital trajectories,27,28 and control systems.29-31 This thesis describes the use of a GA
to optimize the aerodynamic shape for the forebody of a launch vehicle. The GA used for
this study is the IMPROVE? code written by Dr. Murray Anderson.32 This is a binary
encoded tournament based GA and features many advanced techniques such as a pareto
option, nicheing, and elitism and is used in Refs. 17, 19-23, 35, 39, and 40. The
population members for this study are similar geometric shapes which are initially
randomly produced by the GA within a prescribed design space. Once a geometric shape
is defined the GA passes the member to an objective function where a grid for the
geometry is generated and the flow parameters are calculated by a computational fluid
dynamics (CFD) solver. The value of the parameter to be minimized is then sent back to
the GA for evaluation of that member?s performance. The goal of the GA is to find a
geometric shape with minimum pressure fluctuation level or axial force coefficient
throughout several flight conditions in a typical launch trajectory.
The objective function for this effort is a combination of aerodynamic parameters
obtained from CFD solutions for the flow surrounding candidate designs. The use of a
3
CFD model as an objective function for a GA can be very computationally expensive. In
order to shorten run times the CFD solver is run on a Linux cluster of microprocessors.
Also a simple two-dimensional axisymmetric model is employed to further reduce the
computational demands. Various configurations were tested to determine which solver
model to use as well as what grid size to use for the model. This paper describes the
development of the CFD model and the implementation of this model into the GA to
minimize the fluctuating pressure level and axial force coefficient. The procedure for
combining a CFD solver with a GA is similar to that described in Refs. 35, 39 and 40 for
freight truck applications.
4
2 LAUNCH VEHICLE MODEL
2.1 MODEL GEOMETRY
The geometric shape that is of interest for this study is the forebody of a launch
vehicle. The particular vehicle shape used in this study is based on the proposed
geometry for the Ares I Crew Exploration Vehicle. To clearly show the vehicle shape a
three dimensional representation of the geometry is shown in Figure 1. The geometry
consists of a blunted ogive nose tip followed by a slender cylindrical section. The
geometry then expands through three conic sections to a larger cylindrical section.
Figure 1: 3D Representation of Launch Vehicle Geometry
The 2D axisymmetric model used in this study and the variables defining the
geometry are shown in Figure 2. This geometry is axisymmetric about the dotted line.
5
Note that the length of the model shown in Figure 2 is shortened to more clearly show the
geometry features. The geometry is completely defined by the eleven variables shown in
Figure 2, however not all variables are included as design variables. For this study the
only dimensions changed during the GA optimization process are the dimensions
defining the three conic sections. These variables are Rc1, Rc2, LcTot, Lc1, and Lc3 as
shown in Figure 2. The remaining dimensions are held constant but are included in the
model to more accurately capture the flow characteristics. Also, the aft end of the launch
vehicle is not included in the model since only the forebody of the vehicle is of interest in
this study. The actual parameters adjusted by the GA are dimensionless parameters where
the cone radii and total length of the conic sections are relative to the base diameter, Rb,
and the length of each conic section is relative to the total length of the conic sections.
The dimensionless parameters that define the baseline model for this study are shown in
Table 1. This baseline geometry was chosen to resemble the Ares I Crew Launch
Vehicle.
6
Figure 2: Launch Vehicle Geometry with Design Variables
Table 1: Geometric Parameters and Dimensions of Baseline Launch Vehicle
Parameter Dimensionless In meters
Rb 1.00 2.5380
Rc1 0.75 0.9518
Rc2 0.50 1.2690
LcTot 2.00 5.0760
Lc1 0.15 0.7614
Lc3 0.45 2.2842
2.2 FLIGHT CONDITIONS
The goal of this study is to demonstrate a methodology for optimizing a launch
vehicle shape with a minimum axial force coefficient and minimum fluctuating pressure
level over a range of flight conditions seen in a typical launch trajectory. The effects of
axial forces are only significant during the early part of the launch trajectory and
7
problems caused by pressure fluctuations are usually seen during the ascent phase and
during maximum dynamic pressure. For this reason only flight conditions up to
approximately 50,000 ft were considered, where pressure and density are about 11% and
15% respectively of sea level conditions in a standard atmosphere. The launch vehicle
ascent trajectory used for this study is shown in Figure 3, showing altitude and dynamic
pressure as a function of Mach number. This ascent trajectory is similar to that of the
Saturn V launch vehicle.41
Figure 3: Altitude and Dynamic Pressure as a Function of Mach Number
It is important to include the condition of maximum dynamic pressure since axial
force is generally high during this flight condition. For this ascent trajectory the
0
5000
10000
15000
20000
25000
30000
35000
40000
0
20000
40000
60000
80000
100000
120000
140000
160000
0 1 2 3 4 5 6
Dy
na
mi
c P
re
ss
ur
e (
N/
m2
)
Al
tit
ud
e (
ft)
Mach Number
Altitude
Dynamic Pressure
8
maximum dynamic pressure condition occurs at a Mach number of approximately 1.50
and an altitude of about 39,300 ft. Also, since pressure fluctuations are generally high
during transonic flight due to shock instabilities, it is important to include flight
conditions near the transonic flight regime. While it would be best to include all the flight
conditions in the trajectory, it would not be practical due to the extensive amount of time
it would take to run every flight condition through the CFD solver. For this reason only
five conditions are modeled in this study. The exact flight conditions considered for this
study are shown in Table 2.
Table 2: Flight Conditions Considered During the Ascent Trajectory
Flight
Condition Mach Number Altitude Temperature Pressure
1 0.50 7100 274.08 77889
2 0.85 16667 255.13 53445
3 1.15 28600 231.49 32058
4 1.50 39300 216.65 19396
5 2.00 51800 216.65 10636
2.3 CFD MODEL
The CFD solver used for both the pressure fluctuation level and axial force
minimization studies was the Fluent CFD solver. The solver is operated on a Linux
cluster of microprocessors that has a total of 30 nodes where each node consists of two
AMD Opteron 242 (64-bit) chips for a total of 60 processors. Fluent is a robust CFD
software package with a wide range of capability for modeling fluid flow. For both
studies the Fluent CFD software solves the steady-state Reynolds-Averaged Navier-
Stokes (RANS) equations using a cell-centered finite-volume method for integration. The
9
RANS equations allow for a solution of the mean flow parameters with a reduced
computational expense compared to other methods such as Large Eddie Simulation
(LES). The Reynolds-Averaged approach is commonly used for many practical
engineering applications.
Both CFD models use an axisymmetric segregated solver such that the momentum
and continuity equations are decoupled. The fluid was modeled using Fluent?s built-in
properties for air where density was modeled assuming an ideal gas. Also, the energy
equation is activated, since the modeled trajectory goes through a range of high-speed
compressible flow conditions.
To more accurately model the flow around the vehicle, a built-in turbulence model
was used. There are several different options within Fluent for modeling turbulence,
however the k-? turbulence model was deemed suitable for this study as it is widely used
for both incompressible and compressible flows. The k-? model is a two equation
turbulence model that includes the turbulent kinetic energy, k, and the turbulent
dissipation rate, ?. While there are a variety of k-? turbulence models, such as the
Renormalization Group (RNG) and Realizable approaches, the standard k-? model was
employed for this study.
