ABSTRACT 
 
 
 
 
Title of Document: MULTIDISCIPLINARY OPTIMIZATION OF 
NON-SPHERICAL, BLUNT-BODY HEAT 
SHIELDS FOR A PLANETARY ENTRY 
VEHICLE 
 
  
 Joshua E. Johnson 
 Master of Science, 2006 
  
Directed By: Professor Mark J. Lewis 
 Department of Aerospace Engineering 
 
 
Gradient-based optimization of the aerodynamic performance, static stability, and 
stagnation-point heat transfer has been completed to find optimal heat shield geometries for 
blunt-body planetary entry vehicles. In the parametric study, performance trends have been 
identified by varying geometric parameters that define a range of cross-sections and axial 
shapes. Cross-sections considered include oblate and prolate ellipses, rounded-edge polygons, 
and rounded-edge concave polygons. Axial shapes consist of the spherical-segment, 
spherically-blunted cone, and power law. By varying angle-of-attack and geometric 
parameters, the aerodynamics, static stability, and heat transfer are optimized based on 
Newtonian impact theory with semi-empirical shock-standoff distance and stagnation-point 
heat transfer correlations. Methods have been verified against wind tunnel and flight data of 
the Apollo Command Module and are within 15% for aerodynamic coefficients and 
stagnation-point heat fluxes. Results indicate that oblate parallelogram configurations provide 
optimal sets of aerothermodynamic characteristics.   
  
 
 
 
 
 
 
 
 
 
 
MULTIDISCIPLINARY OPTIMIZATION OF NON-SPHERICAL, BLUNT-BODY 
HEAT SHIELDS FOR A PLANETARY ENTRY VEHICLE  
 
 
 
 
By 
 
 
 
Joshua E. Johnson 
 
 
 
 
 
Thesis submitted to the Faculty of the Graduate School of the  
University of Maryland, College Park, in partial fulfillment 
of the requirements for the degree of 
Master of Science 
2006 
 
 
 
 
 
 
 
 
 
Advisory Committee: 
Professor Mark J. Lewis, Chair 
Professor Roberto Celi 
Associate Professor James Baeder 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
? Copyright by 
Joshua Elijah Johnson 
2006 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ii 
 
Dedication 
 To my parents John and Caroline who have given me immeasurable 
knowledge in life and have encouraged me to be a part of furthering human space 
exploration. To my sisters Justine and Lucia Maria who are just starting out in this 
world.  
 
 iii 
 
Acknowledgements 
I would like to express my gratitude to Dr. Mark Lewis for the opportunity to 
conduct this research, who through his achievements in the field of hypersonics, has inspired 
me to strive forward in my studies and aerospace career. I would like to thank Dr. Ryan 
Starkey for his endless patience and time given in assisting me with my research. I would also 
like to thank the other members of my committee Dr. Roberto Celi, who taught me gradient-
based optimization theory, and Dr. James Baeder, who taught me the basic theories of 
computation fluid dynamics.  
I also would like to acknowledge the NASA funding that supported this work. This 
research was supported by the Space Vehicle Technology Institute (SVTI), one of the NASA 
Constellation University Institute Projects (CUIP), under grant NCC3-989, with joint 
sponsorship from the Department of Defense. Appreciation is expressed to Dr. Ken Yu, 
director of the SVT Institute at the University of Maryland, Claudia Meyer of the NASA 
Glenn Research Center, program manager of CUIP, and to Dr. John Schmisseur and Dr. 
Walter Jones of the Air Force Office of Scientific Research, the support of whom is greatly 
appreciated. 
Finally, I would like to thank my fellow SVTI graduate students, who have given me 
friendship and support, and especially to Adam Dissel for the early hours of company and 
learning that has greatly enhanced my graduate school experience.  
 
 
 iv 
 
Table of Contents 
 
 
Dedication?..............................................................................................................ii 
Acknowledgements................................................................................................iii 
Table of Contents.....................................................................................................iv 
List of Tables.............................................................................................................vi 
List of Figures.........................................................................................................vii 
List of Symbols..........................................................................................................x 
Chapter 1. Introduction.......................................................................................1 
1.1. Motivation.....................................................................................................1 
1.2. Previous Work..............................................................................................2 
1.2.1. Heat Shield Geometries........................................................................2 
1.2.2. Heat Transfer......................................................................................12 
1.2.3. Misconceptions...................................................................................17 
1.3. Research Objectives....................................................................................19 
1.4. Thesis Overview.........................................................................................20 
Chapter 2. Blunt-Body Heat Shield Geometries...........................................22 
2.1. Axial Shapes...............................................................................................22 
2.1.1. Spherical-segment...............................................................................23 
2.1.2. Spherically-blunted Cone....................................................................25 
2.1.3. Power Law..........................................................................................25 
2.2. Cross-section Shapes..................................................................................27 
2.3. Generating Blunt-Bodies............................................................................29 
2.4. Geometric Properties..................................................................................32 
Chapter 3. Aerodynamics and Static Stability..............................................37 
3.1. Modified Newtonian Impact Theory..........................................................38 
3.2. Shock-standoff Distance: Kaattari?s method..............................................44 
3.2.1. Method Implementation......................................................................45 
3.2.2. Accounting for Nonaxisymmetric Shapes & Angle of Attack...........53 
3.3. Static Stability.............................................................................................61 
3.4. Correcting Misinterpretations.....................................................................64 
3.4.1. Static Roll Stability Requirement.......................................................64 
3.4.2. Determining the Location of the Center of Pressure..........................66 
Chapter 4. Heat Transfer...................................................................................69 
4.1. Convection..................................................................................................70 
4.2. Radiation.....................................................................................................73 
Chapter 5. Description of Code and Optimization Process........................77 
5.1. Code Layout................................................................................................77 
5.2. Design Optimization Tools (DOT).............................................................80 
5.3. Choosing Objective Functions....................................................................82 
Chapter 6. Design Code Validation................................................................85 
6.1. Aerodynamics and Static Stability..............................................................85 
 
 v 
 
6.1.1. Comparison with Apollo Wind Tunnel Data......................................86 
6.1.2. Comparison with Apollo Flight Test Data..........................................90 
6.1.3. Comparison with Additional Sources.................................................93 
6.2. Stagnation-Point Heat Transfer...................................................................94 
6.2.1. Apollo 4..............................................................................................95 
6.2.2. FIRE II..............................................................................................102 
Chapter 7. Parametric Analysis......................................................................105 
7.1. Selecting a Superelliptical Base................................................................105 
7.2. Heat Shield Shapes and Aerodynamic Performance................................106 
7.3. Hypersonic Aerodynamic Performance and Stability...............................109 
7.4. Effect of Eccentricity on Aerodynamic Performance...............................115 
7.5. Comparison of Aerothermodynamic Performance on Two Heat Shields 119 
Chapter 8. Optimal Configurations..............................................................126 
8.1. Maximizing LV/D and ?VLV/D.................................................................132 
8.2. Maximizing (LV/D)/qs,tot...........................................................................138 
8.3. Minimizing qs,tot........................................................................................141 
8.4. Minimizing Cm,cg,?.....................................................................................146 
8.5. Limitations of this Optimization...............................................................150 
Chapter 9. Conclusions....................................................................................152 
9.1. Summary of Results..................................................................................152 
9.1.1. Parametric Analysis..........................................................................152 
9.1.2. Optimization Results.........................................................................153 
9.2. Suggestions for Future Work....................................................................155 
Appendix A: Optimization Iteration Histories...............................................158 
A.1 Maximizing LV/D and ?VLV/D.................................................................158 
A.2   Maximizing (LV/D)/qs,tot...........................................................................161 
A.3  Minimizing qs,tot........................................................................................163 
A.4   Minimizing Cm,cg,?.....................................................................................165 
Bibliography..........................................................................................................167 
 
 
 
 
 
 
 vi 
 
List of Tables 
 
Table 2.1. Superformula parameters for rounded-edge polygons (n3 = n2)................28 
Table 5.1. Most useful objective functions.................................................................83 
Table 5.2. Questionable objective functions...............................................................84 
Table 5.3. Objective functions to use with caution.....................................................84 
Table 6.1. Percent error of Newtonian computations compared to wind tunnel data, 
M? = 9, in Ref. [9], Percent error averaged over -30o ? ? ? 0o...........................89 
Table 6.2. Comparison of Apollo AS-201 Data and Computations, M? = 14, ? = -
16.5o at 4900 s.....................................................................................................91 
Table 6.3. Comparison of Apollo 4 Data and Computations, M? = 30, ? = -25o at 
30040 s................................................................................................................92 
Table 6.4. Apollo 4 Comparison of Total Heat Transfer..........................................100 
Table 7.1. Variable ranges and constants for each axial shape.................................108 
Table 7.2. Effects of geometric parameters and ? on aerodynamic performance....111 
Table 7.3 Aerodynamic comparison of spherical-segments (? = 5o)........................118 
Table 8.1. Optimization design variables with side constraints................................127 
Table 8.2. Initial and Optimal Designs for maximizing LV/D for different m..........133 
Table 8.3. Aerothermodynamic comparison of LV/D and ?VLV/D results at M? = 
32.8255 (m = 4).................................................................................................136 
Table 8.4. Aerothermodynamic comparison of (LV/D)/qs,tot results at M? = 32.8255 (m 
= 4)....................................................................................................................139 
Table 8.5. Aerothermodynamic comparison of qs,tot results at M? = 32.8255 (m = 4).
...........................................................................................................................143 
Table 8.6. Aerothermodynamic comparison of Cm,cg,? results at M? = 32.8255 (m = 4).
...........................................................................................................................148 
Table A.0.1. Optimized (LV/D)/qs,tot for M? from 30 to 41,  m = 4...........................162 
Table A.0.2. Optimized qs,tot for M? from 30 to 41,  m = 4......................................164 
 
 
 
 vii 
 
List of Figures 
 
Figure 1.1. The two most familiar heat shield geometries............................................3 
Figure 1.2. Apollo 4 Trajectory from Ref. [7]..............................................................4 
Figure 1.3. Viking Landing Capsule System from Ref. [19]........................................6 
Figure 1.4. Raked cone from Ref. [21].........................................................................7 
Figure 1.5. AFE flight vehicle configuration from Ref. [21]........................................8 
Figure 1.6. Bent biconic configuration from Ref. [24].................................................9 
Figure 2.1. Fixed-body coordinate system, spherical-segment, ? = 30o, ?s = 60o, n2 = 
n3 = 2...................................................................................................................22 
Figure 2.2. Spherical axial profiles and heat shields..................................................24 
Figure 2.3. Power law axial shape..............................................................................26 
Figure 2.4. Cross-section shapes produced using parameters from Table 2.1............29 
Figure 2.5. Spherically-blunted cone, rn/d = 0.25, ?c = 70o, e = 0.95, m = 5, n1 = 1.75, 
n2 = 2...................................................................................................................31 
Figure 2.6. Spherical-segment, ?s = 40o, e = -0.85, m = 12, n1 = 10.75, n2 = 1..........31 
Figure 2.7. Power law, A = 3, b = 0.75, e = 0.5, m = 5, n1 = 1.75, n2 =  5..................31 
Figure 3.1. Coordinate systems with positive ? and ? and moment conventions......37 
Figure 3.2. Spherically-blunted cone profile shock-standoff distance variance.........58 
Figure 3.3. Cl,cg,? distribution for spherical-segment, elliptical base (n2 = 2), varying e 
and ?s, ? = 20o, ? = 5o.........................................................................................65 
Figure 3.4. Variation in Cm,cg with angle of attack.....................................................67 
Figure 4.1. Stagnation-point shock-standoff distance wind tunnel data with empirical 
curve-fit, Ref. [48]..............................................................................................75 
Figure 5.1. Diagram of Overall Code.........................................................................78 
Figure 5.2. Diagram of Analysis Code.......................................................................79 
Figure 6.1. Cm,cg comparison between modified Newtonian and wind tunnel data, Ref. 
[8]........................................................................................................................86 
Figure 6.2. Cl,0 comparison between modified Newtonian and wind tunnel data, Ref. 
[8]........................................................................................................................87 
Figure 6.3. Aerodynamic force comparisons between modified Newtonian and wind 
tunnel data, Ref. [8]............................................................................................88 
Figure 6.4. Normal-shock density and specific heat ratios for the high radiative heat 
flux portion of the Apollo 4 trajectory................................................................95 
Figure 6.5. Shock-standoff distance method comparison...........................................96 
Figure 6.6. Apollo 4 reff for stagnation-point radiative heat transfer..........................97 
Figure 6.7. Validation of radiative heat flux correlations for Apollo 4......................98 
Figure 6.8. Apollo 4 radiative heat transfer code validation.......................................99 
Figure 6.9. Apollo 4 Trajectory from Ref. [36]..........................................................99 
Figure 6.10. Convective heat flux distribution of Apollo Command Module at ? = -
25o from Ref. [36].............................................................................................102 
Figure 6.11. FIRE II Total heat flux comparison with flight data from Ref. [47]....103 
Figure 7.1. Cm,0,? distribution varying m-gon and e with spherical-segment axial shape 
and rounded-edge concave cross-sections, ?=-20o, ? = 5o................................106 
 
 viii 
 
Figure 7.2. Chosen heat shield shapes for parametric analysis.................................107 
Figure 7.3. Cm,0,? distribution for power law A and n2, b = 0.75, e = 0.5, ?=-20o, ? = 
5o.......................................................................................................................109 
Figure 7.4. Cm,cg,? distribution for power law A and n2, b = 0.75, e = 0.5, ?=-20o, ? = 
5o.......................................................................................................................110 
Figure 7.5. Effects of ?s, ?c (rn/d = 0.25), A (b = 0.75) on stability coefficients and 
derivatives, ? = -20o, ? = 5o..............................................................................113 
Figure 7.6. Effects of ?s, ?c (rn/d = 0.25), A (b = 0.75), and b (A = 2) on ?v, ? = -20o, ? 
= 5o....................................................................................................................114 
Figure 7.7. ?v distribution for spherical-segment, elliptical base (n2 = 2), varying e 
and ?s.................................................................................................................115 
Figure 7.8. Cl,cg,? distribution for spherical-segment, elliptical base (n2 = 2), varying e 
and ?s, ? = 20o, ? = 5o.......................................................................................116 
Figure 7.9. LV/D distribution for spherical-segment: elliptical base (n2 = 2), varying e 
and ?s, ? = 20o, ? = 5o.......................................................................................117 
Figure 7.10. Spherical-segment  ?s = 90o, e = -0.95 with an increase of 56% in LV/D.
...........................................................................................................................117 
Figure 7.11.  ,totsq&  distributions for m = 4 and 6, spherically-blunted cone axial 
shape, ?c = 55o, rn/d =0.05, varying e and n2 = 1.3 to 2.3, M? = 32.8, ? = -20o.120 
Figure 7.12.  ,totsq&  distributions for m = 4 and 6, spherically-blunted cone axial shape, 
?c = 55o, rn/d =0.05, varying e and n2 = 1.3 to 4.0, M? = 32.8, ? = -20o...........121 
Figure 7.13. LV/D distribution for m = 4 and 6, spherically-blunted cone axial shape, 
?c = 55o, rn/d =0.05, varying e and n2 = 1.3 to 4.0, M? = 32.8, ? = -20o...........122 
Figure 7.14.  ,convsq& and  ,radsq&  distributions for parallelogram base (m = 4, n2 = 1.4 ), 
spherically-blunted cone axial shape, ?c = 55o, varying e and rn/d, M? = 32.8, ? = 
-20o....................................................................................................................123 
Figure 7.15.  ,totsq& distribution for parallelogram base (m = 4, n2 = 1.4 ), spherically-
blunted cone axial shape, ?c = 55o, varying e and rn/d, M? = 32.8, ? = -20o....125 
Figure 8.1. Optimized geometries from Table 8.2....................................................133 
Figure 8.2. Optimized geometries from Table 8.3....................................................137 
Figure 8.3. Maximized (LV/D)/qs,tot  geometries from Table 8.4...............................140 
Figure 8.4. Maximized (LV/D)/qs,tot  results for M? = 30 ? 41..................................141 
Figure 8.5. Minimum qs,tot configurations, spherical-segment axial shape..............142 
Figure 8.6. Minimum qs,tot  geometries from Table 8.5............................................144 
Figure 8.7. Minimized qs,tot  results for M? = 30 ? 41...............................................146 
Figure 8.8. Minimum Cm,cg,? geometries (m = 4) in Table 8.6..................................147 
Figure A.0.1. Iteration history for maximum LV/D hexagonal shape (m = 6) in Table 
8.2......................................................................................................................158 
Figure A.0.2. Iteration history for maximum LV/D power law/parallelogram (m = 4) in 
Table 8.3...........................................................................................................159 
Figure A.0.3. Iteration history for maximum ?VLV/D spherically-blunted 
cone/parallelogram (m = 4) in Table 8.3...........................................................160 
Figure A.0.4. Iteration history for maximum (LV/D)/qs,tot spherically-blunted cone axial 
shape (m = 4) in Table 8.4................................................................................161 
 
 ix 
 
Figure A.0.5. Iteration history for minimum qs,tot spherical-segment axial shape (m = 
4) in Table 8.5...................................................................................................163 
Figure A.0.6. Iteration history for minimum Cm,cg,? power law axial shape (m = 4) in 
Table 8.6...........................................................................................................165 
Figure A.0.7. Iteration history for minimum Cm,cg,? spherical-segment axial shape (m 
= 4) in Table 8.6................................................................................................166 
 
 x 
 
List of Symbols 
 
A = Coefficient of power law 
Ab = Area of the base (m2) 
Ap = Planform area (m2) 
AFE = Aeroassist Flight Experiment 
a = Speed of sound (m/s) 
a1 = Semimajor axis length of a superellipse (m) 
a2 = Semimajor axis length of a blunt body(m) 
b = Exponent of power law 
b1 = Semiminor axis length of a superellipse (m) 
b2 = Semiminor axis length of a blunt body (m) 
C = Aerodynamic coefficient 
CEV = Crew Exploration Vehicle 
CFD = Computational fluid dynamics 
CM = Command Module, refers to Project Apollo unless noted otherwise  
DOT = Design Optimization Tools 
d = Diameter (m) 
d1, d2 = Distances from mesh point to point as indicated in the text 
d(j,k),(j+1,k) = Distance from mesh point (j,k) to point (j+1,k) 
dA = Differential surface area 
dAp = Differential planform area 
dr = Differential distance from mesh point to origin 
dV = Differential volume 
d? = Differential theta  
d? = Differential sweep angle  
E = Search direction for optimization 
e = Eccentricity 
FIRE = Flight Investigation of the Reentry Environment 
Gr   = Vector of constraints  
Gq  = qth constraint 
F = Integration of f with respect to x 
f = Function of x 
g1, g2, g3 = Radiation correlation parameters 
gw = Ratio of wall enthalpy to total enthalpy 
H = Correlation exponent 
h = Altitude (ft) 
 = Step size 
j = Mesh point index in ?-direction 
jmax = Maximum mesh points in ?-direction 
k = Mesh point index in x-direction 
kmax = Maximum mesh points in x-direction 
l = Length of heat shield from nose to base (m) 
lb = Length from differential surface to base (m) 
 
 xi 
 
lM = Length of the semimajor axis of the cross-section 
L/D = Lift-to-drag ratio 
MMFD = Modified Method of Feasible Directions 
M? = Freestream Mach number 
m =  Number of sides of the superellipse in the superformula 
N = Total number of sections in which to divide the conical base  
NCON = Number of constraints  
NDV = Number of design variables 
n = Number of the section of the conical base 
n1, n2, n3 = Superelliptic parameters 
n? = Outward normal of a surface 
OBJ = Objective function 
O(Xr ) = Objective function as a function of Xr  
PEV = Planetary entry vehicle 
p = Index corresponding to a specific design variable Xp 
q = Index corresponding to a specific constraint Gq 
q? = Freestream dynamic pressure (Pa) 
 sq&   =  Stagnation-point heat flux (W/m2)  
RCS  = Reaction control system 
r = Base radius (m) 
rc = Corner radius (m) 
reff = Effective radius of curvature for radiative heat transfer (m) 
rn = Nose radius for spherically-blunted cone axial shape (m) 
rs = Radius of curvature for spherical-segment axial shape (m) 
rst = Length from the central axis to the stagnation point (m) 
rsh = Radius of curvature for the spherical shock (m) 
S = Surface area of cross-section (m2) 
Shs = Surface area of heat shield without base area (m2)  
SCRAMP  = Slotted Compression Ramp 
TPS = Thermal Protection System 
u = Iteration number 
V? = Freestream velocity magnitude (m/s)  
Vhs = Volume of heat shield (m3) 
Vx, Vy, Vz = Freestream velocity components normalized by V? 
v = Exponent of superellipse equation 
WT = Wind tunnel 
w = Distance from point (j,k) to x-axis 
wv = Wind angle 
Xr  = Vector of design variables Xp, in which p = 1 to NDV 
Xp = pth design variable 
x, y, z = Coordinate directions  
xs  = The streamwise distance from the apex of the shock to the point on  
  the shock at distance y* from the central axis 
? = Angle-of-attack (deg) 
?* = Scalar parameter for optimization 
? = Sideslip angle (deg) 
 
 xii 
 
? = Ratio of specific heats 
?so = Shock-standoff distance (ft) 
? = Inclination on forward body surface at the tangent point with corner  
  radius, also mentioned as tangency angle  
?v = Volumetric efficiency (surface area comparison) 
?c = Half-cone angle (deg) 
?s = Half-spherical-segment angle (deg) 
?* = Shock surface inclination at a point opposite of the sonic point on  
  a body with respect to a plane normal to the freestream direction 
?*o = Shock surface inclination at a point opposite of the sonic point on  
  a flat disk  
?  = Lagrange multiplier  
? = Density (kg/m3) 
?* = Sonic angle of a blunt body, which is the body surface inclination of 
   a blunt body at the sonic point with respect to a plane normal  
  to freestream direction 
?*1,  = Sonic angle of a sphere  
? = Sweep angle (rad) 
 = Azimuth angle 
? = Angle from x-axis and in the xy plane (rad) 
 
Subscripts 
0 = Location at tip of nose, stagnation point, moments normalized by 
base diameter d 
1 = Before the normal shock 
2 = After the normal shock 
75 = 75% of value assuming uniform density  
A = Axial force 
cg = Location of center of gravity 
conv = Convective 
cp = Location of center of pressure 
D =  Drag  
eff = Effective 
H = Horizontal component with respect to the body 
j = xy-plane index 
L = Lift  
l = Rolling moment 
lower = Lower constraint value 
m =  Pitching moment 
N = Normal force 
nd = Non-dimensionalized vector 
n = Yawing moment 
R = Resultant 
rad =   Radiative    
sl =    Sea level  
tot = Total 
 
 xiii 
 
UD = Uniform density 
upper = Upper constraint value 
V = Vertical component with respect to the body 
Y = Side force 
? = Derivative with respect to angle-of-attack (per rad) 
? = Derivative with respect to sideslip angle (per rad) 
? = Freestream conditions 
 
 1 
 
Chapter 1.  Introduction 
1.1. Motivation 
A blunt-bodied planetary entry vehicle (PEV) requires a proper combination 
of aerodynamic performance, stability, and heat transfer for it to successfully follow 
its designed planetary entry trajectory. The simplest blunt-bodied planetary entry 
vehicle is the re-entry space capsule. It is covered with thermal protection material, 
with heat shield materials that can withstand very high temperatures on the surface of 
the vehicle that faces the freestream.  
The heat shield is designed to protect a majority of the vehicle from entry 
temperatures, and because the flow typically separates from the vehicle past the heat 
shield, its geometry generally defines the capsule?s hypersonic aerodynamic 
performance. The design also provides passive protection from entry flow 
disturbances that could destabilize the vehicle and endanger the astronauts, and active 
protection from rolling is also provided on human planetary entry vehicles with a 
reaction control system (RCS). The passive protection is acquired through the 
aerodynamic design of the heat shield, and it is common during an initial analysis for 
aerodynamic performance, stability and heat transfer characteristics to be satisfied 
mostly through the design of a vehicle?s heat shield.  
Blunt-bodied capsules, such as the one that will be used by NASA?s Crew 
Exploration Vehicle (CEV), will re-enter Earth?s atmosphere at high hypersonic 
velocities, with projected Mach numbers ranging from 30 to 50. Such high Mach 
numbers deliver high heat fluxes greater than 300 W/cm2 along the frontal surface 
 
 2 
 
and heat loads greater than 30000 J/cm2. Since it is not clear that the classic spherical-
segment1 or spherically-blunted cone2,3 provides optimal aerothermodynamic 
performance, executing an optimization on the aerodynamics, static stability, and heat 
transfer for a broad range of heat shield shapes may assist in ascertaining an optimal, 
stable configuration with higher lift-to-drag (L/D) and lower heat transfer fluxes and 
loads.  
A parametric formulation of the aerodynamic performance and stability of 
heat shields based on Newtonian impact theory is completed in this work, with the 
addition of semi-empirical methods for the calculation of the stagnation-point heat 
fluxes. An optimizer is applied to this formulation to analyze the aerothermodynamic 
performance of heat shield geometries. 
1.2. Previous Work 
1.2.1. Heat Shield Geometries 
The two primary classes of shapes that have been developed by NASA are the 
spherical-segment and the spherically-blunted cone, shown in Figure 1.1(a) and (b) 
respectively. Extensive research on spherical-segments led to its use as the heat shield 
for human re-entry space capsules including the NASA Gemini and Apollo missions. 
The spherical-segment is described by a spherical radius rs and maximum spherical-
segment angle ?s measured from the central axis. The Project Gemini Re-entry 
Module4,5 included a 38o spherical-segment heat shield (?s = 19o) with rs = 3.712 m, 
and a base diameter of 2.32 m. The Project Apollo Command Module (CM) had a 50o 
spherical-segment (?s = 25o) with rs = 4.694 m, and a base diameter of 3.912 m.  
 
 3 
 
 
            a) Spherical-segment heat shield, ?s = 60o.      b) Spherically-blunted cone heat shield,  
       rn/d = 0.25,  ?c = 60o.    
Figure 1.1. The two most familiar heat shield geometries. 
Specific Apollo mission aerodynamic flight data is provided for Apollo 
mission AS-202 and Apollo 4 in Ref. [6] and Ref. [7] respectively. The Command 
Module in AS-202 re-entered Earth?s atmosphere at 8.23 km/s (27,000 ft/s) to 
produce planetary entry conditions that occur when entering from satellite orbit. The 
Apollo 4 (AS-501) followed an elliptical orbit around Earth that produced the 
expected re-entry velocity from lunar return of 10.7 km/s (35,100 ft/s). Rather than 
passing by the Moon, Apollo 4 had a second Service Module Propulsion System 
(SPS) burn at trajectory location 13 marked in Figure 1.2 to produce lunar return re-
entry conditions and reduce the mission time to 8.5 hours (from six days). Hillje7 
provided Apollo flight data which was used for code validation and is presented in 
Chapter 6. Note that most of the aerodynamic data from the Apollo flights can be 
reproduced accurately by this work, except for the normal coefficient CN which can 
have up to 40% difference compared to the available flight-derived data.     
x 
z 
? ?s 
rs 
y 
  
  
rn d 
l 
?c 
 
 4 
 
 
Figure 1.2. Apollo 4 Trajectory from Ref. [7]. 
Moseley analyzed the aerodynamic stability characteristics of the Apollo 
Command Module in Refs. [8] and [9]. Both of these NASA Technical Notes provide 
aerodynamic, static and dynamic stability wind tunnel data on several Command 
Module configurations with their schematics and relative dimensions. From this set, 
only two figures are dedicated to rolling moment stability in Ref. [8], and so a 
discussion of the Command Module?s roll static stability is provided in Chapter 6. 
DeRose10 provides an analysis of the center-of-gravity offset from the central axis of 
the vehicle and also discusses the proper way to compare wind tunnel measurements 
of a non-ablative model to a flight vehicle with an ablative heat shield. Horstman11 
compares the Apollo and Gemini wind tunnel models with center-of-gravity offsets.  
The spherically-blunted cone is another commonly-used blunt-body space 
capsule configuration, as shown in Figure 1.1(b). It is defined by half-cone angle ?c, 
the nose radius rn, and base diameter d. It is common to see the ratio rn/d as a listed 
characteristic. Because convective heat transfer increases while radiative heat transfer 
decreases with a reduction in radius for a blunt-body, there is an optimum rn that 
minimizes the heat transfer to the vehicle. The advantage of this configuration is that 
it can offer designs with the same base radius as a spherical segment without 
 
 5 
 
constraining the nose radius. If the optimal heat shield design for a given set of 
mission requirements has a smaller nose radius than can be offered by a spherical-
segment with base radius r, then that nose radius could be generated with a 
spherically-blunted cone configuration. Also, the equations that render the 
spherically-blunted cone can be setup to account for the entire spherical-segment 
regime. 
Chrusciel3 provides a method for calculating the aerodynamic characteristics 
of spherically-blunted cones. Chrusciel provides the change in the center-of-pressure 
location with an increase in nose radius and was used in this work to determine a 
reasonable range of center-of-pressure locations. A discussion on the 
misunderstanding present in several sources concerning the calculation of the center-
of-pressure location is included in Chapter 3 and those sources are mentioned later in 
this section.  
Jones12 has completed a wind tunnel investigation on model comparison of the 
pressure distributions on sharp-nosed and spherically-blunted cones with large cone 
angles ?c at hypersonic speeds in air (M? = 7.9), helium (M? = 20.3), and 
tetrafluoromethane CF4 (M? = 6.2). The pressure distributions from the three different 
gases for the tested configurations almost always overlap each other at ? = 0o. These 
results are also compared to theoretical methods including Newtonian theory and 
concluded that the Newtonian theory does not predict the surface pressure distribution 
properly near the edges of the configuration. It also shows that there are significant 
differences in the actual distribution and the one rendered by Newtonian theory. It is 
shown in this work that although this may be true, Newtonian theory can predict the 
 
 6 
 
aerodynamic forces and moments of the Apollo Command Module (spherical-
segment) within 15% and the trim angle-of-attack within 1.2o. Additional work on 
spherically blunted cones has been completed by Tauber13 and Bernot.14  
The spherically-blunted cone configuration was used on the Viking2,15 and 
Pathfinder16,17,18 missions to Mars. Two Viking missions consisted of an orbiter and a 
lander. The orbiter mapped the surface of Mars, and the landers of the Viking I and II 
missions had the first successful Mars landings in 1976. The Viking space capsule,19 
shown in Figure 1.3, housed the lander and had a superlight ablative (SLA) heat 
shield of dimensions ?c = 70o, rn/d = 0.25, and d = 3.505 m.  
 
Figure 1.3. Viking Landing Capsule System from Ref. [19]. 
In 1997, the Mars Pathfinder mission had a Mars rover that was thermally protected 
during planetary entry by an aeroshell thermal protection system (TPS). Once on the 
surface, the Pathfinder rover would photograph the immediate vicinity and acquire 
data that would be sent back to Earth for analysis. Although higher heat fluxes were 
expected during planetary entry, the Pathfinder mission had a heat shield with the 
same ?c and rn/d as the Viking capsules. 
 
 7 
 
 More recent heat shield designs include the raked cone, the biconic and bent 
biconic cones, the parashield, and the flare-skirt aft-body. The raked cone 
geometry20,21 is a spherically-blunted cone raked at an angle as shown in Figure 1.4.  
 
Figure 1.4. Raked cone from Ref. [21]. 
This geometry offers a way to produce positive lift at zero angle-of-attack. Since the 
bottom surface of the heat shield has greater surface area than the upper half by 
design, the flow accelerates more over the bottom surface. This produces positive lift 
because a larger portion of the surface pressure contributes to lift from the bottom 
surface than the top surface. The raked cone geometry can be designed to render 
higher L/D at negative angles-of attack than its axisymmetric analogue.  
 An example of the raked cone geometry is the Aeroassist Flight Experiment 
(AFE) that NASA worked on and cancelled in the early 1990s. It had a 14 ft diameter 
and was planned to participate in ten Shuttle-launched experiments. Figure 1.4 shows 
a few of the dimensions of the AFE?s heat shield geometry. It has a 60o half-cone 
angle and a 73o cone rake angle. One additional feature of the AFE?s rendition of the 
 
 8 
 
raked cone is its ellipsoid nose with an ellipticity equal to 2.  The AFE geometry is 
shown in Figure 1.5.  
 
