AERODYNAMIC HEATING OF A HYPERSONIC NAVAL … · AERODYNAMIC HEATING OF A HYPERSONIC NAVAL PROJECTILE ... for their applicability to the problem under study, and outside of the GASP

AERODYNAMIC HEATING OF A HYPERSONIC NAVAL PROJECTILE LAUNCHED AT SEA LEVEL

ARTHUR A. MABBETT

Dissertation Submitted to the Faculty of the Virginia Polytechnic Institute and State University in Partial Fulfillment of the Requirements for the Degree Of

Doctor of Philosophy

In Aerospace Engineering

Dr. Joseph A. Schetz Dr. Robert Walters

Dr. Richard Barnwell Dr. Thomas Diller Mr. Keith Lewis

10 April 2007 Blacksburg, VA

Keywords: hypersonic, EM Railgun, projectile, computational fluid dynamics, finite

element analysis, aerodynamic heating, aeroheating

AERODYNAMIC HEATING OF A HYPERSONIC NAVAL PROJECTILE LAUNCHED AT SEA LEVEL

ARTHUR A. MABBETT

ABSTRACT

Hypersonic flight at sea-level conditions induces severe thermal loads not seen by any

other type of current hypersonic system. Appropriate design of the hypersonic round requires a

solid understanding of the thermal environment. Numerous codes were obtained and assessed

for their applicability to the problem under study, and outside of the GASP CHT module, no

efficient codes are available that can model the aerodynamic heating response for a fully detailed

projectile, including all subassemblies, over an entire trajectory. Although the codes obtained

were not applicable to a fully detailed thermal soak analyses they were useful in providing

insight into ablation effects. These initial trade studies indicated that ablation of up to 1.25

inches could be expected for a Carbon-Carbon nosetip in this flight environment. In order to

capture the thermal soak effects a new methodology (BMA) was required. This methodology

couples the Sandia aerodynamic heating codes with a full thermal finite element model of the

desired projectile, using the finite element code, ANSYS. Since ablation can be treated

elsewhere it was not included in the BMA methodology. Various trajectories of quadrant

elevations of 0.5°, 10°, 30°, 50°, and 80° were analyzed to determine thermal time histories and

maximum operating temperatures. All of the trajectories have the same launch condition, Mach

8 sea-level, and therefore will undergo the same initial thermal spike in temperature at the nose-

tip of approximately 3,100 K (5600°R). Of the five trajectories analyzed the maximum internal

temperatures experienced occurred for the 50° quadrant elevation trajectory. This trajectory

experienced temperatures in excess of 1,000 K (1800°R) for more than 80% of its flight time.

Validation studies of five second flight durations indicated cost savings of over three orders of

magnitude when compared to full CFD solutions.

FOREWORD

The Electromagnetic Railgun Innovative Naval Prototype Program of the Office of Naval

Research funded the Naval Surface Warfare Center, Dahlgren Division (NSWCDD), Railgun

Project Office (G308), to study the aerodynamic heating of hypersonic projectiles launched at

sea level. Hypersonic flight of projectiles subjects their airframes to severe thermal shock and

loading. Traveling at hypersonic velocities at sea-level conditions induces severe thermal loads

not seen by any other type of hypersonic system. The heat soak into internal components such as

temperature-sensitive inertial measurement units further exacerbates the severe flight

environment. Current test assets such as wind tunnels and arc jets are incapable of replicating

Mach 8 sea-level flight conditions for the period necessary to measure thermal soak effects.

National aeroheating designers lack a user-friendly efficient tool to perform detailed aerothermal

design trades. The methodology presented in this study has shown significant cost and

timesaving by coupling a Sandia National Laboratories boundary layer aeroheating code with a

highly detailed finite element model incorporating hot wall effects.

The author gratefully acknowledges the contributions made to this study by Sandia

(Donald Potter, David Kuntz, and Basil Hassan), Aviation and Missile Research, Development,

and Engineering Center (Gerald Russell, Bruce Moylan), and ITT Corporation (Forrest Strobel,

Al Murray). Special thanks go to Reece Neel at Aerosoft, Inc., for the hours of help and support

while obtaining the computational fluid dynamics solutions with GASP and Adam R. Jones of

NSWCDD for generating solid models and trajectory data. Additionally, Vanessa Gentzen, a

Virginia Polytechnic Institute and State University alumna and current University of Maryland

graduate student, helped lay some of the groundwork for the finite element code (ANSYS)

programming. In addition, thanks are due to the author’s NSWCDD co-workers for their

assistance and patience during this endeavor. Similarly, without the aid and guidance of the

author’s doctoral committee the degree would not have been possible. Thanks to Dr. Joseph

iii

Schetz (advisor), Dr. Robert Walters, Dr. Richard Barnwell, and Dr. Thomas Diller. Special

thanks go to the author’s Navy committee representative, Keith Lewis, who helped the author

remain focused on completing the degree objectives. Robert Weaver and Michaele Morton of

DTI Associates edited and reviewed this report.

iv

CONTENTS

Section Page 1.0 INTRODUCTION ............................................................................................................. 1

1.1 PROBLEM DESCRIPTION.......................................................................................... 1

1.1.1 EML System ........................................................................................................... 1

1.1.2 Inherent Challenges ................................................................................................ 4

1.2 HISTORY ...................................................................................................................... 7

1.3 OVERVIEW ................................................................................................................ 12

2.0 CURRENT AERODYNAMIC HEATING CODES....................................................... 15

2.1 AEROPREDICTION CODE 2005.............................................................................. 15

2.2 BLUNTY ..................................................................................................................... 18

2.3 SANDIA ONE-DIMENSIONAL DIRECT AND INVERSE THERMAL CODE..... 26

2.4 MAGIC ........................................................................................................................ 27

2.5 ABRES SHAPE CHANGE CODE (ASCC86) ........................................................... 40

2.6 AEROHEATING AND THERMAL ANALYSIS CODE (ATAC05)........................ 46

2.7 GASP ........................................................................................................................... 46

3.0 AERODYNAMIC HEATING TOOL ASSESSMENT CAPABILITIES AND

LIMITATIONS...................................................................................................................... 50

3.1 AEROPREDICTION 2005/SEMI-INFINITE SOLID CORRELATIONS................. 51

3.2 AEROPREDICTION 2005/ANSYS............................................................................ 54

3.3 SANDIA NATIONAL LABORATORIES: BLUNTY/SODDIT .............................. 61

3.4 SANDIA NATIONAL LABORATORIES: ASCC.................................................... 66

3.5 ITT AEROTHERM: ATAC05 ................................................................................... 72

3.6 CONCLUSIONS ......................................................................................................... 78

4.0 HOT WALL FEA METHODOLOGY ............................................................................ 80

4.1 BLUNTY/MAGIC/ANSYS HOT WALL METHODOLOGY................................... 82

v

4.2 METHODOLOGY VALIDATION ............................................................................ 90

4.2.1 Cold Wall Heat Flux Validation ........................................................................... 92

4.2.2 Passive Nose-Tip Technology Program................................................................ 96

4.2.3 GASP Slender Body Comparison....................................................................... 104

4.3 CONCLUSIONS AND RECOMMENDATIONS .................................................... 126

5.0 ELECTROMAGNETIC RAILGUN HYPERSONIC ROUND ANALYSIS ............... 131

5.1 PROJECTILE AND FLIGHT CONDITIONS.......................................................... 131

5.1.1 ANSYS Finite Element Analysis........................................................................ 133

5.2 RESULTS AND DISCUSSION................................................................................ 137

6.0 CONCLUSIONS............................................................................................................ 147

7.0 REFERENCES .............................................................................................................. 149

APPENDICES

A—PRESSURE-TEMPERATURE-ENTHALPY TABLE FOR NON-ABLATING SURFACE AIR..................................................152

B—PASSIVE NOSE-TIP TECHNOLOGY PROGRAM HEAT DECK .......................155

C—MACH 8 & 10 5-INCH PROJECTILE HEAT DECK.............................................158

D—HYPERSONIC ROUND TRAJECTORY STUDY .................................................161

E—ANSYS CODE ..........................................................................................................170

vi

ILLUSTRATIONS

Figure Page

FIGURE 1. LAUNCHER AND INTEGRATED LAUNCH PACKAGE COMPONENTS......... 2

FIGURE 2. ELECTROMAGNETIC LAUNCH CONCEPT ........................................................ 3

FIGURE 3. VOLUME OF MATERIAL EXPOSED TO HIGH SURFACE TEMPERATURES 5

FIGURE 4. HYPERSONIC PROJECTILE HISTORY (COURTESY BOEING-ST. LOUIS).... 8

FIGURE 5. OPERATIONAL HYPERSONIC TEST FACILITIES (COURTESY AEDC) ...... 10

FIGURE 6. AEDC ARC AND WIND TUNNELS (COURTESY AEDC)................................. 11

FIGURE 7. BLUNTY FLOW FIELD GEOMETRY (REF. 10) ................................................. 19

FIGURE 8. STANDARD EQUATIONS FOR CALCULATING FREE-STREAM PROPERTIES (REF. 10) ...................................................................................................... 21

FIGURE 9. VELOCITY VECTORS ACROSS SHOCK (REF. 10)........................................... 21

FIGURE 10. PRESSURE BEHIND SHOCK, DENSITY, ENTHALPY, AND VELOCITY (REF. 10)............................................................................................................................... 22

FIGURE 11. VELOCITY RATIOS SIMPLIFIED...................................................................... 31

FIGURE 12. SURFACE PRESSURE PARTITIONS (REFERENCE 22) ................................. 41

FIGURE 13. EXPLICIT AND IMPLICIT GRIDS (REFERENCE 22,23)................................. 45

FIGURE 14. FLUID/SOLID ZONAL BOUNDARY ................................................................. 48

FIGURE 15. SUMMARY OF CODE CAPABILITES............................................................... 51

FIGURE 16. HYPERSONIC ROUND APPROXIMATE GEOMETRY ................................... 51

FIGURE 17. AP05 BOUNDARY CONDITIONS...................................................................... 52

FIGURE 18. SEMI-INFINITE SOLID APPROXIMATION ..................................................... 53

FIGURE 19. REFINED AP05 BOUNDARY CONDITIONS.................................................... 54

FIGURE 20. BOUNDARY CONDITION REGIONS................................................................ 55

FIGURE 21. ANSYS MODEL.................................................................................................... 56

FIGURE 22. NODE 1 TEMPERATURE HISTORY.................................................................. 57



vii

FIGURE 25. NODE 1 TEMPERATURE HISTORY (WITH INTERNAL GEOMETRY) ....... 59



FIGURE 28. BLUNTY COLD WALL HEAT FLUX (50 SECONDS) ..................................... 63

FIGURE 29. BLUNTY COLD WALL HEAT FLUX (FULL)................................................... 63

FIGURE 30. BLUNTY, IN-HOUSE, DETRA COMPARISON ................................................ 64

FIGURE 31. BLUNTY/SODDIT THERMAL RESPONSE....................................................... 65

FIGURE 32. TUNGSTEN OXIDE RECESSION, M8, RN = 0.125-INCH FLAT PLATE....... 66

FIGURE 33. ASCC IMPLICIT AND EXPLICIT GRID ............................................................ 67

FIGURE 34. HIGHLY SKEWED GRID CAUSED BY THICK SURFACE LAYER AND ORIGIN LOCATION ........................................................................................................... 68

FIGURE 35. ASCC, MACH 8, QE 0.5, C-C............................................................................... 69

FIGURE 36. ASCC, MACH 8, QE 20, C-C................................................................................ 69



FIGURE 39. ATAC05 TEMPERATURE HISTORIES.............................................................. 74

FIGURE 40. ATAC05 RECESSION .......................................................................................... 75

FIGURE 41. STREAMLINE VARIATION THERMAL HISTORY......................................... 75

FIGURE 42. STREAMLINE VARIATION RECESSION......................................................... 76

FIGURE 43. PATCH VARIATION THERMAL HISTORY ..................................................... 76

FIGURE 44. PATCH VARIATION RECESSION..................................................................... 77

FIGURE 45. AEROTHERMAL DESIGN FLOW...................................................................... 82

FIGURE 46. SAMPLE ANSYS GEOMETRY/MESH............................................................... 83

FIGURE 47. SURFACE LOCATIONS FOR HEAT TRANSFER OUTPUT............................ 84

FIGURE 48. SAMPLE TRAJECTORY INPUT DATA (TIME, ALTITUDE, VELOCITY).... 84

FIGURE 49. SAMPLE HEAT DECK OUTPUT........................................................................ 85

FIGURE 50. SAMPLE HEAT FLUX BOUNDARY CONDITIONS ........................................ 87

FIGURE 51. 2-D ELEMENT NODE AND FACE LOCATIONS.............................................. 88

FIGURE 52. TWO ADJACENT ELEMENTS ........................................................................... 89

FIGURE 53. SAMPLE HEAT FLUX PLOT AT A GIVEN LOAD STEP ................................ 89

FIGURE 54. PARAMETER SPACE OF AVAILABLE EXPERIMENTAL DATA................. 91

FIGURE 55. CLEARY 15° CONE GEOMETRIES ................................................................... 93

FIGURE 56. THERMOCOUPLE LOCATIONS........................................................................ 93

viii

FIGURE 57. CASE 1 CLEARY VERIFICATION..................................................................... 95

FIGURE 58. CASE 2 CLEARY VERIFICATION..................................................................... 95

FIGURE 59. PANT CALORIMETER MODEL......................................................................... 96

FIGURE 60. CALORIMETER THERMOCOUPLE LOCATIONS........................................... 97

FIGURE 61. ANSYS GEOMETRY............................................................................................ 98

FIGURE 62. PANT GRID REFINEMENTS .............................................................................. 99

FIGURE 63. GRID CONVERGENCE FOR BLUNTY/MAGIC/ANSYS PANT VALIDATION CASE .................................................................................................................................... 99

FIGURE 64. BMA, GASP, AND EXPERIMENTAL TEMPERATURE PROFILES AT THERMOCOUPLE NUMBER 2 LOCATION.................................................................. 101

FIGURE 65. BMA, GASP, AND EXPERIMENTAL TEMPERATURE PROFILES AT THERMOCOUPLE NUMBER 13 LOCATION................................................................ 101

FIGURE 66. BMA, GASP, AND EXPERIMENTAL TEMPERATURE PROFILES AT THERMOCOUPLE NUMBER 67 LOCATION................................................................ 102

FIGURE 67. GASP COMPARISON PROJECTILE GEOMETRY ......................................... 105

FIGURE 68. HEMISPHERICAL MESH REFINEMENT STUDY ......................................... 106

FIGURE 69. NODAL TEMPERATURE MESH CONVERGENCE ....................................... 107

FIGURE 70. ANSYS MESH AND NODE LOCATIONS ....................................................... 108

FIGURE 71. NODAL THERMAL HISTORY, LOAD STEP = 0.01 S ................................... 110

FIGURE 72. NODAL THERMAL HISTORY, LOAD STEP = 0.005 S ................................. 110

FIGURE 73. NOSE-TIP TEMPERATURE CONVERGENCE ............................................... 111

FIGURE 74. COARSE MESH USED FOR CONVERGENCE STUDIES (CREATED IN ICEM) ................................................................................................................................. 112

FIGURE 75. COARSE MESH – CLOSE UP VIEW OF NOSE TIP ....................................... 112

FIGURE 76. COARSE MESH – AXIAL RESOLUTION........................................................ 113

FIGURE 77. MEDIUM MESH (23,087 CELLS) ..................................................................... 114

FIGURE 78. MEDIUM MESH – CLOSEUP VIEW OF NOSE TIP........................................ 115

FIGURE 79. MEDIUM MESH – AXIAL RESOLUTION....................................................... 115

FIGURE 80. FINE MESH (44,639 CELLS) ............................................................................. 116

FIGURE 81. FINE MESH – CLOSEUP VIEW OF NOSE TIP ............................................... 116

FIGURE 82. FINE MESH – AXIAL RESOLUTION............................................................... 117

FIGURE 83. GASP GRID CONVERGENCE: FINE MESH IS CONVERGED..................... 118

FIGURE 84. TIME STEP CONVERGENCE INDICATE 1E-4 SECONDS CONVERGED (FINE MESH)..................................................................................................................... 119

ix

FIGURE 85. STEADY STATE MACH 10 FLOW FIELD SOLUTION ................................. 121

FIGURE 86. STEADY STATE MACH 10 FLOW FIELD SOLUTION (SHOCK REGION) 121

FIGURE 87. FLOW FIELD TEMPERATURE PROFILE @ TIME = 5S ............................... 122

FIGURE 88. SOLID MODEL TEMPERATURE PRFILE @ TIME = 5S............................... 122

FIGURE 89. MACH 10 VALIDATION STUDY (BMA: SOLID, GASP: DASHED)............ 123

FIGURE 90. STEADY STATE MACH 8 FLOW FIELD SOLUTION ................................... 124

FIGURE 91. STEADY STATE MACH 8 FLOW FIELD SOLUTION (SHOCK REGION).. 124

FIGURE 92. FLOW FIELD TEMPERATURE PROFILE @ TIME = 5 SECONDS .............. 125

FIGURE 93. SOLID MODEL TEMPERATURE PROFILE @ TIME = 5 SECONDS........... 125

FIGURE 94. MACH 8 VALIDATION STUDY (BMA: SOLID, GASP: DASHED).............. 125

FIGURE 95. HYPERSONIC ROUND APPROXIMATE GEOMETRY ................................. 132

FIGURE 96. NOTIONAL TRAJECTORIES OF EMRG HYPERSONIC ROUND................ 132

FIGURE 97. ANSYS GEOMETRY AND MATERIAL PROPERTIES.................................. 134

FIGURE 98. INTERNAL ASSEMBLY CONVERGENCE STUDY GRIDS.......................... 135

FIGURE 99. INTERNAL GEOMETRY MESH CONVERGENCE STUDY IMPLIES COARSE MESH IS SUFFICIENT .................................................................................... 136

FIGURE 100. ANSYS MESH AND NODE LOCATIONS OF INTEREST ........................... 136

FIGURE 101. APPLIED HEAT FLUX BOUNDARY CONDITIONS ................................... 137

FIGURE 102. QE 0.5° THERMAL HISTORY ........................................................................ 138

FIGURE 103. QE 10° THERMAL HISTORY ......................................................................... 139




FIGURE 107. MAXIMUM DESIGN TEMPERATURES ....................................................... 143

x

TABLES

Table Page TABLE 1. MATERIAL PROPERTIES ...................................................................................... 57

TABLE 2. SAMPLE PRESSURE-TEMPERATURE-ENTHALPY TABLE ............................ 83

TABLE 3. TRAJECTORY INPUT DATA (TIME, ALTITUDE, VELOCITY) ........................ 85

TABLE 4. HEAT DECK OUTPUT ............................................................................................ 85

TABLE 5. HYPERSONIC EXPERIMENTS WITH POTENTIAL RELEVANCE................... 91

TABLE 6. WIND TUNNEL CONDITIONS .............................................................................. 98

TABLE 7. NICKEL THERMAL PROPERTIES ........................................................................ 98

TABLE 8. AVERAGE VARIATION FROM EXPERIMENT................................................. 102

TABLE 9. HYPERSONIC APPROXIMATION ERROR ........................................................ 103

TABLE 10. C-350 CONSTANT THERMAL PROPERTIES .................................................. 105

TABLE 11. NODAL COORDINATES .................................................................................... 108

TABLE 12. CPU HOURS FOR ANSYS SOLUTION ............................................................. 111

TABLE 13. GASP (MEDIUM MESH) CPU HOURS.............................................................. 120

TABLE 14. BMA AVERAGE VARIATION FROM GASP.................................................... 128

TABLE 15. COST COMPARISONS........................................................................................ 129

TABLE 16. TRAJECTORY MAXIMUM TEMPERATURES ................................................ 142

xi

GLOSSARY

ACE Aerotherm Chemical Equilibrium

AEDC Arnold Engineering Development Center

AHSTF Arc-Heated Scramjet Test Facility

AMRDEC Aviation and Missile Research, Development, and Engineering Center

ANSYS Finite Element Code

AP05 Aeroprediction Code 2005

APTU Aerodynamic and Propulsion Test Unit

ASCC ABRES Shape Change Code

ATAC Aeroheating and Thermal Analysis Code

atm atmosphere(s)

BC’s Boundary Conditions

BMA BLUNTY/MAGIC/ANSYS methodology

Btu British Thermal Unit(s)

Btu/ft2-s British Thermal Unit(s) per square foot second

Btu/lbm British Thermal Unit(s) per pound mass

C-C Carbon-Carbon

CFD Computational Fluid Dynamics

CHT conjugate heat transfer

CMA Charring Material Thermal Response and Ablation

CPU Central Processing Unit

CPUH Central Processing Unit Hours

CTE Thermal Expansion Coefficient

DoD Department of Defense

EML Electromagnetic Launch

EMRG Electromagnetic Railgun

FEA Finite Element Analysis

xii

GLOSSARY (Continued)

ft/s foot (feet) per second

ft2-s foot squared second

G acceleration(s) of gravity

GASP CFD computer code

GE General Electric

G308 Railgun Project Office

HAAP Heavy Anti-Armor Projectile

HIBLARG Hypersonic Integral Boundary Layer Analysis of Reentry Geometries

HSR hypersonic round

ILP Integrated Launch Package

in. inch(es)

INP Innovative Naval Prototype

JCS Johnson Space Center

J/kg-K joule(s) per kilogram Kelvin

K degree(s) Kelvin

kg kilogram(s)

kg/m3 kilograms per cubic meter

kJ/kg kilojoules per kilogram

kG a kilo-G, or 1,000 times the acceleration of gravity

km/s kilometer(s) per second

LaRC Langley Research Center

lbm pound-mass

LORN LORN cone criterion

MAGIC CFD computer code

MPa megapascal(s)

NAFF NASA Ames Flow Field

NASA National Aeronautics and Space Administration

NASP NASP transition criteria

nmi nautical mile(s)

NSWCDD Naval Surface Warfare Center, Dahlgren Division

xiii

GLOSSARY (Continued)

ONR Office of Naval Research

PANT Passive Note-Tip Technology

psi pound(s) per square inch

QE Quadrant Elevation

RANS Reynolds-Averaged Napier-Stokes Equations

RV reentry vehicles

s second(s)

SANDIAC SANDia Inviscid Afterbody Code

SI International System of Units

SLBM Submarine/Sea-Launched Ballistic Missile

SODDIT Sandia One-Dimensional Direct and Inverse Thermal Code

TPS Thermal Protection System

TRW TRW transition criteria

UHVP Ultra High-Velocity Projectile

Virginia Tech Virginia Polytechnic Institute and State University

W/m-K watt(s) per meter Kelvin

W/m2 watts per meter squared

W/m2-K watt(s) per meter squared Kelvin

1-D One-Dimensional

2-D Two-Dimensional

3-D Three-Dimensional

3DOF three degrees of freedom

º degree(s)

ºC degree(s) Celsius

ºF degree(s) Fahrenheit

ºR degree(s) Rankine

% percent

" inch(es)

xiv

LIST OF SYMBOLS

VARIABLES a speed of sound, acoustic speed A area alt∞ physical altitude altgp geo-potential altitude B magnetic field vector CD drag coefficient Cf local skin friction coefficient Cfi incompressible skin friction coefficient CH Heat transfer coefficient Ch Stanton number Cp specific heat at constant pressure CPUH CPU hours, CPUH = No. of CPUs × Solution Time erf w error function erfc w inverse error function, 1-erf w F force (denotes Lorentz force in EM launch diagram) f p. 47 fluid F(s) shape factor g gravity h enthalpy, heat transfer coefficient H total enthalpy, boundary layer shape factor I total armature current J current vector k thermal conductivity k(T) temperature dependent thermal conductivity L′ inductance gradient Le Lewis number Lm´ molecular temperature gradient M Mach number P pressure Pr Prandtl number Q* approximate energy dissipation method

qq ,′′ heat flux R nose radius, real gas constant, recovery factor RN stagnation point radius of curvature rN nose-tip radius

xv

VARIABLES rc recovery factor Re Reynolds number s arc length along curvature S entropy T temperature t time Ti,∞ initial and far field temperatures Tm base molecular temperature U, V velocity u,v,w x, y, and z velocity components z axial distance from the start of the aft cone α thermal diffusivity β shock angle γ ratio of specific heats ∆ distance δ Boundary layer thickness δ* displacement thickness ε emissivity, user defined tolerance θ momentum θc cone half angle µ viscosity ν kinematic viscosity ρ density τ shear stress φ energy thickness 0 sea level

SUB/SUPERSCRIPTS ∞ free stream * reference value aw adiabatic wall D dissociation e boundary layer edge o stagnation point r recovery rad radiation s behind the shock wave w wall

xvi

1.0 INTRODUCTION

The Office of Naval Research (ONR) has embarked on an Innovative Naval Prototype

(INP) of an electromagnetic railgun (EMRG) system that will accelerate projectiles to hypersonic

speeds, enabling ranges beyond 200 nautical miles (nmi) in less than 6 minutes of flight time.

1.1 PROBLEM DESCRIPTION

Over the last few decades, significant advances have occurred in electromagnetic

launch (EML) technology. The United States Navy is currently considering an EMRG concept

for a future long-range naval weapons system. Preliminary studies have shown such a system

will have the capability to launch a projectile upward of 2.5 kilometers per second (km/s) and to

a distance greater than 200 nmi. Significant challenges exist in the design of a projectile system

capable of withstanding the severe launch environment produced by an EML weapon system

(Reference 1).

1.1.1 EML System

An EML system consists of an electromagnetic launcher, a pulsed power supply that

delivers electrical current to the system, and an integrated launch package (ILP). The mass

launched by the system, the ILP, consists of the projectile, a pusher plate, an armature, and sabot

petals for additional structural support. The launcher is made up of two parallel conducting rails

within which the current is pulsed. The armature, which touches both rails to complete a circuit,

is located behind the projectile and accelerates the high-strength pusher plate, which drives the

projectile. Figure 1 depicts the launcher and ILP components.

1

Sliding Armature (conducts electrical

current between rails)

Sabot (supports projectile)

Pair of Metal Conducting Rails

(inside barrel)

Source of Stored Electrical Energy

(e.g., capacitor bank)

Low Drag Hypersonic Projectile

Sliding Armature (conducts electrical

current between rails)

Sabot (supports projectile)

Pair of Metal Conducting Rails

(inside barrel)

Source of Stored Electrical Energy

(e.g., capacitor bank)

Low Drag Hypersonic Projectile

FIGURE 1. LAUNCHER AND INTEGRATED LAUNCH PACKAGE COMPONENTS

EML is a relatively simple concept based on the principles of the Lorentz force. The

current flows down one of the rails and conducts through the armature to the opposite rail. This

current loop generates an electromagnetic field around each rail. The current in the armature

interacting with the magnetic fields around each rail produces a strong electromagnetic force that

drives the armature down the rails (Figure 2) and out of the muzzle. The force created by the

current in the magnetic field is known as the Lorentz force and is calculated using Equation (1)

for the specific force per unit volume or Equation (2) for total force:

BJF ×= , (1)

2

21 IL′=F , (2)

where

2

F = Lorentz (axial) force,

J = current vector,

B = magnetic field vector,

L′ = inductance gradient, and

I = total armature current.

Magnetic field generated around rails

as current flows through circuit.

Magnetic field interacts with armature current, generating

axial propulsive force

Switch closes, current flows through rails & armature.

Lorentz force accelerates armature & projectile down barrel.

J

BF

Lorentz Force (F) = Current (J) X Magnetic Field (B)

Magnetic field generated around rails

as current flows through circuit.

Magnetic field interacts with armature current, generating

axial propulsive force

Switch closes, current flows through rails & armature.

Lorentz force accelerates armature & projectile down barrel.

J

BFJ

BFJ

BF

Lorentz Force (F) = Current (J) X Magnetic Field (B)

FIGURE 2. ELECTROMAGNETIC LAUNCH CONCEPT

After the ILP exits the muzzle, aerodynamic forces decelerate the pusher plate and

armature and discard them. Similarly, the sabot petals peel off, allowing the projectile to fly free

to the target (see Reference 2).

The ONR EMRG INP program seeks to mature the technologies necessary to build a

system that accelerates projectiles up to the proposed 2.5-km/s speeds for greater than 200-nmi

ranges. Along with the greatly enhanced range, several other benefits would accrue to the Navy.

Unlike conventional naval guns, the storage, handling, and use of propellant charges will become

unnecessary. Additionally, projectile lethality results from kinetic energy on target, making

additional large amounts of high explosive unnecessary. These advantages greatly simplify the

logistics associated with high explosives. Finally, with no propellant charges and the relatively

small size of the rounds, ship magazines can carry many more projectiles.

3

1.1.2 Inherent Challenges

Each subassembly of the EMRG system has its own inherent challenges. The two

driving challenges for projectile development are the high accelerations of gravity (G) launch

environment and aerodynamic heating concerns. A tactical EMRG will accelerate a projectile in

a 10- to 14-meter barrel, resulting in approximately 35 kilo-Gs (kG) of axial force

(i.e., 35,000 times the acceleration gravity) applied to the ILP. Any rail asymmetries can cause

non-axial application of a fraction of this load, resulting in high bending moments in the ILP as

well.

The notional EMRG provides an initial launch velocity of approximately 2.5 km/s

(8,200 feet per second (ft/s)), or roughly Mach 7.5 for a 20-kilogram (kg) ILP. Rounding up for

a margin of safety to Mach 8 yields significant aerodynamic loads on the projectile of 9

megapascals (MPa) (89 atmospheres, 187,968 pounds per square foot) stagnation pressure and an

enthalpy of 3994.5 kilojoules per kilogram (kJ/kg) (1717 British thermal units per pound-mass

(Btu/lbm)). Using the pressure-enthalpy-temperature tables in NACA TN 4265, the

compressibility effects are negligible, (Z = 1.01) and therefore, the stagnation temperature is

approximately 3,300 degrees Kelvin (K) (5,940 degrees Rankine [°R]).

Unlike space vehicles, reentry vehicles, and high altitude missiles, projectiles launched at

sea level encounter the densest air and highest velocity instantaneously. Due to the very short

time at which the load is applied, large thermal gradients form from the skin, which approaches

the stagnation temperature, inward. These large thermal gradients cause large gradients in

thermal expansion, which in turn yield significant stresses. When the induced internal stresses

become larger than the tensile yield stress of the materials, the material will begin to fail. Figure

3 depicts a volume of material that is exposed to high temperatures on the surface in a rapid

fashion.

