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CONVECTIVE A N D RADIATIVE HEAT TRANSFER TO BLUNT BODIES AT ANGLE OF ATTACK

by H. HOSHlZAKl

https://ntrs.nasa.gov/search.jsp?R=19670007242 2018-07-03T03:09:51+00:00Z

CONVECTIVE AND RADIATIVE HEAT TRANSFER TO BLUNT BODIES

AT ANGIX OF ATTACK

H. Hoshizaki

4- 06- 66- 11

Fina l Report, Par t 11,

Prepared Under Contract NAS 7-386

July 1966

LOCKHEED P A L 0 ALTO RESEARCH LABORATORY L O C K H E E D M I S S I L E S & S P A C E C O M P A N Y A G R O U P D I V I S I O N OF L O C K H E E D A I R C R A F T C O R P O R A T I O N

I'

FOREXORD

The work described i n t h i s repor t w a s completed f o r the

National Aeronautics and Space Administration Headquarters,

under the terms and spec i f ica t ions of Contract NAS 7-386, issued through NASA Resident Office - JPL, 4800 Oak Grove

Drive, Pasadena, Cal i fornia .

The work was performed i n the Aerospace Sciences Laboratory,

R. Capiaux, Manager, of the Lockheed Pa10 Alto Research

Laboratory.

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LOCKHEED P A L 0 ALTO RESEARCH LABORATORY L O C K H E E D M I S S I L E S 6 S P A C E C O M P A N Y

A G R O U P D I V I S I O N O F L O C K R E E D A I R C R A F T C O R P O R A T I O N

ABSTRACT

The convective and rad ia t ive k a t t r a n s f e r t o b lunt bodies

a t large angles of a t t ack i s invest igated. I n the ana lys i s

the shock shape i s assumed t o be a known quant i ty . Approx-

imate solut ions t o the integrated t angen t i a l and azimuthal

momentum equations are used t o determine the shock l aye r

ve loc i ty p ro f i l e s . A f i n i t e difference type method i s

used t o solve the energy equation. A c r u c i a l assumption

made i n the ana lys i s i s t h a t t h e crossflow i s l o c a l l y

similar. This assumption reduces the general three-dimen-

s iona l problem t o a two-dimensional one where the azimuthal

angle appears as a parameter. The solut ion y i e lds the con-

vective and r ad ia t ive heat t r a n s f e r d i s t r i b u t i o n as well as

the de t a i l ed s t ruc tu re of the shock layer . The r ad ia t ive

heat t r a n s f e r i s evaluated spec t r a l ly taking i n t o account

the coupling between the flow f i e l d and the rad ia t ion

t ranspor t . Typical r e s u l t s f o r one f l i g h t condition a re

presented.

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LOCKHEED P A L 0 ALTO RESEARCH LABORATORY L O C K H E E D M I S S I L E S 6 S P A C E C O M P A N Y A G R O U P D I V I S I O N OF L O C K H E E D A I R C R A F T C O R P O R A T I O N

J

ACKNOWDGMENTS

The author would like to express his sincere appre- ciation to A. C. Buckingham and K. H. Wilson of the laboratory f o r their helpful discussions and suggestions

during the course of the work described in this report. Particular thanks are extended to H. R. Kirch for her many contributions in IBM code revisions, programming,

and suggestions on the numerical analysis.

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LOCKHEED P A L 0 ALTO RESEARCH LABORATORY L O C K H E E D M I S S I L E S & S P A C E C O M P A N Y A G R O U P D I V I S I O N O F L O C K H E E D A I R C R A F T C O R P O R A T I O N

Sect ion

FOFEWORD

ABSTRACT

ACKNOWIJ3DGME34TS

N O N E N C L A m

1 INTRODUCTION

2 ANALYSIS

2 . 1 Method of Solution

2.2 Thin Shock-Layer Equations

2.3 In tegra t ion of Momentum Equations

2 .4 Evaluation of In t eg ra l s

2 .5 Energy Equation

2.6 Shock Shape and Shock Boundary Conditions

3 GAS PROPERTIES

3 .1 Thermodynamic and Transport Proper t ies

3.2 Number Densi t ies

3.3 Absorption Coeff ic ients

4 NlTMERICAL METHOD

5 DISCUSSION OF THE “ULTS

6 REFEFiENCES

APPENDlXA. NUMBERDENSITY

AP??ENDM B. AESORPTION COEFFICIENTS

i x

Page iii

v v i i

x i 1

2

2

4 10

16 18 24

27

27

27 28

31 32

39 43 45

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ILUTSTRATIONS

Figure 1 2

3

4 5

6

7

8

9

10

11

Body- Oriented Coordinate System

Azimuthal Distribution of Surface and Heat Transfer Over an Elliptical Cone at Angle of Attack

Azimuthal Distribution of Surface Pressure and Heat Trensfer %nr cc B11I;t’,ez,l CUX ai h g i e of AttacK Summary of Absorption Coefficients Velocity and Enthalpy Profiles for CY = loo, cp = 0, 5 = 0.155 (stagnation point), R = 2.64 ft and rm = 1.03 ft Velocity and Total Enthalpy Profiles for Cy = loo, (p = 30°, 5 = 0.18, = 3.45, h = 1.7, R = 2.64 ft and r = 1.03 ft - Convective Heat Transfer Distribution, CY = 10 , U = 3.45, h = 1.7, R = 2.64 ft, rm = 1.03 ft Radiative Heat Transfer Distribution, CY = 10 , 5 = 3.45, 5 = 1.7, R = 2.64 ft, rm = 1.03 ft - Monochromatic Radiative Flu, CY = 10 , U = 3.45, h = 1.7, R = 2.64 ft, rm = 1.03 ft Quantum-Mechanical Correction Factor 5 (hv) Effective Cross Section of Molecules

= 3.45, 5 = 1.7,

0 m

0

0 -

X

20

21

29

33

34

35

37

38 46 50

NOICENCLATURE

A

a i

bi

Bv

C P

C

Di j T

D, I

E

n E

f

h

hi

h -

k

K

i;

L

Mi

Avogadro ' s number

t angen t i a l ve loc i ty p ro f i l e coe f f i c i en t s

crossflow ve loc i ty p ro f i l e coe f f i c i en t s

spec t r a l Planckian in t ens i ty

t o t a l spec i f i c heat a t constant pressure

l i g h t ve loc i ty

d i f fus ion coe f f i c i en t f o r a multicomponent system

thermal d i f fus ion coef f ic ien t

rad ian t emission per un i t time per u n i t volume

exponential function

t angen t i a l ve loc i ty r a t i o , u/u6

s t a t i c enthalpy, a l so Planck constant

s t a t i c enthalpy of ith species, including enthalpy of formation

a l t i t u d e i n 10 f t .

i n t e g r a l s defined by Eqs. (22), (29) or (33)

crossflow ve loc i ty r a t i o , w/w6

t o t a l thermal conductivity, a l s o Boltzmann constant

defined by Eq. 45

frozen thermal conductivity

reference length

molecular weight of species i

5

xi


MO

N

n

n

P

i

t

pr

R

Re

r

T

Um

Ir

U

V

W

X j

u

X

x, Y, Cp

6

6 - ?1

e

molecular weight of undissociated gas

number densi ty , a l s o defined by Eq. (46)

moles of species i per u n i t volume

t o t a l number of moles per u n i t volume

s t a t i c pressure

t o t a l Prandt l number, Pr = CpiL/k

convective energy f lux

rad ia t ive energy f lux

body radius of curvature

Reynolds number, p6, UmR/p6,

body radius measured from body cen te r l ine

temperature

f ree- stream ve loc i ty

ve loc i ty i n 10 fps 4

veloc i ty component p a r a l l e l t o body i n x -d i r ec t ion

ve loc i ty component normal t o surface

ve loc i ty component p a r a l l e l t o body i n cp-direction

mole f r ac t ion of species j , x dis tance from geometric center , (Fig. 1)

= N ~ / N ~

body-oriented coordinate system

shock detachment dis tance

transformed shock detachment d is tance

Dorodnitzyn var iab le

body angle, a l s o kT

x i i

n m n

v

'p

X

body curvature

1 + ny

absorption coefficient, cm

dynamic viscosity

kinematic viscosity, also frequency

nondimensional surface distance x/R

dens it y

density ratio across shock, p,

Stefan-Boltzmann constant

effective cross-section, O* = nv/N

monochromatic optical thickness

azimuthal angle

-1

/ p 6

\/kT

Sub scripts

a atom

e electron

i

in inviscid

m molecule, also maximum

ith chemical species, also ion

nitrogen molecule N2

O 2

N nitrogen atom

oxygen molecule

0 oxygen atom

S shock

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4

0

total

wall quantities

quantities immediately behind shock

free- stream condition

stagnation point

barred quantities are non-dimensional

xiv

Section 1

INTRODUCTION

In t h i s paper the problem of determining the convective and r ad ia t ive heat

t r a n s f e r t o b lunt bodies a t la rge angles of a t t ack i s considered. Flow

f i e l d s about b lunt bodies a t angles of a t t a c k a r e of i n t e r e s t s ince space

vehicles en ter ing planetary atmospheres w i l l perform l i f t i n g maneuvers t o

l i m i t the decelerat ion and t o minimize aerodynamic heating. Previous inves-

t i g a t i o n s of flow f i e l d s about bodies a t angles of a t t a c k have been r e s t r i c t e d

t o non-radiating, i nv i sc id flows ( R e f . 1-4) or inv isc id , r ad ia t ing flows about

pointed cones (Ref. 5) .

