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S.A CONTRACTOR REP0 RT * NASA CR.73Z23 ,m¢ GPO PRICE $ CFSTI PRICE(S) $ Hard copy (HC) Microfiche (M F) ff 653 July 65 RADIATION TRANSPORT FOR BLUNT-BODY FLOWS INCLUDING THE EFFECTS OF LINES,AND ABLATION LAYER by Jin H. Chin (THRU) (NASA CR O_R TMX OR AD NUMBER) .......... Preparedby LOCKHEED MISSILES & SPACE COMPANY Sunnyvale, Califarnia for Ames Research Center NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. December 1967 https://ntrs.nasa.gov/search.jsp?R=19680014234 2020-04-03T04:04:02+00:00Z

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RADIATION TRANSPORT FOR BLUNT-BODYFLOWS INCLUDING THE EFFECTSOF LINES,AND ABLATION LAYER

by Jin H. Chin

• (THRU)

(NASA CR O_R TMX OR AD NUMBER) ..........

Preparedby

LOCKHEED MISSILES & SPACE COMPANYSunnyvale, Califarniafor Ames Research Center

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. December 1967

https://ntrs.nasa.gov/search.jsp?R=19680014234 2020-04-03T04:04:02+00:00Z

NASACR- 7 5

RADIATION TRANSPORT FOR BLUNT-BODY FLOWS

INCLUDING THE EFFECTS OF LINES AND ABLATION LAYER

By Jin H. "Chin

Distribution of this report is provided in the interest of

information exchange. Responsibility for the contentsresides in the author or organization that prepared it.

Prepared under Contract No. NAS 2-4219 byLOCKHEED MISSILES & SPACE COMPANY

Sunnyvale, California

for Ames Research Center

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

For sale by the Clearinghouse for Federal Scientific and Technical hfforllmtion:Springfield, Virginia 22151 - Prices

FOREWORD

The work described in this report was completedfor the National Aeronautics and SpaceAdministration,Ames Research Center, under Contract No. NAS2-4219,with N. Vojvodich as Technical Monitor.

The author extends his appreciation to L. F.Hearne and K. H. Wilson for their help and discussionduring this study.

ii

CONTENTS

FOREWORD

SUMMARY

INTRODUCTION

1 NOMENCLATURE

FLOW FIE LD ANALYSIS2.1 Governing Equations2.2 StagnationFlow Field

2.2.1 Air Layer2.2.2 Ablation Layer

2.3 Streamtube Formulation

3 RADIATION TRANSPORTANALYSIS3. 1 Basic Radiative Equations3.2 Equations for a Finite Number of S.ublayers3.3 Treatment of Continuum Contributions

3.3.1 The Continuum Absorption Coefficient3.3.2 The Multi-Band Model

3.4 Treatment of Line Contributions3.4. 1 The Line Absorption Coefficient3.4.2 The Line Transmittance and Equivalent Width3.4.3 Absorption Coefficient Notation3.4. 4 Line Calculation Limiting Cases

4 NUMERICAL METHODS4.1 Division of Continuum Bands and Line Groups4.2 Schemeof Flow Field Iteration4.3 Schemeof Line Calculation

5

ii

1

1

3

7789

1013

15

15

17

19

19

20

20

20

22

24

27

3737

3738

RESULTS 43

5. 1 Effects of Exponential Kernal Approximation 435.2 Effects of Continuum Band Models 435.3 Effects of Line Model 43

5.4 Effects of Ablation Layers 445.5 Effects of the Number of Sublayers 445.6 Effects of Environmental Variables 45

5.7 Effects of Perturbation of Radiative Properties 455.8 Effects of Precursor Heating 465.9 Equivalent Width Results 465.10 STRADS Results 47

6 CONCLUDING REMARKS 47

iii

CONTENTS (Continued)

Appendix

A THERMODYNAMIC PROPERTIES OF AIR AND ABLATION VAPOR

B RADIATIVE PROPERTIES OF AIR AND ABLATION SPECIES

B. 1 Absorption Cross Sections of Air SpeciesB. 1.1 Continuum Absorption Cross SectionsB. 1.2 Line Absorption Cross Sections

B. 2 Absorption Cross Sections of Ablation Species

C THE STAGRADS CODE

C. 1 STAGRADS ListingC. 2 Input Data Listing

C. 3 Output Data Listing, Including Details of Line Data Used

REFERENCES

LIST OF TABLES

LIST OF ILLUSTRATIONSi

LISTING OF STAGRADS AND INPUT DATA

!OUTPUT DATA LISTING, INCLUDING DETAILS OF LINE DATA USED

49

5151515152

53535558

65

67

75

99

163

iv

RADIATION TRANSPORT FOR BLUNT-BODY FLOWS

INCLUDING THE EFFECTS OF LINES AND ABLATION LAYER

By Jin H. Chin

Lockheed Missiles & Space Company

SU_CIMARY

Efficient numerical methods are developed for fully radiation-coupled inviseidblunt-body flows. The spectral nature of radiation transport is treated in detail,using as many as 21 spectral regions and accounting for air and ablation vapor con-tinua and 75 discrete atomic lines. A concept of line-group equivalent width andaverage transmittance is introduced in a finite-difference formulation of radiationtransport. This concept is used to account for line overlaps and to formulate the

numerical procedures.

The application of the numerical methods is described. The effects of environ-mental variables, of the numerical methods used, and of the uncertainties in radiativeproperties are discussed. The results demonstrate that the self-absorption and energyloss effects decrease the sensitivity of the heat fluxes to the changes in environmentalvariables and to the uncertainties in radiative properties.

INTRODUCTION

This report presents the analyses and results of a study of radiation transportfor blunt vehicles returning from space missions at superorbital velocities. Thestudy is an extension of the author's previous investigation (refs. 1 and 2). In refs. 1

and 2, the effects of non-grey, self-absorption and energy loss for inviscid blunt-bodyflows were considered. The air continuum spectral absorption coefficients were r_p-

resented approximately by a 2-band (level) and a 6-band model. The contributions ofdiscrete atomic line radiation were not taken into account. The solutions of the flow

field for the stagnation and nonstagnation regions were formulated separately. Asimilarity solution was obtained for the stagnation region and a streamtube methodwas used for the nonstagnation region. The formulation of the present investigationincludes the effects of discrete atomic-line radiation and the presence of an inviscid

ablation layer adjacent to the wall. Due to time limitations, however, these effectsare incorporated in the numerical calculations for the stagnation region only. Anewer set of air continuum absorption coefficients is used for both the stagnation and

nonstagnation regions.

The purpose of this investigation is to develop economical numerical methodsfor fully radiation-coupled, inviscid, blunt-body flows, and to assess the effects ofenvironmental variables, of the numerical procedures used, and of the uncertainties

in radiative properties.

_" The flow-field analysis is reviewed in Section 2. The flow field is assumedinviscid with an interface between the ablation products and the shock layer air.The stagnation region air layer and ablation layer are analyzed using similaritytransformations, with continuity of pressure across the interface. The streamtubeformulation for the nonstagnationregion is very briefly described, as only an exam-ple result (fig. 23) is given.

The detailed analysis of radiation transport is given in Section 3. The diver-genceof the radiative flux is calculated by one-dimensional approximations. Forthe integration of the energy equation, the shock layer and ablation layer are dividedinto a number of sublayers, each with constantproperties. The integrals of the radia-tive divergence over the individual sublayers are used in the finite-difference energybalance. The various terms in the expression bf these integrals are of the samebasic functional form. Studies of the mathematical properties of this basic functionalform lead to a theory of calculating the continuumand discrete (atomic lines) radia-tion contributions. A concept of line-group equivalent width and average transmittanceis used to calculate the contribution of a group of lines, accounting for line overlaps.Readers not interested in the details of radiation transport calculations may skip mostof Section 3.

Section 4 describes the numerical methods, including the division of continuum

bands and line groups, the scheme of flow fielditeration, and the scheme of line

calculations. Some of the equations [e.g., Eqs. (129)- 156)] presented are used

in the computer program and may be omitted by readers not interested in thenumerical details.

The results of calculations are given in Section 5. Except fig. 23 which showsan example of the heat flux distribution for a blunt vehicle, all results are for the

stagnation region obtained using the STAGRADS code.

The air and ablation vapor thermodynamic properties, the radiative properties

of air and ablation species, and the description of the STAGRADS code are given inthe appendix. "

a

A

Ail, i2,m

b

b'

bnn, (v)

B

C

d ,nil

e

E

E n (_)

f

f ,nn

F

g

gn

h

H

I

k

K

_s ' _y

L

m

1 NOMENC LATURE

constant in exponential kernal approximation E 3 (_)a = 4 is used.

1 +Ky

function defined by Eq. (118)

constant in exponential kernal approximation E 3 ( _ )b = 2 is used

b ( 7re2/mc 2 )

line shape factor , Eq. (81)

Planck distribution function

speed of light

line-center shift

electron charge

energy of electronic state above ground state

n-th exponential integral

band system f-number; or A L+I pv [ RN/( L + 1) u r

absorption f-number for transition n --n!

pV/PsV s or A L+I pV/PwV w

h/h s or h/h w

fkgnk , a constant for each line

statistical weight of the n-th state

static enthalpy or Planck's quantum constant

total enthalpy, or enthalpy ratio (Appendix A)

specific intensity

Boltzmann' s constant

body curvature , = 1/R N at stagnation point

directional cosines of vector _2 , Eq. (54)

0 for two-dimensional, 1 for axisymmetric

electron mass

-b_:~ (b/a) e ,

-b_~ (b/a) e .

1/2, Eq. (40)

_e

mw

n

N

Ne

P

q

%qr ' qr

s yQ

r

ro

R

R M

RO

R*

8

g

S

T

Trans

U

Ur

v

Width

Y

Z

Z

"ge

wall blowing mass flux

number of sublayers; state index

k-th line lower state index, see Section 3.4.3

species particle number density

electron particle number density

static pressure

net radiative power gain per unit volume

radiative flux vector

components of radiative flux in s and y directions

radiative power gain for a finite sublayer; partition function

distance from plane or axis of symmetry i

value of r at streamline entry point

ratio of equivalent width to line-group width for isolated lines

maximum radius, fig. 22

nose radius

distance from stagnation point to effective axis of symmetry

ratio of equivalent width to line-group width by integration

distance along body from stagnation point

distance along body from point of symmetry, fig. 22

line strength, Eq. (110)

absolute temperature

line-group transmittance

velocity parallel to body

value of ablation layer u at interface , Ue = Ur (S/RN)

velocity normal to body

line-group equivalent width

normal distance from body surface

normal distance from shock wave

compressibility factor

line effective half-width

k-th line half-width due to 1 electron per unit volume

4

w

Ynn v

5v k

A

AHV

AY i

AZ.i

AY m

AT.]

V

Va

Vo, nn t

P

PSL

(7

T

¢

f_

Subscripts

a

ab

c

d

e

i

i I, i 2

for n-_ n' transition

integration interval for k-th line

total thickness between shock and wall

latent heat of vaporization

Y i+l - Yi

zi+ 1 -zi; Az i = Ay i

integration interval for m-th line group

/zj Az. J

transformed normal distance; dummy variable

transformed normal distance, Eq. (34)

linear absorption coefficient

frequency

average frequency within line group

frequency corresponding to unperturbed transition

transformed distance from stagnation point, Eq.variable

density

sea-level density

cross section; Stefan-Boltzmann constant

optical thickness

angle between flight vector and normal

stream function

solid angle

air layer

ablation layer; with ablation layer effect

continuum contribution

discrete contribution

at interface between air layer and ablation layer

species index; dummy index for sublayer

specific sublayers

n ..-_ n !

(33) ; dummy

5

J

k

nn t

S

sat

w

v

oO

dummy sublayer index; dummy index for frequency integration

dummy index for lines

for transition n --* n'

immediately behind shock; toward shock

at satellite velocity

at wall conditions

at frequency

at ambient conditions

2 FLOW FIELD ANALYSIS

2.1 Governing Equations

To simplify the formulation of the problem, the flow field is assumedquasi-steady; inviscid, non-conducting, and in thermodynamic equilibrium. An interfaceexists between the ablation products and the shock-layer air. The wall is assumedto be at the equilibrium sublimation temperature corresponding to the surfacepressure.

In body-oriented Coordinates (s , y) shownin fig. 1, the conservation equa-tions for a two-dimensional or axisymmetric, inviscid flow may be written as follows-

continuity

0.r Lpu + 8Ar Lpv = 0 (1)as 8y

s-momentum

au au _apu_-_+ Ave+ Kuv = - P as (2)

y-momentum

0v _v A apu_-+ Ave- - Ku 2 = --- (3)p ay

energy

OH OH AU_" + Av_, = _-q (4)

The net radiative power gain per unit volume is related to the divergence of theradiative flux:

q = - div_r = - [ ay + _J (5)

The equations-of-state of equilibrium air and equilibrium ablation vapor maybe expressed in the following form:

p , T , N i , . . - function (p, h) (6)

The boundery conditions are as follows:

At the wall,

.

u = u = 0 (7)w

4v = vw = ;-nw/P w = (qw - O'Tw)/PwAHv (8)

h = h w = h w(pw) (9)

The wall is assumed to be black.

At the shock wave,

u = u s = uoo [sin _sC°S (_bw - _s ) + ¢ cos _s sin(_w - Cs ) ] (10)

v = v s = u [sin_bsSin(_ w - _bs) - EeOSCsCOS(_ w i_bs)l (ii)!

oouP = Ps = p oo (1 - ¢) cos 2 (_s + P_ (12)*

At the interface,

Pa, e = Pab, e (14)

V ,e ab,eUa, e Uab, e

(15)

These equations will be approximated for different regions of the flow field'.

2.2 Stagnation Flow Field

For the stagnation region, s/R_ << 1 , the conservation equations may besimplified. The kinetic energy change" is of a smaller order than the enthalpy changeand the pressure variation normal to the wall is small. Equations (3) and (4) may

then be approximated by Eqs. (16) and (17), respectively.

8p = 0 or P = Ps (16)8y

8

*In superorbital velocities, Poo and hoo are negligible compared to Ps and hs, re-spectively. Precursor heating may increase hoo . However, for the velocities of

interest in this study, the precursor heating is small, as will be discussed inSection 3.1.

Oh _h A

u + Av --Oy= p q (17)

For this study, the shockwave and the interface are further assumed to be con-

centric with the body surface so that _bs = Ce = _w "

2.2.1 Air layer. - For the stagnation air layer, Eqs. (10) to (13) reduce to:

S

u s = u -- (18)

= - ¢ u (19)VS ¢o

Ps = P_ u2_ (1 - _) - + p_ i (20)

1 2 2)h s = _Uoo (1 - + h (21)

For a concentric interface,

v e = 0 (22)

For inviscid flow, the interface conditions for u and h cannot be specified;

they are part of the solution to be obtained.