4

HOT

Room TempNo expansion

Max expansion

FIGURE 3. VOLUME OF MATERIAL EXPOSED TO HIGH SURFACE TEMPERATURES

Numerous methods exist for calculating nose-tip heat flux. Boundary layer codes heavily

use the Fay and Riddell (Reference 3) correlation. By making some assumptions and

approximations, Fay and Riddell can be modified to remove the dependency on the boundary

layer edge properties. Scott, Detra, and the Navy’s submarine/sea-launched ballistic missile

(SLBM) group have various forms.

Equation (3) shows the Fay and Riddell correlation (Reference 3),

⎭⎬⎫

⎩⎨⎧ −+−⎥

⎦

⎤⎢⎣

⎡ −= ∞

HhLehHPP

Rq D

woeo

eowo052.0

25.0

04.01.05.00 )1(1)()(2)()()2.32(94.0

ρρµρµ , (3)

where

= heat flux, q

R = nose-tip radius,

µ = viscosity

w = wall,

o = stagnation point

e = boundary layer edge,

∞ = free stream,

H = total enthalpy,

Le = Lewis number (1.4), and

5

D = dissociation.

Equation (4) shows the Detra modification (References 3 and 4) of the Fay and Riddell

correlation in English units with nose radius in inches,

15.32

1

0 795011030

⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛= ∞∞ V

Rq

ρρ , (4)

where

V = velocity, and

0 = sea level.

Equation (5) shows the Scott modification (References 4 and 5) of the Fay and Riddell

correlation in English units with nose radius in inches,

( )05.3

421

1018300

⎟⎠⎞

⎜⎝⎛= ∞

∞V

Rq ρ . (5)

Equation (6) shows the Navy SLBM (Courtesy of Dr. Bill Dorsey, Naval Surface

Warfare Center (NSWCDD) K08) modification of the Fay and Riddell correlation,

540

15.3

4

21

0 10865

hhhhV

Rq

t

wt

−−

⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛= ∞∞

ρρ , (6)

where all units are English except for radius in inches and,

ht = enthalpy at stagnation,

hw = enthalpy at wall, and

h540 = enthalpy at 540°R.

Due to aerodynamic drag considerations, the notional EMRG hypersonic-round, nose-tip

radius ranges from 0.10 inch up to 0.20 inch. For comparison, the predicted stagnation point

heat flux values from Equations 6 – 8, ranging from 6,600 to 4,700 British thermal units per

square foot-second (Btu/ft2-s), respectively, are an order of magnitude higher than either the

Apollo reentry vehicle or the Mars Pathfinder, which are 437 and 93 Btu/ft2-s, respectively.

6

1.2 HISTORY

With the space race in full swing during the 1960s, National Aeronautics and Space

Administration (NASA) engineers realized that returning a vehicle traveling at hypersonic

speeds through the Earth’s atmosphere presented a significant challenge. Great advances in

material technology and the capabilities to model these systems were achieved throughout the

space race and well into the 1970s and 1980s with the space shuttle program and nuclear reentry

vehicles. By this point, hypersonic flight at altitude was relatively routine with the advent of

Mach 25 intercontinental ballistic missiles, Mach 25 space shuttle reentry, and even the Mach 36

reentry of the Apollo spacecraft (Reference 6).

The advances in the modeling of these hypersonic systems during the 1970s and 1980s

were made for these very specific applications and have been extended to numerous similar

systems. Although the underlying physics is the same, space and reentry vehicles including

Intercontinental Ballistic Missiles (ICBM), differ greatly from hypersonic rounds at sea level.

First, space and reentry vehicles gradually encounter increasingly denser air as they reenter the

atmosphere, thereby minimizing thermal shock and mechanical shock. However, in a sea-level

application, a hypersonic round experiences maximum heating and shear forces at muzzle exit.

Second, geometrically, a notional EMRG projectile is much smaller than any existing developed

and tested hypersonic system. For example, the Apollo capsule had a nose bluntness of

approximately 16 feet, resulting in a heat flux of 437 Btu/ft2-s, which contrasts with the predicted

6,000 Btu/ft2-s of a Mach 8 0.125-inch nose-tip radius projectile. Given these larger nose radii,

larger airframes, and lower heat flux values, a full-up thermal history of the entire system is not

necessary. Current methodologies typically use cold-wall heat transfer assumptions or an

iterative hot-wall approach to look at specific areas of interest. However, in a small slender

system like a hypersonic projectile, thermal history is vital to the survival of temperature-

sensitive internal assemblies depending on the overall mission time.

7

The advances in EML technology during the 1980s, predominantly for strategic defense,

generated a new wave of hypersonic systems. Figure 4 illustrates hypersonic projectile work

since 1983.

CCEMLILP

UHVPDart

SLEKE 1Rodman

Cone

UHVPHAAP

SLEKE 1LongRod

SLEKE 2Long Rod &Rod/Tube

200,

000

kgee ge

e kge

e

70,0

00 k

gee

70,0

00 k

gee

,000

kge

e

100,

000

k

100,

000

70

1983 – 2006Development

Database

70,0

00 k

G

70,0

00 k

G

70,0

00 k

G

100,

000

kG

100,

000

kG

200,

000

kG

FIGURE 4. HYPERSONIC PROJECTILE HISTORY (COURTESY BOEING-ST. LOUIS)

Many of the projectiles shown in Figure 4 were intended for velocities upwards of

4 km/s; however, just a few were ever tested, and those were tested at lower velocities of around

2 km/s. The tested rounds that survived gun launch were only in free flight for ranges up to

about 1 kilometer, which would permit about one-half second of flight time. This short flight

time prohibited the generation of useful data about the thermal response of the system. In

addition, other than imaging the projectile at muzzle exit, no downrange imaging of projectile

structural integrity occurred. It was assumed that if the round made it to target, then the structure

remained intact, and for short-range tests, that was all that was necessary. However, for a

tactical long-range system, the integrity of the nose-tip plays an enormous role in the drag and

dispersion of the round. Few of these systems underwent aerothermal trade studies. Typical

analyses consisted of checking the stagnation temperature against the material’s melt

temperature—usually tungsten tips. For the short-range systems, there were no thermal soak

considerations because of the short flight times.

8

Until now, there has been no need to investigate thoroughly the aerothermal performance

(i.e., thermal shock, nose-tip integrity, and thermal soak) of hypersonic rounds launched at sea

level, in particular sharp-tip, slender-cone-angle projectiles. With the advances recently made in

EML technology, it is now feasible to launch these rounds at hypersonic speeds attempting to

achieve long-range fire support. Consequently, the appropriate tools and models need

development to permit the accurate prediction of the aerothermal performance of these

projectiles.

To further compound the problem, existing test facilities cannot replicate the conditions

of a Mach 8 sea-level launch (Reference 7). Figure 5 shows the currently available facilities and

their operational space. Superimposed on the facilities are typical trajectories for numerous

hypersonic systems of interest (References 8 and 9). As shown in the figure, the trajectory of a

notional naval hypersonic round, HSR M8 QE50, does not overlap any existing test facility for a

majority of its flight, and, most troubling, does not overlap any existing test facility for the most

critical portions of its flight, namely the flight below 20,000 foot altitude. This leaves a

capability gap in material testing and code verification and validation. The currently used codes

in the hypersonic regime have been verified and validated only in the regimes of interest to date:

high Mach, high altitude, and large bodies. None of these codes have been fully exercised or

validated in the sea-level regime.

9

1

10

100

1000

0 2 4 6 8 10 12 14 16 18 20 22 24 26

JSC/AMES

AEDC H1 & H3AEDC HR

AEDC H2LaRCAHSTFF

SCIROCCODe = 76 in De = 36 in.

H2 w/ H3 HTRSCRAMJET

ASCENTq = 1000 psf

ICBM

ICBM Max Heating

Shuttle

HSR M8 QE50JSC/AMES

AEDC H1 & H3AEDC

H1 & H3AEDC HRAEDC HR

AEDC H2AEDC H2LaRCAHSTFF

SCIROCCODe = 76 in De = 36 in.

H2 w/ H3 HTRSCRAMJET

ASCENTq = 1000 psf

ICBM

ICBM Max Heating

Shuttle

HSR M8 QE50

SIMULATED VELOCITY, KFT/S

200 400 600 800 1000 2000 4000 6000 8000 10000TOTAL ENTHALPY, BTU/LBM

Negligible Aeroheating < M5

ALTI

TUDE

kF

T

LENS

FIGURE 5. OPERATIONAL HYPERSONIC TEST FACILITIES (COURTESY AEDC)

Numerous facilities have the capability to replicate the aerodynamic flight environment

necessary for aerodynamic performance by matching Mach number and Reynolds number.

However, no existing facility can adequately replicate a sea-level hypersonic environment to

capture time-accurate, aerothermal performance. By looking at the plotted trajectories in

Figure 5, it is easy to see how the various facilities got their origins, with the NASA facilities

focusing on space vehicles while the Arnold Engineering and Development Center (AEDC)

focused on military applications. Figure 6 shows the arc jet and tunnel facilities available at

AEDC where again, no facility can replicate the low-altitude aerothermal environment. Shock

tunnels have not been considered because of the short test times, which are on the order of

milliseconds. These tunnels are excellent at replicating an aerodynamic test or a short-duration

heat flux, but an extended thermal response study is not possible.

10

Simulated Velocity, kft/s

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6

0.1

1

10

100

1,000

10,000

100,000

200 1000 2000 6000 8000 100004000

Total Enthalpy, Btu/lbm

H1/H3 NormalH1/H3 Mixing

HR FlatHR Peaked

PeakedFlat

Mixing (fabrication required)B, M=8B, M=6

C, M=4C, M=8

C, M=10

APTU

H2

Heat

Flux,

Btu/f

t2 sec

Simulated Velocity, kft/s

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6

0.1

1

10

100

1,000

10,000

100,000

200 1000 2000 6000 8000 100004000

Total Enthalpy, Btu/lbm

H1/H3 NormalH1/H3 NormalH1/H3 MixingH1/H3 Mixing

HR FlatHR FlatHR PeakedHR Peaked

PeakedPeakedFlatFlat

Mixing (fabrication required)Mixing (fabrication required)Mixing (fabrication required)B, M=8B, M=6B, M=6

C, M=4C, M=4C, M=8C, M=8

C, M=10C, M=10

APTUAPTU

H2H2

Heat

Flux,

Btu/f

t2 sec

HSR Launch

FIGURE 6. AEDC ARC AND WIND TUNNELS (COURTESY AEDC)

Several existing or soon to exist test assets may fill the test gap. The Department of

Defense (DoD) and NASA routinely make use of sounding rockets for hypersonic testing.

Although these rockets are excellent at lifting payloads to altitude and providing high speeds at

altitude, they cannot effectively perform at low altitudes in dense air. Rocket sleds can provide

hypersonic speeds at close to sea-level conditions. In particular, the test track at Holloman Air

Force Base, New Mexico, can produce velocities up to 2 km/s in air and 3 km/s in a helium

atmosphere. This test track has been used for lethality testing, ablation, and more recently,

erosion effects in rain fields. Due to the nature of rocket sled testing, severe vibration makes it

difficult to instrument specimens to telemeter data. Additionally, it is quite difficult to avoid

shock impingement issues with the specimens and the sled/rocket. Moreover use of this type of

facility quickly becomes cost prohibitive. Last, a small number of high-velocity chemical guns

are currently under development. These guns will be test assets only (i.e., not tactical) due to the

overwhelming length necessary to achieve high velocities, but they may be capable of providing

velocities of upward of 2.2 km/s. Until a full-up EMRG comes online, these high-velocity guns

may be the only opportunity to simulate the thermal shock and shear of sea-level hypersonic

11

flight. Thermal soak issues may potentially be addressed by these systems if long-range tests can

be achieved with telemetered thermocouple data.

1.3 OVERVIEW

Hypersonic launch and flight subjects projectile airframes to severe in-bore launch

dynamics followed by thermal shock and loading. Traveling at hypersonic velocities at sea-level

conditions induces severe thermal loads not seen by any other type of current hypersonic system.

For example, a Mach 8 slender cone projectile with a nose-tip radius on the order of 0.125 inch

launched at sea level can undergo maximum temperatures near 6,000°R in less than a second

after gun launch. In addition, heat soak further exacerbates the severe flight environment by

affecting temperature-sensitive internal components, such as inertial measurement units.

Existing thermal protection systems (TPS) have historically used expensive, exotic materials, but

expected maximum operating temperatures, in combination with thermal shock and aerodynamic

shear, exceed the capabilities of most readily available high-temperature materials.

Most of the advances in hypersonic system modeling have very specific applications, in

particular, space and reentry vehicles (References 10, 11, 12, and 13). Although the underlying

physics is the same, space and reentry vehicles differ greatly from hypersonic rounds launched at

sea level. Space and reentry vehicles gradually encounter increasingly denser air as they reenter

the atmosphere. In these cases, thermal shock and mechanical shock are minimal; however, in a

sea-level application, maximum heating and shear forces occur at muzzle exit. No existing test

facilities can replicate this setting (Reference 8, Figure 4, and Figure 5), so typical TPS materials

remain untested at these conditions. Similarly, the lack of a test facility means that the available

modeling tools remain unverified and unvalidated for this flight regime (References 10, 11, 12,

and 13).

Appropriate design of the hypersonic round requires a solid understanding of the thermal

environment. Current codes need updating to incorporate new modeling schemes that capture

the sharp, slender bodies of hypersonic rounds. Numerous codes were obtained and assessed for

12

their applicability to the problem at hand, and outside of the GASP (Reference 26) conjugate

heat transfer (CHT) module, no codes are available that can model the aerodynamic heating

response for a fully detailed projectile, including all subassemblies, over an entire trajectory.

Running a full Navier-Stokes computational fluid dynamics (CFD) code for a trajectory is

prohibitively expensive. As will be shown in Section 4.0, a constant Mach 10 flight of 5 seconds

costs approximately 5,000 central processing unit (CPU) hours. A typical flight trajectory would

be closer to minutes of flight time rather than a few seconds. Moreover, a tactical unpowered

projectile has constantly changing boundary conditions that would require constantly changing

boundary conditions in the CFD code as well, which is unpractical. This study develops and

validates a full transient finite element analysis (FEA) tool capable of modeling a fully detailed

hypersonic system through its entire flight.

The intention of this work was to develop a methodology to perform a fully detailed

thermal analysis of a hypersonic projectile system. Considering the complexity of ablation and

erosion phenomena and the availability of codes to partially address this issue these effects will

not be included in the new methodology. The parameters of interest are the overall thermal

profile and the peak temperatures, particularly at the stagnation point since this can be readily

compared to theory for accuracy. Current aeroprediciton tools that are commonly used by the

Navy suggest 10-percent to 20-percent error for certain parameters. Considering that this

capability does not currently exist, it is reasonable to assume some error when compared to the

validation cases. A methodology that falls within 10-percent to 15-percent average variation for

thermal profile would be deemed acceptable considering the current state of the art and cost

savings. Since stagnation point can be accurately calculated via theory, the stagnation point

predicted by this methodology should be approximately 5-percent accurate.

The methodology developed in this study couples the Sandia National Laboratories

aerodynamic heating codes BLUNTY and MAGIC (Section 3.0, Reference 11-13) with a full

thermal finite element model of the desired projectile using the finite element code, ANSYS

(Reference 33). MAGIC solves the boundary layer momentum energy equations, and it provides

recovery enthalpy, heat transfer coefficient, and surface pressure for various points along a given

geometry, as well as each point along the provided trajectory. Using the pressure-temperature-

13

enthalpy tables developed in the Aerotherm Chemical Equilibrium (ACE) code at Sandia

National Laboratories for standard air, a hot wall enthalpy can be obtained using the current local

wall temperature from ANSYS and the wall pressure from MAGIC. An updated heat flux

boundary condition is calculated using the enthalpy-based heat transfer shown in Equation (7):

)( wr hhCHq −=′′ , (7)

where

q" = heat flux,

CH = heat transfer coefficient (MAGIC),

hr = recovery enthalpy (MAGIC), and

hw = wall enthalpy (look up table).

Section 4.0 provides explicit detail about the methodology developed by this study, as

well as the method validation steps. The resulting method fills a capability gap by enabling the

user to establish a thermal time history of a geometrically accurate projectile for a typical

trajectory of interest in an efficient manner. ANSYS can be used to solve full-up system models

to capture the thermal response of temperature-critical subassemblies vital to projectile

performance. In comparison to computational fluid techniques, this method can save orders of

magnitude of CPU hours (CPUH) (CPUH = number of CPUs × hours running solution). A

validation case of 5 seconds at Mach 10 flight conditions with this methodology costs less than

4 CPUH, whereas the CHT CFD solution would take approximately 5,000 CPUH.

14

2.0 CURRENT AERODYNAMIC HEATING CODES

During this study, an assessment of the current aeroheating capabilities and available

codes was performed. Many of these codes were developed for very specific applications and

are exceptional tools for their intended uses. Over the years some attempts have been made to

generalize the codes. However, these codes have typically not included sharp, slender,

hypersonic objects launched at sea level. This section provides general background information

on the various tools readily available nationwide. Section 3.0 further details the capabilities and

limitations of the various codes by exercising them through a range of possible hypersonic

projectile trajectories. This will establish their relevance to the current problem area.

2.1 AEROPREDICTION CODE 2005

Dr. Frank Moore of NSWCDD originally developed the Aeroprediction Code 2005

(AP05) in 1972 (Reference 14). Initially, the code was established for body-alone aerodynamics

of projectile and missile systems at low angle of attack and Mach numbers less than 3. Various

additional modules were subsequently included to build this highly capable aeroprediction tool.

Currently, the code can accurately predict weapon aerodynamics of full configurations—such as

wing, tails, canards, etc.—as well as 3-degree-of-freedom (3DOF) models and, of relevance to

this study, a limited amount of aeroheating.

AP05 is a semi-empirical code that incorporates a large amount of aerodynamic data,

particularly from the tri-services database and theoretical methods. Overall, the aerodynamic

predictions of this code are quite accurate, considering the efficiency of the code. Quick

engineering trade studies can be performed at minimal expense.

15

In the late 1980s and early 1990s, the development of new technologies began pushing

the flight envelope into the high supersonic to hypersonic regimes in response to the

requirements of missile defense programs. By crossing into this flight regime, two important

phenomena become increasingly important: real gas effects and aerodynamic heating both of

which were incorporated into the Aeroprediction code.

Provided with current flight conditions (i.e., velocity, altitude, and wall temperature and

geometry), AP05 can calculate and generate heat transfer data. For the stagnation point heat

transfer rate, AP05 uses standard correlations that can be found in Anderson (Reference 6).

Equation (8) gives the heat transfer coefficient as

waw

w

TTqH−

= , (8)

where

H = total enthalpy,

T = temperature, and

aw = adiabatic wall.

Equation (9) gives the heat transfer rate as

)(Pr763.0 006.0

wawe

w hhdxduq −= − µρ , (9)

where

Pr = Prandtl number,

ρ0 = stagnation density,

µ0 = stagnation viscosity, and

dxdue = velocity gradient.

16

Equation (10) provides the adiabatic wall temperature as

)( 0 eceaw TTrTT −+= , (10)

where

rc = recovery factor,

Te = boundary layer edge temperature, and

To = stagnation temperatures.

Equation (11) provides the stream-wise velocity gradient as

0

0 )(21ρ

∞−=

pprdx

du

N

e , (11)

where

rN = nose-tip radius,

p0, = stagnation pressure, and

p∞ = free stream pressure.

Finally, Equation (12) provides the adiabatic wall enthalpy as

(12) )( 0 eceaw hHrhh −+= ,

where

he = enthalpy at boundary layer edge,

H0 = total enthalpy, and

rc = Pr1/2 (laminar) = Pr1/3 (turbulent) recovery factor.

Once the conditions are known at the stagnation point, the AP05 code solves for the heat

transfer and adiabatic wall temperature at each location. Transition to turbulence is determined

via a user-defined critical Reynolds number. The laminar heat transfer rate and the turbulent

heat transfer rate are calculated using Equations (13) and (14), respectively, and the variables

defined by Equations (15) through (17) (Reference 14):

17

)()(Pr332.0 667.0, waw

Ne

blw hh

NRVq −=∗

∗−∗ ρ , (13)

and

)(

ln

)(Pr185.0 584.2667.0

, waw

Ne

btw hh

NR

Vq −

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

∗

∗−∗ ρ , (14)

where = conditions that refer to values taken at the Eckert reference enthalpy,

(References 14 and 15), ∗ ∗h

)(22.0)(5.0 eawew hhhhh −++=∗ , (15)

)(2 0 eb hHV −= , and (16)

∗

∗

=µ

ρ sVR bNe , (17)

where s = boundary layer running length, and

N = 3 (laminar) = 2 (turbulent) Mangler transformation factor.

AP05 outputs the calculated heat transfer coefficient and the adiabatic wall temperature at

numerous surface locations along the axis of the body for a given flight condition.

2.2 BLUNTY

BLUNTY (Reference 10) was originally developed in the late 1960s at Sandia to capture

the aerodynamic heating rates for standard sphere-cone geometries. The procedure uses curve

fits to shock shape and pressure distributions obtained from tables defined by runs of the NASA

Ames Flow Field (NAFF) code (Reference 10). The tables developed are for cone angles 5

through 9, 9.5, and 10 through 15, at Mach numbers of 2, 4, 6, 8, 10, 15, 20, 25, and 30. Figure 7

depicts the overall geometric basis for the flow calculations.

18

Shock

Boundary Layer

Nose Tiprs

β

y

Free stream properties, ∞

Props immediately behind shock wave, s

Boundary Layer edge props, e

Reference props, *

Wall props, w

Stagnation point, o

Shock

Boundary Layer

Nose Tiprs

β

y

Free stream properties, ∞

Props immediately behind shock wave, s

Boundary Layer edge props, e

Reference props, *

Wall props, w

Stagnation point, o

FIGURE 7. BLUNTY FLOW FIELD GEOMETRY (REF. 10)

The boundary layer edge properties are determined via a stream-tube mass balancing

technique that couples the inviscid flow field to a boundary layer mass flux correlation. Both

ideal gas and thermodynamic equilibrium air models are available. Heat transfer rates are

calculated based on Fay and Riddell (stagnation); Kemp, Rose, and Detra (laminar); and a

modified Rose, Probstein, and Adams (turbulent).

Initially the free stream properties must be calculated based on the 1962 United States

Standard Atmosphere (up to 90 kilometers (295,276 feet)). The atmosphere tables are linear

segments of a geo-potential altitude defined by a base molecular temperature (Tm) and a

molecular temperature gradient, Lm´. Based on the physical altitude, a geo-potential altitude can

be calculated by Equation (18):

∞

∞

+×

=altaltaltgp 208809401

208809401 , (18)

where alt∞ = physical altitude, and

altgp = geo-potential altitude.

19

A distance, ∆, is calculated as the distance between the actual geo-potential altitude and

the nearest tabulated altitude. Temperature and density can be obtained for an altgp less than

295,276 feet using Equation (19) with mTT =∞ :

)exp( 0

mB RT

g ∆−=∞ ρρ ; (19)

Temperature and density can be obtained for an altgp greater than 295,276 feet using the

definition for given in Equation (20) with Equation (21): ∞T

∆′+=∞ mm LTT , (20)

and

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−

′−= ∞

∞mm

B TT

LRg ln1exp 0ρρ , (21)

where g0 = gravity at sea level

B = tabular value, and

Lm´ = molecular temperature gradient.

The remaining free-stream properties are calculated from the standard equations (in

English units) once density and temperature are known, provided in Figure 8.

20

∞

∞∞ = a

UM

∞∞ = RTa γ

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+−=

∞

∞∞ 72.198

)826968.2(23

TTEµ

∞

∞∞∞ = µ

ρ UrRe

∞∞∞ = RTP ρ R = gas constant Pressure:

Sound speed: γ = spec. heat ratio

Mach Number:

Viscosity (low-temp):

r = nose radius Reynolds Number:

h∞ = static enthalpy, tabulated by P, ρ 2

21

∞∞ += UhHTotal Enthalpy:

FIGURE 8. STANDARD EQUATIONS FOR CALCULATING FREE-STREAM PROPERTIES (REF. 10)

Tabulated shock shape data are built into the code, so the properties just behind the shock

can easily be calculated. Figure 9 depicts the velocity vectors across the shock at an angle to the

normal streamline of the shock angle β.

U∞

u∞v∞

Us

us vs

streamline

shock

β

U∞

u∞v∞

Us

us vs

streamline

shock

β

βsin∞∞ =Uu βcos∞∞ =Uv

βsin∞∞ = MM N

FIGURE 9. VELOCITY VECTORS ACROSS SHOCK (REF. 10)

With these parameters, the pressure behind the shock, density, enthalpy, and velocity

vectors can be obtained via an iterative process using the equations in Figure 10.

21

∞∞∞

∞ +−

= uuPPu s

s ρ

21∞∞

∞

∞

−+

=

uPP s

s

ρ

ρρ

∞= vvs22

21

ssss vuhH ++=

( )⎥⎦

⎤⎢⎣

⎡+

−−= ∞

∞ 112 2

γγγMPPs

FIGURE 10. PRESSURE BEHIND SHOCK, DENSITY, ENTHALPY, AND VELOCITY (REF. 10)

If Hs is within a user defined tolerance, ε, of the total enthalpy, H, then the pressure

behind the shock is Ps. Otherwise, the increment Ps´ = 1.05Ps. The process is continued using a

Newton-Raphson method until the tolerance is met. Knowing Ps, the stagnation pressure can be

calculated, Peo, using Equation (22):

1

12

1 2)1(1

−

⎥⎦⎤

⎢⎣⎡ −+=

γγ

γsseo MPP , (22)

where

γ = ratio of specific heats, and

M = Mach number.

Entropy is constant along a streamline, so the values of seo and Peo are known. Using

pressure-entropy tables, the value of the stagnation point enthalpy can be obtained, heo.

Assuming that the velocity at the stagnation point is zero, ueo = 0, then the total enthalpy, Heo,

equals the stagnation point enthalpy, heo. Again an iterative approach is used. If Heo is not

within ε of H, then a new Peo´ = 1.000001Peo is chosen. This process is repeated until the

tolerance is met and the new stagnation point velocity can be calculated as Equation (23), if

(H-heo) > 0,

(23) )(2 eoeo hHu −= ,

and as Equation (24), if (H-heo) < 0,

22

(24) 0=eou .

The viscosity behind the shock is calculated by Cohen (Reference 16) using Equation (25):

992.0

13329.0

3329.0

1

11

10213.11

10213.11

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=PP

hh

hh

ref

ref

ρµρµ , (25)

With href defined by Equation (26),

2

2

8119.2sftEhref = . (26)

Finally, a similar scheme is used to solve for the values along the boundary layer edge for

the rest of the body. The first step is taken just off the stagnation point. It is assumed that the

stagnation streamline is swallowed at this point, and thus, the entropy at this point is the same as

the stagnation point entropy. Knowing Pe from the shock and pressure tables and the entropy at

this new point (call it Point 2), then the remaining properties can be obtained from tables or

calculated with the previously defined equations. With the new enthalpy value, the velocity

component, ue2, can be calculated using Equation (27):

)(2 eoeo hHu −= . (27)

Using this value and conservation of mass principles, a more accurate prediction of

streamline swallowing can be obtained and thus a new entropy value. This procedure is used for

the remainder of the body, using estimates of entropy from the previous step as an initial

estimate.

Before the heat transfer can be calculated, the flow must be defined as laminar or

turbulent. BLUNTY allows two options to determine transition, momentum-thickness Reynolds

number or flight test data. At each station along the body, the Reynolds number is calculated

using the boundary layer edge properties with Equation (28),

23

5.0

0

2664.0Re ⎥⎦

⎤⎢⎣

⎡= ∫

s

eeee

dsrur

µρµθ , (28)

where

r = radius at location,

µ = viscosity,

u = velocity component,

ρ = density, and

s = arc length along curvature.

For the stagnation region, Fay and Riddell theory is used to calculate the heat flux

(Reference 17) using Equation (29),

⎭⎬⎫

⎩⎨⎧ −+−⎥

⎦

⎤⎢⎣

⎡ −= ∞

HhLehHPP

RD

woeo

eowo052.0

25.0

04.01.05.00 )1(1)()(2)()()2.32(94.0

ρρµρµq , (29)

where

R = nose radius,

Le = Lewis number = 1.4,

w,e = wall and edge properties, respectively,

hD0 = atomic dissociation energy times atom mass fraction in Btu/lbm (Reference 18).

Kemp, Rose, and Detra theory are used for laminar heat transfer rates, as calculated in

Equation (30) (Reference 19),

⎭⎬⎫

⎩⎨⎧ −+

⎭⎬⎫

⎩⎨⎧ −+

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛

=

∫ HhLe

HhLe

q

dsrudsdu

ruqD

De

s

ewo

ewo

ewl

052.0

52.0

05.0

0

2 )1(1

)1(1

)()(

)(5.0

ρµρµ

ρµ , (30)

where

24

hDe = hD0 taken at edge properties,

and the velocity gradient is calculated using Equation (31),

eo

o

o

e PPdsdu

ρ)(2 ∞−

=⎟⎠⎞

⎜⎝⎛ . (31)

Last, the turbulent heat transfer theory taken from Rose, Probstein, and Adams

(Reference 20) is given as Equation (32),

[ ]CfiC

sFhHs

q fwr

e

e

t2.08.0

32 )()(Re

Pr

029.0−=

µ , (32)

where Equation (33) defines

⎭⎬⎫

⎩⎨⎧ −+=

HhLe De

fi

f )1(1 β

CC

, (33)

when 1>fi

f

CC

, and Equation (34) defines

2.08.0

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

∗∗

eefi

f

CC

µµ

ρρ , (34)

when 1<fi

f

CC

,

where

Cf,fi = local and incompressible skin friction, respectively;

Pr = Prandtl number;

Hr = recovery enthalpy;

hw = enthalpy at wall; and

25

F(s) = shape factor, defined by Equation (35):

( )∫

= s

eee

eee

sdru

russF

0

41

45

49

41

45

49

µρ

µρ . (35)

Using the previously described procedures, BLUNTY will output a heat deck for each

point along a given trajectory that includes recovery enthalpy, cold wall heat flux, heat transfer

coefficient, edge pressure, and edge velocity for each station along the body. Validation studies

of BLUNTY have shown agreement with stagnation point predictions by using Fay and Riddell

theory; however, BLUNTY tends to under predict off stagnation point conditions. In the

methodology developed (Section 4.0) BLUNTY will be used to obtain the stagnation point

boundary conditions.