2.4 MESHING THE MODEL
The grid generator is a FORTRAN routine that develops a structured mesh based
on the variables that define the geometry. To clearly show the structure of the mesh a
very course mesh for the model is illustrated in Figure 4. The entire mesh is structured
and contained in one zone. Indexing starts at the nose of the model and extends to the
10
right for the increasing i direction and outward from the wall for the increasing j
direction. The most difficult part of programming the grid generator was mapping the
nodal points to provide an adequate mesh while also avoiding grid overlap. Due to the
model geometry and the use of a structured grid a region of potential grid overlap can
occur. This region is indicated by the dashed oval in Figure 4. Grid overlap becomes
more difficult as the angle of the first conic section becomes larger. Grid overlap is
avoided in the grid generator by mapping the nodal points so that the grid lines extending
in the j direction away from the wall gradually curve away from the region of potential
grid overlap.
Figure 4: Example of Launch Vehicle Mesh
Other regions of interest in the mesh are the nose, the conic sections, and the nodal
point distributions along the cylindrical sections in the i direction. The nodes near the
nose and conic sections of the model are kept very dense due to the pressure gradients
seen in these areas. Spacing in the i direction for both the nose and conic sections is held
at the same constant value. This spacing value is determined by specifying the number of
points along the first arc of the nose. The nodal points are kept perpendicular to the wall
Region of Potential
Grid Overlap
11
throughout the nose section; however this is not possible for the small cylindrical section
or the conic sections. A close-up of the very course mesh near the conic sections is shown
in Figure 5. This figure also shows more clearly how the grid lines in the j direction
gradually curve as the nodal points get further away from the wall. The points extending
from the wall near the conic sections are kept near perpendicular and can be seen in
Figure 5.
Figure 5: Close-up of Grid Near Conic Sections
An attempt is made to reduce the number of unnecessary elements by varying the
spacing of points along the cylindrical sections. A cosine distribution is used for the small
cylindrical section such that the nodal points are close together near the nose and first
conic section whereas the points are much more spaced out through the middle of the
section. A cosine distribution is also used for the large cylindrical section however the
function is modified such that the spacing of the points are close together near the end of
12
the conic sections and gradually become more spaced out as the points get near the end.
Since for this study the impact on the axial force coefficient from only the forebody of
the vehicle is of interest, the aft end is not included in the model.
Spacing of nodal points in the j direction is controlled using a hyperbolic tangent
function. This function allows for variable spacing of the nodal points in the j direction.
The function is set up such that nodal points near the wall are spaced very closely
together while the spacing gradually increases as the points get further out from the wall.
This allows for adequate grid resolution near the wall while not having excessive element
density far from the wall. The maximum distance from the wall deemed to be adequate
for this flow field is seven large cylinder diameters in the j direction. To determine the
first point from the wall the Near Wall Model is implemented. This model consists of
approximating the skin friction coefficient to estimate the shear stress at the wall. With
the estimated shear stress the friction velocity can be calculated, which allows the
normalized turbulence length y+ to be determined as a function of the distance y normal
to the wall. Setting y+ equal to one and solving for y gives an appropriate distance for the
first nodal point from the wall. This method is similar to that used in reference 39 and 40.
2.5 GRID REFINEMENT STUDY
A grid refinement study was conducted on the model for a fixed geometry to ensure
accurate results. This fixed geometry is also considered the baseline shape for this study.
The number of nodal points in both the i and j directions were varied to obtain a course
mesh and a fine mesh. Table 3 shows the different mesh sizes and Figure 6 shows a
comparison of the two meshes investigated in the grid refinement study. The two images
13
on the left side of the figure show the nose and conic regions for the course mesh while
the two images on the right show the nose and conic regions for the fine mesh. The goal
of this refinement study was to verify that no substantial change in axial force coefficient
existed for the different meshes.
Table 3: Mesh Sizes Considered for Grid Refinement Study
imax jmax Number of Cells
Course Mesh 476 49 22800
Fine Mesh 660 84 54697
Figure 6: Comparison of Course and Fine Mesh
As mentioned previously, one of the goals of this optimization study is to minimize
axial force over a range of flight conditions. The grid refinement study was carried out by
running both meshes through the CFD solver over the range of flight conditions
14
considered in this study. To compare the results for each grid, the mean axial force
coefficient for all flight conditions was plotted as a function of iteration. Figure 7 shows
the mean axial force coefficient as a function of iteration for both meshes considered in
the refinement study.
Figure 7: Axial Force Convergence for Considered Meshes in Refinement Study
Both meshes were allowed to run for 15000 iterations for all flight conditions.
Figure 7 shows that both meshes converge to a mean axial force coefficient near 0.467
where the fine mesh gives an axial force coefficient just slightly higher than that of the
course mesh. This difference is insignificant for this study since obtaining the exact axial
15
force coefficient is not the primary concern but rather finding a geometric shape with a
minimum axial force is the goal. Also Figure 7 shows that the mean axial force
coefficient for both meshes is sufficiently converged near 1100 iterations. Since the axial
force varies as a function of flight Mach number it is important to ensure that there is also
no significant difference in the axial force coefficient for each flight condition. Figure 8
shows the axial force coefficient through the prescribed flight conditions for both the
course and fine meshes. It is shown from this figure that no significant differences exist
in the axial force coefficient throughout the ascent trajectory for the two meshes. In
addition to Figure 7 and Figure 8 the computation time to compute 15000 iterations for
both meshes at each flight condition is displayed in Figure 9.
16
Figure 8: Axial Force Coefficient Throughout Ascent for Course and Fine Mesh
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.0 0.5 1.0 1.5 2.0 2.5
Ax
ial
F
or
ce
C
oe
ffi
cie
nt
Mach Number
Course Mesh
Fine Mesh
17
Figure 9: Comparison of Computation Times at Each Flight Condition
Figure 9 shows a substantial increase in computation time for the fine mesh
compared to the course mesh. For most flight conditions the fine mesh took over twice as
long as the course mesh to solve. Due to the extensive run time of the fine mesh and
insignificant difference in axial force coefficient between the course and fine mesh, the
course mesh was chosen to be most suitable for this study. Figure 9 also shows that the
computation time for the Mach 0.85 condition took the longest to compute. This indicates
that this flight condition is the most difficult case for the Fluent CFD solver to resolve.
To further validate the use of the course grid over the fine grid the pressure distribution is
compared for both grids. Figure 10 and Figure 11 show contour plots of the pressure
distribution near the conic sections at the Mach 0.85 flight condition for the course and
fine grids respectively. Comparison of these two figures clearly shows that no significant
difference exists in the pressure distribution for the course and fine grids. Note that the
0:00
0:30
1:00
1:30
2:00
2:30
3:00
M=0.50 M=0.85 M=1.15 M=1.50 M=2.00
Co
mp
ut
ati
on
Ti
me
(h
r:m
in)
Flight Condition
Course Mesh
Fine Mesh
18
pressure values displayed in these images is in gauge pressure where zero pressure is the
ambient flight condition.
