CHARACTERISTICS OF COUPLED NONGRAY RADIATING GAS FLOWS

WITH ABLATION PRODUCT EFFECTS ABOUT BLUNT BODIES

DURING PLANETARY ENTRIES (NASA-TN-X-72078) CHARACTERISTICS OF N75- 1097 1

COUPLED NONGRAY RADIATING GAS FLOWS UITH ABLATION PRODUCT EFFECTS FABOUT BLUNT BODIES DURING PLANETARY ENTRIES Ph.D. U n c l a s T h e s i s - (NASA)_ 181 p HC $7.00 CSCL 22C _ G3/13 0 2 3 2 8

by

KENNETH SUTTON

A t h e s i s submitted t o t h e Graduate Faculty of North Carol ina S t a t e Univers i ty a t Raleigh

i n p a r t i a l f u l f i l l m e n t of t h e requirements f o r the degree of

Doctor of Philosophy

DEPARTMENT OF MECHANICAL AND AEROSPACE ENGINEERING

RALEIGH

1 9 7 3

APPROVED BY:

https://ntrs.nasa.gov/search.jsp?R=19750002899 2020-03-18T14:15:00+00:00Z

SUTTON, KENNETH. C h a r a c t e r i s t i c s of Coupled Nongray Radiat ing Gas Flows

wi th Ablat ion Product E f f e c t s about Blunt Bodies during Planetary En-

t r i e s . (Under t h e d i r e c t i o n of FREDERICK OTTO SMETANA).

A computational method is developed f o r the fully-coupled s o l u t i o n

of nongray, r a d i a t i n g gas flows wi th a b l a t i o n product e f f e c t s about

b l u n t bodies during p lane tary e n t r i e s . The t reatment of r a d i a t i o n ac-

counts f o r molecular band, continuum, and atomic l i n e t r a n s i t i o n s wi th

a d e t a i l e d frequency dependence of t h e absorpt ion c o e f f i c i e n t . The

a b l a t i o n of t h e e n t r y body is solved a s p a r t of the s o l u t i o n f o r a

s t eady- s t a t e a b l a t i o n process.

Appl ica t ion of t h e developed method i s shown by r e s u l t s a t t y p i c a l

condi t ions f o r unmanned, s c i e n t i f i c probes during e n t r y t o Venus. The

r a d i a t i v e hea t ing r a t e s along t h e downstream region of t h e body can,

under c e r t a i n condi t ions , exceed the s t agna t ion po in t value. The ra-

d i a t i v e hea t ing t o t h e body is a t t enua ted i n the boundary l a y e r a t t h e

downstream region of t h e body a s w e l l a s a t the s t agna t ion poin t of t h e

body.

Resu l t s from a s tudy of t h e r a d i a t i n g , i n v i s c i d flow about spheri-

cally-capped, con ica l bodies during p l ane ta ry e n t r i e s a r e presented

and show t h a t t h e nondimensional, r a d i a t i v e hea t ing d i s t r i b u t i o n s a r e

nonsimilar wi th e n t r y condi t ions . Therefore, extreme caut ion should be

exerc ised i n at tempting t o e x t r a p o l a t e r e s u l t s from known d i s t r i b u t i o n s

t o o t h e r e n t r y condi t ions f o r which s o l u t i o n s have not y e t been obtained.

BIOGRAPHY

Kenneth Sutton was . He

was reared i n Jacksonv i l l e , F lo r ida and graduated from Andrew Jackson

High School i n 1957. He a t tended Jacksonv i l l e Univers i ty f o r two years

be fo re t r a n s f e r r i n g t o t h e Univers i ty of F lo r ida where he received t h e

degree of Bachelor of Mechanical Engineering i n 1962.

Af t e r graduat ing, he accepted a p o s i t i o n with t h e Langley Research

Center of t h e National Aeronautics and Space Administrat ion i n Hampton,

Virg in ia . He received h i s graduate educat ion through the NASA Graduate

Study Program. I n 1963 he returned t o the Univers i ty of F lo r ida and

rece ived the degree of Master of Engineering i n 1964. He began n i g h t

s t u d i e s i n 1964 with the Tidewater Extension Center of George Washington

Univers i ty and received the degree of Master of Science i n Governmental

Administrat ion i n 1967. He entered North Carol ina S t a t e Univers i ty a t

Raleigh i n 1968 t o begin advanced s t u d i e s i n t h e Department of Vechan-

i c a l and Aerospace Engineering. He re turned t o t h e Langley Research

Center i n 1969 where he performed t h e a n a l y s i s presented i n t h i s thesls.

He i s p resen t ly assigned t o the Advanced Entry Analysis Branch of

t h e Spacecraf t Systems Division a t t h e Langley Research Center. He i s

a member of the American I n s t i t u t e of Aeronautics and Ast ronaut ics and

t h e American Socie ty of Mechanical Engineers. He i s r e g i s t e r e d a s a

P ro fes s iona l Engineer by t h e S t a t e of F lo r ida .

ACKNOWLEDGEMENTS

The author wishes t o express h i s apprec ia t ion t o t h e National

Aeronautics and Space Administration f o r i t s continued support of h i s

graduate s t u d i e s and f o r support ing t h e research presented i n t h e

t h e s i s .

The author wishes t o thank t h e members of h i s advisory committee

f o r t h e i r he lp and cooperat ion. He is e s p e c i a l l y g r a t e f u l t o D r .

Frederick 0. Smetana f o r h i s a s s i s t a n c e and pa t ience during the

au thor ' s s t u d i e s and research inves t iga t ion .

Specia l apprec ia t ion is extended t o Mr. Gerald D. Walberg of the

Langley Research Center f o r proposing the top ic of t h e research inves-

t i g a t i o n , f o r h i s continued support of the i n v e s t i g a t i o n , and f o r h i s

consu l t a t ion on many t echn ica l mat te rs . The author is apprec ia t ive of

t h e t e c h n i c a l consu l t a t ion given t o him by members of t h e Langley

Research Center: D r . Walter B . Ols tad , Mr. Linwood B. C a l l i s , D r .

Robert E. Boughner, M r . Ralph A. Falanga, Mr. John T. S u t t l e s , and

D r . G. Louis Smith. Also, the au thor is g r a t e f u l f o r t h e t echn ica l

a s s i s t a n c e by members of the Aerothem Corporation: M r . William E.

N ico le t , D r . Robert M. Kendall , and Mr. Eugene P. B a r t l e t t .

The author is g r a t e f u l f o r t h e p leasant conversat ion, encourage-

ment, and t e c h n i c a l a s s i s t a n c e extended t o him by h i s o f f i c e co l leagues ,

Mr. Randolph A . Graves, J r . , and M r . E. Vincent Zoby. Specia l g r a t i t u d e

is expressed t o M r . Bennie W . Cocke, Jr . of the Langley Research Center

f o r h i s guidance and h e l p i n t h e au thor ' s e a r l i e r ca ree r . Also,

apprec ia t ion i s extended t o M r . James H. Godwin of Langley Research

Center f o r h i s a s s i s t a n c e and f r i endsh ip during t h e many n igh t s t h e

author spent a t t h e computer complex.

The au thor ' s graduate s tudy was made e a s i e r by t h e work of M r .

Dick Cole and M r . John Witherspoon of t h e Langley Research Center and

Miss Eleanor Br idgers of North Carol ina S t a t e Univers i ty i n e f f i c i e n t l y

handling t h e necessary admin i s t r a t ive arrangements during h i s enrollment

a t North Carol ina S t a t e Univers i ty .

The author wishes t o express h i s apprec ia t ion t o Miss Mary Anne

Monaco f o r h e r e d i t o r i a l a s s i s t a n c e and typing the o r i g i n a l d r a f t of

t h e t h e s i s . Gra t i tude i s a l s o expressed t o M r . Charles R. P r u i t t of

t h e Langley Research Center f o r handling t h e f i n a l p repa ra t ion of t h e

t h e s i s .

Page

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . v i

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . v i i

LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . x i

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

REVIEW OF LITERATURE . . . . . . . . . . . . . . . . . . . . . . . 6

. . . . . . . . . . . . . . . . . . R a d i a t i o n T r a n s p o r t Modeling 7

R a d i a t i n g Flow F i e l d s . . . . . . . . . . . . . . . . . . . . 11 Flow a t S t a g n a t i o n Region . . . . . . . . . . . . . . . . . . 11 Flow abou t B l u n t Bodies . . . . . . . . . . . . . . . . . . . 1 4

. . . . . . . . . . . . . . Thermal Ana lyses f o r Venusian En t ry 19

METHOD OF ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . 24

R a d i a t i n g . I n v i s c i d Flow F i e l d S o l u t i o n . . . . . . . . . . . . 28

Boundary Layer S o l u t i o n . . . . . . . . . . . . . . . . . . . . 42

R a d i a t i v e T r a n s p o r t S o l u t i o n . . . . . . . . . . . . . . . . . . 5 1

RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . . 58

. . . . . . . . . . . . . . Non.Radiating. I n v i s c i d A i r S o l u t i o n 59

. . . . . . . . . R a d i a t i n g I n v i s c i d S o l u t i o n s abou t B l u n t Bodies 60

Fully.Coupled. S t a g n a t i o w P o i n t . S o l u t i o n s f o r E a r t h Reen t ry . . 68

. . . . . . . . . . . Fully-Coupled S o l u t i o n s f o r Venusian E n t r y 70

. . . . . . . . . . . . . . . . . . . . . . SUMMARY AND CONCLUSIONS 78

EXTENSION OF PRESENT RESEARCH . . . . . . . . . . . . . . . . . . . 80

LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . 129

. APPENDIX A Unsteady C h a r a c t e r i s t i c s S o l u t i o n . . . . . . . . . . 137

APPENDIX B . S t a g n a t i o n - P o i n t S o l u t i o n f o r R a d i a t i n g . I n v i s c i d F l o w F i e l d . . . . . . . . . . . . . . . . . . . . . 1 4 1

APPENDIX C . Computer Program f o r R a d i a t i n g . I n v i s c i d Flow F i e l d S o l u t i o n . . . . . . . . . . . . . . . . . . . 144

LIST OF TABLES

Page

i.. Stagnation-point, radiative heating rates for air entry . . . 81

T I . Results from fully-coupled, stagnation-point solutions with mass injection for air entries . . . . . . . . . . . . 82

LIST OF FIGURES

Page

1. Illustration of the mating of the boundary layer solution to the inviscid layer solution . . . . . . . . . . . . . . . . . 83

2. Calculation procedure for the fully-coupled, radiating flow field solution . . . . . . . . . . . . . . . . . . . . . . . 84

3. Flow field coordinate system for axisynunetric blunt body . . . 85

4 . Schematic of use of unsteady characteristics at shock wave . . 86

5. Nonradiating, inviscid flow solution around a blunt body. Present method is compared with the method of Inouye et al. (ref. 88) . . . . . . . . . . . . . . . . . . . . . . 87 -- (a) Free-stream conditions and body shape . . . . . . . . . 87

(b) Shock shape at forward region of body and sonic line location . . . . . . . . . . . . . . . . . . . . 88

(c) Shock shape at downstream region of body . . . . . . . . 89

(d) Shock stand-off distance around body . . . . . . . . . . 90

(el Velocity distribution along body . . . . . . . . . . . . 91

(f) Pressure distribution along body . . . . . . . . . . . . 92

6 . Radiating, inviscid flow solution about a blunt body during Earth reentry. Present method is compared with the method of Callis (ref. 3 ) . . . . . . . . . . . . . . . . . . . . . 93

(a) Free-stream conditions and body shape . . . . . . . . . 93

(b) Pressure distribution along body . . . . . . . . . . . . 94

(c) Shock stand-off distance around body . . . . . . . . . . 94

(d) Radiative heating along body . . . . . . . . . . . . . . 95

(e) Radiative heating along body. Callis' results modified 96

7. Stagnation-point, radiative heating rates in an air atmo-- sphere as a function of nose radius. Present results are compared with the method of Callis (ref. 7) . . . . . . . . . 97

viii

Page

8. Radiative heating distribution for free-stream conditions and body shape given in Figure 6(a) except Rn=2.0t m. Present results are compared with the method of . . . . . . . . . . . . . . . . . . . . . . . Callis(ref.3) 98

9. Radiative heating distributions along a spherically-capped, conical body from radiating, inviscid flow solutions for entries in different atmospheres and different free-stream

. . . . . . . . . . . . . . . . . . . . . . . . . conditions 99

10. A comparison of the stagnation-point, radiative heating rates for entries in air and in a .90 C02 - .10 N2 (by volume) gasmixture . . . . . . . . . . . . . . . . . . . . . . . . . 100

11. The distribution of radiative heating along spherically- capped, conical bodies for entry in air . . . . . . . . . . . 101

12. The distribution of radiative heating along spherically- capped, conical bodies for entry in a .90 CO - .10 N2

2 (by volume) gas mixture . . . . . . . . . . . . . . . . . . . 102

- 3 (b) V_=11.175 km/s ; pm=2.850 x 10 kg/m3 ; pa=117.3 N/m 2

h =-8.43 M/kg ; Rn=.3048 m . . . . . . . . . . . . . . 102 m

13. The effect of nose radius on the distribution of radiative heating along a spherically-capped, conical body for entry in air . . . . . . . . . . . . . . . . . . . . . . . . 103

14. The effect of nose radius on the distribution of radiative heating along a spherically-capped, conical body for entry

. . . . . . . . in a .90 C02 - .10 N (by volume) gas mixture 104 2

(a) Va= 8.740 km/s ; pa=3. 294 x kg/m3 ; p_=130.6 N/m 2

. . . . . . . . . . . . . . . hm=-8.43 w/kg ; ~~-60" Y. 104

(b) V_=11.175 km/s ; p_=2.850 x kg/m3 ; ~2117.3 N/m 2

0 ha=-8.43 MJ/kg ; BC=60 . . . . . . . . . . . . . . . . 105 15. Stagnation-point, radiative heating rates in C02 - N2 gas

mixtures. Rn=.3048 m . . . . . . . . . . . . . . . . . . . . 106 -3 3

(a) Va=12.0 km/s ; pm=2.99 x 10 kg/m . . . . . . . . . . 106

Page

16. Fully-coupled, stagnation-point solution for Earth reentry. Present method is compared with the method of Garrett (ref. 19) . . . . . . . . . . . . . . . . . . . . . . 107

(a) Free-stream conditions and specified wall conditions . . 107

(b) Temperature profile through layer . . . . . . . . . . . 108

(c) Radiative heating profile through layer . . . . . . . . 109

(d) Spectral distribution of radiative heating toward the body due to continuum processes . . . . . . . . . . . 110

(e) Spectral distribution of radiative heating toward the body due to line processes . . . . . . . . . . . . . . 111

17. Trajectories for large and small probes during entry to Venus . . . . . . . . . . . . . . . . . . . . . . . . . . 112

18. Stagnation-point, radiative heating rates along entry trajec- tories for large and small probes. Results from radiating, inviscid flow solutions . . . . . . . . . . . . . . . . . . 113

19. Fully-coupled, radiating flow solution with steady-state ablation of carbon-phenolic heatshield for large probe . . . 114

(a) Free-stream conditions and body shape . . . . . . . . . 114

(b) Comparison of radiative heating . . . . . . . . . . . . 114

(c) Radiative and convective heating distribution . . . . . 115

(d) Ablation rate distribution . . . . . . . . . . . . . . . 115

(e) Aerodynamic shear distribution . . . . . . . . . . . . . 116

(f) Momentum thickness Reynolds number distribution . . . . 116

(g) Spectral distribution of radiative heating toward the body due to continuum processes . . . . . . . . . . . 117

20. Fully-coupled, radiating flow solution with steady-state ablation of carbon-phenolic heatshield for small probe . . . 118

(a) Free-stream conditions and body shape . . . . . . . . . 118

Page

. . . . . . . . . . . . (b) Comparison of r a d i a t i v e hea t ing 118

(c) Radia t ive and convect ive hea t ing d i s t r i b u t i o n . . . . . 119

. . . . . . . . . . . . . . . (d) Ablat ion r a t e d i s t r i b u t i o n 119

(e) Aerodynamic s h e a r d i s t r i b u t i o n . . . . . . . . . . . . . 120

f Momentum th ickness Reynolds number d i s t r i b u t i o n . . . . 120

(g) Spec t r a l d i s t r i b u t i o n of r a d i a t i v e hea t ing toward the . . . . . . . . . . . body due t o continuum processes 121

21 . Comparison between a tu rbu len t and laminar boundary l aye r f o r fully-coupled s o l u t i o n f o r l a r g e probe . . . . . . . . . . . 122

(a) Heating r a t e d i s t r i b u t i o n . . . . . . . . . . . . . . . 122

(b) Ablat ion r a t e d i s t r i b u t i o n . . . . . . . . . . . . . . . 122

(c) Aerodynamic shea r d i s t r i b u t i o n . . . . . . . . . . . . . 123

(d) Spec t r a l d i s t r i b u t i o n a t t h e wa l l o f r a d i a t i v e hea t ing toward t h e body due t o continuum processes . . . . . . 123

22 . Fully.coupled. r a d i a t i n g flow s o l u t i o n with s t eady- s t a t e a b l a t i o n of carbon-phenolic hea t sh ie ld f o r high v e l o c i t y e n t r y . . . . . . . . . . . . . . . . . . . . . . . 124

(a) Free-stream condi t ions and body shape . . . . . . . . . 124

(b) Comparison of r a d i a t i v e hea t ing . . . . . . . . . . . . 124

(c) Radia t ive and convect ive hea t ing d i s t r i b u t i o n . . . . . 125

(d) Ablation r a t e d i s t r i b u t i o n . . . . . . . . . . . . . . . 125

(e) Aerodynamic shea r d i s t r i b u t i o n . . . . . . . . . . . . . 126

. . . . ( f ) Momentum th ickness Reynolds number d i s t r i b u t i o n 126

(g) S p e c t r a l d i s t r i b u t i o n of r a d i a t i v e hea t ing toward t h e body due t o continuum processes . . . . . . . . . . . 127

(h) Spec t r a l d i s t r i b u t i o n of r a d i a t i v e hea t ing toward t h e body due t o l i n e processes . . . . . . . . . . . . . . 128

LIST OF SYMBOLS

D.. - 11 D

parameter defined by eq. 21

sonic velocity, see eq. 49

parameter defined by eq. 22

Planck function

parameter defined by eq. 23

parameter defined by eq. A4

'parameter defined by eq. A5 or eq. 35

parameter defined by eq. A6 or eq. 36

specific heat

specific heat parameter defined by eq. 61

parameter defined by eq. 24

binary diffusion coefficient for it11 and jth chemical species

self-diffusion coefficient of a reference species

parameter defined by eq. 25

exponential integral of order n

exponential integral of order 3

divergence of radiative flux defined by eq. 26

diffusion factor of ith chemical species in a given chemical system. see eq. 57

represents the variables p,u,v, or h in eqs. 37 to 39

stream function for boundary-layer equations, defined by eq. 70

metric coefficient for ith coordinate, used herein as g =A, g =1, g3=r (see eqs. 6 to 8) 1 2

2 total enthalpy, h+u 1 2 , in boundary-layer equations

enthalpy

enthalpy parameter defined by eq. 61

Planck constant

heat of formation of undecomposed ablation material

monochromatic radiation intensity

diffusional mass flux of chemical element i, see eq. 58

body curvature parameter defined by eq. 9

mass fraction of chemical element i

thermal conductivity

differential line element in eqs. 6 and 7

molecular weight

ablation rate of the heatshield

thermodynamic variable defined by eqs. 31 and 48

thermodynamic variable defined by eqs. 32 and 49

pressure

convective heat flux, see eq. 60

radiative heat flux

heat flux conducted into the heatshield

momentum thickness Reynolds number for boundary layer

nose radius of body at s=0, see Figure 3

radial distance from axis of symmetry of body, see Figure 3

body radius from axis of symmetry, see Figure 3

transformed s coordinate for inviscid-layer equations, defined by eq. 15

stream function for inviscid-layer equations, defined by eq. 45

coordina te along t h e body s u r f a c e , s e e Figure 3

transformed t coordina te defined by eq. 1 5

temperature

temperature a t y=6 i n i n v i s c i d flow f i e l d

time

dummy v a r i a b l e of monochromatic o p t i c a l depth i n eqs. 76 t o 78

v e l o c i t y tangent t o body s u r f a c e , s e e Figure 3

free-stream v e l o c i t y

e n t r y v e l o c i t y

v e l o c i t y normal t o body su r face , s e e Figure 3

weight of en t ry probe

b a l l i s t i c c o e f f i c i e n t where t h e drag c o e f f i c i e n t , C d , i s f o r a r e a , A , of t h e probe 's base

genera l coord ina te s , used he re in a s x =s, x =y, and x =$ ( see eqs. 6 and 7)

1 2 3

mole f r a c t i o n of chemical spec ie s j

transformed y coordina te , def ined by eq. 15

coordina te normal t o t h e body su r face , s e e Figure 3

quan t i ty f o r j t h chemical s p e c i e s defined by eq. 59

quan t i ty f o r i t h chemical element defined by eq. 59

d i s t a n c e along a x i s of symmetry of body

func t ion f o r t h e exponential i n t e g r a l of o rde r n, E (2) n

mass f r a c t i o n of chemical element i i n chemical spec ie s j

d i f f e r e n c e between t h e angle of the shock wave and t h e angle of t h e body s u r f a c e , s e e Figure 3

angle between a ray and t h e normal f o r r a d i a t i o n f l u x i n eq. 71

entry angle

shock stand-off distance of inviscid layer

boundary-layer displacement thickness defined by eq. 2

total thickness of shock layer defined by eq. 1

monochromatic emissivity defined by eqs. 76 and 78

emissivity of the heatshield

transformed y coordinate for boundary-layer equations, defined by eq. 69

angle of bodysurface relative to axis of symmetry of body, see Figure 3

cone half angle for a spherically-capped, conical body

curvature parameter defined by eq. 9

value of X at y=6

viscosity in boundary-layer equations

parameter defined by eq. 59

parameter defined by eq. 59

monochromatic linear absorption coefficient corrected for induced emission

uv for molecular band transitions, defined by eq. 81

radiation frequency

transformed s coordinate for boundary-layer equations, defined by eq. 6 9

density

turbulent eddy diffusivity

turbulent eddy conductivity

turbulent eddy viscosity

Stefan-Boltzmann constant

T monochromatic o p t i c a l depth defined by eq. 75 V

T aerodynamic shea r a t t h e wa l l W

4 azimuth angle , s e e Figure 3

n s o l i d angle

s u b s c r i p t s

a r e f e r s t o t h e a b l a t i o n m a t e r i a l

c r e f e r s t o t h e char of t h e decomposed a b l a t i o n ma te r i a l

e r e f e r s t o the boundary-layer edge

g r e f e r s t o t h e py ro lys i s gas of the decomposed a b l a t i o n m a t e r i a l

i 9 j r e f e r s t o the i t h o r j t h chemical spec ie s o r chemical element

o . e f e r s t o s tagnat ion-pQint va lues , s = O

s r e f e r s t o condi t ions behind a normal shock wave

w r e f e r s t o t h e body wa l l

m r e f e r s t o free-stream condi t ions

6 r e f e r s t o condi t ions a t y=6 i n the i n v i s c i d l a y e r

v r e f e r s t o r a d i a t i o n frequency

s u p e r s c r i p t s

c ,L r e f e r s t o continuum t r a n s i t i o n s , inc luding molecular band tran- s i t i o n s , o r l i n e t r a n s i t i o n s i n eq. 79

1 , 2 r e f e r s t o t h e f i r s t o r second s t e p i n t h e two-step numerical method f o r t h e r a d i a t i n g , i n v i s c i d f low s o l u t i o n

+ r - r e f e r s t o t h e d i r e c t i o n of t h e r a d i a t i v e f l u x (+ r e f e r s t o the r a d i a t i v e f l u x toward t h e shock wave and - r e f e r s t o t h e r a d i a t i v e f l u x toward t h e body)

prime r e f e r s t o dimensional q u a n t i t i e s i n t h e s e c t i o n of Radiat ing, Inv i sc id Flow F i e l d Solu t ion

INTRODUCTION

One of t he major s c i e n t i f i c endeavors of the present day i s t he

exploration of t he planets by unmanned and manned spacecraft. The

present technique for planetary entry i s t o use atmospheric f r i c t i on as

a brake t o slow the spacecraft from hypersonic speeds. During hyper-

sonic entry the gas around the body i n the formed shock layer is heated

by the dissipated k ine t ic energy. In order for the spacecraft t o sur-

vive, t he main body of t he spacecraft w i l l be protected f r o m t h i s source

of intense heat by the use of an ablative heat shield material. The

speeds required for planetary en t r i e s w i l l be well i n t he excess of t he

parabolic speed i n order t o reduce t r i p times for interplanetary t rave l .