2.3 SANDIA ONE-DIMENSIONAL DIRECT AND INVERSE THERMAL CODE

The Sandia One-Dimensional Direct and Inverse Thermal Code (SODDIT) (Reference

34) was developed by Sandia to account for in-depth conduction and ablation effects on reentry

vehicles. The code accepts enthalpy-based aeroheating decks from codes such as BLUNTY and

allows them to be applied as boundary conditions. With the heat flux boundary conditions

applied, a one-dimensional (1-D) heat conduction solution is obtained for the nodal network

described by the geometry.

A given control volume is governed by the energy diffusion equation, shown in

Equation (36),

)()()()(

1 TQxTxATk

xxAdtdTCp +⎥⎦

⎤⎢⎣⎡

∂∂

∂∂

=ρ , (36)

where

Cp = specific heat at constant pressure,

A = area, and

k(T) = temperature-dependent thermal conductivity.

26

The boundary condition is expressed as Equation (37) (Reference 34),

[ ])(),()()())(()( 0

40

40000 tTtxTTtqThhCtq radwwradwwwrh −−′′+−≡′′ σεα , (37)

where

q0″ = heat flux per unit area,

Ch = heat transfer coefficient,

α = thermal diffusivity,

hr,w¸ = recovery and wall enthalpy,

qrad″ = radiation heat flux,

ε = emissivity,

T = temperature, and

w0 = wall and space values.

SODDIT can model three geometric options: flat plate, curved surface (i.e., cylinders

and cones), or sphere (Reference 34). The boundary conditions are applied consistently over the

entire surface depending on the geometry option selected.

The code accounts for ablation using an approximate energy dissipation method, Q*,

which approximates melting ablation. Q* is a measurement of the amount of energy required to

melt a one-pound mass of material. These data have been obtained for numerous melting

materials via thermal tests. SODDIT uses these data to calculate the rate of material loss as well

as the approximate total ablation of the material. As the surface reaches the melting temperature,

a certain amount of mass is removed based on the Q* value. The nodal network is adjusted and

the calculation continues repeating the ablation procedure for the entire trajectory.

2.4 MAGIC

The MAGIC (Reference 11-13) suite of tools is a front-end run stream that executes a

nose-tip code, an inviscid afterbody code, and a boundary layer code. The inviscid codes

27

develop the free-stream conditions and the streamline properties. This information then feeds to

the boundary layer code to calculate a boundary layer solution, thus providing aerodynamics,

aeroheating, and flow field properties for complex reentry vehicle configurations. Ideal gas and

equilibrium air models are available.

The default nose-tip code, General Electric Two-Dimensional (2-D) Inviscid Transonic,

solves the non-conservative form of the unsteady Euler equations in a spherical coordinate

system using a radial grid (Reference 11). A centered finite difference algorithm is used to solve

the transformed equations.

SANDia Inviscid Afterbody Code (SANDIAC) (Reference 12) is an extensively updated

version of the General Electric Three-Dimensional (3-D) Inviscid Supersonic Code. Both the

non-conservative and conservative forms of the Euler equations are solved in a cylindrical

coordinate system, (r,φ,z). The non-conservative form of the Euler equations in Cartesian

coordinates are given in Equations (38) through (42),

0=∂∂

+∂∂

+∂∂

+∂∂

+∂∂

+∂∂

zw

zw

yv

yv

xu

xu ρρρρρρ , (38)

01=

∂∂

+∂∂

+∂∂

+∂∂

xp

pzuw

yuv

xuu , (39)

01=

∂∂

+∂∂

+∂∂

+∂∂

yp

pzvw

yvv

xvu , (40)

01=

∂∂

+∂∂

+∂∂

+∂∂

zp

pzww

ywv

xwu , (41)

0=∂∂

+∂∂

+∂∂

+∂∂

+∂∂

+∂∂

zpw

ypv

xpu

zwp

yvp

xup γγγ , (42)

28

where

p = pressure,

u,v,w = x,y,z velocity components,

ρ = density, and

γ = ratio of specific heats.

Equations (43) through (52) are preparatory for Equations (53) through (56). For

simplification, these variables are defined as

zdd∂∂

=′ , (43)

pP ln= , and (44)

ρpF = . (45)

Now, transforming the Cartesian to a cylindrical equation defines the following variables as

22 )( ydxr −+= , (46)

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

= −

dyx1tanφ , and (47)

zz = ,

giving the transformation in Equation (48),

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂∂∂∂∂

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥⎥⎥⎥

⎦

⎤

∂∂∂∂∂∂

Z

Y

X

YXYXYX

z

r

zz

rr

100

φφφ

⎢⎢⎢⎢⎢⎢

⎣

⎡

. (48)

29

Finally, the velocity components of the two coordinate systems must be defined for Equations

(53) through (56). For the Cartesian coordinate system, these variables are defined as

φφ cossin vuu −−= , (49)

(50)φφ sincos vuv −= , and

ww = .

For the cylindrical coordinate system, these variables are defined as:

φφ cossin vuu +−= , (51)

φφ sincos vuv −−= , and (52)

ww = .

Applying these substitutions and the transformation yields Equations (53) through (56):

0sincos =⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −′−++++++ φ

φφ

φφγ wr

wdwru

rv

uPwPrvPu rzrzr , (53)

0~~~ =+−++ rzr FPrvvwuu

rvuu φ , (54)

01~~~ =+−++ φφ FPrr

vuwvvrvvu zr , (55)

and

0sincos~~ =⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −′−+++ φφ

φφ Pr

dPFwwwrvwu zzr . (56)

The entropy equation transforms into the cylindrical coordinate system in Equation (57),

0~~ =++ zr SwSrvSu φ , (57)

where the new velocity components are written as Equation (58) and Equation (59):

φcos~ dwuu ′−= , (58)

30

and φsin~ dwvv ′+= . (59) Equation (60) defines the density derivative,

iSii x

PaP

PxP

x ∂∂

=∂∂

∂∂

=∂∂

2

ρρ , (60)

where Equation (61) defines the acoustic speed, a, as

S

Paρ∂∂

=2 , (61)

and Equation (62) defines the isentropic exponent, γ, as

Pa 2ργ = . (62)

Now, the velocity ratios are simplified as shown in Figure 11.