Figure 1.5. AFE flight vehicle configuration from Ref. [21]. 
At zero angle-of-attack, the L/D is approximately 0.30 while it is approximately 0.43 
at ? = ?10o.22 Wells presents the aerodynamic performance and shock shapes of the 
AFE from wind tunnel results in Ref. [22]. Micol gives the wind tunnel results for its 
hypersonic lateral and directional stability in Ref. [21] and discusses a simulation of 
real gas effects on the AFE in Ref. [23].   
 The biconic and bent biconic bodies have been looked at for an aero-assisted 
orbital-transfer vehicle application, but it can be argued that they can be used for 
planetary entry vehicle applications as well. The biconic heat shield is a spherically-
blunted cone with an additional conical frustum that has a smaller half-cone angle. 
The bent biconic heat shield24,25 has this conical frustrum tilted at an angle with the 
spherically-blunted cone as shown in Figure 1.6(a). Davies and Park24 present the 
aerodynamics characteristics of a bent biconic with a fore half-cone angle of 12.84o 
and an aft half-cone angle of 7o tilted at 7o. This configuration enables a blunt-nosed 
 
 9 
 
body to have L/D > 1 at positive angle-of-attack as shown in Figure 1.6(b). Because 
of its slenderness due to its low half-cone angle, it can be argued that this is an 
atypical example of a blunt-body.  
 
a)  Bent biconic geometry 
 
b) Aerodynamic characteristics 
Figure 1.6. Bent biconic configuration from Ref. [24]. 
The common blunt-body produces positive L/D at negative angle-of-attack. 
Davies and Park acknowledge that a half-cone angle larger than 45o is required to 
produces positive lift values at negative angle-of-attack for an axisymmetric 
 
 10 
 
spherically-blunted cone. The reason for this is that at negative angle-of-attack, the 
axial force has a larger contribution to lift than the normal force. The axial force?s 
contribution is positive while the normal force?s is negative, thereby rendering 
positive lift. Slender designs are usually not considered for planetary entry because 
they produce higher heat transfer rates at the leading edges than blunter designs. 
Although the presented bent biconic configuration is slender, it will become a feasible 
design when more advanced high temperature materials are available. In fact, one 
could argue that this configuration could be feasible today if an ablative material is 
applied and if the nose radius-of-curvature is larger than that of the edge of the 
Apollo CM?s heat shield, which was the location of highest heat transfer (not the 
stagnation point). Either way, its forebody half-cone angle can be easily modified for 
planetary entry applications. 
Other more recent vehicle geometries are the parashield and the Slotted 
Compression RAMP probe. The parashield is a flexible, umbrella-like planetary 
entry, aerobrake, or aeroassist vehicle.26 Magazu, Lewis, and Akin completed an 
analysis of a parashield with a ballistic coefficient of 181 Pa composed of twelve 
radial spars for LEO re-entry. This configuration has a mass of 150 kg and could be 
scaled by at least a factor of ten if desired. They determined that this parashield has a 
hypersonic L/D of 0.18 at ? = 15o and that there is increased pressure at each spar 
whether or not concavity exists.26 The portability of this geometry in closed 
configuration is a feature unique to parashield and inflatable ballute geometries.  
Murbach27 has examined the Slotted Compression RAMP (SCRAMP) probe, 
which has undergone several sub-orbital test flights under his supervision. This 
 
 11 
 
geometry has a long cylindrical forebody with a hemispherical nose and an aft flare of 
high half-cone angle (?70o) attached to the rear of the vehicle. The aft flare creates 
most of the drag on the vehicle but also produces a compression ramp leading to flow 
recirculation. As a result, several slots are placed where the aft flare meets the 
cylindrical fore body to minimize the flow recirculation and thereby increase vehicle 
drag.  
The payload in the SCRAMP probe is located inside the front of the 
cylindrical forebody in order to place the center of gravity far forward. Since the aft 
flare produces most of the drag on the vehicle, the location of the center of pressure is 
far behind this center of gravity, rendering a negative Cm,cg,?. Murbach observed that 
by increasing the slot size, the maximum pressure on the aft shield is increased. This 
configuration has a negative Cm,cg,? which is considerably better than the Apollo 
CM?s value of -0.143/rad,9 rendering an outstanding, longitudinally stable entry 
vehicle. Note that the described configuration does not produce significant lift but 
may be designed to in future flight tests. 
Several of the classic and more recent design configurations were analyzed as 
lunar-return planetary entry applications in the work of Whitmore.28 Whitmore?s 
computational results for aerodynamics and stability are consistent with the results 
presented in this thesis. Bertin29 provides a thorough overview and several 
correlations concerning hypersonic aerothermodynamics, and Rasmussen30 provides 
an in-depth look at hypersonic aerodynamics.  
 
 12 
 
1.2.2. Heat Transfer  
Extremely high heat transfer rates may be experienced during planetary entry. 
Since the PEV undergoes high heat transfer rates for several minutes, the heat transfer 
load is equally important.  
In this section general references for planetary entry heat transfer are first 
mentioned. Since the heat transfer results of this work consist of stagnation-point heat 
fluxes, the corresponding references are mentioned and referred to throughout the 
text. For calculating the heat flux, there are more sources and emphasis in this work 
given to radiation than convection because radiation is an ongoing research topic. The 
drawback of this work is that the heat load is not determined, but it is mentioned as an 
important topic for future work.  
In recent analyses, Park31 and Rochelle32 discuss the aerothermodynamic 
environments for Mars entry and return, as well as lunar return. Park presents an 
analysis of two human missions (one is 330 days and the other is 436 days) to Mars 
and notes that the existence of an optimum nose radius for the tradeoff between 
convective and radiative heat transfer. Rochelle analyzes several capsule geometries 
including the biconic and modified AFE aeroshell capsules for Mars entry and notes 
that with an increase in entry velocity from 10 km/s to 12 km/s, the radiative heat flux 
goes from 13% to 42% of the total heat flux.   
Two classic references from Rose33 and Kemp34 at the Avco-Everett Research 
Laboratory give a general overview of stagnation-point and laminar heat transfer in 
dissociate air. Kemp notes that the maximum heat transfer flux can be larger than the 
stagnation-point heat flux, as is the case for a flat-nosed body in which the convective 
 
 13 
 
heat transfer flux is approximately 30% larger at the corner than that at the 
stagnation-point. This is noticed in the wind tunnel results for the Apollo CM 
presented by Lee,35 in which the corner has a heat flux that is 80% larger than at the 
stagnation-point at ? = 33o at M? = 9.07. The main Apollo missions that included the 
astronauts did not re-enter at 33o, and so the maximum heating rate was lower. This 
wind tunnel test was completed to find out the worst-case scenario with the crew 
compartment (aftbody) being tangent with the freestream flow. Lee also offers a 
thorough explanation of the convection and radiative heat transfer and gives the heat 
flux and pressure distributions about the heat shield. Note that the radiative heat flux 
correlation is analyzed in Chapter 6, but is not recommended for use since there are 
simpler correlations that follow the Apollo flight data better.  
Two Apollo CM were instrumented for aerothermodynamic analysis. Lee36 
compares the flight results from superorbital entry, as is the case in lunar return, with 
predictions  (note that Ref. [35] was written before the Apollo missions were 
completed). Lee also gives an aerothermodynamic evaluation in Ref. [37] that 
presents the highlights of the re-entry aerodynamics and heat transfer for the Apollo 
missions.  
The heat shield of the Apollo CM is the main part of the Apollo?s Thermal 
Protection System (TPS). Pavlosky38 details the history of designing the Apollo TPS 
and the manufacturing process. Also, he includes a summary of the predicted 
maximum heat transfer rates and loads for Apollo missions 8 and 10-16. Park and 
Tauber39 provide a current review of heat shielding problems experienced by the 
Apollo 4, 6, Pioneer-Venus, and Galileo Probe missions. Also, Scotti40 presents a 
 
 14 
 
compilation of TPS technologies that were current in 1992 including the shuttle tiles, 
a TPS design for the cancelled NASA Aeroassist Flight Experiment (AFE), and 
proposed future materials.  
The amount of stagnation-point heat transfer is one of the main 
aerothermodynamic benchmarks for comparing the capability of one entry vehicle to 
another. The stagnation-point heat transfer is not necessarily analogous with the point 
of maximum heat transfer, as was the case for the Apollo CM and the flat-nosed 
body. As a result, a more appropriate benchmark would be to compare the point of 
maximum heat transfer. Because this requires a more computationally expensive 
process that is beyond the focus of this thesis, previous work concerning the 
convective and radiative heat transfer at the stagnation-point has been acquired. 
Lovelace41 provides correlations for both convective and radiative heat 
transfer at the stagnation point; it will be shown in Chapter 6 that the radiative heat 
transfer correlation is one of two that matches the Apollo flight data the closest.  
For convective heat transfer, Tauber42 provides the stagnation-point, laminar 
and turbulent flat plate correlations that he validates against the US Space Shuttle 
heating rates. Tauber applies the well-known Fay and Riddell43 relation that assumes 
that convection is inversely proportional to the square root of the nose radius. This 
would suggest that the convective heat transfer approaches zero as the nose radius is 
increased, but Zoby44 notices that blunt bodies have more enthalpy than expected by 
this theory. Zoby shows that an adjustment that is based on the change in the velocity 
gradient from that of a hemisphere can produce an effective radius that can be applied 
to improve the theory?s accuracy. 
 
 15 
 
For radiative heat transfer, Tauber45 also provides the stagnation-point 
correlations for Earth and Mars entries. This is one of the two correlations that match 
the Apollo flight data the closest. Originally, Tauber?s correlations were not meant to 
be applied to the entire altitude regime on Earth, but this thesis will show that there is 
not much difference from Lovelace?s correlation41 when following the Apollo 4 
trajectory. A review of the stagnation-point radiation from the Apollo 4 is available 
by Park.46 
Cauchon47 provides the radiative heating results from the Second Flight 
Investigation of the Reentry Environment (FIRE II) experiment at the superorbital 
entry velocity of 11.4 km/s (37,400 ft/s), which is slightly larger than the fastest 
Apollo entry at 10.7 km/s (35,000 ft/s) for Apollo 4.7 Cauchon compares the theory to 
the few flight test data points. The FIRE II had three spherical-segment heat shields 
layered over each other. After initiating planetary entry, heat transfer data is obtained 
for the first heat shield. Then it is jettisoned from the entry vehicle at a chosen point 
in the trajectory, leaving the second heat shield surface to face the freestream flow. 
Data is obtained for the second heat shield, and then it is jettisoned, leaving the third 
heat shield surface to protect the vehicle for the remainder of the trajectory. The three 
heat shields had different nose radii: 0.935 m, 0.805 m, and 0.702 m for the first 
through third heat shields respectively. Cauchon shows that the theory is closer to the 
FIRE II calorimeter data by accounting for radiation cooling and coupling; otherwise, 
the theory overshoots the calorimeter data by 30%.  
Ried48 compares the flight measurements and engineering predictions on the 
Apollo CM for mission four. Ried also provides a computational approximation of 
 
 16 
 
the total radiative heating rate near the stagnation point including ultraviolet lines and 
continuum, which were not measured by the radiometers on the heat shield of Apollo 
4. The shock-standoff distance over the time of maximum radiative heat transfer is 
also provided. A correlation that closely matches the behavior of how the stagnation-
point shock standoff distance varies with normal-shock density ratio is included and 
could be used along with Lovelace?s41 or Tauber?s45 correlations to calculate 
stagnation-point heat transfer fluxes. For planetary entry at velocities larger than that 
of the Apollo missions, curve fits of inviscid heating rates and cooling factors have 
been produced by Suttles.49 It is recommended that these curve fits be validated 
before use in future work. 
 Determining how the stagnation-point radiative heat transfer rate varies as a 
function of angle of attack for the Apollo CM is investigated by Walters.50 By 
accounting for the stagnation-point shock-standoff distance, Walters is able to 
approximate this behavior, although the wind tunnel data is noticeably scattered. 
Additionally, he applies Kaattari?s correlation51 to produce an outstanding match of 
the Apollo CM?s experimental shadowgraph of the shockwave shape at Mach 19.5 
and 31.5o angle of attack.   
The Kaattari method approximates the shock-standoff distance of a blunt-
body at zero angle of attack52 and the shock envelope of spherical-segment blunt-
bodies at large angles of attack.51 This highly empirical method uses the normal shock 
density ratio to approximate the location of the sonic line on the body as well as the 
shock and body surface inclinations at the sonic point along with theory to determine 
the ratio of the shock-standoff distance to the nose radius. Since a planetary entry 
 
 17 
 
vehicle enters the atmosphere at high velocities, high temperatures are produced in 
front of the heat shield, and so high temperature gas correlations from Srinivasan53 
(an updated version of the Tannehill54 correlations) can be used to determine the 
effective specific heat ratio to determine the normal-shock density ratio. This thesis 
work applies the Kaattari method, with its implementation described in Chapter 3, to 
apply Lovelace?s41 and Tauber?s45 radiative heat flux correlations to blunt-bodies at 
any angle of attack.  
1.2.3. Misconceptions 
Misunderstandings and discrepancies in publications have been noticed 
throughout the literature survey. The most common misunderstanding is the theory 
behind calculating the x-location of the center of pressure of a blunt body. This 
location is dependent on both the axial force and normal force?s contributions to the 
pitching moment; neither can be ignored. However, several references calculate the x-
location by assuming the axial force?s contribution to the pitching moment is 
negligible. This thesis explains in Chapter 3 that the axial force?s contribution to the 
pitching moment is at least the same order of magnitude as the normal force?s 
contribution and thereby must be accounted for in the case of a blunt body. From 
here, it is possible to match the Apollo CM?s pitching moment wind tunnel data as 
will be shown in Chapter 6.  
Bertin,29 Rasmussen,30 and Regan55 have textbooks that assume the axial 
force?s contribution to the pitching moment is negligible in sections concerning blunt 
bodies. However, Bertin?s textbook29 also has the correct derivation, and the example 
 
 18 
 
in which the assumption is made in the first edition of Regan?s textbook55 has been 
removed from Regan?s second edition. 
Figure 9 in the work of Levine56 provided the x-location of the center of 
pressure of several spherical and blunt cone geometries. However, they are different 
from the values of this thesis by one order of magnitude except for one case. Because 
the work of this thesis was able to match the aerodynamic and stability results of the 
Apollo wind tunnel and flight test data and basic conical shapes, the use of Levine?s 
values of the x-location is not recommended.     
Arora57 maximizes the center of pressure?s x-location for a spherically-blunted 
cone-flare configuration. However, Arora?s work assumes that the axial force?s 
contribution to the pitching moment is negligible. As a result, the optimized x-
location value is infeasible. Because the x-location is incorrect, it is not possible to 
find the correct pitching moment value about the blunt-body?s center of gravity. An 
active researcher concerned with the aerodynamic performance and the stability of 
blunt-body planetary entry geometries should be attentive to this issue.   
Papadopoulos58,59 has figures that are mislabeled as trajectories for Apollo 
missions AS-201, 4, and 6. The only relations between the figures and the Apollo 
missions are the mentioned missions? entry flight-path angles. The plots are meant to 
show code output according to a particular Apollo mission?s entry flight-path angle. 
Because several plots on a single page have legends with Apollo mission numbers, it 
is easy for one to assume that these must be the flight velocities, decelerations, and 
Reynolds numbers experienced by the Apollo missions listed. This is a simple 
 
 19 
 
miscommunication and not suggesting that Papadopoulos?s results are not accurate; 
he has validated his results with POST.  
Several sources that are not suggested for estimating the radiative heat transfer 
at the stagnation-point for an Apollo-like capsule include Barter60 and Gupta.61 It is 
possible that both sources have typographical errors in the equations, but this is not 
certain. However, it is shown in Chapter 6 that these equations do not match the 
behavior of flight test data from Apollo 4. There are other correlations that match the 
flight test data closely.  
1.3. Research Objectives 
The primary objective of this research is to determine if and how the 
aerodynamic performance, static stability, and stagnation-point heat transfer can be 
improved over those of previously developed blunt-body planetary entry vehicles. 
Running a computerized optimization over a range of heat shield geometries is one 
way to efficiently search an entire design space for such an answer. Ideally, this 
design space would be controlled by several geometric parameters that could vary the 
shape of the heat shield to a great extent.   
The classic spherical-segment and spherically-blunted cone heat shield 
geometries were originally developed since these shapes are simple yet satisfy the 
mission requirements of Apollo and Viking respectively. Another common shape is 
the power law that has been applied to reduce drag on the nose of hypersonic 
missiles. One of the goals of this research is to explore a wide design space, one 
beyond that of previous work. This is accomplished by extending these geometries to 
 
 20 
 
vary eccentricity and transform the base cross-section shape from an ellipse to any 
rounded-edge polygon.  
There are indications that better aerothermodynamic performance is possible, 
and using today?s technology, it is possible that these heat shields could be produced 
at costs similar to reproduction costs of previous work. All of the past heat shield 
geometries except for the raked cone and the bent-biconic configuration are 
axisymmetric designs. The main advantages of the raked cone and the bent-biconic 
are their larger lift-to-drag ratios compared to their axisymmetric counterparts, and so 
it is expected that the non-axisymmetric designs will produce higher lift-to-drag 
ratios. It is also possible that an optimum nose radius exists that could reduce the heat 
transfer load on the vehicle during lunar or Mars return. These questions are answered 
in this work, limited to the blunt-body?s hypersonic performance.  
1.4. Thesis Overview 
This thesis is organized to first introduce the investigated blunt-body 
geometries and their geometric parameters in Chapter 2. Then Chapter 3 describes the 
Newtonian aerodynamic theory and a method for approximating the shock-standoff 
distance from the stagnation point. It also includes the equations for determining the 
static stability of a given heat shield geometry and provides explanations for a couple 
misinterpretations of basic theory. Chapter 4 gives the main assumptions and 
correlations for convective and radiative heat transfer at the stagnation point. A 
description of the code?s layout and the chosen optimization process is included in 
Chapter 5. Then the design code is validated against Apollo wind tunnel and flight 
 
 21 
 
test data in Chapter 6. In order to better understand the design space, a parametric 
analysis has been conducted. The results are included in Chapter 7 and happen to 
provide better initial designs for optimization. Chapter 8 presents the optimized 
configurations of this work in detail, and the final conclusions of this thesis along 
with suggested future work are detailed in Chapter 9.  
 
 
  
 
 22 
 
Chapter 2. Blunt-Body Heat Shield Geometries 
 
Each heat shield in this work is defined by two geometries: the base cross-
section of the heat shield and the axial shape that is swept about the central axis and 
modified to match the base cross-section. The coordinate system for this work is 
included in Figure 2.1 with the sweep angle ? and conventional directions for 
positive moments. One change from convention that is not shown occurs when the 
vertical lift coefficient CL,V < 0, in which the direction of the positive rolling moment 
switches in order for a negative Cl,? to still indicate a statically roll stable shape. An 
explanation of the sign reversal is included in the stability section of Chapter 3. 
Figure 2.1. Fixed-body coordinate system, spherical-segment, ? = 30o, ?s = 60o, n2 = n3 = 2. 
2.1. Axial Shapes 
The shape of the heat shield that protrudes from the base is called the axial 
shape in order to easily differentiate the protruded shape from the cross-section. The 
axial shape of the heat shield by itself represents the shield?s profile if its base cross-
x 
y 
z 
Pitching 
Yawing 
Rolling 
? 
? 
 
 23 
 
section were to be circular. If the base cross-section is not circular, then the axial 
shape is at least the top half of the shields profile at sweep angle ? = 0o. Then the 
axial shape is modified to follow the outline of the base cross-section. Three axial 
shapes: the spherical-segment, the spherically-blunted cone, and the power law are 
applied in this work and described in the following sections along with how they are 
generated. 
2.1.1. Spherical-segment 
A general spherical-segment is a region of a sphere that is left after the sphere 
is cut by two parallel planes. A closed spherical-segment is a region of a sphere 
encompassed by spherical-segment angle ?s, in which ? = 90o - ?s, in which only one 
plane, parallel to the yz-plane divides the sphere. A closed spherical-segment, one 
shown in Figure 2.1, is also known as a spherical cap and is the type that is applied to 
previous and present heat shield design.  
For this work, only the profile of the spherical-segment is utilized by using the 
following equations for the xy-plane at zero sweep angle: 
rmax = xmax/(1-cos(?s)),                                (2.1) 
xk = rmax(1-cos(?k )),                  (2.2) 
yk = rmaxsin(?k),      (2.3)     
in which xmax = 1 in order to normalize the geometry to the length of the heat shield    
l = xmax.  This set of equations produces the axial shapes shown in Figure 2.2 (a) for 
 
 24 
 
the Apollo CM (?s = 25o) and a hemisphere (?s = 90o). The corresponding spherical 
heat shields, which have circular cross-sections, are shown in Figure 2.2 (b) and (c). 
 
 
a) Spherical-segment axial profiles at ? = 0o.   
 
                
                   (b) Spherical heat shield, ?s = 25o.        (c) Spherical heat shield, ?s = 90o. 
Figure 2.2. Spherical axial profiles and heat shields. 
 
 
 
?s = 25o 
        ?s = 90o 
 
 25 
 
2.1.2. Spherically-blunted Cone 
The spherically-blunted cone?s geometric parameters have been introduced in 
the previous work section of Chapter 1. Figure 1.1(b) is an example of the 
spherically-blunted cone heat shield. Its axial profile has two parts, a spherical nose 
and the conical body. The spherical nose is generated by producing a spherical-
segment with ?s = pi/2-?c to provide slope continuity from the spherical nose to the 
conical body. If the conical base is divided into N vertical sections that are equally 
spaced along the x-direction, then the spherically-blunted cone profile in the xy-plane 
is generated with the system of equations 
??
???
=+??
<??
= ,
tan)sincos1(
0,)cos1(
scn
sn
yr
r
x ??
???
???
                          (2.4) 
 
??
???
??=?+
<?
= .0 ,sinsin
0sin
2
1
NnnNrdr
r
y
s
n
n
sn
????
???
                   (2.5) 
Also, the spherical-segment can be produced using only the first equation for both x 
and y over the range 0 ? ? ? ?s.  
2.1.3. Power Law 
 The power law offers axial shapes with a wide range of bluntness controlled 
by coefficient A and exponent b with the equation  
y = Axb.          (2.6)  
The effects of varying these two parameters are shown in Figure 2.3. For a fixed 
value of b, increasing the slenderness ratio A increases the bluntness of the shape. As 
 
 26 
 
a result, the effects of increasing the slenderness ratio should be similar to those of 
decreasing the spherical-segment angle ?s or increasing the spherically-blunted cone 
angle ?c.  
     
(a) Axial profile varying A, b = 0.75.                       
 
 (b) Axial profile varying b, A = 1.  
Figure 2.3. Power law axial shape. 
For a fixed value of A, increasing the value of exponent b from 0.01 to 1.0 
transforms the axial shape from a flat nosed body to a sharp cone respectively. As a 
result, a power law shape with b = 1.0 is equal to a sharp cone with a corresponding 
 
 27 
 
angle ?c. The power law can also nearly match the spherical segment for a given set 
of A and b or it could produce profiles that are different from these two classic 
shapes.  
Rasmussen30 mentions that two minimum drag power-law bodies have 
exponent values of two-thirds and three-fourths based on Newton-Busemann and 
Newtonian surface pressure models. 
When calculating the shock-standoff distance, special cases account for this 
heat shield shape having a discontinuity at the tip (x = 0) and are discussed in Chapter 
3. Varying the power law?s slenderness ratio and exponent should render a wide 
range of aerodynamic performance and stability characteristics. 
2.2. Cross-section Shapes 
 The cross-section shape is the geometry of the heat shield?s base. The base 
cross-section is usually chosen to be circular although there have been a couple cases 
of slightly elliptic cross-section including the AFE.22 For optimization, it would be 
ideal to have a base cross-section equation that can generate a wide-range of shapes. 
In order to produce eccentric base cross-sections, the equation of the ellipse can be 
applied. To produce shapes that range from a parallelogram to an ellipse, Sabean62 
uses the following Cartesian equation for a typical superellipse curve: 
1
11
=+
vv
b
y
a
x ,    (2.7) 
By varying v from 1 to 2, the superellipse can transform from a parallelogram to an 
ellipse.  
 
 28 
 
  In 2003, Gielis63 published a more generalized superellipse equation called the 
superformula; it can transform a polygon into an ellipse and then into a rounded-edge 
concave polygon. It defines the cross-section radius for 0 ? ? ? 2pi: 
( ) ( ) 132 1
1
4
1
1
4
1 sincos
)(
nnn
b
m
a
mr
?
?
?
?
?
?
?
?
?
+= ??? ,                       (2.8) 
in which m corresponds to the number of sides of a polygon, n1 and n2 are modifiers, 
and n3 is set equal to n2 to produce sharp or rounded-edge polygons. In the present 
work, n2 was set to 1, and then values of n1 that produced sharp-edged, non-concave 
polygons were determined. Corresponding values of n2 for rounded-edge polygons 
were then determined. Because of the increased heat transfer that would be produced 
on sharp edges, zero radius-of-curvature polygons were not considered. Setting n2 = 2 
produces an ellipse; increasing n2 beyond 2 will produce concavity in the shape. Table 
2.1 presents the values for m, n1, n2, and n3 to produce rounded-edge polygons, both 
straight and concave.  
Table 2.1. Superformula parameters for rounded-edge polygons (n3 = n2). 
m n1 n2 
4 1.00 1.50 ? 4.0 
5 1.75 1.50 ? 4.0 
6 2.30 1.50 ? 4.0 
7 3.20 1.50 ? 5.0 
8 4.00 1.40 ? 6.0 
9 5.50 1.40 ? 6.0 
10 7.00 1.40 ? 7.0 
 
  The cross-sections in this analysis include polygons ranging from four to ten 
sides. Once m and n1 are set, n2 can be varied to transform a polygon into an ellipse 
 
 29 
 
and then into a concave polygons, in which n3 = n2. With the parameters in Table 2.1, 
the cross-sections in Figure 2.4 can be constructed. If a rounded-edge pentagon is 
constructed, as shown in Figure 2.4(d), then n2 can be increased to 2 in order to 
produce the corresponding ellipse in Figure 2.4(b), and then n2 can be increased to 4 
to produce the rounded-edge concave pentagon in Figure 2.4(e).  
 
     
 
     
 
                
a) m = 4, n2 = 1.5.                 b) n2 = 2.0.                 c) m = 4, n2 = 4.0. 
 
        
 
     
 
 
                       d) m = 5, n2 = 1.5.                  e) m = 5, n2 = 4.0.             f) m = 6, n2 = 1.5. 
Figure 2.4. Cross-section shapes produced using parameters from Table 2.1. 
  If a1 = b1 = 1, the cross-section will have no eccentricity, and so n2 = 2 will 
produce a circular cross-section. Because a1 and b1 in Eqn. (2.8) relate differently to 
eccentricity than they do in Eqn. (2.7) for v = 1, it was concluded that it is easier to 
produce an eccentric heat shield by multiplying either a1 or b1 by the cross-section 
radius r when generating the blunt body as described in the next section.  
2.3. Generating Blunt-Bodies 
 Once the axial shape at ? = 0o is rendered, it is swept about the central body 
axis (the x-axis) according to the chosen base cross-section with Eqn. (2.8) according 
to the following three-dimensional equation set 
 
 30 
 
( )
( )??
??
?
=
=
=
,sin
cos
,2,
,2,
,
jkjkj
jkjkj
kkj
rbz
ray
xx
?
?       (2.9) 
in which j and k are indices, r is the radius at a given j,k location, and a2 and b2 are the 
lengths of the semimajor and semiminor axes of the generated blunt body. In this 
work, eccentricity e has a range in-between ?1 and 1, in which e < 0 corresponds to 
oblate geometries and e > 0 corresponds to prolate geometries. The semimajor and 
semiminor axes are determined from the following equation sets 
 ( )
??
???
<?
<<??=
,101
,011 2
1
2
22
e
eeba             (2.10) 
( )??
???
<??
<<?
=
.101
,011
2
1
2
2
2 eea
e
b            (2.11) 
The superformula cross-section equation reflects the shapes made by n2 > 2 about the 
horizontal axis. As a result, to keep consistency when varying n2, the reflection was 
removed by setting yj,k = - yj,k and zj,k = -zj,k.  
 Examples of generated blunt bodies are included to show the variety of shapes 
that can be created from this set of axial and cross-section shapes. Figure 2.5 shows a 
prolate (approximately 4:1 axes length ratio) spherically-blunted cone blunt body. 
Figure 2.6 shows an oblate (approximately 3:1 axes length ratio) 12-sided polygon 
blunt body with a spherical-segment axial profile. Figure 2.7 shows a slightly prolate, 
concave rounded-edge pentagon with a power law axial profile. Note that the angled 
views do not correspond to any specific orientation. 
 
 31 
 
   
a) Front view. b) Side view.      c) Angled view. 
Figure 2.5. Spherically-blunted cone, rn/d = 0.25, ?c = 70o, e = 0.95, m = 5, n1 = 1.75, n2 = 2. 
 
 
      a) Front view.  b) Side view.      c) Angled view. 
Figure 2.6. Spherical-segment, ?s = 40o, e = -0.85, m = 12, n1 = 10.75, n2 = 1. 
 
  
                        a) Front view.  b) Side view.      c) Angled view. 
Figure 2.7. Power law, A = 3, b = 0.75, e = 0.5, m = 5, n1 = 1.75, n2 =  5. 
 