The temperatures of the heated gas behind a shock wave associated with

these high-speed en t r i e s are suff ic ient for gas radiation t o be a major

contributor t o t he heating of the spacecraft. The inclusion of radia-

t i v e heating i n the energy equation of flow f i e l d solutions causes a

strong coupling between the radiat ive t ransport , the inviscid shock

layer , the boundary layer , and the ablative heat shield analyses. Com-

putational methods for t he solution of t h i s complex flow phenomena are

required before heat shields can be designed t o survive during planetary

en t r i e s a t speeds i n excess of parabolic speed.

A t entry speeds l e s s than parabolic speed the radiat ive heating i s

negligible and convection through the boundary layer i s t he dominant

mode of heat t ransport t o t h e body. The neglect of rad ia t ive transport

uncouples the inviscid flow equations and the boundary layer equations;

however, t he boundary layer analysis and the heat shield analysis are

s t i l l coupled. Extensive analyses f o r boundary layers with mass injec-

t i on have been conducted over t he years so tha t the necessary inputs

t o the heat shield analysis for convective heating and mass diffusion

can be determined without a coupled solution of the equations.

A t entry speeds i n excess of parabolic speed the radiat ive trans-

port cannot be neglected and the rad ia t ive heating can be much more

dominant than convective heating. In general, the radiat ive t ransport

i n a gas is governed by in tegra l equations involving the absorption

coeff ic ient integrated over both radiation frequency spectrum and

three-dimensional space. The absorption coefficient i s dependent on

the temperature, density, and the number density of t he species present

i n the gas ( r e f . 1, page 721). Thus, the radiat ive transport i s depen-

dent on the thermodynamic prof i les throughout the gas layer between the

shock wave and body. The absorption of radiation by the cooler gas i n

t he boundary layer w i l l change the thermodynamic prof i les i n the bound-

ary layer and a f fec t the convective heating and mass diffusion t o t he

heat shield. In ear ly s tudies , t he incident radiation t o the spacecraft

was taken as t he radiat ion from the inviscid shock layer. Later s tudies

have shown tha t t he boundary layer , with and without inject ion of abla-

; t i on products, can absorb radiat ion and reduce the incident radiat ive

heating t o the spacecraft . For high-speed planetary entry, the solution

of the flow f i e l d equations and the heat shield analysis is very complex

because of t he coupling of radiat ive transport .

The radiat ive transport t o t he stagnation region of blunt bodies

f o r en t r i e s in to air has been the primary in te res t fo r most s tudies of

radiat ing flow f i e ld s with abiation product effects. The in t e r e s t for

en t r ies in to a i r was due t o Earth reentry a f t e r exploration of t he

Earth 's moon and preliminary s tudies for Earth reentry a f t e r exploration

of Mars. There i s a need for investigation of t he radiating flow f i e l d

i n other gaseous atmospheres because of the increased in te res t i n explor-

a t ion of other planets (Venus, Jup i te r , e.). The emphasis on the

stagnation region was due t o a simplification of the solution and be-

cause the radiat ive heating was considered t o be greates t i n t h i s region.

However, the recent r e su l t s by Olstad ( r e f . 2 ) and by C a l l i s ( r e f . 3) for

an inviscid, radiat ing flow f i e l d have shown tha t the radiat ive heating

t o t he flank regions of blunted-conical bodies can equal or exceed the

rad ia t ive heating t o the stagnation region. A s discussed by Page ( r e f .

41, most of t h e heat shield surface and weight i s downstream of t h e

stagnation region and val id treatment of fully-coupled solutions i s

needed downstream of the stagnation region along cone flanks or over

spherical noses. Anderson ( r e f . 5) s t a t e s the need f o r addit ional so-

lu t ions of the coupled radiat ive problem with ablation product e f fec t s

t o complement the exis t ing resu l t s . These points i l l u s t r a t e t he need

f o r a method of calculating the fully-coupled radiating flow f i e l d , with

ablat ion product e f fec t s , around the e n t i r e body and f o r en t r ies in to

atmospheres of a rb i t ra ry gases.

The purpose of t he present study i s t o investigate and develop a

computational method for a fully-coupled solution of the radiating flow

f i e l d , with ablation products in jec t ion , about blunt bodies for en t r i e s

i n to a rb i t ra ry gas mixtures. Basically, t he method i s t o separate the

r a d i a t i n g flow f i e l d i n t o an ou te r i n v i s c i d l a y e r and an inner boundary

l a y e r with i n j e c t e d a b l a t i o n products; then , t h e two so lu t ions are

coupled by t h e r a d i a t i v e t r a n s p o r t through both l a y e r s and by t h e

boundary l a y e r displacement th ickness f o r mass i n j e c t i o n . The r a d i a t i v e

f l u x i s included i n t h e energy equation of both t h e i n v i s c i d s o l u t i o n

and t h e boundary l a y e r so lu t ion . A s previously s t a t e d , t h e fully-

coupled formulat ion fo r r a d i a t i n g p lane ta ry en t ry i s very complex and

all approaches r e q u i r e varying degrees of approximation i n order t o

ob ta in a so lu t ion . By separa t ing t h e two d i f f e r e n t l a y e r s of flow, a

b e t t e r t reatment should be poss ib le f o r t h e so lu t ion of each l a y e r .

A second-order, time-asymptctic technique i s used f o r t h e so lu t ion

o f t h e i n v i s c i d shock l a y e r . The s e t of unsteady governing equations i s

hyperbol ic , t h u s t h e technique i s v a l i d i n both t h e subsonic and super-

sonic regions of t h e inv i sc id flow about t h e body. A computer program

f o r a numerical s o l u t i o n of t h e technique i s developed i n t h e s tudy.

An equi l ibr ium thermodynamic subroutine and a d e t a i l e d r a d i a t i v e t rans-

por t subroutine a r e included i n the computer program so t h a t a r b i t r a r y

gas mixtures can be considered. Moret t i and Abbett ( r e f . 6 ) used a

time-asymptotic technique f o r an i n v i s c i d flow s o l u t i o n without radia-

t i o n i n a pe r fec t gas and C a l l i s ( r e f s . 3 and 7) used a time-asymptotic

technique f o r a r a d i a t i n g , i n v i s c i d flow s o l u t i o n i n air.

The s o l u t i o n of t h e inner gas l a y e r i s f o r a nonsimilar , m u l t i -

component boundary l a y e r with mass i n j e c t i o n , unequal d i f fus ion coef-

f i c i e n t s , and a r b i t r a r y chemical species . The n u m e r i c d , i n t e g r a l

matr ix method of B a r l e t t and Kendall ( r e f . 8 ) i s used f o r t h e boundary

l a y e r solut ion and t h e r e is an e x i s t i n g computer program c a l l e d BLIMP.

In t h e computer program, t h e thermodynamic p roper t i e s and t r anspor t

p roper t i e s ( v i s c o s i t y , e t c . ) a r e ca lcula ted a s p a r t of the solut ion.

The i n j e c t i o n r a t e of ab la t ion products can be e i t h e r prespeci f ied o r

ca lcula ted during t h e solut ion f o r a steady-state ab la t ion process.

The computer program i s modified i n the present study t o account f o r

r a d i a t i v e t ranspor t within t h e boundary l ayer .

The r a d i a t i v e t r anspor t so lu t ion f o r coupling t h e inv i sc id l a y e r

and t h e boundary l ayer i s f o r a nongray gas with molecular band, contin-

uum, and atomic l i n e t r a n s i t i o n s . The in tegra t ion over t h e r a d i a t i o n

spectrum uses a de ta i l ed frequency dependence of the absorption coef-

f i e i e n t s and t h e tangent s l a b approximation i s used f o r in tegra t ion over

physical space. An e x i s t i n g computer program (RAD/EQUIL) of t h e numeri-

c a l method by Nicolet ( r e f . 9 ) i s used f o r the solut ion of the r a d i a t i v e

t r anspor t . In addi t ion t o using t h i s program f o r coupling t h e flow

f i e l d s , an extensive modification of t h e program is made so t h a t it can

be used e f f i c i e n t l y a s a subroutine i n t h e developed computer program

f o r t h e inv i sc id r a d i a t i n g l a y e r .

The appl ica t ion of t h e present developed method i s shown by a

study of t y p i c a l points on an e n t r y t r a j e c t o r y f o r Venus. To t h e

author ' s knowledge, t h e r e a r e no published r e s u l t s f o r a fully-coupled

rad ia t ing solut ion with ab la t ion products in jec t ion a t e i t h e r t h e stag-

nat ion o r downstream regions on a body f o r Venusian ent ry . The v a l i d i t y

of t h e method i s inves t igated by a comparison with published r e s u l t s f o r

Earth entry.

REVIEW OF LITERATURE

The review of l i t e r a t u r e i s r e s t r i c t ed t o pr ior s tudies which

present t he more pertinent r e su l t s f o r t he radiating flow about blunt

bodies with ablation products inject ion. The fully-coupled solution of

radiat ing flow with ablation analysis is a complex problem and involves

many discipl ines; therefore , a detailed review of the en t i r e f i e l d for

t he thermal analysis during planetary entry i s not presented. A good

review and extensive reference list f o r radiat ing shock layers i s given

by Anderson ( r e f . 5 ) . Similarly, a review of ablat ive heat shields fo r

planetary en t r i e s i s given by Walberg and Sullivan ( r e f . 10) . These two

a r t i c l e s provide a good review of the present technology of radiat ive

heating and ablation analysis for high-speed planetary entry. For

addit ional discussions of the background information presented i n t he

Introduction on planetary gas dynamics and planetary entry, t he reader

i s referred t o reference8 4 and 11 t o 15 . Information on t ra jec tory

analysis for planetary missions is available i n reference 16. A dis-

cussion on types of heat shield materials and ablation mechanisms i s

given i n reference 17. A good analysis on the coupling between the

boundary layer and a charring ablator is given by Kendall etg., ( r e f .

18).

The remaining sections of t he l i t e r a t u r e survey present more

detai led information which i s pertinent t o an analysis of the radiating '-

flow about blunt bodies. The f i r s t section discusses features of radi-

a t ion t ransport modeling. The second section is a discussion on radia-

t i n g Plow f i e l d analyses. In addition t o flows about blunt bodies, a

discussion is presented on significant and applicable r e s u l t s from

stagnation region analyses. Finally, a section i s presented on the

thermal analyses f o r Venusian entry.

Radiation Transport Modeling

An atom or molecule i n a high-temperature gaseous medium can emit

and absorb radiat ion a t a frequency charac te r i s t ic of some t r ans i t i on

from one quantum s t a t e t o another and each specie has an absorption

spectrum which can be expressed i n terms of an absorption coefficient.

Because of quantum t rans t ions , t he absorption coeff ic ient is a function

of frequency. Radiation of a given frequency t rave ls i n physical space

a distance ( radiat ion mean f r ee path) before being absorbed. (The above

description is from reference 1, page 721.) Thus. the rad ia t ive trans-

port i n a gaseous medium i s a function of frequency and distance and the

equation for t he t o t a l rad ia t ive transport is an in tegra l over both

frequency spectrum and physical space.

The solution for a radiating flow f i e l d i s a se t of integro-

d i f f e r en t i a l equations and approximations are used t o solve t h i s complex

system of equations. One of the approximations deals with the spa t i a l

aspect of radiat ive transport . The absorption of radiation a t a point

i n the flow f i e l d i s a function of t he emitted radiat ion from surrounding

f lu id elements; and, since radiat ion t r ave l s a distance before being ab-

sorbed, t he rad ia t ive transport i s a three-dimensional problem i n space.

To reduce the complexity of t h e solution, t he "tangent slab approximation"

is usua l ly used f o r r a d i a t i n g flow f i e l d s . This approximation t r e a t s

t h e gas l a y e r as a one-dimensional s l a b i n ca lcu la t ion of t h e r a d i a t i v e

t r anspor t . The r a d i a t i v e heat f l u x i s neglected except i n t h e d i r e c t i o n

normal t o e i t h e r t h e body o r shock wave.

Garre t t ( r e f . 191, a f t e r a review of +he s tud ies of references 20

t o 22, s t a t e s t h e e r r o r introduced by t h e approximation should be less

than 5 percent a t t h e s tagnat ion point . Page ( r e f . 4 ) , i n a discussion

of t h e three-dimensional e f f e c t s , concludes t h a t t h e tangent s l a b

approximation i s s u f f i c i e n t l y accura te f o r engineering ca lcu la t ions when

t h e shock-layer th ickness i s small compared t o t h e curvature (-1/12).

Page's conclusions a r e based pr imar i ly on t h e s t u d i e s by Chien ( r e f . 23)

and by Wilson ( r e f . 24). The study by Wilson was a comparison of t h e

tangent s l a b approximation wi th an exact three-dimensional numerical

c a l c u l a t i o n at t h e s tagnat ion and downstream regions of a b lunted ,

con ica l body f o r a t y p i c a l Earth r een t ry . The e r r o r f o r t h e tangent

s l a b approximation was 15 percent a t t h e s tagnat ion region and l e s s

e r r o r a t t h e con ica l region. The tangent s l a b approximation i s used i n

t h e present study.

I n r a d i a t i o n t r anspor t modeling, t h e gas may be t r e a t e d as t rans-

parent , gray, o r nongray. A t ransparent gas i s assumed t o emit radia-

t i o n but not absorb rad ia t ion . The treatment of a gray o r nongray gas

allows both emission and absorption. However, t h e absorption coe f f i c i en t

i s assumed constant over t h e frequency spectrum f o r a gray gas. A f r e -

quency dependence of t h e absorption coe f f i c i en t , e i t h e r approximate o r

i n d e t a i l , is included f o r a nongray gas . Anderson ( r e f . 5) t r a c e s t h e

h i s t o r i c a l development of r a d i a t i n g flow f i e l d s and shows t h e progression

from t ransparen t , continuum only , uncoupled ca lcu la t ions used by e a r l y

inves t iga to r s t o t h e nongray, continuum and l i n e , coupled energy C ~ ~ C U -

l a t i o n s which a r e cu r ren t today. A s discussed by Anderson, a gray gas

ana lys i s i s not s u f f i c i e n t l y accura te f o r r een t ry appl ica t ions and t h e

gas should b e t r e a t e d a s nongray.

The frequency dependence of absorption coe f f i c i en t fo r a nongray

gas i s t r e a t e d i n d e t a i l o r by a "stepmodel". I n a s t e p model, t h e

frequency dependence i s broken i n t o a number of d i s c r e t e s teps ( s e e , f o r

example, references 7 and 25 t o 27). The number of s t eps a r e genera l ly

from two t o nine. The c a l c u l a t i o n of t h e r a d i a t i v e heat f lux involves

an i n t e g r a l over frequency and using a step-model w i l l s implify t h e

in teg ra t ion . However, t h e i n j e c t i o n of ab la t ion products i n t o t h e flow

f i e l d increases t h e number of species t o be considered and t h e r e is a

r ap id change i n chemical composition i n t h e boundary l aye r . A step-

model has t o be cons t ructed f o r t h e p a r t i c u l a r gas composition and t h e

add i t ion of ab la t ion products w i l l complicate t h e construct ion. Boughner

( r e f . 25) had d i f f i c u l t y i n developing a general step-model f o r a

CO -N mixture; and, t h e developed step-model w a s only v a l i d over a 2 2

r e s t r i c t e d and narrow range of condit ions. The d i f f i c u l t y was due t o

molecular band rad ia t ion .

The r a d i a t i v e heat f l u x wi th ab la t ion product e f f e c t s has been

ca lcu la ted f o r a nongray gas and t h e frequency dependence of absorption

coe f f i c i en t was considered i n d e t a i l ( see , f o r example, references 1 9 ,

28, and 29) . There a r e computer programs f o r ca lcu la t ing t h e d e t a i l e d .

nongray radiat ive t ransport of which the ones used primarily for reentry

application are RATRAP, developed by Wilson ( r e f . 30) ; SPECS, developed

by Thomas ( r e f . 31) ; and RADIEQUIL, developed by Nicolet ( r e f s . 9 and

32). Su t t les ( r e f . 33) made a comparison of these programs i n a recent

study and concluded tha t t he RAD/EQUIL program depended on fewer approx-

imations, included m r e d e t a i l , and required l e s s computer time than

RATRAP and SPECS. The RAD/EQUIL program is used i n t he present study.

A s discussed by Penner and Olfe ( r e f . 13 ) , radiation i s coupled t o

t he conservation equations by the radiat ion pressure, radiat ion energy

density, and radiat ion energy t ransfer . The radiation pressure and

radiat ion energy density a r e usually negligible and can be neglected even

when the radiation energy t ransfer i s very important. The two terms can

be and are neglectedin solutions which deal primarily with radiat ive heat-

ing ( r e f s . 13 and 34). Therefore, t he coupling between radiation and the

conservation equations i s due t o t h e divergence of the radiat ive heat

f l ux i n t he energy equation.

For addit ional information on radiation and radiat ive t ransport ,

t he interested reader is referred t o the numerous available books. Two

good books with vhich the author is most familiar are by Penner ( r e f . 35)

and by Penner and Olfe ( ref . 13). Radiative heating is considered i n

t he book on gas dynamics by Vincenti and Kruger ( r e f . 34). Also, ref-

erences 36 t o 38 present analyses of radiat ive transport fo r high-

temperature gaaes.

Radiating; Flow F ie lds

The i n t e r e s t of t h e present study i s t h e rad ia t ing flow f i e l d about

blunt bodies; the re fo re , t h e flow i n both the s tagnat ion region and t h e

, downstream region has t o be considered. Also, t h e p r i o r s tud ies which

ificluded ab la t ion product e f f e c t s have been f o r t h e s tagnat ion region.

A discussion and l i t e r a t u r e survey a r e presented f irst on t h e per t inent

r e s u l t s from analyses of stagnation-region flow; then a discussion and

l i t e r a t u r e survey a r e presented f o r flow about blunt bodies.

Flow at S t a m a t i o n Region

The recent doctora l t h e s i s by Garre t t ( r e f . 1 9 ) presents a de ta i l ed

review of t h e var ious methods used f o r solving the r a d i a t i n g flow f i e l d

a t t h e s tagnat ion region. Also, t h e various methods of solut ion a r e

discussed i n t h e papers by Coulard & 2 ( r e f . 39 ) and by Anderson ( r e f .

5 ) . The reader i s re fe r red t o these papers fo r a descr ip t ion of t h e

solut ions . Only t h e pe r t inen t r e s u l t s w i l l be presented here.

The e a r l i e r s t u d i e s by Howe and Viegas ( r e f . 40) and by Hoshizaki

and Wilson ( r e f s . 41 and 42) have shown t h e r e i s a coupling between

r a d i a t i v e and convective heating and t h a t r ad ia t ion from t h e higher-

temperature i n v i s c i d region can be absorbed by t h e lower-temperature

viscous region. Later s tud ies have shown t h a t t h e ab la t ion products

i n j e c t e d i n t o t h e flow f i e l d from a b l a t i v e heat sh ie lds can s igni f icant -

l y reduce t h e r a d i a t i v e heating t o t h e body ( r e f s . 19 , 22, 28, 29, 43,

and 4 4 ) . These s tud ies were f o r Earth reen t ry except a l i m i t e d pa r t

of t h e study by Wilson ( r e f . 4 4 ) w a s f o r Jovian ent ry . Hoshizaki and

Lasher ( r e f . 22) s t a t e t ha t atomic carbon from the ablat ion products i s

t he pr inciple absorber and Smith &&. ( r e f . 28) show tha t carbon monox-

ide can also s ignif icant ly absorb the radiat ion f romthe radiating shock

layer.