τσ

τσ

wvwu

wvwu

====

//

τσ

φττφσσ

~~~~

sin~cos~

wvwu

dd

==

′+=

′−=

FIGURE 11. VELOCITY RATIOS SIMPLIFIED

Substituting the above terms and rearranging and simplifying the yields produce Equations (63)

through (66):

0sincos11~~2 =⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛ −′++++++ zrrzr PP

rPdF

wrrPP

rP φφφ

φφστσγτσ , (63)

0sincos1~~~22 =⎥

⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛ −′+++++ zrrrz PP

rPd

wFFP

wrr φφφφσττστσσσ , (64)

0sincos~~~

2 =⎥⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛ −′+++++ zrrz PP

rPd

wF

rwFP

rr φφ

φφφττστττστ , (65)

31

and

0~~ =++ zr SSr

S φτσ . (66)

The z momentum equation is dropped and w is evaluated in terms of total enthalpy, as

shown in Equation (67):

( )222 121 τσ +++= whHtot . (67)

Equation (68) defines m as

2wFm = , (68)

and the collected terms are then placed in matrix form, as shown in Equation (69),

0ˆˆˆˆ =+++ cdNdMdL rz φ , (69)

where the variables are defined in Equations (70) through (74):

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−

=

10000100010001(

ˆmmm

Lτσγ

, (70)

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

′+′

′+

=

σσφτ

σφσγφγσ

~0000~0cos00~cos00)cos~(

ˆdm

mdmdm

M , (71)

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

′−

′−

′−

=

ττφτ

τφσγφγτ

~0000~0sin00~sin00)sin~(

ˆdmm

dmdm

N , (72)

(73)

32

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡−

=

0

~~1ˆτσττ

σγ

rc , and

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

S

P

dτσ

. (74)

Capturing the discontinuities in the solution requires the conservative form of the Euler

equations (Reference 12), as shown in Equation (75),

0ˆˆˆ=

∂∂

+∂∂

+∂∂

γG

xF

zE , (75)

where the variables are defined Equations (76) through (78).

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+=

wpewvwu

pww

E

)(

ˆ

2

ρρ

ρρ

, (76)

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+=

upeuv

puuwu

F

)(

ˆ 2

ρρρρ

, and (77)

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

++

=

vpepv

vuvwv

G

)(

ˆ2ρρρρ

. (78)

33

Similarly, the non-conservative form of the Euler equations transforms them into the

local cylindrical coordinate system with a moving axis d′, as shown in Equation (79),

0=+∂∂

+∂∂

+∂∂ H

YG

XF

ZE , (79)

where the variables are defined with Equations (80) through (88):

EJ

E 1= , (80)

( )GXFXEXJ

F rz φ++=1 , (81)

( )GYFYEYJ

G rz φ++=1 , and (82)

HJ

H 1= , (83)

where

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+=

wpevwuw

pww

E

)(

2

ρρ

ρρ

, (84)

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

++

′−=

upepuu

vudpwu

u

F

~)(

~~

cos~~

ρρ

φρρ

, (85)

(86)

34

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+

′+=

vpeuv

pvvdpwv

v

G

~)(

~~

sin~~

ˆ

ρρ

φρρ

,

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+−+

′−=

upevvuuuvuv

dpuwu

H

~)()~~()~~(

cos~~

ρρ

φρρ

, and (87)

(88)rr YXYXJ φφ −= ,

where

J = Jacobian of the transformation.

Last, to obtain the boundary layer solution, the Hypersonic Integral Boundary Layer

Analysis of Reentry Geometries (HIBLARG) is used. This code solves the integral forms of the

momentum and energy equations along the inviscid streamlines for a 3-D flow field

(Reference 13). Momentum is given in Equation (89),

( )

211 2

22f

eeeeee

CU

dsd

Udsdh

hdsdp

UH

dsd

+⎥⎥⎦

⎤

⎢⎢⎣

⎡−−= ρρρ

θθ β

β , (89)

and energy is given in Equation (90),

( ) ( )( )⎥⎥⎦

⎤

⎢⎢⎣

⎡−

−+−

−−

= wteewteewt

wrh hhU

dsd

hhUdsdh

hhhhhC

dsd ρ

ρφφ β

β

11

, (90)

where Equations (91) through (94) define the variables as

35

ee

wf U

Cρτ

=,

(91)

( )wree

wh hhU

qC−

=ρ ,

(92)

(93)

∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛−≡

δ

ρρθ

0

1 dYUU

UU

eee , (94)

∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛−≡

δ

ρρφ

0

1 dYhh

UU

eee , and where

Cf = the local skin friction coefficient,

Ch = the Stanton number,

θ = momentum, and

φ = energy thickness.

Equation (95) defines the boundary layer shape factor, H, as

θδ *

≡H,

(95)

and Equation (96) defines the entrainment shape factor, F, as

θδδ *−

≡F,

(96)

where

(97)∫ ⎟⎟

⎠

⎞⎜⎜⎝

⎛−≡

δ

ρρδ

0

* 1 dYUU

ee , where

∗δ = the displacement thickness.

36

Depending on the state of the flow, laminar or turbulent, different correlations are used to

obtain expressions for various parameters in the momentum and energy equations.

For laminar flow, the velocity profile assumed is given in Equation (98),

43

22 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

δδδYYY

UU

e , (98)

with shape factors H and F calculated with Equation (99) and Equation (100),

614.0029.3 −=

e

wl T

TH,

(99)

and

(100)

21

0378.0388.4521.1 ⎟⎟⎠

⎞⎜⎜⎝

⎛++=

e

wl T

TF.

Theory and data for incompressible flow over an isothermal flat plate give the skin

friction number in Equation (101),

θRe245.0

0,, =lfC,

(101)

and give the Stanton number in Equation (102),

φRePr

22.0

340,, =lhC

.

(102)

The turbulent velocity profile assumption is given in Equation (103),

n

e

YUU

1

⎟⎠⎞

⎜⎝⎛=δ ,

(103)

37

where Equation (104) defines n as

θ

θ

Reln141.079.2Reln37.0

−+

=n.

(104)

Currently, HIBLARG (Reference 13) uses values for the shape factors that were derived

during the code upgrade to the ABRES Shape Change Code (ASCC) of 1980. The shape factors

are obtained from curve fits from numerical integration of a modified turbulent power law

velocity profile, shown in Equation (105),

[ ] ( )[ ]θ

θRelog0.60.1 1010Re 6 −+=

=DHHt , (105)

where the reference value is defined by Equation (106) and the variables by Equations (106)

through (111),

( )CBMAH exp610Re

==θ , (106)

where

(107)

00 /0231.00064.0 ATTAA

eT

w

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+=

,

( )⎩⎨⎧

><

=−

5.1817.35.1/35.0 5.0

MMTTB eTw

,

(108)

(109)

4.01−

= eeMM γ

, and

(110)

⎩⎨⎧

><

=5.129.05.125.1

MM

C,

where

( )eTw TTA /20.102.00 +−= . (111)

Equation (112) expresses the entrainment shape factor as

38

( )[ ] ( ) 80.3exp/5.0 2

22.01 += − MBTTBF

eTwt , (112)

where Equations (113) and (114) define the variables as

( )4

101 10Relog5.14.9 −×+= θB , and (113)

( )

eTw TTB /05.007.02 +−= . (114)

Similarly, theory and data for turbulent flow over an isothermal flat plate yield a skin

friction number, shown in Equation (115) (Reference 13),

( ) 5262.1

100,, Relog

Re100Re010742.0

Re245.0

2−

++= θ

θ

θ

θ

tfC

, (115)

and the Stanton number, given in Equation (116),

( )( ) bth

aC −

++= φ

φ

φ

φ

RelogRe100Pr

Re

RePr

22.0105.0

340,,

,

(116)


( )[ ]Pr8.20.1

86.0Pr14.00.1038.0 2

+−−

=a, and

(117)

( )Pr13532.0exp7174.1 −=b . (118)

Four options for transition criteria are available to the user, GE low and high mass

addition criteria, LORN, TRW, and NASP (Reference 13). The four criteria are based on some

type of experimental data, but for most hypersonic systems, the LORN criterion is the preferred

method (Reference 13), which is shown in Equation (119):

)134.0exp(275Re , etr M=θ . (119)

As with BLUNTY, the standard heat transfer correlations are used in Equation (120),

39

( )wrheew hhCUq −= ρ , (120)

and the recovery enthalpy is shown in Equation (121),

2

2e

erURhh +=

, (121)


21

Pr=R (laminar recovery factor (Reference 13)), and (122)

31

Pr=R (turbulent recovery factor (Reference 21)). (123)

Depending on the mode, MAGIC can output aerodynamic data, aeroheating data, or

simple flow field properties. The output for the aeroheating deck contains recovery enthalpy,

cold wall heat flux, heat transfer coefficient, edge pressure, and edge velocity for each station

along the body. Unlike BLUNTY the momentum integral methods used by MAGIC do not

accurately predict stagnation point properties but they are much more accurate for the off

stagnation point locations. MAGIC will be used in conjunction with BLUNTY to establish the

full aeroheating boundary conditions detailed in Section 4.0.

2.5 ABRES SHAPE CHANGE CODE (ASCC86)

ASCC (Reference 22, 23) was developed in the late 1970s to capture the axisymmetric

shape change phenomena and material response of reentry vehicle nose tips. Since its release,

ASCC has continuously undergone improvements and expansions to incorporate the advances in

reentry technology.

ASCC86 uses semi-empirical correlations specialized for certain regions of the nose tip

and afterbody to predict surface pressure distribution. Similar to thin shock layer theory, the

shock shape and standoff distance are predicted with an integral solution of the momentum and

continuity equations. Figure 12 shows the three defined surface pressure partitions.

40

R*

Sonic PointΙ

Ι Ι

Ι Ι Ι

FIGURE 12. SURFACE PRESSURE PARTITIONS (REFERENCE 22)

The region from stagnation to sonic point along the body is Region I. The pressure

calculation is based on the Dahm extension of the Love correlation (References 22 and 23), as

shown in Equation (124) and the variables defined in Equations (125) through (129):

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎥

⎦

⎤⎢⎣

⎡−−

−−=max

*

*

11

1RR

PPPPPP NMN

FDMN (124)

( ) ( ) ( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−+−+−+−⎟⎠⎞

⎜⎝⎛ − ∞∞ *

*2

**2

* 11cos11

21cos11

PPPPP

ssP

ssP

ss MN

FDFD θθ ,

where

0PPP =

, (125)

RN = stagnation point radius of curvature,

Rmax = maximum (RN, R*),

R* = distance from sonic point to body axis,

s = surface stream length from stagnation point,

θ = angle of local tangent with body axis,

* = sonic point conditions,

( ) θ2sin1 ∞∞ −+= PPPMN , (126)

1*

12 −

⎟⎠⎞

⎜⎝⎛

+=

λλ

λP

,

(127)

41

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛−−−= −− λλ e

ssPePFD

2

**

16111

, and

(128)

⎟⎠⎞

⎜⎝⎛= *ln5

ssλ

, (129)

where

P0 = stagnation point pressure at sea level.

From the sonic point to afterbody shoulder is Region II. Modified Newtonian theory is

used for the pressure distribution in this region, MNP , from Equation (126). To account for the

smooth transition between Regions I and II, a blending scheme is used to smooth the pressure

distribution.

Downstream of the shoulder is Region III, which uses a correlation developed at

Aerospace Corporation based on blast wave theory (for Mach 5 and greater flow)

(Reference 24). Below Mach 5, Region II correlations are used, shown in Equation (130),

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ += c

D

cn

c

p

Crzf

Cθθ

θ,1

2

2

, (130)

where

θc = cone half angle,

z = axial distance from the start of the aft cone,

r = radius at start of aft cone, and

CD = drag coefficient.

The function, , is determined from a series of curve fits to exact numerical solutions of

varying bluntness cones.

nf

42

The next step in the procedure is to calculate the bow shock shape. This is done

by solving the integral continuity and axial momentum equations. Similarly, the boundary layer

solution is derived from solving the integral momentum and energy equations,

Equation (131) and Equation (132), respectively:

( ) ( )

dsdP

uH

uuvC

urdsd

ur ee

wfwfe

e22

,22 2

1ρθ

ρρ

θρρ

++=,

(131)

and

( ) ( )( ) ( ) ( )( )wetee

wwftw

wet

wrhwetee

wetee hhuhhv

hhhhChhur

dsd

hhur −−

+−−

=−− ,

,,

,,

,

1ρρ

φρρ .

(132)

Equation (133) expresses the entrainment relation,

( )∫−=∞∞

s

we dsvrrFyu0

2 2Re2 ρµρ θ

, (133)

where

w = wall,

e = boundary layer edge, and

f,w = properties obtained through an isentropic expansion from the stagnation point to the local pressure.

From this point, the boundary layer solution is very similar to that of HIBLARG with

slight modifications to the laminar shape factor, shown in Equation (134) and Equation (135),

614.0029.3 −=

w

elH

ρρ

, (134)

and

21

0378.0388.4521.1 ⎟⎟⎠

⎞⎜⎜⎝

⎛++=

w

elF

ρρ

,

(135)

and to the turbulent shape factor, shown in Equation (136) and Equation (137),

43

( ) 96.02.31285.2 −+= −

w

ent eH

ρρ

, (136)and

( )58.2ln28.5 −+⎟⎟

⎠

⎞⎜⎜⎝

⎛+= nF

w

et ρ

ρ

, (137)

where Equation (138) defines the variable n as

θ

θ

Reln14.079.2Reln37.0

−+

=n.

(138)

Last, the skin friction and Stanton number for laminar flow are the same as used in

HIBLARG, shown in Equation (139) and Equation (140), respectively:

θRe245.0

0,, =lfC ,

(139)

and

φRePr

22.0

340,, =lhC

,

(140)

with some modification for the turbulent values, given in Equation (141) and Equation (142):

( ) 6.1

100,, Relog

Re100Re0123.0

Re245.0

2−

++= θ

θ

θ

θ

tfC

, (141)

and

( )( ) 6.1

105.0340,, Relog

Re100PrRe0123.0

RePr

22.0 −

++= φ

φ

φ

φ

thC

.

(142)

44

Unlike MAGIC, ASCC contains its own in-depth thermal conduction model. With this

the code can account for various phenomena that influence the heat transfer such as transpiration,

surface roughness, fluid properties at varying Mach numbers, proximity to the transition

location, and particle interaction. Detailing these phenomena is beyond the scope of this paper

but can be found in depth in Reference 22.

The in-depth thermal response and ablation modeling incorporate an explicit stationary

grid to capture the in-depth response, while a moving implicit grid captures the ablation front.

Figure 13 displays the grid setup.

FIGURE 13. EXPLICIT AND IMPLICIT GRIDS (REFERENCE 22,23)

A solution is obtained for the moving orthogonal implicit grid by the governing equation

shown as Equation (143),

⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

+⎥⎦

⎤⎢⎣

⎡

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∂∂

+∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=∂∂

sT

rTsc

rTk

rrr

rsTk

rr

rs

rrr

tTc p

cb

c

b

cb

p θρρ cot111

1

,

(143)

and the explicit in-depth grid shown in Equation (144),

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

=∂∂

yTyk

yyxTk

xtTcp

1ρ.

(144)

45

Based on the temperature and pressure of the surface, the recession rates and overall

recession are calculated. Material recession is defined by the thermo-chemistry deck of the

material, which for the most part comes from experimental data defining the chemical behavior

of the material.

2.6 AEROHEATING AND THERMAL ANALYSIS CODE (ATAC05)

ITT developed the ATAC (Reference 25) code from its predecessor ASCC. Both

ASCC86 (Sandia) and ATAC have a similar heritage, but various modifications and tweaks have

distinguished the two. ATAC was developed to analyze non-axisymmetric, 3-D nose-tip shape

change caused by angle of attack and/or non-uniform material roughness (Reference 25).

ATAC05 in particular had been modified to also include the Charring Material Thermal

Response and Ablation (CMA) procedure.

ATAC05 is an extremely useful tool with the ability to analyze phenomena in an efficient

manner. It is among the few codes that have an extensive particle impact model to incorporate

erosion effects.

Because ATAC05 is commercially available, information regarding most of the updates

is proprietary. For the purpose of this work it can be assumed that, for the most part, ATAC05

shares its underlying physics with ASCC. The additional modules are beyond the scope of this

effort.

2.7 GASP

CFD codes are codes of the highest level of fidelity that can be used for calculating the

aerodynamic heating boundary conditions. GASP, developed in 1993 by Virginia Polytechnic

Institute and State University (Virginia Tech), and AeroSoft, Inc., solves the Reynolds-Averaged

Navier-Stokes equations (RANS) to establish a flow field solution (Reference 26). For turbulent

flow a turbulent Prandtl number model is used to calculate the heat flux value across the

46

boundary layer. In contrast the MAGIC suite of tools solves the more simplified inviscid Euler

equations to establish inviscid flow field properties. In order to obtain the viscous heating

solution the inviscid properties are used as the boundary conditions for solving the integral forms

of the momentum and energy equations.

Within the last year, AeroSoft, Inc., with the aid of Dr. J. Schetz and Virginia Tech, has

incorporated a CHT module. By adding this solid conduction solver, the RANS equations can be

coupled with the heat conduction equations of an isotropic solid, as shown in Equation (145):

qzTk

zyTk

yxTk

xtTcv +⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

=∂∂ρ

. (145)

For finite volume implementation the conduction equation can be rewritten in integral

form as Equation (146):

( )( ) dVqdAnTkTdVt

cA

v ∫∫∫∫∫∫∫ +⋅∇=∂∂ ˆρ

. (146)

The conduction equation is solved over the entire mesh for a volume averaged

temperature, shown in Equation (147):

∫∫∫= TdVV1T

. (147)

The solution is iterated over a set number of cycles per time step in order to accurately

couple the fluid and solid solvers. Figure 14 depicts the fluid and solid zonal boundaries.

47

Fluid

SolidTs2

Ts1

Tf1

Tf2

Tw

FIGURE 14. FLUID/SOLID ZONAL BOUNDARY

In Figure 14, Tf and Ts represent the cell center temperatures of the fluid and solid,

respectively. The wall temperature, Tw, is the zonal boundary temperature. Therefore, the fluid

temperature at the wall and the solid temperature at the wall must be equal, as in shown

Equation (148):

wswfw TTT == . (148)

Similarly, from conservation of energy, the heat flux values must also be equal and

opposite, as shown in Equation (149),

swfw qq −= (149)


nTkq ffw ∂∂

−=, and

(150)

nTkq ssw ∂∂

−=.

(151)

If a heat source is present at the wall, then qfw is defined by Equation (152):

0qqq swfw −−= . (152)

48

At each cycle the wall temperature at the boundary, Tw, is calculated by setting the heat

flux values of the fluid and solid at the wall. The fluid zone is computed with this new wall

temperature. Fluxes and fluid properties are calculated at each cell face, boundary, or center,

depending on the parameter. All parameters are updated. The updated parameters are used

again to recalculate the wall temperature using the equal and opposite heat fluxes. The solid

zone is then calculated as defined by the solid conduction equations. The process is repeated for

each time step.

In comparison, the methodology developed (see Section 4.0) is not physically tied to any

outside fluid cells. In the finite element realm the boundary conditions are applied as heat loads

on the exterior face of the solid surface elements. At time equals zero the first heat flux

boundary condition is calculated based on the wall enthalpy (interpolated form enthalpy tables

using pressure and temperature), the heat transfer and fluid enthalpy (outputs from SNL

aeroheating codes). This heat flux is applied for a small increment in time resulting in a

temperature rise over the time increment.

Aerosoft, Inc., of Blacksburg, Virginia, has been in the process of validating the new

CHT model in GASP with experimental data. To date the results look very promising. Some of

these results are discussed in depth in a recent paper published by Aerosoft and Virginia Tech

(Reference 26).

49

3.0 AERODYNAMIC HEATING TOOL ASSESSMENT CAPABILITIES AND LIMITATIONS

During this study, various codes and tools were assessed for their capabilities and

limitations in the hypersonic regime. The overall process began at a very low fidelity level and

built up to the highest fidelity possible with CFD. In the available arsenal of tools, AP05, in

combination with semi-infinite solid correlations, was a good first approximation to the expected

nose-tip temperatures of the projectile. To capture thermal soak effects, the AP05 boundary

conditions were applied in a finite element code (ANSYS) for an in-depth thermal history. With

the acquisition of the various Sandia tools, analyses were performed to obtain an understanding

of the capabilities and limitations of these tools. These analyses primarily focused on nose-tip

temperatures and ablation concerns. Similarly, a trial version of ATAC 3-D was obtained in to

attempt a replication of the Sandia ASCC results.

From the studies performed, it is apparent that a capability gap existed in accurately

predicting thermal soak effects on a detailed system. None of the tools present in the open

source or commercially-off-the-shelf codes had the capability to efficiently couple hot wall

aerodynamic heating with a highly detailed, structural model for an entire flight trajectory.

The following discussion outlines the studies performed in assessing these tools, while

Section 4.0 details the methodology developed and the associated verification. Figure XX

summarizes the various attributes to the codes assessed in this section. As will be seen in

Section 4.0 BLUNTY and MAGIC will be one-way coupled with ANSYS to develop the new

methodology.

50

Code

Minimum Cone Angle (degrees)

Minimum Nose Radius (in.)

Flight Regime (Mach)

Detailed 2D Thermal

ConductionHot Wall Effects Cost

AP 05 None None All No No HighBlunty 5 None 6+ No Yes Low

SODDIT None None All No Yes LowMAGIC None None 6+ No Yes LowASCC Problematic 0.2 All Problematic Yes LowATAC Problematic 0.2 All Problematic Yes LowGASP None None All Yes Yes High

FEA Methodology None None 6+ Yes Yes Low

FIGURE 15. SUMMARY OF CODE CAPABILITES

3.1 AEROPREDICTION 2005/SEMI-INFINITE SOLID CORRELATIONS

At the onset of the EMRG program, initial rough order-of-magnitude estimates of nose-

tip temperature were required to begin defining the projectile trade space. Initially, AP05 was

the only readily available tool that could provide some limited aerodynamic heating data. Using

the 3DOF models available in AP05, a trajectory was established for the projectile based on a

2.5-km/s initial velocity and a launch angle (or quadrant elevation angle, (QE)) of 50 degrees

Figure 16 depicts the notional airframe geometry used for these studies.

30”

0.125” nose tip radius

3.6° cone angle

FIGURE 16. HYPERSONIC ROUND APPROXIMATE GEOMETRY

51

With the trajectory data points in hand, AP05 can estimate the heat transfer coefficient

and adiabatic wall temperature for each flight point for numerous points along the body.

Unfortunately, AP05 has never coupled the aerodynamic heating module with the trajectory

model; therefore, the aerodynamic heating boundary conditions have to be determined at each

point in the trajectory as an individual analysis. Consequently, developing the aeroheating

boundary conditions for a 300-second flight trajectory is a labor intensive process. At the

present time the code does not enable any type of automation and therefore prevents the use of

any type of scripting. Additionally, the boundary conditions are calculated based on a user input

wall temperature. To properly account for hot wall conditions, an inefficient and expensive

iterative process would have to be implemented. Figure 17 shows the boundary conditions

obtained from AP05 that were used in the semi-infinite solid prediction as well as the initial FEA

simulation.

Time (seconds)

Taw

(K)

Heat

Tran

sfer C

oeff.,

h (W

/m2-

K)

FIGURE 17. AP05 BOUNDARY CONDITIONS

The 1D semi-infinite solid correlation assumes that the axial direction (X) is much

greater than the radial direction. Equation (153) gives the resulting correlation with constant

convection boundary conditions from Incopera and Dewitt (Reference 27),

(153)

52

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟

⎠

⎞⎜⎝

⎛=−−

∞ kth

txerfc

kth

khx

txerfc

TTTtxT

i

i αα

αα 2

exp2

),(2

2

,

where

t = time,

Ti,∞ = initial and far field temperatures,

h = heat transfer coefficient,

k = thermal conductivity,

erfc w ≡ 1 – erf w (tabulated in most mathematics text books), and

α = thermal diffusivity.

Using the boundary condition table established in AP05, Equation (153) was solved in a

stepwise manner. Initially, the material starts at a standard temperature of 288 K at sea level.

The first boundary condition is applied for the duration of the step in the table. The resulting

temperature becomes the initial temperature for the second step. This process continues through

the entire flight trajectory. Initial estimates used pure tungsten as the material for the solid

body. Figure 18 shows the resulting term of response with the applied boundary conditions of

Figure 17.

0

500

1000

1500

2000

2500

0 50 100 150 200 250 300 350 400 450

Range (km)

Tem

pera

ture

(K)

Exo-atmosphereRadiation, conduction,

internal heating??

Estimated Nose Temperature Profile @ Maximum Range Launch over Endo- Exo- Endo- Atmospheric

3140oF, 1727oC

FIGURE 18. SEMI-INFINITE SOLID APPROXIMATION

53

Although the results in Figure 18 are approximate, they yield a baseline for future

comparison. The method used here is relatively easy to perform but labor intensive and

inefficient. In addition, the method does not consider hot wall boundary conditions nor does

geometry come into play. This method is not recommended for anything other than rough order-

of-magnitude estimates, but the following sections will show that other methods are more

accurate and efficient, culminating in the final, proposed methodology.

3.2 AEROPREDICTION 2005/ANSYS

The 1D semi-infinite solid method in the previous sub-section analyzed a block mass of

pure tungsten with the nose-tip boundary conditions applied. A further refinement to this

method would involve detailing the boundary conditions based on axial location along the

centerline. Figure 19 depicts such a boundary condition refinement. The new boundary

conditions are considerably more detailed than in the semi-infinite solid correlation; however,

due to the inefficiencies in AP05, further refinement would be exhausting and not much more

beneficial.

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

0 50 100 150 200 250 300 3500

500

1000

1500

2000

2500

3000h (Nose Region)h (Mid-Body)Taw (Nose Region)Taw (Mid-Body)

Time (s)

Heat

Xsf

erCo

eff.,

h (W

/m2-

K) Adiabatic Wall Tem

p (K)

FIGURE 19. REFINED AP05 BOUNDARY CONDITIONS

54

In this 2D (axisymmetric) refinement (Figure 19), the body was broken into a nose region

and a body region, with an average value applied in a “transitional zone,” as depicted in Figure

20. The heat transfer value was taken along the length and applied for a given time step as

defined by the trajectory. In Figure 21, the locations identified in bold represent the nodal

locations of interest that will be plotted as a function of time, while the italicized indicators show

the component and its material. The inset table included in Figure 21 references the actual node

number in ANSYS, as well as the location in the coordinate system. Table 1 lists the material

properties used for the analysis.

Nose BC’s

Transitional BC’s

Body BC’s

X = 1.956 in.

X = 2.2 in.

Mar aging steel airframe

FIGURE 20. BOUNDARY CONDITION REGIONS

55

Y – along centerline

X – radial from projectile axis

Tungsten Nose Fill

Tungsten Carbide InsertN1

N2 N3

Nose_Fill

Pellets


Node ANSYS Node No. X-Coord (m) Y-Coord (m) X-Coord (in) Y-Coord (in)Node_1 3180 0.0025 0.0138 0.0970 0.5421Node_2 1173 0.0196 0.3095 0.7717 12.1850Node_3 644 0.0339 0.5573 1.3346 21.9394

Nose_Fill 1625 0.0046 0.1755 0.1824 6.9094Pellets 481 0.0088 0.4179 0.3450 16.4508

FIGURE 21. ANSYS MODEL

56

TABLE 1. MATERIAL PROPERTIES

Material Thermal

Conductivity, k (W/m-K)*

Specific Heat, Cp

(J/kg-K)** Density (kg/m3)

Tungsten carbide 163.3 134 17,200 C-350 26.5 245 8,000 Tungsten 100 180 18,750 *W/m-K = watts per meter Kelvin **J/kg-K = joules per kilogram Kelvin ***kg/m3 = kilograms per cubic meter

Initially, the analysis was performed with no interior detail such as the nose fill or the

insert. This particular analysis yields an upper bound to the problem because the addition of the

thermal mass will only reduce the airframe temperatures. In addition, the reduction in included

model detail allows for faster solution times for more efficient trade studies.

For the airframe-only configuration, trends of the thermal time-history can be established.

Due to the lack of interior detail (i.e., tungsten nose fill, payload), the heat must be distributed

throughout the airframe, which is the worst-case scenario. Figure 22, Figure 23, and Figure 24

show the airframe-only temperature time histories.

Node_1

Time = 0 to 313 seconds Time = 0 to 10 seconds

Max Temp at Re-entry = 1040 K

1450 K

FIGURE 22. NODE 1 TEMPERATURE HISTORY

57

Node_2

Time = 0 to 313 seconds Time = 280 to 313 secondsMax Temp at Re-entry = 845 K


Node_3



As Figure 22 shows, the maximum temperature experienced at Node 1 occurs within the

first few seconds of flight. At launch, the heat loads experienced by the projectile are the

greatest because of the extremely high velocities and low altitudes. It takes approximately

3 seconds for the maximum heat loads to soak into the airframe, such that the temperatures are

felt at Node 1.

58

At Node 1, a maximum temperature of 1,450 K occurs, which approaches the melting

temperature of maraging steel (1,673 K). More of a concern is that this node is representative of

the interior region. The maximum temperature encountered will be along the surface,

particularly at the stagnation point. As a general rule, the skin temperature will be approximately

0.5 to 0.6 times the adiabatic wall temperature—the theoretical maximum wall temperature,

assuming that no heat transfer occurs at the surface. With this in mind, the adiabatic wall

temperature near the nose reaches approximately 2,600 K. Sixty percent of this temperature is

1,560 K, which again pushes the limits of maraging steel.

At Nodes 2 and 3, the maximum temperatures do not occur until the descent portion of

the flight, more precisely, at strike. The temperatures in the mid-body and aft sections of the

body are highly tied to the heat flux on the nose region and the thermal conductivity. Most of the

heat that enters the body penetrates at the nose region. The rate at which this heat load

distributes through the airframe relates directly to the thermal conductivity of the material, hence

the time lag between the maximum heat rates and the maximum temperatures at Nodes 2 and 3.

By adding the internal detail to the model, the thermal mass—and therefore, the

thermal capacitance—of the system increases; it would be expected that the overall temperatures

are slightly lower in the regions influenced by the additional mass (Figure 25, Figure 26, and

Figure 27.)

Node_1

Time = 0 to 313 seconds Time = 0 to 10 seconds

Max Temp at Re-entry = 1040 K

1450 K

FIGURE 25. NODE 1 TEMPERATURE HISTORY (WITH INTERNAL GEOMETRY)

59

Node_2



Node_3



For Node 1, no noticeable change occurs despite the additional mass that was added to

the system at location well back from the nose region. In essence, no change occurred near the

nose that would influence the rate at which the wall temperature increases. However, both

Node 2 and Node 3 experienced noticeable differences in the maximum temperature achieved:

Node 2 (845 K to 682 K) and Node 3 (725 K to 655 K). With the increase in thermal

capacitance, the system distributed the heat load over a larger amount of mass than in the

60

airframe-only case, thereby reducing the maximum temperature experienced in the middle to aft

regions.

As before, the major drawback to this methodology for obtaining thermal soak effects is

the labor intensive acquisition of boundary conditions in AP05. The boundary conditions are not

based on cold wall assumptions and do not properly account for stagnation heating, laminar

heating, and transition to turbulent heating.

3.3 SANDIA NATIONAL LABORATORIES: BLUNTY/SODDIT

After the limited resources and tools available at NSWCDD were exhausted, attention

was turned to the national experts in the realm of aerodynamic heating. Sandia has a legacy in

hypersonic reentry vehicles. Over the years, many of these programs have spawned various

aeroheating codes of varying fidelity to analyze particular phenomena of interest. Each of these

codes specializes in nose-tip predictions of thermal response and ablation, as well as a minimal

amount of in-depth thermal conduction. However, the manner in which the codes are set up

prohibits extensive geometric detail, as can be achieved in ANSYS.

BLUNTY/SODDIT is the first increment in fidelity of this study. The intent of

BLUNTY/SODDIT is the accurate and efficient prediction of aerodynamic heating boundary

conditions and the resulting material response. Together, these codes constitute an extensive

engineering tool for the performance of numerous, quick trade studies.

BLUNTY uses experimental data to establish shock and pressure parameters. For cases

that fall within the realm of available data, BLUNTY tends to predict aeroheating conditions

very well. However, exercising the code outside of the intended geometries can introduce

significant uncertainty for off-stagnation-point calculations (the smallest cone angle allowed is

5 degrees). Due to the normality of the shock at the stagnation point, BLUNTY, using Fay and

Riddell theory, accurately predicts stagnation-point, heat-transfer rates regardless of the

geometry.

61

The BLUNTY output directly feeds into SODDIT as boundary conditions for a 1-D

material response analysis. For most cases, a 1-D conduction code suffices for material response

trade studies. However, for highly conductive materials, as well as some of the smaller

geometries, the multi-dimensional effects may be considerable. SODDIT makes use of the Q*

ablation method. Numerous materials of interest do not behave according to this model, and

therefore, they cannot be analyzed in SODDIT. Last, SODDIT has the capability to analyze

three geometric orientations: flat plate, curved surface, and sphere. The user defines the

thickness of the layer and the material properties. SODDIT applies the boundary conditions to

the entire surface. As one can imagine, the flat-plate approach is less stringent, whereas the

sphere assumes the heat transfer over the entire surface, resulting in the most conservative

estimates of temperature (and ablation). In larger geometric systems these approaches can

reasonably estimate material response. However, in a system such as the hypersonic round, none

of the approaches truly replicates the geometry (0.125-inch nose tip, 3.6-degree cone angle). A

sphere grossly under-represents the thermal mass of the system, whereas the flat-plate geometry

results in extremely thin layers to capture the system mass.

Despite the limitations, the BLUNTY/SODDIT tool set provides an efficient means to

run trade studies to establish trends and provide insight into the problem. One of the initial

studies compared the cold wall heat flux at the stagnation point for the various trajectories.

Figure 28 plots the BLUNTY cold wall, heat flux values for various gun elevation (also referred

to as quadrant elevation, QE) launches at Mach 8 for the first 50 seconds. Figure 29 extends the

plot to the entire time of flight.

62

0

1000

2000

3000

4000

5000

6000

7000

0 5 10 15 20 25 30 35 40 45 50

QE 85 QE 80 QE 70 QE 60 QE 50 QE 40

QE 30 QE 20 QE 10 QE 5 QE 0.5

Heat

Flux

, q” (

BTU/

ft2-s)

Time (seconds)

H

eat F

lux,

q"

(BTU

.ft2 -s

)

FIGURE 28. BLUNTY COLD WALL HEAT FLUX (50 SECONDS)

Heat

Flux

, q” (

BTU/

ft2-s)

Time (seconds)

0

1000

2000

3000

4000

5000

6000

7000

0 100 200 300 400 500 600

QE 85 QE 80 QE 70 QE 60 QE 50 QE 40

QE 30 QE 20 QE 10 QE 5 QE 0.5

Hea

t Flu

x, q

" (B

TU.ft

2 -s)

FIGURE 29. BLUNTY COLD WALL HEAT FLUX (FULL)

Figure 28 and Figure 29 plot the cold wall, heat flux values from BLUNTY over the first

few seconds of flight, as well as the entire trajectory. The trajectories for QE 0.5 to 20 are such

short flight times they tend to overlay in Figure 29 due to the plot resolution. These plots show

the initial similarity of each trajectory, but over the full time of flight, each trajectory experiences

63

varying degrees of heat flux. From these plots, one can deduce that the area under the curve

should represent the most severe case in terms of ablation; however, it is not clear whether this is

also the worst thermal soak scenario. The 20-degree launch angle case should have the most

material loss from ablation due to the integrated heat load, but, because of the relatively short

