Radiating and absorbing steady flow over symmetric bodies

J. Quant. Spectrosc. Radiat. Transfer. Vol. 8, pp. 119-144. PergamonPress, 1968. Printed in Great Britain

RADIATING

AND

OVER

ABSORBING

SYMMETRIC

STEADY

FLOW

BODIES

K. C. WANG* Martin Company, Baltimore, Maryland Abstract--This work presents an approximate non-linearized solution of the multi-dimensional radiating and absorbing steady flow over smooth symmetric bodies, pointed or blunt, straight wall or curved. Using a series expansion in the density across the shock, the multi-dimensional differential approximation for radiative transfer is reduced to a locally one-dimensional form normal to the body surface in the zeroth approximation. Zeroth order and part of the first order solutions of the radiation-convection coupled fields are then obtained; as specific examples, flows over a wedge, cone, and sphere are discussed in detail. Similar results previously obtained for an unsteady piston problem show satisfactory agreement with exact solutions. I. I N T R O D U C T I O N

THIS paper is concerned with the thermal radiation effects for a multi-dimensional steady flow over symmetric bodies. Both absorption and emission are considered ; i.e., the optical thickness is arbitrary. It is a counterpart of the one-dimensional unsteady piston problem with thermal radiation previously considered by WANG.m The method of approach is similar in many aspects for these cases; namely, one uses series expansions in the density ratio across a shock. This method has been applied to classical hypersonic flow problems by several authors, notably CHERNYI¢2J and FREEMAN.~3) Its extension to include thermal radiation effects has been developed by the author. ") Validity was demonstrated through favorable comparison with exact solutions (WANG¢4)). Previous investigations of the thermal radiation effects on a steady flow over symmetric bodies are restricted either to (a) the transparent limit or (b) the stagnation region of a blunt body where a one-dimensional model of gas slab can be reasonably justified. For some relevant references, the reader is referred to a recent review paper by GOULARDand TRAUGOTT.(5)

To the author's knowledge, only OLFE¢6)and CHENG17) have considered to date a multidimensional steady radiating flow problem with self-absorption. In both cases, the approach is a linearization procedure. Olfe considered a wedge flow behind a shock perturbed by small radiation which is evaluated from the corresponding non-radiating flow. Since his unperturbed states (i.e., non-radiating solutions) are taken to be constant, application of his scheme is limited to a steady wedge flow or an unsteady plane shock propagating with constant speed. Cheng considered a wavy wall flow using the linearized equations derived on the basis of small disturbances. Here the high temperature and hence radiation are not generated by a strong shock due to the body's motion ; the physical picture is, therefore, different from that of Olfe's and the present work. * Now at RIAS, 1450 S. ROlling Rd., Baltimore, Maryland. 119