The recent t hes i s by Garrett ( r e f . 19) presents a comparison of t he

r e su l t s a t t he stagnation region for a i r in ject ion and ablation product

injection. Garrett uses an implicit f i n i t e difference method for a

solution a t t h e stagnation-point of a blunt body and t r e a t s t he e n t i r e

shock layer a s viscous. The r e su l t s of Garrett show t h a t massive in-

jection of ablation products from a carbon phenolic heat shield into' a i r

reduced t h e rad ia t ive heating t o t h e body by 40 percent; but , massive

in jec t ion of air only reduced the radiat ive heating by 1 5 percent. A

comparison of r e s u l t s presented by Garrett f o r solutions from various

studies showed tha t ablation products reduce the rad ia t ive heating; bu t ,

there were s ignif icant differences i n the magnitude of t he reduction.

Smith & &. ( r e f . 28) used a method for the stagnation region which

i s similar t o the method i n the present study. That i s , separating the

inviscid layer and the boundary layer and then coupling the two by the

radiat ive transport . The study was for Earth reentry with ablation

,product e f fec t s . The derivation of t he boundary layer equations included

t h e rad ia t ive heat f l u i n the energy equation and was fo r moderate and

strong blowing. However, t he r e su l t s presented did not include the

radiat ive heat f l ux i n t he energy equation for t he boundary layer and

was only f o r strong blowing. The solution of t he equations was by

;numerical methods and D r . Smith has s ta ted there are d i f f i c u l t i e s i n

extending t h e so lu t ion t o include r a d i a t i v e heat f lux i n t h e boundary-

l a y e r energy equation and t o include moderate blowing. The problem has

not been solved." The study by Smith -- e t al. did show t h a t t h e bas ic

method i s v a l i d and i n t h e present s tudy t h e treatment of t h e boundary-

l a y e r solut ion i s much b e t t e r than t h e approximate method used by Smith

e t a l . -- Hoshizaki and Lasher ( r e f . 22) t r e a t e d t h e e n t i r e shock l a y e r a s

viscous and s t a t e d t h e advantage i s t h a t it el iminates t h e necess i ty of

matching t h e r a d i a t i v e heat f l u x a t t h e inviscid-viscid i n t e r f a c e .

Gar re t t ( r e f . 19) t r e a t e d t h e e n t i r e l a y e r as viscous and t h e r e were

i r r e g u l a r i t i e s i n t h e enthalpy p r o f i l e s a t t h e inviscid-viscid i n t e r f a c e

fo r strong blowing. The enthalpy p r o f i l e s come from t h e energy equation

and t h i s equation i s where t h e r a d i a t i v e t ranspor t i s coupled t o t h e flow

equations. I n t h e present s tudy, t h e coupling i s not a d i r e c t matching

of t h e r a d i a t i v e heat f l u x a t t h e in te r face . The r a d i a t i v e heat f l u x t o

be coupled i n t h e energy equation of t h e boundary l ayer i s taken from t h e

solut ion of t h e r a d i a t i v e t r anspor t over t h e e n t i r e shock l ayer . The

magnitude of t h e r a d i a t i v e heat f l u x from t h e boundary l ayer toward t h e

inv i sc id l ayer i s small compared t o t h e f lux toward t h e boundary l a y e r

from t h e invisc id l a y e r ( see , f o r example, r e f . 28). Thus, i n t h e pre-

sen t method t h e solut ion f o r t h e r a d i a t i n g inv i sc id l ayer i s solved once

and an i t e r a t i o n i s only required between t h e boundary l ayer so lu t ion and

r a d i a t i v e t ranspor t so lut ion.

*Personal communication with D r . G. Louis Smith, Langley Research Center, Hampton, Virginia

Garrett ( r e f . 19) and Rigdon & &. ( r e f . 29) s t a t e the need f o r

properly calculating the thermodynamic properties of the ablation pro-

ducts. In t h e present study, t he thermodynamic properties are calculated

during the solution for both the inviscid region and the boundary layer

and a rb i t ra ry gases can be considered.

As i l l u s t r a t e d by the comparisons presented by Garrett ( r e f . 19),

there are discrepancies for t he magnitude of rad ia t ive heating reduction

due t o ablation e f f ec t s even for t h e stagnation region. The exis t ing

r e s u l t s are primarily for Earth reentry and calculations need t o be

performed for other planetary en t r ies . Anderson ( r e f . 5 ) s t a t e s t ha t

r e a l i s t i c calculations f o r rad ia t ive heating t o entry vehicles should

def in i te ly account for ablation e f fec t s and addit ional rad ia t ive coupled

analyses with ablation e f fec t s should be made i n order t o complement t he

exis t ing r e su l t s .

Flow about Blunt Bodies

There have been a number of s tudies and numerical solutions f o r t he

rad ia t ive heating with ablation e f fec t s a t the blunt-body stagnation

region but solutions for the downstream regions of t he flow f i e l d are

not so advanced. The few studies which have considered the radiat ing

flow about blunt bodies have shown tha t radiat ive heating a t the flank

regions of hemisphere-conical bodies can be a la rge f ract ion and even

exceed the rad ia t ive heating at t he stagnation point ( r e f s . 2, 3 , 45,

and 46). These s tudies were f o r Earth reentry, used a step-model for t he

radiat ive t ransport , and did not include ablation product effects .

The study by Olstad ( r e f . 2) was for an inviscid shock layer with

the solution of t he flow f i e l d by the method developed by Maslen ( r e f .

47) but modified f o r radiat ive transport . The method of Maslen is based

on t h e flow being pa ra l l e l t o the shock wave and uses a von Mises trans-

formation. This method uncouples t he y-momentum equation. However, t he

method is not s t r i c t l y va l id a t the stagnation region. The study by

Olstad was f o r t he Earth reentry of blunted-conical bodies with cone

half angles of 30, 45 , and 60 degrees. A step-model was used for the

rad ia t ive transport . The r e s u l t s showned tha t t he radiat ive heating

along t h e cone flanks was comparable t o the stagnation region values for

t he la rger cone angles. Olstad also presented some preliminary r e s u l t s

for a i r inJection in to t he shock layer. These solutions were performed

by using the Maslen method for an inner inviscid layer. The r e s u l t s

showed tha t t h e a i r in ject ion was more effect ive i n reducing the radia-

t i v e heating a t the stagnation region than a t the downstream region. The

en thdpy prof i les of t he inner inviscid layer give the appearance of a

boundary layer but a r e a r e su l t of absorption of t he radiant energy emit-

t ed by the outer inviscid layer.

Callis ( r e f . 3 ) investigated the radiating, inviscid flow about long

blunt bodies by using a time-asymptotic technique. A discussion of t h i s

technique is given below. The study by Cal l is was for Earth reentry and

a step-model was used for t he radiat ive t ransport . The body shapes were

spherically capped cones. The r e su l t s showed tha t f o r l a rger cone angles

(60 degrees half angle) the inviscid radiat ive heating r a t e s were i n ex-

cess of t he stagnation values over a large portion of the flank region.

Callis s t a t e s t h a t t h i s increase i n heat ing r a t e s along t h e cone f lanks

could pose a ser ious thermal protec t ion problem.

Chou ( r e f . 46) presents an approximate method f o r t h e flow about

blunt bodies i n which t h e e n t i r e l ayer i s considered t o be viscous. The

s o l u t i o n i s based on an approximate method f o r nonsimilar terms, t h e

specie con t inu i ty equation i s neglected, t h e densi ty-viscosi ty product

i s constant , and t h e w a l l condit ions a re f o r constant mass i n j e c t i o n and

temperature along t h e body. An equation i s developed f o r t h e nonsimilar-

i t y of t h e t angen t ia l ve loc i ty and Chou s t a t e s t h a t t h e v a l i d i t y of t h e

equation cannot be defined r igorously and t h e physcial reason f o r t h e

successPul app l i ca t ion i s not c l e a r . The appl ica t ion of t h e method was

f o r Earth reen t ry and t h e in jec ted gas was a i r . However, r a d i a t i v e

t r anspor t was not considered f o r t h e air i n j e c t i o n cases. A comparison

w a s made with t h e method of Olstad ( r e f . 2) f o r inv i sc id r a d i a t i n g flow.

The check cases were supplied by D r . Olstad and Chou s t a t e s t h a t h i s

method is v a l i d when t h e r e s u l t s a r e compared. However, t h e r e is some

question about t h e manner i n which t h e r e s u l t s a r e being compared and

t h e comparison may not be as good as shown by Chou. The disagreement

over the comparison of t h e r e s u l t s has not been resblved.*

P r i o r so lu t ions f o r t h e rad ia t ing flow about b lunt bodies d id not

include ab la t ion product e f f e c t s and were f o r Earth reent ry . The need

f o r a fully-coupled so lu t ion with ab la t ion products f o r a r b i t r a r y gases

can be i l l u s t r a t e d by t h e s tud ies of Tauber ( r e f s . 48 and 49) and Tauber

'Personal communications with D r . W. B. Olstad, Langley Research Center, Hampton, Virginia

and Wakefield ( r e f . 50) . These s tudies were for planetary entry i n t o

Jupi te r , Saturn, Uranus, and Neptune with a nominal atmospheric composi-

t i on of 85 percent hydrogen and 1 5 percent helium (mole f r ac t ion ) . The

basis f o r t he thermal analysis i s presented i n the paper by Tauber and

Wakefield. These s tudies had t o r e ly on stagnation-point r e su l t s for

the thermal analysis a t the downstream regions. Results from the solu-

t ions for Earth reentry and t h e l imited r e su l t s by Wilson ( r e f . 44) for

H -He mixture were used. This recourse had t o be used because of the 2

lack of r e su l t s for flow around bodies with ablation product e f f ec t s and

the lack of r e s u l t s f o r entry into planets other than Earth. Additional

discussion of t h i s problem i s given by Page ( r e f . 4 ) .

There i s no method which is generally accepted a s being bes t , even

when radiat ive transport i s not included, for the shock layer solution

around hypersonic blunt bodies ( r e f . 39). This point i s shown i n the

paper by Perry and Pasiuk ( r e f . 51) i n which numerous solutions are

compared and are compared with experimental data. There i s considerable

s ca t t e r between the d i f fe ren t solutions even i n the region of subsonic

flow. The solution used i n t he present study for the radiat ing, inviscid

layer i s a numerical method of a time-asymptotic technique.

Pr ior studies have used the time-asymptotic technique for t he solu-

t i on of an inviscid flow f i e l d ( see , fo r example, re fs . 3 , 6 , 52 and 53).

The time-asymptotic technique re ta ins the time dependent terms i n the

governing equations of t he flow f i e l d and permits t he time t o increase

u n t i l steady-state conditions a r e reached. By re ta ining the time de-

pendent terms, the s e t of governing equations i s hyperbolic and the same

computational method can be used i n both t h e subsonic and supersonic

regions of the flow along t h e body. The development of the technique is

due t o t h e s tudies by Von Neumann and Richtmyer ( r e f . 541, by Lax ( r e f .

55), and by Lax and Wendroff ( re f . 56). The present method for t he solu-

t i on of t he radiat ing, inviscid flow f i e l d i s based on t h e s tudies Of

Moretti and Abbett ( r e f . 6 ) , C a l l i s ( r e f . 3 ) , and MacCormack ( r e f s . 57

and 58).

Moretti and Abbett ( r e f . 6) applied the time-asymptotic technique

t o flow around blunt bodies without rad ia t ive t ransport . The advantages

of t he method over other methods a r e discussed b r i e f ly by Moretti and

Abbett ( r e f . 6) and more f u l l y by Moretti and Abbett ( r e f . 59). A

comparison of t h e i r r e s u l t s with t he data presented by Perry and Pasiuk

showed the method t o be a s accurate i n matching t h e experimental data as

any other method. Also, it was shown t h a t t h e shock wave can be t r ea t ed

as a discontinuity and thereby reduce the number of mesh points required

i n an accurate nmer ica l solution.

C a l l i s ( r e f . 7 ) , applied the ttme-asymptotic technique t o stagnation-

point solutions and Ca l l i s ( r e f . 3 ) , applied the technique for flow about

long blunt bodies. The solutions were f o r radiat ing, inviscid flow and

showed the time-asymptotic technique t o be va l id f o r radiat ing flow

f ie lds . C a l l i s s t resses t he need f o r t r e a t i n g t h e shock wave a s a dis-

continuity. In addit ion t o reducing the number of mesh points, t he

property prof i les a r e more accurately defined f o r t h e rad ia t ive trans-

port.

19

I n t h e s t u d i e s by C a l l i s , and by Moret t i and Abbett, a Taylor s e r i e s

expansion wi th the f i r s t t h r e e terms re t a ined was used f o r the advance-

ment of t h e flow f i e l d v a r i a b l e s i n time. Thus, t h e so lu t ions a r e of

second-order accuracy i n t ime and r e q u i r e f i r s t - o r d e r and second-order

time d e r i v a t i v e s o f t h e flow f i e l d va r i ab les . A method of second-order

accuracy has been developed by MacCormack ( r e f s . 57 and 58) which re-

qu i re s only t h e f i r s t - o r d e r t ime d e r i v a t i v e s of t h e flow f i e l d v a r i a b l e s .

Bas ica l ly , t h e method involves a two-step process f o r t h e advancement i n

time with t h e s p a t i a l d e r i v a t i v e s of t h e flow f i e l d v a r i a b l e s taken i n

a l t e r n a t e d i r e c t i o n s . The g e r e r a l method of MecCormack i s used i n t h e

present s tudy and i s described f u r t h e r i n t h e sec t ion on Analysis.

Anderson ( r e f . 60 ) compared t,he two methods of time advancement i n a

time-asymptotic technique f o r t h e s t eady-s t a t e , nonequilibrium, flow i n

a nozzle and s t a t e s t h e method developed by MacCormack t o r equ i re fewer

computations and 30 percent l e s s computer t ime f o r i d e n t i c a l r e s u l t s .

The s t u d i e s by Moret t i snd Abbett, by C a l l i s , and by MacCormack

have shown t h e numerical methods f o r t h e time-asymptotic technique t o

be convergent, s t a b l e , and i n s e n s i t i v e t o i n i t i a l guesses. A numerical

method of second-order accuracy f o r t h e time-asymptotic technique and

a discontinuous shock wave i s used i n t h e present s tudy f o r t h e r a d i a t i n g ,

i n v i s c i d flow f i e l d so lu t ion .

Thermal Analyses f o r Venusian Entry

The fully-coupled s o l u t i o n f o r t h e r a d i a t i n g flow f i e l d wi th abla-

t i o n product e f f e c t s has been inves t iga ted i n s e v e r a l s t u d i e s f o r Ear th

reentry ( re fs . 19, 22, 28, 29, 43, and 44) and Wilson ( r e f . 44) investi-

gated a solution for Jupi te r entry. These solutions were for t he stag-

nation region of the body. A t present, there are no published r e su l t s

f o r a fully-coupled solution with ablation product e f fec t s at e i t he r t he

stagnation or downstream regions for Venusian entry. A good review of

the present s t a t e of technology of the- analyses and ablation analyses

for Venusian entry is given by Walberg and Sullivan ( r e f . 10) .

The primary emphasis a t the present time i s for unmanned exploration

of Venus. A description of t h e exploration and the s c i en t i f i c experi-

ments is presented i n references 61 t o 64. Additional information on

t r a j ec to r i e s , body shapes, and body s izes for unmanned en t r ies i s given

i n references 65 t o 69. The entry veloci t ies range from 10 t o 1 5 h / s

with t he ve loc i t ies for peak radiat ive heating being from 7 t o 1 3 h /s .

The range of stagnation-point pressure on the body i s 1 t o 10 atmospheres.

The exploration of Venus is a multi-probe, entry mission with one large

probe and three small probes. The nominal body for t he probes i s a

spherically capped, conical shape with a .34 meter nose radius and a 60

degree half angle for t he la rge probe and a .14 meter nose radius and a

45 degree half angle f o r t he s m a l l probes.

Models of the Venusian atmosphere are given i n references 62, 63,

and 70. The data for t h e models are primarily from the f l i gh t s of

Mariner V and Venera 4. Carbon dioxide is the major gas component for

t h e Venusian atmosphere with up t o 1 0 percent by volume of nitrogen.

Reviews of analybical and experimental studies of t he rad ia t ive

heating f o r Venusian entry and for gas compositions similar t o Venusian

atmosphere a r e presented i n references 10 , 25, 39, and 71. Therefore, a

complete review of t he l i t e r a t u r e w i l l not be given. The r e s u l t s of a

recent experimental study i n a shock tube f o r the convective and radia-

t i v e heating i n a i r and i n a .90 CO - .10 N2 mixture (mole f rac t ion) 2

a r e presented by Livingston and Williard ( r e f . 72). A review of t he

ex is t ing s tudies shows there i s a difference i n the rad ia t ive t ransport

betveen a i r and C02/N2mixtures. The C02/N2 mixtures begin t o rad ia te

at a lower temperature than a i r because of molecular band t r ans i t i ons ,

primarily CN and CO. A t ve loc i t ies l e s s than 10 km/s t h e rad ia t ive

heating can be as much a s a factor of 10 greater than a i r . However, a t

ve loc i t ies above 10 km/s the rad ia t ive heating i s comparable i n t he two

types of gases because of t he domination of the atomic processes present.

A detai led study of radiat ive heating i n C02/N2 mixtures i s pre-

sented i n the t hes i s by Boughner ( r e f . 25). However, atomic l i n e

t r ans i t i ons were neglected and only f l i g h t ve loc i t ies from 8.5 t o 10.5

km/s were considered. Most of t he study was for a mixture of 50 percent

C02 and 50 percent N (mole f rac t ion) and the r e su l t s showed the emission 2

spectrum t o be dominated by the C0(4+) band system and the red and v io le t

bands of CN.

Deacon and Rumpel ( re f . 71) present r e su l t s for the radiat ive

heating of a .80 CO - .20 N2 mixture (by mole f rac t ion) a t f l i g h t 2

ve loc i t ies from 5 t o 16.5 h / s . The important radiators were found t o

be C0(4+) band system, continuum from carbon, and atomic t rans i t ions of

neutral carbon. This study was f o r a constant property layer . Deacon

and Rumpel s t a t e s tha t t o correct for a nonadiabatic layer the corrections

from a i r s tud ies would have t o be applied due t o t h e l ack of d a t a f o r

CO /N mimures. 2 2

Spiegel &. a. ( r e f . 69), present r e s u l t s f o r t h e thermal p ro tec t ion

of a spacecraft during a Venusian ent ry . This inves t iga t ion was f o r a

completely uncoupled solut ion. I n t h e study it was assumed: t h e shock

l a y e r i s isothermal, no coupling between r a d i a t i v e and convective heating,

no rad ia t ion absorption by ab la t ion products , and no rad ia t ion cooling.

The study was f o r a spher ica l ly capped, conical body with a .15 rc nose

rad ius and a 60 degree cone ha l f angle. The heat-shield mate r i a l was

high-density phenolic nylon. The r e s u l t s show t h e r a d i a t i v e heating a t

a locat ion on t h e cone f lank t o be g rea te r by a f a c t o r of two than t h e

stagnation-point value. Also, t h e ab la t ion r a t e s of t h e heat-shield

mate r i a l a r e comparable f o r t h e cone f lank and the s tagnat ion point .

These r e s u l t s need t o be v e r i f i e d by a fully-coupled solut ion.

A recen t , but unpublished, ana lys i s has been conducted by t h e

present author and R. A . Falanga f o r t h e fully-coupled, r a d i a t i n g , flow

f i e l d solut ion with ab la t ion products in jec t ion a t t h e s tagnat ion point

of a blunt body f o r Venusian entry.* The method of solut ion w a s similar

t o t h e present method except t h e r a d i a t i n g , inv i sc id , flow f i e l d solut ion

w a s by t h e method of Falanga and Sull ivan ( r e f . 73) and can only be used

i n t h e subsonic region of flow about a b lunt body. The r e s u l t s of t h i s

recent study were t h a t t h e boundary l ayer with ab la t ion products in jec-

t i o n w i l l reduce t h e r a d i a t i v e heat ing t o t h e body by 35 percent a t

"Publication pending i n Journal of Spacecraft and Rocket.

velocities of 9.86 and 11.18 km/s; but, the reduction was only 14 percent

at a velocity of 8.11 km/s. The injection rates were for the steady-

state ablation of a high-density, phenolic-nylon, material.

24

METHOD OF ANALYSIS

The r a d i a t i n g flow f i e l d about t h e ent ry body i s separated i n t o an

outer l ayer where t h e inv i sc id flow equations a r e appl icable and an inner

viscous layer where t h e boundary l ayer equations a r e applicable. The

divergence of t h e r a d i a t i v e f l u x i s included i n t h e energy equation f o r

the solut ion of each l ayer and t h e boundary l ayer i s coupled t o t h e

i n v i s c i d shock l a y e r by t h e r a d i a t i v e t r anspor t through both l a y e r s and

by t h e boundary l ayer displacement thickness. This coupling of t h e

boundary l ayer solut ion t o t h e inv i sc id shock l ayer solut ion i s i l l u s -

t r a t e d by t h e temperature p r o f i l e shown i n Figure 1. The inv i sc id flow

f i e l d i s displaced from t h e w a l l by theboundary-layer displacement

thickness and t h e boundary-layer p r o f i l e s a re used out t o t h e point

where t h e boundary-layer edge values and t h e i r de r iva t ives equal t h e

inv i sc id l ayer values. For t h e boundary-layer so lu t ion , t h e radia-

t i v e t ranspor t i s ca lcula ted from t h e mated p r o f i l e s f o r t h e e n t i r e

l ayer . The t o t a l thickness of the flow f i e l d now becomes

where 6 i s the thickness of t h e inv i sc id shock l ayer (shock stand-off

d i s t ance) and 6* i s t h e boundary l ayer displacement thickness ca lcu la ted

The f i r s t term on t h e RHS of t h e expression is the usual term for t h e

displacement thickness of a compressible boundary l a y e r without m a s s

i n j e c t i o n and i s due t o t h e re ta rda t ion of t h e flow i n t h e boundary

l a y e r . The second term is t h e increase i n displacement thickness due t o

mass i n j e c t i o n at t h e wal l f o r around t h e body.