Webb (Ref. 1) used the inverse method while Waldman (Ref. 2) employed the

method of i n t e g r a l r e l a t i o n s t o solve the angle-of-attack b lunt body problem.

Bohachevsky (Ref. 3) on the o ther hand, has examined the unsteady, three-

dimensional flow about b lun t bodies where the asymptotic so lu t ion ( i n t ime)

gives a good descr ip t ion of the steady-state so lu t ion .

developed a semi-empirical method f o r pred ic t ing the shock shape of b lunt

bodies a t angle-of -a t tack .

Kaattari (Ref . 4) has

In the present inves t iga t ion the rad ia t ing , viscous shock l aye r on b lunt

bodies a t la rge angles-of-attack i s examined.

i s t o simultaneously determine the convective and r ad ia t ive heat t r a n s f e r .

Energy l o s s by r ad ia t ion , self-absorption and the coupling between the invis-

c i d and viscous regions due t o rad ia t ive energy t r a n s f e r are taken i n t o

account. Mass i n j e c t i o n e f f e c t s a r e cur ren t ly neglected.

The object ive of the ana lys i s

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LOCKHEED PAL0 ALTO RESEARCH LABORATORY L O C K H E E D M I S S I L E S 6 S P A C E C O M P A N Y A G R O U P D I V I S I O N OF L O C K H E E D A I R C R A F T C O R P O R A T I O N

Section 2

ANALYSIS

2 . 1 METHOD OF SOLUTION

The general problem considered i s the rad ia t ing , viscous b lunt body problem

a t angle-of-attack.

Lilt. Tiuw s i ruc iure i n tne shock layer are t o be determined for a spec i f ied

body shape and angle-of-attack.

the general problem.

dimensional shock layer when both the body and shock shape a r e prescr ibed.

The t h i n shock l aye r approximations a r e used t o reduce the e l l i p t i c conserva-

t i o n equations t o parabolic equations. The upstream influence, namely, the

influence of the corner region (see Fig. 1) on the flow near the s tagnat ion

point , i s accounted f o r by the prescribed shock shape.

Kaattari's method w i l l be used t o determine the shock shape.

does not include the e f f e c t of rad ia t ion t r a n s f e r on the shock shape, the

present ana lys i s i s l imi t ed t o conditions where such e f f e c t s are small.

T h i s i s a d i r e c t problem i n tha t the shock shape and

I n t h i s paper we do not attempt t o solve

Instead we inves t iga te the flow s t ruc tu re of the three-

I n t h i s ana lys i s

Since t h i s method

The t h i n shock layer approximation l i m i t s t he present ana lys i s t o the f r o n t

face of b l u n t bodies. This i s not a ser ious l i m i t a t i o n s ince the convective

and rad ia t ive heating a r e most severe i n t h i s region. I n t h i s ana lys i s the

shock l aye r i s divided i n t o inv i sc id and viscous regions. The ex ten t of

t h e viscous region i s unknown and i s t o be determined as p a r t of the so lu t ion .

The shock layer gas i s assumed t o be i n thermodynamic equilibrium. The

emission and absorption of rad ian t energy i s evaluated spec t r a l ly w i t h the

one-dimensional, p a r a l l e l slab approximation.

h

c

I

Fig. 1 Body-Oriented Coordinate System

3


A c ruc ia l part of t he ana lys i s i s the e f f ec t ive decoupling of the crossflow

from the a x i a l flow by the assumption t h a t the shock detachment dis tance i n

the transformed coordinate system and the enthalpy and crossflow ve loc i ty

p r o f i l e s are slowly varying functions of the azimuthal angle.

imation enables solut ions t o be obtained along geodesics, independently Of

t he flow along neighboring geodesics.

dependent on the l o c a l crossflow.

similar" approximation employed i n two-dimensional boundary-layer theory.

I n t h i s ana lys i s we a r e e s s e n t i a l l y assuming that the flow i s l o c a l l y similar

i n the azimuthal d i rec t ion .

T h i s approx-

The solut ion along a geodesic i s , however,

This approximation i s similar t o the f f l o c a l l y

I n order t o obtain sa t i s f ac to ry t angen t i a l ve loc i ty p ro f i l e s , the t angen t i a l

ve loc i ty is assumed t o be l i n e a r i n the inv i sc id region and cubic i n the vis-

cous region. The ve loc i ty i s matched a t the inviscid-viscous in t e r f ace t o

obta in a continuous p r o f i l e . The inv i sc id t angen t i a l ve loc i ty gradient i s

obtained from the in tegra ted (across the shock l aye r ) t angen t i a l momentum

equation. The cross-flow ve loc i ty i s a l s o determined i n a similar manner w i t h

t he tangent ia l momentum equation replaced by the in tegra ted a z i m u t h a l momentum

equation. The edge of viscous l aye r i s determined from the in tegra ted contin-

u i t y equation.

manner r e s t s on the knowledge that t h e inv i sc id ve loc i ty p r o f i l e s a r e near ly

l i n e a r (Ref. 6-8). A fu r the r j u s t i f i c a t i o n i s t h a t approximate ve loc i ty pro-

f i l e s w i l l r e s u l t i n accurate t o t a l enthalpy p r o f i l e s i f the energy equation

i s solved by f i n i t e d i f fe rences or similar numerical methods.

The j u s t i f i c a t i o n f o r construct ing the ve loc i ty f i e l d i n t h i s

2.2 THIN SHOCK-LAYER EQUATIONS

The conservation equations which are v a l i d f o r t h i n shock l aye r s when the

viscous region thickness i s t h e same order-of-magnitude as the shock-detach-

merit d is tance w i l l be presented. The a d d i t i o n a l s impl i f ica t ion which r e s u l t s

when the viscous region i s assumed t o be t h i n w i t h respec t t o the shock-layer

thickness w i l l then be indicated. Since the shock-detachment dis tance i s

O ( P ) and t h e thickness of the viscous region i s O(l '&e) , we have f o r a

4


viscous shock layer , ‘j -l/d-i% . from the conservation equations (Ref. 9) by neglect ing a l l terms of O(p ) and higher. The body-oriented coordinate system used i n t h i s ana lys i s i s

shown i n Fig. 1. Carrying out the order-of-magnitude ana lys i s r e s u l t s i n

t h e following equations va l id t o O(p).

The approximate equations can be obtained -2

x- momentum

y- mome n t um

+ pT a v +--I p aw +&- a (ux) ap + - - a U ap - - -- 1 ap a ( r u ) Y acp ay ax r ay ax

y-momentum

LOCKHEED P A L 0 ALTO RESEARCH LABORATORY L O C K H E E D M I S S ~ ~ C S L S P A C E C O ’ M F ~ N Y A C R O U P D I V I S I O N O F L O C K H E E D A I R C R A F T C O R P O R A T I O N

cont inui ty

a a a - - ( r p u ) + - ( i rpv ) + - (npw) = o ax a Y a'p

energy

(4)

ah pw ah - a p w a p + l a (-k-) 1 a T + pv ay + - - - - - + v - + - - pu ah Fax r a'p E ax ay r a'p ?E rk ay

N

For terms of O ( ! ) i n the above equations we have s e t u = 1 and r = r t o be

consis tent w i t h our order-of-magnitude approximation. These equations when

reduced t o the two-dimensional case, d i f f e r s l i g h t l y from the t h i n shock l aye r

equations presented by Hayes and Probstein (Ref. 10) s ince our ana lys i s i s

not r e s t r i c t e d t o stagnation-point flows.