 32 
 
2.4. Geometric Properties 
The main geometric properties calculated in this work are the surface and 
planform areas, the volume, the volumetric efficiency of the heat shield, and the 
location of the center of gravity. The following is a description of how the areas and 
volume are determined. Each point (j,k) is a part of a quadrilateral with four points 
(j,k), (j-1,k), (j,k-1), and (j-1,k-1). The distance between points (j,k) and (j-1,k) is 
indicated by d1,a,  
( ) ( ) ( ) ,2,1,2,1,2,1,,1 kjkjkjkjkjkja zzyyxxd ??? ?+?+?=                  (2.12) 
and the distance between points (j,k-1) and (j-1,k-1) is d1,b. These two distances are 
then averaged to produce d1. The distance between points (j,k), and (j,k-1) is d2,a, and 
the distance between points (j-1,k) and (j-1,k-1) is d2,b. Likewise, they are averaged to 
produce d2.  
The product of d1 and d2 is the differential surface area dA. All the differential 
surface area components are summed to produce surface area Shs, which is a term in 
determining the volumetric efficiency described later in this section. The differential 
planform area dAp is the product of w1 and w2, which are the distances from and to the 
same point-locations as d1 and d2 if they were assumed to have equal x-components. 
In other words, the planform area dAp, which is equal to part of the heat shield?s base 
area, depends only on the area projected onto the yz-plane, in which   
( ) ( ) ,2,1,2,1,,1 kjkjkjkja zzyyw ?? ?+?=       (2.13) 
 
 33 
 
and likewise for w1,b, w2,a, and w2,b. Then the sum of the differential planform areas is 
equal to the heat shield?s base area. The planform area is used in part to non-
dimensionalize the aerodynamic forces and moments. 
 The differential volume is the product of the differential planform area and the 
distance from the differential surface to the shields base written as lb 
???
?
???
? +++?= ????
4
1,1,11,,
max
kjkjkjkj
b
xxxxxl
.              (2.14) 
Then a summation of all the differential volumes produces the volume of the heat 
shield Vhs.  
 The volumetric efficiency is one way to benchmark how much volume is 
available for a given amount of surface area. In the case of a heat shield shape, which 
is not a closed-shape, the volumetric efficiency is normalized to a hemisphere 
( ) .18 3231
hs
hs
v S
Vpi? =
    (2.15) 
As a result, the most volumetrically efficient shape is the non-eccentric spherical-
segment with ?s = 90o.  
 The following equations calculate the location of the center of gravity of a 
uniform density body 
{ } .,,,,
??
???
??
???=
???
???
???
???
???
???
dV
zdV
dV
ydV
dV
xdV
ZYX cgcgcg
???
                (2.16) 
This equation is applied to a 3D mesh through numerical integration. One numerical 
integration method is Simpson?s rule,64  
 
 34 
 
?
+
=
12
1
)()(
nx
x
dxxfxF       (2.17) 
( ) ( )( )1212532421 ...2...43 +? +++++++++? nnn ffffffffh , 
which is fifth-order accurate in h. For this work, h is a space step size. Additionally, 
this method requires an odd number of points to be integrated, and so the final point 
in the mesh is represented by the index 2n+1. Because the space step h may not 
necessarily be the same for each pair of points, this work applies a modified version 
of Simpson?s Rule that uses individual step sizes.  
?
+
=
12
1
)()(
nx
x
dxxfxF          (2.18) 
( ) ( )( )12121212553322442211 ...2...431 ++?? +++++++++? nnnnnn fhfhfhfhfhfhfhfh . 
It is likely that more than one integration is required to determine each term of the 
center-of-gravity location since the triple integrals indicate a three-dimensional shape 
that requires integration in three directions. It is easier to use the polar coordinate 
system in Figure 1.1(a). This allows one to conduct two integrations instead of three.  
 The method applied in this work for determining the center of gravity?s 
location with numerical integration is detailed in this section. In the applied form, the 
general equation that is equal to Eqn. (2.16) is  
{ } .,,,, 3,3,3,
??
?
??
?=
NVol
f
NVol
f
NVol
fZYX zyx
cgcgcg        (2.19) 
The term NVol is a volume-related term, but it is not necessarily equal to the volume 
of the heat shield. For the spherical-segment axial shape, NVol is equal to Vhs, but for 
 
 35 
 
the spherically-blunted cone and power law axial shapes, NVol is equal to the terms 
that are not cancelled out from being common to the numerator and denominator of 
Eqn. (2.16). NVol is defined as 
,
max
1
max
1
,? ?
= =
=
k
k
j
j
kjdNVolNVol                 (2.20) 
in which dNVolj,k is the differential volume-related term defined as  
?
?
?
?
?
?
?
?
?
?????? ?
=
law,Power ,2
cone blunted-ySphericall,2
segment-Spherical,2cos2
,,
2
,
,,
2
,
,,
2
,
,
kjkj
kj
kjkj
kj
kkjkj
kj
kj
drdw
drdw
drdw
dNVol
?
?
?pi?
  (2.21) 
in which spherical-segment angle ?k only varies with x-location, d?j,k is the 
differential sweep angle in the yz-plane. This differential angle is determined from the 
law of cosines  
,2acos
,1,
2
),(),,1(
2
,1
2
,
, ??
?
?
???
? ?+=
+
++
kjkj
kjkjkjkj
kj ww
dwwd?
            (2.22) 
and drj,k is the average of the distances between the points (j+1,k) and (j+1,k-1) and 
the points (j,k) and (j,k-1), which is the numerical analogue to a differential length, 
( ).21 )1,(),,()1,1(),,1(, ??++ += kjkjkjkjkj dddr        (2.23) 
Since there are two differentials in dNVol, it is noted that it will be integrated twice to 
produce the denominators of Eqn. (2.19). With these equations, the denominator is 
 
 36 
 
fully defined. For the numerator, the subscript letter on the function f corresponds to 
the x, y, and z-components, and the subscript number 3 corresponds to the result after 
the two integrations. The subscript number 2 corresponds to the result after one 
integration, and the subscript number 1 corresponds to the initial case before 
integration. The initial variables fx,y,z,1 that are integrated twice to produce fx,y,z,3 are 
defined as 
{ } { }.,,,, ,,,,,,1,1,1, kjkjkjkjkjkjzyx dNVolzdNVolydNVolxfff =                    (2.24) 
These are integrated first with respect to ?, and there are a kmax-number of values for 
fx,y,z,2 
( )
k
x
x
xkx
n
ff ??
?
?
???
?= ?+12
1
1,2, .         (2.25) 
( ) ( )( )
k
nxnxxxnxxxx ffffffff ??
??
?
? +++++++++?
+? 12,1,12,1,5,1,3,1,2,1,4,1,2,1,1,1, ...2...43
1
, 
and likewise for fy,2 and fz,2. Note that the fx,1 includes the differentials d? and dr, and 
thus they are not written in Eqn. (2.25) and (2.26). The fx,y,z,2 are integrated with 
respect to r, and so there is one value for each component of f3 
?
+
=
12
1
2,3,
nx
x
xx ff                 (2.26) 
( ) ( )( )12,2,12,2,5,2,3,2,2,2,4,2,2,2,1,2, ...2...431 +? +++++++++? nxnxxxnxxxx ffffffff , 
and likewise for fy,3 and fz,3. With these definitions, the location of the center of 
gravity of a given three-dimensional shape can be determined. 
 
 37 
 
Chapter 3. Aerodynamics and Static Stability 
The theory behind the aerodynamics and the static stability is presented in this 
section. The aerodynamic characteristics are then transformed into the quantities that 
are applied in the code of this work. The method for determining the shock-standoff 
distance to the stagnation point is introduced also, and modifications to the method to 
account for angle of attack are stated. Then this chapter finishes with a couple 
corrected misinterpretations of general aerodynamic theory.  
Two coordinate systems applied in this work are the freestream coordinate 
system shown in Figure 3.1(a) with the positive angle of attack and sideslip angle 
conventions, and the fixed coordinate system shown in Figure 3.1(b), which is 
slightly different from Figure 2.1.  
a) Freestream coordinate system with ? and ?.   b) Fixed coordinate system with aerodynamic  
  moment conventions. 
Figure 3.1. Coordinate systems with positive ? and ? and moment conventions. 
 
M? 
? (+) 
? (?) 
x
y 
z 
Pitching 
Yawing 
x 
z 
? ?s 
rs 
y 
Rolling 
CL,V > 0 
CL,V < 0 
 
 38 
 
3.1. Modified Newtonian Impact Theory 
 After the heat shield geometry is generated, its aerodynamic characteristics 
are calculated based on a modified Newtonian surface pressure distribution. Simple 
Newtonian theory is equivalent to the limit of exact oblique shock theory as M? 
approaches infinity and ? approaches one.65 This work is currently being applied to 
Earth atmospheric entry by assuming ? = 1.4, and a similar analysis can be applied to 
a Martian atmosphere of CO2 by assuming ? = 1.3. However, when calculating the 
shock-standoff distance, the effective specific heat ratio after the blunt-body shock is 
determined using the empirical correlations of high temperature air from Tannehill,54 
as explained in Section 3.2 on shock-standoff distance.  
Because Newtonian theory allows aerodynamic performance to be determined 
within a fraction of a second of computation time, it has been chosen over more 
complicated, time-intensive methods for optimization reasons. Results from the code 
that apply modified Newtonian theory are compared to wind tunnel and flight test 
data from Apollo Command Module (CM) in Chapter 6.  
The pressure coefficient, which is the pressure difference normalized by the 
dynamic pressure, is given by the equation 
???
?
???
? ?=?=?=
????
?
?
? 12
2
1 22 p
p
MV
pp
q
ppC
p ?
?
.              (3.1) 
Newtonian theory assumes that the component of a particle?s momentum that is 
normal to the surface is destroyed when impinging on the face of the blunt body 
while its tangential momentum is conserved.29,30,55 The Newtonian model is 
 
 39 
 
2
max,
?
???
?
???
? ?=
?
?
V
nVCC
pp
r
           (3.2) 
 for .0? <?? nV
r
 The surface is in the aerodynamic shadow region when 0? ??? nV
r
 
resulting in Cp = 0.  
 For simple Newtonian, the maximum value of the pressure coefficient is 
assumed equal to two. Modified Newtonian theory accounts for the maximum value 
Cp,max, according to the Rayleigh Pitot tube formula66 that gives the ratio of the 
stagnation pressure after the shock to the freestream pressure  
( )
( )
1
2
222
2,0
124
1
1
21 ?
?
??
?
???
?
???
?
??
+
???
?
???
?
+
+?= ?
?
??
?
?
??
M
MM
p
p .                 (3.3) 
This produces the following relation for Cp,max 
???
?
???
? ?=
??
12 2,02max, ppMCp ? .          (3.4) 
The velocity is modeled as a function of the angles of attack and sideslip 
{ } ( ) ( ) ( ) ( ) ( ){ },sin,cossin,coscos,, ?????==
?
?
zyx VVVV
Vr        (3.5) 
and the local normal vector is approximated by setting it equal to the cross product of 
two local vectors on the differential surface dA. Two local vectors are formed by 
subtracting the values of the x, y, and z-locations at point (j-1,k) from those at point 
(j,k) and likewise for points (j,k-1) and (j,k). The cross product of the two newly-
formed vectors in terms of the x, y, and z values at points (j,k), (j-1,k), and (j,k-1) is 
the numerator of local normal vector and written as 
 
 40 
 
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( ).,1,1,,1,,,1,, 1,,,1,1,,,1,,
,1,1,,1,,,1,,
kjkjkjkjkjkjkjkjzv
kjkjkjkjkjkjkjkjyv
kjkjkjkjkjkjkjkjxv
yyxxyyxxn
zzxxxxzzn
zzyyzzyyn
????
????
????
?????=
?????=
?????=
         (3.6) 
This formulation will work for points with j?1 and k?1 and at the base of the heat 
shield geometry for k = kmax also. For points with j = 1, the vector formed by points 
(j,k) and (j-1,k) is replaced by (j+1,k) and (j,k), and then the numerator of the local 
normal vector is formed. For points with k = 1, which is at the tip of the nose of the 
vehicle, it is assumed that nx = -1, and the other components are zero. The magnitude 
of vector nv is then calculated to produce the normal vector 
( ) ( ) ( )
{ } ,?,?,?,,?
?
,,,
2
,
2
,
2
,
??
???
??
???==
++=
v
zv
v
yv
v
xv
zyx
zvyvxvv
n
n
n
n
n
nnnnn
nnnn
       (3.7) 
and Eqn. (3.2) becomes  
( ) .2max, zzyyxxpp nVnVnVCC ++=            (3.8) 
These equations calculate the pressure coefficient for a general blunt-body geometry 
given in the form of a three-dimensional structured mesh, with each x, y, and z surface 
location determined by sweep angle location j and x-section location k.  
 All the aerodynamic forces and moments are calculated as non-dimensional 
terms. The forces looked at in this analysis are the normal, axial, and side forces with 
coefficients CN, CA, and CY respectively. The coefficients are defined as 
),(),(,),(,1,         ,
1
kjkjpkjyx
S
py
p
N dACnfdACnAC =
?= ?? ,    (3.9)
 
 
 41 
 
),(),(,),(,1,         ,
1
kjkjpkjxy
S
px
p
A dACnfdACnAC =
?= ?? ,            (3.10)
 
),(),(,),(,1,         ,
1
kjkjpkjzz
S
pz
p
Y dACnfdACnAC == ?? ,            (3.11) 
in which dAj,k is the differential surface area defined as 
kjkjkjkj ddrwdA ,,,, ?= ,        (3.12) 
Using the written equations for fx,1 that correspond to each of the force coefficients, 
numerical integration is completed using Simpson?s rule with the process detailed in 
Chapter 2. Then the force coefficients become 
{ } .1,1,1,, 3,3,3, ?
?
???
??
??? ???=
z
p
y
p
x
p
YAN fAfAfACCC             (3.13) 
The lift and drag coefficients can be determined after the normal, axial, and side force 
coefficients are calculated. For the Apollo CM, Hillje6,7 divides the lift coefficient 
into a vertical lift coefficient CL,V and a horizontal lift coefficient CL,H. This work uses 
the following definitions for the lift and drag coefficients  
( ) ( )?? sincos, ANVL CCC ?= ,     (3.14) 
( ) ( ) ( )??? sincoscos, AYHL CCC ?= ,        (3.15) 
( ) ( )2,2, HLVLL CCC += ,           (3.16) 
( ) ( ) ( )VAYND wCCCC cossinsin ++= ?? ,                   (3.17) 
 
 42 
 
in which wV is the wind angle, or the magnitude of the angle from the velocity vector 
to the central-body axis,  
( ) ( )
??
?
?
?
??
?
?
? +
=
x
zy
V V
VVw 22atan
.              (3.18) 
The resulting lift-to-drag ratio equations are 
D
HLH
D
VLV
D
L
C
C
D
L
C
C
D
L
C
C
D
L ,,     ,   , === .       (3.19) 
The aerodynamic moments about the nose that are considered in this analysis 
are the pitching, yawing, and rolling moments with coefficients Cm,0, Cn,0, and Cl,0. 
The equation for the coefficient of the pitching moment about the nose is 
d
YC
d
XCC cp
A
cp
Nm +?=0, ,           (3.20) 
,                  
   ,21in which  
),(),(,),(,,1,
0,,
kjkjpkjykjx
S
py
p
cp
NNm
dACnxf
dACxnrAdXCC
=
=?= ??
      (3.21) 
( )
.                   
   ,21    and
),(),(,),(,,1,
0,,
kjkjpkjxkjy
S
px
p
cp
AAm
dACnyf
dACynrAdYCC
?=
?== ??
       (3.22) 
Cm,N,0 is the normal force?s contribution to the pitching moment, and Cm,A,0 is the axial 
force?s contribution. Similar to integrating fx,1  to determine the force coefficients, the 
fx,1 and fy,1 of each moment coefficient can be integrated separately in order to find fx,3 
 
 43 
 
and fy,3. Those are used to determine the location of the center of pressure after are 
calculated. The equation for the coefficient of the yawing moment about the nose is 
d
XC
d
ZCC cp
Y
cp
An +?=0, ,            (3.23) 
,                  
   ,21  in which  
),(),(,),(,,1,
0,,
kjkjpkjxkjx
S
px
p
cp
AAn
dACnzf
dACznrAdZCC
=
=?= ??
      (3.24) 
( )
.                   
   ,21    and
),(),(,),(,,1,
0,,
kjkjpkjzkjy
S
pz
p
cp
YYn
dACnxf
dACxnrAdXCC
?=
?== ??
                                    (3.25) 
Cn,A,0 is the axial force?s contribution to the yawing moment, and Cn,Y,0 is the side 
force?s contribution. The equation for the coefficient of the rolling moment about the 
nose is 
d
ZC
d
YCC cp
N
cp
Yl +?=0, ,               (3.26) 
,                  
   ,21  in which  
),(),(,),(,,1,
0,,
kjkjpkjzkjx
S
pz
p
cp
YYl
dACnyf
dACynrAdYCC
=
=?= ??
       (3.27) 
( )
.                   
   ,21    and
),(),(,),(,,1,
0,,
kjkjpkjykjy
S
py
p
cp
NNl
dACnzf
dACznrAdZCC
?=
?== ??
          (3.28) 
Cl,Y,0 is the side force?s contribution to the rolling moment, and Cl,N,0 is the normal 
force?s contribution. After fx,3 and fy,3 are determined for each coefficient, the moment 
coefficients become 
 
 44 
 
{ } ( ) ( ) ( ) .2 1,2 1,2 1,, 3,3,3,3,3,3,0,0,0, ?
?
???
??
??? +++=
lyx
Mp
nyx
Mp
myx
Mp
lnm fflAfflAfflACCC   (3.29) 
Note that all moment coefficients are defined as the ratio of the moment to 
q?SlM, in which lM is defined as the length of the semimajor axis. This definition of lM 
makes it possible for the yawing moment coefficient Cn,0 at a given sideslip angle (the 
condition ?1 ? 0, ?1 = 0) for a prolate shape e1 > 0 to be equivalent to the pitching 
moment coefficient Cm,0 for an oblate shape at a given angle-of-attack (the condition 
?2 = ?1, ?2 = 0, e2 = -e1). This definition was chosen over defining lM as the span of 
the heat shield. 
In validation, it is shown that the rolling moment calculations rendered a z-
location of the center of pressure, Zcp, that is inconsistent with the calculations of the 
other two moments, and so the yawing moment coefficient was used to calculate Zcp. 
One observation is that the components of Cl,0 are of such small magnitude that the 
values may have numerical error on the same order, thereby producing inconsistent 
values for Zcp. One reason for this may be that Simpson?s rule has difficulty 
integrating numbers that have values near zero. Since the value of Cl,0 is orders of 
magnitude lower than the pitching and yawing moment coefficients, it is possible that 
Simpson?s rule reached its limit around the value of Cl,0 without affecting Cm,0 and 
Cn,0. 
3.2. Shock-standoff Distance: Kaattari?s method 
Kaattari?s method for calculating the shock-standoff distance to the stagnation 
point is semi-empirical.51,52 Kaattari?s method assumes that the shock shape is 
 
 45 
 
spherical with radius rsh. There are several curves that relate the normal shock density 
ratio ?2/?1 to characteristics empirically derived, including the shock-standoff 
distance at the sonic point for several shapes such as a cone, paraboloid, and sphere. 
Then there are several aerodynamic and geometric theory-based equations applied in 
determining the shock-standoff distance from the chosen geometry. Kaattari?s method 
is not completely explained in this work because it is well detailed for the general 
blunt-body case at zero angle of attack in Ref. (52) and for a spherical-segment blunt-
body at large angles of attack in Ref. (51). To introduce the method, this section does 
include the main equations. Then a description of how this method is implemented on 
the computer is included since it was originally an iterative process completed by 
hand. Afterwards, modifications to the method accounting for angle of attack in Ref. 
(51) are included to estimate the shock-standoff distance for the general blunt-body 
case.  
3.2.1. Method Implementation 
Kaattari gives two similar methods for determining the shock-standoff 
distance, one for conic-section bodies and the other for non-conic-section bodies. 
Kaattari recommends using the conic-section shock equations with ellipsoids and the 
non-conic-section shock equations with the spherically-blunted cone. For this work, 
the non-conic-section shock equations were chosen to be used for all three axial 
shapes because they account for edge bluntness while the conic-section equations do 
not. Also, for the spherical-segment blunt-bodies, it was noticed that the conic section 
equations and non-conic section equations produced similar results for a spherical 
 
 46 
 
body. Additionally, ellipsoid relations assume a complete ellipsoid is present rather 
than a segment of an ellipsoid, and this may become problematic at angles of attack. 
Kaattari originally calculated the shock characteristics of axisymmetric shapes 
by hand using plots of several geometric and aerodynamic characteristics that varied 
with the normal-shock density ratio. He needed to iterate his solution until the method 
found a converged value of the shock-standoff distance. This work uses the computer 
to automate Kaattari?s process. Part of this is accomplished by curve-fitting the 
necessary semi-empirical plots. This section supplies the curve-fit equations that were 
generated for this work. Note that all the given digits are required in order for the 
curve-fits to register the corresponding norm of the residuals or R2 values (which is 
given for most of the curve-fit equations). Also, even though all these digits are 
required to have an accurate curve-fit, this does not mean that the resulting value has 
a correspondingly large number of significant digits. It is suggested that the 
calculated results of the given curve-fits do not have more than three significant 
digits. As noted previously, this section does not detail the entire Kaattari method; it 
only includes the main modifications and additions to the method including curve-fit 
equations.  
This section also refers to figures that are not included in this work, but the 
reference numbers are given. The G function relates the ratio of the shock-standoff 
distance to the shock radius, ?o/rsh, to the blunt-body?s radius of curvature at the nose  
,11
sh
o
shn
sho
sh
o
n
o
rrr
r
rrG
?
???
?
???
? ?+=?
???
?
???
? ?+=                                  (3.30) 
which can be manipulated to become 
 
 47 
 
 .
2
141
???
?
???
?
???
?
?
???
?+
=?
n
sh
n
sh
sh
o
r
r
r
rG
r                                   (3.31) 
These relations52 are based on a given normal-shock density ratio. Figure 1 from Ref. 
[52] provides a relation for the G function to the normal-shock density ratio for ?-
values of 1.0 and 1.4 
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
=+??
?
?
???
?+
???
?
???
?
???
?
???
?
???
?
???
?
=
???
?
???
?+
+??
?
?
???
?
???
?
???
?+
???
?
???
?
=
1.4.
0025566,0.  :residuals of Norm
,0.00062937  0.69906     
 0.047552 -1.4794 - 2.4242
1,0.0033845,  :residuals of Norm
,0.00024476 -0.81267      
  1.4354 - 2.9402  2.2378-
2
1
2
2
1
3
2
1
4
2
1
2
1
2
2
1
3
2
1
4
2
1
???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
G
          (3.32) 
If ?1/?2 < 0.15, then the two curves converge and the curve-fit equation for ? = 1.0 
must be used. Otherwise, interpolation between the two curves can be completed for 
0.15 < ?1/?2 < 0.45.  The shock-standoff distance at the sonic point ?* is non-
dimensionalized by y*, which is the normal distance from the axis of symmetry to the 
sonic point on the body. Figure 2 (Ref. [52]) relates the ?*/y* to the ?1/?2 for two 
specific heat ratio values for 0.01 < ?1/?2 < 0.45, 
 
 48 
 
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
=+??
?
?
???
?+
???
?
???
?
???
?
???
?
=
???
?
???
?+
???
?
???
?
???
?
???
?
=?
1.4.
0.0076548,  :residuals of Norm
,0.00052448 1.9834     
 0.80536  -1.4064 
1,0.0087586,  :residuals of Norm
0.0018741, +2.7174       
  4.3473 -  6.418 
2
1
2
2
1
3
2
1
2
1
2
2
1
3
2
1
*
*
???
?
?
?
?
?
?
?
?
?
?
?
y
                        (3.33) 
Kaattari validates these curve-fits with experimental data for spherical, disk, and cone 
shapes, as well as with theoretical results for spherical and parabolic shapes for zero 
angle of attack. From these two equations, the shock surface inclination at a point 
opposite the sonic point on a flat disk ?*o can be determined 
.atan
*
*
*
??
?
?
?
?
??
?
?
?
?
?=
y
G
o?                     (3.34) 
The sonic angle for a sphere ?*1 is defined as the body surface inclination of a 
sphere at the sonic point, with respect to a plane normal to the freestream direction. 
The variation of the sonic angle for a sphere ?*1 with ?1/?2 is given in Figure 4 from 
Ref. [52] for 0.025 < ?1/?2 < 0.45, 
 
 49 
 
??
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
=
+??
?
?
???
?+
???
?
???
?
???
?
???
?
???
?
???
?
=??
?
?
???
?+
???
?
???
?
???
?
???
?+
???
?
???
?
=?
1.4.
32.589,  61.984     
 99.371 - 7.2046 -   104.8
1,33.383, +80.348      
  380.2 - 908.77  797.1-
(deg)
2
1
2
2
1
3
2
1
4
2
1
2
1
2
2
1
3
2
1
4
2
1
1*
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
          (3.35) 
If ?1/?2 < 0.15, then the curve-fit equation for ? = 1.0 must be used. Otherwise, 
interpolation between the two curves can be completed for 0.15 < ?1/?2 < 0.45. The 
sonic angle for a rounded-corner blunt-body ?* varies with rc/r 
( )( )
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?<?
?
??
?
?
?<
????????????
?<
=
?
?
.00.110.01.01,+ln 0.2159
,10.001.0,0.15140845+           
5.31156637+818.3525394- 
,01.000.0,20
sin
sin
2
1*
*
r
r
r
r
r
rr
r
r
r
r
rrr
cc
c
cc
c
c
          (3.36) 
For each axial shape, ?, which is the inclination on the forward body surface at the 
point tangent with the corner radius, is given by the following set of equations. From 
these, the sonic angle ?* can be determined. For the spherical-segment and power law 
axial shapes, 
 
 50 
 
??
?
?
?
?
??
?
?
?
?
?
???
?
???
??
??
?
?
?
?
?
??
?
?
?
?
?
?
???
?
???
? ?
=
law.Power ,atan2
segment,-Spherical,1
1
asin
1max1max1
1max1max1
-,k_,k_
-,k,k
n
c
c
n
- xx
-yy
r
r
r
r
r
r
pi
?              (3.37) 
For the power law axial shape, the term subtracted from pi/2 can be defined as an 
effective local half-cone angle. There are two cases for the spherically-blunted cone: 
one in which the spherical-segment is the dominant shape, in which the rn/d is large, 
and the other in which the cone shape dominates. This is described by the following 
equation set for the spherically-blunted cone 
?
?
?
?
??
?
?
?
<?
?
??
?
? ?
???
?
???
??
?????? ??<?
=
.2sin21, x-x y-yatan2
,2sin210,2
1-k_max1,k_max1,
1-kmax1,kmax1,
d
r
d
r
n
c
c
nc
?pipi
?pi?
pi
?         (3.38) 
Note that if ? > 37o, then ? is set equal to 37o since that is the limit of Kaattari?s 
dataset. The spherically-blunted cone does not approach this limit since it would 
require a half-cone angle ?c < 53o, and the limit for the half-cone angle in the 
optimizer is set at 55o. Kaattari?s method also notes that if the sonic angle ?* < ?, then 
the sonic point is assumed to be taken at the tangency angle ?, and so ?* is set equal 
to ? for this case. To determine ?*, Kaattari supplies Figure 6(b) from Ref. [52] that 
relates the difference ?* ? ?*o as a function of ?*. This relation varies with the 
normal-shock density ratio, and their corresponding curve-fit equations with ?* limits 
are included  
 
 51 
 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??<=
??
???
????
??<=??
??
????
??<=
??
??
????
????
??<=+??
???
????
??<=??
???
????
??<=??
???
????
??<=??
???
????
=?
.401,20
0.11582,  :residuals of Norm      
0.012458, +0.040357 -0.020612   +     
 0.0014138 -106.7379   +     
 101.4433 -101.2753
405.17,16
0.0795,  :residuals of Norm     
115.89, + 24.509 -2.1177   +     
 0.094787 -0.0023397   +     
 103.0006 -101.5644
400,12
0.067392,  :residuals of Norm    
,0.00075942 -0.0075298+0.0034554 -     
 0.0028275+0.00035965 -     
102.3289 +108.0148-     
101.4038 +109.8112-
404,8
0.0583,  :residuals of Norm     
0.0038209,0.0016152-0.00034058  +     
 0.00053225 +102.7801 -     
 106.9855 +106.0809-
404,6
0.0793,  :residuals of Norm      
0.0085934,-0.019362 +0.0054599-      
0.00087115 +103.7688 -      
108.0483 +106.2549-   
401,4
0.04381,  :residuals of Norm      
0.0031709, +0.01135 +0.01933 -     
0.0021927 +109.1189 -     
 101.7632 +101.3002-
401,3
0.10637,  : residuals of Norm      
0.026591, +0.036272 - 0.01669 -     
 0.0017384 +106.5557-     
 101.1554 + 107.8164-
(deg)
*
1
2
*
2
*
3
*
4
*
5-
5
*
6-6
*
8-
*
1
2
*
2
*
3
*
4
*
5
*
5-6
*
7-
*
1
2
*
2
*
3
*
4
*
5
*
5-6
*
7-
7
*
8-8
*
11-
*
1
2
*
2
*
3
*
4
*
5-
5
*
7-6
*
9-
*
1
2
*
2
*
3
*
4
*
5-
5
*
7-6
*
9-
*
1
2
*
2
*
3
*
4
*
5-
5
*
6-6
*
8-
*
1
2
*
2
*
3
*
4
*
5-
5
*
-66
*
-9
**
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?? o
       (3.39) 
For all equations in this set except ?2/?1 = 12, for 0 < ?* < 1 it is assumed that the 
difference ?* ? ?*o is zero. For this region, the equation for ?2/?1 = 12 is applied for 
normal-shock density ratios of twelve and greater. The equation for ?2/?1 = 16 is not 
 
 52 
 
applied until ?* > 17.5, and if it is used before this, the results could be erroneous 
since the curve fit was designed only for the noted ranges. Linear interpolation 
between two of the curves within this range of ?2/?1 can be exercised to approximate 
the values of the difference for normal-shock density ratios that are not listed. 
 The main non-conic-section body shock equation relates the shock radius rsh 
to the stagnation shock-standoff distance, the corner radius, sonic shock-standoff 
distance and the tangency angle 
( ) ( )( ) ( )( ) ( )
( )( ) ??
?
?
?
?
?
?
?
?
?
?
???
???
?
???
? ???+
???
?
???
? ?+
??+=??
?
?
???
? ?
*
*
*
*
** sin11
coscossin1cos 1costan
r
r
r
r
r
r
yy
x
ry
r
c
cn
s
sh
osh ?
?
?
??
, (3.40) 
in which xs is the streamwise distance from the apex of the shock to the point on the 
shock at distance y* from the central axis. This relation along with the shock solution 
nomograph equation, 
( )???
?
???
? ?=
*** tan
1
?y
r
y
xB shs
s ,         (3.41) 
and the initial value of ?o/rsh set equal to G, determine the values of Bs, y*/r, rsh/r, and 
?o/r, 
( )*2
2
* tan
1
?????
?
???
?=
y
rB sh
s ,        (3.42) 
( )( ),sin11 ** ???= rrry c       (3.43) 
,*
* r
y
y
r
r
r shsh =            (3.44) 
 
 53 
 
,*
* n
sh
n
sh
r
r
r
y
y
r
r
r =            (3.45) 
r
r
rr
sh
sh
oo ?=? .           (3.46) 
From here, the value of ?o/rsh based on Eqn. (3.31) is calculated and the process is 
repeated. Once the value of ?o/rsh converges within 0.001, the iterative process has 
been completed, and the shock properties of the blunt-body have been determined for 
zero angle of attack. The main output variable is ?o/rn. The complete process with 
referenced figures and examples is included in Ref. (52). 
3.2.2. Accounting for Nonaxisymmetric Shapes & Angle of Attack  
Kaattari51 offers a way to account for angle of attack for spherical-segment 
axisymmetric blunt bodies. In this section, a basic method is described for 
determining how ?o/rn changes both for a nonaxisymmetric blunt body and with angle 
of attack based on Kaattari?s method. It also has been modified to account for the 
spherically-blunted cone and power law axial shapes. These modifications are meant 
to produce results that follow expected trends only. Although there is confidence in 
the expected trends, the results should not be accepted as unequivocal since several of 
the more exotic shapes have not been studied before from a re-entry heat transfer 
standpoint. The expectation is that these modifications produce shock-standoff 
distances for nonaxisymmetric geometries within the proper order of magnitude. The 
reason for using this method is to approximate the stagnation-point radiative heat 
flux, and Chapter 4 explains how it can be approximated for a blunt body using the 
shock-standoff distance.   
 