Figure 10: Pressure Distribution Plot for Course Grid
Figure 11: Pressure Distribution Plot for Fine Grid
2.6 PRESSURE FLUCTUATION MODEL
The pressure fluctuation level was
This point is approximately 0.25
This point was chosen because it was suspected to experience the highest levels of
pressure fluctuations on the baseline geometry. The fluctuating pressure level was
calculated at this point
parameters include freestream Mach number,
local velocity, local density, viscosity and Reynolds Number. Note that what is meant by
freestream is actually global freestream, not the condition at the edge of the boundary
layer. For this study, the flow parameters at the edge of the boundary layer are referred to
as the local flow parameters at the point of interest.
Figure 12: Locatio
While the freestream flow parameters are given for each flight condition,
the local flow parameters at the point of interest
that fact that the boundary layer thickness is dependent on the geometry and vehicle flight
speed. So simply investigating the local flow field at this point for a given flight
condition and assigning a height above the surface to sample l
not suffice. For supersonic freestream Mach numbers, an expansion fan develops such
19
calculated at only one point on the geometry.
m aft of the third conic section as shown in
based on several local and freestream flow parameters. The
freestream and local dynamic pressure,
n of Pressure Fluctuation Level Calculation
is a nontrivial exercise
ocal flow parameters will
Figure 12.
extracting
. This is due to
that the flow is accelerated around this corner. During subsonic flight the acceleration of
the flow is not as pronounced. The effect that
is shown by Figure 13 in which the flow velocity is plotted versus the distance from the
wall. Only three of the five flight cond
effect flight speed has on the local velocity profile.
Figure 13
It is desired to take the location of the peak velocity as the
flow parameters from the CFD data as it is believed that this peak velocity has the largest
effect on the fluctuating pressure level.
speed but also on the geometry. Since the GA can generate any combination of geometric
variables within the design space, this point can move relative to the
variables. A method had to be implemented
communicating to the CFD solver the coordinates of this location. This was done by
modifying the batch file used by
In the grid generator routine, the
routine for writing the batch file. The
calculated is 0.25m aft of the corner
extended from the surface at this
20
flight speed has on the local flow velocity
itions are shown in this figure to demonstrate the
: Effect of Flight Speed on Velocity Profiles
sampl
This location is not only dependent on flight
into the objective function
the CFD solver to execute commands for
x and y location of the corner point is passed to another
x location where the pressure fluctuation level is
x location. An equal distribution of 25 points is
x location to a distance of 0.75m above the surface.
e point for local
GA selected
for
each member.
21
Upon solution convergence the CFD solver extracts the local flow parameters at these 25
points and records the data in an output file. This data is then read by another FORTRAN
routine which determines the point at which the peak velocity occurs and calculates the
local flow parameters based on given CFD data at that same point.
The pressure fluctuation prediction model implemented into this optimization study
is shown in the equation below. This is a physics based model that calculates the RMS
pressure fluctuation level and has been developed by curve fitting flight data of
fluctuating pressure levels for several different launch vehicles. This model has been
shown to provide adequate correlation to flight data for a similar location on similarly
shaped launch vehicles. For proprietary reasons, the actual flight data along with the
curve fitting results cannot be provided. It should be noted, however, that it is not the
intent of this study to provide a highly accurate model for calculating pressure
fluctuations. A More robust model for pressure fluctuations can be implemented into this
optimization process with relative ease. This particular model was chosen for this study
as it was readily available to the author.
( ) ??
?
?
???
?
+?+????
?
?
?
?
?
?????? ?+?
??
?
?
22
2
2
)(2
11*Re
MMbq
qMA
q
p
local
N
n
?
??
22
3 LAUNCH VEHICLE OPTIMIZATION
3.1 CODE STRUCTURE
The code structures for the axial force minimization study and the pressure
fluctuation minimization study were very similar. The optimization process for a typical
GA is shown in a diagram in Figure 14. The GA starts with the first generation of
members by randomly selecting variables within the prescribed design space. In this case,
each member represents a particular set of geometric parameters which define the shape
of a launch vehicle resembling the Ares I Crew Launch Vehicle. Each member is passed
one by one to the objective function where a grid is generated based on the geometric
parameters. This grid is then sent to a CFD solver where the axial force coefficient is
calculated for that member. Once a generation is completed each member is ranked
according to its performance relative to other members in the same generation. This
ranking system determines the selection of variables for the next generation. This process
is repeated until all generations are completed. For this study 30 members were evaluated
over 20 generations. The goals of these optimizations are to minimize the average axial
force coefficient and fluctuating pressure level for the launch vehicle throughout several
flight conditions in a typical launch trajectory.
23
Figure 14: Flow Diagram for a Typical Genetic Algorithm
An effort was made to reduce computational expense by having each CFD run
parallelized by the Fluent solver on a Linux cluster maintained by Auburn University.
The Linux cluster houses 30 nodes where each node contains two AMD Opteron 242 (64-
bit) chips for a total of 60 processors. For this study each CFD run was distributed across
two nodes (four processors). With this approach the average computation time for each
member to be solved at all five flight conditions by the CFD solver was approximately
3.25 hours. Running the member sequentially for 30 members over 20 generations would
result in a total GA run time of about 1950 hours or over 80 days. This is far too long of a
run time for this study, so methods had to be implemented in order to substantially reduce
the computational expense.
A method for greatly reducing the GA run time was implemented into this study
and was developed by Doyle.40 This method consists of running multiple members
simultaneously while each member was parallelized by Fluent. This works by essentially
running a script file that executes Fluent after the mesh for a particular member has been
generated. This script file allows Fluent to be executed in the background so that the GA
24
can continue and load in another member instead of waiting for the previous member to
finish. Multiple slots were allotted for members to occupy where no more than one
member could occupy a slot. However, steps must be taken to ensure that multiple
members are not started simultaneously. This is handled by essentially suspending the
program in a loop for a brief period to allow the previous member time to initialize. More
details explaining this modification to the GA is discussed in reference 40. Figure 15
shows a flow chart for the optimization process running multiple members
simultaneously.
Figure 15: Flow Diagram for GA Running Multiple Members Simultaneously
For this study 10 slots were allotted so that 10 members could be run
simultaneously. At the beginning of a generation the first 10 members of the generation
were loaded into all the slots one by one. After the first ten were loaded the GA began to
check for the completion of a member. Once the GA detected that a member was finished
the GA would load in the next member. This process would repeat until all members in
that generation were complete. This method of running multiple members simultaneously
25
significantly reduced the total GA run time. With 10 members running at once and 30
members over 20 generations the total GA run time was cut down to approximately 225
hours or just over 9 days.
Table 4 shows the design space for this study and is the same for both optimization
studies. Limitations had to be placed on the design space to prevent impractical
geometries from being generated, such as having the first cone radius larger than the
second cone radius, or having a very large cone angle. This was partially handled by
defining the second cone radius, Rc2, to be a percentage of the base radius, Rb, and the
first cone radius, Rc1, being a percentage of the Rc2. Preventing large cone angles was
not as easily handled as there are many combinations of design parameters that could
produce a large cone angle. It was important, however, to eliminate the possibility of
producing geometries with large cone angles as it was found that at supersonic Mach
numbers the CFD solver would diverge for large cone angles which would cause the GA
run to crash. To eliminate this possibility a check was placed in the grid generator to
check that the angle of each cone was less than 60?. If a cone angle was found to be
larger than 60? the CFD solver would not be executed and the axial force coefficient
would be forced to a large number so that the GA would see that member as a bad set of
design variables.