Since there are more h,own boundary conditions at the shockwave, it.is more

convenient for the air layer solution to use the shockwave as the datum for the nor-mal distance. Using transformations similar to the Lees-Dorodnitsyn transforma-tions for compressible boundary layers and assuming similarity, the following re-sults were obtained in ref. 1":

_p_ d___z (23)d_ - (L + 1) Ps RN

F - - pv = __pv (24)p_ u Ps vco S

_ d F = u (25)d_ u

S

*The analysis of the stagnation flow field for the ablation layer is similar to that forthe air layer. Consequently, only the analysis for the ablation layer is presented

in detail in the following subsection.9

10

h

S

(26)

F d2Fi Ida)2d_2 = (L + 1) - 2E (1

-E)Ps

(27)

d_ (L + i) pu(28)

or

2F dg - 3 ( q dz)

P_ u_o

(28a)

where z is the distance from the shockwave toward the interface. In Eqs. (23) to(28), the shock layer has been assumed thin compared to the nose radius so thatA_I.

The boundary conditions are:

At the shockwave

dF= 0 , F - d_ - g = 1 (29a)

at the interface,

where g is the value offrom theeolution.

= _e ' F = 0 (29b)

at the interface. The value of _e must be determined

The numerical integration of Eqs. (27) and (28a) is described in Section 4. 2.

2.2.2 Ablation layer. - For the stagnation ablation layer, Eqs. (16) and (20)indicate-

2P = Pe = P_oU (1 - c) + p_ (30)

Ue

Since the layer edge is a streamline and is concentric with the wall, using

= u r (s/RN) one obtains:

dPe _ dUe 2 s

ds Pe Ue ds - PeUrlt 2N

u2 s__

- 2p_ _ (1 - e) R_q

(31)

"Hence,

1/2

Ur = [2P_(1-E)]uoo Pe(32)

where Pe is not known before the solution is obtained.

Equations (1), (2), and (17) may be simplified by introducing the followingtransformations:

S2L

UedS

0

(33)

ii

y

dy (34,_-,/_0

8_ 85 L_yy = -purL ' _ss = Apvr (35)

f(_,77) = (36)

hg(_'_) - h (37)

W

If similarity is assumed so that all dependent variables are functions of 7? only,the transformed equations may be shown as follows:

f d2f 1- df 2 PooU_o K (L + 1)R N fdf

d_?---2= (L + 1) - 2 2 AL+lp u _.pu r r

(38)

f dg = ARNdU (L + 1)PUrh w q

(39)

The transformation also yields the following relations:

f jR. ]AL+lpv (L + 1)u r

1/2

(40)

11

d_ [(L+ l)Ur]

_/2

ALp

Further simplification of Eqs. (38) and (39) is possible by introducing thefollowing:

F

f A L+I_ pv

PwVw "(L + 1)u r

1/2v w

l 2" 1/2

1d_ = 1/2 dq =

OwVw "(L + 1)u r

2P_o p A L dy(L + i)

\pwVw / Pw RN

Equations (38) and (39)become, respectively,

F

fd2F = 1 ;{dF] 2

d_2 (L + i) [\d_/[ 2)iJ2Pw 1 + 1 _PwVw F

P AL+I \2Poo u2oo

1 A L+I q dyF dg = PwVwh w

For Aab/R N << 1, A _ 1.

For conditions with

v 2 /p u 2Pww/_oo

<< 1

one may neglect the last term in Eq. (45).

(41)

(42)

(43)

(44)

(.45)

(46)

12

F d2F 1 "r/dF_ 2 Pw

d_2 (L + 1) [\d_] p(47)

The boundary conditions are as follows:

at _= y= 0

1/2

= - Pe ](48a)

at _ = _e or y = Aab

i/2

dF (Pw_ 2F = 0 , d--_ = \Pe] ' P = p u (1 - e) + p_ (48b)

Because of the last term in Eq. (47), the momentum equation is coupled to theenergy equation. The surface blowing rate and the layer thickness are not knownbefore the solution is obtained.

The numerical integration of Eqs. (46) and (47) is described in Section 4.2.

2.3 Streamtube Formulation

For the nonstagnation region, the streamtube formulation described in ref. 1 is

used. For the inviscid air layer, the conservation equations for an infinitesmalstreamtube may be written as follows:

continuity

L dy _ L dropur ds - P_u_oro d---s (49)

momentum

du dp (50)Pu _-_ = - ds

energy

d +

pu ds = q (51)

13

" where r o is the radial distance at which the streamtube enters the shock. Theentry conditions for the streamtube are given by Eqs. (10) to (13). The applicationof the streamtube method during this study has been limited to air-layer calculationswith continuum radiation only. The integration of Eqs. (49) to (51) follows the pro-cedures described in ref. 1 and will not be discussed further in this report. The

streamtube method may also be applied to the ablation-layer calculations.

14

3 RADIATION TRANSPORTANALYSIS

3.1 Basic Radiative Equations

The net radiative power gain per unit.volume by the gas in local thermodynamicequilibrium is given by

f f - qq(s,y) = dv #v(S,y)[Iv(S,y,_,¢2 ) - Bv(S,y d_2

v _=47r

(52)

where I v is the monochromatic specific intensity (power per unit solid angle per unitarea normal to the direction of propagation), By the Planck distribution function, and#v is the spectral (monochromatic) absorption coefficient accounting for induced emis-sion. The monochromatic specific intensity is governed by the following equation ofradiative transfer.

In body-oriented coordinates, Eq. (58) becomes

where _s and ty are the directional cosines of the vector _ with respect to thecoordinate axes. The flow field is assumed either two-dimensional or axisymmetricso that the gradient of the specific intensity in the circumferential direction vanishes.

The net radiative flux crossing a unit area with unit normal -_, is given by

Then,.

P_= 4r

f •qr = _s Iv (-_) d_2 (56a)

v,s _=4_

r

qr = J ty Iv (_) d_ (56b)v,y _2= 4_r

.15

Integrating Eq. (54) over all solid angles and using Eqs. (52), (56a), and (56b), one

obtains the monochromatic form of Eq. (5)

qr 0 qrv, s + v, y

Aas 0y - divqr = -qv(S'Y) (57)V

Under conditions that the temperature gradients in the s-direction are small com-

pared to that in the y-direction, one may neglect fs[a Iv(_)/A as] in Eq. (54) and

(O qrv, s/A as)

in Eq. (57) to obtain the "tangent slab" or one-dimensional approximation for radia-

tion transport, i

For the environmental condition of interest in this study (uoo< 60,000 ft/sec,

RN < 10 ft) the effectof precursor radiation heating of the ambient air ahead of the

bow shock is to increase the wall heat flux by the order of 10 percent or less (refs. 3and 4). Thus, itdoes not appear that the neglect of precursor heating in the basic

•radiative calculations contributes significantlyto the uncertainty in magnitude of radia-

tion heating. For this investigation, the precursor heating effect and the shock-wave

reflectivityare neglected and the body surface is assumed black.

For one-dimensional radiation transfer, the monochromatic net radiative power

gain or the radiative flux divergence is given by refs. 1 and 5....

0 qrv, y

0y = qv (y)

-47r #v Bv + 27r #v BW,

f ' B' E 1#v p

OY I)yf, " dy"

#v dy' (58)

where E n(_) is the nth exponential integral and A is the thickness of the "tangentslab. "

The succeeding terms on the right-hand side of Eq. (58) can be recognized as (i)

radiative loss, (2) absorption of radiation from body surface attenuated by the medium

between the body and the local point, and (3)absorption of radiation emitted by medi-

um on both sides of the local point.

16

In terms of z = A - y , the monochromatic optical thickness is given by

Z

T v(z) = f #v(Z') dz'

O

(59)

A

_'A,v = f #v(z') dz' (60)

O

Equation (58) may then be rewritten as

T_, Y

qv(rv) = -4_PvBv + 2_#vBw. vE2(TA, v - TV) + 2_.p f Bv(t)EI([T v - tI)dt0

(61)

The monochromatic heat fluxes incident to the wall and leaving the shockwave to-ward the ambient air are given by Eqs. (62) and (63), respectively,

TA, V

I "qw, v = 2_ B v(t) E 2(TA,v - t) dt (62)

O

T&, V

f "qs, v = 27r B v (t) E 2 (t) dt+ 27r Bw, v E3 (TA, V) (63)

O

Integration of Eq. (61) over all frequencies then yields the q required in Eqs. (28),(28a), and (46). Equations (62) and (63) may be integrated over all frequencies to ob-tain the total heat fluxes.

3.2 Equations for a Finite Number of Sublayers

In order to reduce the computation time required for integration of the conservationand radiative transport equations, the shock layer and ablation layer may be dividedinto a number of sublayers. Across the width of the sublaycr, constant properties

are assumed, but radiation traversing through is attenuated by each of the differentialwidths within the sublayer. Equation (61) may be integrated across the width of thei th sublayer from the shockwave (fig. 2).

17

Zi+l

Qi(v) ---f qv(z) dz (64)

Z.I

Carrying the integration, one obtains

Qi(u) = 2n BwIE3(Tn+ 1 - Ti+l) - E 3 (Tn+ 1

n

j=l

For convenience, the subscript v has been omitted in Eq. (65) and the equations

following. Equations relating monochromatic quantities may be recognized by the

presence of at least one term with argument v. The optical thickness for the sub-layers are given by

h_-i(v) = _i Azi (66a)

TI (v) = 0 (66b)

i

7"i+l(v) = _ A'r i , i = 1, 2,..., n (66c)

j=l

The monochromatic radiative heat flux to the wall, qw(V), is given by Eq. (67), ob-tained from Eq. (62).

n

qw(V) = 2_ _ Bi[E3(Tn+ 1 -Ti+l)- E3(Tn+ 1 - Ti)] (67)

i=l

The monochromatic radiative heat flux leaving the shock front may be similarlyderived.

n

qs(V) = 2v _ Bi[E3(_-i)- E3(Ti+I) 1

i=1

+ 2_ Bw E 3 (Tn+l) (68)

The total energy gain or heat fluxes may be obtained by integration of Eqs. (65),(67), and (68) over all frequencies. The spectral integration for the continuum radia-

tion and the line radiation will be discussed separately in the following subsections.

!8

The third exponential integral, E3 (_) , may be calculated using numerical correla-tions and series. However, considerable computation time may be savedby using the"exponential kernal approximation" for the exponential integrals. For instance,

E3(_) ~ b e-b_ (69)a

Matching the values of the approximate function and its slope at zero argument with theexact values E3(0) and El(0 ) =-E2(0) , respectively, one obtains b = 2 and a = 4so that

E 3(_) ~ 0.5e -2_ (70)

The individual terms of Eqs. (65), (67), and (68) are of the general form

Iij k(v) = B kE3([T i - Tj[) r

(71)*

One then can study the spectral integration of Eq. (71) as a basic step for performingthe spectral integration of the more _'nrnplin_t_d _q_ [c_l 1_7_ ._n,_ t_c_x v+ ...-11 ,.^shown in the Section 3.4.2 that the exponential kernal approximation enables a con-venient formulation of line radiation transport.

3.3 Treatment of Continuum Contribution

3.3.1 The continuum absorption coefficient. - The spectral absorption coefficients

and the optical thicknesses may be separated into two parts - the continuum (subscriptc) and the discrete (subscript d).

#(v) = ]ze + #d (72)

= + (73)T rc Td

The discrete part varies with frequency much more rapidly than the continuum part.For the present study, the discrete part corresponds to the contribution due to theatomic lines. The continuum part corresponds to the contribution of molecular bandsystems, free-free, and photo-absorption processes.

*With b = 1 and a = 1 , Eq. (71) and many of the following equations may be usedfor radiation transport calculations for a pencil of radiation.

19

3.3.2 The multi-band model. - For the continuum radiation, the multi-band modelis used. The continuum spectral absorption coefficient is assumed constant within the

individual spectral bands or intervals. The Planck intensity function is integrated using

appropriate series expansions. For instance, consider spectral integration of Eq. (71)over a band with v1_< v<_ v2.

v2 v2

f Iijk (v) dv = E3(17" i - "rjlc) /J Bk(V) dv

vI vI

v2

, ,o)fvI

B k (v) dv (74)

The value of the spectral absorption Coefficient for a given band is calculated byappropriate averages. For bands which are "expected to be optically thin for mostcalculations, the partial Planck-mean defined by Eq. (75) may be used.

(T, v1 , v2) =

/2 o/.d

fv1

/.t(T, v) B (T, v) dv

v2

f (T, v)B dv

v1

(75)

3.4 Treatment of Line Contributions

3.4.1 The line absorption coefficient. - The absorption cross-section of a line(bound-bound) may be written for the transition n--* n' as follows:

Crnn,(V) = (re2-_ 1 ( -fnn' bnn' (v) 1 e -hvnn'/k

\mc /(76)

where

e, m = electron charge and mass, respectively

c = velocity of light

2O

k

fnn' =

Vnn' =

bnn ,(v) =

Boltzmann's constant

absorption f-number for transition n-- n'

line center frequency for transition n--, n'

line shape factor normalized according to

The factor (1 - e -hvnn'/kT)

oO

f bnn,(V ) dv = 1

0

accounts for induced emission.

(77)

To obtain the spectral absorption coefficient, the cross section is multiplied by theoccupation number, N n , of state n (the number density of a given species in state

n). For a monatomic gas in local thermodynamic equilibrium, the occupation numberis given by the Boltzmann formula

where

gn --

Q=

E =n

N =

gn" -(E a/kT)

N n = -_-Ne

• ,

!

statistical weight of the n th state

electronic partition function of monatomic species

energy of n th electronic state above ground state

total number of particles of the given species per unit volume

(78)

The partition function is given by

- (En/kT)

Q = _ gn e (79)

n

The energy levels, partition functions, and statistical weights for nitrogen and oxygenatoms and ions are given in ref. 6.

From Eqs. (76) and (78), the bound-bound absorption coefficient for transitionn _ n' is then given by

2 / gn -En/kT [ (-h Vnn,_ I#nn' : \mc(_re, frm,_.Ne bnn,(V) I - exp_ "k-T ]l(80)

21

For electron-impact broadening, the lines follow a Lorentz shapewith the half-width proportional to the electron particle density:

1 Tnn,bnn, (v) = -

v - (v - dnn,) +O, m'l v TnnV

(81)

Tnn, = Tnn, Ne (82).

where

Vo, nn'

dnn , =

Tnn,-

Nei

= frequency corresponding to unperturbed transition n--* n'

line-center shift

half-width due to 1 electron per unit volume for

electron particle number density

n-_ n I

The normalized half-width 7nn' , as calculated by R. Johnston, for carbon,

nitrogen, and oxygen atoms and ions are tabulated in ref. 7.