The methods of so lu t ion of the inv i sc id flow f i e l d , the boundary

l a y e r , and the r a d i a t i v e t r anspor t a re presented i n sub-sections of t h e

Method of Analysis. The so lu t ion f o r the rad ia t ing , inv i sc id flow f i e l d

about a blunt body was developed i n the present study and t h e develop-

ment of t h e flow f i e l d equations and t h e numerical method i s described

ir. d e t a i l . A second-order, time-asymptotic technique i s used f o r t h e

sulut ion. The solut ion of t h e inner viscous l ayer i s f o r a nonsimilar,

multicomponent boundary l a y e r with mass i n j e c t i o n , unequal d i f fus ion

c o e f f i c i e n t s , and a r b i t r a r y chemical species. An e x i s t i n g computer

program (BLIMP) i s used f o r t h e boundary l ayer so lu t ion with rnodifica-

t i o n s made i n t h e present study f o r coupling the r a d i a t i v e t r anspor t

and mating the solut ion t o t h e inv i sc id flow f i e l d . The r a d i a t i v e

t r anspor t so lut ion i s f o r a nongray gas with molecular band, continuum,

and l i n e t r a n s i t i o n s . An e x i s t i n g computer program (RAD/EQUIL) i s used

f o r the solut ion. In addi t ion t o using RAD/EQUIL f o r coupling t h e radi-

a t i v e t r anspor t through t h e e n t i r e l a y e r , an extensive modification of

t h e program was made i n t h e present study so t h a t it could be used e f -

f e c t i v e l y as a subroutine i n t h e developed computer program f o r t h e

r a d i a t i v e , inv i sc id flow f i e l d .

The rad ia t ing , inv i sc id flow f i e l d solut ion is calcula ted only once

f o r t h e f u l l y coupled so lu t ion ; b u t , an i t e r a t i o n i s required t o couple

t h e boundary l a y e r so lu t ion t o t h e i n v i s c i d flow so lu t ion because of

inc lus ion of t h e r a d i a t i v e heat f lux i n t h e boundary l ayer and loca t ing

the boundary layer edge i n the inviscid flow f i e ld . The boundary layer

solution i s assumed not t o affect the inviscid layer solution through

higher order e f fec t s such as recalculating the inviscid solution for an

"effective" body composed of the or ig ina l body plus the displacement

thickness. Also, the radiat ive heat f lux from the boundary layer toward

the inviscid layer i s small compared t o the f lux from the inviscid layer

toward the boundary layer. Thus, t he inviscid layer solution should be

only s l i gh t ly affected by the boundary layer solution. The boundary

conditions for t he boundary layer edge are the appropriate values and

t h e i r derivatives a t t he match point i n the inviscid flow solution and

a re not the values at the wall condition of the inviscid flow solution.

The calculation procedure f o r the fully-coupled solution i s shown

i n Figure 2. The steps i n the procedure a re :

1. An inviscid , radiat ing solution i s calculated from t ra jec tory data.

2. A boundary layer solution with uncoupled radiat ive transport i s

calculated using the inviscid dl values fo r the boundary layer

edge conditions. For steady-state ablation, the radiat ive heating

t o t he w a l l i s taken as t he inviscid w a l l value.

3. The prof i les from the inviscid solution and the boundary layer

solution are mated and the radiat ive transport i s calculated for the

en t i r e layer.

4. A boundary layer solution with coupled radiat ive transport i s calcu-

la ted using the values a t t he match point of the two layers for the

boundary layer edge values.

5. Steps 3 and 4 are repeated u n t i l t he radiat ive transport through the

layer , the match point condition, and the ablation r a t e s ( for steady-

s t a t e ablation ) converge.

The inviscid flow f i e l d and the boundary layer solutions are for the

flow around a body and the coupling of the two layers and the conver-

gence of the f i na l solution is applied a t each locat ion around the body.

The solutions of the rad ia t ing inviscid flow, the boundary layer ,

and the radiat ive t ransport fo r t he t o t a l layer are calculated by sep-

arate computer programs which are coupled through the i t e r a t i on procedure.

Approximately nine hours of t o t a l computer time are required for a f u l l y

coupled solution about a body and the computer storage requirements for

the radiat ive transport program, the radiating inviscid flow program,

and the boundary layer program a re 56000 1150008, and 256000~, 8 '

respectively. No attempt has been made t o combine the programs i n t o a

single computer program because of the large storage requirements and the

large computat ional time.

The e f fec t of t he boundary-layer displacement thickness on the in-

viscid flow solution and subsequent effect on the boundary layer solution

could be accounted for by repeating the s e r i e s of calculat ions as required

u n t i l convergence. The computational time for such a procedure would be

excessive. Also, the shape of a body i s changing during a high velocity

entry due t o the ablation of the heatshield. Accounting for these shape

changes i n a solution would require an extensive se r i e s of i t e ra t ions .

In t he present study, the body shape i s the shape pr ior t o entry.

The r a t e and elemental mass f ract ions f o r t he mass inject ion of the

ablation products i n t o t he boundary layer can be e i t he r prespecified or

calculated during a solution for steady-state ablat ion of a heatshield.

The equations f o r the wall boundary conditions i n t he boundary layer

solution includes t he necessary mass and energy balances for steady-state

ablation. Steady-state ablation i s a good representation of the ablation

phenomena a t entry conditions where s ignif icant radiat ive heating i s pre-

sent and i s be t t e r than a specif icat ion of t he mass inject ion ra te .

Stesdy-state ablat ion i s used for t he representation of t he ablation of

the heatshield f o r t h e Venusian entry solutions.

Radiating. Inviscid Flow Field Solution

A time-asymptotic technique i s used for the solution of the radia-

t i n g inviscid flow f i e l d ; therefore, the analysis is based on the equa-

t i ons of conservation of mass, momentum, and energy f o r unsteady flow.

The conditions f o r which the present analysis i s carr ied out are t ha t

the flow is axisymmetric, inviscid, and nonconducting; the gas i s i n

l oca l thermodynamic and chemical equilibrium; the tangent s lab approxi-

mation is val id for t he radiat ive t ransport ; and, t he shock wave is a

d i sc re te surface.

The conservation equations expressed i n vector notation are

Mass:

Momentum :

Energy:

ah + pv-vh -+ + v-q *R - & - ; ; . v p = O at a t ( 5 )

Equations 3 t h r u 5 a r e f o r a genera l , or thogonal , c u r v i l i n e a r coordinate

system. The coordina te system f o r high speed e n t r i e s can be e i t h e r body

o r i en ted ( d i r e c t method) o r shock o r i e n t a t e d ( inverse method). The body-

orientated, coordinate system as shown i n Figure 3 i s used i n t h e present

ana lys i s .

Equations 3 t h r u 5 a r e transformed t o t h e p r e s e n t , body-orientated,

??ordina te system by us ing r e l a t i o n s h i p s f o r vec tor opera t ions as given

in reference 74. The necessary metr ic c o e f f i c i e n t s , g i , f o r the coordi-

nate t ransformation a r e determinec? from t h e length of a d i f f e r e n t i a l

l i n e element and t h e genera l expression i s

2 2 2 (an.) = g, (dxl) + gP2(dx2) 2 2 (6)

The expression f o r t h e present s , y , @ coordinate system i s

= ~ ~ ( d s ) ~ + (dy)2 + r2(ci@)2 ( 7 )

Comparing t h i s expression t o t h e genera l expression, t h e requi red metr ic

c o e f f i c i e n t s a r e

where

r = y cos 8 +

A = l + I c y

The angle 0 i s a function of t he coordinate s and the magnitude of the

angle and the functional r e l a t i on t o s i s known from the specification

of the body shape.

By knowing the metric coeff ic ients and using the relationships given

i n reference 74, the vector operations i n equations 3 thru 5 can be car-

r i e d out and the equations determined for the present coordinate system.

Applying the axisymnetric conditions t ha t the velocity and a l l gradients

i n t he @ direct ion are zero, the conservation equations become

Mass:

s-momentum:

y-momentum:

Energy :

(13)

The bracketed term of equation 1 3 i s the divergence of t he radiat ive

f lux with t he application of t he tangent s lab approximation. In the

tangent s lab approximation, t he derivative of the radiat ive flux i n t he

y direct ion i s considered i n the solution but the der ivat ive i n the s

direction is neglected. The parameters in this section of the report

are non-dimensionalized by

r ' r = - Rn'

where the subscript refers to the free stream value and the prime to

dimensional quantities.

The value of y at the shock wave will vary with the s location

around the body and will also vary with time because the shock wave

is moving during the solution until a steady-state solution (final

solution) is obtained. It is convenient in the solution to fix the

location of the shock wave and this is accomplished by the transformation

where

The quantity 6 is the shock stand-off distance and the transformation of

equations 15 gives Y=O for y=O and Y=l for y=6. The inviscid flow field

is now a rectangular coordinate system of length one for the Y-coordinate

and a length for the S-coordinate as specified by the body shape. The

value of S is zero at the centerline of the body. The differential

relationships t o transform equations 1 0 thru 13 from the s, y, t

coordinate system t o t h e S, Y , T system are

The quantity As i s the value of A evaluated a t y equals 6. Hereafter,

subscript notation is used t o denote p e r t i d different ia t ion with

respect t o the variables S, Y, and T.

Applying the transformation of equations 16, the conservation

equations 1 0 thru 13 become

Mass :

where

Energy:

v Y A = - - - 6 - - 6 6~ ii is tan 6

B = g ~ * tan 6

Kuv D = - X

Ku 2

E = - a

h = 1 + KGY

hG = 1 + KG

The mass equation, equation 17, i a now changed from density as the de-

pendent var iable t o pressure as the dependent variable. This i s accom-

plished by using the s t a t e equation i n the fUnctional form

h = ~(P,P)

t o obtain the d i f f e r en t i a l re la t ions

Using the d i f fe ren t ia l re la t ions of equations 29 and combining equations

17 and 20, a new equation for the conservation of mass i s obtained i n

t he form

Mass:

where

The values for the thermodynamic variables of p, P1, and P a re 3

determined by the conditions of chemical equilibrium f o r t he gas with

the specification of the elemental mass f rac t ions of the gas and the

two thermodynamic s t a t e variables of pressure and enthalpy. The depen-

dent variable i n t he mass equation was changed from density t o pressure

as the dependent variable because density cannot be used as one of t he

specified thermodynamic s t a t e variables i n t he chemical equilibrium

program used i n t he present study. Any two of t he equations 17, 20, o r

30 can be used i n the inviscid flow solution depending on the method

used for the thermodynamic s t a t e .

The basic equations for the solution of t he radiat ing, inviscid

flow f i e l d are equations 18 thru 28 and equations 30 thm 32. The terms

pu s in 0 pv cos 0 cos 8 r I r

which appear i n equations 23 and 26 a re indeterminant at the center l ine ,

S=O, and are evaluated by the L'Hospital rule.

The solution f o r t he location of t he shock wave and the flow f i e l d

variables along the shock wave (Y-1) i s by a method using a quasi, one-

dimensional, unsteady charac te r i s t ic solution and the Rankine-Rugoniot

equations for a moving shock. This method i s discussed by Moretti and

Abbett (ref. 6) and by Callis (ref. 7). A derivation of the equations

for the characteristic solution is presented in Appendix A. The equation

for a right running characteristic is

and the compatibility equation is

where

The set of derived equations for the radiating, inviscld flow

field about a blunt axisymmetric body is solved by the numerical methods

presented below. In addition to the complete body solution, a se&kate

solution for only the stagnation point of a blunt body was obtained

in the present study and the equations are presented in Appendix B.

The concept of the time-asymptotic technique is to assume initial

values for the flow field variables and then to increment the variables

in time until the variables are invarient with time. Thus, the evolution

of the flow in time is followed until the flow obtains a steady-state

condition. The numerical method used in the present study to obtain

a solution is based on the two-step method of MacCormack and a description

of the numerical analysis for the method is given in references 57 and

58.

A t time T the value of a var iable is known a t a l l locations i n" the

" flow f i e ld . The advancement of a var iable from time T t o T+B (one

i t e r a t i o n ) i s given by

~ ( S , Y , T + A Q = ~ ( s , Y , T ) + FAT where

The symbol f represents t he var iable p , u, v , or h and the superscripts

'3 1 and 2 re fe r t o t he first and second s tep i n the two-step process. The

time derivative, fT, of the variable i s calculated from the unsteady

equations (equations 18, 19, 20, o r 30) with f i n i t e differences used

for the spa t i a l derivatives. The calculation of fr and the advancement

from T t o T+AT is a two-step process as follows:

1. Compute f k a t time T from the unsteady equations using forward

differences for t he s p a t i a l derivatives. Advance f by

2 2. Compute fT a t n + l from the unsteady equations using the variables

n+l and backward differences fo r t he spa t i a l derivatives. Then,

ave compute the average value of t he time derivative, fT , by equation

38 and advance the variable, f , t o time T+AT by equation 37.

Step 1 is applied t o each var iable p , u , v and h a t all S, Y locations

,before proceeding t o s tep 2.

A modification t o the basic method of MacCormack for t h e s p a t i a l

derivatives is used i n the present study. In t he basic method, two-

point, forward f i n i t e differences a r e used f o r t he first s t ep and two-

point, backward differences are used f o r t h e second s t ep along all

coordinates. In t h e present method, three-point, c en t r a l f i n i t e dif fer-

ences are used for t he derivatives with respect t o t h e S-coordinate for

both s teps i n the two-step process. Three-point, forward differences on

the first s t ep and three-point, cen t ra l differences on the second s tep

ere used for the derivatives of h and u with respect t o the Y-coordinate.

The derivatives of p and v with respect t o t he Yzcoordinate a r e the basic

method of two-point forward on t h e f i r s t s tep and two-point backward on

the second step. A Taylor se r ies expansion i s used f o r t he derivation

of t he f i n i t e difference equations f o r unequal spacing of t he nodal

points along each coordinate. The same equations are also val id when

equally spaced points a r e specified.

The above numerical method i s used f o r al l nodal points except the

nodal points along the shock wave (Ypl). The flow f i e l d variables a t

t h e nodal points along the shock wave and the location of the shock

wave are calculated by a numerical method using the equations from the

quasi, one-dimensional, character is t ics analysis and the Rankine-Hugoniot

equations for a mving shock wave. The numerical method is i l l u s t r a t e d

by the diagram i n Figure 4. A t time T a l l the quant i t i es a r e known

and the properties a t D. 6, and 6T a re desired a t time T+AT. The

solution procedure i s a s follows:

1. Assume a 6T a t time T+AT and use trapezoidal integration t o

determine 6 by

G(T+AT) = 6(T) + + 6T(T+AT) 2 I AT

The angle B is calculated by

2. The properties of p , u , v, and h at point D a r e cdcu la t ed from the

Rankine-Hugoniot equations f o r a moving shock with t he known free-

stream properties, t h e assumed 6 , and calculated 0 . T

3. Point B i s located from the character is t ics equation (eq. 33) by

where averaged properites of points B and D are used for the inte-

gration. The properties a t point B are averaged properties of points

A and C.

4. A value of v a t point D i s computed from the integrat ion of the

compatability equation (eq. 34) from point B t o point D using

averaged properties

vD = vB + (.* + i, AT -[%) (pD-pBj (43) 0% ave aye

The value for pD i s the value calculated by the Rankine-Hugoniot

equations i n s t ep 2. t 5. The computed value of vD Prom the compatibility equation is compared

with t he value calculated from the Rankine-Hugoniot equations i n

s tep 2. I f the values for vD do not match, then a new value for

6T(T+AT) i s assumed and s teps 1-5 a r e cycled u n t i l convergence.

The numerical method i s applied a t each time s tep for each S-coordinate

location beginning with the S=O location. The method is converged at

each S location before proceeding to the next location around the body.

A Newton-Raphson method is used for successive guesses for 6T after the

first two guesses.

The value of the shock stand-off distance, 6, at each S location is

checked by a mass balance. For the mass balance, the mass flow through

a plane in the flow field at the particular S location is equal to the

mass flow from the free stream through the shock wave up to the particular

S location. The expression for 6 from the mass balance is

I~b ' = 2(SF - rb cos 0 )

where

To insure a mass balance for the converged solution, the shock stand-off

distance at each S location is adjusted if necessary to the value from

equation 44 as the solution approaches convergence.

The size of the time step for the advancement at each iteration

is calculated by the use of the characteristic equation (eq. 33) in

.the form

.This stability criterion is based on the Courant-Friedricks-Lewy

condition and insures that the net &ope is less than the characteristics

islope for hyperbolic equations (ref. 75, chap. 9). Equation 46 is a

one-dimensional equation being used for a two-dimensional problem;

however, the AY i s smaller than AS i n the problem and no s t a b i l i t y

problem was encountered. A time s tep i s calculated a t each nodal point

i n the flow f i e l d and the minimum value for the nodal points along a

ray a t a S-coordinate i s used f o r all the points along tha t par t icu la r

S-coordinate. There i s a l a rge var ia t ion i n time s teps around the body

and using a varying time s tep around the body requires l e s s i t e r a t i ons

for convergnece than using a minimum time s tep for t he en t i r e flow

f i e ld . Having a d i f fe ren t time plane through the flow f i e l d did not

affect t he converged steady-state solution.

The radiat ive heat f lux, the thermodynamic proper t ies , and the

solution of the Rankine-Hugoniot equations are calculated by using the

computer program of Nicolet ( r e f s . 9 and 32). A modified version of

t h i s program i s used as subroutines t o the computer program developed

for the radiat ing, inviscid flow f i e ld . A discussion of the method of

calculating the rad ia t ive heat f lux and the modifications a r e presented

i n the sub-section of Radiative Transport. Large computational times

are required for the calculation of t he radiat ive heat f lux and the

thermodynamic properties; however, these parameters do not have t o be

updated a t each i t e r a t i on . A study was made and it was determined tha t

the thermodynamic properties and rad ia t ive heat f l ux can be calculated

every 50 t o 100 i t e r a t ions without affect ing the converged solution but

great ly reducing the computational time. For the i t e r a t i ons when the

thermodynamic subroutine i s not used, t he density i s calculated by

with the thermodynamic properties P1 and P held constant. The properties 3

P and P a r e calculated by 1 3

where T i s temperature, a i s sonic velocity, M i s molecular weight, and

c is specif ic heat. The non-dimensionalization of the properties i s P

given by

where the subscript s re fe rs t o the conditions behind a normal shock.

The expressions are derived from equations 31 and 32 by using thermo-

dynamic re la t ions . The necessary properties for calculating P1 and P 3

by equations 48 and 49 are available from the thermodynamic subroutine.

Pressure and enthalpy which are calculated by the unsteady equations are

used as t he two specif ied, thermodynamic s t a t e variables for the thermo-

dynamic subroutine and the rad ia t ive heat f lux subroutine. The solution

of the Rankine-Hugoniot equations for a moving shock wave is calculated

by the usual Rankine-Hugoniot equations with the ve loc i t ies on e i t he r

s ide of t he shock made r e l a t i ve t o a fixed shock location. As mentioned,

a solution for t he Rankine-Hugoniot equations i s included i n the program

of Nicolet . The i n i t i a l values t o begin a solution a re calculated by assuming

the shock wave is s ta t ionary and the shock stand-off distances around

t he body are equal t o the shock stand-off distance a t the center l ine

(S=O). The value a t the center l ine i s calculated by a correlation from

reference 76 of 6 = .78/p where p i s the nondimensional density S s

behind a normal shock and 6 i s the nondimensional stand-off distance. The

flow f i e l d variables for t he nodal points along the shock wave are calcu-

l a t ed from the Rankine-Hugoniot equations. The var iables for the w a l l nodal

points are calculated by assuming an isentropic expansion along t h e body

with a Newtonian pressure d i s t r ibu t ion . Values for intermediate points

i n t he flow f i e l d are calculated by l i nea r interpolation of the shock

and body values.

The convergence of a solution is determined by checking the change

i n enthalpy between i t e r a t ions . The percentage change i n enthalpy

between two successive i t e r a t i ons must be l e s s than a specified s m a l l

value and t h i s c r i t e r ion must be net for a specified number of i t e ra t ions .

The enthalpy a t the body, shock, and midpoint along a ray a t each S

locat ion i s checked.

A computer program was developed for t he numerical solution of the

radiat ing, inviscid flow f i e ld . The program is writ ten i n FORTRAN I V

language for CDC 6000 se r i e s computers. A l i s t i n g of the program i s

given i n Appendix C.

Boundary Layer Solution

The boundary layer equations axe for a nonsimilar, multicomponent

boundary layer with mass in jec t ion , unequal diffusion coeff ic ients , and

a rb i t r a ry chemical species. The solution of the equations i s by a

numerical, in tegra l matrix method and there i s an exis t ing computer

program for t he solution cal led BLIMP. Many types of complex, react ive

boundary layer problems can be solved by the program and numerous pub-

l i ca t ions a r e required f o r a complete description of t he program and the

concepts and methods used i n a solution. A brief discussion i s presented.

of t he basic method and of the modifications t o the program made i n the

present study. A l i s t of the s ign i f ican t publications i s given a f t e r

t he discussion of t he basic method.

The present discussion of t he basic method i s from reference 8. The

boundary layer equations f o r an s, y coordinate system of a blunt, axi-

symmetric body (see Figure 3 ) a re

Mass:

s-momentum :

y-momentum :

Energy :

where

Species : aKi i a aEi

pu as + p v - = - ay a~ ( P E ~ - ~ i )

The species' conservation equation is solved in terms of elemental mass

fraction, Ki, rather than the species1 mole fraction by the application

of the Shvab-Zeldovich transformation. This transformation eliminates

the chemical source term for chemical equilibrium and reduces the number

of species' conserpation equations from the number of species to the

number of elements. For a boundary layer with ablation products injec-

tion, the number of species is typically from 20 to 50 whereas the number

of elements is only 2 to 6.