flight time, the heat may not have enough time to propagate rearward into the round. The longer

flight times that retain considerable heat loads are likely the most severe thermal soak cases.

Figure 30 provides a closer look at the cold wall, heat flux calculations from BLUNTY

compared to those done with an in-house methodology (provided by Dr. Dorsey of NSWCDD

K08), as well as the Detra correlation. The figure plots only the first few seconds to show some

small variation, but as expected, all three plots match very well.

Heat

Flux

, q” (

BTU/

ft2-s)

Time (seconds)

0

1000

2000

3000

4000

5000

6000

7000

0 2 4 6 8 10 12 14 16 18 20

In House CorrelationDetra CorrelationBlunty

Hea

t Flu

x, q

" (B

TU.ft

2 -s)

FIGURE 30. BLUNTY, IN-HOUSE, DETRA COMPARISON

SODDIT has limited material options that are of interest in this application. In particular,

the nature of the projectile makes heavy refractory metals not available in SODDIT beneficial for

both flight stability and penetration mechanics. An initial study considered pure tungsten due to

its high melt temperature to determine the approximate maximum temperatures experienced at

the stagnation point. To be conservative, this analysis used a sphere geometry with a radius

64

matching that of the nose tip (0.125 inch). The longest range case was analyzed (i.e., QE = 50

degrees).

Figure 31 plots the temperatures at various depths into the surface. The sphere is a mere

0.25-inch diameter, so virtually no temperature difference occurred through the thickness.

BLUNTY/SODDIT predicted a maximum temperature of approximately 2,888 K (5,200°R),

which occurs less than a second after launch. The overall temperature profile spikes at launch

and quickly dissipates as the projectile slows and increases in altitude. Upon reentry some

additional heating occurs but not nearly as severe as upon launch. Pure tungsten has an

extremely high melting temperature, as do most refractory metals. The mode of ablation in these

materials is melting ablation, but the maximum temperature experienced is well below the

melting point (3,672 K (6,611°R)). In reality, tungsten undergoes a very slow oxidation process

at room temperature. Above 777 K (1,400°R), the process significantly increases. When the

material reaches temperatures in excess of 1,445 K (2,600°R), the oxide sublimates as rapidly as

it forms, and this results in substantial material loss. Figure 32 shows the amount of material

loss expected during the long-range case for a more realistic tungsten tip.

Mach 8, 50° QE, Rn = 0.125”, sphere tip Pure Tungsten (Tm = 3673K = 6611 °R)

5250 °R, 2916K

4000 °R, 2222K

0

1000

2000

3000

4000

5000

6000

0 50 100 150 200 250 300 350 400

X = 0 X = 0.0025 X = 0.005X = 0.01 X = 0.02 X = 0.05X = 0.075 X = 0.1 X = 0.125

Temp

erat

ure (

R) Temperature (K)

Time (sec)

Depth into Surface

555 K

1111 K

1666 K

2222 K

2778 K

5250 °R, 2916K

FIGURE 31. BLUNTY/SODDIT THERMAL RESPONSE

65

FIGURE 32. TUNGSTEN OXIDE RECESSION, M8, RN = 0.125-INCH FLAT PLATE

The BLUNTY/SODDIT results do not provide the final word on the analyses, but they do

give insight into some of the concerns and challenges facing this study. The severe thermal

shock environment is unlike that of any other system. The temperatures expected have been seen

before in other systems, but the material solutions are not trivial. Last, understanding how

materials behave at these temperatures, pressures, and shock needs considerable attention.

3.4 SANDIA NATIONAL LABORATORIES: ASCC

ASCC86 is a self-contained aeroheating and material response code. ASCC takes an

approach similar to BLUNTY and MAGIC for solving the boundary conditions; however, the

material response code incorporates 2-D effects as well, and it provides more realistic geometric

modeling. Both ASCC and ATAC05 evolved out of the same code and are two of the most

widely used tools. Both codes have exceptional capabilities, as well as some limitations.

Rather than attempting to develop graphical user interfaces, Sandia has developed an

interactive script that prompts the user for the input necessary to run the code. As an added

66

benefit, the user has a high level of control on the input deck and is given tips and suggestions

for defining various parameters.

ASCC uses two overlapping, orthogonal coordinate systems: a moving-body oriented

coordinate system that covers the heated surface layer and a fixed cylindrical system covering

the computational domain. A finite difference approach is used in both regions. However, the

heated layer region uses an implicit scheme, whereas the computational domain is explicit. The

heated layer captures the surface effects, while the computational domain calculates the

temperature response of the material (Figure 33) (Reference 28).

FIGURE 33. ASCC IMPLICIT AND EXPLICIT GRID

The implicit “gridding” scheme requires a single point of origin in the explicit grid from

which the grid rays emanate. This scheme has been well validated and verified for numerous

applications and larger scale bodies, but some inherent risks with smaller, more slender systems

do exist. A basic assumption with finite difference formulation in the implicit region is that the

lateral diffusion term is small compared with the normal diffusion term. This requires that the

rays need to be as ortho-normal to the surface as possible. Some corrections have been built in

to ASCC, but an extreme lack of orthogonality causes significant numerical instabilities. When

the origin is placed too close to the nose tip, the rays become highly skewed, invalidating the

orthogonality requirement. In addition, in extreme cases, the surface recession can actually

67

overtake the origin, causing the grid to be invalid. On the other hand, placing the origin point

too far rearward can cause a geometric cusp in the grid, also causing instabilities. ASCC86 has

for the most part resolved the cusp issue by implementing a scheme to use vertical lines past the

origin to ensure orthogonality. Some general rules of thumb are suggested for selecting the

origin, but at times, particularly in the more complex geometries, an iterative approach must be

taken. Figure 34 illustrates a highly skewed grid due to a thick surface layer and origin location.

FIGURE 34. HIGHLY SKEWED GRID CAUSED BY THICK SURFACE

LAYER AND ORIGIN LOCATION

In this particular analysis, the skewness limitation was encountered often. The design

nose tip, 0.125 inch, was much too small to establish an orthogonal grid, and the code

discontinued when it could not converge to a solution. The most promising approach was when

the origin was moved close to the tip, thus setting up a more orthogonal grid; however, after only

a few time steps into the solution, the ablation effects surpassed the origin. Various parameters

were systematically tweaked in an attempt to converge to a solution, but to no avail. The last

attempt slowly increased the nose-tip radius until finally convergence would be reached. The

smallest nose-tip radius for this geometry converged by ASCC was 0.20 inch, so the results

obtained are for this nose-tip radius and not the design nose tip. It would be expected that a

slight increase in ablation and maximum heat flux would be seen for the smaller nose tip.

With the improved modeling capability of ASCC versus BLUNTY/SODDIT, a more

robust look at various configurations became possible, as well as a search for more suitable

material candidates such as carbon-carbon (C-C). Presumably, the direct-fire, low-QE

trajectories will produce the most severe temperatures and ablation rates. For that reason, a QE

of 0.5 degree was analyzed as well as the 20-, 50-, and 85-degree cases. Figure 35, Figure 36,

Figure 37, and Figure 38 show the thermal history and ablation recession histories for those four

68

cases. The thermal history plots exhibit some oscillation during the solution, and eventually they

converge to a more behaved trace. This variation is on the order of 8 percent at most, which is

acceptable. Figure 35 is an ASCC graph that plots temperature and recession against time for a

Mach 8 launch at a QE of 0.5 degree. Figure 36, Figure 37, and Figure 38 are ASCC graphs that

plot temperature and recession against time for Mach 8 launches at a QEs of 20, 50, and 85

degrees, respectively.

Mach 8, 0.5° QE, Rn = 0.20” C-C

Temp

(R)

Time (s)

Recession (in)

0

1000

2000

3000

4000

5000

6000

7000

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6QE 0.5 TempRecession

5700 °F

FIGURE 35. ASCC, MACH 8, QE 0.5, C-C

Mach 8, 20° QE, Rn = 0.20” C-C

Temp

(R)

Time (s)

Recession (in)

0

1000

2000

3000

4000

5000

6000

7000

0 20 40 60 80 100 120 140 1600

0.2

0.4

0.6

0.8

1

1.2

1.4QE 20 TempRecession

FIGURE 36. ASCC, MACH 8, QE 20, C-C

69

Mach 8, 50° QE, Rn = 0.20” C-C

Temp

(R)

Time (s)

Recession (in)

0

1000

2000

3000

4000

5000

6000

7000

0 50 100 150 200 250 300 350 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7



Mach 8, 85° QE, Rn = 0.20” C-C

Temp

(R)

Time (s)

Recession (in)

0

1000

2000

3000

4000

5000

6000

7000

0 100 200 300 400 500 6000

0.1

0.2

0.3

0.4

0.5



In general, the trajectories represented by Figure 35, Figure 36, Figure 37, and Figure 38

experienced the same approximate maximum temperature, 3,444 K (6,200°R), because with a

Mach 8 sea-level launch, the heat flux on the round is greatest right at launch. The maximum

nose-tip temperature occurs shortly after launch, on the order of a tenth of a second. In reality,

70

the maximum temperatures would be reached almost instantaneously. At the moment the

projectile exits the muzzle a strong shock wave immediately develops at the nosetip. The

temperature jump across the shock and the stagnation of the flow at the nosetip region results in

an instantaneous stagnation fluid temperature of 3,300 K (5,940 °R) for a Mach 8 sea-level flow.

Depending on the materials ability to conduct and radiate the heat away the temperature rise will

be on the same time order. Notably, the temperature predicted of 3,444 K (6,200°R) is higher

than the theoretical maximum fluid temperature. The best explanation for this effect, which was

not seen in tungsten or other refractory metals, is that C-C ablation is an exothermic reaction.

Therefore, the ablation of the material actually results in additional heat being generated, much

like the use of charcoal in a barbecue.

The reentry vehicle community has flown C-C nose tips to these temperatures (i.e.,

3,444 K (6,200°R)) in flight demonstrations, as well as exposed them to similar high pressure

and shear stresses. The major difference, however, is the time required to reach the maximum

temperatures and pressures. Reentry vehicles enter from extreme altitudes with no heating or

pressure effects. The material begins to experience the shear forces and the heat loads gradually.

In a sea-level-launched application the loads and temperatures are experienced instantaneously.

The resulting thermal shock and shear forces could easily rip apart materials. Current facilities

cannot replicate such environments. Until these rounds can be fired at those speeds in a sea-level

environment, the extrapolation of data collected from arc jets, laser tests, and light-gas guns to

this regime is the best that can be done.

After muzzle exit and reaching maximum temperatures, the remaining temperature

profile and recession depends entirely on the trajectory—neglecting particle impact erosion. The

trajectories with low altitude and high velocity will maintain the higher heat flux values. This

explains why a direct fire trajectory (i.e., sea-level flight only) is less severe from a thermal soak

perspective, and a mid-range trajectory is the most stringent.

The direct-fire trajectory, QE 0.5 degree (Figure 35), hits a peak temperature of roughly

3,444 K (6,200°R). Despite the short flight time, the nose tip ablates at a considerable rate with

71

an expected 0.5-inch recession. The recession discussed in this context is ablation effects only.

Particle impact analysis will await future studies.

From Figure 29, one could assume the trajectory with the highest integrated heat flux

should be the most stringent case regarding ablation effects. The ablation process acts to absorb

and remove some of the heat load that is applied to the nose-tip. The total amount of energy into

the nose region will define the amount of ablation seen. As expected, the trajectory around the

20-degree QE (Figure 36) has the highest total recession due to an ablation of approximately

1.25 inches. On a large-scale vehicle, this number is small compared to the overall dimensions.

However, on a sharp 3.6-degree cone, on the order of 30 inches, a 1.25-inch recession is

considerable (almost 5-percent body length). In particular, the aerodynamic effects caused by

the shape change will play a significant role in the guidance and overall range of the round.

The 50- and 85-degree QE trajectories (Figure 37 and Figure 38) behave similarly

because of the endo-, exo-, endo-atmospheric nature of the flight. Both trajectories undergo a

period of ablation while ascending and descending through the atmosphere. During the exo-

atmospheric transit no ablation occurs. Heat flux is a function of density, so gaining altitude

greatly reduces the heat flux on the round. Both trajectories display this with a steep ablation

curve initially, tapering off to nothing, and then upon reentry they begin to recess again. Total

recession amounts are approximately 0.7 inch and 0.55 inch for the 50- and 85-degree cases,

respectively.

3.5 ITT AEROTHERM: ATAC05

For comparison, ATAC05 was intended to be used to perform a study identical to the

ASCC86 analysis. Unfortunately, significant limitations prevented the comparison. ITT intends

to look further into the issues outlined in the following paragraphs.

Unlike ASCC 86, ATAC05 is relatively easy to get a solution. The major difference is

that ASCC86 will not provide a solution if convergence is not obtained. However, ATAC05 will

72

provide a solution even when poor convergence occurs. Convergence studies showed many

inconsistencies with the code. Three studies using ATAC05 established where the

inconsistencies occurred—time step variation, streamline variation, and patch variation. A nose

tip of 0.125 inches was used as the baseline for the study while a 0.20-inch nose tip was briefly

examined to determine if similar trends occurred. Not until the nose-tip radius was significantly

increased did the inconsistencies die out.

By varying the time step, one would assume that with decreasing time steps there would

be some trend to convergence. For the small, slender objects, ATAC05 could not come to such a

trend. Unlike ASCC86, which could come to a relatively stable temperature profile, ATAC05

would have variations between time steps on the order of 555 to 833 K (1,000 to 1,500°R). The

code was unable to converge to a stable temperature profile. As a step further, time steps were

refined to potentially dampen out the wild variations, as well as come to some converged value.

Instead, no convergence was encountered and the variations continued. The actual value in the

code, DTAR, is a user-defined, initial time step size. Depending on how the solution is

proceeding, the code can vary the increments throughout the solution. Figure 39 shows the

temperature fluctuations. The time step of 1.0 second had a final temperature of 2,400 K

(4,320°R); 0.5 second was at 2,055 K (3,700°R); 0.2 second was at 2,666 K (4,800°R); and 0.1

second was at 3,055 K (5,500°R). Further reductions of an order of magnitude made no

difference or trend.

73

Mach 8, 0.5° QE, Rn = 0.125” C-C

Temp

(R)

Time (s)0

1000

2000

3000

4000

5000

6000

7000

0.0 1.0 2.0 3.0 4.0 5.0 6.0

BaselineDTAR = 0.1DTAR = 0.20DTAR = 0.50DTAR = 1.0DTAR = 0.05DTAR = 0.01DTAR = 0.001

FIGURE 39. ATAC05 TEMPERATURE HISTORIES

As a result of the variations in the predictions of temperatures, the predicted recession

was also inconsistent. Depending on whether that particular thermal history was higher or lower

than the next time step within the run, recessions varied as much as 50 percent. Again, as Figure

40 shows, no particular trend emerges as time steps are refined.

74

Mach 8, 0.5° QE, Rn = 0.125” C-C

Rece

ssio

n (in

ch)

Time (s)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.0 1.0 2.0 3.0 4.0 5.0 6.0

BaselineDTAR = 0.1DTAR = 0.20DTAR = 0.50DTAR = 1.0DTAR = 0.05DTAR = 0.01DTAR = 0.001

FIGURE 40. ATAC05 RECESSION

Similar studies varying the number of streamlines per patch, as well as number of patches

on a given segment, produced similar inconsistencies. The temperature fluctuations continued in

all cases as before. Figure 41, Figure 42, Figure 43, and Figure 44 show the resulting thermal

histories and recession predictions for the studies.

Mach 8, 0.5° QE, Rn = 0.125” C-C

Time (s)

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

6000

6500

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Baseline

ISTRM = 2

ISTRM = 3

Tem

p (R

)

FIGURE 41. STREAMLINE VARIATION THERMAL HISTORY

75

Mach 8, 0.5° QE, Rn = 0.125” C-C

Time (s)

0.000.050.100.150.200.250.300.350.400.450.500.550.600.650.700.75

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Baseline

ISTRM = 2

ISTRM = 3

Rece

ssio

n (in

ch)

FIGURE 42. STREAMLINE VARIATION RECESSION

Mach 8, 0.5° QE, Rn = 0.125” C-C

Time (s)

Tem

p (R

)

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

6000

6500

7000

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Baseline, NPATCH = 4

NPATCH = 3

NPATCH = 5

NPATCH = 6

FIGURE 43. PATCH VARIATION THERMAL HISTORY

76

Mach 8, 0.5° QE, Rn = 0.125” C-C

Time (s)

Rece

ssio

n (in

ch)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Baseline

NPATCH = 3

NPATCH = 5

NPATCH = 6

FIGURE 44. PATCH VARIATION RECESSION

In the thermal history plots, Figure 41 and Figure 43, the temperatures are all fairly close

to each other. Obviously wide variation still exists from time step to time step, but on average,

the results are close. In contrast, the recession predictions show much more variation than

alluded to by the temperature histories.

The baseline case (Figure 41 and Figure 43) used one streamline per patch. Increasing

from one to two streamlines changes the recession solution slightly, as expected. Intuitively,

increasing the streamlines even more should also result in a slight change, eventually coming to

some convergence. Instead, increasing from two to three streamlines changes the recession

significantly.

Patch variation behaved much more consistently with expectations, as shown in

Figure 42 and Figure 44. The baseline number of patches per segment is four. Reducing this

number to three significantly changed the recession predications. In the other direction, as the

number of patches increased—thus giving finer resolution—the recession prediction tended

toward a converged value.

77

Further studies should be performed with ATAC05 to ascertain where the inconsistencies

exist. By doing so, these particular areas can be reevaluated and corrected if needed. From

discussions with AMRDEC, ITT intends to bring some resolution to these outstanding concerns.

3.6 CONCLUSIONS

The preceding studies used existing tools to shed light on the design requirements of a

hypersonic projectile airframe. In addition, an assessment of the available tools and the

capability gaps present was completed.

Temperatures in excess of 3,000 K (5,400°R) are not unreasonable and are to be expected

for this application. Furthermore, recession amounts to about 1.5 inches, or more importantly,

almost 5-percent loss of the entire airframe length. These temperatures are fairly typical of high-

performance reentry vehicles. The reentry vehicle community has flown materials such as C-C

at these temperatures with good survival rates. The major difference in this particular case is

time duration for the onset of the maximum temperatures and pressures. Reentry vehicles,

although operating at extreme velocities, do not encounter significant air density for a period of

time. The heating and loading profiles are much less severe than the instantaneous thermal and

pressure loading seen from a hypersonic system launched at sea level. Test facilities do not have

the capabilities to replicate these flight conditions. Until the ability to fire these rounds at the

desired sea-level velocities exists, testing will have to be done piecemeal to acquire some data at

close to design conditions. All conditions may not be tested at once, but via arc jet, tunnel, sled

track, and laser facilities, many of the design conditions can be approximated. The all-up testing

will come with a gun system capable of firing at these velocities.

The various codes have their places within the fidelity levels necessary; however, there

are still significant limitations. Some of the basic tools, such as BLUNTY/SODDIT, are

excellent at giving quick, first-cut temperature and ablation predictions. They provide a sanity

check before continuing with higher fidelity analyses. From the analyses presented within this

section, BLUNTY/SODDIT provided very similar numbers to those predicted by the higher

78

fidelity codes. The geometric modeling scheme used and the 1-D nature of SODDIT do limit the

capabilities, thus requiring the need for higher fidelity codes for more stringent design studies.

Similarly, ASCC86 and ATAC05 are exceptionally useful tools for their intended use.

Both codes have a much broader range of applicability than BLUNTY/SODDIT, but they are not

without limitations. Both codes have geometric limitations, not surprisingly since they are built

from the same original source. The ASCC86 analysis clearly shows that the gridding scheme

may not be appropriate for slender, sharp bodies. Increasing the nose-tip radius helped in

converging the solution, which also points to the geometric issues. On the other hand, ATAC05

seemed too prone to producing a solution even if it was incorrect. Nothing obviously pointed to

geometric concerns in ATAC05, but considering the known geometric limitations of ASCC86,

further study is warranted.

It is clear that there is a capability gap in efficiently modeling the boundary layer coupled

to in-depth thermal response. The codes that were assessed in the preceding section have their

place of applicability; however, none can efficiently predict the thermal response of an entire

detailed system.

79

4.0 HOT WALL FEA METHODOLOGY

The preceding sections have demonstrated the capabilities and limitations of the

aerodynamic heating codes that are readily available in the United States. Clearly, a large

capability gap exists in the development of a high-fidelity, geometrically accurate, thermal soak

model. Presently, GASP has the capability to capture the CHT and associated thermal response.

GASP is an industry standard for high accuracy steady-state or short time-accurate systems, but,

for anything much greater than a few seconds of flight time, GASP becomes very expensive.

It is the intent of this study to develop a new methodology that will use the aerodynamic

heating tools of Sandia to develop the boundary conditions for a highly detailed finite element

model. The methodology will be capable of solving for the thermal soak of an entire flight

trajectory of many seconds to minutes in an efficient, inexpensive manner, which will be coupled

in to the overall design process for hypersonic projectiles. This section intends to detail the

development of the methodology and the associated validation studies performed.

BLUNTY and MAGIC will be used to establish the aerodynamic heating boundary

conditions necessary for the finite element thermal model. BLUNTY, using Fay-Riddell theory

for the stagnation point, has been well validated at the stagnation point but tends to under predict

values off stagnation point. MAGIC however, which uses momentum and energy relations, is

much more accurate for off stagnation point. The combination of these two codes will cover the

entire geometric regime of interest. In addition both codes were developed by the same group at

SNL and therefore have very similar input and output decks making the coupling process much

more efficient. The outputs of these two codes can be directly imported to a finite element

analysis (FEA) package to establish the transient heat load boundary conditions.

80

ANSYS was the FEA code of choice for numerous reasons. The code was initially

developed in the 1970’s and has continuously undergone upgrades and validation studies up to

its current version 11.0. Within the EMRG team at the Naval Surface Warfare Center Dahlgren

(NSWCDD) ANSYS version 10.0 is the standard finite element analysis tool. In addition the

thermal module within ANSYS was one of the more user-friendly and customizable tools

available. Using the ANSYS programming language a tool was created implementing this

methodology. The boundary conditions from BLUNTY and MAGIC are coupled with the

thermal model in ANSYS to calculate a hot wall heat flux for each time step in the trajectory of

interest resulting in a time accurate thermal solution of the projectile over its trajectory.

Initially, mission goals drive the projectile geometry in terms of drag characteristics,

mass, and payload. In an iterative process the geometry is roughly defined and resulting

trajectory models are established determining if the design meets mission goals. The geometry is

iterated upon until such goals are met. With the notional geometry and trajectory models, the

Sandia codes are used to establish the boundary conditions for this baseline study. Similarly,

material properties can be notionally selected. The geometry, material properties, and boundary

conditions are input to the ANSYS method resulting in a full-time accurate-thermal history of the

projectile. Figure 45 depicts the design flow for a typical hypersonic projectile.

81

Aerodynamic Design/Trade Study

3 DOF Trajectory

BLUNTY/MAGIC for Heat Deck

Geometry Material Properties

Initialize Solution

Calculate Heat Flux at Each Axial

Station

Apply Heat Flux to Each Axial

Location

Apply Scaled Heat Flux to Each

Intermediate Elementsolve

Delete All heat Flux Boundary

Conditions

Query Nodal Temp at Each Axial Location

Iterate for Number of Load Steps

Time Accurate Thermal Profile of System

ANSYS APDL

Aerodynamic Design/Trade Study

3 DOF Trajectory

BLUNTY/MAGIC for Heat Deck

Geometry Material Properties

Initialize Solution

Calculate Heat Flux at Each Axial

Station

Apply Heat Flux to Each Axial

Location

Apply Scaled Heat Flux to Each

Intermediate Elementsolve

Delete All heat Flux Boundary

Conditions

Query Nodal Temp at Each Axial Location

Iterate for Number of Load Steps

Time Accurate Thermal Profile of System

ANSYS APDL

FIGURE 45. AEROTHERMAL DESIGN FLOW

4.1 BLUNTY/MAGIC/ANSYS HOT WALL METHODOLOGY

Currently, the BLUNTY/MAGIC/ANSYS methodology (BMA) applies to 2-D

axisymmetric bodies only. A future upgrade to the tool may have the capability to analyze

arbitrary shapes, including fins, wings, and canards. CAD software is typically used to create a

detailed 2-D axisymmetric slice which can be exported to ANSYS. This allows for easy

inclusion of geometric detail, including subassemblies and components.

With the geometry imported, a material database needs to be created to define each

different material present in the model. With this, as the model is meshed, each material model

associated with a component will be assigned to the element being used for the meshing

operation of that component. For example, in Figure 46, the blue elements represent C-350

maraging steel; the red elements, tungsten carbide; and the green elements, pure tungsten.

82

X

YAxis of Symmetry

FIGURE 46. SAMPLE ANSYS GEOMETRY/MESH

The heat deck boundary conditions needs to be read into the database (i.e., the ANSYS

file containing the geometry, mesh, etc.), as well as the pressure-temperature-enthalpy tables for

air, before implementing the ANSYS aeroheating tool. These tables are derived using a thermo-

chemistry tool, such as the ACE code (Reference 29). ACE is a thermo-chemistry analysis tool

that can calculate the chemical and thermodynamic state of any chemical system. ACE uses the

principles of conserving the chemical elements and energy at a given surface. In this case, air is

the gas of concern. Table 2 shows a small sample of the output of an ACE run. The air enthalpy

tables were provided by Sandia; ACE was not included in this study.

TABLE 2. SAMPLE PRESSURE-TEMPERATURE-ENTHALPY TABLE

Pressure (atm)

Temperature (°R)

Enthalpy (Btu/lbm)

0.001 2,800 1,449.78 0.001 2,600 1,153.4 0.01 2,800 1,125.896 0.01 2,600 868.233 0.1 2,800 887.674 0.1 2,600 740.271

Inputting these data as an ANSYS table allows for 2-D lookup within ANSYS. For

example, by defining a pressure and temperature, ANSYS will interpolate between both

parameters and calculate an enthalpy value. Appendix A contains the full pressure-temperature-

enthalpy table for air, which is being used by the methodology.

The inputs are slightly different between BLUNTY and MAGIC. BLUNTY relies on

experimental data to calculate pressure coefficients and shock shape for varying cone angles.

Regardless of the cone angle, the stagnation point values are concerned only with the trajectory

and the nose bluntness. MAGIC, on the other hand, allows for user-defined geometries rather

83

than tabulated experimental data. Hence, more care must be taken in MAGIC to ensure the

proper geometry is being modeled. In addition, the user must define axial locations where

heat transfer results will be output. For example, assuming a projectile length of 10 inches

(Figure 47), the output locations on the surface in the axial direction may be defined as 0, 0.25,

0.5, 1.0, 3.0, 5.0, 7.0, and 10.0 inches. It is suggested to cluster more values near the stagnation

point because of the high heating rates and to capture the laminar/turbulent transition.

Axis of Symmetry

0.0 10.05.0

0.25”0.5”

1.0”

3.0”5.0”

7.0” 10.0”

Axis of Symmetry

0.0 10.05.0

0.25”0.5”

1.0”

3.0”5.0”

7.0” 10.0”

FIGURE 47. SURFACE LOCATIONS FOR HEAT TRANSFER OUTPUT

After the geometry is defined, the trajectory must be provided to the codes. Both codes

treat the trajectory as a series of single data points representing altitude and velocity. Time is

included in the input for tracking purposes only and is output with the data, but it does not come

into consideration for the calculations. At each data point, numerous parameters are calculated

and tabulated for each axial location as a function of time to coincide with the trajectory.

Figure 48 is an actual sample of trajectory input (time, altitude, and velocity); these data are

tabulated with column headings in Table 3 for clarity. Likewise, Figure 49 shows a sample of

heat deck output, and Table 4 shows the output tabulated with column headings. All units are in

English units. The heat deck table can be read into ANSYS as an array, which will drive the heat

flux boundary conditions as a function of the trajectory.

0 0 8930.78

0.5 418.77 8756.121 784.93 8598.14

1.5 1136.62 8445.04

FIGURE 48. SAMPLE TRAJECTORY INPUT DATA (TIME, ALTITUDE, VELOCITY)

84

TABLE 3. TRAJECTORY INPUT DATA (TIME, ALTITUDE, VELOCITY)

Time (s)

Altitude (ft)

Velocity (ft/s)

0 0 8,930.78 0.5 418.77 8,756.12 1 784.93 8,598.14 1.5 1,136.62 8,445.04

0.000 0.000 1587.7 -5822.52 3.65E+00 8.46E+01 0 BASELINE 0 S 00.000 1.000 1587.7 -5822.52 3.65E+00 8.46E+01 0 BASELINE 0 S 00.000 0.000 1436.54 -379.69 0.26424 3.58814 6817.58 QE50RG 0 0.25 L 00.000 1.000 1329.4 -304.59 0.22906 2.67567 6606.81 QE50RG 0 0.25 L 6804.90.000 0.000 1473.57 -939.56 0.63745 2.42551 7151.49 QE50RG 0 0.5 T 00.000 1.000 1364.71 -712.4 0.52187 1.85621 6917.93 QE50RG 0 0.5 T 6804.9

FIGURE 49. SAMPLE HEAT DECK OUTPUT

TABLE 4. HEAT DECK OUTPUT

Not Used

Time (s)

Recovery Enthalpy

(hr) (Btu/lbm)

Cold Wall Heat Flux* (Btu/ft2-s)

Heat Transfer

Coefficient (CH)

Edge Pressure

(Pe) (atm)

Edge Velocity

(Ue) (ft/s)

User-Defined

Case Identifier

Axial Location**

(inch)

Flow Regime

0.0000 0.000 1,587.7 -5,822.52 3.65E+00 8.46E+01 0 Baseline 0 S 0 0.0000 1.000 1,587.7 -5,822.52 3.65E+00 8.46E+01 0 Baseline 0 S 0 0.0000 0.000 1,436.54 -379.69 0.26424 3.58814 6,817.58 QE5ORG ( 0.25 L 0 0.0000 1.000 1,329.4 -304.59 0.22906 2.67567 6,606.81 QE5ORG ( 0.25 L 6,804.9 0.0000 0.000 1,473.57 -939.56 0.63745 2.42551 7151.49 QE5ORG ( 0.5 T 0 0.0000 1.000 1,364.71 -712.4 0.52187 1.85621 6917.93 QE5ORG ( 0.5 T 6,804.9

*Cold wall is denoted with the negative sign **S = Stagnation, L = Laminar, T = Turbulent

Referring to Figure 45 all the preprocessing outside the ANSYS loop has been completed

and the BMA method can begin. The code initializes the entire mesh to a user-defined initial

temperature, which allows for a starting point in the heat flux calculations. The sequence of

events will follow the steps defined in the heat deck array. Each time increment in the heat deck

corresponds to one load step and time step.

Recalling that the heat deck is associated with a predefined number of axial locations, the

ANSYS model also needs to associate these loads with the specified axial location. The

geometry does not change during the simulation, so each axial location can be associated with a

specific node along the exterior surface of the projectile. For example, the axial location of zero

directly corresponds to the node at the nose tip.

85

Starting with load step one, the heat transfer parameters in the heat deck define the

parameters that ANSYS will use to calculate the heat flux values. At high temperatures the heat

transfer correlations are based on enthalpy values rather than temperature due to the varying

specific heats. The heat transfer equation for heat flux is given as Equation (154),

( )wrheew hhCUq −= ρ , (154)

where

ρ = density,

U = velocity,

Ch = Stanton number,

hr = recovery enthalpy,

hw = wall enthalpy, and

sub e = edge properties.

In BLUNTY and MAGIC, the heat transfer coefficient is calculated using Equation

(155):

hee CUCH ρ= . (155) Combining (154) and Equation (155) gives Equation (156):

( )wrw hhCHq −= . (156)

At each axial location, 1 through N, the code will loop through the N nodes and define an

array of heat flux values for each of those nodes. Starting with node location one, the recovery

enthalpy, hr, the heat transfer coefficient, CH, and the edge pressure, Pe, are obtained from the

heat deck and stored in temporary variables, HR_, CH_, and PE_, respectively. The node at this

location is known—at load step one, it is the initial user-defined temperature,—so it can be

queried and stored as well, in a temporary variable. Knowing the edge pressure and the wall

86

temperature at the node, the associated wall enthalpy can be obtained from the lookup pressure-

temperature-enthalpy table. Now, all the variables needed to calculate the heat flux value are

known: the heat transfer coefficient, CH; the recovery enthalpy, hr; and the wall enthalpy, hw.

Using Equation (156), the heat flux value can be calculated for this particular node and load step.

This process is repeated for the remaining N axial positions. Figure 50 depicts the heat flux

boundary conditions (colored arrows) as applied by this procedure.

2mW

FIGURE 50. SAMPLE HEAT FLUX BOUNDARY CONDITIONS

At each defined axial location, a heat flux boundary condition has been applied. Now, to

capture the entire surface, a marching scheme is employed that will apply an interpolated heat

flux value to each node between axial locations. In addition, ANSYS applies heat flux values to

the faces of elements. Depending on the orientation of the element, the face numbers do not

necessarily line up. For example, one face can have Face 2 as its exterior face and the next face

can have its Face 3 as the exterior face. Figure 51 depicts the face orientation of a 2-D element