Excited electronic states having the same core configuration, principal and orbitalquantum numbers but with different spin multiplicity and total orbital angular momen-tum for L-S c6upling have nearly the same En. Therefore, it is convenient to com-

bine these States into a single group so that only the occupation nmnber of this groupof states need be calculated. However, the f-numbers of the individual transitionsmust be multiplied by the ratio of the statistical weight of the individual lower state to

¢

that of the group of states. In other words, one may consider in Eq. (80) that gn isthe statistical weight of a group of lower states of similar energies and fnn' is aneffective f-number which is equal to the f-number of the particular transition multi-

plied by a statistical-weight ratio.

The total bound-bound absorption coefficient is the sum of all #nn' due to differentlower-state groups of different species.

3.4.2 The line transmittance and equivalent width. - Consider the spectral integra-tion of Eq. (71) over a finite frequency interval Av across which the Planck intensityfunction and the continuum contribution may be approximated by constant values.Then,

Iij k =_ f Iijk(V)dv

AV

-blTi-rjl

_ - B,_ ea

Av

_Bk(Va)a exp[-blTi rJl c, va AV

-bl i- jlc d

e dv

dv

(83)

22

where va is an average frequency (e.g., center frequency) within Av.

Define the average transmittance as follows:

f _b] Ti__.jTTransij -= A----V e

Av

Then,

-b[ ri-Tj[b

Iij k -_ _ Bk(Va) e

c, vaAp

Transij

Alternately:,ii

2_v

!i

i

Cijk

one can write Eq. (83) as follows:

c bf Bk ee dv -

Av

-b[ Ti-Tj ]

_ b Bk (Va) ea

C,Pa

f (1 .. e-bITi-TJld)dvAv

where

cij k aAv

-b Iri-Tj]C d vB k e

is the continuum contribution.

Define the equivalent width over Av as follows:

Widthii = - e du

Av

Then, Eq. (86) may be rewritten as

(84)

(85)

(86)

(87)

(88)

• 23

-b] _'i-_'j ]b c, v a

Iij k _ Icijk -_Bk(Va)e Widthij (897

The use of Eqs. (87) and (88) permits different spectral divisions for the calculation

of the two terms on the right-hand side of Eq. (89).

From Eqs. (84) and (88)

Widthij = Av(i - Transij ) (90)

The study of line transport then reduces to the calculation of Trans.. or Width...Ij Ij

3.4.3 Absorption coefficient notation. - In order to avoid using too many indices

to describe the variables, the following system of notation will be used: i

i = dummy index for sublayers

i1 , i 2 = specific sublayers

k = dummy index for lines (except in product kT7

m = dummy index for line frequency regions

n. = kth line lower state iridex (The species index is eliminated and the

value of n k specifies the species. The term Nnk will be used todenote the total particle number density for the species specified bynk. )

Thus,

(v) = 2tF gnk -Enk/kTi (i -hUk/kTi) .'7re -- N. e

(_c2]_.Z fk Qi, n k 1, n k - e bi, k (v) (91)

k

represents the spectral absorption coefficient of the i th sublayer at v. For many

situations, the line-center frequency vk in [1 - exp (-hvk/kT)] may be approximatedby vm , an average frequency for the mth frequency region.' It should be noted that

lines outside the m th frequency region may contribute to the value of #i(v) at vwithin the mth frequency region.

In terms of the new notation, one has the following:

1 Ti, k-- (927

bi, k (v7 = 7r 2 2

[v- (v k - di, k) ] + 'Yi, k

24

if[ii 2Transil, i2, m - Avm exp -bi

Avm

Widthil, i2, m = AVm(1 - Transil, i2, m )

J_i(v) Az i dv

)x_I [-b il' i2

1 - exo

L L

The sum in Eq. (93) represents the following operation:

(93)

d_

(94)

"0 , i 2 = i I

11 , 12

i

_i =

12-I

i--t 1

_i ' i2 > il (95)

iI-I

i= i2

i2 < iI

In order to simplify the calculations, the effects of line shiftwill be neglected. For

the conditions of interest, it is reasonable to consider only lines due to neutral nitrogen

atoms with an effectivedensity equal to the sum of the actual NI and OI values.*

With this approximation, Ni, nk may be replaced by N i and Qi, nk by Qi. The

numerical computation is thus considerably simplified. One may now write

*One may also neglect the contribution of the OI vacuum ultravioletlines.

25

Width.11, i2, m = - exp - b' _ 1

Avm

AZi _ gfk exp (-

k

- exp

%kTi/ Yi, k k}/2 : y2

(v- Vk) + i,

where

b' = b(_e2/mc 2)

gfk = fkgnk ' a constant for each line

The expression for the average transmittance now becomes:

fili2{I ]7 exp -b' NiTransi l,i2,m - Avm i Qii

Avm

/

exp1 _/i, k

(v - Vk)2 +

dv

(96)

dv

(97)

26

where

i1, i2

/7i

I-

_i =

i2-i

/7

i 1-1

/7i2

, i 2 = i 1

_i ' i2>il

_i ' i2 < _il

(98)

In the conditions of interest, itis reasonable to assume that _'nn'-__i,k is aslow-varying function of temperature. Then, the following approximation may beused:

Ti,k _ Tk Nei (99)

3.4.4 Line calculation limiting cases. - The study of line transport now reduces to

the calculation of Widthil, i2,m or Transil, i2,m according to Eqs. (96) and (97).

In general, the integral in Eqs. (96)or (97)can not be expressed in terms of simple

combinations of analytic functions. Although numerical integrations can always be

used to evaluate these integrals, the computation time required may become exces-

sive. One should then look for appropriate approximations applicable to different"

asymptotic situations. For situations where numerical integrations must be used,the numerical schemes should be selected with a consideration for the minimizationof computation time.

The approximations for different asymptotic situations are discussed below:

(1) Thin-line approximation. -

For _ <<1, e-_ -_ 1 - _. Now,

27

b'

iI,i2

i

- exp

- kTi/j azi gfk exp - kTi) _ (v - Vk)2 + T 2i, k

-< b'

i1,i2

i_-ex_ - G)]_z_ _*kOX_- - _ _.

k kTi] 7r 0 + yi,2 k 11'i2'm

(1oo)

where the identitydefines Til'i2,m , an effectiveoptical thickness evaluated at the

line centers. Then, for Til,i2,m << 1 :

Widthil, i2, m

iI,i2

I : [ (- i__ _, _ _ _I _"_ _,_2

AV m i Qi 1 exp kTi/JAzi k gfkexp kTi/_ (v - Vk)2 + 7i, k

dv

iI,i2

i Qii 1- exp - kTi]jAz i gfk exp kTi/ 7r (v Vk)2 2k Av - + 3_i, km

dy

iI,i2

i_ Nil ( hVl:n_] ( Enk_= b' Qii 1 -exp - kTi]jAzi _ gfk exp • kTi]G(Vum• k

, V_m, vk, 7i, k ) (101)

where

=i[ ru .: t=-, <io ,G<Vum,V_m,Vk,Yi, k ) _ tan -1 ('_m-_Q]\ Ti, k \ _i_k ]J

with rum and V_m as the upper and lower limits, respectively, of the mth frequencyinterval.

28

i

When the contributing lines are well within the frequency region (notnear the twoboundaries), Eq. (102) may be approximated by:

G (rUm, V£m , Vk, Ti ' k) = i (103)

With the above approximation, the equivalent width then becomes the sum of the so-called strengths* of the contributing lines.

Equation (101) is valid for both isolated and overlapped lines, provided

Til,i2,m << 1.

(2) Isolated-line approximation

When Av m is large compared with 7i k and the contributing lines within thefrequency region may be considered isolat'ed, the integral in Eq. (96) may be approxi-mated by the sum of integrals for the individual line. Thus,

W{A+_.

.....Ii,i2,mI { if'i2 [

N i hVm_ ]

1 - exp -b' 1 - exp (- k-- iJ]ahk 6v k i

gfk exp - kTi/Tr })(v - Vk)2 + T 2i, k

dv (104)

where 6v k is the integration interval for the kth line.

The integral in Eq. (104) for a single value of the index k represents the equivalentwidth of a single line for a nonisothermal path. The use of an effective half-width

(ref. 8) may enable representation of the equivalent width in terms of the Ladenburg-Reiche function (ref. 9).

Consider the following integral:

I = ; [1- exp /-1 _ Si--Ti- ._]

oO

s[, /-'-'-_ - exp 2 e27r_ +,y_ °

d_ (105)

*The strength of a single line over an isothermal path (L)

(Ni/Qi) [1 - exp (- hvk/kTi) ] Lgf k exp (- Enk/kWi)

may be defined as

29

where Te is an effective half-width.later. )

(Thedetermination of Te

Equation (105)may be integrated to yield:

will be discussed

(106)

where

f(_) = _ e-_ [Io(_) + 11(_)] (1o7)

is the Ladenburg-Reiche function with Ifunctions. _ o

Two asymptotic expressions of f(_ )

! f(} ) -_

(_) and I 1 (})

exist:

, _ <<I

, _ >>1

being the modified Bessel

(lO8)

2at } = -

"/r'the two asymptotic expressions become equal.

Since the lines are considered isolated, the interval of integration 6k may be ex-tended to (-oo , oo ). Therefore, one may rewrite Eq. (104) as follows:

i 'i2 1

• _ Si, k, mTi,

X "- 2_ Tekf 1 ____ . (109)Widthi 1 , i 2 , m k 2_e k

where

t hVm__ Eak

5 --qi - kT---:S. = b' I

I, k,m Qi - e ZXzi gfk e (110)

30

4

Now, consider the determination of "ye. Comparing Eqs. (101) and (109) and usingthe first of Eq. (108) for isolated thin lines, one obtains:

i 1 , i 2

Si,k,mTi,ki

Te k = il ,i2 (111)

Si k mi ' '

which is mean weighted according to the line strengths of the individual sub-layers.

Using the second of Eq. (108) for isolated strong lines, one obtains the followingequation:

172i 1 0i 2

Widthi 1,i 2,m _- _ 2_r_/e kk

Si,k, m _/i, k2 i

27r _/ek

. . 1/2

= 2 Si k m "Yi k[isolated strong line

' \square-root approximation]

(112)

The effective half-width does not appear in Eq. (112)• Therefore, it is reasonable to

assume that the effective half-width defined by Eq. (111) may be used, as an approxi-mation, in Eq. (109) for situations between the thin and strong limits for isolatedlines. *

For numerical calculation of the Ladenburg-Reiche function, the following approxi-mations are useful:

0.685 }3/4

-< (0.685) 4

, -(0.685) 4 < _ <1

- 27r (0.685) 4. (113)

4 1

2 )4(0.685

*This may be compared with the Curtis-Godson approximation (ref. 10).

31

(3) Effective isothermal region. Extending the idea of the effective half-width tonon-isolated lines, one may rewrite Eq. (96) as follows:

Width.11 , i 2 , m

Avn

1 - exp _ Ni-b' Qi - e A_. i

i

(114)

E

kTi 1 vi, k- '2 _dv

gfk e 7r(v _ Vk)2+ Ye k

k

!

where 7e k is an effective half-width of the k th line.

Under certain situations, a single lower state or a small number of lower states ofnearly the same electronic energy contribute to the mth frequency region. One may

then introduce the following approximations:

E N E (115)_k m

Yi k _- Yk Ne. (116), 1

Equation (114) may then be written as follows:

Width.11 , i 2 , m

z_

• 1 - exp

m

-b, o- e- Az i Ne i

i

gfk_-(v - Vk)2 + ye k

(I17)

32

The two summations in Eq. (117) may be performed separately, thus saving aconsiderable amount of numerical calculations. Let

iI,i2

Ail'i2'm = Qi 1 - exp/--_i)JAz iexp - _ Ne ii

(118)

TJhen, Eq. (117) becomes

Widthi 1,i2,m -_ I

AP m

i - exp_ b' 1-All, i2,m gfkk (P - Vk)2 + "Ye

dl)

(119)

The fact that All ' i2, m is a sum, independent of the dummy sublayer index i ,implies that the equivalent width given by Eq. (119) may be considered to be that due to

an effective isothermal region. -

Equation (119) may be further simplified, if the thin-line or the isolated strong-lineapproximation is used.

From Eq. (100),

i 1 , i 2

[ ( hVrn/] _ ( Enk_l 1Ni 1 exp Azi gfk exp kTi/_'il, i2, m = b' - -i Qii kTi]J k 7r 7i, k

i I , i 2

i k _ 7i' k

Then, for Til,i2,m <<I , Eq. (I01) yields with G = 1:

iI,i2

[ ( ( Em )Widthil, i2, m __ _ _i 1 - exp - kTi]jzkziexp _b' gfk• i

(120)

(121)

33

Now, consider Eq. (119) for isolated thin lines, one has in analogy to Eq. (109).

Widthil, i2, m -_k

All, 12,m27r7e k ---,_2

2,__ek

X" _kA

i 1,i2,m Z b'g_k ek

(122)

Subtracting Eq. (121) from Eq. (122), one obtains the following

i 1, i 2

yu' k 1-k i

= 0 (123)

For arbitrary variation of gfk and of the number of k-terms, an expression of theeffective half-width satisfying Eq. (123) is given below:

• o

hVm_ E m

• "2 hVm_ ] Tk (124)

Equation (124) may also be derived directly from Eq. (111), using Eqs. (115) and(116).

The equivalent width for isolated strong lines for the effective isothermal region,

according to the square-root approximation, may be obtained from Eq. (112).

34

2 A1/2 mi I , i 2,

k

1/2

'_' gfk _;

(b' gfk Tk )i/2 (125)

The effective half-width does not appear in Eq. (125). From the above consideration,it is assumed that the effective half-width defined by Eq. (111) (or Eq. (124) as a specialcase) may be used as an approximation along the whole "curve of growth. "

(4) Criteria for line isolation. From the above discussions, itis apparent that the

calculations are much simplified ifthe isolated-line approximation may be made. With,

the Lorentz line shape, the wings of the lines extend to +_ and -_. Therefore, the

lines always overlap to a certain degree. One needs then to qualify the description "iso-latedlines" and to establish suitable criteria for line isolationfor numerical calcula-tions.

Let

Then, if

quantity calculated using the isolated-line approximations

quantity calculated by numerical integration without the isolated-line

approximations

IZ Z*] < £ << 1 ,]

Z l(126)

the isolated-line approximations may be used.

NOW,

1Trans. - Trans.* -11, i2, m _1' i2' m A_ m (Widthil, i2,m (127)

35

The following criterion is more stringent than Eq. (126):

Transil, i2, m - Trans*l, i2, m

Transil Widthil, i2, m)Smaller of i2, m' Av m

-<_ <<1 (128)

Unfortunately, the usefulness of the isolated-line approximations is lost if

Trans*il, i 2 , m or (Width*J1, i2, m/Avm) must be calculated for application in

Eq. (128). Therefore, other more convenient approaches should be used.