26
Table 4: Prescribed Design Space for Both Optimization Studies
Parameter Minimum Maximum Increment
Rc1 0.50 1.00 0.02
Rc2 0.40 0.99 0.02
LcTot 1.50 3.00 0.05
Lc1 0.05 0.35 0.01
Lc3 0.10 0.60 0.01
3.2 PRESSURE FLUCTUATION MINIMIZATION STUDY
3.2.1 CONVERGENCE CRITERIA
The RMS fluctuating pressure level is the parameter of interest for this study and it
would be ideal to monitor CFD solution convergence based on this parameter. However,
Fluent does not have the option of calculating and monitoring the pressure fluctuation
level so the local pressure at this point of interest was monitored. Fluent is unable to
monitor solution convergence based on pressure so it was necessary to investigate the
value of the residuals for continuity, x-momentum, y-momentum, energy, k, and ? when
the local pressure was sufficiently converged. From the grid refinement study in the
previous chapter it was shown that the course mesh was deemed suitable for both
optimization studies. For this reason, only the convergence criteria for the course mesh
using the baseline geometry will be investigated.
As indicated in the previous chapter by Figure 9 the Mach 0.85 flight condition was
the most difficult condition for Fluent to resolve, so this condition was used to determine
the convergence criteria. This flight condition was run for 15000 iterations to allow for
more than enough time for the CFD solver to obtain a converged solution. Figure 16
27
shows the residuals and local pressure plotted against the number of iterations. This
figure shows that the residuals have completely converged around 6000 to 7000
iterations. The local pressure, however, converged much sooner at less than 2000
iterations. At 2000 iterations all the residuals had converged to 10-6 or lower, except for
the continuity residual, which had converged to approximately 10-5. To ensure adequate
convergence of the local pressure, the solution was deemed converged once all the
residuals reached 10-6 within a maximum of 10000 iterations. If a particular member did
not meet this convergence criterion then that member was disqualified by setting the
RMS pressure fluctuation level to an extremely high value of 1000.
28
Figure 16: Residuals and Local Pressure Convergence for the Course Mesh and
Mach 0.85 Flight Condition
Setting an extremely high performance penalty for a particular member of
externally understandable characteristics essentially eliminates the traits that cause the
CFD solver difficulty in obtaining converged solutions. The negative impact of this
disqualification is that it eliminates members due to problems with solution convergence,
and not because of high pressure fluctuations. This means that there is a possibility that
the GA could eliminate a potentially good performer. Further study could be done to
determine if the disqualified members could produce good performance; however this
would require a great deal more time that is beyond the scope of this study as this study
29
aims to demonstrate a methodology for reducing the fluctuating pressure level of launch
vehicles.
During trial GA runs it was recognized that the aforementioned convergence
criteria was not sufficient for all the possible geometries allowed in the prescribed design
space. It was noticed that for some members at certain flight conditions, the residual for
continuity would converge to a value less than 10-6 thereby never actually meeting the
convergence criteria. For these cases, however, it was noticed that the local pressure was
sufficiently converged. This resulted in the GA eliminating potential good members that
had converged solutions. The solution to this problem was to develop an additional
convergence criterion. This criterion was a check to see if the local pressure had
sufficiently converged even if the residual convergence criterion was not satisfied. The
local pressure was deemed converged if the value was found to not differ by more than
0.1% over the final 2000 iterations. If the member failed this pressure convergence
criterion, then the member was disqualified by setting the RMS pressure fluctuation level
to a high value. This essentially disqualifies the member as discussed previously.
3.2.2 PRESSURE FLUCTUATION MINIMIZATION RESULTS
The average fluctuation pressure level throughout the ascent trajectory for the
baseline model was calculated to be 0.0216. The optimized geometry gave a pressure
fluctuation level of just 0.0178. This was approximately a 17.5% reduction from the
baseline geometry. The GA was not able to significantly improve performance after the
1st generation however.
30
Figure 17 shows the fluctuating pressure level throughout the ascent trajectory for
the baseline and optimized geometries. This figure shows that the pressure fluctuations
for both geometries follow the same trend as a function of free stream Mach number
where the fluctuation level for the optimized geometry stays fairly constant after Mach
1.15. After Mach 1.15 is where the optimized geometry sees the most improvement in
performance over the baseline geometry. The Mach 0.5 and 0.85 flight conditions see
little change where the optimized geometry actually has a slightly higher pressure
fluctuation level for the first flight condition. The largest reduction in fluctuating pressure
level occurs during the last flight condition of Mach 2.0, in which a 36.6% reduction was
achieved. While a 17.5% reduction of the average RMS pressure fluctuation was
achieved, there was only a 7.7% reduction in the peak RMS pressure fluctuation level.
This result could be improved in a couple of different ways. One way would be to open
up the design space to include more surfaces throughout the conic sections, thereby
allowing more flexibility in the geometry. Another way that would likely improve this
result would be to monitor only the Mach 0.85 flight condition where the peak fluctuation
level occurs.
31
Figure 17: Fluctuating Pressure Level throughout Ascent for Baseline and
Optimized Geometries
Figure 18 and Figure 19 show the evolution of the pressure fluctuation level
throughout the total number of generations allowed for this optimization. Figure 18
shows how the maximum, average, and minimum pressure fluctuation members change
for each generation. The erratic behavior of the maximum pressure fluctuation member is
due to the GA?s random manipulation of some of the population members. Figure 18
does not show any members that were disqualified. This is not because there were no
disqualifications. The disqualified members were excluded from Figure 18 because there
were so many. There were a total of 20 members that were disqualified, however not
every generation produced a disqualified member. Several generations produced more
than one member that was disqualified. The reason for the large number of
disqualifications is due to the geometric shape the GA tended to favor. The GA had a
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25
Flu
ctu
ati
ng
P
re
ss
ur
e L
ev
el
Free Stream Mach Number
Baseline Geometry
Optimized Geometry
32
tendency to produce members with large first cone angles and very small second and
third cone angles. So these disqualifications were due to the GA?s selection of variables
resulting in cone angles that exceeded the maximum angle allowed. No member was
disqualified due to failing to meet the convergence criteria.
Figure 18: Maximum Average and Minimum Pressure Fluctuation Level Evolution
The results shown in Figure 18 show practically no change in the best performer
throughout the generations. To more clearly show the improvement of the best performer
throughout the generations, the best performer is shown separately in Figure 19. The GA
was able to improve the best performer by only 0.66% from the first to the last
generation. This indicates that the GA was able come very close to the optimized
geometry in the initial generation by the random selection of members within the design
space. The plot of the average performance member shows that the GA was able to
0.017
0.019
0.021
0.023
0.025
0.027
0.029
0 2 4 6 8 10 12 14 16 18 20
Flu
ctu
ati
ng
P
re
ss
ur
e L
ev
el
Generation
Max
Avg
Min
33
quickly learn what combination of parameters produced good performers. By about the
eighth generation the average member performed nearly as well as the best performer.