W

The thermal conductivity which appears i n the energy equation i s the t o t a l

thermal conductivity defined as the sum of the t r a n s l a t i o n a l and reac t ion con-

d u c t i v i t i e s . This very usefu l concept of combining the t ranspor t of energy

by ordinary conduction and d i f fus ion of r eac t ing species has been discussed

by Hirschfelder (Ref. 11) and employed by severa l authors (Ref. 12-14) i n

obtaining solut ions t o the boundary-layer equations. The t o t a l thermal con-

duc t iv i ty i s defined by the following equation.

2 M. M . D i j Ej dX - D T ]

k = ' k - l h i [ $ t 1 1 J i j .+ i

The rad ia t ion term E i n the energy equation s p e c i f i e s the ne t emission or absorption of radiant energy per u n i t volume per u n i t time. It i s ca lcu la ted

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LOCKHEED PAL0 ALTO RESEARCH LABORATORY L O C K H E E D M I S S I L E S 6 S P A C E C O ' M C X N Y A G l O U P D I V I S I O N O F L O C K M E E D A I R C R A F T C O R P O R A T I O N

assuming the shock layer is locally one-dimensional. That is, f o r any value

of x and cp , temperature and density gradients in the x and 'p directions at

all points across the shock layer are neglected insofar as radiation transport

is concerned. In addition, the geometric approximation is made that the shock

layer geometry is that of an infinite plane slab.

It is clear that without the one-dimensional approximation the radiation

transport calculation is much more difficult to perform. More significantly,

the parabolic character of the thin shock-layer equations is destroyed as

influence can now be propagated upstream via the radiation field. imation of replacing the shock-layer geometry by an infinite plane slab requires,

for optically thin shock layers, that the shock-layer thickness be small relative

to the local radius of curve, i.e., SIR<< 1. layers the requirement for the plane slab approximation is that the mean free

path for radiation xR be small compared to the local radius of curvature, i.e., hR/R<< 1.

The approx-

For optically thick shock

Hence, the plane slab approximation becomes more accurate in the important

spectral ranges when the shock layer is optically thick. Indeed, Kennet, and

Strack (Ref. 15) show that for optical depth unity, the plane slab approxima-

tion is in error by only 5% for an isothermal, spherical shock layer. Concern-

ing the approximation of one-dimensional temperature and density fields, Koh (Ref. 16) has investigated the effects of the non-isothermal temperature on the radiation flux at the stagnation point of a sphere and found only a 3$ difference from an isothermal solution when the shock layer has an optical depth of

unity. Although these results are for axially symmetric flows, the conclusion

that the one-dimensional plane-parallel slab model can be used to obtain quan-

titatively valid results will hold for asymmetric flows as long as the azimuthal

and axial variations are comparable.

Except for the locally one-dimensional treatment, the radiation term is eval-

uated in an exact manner. No approximations restricting the analysis to opti-

7

LOCKHEED P A L 0 ALTO RESEARCH LABORATORY L O C K H E E D M I S S I L E S L S P A C E C ' O , M P ~ N Y A G Q O U P D I V I S I O N O F L O C K H E E D A I R C R A F T C O R P O R A T I O N

c a l l y th in or o p t i c a l l y th ick shock layers a r e made.

treatment Of t he emission and absorption processes i s made. Moreover, t he

rad ia t ion t ranspor t i s performed i n t e r m s of a de t a i l ed s p e c t r a l ana lys i s using

r e a l i s t i c absorption coef f ic ien ts . The s p e c t r a l t ranspor t solut ion shows t h a t

grey gas approximations using a Planck mean opaci ty produce r e s u l t s which are

i n e r r o r by as much as an order of magnitude (Ref. 17).

Instead a f u l l i n t e g r a l

The rad ia t ion f l u x divergence can be expressed as

43 I. I.

I n Eq. ( 7 ) , UV i s the volumetric ( spec t r a l ) absorption coe f f i c i en t ; I, i s the

l o c a l rad ia t ion in tens i ty ; Bv i s the equi l ibr ium (i .e., Planckian) r ad ia t ion

in t ens i ty .

Solving f o r the i n t e n s i t y under the one-dimensional approximation and perfom-

i n g the in tegra t ion over s o l i d angle ind ica ted by. Eq. (7) leads t o the follow-

ing expression f o r the f lux divergence ( R e f . 18').

where Tv i s the o p t i c a l depth a t frequency v

8

LOCKHEED P A L 0 ALTO RESEARCH LABORATORY t O C K H E E D M I S S I L E S 6 S P A C E C O ' M P A N Y A G Q O U P D I V I S I O N O F L O C K H E E D A I R C R A F T C O R P O R A T I O N

B, i s the spec t r a l Planckian in tens i ty ,

2hJ 1 2 exP(hv/BT)-l B, =

C

E n ( t ) i s the exponent ia l - integral of order n,

00

dz exp(-tz) E , W = so Z

In a r r i v i n g a t Eq. (8) it has been assumed t h a t t h e medium outside the shock

wave ne i the r emits, absorbs, o r r e f l e c t s rad ian t energy and the body surface

a c t s as a per fec t absorber and does not r e rad ia t e .

for a given shock layer temperature d i s t r ibu t ion , are evaluated numerically

using the spec t r a l absorption coef f ic ien ts described i n Subsection 3.3.

The quadratures i n Eq. ( 8 ) ,

The t h i n shock layer equations (Eqs. 1-5) were derived on the b a s i s t h a t t he

th ickness of t he viscous region i s comparable t o t h e shock layer thickness.

S igni f icant en t ry heating usual ly occurs when the Reynolds number i s between

10 and 10 so t h a t the viscous layer i s much thinner than the shock layer .

For these conditions we can fur ther simplify the conservation-equations by

s e t t i n g N. = 1 and r = rw i n a l l of the viscous terms.

4 6

-

The energy equation can be more conveniently d e a l t with when wr i t ten i n terms of

t o t a l enthalpy. This i s accomplished i n the usual manner by combining the

momentum equations with the energy equation wr i t ten i n terms of t he s t a t i c

enthalpy (Eq. 5) . given by Eqs. (1) and (3) with E = 1 and r = r

The appropriate form of the x and cp-momentum equations a r e

i n the viscous terms only. W

The appropriate y-momentum i s obtained by neglect ing a l l terms which are of higher order than aP/ay since the pressure gradient term i n the energy equa-

9

LOCKHEED P A L 0 ALTO RESEARCH LABORATORY L O C K H E E D M l S S l l E S & S P A C E C O M P A N Y A C Q O U P D I V I S I O N O F L O C K H E E D A I R C R A F T C O R P O R A T I O N

t i o n i s O ( p ) .

t i o n t o

Neglecting the higher order term reduces the y-momentum equa-

The j u s t i f i c a t i o n for the add i t iona l s impl i f ica t ion can perhaps be b e t t e r

explained by point ing out tha t the o r i g i n a l form of the y-momentum equation

(Eq. 2) i s cor rec t t o O ( p ) with respect t o i t s e l f .

t i on i s used i n cnngiinrl.ticn ~ t t k t k ~ C z C i - g i - c y u a i i u n ine nigher order terms

i n the y-momentum become of O ( P ) and can be neglected s ince the pressure

gradient terms i n the energy equation a r e of O ( p ) .

When the y-momentum equa-

-2

Combining the momentum equations with the energy equation (Eq. 5) resu l t s i n

u2 w2 where the t o t a l enthalpy i s H = h + ~ + z and Pr i s the t o t a l Prandt l

number.