in ANSYS. The numbers on the corners represent the node locations and the numbers in the

boxes represent the face number. For example, the edge from node position 1 to 2 is Face 1. As

the marching scheme progresses, the orientation of each element has to be established so that the

heat flux can be applied in the correct direction.

87

1 2

34

1

2

3

4

FIGURE 51. 2-D ELEMENT NODE AND FACE LOCATIONS

Due to the large numbers of elements, careful bookkeeping is required to insure proper

application of the heat flux boundary conditions. The first element is easily selected, because the

nose-tip node location is known. By selecting the nose-tip node and selecting the only element

attached to it, the tip element is selected. From here the remaining nodes, which are attached to

the tip element, are selected. Temporary variables are used to hold the node numbers at each

corner. These numbers are then checked to see what nodal position they occupy on the element:

1, 2, 3, or 4. By comparing the value at the nose-tip node to the element nodal positions, the

exterior face can be determined, as well as the face being shared by the next element. For

example, Figure 50 assumes the tip node is located in the one position. (As a side note, the

geometry has to be set so that Y is the axis of symmetry for ANSYS axisymmetric problems.)

According to the geometry schematic in Figure 50, if the nodes progress in the first quadrant

(positive x, y), then the face on the exterior has to be Face 1. Likewise, the face being shared by

the next element would be Face 2. Unfortunately, just because a face is shared does not mean it

shares the same face number. One element can share Face 2 with its adjacent element, but the

orientation of the adjacent element may call it Face 3. Figure 52 depicts this issue in better

detail. The gap in the figure is included for clarity and is not there in reality. The faces being

shared are actually one face. Using Figure 52 as an example, assuming the exterior line is along

the bottom of the elements, then the faces along the exterior would be 1 and 4, respectively.

88

1 2

34

1

2

3

4

1

23

4

1

2

3

4

FIGURE 52. TWO ADJACENT ELEMENTS

Once the exterior face is determined for the specific element, the heat flux value needs to

be applied. Only the values at the axial locations are known, so a scaling routine applies a scaled

load based on the values at the two axial locations; the element falls between it and its y location.

If a plot were created for heat flux versus the y location (axial), it would look something like

Figure 53.

0

1000

2000

3000

4000

5000

6000

7000

8000

0 1 2 3 4 5 6 7 8 9 1

Axial Location From Tip (y)

Hea

t Flu

x (q

")

N-1

N

0

N*

FIGURE 53. SAMPLE HEAT FLUX PLOT AT A GIVEN LOAD STEP

The heat flux at a point N* (Equation (157)) would be

89

bmyq NN +=′′ ** , (157)

where, with slope, m (Equation (158), and intercept, b (Equation (159)),

1

1

−

−

−′′−′′

=NN

NN

yyqqm

, and (158)

11 −− −′′= NN myqb . (159)

This scheme is continued along the entire exterior of the projectile until all exterior

elements have a heat flux value applied. With the boundary conditions applied for the given load

step—Figure 50 as a sample boundary condition plot—the solution can commence. Recalling

that the heat deck input defines the load step and the time steps, the solution is run up to the

defined end time of the load step. At this point, the process continues on to the next load step

and recalculates all the heat flux boundary conditions because the wall temperature has changed.

This procedure is repeated for each load step for the entire flight trajectory.

4.2 METHODOLOGY VALIDATION

After extensive research, there seemed to be no readily available experimental data in the

open or classified literature pertaining to the specific flight regime and geometries of interest. As

Figure 5 in Section 1.2 clearly illustrates, no existing facility can replicate the flight environment

necessary to perform a full transient thermal soak experiment. However, this methodology

predicts overall time-accurate thermal soak of a projectile system rather than just heat transfer

boundary conditions. Figure 54 depicts an overview of the readily available data for hypersonic

applications while Table 5 lists the experiments that had potential relevance to the flight

conditions of interest.

90

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 20 40 60 80 100 120 140Stagnation Pressure (atm)

Stag

natio

n Te

mpe

ratu

re (°

R)

• Numerous configurations

• All cold wall heat flux

• At high altitude conditions

• PANT Run 9, Mach 5

• 2.5” nose radius

• Numerous PANT Cases

• Mach 10 Sea Level

• GASP benchmark


• Hypersonic Round (M8 Sea Level)

• Mach 8 Sea Level GASP Benchmark


• Strategic Systems – high M, alt.

• Recession Data Only

• Missile Defense applications (classified) -Sprint, Nike, SM

• Cleary, Mach 10 High Alt.• Cold Wall Heat Flux Verification

FIGURE 54. PARAMETER SPACE OF AVAILABLE EXPERIMENTAL DATA

TABLE 5. HYPERSONIC EXPERIMENTS WITH POTENTIAL RELEVANCE

Experiment Mach Alt. (kft)

Stag. Press. (psi)

Stag. Temp. (°R)

Nose Radius

(in.)

Cone Angle Type of Data/Commemts

PANT Wind Tunnel H 4.95-5.0 0-30 85-503 805-852 2.5 0-17 Thin-walled Calorimeter of Ablated Shapes PANT Wind Tunnel I 4.95-5.0 0-30 117-808 953-1473 0.75-3.5 0-17 Shape Change Study PANT Wind Tunnel A 4.95-5.0 0-30 70-829 973-1364 0.75-3.5 0-17 Shape Change Study PANT Wind Tunnel J .022-.075 100+ 0.21-2.6 1660-1905 2.5 0-17 Shape Change Study

PANT Wind Tunnel C 4.94-5.0 0-30 117-1477 1350-1500 1.5-2.5 0-17 Shape Change with some Thin-walled Calorimeter

NOL May 1968 5.93-6.0 0-30 147-1470 960-1010 0.03 5 Cold Wall Heat Flux Royal Armament March 1970 5.2 100 3.1 - 0.04 5 Cold Wall Heat Flux

Cleary 10.6 150 1.57 7920 0.375-1.1 15 Cold Wall Heat Flux Sprint 10.0 40 200 7920 0.5 6 Cold Wall Heat Flux Koyama, Japan March 1996 7.1 0 580 1260 0.08-1.0 5-15 No available data in open literature.

Classified ablation tests. Silton, Goldstein, Univ. of Texas, 1998 4.92 10 333 666 0.25 0 Forward Facing Cavity Experiments

The most common experimental data that can be found fall within the low pressure (i.e.,

high altitude) and low stagnation temperature regime. Numerous configurations of slender

bodies with nose radii on the order of 1/2 inch to 1 inch can be found in Moore (Reference 14),

as well as Cleary (Reference 30); however, the experimental data consist of cold wall heat flux

values only that cannot be readily used for validation of the in-depth conduction. Instead this

91

data can serve as validation that the BMA methodology is calculating heat flux values accurately

and appropriately applying these to the geometry. Section 4.2.1 discusses the validation results.

Throughout the development of the space and missile programs, numerous systems have

been tested at very high temperatures and moderate pressures. All of the readily available open

and classified literature that was researched dealt with material response via coupon testing and

ablation/erosion effects. Some of the recession experiments from the Passive Nose-Tip

Technology Program (PANT) (Reference 31) begin to approach the regime of interest for the

hypersonic round; however, the data available are for recession rates only and not in-depth

thermal conduction. Along with the high enthalpy recession tests, the PANT program performed

numerous wind tunnel tests approaching 2,000°R and 100-atmosphere stagnation temperature

and pressure. In particular, the ninth test run of the PANT series is the highest stagnation

temperature test with the correct stagnation pressure that had in-depth thermocouple data for

significant test duration (10 seconds). With this experiment, pressure has been matched exactly;

however, the temperature is significantly lower, and the geometry is much more blunt (2.5-inch

nose radius) as well as thin walled. The corresponding Mach number of the experiment, Mach 5,

is just outside the regime of intended use for the Sandia boundary layer codes and therefore may

not provide accurate boundary conditions for BMA method; however, this should be an

acceptable validation case for GASP.

In addition to the PANT validation case GASP has been validated with numerous CHT

cases as well by Aerosoft, Inc during the CHT module development. By demonstrating another

case which validates GASP, it would be safe to assume that GASP can be used as the benchmark

for developing a validation case closer to the actual flight regime and geometry. For simplicity,

cases were run at Mach 8 and 10 at sea-level with a 0.20-inch nose radius.

4.2.1 Cold Wall Heat Flux Validation

Prior to validating the BMA methodology solutions it would be beneficial to validate that

the methodology is indeed calculating and applying the appropriate heat flux boundary

conditions. Cleary, of NASA Ames Research Center, published his works in October of 1969

92

(Reference 30) regarding “The Effects of Angle of Attack and Bluntness on Laminar Heating-

Rate Distributions of a 15-degree Cone at a Mach Number of 10.6.” In Cleary’s study,

numerous cases were examined at angle of attack and varying bluntness but for this verification

only two cases were used. Both cases were run in the Ames 3.5 ft. Hypersonic Wind Tunnel at a

Mach number of 10.6 with corresponding stagnation temperature and pressures of 89.971 °R and

2.66 pounds per square foot respectively. Figure 55 depicts the geometries of the two cases. Case 1

Nose Radius = 0.375 inch

Cone Half Angle = 15°

Length = 21.32 inch

Base Radius = 6 inch

Case 2

Nose Radius = 1.1 inch

Cone Half Angle = 15°

Length = 19.24 inch

Base Radius = 6 inch

FIGURE 55. CLEARY 15° CONE GEOMETRIES

The models were fitted with ten thermocouples spaced along the length of the surface as seen in

Figure 56.

1 2 3 4 5 6 7 8 9 10

Axial Location 1 2 3 4 5 6 7 8 9 10Xs/L 0.207 0.25 0.293 0.38 0.466 0.552 0.638 0.724 0.811 0.897

Case 1 3.56 4.53 5.49 7.43 9.36 11.30 13.23 15.15 17.09 19.01Case 2 1.49 2.45 3.41 5.35 7.28 9.21 11.14 13.08 15.00 16.94

Thermocouple

FIGURE 56. THERMOCOUPLE LOCATIONS

Before inserting the model into the flow stream they were allowed to equilibrate to the

ambient temperature of 530 °R. Once inserted the models were held in the flow stream for

93

approximately three seconds while the thermocouples recorded the transient temperature profile

on the surface. The resulting temperature curves were used to determine the cold wall heat flux

at each thermocouple location according to Equation (162):

dtdTcq w

www τρ= (162)

where, =dt

dTw slope of the temperature versus time curves.

The accuracy of the data recorded was established by taking repeated measurements of

the stagnation point heat transfer rate and comparing them to each other and theory (Fay and

Riddell). Cleary suggests that the accuracy of the lowest rates presented to be approximately

±20 percent.

For this validation study the heat deck was established in BLUNTY and MAGIC for the

stagnation temperature and pressure conditions used in the Cleary experiment. In the BMA

method the boundary condition calculated at the initial load step is essentially the cold wall heat

flux, seeing how it is applied to an ambient wall. Using the same geometric models as Cleary the

BMA method should calculate and apply the same heat flux values that Cleary experimentally

recorded at the various thermocouple locations. Figure 57 and Figure 58 show the resulting

BMA heat flux values compared to the Cleary results.

94

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12 14 16 18 20

Cleary Exp.

ANSYS/BLUNTY/MAGIC

Axial Location, X (in.)

Col

d W

all H

eat F

lux,

q”

(BTU

/sq.

ft-s)

FIGURE 57. CASE 1 CLEARY VERIFICATION

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12 14 16 18 20

Cleary Exp.

ANSYS/BLUNTY/MAGIC

Axial Location, X (in.)

Col

d W

all H

eat F

lux,

q”

(BTU

/sq.

ft-s)

FIGURE 58. CASE 2 CLEARY VERIFICATION

In both plots the BMA heat flux values track very closely with those recorded by Cleary

and well within the ±20 percent experimental error Cleary suggests (denoted by the error bars).

Case 1 had an average variation of 0.35 percent from experiment and Case 2 slightly higher at

2.6 percent average variation, but again this is well within the experimental error. Although this

experimental comparison does not validate the entire method it does validate that the BMA

methodology is calculating the heat flux values accurately and applying them accordingly.

95

4.2.2 Passive Nose-Tip Technology Program

“The overall objective of the PANT program . . . is to improve and validate the accuracy

of existing nose-tip design analysis computer codes and to evaluate the effects of angle of attack

on nose-tip shape change” (Reference 31). During this program, numerous wind-tunnel

experiments were conducted to develop calorimeter, pressure distribution, and shape change data

for code comparison. Of particular interest to this methodology was a test performed to calibrate

the tunnels that tracked various thermocouple readings on the inside of a thin-walled, blunted-

sphere cap.

The model used for the calibration tests, shown in Figure 59, was an electroformed-

nickel-sphere-cap conical model. The nose radius was 2.5 inches, with an 8-degree cone half

angle extending out to a 7.5-inch base diameter. The electroformed nickel was nominally

0.080-inch thick.

FIGURE 59. PANT CALORIMETER MODEL

On the inside wall, 75 chromel/alumel thermocouples were spot welded to the surface, as

shown in Figure 60.

96

FIGURE 60. CALORIMETER THERMOCOUPLE LOCATIONS

Of the data collected by Baker (Reference 31), Thermocouples 2, 13, and 67 were

presented for one of the calibration test series. From Figure 60, Thermocouple 2 is located on

the interior surface of the nose cap, 12 degrees from the tip, at a zero-degree circumferential

location. Similarly, Thermocouple 13 is located at 36 degrees from centerline and at a

225-degree circumferential. Last, Thermocouple 67 is at an axial location in the conical section

of 9 inches at a 180-degrees circumferential location. Table 6 lists the tunnel conditions for this

particular test.

97

TABLE 6. WIND TUNNEL CONDITIONS

Mach Number 5.02 Feed Pressure 1,477 psi Feed Temperature 772 °F Duration 10 seconds

For modeling considerations, no material data was presented in the report. Therefore,

Table 7 lists the material properties of plain nickel, which were taken from Incropera and Dewitt

(Reference 27).

TABLE 7. NICKEL THERMAL PROPERTIES

Density, ρ 8912 kg/m3

Specific Heat, Cp 460 J/kg-K

Thermal Conductivity, k 60.7 W/m2-K

Figure 61 depicts the model generated in ANSYS to represent the calorimeter test article.

The model is axisymmetric and in SI units. The y-axis is the axis of symmetry as required by

ANSYS. The numbers indicate points along the curve (X, Y), which define the curve. The

boundary conditions for the BLUNTY/MAGIC/ANSYS methodology were obtained from

BLUNTY and MAGIC with a slight variation. Instead of inputting a trajectory, the codes have

an option to input wind-tunnel parameters. The heat deck established for the PANT ANSYS

model can be found in Appendix B.

0,00,0.0026

0.0635, 0.06223

0.061477, 0.062421

0.09525, 0.31496

0.093227, 0.31515

FIGURE 61. ANSYS GEOMETRY

98

Initially three cases were run to verify grid convergence. Grids of 120, 324, and

1,127 nodes were examined. Figure 62 depicts the three cases examined.

80

80

6

40

403

20

40

2

Number of element divisions

Case 3: 1127 nodes

Case 2: 324 nodes

Case 1: 120 nodes

FIGURE 62. PANT GRID REFINEMENTS

All three cases were run for the full ten seconds of the experiment. Figure 63 depicts the

solutions for the 10-second run time for all three thermocouple locations.

Tem

pera

ture

(K)

Time (s)

LS = 1.0s

293

343

393

443

493

543

593

643

693

743

0 2 4 6 8 10

TC 2 N#1127

TC 13 N#1127

TC 67 N#1127

TC 2 N#324

TC 13 N#324

TC 67 N#324

TC 2 N#120

TC 13 N#120

TC 67 N#120

Thermocouple No. 2

Thermocouple No. 13

Thermocouple No. 67

Increasing refinement

FIGURE 63. GRID CONVERGENCE FOR BLUNTY/MAGIC/ANSYS PANT VALIDATION CASE

99

In Figure 63, close attention should be paid to each set of lines for each thermocouple

location. Results for the 120-node case are in green, red for the 324-node case, and blue for the

1,127-node case. As the mesh size is refined, the plot lines begin to overlay, showing

convergence. For this study the highest refinement was used.

Reece Neel used GASP at Aerosoft to run this model in a similar fashion. The inlet

conditions were specified as a Mach number and fluid temperature. A solid material model was

input to match the thermal properties used in ANSYS for the electroformed nickel. The flow

field solution was set up using Roe’s flux difference scheme with a second-order upwind bias,

Kang and Dunn’s air model with real gas chemistry, and a Spalart-Allmaras turbulence model. It

was assumed that the wind tunnel had established a flow field, and then the test article was

injected rather than allowing the flow to develop with the model in place. This assumption leads

the GASP solution to solve for a steady-state Mach 5.02 flow first, excluding the CHT (i.e., a

solid model is not included). Once convergence is reached, the solid model is included in the

solution and run for a time-accurate 10 seconds. The gridding scheme used by Mr. Neel was

based on previous experience at Aerosoft with similar simulations. For the studies performed

here, the solid model meshes were of higher resolution than the finite element mesh used in

ANSYS. Based on the ANSYS mesh refinement study, an even finer mesh should be sufficient

for GASP.

In both solutions, nodes were selected as close as possible to the actual thermocouple

locations from the experiment. These three nodes from each solution were plotted against the

experimental results found in Reference 31, pages 5-4 through 5-6. Figure 64, Figure 65, and

Figure 66 show these results.

100

050

100150200250300350400450500550600650700750

0 2 4 6 8 10 12

ANSYSTC 2

GASPTC 2

Exper.TC 2

Time (s)

Tem

pera

ture

(K)

FIGURE 64. BMA, GASP, AND EXPERIMENTAL TEMPERATURE

PROFILES AT THERMOCOUPLE NUMBER 2 LOCATION

Time (s)

Tem

pera

ture

(K)

050

100150200250300350400450500550600650700750

0 2 4 6 8 10 12

ANSYSTC 13

GASPTC 13

Exper.TC 13



101

Time (s)

Tem

pera

ture

(K)

050

100150200250300350400450500550600650700750

0 2 4 6 8 10 12

ANSYSTC 67

GASPTC 67

Exper.TC 67



The results for the three thermocouple comparisons are not as good as one would have

hoped. Table 8 shows the average variation from the experimental data for each method, BMA

and GASP.

TABLE 8. AVERAGE VARIATION FROM EXPERIMENT

Thermocouple No. ANSYS (%)

GASP (%)

2 3.6 0.28

13 -12.0 -4.4

67 -11.0 2.3

From Figure 64 through Figure 66, GASP predicts the temperature profile very well, less

than five percent variation for all three nodes. By looking at the slope of the experimental curve

and the GASP curve, it is apparent that the overall thermal profile is captured as well. At first

glance the average variation the BMA methodology is within 15 percent, however, this is not the

only criteria that should be used to establish validity of the methodology. Unlike GASP, the

BMA methodology does not capture the slope of the experimental temperature versus time

102

curve. Appropriate prediction of the overall thermal profile is a critical aspect of the aerothermal

design process. Therefore in order to establish a level of validation, not only does the average

variation need to be assessed, but on a qualitative level so does the overall profile.

From the experimental data, Thermocouple 13 showed higher temperatures than

Thermocouple 2, which occurs quite often in blunt body aeroheating. Near the stagnation point

the flow is laminar. As the flow moves downstream it will eventually transition to turbulence.

This jump to turbulence will increase the heat transfer rate dramatically, and therefore, the

hottest temperatures tend to be seen just off the nose tip. This phenomenon is apparent on

hypersonic flight vehicles by the shape change at the nose tip. “Turbulent gouging” defines the

increased ablation at the transition point commonly seen on hypersonic vehicles.

Unfortunately the PANT validation study did not show good agreement between the

BMA methodology and experiment. After further investigation the most likely cause of the

disagreement lies in the boundary conditions established by BLUNTY and MAGIC. Both codes

use numerous hypersonic approximations to establish boundary layer edge properties. These

approximations are based on the assumption that the Mach number squared is large. For

example, looking at Table 9, Mach numbers less than six begin to make the assumption’s

validity questionable considering the error build-up.

TABLE 9. HYPERSONIC APPROXIMATION ERROR

Mach Number MMK ⇒−= 12 12 −= MK % Error

4 4 3.87 3.3 %

5 5 4.89 2.1 %

6 6 5.92 1.4 %

7 7 6.93 1.03 %

10 10 9.95 0.5 %

103

Other sources of possible error are the material properties and thermocouple spot welding

considerations. Neither method takes into consideration the thermal resistance that spot welding

introduces. The additional resistance may cause the thermocouple to report a slightly lower

temperature than the actual. Without having tested this specifically during the experiment, it is

impossible to make any claims to the actual influence. It would be assumed that some standard

procedures were in place to ensure the spot welding process was consistent from weld to weld.

With this in mind, the trends should have shown that the numerical predictions would always be

higher than the experiment, which was not the case. Similarly, assumptions were made

regarding material properties. It is safe to assume that the nickel used in the experiment is close

enough for comparisons at room temperature; however, temperature-dependent properties were

not considered. Standard nickel in this temperature regime has relatively stable properties. The

diffusivity times the density, or in other terms, the conductivity divided by specific heat, changes

by about 2 W/m-K divided by J/kg-K from 65.6 to about 63.6. This should not influence the

results significantly. Moreover, both GASP and BMA use the same material models and

therefore this does not explain the variation between GASP and BMA. Therefore there must be

other sources of error not apparent causing the variation.

Although the BMA method did not show agreement to the PANT study, GASP showed

excellent agreement. Thus far GASP has been validated on numerous conjugate heat transfer

applications and with this additional study has demonstrated it’s validity over a large range of

applications.

4.2.3 GASP Slender Body Comparison

GASP has been well validated for numerous CHT studies as well as the previous PANT

comparison. Considering the lack of experimental data in the flight regime of interest, GASP

will be used as a baseline to validate the BMA methodology in the appropriate regime. For this

comparison two cases were considered. A slender projectile was “flown” at Mach 10 sea-level

conditions for a 5-second flight time as an upper bound to the flight regime of interest, and a

second case at Mach 8 representing the actual initial flight regime was also run. As a reference,

104

Mach 10 sea-level flight has a stagnation temperature of approximately 4,400 K (≈8,000°R) and

Mach 8 of 3,300 K (≈5900°R). The projectile is a sharp 3.6-degree, half-cone angle, 5 inches

long, with a nose-tip radius of 0.20 inch. The baseline design is for a 0.125-inch nose radius but

considering all the scoping studies done in section three a 0.20-inch radius was selected for

consistency. A solid body was used for the validation study only and was not internally

representative of a tactical projectile. Figure 67 depicts the 2-D axisymmetric geometry of the

solid body.

FIGURE 67. GASP COMPARISON PROJECTILE GEOMETRY

The material was assumed to be standard C-350 steel taken at constant room temperature

(Table 10). Temperature-dependent properties were not included because this case is for

validation purposes only. The same constant material properties will be used in both the BMA

method and GASP. If temperature-dependent material properties were used at each time step,

the solution would iterate to determine the correct material property values. This would increase

solution times, and, for the purpose of validation, as long as the same material properties are

used in both the BMA method and GASP, the material property influence will be identical.

Notably, using C-350 steel for this case is for comparison purposes only. Due to the maximum

temperatures experienced during hypersonic flight, it is unrealistic to assume that C-350 steel

will be used for the entire airframe without being thermally protected. However, for the

comparisons, ablation considerations and structural design were not considered.

TABLE 10. C-350 CONSTANT THERMAL PROPERTIES

Density, ρ 8,000 kg/m3

Specific Heat, Cp 245 J/kg-K

Thermal Conductivity, k 26.5 W/m2-K

Due to the large heat flux values, it is important to have a fine mesh along the surface of

the projectile to capture the appropriate thermal gradients. To determine the appropriate grid

105

size, a short study was performed on a simplified representative of the projectile nose tip and

associated heat flux values. The model consisted of a 2-D axisymmetric hemispherical nose-tip

radius of 0.125 inch. For simplicity, 1E8 watts per meter squared (W/m2) constant heat flux was

applied for three seconds. This particular value is the most severe cold wall heat flux boundary

condition found in any of the analyzed cases in this work. A typical cold wall heat flux value for

a Mach 8 projectile with a nose radius of 0.125 inch is approximately 7E7 W/m2 and 1E8 W/m2

for the Mach 10 case. Considering the nose tip is the most severe case, if convergence is met in

the nose-tip region and the mesh density is consistent throughout the projectile surface region,

then this study can be used to define the overall mesh for the remainder of the round.

The hemisphere was meshed by specifying the number of divisions along each boundary

line. The number of divisions defines the number of elements along the edge. Initially, a coarse

mesh was created with 10 divisions along each edge followed by refinements of 20, 40, and

80 divisions. Figure 68 depicts the resulting meshes.

Divisions = 10

Divisions = 40

Divisions = 20

Divisions = 80

FIGURE 68. HEMISPHERICAL MESH REFINEMENT STUDY

106

Each case was run with the constant heat flux value for a period of three seconds. At the

end of each run, four nodal temperatures were recorded. These nodes are located on the surface

at the nose tip or 0 degree, 15 degrees, 30 degrees, and 45 degrees. Due to the small thermal

mass of the tip and the constant heat flux value, the material does nothing but continuously heat

for three seconds, resulting in excessive temperatures; however, this study is for convergence

criteria only and not for temperature magnitude. Figure 69 shows the nodal temperatures plotted

against the number of divisions for each of the cases.

1.E+03

1.E+04

1.E+05

0 10 20 30 40 50 60 70 80

Tip Temp15 degree30 degree45 degree

Tem

pera

ture

(R)

Divisions Along Edges FIGURE 69. NODAL TEMPERATURE MESH CONVERGENCE

As the number of divisions increases beyond 40, the nodal temperatures begin to roll off.

This flattening of the curves indicates little change in the predicted temperature with grid

refinement. To err on the conservative side, 50 divisions along the edge will be used in all

studies, which equates to about 2,000 elements in the nose-tip region. Overall the grid size in

Figure 70 is 5,875 elements. Figure 70 and Table 11 provide the node-locations and associated

coordinates, recalling that the y-axis is the axis of symmetry.

107

22726222582

2645 2677

824813

799788 32

50

22726222582

2645 2677

824813

799788 32

50

FIGURE 70. ANSYS MESH AND NODE LOCATIONS

TABLE 11. NODAL COORDINATES

Node X-coordinate (m) Y-coordinate (m) 2 0 0 27 0 8.47E-4

2582 0 1.32E-1 2622 0 6.86E-2 50 4.11E-3 2.09E-3 32 5.08E-3 5.08E-3 788 5.23E-3 7.47E-3 799 5.53E-3 1.22E-2 813 5.96E-3 1.91E-2 824 6.35E-3 2.52E-2 2677 1.28E-2 7.65E-2 2645 9.57E-3 1.27E-1

The boundary conditions are obtained as before using BLUNTY for the stagnation point

and MAGIC for the remainder of the projectile. Nine stations were selected along the surface in

the axial direction: 0, 0.10, 0.20, 0.30, 0.50, 0.75, 1.0, 3.0, and 5.0 inches. Four cases were run

at the Mach 10 flight conditions to establish convergence with time step, which results in four

levels of refinements in the heat decks as well. The coarsest heat deck has the largest time step

of 1 second, followed by 0.1 second, then 0.01 second, and the finest time step of 0.005 second.

The time step defines the load step and, by doing so, the time at which the heat-flux boundary

condition is applied. The resulting heat decks can be found in Appendix C. During the solution,

thermal histories were output for several nodes throughout the body as listed in Table 11.

The initial large time step of one second grossly over-predicts the temperatures. The

maximum expected nose-tip temperature should approach the stagnation point fluid temperature

of approximately 4,400 K. Initially, the cold wall heat flux value is applied because the body is

108

at ambient temperature. This large heat flux value is applied for the full 1-second duration of the

load step, thus resulting in an excessive overshoot. At the change in load step, due to the high

temperature, the next calculated heat flux value applied to the surface will be zero because the

wall and recovery enthalpy will be the same. Throughout the remainder of the 5 seconds, the

wall temperature at the nose region remains hotter than the fluid, and therefore, the temperatures

continuously decline as the heat is conducted into the body.

By refining the load step size, the heat flux value is recalculated based on the wall

temperature at shorter intervals, better capturing the heat rise. This implies that, at each 0.10

second, the heat flux boundary condition is recalculated and applied for 0.10 second; by doing

so, the predicted temperatures come down to more reasonable levels. However, some significant

oscillations still exist in the temperature profile near the nose tip. This indicates that this time

step size is right on the line of causing the over-predictions. The material overshoots the “real”

temperature at one time step. Seeing how the enthalpies are now equal, the new calculated heat

flux at the next step is zero. The heat is conducted away into the body, and this reduces the

surface temperature. When the next step calculates the heat flux, an enthalpy difference now

exists and the material heats up again. The solution continues to oscillate about the true

predicted value.

Figure 71 and Figure 72 behave much more as expected, with both approaching the

stagnation temperature at the nose tip. The time step is much smaller than the previous two

cases, which results in a smoother transition between load steps. The smooth boundary condition

application prevents the predicted temperatures from overshooting and oscillating as shown in

the previous figures. As the skin temperature increases, the wall enthalpy increases resulting in a

smaller and smaller change between wall and fluid enthalpy, eventually reaching a state where

no more heat transfer occurs across the boundary.

109

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Node 2

Node 50

Node 32

Node 788

Node 799

Node 813

Node 824

Node 2677

Node 2645

Tem

pera

ture

(K)

Time (s)

7200 °R

6300 °R

5400 °R

4500 °R

LS = 0.01s

3600 °R

2700 °R

1800 °R

900 °R

FIGURE 71. NODAL THERMAL HISTORY, LOAD STEP = 0.01 S

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Node 2

Node 50

Node 32

Node 788

Node 799

Node 813

Node 824

Node 2677

Node 2645

Tem

pera

ture

(K)

Time (s)

8100 °R

7200 °R

6300 °R

5400 °R

4500 °R

3600 °R

2700 °R

1800 °R

900 °R

FIGURE 72. NODAL THERMAL HISTORY, LOAD STEP = 0.005 S

Because the stagnation point temperature is known at the nose tip, nodal location was

used to determine time step convergence. From theory, at Mach 10 the nose-tip temperature

should be very close to the fluid stagnation temperature of 4400 K. As the time step size

decreases, the curves at the nose tip should smoothly approach the stagnation temperature.

Figure 73 depicts the time step convergence.

110

0

1000

2000

3000

4000

5000

6000

7000

0.0 1.0 2.0 3.0 4.0 5.0

LS = 0.1 s LS = 0.01 s LS = 0.005 sTe

mpe

ratu

re (K

)

Time (s)

Load Step Size (sec.)

FIGURE 73. NOSE-TIP TEMPERATURE CONVERGENCE

Figure 73 shows that a load step of 0.01 second is adequate. For computation time, this

load step is sufficient (Table 12, CPUH = number of CPUs × solution time).

TABLE 12. CPU HOURS FOR ANSYS SOLUTION

Load Step Size (s) No. of Steps in Run CPUH

(#CPUs × solution time)

1.0 6 <1

0.10 51 2

0.01 501 4

0.005 1,001 10

For comparison, the identical case was simulated in GASP. As in the PANT validation,

the flow field solution was set up using Roe’s flux difference scheme with a 2nd-order upwind

bias, Kang and Dunn’s air model with real gas chemistry, and the Spalart-Allmaras turbulence

model. Again, a steady-state solution was obtained first without the solid model included in the

solution and then a time accurate solution was obtained out to five seconds while including the

solid model.

111

In order to establish grid convergence, three grid sizes were examined. Figure 74 through

Figure 76 illustrate the GASP fluid and solid grid for the coarse study (containing 4200 cells),

with the blue grid being the fluid and green the solid.

FIGURE 74. COARSE MESH USED FOR CONVERGENCE STUDIES (CREATED IN ICEM)

1.2E-2 m1.2E-2 m

FIGURE 75. COARSE MESH – CLOSE UP VIEW OF NOSE TIP

112

FIGURE 76. COARSE MESH – AXIAL RESOLUTION

For comparison purposes, four nodes were selected in ANSYS that could easily be

matched with a corresponding point in GASP. The main purpose of the BMA method is to

calculate thermal soak effects, therefore nodes were selected along the axis of symmetry at the

nose tip, just past the nose radius (0.03 inch), mid-body (2.6 inches), and the base plane: Nodes

2, 27, 2582, and 2622 in ANSYS, respectively.

With the coarse mesh, Nodes 27, 2622, and 2582 did not experience any increase in

temperature. Similarly, the node at the nose-tip should be approaching the stagnation

temperature, but it only reaches 2,000 K rather than the 4,400 K expected. After discussions

with Mr. Reece Neel at Aerosoft, it is evident that the boundary layer resolution was not fine

enough to appropriately capture the thermal transport to the wall. Without a refined grid in the

boundary layer and just inside the solid wall, thermal transfer effects are poorly predicted. A

good indicator of boundary layer resolution is to plot the value of y+ along the wall. The term y+

is a dimensionless length in the y direction relating wall shear, density, and viscosity as seen in

Equation (160):

ρτ

υwy

≡y + (160)

113

The laminar sub-layer is typically found below y+ = 7 (Reference 13). To get accurate resolution

in the turbulent boundary layer points are required within this sub-layer (Reference 32). A value

of one along the surface near the end of the projectile is a good indicator that the boundary layer

resolution is fine enough such that transport properties in the boundary layer are considered. The

coarse mesh used for the preliminary study had a y+ value of approximately 20. After numerous

refinements, a y+ value of 1.16 was achieved. Similarly, the solid model grid was selected with

two considerations in mind. First and foremost, the radial mesh density along the surface needs

to be similar to the boundary layer radial mesh density to avoid any large jumps in element size.