Figure 19: Minimum Fluctuating Pressure Level Evolution
Figure 20 shows the comparison of the optimized and baseline geometries. The first
thing to notice about the shape optimized for minimizing the pressure fluctuation level is
the large angle for the first conic section. While the shape still somewhat resembles a
parabolic or blunted ogive shape the presence of the large cone angle makes this design
undesirable overall. The reason for this result is because the fluctuating pressure level
was monitored from only one critical point in the geometry. This point lies just aft of the
point where the third conic section meets the large cylinder base section. As the
optimized shape shows in Figure 20, the GA produced a shape that had a very small
angle for the third conic section. With smaller cone angles for the third section, this
would result is less flow separation which is a primary cause for high pressure
0.0175
0.0177
0.0179
0.0181
0.0183
0.0185
0 2 4 6 8 10 12 14 16 18 20
Flu
ctu
ati
ng
P
re
ss
ur
e L
ev
el
Generation
34
fluctuations. A new critical point would arise, however, just aft of the first conic section.
Due to the large difference in the first and second cone angles flow separation occurs at
this point for several of the flight conditions. While the overall optimized geometry
would not be an ideal design due to the large angle of the first conic section, the
methodology employed here has proved useful in minimizing the fluctuating pressure
level at a critical point.
Figure 20: Pressure Fluctuation Level Study Geometry Comparison
Shown in Figure 21 through Figure 25 are the variable distribution plots for each
GA variable. These plots show the GA?s selection of each variable for all the members in
each generation within each parameter?s design space. Through these plots one can see
the evolution of each design variable throughout the optimization process. For these
figures the solid line represents the best performance member for each generation.
35
Figure 21 shows the variable distribution of the normalized Rc1 design variable. It
can be seen that the GA begins in the initial or 0th generation with a fairly uniform
distribution across the members. It appears that the majority of the members have
normalized Rc1 variables in the upper half of the design space. This is just a coincidence
since the population members for the initial generation are randomly generated by the
GA. However, the GA does tend to produce members in the upper region of the design
space. As can be seen from the solid line, the best performer is in the upper region for the
entire optimization run. By the 8th generation the best perform has a normalized Rc1
variable of 0.855 and does not change for the remainder of the GA run. Such a large
radius for the first conic section is the primary reason for such a large cone angle. This
was an undesired result since such a large cone angle would generation high pressure
fluctuation levels at a point just aft of the first conic section. This result could be avoided
if fluctuating pressure levels were calculated for several points along the geometry.
36
Figure 21: Pressure Fluctuation Study Variable Distribution for Rc1
The variable distribution for the normalized second cone radius, Rc2, is shown in
Figure 22. The GA quickly narrows the selection of the Rc2 variable to the upper region
of the design space. By the 5th generation the majority of the members are between 0.80
and 0.85. The best perform does not change after the 5th generation where the best
performer had a normalized second cone radius of 0.838. Such a large radius for the
second conic section was expected since the larger this radius is the smaller the angle is
for the third conic section. With smaller angles for the third conic section, the flow is not
accelerated as much over the point where the pressure fluctuation is calculated, which
leads to lower fluctuating pressure levels.
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
0 2 4 6 8 10 12 14 16 18 20
No
rm
ali
ze
d R
c1
G
A
Va
ria
ble
Generation
37
Figure 22: Pressure Fluctuation Study Variable Distribution for Rc2
Figure 23 shows the variable distribution for the normalized total length of the
conic sections, LcTot. The GA?s selection of the LcTot variable tends to be fairly spread
out in the design space for the first several generations. After the 5th generation a large
majority of the members have a normalized LcTot parameter of between 2.20 and 2.40.
The LcTot variable for the best performer stays constant after the 5th generation with the
exception of the best performer in the 17th generation. The best performer had a
normalized LcTot variable of 2.37. It was anticipated that the GA would maximize this
parameter since longer conic sections with a given cone radius would result in smaller
cone angles, thus resulting in a more stream-line shape. Further work is recommended to
investigate the reason for this unexpected result.
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
0 2 4 6 8 10 12 14 16 18 20
No
rm
ali
ze
d R
c2
G
A
Va
ria
ble
Generation
38
Figure 23: Pressure Fluctuation Study Variable Distribution for LcTot
The variable distribution of the normalized first cone length, Lc1, is shown in
Figure 24. The variables tend to be distributed throughout the upper half of the design
space for the first few generations. By about the 7th generation however, most members
have normalized first cone lengths between 0.15 and 0.20. The best performer, however,
does not change after the 5th generation. The best performer had a normalized Lc1
parameter of 0.176.
1.50
1.60
1.70
1.80
1.90
2.00
2.10
2.20
2.30
2.40
2.50
2.60
2.70
2.80
2.90
3.00
0 2 4 6 8 10 12 14 16 18 20
No
rm
ali
ze
d L
cT
ot
G
A
Va
ria
ble
Generation
39
Figure 24: Pressure Fluctuation Study Variable Distribution for Lc1
Shown in Figure 25 is the distribution for the normalized length of the third conic
section, Lc3. This figure shows a very sparse selection of the Lc3 parameter throughout
the optimization run. The GA begins to slightly narrow its selection by the 5th generation
to the upper region of the design space; however the variable selection is still very spread
out compared to the other variable distribution plots. The reason for this could be related
to the impact this variable has on the cone angle for the third conic section. The angle of
the third conic section has a substantial impact on the flow characteristics at the point
where the pressure fluctuation is calculated. By about the 14th generation the selection
falls mostly between 0.40 and 0.50, with the best performer of the last generation having
a normalized Lc3 variable of 0.402.
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0 2 4 6 8 10 12 14 16 18 20
No
rm
ali
ze
d L
c1
G
A
Va
ria
ble
Generation
40
Figure 25: Pressure Fluctuation Study Variable Distribution for Lc3
Shown in Figure 26 is a comparison of the pressure distributions for the baseline
and optimized geometries at the Mach 0.85 flight condition, which was the condition
where the peak pressure fluctuation level was calculated in this study. This figure shows
a clear difference in the pressure distribution throughout the conic sections for the two
geometries. In particular, there is a significant difference in the pressure field near the
point at which the pressure fluctuation was calculated. The pressure in the region of the
flow near this point is substantially higher pressure for the optimized geometry than for
the baseline geometry. Note that the pressure scale shown in Figure 26 is gage pressure in
pascals, thus 0 is the standard pressure at altitude.
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0 2 4 6 8 10 12 14 16 18 20
No
rm
ali
ze
d L
c3
G
A
Va
ria
ble
Generation
41
Figure 26: Pressure Fluctuation Study Pressure Distribution Plot for Baseline and
Optimized Geometries for Mach 0.85 Flight Condition
A comparison of the dynamic pressure distribution for the baseline and optimized
geometries is shown in Figure 27. Since the ascent trajectory used for this study
experiences maximum dynamic pressure near the Mach 1.50 flight condition, this is the
condition shown in the figure for comparison. This figure shows a moderate difference in
the dynamic pressure field near the point at which the pressure fluctuation level is
calculated. The most significant difference in the dynamic pressure occurs near the first
conic section. The dynamic pressure is so low due to the low flow velocity in this region.