The complexity of using unequal diffusion coefficients for a

multicomponent mixture is reduced by a bifurcation approximation for the

binary diffusion coefficients. The approximation is expressed as

where 5 is the self-diffusion coefficient of a reference specie and

Fi, F are the diffusion factors for species i and j, The Fi of a 'I

specie for a given chemical system can be determined by finding the set

of diffusion factors that gives the minim total system residual error

and a method is presented in reference 77. The bifurcation approximation

permits the diffusive flux of element i to be expressed explicitly in

terms of properties and gradients of element i and of the system as a

whole, but not of other elements. Also, the I(1-1)/2 binary diffusion

coefficients are replaced by I diffusion factors where I is the number

of chemical species.

The diffusional mass flux of element i in the species' conservation

equation (eq. 56) can be expressed as

by use of the bifurcation approximation end the definition of the new

quantities -

The subscript J refers to species and the subscript i refers to elements.

The quantity a is the mass fraction of element i in species j and Z iJ J

is a quantity which for unequal diffusion lies between a mass fraction

and a mole fraction for species j.

C The convective heat flux term q in the energy equation (eq. 54)

is the sum of heat flux due to conduction, diffusion, turbulence, and

dissipation of kinetic energy. Using the bifurcation approximation for

the heat flux due to diffusion, the convective heat flux can be ex-

pressed by

where

The turbulent transport terms in the equations are expressed in

the Boussinesq form of eddy viscosity, eddy diffision, and eddy

conductivity. I f turbulent transport i s considered, t he turbulent

transport properties are included i n the boundary layer equations when

a prespecified Reynolds number i s exceeded. The Reynolds number i s

based on edge properties and momentum thickness. A t r ans i t i on region

between laminar flow t o turbulent flow i s not included.

The boundary conditions a t the boundary layer edge a re t he specif i -

cations of u, H , and zi and t h e i r derivatives with respect t o the y-

coordinate. The w a l l boundary conditions allow for the poss ib i l i ty of

solving numerous types of problems and thus specification of dif ferent

variables. However, all the var ia t ions include zero s l i p a t the w a l l ,

u =0, and involves i n some form the assignment of values for mass in- W

jection r a t e , ( p ~ ) ~ ; w a l l enthalpy o r w a l l temperature, hw or Fw; and

elemental mass f ract ions a t the w a l l , k i ,w'

The equations of the w a l l boundary conditions for energy balance

and elemental mass balance allow a solution for steady-state ablation

of a heatshield. The energy balance i s

where t he subscripts g and c re fe r t o the pyrolysis gas and char for

ablation and qcond i s t he heat conducted in to the ablation material at

t h e w a l l surface. For steady-state ablation

where m i s the t o t a l ablat ion r a t e , m plus m and h0 i s the heat of a g C ' a

formation of t h e undecomposed ablation material. Noting tha t

(PV), = ma

then the surface energy balance f o r steady-state ablation i s

The elemental mass balance i s

and, fo r steady-state ablation

thus, t h e elemental mass balance for steady-state ablatton is

A solution fo r steady-state ablation does not require knowledge of

the in-depth response of the ablation material but only the surface

interact ions between the boundary leyer and the ablation material. From

equations 65 and 68, it i s shown t h a t the additional properties of t h e

elemental mass f ract ions and heat of formation of the undecomposed

'ablation material and the w a l l emissivity are a l l t h a t is required for

a steady-state ablation solution. The solution of the surface balances

gives the basic boundary conditions of injection r a t e , enthalpy, and

elemental mass f ract ions a t the wall ' . -

An 5, q coordinate system is used for the actual solution of the

boundary-layer equations by the transformations of

and a streem function of

The transformed equations will not be presented.

The numerical procedure for the solution of the boundary layer

equations is an integral matrix method. The boundary layer is divided

into strips in the normal direction (n coordinate) and the conservation

equations are integrated across the strips. Across a strip, the primary

dependent variables of velocity, total enthalpy, and elemental mass

fractions and their derivatives with respect to n are related by Taylor

series expansions with cubic or quadratic fits. Implicit, finite dif-

ference expressions are used for the derivatives in the streamwise di-

rection ( E coordinate). The set of simultaneous equations wbich

includes the Taylor series, the conservation equations, and the boundary

conditions are solved by general Newton-Raphson iteration using a matrix

inversion technique involving successive matrix reductions.

Many aapects of the BLIMP program have been left out in the above

discussion. Further information for the boundary layer method and the

BLIMP computer program is presented i n t he following references. The

solution f o r t h e boundary layer equations i s given i n references 8 , 18,

78, and 79. The solution i s for an a rb i t r a ry , multicmponent gas mix-

t u r e and the method for t he thermodynamic properties f o r chemical equi-

l ibrium i s given i n reference 80 and the method for t he mixture transport

properties i s given i n reference 81. A method for considering a boundary

layer for a blunt body solution with entropy layer i s presented i n

reference 82. Information on the turbulent transport model i s given i n

references 79, 82, and 83. User's manuals for the BLIMP program are

references 84 t o 86. The description of a companion analysis for in-

depth response of an ablative heatshield i s given i n reference 87.

Modifications t o the BLIMP program were necessary i n the present

study t o c l a r i f y t he coupling of radiat ive heating and t o make the

boundary layer edge conditions compatible with the inviscid flow f i e l d .

The radiat ive heating was included i n t he fo rmla t ion of the energy

equation and i n t he coding of t he BLIMP program; however, the values

f o r the radiat ive heating were s e t t o zero i n the program and no pro-

C R vis ion was made t o input the values. The term (q + q ) was considered

R a s being only convective heating, since q was s e t t o zero, and was used

i n expressions i n t he program which are only applicable for convective

R heating. When the radiat ive heating, q , is included i n the term

R (qC+ q ) these expressions are invalid. There was a provision for

inputing radiat ive heating t o the w a l l , unattenuated by the boundary

layer , fo r use i n t he w a l l energy balance for steady-state ablation.

The convective heating for t he energy balance was represented by the

C R term (q + q ) a t t he w a l l and i f t he radiat ive heating through the layer

i s included i n t he term by qR then the radiat ive heating a t t he w a l l i s

accounted for twioeand the energy balance i s invalid. The BLIMP program

was modified by re-defining cer ta in expressions t o c l a r i f y the difference

between convective and rad ia t ive heating and t o properly input t he radi-

a t i ve heating a t the nodal locations.

The derivatives for t he edge boundary conditions are s e t i n t he

BLIMP program a s t he usual boundary layer edge condition of

These boundary conditions consider t h a t u, H, and zi a re invariant with

y i n the inviscid layer. The edge conditions of zero for the veloci ty

and t o t a l enthalpy gradients may not be of sufficient accuracy and may

give incorrect r e s u l t s f o r t he boundary layer solution because of non-

adiabatic e f fec t s when rad ia t ive transport is present and when an en-

tropy layer i s present i n t he inviscid flow f i e l d of a blunt body.

There are several methods i n BLIMP for considering the effect of an

entropy layer for a blunt body and for these methods the derivatives are

specified by appropriate values and not s e t t o zero. However, the me-

thods did not consider t he e f f ec t on the derivatives for a non-adiabatic

inviscid flow f i e l d from rad ia t ive transport and were not appropriate for

the present study. The BLIMP program was modified i n t he present study

so tha t the derivatives of veloci ty and t o t a l enthalpy with respect t o

the y coordinate can be specified by inputs and correct ly transformed t o

t h e 11 coordinate. Also, a modification was made for specifying the t o t a l

enthalpy at t h e boundary l a y e r edge around t h e body. These modificat ions

al low f o r equating t h e boundary l a y e r edge values with t h e values of t h e

r a d i a t i n g , i n v i s c i d f low f i e l d a t t h e match point i n t h e flow f i e l d .

Radiat ive Transport Solut ion

The so lu t ion of t h e r a d i a t i v e t r anspor t i s f o r a nongray gas with

molecular band, continuum, and atomic l i n e t r a n s i t i o n s . A d e t a i l e d

frequency dependence of t h e absorption c o e f f i c i e n t s i s used i n t h e in-

t e g r a t i o n over t h e r a d i a t i o n spectrum and t h e tangent s l a b approximation

i s used f o r i n t e g r a t i o n over physica l space. An e x i s t i n g computer pro-

gram ( W / E Q U I L ) by Nicolet ( r e f s . 9 and 32) i s used f o r t h e ca lcu la t ion

of t h e r a d i a t i v e t r a n s p o r t . The present sec t ion presents a b r i e f d i s -

cussion of t h e method and t h e modificat ions t o t h e computer program when

used a s subroutines f o r t h e r a d i a t i n g , inv i sc id flow f i e l d so lu t ion .

The tangent s l a b approximation t r e a t s t h e r a d i a t i v e t r anspor t a s a

one-dimensional problem i n t h e d i r e c t i o n normal t o t h e body. Thus, t h e

equations f o r t h e r a d i a t i v e t r anspor t a r e derived f o r a gas confined

between two i n f i n i t e p a r a l l e l boundaries and t h e property gradients of

t h e gas a r e considered a s zero except i n t h e normal d i r e c t i o n between

t h e boundaries. For t h e present problem, t h e boundaries a r e t h e shock

and t h e body and t h e normal d i r e c t i o n i s perpendicular t o t h e body. The

tangent s l a b approximation is appl ied l o c a l l y a t each loca t ion around

t h e body. Note, t h e tangent s l a b approximation i s used only f o r t h e

r a d i a t i v e t r anspor t c a l c u l a t i o n s and not f o r t h e gradients of t h e thermo-

dynamic s t a t e and flow p r o p e r t i e s f o r t h e flow f i e l d so lu t ion .

The r a d i a t i v e heat f l u x across a surface a t a point i n t h e gas l ayer

i s R dq , = I cos y dQ v (71 )

where I i s t h e i n t e n s i t y , y i s t h e angle between a ray and the normal, v and fL i s t h e s o l i d angle. The bas ic equation f o r t h e i n t e n s i t y gradient

along a ray f o r a gas i n l o c a l thermodynamic equil ibrium i s

where p i s t h e l i n e a r absorption coef f i c ien t correc ted f o r induced emis- v

s ion and B i s the Planck function. The so lu t ion of equations 71 and 72 v

a t a point y i n the gas l ayer f o r t h e d i r e c t i o n a l , r a d i a t i v e heat f lux

i s given by Nicolet i n the form

where heat f luxes i n t o t h e s l a b at the boundaries a r e neglected. The

net f l u x at point y i s

The plus sign i n t h e supersc r ip t s r e fe res t o t h e heat f lux i n the direc-

t i o n toward t h e shock and t h e negative s ign f o r t h e f l u x toward t h e body.

Nicolet solves t h e f l u x equation i n a Planck funct ion, By, and

emiss iv i ty , E coordinate system where v '

where t a re t h e dummy values of o p t i c a l depth and E (2) i s the exponen- v n

t ie . i n t e g r a l of order n. For t h e so lu t ion by Nicole t , the exponential

i n t e g r a l of t h i r d order i s approximated by

and t h e expressions f o r emiss iv i ty become

+ - E - 1 - exp [ 2 ( t v - T ~ ) ] v - -

E - 1 - exp [2(TV - t,,)] V

The s p e c t r a l absorption coef f i c ien t fo r a gas mixture i s i n general

where t h e f i r s t term i s t h e continuum contr ibut ion with a summation over

all continuum t r a n s i t i o n s and t h e second term i s t h e l i n e contr ibut ion

with a summation over a l l l i n e t r a n s i t i o n s . The absorption Coefficients

a r e ca lcula ted i n t h e RAD/EQUIL program and t h e t h e o r e t i c a l expressions,

approximations, and experimental data of the various t rans i t ions for the

species are given i n d e t a i l i n t he report of Nicolet ( r e f . 9) . Transi-

t i ons of t he species N , C , H , 0 , H-, C-, N-, 0-, Hg, C2, N2, 02, NO, C O ,

CN, and N+ are considered for t he radiat ive t ransport . 2

In the computer program, t he rad ia t ive heating i s separated in to

heating due t o continuum t r ans i t i ons and l i n e t rans i t ions . The continuum

contribution is calculated by using only the f i r s t term of equation 79

for t he spectra l absorption coeff ic ient and then the l i n e contribution

i s calculated by

R where the t o t a l absorption coefficient i s used for the 4, value. The

molecular t rans i t ions a r e t rea ted as an "equivalent" continuum process

by a bandless model i n which the bands within each band system are

smeared by

- where the frequency increments Av a re selected so tha t Y, varies smoothly

over the frequency spectrum of the band system. The molecular t rans i t ions

are included i n the continuum contribution t o the rad ia t ive f lux.

For t he continuum calculat ions , the frequency spectrum i n terms of

photon energy (hYv) i s divided i n t o 48 frequency nodal points over t h e

range 0 t o 1 5 ev. The frequency nodal points a r e not equally spaced

over t he spectrum but a r e spaced t o resolve the frequency var ia t ion of

t he absorption coeff ic ient especial ly a t photoionization edges. The

55

number of node points, frequency range, and spacing a re inputs t o the

program and the above values were used i n the present study.

The l i n e grouping technique i s used for the atomic l i n e t rans i t ions .

In t h i s technique, t h e l i n e t rans i t ions near a specified frequency value

are grouped together and the radiat ive heating i s given as tha t from the

l i n e group. However, each l i n e within the group i s t reated individually

and can assume unshifted Lorentz o r Doppler l i n e shapes or a combination

of t he two with t he ha l f widths including the Stark, resonance, and

Doppler effects . The frequency gr id for each l i n e i s composed of 1 3

nodal points and the gr id i s dependent upon the character is t ics of the

gas layer as well as the individual l i n e . Twenty l i n e groups with a

t o t a l of 134 l i n e s are used i n t he present study. The l i n e groups a r e

not connected over the frequency spectrum, but are spaced according t o

t he frequency locations of the l i n e t rans i t ions .

The thermodynamic properties and species concentrations required

f o r the calculat ion of t he spectra l absorption coeff ic ients and Planck

functions a r e calculated i n t he computer program by a chemical equi l i -

brium method from the specif icat ion across the spa t i a l grid of t he

enthalpy (or temperature), pressure, and elemental mass f ract ions . Only

the species a s given above a re used for the radiat ive transport calcula-

t ions ; bu t , addit ional species can be included i n t he chemical equi l ibr i -

um calculation. A method for t he solution of the Rankine-Hugoniot equa-

t i o n s for shock waves i s a l so included i n the program. The methods for

the chemical equilibrium solution and the Rankine-Hugoniot solution are

given i n t h e report by Nicolet. The calculation of t he thermodynamic

s t a t e i s consistent throughout t h e fully-coupled solution because the

chemical equilibrium method is the same i n both the RAD/EQUIL and t h e

BLIMP programs.

BriefLy, the solution procedure fo r the radiat ive transport i s t h a t

the chemical equilibrium calculations are made fo r t h e specified spa t i a l

grid. The spectral absorption coefficients, Planck function6 and opt ica l

depths are calculated. Then t h e emissivit ies are calculated and the

transformation t o t h e By, fy coordinates i s made by knowing Bv and Ev

a s a function of y. The heat f lux equations are then integrated. Cubic

integration formulas a re used f o r all integrations except fo r the fre-

quency integration of the continuum flux. These in tegra ls a re evaluated

by l i nea r formulas because of the discontinuity a t photoionization

edges.

The RAD/EQUIL computer program i s used for the calculation of the

radiat ive transport through the t o t a l layer frcm t h e coupling of the

:boundary layer t o t h e inviscid layer. This is s tep 3 i n the calculation

procedure. ALSO, a modified version of the RAD/EQUIL program i s used as a

subroutine i n the computer program for the radiating, inviscid flow

' f i e l d solution. The following coments r e fe r t o the modified version.

A separation between the subroutines for the radiat ive transport calcu-

la t ions and the chemical e q u i l i b r i d calculations was made i n order t h a t

the chemical equilibrium calculations can be performed without calcu-

' l a t i ng the radiat ive transport . The chemical equilibrium subroutine

was modified by the addition of a rcjutine t o calculate t h e necessary

thermodynamic properties for the evaluation of the P1 and P properties 3

;required for t h e inviscid flow solution. The required addition was . ..

taken from t h e chemical equil ibrium vers ion i n t h e BLIMP computer pro-

gram. The method f o r c a l c u l a t i n g thermodynamic de r iva t ives f o r t h e chem-

i c a l equil ibrium vers ion i n BLIMP i s given i n reference 80. Also, t h e

s to rage requirement was reduced by d e l e t i n g opt ions which were not re-

qui red , by reducing t h e maximum number of s p a t i a l nodal po in t s , and by

reducing t h e maximum number of chemical species i n t h e chemical e q u i l i -

brium so lu t ion . The modified ve r s ion of t h e FW/EQUIL program was then

mated t o t h e computer program f o r t h e r a d i a t i n g , i n v i s c i d flow f i e l d

so lu t ion .

RESULTS AND DISCUSSION

A computational method has been developed f o r t h e fully-coupled

s o l u t i o n of nongray, r a d i a t i n g gas flow wi th a b l a t i o n product e f f e c t s

about b l u n t bodies during p lane tary e n t r i e s . Appl ica t ion of t h e devel-

oped method is shown by r e s u l t s f o r Venusian e n t r i e s . There a r e no

published r e s u l t s f o r a fully-coupled s o l u t i o n of t h e r a d i a t i n g flow

about a b l a t i n g , p lane tary en t ry bodies; t he re fo re , the v a l i d i t y of t h e

present method is shown by comparing the r e s u l t s from so lu t ions t o sub-

components of t h e method wi th published r e s u l t s of corresponding

s o l u t i o n s .

The present method f o r t h e s o l u t i o n of an i n v i s c i d flow f i e l d i s

cumpared with e x i s t i n g methods f o r both non-radiating and r a d i a t i n g gas

flows about b lun t bodies during Earth r een t ry . Resul t s from s o l u t i o n s

by the p resen t method f o r the r a d i a t i n g , i n v i s c i d flow about b l u n t

bodies a r e a l s o presented f o r p lane tary e n t r i e s . The present method

f o r a fully-coupled s o l u t i o n with a b l a t i o n product e f f e c t s i s compared

wi th an e x i s t i n g method f o r the s t agna t ion point of b l u n t bodies f o r

Ear th r een t ry . Resu l t s from the present method a r e then presented f o r

t h e fully-coupled s o l u t i o n about b l u n t bodies f o r Venusian e n t r i e s .

Severa l comments a r e given about t h e p resen ta t ion of t h e r e s u l t s .

I n t h e present coordina te system, t h e r a d i a t i v e hea t ing d i r ec t ed toward

t h e shock wave is p o s i t i v e and t h e r a d i a t i v e hea t ing d i r e c t e d toward t h e

body is negat ive . However, t h e r a d i a t i v e hea t ing r a t e s a t t h e wal l a r e

given by p o s i t i v e va lues i n t h e p resen ta t ion of r e s u l t s . A s s t a t e d i n

t h e s e c t i o n of Radia t ive Transport So lu t ion , the r a d i a t i o n con t r ibu t ions

by molecular band t r a n s i t i o n s a r e included i n t h e continuum con t r ibu t ion

t o t h e r a d i a t i v e hea t ing . The t o t a l con t r ibu t ions of molecular band

t r a n s i t i o n s and continuum t r a n s i t i o n s a r e r e f e r r e d t o by continuum

processes. The l i n e grouping technique is used f o r t h e atomic l i n e

t r a n s i t i o n s and t h e s p e c t r a l d i s t r i b u t i o n s f o r the l i n e t r a n s i t i o n s a r e

given by t h e c o n t r i b u t i o n of t h e l i n e group and not by t h e ind iv idua l

l i n e s .

Non-Radiating, Inv i sc id A i r Solu t ion

The p resen t method f o r an i n v i s c i d flow f i e l d has been used t o

ob ta in a s o l u t i o n f o r t h e nonradia t ing flow about a s p h e r i c a l l y capped,

con ica l body during an Earth r een t ry . The r e s u l t s a r e presented i n

F igure 5 and a r e compared with r e s u l t s by t h e method of Inouye 4.

( r e f . 88) . The method of re ference 88 is considered t o be one of t h e

more accura t e methods and provides a good means of eva lua t ing t h e accu-

racy of t h e present method f o r an i n v i s c i d flow f i e l d so lu t ion .

The s o l u t i o n is f o r the free-stream condi t ions and body shape

given i n F igure 5(a) . The r e s u l t s f o r t h e shock shape a t the forward

region of t h e body and t h e l o c a t i o n of t h e sonic l i n e a r e presented i n

F igure 5(b) . Figure 5(c) p re sen t s t h e r e s u l t s f o r t h e shock shape a t

t h e downstream region of t h e body. Because t h e thickness of t h e shock

l a y e r i s much smal le r than t h e dimensions of the body, a b e t t e r means

of comparing t h e shock shape i s t h e shock stand-off d i s t ance and these

r e s u l t s a r e presented i n F igure 5 (d ) . The r e s u l t s f o r t h e shock stand-

off d i s t ance i l l u s t r a t e t h e i n f l e c t i o n po in t s i n the shock shape around

t h e body. The r e s u l t s from t h e p resen t method a r e i n good agreement

with t h e r e s u l t s by t h e method of re ference 88 f o r t h e shock shape,

e s p e c i a l l y cons ider ing t h e i n f l e c t i o n po in t s , and f o r the son ic l i n e

loca t ion .

The r e s u l t s f o r t h e d i s t r i b u t i o n of t a n g e n t i a l v e l o c i t y and pres-

s u r e along the body a r e presented i n Figures 5(e) and 5 ( f ) , r e spec t ive ly .

The r e s u l t s from t h e p resen t method a r e i n good agreement wi th t h e

r e s u l t s by t h e method of r e fe rence 88 even i n t h e region of t h e over-

expansion of t h e flow a t t h e sphere-cone junct ion.

The good agreement between t h e present r e s u l t s and those by t h e

method of re ference 88 i n d i c a t e s t h e present method provides a good

s o l u t i o n f o r t h e i n v i s c i d flow about a b lun t body. Radiat ing flow is

n o t t r e a t e d by t h e method of re ference 88.