 54 
 
 For an axisymmetric spherical-segment geometry, Kaattari provides the 
process for estimating the shock-standoff distance in Ref. (51), and the equation  
( ) ( )??? sincos
3
max
3
max
1 ?
?
?
?
?
?
?
?
???
?
???
?+??=?
r
rc
r
rc
dd
ststo             (3.47) 
relates the shock-standoff distance at angle of attack to that at zero angle of attack. 
Note that for this work, the angle of the azimuth ? is assumed equal to zero. To 
determine how the value of ?o/rn varies with angle of attack, semi-empirical data of 
shock correlation functions from Kaattari in Ref. (51) have been curve-fit. Applying 
these curve-fits is described in the next section, including the modifications required 
to use Kaattari?s method to approximate how the shock-standoff distance changes 
with angle of attack. Figure 9(a) from Ref. [51] shows how the shock correlation 
coefficient c1 varies with the normal-shock density ratio and the tangency angle ? and 
is curve-fitted (input ? in degrees) with indicated limits of applicability  
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?<=??
??
?+?
?<=?
??
?<=?
??
=
,450,5
0.9999, = R                        
 20.23026450 +     
102.21715933 -101.15717926 -    
104.78956150 +102.31598093 -    
105.39975415 104.12212802-
,450,4
 
9998, 0.= R                     
 10.22140328 +107.60025063-  
108.78810663 -109.63317384 
,450,3
0.9987, = R                      
 30.21438038 +107.64482284-  
 101.16661976 + 101.93013729-
1
2
2
3-24-
36-47-
59-611-
1
2
2
4-
25-37-
1
2
2
4-
2-53-7
1
o
o
o
c
?????
??
??
????
??
????
??
  (3.48) 
 
 
 55 
 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?<=
?
??
??
?<=?
??
??
?<=?
??
??
?<=
+
??
??
??
?<=??
??
??
?<=?
??
??
?<=?
??
??
=
.400,22
0.9990, = R                        
10.20014809 +109.52531793-     
103.29553862 -102.95290942+    
106.46122093 -104.60808129
,400,20
0.9991, = R                         
 30.21011300 +109.44613956-     
103.50334207 -103.16638882+     
 107.49638803 -106.06112566
,400,16
0.9997, = R                        
 40.22155108 +108.09283123 -    
103.58846154 -102.54224815+    
105.34031912 -104.06700008
,400,12
0.9998,= R                         
 232157648 .0     
109.22392526 -102.01204775 +     
 102.69805855 -101.52078813 +     
103.32278424 -102.56915093
,400,10
0.9999, = R                        
 30.23240649 +   
 108.40015875 -103.31432550 +   
103.65809838 -101.62858238 +   
 102.99995789 -102.00439718
,5.420,8
0.9999, = R                        
 40.23184113 +104.54631057-    
101.56690529 -102.18083922 +    
 106.32411067 +109.07434828-
,5.420,6
0.9999, = R                         
232732751 0.+ 102.50199798-     
 102.09241117 -104.08148122+     
 102.69993388 -102.92464389-
1
2
2
3-
24-35-
47-59-
1
2
2
3-
24-35-
47-59-
1
2
2
3-
24-35-
47-59-
1
2
2
3-24-
35-46-
58-610-
1
2
2
3-24-
35-46-
58-610-
1
2
2
3-
24-36-
48-510-
1
2
2
3-
24-36-
4-95-10
1
o
o
o
o
o
o
o
c
???
?
??
??
????
??
??
????
??
??
?????
??
??
?????
??
??
????
??
??
????
??
??
(3.49) 
 
 56 
 
Note that all the digits have to be included in order to maintain accuracy. If digits are 
removed, it is strongly recommended that the modified curve-fit be plotted to verify 
that the curve-fit is still valid throughout the entire range and to verify none of the 
curve-fits intersect each other. Interpolation for cases with normal-shock density 
ratios in-between the given curve fits has been tested and is completely feasible. The 
correlation constant c3 is also supplied in Figure 9(b) from Ref. [51], and the curve-fit 
equations (input ? in degrees) have been generated in this work 
( )
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?<=
?
??
?+?
?<=?
??
??
?<=?
?+?
??
?<=
?
??
?+?
??
?<=?
??
?<=?
??
=??
?
?
???
?
,400,8
0.9994, = R                         
480.01414886 +101.32633841 -      
102.21510280 +102.15386203 -      
106.22306898 105.69532428-
,450,7
0.9995, = R                         
149288590 0.0+ 101.21889535 -      
101.66417857 +101.56325253-      
104.33744906 +103.80921614-
,450,6
0.9995, = R                         
440.01593340 +101.27550983 -     
109.92739734 107.24637110-     
101.63488703 +101.13917530-
,450,5
0.9998, = R                         
101.77036651 +      
101.14702612 -104.96819914-      
108.56040487 104.49628934-      
109.54110294 +107.15150540-
,450,4
0.9995, = R                      
300.02011174 + 101.35288319 -        
101.72390334 +101.48406667- 
,450,3
9994, 0.= R                      
430.02318882 + 101.82847314-     
103.89964215 +103.39744191-
sin
1
2
2
4-
25-36-
48-510-
1
2
2
4-
25-36-
48-510-
1
2
2
4-
26-37-
48-510-
1
2
2
2-
4-26-
37-48-
510-612-
1
2
2
4-
26-38-
1
2
2
4-
2-63-8
2
3
o
o
o
o
o
o
n
d
rc
???
?
??
??
????
??
??
????
??
??
?????
??
??
????
??
????
??
?
   (3.50) 
 
 57 
 
( )
??
?
?
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?
?
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?
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?
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?
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?
?
?
?
?
?
?
?
?
?<=
?
??
??
?<=
?
??
??
?<=
?
?+?
??
?<=
?
?+?
??
?<
?<
=
=
?
??
?+?
?
??
??
=?
?
??
?
?
.400,22
0.9996, = R                         
109422874 0.0+ 104.37257752-     
102.90105343 +101.56703453 -     
106.12012384 + 106.61739779-
,400,20
0.9996 = R                         
740.01115478 +104.37257752-     
102.90105343 +101.56703453 -     
106.12012384 + 106.61739779-
,400,18
0.9996 = R                         
740.01137978 +104.37257752 -     
102.90105343 101.56703453 -     
106.12012384 +106.61739779-
,400,16
0.9996, = R                         
116297874 0.0+ 104.37257752-     
102.90105343 101.56703453 -     
106.12012384 +106.61739779-
,400
,400
,14
,10
0.9993, = R                         
119228070 0.0+ 104.76005398-     
105.67714535 +101.73419068 -     
106.54759070 107.03151544-
0.9995, = R                         
160.01314489 +101.27943518 -      
102.06298722 +102.32498214-      
107.25696594 +107.04675716-
sin
1
2
2
5-
26-36-
48-510-
1
2
2
5-
26-36-
48-510-
1
2
2
5-
26-36-
48-510-
1
2
2
5-
26-36-
48-510-
1
2
1
2
2
5-
26-36-
48-510-
2
4-
25-36-
4-85-10
2
3
o
o
o
o
o
o
n
d
rc
???
?
??
??
???
?
??
??
???
?
??
??
???
?
??
??
?
?
?
?
?
?
?
??
??
?
??
??
?
(3.51) 
Because density ratios over sixteen may occur in the regions of maximum heat 
transfer during planetary entry, it was decided to extrapolate the curve-fit for ?2/?1 > 
16 listed in Eqns. (3.49) and (3.51). With all of these equations, it is possible to 
approximate the shock-standoff distance. This is the full extent of Kaattari?s method.  
 To account for nonaxisymmetric geometries, it is assumed that the change in 
the shock-standoff distance for shape variance and an angle of attack is dependent on 
 
 58 
 
eccentricity and the axial shape of the upper half at ? = 0o (j = 1) and the lower half at 
? = 180o (j = (jmax-1)/2). An example geometry is shown in Figure 3.2 to show how 
Kaattari?s method is applied.  
 
Figure 3.2. Spherically-blunted cone profile shock-standoff distance variance. 
Kaattari?s method is applied to determine the radii of the shock for the upper 
profile rsh,upper = 4.827 and the lower profile rsh,lower = 11.30 separately. This is 
accomplished by accounting for the different half-cone angles for the upper and lower 
surfaces, ?c,upper = 70o and ?c,lower = 76o respectively. Also the different upper and 
lower base radii, rupper = 1.958 m, and rlower = 2.864 m respectively are accounted for 
and added together to produce the base diameter. The shock-standoff distances for 
each case is calculated ?o,upper = 0.4175 m and ?o,lower = 0.9215 m. After the shock-
standoff distances are determined this work assumes that this cone shape must have a 
zero angle of attack shock-standoff distance ?o that is in-between the two calculated 
shock-standoff values. It is assumed that each has equal effect on ?o, and so it is set 
 
 59 
 
equal to the average of the two and produces the open circle in Figure 3.2 
corresponding to ?o = 0.6695 m.  
Originally, to account for angle of attack, Kaattari?s Eqn. (3.47) was applied, 
but for the case of Figure 3.2, ??<0 > ??>0, which does not follow the most probable 
trend. At ?<0, the upper profile faced the freestream more than the lower profile, and 
so it should have more effect on ?? than the lower profile. Because rsh,lower > rsh,upper, it 
is expected that ??=-(90-?) < ??=90-?, which is the opposite of the trend produced by 
Eqn. (3.47). As a result, the following basic method is applied to guarantee that this 
trend is held.  
This work assumes that the shock-standoff distance at ? = 90o - ?c,lower, which 
is ?>0o, can be approximated by the distance from the lower shock with radius rsh,lower 
that is perpendicular to the lower face. This distance is the length between the two 
inverted triangles shown in Figure 3.2. For ?<0, the shock-standoff distance at ? = -
(90o - ?c,upper), can be approximate by the distance from the upper shock with rsh,upper 
that is perpendicular to the upper face. This distance is the length between the two 
triangles shown in Figure 3.2. As a result, this also means that if rsh,lower < rsh,upper, 
then ??=-(90-?) > ??=90-?, thus this method accounts for either case. Linear interpolation 
between ?o and ?? is applied once the endpoints ??=-(90-?.upper) and ??=90-?,lower are 
determined. This simple method is implemented only so that the shock-standoff 
distance is varied and follows an expected trend.        
 To account for eccentricity, the average base radius is determined; then it is 
divided by the maximum base radius to produce the non-dimensional average base 
radius ravg,nd. This term is then used to calculate the upper and lower shock radii 
 
 60 
 
?
?
?
???
?
<
?
=
,0,
,0,
,
,
,
,
,
errrrrr
errrrrr
r
ndavg
upper
upper
n
n
uppersh
ndavgupper
upper
n
n
uppersh
uppersh         (3.52) 
and 
?
?
?
???
?
<
?
=
.0,
,0.
,
,
,
,
,
errrrrr
errrrrr
r
ndavg
lower
lower
n
n
lowersh
ndavglower
lower
n
n
lowersh
lowersh               (3.53) 
Eccentricity is accounted for through this approach so that the calculated shock radii 
of a prolate shape with eccentricity e at ? and those of an oblate shape with the same 
eccentricity are affected equally. This is one way of producing consistency 
throughout the results. Note that this addition does not provide true consistency when 
calculating the radiative heat transfer, as the value of qrad varies up to 33% at 
eccentricity |e| > +/- 0.9 and up to 20% at |e| > +/- 0.8. As a result, any calculations 
for highly eccentric shapes should be seen only as approximate solutions.  
 For rounded-edge concave shapes such as that shown in Figure 2.7, the profile 
could look similar to an inverted Figure 3.2. In this case, it is assumed that only the 
surface with the larger r (in the case of Figure 2.7 it would be rupper) is applied rather 
than using the average of shock-standoff distances because the surface with the larger 
r represents the primary radius of the shock-shape. This is assumed for the rounded-
edge concave shapes since they have not been investigated experimentally or through 
computational fluid dynamics (CFD). It should be noted that the shock shape about 
rounded-edge concave shapes may have a radius that varies with sweep angle, and so 
only basic assumptions can be made at the level of this work?s analysis.  
 
 61 
 
3.3. Static Stability 
The stability of a planetary entry vehicle is one of the main factors that 
determines whether a particular design is feasible. If the PEV requires quasi-steady 
flow in order for it to follow the designed trajectory, then the vehicle must be able to 
remain or return to the designed trim angle of attack ?trim after encountering flow 
disturbances. This can be determined by looking at the static and dynamic stability of 
the vehicle. This work analyzes the heat shields? static stability only for a first-look 
analysis. An in-depth, full vehicle analysis would require the study of both the static 
and dynamic stability since it is possible for a statically stable vehicle to be 
dynamically unstable. If a vehicle is statically stable, then it possesses the 
aerodynamic moments required to restore the vehicle to an equilibrium state after 
encountering a flow disturbance.67  
One way to measure a vehicle?s static stability is to analyze its aerodynamic 
moments about the vehicle?s center of gravity. In this work, the aerodynamic moment 
coefficients Cm,0, Cn,0, and Cl,0 about the nose of the vehicle and the force coefficients 
CN, CA, and CY are calculated using modified Newtonian Impact Theory. Then the 
center of pressure location is determined from this information as previously 
explained. The center of pressure is the location at which the aerodynamic forces are 
applied. As a result, the aerodynamic moments, which are produced by the 
aerodynamic forces, are zero about the center of pressure.  Once a center of gravity 
location is either chosen or calculated, moments produced by the aerodynamic forces 
about the center of gravity are calculated.  
 
 62 
 
For a fixed body-axis moment with conventions for positive moments given in 
Figure 3.1(b), there are twelve possible center of pressure locations, with respect to 
the vehicle?s center of gravity and the central axis. For example, in longitudinal 
stability, the contributions of the normal and axial forces to the pitching moment 
about the center of gravity are determined. If the center of gravity is below the central 
axis, then the center of pressure can be below the center of gravity, above the center 
of gravity in the same quadrant (below the central axis), or above the center of gravity 
in a different quadrant (above the central axis). In each of these cases, the center of 
pressure could be in front of, aligned with, or behind the center of gravity. If the 
aligned case is grouped with either the in front or behind cases, then there are six 
different geometric cases. The other six cases correspond to the inverted case in 
which the center of gravity is above the central axis, thereby producing a total of 
twelve cases.  
After deriving each of the twelve cases for the pitching, yawing, and rolling 
moments about the center of gravity, it was observed that all twelve cases for each 
moment could be reduced into one equation, producing the following three equations 
,, ??
?
?
???
? ?+
???
?
???
? ?=
d
YYC
d
XXCC cgcp
A
cpcg
Ncgm           (3.54) 
,, ??
?
?
???
? ?+
???
?
???
? ?=
d
ZZC
d
XXCC cpcg
A
cgcp
Ycgn           (3.55) 
?
?
?
???
?
<??
?
?
???
?
???
?
???
? ?+
???
?
???
? ??
???
?
?
???
? ?+
???
?
???
? ?
=
.0,
,0,
,
,
,
VL
cpcg
Y
cgcp
N
VL
cpcg
Y
cgcp
N
cgl
CdYYCd ZZC
CdYYCd ZZC
C              (3.56) 
 
 63 
 
In order for the definition of rolling moment static stability to remain constant, the 
rolling moment Cl,cg equation accounts for the change in positive moment convention 
that occurs when the vertical component of the lift coefficient becomes negative. The 
explanation for this is included in the next section, which comments on two 
misinterpretations of basic aerodynamic and stability theory.  
With the moments about the vehicle?s center of gravity determined, its static 
stability can be determined through an analysis of its moment derivatives with respect 
to angle of attack and sideslip angle. These derivatives Cm,cg?, Cn,cg?, and Cl,cg? are 
known as the static stability derivatives. For longitudinal stability, the derivative of 
the pitching moment with respect to angle of attack must be negative. For yaw 
stability, the derivative of the yawing moment with respect to sideslip angle must be 
positive. For roll stability, the derivative of the rolling moment with respect to 
sideslip angle must be negative. If the positive moment convention for Cl,cg were kept  
constant for positive and negative lift, then roll stability requires the rolling moment 
derivative to be positive for negative lift.  
To calculate the stability derivatives numerically, the code determines the 
Cm,cg, Cn,cg, and Cl,cg at ? - 0.25o, ? + 0.25o, ? - 0.5o, ? + 0.5o in order to use the 
following finite-differencing approximate of the first derivative 
,25.0,,25.0,,,,, ?? ??? ?
?
== ?+
oo cgmcgmcgm
cgm
CC
d
dCC          (3.57) 
,5.0,,5.0,,,,, ?? ??? ?
?
== ?+
oo cgncgncgn
cgn
CC
d
dCC         (3.58) 
.5.0,,5.0,,,,, ?? ??? ?
?
== ?+ oo cglcglcglcgl
CC
d
dCC
        (3.59) 
 
 64 
 
These numerical approximations are second-order accurate in space. With the static 
stability derivatives calculated, an analysis of the vehicle?s static stability can be 
completed. 
3.4. Correcting Misinterpretations  
Two common misconceptions are detailed in this section to emphasize the 
importance of understanding these basic aerodynamic concepts. The first concerns the 
static roll stability requirement on a given aerodynamic vehicle. The second involves 
calculating the location of the center of pressure on a passive aerodynamic vehicle.  
3.4.1. Static Roll Stability Requirement 
When either a disturbance in the flow or a control input generates a rolling 
moment about the center of gravity of a lift-generating vehicle, the direction of the lift 
vector relative to the horizon is no longer perpendicular, causing the vehicle to 
sideslip. In general, a lift-generating vehicle sideslips as it rolls, and general stability 
theory concludes that there is a coupled effect that can be related to the vehicle?s roll 
angle and the freestream sideslip angle. To make a lift-generating vehicle statically 
stable when it encounters a flow disturbance that brings the vehicle away from its 
desired orientation and path, the vehicle must be able to produce a counter-moment to 
bring it to its initial orientation (usually zero-roll angle or the designed trim position).    
Aircraft stability assumes that the lift vector is always positive for an aircraft 
in wings-level attitude; this leads to the standard convention that a positive rolling 
moment renders a positive change in sideslip. As a result, the rolling stability 
 
 65 
 
derivative Cl,? is required to be negative for static stability to counter flow 
disturbances.67  
Spacecraft stability must also account for the possibility that the vehicle may 
be designed to produce negative lift during a portion of its trajectory. If the lift vector 
is negative, then a positive rolling moment renders a negative change in sideslip, 
producing the reverse relationship between roll angle and sideslip angle, in which a 
positive roll angle produces a negative sideslip angle. In this case, the rolling stability 
derivative Cl,? is required to be positive for static roll stability. Since a negative Cl,? is 
commonly associated with a statically roll stable vehicle, this convention is 
maintained in this work by reversing the direction of the positive rolling moment 
convention when the vehicle produces negative lift as shown in Figure 3.1(b). The 
sign reversal of the positive rolling moment produces the discontinuity shown in 
Figure 3.3 in order for all roll stable configurations to have Cl,? < 0.  
 
Figure 3.3. Cl,cg,? distribution for spherical-segment, elliptical base (n2 = 2), varying e and ?s, ? = 
20o, ? = 5o. 
 
 
 66 
 
As a result, if the positive rolling moment direction were kept constant, then 
there would be stable configurations with Cl,? > 0, and there would not be a 
discontinuity. With the sign reversal, Figure 3.3 clearly divides the stable and 
unstable oblate geometries. Phillips68 includes a good discussion on roll stability in 
his flight mechanics textbook. 
3.4.2. Determining the Location of the Center of Pressure 
  One of the assumptions of aircraft stability, CL >> CD, cannot be followed in a 
blunt-body analysis since blunt bodies usually have an L/D < 1. The assumption of CL 
>> CD leads to the conclusion that the center of gravity must lie in front of the 
aircraft?s neutral point (or the vehicle?s center of pressure) to satisfy the requirement 
of Cm,cg,? < 0 for longitudinal static stability.31 Since this assumption does not apply 
to blunt-bodies, it may be possible to produce longitudinal static stability with the 
center of pressure in front of the center of gravity. From modified Newtonian results 
for Cm,cg shown in Figure 3.4 for -30o ? ? ? 0o, it is determined that the Xcp/l = 0.6556 
and Ycp/l varied from 0.0000 to 0.5530 for a spherical-segment of ?s = 25o with a non-
eccentric base. As a result, the code suggests that the Apollo CM with a Xcg/l = 2.171 
is one successful example of maintaining longitudinal static stability with the center 
of pressure in front of the center of gravity.  
  Similar to Xcp/l being constant over a range of ? for a spherical-segment e = 0, 
?s = 25o, it has been proven that Xcp/l = 0.6667 for a sharp cone in a Newtonian 
surface pressure field, suggesting that Xcp/l is independent of half-cone angle ?c.29 
Note that not all spherical-segments have Xcp/l = 0.6556, but at least one does with ?s 
 
 67 
 
= 25o, e = 0 including the Apollo CM. The general pitching moment equations that 
relate Xcp and Ycp to Cm,0 are given as Eqns. (3.20-3.22). These equations follow the 
coordinate system shown in Figure 3.1(b) that has a positive Cm corresponding to a 
nose-up moment. Bertin29 notes that the axial force?s contribution to the pitching 
moment is commonly neglected in the definition of the center of pressure location at 
and near zero angle-of-attack. However, this assumption does not apply to blunt-body 
aerodynamics because Cm,A,0 is commonly the dominant term for a blunt-body.  
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
-35 -30 -25 -20 -15 -10 -5 0
Angle of Attack (deg)
Cm
,cg
Modified Newtonian
Wind Tunnel Data
Apollo Wind Tunnel Data, M = 6
 
Figure 3.4. Variation in Cm,cg with angle of attack. 
  Cm,N,0 is not usually the dominating term for slender bodies, in which case the 
resulting formula29,30,55 for a circular cone at zero angle-of-attack Xcp/l = 2/(3cos2?c) 
can be used, but this equation complicates the analysis since Xcp/l is constant at 
0.6667 without the assumption. When this assumption is applied to a blunt circular 
cone with ?c = 70o, then Xcp/l = 5.7, which is not close to the actual Xcp/l = 0.6667. 
This assumption is only close to the exact solution for small ?c, but a blunt-body cone 
 
 68 
 
does not have a small ?c. For a general blunt-body shape, the following relation does 
not determine the x-location of the center of pressure  
.0,
N
mcp
C
C
d
X ??
           (3.60)  
Instead, Xcp/l for a given blunt-body shape can be determined from the normal force?s 
contribution to the pitching moment Cm,N,0 as shown in Eqn. (3.21).  
  Arora?s 2003 aerodynamic shape optimization work57 maximizes Xcp. 
Unfortunately, he equates Xcp to Cm,0/CN, which contradicts Eqn. (3.60). Since he does 
not account for the axial force?s contribution to the pitching moment, his work is an 
example of recent research that has made this incorrect assumption on basic blunt-
body aerodynamics, rendering the published optimization results unusable. His 
conclusion is that the optimized Xcp value is 1.213 m, but a quick analysis of the 
geometry he chose accounting for the axial force?s contribution to the pitching 
moment suggests that the actual Xcp is less than approximately 1.1 m, suggesting that 
his optimization results are within an infeasible region of x-locations due to the 
incorrect definition of Xcp. Additionally, his optimization results could be 
significantly different when accounting for the general relation of Xcp and Cm,0 given 
by Eqn. (3.20). Note that the incorrect definition of Xcp for a blunt body is included in 
Regan?s first edition55 and Rasmussen.30 These are all examples of published work 
that show what Bertin29 mentions is the conventional definition of Xcp, not the true 
definition of Xcp that is required to complete an aerodynamic analysis of blunt-bodies.  
 
 
 
 
 69 
 
Chapter 4. Heat Transfer  
Two heat transfer modes, convection and radiation, are considered in this 
work. The two arguably most important heat transfer characteristics are the heat flux 
and the heat load. The heat flux is power density in the form of heat per unit area. The 
heat load is equal to the heat flux integrated over the trajectory in time. It is common 
in the first analysis of a vehicle to look at the heat fluxes and heat loads at the 
stagnation point because this is usually accepted as the point of highest heat transfer.  
This is not true for the Apollo CM, which re-entered at an angle of attack that 
brought the stagnation point near the corner radius. In such cases, as noticed in Ref. 
(35), the heat transfer is higher at the edge of the heat shield than at the stagnation 
point. In fact, the Apollo Command Module at ? = ?33o had a heat transfer flux at the 
edge that is 92% larger than that at the stagnation point corresponding to ? = 0o, 
according to wind tunnel results,35 but the actual CMs did not necessarily travel 
trajectories that required this high an angle of attack. The most likely reason that this 
was one of the highest angles of attack tested is because the conical shaped crew 
compartment that connected to the heat shield was at a 33o angle with the horizontal. 
As a result, the flow would pass by the crew compartment flush at ? = ?33o.  The 
Apollo 4 CM traveled with a maximum angle of attack of ?25o, in which the heat 
transfer at the edge is around 60% higher than at the stagnation point corresponding 
to ? = 0o, according to other wind tunnel results.36 In both the cases of ? = ?25 and 
?33o, the stagnation-point heat flux is approximately 15% larger than it would be at ? 
 
 70 
 
= 0o.35,36 As a result, the heat flux is higher at the edge of the heat shield than that at 
the stagnation-point of the Apollo CM.  
Both the heat flux and heat load are equally important; however, for this work, 
only the heat flux is analyzed as it has been decided that the heat load would be 
determined with use of a trajectory code at a later time. This chapter explains the 
correlations applied in this work and their assumptions. The correlations were 
originally designed to calculate the stagnation-point heat flux on a sphere. Applying 
these correlations allows for the heat flux to vary with the radius of curvature of a 
given blunt body. Explanations on how these correlations are applied to blunt-bodies 
are included in the following sections. These correlations also vary with altitude 
through a freestream density factor and with speed through a freestream velocity 
factor. For this work the ARDC 1959 Model69 and US Standard Atmosphere of 
197670 have been applied. The 1976 standard atmosphere is applied for results, but 
the 1959 model atmosphere is used as noted throughout the text for the application of 
older correlations and code validation of results that used standard atmospheres 
before 1976. 
4.1. Convection 
  The stagnation-point convective heat transfer correlation of Tauber42 is 
applied in this work. Tauber assumes equilibrium flow conditions and a flight regime 
where boundary-layer theory is valid. Tauber produces a specific equation for 
planetary entry from satellite speed, but it is the objective of this work to approximate 
the heat transfer flux for planetary entry from both satellite and superorbital entry 
 
 71 
 
speeds. To account for this, the general form of his correlations, which is given as 
Eqn. (4) of his article42, is applied to produce the following correlation for the 
stagnation-point convective heat flux  ,convsq&   
( ) 35.05.08, )1( 1083.1 ???? ??= Vgrq wnconvs ?& ,               (4.1) 
in which gw is the ratio of wall enthalpy to total enthalpy. It is assumed that gw << 1, 
and so gw is zero in this work. This correlation assumes a fully catalytic surface. This 
correlation also holds true to the Fay and Riddell43 derivation that states the  ,convsq& is 
inversely proportional to the square root of the nose radius. Zoby44 presents evidence 
that this relation breaks down for blunt bodies with ratio values of base radius r to 
nose radius rn less than 0.6, in which rn > r. He suggests that this is due to the velocity 
gradient being higher than would be otherwise expected by the deriving ? ,convsq& rn-0.5. 
If possible, this would be a good addition for future work. This is only one example 
that these heat transfer derivations are not accepted as fact in their application to blunt 
bodies, but they generate trends that are accepted as generally true for stagnation-
point heat transfer.  
 The form of the correlation shown in Eqn. (4.1) was originally designed for 
calculating stagnation-point heat flux on a sphere. Since the stagnation-point 
convective heat flux relies mainly on the geometry nearby the stagnation-point, as 
opposed to the full body shape and size, it can be approximated by setting the radius 
of curvature equal to the nose radius. The nose radius is the term that relates the heat 
flux to the geometry in Eqn. (4.1). For the spherical-segment and spherically-blunted 
cone axial shapes, determining rn is a trivial calculation. For the power law axial 
 
 72 
 
shape, the slope of the shape at the tip of the nose may not necessarily be continuous, 
as in the case of setting b = 1, in which a sharp cone axial shape is generated.  
As a result, the blunt-bodies with a power law axial shape must have an 
equivalent nose radius term produced. This equivalent nose radius is only an 
approximation and should not be accepted as a complete model for power law shapes. 
First, it is assumed that the power law shape?s nose tip is blunted if its slope is 
discontinuous. Then the profile of the power law shape is examined; for this analysis, 
a line that is normal to the power law profile and that produces a 15o angle with the 
horizontal is generated. A segment of that line that begins at the line?s intersection 
with the power law profile and ends at its intersection with the horizontal central axis 
is produced. The effective radius is assumed to be equal to the average of the length 
of that line segment and the distance to the nose from the end point of that line 
segment on the horizontal axis. If the normal line intersects the power law curve 
beyond the base of the geometry (the intersection occurs outside of the heat shield 
shape), then the geometry?s curvature is assumed sharp and given an effective radius 
value of 0.001 m. To produce a true method for determining the effective radius of 
the general power law shape for convective heat transfer is a research topic in itself. 
 
 
 
 
 
 
 
 73 
 
4.2. Radiation 
 Because radiation over a blunt-body can be primarily modeled as an elliptic 
problem, the radiation at the stagnation point depends on the body of the vehicle in 
addition to the nose radius. Also, the radiative heat flux is significantly more sensitive 
to the angle-of-attack of the blunt body than the convective heat flux. In this work, all 
these effects are incorporated into the term called the effective radius. In order to 
apply  ,radsq& correlations for spheres to non-spherical blunt-bodies, the effective radius 
in this work is directly related to the spherical radius for a given set of freestream 
conditions. This is accomplished by estimating the shock-standoff distance ?so across 
from the stagnation point at a given normal shock density ratio ?2/?1 of the blunt 
body.  
 To calculate the normal shock density ratio ?2/?1, high temperature properties 
of air must be determined in order to calculate an effective specific heat ratio after the 
normal shock ?eff,2. Tannehill54 supplies high temperature air correlations for this 
work, and they are also located in Chapter 11 of Anderson.65 The effective specific 
heat ratio after the normal shock is determined through an iterative process. First, a 
test variable for ?eff,2 is called ?test,2 and set equal to 1.4 as an initial condition. Then the 
corresponding pressure p2 and density ?2 after the normal shock are calculated based 
on ?test,2 
( ) ,1121 21
2,
2,
12 ???
?
???
? ?
++= Mpp test
test
?
?              (4.2) 
 
 74 
 
( )
( ) ,21
1
2
12,
2
12,
12 ??
?
?
???
?
+?
+=
M
M
test
test
?
???             (4.3) 
and these two variables are entered into Tannehill?s Fortran code that returns the 
corresponding enthalpy h2 and ?eff,2 
,1
2,
2,
2
2
2 ??
?
?
???
?
?= eff
effph
?
?
?
        (4.4) 
( )( ),exp1 11109 876543212, cYcXc
YZcZcYccYZcZcYcc
eff +++
++++++++=?       (4.5) 
in which ,  ,
10013.1log  ,292.1log 5
2
10
2
10 YXZY
pX ?=?
?
??
?
?
?=??
??
?
?= ?           (4.6) 
and coefficients c1 through c11 are curve-fit values tabulated in Tannehill.54 For this 
work, only the ?eff,2 is used. The value of ?eff,2 is compared to ?test,2; then ?test,2 is set 
equal to the calculated ?eff,2, and the process is iterated until the absolute value |?eff,2 ? 
?test,2| is less than 0.01. Once a converged value of ?eff,2 is determined, then the 
corresponding normal shock density ratio ?2/?1 has been calculated and could be used 
in Kaattari?s method described in Chapter 3 to approximate the shock-standoff 
distance.   
 It is assumed that the effective nose radius for stagnation-point radiative heat 
transfer, reff, for a given blunt body is equal to the radius of a particular sphere that 
maintains an equal shock-standoff distance. After calculating the normal shock-
standoff distance, the corresponding spherical radius still would have to be 
determined.  
 
 75 
 
 According to wind tunnel results shown in Figure 4.1, the ratio of the normal 
shock-standoff distance to a sphere of radius reff is constant for a given normal-shock 
density ratio.  
 
Figure 4.1. Stagnation-point shock-standoff distance wind tunnel data with empirical curve-fit, 
Ref. [48]. 
Ried48 offers an empirical curve-fit that renders an acceptable approximation, also 
shown in Figure 4.1:  
.1
12
1
1
1
2
1
2
2
1
2
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
???
?
???
? ?
=?
?
?
?
?
?
?
eff
so r                   (4.7) 
Then the reff from this equation replaces the sphere?s radius in the stagnation-point 
radiative heat flux correlations. Three  ,radsq& correlations are applied over a range of 
velocities. For V? < 9000 m/s the correlation of the following form is applied 
 
 76 
 
 ,10000
32
1,
gg
sl
effrads
Vgrq ?
?
??
?
?
???
?
???
?= ??
?
?&     (4.8)  
in which g1 = 100, g2 = 1.6, and g3 = 8.5 from Ref. [29] for velocities below 7620 m/s 
and g1 = 6.8, g2 = 1.78, and g3 = 12.5 for velocities from 7620  to 9000 m/s from Ref. 
[41]. For velocities above 9000 m/s, the Tauber and Sutton45 equation is applied   
)(10168.4 22.14, Vfrq Heffrads ??= ?& ,            (4.9) 
in which ,10072.1 325.088.16 ?????= ?VH           
and    
?
?
?
?
??
?
?
?
?<
?
??
???
??
=
?
??
?
?
?
?
?
???
?
?
?
?
?
m/s 1600011500
.17168.3333 - V6.25525796 + V108.42982517 -
 V104.89774670 + V101.00233100-
m/s 115009000,39494.8753 -  V16.1078691 + V102.43598601 -
 V101.61370008 + V103.93206793-
)(
24
38412
23
37412
V
V
Vf  
These curve-fit equations for f(V) have < 2% difference with the published values.45 
Note that these curve-fit equations have coefficients with a high number of digits in 
order to produce the <2% error. It is suggested that all of these digits are maintained; 
if they are not, the values of f(V) may go below zero for low V? or produce extremely 
large values for high V?, either case producing erroneous results. The high number of 
digits does not correspond to the number of significant figures from this correlation. It 
is suggested that no more than three digits should be specified as significant for all 
three radiation correlations.  
 