42
Figure 27: Pressure Fluctuation Study Dynamic Pressure Distribution Plot for
Baseline and Optimized Geometries for Mach 1.50 Flight Condition
To further show differences in the flow field for the baseline and optimized
geometries, the distribution of the x component of velocity is shown in Figure 28. This
figure shows a field plot of the axial velocity component for the Mach 0.85 flight
condition. This figure shows a significant difference in the velocity distribution near the
conic sections for both geometries. The most important difference of interest in this study
is the substantial reduction in the flow velocities for the optimized geometry near the
point aft of the third conic section. This reduction in flow velocity near this point is the
result of the GA?s selection of geometric parameters. Since the third cone angle of the
optimized geometry is small compared to that of the baseline geometry the flow is not
accelerated as much as it passes over the corner.
43
Figure 28: Pressure Fluctuation Study Velocity Distribution Plot Comparison at
Mach 0.85 Flight Condition
To show more detail in the x velocity distributions for the baseline and optimized
geometries, a close-up of the flow velocity near the conic sections at the Mach 0.85 flight
condition is show in Figure 29 including streamlines. This figure again shows the
substantial decrease in flow velocity near the point where the pressure fluctuation is
calculated. This figure also shows more clearly the new problematic area aft of the first
conic section that arises in the optimized geometry. Figure 29 shows that a separated
region of flow develops aft of the first conic section. It is suspected that this would result
in higher fluctuating pressure levels at this point than at a similar point for the baseline
geometry, thus making the optimized geometry not ideal as a practical design for a
launch vehicle. As mentioned previously however, the objective of this study is to
44
demonstrate a methodology for reducing the pressure fluctuation level, which has been
accomplished by the presented results.
Figure 29: Velocity Distribution Close-Up for Baseline and Optimized Geometries
for Pressure Fluctuation Study at Mach 0.85
3.3 AXIAL FORCE MINIMIZATION STUDY
3.3.1 CONVERGENCE CRITERIA
Since the axial force coefficient is the parameter of interest for this study, it is
desirable to monitor CFD solution convergence based on the axial force coefficient.
Fluent does have the option of calculating the axial force coefficient, however Fluent is
not able to monitor solution convergence based on the axial force coefficient. As with the
pressure fluctuation study, it was necessary to investigate at what point the residuals for
continuity, x-momentum, y-momentum, energy, k, and ? were at when the axial force
45
coefficient was sufficiently converged. The course grid and Mach 0.85 flight condition
was used for determining the convergence criteria for the same reason as discussed
previously for the pressure fluctuation study.
This Mach 0.85 flight condition was run for 15000 iterations to allow for plenty of
time for the CFD solver to obtain a converged solution. Figure 30 shows the residuals and
axial force coefficient plotted against the number of iterations. This figure shows that the
residuals have completely converged around 6000 to 7000 iterations. The axial force
coefficient, however, converged much sooner at less than 2000 iterations. At 2000
iterations all the residuals had converged to 10-6 or lower, except for the continuity
residual, which had converged to approximately 10-5. To ensure adequate convergence of
the axial force coefficient, the solution was deemed converged once all the residuals
reached 10-6 within a maximum of 10000 iterations. If a particular member did not meet
this convergence criterion then that member was disqualified by setting the axial force
coefficient to an extremely high value of 1000. This disqualification is similar to that
discussed in section 3.2.1 for the pressure fluctuation study.
46
Figure 30: Residuals and Axial Force Coefficient Convergence for the Course Mesh
and Mach 0.85 Flight Condition
As with the pressure fluctuation study, it was realized that this convergence criteria
was not sufficient for all the possible geometries allowed in the prescribed design space.
For some members at certain flight conditions, the residual for continuity would converge
to a value less than 10-6 thereby never actually meeting the convergence criteria. The
axial force coefficient was found to be sufficiently converged for these cases however. To
prevent the possibility of the GA eliminating these members which may have good
performance, an additional convergence criterion was implemented similar to that of the
47
pressure fluctuation study. This criterion was a check to see if the axial force coefficient
had converged even if the residual convergence criterion was not satisfied. The axial
force coefficient was deemed converged if the value was found to not differ by more than
0.1% over the final 2000 iterations. If the member failed this axial force convergence
criterion, then the member was disqualified by setting the axial force coefficient to a high
value as mentioned previously.
3.3.2 AXIAL FORCE MINIMIZATION RESULTS
As shown in Figure 7 the average axial force coefficient throughout the prescribed
ascent trajectory for the baseline geometry is approximately 0.467 For comparison, the
V-2 missile at zero angle of attack has an average axial force coefficient of about 0.26.
This value was obtained from an axial force coefficient versus Mach number plot for the
V-2 missile in Sutton.42 The axial force coefficient varies as a function of flight Mach
number and this is shown for the baseline geometry in Figure 8. The average axial force
coefficient for the optimized geometry is approximately 0.204. This was a reduction of
about 56% from the baseline geometry. The GA arrived at this optimized solution after
20 generations.
Figure 31 shows the axial force coefficient throughout the prescribed ascent
trajectory for the baseline and optimized geometry. The figure shows that the axial force
profile during ascent follows approximately the same trend for both geometries where the
axial force profile for the optimized geometry is substantially lower. Also, there is not as
much of a change in axial force from the lowest point at Mach 0.5 to the peak at Mach
1.5. The baseline geometry peak axial force at Mach 1.5 is about 5 times that of the Mach
48
0.5 condition, while the optimized peak axial force at Mach 1.5 is about 4.5 times the
Mach 0.5 flight condition. The important result of Figure 31, however, is the substantial
decrease in the axial force coefficient for all five points in the ascent trajectory. For most
flight conditions the axial force coefficient is reduced by about 50%. The largest axial
force reduction was achieved for the Mach 0.85 flight condition, in which a 70%
reduction in the axial force coefficient was achieved.
Figure 31: Axial Force Coefficient throughout Ascent for Baseline and Optimized
Geometries
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25
Ax
ial
F
or
ce
C
oe
ffi
cie
nt
Mach Number
Baseline Geometry
Optimized Geometry
49
Figure 32: Maximum Average and Minimum Axial Force Coefficient Evolution
The evolution of the axial force coefficient throughout the total number of
generations allowed for this study is shown in Figure 32 and Figure 33. Figure 32 shows
how the maximum, average, and minimum axial force coefficient members change for
each generation. The erratic behavior of the maximum axial force members is expected
due to the GA?s random manipulation of some members. It is also important to note in
Figure 32 that the maximum axial force coefficient members in the initial, first, and third
generations are off the graph. This is because these generations generated members that
were disqualified in which the axial force coefficient was set to 1000. Further
examination showed that these members were disqualified due to a ?bad geometry? as
discussed in section 3.1. There were a total of five members that were disqualified, three
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0 2 4 6 8 10 12 14 16 18 20
Ax
ial
F
or
ce
C
oe
ffi
cie
nt
Generation
Max
Avg
Min
50
of which were in the initial generation. All five of these disqualifications were due to
geometry issues. No member was disqualified due to solution convergence.