Radiat ing, I n v i s c i d Solu t ions About Blunt Bodies

Resul t s a r e presented from s o l u t i o n s of t h e r a d i a t i n g , i n v i s c i d gas

flow about b lun t bodies dur ing p lane tary en t ry . The present r e s u l t s f o r

s o l u t i o n s of e n t r i e s i n t o the E a r t h ' s atmosphere a r e compared wi th

e x i s t i n g methods.

The present method i s compared wi th t h e method of C a l l i s ( r e f . 3)

f o r an Ear th r een t ry and t h e r e s u l t s a r e presented i n F igure 6 . The

free-stream condi t ions and body shape a r e given i n Figure 6 (a ) . The

d i s t r i b u t i o n s of t h e wa l l p re s su re and t h e shock stand-off d i s t ance a r e

presented i n F igures 6(b) and 6 ( c ) , r e spec t ive ly . The two methods a r e

i n good agreement f o r t h e s e parameters. The d i s t r i b u t i o n of w a l l , ra-

d i a t i v e hea t ing r a t e i s presented i n Figure 6(d) . The r e s u l t s by t h e

two methods a r e i n reasonable agreement a t t h e forward region of t h e

body; however, t h e r e is a cons iderable d i f f e rence between t h e methods

a t t h e downstream region of t h e body. The r e s u l t s by C a l l i s a r e g r e a t e r

by a f a c t o r of two than t h e present r e s u l t s a t the downstream region.

An a n a l y s i s was done t o e x p l a i n t h e d i f f e r e n c e between t h e methods.

The thermodynamic p r o f i l e s of temperature and pressure and t h e shock

stand-off d i s t a n c e s from t h e r e s u l t s by C a l l i s were used a s inpu t s t o

t h e r a d i a t i v e t r a n s p o r t program, (RAD/EQUIL), used i n the present

method. The r e s u l t s of t h i s c a l c u l a t i o n a r e compared wi th t h e present

r e s u l t s i n F igure 6 ( e ) . As shown by t h e d a t a , t h e d i s t r i b u t i o n s of w a l l ,

r a d i a t i v e hea t ing r a t e is now i n good agreement a t t h e downstream region.

A s i m i l a r type of a n a l y s i s had t o be used by S u t t l e s ( r e f . 89) i n com-

par ing h i s r e s u l t s wi th C a l l i s f o r l a r g e stand-off d i s t ances a t t h e

s t agna t ion po in t of b l u n t bodies . The r a d i a t i v e hea t ing r a t e s by C a l l i s

were g r e a t e r than those of S u t t l e s f o r l a r g e r stand-off d i s t ances .

Resul t s from t h e p resen t method and t h e method of C a l l i s a r e i n I

good agreement f o r nonradia t ing flow so lu t ions . The d i f f e r e n c e between

t h e methods f o r r a d i a t i n g flow is a t t r i b u t e d t o t h e r a d i a t i o n models.

The r a d i a t i o n model used by C a l l i s ( r e f s . 3 and 7) was developed by

Olstad ( r e f s . 2 and 26) and is based on a "step-model" approximation

f o r t h e frequency dependence of t h e absorpt ion c o e f f i c i e n t . It is

poss ib l e t h a t t h i s r a d i a t i o n model by Olstad t r e a t s t h e r a d i a t i n g gas a s

being too t ransparent .* Thus, more r a d i a t i o n i s t ransmi t ted t o t h e body

and t h i s e f f e c t would be more n o t i c e a b l e t h e g r e a t e r the shock stand-off

d i s t ance .

The present method was used t o c a l c u l a t e the r a d i a t i v e hea t ing r a t e

t o the s t agna t ion po in t of a b l u n t body f o r a range of nose r a d i i and

t h e r e s u l t s a r e compared wi th t h e method of C a l l i s ( r e f . 7) i n Figure 7 .

The same r a d i a t i o n model was used by C a l l i s i n re ferences 3 and 7. The

r e s u l t s by C a l l i s a r e g r e a t e r than t h e present r e s u l t s and t h e d i f f e r e n c e

inc reases with t h e shock stand-off d i s t a n c e (nose r a d i u s ) .

A s o l u t i o n was obta ined f o r t h e same condi t ions a s previous ly given

i n Figure 6 ( a ) , except t h e nose r ad ius was increased t o 2.0 meters . The

d i s t r i b u t i o n of r a d i a t i v e hea t ing r a t e i s presented i n Figure 8. The

r e s u l t s of C a l l i s a r e 50 percent g r e a t e r than the present r d s u l t s a t t h e

forward region of t h e body a s w e l l a s the downstream region.

The s tagnat ion-point hea t ing r a t e s from s o l u t i o n s by t h e present

method and t h e method of C a l l i s a r e compared wi th t h e methods of r e f -

erences 73 and 89 i n Table I f o r two nose r a d i i . The methods of re f -

erences 73 and 89 can only be used f o r t h e flow i n t h e subsonic reg ion

of a b lun t body. For t h e smal le r nose r ad ius , t h e r e s u l t s agree wi th in

15 percent . For t h e l a r g e r nose r ad ius , t h e present r e s u l t s and those

by references 78 and 89 agree w i t h i n 5 percent ; b u t , t h e r e s u l t by C a l l i s

is 50 percent g r d a t e r . The r a d i a t i o n model (RATRAP, r e f . 30) used by

*Personal Communications wi th D r . W . B. Ols tad , Langley Research Center, Hampton, V i rg in i a .

re ferences 73 and 89 t r e a t s t h e frequency dependence of absorpt ion co-

e f f i c i e n t i n d e t a i l as does t h e p resen t r a d i a t i o n model (RAD/EQuIL, r e f .

9 ) . A s s t a t e d i n t h e s e c t i o n Radiat ion Transport Modeling, S u t t l e s ( r e f .

32) made a comparison of these r a d i a t i o n models and concluded t h a t t h e

RADIEQUIL program was t h e b e t t e r model. The p resen t method is i n agree-

ment with o t h e r methods which use a d e t a i l e d r a d i a t i o n model.

The previous ly a v a i l a b l e r e s u l t s f o r t h e r a d i a t i n g , i n v i s c i d flow

about b lun t bodies have been provided by t h e a n a l y s i s of C a l l i s ( r e f . 3)

and Olstad ( r e f . 2 ) . The step-model approximation developed by Olstad

was used i n both methods f o r t h e t reatment of r a d i a t i o n . Olstad uses

an approximate method f o r t h e s o l u t i o n of t h e flow f i e l d and t h e method

is inve r se , shock shape s p e c i f i e d , r a t h e r than d i r e c t , body shape speci-

f i e d , a s t h e p resen t method. For t h e s e reasons, t h e p resen t method was

not compared wi th t h e method of Olstad. Since t h e same r a d i a t i o n model

was used by Olstad and C a l l i s , t h e r e s u l t s of Olstad should not agree

with r e s u l t s by t h e present method.

The d i f f e r e n c e i n t h e r e s u l t s between t h e present method and t h e

method of C a l l i s ( r e f . 3) f o r t h e r a d i a t i v e hea t ing r a t e s is a t t r i b u t e d

t o t h e d i f f e r e n t r a d i a t i o n models used i n t h e methods. The r a d i a t i o n

model used i n t h e p resen t method t r e a t s t h e frequency dependence of t h e

absorpt ion c o e f f i c i e n t i n d e t a i l and is a more accura t e model than t h e

s t e p model used by Callis. From t h e above a n a l y s i s , i t is concluded

t h a t the p resen t method provides a good s o l u t i o n f o r t h e r a d i a t i n g ,

i n v i s c i d gas flow about b lun t bodies .

C a l l i s ( r e f . 3) and Olstad ( r e f . 2) s t a t e t h a t t h e nondimensional,

r a d i a t i v e hea t ing r a t e d i s t r i b u t i o n s along b lun t bodies a r e nonsimilar

with respect to body angle and nose radius. Suttles (ref. 89) states

that the nondimensional, radiative heating rate distribution along the

body at the subsonic region of flow for hemispheres to be relatively

insensitive to the size of the nose radius. These results are based on

entries into an air atmosphere. The heating rates along the body are

R R nondimensionalized by the stagnation-point value, q 1% , and the dis-

w ,o

tance along the body is nondimensionalized by the nose radius, s/Rn.

The present method was used to investigate the above mentioned

trends. The results presented are for spherically-capped, conical

bodies since this is the body shape presently being considered for plan-

etary entries. However, the present method is not restricted to this

type of body shape. Results from stagnation-point solutions are also

presented.

Results of radiative heating distributions from solutions for

entries into different atmospheres and different entry conditions are

presented in Figure 9. The results for the two entry conditions of the

CO -N mixture are representative of entry to Venus and the results for 2 2

the H -He mixture are representative of entry to Jupiter. Note that the 2

distributions are highly nonsimilar for the different cases. The results

for case C and D are for the same atmosphere but different entry condi-

tions; but, the distributions are nonsimilar. The present method can

be readily used for entries into atmospheres other than air.

A comparison of the stagnation-point, radiative heating rates for

entries in air and in a .90C02-.ION2 mixture (by volume) is presented

in Figure 10. The radiative heating in the CO -N mixture is much 2 2

greater than in air at the lower velocities; but, the results are com-

parable at the higher velocities. The difference at the lower veloc-

ities is due to the molecular band radiation from the red and violet

bands of cynrogen, CN(R) and CN(V), and the fourth positive band of

carbon dioxide, CO (4+) , in the CO -N mixture. The atomic species are 2 2

dominant at the higher velocities, greater temperature behind the shock

wave, and the radiation from the continuum and line transitions is great-

er than from the molecular band systems.

As the radiating gas flows around a body there are two effects which

can influence the radiative heating to the body. The volume of radiating

gas increases, increased shock stand-off distance, which will increase

the radiative heating. This effect is offset by the decrease in temper-

ature as the gas expands around the body. The distributions of radia-

tive heating along spherically-capped, conical bodies for entry in an

air atmosphere are presented in Figure 11 for different body angles.

The distributions are nonsimilar and the nondimensional radiative heat-

ing is greatest for the largest body angle. The shock stand-off dis-

tances and temperatures at a location on the conical portion of the body

are also presented. Both the temperature and shock stand-off distance

are greater for the larger body angle; thus, the greater the radiative

heating. Note that the dimensional, stagnation-point heating rate is

insensitive to the downstream body angle for spherically-capped, conical

bodies. These results are in agreement with the trends presented by

Callis (3) and Olstad (4).

The effect of body angle on the nondimensional, radiative heating

distribution for entries to Venus is presented in Figure 12 for two

entry conditions. As for air entry, the radiative heating rates along

the conical region of the body are greater for the larger cone angle.

Also, the stagnation-point, radiative heating rate is insensitive to

the cone angle. As previously stated, the nondimensional, radiative

heating distribution is nonsimilar with respect to entry conditions.

0 For the 60 body, the radiative heating rates along the downstream re-

gion of the body exceed the stagnation-point value at the lower velocity

entry. At the higher velocity entry, the radiative heating rate in-

creases along the downstream region but does not exceed the stagnation-

point value.

The effect of nose radius on the nondimensional, radiative heating

distributions for an entry in air is presented in Figure 13. The dis-

tributions are slightly nonsimilar for the downstream region of the

body. Callis (ref. 3) and Olstad (ref. 2) state the distributions to

be nonsimilar and the present results are in agreement. However, the

results of Callis and Olstad indicate a greater nonsimilarity than shown

by the present results for an air entry. The reason is probably due to

the radiation model used by Callis and Olstad which might be too trans-

parent as previously discussed. In which case, the nose radius would

have a greater effect than shown by the present results. Suttles (ref.

89) states the distributions along the subsonic flow region of the body

to be similar with respect to nose radius. The present results are in

agreement with Suttles since the distributions are similar for the

forward region of the body, s/Rn<0.4.

The e f f e c t of nose r ad ius on t h e nondimensional, r a d i a t i v e hea t ing

d i s t r i b u t i o n s f o r e n t r i e s t o Venus is presented i n F igure 14. The dis-

t r i b u t i o n s a r e s l i g h t l y nons imi lar f o r t h e downstream region of t h e body

f o r t h e lower v e l o c i t y e n t r y . However, t h e d i s t r i b u t i o n s a r e highly non-

s i m i l a r f o r t h e h igher v e l o c i t y e n t r y . For both e n t r i e s , t h e d i s t r i b u *

t i o n s a r e s i m i l a r a t t h e forward region of t h e body up t o approximately

t h e sphere-cone junct ion .

The present r e s u l t s a r e based on a l imi t ed range of en t ry condi t ions

and body shapes. This same s i t u a t i o n a p p l i e s t o t h e s t u d i e s by C a l l i s

( r e f . 3 ) , Olstad ( r e f . 2), and S u t t l e s ( r e f . 89) f o r e n t r i e s only i n an

a i r atmosphere. Based on t h e present r e s u l t s , t h e fol lowing t rends seem

t o be gene ra l ly p reva len t . The nondimensional, r a d i a t i v e hea t ing dis-

t r i b u t i o n s ( </{,o versus s/Rn) a r e nonsimilar wi th r e spec t t o e n t r i e s

i n d i f f e r e n t atmospheres and with r e spec t t o e n t r i e s i n the same atmos-

phere a t d i f f e r e n t e n t r y condi t ions . The d i s t r i b u t i o n s a r e a l s o non-

s i m i l a r w i th r e spec t t o body ang le and nose r ad ius . Extreme cau t ion

should be exerc ised i n at tempting t o ex t r apo la t e t h e r e s u l t s from known

d i s t r i b u t i o n s t o o t h e r e n t r y condi t ions o r body shapes f o r which solu-

t i o n s have not y e t been obtained, For l a r g e body ang les , t h e r a d i a t i v e

hea t ing r a t e s t o the downstream region of a body can exceed t h e stagna-

t i o n po in t va lue f o r c e r t a i n e n t r y condi t ions .

The exac t composition of t h e Venusian atmosphere is unknown a t

present . The most r ecen t d a t a i n d i c a t e s t h e composition t o be predomi-

n a t e l y CO wi th up t o 10 percent by volume of N2. The e f f e c t of gas 2

composition for CO -N mixtures on the radiat ive heating t o the stagna- 2 2

tion-point of a body is presented i n Figure 15 for two entry conditions.

For the higher velocity entry, the radiat ive heating r a t e var ies by only

25 percent over the range of C02 content with the higher values occurring

f o r the greater percentage of CO For the lower velocity entry, the 2'

radiat ive heating r a t e can vary by a factor of 6 over the range of CO 2

content. This is due to the domination of radiation by the molecular

band t ransi t ions of C0(4+), CN(V), and CN(R) a t the lower velocity. For

the composition of present i n t e r e s t for Venusian entry ( l e s s than 10

percent N2), the e f f ec t of composition on the stagnation-point, radia-

t ive heating r a t e is negligible a t the higher velocity; but , a 22 per-

cent difference occurs with composition a t the lower velocity.

Fully-Coupled, Stagnation Point, Solutions f o r Earth Reentry

The present method has been used to obtain fully-coupled solutions

a t the stagnation point of blunt bodies for Earth Reentry. The solutions

are for mass inject ion a t the wall of a i r and ablation products. The

resu l t s from the present method a re compared with the r e su l t s given by

Garrett ( re f . 19).

The solutions were obtained f o r the free-stream conditions and mass

inject ion ra tes presented i n Table 11. These conditions are representa-

t i ve of a manned spacecraft eeentering the Earth's atmosphere a f t e r ex-

ploration of Mars. For the solutions, the wall conditions of the tem-

perature and the r a t e and composition of the injected gas were prespeci-

f ied. This procedure was used to be consistent with the method of

G a r r e t t . The r e s u l t s from the present method and t h e r e s u l t s given by

Gar re t t a r e presented i n Table 11.

The present r e s u l t s agree with t h e r e s u l t s of G a r r e t t w i th in 5 per-

cent f o r t h e mass i n j e c t i o n of a b l a t i o n products and wi th in 10 percent

f o r t h e mass i n j e c t i o n of a i r . This good agreement is b e t t e r than has

been previously s h a m i n comparing any two methods ( s e e r e f . 19 ) . As

s h a m by t h e r e s u l t s , t h e mass i n j e c t i o n of a b l a t i o n products i s twice

a s e f f e c t i v e a s t h e mass i n j e c t i o n of a i r i n reducing t h e r a d i a t i v e heat-

inging t o t h e wa l l . The mass i n j e c t i o n of a b l a t i o n products reduced t h e

r a d i a t i v e hea t ing t o the wa l l by 40 percent a t t h e l a r g e r blowing r a t e .

The present r e s u l t s f o r the temperature p r o f i l e and t h e r a d i a t i v e

f l u x p r o f i l e through t h e gas l aye r f o r a s o l u t i o n a r e compared wi th t h e

r e s u l t s of Gar re t t i n Figure 16. The spec i f i ed condi t ions a r e given i n

Figure 16(a) and t h e temperature p r o f i l e and r a d i a t i v e f l u x p r o f i l e a r e

given i n Figure 16(b) and 1 6 ( c ) , r e spec t ive ly . The present r e s u l t s a r e

i n good agreement with t h e r e s u l t s of G a r r e t t . For t h e present so lu t ion ,

s p e c t r a l d i s t r i b u t i o n s of r a d i a t i v e hea t ing toward t h e wa l l due t o con-

tinuum processes and due t o l i n e processes a r e presented i n Figures 16

(d) and 1 6 ( e 3 ~ r e s p e c t i v e l y . S p e c t r a l d i s t r i b u t i o n s were not given i n t h e

t h e s i s by Gar re t t .

For t h e continuum processes , p r a c t i c a l l y a l l r a d i a t i o n beyond 11 eV

is absorbed wi th in t h e boundary l a y e r and i s a t t r i b u t e d t o t h e atomic

spec ie s being a t a lower temperature i n t h e boundary l a y e r than i n t h e

i n v i s c i d l aye r . There i s a s l i g h t absorpt ion of r a d i a t i o n wi th in the

boundary l a y e r between t h e s p e c t r a l range of 4 t o 10 eV but the e f f e c t

is o f f s e t by increased r a d i a t i o n emission i n t h e s p e c t r a l range of 1 t o

4 eV. For t h e l i n e processes, t h e r a d i a t i o n is sharp ly reduced i n t h e

s p e c t r a l reg ion of 7 t o 11 eV. P a r t of t h i s reduct ion f o r t h e l i n e

groups a t 7.1, 8 .4 and 9 . 4 eV i s due t o the absorpt ion by t h e f o u r t h

p o s i t i v e band system of carbon monoxide, C0(4+ ) . The carbon monoxide i s

p resen t i n t h e boundary l a y e r from the mass i n j e c t i o n of a b l a t i o n prod-

uc t s . The present r e s u l t s f o r t h e s p e c t r a l regions of absorpt ion , o r

increased emission, w i t h i n t h e boundary l aye r a r e i n agreement with t h e

r e s u l t s given i n re ference 28.

The good agreement of t h e present r e s u l t s with those given by

G a r r e t t ( r e f . 19) shows t h a t t h e present method f o r a fully-coupled solu-

t i o n wi th a b l a t i o n product e f f e c t provides a good s o l u t i o n a t t h e s tag-

n a t i o n poin t of a b lun t body f o r Earth r een t ry . The method by G a r r e t t

can only be used a t t h e s t agna t ion poin t of a b lun t body and has not

been used f o r e n t r i e s i n atmospheres o the r than a i r .

Fully-Coupled Solu t ions f o r Venusian Entry

The present method was used t o obta in so lu t ions a t t y p i c a l condi-

t i o n s f o r unmanned, s c i e n t i f i c probes during Venusian en t ry . One mission

c u r r e n t l y under s tudy i s a multi-probe en t ry with one l a r g e probe and

t h r e e small probes which a r e spherical ly-capped, con ica l bodies. A

desc r ip t ion of t h i s type of mission i s presented i n re ferences 61 t o 64.

The e n t r y probes a r e r e l eased from a spacec ra f t and a r e t a rge ted t o

d i f f e r e n t l o c a t i o n s on Venus.

The en t ry parameters and body shapes l i s t e d below were used t o

c a l c u l a t e nominal t r a j e c t o r i e s f o r use i n t h e present ana lys i s . The

only small probe considered h e r e i n i s t h e one en te r ing a t the s t e e p e s t

en t ry angle.

Large Probe

Entry v e l o c i t y , kmls 11.06 Entry angle , degrees -42

W/CdA, kg/m 2

78.5

Weight, kg 158.8 Nose r a d i u s , m .3429 Body half-angle, degrees 60 Base diameter , m 1.3716

Small Probe

11.06 -56

125.7

22.7 .I397 45 .4064

The atmospheric model used f o r t h e Venusian atmosphere is t h a t presented

i n re ference 62 and the gas composition i s 97 percent carbon d ioxide and

3 percent n i t rogen by volume. The ca l cu la t ed e n t r y t r a j e c t o r i e s a r e

presented i n Figure 17.

The r a d i a t i v e hea t ing r a t e s a t the s t agna t ion po in t of the bodies

were ca l cu la t ed along the t r a j e c t o r i e s t o determine t h e l o c a t i o n of peak

r a d i a t i v e hea t ing . The "stagnat ion-point" vers ion of t h e developed,

r a d i a t i n g i n v i s c i d flow program was used f o r these ca l cu la t ions . The

r e s u l t s a r e presented i n Figure 18 f o r t h e l a r g e and smal l probes. Even

though t h e nose r ad ius i s sma l l e r , t h e r a d i a t i v e hea t ing t o t h e small

probe i s g r e a t e r due t o t h e deeper pene t r a t ion i n t h e denser reg ions of

t h e atmosphere a t h igher v e l o c i t i e s . Solu t ions f o r t h e r a d i a t i n g , in-

v i s c i d flow around t h e bodies were then obtained a t t h e peak hea t ing

condi t ions and t h e s tagnat ion-point va lues of r a d i a t i v e hea t ing r a t e s

from these s o l u t i o n s a r e a l s o presented i n Figure 18. It i s a t t hese

condi t ions f o r peak r a d i a t i n g hea t ing t h a t fully-coupled s o l u t i o n s wi th

a b l a t i o n product e f f e c t s were obtained f o r t h e r a d i a t i n g gas flow around

t h e l a r g e and small probes. Herea f t e r , t h e s o l u t i o n s a r e r e f e r r e d t o a s

l a r g e and small probes.