Second, the BMA grid study discussed earlier in this section was used as a baseline for level of

refinement. Recalling that the BMA grid size was 5,875 elements, the GASP solid grid size

was 8,500 (23,087 cells total). Both ANSYS and GASP use a basic thermal conduction model,

so it is safe to assume that a finer, solid-body mesh used in GASP than in the BMA method is

more than sufficient to capture the in-depth thermal response. Figure 77 through Figure 79

depicts the new medium mesh with the y+ value of 1.16.

FIGURE 77. MEDIUM MESH (23,087 CELLS)

114

2E-4 m

FIGURE 78. MEDIUM MESH – CLOSEUP VIEW OF NOSE TIP

FIGURE 79. MEDIUM MESH – AXIAL RESOLUTION

Note in Figure 78, what seems to be a solid blue line along the solid/fluid boundary is a

series of finer and finer cells capturing the boundary layer. Similarly this same resolution was

copied into the solid to ensure there are no major jumps in cell size along through the solid/fluid

boundary.

115

Lastly, as a final mesh refinement more attention was paid to the boundary layer, shock

region, and the overall axial resolution. In the fine mesh, the cell count was doubled compared

to the medium mesh – 44,639 total cells. As before, what seem to be solid blue and green lines

is a clustering of cells that could not be zoomed in close enough to truly get a perspective of the

grid size. Figure 80 through Figure 82 depict the fine mesh.

FIGURE 80. FINE MESH (44,639 CELLS)

2.2E-5 m

FIGURE 81. FINE MESH – CLOSEUP VIEW OF NOSE TIP

116

FIGURE 82. FINE MESH – AXIAL RESOLUTION

In order to replicate the most realistic flight environment via CFD the following

procedure was used to establish the flow field solution and resulting thermal conductive model as

done in the PANT case. Initially, the flow field solution was established by obtaining a steady

state solution without including the solid model. In a subsequent run the solid model was now

included in the solution and run for a time accurate five second duration. Roe’s flux difference

scheme with a second-order upwind bias, Kang and Dunn’s air model with real gas chemistry,

and a Spalart-Allmaras turbulence model was used for the solution. A time step of 1E-4 seconds

was determined to be the converged time step necessary for these types of problems and will be

discussed later. For consistency the values tracked in the GASP solution coincide with the

locations of nodes 2, 27, 2622, and 2582 from Figure 70 to emphasize the stagnation point

solution and the in-depth thermal conduction.

117

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0.0 1.0 2.0 3.0 4.0 5.0

GASP (F)Node 2

GASP (F)Node 27

GASP (F)Node 2622

GASP (F)Node 2582

GASP (M)Node 2

GASP (M)Node 27

GASP (M)Node 2622

GASP (M)Node 2582

Medium = 23,087 cellsFine = 44,639 cells

Time (s)

Tem

pera

ture

(K)

FIGURE 83. GASP GRID CONVERGENCE: FINE MESH IS CONVERGED

In Figure 83, solid lines represent the fine mesh, long dashes the medium mesh, and short

dashes the coarse mesh, while blue is at the nose tip, brown aft of the nose tip, green about mid-

body, and red at the base. Recalling the coarse mesh did not have sufficient boundary layer

resolution (y+ large), thermal transport across the boundary layer was virtually non-existent. In

the second iteration of the mesh refinement a y+ value on the order of one was achieved

appropriately capturing the thermal transport across the boundary layer. In order to establish

convergence, a third refinement was done doubling the cell count and concentrating the

refinement on the boundary layer and axial resolution. The thermal profiles depicted in the

above figure clearly show that there is not a significant change from the medium mesh to the fine

mesh – less than 2 percent at nodes 2, 27, less than 4 percent at node 2622, and less than 1

percent at node 2582. With this in mind, the medium mesh would be sufficient considering the

cost savings and therefore was used for the time step convergence study; however, seeing how

the solution was already run for the fine mesh these results were used as a comparison for the

BMA methodology.

118

Considering that the stagnation point temperature is easily predicted by theory, it is a

convenient point to evaluate for convergence. Figure 84 depicts the time step convergence

results for the stagnation point solid temperature (BMA Node 2).

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0.0 1.0 2.0 3.0 4.0 5.0

dt = 1E-3

dt = 1E-4

dt = 5E-5

Time (s)

Tem

pera

ture

(K)

T0 thry. = 4400 K

Time Step (sec.)

FIGURE 84. TIME STEP CONVERGENCE INDICATE 1E-4 SECONDS CONVERGED (FINE MESH)

The large time step of 1E-3 seconds does not come close to the theoretical stagnation

point temperature expected, which was predicted by the default value of 1E-4 seconds. Due to

the low nose-tip temperature, there is very little in-depth conduction as well. With a time step of

1E-3 seconds the nose-tip temperature approaches the expected value of 4400 K but never quite

gets there over the five-second duration. Lastly, to verify that there is no significant change with

refining the time step further, a value of 5E-5 was used. As can be seen in the figure the dotted

yellow line (5E-5 seconds) tracks almost identically on the solid blue line (1E-4 seconds)

indicating that convergence has been reached. Table 13 lists the GASP medium mesh CPU

hours solution time.

119

TABLE 13. GASP (MEDIUM MESH) CPU HOURS

Time Step (s)

CPUH (#CPUs × solution time)

5E-3 48

1E-3 ≈400

1E-4 ≈5400

5E-5 ≈7000

GASP has been validated in numerous flight environments as well as the PANT case

presented herein thus establishing GASP as an acceptable baseline validation for the BMA

methodology. Having established convergence for both the BMA and GASP analyses, the two

can be compared for the Mach 10 and Mach 8 cases respectively. Recalling from Figure 70, the

nodes tracked in BMA - 2, 27, 2622, and 2582 – were also tracked in the GASP analyses in order

to capture the stagnation point as well as in-depth conduction.

Initially the flow field solution was established by obtaining a steady state solution

without including the solid model. Establishing this steady state solution initializes the flow

field to a Mach 10 flight condition. Figure 85 and Figure 86 depict the steady state Mach 10

flow field solution without including the solid model (depicted in gray). Due to Tecplot’s

convention in the contour scale, the minimum and maximum values are not depicted; however,

they follow the numerical progression of the scale (zero being dark blue and ten being red).

120

Projectile

Mach 10 Flow Field

FIGURE 85. STEADY STATE MACH 10 FLOW FIELD SOLUTION

Projectile

Mach 10 Flow Field

FIGURE 86. STEADY STATE MACH 10 FLOW FIELD SOLUTION (SHOCK REGION)

In a subsequent run, the solid model was now included in the solution and run for a time

accurate five-second duration. This process attempts to replicate the sudden projectile insertion

into a flight environment from gun launch. Roe’s flux difference scheme with a second-order

upwind bias, Kang and Dunn’s air model with real gas chemistry, and a Spalart-Allmaras

121

turbulence model was used for the solution. Referring to Figure 86 there is an unexplained bump

in the flow field at the stagnation point. It is unclear as to the origins of this oddity but some

suggest that Roe’s scheme may not be the most appropriate to use in the axial direction for high

Mach number flows. It may be helpful to use Van Leer’s axially and Roe’s in the radial

direction. Figure 87 and Figure 88 depict the resulting flow field and solid model thermal

profiles at five seconds.

Projectile4339 K

FIGURE 87. FLOW FIELD TEMPERATURE PROFILE @ TIME = 5s

4339 K

FIGURE 88. SOLID MODEL TEMPERATURE PRFILE @ TIME = 5s

Considering GASP’s convention of assigning the fluid wall temperature and solid wall

temperature equal, it is expected that at the stagnation point the fluid and solid models will

indeed be equal as seen in Figure 87 and Figure 88. Throughout the solution, the nodes of

interest were tracked for comparison to the BMA methodology. Figure 89 depicts the Mach 10

BMA, GASP validation comparison.

122

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0.0 1.0 2.0 3.0 4.0 5.0

BMANode2

BMANode27

BMANode2622

BMANode2582

GASP (F)Node 2

GASP (F)Node 27

GASP (F)Node 2622

GASP (F)Node 2582

Time (s)

Tem

pera

ture

(K)

T0 thry. = 4400 K

FIGURE 89. MACH 10 VALIDATION STUDY (BMA: SOLID, GASP: DASHED)

As seen in the above figure, both the BMA method and GASP asymptotically approach

the theoretical stagnation point of 4400 K. The BMA method slightly under predicts the

stagnation point temperature at 4223K (≈3.5 percent error); however, GASP is very close to

theory with a predicted value of 4339 K (≈1 percent error). Since the stagnation point

temperature can be readily calculated via theory, it makes a good data point for comparison.

Also of significant importance in determining the validation is the overall trends. The BMA

methodology tracks closely with the GASP thermal time history; unlike the PANT case

discussed earlier, the slopes of the BMA curves and the GASP curves are very similar. The

overall average variation of BMA compared to the GASP results are all within 10 percent with a

time savings of over three orders of magnitude.

Similarly, as a secondary validation, a closer flight regime was selected to the notional

Electromagnetic Railgun hypersonic round of Mach 8. This study will serve as an additional

data point to the Mach 10 case while the Mach 10 case acts as an upper bound to the problem of

interest. For this scenario, the same fine grid and solution method was used in GASP as in the

previous Mach 10 case. Figure 90 through Figure 93 depict the steady state flow field solutions

for the Mach 8 study and the resulting temperature profiles at five seconds.

123

Projectile

Mach 8 Flow Field

FIGURE 90. STEADY STATE MACH 8 FLOW FIELD SOLUTION

Projectile

Mach 8 Flow Field

FIGURE 91. STEADY STATE MACH 8 FLOW FIELD SOLUTION (SHOCK REGION)

124

Projectile3329 K

FIGURE 92. FLOW FIELD TEMPERATURE PROFILE @ TIME = 5 SECONDS

3329 K

FIGURE 93. SOLID MODEL TEMPERATURE PROFILE @ TIME = 5 SECONDS

Figure 94 depicts the results for the Mach 8 validation study. BMA results are denoted in

solid lines while GASP results are depicted as dashed.

0

500

1000

1500

2000

2500

3000

3500

0.0 1.0 2.0 3.0 4.0 5.0

BMANode2

BMANode27

BMANode2622

BMANode2582

GASP (F)Node 2

GASP (F)Node 27

GASP (F)Node 2622

GASP (F)Node 2582

Time (s)

Tem

pera

ture

(K)

T0 thry. = 3300 K

FIGURE 94. MACH 8 VALIDATION STUDY (BMA: SOLID, GASP: DASHED)

125

Again both the BMA method and GASP predict within reason the theoretical stagnation

point of 3300 K. The BMA method slightly over predicts the stagnation point temperature at

3382 K (≈2.4 percent error); GASP is again very close to theory with a predicted value of 3329

K (≈0.8 percent error). The overall average variation of BMA compared to the GASP results are

within 7 percent with a time savings of over three orders of magnitude.

4.3 CONCLUSIONS AND RECOMMENDATIONS

The EM Railgun hypersonic round has a unique flight environment, which has not been

encountered by any other systems. The combination of high Mach numbers at sea level induces

severe aeroheating loads that in turn cause significant thermal soak concerns. Traditionally,

vehicles and missile systems that travel at these Mach numbers or higher do so at much greater

altitudes, thus reducing the thermal shock and high heating rates. More importantly, due to the

size and geometries, these systems have never had to perform significant full system thermal

studies. With this in mind, there is no readily available data in either the open and classified

literature that can provide relevant experimental in-depth thermal history values for a sea level

launched hypersonic vehicle. The closest data compared was that of a cold wall heat flux

experiment at Mach 10.6 and a blunt Mach 5 case from the PANT series.

The Cleary experiment from Ames Research Center in October of 1969 provides a basis

for verification that the BMA methodology is calculating and applying cold wall heat flux values

accurately. The two cases considered were relatively sharp slender bodies, 0.375 inch and

1.1 inch at 15-degree cone angle. Notionally, the hypersonic round will have a nose radius

somewhere around 0.10 inch to 0.20 inch at a 3.6-degree cone angle. Using the BMA

methodology, the heat flux values were calculated at time equals zero thus corresponding to a

cold wall heat flux condition. According to Cleary, the experimental error in the test was around

20 percent, which was calculated by the variations of numerous runs from theory. In the first

case, 0.375-inch nose radius, BMA had an average variation from the Cleary experiment by

2.6 percent and the second case, 1.1-inch nose radius varied by less than 0.40 percent, which are

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well within the experimental error. Notably both curves display similar trends over the axial

stations. This verification study has demonstrated that the BMA methodology accurately

calculates and applies the heat flux values.

Without a representative experiment that included in-depth thermal response, the next

closest was used. The PANT program was specifically targeting the analysis and validation of

tools to predict shape change phenomena of ablating materials. From this data, some non-

ablating calorimeter data was obtained at high stagnation pressures but low temperatures and

Mach numbers. In particular, the closest case in the PANT series was at Mach 5,

100 atmospheres, and just under 2000°R stagnation temperature. Unfortunately, the BMA

methodology showed poor performance predicting the overall thermal response of the test

model. At the three nodes of interest, variations of approximately 4 percent, 12 percent and

11 percent were seen; however, more discouraging was the overall slope of the curve compared

to experiment. After further investigation into the sources of error it has been determined that

the aeroheating codes used to establish the boundary conditions are not valid at low Mach

numbers. The hypersonic assumptions made in the codes begin to introduce significant sources

of error as the Mach numbers get much below six.

Although the PANT experiment was not useful in validating the BMA methodology, it

was useful in two other aspects. An extremely important fact that is often times overlooked is

the realm of applicability for a particular method. All too often analyses are performed without

consideration for the validity of the method in the regime of interest. Use of these predictions

can lead to erroneous results incurring significant time and money. More importantly for the

purposes of this methodology development, GASP predicted the thermal response quite

accurately and tracked very well with the overall slopes of the experimental data. Considering

GASP has been well validated on numerous other applications (see Schetz, Neel, Marineau

Reference 26), this additional validation further justifies the use of GASP as a baseline analysis

tool for validating the BMA methodology. Two cases were analyzed using both GASP (as the

baseline) and the BMA methodology. A sharp slender projectile was “flown” at Mach 10, as an

upper bound, and at Mach 8 for five-second durations.

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In both cases, GASP accurately predicted the theoretical stagnation points to within about

1 percent and the BMA method approximately 3.5 percent. The overall average variation seen

was 10 percent or less as seen in Table 14.

TABLE 14. BMA AVERAGE VARIATION FROM GASP

Mach 10 Mach 8

Node 2 -7% -4.5%

Node 27 -5% -3.8%

Node 2622 -0.90% -0.79%

Node 2582 -10% -7%

More importantly, the overall trends were quite similar with one exception. GASP

always tended to have a slightly steeper slope, or in other words responded to the heat load more

rapidly. Referring to Figure 89 and Figure 94, in every instance GASP was always ahead of the

BMA predictions. The primary cause to this is the method that the boundary conditions are

applied. GASP assumes that the fluid wall and solid wall boundary temperatures are equal. In

order to most accurately replicate the gun launch environment, the projectile must be inserted

into a fully developed flow stream. This flow stream has to be established by running the CFD

solution out to steady state with no solid model included. Once the solid model is included in the

solution, the solid wall boundary takes on the fluid wall temperatures in the next time step (1E-4

seconds). This approach is the most accurate replication of a true environment that can be

obtained from the CFD methods. The true physics would require a resting projectile at time

equals zero. From zero to a few milliseconds the projectile is accelerated to full muzzle exit

velocity and shortly thereafter experiences a fully developed flow field. This would not be

possible to numerically replicate in CFD.

In the BMA methodology the boundary conditions are not physically tied to any outside

fluid cells. In the finite element realm the boundary conditions are applied as heat loads on the

exterior face of the solid surface elements. At time equals zero the first heat flux boundary

condition is calculated based on the wall enthalpy (interpolated form enthalpy tables using

pressure and temperature), the heat transfer and fluid enthalpy (outputs from

BLUNTY/MAGIC). This heat flux is applied for a small increment in time, in this case 1E-2

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seconds, resulting in a temperature rise over the time increment. Depending on the frequency of

the output and the substeps taken in between each load step, the slope can be slightly varied.

Recalling the time step convergence study 1E-2 seconds was sufficient to achieve convergence.

Considering the intended use of this method, a small variation in curve slope is acceptable if the

final temperatures and overall profiles are accurately predicted.

The validation studies presented herein represent a mere fraction of the overall flight

durations expected for the EMRG hypersonic round. Flight times of six to eight minutes are not

uncommon. Table 15 displays the overall cost in CPU hours required for the five-second

duration validation studies.

TABLE 15. COST COMPARISONS

Case Processors Hours CPUH (CPU x Hours)

Mach 10 BMA dt = 1E-2 s

2 2 4

Mach 10 GASP Medium Mesh (23,087 cells)

dt = 1E-4 s 32 168 5,376

Mach 10 GASP Fine Mesh (44,639 cells)

dt = 1E-4 s 32 420 13,440

Based on the validation studies, it would be cost prohibitive to attempt to run a realistic

flight trajectory using CFD methods. Additionally, a flight trajectory has continuously changing

conditions that would be unrealistic to change in a CFD tool. The BMA method allows for

continuously changing flight conditions as well as significant cost savings of over three orders of

magnitude.

At the onset of this dissertation, a goal of 15 percent accuracy was suggested.

Considering this capability does not currently exist, it is reasonable to assume some error when

compared to the validation cases. A methodology that falls within 10 percent - 15 percent

average variation for thermal profile would be deemed acceptable considering the current state of

the art and cost savings. The data thus presented in the validation study suggests errors on the

order of 10 percent, which meets the goals set forth. In the following section the BMA method

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will be used to demonstrate its utility in analyzing a full projectile system over numerous

relevant flight trajectories.

130

5.0 ELECTROMAGNETIC RAILGUN HYPERSONIC ROUND ANALYSIS

The development and validation of the BMA methodology allows an analysis of the

various trajectories of the EMRG hypersonic round. To this point, the only transient full FEA

models developed used AP05 with cold wall boundary conditions. Comparing past results with

AP05/ANSYS to nose-tip solutions from the Sandia codes makes it clear that the AP05 greatly

underpredicted the tip temperatures. The nose-tip temperature predicted in the initial analyses

was approximately 2000 K, but it should be much closer to the real gas stagnation temperature of

3,300 K at a Mach 8 sea-level launch. AP05/ANSYS did not include realistic hot wall heat flux

values, but it also underpredicted the boundary conditions. BMA methodology will be used to

study the effects of trajectory launch angles on the overall thermal performance of a notional

hypersonic round. These initial studies do not include thermal protection systems studies nor

ablation/erosion effects. Future studies using the BMA methodology will be performed

analyzing the overall parameter space.

5.1 PROJECTILE AND FLIGHT CONDITIONS

The notional concept for the hypersonic round is a sharp, slender body with a nose radius

near 0.125 inch. This would be sufficient to minimize drag and provide some mitigation to the

aeroheating. As the nose tip increases, the heating decreases (recalling q″ ~ 1/sqrt(R)).

However, the drag increases at a much greater rate. Currently, the studies have assumed a mass

constraint based on the available energy at muzzle exit. This in turn has defined the necessary

interior volume and overall dimensions of the round. For drag purposes, a cone configuration is

used, although ogive profiles have a larger interior volume. Figure 95 depicts the notional

hypersonic round exterior geometry.

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30”

0.125” nose tip radius

3.6° cone angle

FIGURE 95. HYPERSONIC ROUND APPROXIMATE GEOMETRY

Initial trade studies of the EMRG system indicate that launch velocities of approximately

2.5 km/s are achievable (Reference 1). As a safety margin, the aeroheating analyses have

assumed a Mach 8 launch condition rather than the Mach 7.5 that 2.5 km/s yields; Mach 8 at sea

level is approximately 2.7 km/s (8,900 ft/s). Depending on the mission and range required,

various trajectories can be achieved by changing the QE, the angle at which the gun is positioned

for launch. Notional elevations range from a slight negative angle (direct fire applications) to an

angle near 80 degrees. Figure 96 depicts the various trajectories of interest, and each curve

denotes a launch angle; the figure does not show a direct fire trajectory because of the short

range.

Range (nmi)Range (nmi)

Altit

ude (

kft)

Altit

ude (

kft)

1010°°2020°°

4040°° 5050°°

6060°°

7070°°

8080°°

3030°°

FIGURE 96. NOTIONAL TRAJECTORIES OF EMRG HYPERSONIC ROUND

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The number of potential trajectories militates against the use of visual inspection to

determine the worst aeroheating case. The ablation studies performed with the Sandia National

Laboratories codes determined the case for which the velocity and density remain the highest

during the trajectory will experience the most severe ablation effects (around 20° QE). However,

the most severe in depth thermal soak effects will more than likely occur on a trajectory that

experiences significant heating during ascent and a considerable overall flight time. For the

purposes of this study trajectories of 0.5°, 10°, 30°, 50°, and 80° were analyzed. The trajectories

analyzed along with the resulting thermal time histories can be found in Appendix D.

5.1.1 ANSYS Finite Element Analysis

To be consistent, the same solid model was used as that from the original AP05/ANSYS

study. As the design of the EMRG hypersonic round progresses, the level of detail contained in

this model will increase. At the present time only representative mass fills were considered. It

was assumed that no thermal protection system is included and all masses are in perfect contact.

The resulting analyses will therefore be quite conservative however it does provide the designer

with a starting point for the thermal considerations. Realistically higher fidelity models will

eventually be incorporated to model contact resistances as well as thermal protection and

mitigation concepts. Figure 97 depicts the C-350 maraging steel airframe, the tungsten nose-tip

fill, and the tungsten carbide fill and their associated material properties.

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Y – along centerline

X – radial from projectile axis

Tungsten Nose Fill

Tungsten Carbide Fill


18750180100Tungsten

800024526.5C350

17200134163.3Tungsten Carbide

Density (kg/m3)C (J/kg-K)K (W/m-K)Material

18750180100Tungsten

800024526.5C350

17200134163.3Tungsten Carbide

Density (kg/m3)C (J/kg-K)K (W/m-K)Material

30”

3.6° Cone Half Angle

0.125” Nose Radius

FIGURE 97. ANSYS GEOMETRY AND MATERIAL PROPERTIES

Particular detail was paid to the meshing nose-tip region, as well as the region just inside

the wall. Past experience shows that these regions must have very fine meshes to capture the

large thermal gradients present along the skin. The convergence study presented in Section 4.2.3

discusses how the nose-tip mesh was selected. The internal geometries are much more stable

because they are not dealing with as large thermal gradients as the nose tip and surface layers.

As long as the nose tip and surface elements capture the appropriate gradients, the rest of the

internal model can be meshed relatively coarsely since the heat conduction model is a simple

linear solution. To ensure that this is indeed the case, a convergence study was performed

varying the level of refinement in the internal assemblies. The nose tip and surface element

resolution was determined from the convergence studies in Section 4.2.3. Figure 98 depicts the

three levels of mesh refinement for the internal assemblies.

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Case 1: 840 elements (Tungsten Fill+Dispense)



Dispense Tungsten Fill

FIGURE 98. INTERNAL ASSEMBLY CONVERGENCE STUDY GRIDS

The numbers of elements denoted in the figure are those for the internal assemblies only.

The surface mesh and nose tip is the same refinement as defined in Section 4.2.3. As done in the

nose-tip convergence study, a constant heat flux of 1E8 W/m2 was applied to the entire surface

of the projectile for 2 seconds. The selected heat flux is much larger than any that will ever be

experienced in this particular study; therefore, convergence at a more severe case will be

sufficient to establish convergence for the case in question. The resulting thermal profiles

indicate that, for each refinement, little change occurred in the predicted temperatures, as

expected. At the base of the tungsten fill, less than 1-percent variation occurred over the three

meshes, and similarly for the mid-dispense location, less than 0.25-percent variation occurred.

Figure 99 depicts the convergence of the three mesh resolutions and implies that the coarsest

mesh is sufficient for the study.

135

Time (s)

Tem

pera

ture

(K)

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 0.5 1 1.5 2

W BASE #1

MID-DISPENSE #1

W BASE #2

MID-DISPENSE #2

W BASE #3

MID-DISPENSE #3

Tungsten Fill Base

< 1% variation

Mid-Dispense

< 0.25% variation

FIGURE 99. INTERNAL GEOMETRY MESH CONVERGENCE STUDY

IMPLIES COARSE MESH IS SUFFICIENT

With the mesh sizes established, the entire model can be meshed accordingly. Figure 100

depicts the meshed geometry with the node locations that will be tracked.

Y

X

Node 17381

Node 21217

Node 9912Node 10778Node 13556

Node 17196

Node 16823

Node 16283

Node 2

Node 48

Node X Y Location DescriptionSURFACE

2 0 0 nose tip48 2.98E-03 1.27E-03 45 degree tip

17381 5.37E-03 2.53E-02 1 inch back21217 6.91E-03 5.18E-02 2 inch back9912 1.12E-02 1.26E-01 5 inch back

10778 2.40E-02 3.46E-01 Surface 13.5"13556 4.59E-02 7.24E-01 Surface Base 28.5"

CENTERLINE16283 0 2.30E-01 Tungsten Fill Base16823 0 5.55E-02 Mid-Dispense17196 0 5.60E-01 Mid-Bulkhead

FIGURE 100. ANSYS MESH AND NODE LOCATIONS OF INTEREST

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Once the mesh and heat deck have been input the BMA methodology is implemented.

The program initializes the entire geometry to a standard temperature of 293 K and begins

looping through the heat loads via the heat deck provided. Figure 101 shows a sample of the

heat flux boundary conditions applied to the nose-tip region for a Mach 8 cold wall (time equals

zero) case.

2mW

FIGURE 101. APPLIED HEAT FLUX BOUNDARY CONDITIONS

Notably, the heat flux drops off about one order of magnitude around the nose-tip curvature.

This clearly indicates how crucial a role the nose tip plays into the thermal performance of the

round.

5.2 RESULTS AND DISCUSSION

The BMA method was implemented on the five trajectories using standard dual-

processor workstation with a large amount of storage space. It was found that the number of

processors used did not greatly affect the overall computation time since the driving parameters

were actually the calculation and application of the boundary conditions rather than the solver.

Unfortunately, ANSYS writes out extremely large result files. In this particular application there

were trajectories containing over 12,000 load steps of which each step was written out to the file.

This caused resulting file sizes on the order of 75 gigabytes which significantly limited where the

cases could be run. By attaching a large portable backup drive to the machine these files could

easily be written allowing for the analysis of the full trajectories intended.

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A direct fire application is often mistaken as the thermally worst trajectory experienced

by the hypersonic round. From a cursory look, this trajectory does experience the fastest

deceleration in the densest environment; however, much of this energy loss is actually dissipated

in gas effects of the flow stream rather than into all aeroheating. Figure 102 depicts the thermal

time history for the 0.5° QE trajectory.

0

500

1000

1500

2000

2500

3000

3500

0 1 2 3 4 5 6

Node 2

Node 48

Node 17381

Node 21217

Node 9912

Node 10778

Node 13556

Node 16283

Node 16823

Node 17196

Time (s)

Tem

pera

ture

(K)

(°R)

6300

5400

4500

3600

2700

1800

900

FIGURE 102. QE 0.5° THERMAL HISTORY

In roughly a half second, the nose-tip temperature reaches almost 2900 K (5219°R) thus

imposing significant thermal shock to the nose-tip material. As expected, the maximum

temperature experienced at the nose tip approaches 3100 K, just shy of theoretical stagnation

point temperature of Mach 8. Since this trajectory decelerates so rapidly, it quickly falls below

the hypersonic regime thus significantly reducing the aeroheating effects. In combination, a

flight time of only five seconds never allows the large heat loads to conduct into the internal

assemblies resulting in lower internal temperatures than will be seen in the longer-flight-time

trajectories.

138

Unexpectedly, the maximum temperature seen at the aft node, 13556, is actually higher

than that experienced at the 13.5 inch surface station. The primary reason for this is not due to

the actual physics since the heat rates are indeed higher at 13.5 inches, but rather due to the solid

model. At the present time no internal mass is included in the aft cavity. The additional thermal

mass present in the main section of the body acts as a sink helping reduce the temperatures at the

mid-body node whereas the aft node has a very small amount of wall mass only.

As the quadrant elevation begins to incline much above five degrees, flight times begin to

dramatically increase and eventually endo-exo-endo atmospheric flights are achieved above forty

degree quadrant elevations. Figure 103 depicts the thermal time history for the 10° QE

trajectory.

0

500

1000

1500

2000

2500

3000

3500

0 10 20 30 40 50 60 70 80

Node 2

Node 48

Node 17381

Node 21217

Node 9912

Node 10778

Node 13556

Node 16283

Node 16823

Node 17196

Time (s)

Tem

pera

ture

(K) (°R)

6300

5400

4500

3600

2700

1800

900

FIGURE 103. QE 10° THERMAL HISTORY

Again, the nose-tip temperature jumps to 3100 K (5600°R) in less than a second as seen

in the 0.5 degree case. Regardless of the quadrant elevation, the launch environment is

consistent among all the trajectories right at muzzle exit, Mach 8 sea-level. It is expected that

139

each of these trajectories should reach about the same maximum temperature at the nose tip in

less than second. The amount of time spent in the atmosphere is what dictates the internal soak

effects, which will greatly vary between the trajectories. Figure 104 through Figure 106 depict

the thermal time histories for the remaining three trajectories, 30°, 50°, and 80°.

0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200 250

Node 2

Node 48

Node 17381

Node 21217

Node 9912

Node 10778

Node 13556

Node 16283

Node 16823

Node 17196

Time (s)

Tem

pera

ture

(K) (°R)

6300

5400

4500

3600

2700

1800

900


0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200 250 300 350 400

Node 2

Node 48

Node 17381

Node 21217

Node 9912

Node 10778

Node 13556

Node 16283

Node 16823

Node 17196

Time (s)

Tem

pera

ture

(K) (°R)

6300

5400

4500

3600

2700

1800

900


140

0

500

1000

1500

2000

2500

3000

3500

0 100 200 300 400 500

Node 2

Node 48

Node 17381

Node 21217

Node 9912

Node 10778

Node 13556

Node 16283

Node 16823

Node 17196

Time (s)

Tem

pera

ture

(K) (°R)

6300

5400

4500

3600

2700

1800

900


From the previous figures general trends can be seen throughout. All trajectories reach

3100 K (5600°R) very rapidly due to the Mach 8 sea-level launch condition of each. As the

trajectories increase in quadrant elevation more time is spent in the exo-atmosphere. As the

projectile traverses the endo-atmosphere the large heat loads at muzzle exit dissipate as the

altitude increases. This results in an immediate spike in temperatures until a point is reached

where the wall enthalpy becomes equal to the air enthalpy. At this point the heat transfer stops

until the wall enthalpy once again drops below the air enthalpy. In addition to this effect, the

continuous in-depth conduction is always acting to equilibrate the temperatures throughout the

body. The general trends can be separated into two categories for the endo-exo-endo trajectories,

trends at the nose tip and surface regions and trends for internal assemblies.

The nose tip and surface regions are driven by the aeroheating loads experienced during

atmospheric flight while the internal assemblies experience the resulting in-depth conduction of

these heat loads. The locations near the nose tip and forward regions tend to spike in

temperature just after muzzle exit. As the heat loads dissipate due to deceleration and altitude

the temperatures begin to decline since the in-depth conduction now becomes the dominant

141

mode of heat transfer. During the exo-transit these temperatures continuously decline until they

reach a state of thermal equilibrium where no more conduction occurs. As the projectile begins

to re-enter the atmosphere the aeroheating effects begin to dominate once more resulting in a

secondary temperature spike.

The internal assemblies follow a slightly different trend than do the nose tip and surface

regions. Since internal heat generation from the electronics is not being considered in this study

the only source of heat results from the in-depth conduction of the aeroheating effects. This

causes a lag in temperature response from the surface regions due to the conduction path. Unlike

the surface regions the internal assemblies have a much slower and smoother temperature rise,

which extends into the exo-atmospheric transit. Although the surface regions are no longer

being heated, the heat loads are still being conducted into the body of the round. Eventually the

internal assemblies reach a thermal equilibrium and no more conduction takes place until the