 77 
 
Chapter 5. Description of Code and Optimization 
Process  
All of the presented theory has been implemented into a Fortran code for this 
work. For optimization, this code is linked to VR&D?s Design Optimization Tools 
(DOT).71 This chapter presents the layout of the code written for this work, a brief 
introduction to DOT and a summary of the chosen optimization method, and a list of 
studied objective functions with each one?s value to this work.   
5.1. Code Layout 
  The overall code layout is composed of three main components: the operating 
code, analysis code, and DOT. The overall code layout is shown in Figure 5.1. The 
operating code is the primary program from which all the other programs are 
managed. The analysis code determines the aerothermodynamic characteristics of a 
chosen blunt-body shape at given freestream conditions. Then the operating code 
calls DOT to determine the values of the design variables to be sent into the analysis 
code for each iteration of the optimization process.  
 The operating code includes all the settings and inputs required to operating 
the analysis and DOT codes. There are three operating modes available in this code. 
The single case mode can run the analysis code once at the given ? and ? to get the 
aerodynamic coefficients and the heat transfer determined. It can also run the analysis 
code five times in order to also calculate Cm,cg,?, Cl,cg,?, and Cn,cg,?, which require the 
values of Cm,cg, Cl,cg, and Cn,cg at four neighboring conditions (?+0.5o, ?), (?-0.5o, ?), 
 
 78 
 
(?, ?+0.5o), and (?, ?-0.5o) for the second-order accurate finite-difference scheme 
mentioned in Chapter 3.    
 
Figure 5.1. Diagram of Overall Code. 
The second operating mode allows the user to conduct a parametric study of both 
geometric and freestream parameters for any geometric shape. It is possible to vary 
these variables at any resolution desired. Finally the third operating mode is the 
optimization setup that calls the DOT optimization routine. DOT determines the 
values of the design variables at which to run the analysis code based on a 
constrained method. DOT is setup in this work to use the modified method of feasible 
directions (MMFD) constrained method for optimization.  
 
Operating Code 
Single Case 
? Analysis code 
single run 
? Analysis code 5 
runs 
Parametric 
Study 
? Analysis code 
10-10000 runs 
Optimization 
Setup 
? Design 
variables and 
constraints 
Design  
Optimization Tools  
(DOT) 
 
Constrained method: 
Modified Method of 
Feasible Directions 
(MMFD) 
Analysis Code 
Required subroutines: 
 ARDC 1959 Model Atmosphere 
US 1976 Standard Atmosphere 
Tannehill?s High Temp. Air Code
Design results 
 
 79 
 
The analysis code has three objectives: to generate the blunt-body geometry, 
to determine its geometric properties, and to calculate the aerodynamics, static 
stability, and the stagnation-point heat transfer. A diagram of the analysis code is 
given in Figure 5.2. Given the geometric parameters of a chosen base and axial shape, 
the shape generator produces a 3D mesh of the heat shield geometry. The 
aerodynamics calculator determines the aerodynamic characteristics of a given shape 
based on modified Newtonian surface pressure distribution at a given angle-of-attack, 
sideslip angle, and Mach number. The primary variables that are calculated in the 
aerodynamics code are M?, ?, CL, CD, L/D, CN, CA, Cm,0, Cl,0, Cn,0, Xcp/L, Ycp/L, Zcp/L.  
 
Figure 5.2. Diagram of Analysis Code. 
The aerodynamics calculator is acceptable for determining blunt-body shape 
hypersonic aerodynamics at fine mesh sizes with extremely low run times (usually a 
fraction of a second for a jmax = 121, kmax = 203 mesh). Additionally, the 
aerodynamics code uses Tannehill?s code to determine ?eff,2 and ?2/?1, and it uses 
Kaattari?s method to determine the shock-standoff distance ?so to the stagnation-point 
Cross-section Shape 
Shape Generator 
Aerodynamics  Geometric Properties 
Aerodynamic Stability 
Design Results  
?s, rn,/D, ?c, A, b m, n1, n2, n3, e 
Axial Shape 
Heat Transfer  
M?, ??, ?, ? 
  
Freestream conditions 
 
 80 
 
that are used in the heat transfer portion of the code. The primary output of the heat 
transfer portion of the code are reff,   ,,convsq& and  .,radsq&  
  The analysis code applies a 5th-order accurate Simpson?s Rule integration 
method to determine the aerodynamic characteristics and center of gravity location. 
The center of pressure location is then determined. Geometric properties such as the 
volume, surface area, and planform area are calculated in the analysis code. 
Additionally, a center of gravity can be either chosen or calculated assuming a 
uniform density volume. The static stability calculator uses the aerodynamics in 
conjunction with the location of the center of gravity to determine the moment 
coefficients about the center of gravity and the pitch, yaw, and roll stability 
derivatives. Note that the analysis code requires two atmospheric models and 
Tannehill?s high temperature air code. Two atmospheric models are programmed in 
order to use the older atmospheric model for part of the validation process of the heat 
transfer correlations.   
5.2. Design Optimization Tools (DOT) 
  Vanderplaats Research & Development, Inc.?s Design Optimization Tools 
(DOT)71 is professional software program that varies design variables based on a 
gradient-based minimization method to determine an optimum value of an objective 
function. DOT offers both unconstrained and constrained minimization methods. 
Broydon-Fletcher-Goldfarb-Shanno (BFGS) and Fletcher-Reeves (F.R.) are the two 
unconstrained minimization methods available in DOT. When DOT refers to 
unconstrained methods, it means that there are no constraints present except those on 
 
 81 
 
the design variables that DOT varies. The constrained methods offer the ability to 
restrict values on non-design variables or a combination of variables based on theory 
limits or other reasons. The constrained minimization methods available in DOT 
include the Modified Method of Feasible Directions (MMFD), Sequential Linear 
Programming (SLP), and Sequential Quadratic Programming (SQP). MMFD is the 
most reliable of the three and requires less memory than the other two methods.71 
  The following optimization problem is solved by DOT: find the values of 
NDV design variables contained in Xr  that will minimize or maximize )(OBJ XO r= , 
in which the OBJ is the objective function, subject to constraints Gq(Xr ) ? 0 for q = 1 
to NCON, in which NCON is the number of constraints, and design variables Xp have 
side constraints UppLp XXX ?? for p = 1 to NDV, in which NDV is the number of 
design variables.71 
  For this work, DOT has been setup to use MMFD to vary the design variables 
to find an optimum value of an objective function, in this case, an 
aerothermodynamic parameter or combination of parameters. MMFD uses the 
following overall process. The objective function and constraints are first evaluated at 
the user inputted initial values of the design variables. Then the gradient of the 
objective function and constraints are calculated, and a search direction E is created. 
Then a one-dimensional search is completed to find the scalar parameter ?* that 
minimizes )(XO r . Scalar parameter ?* is used to find a new Xr  that is set equal to the 
sum of the initial Xr  and the product of ?* and the search direction,  
,*1 uuu EXX ?+= ?rr          (5.1) 
 
 82 
 
in which u is the iteration number. If convergence is not satisfied, then iterations of 
the following process are completed until convergence requirements including the 
Kuhn-Tucker conditions are satisfied.71   
  The three Kuhn-Tucker conditions71 must be satisfied for obtaining 
convergence using MMFD. The first is that optimum design *Xr must be feasible, or 
produce constraint values Gq( *Xr ) ? 0 for q = 1 to NCON. The second condition is 
that the product of the Lagrange multiplier ?q and Gq( *Xr ) must be zero. The third 
condition is that the gradient of the Lagrangian becomes zero, in which the gradient 
of the Lagrangian is 
.0*)(*)(
NCON
q =?+? ? XGXO
q
q
rr ?               (5.2) 
Detailed descriptions of MMFD and the Kuhn-Tucker conditions are included in 
Appendix E of Ref. (71). A quick analysis of several different objective functions is 
discussed in the next section.   
5.3. Choosing Objective Functions 
A quick analysis of the results of 57 unconstrained optimizations on 57 
different objective functions has been completed to determine which objective 
functions to use, to observe any trends in the results, and to notice any unexpected 
results. For this analysis, each objective function was maximized. The design 
variables and their initial designs for this preliminary optimization for a hexagonal m 
= 6, n1 = 2.3 superellipse cross-section shape and spherical-segment axial shape are  
 
 83 
 
.
2.0
5.2
35
2
0
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
=
o
s
e
nX
?r
              (5.3) 
The ranges of values for these design variables are 5o ? ?s ? 89o, -0.95 ? e ? 0.95, and 
1.50 ? n2 ? 4.0. The freestream conditions for the Newtonian flow model include M? 
= 30 and negative ? = 25o at an altitude of 200,000 ft.  
In addition to single term objective functions, the fifty-seven are composed 
also of linear combinations or products of aerodynamic coefficients, stability 
coefficients, and geometric parameters. The objective functions included in Table 5.1 
rendered the better aerodynamic results out of the 57 objective functions and are 
thereby listed as the most useful objective functions. Each number in the left column 
corresponds to the number of variables in each objective function.  
Table 5.1. Most useful objective functions. 
Objective Functions 
1 CL, CD, L/D, LV/D, LH/D, CL,V, CL,H, Cm,cg, -Cm,cg,?, C
n,cg,?, -Cl,cg,?, ?v 
2 CLCD, CL + L/D, Cn,cg,? -Cm,cg,?, -?v Cm,cg,?,  ?vL/D 
3 -Cm,cg,? + CL + L/D, -Cm,cg,? + Cn,cg,? - Cl,cg,? 
4 -Cm,cg,? + Cn,cg,? - Cl,cg,? + L/D 
 
The better objective functions that are made up of three or four variables were noticed 
to not be multiplied, only added. The objective functions that produced results that 
were either not as good but still acceptable or produced mediocre results are as 
questionable objective functions in Table 5.2. 
 
 84 
 
Table 5.2. Questionable objective functions. 
Objective Functions 
2 
-Cm,cg,? +CL, -(Cm,cg,?+Cl,cg,?), 
L/D-Cm,cg,?, Cm,cg - Cm,cg,?, 
Cm,cg Cm,cg,?, -Cm,cg,?Cl,cg,? 
3 CL +CD +  L/D, -(C
m,cg,?+Cl,cg,?) Cn,cg,? 
 
Then there are several objective functions that produced results that were not 
acceptable from an aerodynamic and static stability standpoint. These objective 
functions are listed in Table 5.3. 
Table 5.3. Objective functions to use with caution. 
Objective Functions 
1 CL2, Shs, Vhs 
2 
CL + CD, CL L/D, CD + L/D, -Cm,cg,?CL, 
-Cm,cg,? L/D, -Cm,cg,? Shs, -Cm,cg,? ?v, ?v Shs, 
L/D Shs, CL Shs, CD Shs, L/D Vhs 
3 
CL L/D+CD, -Cm,cg,? + CL L/D, 
Cm,cg,?Cn,cg,?Cl,cg,?, Cm,cg,?Cl,cg,? + Cn,cg,?, 
CL L/D((Cm,cg,?)2)1/4, -Cm,cg,? ?v Shs 
4 Cm,cg,?Cl,cg,? + Cn,cg,? + L/D, -(C
m,cg,?+Cl,cg,?)Cn,cg,? L/D 
 
Some of these results may not be intuitive. For example, this analysis suggests that 
maximizing surface area or volume individually produces results that are not as 
acceptable from an aerodynamic and static stability standpoint as maximizing 
objective functions from the other two sets. Note that this is not meant to be a perfect 
test of objective function effectiveness but just a way to determine which objective 
functions would be worth working with first.  
 
 85 
 
sdff 
Chapter 6. Design Code Validation  
To validate the design code, and the corresponding implemented theories, 
results of the design code are compared to data from the Apollo and FIRE II missions. 
The results are divided into two areas: (1) aerodynamic performance and static 
stability and (2) stagnation-point heat transfer. The first section notes that the 
aerodynamic coefficients and stability derivatives match the Apollo data within 15%. 
The second section observes that the maximum heat transfer heat flux and heat load is 
within 15% of actual predictions after the stagnation-point heat transfer heat flux is 
related to the maximum heat flux according to wind tunnel data. 
6.1. Aerodynamics and Static Stability 
The aerodynamic performance and static stability are compared to both 
Apollo wind tunnel data and Apollo flight test data. There is more certainty in the 
Apollo wind tunnel data than in the Apollo flight test data, and most of the wind 
tunnel data is for M? = 6 while the flight test data is for M? = 36. To convert from the 
listed angle of attack value of the NASA Apollo reports to the angle of attack value 
used in this work, defined in Figure 3.1(a), subtract 180o from the NASA reported 
value. This should usually convert the NASA reported value to a negative angle of 
attack. Note that there is not any truly acceptable experimental data on the rolling 
moment stability coefficient, but predictions are compared to Whitmore?s recent 
computational work.28  
 
 86 
 
6.1.1. Comparison with Apollo Wind Tunnel Data  
 Results from the code based on modified Newtonian theory, have been 
compared to wind tunnel8,9 data of the Apollo Command Module (CM). The center of 
gravity is offset from the central body axis in order to trim the Apollo CM at a 
specific angle-of-attack during re-entry.10 Different center of gravity locations were 
considered in the wind tunnel models during the design of the CM. 8,9,11 In Figure 6.1, 
the center of gravity location is Xcg/l = 2.171, Ycg/l = 0.3158, and Zcg/l = 0.0 according 
to the body-fixed coordinate system in Figure 3.1(b).  
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
-35 -30 -25 -20 -15 -10 -5 0
Angle of Attack (deg)
Cm
,cg
Modified Newtonian
Wind Tunnel Data
Apollo Wind Tunnel Data, M = 6
 
Figure 6.1. Cm,cg comparison between modified Newtonian and wind tunnel data, Ref. [8]. 
Additionally, because Xcg/l > 1, the center of gravity location of the CM is past the 
heat shield. The Newtonian results for the pitching moment Cm,cg at M? = 6 in Fig. 5 
produce Cm,cg,? = -0.16/rad; the modified Newtonian results follow the behavior and 
closely agree with the values of the Apollo wind tunnel (WT) data.8 As a result, this is 
evidence that the modified Newtonian results can match the pitching moment closely 
and thus predict ?trim, and it is well-known to match the lift-to-drag ratio for a blunt 
body better than the lift or drag coefficients.  
 
 87 
 
 The wind tunnel data for the rolling moment Cl was scattered near zero and 
has values that are two orders of magnitude smaller than those measured for Cm. 
Although the data accuracy of Cl and Cm is not reported for this wind tunnel data, the 
scattering and smaller values of the Cl data points in Fig. 6 suggest that the 
measurement instruments did not have the precision required to obtain a clear data set 
of the CM?s rolling moment behavior.  
-1.00E-03
-7.50E-04
-5.00E-04
-2.50E-04
0.00E+00
2.50E-04
-6 -4 -2 0 2 4 6
Sideslip Angle (deg)
Cl
,0
M. Newtonian
Wind Tunnel Data
Apollo Wind Tunnel Data, M = 6
 
Figure 6.2. Cl,0 comparison between modified Newtonian and wind tunnel data, Ref. [8]. 
The wind tunnel data suggest a neutrally stable spherical-segment at -20o angle-of-
attack, and the modified Newtonian results agree. A mesh with jmax = 203 and kmax = 
121 has been chosen based on a grid convergence study. Since the Apollo CM is 
axisymmetric, the yawing moment coefficient Cn,0 at a given sideslip angle (the 
condition ?1 ? 0, ?1 = 0) would be equivalent to the pitching moment coefficient Cm,0 
at a given angle-of-attack (the condition ?2 = ?1, ?2 = 0). This is one reason that no 
specific Cn,0 data exists in the references.  
 
 88 
 
0
0.1
0.2
0.3
0.4
0.5
0.6
-40 -30 -20 -10 0
Angle of Attack (deg)
CL
M. Newtonian
M = 15.8 WT
M = 18.73 WT  
a) Validation of CL results. 
 
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
-40 -30 -20 -10 0
Angle of Attack (deg)
CD
M. Newtonian
M = 15.8 WT
M = 18.73 WT  
b) Validation of CD results. 
0
0.1
0.2
0.3
0.4
0.5
0.6
-40 -30 -20 -10 0
Angle of Attack (deg)
L/D
M. Newtonian
M = 15.8 WT
M = 18.73 WT  
c) Validation of L/D results. 
Figure 6.3. Aerodynamic force comparisons between modified Newtonian and wind tunnel data, 
Ref. [8]. 
 
 89 
 
  The modified Newtonian results in Figure 6.3 were calculated at M? = 18.73, 
but the difference between these results and those at M? = 15.8 is negligible (0.05% 
difference at ? = -20o). Modified Newtonian results for the lift and drag coefficients 
are larger than values from both datasets shown in Figure 6.3(a) and (b). However, 
the increase in lift and drag with an increase in Mach number in the wind tunnel data 
is more significant than expected; this may suggest the presence of significant wind 
tunnel effects. At M?=18.73, CL and CD from Newtonian theory are at most 9.6% and 
7.2% larger than the wind tunnel data respectively, and they are within the 
uncertainty of the wind tunnel data of ?0.114 for CL and ? 0.10 for CD.8 L/D 
Newtonian results shown in Figure 6.3(c) agree very well with the wind tunnel data 
(better than for the individual CL and CD as expected).  
  At lower freestream Mach numbers, such as M?=9, the errors are larger than 
10% for the lift coefficient as shown in Table 6.1. While the data presented for 
M?=15.8 and 18.73 are based on Apollo CM models with rc/d of approximately 0.1, 
the data in Table 6.1 corresponds to rc/d = 0.0. Moseley conducted a survey of the 
effect of increasing rc/d, and his wind tunnel results for M?=9 at  rc/d = 0.1 would 
increase the errors from those values in Table 6.1, for CL to approximately 22% and 
CD to approximately 13% while the error in L/D was constant at 8% at ? = -15o.  
Table 6.1. Percent error of Newtonian computations compared to wind tunnel data, M? = 9, in 
Ref. [9], Percent error averaged over -30o ? ? ? 0o. 
 Average Percent Error Maximum Percent Error ? of Maximum Percent Error 
CL 0.45 0.534 18.6% 
CD 1.25 1.296 3.70% 
L/D 0.37 0.412 11.3% 
 
 
 90 
 
As the code used in this work does not account for corner radius in determining the 
surface pressure distribution, this is one reason for the increase in error. The 
Newtonian surface pressure distribution is known to become more accurate with an 
increase in Mach number, and the validation results are consistent with this 
understanding. However, it is shown in the next section that comparisons with the 
flight test data from Apollo AS-202 result in an amount of error similar to that for 
M?=9. Overall, the modified Newtonian results are within 10% of the wind tunnel 
data with corner radius for M? ? 18.73, within 15% of the wind tunnel data without 
corner radius for M? ? 9, and within 25% of wind tunnel data with corner radius for 
M?  ? 18.73. 
6.1.2. Comparison with Apollo Flight Test Data 
 Results from the code have been compared to flight test6,7 data for the Apollo 
Command Module (CM) for mission AS-202 and Apollo 4 (also known as AS-501). 
The CM in AS-202 re-entered Earth?s atmosphere at satellite orbit speed 8.23 km/s 
(27,000 ft/s) while the Apollo 4 CM produced the expected re-entry velocity from 
lunar return of 10.7 km/s (35,000 ft/s).  
 The uncertainty in the flight data varies throughout the trajectory, and so the 
more steady aerodynamic data was identified and utilized. Of the two datasets, the 
flight data from AS-202 had the smaller uncertainty in the flight coefficient data of 
?9% at 4900 s into the mission. The coefficients of the normal force, lift, and lift-to-
drag ratio have percentage errors around this error. However, the coefficients of the 
 
 91 
 
axial force and drag have higher errors around 17%. The Newtonian results are 
compared to the AS-202 flight data in Table 6.2.  
Table 6.2. Comparison of Apollo AS-201 Data and Computations, M? = 14, ? = -16.5o at 4900 s. 
 AS-201 [ref. 6] Mod. Newtonian % Error 
CN -0.05 -0.0454 -9.2% 
CA 1.34 1.56 16.2% 
CL 0.37 0.399 7.8% 
CD 1.28 1.51 17.6% 
L/D 0.289 0.265 -8.4% 
 
The trend in the percentage error being higher for CL than CD seen in the wind tunnel 
data comparison is the opposite for AS-201. Because the corner radius is not 
accounted for, it is expected that the error in L/D to stay constant at 8%, for CL, 
according to wind tunnel results, to have approximately 22% error instead of 7.8% 
and CD to have approximately 13% instead of 17.6%. Note that these percentage 
errors are nearly constant, at least over the range of angle of attack values from ?30o 
to 0o. It is completely feasible for CD to have a difference in error of 4.6% since the 
uncertainty is a higher percentage, and so the only surprising trend is that the lift 
coefficient has 14% less error than expected.  
  From the wind tunnel data, it is suggested that the corner radius affects the lift 
and drag coefficients less at higher freestream Mach numbers, but this is countered 
with the higher errors present in the AS-201 flight test data. In fact, wind tunnel 
effects could be changing the trends also, and so it is apparent that there is not 
consistency throughout this wind tunnel and flight test data to the resolution required 
to reason for the different percentage errors. Therefore, no conclusions could be made 
concerning whether the modified Newtonian results are less than 18?9% accurate.  
 
 92 
 
 Compared to flight data for Apollo 4 at M? = 30, Newtonian theory produces 
a CD that is 3.7% larger and a CL that is 18.6% larger as shown in Table 2.7 According 
to Hillje, the best flight-derived data for CN (near maximum freestream dynamic 
pressure) has an uncertainty of ?0.048.7 Because the normal force coefficient has a 
small magnitude, small precision errors in CN strongly affect the calculation of CL. 
The contribution of CN to CL is one order of magnitude less than the contribution of 
CA to CL. However, the contribution of CN to CD is two orders of magnitude less than 
the contribution of CA to CD, and so an error in CN will not affect CD as much as CL. 
This produces significant increase in accuracy of the Newtonian results for CD at 
3.7% compared to CL at 18.6%. 
Table 6.3. Comparison of Apollo 4 Data and Computations, M? = 30, ? = -25o at 30040 s.  
 Apollo 4 [ref. 7] Mod. Newtonian % Error 
CN -0.11 -0.06387 -41.9% 
CA 1.32 1.400 6.1% 
CL 0.45 0.5337 18.6% 
CD 1.25 1.296 3.7% 
L/D 0.37 0.4119 11.3% 
 
 Additionally, Newtonian theory produces results that trim the CM within 1.2o 
for both Apollo missions AS-202 (? = 17.5o ? 0.5o)6 and Apollo 4 (? = 25.5o ? 3o).7 
For all these reasons, it is concluded that the CN flight data is probably inaccurate, 
rendering the higher percentage errors in CL and L/D. Since the percentage 
differences between Newtonian theory and the acceptable CM experimental wind 
tunnel and flight data is less than 15%, and since the theory follows the behavior of 
the wind tunnel data, modified Newtonian flow is considered acceptable for 
comparing the basic hypersonic aerodynamic characteristics of the investigated blunt-
 
 93 
 
body heat shield shapes with low computational time. Only the rolling moment values 
and stability derivatives have not been completely validated since a lack of this data 
exists, but it is partly validated in the next section. 
6.1.3. Comparison with Additional Sources 
Whitmore28 offers a recent analysis of the Apollo capsule as well as other 
human-rated lunar return vehicles such as a flattened bi-conic with trim flaps and an 
HL-20-derived lifting body configuration. His numerical results on the stability 
characteristics of the Apollo capsule closely match the results of this work?s code. 
Whitmore also uses a modified Newtonian surface pressure distribution to determine 
the aerodynamics of each vehicle.  
 Both the results of this work and of Whitmore suggest that the Apollo capsule 
is slightly statically unstable in the roll direction. In this case, to be statically roll 
stable, the vehicle would have a negative value of Cl,cg,? since vertical lift is positive 
at negative angles of attack for the Apollo CM. Both works indicate that the Apollo 
capsule would have slightly positive values for the Cl,cg,?, if the center of gravity is 
above the central axis during planetary entry.  
Since the Apollo CM had a Reaction Control System (RCS) that could control 
the Command Module?s roll angle, one guess is that the RCS may have been used 
once in a while to fix the CM?s roll alignment. Another guess is that the Cl,cg,? had a 
negligible value for the CM. Whitmore reports a value of Cl,cg,? = 0.0065/rad while 
this work produces a value of Cl,cg,? = 0.00541/rad at ? = -16o to produce L/D = 0.25. 
Since both of our works concur and Whitmore?s work is the only source in the 
 
 94 
 
literature search that offered a value to compare, this is the extent that the Cl,cg,? is 
validated in this work.  
Magazu26 investigated the feasibility and aerothermodynamic performance of 
a 12-sided parashield re-entry vehicle that has a shape similar to a 12-sided umbrella 
with no more than 7% concavity. The reproduction of this heat shield shape is defined 
as having a spherical-segment axial shape with ?s ? 45o and a dodecagon cross-
section without any concavity. The superformula of the superellipse Eqn. (2.8) can 
approximate a sharp dodecagon with the following parameters m = 12, n1 = 10.75, n2 
= n3 = 1.  With these parameters, this work?s code produced a lift-to-drag ratio of 0.19 
compared to the reported 0.18 in Ref. [26] at ? = -15o. 
6.2. Stagnation-Point Heat Transfer  
Validating the results of this work?s stagnation-point heat transfer methods 
against Apollo and FIRE II experimental and flight data is completed in this section. 
It is shown that the methods produce results with < 15% error. Note that it is not the 
purpose of this work to imply from these low errors that heat transfer during planetary 
entry is well understood. CFD would also produce results with errors of 
approximately 10-15% also. Additionally, although the percentage error is low for the 
stagnation-point heat transfer methods of this work, the error would probably increase 
dramatically for entry velocities greater than 12 km/s, in which it is expected that 
radiation cooling and convection coupling would lower the radiative heat flux. 
However, it is unknown precisely how much reduction there would be since no 
instrumented flight tests have been completed on flights with entry speeds greater 
 
 95 
 
than 11.4 km/s from the FIRE II. Several questions still exist on laminar, transitional, 
and turbulent boundary layer heat transfer during planetary entry. As a result, 
experimental research in this area would be especially worthwhile as future work.    
6.2.1. Apollo 4  
The peak radiative heat flux for the Apollo 4 mission occurred at an altitude of 
approximately 200000 ft around 30030 s into the mission at which point the 
Command Module was moving at a speed of 34000 ft/s. For the portion of the 
trajectory with high radiative heat flux, Figure 6.4 shows the calculated normal-shock 
density ratio and corresponding specific heat ratio using Tannehill?s correlation set.54  
 
Figure 6.4. Normal-shock density and specific heat ratios for the high radiative heat flux portion 
of the Apollo 4 trajectory. 
The altitudes and velocities during this portion of the Apollo 4 trajectory are also 
shown in Figure 6.4. The stagnation point on a blunt-body is usually across from the 
part of the bow shock that is normal to the freestream. As a result, the normal-shock 
density ratio ?2/?1 and corresponding effective specific heat ratio after the shock can 
 
 96 
 
be used to approximate the effective radius-of-curvature at the stagnation point. 
Kaattari?s method requires ?2/?1 and ?eff,2 to determine reff. To validate the 
implementation of Kaattari?s method in this work, in Figure 6.5 it is compared to 
other methods of determining the shock-standoff distance for the case of a sphere. 
This figure is partially a reproduction of Figure 4.1 in which the empirical curve-fit 
Eqn. (4.8) is compared to wind tunnel data. As a result, it can be observed that 
Kaattari?s method in Figure 6.5 follows the experimental data closer than the 
empirical curve-fit at the lower values of ?2/?1. Rasmussen30 provided the solutions to 
the Vorticity method and parabolic thin shock layer approximation in his textbook. 
The solution from the Vorticity method follows the behavior of Eqn. (4.8) almost 
perfectly while the parabolic thin shock layer approximation produces shock-standoff 
distances that are at least 25% larger than wind tunnel results.  
 
Figure 6.5. Shock-standoff distance method comparison. 
 
 97 
 
It is noticed that Kaattari?s method and Eqn. (4.8) bracket most of the wind tunnel 
results shown in Figure 4.1. After Kaattari?s method determines the shock-standoff 
distance at zero angle of attack, the modified method for finding the shock-standoff 
distance at the prescribed angle of attack is accounted for through the effective radius 
term. For the Apollo 4, Ried48 generated predictions with early 1970 computer 
technology using CFD. Ried produced an effective radius at the stagnation point that 
would apply for radiative heat transfer, and it is compared to the results of Kaattari?s 
modified method in Figure 6.6. 
 
0
5
10
15
20
25
30
35
7 8 9 10 11 12 13 14 15 16
Effective Radius for Radiation (ft)
An
gle 
of
 A
ttack 
(d
eg
)
Apollo 4 predictions
Code results  
Figure 6.6. Apollo 4 reff for stagnation-point radiative heat transfer. 
Kaattari?s modified method varies no more than 10% from Ried?s predictions. Then 
the reff is applied in the radiative heat flux correlation set. To determine which 
correlations would be best to apply for this work, a plot of the Apollo 4 mission?s 
radiative heat flux shown in Figure 6.7 is used to compare correlations. Figure 6.7 
shows these results for the portion of the Apollo 4 trajectory with high radiative heat 
flux. Both radiometer and calorimeter measurements were made on the Apollo 4 at 
 
 98 
 
the point of maximum heating and the stagnation point, although measurement 
uncertainties were not recorded. Ried shows that his calculations match the 
radiometer results that measured only the visible and infrared radiation. He also 
calculated the UV continuum and UV line radiation. 
 
Figure 6.7. Validation of radiative heat flux correlations for Apollo 4. 
It made the most sense to compare the values of the correlations to the total radiative 
heat flux. As a result, the total radiative heat transfer that includes the visible, 
infrared, UV continuum, and UV lines is shown in Figure 6.7 as the Apollo 4 
predictions. The most recent correlation from Tauber and Sutton45 matched the 
Apollo 4 results for most of the region. However, the results of this correlation do not 
match the predictions for speeds less than 9000 m/s.  
In fact, both Lovelace41 and Bertin29 provide correlations that produce 
conservative results that are not far from the results for speeds less than 9000 m/s. 
Since the correlation from Bertin was originally designed for speeds less than 7620 
m/s (25000 ft/s), Lovelace?s correlation is applied for speeds less than 9000 m/s. 
 
 99 
 
Along with a method to transition between the two correlations, Lovelace and 
Tauber?s correlations are used in this code and produce the results shown in Figure 
6.8. Together, these correlations produce results that are conservative but close 
enough for first-order optimization results. 
 
Figure 6.8. Apollo 4 radiative heat transfer code validation. 
 
Figure 6.9. Apollo 4 Trajectory from Ref. [36]. 
 