Consider the calculation of the equivalent width of two overlapping lines, as shownin fig. 3a. The values of the integrand in Eq. (96), calculated by summing the twoisolated-line contributions exceed that based on two overlapping lines. For the profilesin fig. 3a

R -= Av---'m Widthk, m ~ 0.3 t

1R* -- _ Width*

Avm m

R - R*ZR - R* ~ 0. i

In fig. 3b, the line-center separation is three times that in fig.Z R is much smaller than 0.1 for the same value of R.

3a and the value of

Qualitatively, the value of R may indicate the degree of overlapping. As the valueof R increases, the value of R* approaches unity, as illustrated in fig. 4: At

small values of R, the values of Z R are small so that the lines may be considered

isolated. For a given group of lines (hence a particular pattern of line spacing) withinAv m , it may be possible to obtain an approximate relation R* = F (R) by correlatingthe numerical results of some typical distributions of temperature and species con-

centration (with or without using the approximation of effective isothermal region).This approximate relation may then be used in radiation-coupled flow field calcula-

tions. In view of the uncertainties in f-number and half-width values, the aboveapproach appears reasonable.

IThe value of R is inversely proportional to Avm.

2_

4. NUMERICAL-METHODS

The division of the continuum bands and line groups used in the computer code

STAGRADS (STAGnation point, RADiation-coupled, with external Source*) is brieflydescribed in Section 4.1. Details of the spectral divisions are given in AppendixesB and C. The schemes of flow field iteration and of line calculations are discussed

in Sections 4.2 and 4.3, respectively.

4.1 Division of Continuum Bands and Line Groups

For the continuum air radiation, eight different models are provided before thelines are incorporated. These models consist of from two to nine spectral bands

(see Table 1). The first option (LINEUP = 1) of radiation transport calculationsaccounts for only the air continuum contributions using one of the eight models. Forcalculations with lines, two options are provided. In one option (LINEqSP = 2), the

air continuum contribution is first calculated using one of the eight models of banddivisions. The continuum contribution is then corrected for the presence of thelines [ see Eqs. (86) to (89)] . In another option (LINEq_P = 3), the spectral divi-sion nf the continuum bands is made _'_-* - "_ *_-_ um..... _,_,_,_,,_ w,_** _,,_u of .... line groups and the

total contribution is calculated according to Eq. (85).

A total of 21 line groups are used (see Appendix C). Three of the groups containonly a single line per group. Three of the groups contain no lines at all so that thelast option (LINEUP = 3) of transport calculations may be used.

For calculations with ablation layer, only the continuum contributions of the abla-tion species are included in order to reduce the requirement of computer storageand time. To account for the variation of the ablation species absorption coeffi-cients over a wide spectrum, the division of the continuum bands for these species

is made coincident with-that of the line groups. For calculations with ablation .layer, the last option (LINEfltP = 3) is used so that the contributions of the airatomic lines are included.

4.2 Scheme of Flow Field Iteration

The integration of the conservation equations is for the air layer from the shock-wave toward the interface and for the ablation layer from the body surface toward

the interface. For the integration of the radiative terms and the energy Eqs. (28a)and (46), the air layer and the ablation layer are divided into a number of sublayers.

The air layer is divided in equal increments of the actual distance, but the ablationlayer is divided in equal increments of the transformed distance [see Eq. (44)] . At

*In ref. 1, the radiation transport formulation includes the presence of external

radiation sources. For this study, no e_ernal radiation sources are considered.

37

_J

present, a maximum of ten air sublayers and ten ablation sublayers is allowed in

the computer code in order to reduce the requirement of computer storage and time.

The integration of the momentum Eqs. (27) and (47) requires much less com-

puter time and storage. Consequently, their integration may be performed with

finer increments than that for the energy equations. The density ratios in Eqs. '(27)

and (47) are calculated by interpolations from values obtained using the energyequations.

The flow field calculation begins with the integration of the air layer momentum

equation, using an assumed distribution of Ps/P _- h/hs • The integration isstopped when Eq. (29b) is satisfied at the interface, thus determining the air layerthickness. The air layer is then divided into a number of sublayers and the energyEq. (28a) integrated. Across a sublayer, an average value of the nondimensional

normal velocity F is used, and the net radiative energy gain is given by the spec-trally integrated Q [see Eq. (65)] . Iteration is made to obtain a consistent set

of velocity and enthalpy distributions.

For calculations with ablation layer, an initial enthalpy distribution, linear in

the transformed normal distance between the wail enthalpy and an enthalpy at theinterface estimated from the air layer results, is assumed. The density distribu-tion is calculated using the equations of state of the ablation vapor. The initial

blowing rate is estimated using the air layer results, a surface heat balance, andan estimate of the vapor enthalpy rise. The momentum and energy equations are

integrated in the same manner as for the air layer. The air layer temperaturedistribution is maintained unchanged during ablation layer calculations. Iterationon the wall heat flux is also made to obtain a consistent set of wall heat flux, blow-ing rate, and velocity and enthalpy distributions.

The effect of the presence of the ablation layer on the air layer enthalpy distri-bution is then checked. If necessary, the air layer calculation is repeated while •

the ablation temperature distribution is maintained unchanged. Further repetitionsof ablation layer and air layer calculation are then made, if required, to obtain a

consistent set of wall heat flux, blowing rate, and velocity and enthalpy distributionsfor both layers.

At present, the criteria for convergence used in STAGRADS are: enthalpy dis-tribution, 1 percent and heat flux to the wall, 2 percent. The convergence of theblowing rate is governed by that of the wall heat flux. The convergence of thevelocity distribution is governed by that of the enthalpy distribution.

4.3 Scheme of Line Calculation

When numerical integrations are required to evaluate Widthil i2,m orTransil, i2,m , the selection of the computation schemes should f_e carefully made

to minimize the computer time. For example, the calculation of exponential func-tions requires much more time than for performing multiplications. Therefore, itmay be more economical to calculate

38

exp _ il'i2 il1

according to

i 1 , i 2

ff exp (_i)

i

ff there are many variations of i I and i 2 . Computation time may also be re-duced by using common factors and by avoiding calculating the same variables manytimes (with increasing computer storage requirement, however), i

Consider the numerical integration of the integral in Eqs. (96) or (97). The

integrand is to be evaluated for many values of u, say v], within A_, m . Let jb'e the dummy hidex for _. In terms of the .... _-_ "_ _,_;_,,v _+_n _l_t_d th

important variables are summarized as follows: (Note Ay i = Azi)

Bgfk: b'gf k (129)

N.1

Dynqi = Qq AYi(130)

1

Dynqei = Qii Ayi Nei = Dynqi Nei(131)

Dynqii, m : QqAy i 1- exp kTi/j = Dynq i 1- exp -k_-i/](132)

[_ 1 1 - exp Ne. = Dynqii, Ne.Dynqiei' m Qi Ayi -_i/J 1 m 1

(133)

Si,k, m = Q--_.Ay i - exp _i)]exp Enk b'gfk ,mk_i = Dynqii

= _ ( -Enk_s. = si • exp1,m ,k,m Dynqli,m

k k

exp _Enk _ .

(134)

(135)

39

11,i2,k,m

iI ,i2 iI ,i2

= _ Si,k,m = Bgfk

i iDynqi i exp

,Ill

iI ,i2

Si],i2,m = _ Si = _ S.,//l

i k il'i2'k'm

?i,k = Nei'Y k

Si'k,------m _i_.L__kTi,k,m,j = _Tr

(vj - Vk)2 2+"Yi,k

(136)

(137)

(13s)

(139)

z,m,j _-i,k,m,jk

= .Si, k; m 1

1,k,m rr Yi,k

(14o)

(z4z)

_'i,m = _ Ti,k, m

k

i 1 , i 2

Ti1,i 2,k,m = _ Ti,k,rn

i

(142)

(143)

iI,i2

Til'i2'ra = _k "ril'i2'k'm = _'_ Ti'mi

SYi,k, m = Si,k,rn_/i,k

S_/i, rn = _ SVi, k

k,r/1

(144)

(145)

(146)

4O

SWil, i2 , k, m

iI ,i2

iSYi k m

9 (147)

SYi1 , i2 , m

Yei I , i2 ,k, m

11 , i 2 , k, m

Em'ri,k, m, j

Em'r.1,m,j

]h_.m T.

ll,12,m,j

Width.11,i2,k,m

= _ STil,i2,k,mk

SYiI,i 2 , k, m

S.

11,i2,k,m

SYil, i2,k,m

2_Ye 2ll,io,k,m

: exp (-_,k,m,j)

= exp (- Ti, )m,j

iI,i2

= H Erll T.

1,m,ji

= 2_rYeiI,i2 ,k,m f(}il,i2,k,m )

048)

(149)

(15o)

(151)

(152)

(153)

(154)

Widthil,i2, m : _ 27rnil,i2,k,mf(_i1,i2,k, m) : _ Widthil,kk 12,k, m

[For isolated lines; f(_) is given by Eq. (113).]

J- Emril,i2,m,j) 5v.J

(according to appropriate numerical integration scheme)

(155)

(156)

41

\

= F/-A_---- Widthil i2 m , LINE SPACING PATTERN)(157)1 Width_ 1 , i 2 m Vra ,/kv m , ,

(approximate, empirical-numerical correlation).

The numerical integration of the integral in Eq. (96) then corresponds to the

application of Eq. (156), with a suitable scheme of evaluating Emql ,i2,m,j accord-ing to Eq. (153).

In the numerical scheme of line calculations for each line group, the equivalent

width is first calculated assuming that the lines are isolated. If this value exceeds1 percent of the spectral width for the line group, Eq. (156) is then used to calculatethe equivalent width with line overlaps. The correlation discussed in connectionwith fig. 4 may be used for radiation-coupled flow field calculations. However, the

computer time for typical flow field cases is found to be of the order of only oneminute (UNIVAC 1108). Consequently, efforts in obtaining equivalent-width correla-tions appear unessential (though desirable) at present.

The numerical calculation of Eq. ,1_ [or " " .........._v! in_g_Liu_, of Eq. (96)] is based ona variable inte/'val Simpson's Rule, modified in sttoh a way as to yield results twoorders higher in accuracy than the basic Simpson's Rule. Depending upon the

pattern of line ispacings, the degree of overlap, and the accuracy criterion imposed,the number of evaluations of the integrand [of Eq. (96) or (1 - EmTil,i2,m,])] fora line group may vary over a wide range say from 20 to 800 for a single equivalent

i

width.

42

j

5. RESULTS

Results of numerical calculations are presented in this section. Almost all of

the results presented are obtained using STAGRADS. Only an example of the heat

flux distributions obtained using STRADS is given. Additional results from STRADS

may be found in refs. 11 and 12. A reference case (u_ = 15.24 km/sec, P_/PSL= 1.66 × 10 -4 , RN = 256 cm) is selected for uncertainty studies.

5.1 Effects of Exponential Kernal Approximation

In ref. 1, the exponential integrals used in the radiation transport calculations

were calculated using series representation and numerical correlations. InSection 3.2, the exponential kernal approximation, E3 ([) -_ 0.5 exp (- 2_), isintroduced. This approximation enables the formulation _f the concept of line

transmittance and equivalent width, as discussed in Section 3.4.2. The effectsof the exponential kernal approximation on the shock layer enthalpy distribution,the radiative flux to the wall and to the shock, and the air layer thickness are shownin fig. 5, for a representative environmental condition (the reference case). The

air continuum contributions alone are considered, using band Model No. 4 (5 bands)and 10 sublayers. The close agreement of the enthalpy, heat fluxes, and air layer

thickness can be seen. The enthalpies calculated with actual E 3 (_) are slightlyhigher than that with approximate E 3 (_). However, the optical thicknesses of thefour of the five bands with actual E 3 ([) are slightly lower than that with approxi-mate E 3 ( _ ) . The net results are slightly higher heat fluxes with the exponentialkernal approximation. The approximation E 3 (_) -_ 0.5 exp ( - 2_ ) is used in allresults presented below.

5.2 Effects of Continuum Band Models

The effects of air continuum band models on the radiative flux to the wall can be

seen in Table 1. Only air continuum contributions are considered. The resultsindicate that the two-band model does not account fully for the self-absorption effectin the vacuum ultraviolet. Band Model 4 (Planck-mean for 0.5 -< u -< 10.95 eV

plus 4 vacuum UV bands) provides results sufficiently close to that obtained withmore complicated band models and is adopted for "nominal calculations2' Figure 6shows further comparison of results obtained using Models 1 and 4, including linecontributions.

5.3 Effects of Line Model

Two options (LINEOP = 2 or 3) of line calculations are discussed in Sec-tions 3.4.2 and 4.1. When air layer calculations alone are considered, either ofthe two options mav be used. When the effects of ablation layer are examined, theoption LINEOP = 3 is used. Figure 7 shows the nondimensional enthalpy distri-

butions, heat fluxes, and air layer thicknesses for the two line calculation options.

43.

"The results are for the sameenvironmental condition (the reference case) as forfigs. 5 and 6. The two line options yield nearly the same results.

5.4 Effects of Ablation Layers

The presence of a relatively high density ablation vapor adjacent to the body sur-face further attenuates the radiative flux to the wall. In an inviscid analysis, thechemical reaction betweenthe air species and the wall is not considered. The wallis assumedto be at the equilibrium sublimation conditions corresponding to thesurface pressure. The surface material considered Ln_this study is 70/30 carbonphenolic (see Appendix A for its thermodynamic properties). A 21-band model ofcontinuum absorption coefficients, with the same spectral divisions as that for theair line groups, is used (see Appendix B. 2).

For the reference case, the air layer temperatures are increased slightly near

the interface when the effects of the ablation layer are included, as shown in fig. 8.The heat fluxito the shock is increased slightly (3 percent) whereas that to the wallis reduced g_eatly (41 percent). The results are for the nominal f numbers for the

molecular band systems given in Table.B-2, unless indicated otherwise. Figure9shows the temperature and tangential velocity distributions across the air and abla-tion layers, lit is seen that the ablation vapor is heated slowly at first near the walland very rapi_lly near the interface. The tangential velocity distribution is replotted

in terms of aitransformed "mass" distance in fig. 10. The ablation layer is seen tocarry about the same mass as the air layer, although it is physically thin compared

to the air layer.

The particle number densities in the air and ablation layers are given in fig. 11.*The continuum optical thicknesses are given in fig. 12. The ablation layer is opti-cally thicker than the air layer over all frequencies shown.

Figure 13 shows the spectral heat fluxes. Actually, the heat fluxes should be a

series of steps, being constant over a line group frequency region as shown for theoptical thickness in fig. 12. In order to show the effects of ablation layer in thesame figure, the spectral heat fluxes are "assigned" to a selected point of each linegroup and straight lines are drawn between adjacent points. The results indicate

that the ablation layer effectively blocks all radiation fluxes with frequencies greaterthan 10.95 eV, due to the photo absorption of C ground and excited states, C+ ex-

cited states, and H Lyman (see Table B-l!.