2 .3 INTEGRATION OF MOMENTUM EQUATIONS

The veloci ty f low f i e l d w i l l be determined according t o the method out l ined

i n Subsection 2.1. I n accordance with t h i s method the viscous x-momentum equa-

t i o n (Eq. l) i s in tegra ted across the shock l aye r and i s used t o determine

the inviscid tangent ia l ve loc i ty gradient .

thus s a t i s f i e d i n the la rge .

across the shock layer and i s used t o determine the inv isc id , crossflow V e l O -

c i t y gradient. With the a id of t he cont inui ty equation, the x and cp-momentum

equations can be manipulated i n t o forms convenient for i n t eg ra t ion . Carrying

The viscous x-momentum equation is

The azimuthal momentum equation i s a l s o in tegra ted

10


out the manipulations r e s u l t s i n the following equations.

x-momentum

cp- momentum

In tegra t ing the x and cp-momentum equations across the shock l aye r y ie lds

x- moment um

2 a6 6 6

2 a Jo f2 dy + 2 u au6 Jo ps P 2 f dy - u6 ax + (vu), 6 ax 6 u6 ax

11


A G R O U P D I V I S I O N O F L O C K H E E D A I R C R A F T C O R P O R A T I O N

y - mome n tun

u W and j = - W f = -

u6 6 where

The densi ty r a t i o across the shock wave has been assumed t o be a constant

along the shock surface.

Stream normal veloci ty component remains hypersonic (Ref. 19). it has been assumed t h a t t he pressure grad ien t i n the azimuthal d i r ec t ion i s

independent of y.

This i s a good approximation as long as the free- I n addi t ion

This approximation i s j u s t i f i e d on the basis t h a t the

12

pressure gradient is a second order term in the momentum equation.

The compressibility effects can be reduced considerably by introducing the

Dorodnitsin transformation.

T

The shock detachment distance is related to the transformed shock detach-

ment distance by

1 1

Transforming the momentum equations (Eqs. 16-17) from x, y, cp to 5 , 1, cp

results in

x-momentum

LOCKHEED P A L 0 ALTO RESEARCH LABORATORY L O C K H E E D M 4 S S I L E S 6 S P A C E C O M P A N Y


'p- momentum

where primes denote d i f f e ren t i a t ion w i t h respect t o and

I I = 6 f d v

0 L o

E, = rw/L

F = 6/L

5 = x / L

In the x-momentum equation we have made the approximation t h a t

The coupling between the crossflow and t h e a x i a l flow i n the momentum equations

i s represented by t h e underlined terms i n Eqs. 20 and 21. Neglecting these

terms i s equivalent t o assuming t h a t % and both the crossflow and a x i a l ve loc i ty

p r o f i l e s a r e slowly varying functions of t he azimuthal angle.

momentum equations a r e s t i l l dependent on the l o c a l crossflow. By neglecting

t h e underlined terms, t he momentum equations can be solved along geodesics by

t r e a t i n g the azimuthal angle a s a parameter.

e n t i r e asymmetric flow f i e l d by obtaining a s u f f i c i e n t number of so lu t ions

along geodesics.

Note that the

One can then construct the

Additional approximations can be made t o the in tegra ted cp-momentum equation

(Eq. 21). The fou r th term on the L.H.S. can be approximated by

By considering l i n e a r and constant ve loc i ty p r o f i l e s , t h e value of t h e integ-

ra l i s estimated t o be between 1/3 and 1. Since 6 i s s l i g h t l y g rea t e r than

6 f o r a cold w a l l (see Eq. 18), the two terms i n parentheses tend t o cancel

each o ther , and hence, can be neglected. I n the s i x t h term the following

-

approximation i s made

LOCKHEED P A L 0 ALTO RESEARCH LABORATORY L O C K H E E D M I S S I L E S L S P A C E C O M P A N Y A G R O U P D I V I S I O N O F L O C K H E E D A I R C R A F T C O R P O R A T I O N

since the term i n parentheses i s iden t i ca l ly zero f o r spher ica l bodies

and small f o r near spherical bodies.

x/(r/rw) =: 1 w a s employed.

I n the s i x t h term the approximation -

With these approximations, the cp-momentum equation reduces t o

- - - - -

The solut ion t o the integrated momentum equations requi res the evaluat ion

of the in tegra ls and the boundary conditions a t the shock.

a r e discussed i n subsequent sect ions.

These two items

2.4 EVALUATION OF INTEGRALS

The in t eg ra l s i n the in tegra ted momentum equations (Eqs. 20, 23) can be

evaluated by specifying the form of the ve loc i ty p r o f i l e s .

region, the ve loc i ty p r o f i l e s are assumed t o be cubics .

I n the viscous

16

In the inv isc id flow, the p ro f i l e s are assumed t o be l i n e a r .

1

f i n = 1 - f i n ( l ) ( L Y )

The following boundary conditions are used t o determine the viscous ve loc i ty

p r o f i l e coe f f i c i en t s .

(1) q = o ; u = w = O

(2) 1 = 0 ; momentum equations evaluated a t the wall

w = w v i n

LOCKHEED P A L 0 ALTO RESEARCH LABORATORY L O C K H E E D M I S S ' I L E a 6 S P A C E C O M P A N Y A G R O U P D I V I S I O N O F L O C K H E E D A I R C R A F T C O R P O R A T I O N

By means of the above boundary conditions, the ve loc i ty p r o f i l e coe f f i c i en t s

and the in tegra ls can be expressed i n terms of three unknowns, namely, t he

inv i sc id a x i a l and azimuthal ve loc i ty gradients and the thickness of t h e

viscous layer , 'fl - These unknowns a r e determined by means of the in tegra ted

tangent ia l and azimuthal momentum equations (Eqs. 20 and 23) and the

cont inui ty equation integrated across the shock layer . Sa t i s fy ing the

integrated cont inui ty equation insures t h a t mass i s conserved along a

geodesic. The integrated cont inui ty equation can be wr i t t en as

V

- - a& - - - + n6 w6 acp - n6 r6 v6 - - a i - - 'bU6

The underlined term i n Eq. (28) w i l l be neglected t o be cons is ten t with

the assumptions made previously.

2 . 5 ENERGY EQUATION

I n the energy equation (Eq. 13) only one crossflow term appears e x p l i c i t l y .

Convincing arguments can be p u t f o r t h which j u s t i f y the neglect of t h i s term.

We note, f i r s t of a l l , that the t o t a l enthalpy behind the shock wave i s a

constant .

a l s o a constant a t the wall. We therefore suspect t h a t the va r i a t ion of t he

t o t a l enthalpy i n the azimuthal d i r e c t i o n w i l l be r a t h e r small.

reasons f o r suspecting t h a t t h i s i s t h e case a r i s e by inspect ion Of the

expression f o r t he convective hea t t r a n s f e r . I n the transformed coordinate

Furthermore, for a constant wall temperature the t o t a l enthalpy i s

Additional

18

LOCKHEED PAL0 ALTO RESEARCH LABORATORY L O C K H E E D M I S S - l L E S " h S P A C E C O M P A N Y A G R O U P D I V I S I O N O F L O C K H I E D A I R C R A F T C O R P O R A T I O N

system the convective heat t r a n s f e r can be wr i t t en as

When normalized t o some reference point such as the stagnation point , the

heat t r a n s f e r d i s t r ibu t ion poin t i n e i t h e r the a x i a l or az imutha l d i r ec t ion

takes on the following form:

Again f o r a constant wall temperature the pp r a t i o i s proport ional t o the

pressure r a t i o so t h a t the hea t t r ans fe r d i s t r i b u t i o n can be wr i t t en as

s ince the term i n the bracket i s nearly uni ty .

Now i f the t o t a l enthalpy gradient a t the wall i s a weak function of the

azimuthal d i rec t ion , as suspected, then the heat t r a n s f e r and pressure

d i s t r i b u t i o n s should be qui te similar. I n order t o check t h i s hypothesis

t h e pressure and heat t r a n s f e r d i s t r ibu t ion around e l l i p t i c cones a t angle-

of -a t tack were examined. I n Figs . 2 and 3, the normalized heat t r a n s f e r and

surface pressure d i s t r ibu t ions are compared. It i s seen that the re i s a close

s i m i l a r i t y up t o the poin t where the flow separates (Cp = .6n). t r a n s f e r and pressure da ta were obtained from two separate t es t s c a r r i e d out

a t AEDC (20, 21) .

the heat t r a n s f e r model the differences are ins ign i f i can t a t la rge angles-of-

a t t a c k .