 100 
 
To validate the convective heat flux, radiative heat flux, and the 
corresponding heat load values, results have been generated using the Apollo 4 
trajectory shown in Figure 6.9. Apollo 4?s maximum Mach number7 during Earth 
entry was Mach 40, and because it had the highest entry velocity of all the Apollo 
missions, it also had the largest heat load. The total heat flux and heat loads are 
calculated using both Lovelace?s Eqn. (4.9) and Tauber?s Eqn. (4.10), and so Table 
6.4 includes two columns of results. The equations used to produce each results are 
listed in the title of each column. These results are within 15% of the reported values. 
Table 6.4. Apollo 4 Comparison of Total Heat Transfer. 
Parameter Apollo 4, 
Ref. [38] 
Results from 
[1.6*1.06*Eqn. 
(4.1)+Eqn. (4.9)] 
Results from 
[1.6*1.06*Eqn. 
(4.1)+Eqn. (4.10)] 
qmax,tot  (W/cm2) 483 542 (+12%) 469 (-2.9%) 
Qmax,tot  (J/cm2) 42600 46200 (+8.5%) 38700 (-9.2%) 
 
NASA reported the values of the heat flux and heat load at the point of maximum 
heating, which in the case of the Apollo CM was not at the stagnation point. Although 
this work calculates the stagnation-point heating only as explained in Chapter 4, these 
reported values can still be used for validation. As shown in Figure 6.10, the 
maximum convective heating for the Apollo CM at ? = -25o was 60% larger than the 
stagnation-point convective heat flux at zero angle of attack.  
Maximum heating is located at S/R = 0.9 while the stagnation point at ? = -
25o is located at S/R = 0.74. Since the stagnation point at ? = -25o has a 10% higher 
heat flux than that at zero angle of attack, the maximum heat flux is 45% larger than 
the stagnation-point heat flux at ? = -25o. Although angle of attack has been 
 
 101 
 
accounted for by this work?s radiative heat flux calculations, it has been assumed that 
the convective heat flux would be kept constant at the nose.  
Due to the Apollo CM?s low spherical-segment angle of 25o and corner 
geometry, the stagnation-point does not have the highest convective heat flux at ? = 
25o. Bertin29 notes that a correction factor of 1.06 to the correlation for a sphere can 
be used to account for the change in the sonic line location from 45o to approximately 
25o for the CM. As a result, after multiplying the convective heat flux by 1.06 and 
then 1.60 to account for the corner radius? effect that produces maximum heating, the 
convective heat flux can be added to the radiative heat flux to produce maximum heat 
flux and heat load results within 12% of the reported values. 
 
 102 
 
 
Figure 6.10. Convective heat flux distribution of Apollo Command Module at ? = 25o from Ref. 
[36]. 
6.2.2. FIRE II 
In the case of the FIRE II flight, the entry vehicle?s stagnation point was equal 
to the point of maximum heating, and it traveled mainly at zero angle of attack. As a 
result, it would be expected that this work would match the FIRE II data more closely 
than the Apollo 4 data. However, the FIRE II had an entry velocity of 11.4 km/s 
 
 103 
 
(37400 ft/s), which is slightly faster than the entry velocity of Apollo 4 at 10.7 km/s. 
Because FIRE II had an entry Mach number greater than forty, it is possible for there 
to be coupled effects between convection and radiation that would reduce the total 
heat flux. The FIRE II had three heat shields of different radii placed on top of each 
other. One heat shield would be jettisoned at a time to acquire heat transfer data for 
each heat shield. Since the heat shields have different radii, discontinuities in the 
flight data are expected. As shown in Figure 6.11, flight data from the calorimeter 
produced a noticeably smaller heat flux value than the theory that does not assume 
coupling.  
 
Figure 6.11. FIRE II Total heat flux comparison with flight data from Ref. [47]. 
The correlations of this work, which are labeled as code results, also do not assume 
coupling. The maximum total heat calculated by this work is 9% larger than the flight 
data. The interesting part is that the convective theory curve that assumes coupling 
and the code results nearly match perfectly for the first shield and do not vary greatly 
for the other two. However, it is apparent that the correlations fall approximately 
 
 104 
 
halfway in-between the theory that assumes no coupling and the calorimeter data. It is 
believed that the FIRE II had some coupling, and that this is the reason why the total 
heat flux theory without coupling and the code results produce a peak at a different 
time than the FIRE II calorimeter data. As a result, it would be expected that this 
work?s accuracy would begin to disappear at slightly higher velocities.  
 
 
 
 
 105 
 
Chapter 7. Parametric Analysis 
In order to become familiar with the design space that will be used in 
optimization, a parametric analysis has been conducted. Familiarity with the design 
space and previous work allows one to know whether optimization results are 
reasonable and provides simple test cases for verifying that the optimizer is working 
properly. Also, better initial designs for optimizing a given objective function may 
also be discovered throughout the parametric analysis. Most of this parametric 
analysis is included in Ref. [72]. 
7.1. Selecting a Superelliptical Base  
 Aerothermodynamic results were acquired for several axial and base shape 
combinations with the parameters in Table 2.1. One of the main considerations in 
choosing a polygon of m-sides is for it to be passively longitudinally stable over a 
range of cross-sections, and so one of the most longitudinally stable shapes is 
desirable. In order to compare the value of the pitching moment consistently 
throughout the given shapes, moments about the tip of the nose are considered. 
Moments about the shape?s center of gravity are accounted for later in this analysis. 
The derivative of the pitching moment coefficient Cm,0,? shown in Figure 7.1 varies 
slightly with the rounded-edge concave m-gon compared to its variance with 
eccentricity. Because the hexagon has the most stable value for pitching and yawing 
moments, it was decided that a superellipse with m = 6 would be featured in this 
 
 106 
 
analysis. The results in this section focus on heat shield geometries with the rounded-
edge and concave hexagonal (m = 6) and elliptical cross-sections. 
 
Figure 7.1. Cm,0,? distribution varying m-gon and e with spherical-segment axial shape and 
rounded-edge concave cross-sections, ?=-20o, ? = 5o. 
 The Cm,0,? distribution also suggests that eccentricity always decreases 
longitudinal static stability for the shapes considered. Positive values of the 
eccentricity correspond to the prolate shapes, and negative values to the oblate ones. 
A prolate eccentricity value of 0.5 was chosen as the constant eccentricity at which to 
present results because it does not considerably lower vehicle stability and since data 
on eccentric heat shields is scarce. Note that a particular oblate shape with a larger 
L/D but smaller CL than the common non-eccentric spherical-segment will be 
discussed later in this analysis.  
7.2. Heat Shield Shapes and Aerodynamic Performance 
  The results are acquired at M? = 36, the re-entry Mach number from lunar 
return for the trajectory of the Apollo CM. The code was run over several ranges of 
geometric parameter values and angles-of-attack for each of the axial shapes and 
 
 107 
 
cross-sections. The set of superelliptic bases included a rounded-edge hexagon, an 
ellipse, and a rounded-edge concave hexagon, each with an eccentricity of 0.5 as 
shown in Figure 7.2, along with examples of axial shapes.  
             
         (a) n2 = 1.5,   Spherically-blunted cone,            (b) n2 = 2.0,  Spherical-segment,  
                          rn/d = 0.25, ?c = 70o.                           ?s = 25o.           
 
 
(c) n2 = 4.0, Power law shape,  A = 2, b = 0.75. 
Figure 7.2. Chosen heat shield shapes for parametric analysis. 
The ranges of the geometric variables for the axial shapes are included in Table 7.1; 
this table has ranges that are applied to most of the analysis, including the results in 
Table 7.2. The axial shapes shown in Figure 7.2 are those applied when varying ?; of 
these three axial shapes, the one with the smallest geometric l/r is the spherical-
segment, followed by the spherically-blunted cone, and then the power law shape.  
 
 108 
 
  As used in previous work, a sideslip angle of five degrees has been chosen in 
calculating the values of Cn,? and Cl,?. Apollo 4 had a maximum sideslip of ? = 2o, but 
wind tunnel tests have tested the Aeroassist Flight Experiment32,33 (AFE) at sideslip 
values up to ? = ? 5o.32 Note that it is convention to combine the three normal vectors 
CA, CN, and CY into two vectors CL and CD; this produces vertical and horizontal (with 
respect to the body axis) components of lift, represented by LV and LH respectively for 
cases with ? ? 0.20,21 Even though LH is negligible in this particular analysis since ? is 
small compared to ?, LV is listed in results with ? ? 0 in order to be consistent with 
general theory.  
Table 7.1. Variable ranges and constants for each axial shape. 
Axial Shape Variable Range Constants 
-30o ? ? ? 0o ?s = 25
o  
(Apollo Command Module) Spherical- segment 
5o ? ?s ? 90o ? = -20o 
-30o ? ? ? 0o ?c = 70
o, rn/d = 0.25  
(Viking Mars Lander) 
30o ? ?c ? 89o ? = -20o, rn/d = 0.25 
Spherically-
blunted Cone 
0.01 ? rn/d ? 1.00 ? = -20o, ?c = 70o 
-30o ? ? ? 0o A = 2, b = 0.75 
1 ? A ? 6 ? = -20o, b = 0.75 Power Law Shape  
0.1 ? b ? 0.999 ? = -20o, A = 2 
 
  The coordinate system shown in Figure 3.1(a) has the direction of positive 
sideslip and negative angle-of-attack. The direction of positive moments is shown in 
Figure 3.1(b) for both positive and negative LV, which only changes the convention of 
the rolling moment for reasons explained in Chapter 3. Since blunt-bodies produce 
positive lift at negative angles-of-attack (for ?c > 45o), this analysis mainly focuses on 
how geometric variables vary with a constant negative angle-of-attack. Since the 
 
 109 
 
Apollo CM had trim angles of attack equal to -17.5o and -25.5o during orbital re-entry 
and lunar return respectively, the effects of the geometric parameters are analyzed at 
an angle of attack in-between them at -20o.   
7.3. Hypersonic Aerodynamic Performance and Stability  
 The effects of the power law?s slenderness ratio, A, and cross-section 
parameter, n2, on Cm,0,? are shown in Figure 7.3. This plot is typical of the results for 
all of the aerodynamic parameters in that they are much more sensitive to a change in 
axial shape than to a change in cross-section. As a result, the effects of varying the 
cross-section at a fixed eccentricity are not usually as important compared to the 
effects of changing the axial shape for longitudinal static stability.  
 
 
Figure 7.3. Cm,0,? distribution for power law A and n2, b = 0.75, e = 0.5, ?=-20o, ? = 5o. 
However, a cross-section?s eccentricity can affect the vehicle?s static roll stability 
significantly and is discussed later in this analysis.  Figure 7.4 shows a similar trend 
for Cm,cg,? but also that there is a minimum of ?0.22/rad near A = 1.5 and n2 = 2. Both 
 
 110 
 
Cm,0,? and Cm,cg,? have maximum absolute values at n2 = 2, and this is also 
characteristic of the yawing moment stability derivative. The minimum Cm,cg,? may 
change in value and occur at a different (A, n2) with a base of different eccentricity, or 
different values of b, ?, or ?. Because there are several combinations of variables that 
affect it, the process of optimization, which can vary any or all of the variables, will 
be an effective tool for locating extrema.  
 
Figure 7.4. Cm,cg,? distribution for power law A and n2, b = 0.75, e = 0.5, ?=-20o, ? = 5o. 
 Table 7.2 offers a summary of the sensitivity of main aerodynamic parameters 
(no heat transfer) by arranging the values of percentage difference of the parameter?s 
minimum and maximum into six groups, labeled 0 through 5, in which coefficients in 
category five have the largest percentage difference. Each group corresponds to a 
range of percent difference values. Since the aerodynamic parameters do not vary 
considerably with a change in cross-section, as discussed earlier and shown in Figure 
7.3 and Figure 7.4, the percentage differences as well as the actual values are close to 
each other and the same in sign when varying the cross-section. As a result, Table 7.2 
 
 111 
 
includes one column of data for the set of three base cases instead of three separate 
columns.  
Table 7.2. Effects of geometric parameters and ? on aerodynamic performance. 
 
Spherical 
Segment 
Spherically-blunted 
Cone 
Power Law 
Shape  
? ?s ? ?c rn/d ? A b 
CL,V 5 5 5 4 1 5 5 4 
LV/D 5 5 5 3 0 5 4 3 
Cm 5 5 5 5 0 5 4 1 
Cm,? 2 5 2 3 0 2 3 1 
Cn 1 5 2 5 0 1 4 1 
Cn,? 1 5 1 5 0 1 4 1 
Cl 1 5 1 5 0 3 5 1 
Cl,? 0 5 0 5 0 4 5 5 
Xcp/l 0 2 0 1 1 0 3 2 
Ycp/l 5 3 5 1 1 5 2 1 
Zcp/l 0 2 1 1 1 0 1 1 
?v N 4 N 4 1 N 3 1 
           Note: Ranges of ?, ?s, ?c, rn/d, A, b given in Table 7.1. 
Percent Difference Key 
0:  < 14%  
1: 15% - 39% 
2: 40% - 75% 
3: 76% - 400% 
4: 401% - 999% 
5: 1000% + 
N: Not applicable. 
 Of the three moment coefficients, only Cm varies over two orders of 
magnitude with ? while Cm,? only varies by approximately 50%. Cm also varies in 
two orders of magnitude with ?, ?s, ?c, and A. In general, the moment coefficients and 
stability derivatives varied strongly with ?s, ?c, and A for the given range. For the 
 
 112 
 
spherically-blunted cone, rn/d only had a small effect, by varying the aerodynamic 
parameters ? 5-10% at most. The exponent b for the power law shape varied them ? 
25% except for Cl,? that was affected strongly, although this does not necessarily 
mean that an extremely high value of Cl,? was acquired; it only means that it varied 
strongly with b from a possibly negligible to significant value. As a result, a high 
percentage does not necessarily mean varying strongly from one significant value to 
another one, and some high percentages were produced since the smaller of the two 
absolute values was near zero (<10-5).  In the least, these results can be compared to 
more advanced analyses in future investigations. 
 A human planetary entry vehicle may require a reaction control system (RCS) 
to control the roll angle for human safety and follow the entry trajectory. If the rolling 
moment stability derivative is unstable but small, a less powerful rolling moment 
control system would be needed. Also, the units of the stability derivatives are per 
radian rather than per degree, and so reasonable rolling moment derivative and 
yawing moment derivative values from the AFE include Cl,?=-0.04584/rad = -
0.0008/o and Cn,? = 0.1318/rad = 0.002300/o.21 The AFE has a circular cross-section 
with a blunted-nose, raked-cone axial shape that produces values of Cl,? that are 
within the range of values acquired in this analysis, and the AFE?s values of Cn,? are 
within the mid-range of values acquired in this analysis. 
 Moment coefficients and stability derivatives shown in Figure 7.5 at ? = -20o, 
? = 5o, are affected strongly by varying the geometric parameters. Although the 
magnitudes are not equal, the plot of Cm,? in Figure 7.5(b) would show the same 
behavior as that of Cm in Figure 7.5(a) if the data were reflected about the horizontal 
 
 113 
 
axis. Figure 7.5(c) and (d) show Cn and Cn,? have the same behavior, although Cn is 
an order of magnitude smaller.  
  
            a) Variations in Cm.                   b) Variations in Cm,?. 
 
     c) Variations in Cn.                d) Variations in Cn,?. 
Figure 7.5. Effects of ?s, ?c (rn/d = 0.25), A (b = 0.75) on stability coefficients and derivatives, ? = -
20o, ? = 5o. 
All four variables show that the center of gravity moments and derivatives first 
increase in magnitude with an increase in the geometric parameter, then produces a 
minimum or maximum, and then decreases in magnitude, which is a different 
behavior from that of the moments about the nose that usually have maximum and 
 
 114 
 
minimum values near or at the end points of the range. As a result, it is important to 
note that the values of the moments about the nose may neither be near the values of 
the moments about the center of gravity nor share the same behavior. It is noticed that 
Cm,cg, Cm,cg,?, Cn,cg, and Cn,cg,?  all have maximum magnitudes near the same value, 
suggesting that a value of ?0.22/rad for Cm,cg,? for a power law shape with A = 1.5 
would also be produced with both a spherically-blunted cone of ?c = 42.5o and a 
spherical-segment of ?s = 52.5o. Heat shields with equivalent values of Cn,cg,?, which 
has a maximum magnitude of 0.24/rad, can also be determined. Both the Cl and Cl,? 
have values within those acquired in wind tunnel for the AFE that are of the order of 
0.01/rad for Cl,? and 0.001 for Cl.21  
 
Figure 7.6. Effects of ?s, ?c (rn/d = 0.25), A (b = 0.75), and b (A = 2) on ?v, ? = -20o, ? = 5o. 
  Figure 7.6 shows how all five geometric variables affect the volumetric 
efficiency of the heat shields. The volumetric efficiency in this case is normalized to a 
 
 115 
 
hemisphere in Eqn. (2.15), and this results in the most volumetrically efficient shape 
being the non-eccentric spherical-segment at ?s = 90o. The volumetric efficiency is 
varied with eccentricity and ?s for a range of spherical-segments in Figure 7.7.  
 
Figure 7.7. ?v distribution for spherical-segment, elliptical base (n2 = 2), varying e and ?s. 
For ?s = 90o, an increase in eccentricity decreases the volumetric efficiency, but a 
saddle point exists at ?s = 35o; as a result, below ?s = 35o eccentricity increases the 
volumetric efficiency by up to 10%. 
7.4. Effect of Eccentricity on Aerodynamic Performance 
  The stability derivative of the rolling moment about the center of gravity, in 
Figure 7.8 does not become significantly affected by eccentricity until ?s > 20o. 
Beyond this value of ?s, it is observed that the magnitude of Cl,cg,? increases. For 
clarity purposes, the stable configurations are shown as Cl,cg,? ? 0 in Figure 7.8. While 
some of the oblate shapes are stable or unstable, the non-eccentric and prolate shapes 
(e > 0) are statically roll stable.  
 
 116 
 
  An analysis of the effect of eccentricity on LV/D was conducted to determine 
if there were any possible shapes that were not completely stable but could give a 
noticeable increase in LV/D. Figure 7.9 shows the LV/D distribution for varying ?s and 
eccentricity.  
 
Figure 7.8. Cl,cg,? distribution for spherical-segment, elliptical base (n2 = 2), varying e and ?s, ? = 
20o, ? = 5o. 
Shown in Figure 7.10, a positive ? dataset produced the positive LV/D value of 0.584, 
a 56.1% increase, at ?s = 90o for an oblate shape with e = -0.95, in which b1 = 1.0 and 
a1  = 0.311 at ? = 20o compared to a less round, more blunt ?s = 5o with e = 0.0 with a 
positive LV/D value of 0.374 at ? = -20o. However, CL,V drops to 0.282 from 0.56, a 
49.6% decrease.  
 Table 7.3 lists the aerodynamic and stability characteristics based on modified 
Newtonian flow and uniform density heat shield geometries. The characteristics of 
the following geometries are listed in Table 7.3: ?s = 5o & e=0.0, ?s = 25o & e = 0.0 
(similar to the Apollo CM heat shield), and ?s = 90o & e = -0.95. All three have stable 
 
 117 
 
yawing moments also, with ?s = 25o, e = 0.0 at the largest value of 0.0138 for Cn,cg, 
and the other two have smaller yawing moment coefficients on the order of 0.003.  
 
Figure 7.9. LV/D distribution for spherical-segment: elliptical base (n2 = 2), varying e and ?s, ? = 
20o, ? = 5o. 
While the yawing moment derivative Cl,0,? for ?s = 5o & e = 0.0 is only 27.7% of the 
Cl,0,?  for the AFE, both the oblate ?s = 90o, e = -0.95 and non-eccentric ?s = 25o have 
values of the same order as the AFE?s. The rolling moment derivative about the 
center of gravity for ?s = 5o, e = 0.0 is negligible, but the oblate ?s = 90o, e = -0.95 has 
a Cl,0,? = -0.0924 that is twice the stability derivative of the AFE. The volumetric 
efficiency is lower for the blunt shapes ?s = 5o and ?s = 25o with values of 0.204 and 
0.583 respectively compared to a value of 0.922 for ?s = 90o, e = -0.95. The two non-
eccentric cases have stable pitching moments. 
 
    a) Front view.                                      b) Side view. 
Figure 7.10. Spherical-segment  ?s = 90o, e = -0.95 with an increase of 56% in LV/D. 
 
 118 
 
Table 7.3 Aerodynamic comparison of spherical-segments (? = 5o). 
e = 0, n2 = 2 e = -0.95  n
2 = 2  
?s = 5o 
? = -20o 
?s = 25o 
? = -20o 
?s = 90o 
? = 20o 
CL,V 0.560 0.468 0.282 
LV/D 0.374 0.335 0.584 
(LV/D)? -1.10 -0.961 0.469 
Cm,0 0.0126 0.0613 -0.105 
Cm,0,? -0.0299 -0.145 -0.363 
Cn,0 0.00302 0.0147 0.0133 
Cn,0,? 0.0365 0.177 0.161 
Cl,0 0.000 -3.38E-5 -0.00808 
Cl,0,? ? -4.06E-5 -0.0924 
Cm,cg 0.0126 0.0575 -0.0195 (0.0310)  
Cm,cg,? (/rad) -0.0299 -0.136 -0.0990 (0.0566) 
Cn,cg 0.00301 0.0138 0.00785 (0.00488) 
Cn,cg,? (/rad) 0.0364 0.166 0.0916 (0.0541) 
Cl,cg 0.000 -3.38E-5 -0.00808  (-0.00808) 
Cl,cg,? (/rad) ? -4.06E-5 -0.0924  (-0.0924) 
Xcg/l 0.663 0.662 0.400 (0.635) 
?v 0.204 0.583 0.922 
      
 
With the high eccentricity of e = -0.95 comes the unstable pitching moment derivative 
Cm,cg,? = 0.0566/rad if a uniform density heat shield is assumed. However, if the 
uniform density value Xcg/l = 0.635 is switched to Xcg/l = 0.400, then the pitching 
moment becomes stable with Cm,cg,? = -0.099/rad. In addition to values of 
aerodynamic moments at Xcg/l = 0.400, Table 7.3 includes moment and derivative 
values for this geometry with Xcg/l = 0.635 in parentheses. As a result, it may be 
advantageous to further study oblate geometries that produce higher LV/D since an 
 
 119 
 
increase in trim LV/D opens a larger range of entry flight path angles available for 
planetary entry and significantly increases the range capability of the space capsule.1 
7.5. Comparison of Aerothermodynamic Performance on 
Two Heat Shields   
 
After conducting the optimization cases discussed in the next chapter, it was 
determined that a heat shield with a parallelogram cross-section (m = 4) produces a 
higher L/D than one with a hexagonal cross-section. As a result, a brief parametric 
comparison of the m = 6 and the m = 4 shapes based on stagnation-point heating and 
L/D is conducted. This is completed with a spherically-blunted cone axial shape with 
a 55o half-cone angle and rn/d = 0.05 at ? = -20o. As noted in the next chapter, the 
minimum rn/d for this work is 0.05, and the minimum half-cone angle is 55o; this 
parametric study attempts to show some of the effects of one of the expected worst 
cases from a stagnation-point heating approach. 
The total stagnation-point heating varies with n2 and e for m = 6 in Figure 
7.11(a). Since the method of determining the shock-standoff distance is a piecewise, 
the plot of  ,totsq& is expected to have discontinuous slopes. This ripple effect is more 
pronounced in  Figure 7.11(b) for m = 4. From an optimization standpoint, this 
indicates that the optimizer will have to run numerous sets of initial designs to find a 
set of local minima that may exist in-between the ripples and then determine the 
global minimum. Note that in these cases, the rn/d solely determines the convective 
heat transfer while reff and ? determine the radiative heat transfer. As a result, the 
values for the convective heat transfer are constant for  Figure 7.11(a) and (b) at 961 
W/cm2. Since the convective heat transfer correlations are accepted as reasonably 
 
 120 
 
accurate and since radiation is a current line of research, only the radiative heat 
transfer correlations were explored further. 
 
a) Hexagonal base (m = 6). 
 
b) Parallelogram base (m = 4). 
Figure 7.11.  ,totsq&  distributions for m = 4 and 6, spherically-blunted cone axial shape, ?c = 55o, 
rn/d =0.05, varying e and n2 = 1.3 to 2.3, M? = 32.8, ? = -20o. 
 
 121 
 
Each of these figures includes a drop of 1-5 W/cm2 in at least the n2 < 2 
region, due to using a piecewise method. Since the drop is small compared to the 
magnitude of the total heat flux, it is not significant in this case. For additional plots, 
it is shown that this drop becomes larger but does not exceed 20% of the radiative 
heat flux.  
 
a) Hexagonal base (m = 6). 
 
a) Parallelogram base (m = 4). 
Figure 7.12.  ,totsq&  distributions for m = 4 and 6, spherically-blunted cone axial shape, ?c = 55o, 
rn/d =0.05, varying e and n2 = 1.3 to 4.0, M? = 32.8, ? = -20o. 
 
 122 
 
If the parametric analysis is furthered to n2 = 4, then the  ,totsq& varies for m = 6 in 
Figure 7.12(a) better than it does for m = 4 in Figure 7.12(b) since the method begins 
to break down around n2 = 3 for the parallelogram. This breakdown is shown in 
Figure 7.12(b) with the troughs that are seen through the main slope. This is not 
present in the hexagonal cross-sectional case.  
 
a) Hexagonal base (m = 6). 
 
a) Parallelogram base (m = 4). 
Figure 7.13. LV/D distribution for m = 4 and 6, spherically-blunted cone axial shape, ?c = 55o, rn/d 
=0.05, varying e and n2 = 1.3 to 4.0, M? = 32.8, ? = -20o. 
 
 123 
 
LV/D is shown to have a maximum magnitude of 0.61 with an oblate, rounded-
edge concave hexagon in Figure 7.13(a). In this case the value is negative, which 
means that the hexagonal shape could produce a positive LV/D of equal magnitude at 
positive angle of attack since it is symmetric about the horizontal axis (z-axis). The 
main result found through optimization is that the magnitude of LV/D as shown in 
Figure 7.13(b) could be increased past 1.0 for the oblate rounded-edge parallelogram.     
 
a)  ,convsq&  distribution.  
 
b)  ,radsq&  distribution.  
Figure 7.14.  ,convsq& and  ,radsq&  distributions for parallelogram base (m = 4, n2 = 1.4 ), 
spherically-blunted cone axial shape, ?c = 55o, varying e and rn/d, M? = 32.8, ? = -20o. 
 
 124 
 
For m = 4, the variation in convective heat flux with eccentricity and rn/d is 
included in Figure 7.14(a). The eccentricity does not modify the nose radius 
dramatically, and so the convective heat flux does not vary with eccentricity. 
However, an increase in nose radius alleviates much of the convective heat flux. The 
radiative heat flux shown in Figure 7.14(b) shows the opposite trend: an increase in 
nose radius produces more radiative heat flux. However, the prolate cases show a 
maximum at rn/d = 1.3 while the oblate cases continue to increase. Some uncertainty 
is present with these results since prolate and oblate heat transfer analyses have not 
been researched in-depth. This would be a good topic of future research. It is believed 
that the maximum heat flux of the oblate cases provides a conservative limit on the 
heat flux expected on an actual heat shield based on the way the eccentricity 
correction factor is applied in Eqns. (3.52) and (3.53).  
The trade-off between convection and radiation produce a total heat flux that 
has a local minimum at rn/d = 0.5 for e = -1 and rn/d = 0.8 for e = 1, as shown in 
Figure 7.15(a). Figure 7.15(b) provides another view of the total heat flux for clarity. 
For rn/d > 1, some of the oblate cases increase in total heat flux while all the other 
cases continue to decrease. However, even with this increase, the overall change in 
heat flux is small and suggests that heat shields at 200000 ft and Mach number 32.8 
should either have an rn/d between 0.5 and 0.8 or greater than 1. The Apollo CM?s 
design had a rn/d = 1.18 and thus is in the lower region of heat transfer shown in 
Figure 7.15(a). For an increase in freestream Mach number, the radiative heat transfer 
becomes greater, and the results of optimizing the total heat transfer throughout the 
Mach numbers is included in Chapter 8.  
 
 125 
 
 
a) Emphasis on the local minima. 
 
b) Emphasis on overall distribution. 
Figure 7.15.  ,totsq& distribution for parallelogram base (m = 4, n2 = 1.4 ), spherically-blunted cone 
axial shape, ?c = 55o, varying e and rn/d, M? = 32.8, ? = -20o. 
 
 
 
 126 
 
Chapter 8. Optimal Configurations 
The optimization results of several objective functions are provided for the 
geometry whose axial shape produces the most ideal results. The main optimization 
results are provided as iteration histories of the objective function, design variables, 
and constraints, a 3D image, and a table of its aerothermodynamic characteristics.  
The iteration histories are located in the appendix. The objective functions in this 
analysis consist of maximizing LV/D, ?v(LV/D), LV/D/ ,,totsq&  and of minimizing totsq ,&  
and Cm,cg,?. A separate optimization is run for each of the three axial shapes, and this 
work provides iteration histories of the design variables in non-dimensionalized form 
so that all of them can be shown on the same figure. For each axial shape, the design 
variables with their lower and upper constraint values are given along with their non-
dimensional value counterparts in Table 8.1. For the geometries with a spherical 
segment axial shape, the limit of 5o on ?s was chosen as the lower limit in order to 
provide a blunt-body that is not completely flat, and 89o was chosen as an upper limit 
since the code would produce a zero in the denominator in some equations for ?s = 
90o and is a close approximation from a numerical standpoint.  
The n2 variable from Eqn. (2.8) controls whether the cross-section is a 
polygon, ellipse, or concave polygon. For this analysis, zero radius-of-curvature 
designs are not considered since they produce high heat concentrations. As a result, 
the n2 lower limit of 1.3 produces slightly rounded-edge polygons. As n2 approaches 
2.0, the polygon transforms into an ellipse, and increasing n2 further transforms the 
 
 127 
 
ellipse into a rounded-edge concave polygon. An upper limit of 4.0 on n2 generates 
shapes that still have a reasonable radius-of-curvature.  
Table 8.1. Optimization design variables with side constraints. 
Spherical Segment 
Dimensional Non-dimensional 
5o ? ?s ? 89o 0.0556 ? ?s/90o ? 0.989 
1.3 ? n2 ? 4.0 0.325 ? n2/4 ? 1.0 
-0.968246 ? e ? 0.968246 -0.968246 ? e ? 0.968246 
-30o ? ? ? 30o -1.0 ? ?/30o ? 1.0 
Spherically-blunted Cone 
Dimensional Non-dimensional 
55o ? ?c ? 89o 0.611 ? ?c/90o ? 0.989 
0.15 ? rn/d ? 2.0 0.075 ? (rn/d)/2 ? 1.0 
-0.968246 ? e ? 0.968246 -0.968246 ? e ? 0.968246 
1.3 ? n2 ? 4.0 0.325 ? n2/4 ? 1.0 
-30o ? ? ? 30o -1.0 ? ?/30o ? 1.0 
Power law 
Dimensional Non-dimensional 
0.2 ? b ? 0.9999 0.2 ? b ? 0.9999 
0.9 ? A ? 10.0 0.09 ? A/10 ? 1.0 
-0.968246 ? e ? 0.968246 -0.968246 ? e ? 0.968246 
1.3 ? n2 ? 4.0 0.325 ? n2/4 ? 1.0 
-30o ? ? ? 30o -1.0 ? ?/30o ? 1.0 
 
The eccentricity limits of ?0.968246 were chosen to limit the ratio of the 
semi-major axis? length to the semi-minor axis? to 4-to-1. This was decided from 
inspection, by what appeared to be a liberal choice of a reasonable shape by intuition.  
 