The a b l a t i o n of t h e h e a t s h i e l d f o r t h e ful ly-coupled s o l u t i o n i s

represented by s t eady- s t a t e a b l a t i o n of a carbon-phenolic m a t e r i a l wi th

elemental mass f r a c t i o n s of carbon, 0.851; oxygen, 0.110; hydrogen,

0.035; and n i t rogen , 0.004. The a b l a t i o n r a t e s a r e solved f o r a s p a r t

of t h e fully-coupled s o l u t i o n and a r e not p re spec i f i ed .

The r e s u l t s from t h e fully-coupled s o l u t i o n s f o r t h e l a r g e and

smal l probes a r e presented i n Figures 19 and 20, r e s p e c t i v e l y . The d i s -

t r i b u t i o n along t h e body of r a d i a t i v e hea t ing , convect ive hea t ing , abla-

t i o n r a t e , aerodynamic s h e a r , and momentum th ickness Reynolds number a r e

presented. The s p e c t r a l d i s t r i b u t i o n of r a d i a t i v e hea t f l u x due t o

continuum processes is a l s o presented f o r two body loca t ions .

For t h e l a r g e probe, t h e a b l a t i o n r a t e decreases along t h e body i n

t h e nose region and reaches a nea r ly cons tant va lue along t h e a f te rbody

(Figure 19(d) ) . A s shown i n Figure 1 9 ( c ) , t h e convect ive hea t ing r a t e

decreases along t h e a f te rbody b u t t h e r a d i a t i v e hea t ing r a t e inc reases

t o va lues which exceed t h e s tagnat ion-point value. The r a d i a t i v e heat-

ing t o t h e wa l l is g r e a t e r than t h e convective hea t ing along t h e e n t i r e

body. The t o t a l hea t ing r a t e , r a d i a t i v e p lus convect ive, along t h e

af terbody i s 30 t o 40 percent l e s s than t h e s tagnat ion-point va lue .

Thus, t h e decrease i n convect ive hea t ing r a t e along t h e af terbody ne-

g a t e s t h e e f f e c t of t h e inc rease i n r a d i a t i v e hea t ing r a t e . The wa l l

temperature r e s u l t i n g from t h e s teady-s ta te a b l a t i o n i s 3600°~ , w i th in

2 percent , a long t h e body.

For t h e small probe, t h e r e i s a decrease i n t h e a b l a t i o n r a t e along

t h e a f te rbody (Figure 20(d)) and both t h e convect ive and r a d i a t i v e heat-

i ng r a t e s decrease along t h e af terbody (Figure 20(c)). The convective

hea t ing r a t e is g r e a t e r than the r a d i a t i v e hea t ing r a t e along t h e e n t i r e

body. This r e s u l t is oppos i te t o t h e r e s u l t obtained f o r t h e l a r g e

probe. The t o t a l hea t ing r a t e along t h e a f te rbody of t h e small probe

is 40 t o 60 percent l e s s than t h e s tagnat ion-point value. The w a l l

0 temperature r e s u l t i n g from t h e s t eady- s t a t e a b l a t i o n is 3700 K , w i th in

2 percent , along the body.

The r a d i a t i v e hea t ing r a t e s from i n v i s c i d , r a d i a t i n g s o l u t i o n s a r e

compared wi th t h e r e s u l t s f o r t h e fully-coupled s o l u t i o n s wi th steady-

s t a t e a b l a t i o n i n F igures 19(b) and 20(b) f o r t h e l a r g e and small probes,

r e spec t ive ly . The r a d i a t i v e hea t ing r a t e s from t h e fully-coupled solu-

t i o n s a r e l e s s than t h e va lues from t h e i n v i s c i d , r a d i a t i n g s o l u t i o n s

by 9 and 17 pe rcen t , r e spec t ive ly , f o r t h e l a r g e and small probes a t

t h e s t agna t ion po in t o f t h e bodies . These percent reduct ions a r e nea r ly

cons tant along t h e bodies . The reduct ion i n r a d i a t i v e hea t ing i s due

t o absorpt ion of r a d i a t i o n wi th in t h e boundary l a y e r .

Spec t r a l d i s t r i b u t i o n s of r a d i a t i v e hea t ing toward t h e body due t o

continuum processes a r e presented i n F igures 19(g) and 20(g) , respec t ive-

l y , f o r the l a r g e and sma l l probes. The s p e c t r a l d i s t r i b u t i o n s a r e pre-

sented f o r the s t agna t ion poin t and an af te rbody loca t ion . A s shown by

t h e r e s u l t s , t h e absorpt ion of r a d i a t i o n wi th in t h e boundary l a y e r

occurs i n t h e s p e c t r a l range of 5 t o 10 eV and is due t o s e l f absorpt ion

74

of t h e fou r th p o s i t i v e band system of carbon monoxide, C0(4+). A t t h e

s t agna t ion poin t of the small probe t h e r e is a s l i g h t i nc rease i n radi-

a t i v e hea t ing wi th in the boundary l a y e r a t 2.5 eV and i s due t o t h e

Swan band system of diatomic carbon, C (S) . A s i m i l a r e f f e c t was noted 2

a t t h e o ther body l o c a t i o n s f o r both t h e l a r g e and smal l probes, but t h e

e f f e c t was too smal l t o be presented i n t h e f igu res . The r a d i a t i v e

hea t ing due t o l i n e processes was l e s s than t en percent of t h e t o t a l

r a d i a t i v e heat ing a t t hese en t ry condi t ions and s p e c t r a l d i s t r i b u t i o n s

a r e not presented.

The fully-coupled s o l u t i o n s assumed laminar flow i n t h e boundary

l a y e r . Shown i n Figures 1 9 ( f ) and 20(f ) a r e t h e d i s t r i b u t i o n s of mo-

mentwn thickness Reynolds number around t h e bodies . I f t h e c r i t i c a l

va lue f o r t r a n s i t i o n t o tu rbu len t flow i s about 200, then t h e boundary

l a y e r would be tu rbu len t along t h e a f te rbody f o r both t h e l a r g e and

small probes. This could have an apprec iable e f f e c t on t h e d i s t r i b u t i o n

of convective hea t ing , a b l a t i o n r a t e , and aerodynamic shear .

A fully-coupled s o l u t i o n f o r t h e l a r g e probe wi th a tu rbu len t

boundary l aye r was obtained and t h e r e s u l t s a r e compared wi th t h e lami-

n a r case i n Figure 21. A c r i t i c a l va lue of 200 f o r t h e momentum th ick-

ness Reynolds number was used f o r t r a n s i t i o n t o tu rbu len t flow. As

expected, the convect ive hea t ing rates, a b l a t i o n r a t e s , and aerodynamic

shears increased wi th t r a n s i t i o n t o tu rbu len t flow. The d i s t r i b u t i o n

along t h e body of r a d i a t i v e hea t ing was e s s e n t i a l l y t h e same f o r b o t h

laminar and tu rbu len t flow; thus , t h e tu rbu len t boundary l a y e r d i d n o t

appear t o f u r t h e r a t t e n u a t e t h e r a d i a t i v e hea t ing even though t h e

boundary-layer th ickness was 80 percent g r e a t e r than f o r laminar flow.

This phenomenon can be explained with the wal l s p e c t r a l d i s t r i b u t i o n s

presented i n Figure 21(d) f o r t h e tu rbu len t and laminar s o l u t i o n s . For

t h e tu rbu len t boundary l a y e r , t h e r e is an inc rease i n t h e s e l f absorp-

t i o n of t h e C0(4+) band system i n the s p e c t r a l range of 5 t o 8 eV bu t

t h i s e f f e c t is o f f s e t by an increased emission of t h e C ( S ) band system 2

a t 2.5 eV and t h e red band system of cynrogen, CN(R), a t .6 t o 2 eV.

The s p e c t r a l d i s t r i b u t i o n a t t h e boundary l aye r edge is e s s e n t i a l l y t h e

same f o r both t h e tu rbu len t and laminar boundary l a y e r s and was presen-

ted i n Figure 19(g) . While t h e r a d i a t i v e hea t ing r a t e s along t h e a f t e r -

body of the l a r g e probe a r e t h e same f o r a turbulent o r a laminar boun-

dary l a y e r , t h e w a l l s p e c t r a l d i s t r i b u t i o n of the r a d i a t i o n i s d i f f e r e n t

f o r t h e two cases . I t should not be i n f e r r e d t h a t t h e r a d i a t i v e hea t ing

t o the body w i l l be t h e same f o r a laminar o r t u rbu len t boundary l aye r

f o r o the r e n t r y condi t ions o r body shapes.

A fully-coupled s o l u t i o n was obtained f o r free-stream condi t ions

r e p r e s e n t a t i v e of a h igher e n t r y v e l o c i t y than t h e nominal t r a j e c t o r i e s

and t h e r e s u l t s a r e presented i n Figure 22. The free-stream condi t ions

and body shape a r e given i n Figure 22(a) . The body shape is t h e same

a s t h e l a r g e probe except t h e nose rad ius i s s l i g h t l y smal le r . Herer!

a f t e r , t h i s case i s r e f e r r e d t o a s t h e high v e l o c i t y e n t r y .

A s shown i n Figure 22(c) , t h e r a d i a t i v e hea t ing r a t e i s g r e a t e r

than t h e convect ive hea t ing r a t e along the e n t i r e body. Unlike t h e

l a r g e probe, t h e r e is only a s l i g h t i nc rease i n t h e r a d i a t i v e hea t ing

along t h e af terbody. The convect ive hea t ing r a t e a t t h e s t agna t ion

po in t i s s l i g h t l y less than t h e va lues j u s t downstream of t h e s tagnat ion

po in t and is due t o an increased e f f e c t of blockage of convective heat-

ing by t h e m a s s i n j e c t i o n of t h e a b l a t i o n products . A s shown i n Figure

22(d) , t h e a b l a t i o n r a t e decreases along t h e body i n t h e nose region.

The a b l a t i o n r a t e and t o t a l hea t ing r a t e a r e nea r ly cons tant along t h e

af terbody. The w a l l temperature r e s u l t i n g from t h e s teady-s ta te abla-

0 t i o n i s 3750 K , w i th in 1 percen t , along t h e body.

There i s s i g n i f i c a n t absorpt ion of r a d i a t i o n wi th in t h e boundary

l a y e r f o r t h e high v e l o c i t y e n t r y a s shown by a comparison of r e s u l t s

between t h e fully-coupled s o l u t i o n and an i n v i s c i d , r a d i a t i n g s o l u t i o n

i n Figure 22(b) . The reduct ion i n r a d i a t i v e hea t ing r a t e is 25 percent

a t t h e s t agna t ion po in t and 25 t o 30 percent along t h e af terbody. Spec-

t r a l d i s t r i b u t i o n s of r a d i a t i v e hea t ing toward t h e body due t o continuum

processes a r e presented i n Figure 22(g) f o r t h e s t agna t ion poin t and an

af te rbody loca t ion . For t h e continuum processes , most of t h e absorpt ion

wi th in t h e boundary l a y e r occurs i n t h e s p e c t r a l reg ion of 5 t o 10 eV

and is due mainly t o s e l f absorpt ion of t h e C0(4+) band. There i s an

inc rease i n r a d i a t i v e f l u x a t 2.5 eV and is due t o the C (S) band. A t 2

t h e s t agna t ion po in t t h e r e is absorpt ion a t photon ene rg ie s g r e a t e r than

10 eV and t h i s absorpt ion is a t t r i b u t e d t o atomic s p e c i e s wi th in t h e

coo le r reg ions of t h e boundary l a y e r . There is a l s o absorpt ion a t t h e

photo ioniza t ion edge of carbon a t 8.5 eV.

The r a d i a t i v e f l u x due t o l i n e processes is 50 percent of t h e t o t a l

r a d i a t i v e f l u x a t t h e nose reg ion of t h e body and decreases t o only 20

percent along t h e af terbody. S p e c t r a l d i s t r i b u t i o n s of t h e r a d i a t i v e

heat ing due t o l i n e processes f o r two l o c a t i o n s a t t h e nose reg ion a r e

presented i n Figure 22(h). Absorption of l i n e r a d i a t i o n wi th in t h e

boundary l a y e r occurs only f o r t h e l i n e groups i n t h e u l t r a v i o l e t a t

7 . 1 , 8 . 4 , and 9 . 4 eV. The l i n e r a d i a t i o n i n the v i s i b l e and i n f r a r e d

reg ions of t h e s p e c t r a l , l e s s than 2 eV, i s not a t t enua ted wi th in t h e

boundary l a y e r .

The present r e s u l t s have shown t h a t the r a d i a t i v e hea t ing toward

t h e body is a t t enua ted i n t h e boundary l a y e r f o r Venusian e n t r i e s . This

a t t e n u a t i o n w i l l reduce the r a d i a t i v e hea t ing t o t h e body by 10 t o 20

percent a t en t ry cond i t ions and body shapes which a r e p resen t ly consid-

e red a s nominal. The reduct ion can be a s l a r g e a s 30 percent a t higher

v e l o c i t y e n t r i e s . The r a d i a t i v e hea t ing toward t h e body is a t t enua ted

i n t h e boundary l a y e r a t t h e downstream region of the body a s we l l a s

a t t h e s t agna t ion po in t of t h e body.

SUMMARY AND CONCLUSIONS

A method is presented f o r t h e s o l u t i o n of t h e ful ly-coupled, nongray

r a d i a t i n g gas flow about an a b l a t i n g , p lane tary e n t r y body. The s o h -

t i o n is f o r a gas i n chemical equi l ibr ium and a r b i t r a r y gas mixtures

can be considered. The t reatment of r a d i a t i o n accounts f o r molecular

band, continuum, and atomic l i n e t r a n s i t i o n s with a d e t a i l e d frequency

dependence of the absorpt ion c o e f f i c i e n t . The a b l a t i o n of t h e en t ry

body is solved a s p a r t of t h e s o l u t i o n f o r a s t eady- s t a t e a b l a t i o n

process.

Appl ica t ion of the developed method i s shown by r e s u l t s a t t y p i c a l

condi t ions f o r unmanned, s c i e n t i f i c probes during en t ry t o Venus. The

r a d i a t i v e hea t ing toward the body i s a t tenuated i n t h e boundary l a y e r

f o r Venusian e n t r i e s . This a t t e n u a t i o n w i l l reduce the r a d i a t i v e heat-

ing t o the body by 10 t o 20 percent a t e n t r y condi t ions and body shapes

which a r e present ly considered a s nominal. The reduct ion can be a s

l a r g e a s 30 percent a t h igher v e l o c i t y e n t r i e s . The a t t e n u a t i o n of ra-

d i a t i o n wi th in t h e boundary l a y e r is pr imar i ly due t o the s e l f absorp-

t i o n of t h e fou r th p o s i t i v e band system of carbon monoxide, C0(4+), i n

t h e r a d i a t i o n s p e c t r a l reg ion of 5 t o 10 eV. A t high v e l o c i t y e n t r y ,

t h e a t t e n u a t i o n of r a d i a t i o n a l s o occurs f o r l i n e t r a n s i t i o n s i n t h e

r a d i a t i o n s p e c t r a l reg ion of 7 t o 10 eV.

P r i o r s t u d i e s of fully-coupled s o l u t i o n s wi th a b l a t i o n product

e f f e c t s have been f o r t h e s t agna t ion po in t of a body f o r Earth r een t ry

and have shown t h a t t h e boundary l a y e r with i n j e c t i o n of a b l a t i o n prod-

u c t s is e f f e c t i v e i n reducing t h e r a d i a t i v e hea t ing t o t h e body. The

present results show that the boundary layer over an ablating body will

reduce the radiative heating to the body for entries to Venus. Further-

more, the attenuation of radiation within the boundary layer occurs at

downstream regions of the body as well as the stagnation point of the

body.

Present results from a study of the radiating, inviscid flow about

spherically-capped, conical bodies during planetary entries show that

the nondimensional, radiative heating distributions (</qR versus w,o

s/Rn) to be nonsimilar with respect to entries in different atmospheres

and with respect to entries in the same atmosphere at different entry

conditions. The distributions are also nonsimilar with respect to nose

radius and downstream body angle. Therefore, extreme caution should be

exercised in attempting to extrapolate the results from known distribu-

tions to other entry conditions for which solutions have not yet been

obtained. The radiative heating rates to the downstream region of the

body can exceed the stagnation-point value for certain entry conditions

and body shapes.

EXTENSION OF PRESENT RESEARCH

An immediate ex tens ion of t h e present research can be made i n t h e

following a reas :

1. A4dit ional ful ly-coupled s o l u t i o n s a r e needed f o r a b e t t e r

eva lua t ion of the thermal environment f o r Venusian e n t r i e s . Solu t ions

with a tu rbu len t boundary l a y e r a r e needed f o r the small probe and f o r

h igher ve loc i ty e n t r i e s .

2 . Solu t ions a r e needed f o r body shapes o the r than sphe r i ca l ly -

capped, con ica l bodies f o r Venusian e n t r i e s . Future design may d i c t a t e

a d i f f e r e n t shape and t h e r e s u l t s would be use fu l i n eva lua t ing t h e

e f f e c t s of such a change i n shape.

3 . A study is needed f o r t h e e f f e c t of composition f o r CO -N 2 2

mixtures on t h e r a d i a t i v e hea t ing d i s t r i b u t i o n along a body f o r Venusian

e n t r i e s .

4 . Use the present r e s u l t s and r e s u l t s from add i t iona l s o l u t i o n s

t o develop c o r r e l a t i o n s which can be e a s i l y used i n parametr ic des ign

s t u d i e s f o r Venusian e n t r i e s .

5. The present computational method needs t o be extended t o inc lude

a t r a n s i e n t a b l a t i o n a n a l y s i s .

6 . Use t h e present method t o begin s t u d i e s of e n t r i e s t o Saturn.

Saturn i s c h a r a c t e r i s t i c of t h e g i a n t p l ane t s ( J u p i t e r , Saturn, Uranus,

and Neptune) and probably w i l l be t h e f i r s t chosen f o r exp lo ra t ion by

en t ry probes.

TABLE I. - STAGNATION-POINT, RADIATIVE HEATING RATES FOR AIR ENTRY.

Nethod

Present method

Callis, Ref. 3

Suttles, Ref. 89

Falanga and Sullivan, Ref. 73

R , m/mz

Rn = .3427 m

16.17

18.71

18.38

18.73

Rn = 3.427 m

28.80

45.47

27.26

28.44

TABLE 11. - RESULTS FROM FULLY-COUPLED, STAGNATION-POINT

SOLUTIONS WITH MASS INJECTION FOR A I R ENTRIES.

(a ) V_ = 15.25 km/s ; pm = 2.72 x kg/m3 ; pm = 17.76 N/m 2

- Rn = 3.048 m ; Tw = 3600°K

In jec t ed gas

a i r

2 ( P V ) ~ , kg/m -S

0

415

.415 I a b l a t i o n products*

Composition

-

a i r

830 I a b l a t i o n products*

Present method

3980

G a r r e t t , Ref. 19

4150

In j ec t ed gas tm/m2 %,o*

2 (pv)" , kg/m -s

0

.205

Present method

2307

1710

Composition

-

a b l a t i o n products*

G a r r e t t , Ref. 19

2540

1620

/ I n v i s c i d shock l a y e r I - - - Bo,:riuary l a y e r I I I

Figu re 1. - I l l u s t r a t i o n of t h e i nu t i ng of t h e boundary l a y e r s o l u t i o n t o t h e i n v i s c i d l a y e r s o l u t i o n .

Tra jec to ry da ta . + I n v i s c i d , r a d i a t i v e , * flow f i e l d s o l u t i o n .

-T- I Boundary l a y e r s o l u t i o n with I Luncoupled r a d i a t i v e t r a n s p o r t

I - J

L Inv i sc id and boundary l a y e r p r o f i l e s ] mated. Radia t ive t r a n s p o r t ca l cu la t ion I f o r e n t i r e l a y e r .

-layer b s o l u t i o n with

coupled r a d i a t i v e t r anspor t . + t

Radia t ive t r a n s p o r t , match poin t

I condi t ion , and a b l a t i o n r a t e s

converge7

1 Yes

1 Fully-coupled so lu t ion . 1

Figure 2. - Calcula t ion yror?rdur~ f o r t h e ful ly-coupled, r a d i a t i n g , flow f i e l d s o l u t i o n .

Shock

Body sur f ace

of symmetry

Coordinates

s - Distance along body surface from a x i s of symmetry.

y - Distance perpendicular t o body surface.

+ - Angle measured i n plane normal t o ax i s of symmetry.

Figure 3. - Flow f i e l d coordinate system f o r axisymmetric b lunt body.

Shock

$ - Typical nodal point

Figure 4. - Schematic of use of unsteady character is t ics

at shock wave.

Gas composition by volidne

.79 N 2 - . 21 0 ( a i r ) 2

V_ = 12.192 km/s

- 4 Om = 3.15 x 10 kg/m3

pm = 22.6 N/m 2

h_ = 0.0 J/k&

Rn = 2.0 m

B c = 40'

( a ) Free-stream condit ions and body shape.

Figure 5 . - Nonradiating, i n v i s c i d flow s o l u t i o n around a b l u n t body. Present method i s compared with t h e method of Inouye eta. ( r e f . 8 8 ) .

Shock Sonic

( b ) Shock shape at forward region o f body and son ic line l oca t ion .

Figure 5 . - Continued.

0 Present Method

- Inouye &. , Ref. 88

(c) Shock shape at downstream region of body.

Figure 5. - Continued.

r 0 Present method

(d) Shock stand-off distance around body.

Figure 5. - Continued.

.3

.2

0 Preser l t method .1

- Inouye & &., R e f . 88

0

( e ) Velocity d i s t r i b u t i o n along body.

Figure 5. - Continued.

0 Present method

- 1nouj.e fi s. , Ref. 88

(f) Pressure distribution along body.