The heat

Although the pressure model i s s l i g h t l y d i f f e r e n t from

LOCKHEED PAL0 ALTO RESEASCH LABORATORY L O C K P E E D M I S S I l ' E S ' 6 S P A C E C O M P A N Y A G R O U P D I V I S I O N O F L O C K H E E D A I R C R A F T C O R P O R A T I O N

3 E

P;-

.4 W I% m z 4 P; P H

E 3

2 n

3 E a u \ a u

1. 0

.̂ 10-A w P;

2 m W P; & La 0

10-

Mm = 10 ; STATION - 44% O F BODY LENGTH QJ

v v

v 4 = 0 V

v

R RATIO O F MAJOR T O MINOR AXES (ELLIPTICAT, CROSS SECTION)

IN PITCH PLANE ,q = rpp'pCyX.rE CS::~ ; ; A ~ F A;<GLE

C v

V HEAT TRANSFER V PRESSURE ff = 30"

-

V

V V

PRESSURE COEFFICIENT HEAT TRANSFER COEFFICIENT

. (Y *c 'Pmax SOURCE a ' c hmax R SOURCE - v _ _ - 30" 10" 8 . 5 - 1 1 . 5 (v) R E F . 21 30" 7 " 4 . 3 - 3 1 . 4 3 ( v ) R E F . 20

v v I I 1 1 1 1 I

0 . 2 0.4 0.6 0. 8 1. 0

AZIMUTHAL ANGLE, $/n

Fig. 2 Azimuthal Distribution of Surface and Heat Transfer Over an Elliptical Cone at Angle of Attack

20

v v

v

v v Mm = 10 ; STATION-60% O F BODY LENGTH

V

v

V

PRESSURE COEFFICIENT

(Y ' c '',ax R SOURCE

30" loo 8.5-1 1.5 (v) REF. 21

# = O 1 2

c p ? P w - pm'2pmUm

TW) h P qJ(T, -

R RATIOOFMAJOR TO MINOR AXES (ELLIPTICAL CROSS SECTION)

IN PITCH PLANE ec = EFFECTIVE CONE HALF ANGLE

V HEAT TRANSFER V PRESSURE (Y = 30"

V

V HEAT TRANSFER COEFFICIENT

- (Y - ' c - hrnax SOURCE

30" 7" 3.5-3 1.43 ( v ) R E F . 20

v v I 1 I 1 I 1

AZIMUTHAL ANGLE, # / T

0.2 0.4 0.6 0.8 1 .-0

Fig. 3 Azimuthal Distribution of Surface Pressure and Heat Transfer Over an Elliptical Cone at Angle of Attack

21



The r e s u l t s f o r the pressure and heat t r a n s f e r d i s t r i b u t i o n i n the azimuthal

d i rec t ion ind ica te that the t o t a l enthalpy gradient a t t he w a l l i s a weak

function of the crossflow ve loc i ty as suspected. I n o ther words, the cross-

flow ve loc i ty has a small e f f e c t on the convective heating. For t h i s reason

the crossflow term i n the energy equation w i l l be neglected. However, the

crossflow ve loc i ty can play an important r o l e i n determining the r ad ia t ive

heat t r ans fe r and the thermodynamic and t ranspor t p roper t ies through i t s

influence on the s t a t i c enthalpy p r o f i l e s .

where the mass f l u x normal t o the surface w a s replaced by the following

expression obtained by in t eg ra t ing the g loba l cont inui ty equation.

I n deriving Eq. (32), it w a s assumed that, aq/acp = 0.

consis tent w i t h the assumption t h a t 6 and the t o t a l enthalpy and Velocity

This assumption i s a

22

p r o f i l e s are slowly varying functions of the azimuthal angle.

The quan t i t i e s I and I3 are defined by o,n

Following the general procedure of Smith and Ja f f e ( R e f . 22) f o r obtaining

nonsimilar solut ions, the streamwise gradients a r e replaced by backward

differences.

It i s again assumed t h a t the crossflow ve loc i ty p r o f i l e s are slowly varying

functions of cp so t h a t t he underlined term i n Eq. 32 can be neglected.

In order t o determine the gas propert ies and r ad ia t ive f l u x divergence, t he

pressure var ia t ion across the shock layer mus t be known.

pressure var ia t ion i s obtained from an approximate solut ion t o the inv isc id

y-momentum equation.

The required

Integrat ion of t he inv i sc id y-momentum equation y i e lds

6 2 a v a v P = P6 + Jy pu ax dy + sy pu ay dy - sy * N dy

U

It i s now assumed f o r purposes of evaluating the pressure p r o f i l e only t h a t

t he v -ve loc i ty p r o f i l e s a re l i n e a r and only a function of q, i . e . , v/v6 = 11. Transforming t o 5 , ll r e s u l t s i n

23

LOCKHEED PAL0 ALTO RESEARCH LABORATORY L O C K H E E D M l S S I C ~ I ? S 6 S P A C E C O M P A N Y A G R O U P D I V I S I O N O F L O C K H E E D A I R C R A F T C O R P O R A T I O N

The densi ty p r o f i l e i n the t h i r d t e r m on the R.H.S. of Eq. (36) i s approxi-

mated by

which i n e f f ec t means that we are ignoring the e f f e c t of pressure va r i a t ion

on the densi ty p r o f i l e . I n a l l o ther p a r t s of t h i s ana lys i s , the e f f e c t of

pressure on the dens i ty p r o f i l e i s taken i n t o account.

The energy equation (Eq. 31) i s seen t o be of the form

The solut ion for g i s

where the constants of i n t eg ra t ion of C and C are determined from the 1 2

known values of enthalpy a t the w a l l and shock.

2 .6 SHOCK SHAPE AND SHOCK BOUNDARY CONDITIONS

The shock shape can be obtained from e i t h e r theory o r experiment.

ana lys i s the semi-empirical method of Kaattari ( R e f . 4) i s used t o determine

In t h i s

24

LOCKHEED P A L 0 ALTO RESEARCH LABORATORY L O C K H E E D M I . S S I d ' € S & S P A C E C O M P A N Y A G R O U P D I V I S I O N O F L O C K H E E D A I R C R A F T C O R P O R A T I O N

the shock shape. According t o Kaat tar i , t he shock detachment dis tance f o r

blunt bodies (see Fig. 1) a t angle-of-attack can be expressed a s

where C and C a r e known constants. The shock shape can be r e l a t e d conven-

i e n t l y t o the body geometry by the radiat ion 1 2

r6 = r + sine W

where 8 i s the body surface angle.

The boundary conditions on the shock w i l l be expressed. in terms of t he shock

slope i n the r and ‘p d i rec t ion . The displacement of the shock f ront from

t h e shock f ron t locat ion a t t h e geometric center , x , (see Fig. 1) i s given N

by

3 - x = R [l-cos(sin-’ T6) ] + d [C + C3 ($6) ] cosxpsinCr

s. Rs cy=o rm m

from which the required slopes can be determined.

I n our body-oriented coordinate system the required veloci ty components a r e

t h e normal and tangent ia l components i n the x and ‘p di rec t ion . These

components can be r ead i ly obtained from the ana lys i s presented i n Ref. (1).

LOCKHEED P A L 0 ALTO RESEARCH LABORATORY L O C K H E E D M I S S I 4 , ’ E S 6 S P A C E C O M P A N Y A G R O U P D I V I S I O N O F L O C K H E E D A I R C R A F T C O R P O R A T I O N

- - ax siny sina, K = cos@ + - coscp sincr - - ax

ar racp

The pressure a t t he shock i s given by

(45)

(46)

The solutions t o the momentum equations (Eqs. 20, 23) must proceed from the

po in t where u6 = o t o eliminate t h e streamwise de r iva t ive of the i n t e g r a l s I1 and I*. Eq. (42) .

expressed as polynomials of t h e normal ( t o t h e sur face) var iab le q, the stag-

na t ion point on the body i s on the normal which i n t e r c e p t s t he shock a t the

I n o ther words, t he s tagnat ion streamline i s assumed t o poin t where u

be a s t r a i g h t l i n e . Although it i s r e a l i z e d t h a t t h i s i s not co r rec t (Ref. 1,

23), the inaccuracy introduced i s not se r ious s ince both the convective and

r a d i a t i v e fluxes vary slowly i n the immediate v i c i n i t y of t he s tagnat ion

po in t .