5.5 Effects of the Number of Sublayers

In order to test the accuracy of the finite-difference method of integrating theradiation transport and energy equations, the reference case is recalculated with

fewer sublayers, from 10 + 10 (10 subdivisions in both the air and ablation layers)to 6 + 6. The results are compared in fig. 14. It is seen that the effects on theheat fluxes and total layer thickness are small. The surface reradiation is of the

same order as the heat flux to the wall, causing a significant deviation in the non-dimensional surface blowing rate.

*Only species used in radiative property calculations are plotted (see Appendix B).

44

For the study of the effects of environmental variables, the combination of 6 airsublayers and 6 ablation sublayers will beconsidered adequate.

5.6 Effects of Environmental Variables

The effects of nose radius variation on the radiative fluxes, for constant velocityand altitude, are shownin fig. 15. Increasing the nose radius from I to 8.4 ft (30.48to 256 cm) only increases the heat flux to the wall by 68 percent without ablation layereffects and 24percent with ablation laver effects. Increasing the nose radius increasesthe heat loss to the ambient air more rapidly.

Further results for different enviror/mental variables are given in Table 2 and figs.16, 17, and 18. At the highest altitude (250kft) considered, the surface reradiationexceedsthe heat flux to the wall. The assumption that the surface is at the equilibruimsublimation temperature is not physically meaningful. However, the results may stillbe presented with the understanding that zero ablation is considered. The number of cal-culated points are insufficient to construct figs. 16, 17, and 18 accurately. Someof thepoints are obtained by "interpolation" of datapoints in other figures.

The results indicate that the heat flux to the wall is changedaccording to the fol-lowing:

(1) It increases as the velocity increases; the ablation layer tends to decreasethe velocity effect.

(2) It increases only slowly as the noseradius increases; the ablation layerreduces the nose-radius effect further.

(3) It decreases due to the presence of the ablation layer, more for loweraltitudes, larger nose radii, and higher velocity.

In general, it canbe stated that the ablation layer becomes more efficient in re-ducing the zero-ablation radiative flux to the wall as the latter increases. The analogybetween the ablation layer effectiveness in reducing radiative heat flux to the wall andthe convective blowing effectiveness is noted.

5.7 Effects of Perturbation of Radiative Properties

The effects of uncertainties of the radiative properties are illustrated in figs. 19 and20. In fig. 19, curve 3 corresponds to the reference case when nominal (N) values ofthe radiative properties are used. Curves 1, 2, 4, and 5 represent results withoutthe ablation layer. Curves i and 2 are results of calculation (LINEUP = 1) with aircontinuum contributions only, with nominal cross sections for curve 1 and 2 × thefirst-band cross sections (u < 10.95 eV) for curve 2. The effects of air lines may

then be seen by comparing curves 1 and 3 and the corresponding heat fluxeS. Doublingthe first-band cross sections alene is not sufficient to "match" the line effects.

For curve 4, the line half widths used are twice the nominal value, while for curve 5the line f-numbers are doubled. The effects are to increase both the heat fluxes to thewall and to the shock, but by less than 10 percent.

45

JIA

For curve 6, the line half-widths, line f-numbers, and the ablation species molec-ular band system f-numbers are twice the nominal values. Again the heat flux increaseis relatively small, being less than 17 percent.

The effects of increasing the ablation species molecular band system f-numbers

alone by a factor of two may be seen in fig. 20. The heat flux to the wall is onlyslightly reduced. Attempts were made to study the effects of increasing the f-numbersby a factor of the order of 10. However, because of the rapid temperature rise nearthe wall, convergence difficulties were encountered in iteration of the ablation layer

enthalpy distribution. Approximate energy conservation calculations for the enthalpyrise for a one-dimensional ablation vapor flow indicates, however, the enthalpy riseis nearly the maximum. Consequently, increasing the f-nmnbers further may notchange the heat flux to the wall appreciably, although the enthalpy distribution maychange significantly.

5.8 Effects of Precursor Heating

No detailed analysis of the precursor heating effects is performed in this study.However, using the results of spectral heat fluxes to the shock obtained from STAGRADSand the results of refs. 3, 4, and 13, a first order estimate of the preheating effect onthe radiative flux to the wall may be made. For instance, using fig. 11 of ref. 13 and

assuming all radiation leaving the shock with _ > 12.15 eV* (see Table 2) is absorbed

by the cold air near the shockwave, for u_ = 15.24 km/sec and po_/PSL = 1.66 x 10 -4,the increase in radiative flux to the wall may be estimated to be in the order of 5 per-cent for the nose radii considered in Table 2. For velocities higher than 60 kft/secthe preheating effects may become more significant, however.

5.9 Equivalent Width Results

Figure 21 illustrates the results of equivalent widths of the line groups, for isother-

mal air layers of different thickness. The temperature and density correspond to thatimmediately behind the normal shock for the reference case. As discussed in Section4.3, the equivalent width is first calculated assuming that the lines are isolated. " If

this value exceeds 1 percent of the spectral width of the line group, the equivalentwidth is recalculated by numerical calculation of Eq. (156) to account for line over-

laps. As the layer thickness increases, the equivalent widths of line groups withstrong line overlaps approach the group width asymptotically, for instance line groups19 and 15 in fig. 21. When the ratio of equivalent width to line-group width is much

less than unity, the slope of the line may indicate whether the lines are thin or withinthe strong-line, square-root regime (see Eq. (113)). For instance, the lines withingroup 1 are thin whereas that within group 13 are within the square-root regime, formost of the values of A in fig. 21.

*Cold N2 has a multitude of narrow molecular bands starting at 12.4 eV, but strongabsorption may not begin until the N 2 photo-ionization continua are reached at about

15.5 eV (ref. 14).

46

The results of fig. 21 indicate that line overlaps must be taken into account forblunt-body radiation transport calculations.

5.10 STRADSResults

Downstream of the stagnation region., the stream-tube formulation is used to calcu-late the flow field and radiative fluxes. In this study, the computer code STRADSde-veloped in ref. 1 is modified to incorporate the air continuum band models used inSTAGRADS. The effects of ablation layer are not included. The air line contributionsare "simulated" approximately by multiplying the first-band cross sections with anappropriate factor.

The configuration considered is a blunt vehicle flying at a 33 deg angle-of-attack,as shown in fig, 22. The formulation of STRADSis based on two-dimensional andaxisymmetric configurations. In order to simulate the actual attitude, the effectiveaxis of symmetry is displaced at a distance Ro from the stagnation point, as shownin fig. 22. !

The radiation heating distributions, normalized with respect to the stagnation radia-tive flux to the wall calculated by STAGRADS,are given in fig. 23. The dimensionalsymbols are defined in fig. 22. Varying the value of the first-band cross section mul-tipliers from 2 to 4 produces no significant changesin the normalized heating distri-butions. The radiation heating distribution is seen to exhibit a strong velocitydependency. Further results from STRADSmay be found in refs. 11 and 12.

6. CONCLUDINGREMARKS

An efficient numerical procedure for fully radiation-coupled blunt-body flows hasbeen formulated in this study. The effects of air continuum, atomic line, and ablationlayer radiations are taken into account for determining the stagnation point velocity andtemperature fields, the heat fluxes to the wall andto the shock, and the surface blowingrate. The efficiency of the numerical methods is due to the finite-difference integra-tion of the radiation-transport terms andthe energy equation, to the formulation of theconcept of line-group equivalent width andaverage transmittance, and to the procedureof calculating the equivalent width with line overlaps.

Using the computer code developed, the effects of environmental variables, of thenumerical methods used, and of the uncertainties in radiative properties have beendelineated. The results indicate the following conclusions:

(I) The exponential kernal approximation of the exponential integral yields results

for enthalpy distributions and heat fluxes very nearly the same as that with

actual exponential integrals.

(2) The two-band model of air continuum radiation underpredicts the self-absorptioneffects in the vacuum ultraviolet. Increasing the vacuum UV band numbers

from 1 to 4 (Model 4) provides results sufficiently close to that obtained with

more complicated band models.

47

(3) The two options of line calculations, LINEOP = 2 or 3, yield nearly thesame results.

(4) The ablation layer is very effective in reducing the heat flux to the wall and

thus the surface ablation rate. It practically blocks all radiative fluxes withfrequency greater than 10.95 eV. The ablation layer becomes more efficientin reducing the zero-ablation radiative flux to the wall as the latter increases.

(5) Reducing the number of sublayers from 10 + 10 to 6 + 6 still yields resultsadequate for the study of the effects of environmental variables.

(6) The self-absorption and energy loss effects decrease the sensitivity of theheat fluxes to the changes in environmental variables and to the uncertainties

in radiative properties. An order of magnitude change of the nose radius(from 1 to 10 ft) only increases the heat flux to the wall by 30 percent. Con-

sidering the change of convective heat rates with nose radius, blunt configu-rations may prove to be more effective than slender configurations in reducingthe total wall heat fluxes.

(7) For velocities less than 18 km/sec (_ 60 kft/sec), the effect of precursorheating on the radiative flux to the wall is small.

In future work, the following studies are recommended:

(1) The formulation of radiation transport may be extended to include the pre-cursor radiation and the spectral emissivity of the body surface.

(2) The contributions of additional radiative systems, such as the lines due to

nitrogen ions, and oxygen and carbon atoms and ions, may be inclu4ed.

(3) The formulation of the flow field may be extended to include the precurso_heating effect and the viscous effect (including conduction and diffusion).

(4) The radiation transport formulation developed may be used for calculationsaround the body.

(5) The accuracy of the one-dimensional radiation transport may be investigated,particularly when the temperature gradient in the direction parallel to thebody is large.

(6) The numerical procedures used in STAGRADS may be further improved toincrease the accuracy and to reduce the computation time.

48

APPENDIX A

THERMODYNAMIC PROPERTIESOF AIR AND ABLATION VAPOR

For a given composition of elementary species, two state variables are sufficientto specify the thermodynamic properties of a gaseous mixture in thermodynamic

equilibrium. The state properties may be put in a tabular form so that property cal-culations may be made by Lable look-up and interpolation. Alternately, curve fits of

one state variable versus a second at several levels of a third may be used. Thelatter method is particularly convenient when many calculations are made at a fixedlevel of a certain state variable. For instance, for abluntbody in hypersonic flow, theshock layer normal pressure gradient may be neglected. Curve fits or correlationsof temperature and other state propertiesversusenthalpy at several constant levelsof pressure will be very convenient for calculations along a body normal. Correla-

tionsof this kind for air may be found in refs. 15, 16, and 17.

For air calculations in the present study, correlations of refs. 15 to 17 may be

.used. However, for the purpose of reducing the computer time required, new cor-relations with a more suitable set of units are generated. The division of enthalpyranges is also selected in such a manner that-interpolations of the coefficients in the

correlations between adjacent pressure levels may be more conveniently made.

The units selected for correlation purposes are:

P , pressure in atm

T , temperature ineV, leV =11605.7 °K

H , enthalpy ratio, h/hsatellite , hsatellite = 12,484 Btu/Ib m

PV/VsL , in atm, VSL = sea-level (i atm, 288.16 °K) air specific volume

The correlation formula coefficients for air are given in Table A-1.

With the values of the compressibility factor available, the concentrations of airspecies may be readily calculated ff Hansen's (ref. 18) simplified model of air chem-istry is used. However, in this study, the particle densities (their logarithm) ofneutral nitrogen and oxygen atoms and elections are calculated by double interpola-tion with respect to temperature and the logarithm of density using the results of

Gilmore (ref. 19) and Moeckel and Weston (ref. 20).

The surface material considered in ablation layer calculations is 70/30 carbon-phenolic. The 70/30 carbon-phenolic is assumed to be at the equilibrium sublimationconditions corresponding to the surface pressure. The sublimation latent heat is

4,270; 4,220; 3,900 cal/grn at 0.1, 1, and 10 aim, respectively. If the material sen-sible heat is included in the energy balance, the effective heat of sublimation will behigher.

49

The thermodynamic properties (temperature, enthalpy, density, species particle

concentrations, and pressures) of 70/30 carbon-phenolic vapor in thermodynamicequilibrium were calculated by a free-energy minimization program, and are put ina table form for double interpolation with respect to pressure and enthalpy. Betweentwo consecutive table values, the temperature and the logarithm of specific volume

and of species concentrations are assumed to be linear in enthalpy and in the logar-

ithm of pressure.

50

APPENDIX B

RADIATIVE PROPERTIESOF AIR AND ABLATION SPECIES

The absorption coefficient of a given species is the product of its particle numberdensity and absorption cross sections. For a given mixture of species, the absorp-tion coefficient is the sum of theseproducts. The cross sections of air species andablation vapor species are discussedbelow.

B. 1 Absorption Cross Sections of Air Species

B. 1.1 Continuum absorption cross sections. - The effective cross sections ofneutral nitrogen (NI) and neutral oxygen (OI) given in Tables 4-3 and 4-5 of ref. 7

are used. These cross sections include the bound-free and free-free photon absorp-tion contributions and are per particle of NI or OI. For the conditions of interest,

the relative contribution due to NII is small. The incorporation of absorption contri-butions due to NII and N- continuum has not been made.

For the eight continuum band models described in Section 4. i, partial Planck mean[see Eq. (75)] cross sections are used for bands with frequency less than i0.95 eV.The NI cross sections increase in steps for frequency above i0.95 eV and average

values are used for each of the steps. The OI cross sections do not change apprecia-bly between 4.0 and 13.61 eV but increase in steps above 13.61 eV. For Of, partialPlanck means are used up to 13.61 eV and average values are used for each of thesteps above 13.61 eV. The selection of the band boundaries is based on the locations

of the step changes in cross sections. The band boundaries for the eight models aregiven in Table I. For frequencies above I0.95 eV, the cross sections for the highest-frequency band correspond to that of the lowest-frequency step within the band. For

example, in Model I, the average cross sections between i0.95 and 12.15 eV areused for the second band with boundaries at I0.95 and 30.0 eV.

For calculations with lines and ablation layer, the frequency regions are relatively

narrow. The value of the cross section at a specified point within the frequency re-gion is used. The logarithms of the cross sections are assumed to be linear in fre-quency and temperature between tabulated values.

B. 1.2 Line absorption cross sections. - Only the contribution of NI lines areincluded in the present STAGRADS code. The effective NI particle density for linecalculations is taken as the sum of the actual NI and OI particle densities. A total

of 75 NI lines are considered. Some of the lines are "effective" lines formed bytreating two or more closely situated lines as single lines. The f numbers and thehalf-width (at T = 10,000 ° K) of the NI lines given in Tables 4-9 and 4-15, respec-

tively, of ref. 7 are used. The line shifts are neglected and the half-widths are con-sidered independent' of temperature [see Eq. (99)]. See Appendix C. 3 for details ofline data used.