Figure 5. - Concluded.

Gas composition by volume

.79 N2 - .21 0 ( a i r ) 2

v, = 12.20 km/s

p, = 2.74 x kg/m3

p, = 20.0 N/m 2

h, = 0.0 J/kg

Rn = 0.20 m

O c = 60°

( a ) Free-stream conditions and body shape.

Figure 6. - Radiating, inv i sc id flow so lu t ion about a b lunt body during Earth reent ry . Present method 1s compared with t h e method of C a l l i s ( r e f . 3 ) .

- Present method

0 Cal l i s , Ref. 3

h - 0 V V

(b) Pressure d i s t r ibu t ion along body.

( c ) Shock stand-off distance around body.

Figure 6. - Continued.

- Present method --- Callis, Ref. 3 /

(d) Radiative heating along body.

Figure 6. - Continued.

6 r

\ Present method

--- Callis' r e s u l t s modified by

present r a d i a t i o n model.

( e ) Radiat ive hea t ing along body. C a l l i s ' r e s u l t s modified.

Figure 6. - Concluded.

Gas composition by volume - .79 N2 - .21 O2 (air) V_ = 11.19 km/s

-4 P_ = 2.717 x 10 kg/m3 4 --

/

P, = 27.53 N/m 2 / /--

/ /-

/ #- - 0

0 Callis, Ref. 7

/ 0

/ J

0 /

. I I I I I 0 2 4 6 8 10

Rn, m

Figure 7. - Stagnation-point, radiative heating rates in an air atmosphere as a function of nose radius. Present results are compared with the method of Callis (ref. 7).

/ / -

/ /

/ N - <Callis, Ref. 3

Present method

I- Figure 8. - Radiative heating distribution for free-stream

conditions and body shape given in Figure 6(a) except Rn=2.0 m. Present results are compared with the method of Callis (ref. 3).

* by volume Case

Case

A

B

C

D

Figure 9. - Radiative heating distributions along a spherically-capped, conical body from radiating, inviscid flow solutions for entries in different atmospheres and different free-stream conditions.

Gas ~om~osition*

.79 N2 - .21 O2

.85 H2 - .15 He

.90 C02 - .10 N 2

.90 C02 - .lo N 2

V_, km/s

12.200

50.000

8.740

11.175

p_, kg/m 3

2.744 x

5 .TOO x 3.294 x

2.850 x loe3

2 P,,N/~

20.01

0

130.59

117.30

h_,W/kg

O

0

-8.43

-8.43

R 2 ~ , ~ , M f / m

5.78

9668.00 4.44

35.70

Figure 10. - A comparison of the stagnation-point, radiative heating rates for entries in air and in a .90 C 0 2 - .10 N (by volume) gas mixture.

2

Figure 11. - The d i s t r i b u t i o n of r a d i a t i v e hea t ing along spher ica l ly- capped, con ica l bodies f o r en t ry i n a i r .

Figure 12. - The distribution of radiative heating along spherically-capped, conical bodies for entry in a .90 CO - .l0 N (by volume) gas mixture.

2 2

0 I 1 I I I 0 1 2 3 4 5

s /Rn

(a) V_ = 8.740 km/s ; p _ = 3.294 x kg/m3 ; p_ = 130.6 N/m 2

hm = -8.43 MJ/kg ; Rn = .3048 m

Figure 13. - The e f f e c t of nose radius on t h e d i s t r i b u t i o n of r a d i a t i v e hea t ing along a spherically-capped. con ica l body f o r en t ry i n air.

Figure 14. - The effect of nose radius on the distribution of radiative heating along a spherically-capped, conical body for entry in a .90 CO - .10 N (by volume) gas mixture.

2 2

Figure 14. - Concluded.

60 1 I I I I 1 o 20 40 60 80 l o o

percent CO (by volume) 2

0 I I I I 1 1 o 20 40 60 80 100

percent C02 (by volume)

Figure 15. - Stagnation-point, radiat ive heating ra tes i n C02 - N gas mixtures. Rn=.3048 m

2

Gas composition by volume

.79 N2 - .21 O2 ( a i r )

Vm = 15.250 km/s

pm = 1.77 x k g / m 3

pm = 12.3 N/m 2

hm = 0.0 J/kg

Rn = 2.56 m

Specified w a l l conditions

- Tw = 3840 O K

2 ( p ~ ) ~ = .205 kg/m -s

Elemental mass f ract ions of injected gas :

Carbon -923

Oxygen .058

Hydrogen .018

Nitrogen .001

( a ) Free-stream conditions and specified w a l l conditions.

Figure 16 . - Fully-coupled, stagnation-point solution for Earth reentry. Present method i s compared with the method of Garrett ( r e f . 19).

(b) Temperature prof i le through lqfer .

16 x 103

Figure 16. - Continued.

1 4

12

10

?4 0

I H" 8

6

t - Present method ; 6 = 10.26 cm

--- Garret t , Ref. 19; 6t = 10.34 cm -

-

-

-

-

4

2 I I I I I

0 .2 . 4 .6 .8 1.0

y/st

( c ) Radiat ive heat ing p r o f i l e through layer .

Figure 16. - Continued.

Boundary l a y e r edge 11 \ I \ ,

Photon energy (h'v), eV

(d) Spec t ra l d i s t r i b u t i o n of r a d i a t i v e hea t ing toward t h e body due t o continuum processes.

Figure 16. - Continued.

3 - Boundary layer edge \

N~ 2 -

G .. '"

ffi 7 e 1 -

0 I I I I I I I I I I

Photon energy (h'v), eV

(e ) Spectral dis t r ibut ion of radiat ive heating toward the body due t o l i n e processes.

Figwe 16. - Concluded.

Figure 17. - Trajectories for large and small probes during entry t o Venus.

120

110

100 "

0 a 2 + .rl

* 90 2

80

70

- Large probe Small probe

V = 11.06 h / s e 11.06 h / s

- Ye

= -420 -56'

W/CdA = 78.5 kg/m2 125.7 kg/m2

, W = 158.8 kg 22.7 kg

-

-

4 -4

I I I I I I 0 2 4 6 8 10 12 1 4

Velocity, km/s

- Stagnation-point so lu t ion

0 Around body so lu t ion

F i m r e 18. - Stagnation-point, r a d i a t i v e heating r a t e s along ent ry t r a j e c t o r i e s f o r l a rge and small probes. Results from r a d i a t i n g , inv i sc id flow solut ions .

Gas composition by volume .97 C02 - .03 N2

Vm = 8.803 km/s

- 5.79 x l o b 3 kg/m 3 P, - p_ = 200.1 N/m 2

h = -8.86 W/kg m

Rn = 0.3429 m

ec = 60'

( a ) Free-stream condit ions and body shape.

( b ) Comparison of r a d i a t i v e heat ing.

I n v i s c i d , r a d i a t i n g s o l u t i o n

Fully-coupled s o l u t i o n

Figure 19. - Fully-coupled, r a d i a t i n g flow s o l u t i o n with s teady-s ta te ab la t ion of carbon-phenolic hea t sh ie ld f o r l a r g e probe.

0 - I I I I 1

0 0.5 1 . 0 1 .5 2.0 2.5

s/Rn

Radiative heating

--- -- /Convective heating ---_ ----__

(c) Radiative and convective heating distribution.

(d) Ablation rate distribution.

Figure 1 9 . - Continued.

ae) Aercdynemic shear distribution.

0 0.5 1.0 1.5 2.0 2.5

s /Rn

(f) Momentum thickness Reynolds number distribution.

Figure 19. - Continued.

3 - s/Rn = 2.0 Boundary-layer edge

% 2 - I

N El

iz - ? 1 -

0 I 0 2 4 6 8 10 12

Photon energy (h'v) , eV

3 - s/Rn = 0

% 2 - N

undary-layer edge El s '" 1 -

E 2 0'

0 . 1

Photon energy (h*v), eV

(g) Spectral d i s t r ibu t ion of radiat ive heating toward the body due t o continuum processes.

Figure 19. - Concluded.

Gas composition by volume -97 C02 - .03 N2

( a ) bee-stream conditions and body shape.

(b ) Comparison of radiat ive heating.

1 5 -

Figure 20. - Fully-coupled, radiat ing flow solution with steady-state ablation of carbon-phenolic' heatshield for small probe.

10 - --. \ \

Inviscid, radiating solution

--- - - - - Fully-coupled solution

0 I I I I I O 0.5 1.0 1.5 2.0 2.5

Radiative heating L

( c ) Radiative and convective heating d is t r ibu t ion .

(a) Ablation r a t e dis t r ibut ion.

Figure 20. - Continued.

s/Rn

( e ) Aerodynamic shear distribution.

( f ) Momentum thickness Reynolds number distribution.

Figure 20. - Continued.

0 2 4 6 8 10 12

Photon energy (h*v), eV

$ 2 - N' a is .. '" 1

' \ ,r~~undary-layer edge

Boundary-layer edge

- /' \

0 2 4 6 8 10 12 Photon energy (hwv), ev

"2'

0 ' I 1

(g) Spectral distribution of radiative heating toward the body due t o continuum processes.

Figure 20.- Concluded.

( a ) Heating r a t e d i s t r i b u t i o n .

Turbulent

/

(b) Ablation r a t e d i s t r i b u t i o n .

Figure 21. - Comparison between a turbulent and laminar boundary l a y e r f o r fully-coupled s o l u t i o n for l a r g e probe.

( c ) Aerodynamic shear d i s t r ibu t ion .

N 8 -. 5 " .5

r 3

0 2 4 6 8 10 12

Photon energy (h*v), eV

Turbulent

- 5

(d) Spectral d i s t r ibu t ion at the w a l l of rad ia t ive heating tovard the body due t o continuum processes.

0 I I 0 0.5 1.0 1.5 2.0 2.5

Figure 21. - Concluded.

Gas composition by volume -97 C02 - .03 N,

V - = 11.175 km/s

- 2.85 x kg/m 3 Pm - p_ = 117.3 N l m 2

h m = -8.86 W/kg

Rn = 0.3048 m

8, = 60°

( a ) Free-stream conditions and body shape.

(b ) Comparison of radiat ive heating.

40 -

Figure 22. - Fully-coupled, radiat ing flow solution with steady-state ablation of carbon-phenolic heatshield fo r high velocity entry.

N

-. \ \ \

El \ Inviscid, r a d i a t i w solution

\ Fully-coupled solution

- .- _ - - - - -.- --- /

30 -

N 20 - e

ii Radiative heating

3 __C

- _/-Convective heating ---- ---------- --- - - ---

0 I I 1 I L

( c ) Radiative and convective hea t ingd i s t r i bu t ion .

( d ) Ablation r a t e dis t r ibut ion.

Figure 22. - Continued.

(el Aerodynamic shear distribution.

( f ) Momentum thickness Reynolds number distribution.

Figure 22. - Continued.

* I

\ J I 12 1 4

Photon e n e r a ( h * ~ ) , e~

(a ) spec t r a l d i s t r ibu t ion of radiat ive heating toward the body due t o continuum processes.

Figure 22. - Continued.

2 4 6 8 10 12

Photon energy (h'v) , eV

o 2 4 6 8 10 12

Photon energy (h*v) , eV

(h ) Spectral d i s t r ibu t ion of radiat ive heating toward the body due t o l i n e processes.

Figure 22. - Concluded.

LIST OF REFERENCES

1. Hirschfelder, Joseph 0.; Curt is , Charles F.; and Bird, Byron R. 1964. Molecular Theory of Gases and Liquids. John Wiley and Sons, Inc . , New York.

2. Olstad, Walter B. 1971. Nongray Radiating Flow about Smooth Symmetric Bodies. A I M J . , vol. 9, no. 1: 122-130.

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Chin, J in H. 1968. Radiation Transport for Stagnation Flows In- cluding the Effect of Lines and Ablation Layer. AIAA Paper No. 68-664.

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Burns, Raymond K. ; and Oliver, Calvin C. 1968. Downstream Radiation F l u t o Blunt Entry Vehicles. AIAA J., vol. 6, no. 12: 2452-2453.

Chou, Y. S. 1971. Locally Nonsimilar Solutions f o r Radiating Shock Layer about Smooth Axisynnnetric Bcdies. LMSC N-EE-71-2, Lockheed Missiles and Space Co., Sunnyvale, California.

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Bohachevsky, Ihor 0.; and Mates, Robert E. 1966. A Direct Method for Calculation of the Flow about an Axisymmetric Blunt Body a t Angle of Attack. AIAA J. , vol. 4 , no. 5: 776-782.

VonNeumann, J.; and Richtmyer, R. D. 1950. A Method for t h e Numer- i c a l Calculation of Hydrodynamic Shocks. J. Appl. Phys., vol. 21: 232-237.

Lax, Peter D. 1954. Weak Solutions of Nonlinear Hyperbolic Equations and t h e i r Numerical Computation. Commun. Pure Appl. Math., vol. VII, no. 1 : 159-193.

Lax, Peter D . ; and Wendroff, Burton. 1964. Difference Schemes for Hyperbolic Equations with High Order of Accuracy. Commun. Pure Appl. Math, vol. X V I I , no. 3: 381-398.

MacCormack, Robert W. 1969. The Effect of Viscosity i n Hyper- veloci ty Impact Cratering. ALAA Paper No. 69-354.

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National Academy of Sciences. 1970. Venus, Strategy for Exploration. Report of a Study by the Space Science Board. National Academy of Sciences, Washington, D. C.

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Marcotte, Paul G. 1971. Planetary Explorer, Phase A Report and Universal Bus Description. Goddard Space Fl ight Center, Greenbelt, Maryland.

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Nagler, Robert G. 1970. A Systematic Review of Heat-Shield Tech- nology for Ex t r a t e r r e s t r i a l Atmospheric Entry. Technical Report 32-1436, Jet Propulsion Laboratory, Pasadena, California.

Norman, Herbert C. ; and Hart, Paul M. 1968. Mission Influence on the Aerothermodynemic Environment for a Venus Entry Vehicle. AAS Paper No. 68-4-6.

Strauss, Fkic L. ; and Sparhawk, H. E. 1968. Ablative Heat Shields f o r Planetary Entry Vehicles. AAS Paper No. 68-8-5.

Spiegel, Joseph M.; Wolf, Fred; and Zeh, Dale W. 1968. Simulation of Venus Atmospheric Entry by Earth Re-Entry. AIAA Paper No. 68- 1148.

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Deacon, H. J., Jr.; and Rumpel, W. F. 1971. Radiative Transfer Properties of a Shocked Venus Model Atmosphere. J. Spacecraft, vol. 8, no.. 2: 177-182.

Livingston, F. R.; and Will lard, J. 1971. Planetary Entry Body Heating Rate Measurements i n A i r and Venus Atmospheric Gas up t o T=15,000 K. AIAA J . , vol. 9 , no. 3: 485-492.

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Hughes, W i l l i a m F. ; and Gaylord, Eder W. 1964. Basic Equations of Engineering Science. Schaum Publishing Company, New York.

Isaacson, Eugene; and Keller, Herbert Bishop. 1966. Analysis of Numerical Methods. John Wiley and Sons, New York.

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Graves, Randolph A. , Jr. 1970. Diffusion Model Study i n Chemically Reacting A i r Couette Flow with Hydrogen Injection. NASA Technical Note, NASA TR R-349.

Kendall, Robert M.; and B a r t l e t t , Eugene P. 1968. Nonsimilar Solu- t i on of the Multicomponent Laminar Boundary Layer by an Integral- Matrix Method. AIAA J . , vol. 6, no. 6: 1089-1097.

Kendall, Robert M; Anderson, Larry W . ; and Aungier, Ronald H. 1972. Nonsimilar Solution for Laminar and Turbulent Boundary-Layer Flows over Ablating Surfaces. AIAA J . , vol. 10, no. 9: 1230-1236.

Kendall, Robert M. 1968. A General Approach t o the Thermochemical Solution of Mixed Equilibrium-Nonequilibrium, Homogeneous o r Heterogeneous Systems. Pt. V - An Analysis of the Coupled Chemically Reacting Boundary Layer and Charring Ablator. NASA Contractor Report, NASA CR-1064.

Ba r t l e t t , Eugene P.; Kendall, Robert M; and Rindal, Roald A. 1968. A Unified Approximation f o r Mixture Pransport Properties for Kulticomponent Boundary-Layer Applications. Pt. I V - An Analysis of t he Coupled Chemically Reacting Boundary Layer and Charring Ablator. NASA Contractor Report, NASA CR-1063.

Grose, Ronald D . ; and Kendall, Robert M. 1970. A Nonsimilar Solu- t i o n fo r Laminar and Turbulent Boundary Layer Flows Including Entropy Layers and Transverse Curvature. NASA Contractor Report, ,

NASA CR-73481.

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Anderson, Larry W. ; and Kendall, Robert M. 1970. User's Manual - Volume I1 - Boundary Layer Integral Matrix Procedure (BLIMP). Technical Report No. kFWL-TR-69-114, Vol. 11. Air Force Weapons Laboratory, Kirtland AFB, New Mexico.

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APPENDIX A

UNSTEADY CHARACTERISTICS SOLUTION

A quasi, one-dimensional unsteady charac te r i s t ics solution is used

for t he shock points i n t h e rad ia t ing , inviscid flow f i e l d solution.

The present method f o r t h i s solut ion i s similar t o the methods given by

C a l l i s ( r e f . 7 ) and by Moretti and Abbett ( r e f . 6 ) . A br ie f development

of t he equations i s given below and fur ther de t a i l s are available i n the

above mentioned reports .

I n t h e analysis , t h e pertinent quant i t i es considered are t he normal

component of the veloci ty and the Y-coordinate. Since the tangent ia l

veloci ty i s unchanged across a shock wave, the tangent ia l veloci ty i s

not considered a pertinent quanti ty and the S-momentum equation i s

neglected. Consequentially, the equations f o r mass, Y-momentum, and

energy are considered as quasi, one-dimensional equations modified by

forcing terms on the RHS of the equations. The forcing terms a re the

terms i n t he equations which do not contain der ivat ives , which contain

der ivat ives with respect t o the S-coordinate, and which contain t he

tangent ia l velocity.

The mass equation (eq. 17 ) i s writ ten as

where t he as te r i sk re fe rs t o t he natural logarithm of p t h a t i s

different ia ted. The Y-momentum equation (eq. 19) i s writ ten as

I n t he inviscid flow f i e l d analysis , the mass equation i n terms of

density and the energy equation i n terms of enthalpy were combined t o

form a new equation i n terms of pressure. This combined equation re-

placed the mass equation; however, i n the character is t ics development,

t he combined equation i s used for the energy equation. Thus, the energy

equation (eq. 30) i s writ ten as

The forcing terms are

The forcing terms a r e considered as known quant i t ies and the character-

i s t i c s equations and the compatibility equations are solved for by

standard procedure.

The d i f f e r en t i a l re la t ions for the variables p , p , and v of the

independent variables Y and T are

p! dT + p! dY = dp*

The equations A l t o A3 and A7 t o A9 writ ten i n matrix form are

The charac te r i s t ics equations a r e found by se t t i ng the associated co-

e f f ic ien t determinant equal t o zero.

( A l l )

The solution of equation All i s a cubic i n dY/dT and the charac te r i s t ics

equations are

Equation A12 is a t r i v i a l solution and i s neglected. The compatibility

equations are found by subst i tut ing a column of t h e associated coeff ic i -

ent determinant by the RHS of equation A10 and the determinant i s s e t

equal to zero.

dT dY 0 dp* 0 0

0 O d T d V O 0

0 0 0 dp dT dY

l A O C 1 0 0

0 0 v, c, 1 0 -

~6 0 0 0 ' C 3 1 A

The solution of equation A14 is

dY and substituting for - from equation A13, the compatibility equations

dT

are

From equations A13 and u 6 , the right running characteristic is used in

the solution for the shock points and the equations a r e

APPENDIX B

STAGNATION-POINT SOLUTION FOR RADIATING,

INVISCID FLOW FIELD

The equations for t he radiat ing, inviscid flow f i e l d around an

axisymmetric blunt body a re reducedso t h a t they are applicable f o r

only a stagnation-point solution. Note, the equations presented i n

t h i s section a re not used for t h e stagnation point of the complete body

solution but are used i n a separate program when okly a stagnation-

point solution i s required.

The following conditions, fron symmetry of the body, ex i s t a t t he

stagnation l i n e .

u = 0 v = O I S B = 0

u = o Y h = O S 8 = n/2 t

Ps = 0 Ps = 0 K = l I With these conditions, the equations for mass, Y-momentum, and energy

(eqs. 30, 19, and 20) become

mass :

energy :

The equation for S-momentum (eq. 18) is degenerate at the stagnation

line; but, by differentiation of equation 18 with respect to S and

applying the symmetry conditions the following equation is obtained

The S derivative of the tangential velocity, u is used in the stag- s' nation-point solution rather than the tangential velocity. The parame-

ters in the equations are

In order to solve the set of equation an expression is needed for p SS '

An approximation for this parameter is taken from Newtonian theory

Pss = -2 ( B 9 )

The characteristic equation and the compatibility equation for the

unsteady characteristic solution at the shock location become

where

The equations presented i n t h i s section are non-dimensionalized by

the expression given i n equation 1 4 . Indeterminate terms were evaluated

by the L'Hospital ru le .

APPENDIX C

COMPUTER PROGRAM FOR RADIATING, INVISCID

FLOW FIELD SOLUTION

A program l i s t i n g and a description of inputs are given f o r the

computer program developed f o r t he radiat ing, inviscid flow f i e l d solu-

t ion. The program i s wri t ten i n FORTRAN I V language f o r CDC 6000 se r i e s

computers. I n order t o run the program, all variables have t o be i n i t i -

a l l y s e t t o zero by means external t o the program. A system subroutine

cal led by a SETCORE control card i s used for t h i s external i n i t i a l i z a t i o n

for the computer system used i n the present study.

The inputs and printed outputs f o r a sample case are a lso given.

The sample case i s a radiat ing solution for a .97 C02 - .03 N2 (by

volume) gas composition. The body shape i s a hemispherical nose,

.3048 meter radius, with a 60' half-angle, conical afterbody.

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