 128 
 
The angle of attack is limited to ?30o since human space capsules usually 
enter at or below 25o and since the heat shield may be only half of the main vehicle?s 
shape depending on the value of ?s. If ?s is large, then it is possible that the entire 
space capsule could fit within the heat shield or in the case of a ballute-type heat 
shield. These limits could be modified easily for future work. 
A non-eccentric heat shield with ?c ? 45o is the interface at which the 
spherically-blunted cone begins to produce positive LV at negative ?. If ?c is less than 
this value, then negative LV at negative ?. The term blunt-body for re-entry usually 
insinuates that the vehicle produces positive LV at negative ? and that the shock-
standoff distance is substantial. Since Ried48 approximated that the shock-standoff 
distance of the Apollo 4 was 14 cm and this work predicts 12.4 cm (for t = 30030 s, 
M? = 32.8, ? = 25o at h = 200000 ft), it was decided that the order of accuracy of 
Kaattari?s method is 2.54 cm. As a result, the spherically-blunted cone that has a 
shock-standoff distance of 2.54 cm would be designated as the lower limit for ?c and 
rn/d based on these method limitations. For a non-eccentric heat shield, this work 
predicts a 2.54 cm shock-standoff distance for a spherically blunted cone with ?c = 
55o and rn/d = 0.25.  
Although any solutions that suggest a shock-standoff distance around one to 
five inches should be reinvestigated, the lower limit on the nose radius-to-diameter 
ratio does not exceed 0.25 since previous work such as the Mars Viking missions 
included heat shields with rn/d = 0.25. The chosen lower limit on rn/d is 0.15 to widen 
the design space. For numerical reasons, ?c is limited to 89o rather than 90o. The upper 
limit to rn/d is set at two in order to include some shapes that are pure spherical-
 
 129 
 
segments; the Apollo Command Module can be approximated with ?c = 75o, rn/d 
=1.1831.   
For the geometries with a power law axial shape, the lower limit value of b is 
0.2 since it is both blunt and has a slope change in the shape that is less extreme than 
that for b = 0.1 or 0.01. Newtonian Impact Theory may have a problem with the quick 
slope changes shown in Figure 2.3 for b = 0.1 or smaller. The upper limit on b 
produces a nearly linear line, but the code requires the slope of the power law profile 
to vary at least slightly for numerical reasons. A value of b greater than one would 
produce concave axial shapes with infinitely sharp noses, which are not included in 
this analysis. Since previous work blunt-bodies usually have l ? d (i.e., Apollo CM l/d 
? 1/9), it was decided in this work to widen the design space by including A = (l/d)-1 
= (10/9)-1. Then the upper limit of A was decided to be ten since the power law shape 
loses most of its uniqueness at high A, in comparison with the cone or the spherical-
segment axial shapes.  
Note that although ? is listed as one of the design variables, some of the 
objective functions may be optimized with fixed angle of attack (this is mentioned if 
applicable).  
 This optimization uses the following constraint vector, :Gr    
 { }654321 ,,,,, GGGGGGG =r     (8.1) 
.30.0/1,30001
,0,01.01
,0,01.01
,001.011001,001.011001,11 ,
,
,,
,
,,
,,,,
???
???
?
???
???
?
?+?
<??
?+?
???
?
???
? ?
???
?
???
? +
++?=
DLq
CC
CCCC tots
VL
cgl
VL
cgl
cgncgm &
?
?
??
?
?  
G1 is the only constraint on the geometry that is affected by freestream conditions. 
This constraint relates the angle of attack to the tangency angle ?, which is the angle 
 
 130 
 
produced by the heat shield?s edge surface and the vertical axis. Because the heat 
shield shape is not necessarily and not usually the entire vehicle shape, a limit on the 
angle of attack at which a given heat shield can be analyzed must be chosen in order 
to keep the assumption that the flow separates before passing over the crew 
compartment (aft body). This constraint basically requires that a given heat shield 
must not be placed at an angle of attack more than one degree larger than the 
tangency angle. In this way, the heat shield?s edge is normal to the freestream flow 
when ? = ?. The one-degree above ? was chosen as a small relief factor. 
G2 and G3 are longitudinal and yaw static stability requirements, respectively; 
because this is a numerical analysis, the magnitude value of 0.001 is deemed 
significant rather than 0.000. G4 is the roll static stability requirement, but it is 
different from G2 and G3 in that it allows for a slight instability up to 0.01/rad in order 
to produce heat shields such as the Apollo CM, which was slightly unstable at 
0.005/rad. This keeps the design space open to previous work. The requirement for 
roll static stability changes sign when the CL,V changes sign as explained in Chapter 3. 
G5 is an upper limit constraint on  ,totsq& of 3000 W/cm2 and is three times the 
maximum heat flux of 1000 W/cm2 at which NASA is designing the CEV. This 
allows for designs that will become feasible in the next couple decades. G6 is a 
constraint on the lift-to-drag ratio that requires improvement over previous work. The 
Apollo CM had a lift-to-drag ratio of 0.34 during re-entry, which was used as the 
benchmark to improve upon in this work. Note that not all of these constraints are 
used in the following results. The constraints from vector Gr that are applied for each 
 
 131 
 
optimization are noted for each optimization as well as the constraint iteration 
histories. 
 The design point conditions chosen for this analysis include an altitude of 
200000 ft and Mach number M? = 32.8. These were chosen because they are the 
conditions of Apollo 4 when it experienced the maximum total heat flux. Also, this is 
a point of high radiative heat flux, and for minimizing the stagnation-point heat 
transfer flux, the Mach number is increased to show how the optimum nose radius 
changes due to an increase in radiative heat flux. Although the aerodynamic 
performance and static stability characteristics can be given in non-dimensional 
terms, the heat transfer fluxes require the size of the heat shield to be chosen, and so 
the base diameter of 3.9166 m equal to that of the Apollo Command Module is 
prescribed. For some of the results, the center of gravity is chosen based on a uniform 
density distribution. However, after noticing from the parametric analysis that a 
longitudinally unstable heat shield could become stable by shifting the center of 
gravity forward a reasonable amount, it was decided that both cases would be tested. 
It is mentioned for each of the optimization results whether a uniform density 
distribution is assumed or a center of gravity location is prescribed. The prescribed 
center of gravity location for these results is equal to that of uniform density except 
that the prescribed Xcg is set to 75% of the uniform density value of Xcg. This allowed 
for a larger design space to be accessed. 
 This analysis is based on a total of 184 optimization runs, in which all of the 
initial designs of the design were varied significantly. Each optimization run has a 
different initial design. The number of sides of the superellipse cross-section has been 
 
 132 
 
considered, and results between geometries with different cross-sections are 
compared. All three of the axial shapes have been considered, and the engineering 
global optimum is reported for each in the following sections.  
8.1. Maximizing LV/D and ?VLV/D 
The vertical lift-to-drag ratio by itself and the product of LV/D and volumetric 
efficiency have been maximized. Since the parametric analysis originally 
encompassed only hexagonal superellipse cross-sections (m = 6), the optimization 
analysis began with m = 6. For the spherical-segment axial shape, it is shown in Table 
8.2 that LV/D generally increases with a decrease in the number of sides. The 
parallelogram cross-section has the minimum number of sides examined and thus 
offers the optimum lift-to-drag configurations in this analysis. The optimized m = 4, 
5, and 6 shapes are shown in Figure 8.1. 
Nineteen optimization runs of the spherical-segment, four of the spherically-
blunted cone, and twenty-eight of the power law geometries were completed for 
maximizing LV/D. Constraints G1 through G5 have been applied. The initial designs 
have values of n2 usually equal to 1.6, 2.0, and 3.0. Ideally, the optimizer would be 
able to check if there is a local maximum near the 1.3 side constraint when starting 
from 1.6, in case the local maximum is overlooked when starting from the other two 
initial designs.  
The spherical-segment angle constraint is active for these optimized shapes at 
?s = 89o. This high spherical-segment angle creates geometries that produce positive 
lift at positive angles of attack because the geometries have normal forces that 
 
 133 
 
contribute more to the lift force than the axial force. In the classic case of the Apollo 
Command Module with ?s = 25o, the axial force contributed more to the lift force, 
thereby making a negative angle of attack required for positive lift.  
Table 8.2. Initial and Optimal Designs for maximizing LV/D for different m. 
Spherical Segment Axial Shape 
Initial Design  Optimal Design  Objective Functions 
?s n2 e ? ?s n2 e ? LV/D 
85o 1.4 -0.9 25o 89.0o 1.3 -0.968 21.2o 0.822 (m = 8) 
85o 1.4 -0.9 25o 89.0o 1.3 -0.968 19.8o 0.755 (m = 7) 
85o 1.4 -0.9 25o 89.0o 2.61 -0.968 22.0o 0.753 (m = 6) 
85o 1.4 -0.9 25o 89.0o 1.3 -0.968 24.1o 0.878 (m = 5) 
85o 1.5 -0.9 25o 89.0o 1.3 -0.968 17.6o 1.10 (m = 4) 
 
 
a) Hexagonal cross-section, LV/D = 0.753. 
 
b) Pentagonal cross-section, LV/D = 0.878. 
 
c) Parallelogram cross-section, LV/D = 1.10. 
Figure 8.1. Optimized geometries from Table 8.2. 
The iteration histories of the objective function, design variables, and 
constraints are shown for the hexagonal cross-section spherical-segment geometry in 
 
 134 
 
Figure A.0.1 in the appendix. The main increase in LV/D occurs in the first iteration in 
which the geometry becomes more oblate and ?s is increased to 89o, in which the 
minimum e and the maximum ?s constraints become active, respectively. During the 
entire optimization, the G constraints are neither active nor violated. Because LV/D is 
one of the most important aerodynamic characteristics of a lifting re-entry vehicle, 
and because a high LV/D is desired for several reasons, such as increasing the range of 
available landing sites, the remainder of this optimization analysis is completed with 
parallelogram cross-sections (m = 4). 
The spherically blunted cone with maximum LV/D also has a four-sided cross-
section, shown in Figure 8.2(a). Originally, the optimum was at LV/D = 1.14, but the 
calculated shock-standoff distance was 2 cm, which is smaller than the estimated 
error of 2.54 cm. As a result, the side constraints on the spherical-segment angle and 
the nose-radius-to-diameter ratio were increased to 59o and 0.15 from 55o and 0.10 
respectively. As a result, the calculated shock-standoff distance became 3 cm with 
LV/D = 0.95. However, the shock-standoff distance of the optimum spherical-segment 
is 11 cm. This means that the spherical-segment shock-standoff distance is probably 
more reliable than the 3 cm calculation and may make the spherical-segment option a 
more desirable configuration.  
A total of seventy-two function evaluations of the power law geometry were 
completed over seven iterations to find a maximum LV/D of 2.10. The optimum 
power law geometry has a significant increase in heat transfer flux over the other two 
geometries, and also has heat flux constraint G5 active. The power law shape is 
allowed to have radii as small as 0.001 m while the spherically-blunted cone is 
 
 135 
 
restricted to 0.587 m due to limitations in calculating the shock-standoff distance. As 
a result, only the power law shape can produce 1000+ W/cm2 fluxes in this work at 
the given design point using the variable limits listed in Table 8.1. The increase in 
LV/D is based on the different curvature that is offered by the power law compared to 
the spherically-blunted cone and the spherical-segment. The iteration histories of the 
power law geometry are included in Figure A.0.2 and show that all of the design 
variables except for coefficient A are modified significantly.  
Three optimization runs of the spherical-segment, two of the spherically-
blunted cone, and eleven of the power law geometries were completed for 
maximizing ?VLV/D. Fewer optimization runs were seen as necessary since the initial 
designs for maximum LV/D had already been located. The calculated 
aerothermodynamic characteristics of the best configurations for each of the three 
axial shapes is shown for both objective functions in Table 8.3. Note that the initial 
designs for each optimization are written in the last lines of each table. The final 
results for the spherical-segment did not change between the two objective functions 
and has the highest LV/D equal to 1.10. Although the same is true for the power law 
axial shape, different local minima exist for the two objective functions, and so a 
different initial design for maximizing ?VLV/D was required to find the same 
minimum. The spherical-segment has the smallest total heat flux, followed by the 
spherically-blunted cone that is 14% greater and the power law that is five times 
larger. Note that the shock-standoff distance given for the two power law cases may 
be inaccurate on the scale of comparing them to the other two axial shapes. The 
power law case had a method written in this work that is meant to give rough answers 
 
 136 
 
rather than results precise enough to compare shapes with values closer than ? 100 
W/cm2 at the given design point. 
Table 8.3. Aerothermodynamic comparison of LV/D and ?VLV/D results at M? = 32.8255 (m = 4). 
Maximum LV/D Maximum ?VLV/D 
 
SS* 
?s = 89o 
n2 = 1.3 
e = -0.968 
? = 17.6o 
SC 
?c = 59o 
rn/d=0.15 
n2 =1.3 
e =-0.968 
? = 15.1o 
PL 
b = 0.642 
A = 0.900 
n2=1.3 
e =-0.968 
?= 9.51o 
SS 
?s = 89o 
n2 = 1.3 
e = -0.968 
? = 17.6o 
SC 
?c = 56.7o 
rn/d=0.24 
n2 =1.57 
e =-0.968 
? = 20.0o 
PL 
b = 0.642 
A = 0.900 
n2=1.3 
e =-0.968 
?= 9.51o 
CL,V 0.262 0.196 0.155 0.262 0.235 0.155 
LV/D 1.10 0.946 2.10 1.10 0.752 2.10 
(LV/D)? -0.0184 -0.0233 -1.69 -0.0184 -0.582 -1.69 
Cm,cg,? (/rad) -0.0806 -0.163 -0.0377 -0.0806 -0.00096 -0.0377 
Cn,cg,? (/rad) 0.0302 0.0523 0.00652 0.0302 0.0778 0.00652 
Cl,cg,? (/rad) -0.0719 -0.0754 -0.0183 -0.0719 -0.119 -0.0183 
qs,tot (W/cm2) 491 666 3000 491 558 3000 
qs,conv (W/cm2) 304 555 2820 304 435 2820 
qs,rad (W/cm2) 187 110 180 187 123 180 
?so (cm) 11.0  3.3 10.3 11.0  4.2 10.3 
Xcp/l 0.423 0.636 0.562 0.423 0.618 0.562 
Xcg/l  0.641 0.723 0.683 0.641 0.705 0.683 
CG type** X75 X75 X75 X75 X75 X75 
?v 89.0% 83.6% 84.7% 89.0% 85.2% 84.7% 
 Initial designs 
 
?s = 85o 
n2 = 1.5 
e = -0.9 
? = 25o 
?c = 60o 
rn/d=0.25 
n2 =1.6 
e =-0.9 
? = 20o 
b = 0.80 
A = 1.5 
n2=1.8 
e =-0.01 
? = 20o 
?s = 85o 
n2 = 1.5 
e = -0.9 
? = 25o 
?c = 60o 
rn/d=0.25 
n2 =-1.6 
e =-0.9 
? = 20o 
b = 0.6 
A = 0.9 
n2 = 1.3 
e = -0.968 
? = 9.51o 
  *Axial shape key: SS: spherical segment, SC: spherically-blunted cone, PL: power law.  
** CG type key: UD: uniform density, X75: uniform density except Xcg,75 = 75% Xcg,UD. 
 
Additionally, the power law shape has a slope that is discontinuous at the nose, which 
could render an attached shock rather than a bow shock if it is not blunted. Resolving 
such effects would be a good topic for future work.  
 
 137 
 
These results can be compared to those of the parametric analysis in Table 
7.3. Of the spherically-blunted cone and power law geometries shown in Table 8.3, 
the better of the two of each are plotted in Figure 8.2 for comparison against Figure 
8.1(c).  
                    
a)  Spherically-blunted cone axial shape with LV/D = 0.946. 
 
        
b) Power law axial shape with LV/D = 2.10. 
Figure 8.2. Optimized geometries from Table 8.3. 
Once the shapes are visually compared, the reason for the power law having an 
extremely high convective heat transfer flux is due to its small nose radius. The 
results in Table 8.3 are consistent with this understanding.  
The iteration histories for the spherically-blunted cone axial shape with 
maximum ?VLV/D are included in Figure A.0.3. The iteration history of the objective 
function indicates an increase of more than 400% over the initial design. Although the 
design variables appear to barely change at first glance, an eccentricity change from    
?0.90 to ?0.968 changes the semi-major to semi-minor axis length ratio from two to 
four, which is a substantial change from an aerodynamic standpoint.  
To determine if Mach number independence existed for the aerodynamics, the 
spherically-blunted cone was run at initial designs of ?c = 60o, rn/d = 0.25, n2 =1.6, e 
=-0.9, ? = 20o for M? = 30 through 41 at an altitude of 200000 ft. For M? > 42, 
 
 138 
 
Tannehill?s correlations54 are out of their pressure and density ranges at 200000 ft, 
producing a limit on this code. The solution for maximizing LV/D is the same 
throughout the hypersonic freestream Mach number range 30 through 41. The 
spherically-blunted cone solution has an LV/D = 0.946 at ? = 15.1o and is included in 
Table 8.3. 
8.2. Maximizing (LV/D)/qs,tot 
Seven optimization runs of the spherical-segment, five of the spherically-
blunted cone, and seven of the power law geometries were completed for maximizing 
(LV/D)/qs,tot. Constraints G1 through G5 have been applied. Although different initial 
designs were tested in this analysis, the initial designs for the spherical-segment and 
spherically-blunted cone for this optimization are the same as that required for 
maximizing LV/D. The aerothermodynamic performance and stability characteristics 
of the heat shield shapes with maximum (LV/D)/qs,tot are included in Table 8.4, with 
the initial designs listed at the bottom.  
 The maximized (LV/D)/qs,tot  spherical-segment configuration is identical to its 
maximized (LV/D) configuration. Since the spherical-segment configuration is shown 
in Figure 8.1(c), the other two configurations are shown in Figure 8.3. The 
spherically-blunted cone cases have nearly identical LV/D, but the configuration in 
this optimization has a 27.4% lower qs,tot than the maximized (LV/D) configuration. 
The optimizer accomplishes this by increasing the nose-radius-to-base-diameter ratio 
and increasing angle of attack, as shown in the iteration histories included in Figure 
A.0.4. This configurations rn/d is 145% larger than that of the maximized (LV/D) 
 
 139 
 
configuration. The optimizer also found a slighter higher LV/D in this optimization, 
but they are within 8% of each other. 
Table 8.4. Aerothermodynamic comparison of (LV/D)/qs,tot results at M? = 32.8255 (m = 4). 
Maximum (LV/D)/qs,tot 
 
SS* 
?s = 89o 
n2 = 1.3 
e = -0.968 
? = 17.6o 
SC 
?c = 55o 
rn/d=0.367 
n2 =1.3 
e =-0.968 
? = 15.5o 
PL 
b = 0.426 
A = 0.900 
n2=1.3 
e =-0.968 
?= 14.3o 
CL,V 0.262 0.198 0.221 
LV/D 1.10 0.942 1.35 
(LV/D)?  (/rad) -0.0184 -0.118 0.0429 
Cm,cg,? (/rad) -0.0806 -0.141 -0.0463 
Cn,cg,? (/rad) 0.0302 0.0468 0.00627 
Cl,cg,? (/rad) -0.0719 -0.0682 -0.0134 
qs,tot (W/cm2) 491 483 916 
qs,conv (W/cm2) 304 355 600 
qs,rad (W/cm2) 187 129 316 
?so (cm) 11.0  4.6 36.8 
Xcp/l 0.423 0.591 0.490 
Xcg/l  0.641 0.679 0.630 
CG type** X75 X75 X75 
?v 89.0% 86.4% 84.6% 
 Initial designs 
 
?s = 85o 
n2 = 1.5 
e = -0.9 
? = 25o 
?c = 60o 
rn/d=0.25 
n2 =1.6 
e =-0.9 
? = 20o 
b = 0.5 
A = 1.0 
n2=1.4 
e =-0.9 
? = 10.0o 
  *Axial shape key: SS: spherical segment,  
SC: spherically-blunted cone, PL: power law.  
** CG type key: UD: uniform density,  
X75: uniform density except Xcg,75 = 75% Xcg,UD. 
 
 
 
 140 
 
                
a)  Spherically-blunted cone axial shape with LV/D = 0.942. 
       
b) Power law axial shape with LV/D = 1.35. 
Figure 8.3. Maximized (LV/D)/qs,tot  geometries from Table 8.4. 
The optimum power law geometry produced the largest LV/D of 1.35 and total heat 
flux of 916 W/cm2. The stagnation-point heat fluxes for these vehicles are within the 
feasibility range of modern materials such as the Crew Exploration Vehicle, which is 
being designed for 1000 W/cm2. The variance in shock-standoff distance shows the 
limitations of the method. It should not be used to compare power law and 
spherically-blunted cone axial shapes.  
A short analysis of the objective function  (LV/D)/(qs,tot)n was completed with 
n = 0.25, 0.50, 0.75, 1.0, 2.0, and 3.0 using the spherical segment. The results did not 
change except at n = 3.0. For this case, the LV/D was reduced from 1.10 to 0.54, and 
qs,tot  was slightly reduced from 493 to 479. Since there is no significant difference 
between the total heat flux values even though the lift-to-drag ratio was halved, it was 
decided that the n = 3.0 configuration had lower performance. Overall, the 
optimization is not sensitive to a change in n < 3.0, and when it is sensitive, there is 
evidence that it lowers aerodynamic performance. Since the proportions of convective 
and radiative heat flux to the total heat flux change with velocity, the optimization has 
been run over a range of Mach numbers to observe how much the configurations 
change. The spherically-blunted cone axial shape is chosen for this study since it is 
 
 141 
 
directly connected to the convective heat flux correlation used in this work. The rn/d 
generally decreases with an increase in Mach number as shown in Figure 8.4. This is 
expected since the proportion of radiative heat transfer to the total increases 
dramatically over this range and surpasses the convective have flux around Mach 40. 
While the total heat flux steadily increases with Mach number, the optimized 
configurations have LV/D values that increase and decrease. Table A.0.1 in the 
appendix provides complete configuration information for this Mach number study.    
 
   a) Heat transfer flux qs,tot results.                      b) LV/D results. 
Figure 8.4. Maximized (LV/D)/qs,tot  results for M? = 30 ? 41. 
8.3. Minimizing qs,tot 
Since qs,tot was the first objective function optimized, fifty-two optimization 
runs were completed of the spherical-segment to minimize qs,tot. Also, twelve runs of 
the spherically-blunted cone and six runs of the power law geometries were 
completed. Constraints G1 through G5 were applied. When changing m to vary the 
number of sides of the cross-section, the minimum qs,tot was nearly constant. For m = 
5, 6, and 7, the minimum total heat flux was between 375 and 390 W/cm2. The 
 
 142 
 
parallelogram m = 4 cross-section had a slightly lower minimum at 339 W/cm2, but 
the employed methods should not be taken as accurate to the extent of differentiating 
between shapes with ?50 W/cm2 of each other. Instead it is concluded that all these 
cases have approximately the same minimum within 10-15% of each other.  
Not all of the shapes with local minimum necessarily look similar to each 
other. As shown in Figure 8.5, the optimal solution for the parallelogram is concave 
while the hexagonal geometry is not although they both were run at the same initial 
designs. Still these shapes have similar stagnation-point heat transfer fluxes. For the 
optimized parallelogram, the radiative heat flux is 180 W/cm2 while the convective 
heat flux is 160 W/cm2 since the rn/d is much greater than one.  
                              
a) m = 4, ?s = 16.1o, n2 = 4.0, e = 0.968, ? = -12.6o.        b) m = 6, ?s = 15.9o, n2 = 1.67, e = 0.968,  
                               ? = -10.9o. 
Figure 8.5. Minimum qs,tot configurations, spherical-segment axial shape. 
 Table 8.5 provides the optimized results for parallelogram geometries with 
each of the three axial shapes. While LV/D is approximately the same for each axial 
 
 143 
 
shape, the vertical lift coefficient of the power law shape is three times greater than 
that of the other two. The spherical-segment and spherically-blunted cone results are 
similar because that they are similar shapes. 
Table 8.5. Aerothermodynamic comparison of qs,tot results at M? = 32.8255 (m = 4). 
Minimum qs,tot 
 
SS* 
?s = 16.1o 
n2 = 4.0 
e = 0.968 
? = -12.6o 
SC 
?c = 55o 
rn/d=2.0 
n2 =4.0 
e = 0.968 
? = -12.3o 
PL 
b = 0.485 
A = 7.71 
n2=1.56 
e = -0.0771 
? = -12.5o 
CL,V 0.113 0.117 0.347 
LV/D 0.215 0.212 0.211 
(LV/D)?  (/rad) -1.01 -1.01 -0.998 
Cm,cg,? (/rad) -0.0276 -0.0259 -0.103 
Cn,cg,? (/rad) 0.0173 0.0178 0.0553 
Cl,cg,? (/rad) -0.00264 -0.0021 0.00000235 
qs,tot (W/cm2) 339 393 349 
qs,conv (W/cm2) 160 152 154 
qs,rad (W/cm2) 179 242 195 
?so (cm) 9.90 19.5 12.4 
Xcp/l 0.641 0.644 0.661 
Xcg/l  0.664 0.596 0.589 
CG type** X75 X75 X75 
?v 40.2% 37.6% 44.0% 
 Initial designs 
 
?s = 25o 
n2 = 1.3 
e = -0.8 
? = -10o 
?c = 65o 
rn/d=1.183 
n2 =1.8 
e =-0.8 
? = -10o 
b = 0.35 
A = 7.0 
n2=1.5 
e =-0.10 
? = -12.0o 
  *Axial shape key: SS: spherical segment,  
SC: spherically-blunted cone, PL: power law.  
** CG type key: UD: uniform density,  
X75: uniform density except Xcg,75 = 75% Xcg,UD. 
 
 
 144 
 
The blunted cone has an ? = 14.4o compared to the 16.1o of the spherical-segment. 
The high value of rn/d indicates that this blunted cone is a disguised spherical-
segment. The iteration histories for the optimized spherical-segment are included in 
Figure A.0.5. The geometries with the spherically-blunted cone and power law axial 
shapes are shown in Figure 8.6. 
             
a) Spherically-blunted cone axial shape with qs,tot = 393 W/cm2. 
     
b) Power law axial shape with qs,tot = 349 W/cm2. 
Figure 8.6. Minimum qs,tot  geometries from Table 8.5. 
The trend that is suggested from this observation is that if a concave rounded-edge 
polygon and a rounded-edge polygon have the same LV/D, the vertical lift coefficient 
of the rounded-edge polygon is probably greater than that of its concave counterpart. 
 
 145 
 
Because this spherically-blunted cone has an rn/d = 2, it is a spherical-segment. One 
could approximate the Apollo CM spherical segment with a larger nose-radius-to-
diameter ratio rn/d = 1.1831 and a half-cone angle of 65o. The initial designs for the 
blunted cone are the dimensions of the Apollo CM except for the oblate eccentricity.  
The spherically-blunted cone axial shape allows the optimization to cover some of the 
spherical-segment shapes in its own design space. 
 The heat shield with the spherical-segment axial shape has the smallest total 
heat flux, but as mentioned before, all that can be concluded from these results is that 
the three axial shapes have approximately the same minimum. It would be expected 
that the spherical-segment axial shape would produce the smallest total heat flux 
since it can produce the highest nose radius in a region where convection easily 
dominates if the nose radius is not large. 
 Since the heat flux increases with an increase in velocity, the optimal 
geometry may change. To observe any geometric changes to the optimal 
configurations, optimization throughout the freestream re-entry Mach number range 
from 30 to 41 has been completed. It is expected for the radiative heat flux to begin to 
dominate as the freestream Mach number is increased from M? = 30. Because the 
radiative heat flux for blunt bodies increases with an increase in nose radius, 
according to the correlations, the nose radius of the optimized vehicle should 
generally drop when the freestream Mach number is increased. Figure 8.7 shows how 
the nose-radius-to-diameter radius drops with increasing M?, and complete geometric 
characteristics of the optimized configurations are listed in Table A.0.2. For this 
objective function at the given initial designs, the major drop in nose radius occurs 
 
 146 
 
between M? = 33 and 34. In this case, it could be that the spherically-blunted cone is a 
spherical-segment at M? = 33 that changes into a spherically-blunted cone at M? = 34.  
 
Figure 8.7. Minimized qs,tot  results for M? = 30 ? 41. 
It can be concluded that the spherically-blunted cone would be the choice for 
missions with highest heat fluxes present at an altitude of 200000 ft moving at M? ? 
34. At this altitude, the Apollo CM was moving at  M? = 32.8255, and strictly from a 
heat transfer standpoint, the results of this optimization are consistent with the 
decision to use a spherical-segment for Project Apollo.     
8.4. Minimizing Cm,cg,? 
Nineteen optimization runs of the power law, six of the spherical-segment, 
and three of the spherically-blunted cone were completed for minimizing the pitching 
moment derivative Cm,cg,?. If the blunt-body is entering an atmosphere at angle of 
attack, then the magnitude of the pitching moment derivative is probably the largest 
out of the three stability derivatives. In this case, minimizing Cm,cg,? generates 
 
 147 
 
negative values with large magnitudes, and this corresponds to the most statically 
stable configuration in the longitudinal direction. Constraints G1 through G5 have 
been applied. Since the pitching moment stability requirement constraint G3 is being 
used, MMFD requires the initial configuration to be longitudinally stable. As a result, 
Cm,cg,? is already negative before the optimization begins, and so the optimizer is 
searching for the stable configuration with a larger |Cm,cg,?|.  
The three configurations that are the most statically stable longitudinally are 
shown in Figure 8.8.  
       
a) Power law axial shape                         b) Spherical-segment axial shape 
 
 
c) Spherically-blunted cone axial shape 
Figure 8.8. Minimum Cm,cg,? geometries (m = 4) in Table 8.6. 
The optimized power law geometry has a pointed nose, and the heat flux constraint is 
active. The spherically-blunted cone has a smaller nose radius than the spherical-
segment. Neither the blunt cone nor the power law have the eccentricity of the 
spherical-segment. The aerothermodynamic performance and stability characteristics 
of these shapes are listed in Table 8.6. All three shapes have a minimum value of 
 
 148 
 
approximately ?0.30/rad for the pitching moment stability derivative, and the power 
law shape has value that is at least 27% larger in magnitude than the other two 
geometries.  
Table 8.6. Aerothermodynamic comparison of Cm,cg,? results at M? = 32.8255 (m = 4). 
Minimum Cm,cg,? 
 
SS* 
?s = 47.9o 
n2 = 2.24 
e = -0.64655 
? = -2.68o 
SC 
?c = 55o 
rn/d=0.403 
n2 =2.25 
e = 0.0149 
? = -0.656o 
PL 
b = 0.78 
A = 0.90 
n2=2.28 
e = 0.119 
? = 26.8o 
CL,V 0.0269 0.00918 0.0139 
LV/D 0.0219 0.00705 0.0184 
(LV/D)?  (/rad) -0.452 -0.474 0.379 
Cm,cg,? (/rad) -0.285 -0.276 -0.363 
Cn,cg,? (/rad) 0.165 0.138 0.00369 
Cl,cg,? (/rad) 0.00170 -1.6E-5 0.0000888 
qs,tot (W/cm2) 452 469 3000 
qs,conv (W/cm2) 262 338 2890 
qs,rad (W/cm2) 190 131 110 
?so (cm) 11.3 4.87 3.56 
Xcp/l 0.617 0.622 0.636 
Xcg/l  0.656 0.669 0.707 
CG type** X75 X75 X75 
?v 85.9% 79.8% 93.2% 
 Initial designs 
 
?s = 85o 
n2 = 2.0 
e = -0.8 
? = 25o 
?c = 60o 
rn/d=0.25 
n2 =1.6 
e =0.01 
? = -20o 
b = 0.75 
A = 2.0 
n2=2.0 
e = 0.5 
? = -20o 
*Axial shape key: SS: spherical segment,  SC: spherically-blunted cone, PL: power law.  
** CG type key: UD: uniform density, X75: uniform density except Xcg,75 = 75% Xcg,UD. 
The value of ?0.30/rad is approximately twice the pitching stability derivative of the 
Apollo Command Module.8 This can also be noticed when comparing the results of 
 
 149 
 
this work in Table 7.3. The optimum angle of attack for each is near zero and due to 
the fact that these shapes are symmetric about the horizontal body axis, the lift-to-
drag ratio is nearly zero.  
Similar to the Apollo Command Module, the spherical-segment has a slightly 
unstable rolling moment. This is allowed in the constraints on purpose in order to 
allow the optimizer to choose heat shields similar to the Apollo Command Module?s 
if desired. The Apollo CM is estimated to have had a Cl,cg,? of 0.005/rad (discussed in 
Chapter 6). 
The iteration histories for the optimized power law configuration are included 
in Figure A.0.6. The optimizer produced a geometry with greater stability by 
decreasing coefficient A and the eccentricity, and greatly decreasing the magnitude of 
? to 0o. The iteration histories for the spherical-segment configuration with maximum 
longitudinal static stability are included in Figure A.0.7. The optimizer decreased the 
magnitude of ? and ?s dramatically and originally changed the eccentricity greatly, 
but then it brought the eccentricity back to nearly its original value.  
Most of the results from optimizing other objective functions have magnitudes 
of Cm,cg,? less than 0.10/rad, and only some of the spherically-blunted cone and power 
law results for other objective functions have magnitudes greater than 0.10/rad. The 
maximum ?VLV/D blunted cone configuration is neutrally longitudinally stable. As a 
result, stability has to be traded-off with LV/D and qs,tot to produce an optimal-set 
geometry.  
The configurations with optimal sets of aerothermodynamic characteristics 
have cross-sections that are rounded-edge parallelograms and are generated by 
 
 150 
 
maximizing (LV/D)/qs,tot. These configurations have the characteristics listed in Table 
8.4. All of them have high lift-to-drag ratios (above 0.5) for space capsules, total heat 
fluxes less than or at most equal to the design requirement for NASA?s CEV of 1000 
W/cm2, and stable stability derivatives. The spherically-blunted cone configuration 
has a pitching stability derivative value of ?0.14/rad that is near the ?0.16/rad of the 
Apollo CM, but the required magnitudes for this and the other stability derivatives 
have yet to be determined. Most of this optimization analysis is included concisely in 
Ref. [73]. 
8.5. Limitations of this Optimization  
Several optimization runs were completed since different minima were 
encountered with different initial designs. The design space does not have only one 
minimum for each objective function. These optimization runs found local minimum, 
and the one that had the lowest value is identified as the engineering global minimum. 
The design space has several minima. For example, if the initial design for 
maximizing LV/D for the spherical segment were changed from ?s = 85o, n2 = 1.5, e = 
-0.9, and ? = 25o to ?s = 25o, n2 = 1.4, e = -0.01, and ? = -10o, the optimizer finds a 
local optimum value of 0.320 instead of 1.10 for LV/D. If the optimizer runs without 
constraint vector Gr , then it finds a value of 0.575 instead of 1.10. It is understandable 
for the design space to have more than one maximum, but one has to accept that there 
could be a better optimum in the design space that was not found. There are other 
optimization methods for finding the global maximum, but it is unknown to the 
author whether those methods would provide better results. 
 