5l

B. 2 Absorption Cross Sections of Ablation Species

Only the continuum contributions of ablation species are included in the present

STAGRADS code. The radiative systems considered include the Swan, Fox-Herzberg,

Deslandres-D'Azambuja, Freymark, and Mulliken band systems of C 2 , the CO4+system, and the continuum processes due to the C + free-free contribution and the

photo-absorption of CO, H Balmer, H.Lyman, C ground and excited states, and C +excited states.

As discussed in Section 4.1, the division of the continuum bands for the ablation

species is made coincident with that of the line _,'_,,,_¢

The spectral ranges of the 21 bands and the average cross sections of the absorp-

tion processes are given in Table B-1. The cross section cro of the molecular bandsystems were obtained by wavelength averaging the results obtained by John Weisner(ref. 21) and by normalizing to unity f-number. The temperature dependence isdetermined from the cross-section values at two temperatures (3,000 ° K and 9,000 ° K

or 10,000 ° K) and is very approximate. The cross-sections of the continuum proces-ses are given by the same simple correlation, with f taken as a unity. No perturba-tion of the continuum cross-sections is considered in this study. The f-numbers ofthe band systems are given in Table B-2.

The C 2 Swan systems are the only band contributors considered within the first

five bands. Preliminary results of calculation indicated that the optical thickness forthese bands ( _ 0.62t_ ) was too low. As the absorption in the IR is generally strong due

to the presence of the polyatomic molecules adjacent to the wall and due to systems notincluded (e.g., C 2 Phillips and Ballik-Ramsey Band systems and H Paschen andBrackett continua), the C 2 Swan contribution within Band 1 is raised by a factor of 10 to

compensate for the neglected absorbing systems.

52

APPENDIX C

THE STAGRADS CODE

The listing of the STAGRADS code, for calculation in UNIVAC 1108 at LMSC, isgiven after the figures near the end of the report. The input data required for the

reference case (u_o = 15.24 km/sec , Poo/PSL = O. 166 x 10 -3 , R N = 256 cm)are then listed. Brief explanations are given for the code and the input. The programis very complex. The purpose of presenting the listing is for the benefit of those who,l_ve _l_v desire to modify the program and to make use of some of the subroutines forother related problems.

Representative output data, including details of the line data used, are also givenat the end of the report with brief explanations ....

}C. 1 STAGRADS Listing (See page 99. )

The listing of STAGRADS given is in FORTRAN IV. STAGRADS consists of a mainprogram and a number of subroutines and fun.ctions. In the order of appearance in the

listing, they will be explained briefly below.

C. 1.1 Main Program. The main program serves the following functions:

(1) Read all input data.

(2) Write a majority of the output results.

(3) Integrate the energy equation and iterate the enthalpy distribution for theair layer.

(4) Call the appropriate subprograms (i. e., Subroutine VEL for integration ofthe momentum equation to obtain the velocity distribution).

The built-in data TC (I, J) are used only in Subroutine ABSORP. They are repeatedhere for the purpose of checking (see listing near statement no. 72).

C. 1.2 Subroutine STAGAB, STAGAB (STAGrads ABlation) is for calculation ofstagnation ablation layer. It serves the following functions:

(1) Integrate the energy equation and iterate the enthalpy distribution, wall heat

flux and surface blowing rate for the ablation layer.

(2) Write output of ablation layer results.

(3) Call the appropriate subprograms (i. e., Subroutine VEL for calculation ofthe velocity distribution).

C. 1.3 Subroutine NGGBSP. NGGBSP (Non-Grey Gas) is used for radiation trans-port calculations involving finite bands of the air continuum contributions. As listed,

the exponential kernal approximation is used for E 3 (_). However, the actual

E 3 (_) may be used if only the air continua are considered.

53

C. 1.4 Subroutine CRLINE. CRLINE (CoRrection for LINE) is used for radiationtransport calculations when the contributions of the air lines and ablation speciescontinua are included. It is used for either of the two line-calculation options dis-cussed in Section 4.1.

C. 1.5 Subroutine ABSORC. ABSORC (ABSORption for Continua) calculates theair continuum absorption coefficients (NI and OI) for a specified frequency within theindividual line groups.

C. 1.6 Subroutine lINE. L_rNE calculates the equivalent widths or transmittance

for the line groups. It writes out the calculated results if requested by appropriatecontrols.

C. 1.7 Function FLR. FLR calculates the Ladenburg-Reiche function accordingto the approximations given by Eq. (113).

C. 1.8 Subroutine INTEG. INTEG (INTEGration) performs integration to obtain theequivalent width. It is based on a modified Simpson's rule. It writes out the smallest

integration step and other information if requested by the control.

C. 1.9 Subroutine INTRND. INTRND (INTegRaND) calculates the integrands re-quired for equivalent width calculations.

C. 1.10 Subroutine DOUBL. DOUBL (DOUBLe interpolation) is used to calculateparticle concentrations of NI, OI and e by double interpolation with respect to tem-perature and the logarithm of specific volume.

C. 1.11 Subroutine PROPT. PROPT (PROPerty) calculates air temperature, PVproduct, and compressibility factor for given pressure and enthalpy, using the corre-lation formulas given in Table A-1.

C. 1.12 Subroutine ABSORP. ABSORP (ABSORPtion) calculates the air continuumabsorption coefficients (NI and OI) for the individual bands for the eight band models.

C. I. 13 Subroutine SPABCO. SPABCO (SPectral ABsorption COefficient) calcu-

lates the ablation species continuum absorption coefficients for the 21 spectralregions.

C. 1.14 Subroutine PROT. PROT (PROperTy) calculates for the ablation vapor the

specific volume, temperature, and particle densities of C 2 , C, CO, H, and C+ bydouble interpolation with respect to enthalpy and the logarithm of pressure. Theproperty tables are input in the main program.

C. 1.15 Subroutine SESB. SESB (State Equation for Surface Blowing) calculatesthe surface enthalpy for 70/30 carbon phenolic in sublimation equilibrium at the sur-

face pressure. The surface specific volume and temperature calculated in SESB are

actually not used in STAGAB; they are recalculated by PROT for consistency with thevapor properties.

54

C. 1.16 Subroutine PLANCK. PLANCK calculates the fraction of black-bodypowers within two frequency limits and for a given temperature.

C. 1.17 Function EXPF. EXPF limits the absolute value of the argument for EXPto 85 to avoid numerical overflow.

C. 1.18 Subroutine VEL. VEL (VELocity) integrates the momentum equation forthe air layer and ablation layer.

C. 1.19 Subroutine TBLP4. TBLP4 is used for interpolation of variables such as

specific volume ratio, transformed distances and velocities in VEL.

C.1.20 Subroutine MODLI_ MODL2, ... MODL8. These subroutines specify thefrequency limits for the 8 air continuum band models. They also select the propercross section indices for calculations in ABSORP.

C. 1.21 Function E2F. (listed after input data) E2F calculates the second expo-nential integral by series and numerical correlations. It may be used in conjunction

with E3F in NGGBSP and CRLINE, when the actual exponential integral instead of theexponential kernal approximation is employed.

C. 2 Input Data Listing (See page 155. )

The input data for an example case (the reference case)are listed immediately

following the STAGRADS listing. All input data are read in within the main program.

The input line and line group data are first read. The 10 read statements after the

comment "Input Line and Line-Group Data" read in the values of the followingvariables:

NHV

NLST

GEE(I)

EPS(I)

FHVM(1)

FHVP

FHV(1)

ISOE(I)

Total number of line groups

Total number of lower states

Statistical weight of I-th lower state

Electronic energy of I-th lower state, eV

Lower frequency limit of I-th line group, eV

NUMINT(I)

NU(O

Upper frequency limit of I-th line group, eV

A selected (e. g., central) frequency within I-th line group, eV

> 0 if the approximation of effective isothermal region may be used forthe I-th line group; otherwise <_ 0.

> 0 if numerical integration is to be used (when required) to calculatethe equivalent width of the I-th line group; otherwise < 0.

Number of lines in I-th line group

55

NLINE

ND )

HVL(I)

FF(1)

GAMBA(I)

Total number of lines, calculated by summing all NU

Lower state index for I-th line

Center frequency of I-th line, eV

Effective f-number of I-th line (f-number of the particular transition

multiplied by a statistical-weight ratio, see Section 3.4.1)

Half-width (eV) of I-th line for 1 electron/em 3.

Next, some of the 4 input tables are read, between Statements 8 and 99 in the list-ing. Each of the 4 tables is headed by a leader card.

Table 01

L

CASE

PA

RHOA

UA

RN

NS

ES

HA

TW

MODEL

LINEOP

FCONTN

FDHAB

NRITE1

NRITE2

NRITE3

NRITE4

(Leader Card with value of NC = 0 1; near statement 10)

0 for two-dimensional and 1 for axisymmetric

Case identificationi

Ambient pressure, atm

P_/PsL

u , km/sec

_a N , em

•_ 10 number of air sublayers

/Ps

h/hsa t

T w , eV (ff < 0.02, reset to 0.02; recalculated as equilibrium surfacetemperature if ablation layer effects are considered)

Air continuum band model number, = 1, 2, ... , 8

{ •1 , only air continua with one of the eight band models2 , calculate air continua with one of the eight band models and then

correct for air lines

3 , 21 spectral regions with air continua and lines, with or withoutablation continua

Multiplying factor for air continuum cross sections (v < 10.95 eV ) forthe 8 band models

Multiplying factor for surface material latent heat of vaporization

> 0 print nondimensional entha]py distributions during iteration; < 0

no print

> 0 print equivalent-width information, final result only; _ 0 no print

> 0 print equivalent-width information during iteration; < 0 no print

> 0 print equivalent-width integration information during iteration; = 0

final result only; < 0 no print

56

NRITE5

NRITE6

NRITE7

CMAG

ERR

> 0 print line strength of sublayer, etc. during iteration; = 0 final re-sult only; < 0 no print

> 0 print line strength of air-layer regions, etc. during iteration; = 0final result only; < 0 no print

> 0 print nondimensional enthalpy distributions, QWT, etc,, when either

(but not both) the air or ablation layer iteration converges; -< 0 noprint

0. 0001 times absolute value of the average value of integrand

Integration accuracy control constant

With err = 0.1 and 0.01, the result is said to be accurate to 4-6 places and 5-7

places, respectively. Decreasing the value of err will increase the computing timerequired.

From the results of equivalent-width calculations for a variety of conditions, avalue of err = 0.1 is found adequate for most situations. Under certain situationswhen the lines are very narrow and with a small degree of overlapping, it is foundthat a value of err in the order of 0.01 is required for line groups 4 and 5 in order toobtain accurate results in equivalent widths by numerical integration. However, avalue of err in the order of 0.1 is adequate for flow field and total heat flux

calculations, ii

FRAC First integration step size divided by total integration interval

NSAB <_ 10, number of ablation sublayers

Table 01 must be input for each run of flow field calculations. For flow field cal-culations without the ablation layer, Table 01 ends with a card with -1 (value of NC)in the first two columns. For flow fields calculations with ablation layer, Tables 03

and 04 are also required; the "-1" card is then put at the end of Table 04.

For flow field calculations, each case must end with a "-1" card. For subsequent

cases of the same computer run, Tables 03 and 04 may be omitted if they are identi-cal with the corresponding ones in the preceding case.

Table 02 (Leader card with value of NC = 02; near Statement 80)

Table 02 is input for equivalent width calculations with given layer thickness, num-

ber of sublayers, and distributions of temperature and specific volume. No "-1" cardis required to end each case.

N

DE LTA

T(1)

v(1)

Total number of sublayers

Total thickness of layer, cm

Temperature of I-th sublayer, eV

PsL/P of I-th sublayer

• 57

CMAG, ERR, FRACNRITE1 --*NRITE6 ! see Table 01

Table 03 (Leader card with value of NC = 03; near Statement 7010)

IP

IT

Table 03 inputs the thermodynamic properties of the ablation layer vapor.

PRTAB(I)

TTAB(I, J)

ENTAB(I , J)

RHOTAB(I , J)

C2TAB(I , J)

C1TAB(I , J)

COTAB(I , J)

HTAB (I , J)

CPTAB(I , J)

Total number of pressure values in table (3 allowed in presentprogram)

Total number of temperature values in table (20 allowed in presentprogram)

The i-th pressure value, in lOgl0 (p , atm)

J-th temperature value for I-pressure value, °K

Enthalpy for I-th pressure and J-th temperature, cal/gm

Density for I-th pressure and J-th temperature, gm/cm 3

Particles/cm 3 for I-th pressure and J-th temperature, for C 2

For C 1

For CO

For H

For C+

EMTAB(I , J) For e

Table 04 (Leader card with value of NC = 04; near Statement 7020)

Table 04 inputs the f-numbers and continuum cross section multipliers for ablationlayer radiating systems.

ICOSR

COSR(I)

Total number of input COSR values

f-numbers for band systems. For continua, COSR = 1.0 if the standard

values of the cross sections are used. The order of COSR(I) is given inTable C-1.

HV

HV+

HV-

N

C: 3 Output Data Listing, Including Details of Line Data Used (See page 163.)

The data for the 21 line groups are first listed. The headings of the data are:

Frequency within a line group for calculating Planck intensity functionand continuum cross section, eV

Upper frequency limit of line group, eV

Lower frequency limit of line group, eV

Number of lines within line group

58

I

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

TAB LE C- 1

COSR(I) VALUES

Absorption Processes

C 2 Swan

C 2 F-H

C2 D-D

C 2 Frey

C 2 Mull

Unas signed

Unassigned

Unassigned

CO4 +

CO Photo

Unassigned

Unassigned

Unassigned

H Balmer

H Lyman

Unassignecl

Unassigned

C Photo

Unassigned

Unassigned

C+ Photo

Unassigned

C + + e____C + hv , (C + free-free)

Remarks

Use fvalue

Use f value

Use f value

Use f value

Use f value

Use 0 value

Use 0 value

Use 0 value

Use f value

Use 1 for standard value

Use 0 value

Use 0 value

Use 0 value

Use 1 for standard value

Use 1 for standard value

Use 0 value

Use 0 value

Use 1 for standard value

Use 0 value

Use 0 value

Use 1 for standard value

Use 0 value

Use 1 for standard value

LINE

NLST

HV(1)

FG)

GAM(1)

Line index

Lower state index

Line center frequency, eV

Effective f-number of line, same as FF(I) in Section C.2

Line half-width for 1 electron/cm 3 , eV/(e/cm 3 )

59

BFG(1)

ISOE

NUMINT

GEE(1)

EPS(I)

b'gf , Eq.(129) eV-cm2/particle

> 0, if effective isothermal region may be assumed; < 0 otherwise

> 0, if equivalent width is calculated by integration when its value

according to isolated lines exceeds 1 percent of line-group width; < 0if calculated according to isolated lines only

Statistical weight of I-th lower state

Electron energy of I-th lower state, eV

The next group of output data is the air continuum cross sections TC(I , J) usedin ABSORP. In ABSORP, TC(1 --* 14, 1) are temperatures in eV and TC(1 --* 14,2 -* 27) are values of 38 + logl0 (cross-section, cm2). These values have been

converted to °K and cm2, respectively, in the main program before being printed

out. Each index J (2 --* 27 ) is for a particular frequency interval (either a partialPlanck mean or a representative average).