surface regions begin to heat up once again at re-entry. Notably, only the nose-tip region

experiences its maximum temperature just after muzzle exit. With the increase in heat loads

upon re-entry the remaining regions will approach or surpass the initial high temperatures.

Table 16 denotes the maximum temperatures experienced for each trajectory studied while

contour plots of the various trajectories can be found in Appendix D.

TABLE 16. TRAJECTORY MAXIMUM TEMPERATURES Geometric Location Node QE 0.5

(K) QE 0.5

(ºR) QE 10

(K) QE 10 (ºR)

QE 30 (K)

QE 30 (ºR)

QE 50 (K)

QE 50 (ºR)

QE 80 (K)

QE 80 (ºR)

Nose tip 2 3109 5597 3109 5596 3109 5596 3108 5595 3108 5595 45-degree tip 48 2893 5208 2883 5190 2865 5157 2848 5127 2834 5101 1-inch surface 17381 2390 4301 2377 4278 2342 4216 2294 4129 2544 4578 2-inch surface 21217 2176 3916 2185 3934 2106 3790 2019 3635 2414 4344 5-inch surface 9912 1662 2991 1740 3132 1599 2877 1709 3077 1794 3230 13.5-inch surface 10778 1335 2403 1287 2316 1266 2280 1395 2511 1343 2417

28.5-inch surface (base) 13556 1470 2646 1421 2558 1505 2708 1598 2876 1513 2724

Tungsten fill base 16283 961 1730 1351 2433 1341 2413 1359 2447 1244 2240

Mid-dispense 16823 731 1315 1170 2107 1238 2229 1198 2157 1069 1923 Mid-bulkhead 17196 371 667 1201 2162 1247 2245 1099 1978 968 1743

Due to the various trajectories, it is apparent that the hypersonic round will experience

drastically different thermal environments depending on the mission goals. With this

142

consideration, it is beneficial from a manufacturing and logistics standpoint to design one

projectile that can survive all of the various thermal environments. Figure 107 depicts the

maximum temperatures experienced at various geometric locations over all trajectories.

2544 K (4578 °R)

2414 K (4344 °R)

1794 K (3230 °R)

1395 K (2511 °R)

1598 K (2876 °R)

1247 K (2245 °R)

1238 K (2229 °R)

1359 K (2447 °R)

3109 K (5597 °R)

2893 K (5208 °R)

Electronics/ GNC

Dispense Mechanism / Lethal Mechanism

• MIL STD 250°C (523 K)• Common elec. 125 °C

• RXN Jets

• Mechanisms• Electronics• Safe & Arms• Det Chord

• Minimizing nosetip ablation will be difficult• Thermal shock must be considered• Thermal Soak Mitigation

Airframe Material• Minimize thermal soak• RXN Jet locations

FIGURE 107. MAXIMUM DESIGN TEMPERATURES

Starting from the nose tip, it is apparent that the temperatures expected will pose a

significant challenge to the overall integrity of the nose tip as well as the thermal performance of

the entire round. Notably, the nose-tip predictions of BLUNTY/SODDIT and ASCC, 5,250°R

and 6,000°R, respectively are quite close to those predicted by BMA. The results plotted for

ASCC are for that of an ablating C-C nose tip. The ablation process of C-C is exothermic,

meaning it gives off heat during the process. By doing so the temperature at the tip tends to

become hotter than the stagnation temperature. This same phenomenon occurs in a charcoal

grill.

The nose-tip temperatures predicted by BMA are typical of high-performance reentry

vehicles (RV). The RV community has flown materials such as C-C at these temperatures with

143

good survival rates. The major difference in this particular case is time duration for the onset of

the maximum temperatures and pressures. Reentry vehicles, although operating at extreme

velocities, do not encounter significant air density for a period of time. The heating and loading

profiles are much less severe than the instantaneous thermal and pressure loading seen from a

hypersonic sea-level system. Therefore, thermal shock considerations are of greater importance

to this system than typical RVs which historically have had thermal shock issues already

mitigated. At the present time, test facilities do not have the capabilities to operate at these flight

conditions. Until these rounds can be fired at the desired sea-level velocities, testing will have to

be done piecemeal to obtain some data at close to design conditions. All conditions may not be

tested at once, but via arc jet, tunnel, sled track, and laser facilities, many of the design

conditions can be approximated and show traceability to a field able system. The all-up testing

will come with a gun system capable of firing at these velocities.

Moving back from the nose tip, the temperatures decrease rapidly but are still pushing the

limits of many standard materials. At these elevated temperatures many metals will be past their

useful limits—a temperature at which the strength significantly deteriorates even though the

material has not melted. Depending on the type of control system implemented, it would be

beneficial from an aerodynamic standpoint to place canards or reaction jets forward of the center

of gravity. With these types of temperatures it would be difficult to place anything that extends

into the flow as well as any type of functional components just under the skin. Many jet

propellants would not be able to survive in these temperatures. By about mid-body, the

temperatures begin to approach levels that can be readily handled by many of the high-strength

steels. An interesting, counterintuitive observation is that the aft end is actually hotter than the

mid-body temperature. With the current model being used, this is understandable because there

is very little thermal mass at the tail end, whereas the mid-body has a large amount of tungsten

carbide inside to act as a heat sink. What this does tell the designer, though, is that the aft end

has to be considered and cannot be disregarded due to its location.

Additionally another aspect at this location that is not readily addressed by many codes,

other than some proprietary codes and CFD, is the base heating effect. As the flow passes over

the aft end of the projectile, it will again cause a stagnation point just off the base. The large

144

temperature jump at this stagnation point can heat up the base of the projectile as well. Luckily,

in flight, the base area does not have to support the loads that it did during launch.

Structural survivability and overall aerodynamic performance are the main focus of the

airframe. Additionally, the airframe or components of the airframe will serve as thermal

protection for the internal assemblies. By investigating the predicted temperatures of the internal

assemblies without a TPS, the designer can begin to properly select and size out the protection

necessary. From Figure 107, the internal assemblies reach temperatures of 1359 K (2447°R),

1238 K (2229°R), and 1247 K (2245°R) for the tungsten nose fill, the mid-dispense, and the aft

bulkhead respectively.

The tungsten nose fill does not serve any load bearing or system role other than aid in

center of gravity positioning and final impact dynamics. With its containment within the round

there should be no major concerns with the insert at these temperatures. Of more critical concern

is the dispense region and the aft bulkhead. Both regions will house components, potentially

electrical, chemical, and mechanical, needed for lethal effects as well as overall guidance and

navigation. It is the intent of the hypersonic round to have little to no high explosive for the

dispense event, but if explosives are necessary these high temperatures must be considered when

designing the explosive mechanisms. Many explosives may not necessarily ignite at these

temperatures but can in fact degrade over time.

Lastly, of critical importance is the operating environment the guidance, navigation, and

control system will experience containing numerous small mechanical devices, batteries, and

electronics. The aft bulkhead temperatures predicted with no form of thermal protection are on

the order of 1250 K (2245°R). In this area it is expected that some sort of “canister” will be

mounted containing many of these sensitive components. Military standard electronic

components, which are costly and difficult to purchase in quantity, call for operating

temperatures of 250°C (523 K, 941°R) while most commercial components have maximum

operating temperatures of 125°C (398 K, 716°R). The temperatures predicted for the canister

mounting points are over three times the operating temperatures of the electronics. Extreme

145

caution must be taken in the design of the thermal protection system implemented in this region

to ensure these temperature sensitive components remain within the operating limitations.

146

6.0 CONCLUSIONS

Hypersonic flight at sea-level conditions induces severe thermal loads not seen by any

other type of current hypersonic system. A Mach 8 slender cone projectile with a nose-tip radius

on the order of 0.125 inch launched at sea level can undergo maximum temperatures near

3,300K (5,940°R) in less than a second after gun launch. In addition, heat soak further

exacerbates the severe flight environment by affecting temperature-sensitive internal

components, such as inertial measurement units, mechanical actuator systems, and electronics.

Appropriate design of the hypersonic round requires a solid understanding of the thermal

environment. Numerous codes were obtained and assessed for their applicability to the problem

under study, and outside of the GASP CHT module, no efficient codes are available that can

model the aerodynamic heating response for a fully detailed projectile, including all

subassemblies, over an entire trajectory.

Although the codes obtained from others were not applicable for a fully detailed, thermal

response analysis, they were useful in providing some insight into expected maximum nose-tip

temperatures and ablation trends. For a Mach 8 sea-level-launched naval projectile, expected

nose-tip temperatures can approach 3,100K (5,580°R) and ablation of approximately 1.25 inches.

Additionally these maximum temperatures experienced occur within less than a second after gun

launch suggesting thermal shock will have to be addressed in detail when selecting nosetip

materials.

In order to capture the thermal soak effects of a fully detailed hypersonic round over a

typical flight trajectory, a new methodology was required. This methodology couples the Sandia

aerodynamic heating codes (Section 3.0) with a full thermal finite element model of the desired

projectile, using the finite element code, ANSYS. The resulting methodology (BMA) fills a

capability gap by enabling the user to establish a thermal time history of a geometrically accurate

projectile for a typical trajectory of interest, in an efficient manner. In comparison to

computational fluid techniques, this method was demonstrated to be within 10 percent of the

147

CFD results at a cost savings of over three orders of magnitude. Ablation effects were not

considered in this methodology development.

With the BMA methodology, cost-effective, highly detailed analyses of hypersonic

projectiles can now be done. Various trajectories of quadrant elevations of 0.5°, 10°, 30°, 50°,

and 80° were analyzed to determine thermal time histories and maximum operating

temperatures. All of the trajectories have the same launch condition, Mach 8 sea-level, and

therefore will undergo the same initial thermal spike in temperature at the nose-tip of

approximately 3,100 K (5600°R). This will significantly influence the material selection at the

nose tip. From an aerodynamic standpoint it would be beneficial to have a non-ablating material;

however, ablation greatly aids in the overall thermal soak mitigation. Progressing rearward from

the nose tip the temperatures begin to drop rapidly, over 1300 K (2300°R) in five inches. The

large gradient present over a mere five inches can induce significant thermal stress.

Of the five trajectories analyzed the maximum internal temperatures experienced

occurred for the 50° quadrant elevation trajectory. This trajectory experienced temperatures in

excess of 1,000 K (1800°R) for more than 80% of its flight time. The two major areas of

concern are the dispense/lethality region and the electronics. Both regions are expected to

experience temperatures over 1200 K (2160°R). All mechanical systems must be able to

withstand extended periods of operation at these elevated temperatures. This includes

electronics for the dispense mechanism or guidance, navigation, and control system; the lethal

payload; and any type of shaped charges used for airframe separation.

The BMA methodology has shown an acceptable range of error within 10 percent with

outstanding cost savings to the alternative CFD methods. Furthermore, this methodology adds a

capability that has not readily existed for the detailed thermal design of a hypersonic projectile.

The methodology is not intended to replace the importance of computational fluid methods for

high fidelity analyses of single point events, but this does enable an efficient method for

exploring the overall thermal trade space. The analyses presented herein increase the

fundamental understanding of the deign challenges associated with such a system.

148

7.0 REFERENCES

1. Ellis, R. L.; Poynor, J. C.; and McGlasson, “Influence of Bore and Rail Geometry on an Electromagnetic Naval Railgun System,” IEEE Transactions on Magnetics, Vol. 41, January 2005.

2. Ellis, R. L., and Poynor, J. C., “Sizing an Electromagnetic Railgun for Naval Application,” ASNE: Engine as a Weapon Conference, Annapolis, MD, 2005.

3. Dahm, T. J., et al., “Passive Nosetip Technology (PANT II) Program,” SAMSO-TR-77-11, Volumes I and II, Acurex Corporation/Aerotherm Division, October 1976.

4. Bertin, J. J., Hypersonic Aerothermodynamics, AIAA text publications, AIAA, Washington, DC, 1994.

5. Scott, C. D.; Ried, R. C.; Maraja, R .J.; Li, C. P.; and Derry, S. M., “An AOTV Aeroheating and Thermal Protection Study,” Thermal Design of Aeroassisted Orbital Transfer Vehicles, Vol. 96 of Progress in Astronautics and Aeronautics, AIAA, NY, 1985, pp. 198-229.

6. Anderson, J. D., Hypersonic and High Temperature Gas Dynamics, AIAA text publications, AIAA, Reston, VA, 1989.

7. ABM Research and Development at Bell Laboratories, Project History, Chapter 9, “Sprint Missile Subsystem,” prepared by Bell Laboratories on behalf of Western Electric, for the U. S. Army Ballistic Missile Defense Systems Command under Contract DAHC60-71-C-0005, Whippany, NJ, October 1975.

8. Lu, F. K., Marren, D. E., Advanced Hypersonic Test Facilities, AIAA text publications, AIAA, Reston, VA, 2002.

9. Arnold Engineering Development Center, Test Facilities Handbook, Thirteenth Edition, Arnold Air Force Base, Tennessee, May 1992.

10. Hochrein, G. J., A Procedure for Computing Aerodynamic Heating on Sphere-Cones – Program BLUNTY, Volume 1: Concept and Results, Sandia National Laboratories, Albuquerque, New Mexico, SC-DR-69-243, CFRD, June 1969.

11. Noack, R. W.; Walker, M. A.; Lopez, A. R., PURMAGIC: Processing Utility for Running Multiple Aerodynamic Codes and Geometries Using an Interactive Computer, Sandia National Laboratories, Albuquerque, NM, SAND89-0561, March 1989.

12. Noack, R. W., and Lopez, A. R., Inviscid Flow Field Analysis of Complex Reentry Vehicles: Volume I, Description of Numerical Methods, Sandia National Laboratories, Albuquerque, NM, SAND87-0776, October 1988.

13. Polansky, G. F., Hypersonic Integral Boundary Layer Analysis of Reentry Geometries (HIBLARG) Code Description and User’s Manual Version 2.0, Sandia National Laboratories, Albuquerque, NM, SAND89-0552, March 1989.

149

14. Moore, F. G, Approximate Methods for Weapon Aerodynamics, AIAA, 2000.

15. Eckert, E. R. G., “Engineering Relations for Heat Transfer and Friction in High-Velocity Laminar and Turbulent Boundary Layer Flow over Surfaces with Constant Pressure and Temperature,” Transcript of the American Society of Mechanical Engineers, Vol. 78, No. 6, 1956.

16. Cohen, N. B., Correlation Formulas and Tables of Density and Some Transport Properties of Equilibrium Dissociating Air for Use in Solutions of the Boundary Layer Equations, Langley Research Center, Langley Field, VA, NASA-TN-D-2780, May 1960.

17. Fay, J. A., and Riddell, F. R., “Theory of Stagnation Point Transfer in Dissociated Air,” Journal of the Aeronautical Sciences, Vol. 25, No. 2, February 1958, pp. 73-85.

18. Hord, R. S., Approximate Composition and Thermodynamic Properties of Non-ionized Nitrogen-Oxygen Mixtures, Langley Research Center, Langley Field, VA, NASA-TN-D-2, August 1959.

19. Kemp, N. H.; Rose, P. H.; and Detra, R. W., Laminar Heat Transfer Around Blunt Bodies in Dissociated Air, AVCO Research Report No. 15, Everett, MA, January 1958.

20. Rose, P. H.; Probstein, R. F.; and Adams, M. C., Turbulent Heat Transfer Through a Highly Cooled Partially Dissociated Boundary Layer, AVCO Research Report No. 14, Everett, MA, January, 1958.

21. Schetz, J. A., Boundary Layer Analysis, Virginia Polytechnic Institute and State University, Prentice Hall, NJ, 1993.

22. Rafinejad, D.; Dahm, T. J.; Brink, T. F.; Abbett, M. J.; Wolf, C. J., Passive Nosetip Technology (PANT II) Program, Volume II, “Computer User’s Manual: ABRES Shape Change Code (ASCC),” Acurex Corporation/Aerotherm Division, Mountain View, CA, SAMSO-TR-77-11, October 1976.

23. Abbett, M. J., and Davis, J. E., “Passive Nosetip Technology (PANT) Program Interim Report,” Vol. IV, “Heat Transfer and Pressure Distributions on Ablated Shapes,” Part II – “Data Correlation and Analysis,” Aerotherm Report 74-90, Acurex Corporation/Aerotherm Division, January 1974.

24. As cited in Reference 23, Dr. P. G Crowell, Aerospace Corp, 1971.

25. Murray, A. L., User’s Manual for the Aeroheating and Thermal Analysis Code (ATAC05), ITT Industries, Huntsville, AL, July 2005.

26. Neel, R. E.; Schetz, J. A.; and Marineau, E. C., “Turbulent Navier-Stokes Simulations of Heat Transfer with Complex Wall Temperature Variations”, 9th AIAA/ASME Joint Thermophysics and Heat Transfer Conference, AIAA-2006-3087, San Francisco, CA, June 2006.

27. Incropera, F. P., and DeWitt D. P., Introduction to Heat Transfer, John Wiley & Sons, New York, NY, 1996.

28. King, H C.; Muramoto, K. K.; Murray, A. L.; Pronchick, S. W., ABRES Shape Change Code (ASCC86), Technical Report and User’s Manual, BMO TR-87-57, Acurex Corporation/Aerotherm Division, December 1986.

150

29. Powars, C. A., and Kendall, R. M., User’s Manual: Aerotherm Chemical Equilibrium Computer Program, Aerotherm Corporation, Mountain View, CA, May 1969.

30. Cleary, J. W., “Effects of Angle of Attack and Bluntness on Laminar heating Rate Distribution of a 15° Cone at a Mach Number of 10.6”, NASA TN D-5450, 1969.

31. Baker, D. L.; Wool, M. R.; Powars, C. A.; and Derbidge, T. C., Interim Report Passive Nosetip Technology (PANT) Program: Volume V. Definition of Shape Change Phenomenology from Low Temperature Ablator Experiments, Acurex Corporation/Aerotherm Division, SAMSO-TR-74-86, January 1974.

32. Discussion with Reece Neel at Aerosoft, Inc., Fall 2006.

33. ANSYS User Guide Version 10, ANSYS, Inc., Canonsburg, PA.

34. Blackwell, B.F., Douglass, R.W., Wolf, H., A User’s manual for the Sandia One-Dimensional Direct and Inverse Thermal (SODDIT) Code, SAND-85-2478, Sandia National Laboratories, Albuquerque, NM, May 1987.

151

APPENDIX A

PRESSURE-TEMPERATURE-ENTHALPY TABLE

FOR NON-ABLATING SURFACE AIR

152

TABLE A-1. PRESSURE-TEMPERATURE-ENTHALPY TABLE FOR NON-ABLATING SURFACE AIR

Pressure (atm) Temperature (ºR) Enthalpy (Btu/lbm) 0.001 2800 1449.78 0.001 2600 1153.4 0.001 2400 834.18 0.001 2200 619.95 0.001 2000 498.29 0.001 1800 419.96 0.001 1600 356.29 0.001 1400 296.92 0.001 1200 239.28 0.001 1000 182.95 0.001 800 128.1 0.001 600 75.487 0.001 400 24.606 0.001 300 0.533 0.001 200 -23.406 0.01 2800 1125.896 0.01 2600 868.233 0.01 2400 685.626 0.01 2200 567.318 0.01 2000 484.684 0.01 1800 417.45 0.01 1600 355.988 0.01 1400 296.902 0.01 1200 239.282 0.01 1000 182.95 0.01 800 128.097 0.01 600 75.487 0.01 400 24.606 0.01 300 0.533 0.01 200 -23.406 0.1 2800 887.674 0.1 2600 740.271 0.1 2400 632.78 0.1 2200 550.096 0.1 2000 480.345 0.1 1800 416.656 0.1 1600 355.893 0.1 1400 296.897 0.1 1200 239.282 0.1 1000 182.954 0.1 800 128.097 0.1 600 75.487 0.1 400 24.606 0.1 300 0.533 0.1 200 -23.406

153

TABLE A-1. PRESSURE-TEMPERATURE-ENTHALPY TABLE FOR NON-ABLATING SURFACE AIR (Continued)

Pressure (atm) Temperature (ºR) Enthalpy (Btu/lbm) 1 2800 790.952 1 2600 695.606 1 2400 615.48 1 2200 544.595 1 2000 478.972 1 1800 416.407 1 1600 355.865 1 1400 296.896 1 1200 239.283 1 1000 182.955 1 800 128.097 1 600 75.487 1 400 24.606 1 300 0.533 1 200 -23.406

10 2800 758.105 10 2600 681.068 10 2400 609.959 10 2200 542.858 10 2000 478.545 10 1800 416.334 10 1600 355.86 10 1400 296.898 10 1000 182.956 10 800 128.098 10 400 24.606 10 300 0.533 10 200 -23.406

100 2800 747.525 100 2600 676.462 100 2400 608.237 100 2200 542.335 100 2000 478.432 100 1800 416.329 100 1600 355.872 100 1400 296.909 100 1200 239.291 100 1000 182.959 100 800 128.099 100 600 75.487 100 400 24.606 100 300 0.533 100 200 -23.406

154

APPENDIX B

PASSIVE NOSE-TIP TECHNOLOGY PROGRAM (PANT) HEAT DECK

155

PASSIVE NOSE-TIP TECHNOLOGY PROGRAM (PANT) HEAT DECK

Table B-1 displays the test matrix performed during the PANT Program Volume V,

Part 1 of the Experimental Data Series C. The test matrix dates back to October 1971. Test Run

No. 9 was selected for comparison with GASP.

TABLE B-1. PANT SERIES C TEST MATRIX

156

Considering that the tunnel inlet conditions were constant for the duration of the test, the

established heat deck does not vary. At each load step (and thus each time step), the heat flux

boundary condition is recalculated based on the new wall temperature; however, the recovery

enthalpy, heat transfer coefficient, and edge pressure remain constant at each defined axial

station. Table B-2 depicts the constant values at each axial station. The load steps defined in

ANSYS repeated these constant values at a step of 0.01 second for the 10-second duration.

TABLE B-2. HEAT DECK OUTPUT FROM BLUNTY/MAGIC

Time (s) Hr q"c CH Pe Ue X

0.00 1619.72 -43.42 0.0268 0.1597 0.00 0.000.00 1613.32 -60.72 0.0376 0.1456 1318.33 0.100.00 1604.01 -51.83 0.0323 0.1252 2136.54 0.250.00 1588.82 -42.88 0.0270 0.0972 3020.76 0.500.00 1558.23 -28.98 0.0186 0.0572 4225.25 1.000.00 1491.72 -6.97 0.0047 0.0129 6269.20 3.000.00 1483.81 -5.53 0.0037 0.0109 6461.23 5.000.00 1478.40 -4.77 0.0032 0.0097 6587.35 7.000.00 1473.06 -4.16 0.0028 0.0087 6710.97 10.00

157

APPENDIX C

MACH 8 & 10 5-INCH PROJECTILE VALIDATION HEAT DECK

158

MACH 8 & 10 5-INCH PROJECTILE VALIDATION HEAT DECK

In these particular validation studies, the flight conditions were chosen to be Mach 8 and

10, respectively, at sea-level flight continuously for 5 seconds. At each load step (and thus each

time step), the heat flux boundary condition is recalculated based on the new wall temperature;

however, the recovery enthalpy, heat transfer coefficient, and edge pressure remain constant at

each defined axial station. Table C-1 depicts the constant values at each axial station for the

Mach 10 case and Table C-2 for the Mach 8 case. The load steps defined in ANSYS repeated

these constant values depending on the time step analysis that was being run (recalling runs of 1-

, 0.1-, 0.01-, 0.005-second steps were used for convergence studies).

TABLE C-1. MACH 10 HEAT DECK OUTPUT FROM BLUNTY/MAGIC

Time (s) Hr q"c CH Pe Ue X0.00 2484.10 -9368.95 3.69 132.20 0.00 0.000.00 2378.24 -3826.67 1.61 34.83 5692.61 0.100.00 2345.80 -3281.63 1.40 7.33 7867.28 0.200.00 2339.90 -2834.49 1.21 6.35 8068.16 0.300.00 2328.79 -2192.08 0.94 4.79 8369.19 0.500.00 2319.21 -1748.81 0.75 3.72 8621.47 0.760.00 2312.20 -1473.48 0.64 3.07 8798.79 1.000.00 2287.64 -861.16 0.38 1.66 9400.25 3.000.00 2276.60 -777.95 0.34 1.45 9655.59 5.00

159

TABLE C-2. MACH 8 HEAT DECK OUTPUT FROM BLUNTY/MAGIC

Time (s) Hr q"c CH Pe Ue X0.00 1583.14 -5796.29 3.65 84.32 0.00 0.000.00 1464.44 -1151.05 0.79 8.73 5980.47 0.100.00 1436.46 -437.70 0.30 4.03 6718.22 0.200.00 1429.89 -348.20 0.24 3.31 6864.75 0.300.00 1469.70 -952.01 0.65 2.47 7131.57 0.500.00 1459.95 -671.40 0.46 1.68 7425.12 1.000.00 1452.33 -548.82 0.38 1.34 7656.73 2.000.00 1447.34 -526.24 0.36 1.27 7800.00 3.000.00 1437.77 -557.93 0.39 1.31 8053.87 5.00

160

APPENDIX D

HYPERSONIC ROUND TRAJECTORY STUDY

161

HYPERSONIC ROUND TRAJECTORY STUDY

Due to the extremely large amounts of data, the tables are presented in graphical form.

Values of interest are the velocity, the altitude, the temperature, and the thermal contour plots at

various Quadrant Elevations (QE).

Figures D-1 through D-3 show the information for 0.5° QE. Figures D-4 through D-6

show the information for 10° QE. Figures D-7 through D-9 show the information for 30° QE.

Figures D-10 through D-12 show the information for 50° QE. Figures D-13 through D-15 show

the information for 80° QE.

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 1 2 3 4 5 60

0.02

0.04

0.06

0.08

0.1

0.12

0.14 Vel(ft/s)

Alt(kft)

Time (s)

Velo

city

(ft/s

) Altitude (kft)

FIGURE D-1. QE 0.5° VELOCITY AND ALTITUDE TIME HISTORY

162

Time (s)

Tem

pera

ture

(K)

0

500

1000

1500

2000

2500

3000

3500

0 1 2 3 4 5 60

1000

2000

3000

4000

5000

6000Node 2

Node 48

Node 17381

Node 21217

Node 9912

Node 10778

Node 13556

Node 16283

Node 16823

Node 17196

(°R)

FIGURE D-2. QE 0.5° THERMAL TIME HISTORY

K

KTime = 5 seconds

Time = 1 seconds

FIGURE D-3. QE 0.5° THERMAL CONTOUR PLOT

163

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 10 20 30 40 50 60 70 800

5

10

15

20

25 Vel(ft/s)

Alt(kft)

Time (s)

Velo

city

(ft/s

) Altitude (kft)

FIGURE D-4. QE 10° VELOCITY AND ALTITUDE TIME HISTORY

Time (s)

Tem

pera

ture

(K)

0

500

1000

1500

2000

2500

3000

3500

0 10 20 30 40 50 60 70 800

1000

2000

3000

4000

5000

6000Node 2

Node 48

Node 17381

Node 21217

Node 9912

Node 10778

Node 13556

Node 16283

Node 16823

Node 17196

(°R)

FIGURE D-5. QE 10° THERMAL TIME HISTORY

164

K

KTime = 79 seconds

Time = 0.75 seconds

FIGURE D-6 QE 10° THERMAL CONTOUR PLOT

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 50 100 150 200 2500

50

100

150

200

250 Vel(ft/s)

Alt(kft)

Time (s)

Velo

city

(ft/s

) Altitude (kft)


165

0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200 2500

1000

2000

3000

4000

5000

6000Node 2

Node 48

Node 17381

Node 21217

Node 9912

Node 10778

Node 13556

Node 16283

Node 16823

Node 17196

Time (s)

Tem

pera

ture

(K)

(°R)


K

KTime = 222 seconds

Time = 1 seconds

FIGURE D-9. QE 30° THERMAL CONTOUR PLOT

166

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 50 100 150 200 250 300 350 4000

100

200

300

400

500

600 Vel(ft/s)

Alt(kft)

Time (s)

Velo

city

(ft/s

) Altitude (kft)


0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000Node 2

Node 48

Node 17381

Node 21217

Node 9912

Node 10778

Node 13556

Node 16283

Node 16823

Node 17196

Time (s)

Tem

pera

ture

(K)

(°R)


167

K

KTime = 367 seconds

Time = 1 seconds


0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 100 200 300 400 500 6000

200

400

600

800

1000

1200 Vel(ft/s)

Alt(kft)

Time (s)

Velo

city

(ft/s

) Altitude (kft)


168

0

500

1000

1500

2000

2500

3000

3500

0 100 200 300 400 5000

1000

2000

3000

4000

5000

6000Node 2

Node 48

Node 17381

Node 21217

Node 9912

Node 10778

Node 13556

Node 16283

Node 16823

Node 17196

Time (s)

Tem

pera

ture

(K)

(°R)


K

KTime = 487 seconds

Time = 1 seconds


169

APPENDIX E

ANSYS CODE

170

ANSYS CODE

Below is a copy of the ANSYS programming language code that was used to implement

the methodology. Comments are denoted with an exclamation point.

!-------------------------------------------------

! NOTE:

! MODEL HAS TO BE BRICK MAPPED OVER ENTIRE EXTERIOR

!------------------------------------------------

FINISH

!-----------------------------------------------

! FOR HPC ONLY

!-----------------------------------------------

/FILNAME,HSR_FULL,1

RESUME

/config,nres,2500

/config,nproc,6

!--------------------------------------------------------

!--------------------------------------------

171

!specify how many load steps (rows)

LS = 1332

!LS=200

!--------------------------------------------

!SPECIFY NUMBER OF Y LOCATIONS

YS_ = 9

*DO,B_,1,YS_,1

HEAT_DECK(1,1,B_) = 1E-5

*ENDDO

!--------------------------------------------

!SPECIFY TIME STEP

TSTEP = 1E-3

!--------------------------------------------

!--------------------------------------------

!YCOORDS INCHES OR METERS, YC_ = 1 INCHES

YC_ = 1

*IF,YC_,EQ,1,THEN

CF_ = .0254

*ELSE

CF_ = 1

*ENDIF

!--------------------------------------------

!--------------------------------------------

! Y LOCATIONS WHERE HFLUX DATA OBTAINED

!--------------------------------------------

Y_(1,1,1) = 0

172

USER_DEF = 0

*IF,USER_DEF,EQ,0,THEN

Y_(2,1,1) = 0.25*CF_

Y_(3,1,1) = 0.50*CF_

Y_(4,1,1) = 1.0*CF_

Y_(5,1,1) = 2.0*CF_

Y_(6,1,1) = 5.0*CF_

Y_(7,1,1) = 10.0*CF_

Y_(8,1,1) = 25.0*CF_

Y_(9,1,1) = 28.5*CF_

*ELSE

*DO,B_,2,YS_,1

Y_(B_,1,1) = 0.0254*(B_-1)

*ENDDO

*ENDIF

!--------------------------------------------

! INITIALIZE SOLUTION TO SET ROOM TEMP

!--------------------------------------------

ALLS

CSYS,0

WPAVE,0,0,0

alls

nsel,s,ext

cm,EXT_NODES,node

ALLS

173

CMSEL,S,EXT_NODES

NSEL,U,LOC,X,0

CMSEL,U,BULK_N

NSEL,A,LOC,Y,0

CM,EXT_NODES,NODE

ALLS

*DO,B_,1,YS_,1

NSEL,U,LOC,Y,-1,Y_(B_)-(Y_(B_)/4)-1E-6

NNUM_A(B_,1,1) = NODE(Y_(B_)/10,Y_(B_),0)

*GET,NNUM_Y(B_,1,1),NODE,NNUM_A(B_,1,1),LOC,Y

CMSEL,S,EXT_NODES

*ENDDO

/SOLU

ANTYPE,TRANS,NEW

OUTPR,BASIC,LAST

TREF,293

INIT = 1

*IF,INIT,EQ,1,THEN

alls

sfdele,all,all

sfdele,all,conv

sfldele,all,all

174

dadele,all,all

dldele,all,all

ddele,all,all

d,all,temp,293

time,1E-6 ! sets load step end time

autots,OFF ! program chosen auto time stepping

nsubst,10,10,10

kbc,1 ! 1...stepped load steps

tsres,erase ! reinitializes the time stepping

alls

solve

alls

sfdele,all,all

sfdele,all,conv

sfldele,all,all

dadele,all,all

dldele,all,all

ddele,all,all

*ENDIF

!--------------------------------------------

autots,off ! program chosen auto time stepping

SOLCONTROL,ON

outres,basic,-2

kbc,1 ! 1...stepped load steps, 0 ramped

tsres,erase

!------------------------------------------------------------

175

! Select all exterior nodes and make sectional components

!------------------------------------------------------------

!------------------------------------------------------------

! SELECT TIP NODE

!------------------------------------------------------------

NSEL,R,LOC,Y,0

CM,N_TIP,NODE

!------------------------------------------------------------

NSEL,R,LOC,Y,NNUM_Y(1,1,1),NNUM_Y(2,1,1)

CM,SEG1,NODE

CMSEL,S,EXT_NODES


CM,SEG2,NODE

CMSEL,S,EXT_NODES


CM,SEG3,NODE

CMSEL,S,EXT_NODES


CM,SEG4,NODE

CMSEL,S,EXT_NODES


CM,SEG5,NODE

CMSEL,S,EXT_NODES

176


CM,SEG6,NODE

CMSEL,S,EXT_NODES


CM,SEG7,NODE

CMSEL,S,EXT_NODES


CM,SEG8,NODE

!------------------------------------------------------------

*DO,LS_,1,LS,1

LSCLEAR,ALL

alls

sfdele,all,all

sfdele,all,conv

sfldele,all,all

dadele,all,all

dldele,all,all

ddele,all,all

!--------------------------------------------

!--------------------------------------------

!--------------------------------------------------------------------------

! CALCULATE HEAT FLUX

177

!--------------------------------------------------------------------------

*DO,B_,1,YS_,1

NTEMP_K(B_,LS_,1) = TEMP(NNUM_A(B_,1,1))

NTEMP_R(B_,LS_,1) = 1.8*NTEMP_K(B_,LS_,1)

HR_ = HEAT_DECK(LS_, 2,B_)

CH_ = HEAT_DECK(LS_, 4,B_)

PE_ = HEAT_DECK(LS_,5,B_)

HW_ = ENTHALPY_TABLE(PE_,NTEMP_R(B_,LS_,1),1)

*IF, (HR_-HW_),LT,0,THEN

Q_R(B_,LS_,1) =0

*ELSE

Q_R(B_,LS_,1) = CH_*(HR_ - HW_)

*ENDIF

Q_K(B_,LS_,1) =11356.5*Q_R(B_,LS_,1)

*ENDDO

!--------------------------------------------------------------------------

*DO,B_,1,YS_-1,1,

DIFF_(B_,1,1) = Q_K(B_+1,LS_,1) - Q_K(B_,LS_,1)

*ENDDO

LTIME = HEAT_DECK(LS_,1,1)

!------------------------------------------------------------

!apply q1 to element at nose tip

178

!------------------------------------------------------------

CMSEL,S,N_TIP

ESLN,S,0

!-----

!ATTEMPT TO PICK FACE_

!----

*GET,N_X,NODE,0,NUM,MAX

*GET,EL_NO,ELEM,0,NUM,MAX

*GET,N1_,ELEM,EL_NO,NODE,1




*IF,N_X,EQ,N1_,THEN

SFE,ALL,1,HFLUX,,Q_K(1,LS_,1),,,

FACE_ = 1

NFACE_ = 2

*ELSEIF,N_X,EQ,N2_,THEN


FACE_ = 2

NFACE_ = 3



FACE_ = 3

NFACE_ = 4



179

FACE_ = 4

NFACE_ = 1

*ENDIF

!------------------------------------------------------------

!get node number at q location, store as scalar

!------------------------------------------------------------

!--------------------------------------------------

! GET NUMBER OF ELEMENTS ALONG EDGE

!--------------------------------------------------

ALLS

CMSEL,S,EXT_NODES

*GET,MAXY_,NODE,0,MXLOC,Y

NSEL,U,LOC,Y,MAXY_

ESLN,S,0

*GET,ELNUMS_,ELEM,0,COUNT

ALLS

!--------------------------------------------------

! LOOP FOR REMAINING ELEMENTS

!--------------------------------------------------

FINAL = ELNUMS_-1

*GET,EL_NO,ELEM,EL_NO,ADJ,NFACE_ !GET THE NUMBER OF

NEXT ADJACENT ELEMENT

ESEL,S,,,EL_NO

!-----------------------------------------------------------------------

180

CMSEL,S,EXT_NODES

NSLE,R,ALL

*GET,MIN_,NODE,0,MNLOC,Y

N_X = NODE(0,MIN_,0)

NTEMP_ = TEMP(N_X)

NTEMPS(2,1,1) = NTEMP_

!-----------------------------------------------------------------------

*GET,YVAL_,ELEM,EL_NO,CENT,Y !GET Y VALUE OF CENTROID





*IF,N_X,EQ,N1_,THEN

FACE_ = 1

NFACE_ = 2


FACE_ = 2

NFACE_ = 3


FACE_ = 3

NFACE_ = 4


FACE_ = 4

NFACE_ = 1

*ENDIF

181

*DO,COUNTER,1,FINAL,1

INDEX = 1 + COUNTER

FLAGGER_ = 0

!--------------------------------------------------------------------------------------------------------------------

--------------------------

!--------------------------------------------------------------------------------------------------------------------

--------------------------

*IF,YVAL_,LE,NNUM_Y(2,1,1),THEN

YDIFF_ = NNUM_Y(2,1,1)-NNUM_Y(1,1,1)

SLOPE_ = DIFF_(1,1,1)/YDIFF_

VAR_ = YVAL_*SLOPE_

INT_ = SLOPE_*NNUM_Y(1,1,1)

Q_ = VAR_+Q_K(1,LS_,1)-INT_

SFE,ALL,FACE_,HFLUX, ,Q_, , , !APPLY NEW FLUX

*ELSEIF,YVAL_,LE,NNUM_Y(3,1,1),THEN



VAR_ = YVAL_*SLOPE_







182

VAR_ = YVAL_*SLOPE_







VAR_ = YVAL_*SLOPE_







VAR_ = YVAL_*SLOPE_







VAR_ = YVAL_*SLOPE_




183




VAR_ = YVAL_*SLOPE_







VAR_ = YVAL_*SLOPE_




*ENDIF

!--------------------------------------------------------------------------------------------------------------------

--------------------------

!--------------------------------------------------------------------------------------------------------------------

--------------------------

*GET,EL_NO,ELEM,EL_NO,ADJ,NFACE_ !GET THE NUMBER OF

NEXT ADJACENT ELEMENT

ESEL,S,,,EL_NO !SELECT THE NEXT ELEMNT

184

!-----------------------------------------------------------------------

CMSEL,S,EXT_NODES

NSLE,R,ALL

*GET,MIN_,NODE,0,MNLOC,Y

N_X = NODE(0,MIN_,0)

NTEMP_ = TEMP(N_X)

NTEMPS(INDEX,1,1) = NTEMP_

!-----------------------------------------------------------------------

*GET,YVAL_,ELEM,EL_NO,CENT,Y !GET Y VALUE OF

CENTROID





*IF,N_X,EQ,N1_,THEN

FACE_ = 1

NFACE_ = 2


FACE_ = 2

NFACE_ = 3


FACE_ = 3

NFACE_ = 4


FACE_ = 4

NFACE_ = 1

185

*ENDIF

*ENDDO

ALLS

!-------------------------------------------------------------------------------------

! SETS UP SOLVER

!-------------------------------------------------------------------------------------

*IF,LTIME,LE,60,OR,LTIME,GE,305,THEN

TSTEP = 1E-3

*ELSE

TSTEP = 5

*ENDIF

DELTIM,TSTEP,TSTEP,TSTEP,ON

TIME,LTIME

ALLS

SOLVE

SAVE

!-------------------------------------------------------------------------------------

!-------------------------------------------------------------------------------------

*ENDDO

!-------------------------------------------------------------------------------------

!-------------------------------------------------------------------------------------

186

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AERODYNAMIC HEATING OF A HYPERSONIC NAVAL … · AERODYNAMIC HEATING OF A HYPERSONIC NAVAL PROJECTILE ... for their applicability to the problem under study, and outside of the GASP