The curve on the shock surface where u6 = o can be obtained from

Note t h a t since w e have assumed t h a t t h e ve loc i ty p r o f i l e s can be

6 =

26

LOCKHEED P A L 0 ALTO RESEARCH LABORATORY L O C K H E E D M I S ' S I L ' E S 6 S P A C E C O M P A N Y A G R O U P D I V I S I O N OF L O C K H E E D A I R C R A F T C O R P O R A T I O N

Section 3

GAS PROPERTIES

3.1 THERMODYNAMIC AND TRANSPORT PROPERTIES

The ana lys i s up t o t h i s point i s va l id f o r any multicomponent gas i n

thermodynamic equilibrium. Numerical so lu t ions can be obtained i f t he AL-.--A7T*mn17 ~ I I ~ ~ ~ ~ ~ ~ ~ ~ lI-LL:L, n trazspcrt, and- rad ia t ive proper t ies are known. The thermo-

dynamic and t ranspor t p roper t ies , as computed by Hansen (Ref. 24) and

cor re la ted by Viegas and Howe (Ref. 25), are used i n the numerical so lu t ions .

The cor re la t ion formulas a r e va l id for temperatures between 1000 and

15,000°K and for pressures between 0 . 1 and 100 atmospheres.

0

3.2 WER DENSITIES

The absorption coe f f i c i en t s which a re discussed i n Subsection 3.3, are

most conveniently expressed i n terms of t h e gas-par t ic le number dens i t i e s .

The number dens i t i e s can be e a s i l y obtained from the a i r equation of state

if an idea l ized a i r chemistry model i s adopted.

d i ssoc ia t ion i s assumed t o begin f i r s t as the a i r temperature i s increased.

Af te r a l l the oxygen molecules are dissociated, the ni t rogen molecules

commence d issoc ia t ing . When a l l of the molecules a r e dissociated, ion iza t ion

of t h e atoms begins and no d i s t inc t ion i s made between the ion iza t ion of

the oxygen and ni t rogen atoms.

In t h i s model, oxygen

The number dens i ty of the molecules, atoms, and ions can be expressed i n

terms of the t o t a l number densi ty and the compressibi l i ty Z, by means of the

d e f i n i t i o n s of the t o t a l mass and t o t a l number dens i ty .

27

LOCKHEED P A L 0 ALTO RESEARCH LABORATORY L O C K H E E D M I E ’ S I L E S 6 S P A C E C O M P A N Y


Pt = Pm + Pa + Pi + Pe

N t = N + N + N i + N e m a

With the equation of state

Equation 48 can be rewritten as

N t = N m % Z + N a s Z + N i M , & z Mo M,

(48)

(49)

By considering particular intervals of the compressibility factor Z in which the gas chemistry is specified, the number densities can be obtained in terms

of the total number density and the compressibility factor by means of Eqs. 49-51. tervals have been previously presented in Ref. (26j and are given in Appendix A for convenience.

The number density equations for the specified compressibility in-

3 .3 ABSORPTION COEFFICIEWTS

The radiative energy transfer within the shock layer depends critically on

the absorption coefficient of the gas particles.

is dependent on the nature of the gas particle, its number density, the gas temperature, and the radiation frequency. The absorption coefficients utilized

in the present analysis are summarized in Fig. 4. cular absorption coefficients for air given in Refs. 27, 28, and 29 shows that the only important air molecular absorption coefficients are the O2

Schumann-Runge continuum and the N Birge- Hopf ield band.

The absorption coefficient

An examination of the mole-

2

28


I I

01 2 ;

I I

1

29


A l l of the important absorption coe f f i c i en t s summarized i n Fig. 4 a r e given.

by simple a n a l y t i c a l expressions i n terms of the p a r t i c l e number densi ty ,

temperature, and the r ad ia t ion frequency. The equations a r e used t o

evaluate the r ad ia t ion f l u x gradient term i n the energy equation t o obtain

numerical solut ions.

The continuum absorption coe f f i c i en t s due t o bound-free and f ree- f ree

t r ans i t i ons f o r the neu t r a l and s ingly ionized atoms of ni t rogen and oxygen

were obtained from the theory of Biberman (Ref. 3 0 ) and Armstrong e t a l

( R e f . 31). t r a n s i t i o n s from the higher exc i ted states while Armstrong's d e t a i l e d

quantum-mechanical ca lcu la t ions are used t o provide cross-sect ions f o r

photon absorption due t o t r a n s i t i o n s from the ground and low ly ing exc i ted

states.

Riherman's r e s l l l t ~ zx :sel l Tor tne r'ree-free and bound-free

The absorption coe f f i c i en t f o r a i r i s wr i t t en as the sum of the ind iv idua l

species absorption coe f f i c i en t s

where the term i n the f i r s t bracket accounts f o r induced emission. The

absorption coe f f i c i en t of the oxygen ion i s assumed t o be equal t o tha t of

the nitrogen ion. This approximation i s acceptable s ince the ions cont r i -

b u t e l i t t l e t o the t o t a l absorption coe f f i c i en t except a t the higher f r e -

quencies where the rad ia t ion from the shock l aye r gas i s l i k e l y t o be black

body.

The absorption coe f f i c i en t s for the var ious species a r e presented i n Ref. (26) and i n Appendix B.

LOCKHEED P A L 0 ALTO RESEARCH LABORATORY L O C K H E E D M I S 3 1 C E S & S P A C E C O M P A N Y A G R O U P D I V I S I O N O F L O C K H E E D A I R C R A F T C O R P O R A T I O N

Section 4

NUMERICAL METHOD

The in tegra ted momentum and energy equations a r e solved by an i t e r a t i v e

method. A so lu t ion i s f i rs t obtained a t the point where u6 = 0 .

on the body where u6 = o depends on the value of azimuthal angle which i s

t r e i t e d 8 s 8 pa.ra.md-.er i n the solut ion.

p r o f i l e and a value for the transformed shock detachment dis tance, 6 , a r e

assumed. With these two quan t i t i e s known, Eqs. 20, 23, and 28 a r e solved

simultaneously f o r the inv isc id a x i a l and azimuthal ve loc i ty gradients and

the viscous l aye r thickness, VV.

The point

A t t h i s point a t o t a l enthalpy N

With these three quan t i t i e s known, the ve loc i ty p r o f i l e s can be evaluated.

The pressure var ia t ion across the shock layer can now be determined from Eq. (36). Equation (19) i s now used t o check the assumed value of 6 . We

now i t e r a t e on

p r o f i l e . With t h i s solut ion f o r the ve loc i ty f i e l d , the energy equation

(Eq. 38) i s solved t o obtain a new enthalpy p r o f i l e .

repeated u n t i l a converged so lu t ion for the enthalpy p r o f i l e i s obtained.

N

u n t i l we have a converged solut ion f o r t he assumed enthalpy

The procedure i s

To obta in solut ions around the body the equations are in tegra ted both i n

t h e windward and leeward d i rec t ion .

i s i d e n t i c a l t o t h a t used a t the s t a r t i n g poin t .

The procedure for solving the equations

LOCKHEED P A L 0 ALTO RESEARCH LABORATORY L O C K H E E D M I S ? i I L E S 6 S P A C E C O M P A N Y A G R O U P D I V I S I O N O F L O C K H E E D A I R C R A F T C O R P O R A T I O N

Section 5

DISCUSSION OF THE RESULTS

Numerical r e s u l t s have been obtained f o r one f l i g h t condi t ion (U, =

34,500 fps , h = l70,OOO ft) for the body shape shown i n Fig. 1 with

R = 2.64 f t and rm = 1.03 f t .

shown i n Fig. 5. The t angen t i a l and azimuthal ve loc i ty p r o f i l e s should

be i d e n t i c a l a t the s tagnat ion point s ince the body being considered i s

a spher ica l segment. The s l i g h t d i f fe rence i n the ve loc i ty p r o f i l e s i s

due t o the approximations made i n the so lu t ions t o the x and cp-momentum

equations.

t o be s l i g h t l y l e s s than the value behind t h e shock. This decrease i n

t he inv i sc id l aye r i s due t o r ad ia t ion cooling. The la rge gradients near

the w a l l are, of course, caused by viscous e f f e c t s . Similar r e s u l t s are

presented i n Fig. 6 f o r cp = 30 and 5 = 0.18. The ve loc i ty p r o f i l e s are

again seen t o be qui te similar although i n t h i s case the c ross flow

ve loc i ty of w6 i s approximately twice the t a n g e n t i a l ve loc i ty u6.

t o t a l enthalpy p r o f i l e a t t h i s body s t a t i o n i s a l s o qu i t e similar t o the

stagnation-point p r o f i l e . These r e s u l t s ind ica te t h a t t he flow over the

face of the b lunt body considered i s not t o o d i f f e r e n t from the flow a t the stagnation-point.