Another notable result shown in Figure 32 is the small change in the best performer
throughout the generations. The best performer is shown by itself in Figure 33 in which
the details of how it changes through the generations can be more clearly seen. The GA
was able to reduce the axial force coefficient by only 3% from the first to the last
generation. This result indicates that the GA was able come close to the optimized
geometry in the initial generation by the random selection of members within the design
space. Shown in Figure 32, the plot of the average member for each generation shows
that the GA was able to quickly learn what combination of parameters produced good
performers. By the sixth generation the average member performed nearly as well as the
best performer.
Figure 33: Minimum Axial Force Coefficient Evolution
Table 5 shows the optimized geometric parameters and the minimum and
maximum values allowed for each parameter. The only parameter that was maximized by
0.200
0.202
0.204
0.206
0.208
0.210
0.212
0.214
0 2 4 6 8 10 12 14 16 18 20
Ax
ial
F
or
ce
C
oe
ffi
cie
nt
Generation
51
the GA was the overall length of the conic sections, LcTot. All other parameters fell
within the prescribed design space. This result indicates that the solution obtained from
the GA is a near optimal geometry to minimize axial force. It is plausible that further
reduction in axial force could be obtained by allowing the overall length to increase.
However, as indicated by the change of the best performer over the generations as shown
in Figure 32 and Figure 33, it is not expected that a significant reduction in axial force
would occur. Also, while increasing the overall length might reduce axial force, it
becomes impractical to design longer and longer conic sections.
Table 5: Optimized Parameters and Design Space
Parameter Minimum Maximum Optimized
Rc1 0.50 1.00 0.629
Rc2 0.40 0.99 0.857
LcTot 1.50 3.00 3.000
Lc1 0.05 0.35 0.302
Lc3 0.10 0.60 0.283
To clearly show how the optimized geometry differs from the baseline geometry, a
comparison of the geometries is shown in Figure 34. It was interesting to notice that the
optimized geometry consists of conic sections that have a more parabolic or blunted
ogive shape. This result was expected as the blunted ogive shape is a commonly used
geometric shape in aerodynamics since it has very good aerodynamic performance.
52
Figure 34: Best Performer and Baseline Geometry Comparison
Figure 35 shows the distribution of first cone radius, Rc1, by generation. Initially
the members are distributed somewhat uniformly throughout the design space due to the
random selection by the GA. By the sixth generation it is clear that the GA has narrowed
the selection of the Rc1 radius to between 75% and 63% of Rc2. By the 15th generation,
the GA had further narrowed the radius to between 67% and 63% with the exception of
the one outlier in the 19th generation. This outlier is the result of the mutation procedure
the GA conducts to include randomness in the member generation process. The
optimized geometry had an Rc1 variable of 62.9% of the second cone radius, Rc2.
53
Figure 35: Axial Force Study Variable Distribution for Rc1
The distribution of the second conic section radius, Rc2, is shown in Figure 36. As
with the Rc1 variable, the members are distributed somewhat uniformly throughout the
design space initially. By the sixth generation the GA had significantly narrowed down
the range to between 85% and 90% of the base radius, Rb. The majority of members
continue to fall in this range for the remainder of the optimization run with the exception
of the outliers in generations 10,12,13,15, and 17. The Rc2 variable was 85.7% of the
base radius for the optimized geometry.
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
0 2 4 6 8 10 12 14 16 18 20
No
rm
ali
ze
d R
c1
G
A
Va
ria
ble
Generation
54
Figure 36: Axial Force Study Variable Distribution for Rc2
The total length of the conic section, LcTot, variable distribution by generation is
shown in Figure 37. This figure clearly shows the tendency for the GA to generation
members that maximized the total length of the conic sections. With the exception of the
few outlying members in generations 15, 16, 17, and 20, most all the members after the
6th generation had a total conic section length at the upper limit of the design space. This
result was no surprise as the effects of pressure were expected to be much more important
than viscous effects on the axial force coefficient. This indeed turned out to be the case
for the optimized design. The optimized geometry had a total length of the conic sections
of 300% of the base radius, Rb.
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
0 2 4 6 8 10 12 14 16 18 20
No
rm
ali
ze
d R
c2
G
A
Va
ria
ble
Generation
55
Figure 37: Axial Force Study Variable Distribution for LcTot
Figure 38 shows the variable distribution of the first cone length, Lc1. As with the
total length of the conic sections, the GA produces members that mostly have a first cone
length at the upper limit of the design space. In the 7th generation the GA only generated
members that had first cone lengths of 34% to 35% of the total length of the conic
sections. This indicates that the optimized geometry would have the longest allowable
first cone length. However, after the 7th generation the distribution becomes more spread
out, increasing the range to about 30% to 35%. This happened because the best performer
for the 8th generation had an Lc1 slightly less than the maximum allowed, whereas the
best performers for several generations prior had a maximum Lc1. The GA continued to
generate best performers with smaller first cone lengths, so the GA diversified the
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
2.00
2.20
2.40
2.60
2.80
3.00
0 2 4 6 8 10 12 14 16 18 20
No
rm
ali
ze
d L
cT
ot
G
A
Va
ria
ble
Generation
56
selection of this parameter. The optimized geometry had a first cone length of 30.2% of
the total length of the conic sections.
Figure 38: Axial Force Study Variable Distribution for Lc1
The distribution of the third cone length, Lc3, is shown in Figure 39. As with the
rest of the parameters the initial distribution of the Lc3 variable is fairly uniformly
distributed throughout the prescribed design space. Also similar to the other parameters,
the GA quickly narrows the selection of the third cone length by about the 6th generation.
From the 7th generation on the majority of members have a third cone length of between
25% and 30% of the total length of the conic sections. The optimized geometry had a
cone length of 28.3% of the total length of the conic sections.
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0 2 4 6 8 10 12 14 16 18 20
No
rm
ali
ze
d L
c1
G
A
Va
ria
ble
Generation
57
Figure 39: Axial Force Study Variable Distribution for Lc3
Figure 35 through Figure 39 show the distribution of the normalized GA
parameters throughout the optimization run. The results shown in these figures indicate
that the total length of the conic sections plays an important role in the minimization of
the axial force coefficient. Since the GA maximized LcTot, this is another indication that
pressure is more important to the total axial force than viscous effects, as expected for
this geometry and flight conditions. Another feature of the geometry that plays an
important role in minimizing axial force is the angle for each conic section, which is
dependent upon the radius and length of each conic section. It is evident from the results
of this study that an optimal shape for minimizing axial force would be as long as
possible while having a parabolic or blunted ogive-type shape.