J: i 2 3 4 5 6 7 8 9

Species: NI NI NI NI NI NI NI NI

vI , eV: 0.5 2.0 4.0 6.0 9.50 10.95 12.15 13.61

uu , eV: 2.0 4.0 6.0 9.5 10.95 12.15 13.61 14.50

10 _1 12 13 14 15 16 17 18

NI Ol Ol Ol Ol NI OI NI Ol

14.50 0.5 2.0 4.0 13.61 0.50 0.50 2.00 2.00

10.9530.00 2.0 4.0 (13.61) 30.00 10.95 10.95 10.95 10.95

19 20 21 22 23 24 25 26 27

NI NI NI NI NI OI OI OI OI

0.5 1.0 2.0 3.0 4.00 0.5 1.0 2.0 3.0

1.0 2.0 3.0 4.0 10.95 1.0 2.0 3.0 4.0

The next group of output consists of data of particle per air atom for equilibrium

air. Following a column of 14 temperatures (3,000 --* 24,000 °K), the particles per

air atom for NI, OI, and e (3 column as a group for one density) for lOgl0 (PsI_/P) =1.97634, 2.97634, and 3.97634 are given. The values of SPTABO (I, J, K) with the

main program are in - lOgl0 (particles/air atom). These values have been convertedto particles/air atom before being printed out.

The next group of output corresponds to Table 01 of input. The units of the vari-ables and the definitions of the symbols are given there.

6O

If Table 03 is input, the data will beprinted out. The data printed for the refer-ence case are for three pressures (lOgl0 p, atm = -1.0, 0.0, 1.0). Columns 1 to

9 are respectively temperature (°K), enthalpy (cal/gm), density (gm/em3), and

particles/cm3 for C2, C1, CO, H, C+, and e.

If Table 04 is input, the values of COSR(I) are printed out. See Table C:l foridentification of the index.

The next group of output repeats the information for the line groups including the

selected frequency and the lower and upper line indices.

The symbols appearing in the output are defined below:

NCOUNT

DELTA

PO

HO

TO

QWT

QSHOKT

I

YC

UC

VC

HC

T

V

Z

HV

HV+

HV-

NLINE

TAUC

TAUCT

LINEOP

EMD

Number of iterations for enthalpy distribution, one counter

for the air layer and another counter for the ablation layer

Layer thickness for air or ablation layer, cm

Air and ablation layer pressure, arm

hs/hsa t

T , eVS

Total heat flux to wall, watts/cm 2

Total heat flux to shock, watts/cm 2

Sublayer index from shock

z/A for air layer; y/z_b for ablation layer (measuredfrom center of sublayers)

Normalized tangential velocity

Normalized normal velocity

h/h s for air layer; h/h w for ablation layer

Temperature, eV

PSL/p

Compressibility

Selected (e.g., central) frequency of line group, eV

Upper limit frequency of line group, eV

Lower limit frequency of line group, eV

Number of lines in line group, eV

Continuum optical thickness for air layer

Continuum optical thickness from shock to wall

Line-calculation Option (See Section 4.1)

PwVw / p_ u

61

NQWT

NCONH

QWTAIQWTA2 !

QWTC1QWTC2 !

QWT1

HW

TW

VW

VCW

UR

RMOM

URMAS

IAB

NCI, NC2, NCONH, NC +

M

IPI, I

ISO. WIDTH

ACT. WIDTH

DNU* TRN

ISO-ACT WIDTH

WIDTH (IPl, I)

DELTAB

DELTAT

US

ZETA

NCNTRI

Number of iterations for heat flux to the wall for each

ablation-layer calculation

Counter for enthalpy distribution iteration for each as-sumed wall heat flux

Previously' assumed wall heat flux, watts/cm 2

Previously Calculated wall heat flux, watts/cm 2

New assumed wall heat flux, watts/era 2

hw/hsa t

Tw , eV

PsL/Pw

Vw/U _

Ur/U calculated based on interface streamline momen--

turn consideration

2 2PwV/p u

Ur/U calculated based on mass balance for a finitenum-ber o_ sublayers

Ablation sublayer index from wall

Particle/cm 3 for C 1 , C2 , CO, H, and C + , respectively

Line group index

Upper and lower index for region with indices IP1 and I

Equivalent width assuming isolated lines, eV

Equivalent width with overlap, eV

_v TRANS, eV

ISO - WIDTH-ACT • WIDTH, eV

Equivalent width _ eV

Aab , ablation layer thickness, cm

, thickness between shock and wall, em

US

Z

_p_dzPs A'

o

Counter for number of air layer enthalpy iterationconvergence

62

NCNTR2

NCNTRS

FNUL(KK)

FNUH(KK) (FNI/U)

FNU

AB

QW(KK)

QWTC

QDEL(I_)

QDELT

SQW(KK)

SQDEL(KK)

TAU(I, KK)

K,KM

JD

I

IPl , J

C-AM(I)

ss(I)

SST(I + 1)

SGAM(I, KM)

SGAMT ( I + 1)

TAUD(I)

TAUDT(I + i)

SS12

Counter for number of ablation layer calculationconvergence

Counter for number of either (but not both) air layer orablation layer convergence

Lower frequency limit of KK-th air continuum band, eV

Upper frequency limit of KK-th air continuum band, eV

Central frequency of air continuum band, eV

1.0, signifies with self-absorption

Continuum heat flux to wall for KK-th band, watts/cm 2

Total continuum heat flux to wall, watts/cm 2

Continuum heat flux to shock for KK-th band, watts/cm 2

Total continuum heat flux to shock, watts/cm 2

Spectral continuum flux to wall for KK-th band,watts/cm2 eV

Spectral continuum flux to shock for KK-th band,watts/cm2 eV

Continuum optical thickness for KK-th band from I-thboundary to shock (I = 1)

Line index and line index within line group, respectively

Lower-state index

Sublayers index from shock

Boundary upper and lower indices

Line half-width, eV

Line strength, Eq. (110), eV

I

SS(J), eVJ=l

SS(I) * GAM(I) , (eV) 2

I

__ SGAM (J KM ) (eV)21

Line-center optical thickness, SS(I )/7 GAM(I )

I

TAUD(J )J=l

Line strength between boundaries IPl and J, eV

63

SGAM12

GAME12

ZETA12

WID12K

WID12(IP1 , J)

TAU12

NCOUNT, NREJE

DNU

DXX(I)

Normalized Smallest Step

Value of SGAM between boundaries IP1 and J, (eV) 2

Effective half-width, SGAM12/SS12, eV

Argument of the Ladenburg-Reiche function, Eqs. (106) and

(107)

K-th line contribution to isolated line equivalent width forregion between boundaries IP1 and J, eV

Equivalent width according to isolated lines for region be-tween boundaries iPi and J, eV

Line-center optical thickness for region between boundariesIPI and J

Counter for advancing to next integration interval or for

rejecting existing integration step size, respectively. In

Subroutine EqTEG. Approximate number of evaluations of

integrand for each region between two boundaries =

(NCOUNT + NREJE) × 4

Line group spectral width, eV

Some integration step sizes, eV

= DXX/DNU

64

REFERENCES

lo

.

.

4.

,

.

.

8

.

Chin, Jin H. ; Inviscid Radiating Flow Around Bodies, Including the Effect of

Energy Loss and Non-Grey Self-Absorption, Lockheed Missiles & Space Company,Report LMSC-668005, October 1965.

Chin, Jin H. ; Effects of Non-Grey Self-Absorption and Energy Loss for Blunt BodyFlows, Presented at the Symposium on Interdisciplinary Aspects of RadiativeEnergy Transfer, February 24-26, 1966, Philadelphia, Pennsylvania, (to bepublished in J. Quant. Spectrosc. Radiat. Transfer, January, 1968).

Wilson, K.H. ; Private Communication, June, 1967.

Coleman, W.D., Lefferdo, J.M., Hearne, L.F., and Gallagher, L.W.; A Studyof the Effects of Uncertainties on the Performance of Thermal Protection Systemsfor Vehicles Entering the Earth's Atmosphere at Hyperbolic Speeds, Lockheed

Missiles & Space Company, Sunnyvale, California, (Monthly Progress Report forthe Period 1 June 1967-30 June 1967, Contract NAS 2-4219).

Sparrow, E.M. and Cess, R. D. ; Radiation Heat Transfer, Brooks/Cole Publish-

ing Company, Belmont, California, 1966, Chapter 7•

Gilmore, F.R.; Energy Levels, Partition Functions and Fractional Electronic

Populations for Nitrogen and Oxygen Atoms and Ions to 25,000°K, RANDRM-3748-PR, August 1963.

Wilson, K.H., and Nicolet, W. E. ; Spectral Absorption Coefficients of Carbon,Nitrogen, and Oxygen Atoms, LMSC 4-17-66-5, November, 1965.

Simmons', F.S. ; Radiances and Equivalent Widths of Lorentz Lines for Noniso-

thermal Paths, J. Quant. Spectrosc. Radiat. Transfer vol. 7, 111-121, 1967:

Penner, S.S. ; Quantitative Molecular Spectroscopy and Gas Emissivities,Addison-Wesley, 1959, pp. 42 -45.

10. Goody, R.M.; Atmospheric Radiation Vol. I Theoretical Basis, Oxford ClarendonPress 1964, pp. 237-239.

11. Hearne, L.F_ '• ,_Coleman, W.D., Lefferdo, J.M., Gallagher, L.W., and Chin, _J.H. i A Study of the Effects of Environmental and Ablator Performance Uncertain-

ties on Heat Shielding Requirements for Hyperbolic Entry Vehicles, NASA CR-Lockheed Missiles & Space Company, Sunnyvale, Calif., Dec 1967.

12. Hearne, _L.F. ,Coleman, W.D., Lefferdo, J.M., Gallagher, L.W., and Chin,J. H. ; A Study of the Effects of Enviromnental and Ablator Performance Uncertain-

ties on Heat Shielding Requirements for Hyperbolic Entry Vehicles, NASA CR-Lockheed Missiles & Space Company, Sunnyvale, Calif., Dec 1967 - Data Volume.

65

4

13.

14.

15.

16.

Yoshikawa, Kenneth K. ; Analysis of Radiative Heat Transfer for Large Objects atMeteoric Speeds, NASA TN D-4051, June, 1967.

Huffman, R.E., Tanaka, R.E. and Larrabee, J.C. ; Absorption Coefficients of

Nitrogen in the 1000-580A Wavelength Region, J. Chem. Phys. vol. 39, 4, 1963.

Viegas, J.R. and Howe, J.T.; Thermodynamic and Transport Property Correla-tion Formulas for Equilibrium Air From 1,000°K to 15,000°K, NASA TN D-1429,October, 1962.

Lockheed Missiles & Space Company, Final Report, Study of Heat Shielding Re-quirements for Manned Mars Landing and Return Missions, LMSC Report4-74-64-1, December 1964, Contract NAS 2-1798, p. 3-30.

17. Howe, J.T. and Sheaffer, Y.S. ; Effects of Uncertainties in the Thermal Conduc-

tivity of Air on Convective Heat Transfer for Stagnation Temperature up to30,000°Ki NASA TN D-2678, February, 1965.

18. Hansen, C. F. ; Approximations for the Thermodynamic and Transport Propertiesof High Temperature Air, NASA TR R-50, 1959.

19. Gilmore, _. R. ; Equilibrium Composition and Thermodynamic Properties of Airto 24,000_K, Rand Corporation Memorandum RM-1543, August, 1955.

T

20. Moeekel, W.E. and Weston, Kenneth C. ; Composition and Thermodynamic Prop-erties of Air in Chemical Equilibrium, NACA TN 4265, April, 1958.

Weisner, John D.; private Communication, April, 1966.

66

LIST OF TABLES

Table

i

2

A-1

B-1

Effect of Continuum Band Model on StagnationRadiative Flux

Effect of Environmental Variables

Correlation Formula Coefficients, Air

Spectral Range and Average Cross Sections for 70/30Carbon-Phenolic "21 Band" Model

Ablation Species Band System f-Numbers

COSR(1) Values

Page

68

69

70

72

73

59

67

_qm.<

X

0

Z .o ,r.D -_

°_ _ ,_0 I

0 o

._ x

l_ II

_o _

I11

°_.,i

O o

_ II

0

I O

I

i

I -To

I _i il i

o_O _

_m

o_

O

I OO

I OO

IOO

I OO

i

'r

I

' _-.I Ir " 1_ '

Ij_.£ £

I rm

'--]_

Ii

! i

i

I

,! :!

0,1 ¢¢

J

_'L .'L__' iI 0,1

!

'-I-, _

I

iI r

I n

i

m

O°r4

"O

o.

2_

0"0

"g

m

68

.<[-.,

"O

u

u'J

_ .-. ^

_ Cr _

_ v

_ ^

_ U • . o.

c_

<I

,.0

b

"_ c_

o)

C:L

g

ZoO

U C)

"0Q_

c_

0

:<Q_

.,.4

6_

I

<

<

Z

0

00

<;..-1

Z0N<

N

0

_A

...... _ ..... _ ._ .. ..... . ....... Z_ . _ .....

I|l

III I_

,,_ _

_'_d _ ..........

¢

+

........ _ .........._ _o

o

_d_ dddd d_ "_ Mdd_ _M_ M

IIII I I I II I

t_

..... , . , . , .

II IIII I I 141

III I

,,,, _ ,,, 1'

_._ _ .1_

_ ........... _ .......

°°_ _ __ •

;..1

7O

"13

::1

O

I=1

E_o

N

,,o

o

i o

????

**o.

_ ,-_ ..,.,.

o

o

qo

k

o

§

_q

IH.Q

m

N

+ =

m _1o e_ iN

+

N m

+ _

. no ,, ,_. _

Vl

1.,

VI

I,i

_v

7!

Band

No.

6

7

9

10

11

12

13

14

TABLE B-1

SPECTRAL RANGES AND AVERAGE CROSS SECTIONS FOR 70/30

_ARBON-PHENOLIC "21 BAND" MODEL

Frequency (eV)

Lower Upper

0.50 0.800.80 0. 965

0.965 1.20

1.20 1.3951. 395 1.60

I.60 2.35

2.35 3.25

}

3.2:5 4.10

i,

i

4.10 6.20

6.20 8.00

8.00 8.60

8.60 9.00

9.00 9.70

9.70 10.95

15 10.95 12.1516 12.15 12.7017 12.70 13.35

i8 13.35 13;85

19 i3.85 14.50

20 14.50 16.50

21 16.50 20.00

System

C2 SwanC+f-f

1.86(-161 1.2

C 2 Swan 1.14(-16) 0C 2 F-H 3.94(-18) 0.29C 2 D-D 3.26(-18) 0.65

C2 Swan 3.19(-18) 0.27

C2 F-H 2.44(-17) 0C 2 D-D 1.31(-17) 0.79C2 Frey 2.41(-16) 1.52H Balmer i 3.38(-17) 10.2

Band No.