These r e s u l t s a r e f o r an angle a t t a c k

sf 120. TT-- I vLLuL~t,y - - 2 L rniiialpy p r o f i l e s a t the s tagnat ion point are

The t o t a l enthalpy near the edge of the viscous l aye r i s seen

0

The

Convective heat t r a n s f e r d i s t r i b u t i o n s are shown i n Fig. 7. The stagna-

t i o n point i s located a t < = 0.155. The convective heat transfer increases

on the windward side and decreases on t h e leeward s ide of the s tagnat ion

poin t . The increase i n the convective h e a t . o n the windward s ide i s due t o

the decrease i n the shock-detachment d is tance which r e s u l t s i n an increase

i n the w a l l enthalpy gradien t .

more than compensates for the decrease i n dens i ty which r e su l t s i n an

This increase i n the wall enthalpy gradien t

32

LOCKHEED P A L 0 ALTO RESEARCH LABORATORY L O C K H E E D M I S S I L E S h S P A C E C O M P A N Y A G R O U P D I V I S I O N O F L O C K H E E D A I R C R A F T C O R P O R A T I O N

I I I I

1 I I I

I

1 I I I

0 00 (0 * N 0

r( 0 0 0 0

9 / f f '33NtrLSIa TVNHON

0

?I

a3 0

W

0

* 0

c\l

0

0

0

7 4

00

0

(0

0

* 0

N

0

0

n

& z

n

M

v) v) I+

0 I1 I*I,

0

I I 9- 0 0 rl

II a k 0 +i

m a,

0

4

r;'

33 LOCKHEED P A L 0 ALTO RESEARCH LABORATORY L O C K H E E D M I S S I L E S & S P A C E C O M P A N Y A G R O U P D I V I S I O N O F L O C K H E E D A I R C R A F T C O R P O R A T I O N

I I 1 I 0 a) co * 04

rl 0 0 0 0

1 1 I I a) * @a

0 0 0 0

Q/A ‘33NV,T,SI(I TVINXON

0 a, tll

CJ t- W II

0 0

h

- 7 3 v

I I I I I 0 a3 W TY N

4 0 0 0 0

35 LOCKHEED P A L 0 ALTO RESEARCH LABORATORY L O C K H E E D M I S S I L E S 6 S P A C E C O M P A N Y A G R O U P D I V I S I O N O F L O C K H E E D A I R C R A F T C O R P O R A T I O N

M

0

N

0

l-4

0

0

d

0 I

M 0

l-i

0 0 rl

..

h rn 3

Y cd s

0 u c-

bb .d Fr

increase i n convective heating.

gradient and densi ty decrease which causes convective heating t o decrease.

The convective heat t r a n s f e r i s seen t o be a weak funct ion of the azimuthal

angle. T h i s i s not surpr i s ing since the angle a t t ack was taken t o be 10 . A s the angle a t t ack approaches zero, t he d i s t r ibu t ions should become independent

of Cp. If the angle a t t ack i s increased, the e f f e c t of t he azimuthal angle

on the d i s t r ibu t ions w i l l become more pronounced.

On the leeward s ide both the wall enthalpy

0

Radiative heat transfer d i s t r ibu t ions are shown i n F ig . 8. t i ons are seen t.n ~ . ~ C ~ P P _ S P Tr. 52th t h e ~ i ~ i k ~ i - 6 iccwatr.6 d i rec i ions .

The reason f o r t h i s behavior i s that the r ad ia t ive heat t r a n s f e r i s dependent

on the e n t i r e temperature p r o f i l e s i n the shock layer rather than on l o c a l

quan t i t i e s . The temperature p r o f i l e s a r e qui te insens i t ive t o body loca t ion

on the face of a blunt body. The temperature behind the shock, however,

fa l ls off s l i g h t l y from the stagnation poin t value i n both the leeward and

windward d i rec t ions . This decrease i n the temperature behind the shock i s

responsible f o r t he decrease i n the r ad ia t ive heat t r a n s f e r away from the

stagnation point .

function of t he azimuthal angle f o r t h e angle of a t t a c k considered.

l a r g e r angles of a t t ack we would again expect a pronounced e f f e c t of azi:

muthal angle.

These d i s t r ibu -

The r ad ia t ive heat t r a n s f e r d i s t r i b u t i o n i s a l s o a weak

For

The monochromatic r ad ia t ive f lux f o r two body s t a t i o n s i s shown i n Fig. 9. These r e s u l t s show that approximately half t he r ad ian t energy i s i n the

u l t r a v i o l e t region with the remaining ha l f i n the v i s i b l e and inf ra red

region. The difference between 'p = 0 and cp = 45 small which i s cons is ten t w i th the r e s u l t s shown i n Fig. 8. and t h e d i s t r ibu t ions shown i n Fig. 8 do not include l i n e r ad ia t ion .

0 i s seen t o be qui te

These r e s u l t s

36

LOCKHEED PALO ALTO RESEARCH LABORATORY L O C K H E E D M I S S I L E S 8, S P A C E C O M P A N Y A G R O U P D I V I S I O N O f L O C K H E E D A I R C R A F T C O R P O R A l I O N

0 Q) rn I

3. !z (D t- r(

It

0 k

h

.@ v

I I I I I 0 a3 CD TY c.1

4 0 0 0 0

37

n 3

N

3

4

0

0

4

0 I


5 = 0.155

ED VISIBLE U TRAVIOLET I I I I I I I I I

0 2 4 6 8 10 12 14 16 18

RADIATION FREQUENCY, hv (eV)

0

- Fig. 9 Monochromatic Radiative Flux, (y = IO", U = 3.45, % = 1.7, R = 2.64 f t , and

1: = 1.03 f t m

38

Section 6

REFERENCES

1. Webb, H. G. , Jr., Dresser, H. S., Adler, B. K. and Waiter, S. A . , "An

Inverse Solution f o r the Determination of Flow Fie lds About Axisymmetric

Blunt Bodies a t Large Angle of Attack", AIAA Paper No. 66-413, AIAA 4th

Aerospace Sciences Meeting, Los Angeles, Cal i forn ia , June, 1966.

2. Waldman, G. W . , " In tegra l Approach t o the Yawed Blunt Body Problem",

AIAA Paper No. 65-28, AIAA 2nd Aerospace Sciences Meeting, New York,

New York, January 1965.

3. Bohachevsky, I. 0. and Mates, R. E . , "A Direct Method for Calculation of

t he Flow About an Axisymmetric Blunt Body a t Angle of Attack", CAL Report

No. A l - 1972-A-4, Cornell Aeronautical Laboratory, Cornell University,

Buffalo, New York, December 1965.

4. Kaat ta r i , G. E . , "Shock Envelopes of Blunt Bodies a t Large Angles of

Attack", NASA TN D-1980, 1963.

5. Chin, J. H. and Hearne, L. F., "Shock-Layer Radiation for Yawed Cones

With Radiative Decay", AIAA Journal, Vol. 3, No. 6, June 1965, pp.

1205- 1207.

6. L igh th i l l , M. J . , "Dynamics of a Dissociat ing G a s , Par t I, Equilibrium

Flow", J. Fluid Mech., Vol. 2, 1957, p. 1.

7. Xeriokos, J. and Anderson, W. A. , "A C r i t i c a l Study of the Direct Blunt

Body In t eg ra l Method", DAC-SM-42603, Douglas Ai rc ra f t Corporation,

December 1962.

39

LOCKHEED P A L 0 ALTO RESEARCH LABORATORY L O C K H E E D M I S S I L E S 6 S P A C E C O M P A N Y A G R O U P D I V I S I O N OF L O C K H E E D A I R C R A F T C O R P O R A T I O N

8.

9.

10.

11.

12.

13 *

14.

15 -

16.

17 *

Swigart, R. J . , "A Theory of Asymmetric Hypersonic Blunt Body Flows",

ALAA Journal, Vol. No. 5, 1963, pp. 1034-1042, and p r iva t e communication.