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0 2 4 6 8 10 12 14 16 18 20
No
rm
ali
ze
d L
c3
G
A
Va
ria
ble
Generation
58
The parabolic or blunted ogive shape the GA generated was expected to be the
optimized geometry since similar shapes have shown to have good aerodynamic
characteristics for generating low axial force. The optimization study produced a shape
that is very similar to a blunted ogive in that the first conic section is relatively short with
a larger angle compared to the second and third conic sections. The other conic sections
are much longer with smaller cone angles producing the blunted ogive-like shape. This
shape serves to give good aerodynamic characteristics by having little flow separation
through the conic sections. While the cone angles became smaller in each conic section it
was not expected that the third conic section would have a shorter length than the second
conic section.
A pressure distribution plot of the flow field for both the baseline and optimized
geometry is shown in Figure 40. Again, it is important to note that the pressure values
indicated in this figure are relative to the freestream static pressure, thus 0 pressure is the
ambient static pressure. Considering just the conic sections, the pressures are much
higher for the baseline geometry than for the optimized geometry. The significant
reduction in pressure near this region of the optimized geometry is the primary reason for
the large reduction in axial force on the vehicle.
59
Figure 40: Axial Force Study Pressure Distribution for Baseline and Optimized
Geometries at Mach 1.50 Flight Condition
The results of the axial force minimization study have shown a significant
reduction in the vehicle?s axial force coefficient with a modification of the geometry of
the conic sections. A key geometric parameter of the section that had a great impact on
the axial force coefficient was the total length of the conic sections, LcTot. Since the GA
maximized this parameter, it can be assumed that perhaps further extending the design
space for this parameter would result in a further reduction of the axial force coefficient.
However, the longer the conic sections become, the more impractical the vehicle design
becomes. Longer sections would ultimately result in a heavier vehicle, which would
reduce the amount of payload the vehicle would be able to launch to orbit. So there is a
tradeoff that must be done in order to determine how much reduction in axial force can be
obtained while not adding too much weight, such that the amount of payload-to-orbit is
60
increased. Since this thesis is set out to demonstrate a method for reducing axial force,
determining this tradeoff is beyond the scope of this study; however further work
regarding this topic is encouraged.
61
4 CONCLUSIONS AND RECOMMENDATIONS
An axisymmetric CFD model has been combined with a GA to optimize the
geometric shape of a launch vehicle. This vehicle shape was optimized to minimize both
the pressure fluctuation level at a critical point and the axial force coefficient. This model
was meshed with a structured grid using a custom written FORTRAN routine to develop
the mesh based on the GA selected variables that define the geometry. Measures were
taken to ensure that no grid overlap would occur while appropriately distributing nodal
points. A grid refinement study was performed to ensure sufficiently accurate results.
This refinement study showed that the course grid effectively computed the aerodynamic
parameters in a reasonable amount of time. This grid was used by the CFD solver in
combination with a GA for both axial force and pressure fluctuation minimization
studies.
Further work was conducted to determine the convergence criteria for each
optimization study. For the axial force minimization study, this consisted of determining
the appropriate residual values for which the axial force coefficient was sufficiently
converged. Similarly, the pressure fluctuation study convergence criterion was developed
by determining the residual values for which the local pressure was sufficiently
converged. Upon trial runs for both studies, it was found that a residual based
convergence criterion was not sufficient for the entire design space. An additional
62
convergence criterion was implemented to determine solution convergence based on the
local pressure or the axial force coefficient for the respective optimization studies.
The fluctuating pressure level minimization study showed that the average RMS
pressure fluctuation level was reduced by about 17.5% over the five flight conditions
investigated. The optimized geometry for the fluctuating pressure level study resembled
the shape of a blunted ogive, however with the large first cone angle, it is apparent that
such a design would not perform desirably aerodynamically. The goal of this study
however, was to demonstrate a methodology for using a GA combined with a CFD solver
to minimize pressure fluctuations at a critical point. This goal was achieved by the clear
reduction in pressure fluctuations at the point on the geometry investigated in this study.
The axial force minimization study also showed a reduction in axial force from the
baseline geometry. The optimized geometry had an average axial force coefficient that
was approximately 56% lower than the baseline geometry through the ascent trajectory
using a simple axisymmetric CFD model. The GA was able to accomplish this by
generating a geometry resembling the shape of a blunted ogive, which has historically
given desirable aerodynamic characteristics. While the optimized geometry in the axial
force minimization study significantly reduced axial force, it is not entirely clear whether
or not this design would be practical. The significant increase in total length of the conic
sections could result in additional weight, partially negating the benefits of reduced axial
force. Nevertheless, a method for reducing the axial force coefficient of a launch vehicle
through a typical ascent trajectory has clearly been demonstrated.
This work is considered to the first step toward a much more comprehensive
aerodynamic optimization for launch vehicles and missiles. In light of the results of this
63
study, further work is strongly recommended. Concerning the pressure fluctuation study,
it is suggested that additional research be carried out to develop more robust methods for
minimized pressure fluctuations at several points along the geometry. This could be
conducted in a very similar manner to the method discussed in this study. The additional
work would mostly include modifications to the objective function such that flow
parameters at multiple points are monitored by the CFD solver. Perhaps another
investigation of the pressure fluctuation study would be an additional optimization of the
geometry that only monitors the flight condition at which the peak pressure fluctuation
occurs. This would be a relatively simple modification of the presented work such that
only the Mach 0.85 flight condition is included in the CFD batch file. Also, increasing the
number of conic sections or representing the expansion section with a curved line such as
a third order polynomial would greatly improve the variability of the design space.
However, increased flexibility in the geometry would lead to much more complexity in
the custom grid generator that was written specifically for this study. A more robust grid
generator such as GAMBIT is strongly encouraged for further work using more complex
design spaces.
Additional work is also recommended to expand upon the results of the axial force
minimization study, particularly concerning the tradeoff between reducing axial force and
increasing vehicle weight. Perhaps an optimization study could be carried out in which
the goal is maximizing payload instead of minimizing axial force. This of course would
involve the addition of various models that could determine vehicle mass as a function of
geometric shape among other things. It is also conceivable that both of these optimization
studies could be combined as one optimization study that maximizes payload and
64
minimizes pressure fluctuations during a launch. Such a study would be a much more
comprehensive aerodynamic optimization study of a launch vehicle or missile.
65
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69
APPENDIX A: GA Input File
.false. ; micro
.false. ; pareto
.true. ; steady_state
.false. ; maximize
.true. ; elitist
.false. ; creep
.false. ; uniform
.false. ; restart
.true. ; remove_dup
.false. ; destroy_elite2
.false. ; niche
.false. ; phenotype
2531 ; iseed
0.9 ; pcross
0.001 ; pmutation
0.05 ; pcreep
1 ; ngoals
1.0 ; xgls(j)
.0 ; domst
2550 ; convrg_chk
5 ; no_para
'par1', 1.0, 0.5, .02, .false. ; xmax xmin resolution niche_par
'par2', 0.99, 0.4, .02, .false. ; xmax xmin resolution niche_par
'par3', 3.0, 1.5, .05, .false. ; xmax xmin resolution niche_par
'par4', 0.35, 0.05, .01, .false. ; xmax xmin resolution niche_par
'par5', 0.6, 0.1, .01, .false. ; xmax xmin resolution niche_par
1 ; ifreq
30 ; mempops
20 ; maxgen