8 plusC 2 D-D 4.5(-19) 1.17C 2 Frey 3.98(-16) 0.72

C 2 Mull I.5 (-17) 0

CO 4+ 3.52(-17) I.6

H Balmer 1.19(-17) 10.2

CO 4+ 6.23(-17) 0

H Balmer 5.55(-18) 10.2

C Photo 2.0(-19) 7.46

CO 4+ 6.75(-17) 0

H Balmer 3.09(-18) I0.2

C Photo I.3(-18) 2,67

CO 4+ 6.35(-18) 0.46

H Balmer 2.1(-18) 10.2

C Photo 2.05(-17) I.26

H Balmer 1.32(-18) 10.2C Photo 2.05(-17) 1.26

-- 1.8(-17) 0

CO Photo 1.{)(-17) 0H Lyman 4.85(-18) 0C Photo 2.05(-17) 1.26

- 1.8(-17) oC + Photo - -

NOTE: _, cm2 ;/_, cm-1 ;T, eV; (-16)=

Remarksi

= 2.43(-36)(C+) 2 T-l/2

3rd excited state

2nd excited state

1st excited state

1st excited state

ground state

1st excited state

ground state•excited state

= 3.4 × 10 -20 T-le -5" 4/T

10 -16 ; o" = fqO e-B/T

72

_J

TABLE B-2

ABLATION SPECIES BAND SYSTEM f NUMBERS

System

C 2 Swan

C 2 Fox-Herzberg

C 2 Deslandres-D' Azambuja

C 2 Freymark

C 2 Mulliken

CO 4+

f-Number

0. 0059

0.02

0. 006

0. 002

0.02

0. 017

• 73

_EDING PAGE BLANK NO'T.FI_.

LIST OF ILLUSTRATIONS

Figure

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

Geometry of Inviscid Flow Field

Geometry for a Finite Number of Sublayers

Calculation for Two Overlapping Lines

Qualitative Variation of R* and R

Effects of Exponential Kernal Approximation

Effects of Air Continuum Band Models

Effects of Line-Calculation Options

Effects of Ablation Layer

Temperature and Tangential Velocity Distribution, in PhysicalNormal Distance 84

Tangential Velocity Distribution in Transformed Mass NormalDistance 85

Particle Density Distributions 86

Continuum Optical Thickness 87

Spectral Heat Flux to Wall and Shock 88

Effects of 89

Effects of 90

Effects of 91

Effects of 92

Effects 93

Effects 94

Effects

by Perturbing Ablation Molecular Species f-Numbers

Equivalent Widths for Isothermal Layers of Different Thicknesses

Blunt Vehicle Cmffiguration for STRADS C alculations

Effect of Velocity on Radiation Heating Distribution, Blunt Vehicle

Page

76

77

78

9

80

81

82

83

Number of Sublayers

Nose Radius, Constant Velocity and Altitude

Velocity, Constant Nose Radius

Nose Radius., Constant Altitude

of Nose Radius, Constant Velocity

of Perturbation of Radiative Properties

on Ablation Layer Enthalpy Distribution and Heat Flux to Wall95

96

97

98

75

_ Shock.

\ Edge of Ablation Layer _

Fig. 1 Geometry of Inviscid Flow Field

76

Shock Wave

fAZ 1

Iui

1

Zl.= 0

Z 2

AZ il I

!

Uo

I I

Z i '

Zi+l

Z n

/_Z °kI1 I : I_

Wall _ -Zn+ 1

Total Number of Sublayers = n

Fig. 2 Geometry for a Finite Number of Sublayers

77

?8

0v

I0 _

.-- c_

I I

I I 1 !

I I I

puoJBa,tu I

¢)

0

0

r_

c_

-" _1

4_

_g

1.0

0

R1.O

Fig. 4 Qualitative Variation of R* and R

79

o9I-

Air Continuum Only "_

0.8

0.7

0.6

0.5

Band Model 4I0 Sublayers

Uao

P_o/PSL

RN

P_/Ps

h/hsa t

Ps

TW

qw qs _a

Actual watts/cm 2 cm

E3( _ ) 2,087 4, 326 9.34

I I I I0.2 0.4 0.6 0.8

z/AO

= 15.24 knV'sec

= 1.66 x 10-4

= 256 cm

= O. 058

= 3.98

= 0.44 atm

= 0.331 eV

(3840 ° K

1.0

Fig. 5 Effects of Exponential Kernal Approximation

8O

\\

\\

Air ContinuumWith Line Correction

LINEOP = 2, 10 Sublayers

\\

Band qw qs

Model watts/cm 2

-- -- -- 1 3,056 5,546

4 2,903 6, 339

u = 15.24 km/secoO

poo/Ps L = 1.66 x 10-4

RN = 256 cm

P_o/#s = 0.058

h/hsa t = 3.98

Pa = 0.44 atm

TW

\

! I I0 0.2 0.4 0.6

z/_ a

= 0.331 eV

(3840°K)\

\

i

.0

Fig. 6 Effects of Air Continuum Band Models

81

1.0

e-

0.9

0.8

0.7

0.6

0.50

-- Air With Continuum_,qand Line _k

10 Sublayers

u = 15.24 km/secOO

poo/PSL = 1.66 x 10-4

RN = 256 cm

Poo/Ps = O. 058

h/hsa t = 3.98

Ps = 0.44 atm

TW = 0.331 eV

Band Model 4 for LINEOP = 2_"_ (3840OK)

qw qs _ _

LINEOP watts/cm 2

- 2 2,909 6, 339 8.59 _'_"x

3 2,969 6;581 8.53 "_,_

I I I I

0.2 0.4 0.6 0.8 1.0

z/A a

Fig. 7 Effects of Line-Calculation Options

82

1.3

b.

I--

Air With Continuumand Line

10 Sublayers

Ablation Layer With Continuum

I 0 Sublayers

Ablation

Layer

No

, Yes

u = 15.24 km/secoo

poo/PSL = 1.66 x 10-4

RN = 256 cm

P,,o/Ps = 0.058

h/hsa t = 3.98

Ps = 0.44 atm

TW = 0.331 eV (3840 °K)

Carbon Phenolic Surface

1 eV = 11605.7°K

!

qw ,qs zx,,a

watts,/cm 2 cm

2,969 6, 581 8.53

1,749 6, 798 8.55

I I I I0.2 0.4 0.6 0.8

z/AQ

1.0

Fig. 8 Effects of Ablation Layer

83

1.4u = 15.24 km,/sec

P_o/PsL = 1.66 x 10 -4

RN = 256 cm

! eV - 11605.7°K

AirLayer

AblationLayer

I-.

8i#1

_ Air With Continuumand Line

10 Sublayers

Ablation Layer With Continuum

_ 10 Sublayers

Carbon Phenolic Surface

= 0.44 atm

= 10.94 cm

0.097

0

I I I ....0.2 0.4 0.6

z/z_

0.8 1.0

Fig. 9 Temperature and Tangential Velocity Distribution inPhysical Normal Distance

84

1

1.0

0.8

0.6

0.4

0.2

0

Air

Layer

10 Air Sublayers

10 Ablation Sublayers

Carbon Phenolic Surface

Ablation

Layer

U

P_/PsL

RN

= 15.24 km/sec '

= 1.66 x 10-4

= 256 cm

I I I J0 0.4 0.8 1.2 1.6 2.0 2.4

Z

fp dz_ = ps A

0

Fig. 10 Tangential Velocity Distribution in TransformedMass Normal Distance

85

1018 t

Air

Layer

Ablation

LayerC2

C

H

1017N

O

CO

EU

¢)

U

,m

1016

1015

10 Air Sublayers

10 Ablation Sublayers

u : 15.24 km/sec

: 1.66 x 10-4P_/P SL

RN : 256 cm

Carbon Phenolic Surface

C ÷

1014

0

1 10.4 0.8 1.2

Z

f p dz: Ps AO

l1.6

Fig. 11 Particle Density Distributions

86

t

102

k

101

100

Be-

U

e-l'-

OU

"_- --1a. 10O

ED2_c-

e-Ou

10.2

10-3

10-4

u = 15.24 km/sec

p_o/PSL = 1.66 x I0 -4

RN - = 256 c-m_--_ ....

Shock to Wall

-.---.4"-

°

ocktoj Interfacef

10Ai_s_b!_yer_10 Ablation Sub!ayer_s

Carbon Phenolic Surface

! I I !0 4 8 12 16

/_i eV

20 ¸

Fig. 12 Continuum Optical Thicknesses

87

104

103

>

Eu

a 02

1I0

1000

U

P_/PsL

RN

: 15.24 km,/sec

= 1.66 × 10-4

= 256 cm

10 Air Sublayers

10 Ablation Sublayers

Carbon Phenolic Surface

I

I

II

I\

\\

--qwtv,

m" _ "" qw t l.J,

(s,.2.ght_yqAbove) s Iv i

1356 78

'111111 I I i

4il

II ./I

i

With Ablation Layer i,749

Without Ablation Layer 2,969

Without Ablation Layer 6,581

\\ \

o

Total

watts/cm 2

With Ablation Layer 6, 798

9 15 17 19

IIIIIRegion

I I4 12 16

I0 II 13

I II" 'IFrequency

I8

2O

\

2.1

v, eV

2O

88

Fig. 13 Spectral Heat Flux to Wall and Shock

>

I--

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

%o = 15.24 km//sec

Poo/PsL = 1.66 × 10-4

RN = 256 cm

uT

e-

--II

"_,, u/us

-- 1 eV = 11605.7 °K _

,

-- Carbon Phenolic Surface _.

-- C_bon Phenolic Su'r/ace _\ I

qw q_ A OwVw_J

I I I ! _0 0.2 0.4 0.6 0.8 1.0

Fig. 14 Effects of Number of Sublayers

89

olIOi.---

X

EL)

(3

x"

LI.

>

.¢,.

14

"(3

7

6

5

4

3

2

u_o = 15.24 km,/sec) (50 kft/sec)

poo/PSL = 1.66 × 10 -4 _212_ kft)

6 Air Sublayers

6 Ablation Sublayers

Carbon Phenolic Surfaceqs ab9

q$

qW

-g..e-----'----

0 I i I I0 2 4 6 8

RN , FT

0

9O

Fig. 15 Effects of Nose Radius, Constant Velocity and Altitude

105

104

E

a

o-

103 -

102

RN = 256 cm

• 0

• a

x A

P_/PSL

5.94 x 10-4

1.66 × 10-4

3.26 × 10-5

Altitude, kft

175 •

212

250

J

JJ

J

JJ

f

JJ

JJ

J mJ

JJ

J

6 Air Sublayers

• 6 Ablation Sublayers

Carbon Phenolic Surface

JJ

JJ

J

I xj I

JJ

JJ

12 13

J

Opened Symbols, Without Ablation Layer Effect

Filled Symbols, With Ablation Layer Effect

Tacked Symbols, Zero Ablation

I I I I14 15 16 17

u_o , km/sec

18

Fig. 16 Effects of Velocity, Constant Nose Radius

91

E

105

104

103

102

0

Poo/PsL = 1.66 × 10 -4 (212 kft)

• O

• 13

•( A

u km/sec KFT/secao •

17.38 57

15.24 50

12.20 40

Carbon Phenolic Surface

Opened Symbols, Without Ablation Layer Effect

Filled Symbols, With Ablation Layer EFfect

Tacked Symbols, Zero Ablatton

( i ) Interpolation

0 _

(i)

n_

_l_mm _ _ amJm _

a

• ...._....-iI

Jf

fI

(i)

_g°

I I I I I50 i 00 150 200 250

RN , cm

3OO

Fig. 17 Effects of Nose Radius, Constant Altitude

92

i0 4 --

E..u

.8--

0"

103 --

1020

P_ /Ps L u

• ® 5.94 × 10-4

• !_! 1.66 × 10-4

J( A 3.26 × 10-5

Opened Symbols, Without Ablation Layer Effect

Filled Symbols, With Ablation Layer Effect

Tacked Symbols, Zero Ablation

(i) Interpolation ._.

= 15.24 km/sec (50 kft/sec)

Carbon Phenolic Surface

(;)

©

----Ig-..----------Ig

I

I I I 1 I50 100 150 200 250

RN , cm

100

Fig. 18 Effects of Nose Radius, Constant Velocity

93

.j_

1.0

0.95

0.90'

0.85

0.80

0.75

0.70

0.65

0.60

0.55

0.500

*For band systems

I

u o = 15.24 km/sec

Poo/PSL = 1.66 x 10-4

RN "= 256 cm

10 Air Sublayers.

10 Ablation Sublayers

No Ablation Layer Effect

With Ablatlon Layer Effect

Carbon Phenolic Surface I..

o" * f_ "Y_ qw qsac abp < 10.95 watts,/cm2

n

3 N N N N 2,969 6,5811,749 6, 798

2 2N - - _ 2,573 4, 989

1 N - - _ 2,118 4,481

4 N - N 2N 3,108 6, 985

5 N - 2N N 3,208 7, 220m

6 N 2N 2N 2N 3,361 7, 7371,926 7, 988

"-" Not ConsideredI I

0.2 0.4 0.6 0.8 1.0

Fig. 19 Effects of Perturbation of Radiative Properties

94

4.5j

4.0

3.5

3.0

2.5

2.0

1.5

1.0

%o = 15.24 km/sec

#_o/PSL = 1.66 x 10-4

RN = 256 cm

10 Air Sublayers

10 Ablation Sublayers

Band Systemqw

watts/cm 2

Nominal Crab 1,749

2XNominal Crab 1,738

Carbon Phenolic Surface ?

'1IIIIiIIIIIIIIIIII.

II!

I!

1.0

Fig. 20 Effects on Ablation Layer Enthalpy Distribution and HeatFlux to Wall by Perturbing Ablation MolecularSpecies f-numbers

95

100

t = 1.24s eV (14450 oK)

P/PSL = 2.87 × 10-3

NI = 0.401 x 1017

01 = 0.148 x 1017

e = 0.806 x 1017

I0"I

4 4 0.195

10'3

lO-4 I 10.1 1.0 .... _0.0

_a ' cmI00

Fig. 21 Equivalent Widths for Isothermal Layers of Different Thickness

96

UoO

Effective Axls of

_" Symmetry

R0

\

J

Fig. 22 Blunt Vehicle Configuration for STRADS Calculations

97

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\\

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C_

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"0

0

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ro

cq

98

C. 1 Listing of STAGRADS and Input Data

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