Scala, S. M., "The Equations of Motion i n a Multicomponent Chemically

Reacting Gas", Aerophysics Operation Research Memo. 5, General E l e c t r i c

Co., Missile and Ordnance Systems Dept., Philadelphia, Pennsylvania, 1957.

Hayes, W. D. and Probstein, R. F. , Hy-personic Flow Theory, Academic Press,

New York, 1959, pp. 388-389.

Hirschfelder, J. O. , "Heat Transfer i n Chemically Reacting Mixtures, I",

J. Chem. Phys. - 26, 274-285 (1957).

Hoshizaki, H. , "Heat Transfer i n Planetary Atmospheres a t Super -sa te l l i t e

Speeds", ARS J. (October 1962).

Cohen, N . B . , "Boundary-Layer Similar Solutions and Correlat ion Equations

f o r Laminar Heat Transfer Dis t r ibu t ion i n Equilibrium A i r a t Veloci t ies

up t o 41,000 Feet Per Second", NASA TR R-118, 1961.

Pallone, A. and Van Tassel, W . , "Stagnation Point Heat Transfer f o r A i r

i n the Ionizat ion Regime", ARS J. - 32, 436-437 (1962).

Kennet, H. and Strack, S. L . , "Stagnation Point Radiative Transfer",

ARS J. - 31, NO. 3, 370-372 (1961).

Koh, J. C . Y . , "Radiation From Nonisothermal Gases t o the Stagnation Point

of a Hypersonic Blunt Body', ARS J. - 32, 1374- ( 1962) .

Hoshizaki, H . , "Some Aspects of Radiation Transfer During Hypervelocity

Reentry", Proceedings of t he Workshop on the In t e rd i sc ip l ina ry Aspects

Of Radiation Transfer, J o i n t I n s t . for Lab. Astro. , Univ. of Colo., Vol. 11, 12 February 1965, p. 6.

18. Vincenti, W. G. and Kruger, C . H. , Jr., Introduction t o Physical Gas-

dynamics, John Wiley & Sons, Inc . , New York, 1965.

19. Marrone, P. V. , "Normal Shock Wave i n A i r : Equilibrium Composition and

Flow Parameters f o r Veloci t ies From 26,000 t o 50,000 fps" , CAL Report

No. AG-1729-A-2, Cornell Aeronautical Laboratory, Buffalo, New York,

Aug. 1962.

20. Hillsamer, M. E . , and Rhudy, J. P., "Heat-Transfer and Shadowgraph Tests

of Several E l l i p t i c a l L i f t i ng Bodies a t Mach lo", AFDC-TDR-64-19, Feb. 1964.

21. Palko, R. L., and Ray, A. D . , "Pressure Dis t r ibu t ion and Flow Visualiza-

t i o n Tests of a 1 . 5 E l l i p t i c Cone a t Mach lo", AEDC-TDR-63-163, Aug. 1963.

22. Smith, A. M. 0. and Ja f f e , N. A . , "General Method f o r Solving the Laminar

Nonequilibrium Boundary-Layer Equations of a Dissociating Gas", AIAA J.

- 4, No. 4, (1966).

23. Hayes, W . , "Rotational Stagnation Point Flow", J. Fluid Mech., Vol. 19, No. 3, 1964, p. 366.

24. Hansen, C . F . , "Approximations for the Thermodynamic and Transport Prop-

e r t i e s of High-Temperature A i r " , NASA TR R-50, 1959.

25. Viegas, J. R. and Howe, J. T . , "Thermodynamic and Transport Property Cor- 0 0 r e l a t i o n Formulas f o r Equilibrium A i r From 1,000 K t o 15,000 K , " NASA TN

D-1429, October 1962.

26. Hoshizaki, H. and Wilson, K. H . , "Convective and Radiative Heat Transfer

During Superorbi ta l Entry", LMSC Report No. 4-43-65-5, Lockheed Missile &

Space Co., November 1965, (Revised June 1966).

41


27. Churchill, D. R. , Armstrong, B. H. , and Mueller, K. G . , "Absorption Co-

e f f i c i e n t s of Heated A i r :

Vol. 1, Lockheed Missi les & Space Co., October 1965. A Compilation t o 24,000 K", AFWL TR-65-132,

28. Evans, J. S. and Schexnayder, C . J . , Jr., "An Inves t iga t ion of the Effect

of High Temperature on the Schumann-Runge Ul t r av io l e t Absorption Continuum

of Oxygen", NASA Technical Report R-92, 1961.

29. Allen, R. A . , Textor is , A. , and Wilson, J . , "Measurements of the Free-

Bound and Free-Free Continua of N i t r e g P C , !?x;.;yc~ Afi -" , ;. &a~i,.

Spec. Rad. Trans. (January - February 1965).

30. Biberman, L. M. and Norman, G. E. , "Recombination Radiation and Brehmstra-

lung of a Plasma", J. Quant. Spec. Rad. Trans. - 3, 221-245 (1963).

31. Armstrong, B. H . , Brush, S. , DeWitt, H. , Johnston, R. R . , Kelley, P. S . ,

and Platas , 0. R . , "Opacity of High Temperature A i r " , A i r Force Weapons

Laboratory Report, AF W L T R 65-17, June 1965.

42

Appendix A

NUMBER DENSITY

The expressions which give the number density as a function of the total

number density and the compressibility factor are easily derived from Eqs.

49-51 and are listed below:

Dissociation of Oxygen: 1.0 < Z 5 1.2

N = Nt 0.8 - N2 Z

Equation A-3 states that the number density of N2 is equal to 0.8 times the total number density of air calculated, assuming no dissociation.

Dissociation of Nitrogen: 1.2 < Z 5 2.0

- 0.4 No - Nt z

43


Single Ionization of Atoms: 2 <-Z 5 4

4- z N 0 + NN = Nt (y)

N. = Nt (7) z- 2 1

Double Ionization of Atoms: 4 < ZS 6

Nii = Nt (1-E) 4

44

(A-8)

(A-9)

(A-10)

(A- 11)

.

Appendix B

AI3 SORPTION COEFFIC IEDTTS

The absorption coe f f i c i en t s for t he various species a r e l i s t e d below f o r the

indicated in t e rva l s .

where 16 2 2 - a = 7.25 x 10 cm -ev

45

LOCKHEED PAL0 ALTO RESEARCH LABORATORY

(B- 3 1

03-41

(B-51

L O C K H E E D M I S S I L E S 6 S P A C E C O M P A N Y A G R O U P D I V I S I O N OF L O C K H E E D A I R C R A F T C O R P O R A T I O N

\ \ \ \ \

46

LOCKHEED P A L 0 ALTO RESEARCH LABORATORY


L O C K H E E D M I S S I L E S 6 S P A C E C O M P A N Y

47


where

The absorption coe f f i c i en t s f o r molecular oxygen Schumann-Runge continuum

were obtained from the r e s u l t s of Evans and Schexnayder (Ref. 28).

28 the authors discuss the approximate formula of Sulzer-Wieland and poin t

out i t s deficiency a t e levated temperatures (T z 10,OOO°K). A t lower temp-

e ra tu re s where the molecular oxygen i s present i n a gas mixture i n chemical

equilibrium, the approximate r e s u l t s of SulzeG-Wieland given below are adequate

f o r present purposes.

I n Ref.

N. = 1.49 x 10- [ tanh (q ) ]112 exp [- tanh (7) 0.0975 V

48

(B- 11)

The absorption coe f f i c i en t f o r the nitrogen Birge-Hopfield band was ob-

ta ined from r e s u l t s of R. Allen ( R e f . 29). The c ross sect ion f o r T = 700OoK w a s approximated and used t o obtain the following expression for t he absorp-

t i o n coe f f i c i en t .

n = 1.2 x 10-17 N exp [ - I hv - 13.6 + (1- 7) 0.603 (B-l2) N2

V

The e f f ec t ive c ross sect ions, defined by

n

N V o* = -

for the two molecules discussed above are presented i n Fig. 11.

49

(B- 13)



n- b ..

z I3 u 0,

10-18 m 0" P; u w I3 u w crc crc w

2

lo-l!

N2 BIRGE- HOP FIELD - 0 2 (S-R) CONTINUUM

e = 0.25 I EQ. (B-11)

1 I I I I I I I

6 7 I 1 I

FREQUENCY, hv (eV)

Fig. 11 Effective Cross Section of Molecules

50

convective and radiative heat transfer to … · convective and radiative heat transfer to blunt...

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