 151 
 
In finding the maximum LV/D, the optimizer had to be given initial designs 
based on the parametric analysis in order to find the global minimum. The parametric 
analysis gave the best starting point for the optimizer in maximizing LV/D by setting 
some of the initial designs equal to the maximum in Figure 7.9. Then the optimizer 
was able to find the maximum. This exemplifies the importance of a well-constructed 
parametric analysis.  
In a few cases, the optimizer did not vary one or two of the design variables. 
In maximizing ?VLV/D, there was no variance in angle of attack for the spherically-
blunted cone case. This work uses a semi-empirical method for varying the radiative 
heat flux with angle of attack, and this method is modified to account for n2 and 
eccentricity, as well as the power law and blunted cone axial shapes. It is shown in 
Figure 7.11 that this modified method produces several ripples in the design space. 
Aside from the ripples, it is also nearly flat for n2 < 2, and it is believed that these are 
the reasons that the optimizer does not find an optimum angle of attack far from the 
initial angle of attack when minimizing qs,tot. Better optima may be found in future 
work.  
 
 
 152 
 
 
Chapter 9. Conclusions 
9.1. Summary of Results 
9.1.1. Parametric Analysis 
  A parametric analysis of candidate blunt-body heat shield shapes for a Crew 
Exploration Vehicle has been conducted to determine the main effects of several 
geometric parameters on the aerothermodynamic performance and stability of the 
vehicle. This analysis is completed by picking a cross-section and an axial shape to 
generate a heat shield mesh, and then by placing the mesh into an aerodynamics 
program based on modified Newtonian flow. The results were validated with wind 
tunnel and flight test data, and Cm,cg and L/D matched the values of the Apollo 
Command Module (CM) closer than the results of CL and CD.  
 Although the hexagon is the most aerodynamically stable of the polygons (m 
= 4?10), it did not have the highest LV/D, and LV/D became larger as n2 approaches 
the elliptical cross-section (n2 = 2). There is a tradeoff between LV/D and the pitching 
moment stability derivative. Although the increase in LV/D of 56.1% may be 
beneficial by making the cross-section strongly oblate at e = -0.95, the heat shield 
would be longitudinally statically unstable assuming uniform density. One way to 
remove this instability is by moving the center of gravity forward by 23.5% to Xcg/l = 
0.400, and moving the x-location of the center of gravity forward by 25% (was 
applied in some of the optimization runs to further open up the design space). 
 
 153 
 
 A change in cross-section did not affect the aerodynamic performance of the 
heat shield as significantly as a change in axial shape. A change in axial shape 
rendered a larger range of aerodynamic characteristics. It was also observed that the 
magnitudes of the longitudinal and directional stability derivatives are maximized 
with an elliptical base as opposed to a rounded-edge hexagonal concave base. Table 
7.3 summarized the sensitivities of the aerodynamic parameters to the geometric 
parameters. Cm varied strongly with ?, ?s, ?c, and A while the other moment 
coefficients and stability derivatives varied strongly with ?s, ?c, and A. The results of 
Cl,? are of the same order of magnitude as other blunt-body designs. For ?s < 35o, an 
increase in eccentricity from 0.0 to 0.95 increases the volumetric efficiency by up to 
10%.  
9.1.2. Optimization Results 
Incorporating all of these parameters into an optimization process has assisted 
in producing heat shield shapes with improved combinations of aerodynamic 
characteristics. By maximizing LV/D it was noticed that the parallelogram base cross-
section (m = 4) provides the highest LV/D. Because this is one of the two most 
important characteristics, the other being qs,tot, the optimization continued with m = 4 
rather than at the m = 6 that produced the most longitudinally stable configurations. 
Minimizing the heat transfer flux qs,tot  produced polygon bases with and 
without concavity, and the trend was noticed that even though two shapes may have 
the same LV/D, they do not necessarily have the same CL,V. In this case, the rounded-
edge polygon had a greater vertical lift coefficient by 300%.    
 
 154 
 
The results of maximizing LV/D or minimizing qs,tot, or Cm,cg,? was not as 
promising as maximizing a combinations of these parameters. Each of these has 
trade-offs with the other. The maximum LV/D was usually located between an |?| of 
20o and 30o while the minimum Cm,cg,? would be present near zero angle of attack. 
Most of the optimum heat shields shapes are symmetric about the horizontal axis, and 
thus produce no lift at zero angle of attack, rendering a low lift-to-drag ratio. The 
minimum qs,tot configurations all had LV/D ? 0.2, which is lower than the Apollo 
CM?s 0.34. The maximum LV/D would produce high qs,tot especially for the 
spherically-blunted cone and power law axial shapes. However, by choosing to 
maximize (LV/D)/qs,tot, the optimizer was able to produce a better, more optimal set of 
aerothermodynamic characteristics. In this case, the geometries with rounded-edge 
parallelogram cross-sections and maximum (LV/D)/qs,tot provide the most desired set 
of characteristics (listed in Table 8.4) for the design point altitude of 200000 ft and 
M? = 32.8255.  
Since the heat transfer increases with freestream Mach number, an analysis 
was conducted on how the optimum shape changes for M? = 30 ? 41. When 
minimizing qs,tot by itself, a large drop in the optimum rn/d ratio occurs between Mach 
33 and Mach 34. The value of rn/d at Mach 33 generates a spherical-segment while 
the value at Mach 34 generates a spherically-blunted cone, and this drop shows at 
which Mach number it would be ideal to switch between the two. Since the Apollo 4 
CM was traveling at M? = 32.8255, and the manned Apollo Command Modules 
traveled slower at this altitude, this research agrees with the choice to use a spherical-
segment instead of a spherically-blunted cone for Project Apollo from a heat transfer 
 
 155 
 
standpoint. However, when maximizing (LV/D)/qs,tot  over this range of freestream 
Mach numbers, all the optimum results suggest a spherically-blunted cone over the 
spherical-segment. This renders a higher qs,tot than produced during the Apollo 4 
trajectory and may have created a technology difficulty at the time.  
Due to the fact that the design space had several local minima, perhaps 
another optimization method would have been better suited for locating a global 
minimum. Since several optimization runs have been completed, it is certain that 
these results are near the value of the global minimum, but there is the possibility that 
the true global minimum has not been found. However, several optimal heat shield 
designs with improved characteristics over previous work have been discovered.    
9.2. Suggestions for Future Work 
A substantial amount of future work in planetary entry vehicles can be completed 
to improve upon both this work and the general understanding of the field. To 
increase the accuracy of the aerothermodynamics code, the following additions could 
be made: 
? Account for corner radius. The work of Zoby44 may also assist in 
distinguishing heat shields of the same axial shape and cross-section but 
different corner radii. The corner radius geometry can also be added to the 3D 
mesh so that the surface pressure distribution accounts for it.   
? Improve the method for estimating the shock-standoff distance for angles of 
attack, especially for the power law axial shape and in accounting for base 
eccentricity.  
 
 156 
 
? Use a more accurate method to determine the surface pressure distribution. 
For this work Newtonian Impact Theory was validated, but if the scope of the 
project were to include analyzing the boundary layer, advanced CFD would 
need to be utilized. 
To increase the scope of the aerothermodynamics code, the following additions could 
be made: 
? Include a second-order trajectory for determining non-oscillatory trajectories. 
Loh74 offers a way to modify second-order trajectory theory to approximate 
oscillatory trajectories. A second-order trajectory code does not require the 
computational time of full trajectory packages such as POST or OTIS. As a 
result, the heat transfer correlations can be applied at each point in the 
trajectory to approximate the heat transfer load on the vehicle. 
? Incorporate more axial shapes: the raked, biconic, and bent-biconic cones.  
? Include thermal material properties and temperature constraints.  
? Include a model for high temperature gas properties at 200000 ft, M? > 42. 
The Tannehill correlations are outside their range for these freestream Mach 
numbers.  
? Develop a method of determining the point of maximum heating for a general 
3D body. The velocity gradient could be modeled to determine the point of 
maximum convective heating for a general 3D body. Zoby44 provides some 
results based on the change in the velocity gradient in order to calculate the 
convection for values of rn/d > 2 more accurately than using the inverse 
square-root of the nose radius.    
 
 157 
 
? Validation of the stagnation-point radiative heat transfer curve-fits of Suttles75 
for hyperbolic earth entry with velocities from 11 to 16 km/s. 
Future work concerning planetary entry vehicles outside this computational code that 
would benefit the field: 
? Study laminar, transition, and turbulent boundary layer heat transfer on blunt 
bodies for M? > 25. Wind tunnel analysis of rounded-edge concave heat 
shields would help in investigating these phenomena. The Apollo Command 
Module?s heat shield experienced laminar heat transfer. Turbulent heat 
transfer is unknown at M? > 40, especially the effects of radiation cooling and 
convective-radiative coupling that reduce the total heat transfer flux. For a 
phenomena that increases exponentially with Mach number, the radiative heat 
transfer can easily be miscalculated for M? > 40.   
? Static and dynamic stability guidelines for human space capsules.  
? Flight tests and wind tunnel experiments for M? > 40 ranging up to M?  = 55. 
This could provide arguably the most useful results concerning planetary entry 
at M? > 40. Current aerothermodynamic understanding of this region of 
freestream Mach numbers is modest. Since rolling moment experimental and 
flight data is almost nonexistent, measuring the rolling moment on both 
axisymmetric and non-axisymmetric heat shields would be beneficial. High 
temperature correlations, radiation and general heat transfer models could be 
produced for the first time. Hypersonic aerothermodynamic models can be 
improved. Some models do exist, but they do not apply well to M? > 40.  
 
 158 
 
 Appendix A: Optimization Iteration Histories 
A.1 Maximizing LV/D and ?VLV/D 
  
a) Objective function iteration history.          b) Design variable iteration history 
 
 
c) Constraints iteration history. 
Figure A.0.1. Iteration history for maximum LV/D hexagonal shape (m = 6) in Table 8.2. 
 
 
 159 
 
 
a) Objective function iteration history. 
 
b) Design variable iteration history 
 
c) Constraints iteration history. 
Figure A.0.2. Iteration history for maximum LV/D power law/parallelogram (m = 4) in Table 8.3. 
 
 
 
 160 
 
 
 
a) Objective function iteration history. 
 
b) Design variable iteration history 
 
c) Constraints iteration history. 
Figure A.0.3. Iteration history for maximum ?VLV/D spherically-blunted cone/parallelogram (m 
= 4) in Table 8.3. 
 
 161 
 
A.2   Maximizing (LV/D)/qs,tot 
 
a) Objective function iteration history.       b) Design variable iteration history. 
 
c) Constraints iteration history. 
Figure A.0.4. Iteration history for maximum (LV/D)/qs,tot spherically-blunted cone axial shape (m 
= 4) in Table 8.4. 
 
 162 
 
Table A.0.1. Optimized (LV/D)/qs,tot for M? from 30 to 41,  m = 4. 
Spherically-blunted Cone Axial Shape 
Initial Design: ?c = 60o, rn/d=0.25, n2 =1.6, e =-0.9, ? = 20o 
Optimal Design 
(all cases have ?c = 55o,  
e = -0.968, n2 = 1.3) 
Split Objective Function  Objective Function 
M? rn/d ? LV/D qs,tot (W/cm
2) 
(qs,rad, qs,conv ) 
(LV/D)/qs,tot 
(cm2/W) 
30 0.379 20.5o 0.872 280 (17.0, 263) 0.00311 
31 0.327 14.8o 0.986 359 (44.9, 314) 0.00275 
32 0.292 15.2o 1.02 452 (85.4, 367) 0.00226 
33 0.316 16.8o 0.984 543 (154, 389) 0.00181 
34 0.184 16.1o 1.10 738 (179, 559) 0.00149 
35 0.150 13.6o 1.14 911 (232, 679) 0.00125 
36 0.206 20.2o 0.998 1015 (381, 633) 0.000983 
37 0.188 20.1o 1.01 1200 (480, 720) 0.000842 
38 0.169 13.5o 1.13 1430 (600, 830) 0.000790 
39 0.150 13.8o 1.14 1670 (710, 960) 0.000683 
40 0.150 13.6o 1.14 1910 (880, 1030) 0.000597 
41 0.150 13.5o 1.14 2170 (1050, 1120) 0.000525 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 163 
 
 
A.3  Minimizing qs,tot 
 
 
a) Objective function iteration history.            b) Design variable iteration history 
 
 
c) Constraints iteration history. 
Figure A.0.5. Iteration history for minimum qs,tot spherical-segment axial shape (m = 4) in Table 
8.5. 
 
 164 
 
Table A.0.2. Optimized qs,tot for M? from 30 to 41,  m = 4. 
Spherically-blunted Cone Axial Shape 
Initial Design: ?c = 65o, rn/d=1.1831, n2 =1.8, e =-0.8, ? = -10o 
Optimal Design   Objective Function 
M? ?c rn/d n2 e ? LV/D qs,tot (W/cm
2) 
(qs,rad, qs,conv )  
30 55o 2.00 4.00 0.968 -11.7o 0.201 149 (35.1, 114) 
31 55o 2.00 4.00 0.968 -12.4o 0.213 207 (80.2, 127) 
32 55o 2.00 4.00 0.968 -12.4o 0.213 298 (158, 140) 
33 75.6o 1.80 4.00 0.968 -11.9o 0.204 326 (163, 163) 
34 55o 0.593 2.06 -0.171 -10.5o 0.122 596 (285, 312) 
35 55o 0.615 2.81 0.356 -10.3o 0.129 735 (400, 335) 
36 55o 0.384 1.98 -0.223 -10.2o 0.0992 896 (432, 464) 
37 55o 0.368 1.83 -0.013 -10.3o 0.0978 1073 (556, 517) 
38 55o 0.332 1.84 -0.009 -10.3o 0.0956 1279 (688, 592) 
39 55o 0.313 2.65 0.041 -10.6o 0.115 1456 (795, 661) 
40 55o 0.344 2.61 0.156 -10.3o 0.112 1680 (997, 683) 
41 55o 0.292 2.74 0.073 -12.0o 0.130 1909 (1108, 801) 
 
 
 165 
 
 
A.4   Minimizing Cm,cg,? 
    
a) Objective function iteration history.      
      
b) Design variable iteration history 
 
c) Constraints iteration history. 
Figure A.0.6. Iteration history for minimum Cm,cg,? power law axial shape (m = 4) in Table 8.6. 
 
 166 
 
 
 
 
a) Objective function iteration history. 
 
b) Design variable iteration history 
 
c) Constraints iteration history. 
Figure A.0.7. Iteration history for minimum Cm,cg,? spherical-segment axial shape (m = 4) in 
Table 8.6. 
 
 167 
 
 
 
Bibliography 
  
 
1 Crowder, R. S. and Moote, J. D., ?Apollo Entry Aerodynamics,? Journal of 
Spacecraft and Rockets, Vol. 6, No. 3, 1969, pp. 302-307.  
 
2 Kirk, D. B., Intrieri, P. F., and Seiff, A., ?Aerodynamic Behavior of the Viking 
Entry Vehicle: Ground Test and Flight Results,? Journal of Spacecraft and Rockets, 
Vol. 15, No. 4, 1978, pp. 208-212.  
3  Chrusciel, G. T. and Hull, L. D., ?Theoretical Method for Calculating Aerodynamic 
Characteristics of Spherically Blunted Cones,? AIAA Paper 1968-674, Jun. 1968. 
4 Faget, M. A., Garland, B. J., and Buglia, J. J., ?Preliminary Studies of Manned 
Satellites, Wingless Configuration: Nonlifting,? NACA RM L58E07a, 1958. 
 
5  Kruse, R. L., Malcolm, G. N., and Short, B. J., ?Comparison of Free-Flight 
Measurement of Stability of the Gemini and Mercury Entry Capsules at Mach 
Numbers 3 and 9.5,? NASA TM-X-957, 1964. 
6 Hillje, Ernest R., ?Entry Flight Aerodynamics from Apollo Mission AS-202,? 
NASA TN D-4185, 1967. 
 
7 Hillje, Ernest R., ?Entry Aerodynamics at Lunar Return Conditions Obtained from 
the Flight of Apollo 4 (AS-501),? NASA TN D-5399, 1969. 
 
8 Moseley, W. C., Jr., Moore, R. H., Jr., and Hughes, J. E., ?Stability Characteristics 
of the Apollo Command Module,? NASA TN D-3890, 1967.  
 
9 Moseley, W. C., Jr., Graham, R. E., and Hughes, J. E., ?Aerodynamic Stability 
Characteristics of the Apollo Command Module,? NASA TN D-4688, 1968. 
 
10 DeRose, Charles E., ?Trim Attitude, Lift and Drag of the Apollo Command 
Module with Offset Center-of-Gravity Positions at Mach Numbers to 29,? NASA TN 
D-5276, 1969. 
 
11 Horstman, C. C., and Kussoy, M. I., ?Free-flight Measurements of Aerodynamic 
Viscous Effects on Lifting Re-Entry Bodies,? Journal of Spacecraft and Rockets, Vol. 
4, No. 8, 1967, pp. 1064-1069. 
 
 
 
 168 
 
 
12 Jones, Robert A., Hunt, James L., ?Measured Pressure Distributions on Large-
Angle Cones in Hypersonic Flows of Tetrafluoromethane, Air and Helium,? NASA 
TND-7429, Dec 1973.  
  
13 Tauber, M., Chargin, M., Henline, W., Chiu, A., Yang, L., Hamm, K. R., Jr., 
Miura, H., ?Aerobrake Design Studies for Manned Mars Missions,? Journal of 
Spacecraft and Rockets, Vol. 30, No. 6, 1993, pp. 656-664. 
 
14 Bernot, Peter T., Jordan, Bruce C., ?Static Stability Charcteristis of Cone and Half-
Cone Pyramid Configurations at Mach 6.83,? NASA TND-3544, Aug 1966. 
 
15 Sammonds, R. I. and Kruse, R. L., ?Viking Entry Vehicle Aerodynamic at M = 2 in 
Air and Some Preliminary Test Data for Flight in CO2 at M = 12,? NASA TN D-
7974, 1975. 
 
16 Willcockson, W. H., ?Mars Pathfinder Heatshield Design and Flight Experience,? 
Journal of Spacecraft and Rockets, Vol. 36, No. 3, 1999, pp. 374-379. 
 
17 Moss, J. N., Blanchard, R. C., Wilmoth, R. G., Braun, R. D., ?Mars Pathfinder 
Rarefied Aerodynamics: Computations and Measurements,? AIAA Paper 1998-298, 
Jan. 1998. 
 
18 Milos, F. S., Chen, Y. K., ?Mars Pathfinder Entry Temperature Data, Aerothermal 
Heating, and Heatshield Material Response,? AIAA Paper 1998-2681, Jun. 1998. 
 
19 Ezell, Edward Clinton, and Linda Neuman Ezell. On Mars: Exploration of the Red 
Planet, 1958-1978. NASA SP-4212, 1984. 
 
20 Shapland, D. J., Price, D. A., Hearne, L. F., ?A Configuration for Re-Entry from 
Mars Missions Using Aerobraking,? Journal of Spacecraft and Rockets, Vol. 2, No. 
4, 1965, pp. 501-507. 
 
21 Micol, J. R. and Wells, W. L., ?Hypersonic Lateral and Directional Stability 
Characteristics of Aeroassist Flight Experiment Configuration in Air and CF4,? 
NASA TM 4435, 1993.  
 
22 Wells, William L., ?Measured and Predicted Aerodynamic Coefficients and Shock 
Shapes for Aeroassist Flight Experiment (AFE) Configuration,? NASA TP-2956, 
1990. 
 
23 Micol, John R., ?Simulation of Real-Gas Effects on Pressure Distributions for 
Aeroassist Flight Experiment Vehicle and Comparison With Prediction,? NASA TP-
3157, Apr 1992. 
 
 
 
 169 
 
 
24 Davies, Carol B., Park, Chul, ?Aerodynamic Characteristics of Generalized Bent 
Biconic Bodies for Aero-Assisted Orbital-Transfer Vehicles,? NASA TM 84362, 
May 1983. 
 
25 Davies, Carol B., Park, Chul, ?Optimum Configuration of High-Lift 
Aeromaneuvering Orbital Transfer Vehicles in Viscous Flow,? Journal of Spacecraft 
and Rockets, Vol. 25, No. 3, Jun 1988. 
 
26 Magazu, Harry K., Lewis, Mark J., and Akin, David L., ?Aerothermodynamics of a 
Parashield Re-Entry Vehicle,? Journal of Spacecraft and Rockets, Vol. 35, No. 4, 
1998, pp. 434-441. 
 
27 Murbach, Marcus S., ?SCRAMP: The Development of an Advanced Planetary 
Probe from CFD to Re-entry Test Flight,? 3rd International Planetary Probe 
Workshop, Athens, Greece, June 27-July 1, 2005.  
 
28 Whitmore, Stephen A., Banks, D., Andersen, B., and Jolley, P., ?Direct-Entry, 
Aerobraking, and Lifting Aerocapture for hnuman-Rated Lunar Return Vehicles,? 
AIAA Paper 2006-1033, AIAA Aerospace Sciences Meeting, Jan 2006. 
 
29 Bertin, J., Hypersonic Aerothermodynamics, AIAA Education Series, AIAA, New 
York, 1993. 
 
30 Rasmussen, M., Hypersonic Flow, John Wiley & Sons, Inc. New York, 1994. 
 
31 Park, C., Davies, C. B., ?Aerothermodynamics of Manned Mars Missions,? AIAA 
Paper 1989-0313, Jan. 1989. 
 
32 Rochelle, W. C., Bouslog, S. A., and Ting, P. C., ?Aerothermodynamic 
Environments for Mars Entry, Mars Return, and Lunar Return Aerobraking 
Missions,? AIAA Paper 1990-1791, Jun 1990.  
 
33 Rose, P. H., Stark, W. I., ?Stagnation Point Heat-Transfer Measurements in 
Dissociated Air,? Journal of the Aeronautical Sciences, Vol. 25, No. 1, Feb 1958. 
 
34 Kemp, Nelson H., Rose, P. H., Detra, R. W., ?Laminar Heat Transfer Around Blunt 
Bodies in Dissociated Air,? Journal of the Aero/space Sciences, Vol. 26, No. 7, Jul 
1959, pp 421-430.  
 
35 Lee, D. B., Bertin, J. J., Ried, R. C., ?Apollo Reentry Heating,? NASA TM X-
66780, Sept 1963. 
 
36 Lee, D. B., Goodrich, W. D., ?The Aerothermodynamic Environment of the Apollo 
Command Module During Superorbital Entry,? NASA TN D-6792, Apr 1972. 
 
37 Lee, D. B., ?Apollo Experience Report ? Aerothermodynamics Evaluation,? NASA 
TN D-6843, Jun 1972.  
 
 
 170 
 
 
 
38 Pavlosky, J. E., Leger, L. G. St., ?Apollo Experience Report ? Thermal Protection 
Subsytem,? NASA TN D-7564. 
 
39 Park, C., Tauber, M. E., ?Heatshielding Problems of Planetary Entry, A Review,? 
AIAA Paper 1999-3415, Jun 1999. 
 
40 Scotti, S. J., ?Current Technology for Thermal Protection Systems,? NASA CP-
3157, Feb 1992. 
 
41 Lovelace, Uriel M., ?Charts Depicting Kinematic and Heating Parameters for a 
Ballistic Reentry at Speeds of 26,000 to 45,000 Feet Per Second,? NASA TN D-968, 
Oct 1961. 
 
42 Tauber, M. E., and G. P. Menees, ?Aerothermodynamics of Transatmospheric 
Vehicles,? AIAA Journal of Aircraft, Vol. 24, No. 9, 1987, pp. 594-602. 
 
43 Fay, J. A., Riddell, F. R., ?Theory of Stagnation Point Heat Transfer in Dissociated 
Air,? Journal of Aeronautical Sciences, Vol. 25, No. 2, Feb 1958. 
 
44 Zoby, E. V., Sullivan, E. M., ?Effects of Corner Radiuus on Stagnation-Point 
Velocity Gradients on Blunt Axisymmetric Bodies,? NASA TM X-1067, Mar 1965. 
 
45 Tauber, M. E., Sutton, K., ?Stagnation-Point Radiative Heating Relations for Earth 
and Mars Entries,? AIAA Journal of Spacecraft & Rockets, Vol. 28, No. 1, 1991, pp. 
40-42. 
 
46 Park, C., ?Stagnation-Point Radiation for Apollo 4 ? A Review and Current 
Status,? AIAA Paper 2001-3070, Jun 2001. 
 
47 Cauchon, D. L., ?Radiative Heating Results form the FIRE II Flight Experiment at 
a Reentry Velocity of 11.4 Kilometers per Second,? NASA TM X-1402, Jul 1967. 
 
48 Ried, Jr., R. C., Rochelle, W. C., Milhoan, J. D., ?Radiative Heating to the Apollo 
Command Module: Engineering Prediction and Flight Measurement,? NASA TM X-
58091, Apr 1972. 
 
49 Suttles, J. T., Sullivan, E. M., Margolis, S. B., ?Curve Fits of Predicted Inviscid 
Stagnation-Point Radiative Heating Rates, Cooling Factors, and Shock Standoff 
Distances for Hyperbolic Earth Entry,? NASA TN D-7622, Jul 1974. 
 
50 Walters, Edward E., ?Free-Flight Measurements of Radiative Heating to the Front 
Face of the Apollo Reentry Capsule as a Function of Angle of Attack,? NASA TM X-
851, Feb 1964. 
 
51 Kaattari, George E., ?Shock Envelopes of Blunt Bodies at Large Angles of Attack,? 
NASA TN D-1980, Dec 1963. 
 
 
 171 
 
 
 
52 Kaattari, George E., ?A Method for Predicting Shock Shapes and Presssure 
Distributions for a Wide Variety of Blunt Bodies at Zero Angle of Attack,? NASA 
TN D-4539, Apr 1968. 
 
53 Srinivasan, S., Tannehill, J. C., Weilmuenster, K. J., ?Simplified Curve Fits for the 
Thermodynamic Properties of Equilibrium Air,? NASA-CR-181245, Jun 1986. 
 
54 Tannehill, J. C., and P. H. Mugge, ?Improved Curve Fits for the Thermodynamic 
Properties of Equilibrium Air Suitable for Numerical Computation Using Time-
Dependent or Shock-Capturing Methods,? NASA CR-2470, Oct 1974. 
 
55 Regan, F., Re-Entry Vehicle Dynamics, 1st ed., AIAA Education Series. AIAA, 
New York, 1994, Chap. 9.  
 
56 Levine, P., Ellis, T.R., Georgiev, S., ?Factors Influencing the Design and 
Performance of Mars Entry Vehicles,? Journal of Spacecraft and Rockets, Vol. 2, No. 
2, 1965, pp. 130-146. 
 
57 Arora, R. K., Kumar, P., ?Aerodynamic Shape Optimization of a Re-entry 
Capsule,? AIAA Paper 2003-5394, AIAA Atmospheric Flight Mechanics 
Conference, Aug 2003.   
 
58 Papadopoulos, Periklis and Prabhakar Subrahmanyam, ?Computational 
Investigation and Simulation of the Aerothermodynamics of Reentry Vehicles,? 
AIAA 2005-3206, AIAA/CIRA 13th International Space Planes and Hypersonics 
Systems and Technologies, May 2005. 
 
59 Papadopoulos, Periklis and Prabhakar Subrahmanyam, ?Trajectory Coupled 
Aerothermodynamics Modeling for Atmospheric Entry Probes at Hypersonic 
Velocities,? AIAA 2006-1034, 44th AIAA Aerospace Sciences Meeting and Exhibit, 
Jan 2006. 
 
60 Barter, Neville J., Space Data, Northrop Grumman Space Technology, 5th ed, Mar 
2003, Section 7. 
 
61 Gupta, R. N., Jones, J. J., Rochelle, W. C., ?Stagnation-Point Heat-Transfer Rate 
Predictions at Aeroassist Flight Conditions,? NASA TP-3208, Sept 1992.  
 
62 Sabean, John W., ?Optimization of a Hypersonic Inlet with a Rectangular to 
Circular Transition Duct,? Ph.D. Dissertation UM-AERO 99-09, Department of 
Aerospace Engineering, University of Maryland, College Park, MD, 1999. 
 
63  Gielis, J., ?A Generic Geometric Transformation that Unifies a Wide Range of 
Natural and Abstract Shapes,? American Journal of Botany, Vol. 90, No. 3, 2003, pp. 
333-338. 
 
 
 
 172 
 
 
64 Eric W. Weisstein. "Simpson's Rule." From MathWorld--A Wolfram Web 
Resource. http://mathworld.wolfram.com/SimpsonsRule.html. 
 
65 Anderson, J. D., Hypersonic and High Temperature Gas Dynamics, AIAA, 
Virginia, 2000, Chap. 3.  
 
66 Anderson, J. D., Fundamentals of Aerodynamics, 3rd ed., McGraw-Hill Book Co., 
New York, 2001, Chap. 8.  
 
67 Nelson, Robert. C., Flight Stability and Automatic Control, 2nd Ed., McGraw-Hill 
Book Co., New York, 1997, Chap. 1, 2. 
 
68 Phillips, W. F., Mechanics of Flight, John Wiley & Sons, Inc., Hoboken, NJ, 2004. 
 
69 Anderson, J. D., Introduction to Flight, 4rd ed., McGraw-Hill Book Co., New York, 
2000, Chap. 3.  
 
70 NOAA, NASA, USAF, ?U.S. Standard Atmosphere, 1976,? NASA-TM-X-74335, 
NOAA-S/T-76-1562, Oct 1976. 
 
71 DOT, Design Optimization Tools, Software Package, Ver. 4.20, Vanderplaats 
Research & Development, Inc., Colorado Springs, CO, 1995.  
 
72 Johnson, J. E., Starkey, R. P., and Lewis, M. J., ?Aerodynamic Stability of Re-
Entry Heat Shield Shapes for a Crew Exploration Vehicle,? AIAA Paper 2005-4112, 
July 2005. 
 
73 Johnson, J. E., Starkey, R. P., and Lewis, M. J., ?Aerothermodynamic Optimization 
of Re-Entry Heat Shield Shapes for a Crew Exploration Vehicle,? AIAA Paper 2006-
6273, Aug 2006. 
 
74 Loh, W. H. T., Re-entry and Planetary Entry Physics and Technology, Springer-
Verlag New York Inc., New York, 1968, Chap. 2.  
 
75 Suttles, J. T., Sullivan, E. M., Margolis, S. B., ?Curve Fits of Predicted Inviscid 
Stagnation-Point Radiative Heating Rates, Cooling Factors, and Shock Standoff 
Distances for Hyperbolic Earth Entry,? NASA TN D-7622, Jul 1974.