120

K . C . WANG

In contrast we shall not assume here that either the radiation or disturbances is small, and our scheme is applicable to any smooth symmetric body. Mathematically we shall deal with a non-linear problem rather than a linearized problem. Instead, the only basic assumption in the expansion is that the density everywhere in the flow field should be much higher than the density ahead of the shock. The validity of this assumption has been well demonstrated in the classical hypersonic flow theory, and should be even improved in the present case because thermal radiation like non-equilibrium relaxation generally increases the density of a shock-heated gas layer. With respect to the radiation properties, certain common simplifications previously used are also retained here. These include local thermodynamic equilibrium, gray gas, transparent shock, black and cool body, and negligible upstream absorption. In Sections IIA and liB we show that as in the unsteady case the pressure p~O)and velocity u ~°)are not affected by radiation. This fits into the idea that to simplify a radiating flow problem, one may consider the pressure and velocity as known from the corresponding non-radiating solutions. Radiation affects the next order of approximation for pressure and velocity, i.e. p¢lj, uCl), v¢O),only indirectly through the density p¢o). In Section IIC, the multi-dimensional differential approximation for radiative transfer (TRAtJGOX'r~81 and CrmNG~9)) is shown to be reduced to a locally one-dimensional form containing only derivatives normal to the body surface. This is a crucial point of the present work. This comes as the zeroth approximation in our series expansion, but also follows directly from the classical boundary layer concept. It is a consequence of the hypersonic narrow shock layer approximation and, therefore, has nothing to do with radiation as such. In Section liD, we carry out the zeroth order solution of the radiation-coupled temperature field. In doing so, we discuss and use the so-called local temperature approximation which is not only generally good in the thick limit, but also recovers the thin limit exactly for V. ~. The formulation is first developed without invoking any particular form for the state equation and the absorption coefficient. Restrictions on the latter and the perfect gas relation are introduced only at a later stage for the purpose of obtaining simple analytical solutions. Reasonably good accuracy of such zeroth approximate solutions has been demonstrated in the unsteady piston problem (WANG~IJ). The rest of the present work is devoted to detailed discussions of flows over a wedge and cone and a sphere, chosen respectively as the typical examples of pointed body and blunt body problems. Sections IIIA-IIIB give the results for the transparent limit where no restriction on the absorption coefficient exponents is imposed and higher order pressure and velocities are also readily obtained. Sections IVA-IVB give the same for arbitrary opacity and contain the main contributions of the present work. The essence of this work appears in two recent reports (WANGH~t°~) where more details can be found. II. F O R M U L A T I O N

A. Basic equations Written in the yon Mises coordinates (x, ~b), the relevant continuity and momentum equations take the form (CHERNYIt2)), Cont.

~ dy

purJ'

d~ dx

- ( 1 + y/Rb)pvr J,

(la)

Radiatingand absorbingsteadyflowoversymmetricbodies

121

x-Mom.

Ou Ov U-~x+V~x

=

1 Op p Ox'

tic)

(lb)

y-Mom.

1 By 1 _d~p l + y/R b ax Y + RbU = -~

where ~/is the stream function, p pressure, p density, x and y boundary layer coordinates (Fig. 1), u and v the velocity component along x and y, Rb radius of curvature of the body, r distance from the axis of symmetry, j = 0 and 1 for two-dimensional and axisymmetric bodies, y is considered here as a function of (x, ~b).

/ Os(X~~ u /

Body

x

Pl' Os(~~ V

shock~

shock~

~X~---Axisof symmetry FiG.1. Flowoversymmetricbodies.

The radiation effects enter only in the energy equation and, of course, the transfer equation Energy

pu

[a_I [ h + ~ u e + v 2 ) ] = V . # , ,

1 + y/R~,lax]¢ ,

(ld)

Trans~r q,-

a K '

(le)

122

K.C. WANG

where h is the enthalpy, ~, radiation flux vector, T temperature, a Stefan-Boltzmann constant, K volumetric absorption coefficient. The present multi-dimensional differential approximation for radiative transfer equation (le), was derived by TRAUGOTTta~ and CHENG.(9) For brevity, we have not yet written the vector terms in equations (ld,e) in terms of von Mises variables. To complete the system of equations we write for the time being Thermal state p = p(p, T),

(lf)

h = h(p, T),

(lg)

K = K(p, T).

(lh)

Caloric state

Absorption coefficient

B. Pressure, velocities and shock shape Following CHERNYI(2) we expand in terms of the density ratio across the shock, e, y = ey(O)+ g2ytl) + . . . ,

(2a)

u = u (°) + eu (1) + . . . ,

(2b)

p(1) + . . . .

(2c)

p = p(O)18 +

We also expand v in the same form as y, p and 0s in the same form as u. Substituting (2a, b, c) into (la, b, c) and the usual hypersonic shock jump conditions, one finds (CHERNYI (2))

u(°)(x, 0)

= u(~°)(0) = v c o s

Oh(O),

(3a)

O

P(°)(x, 0) = Pl V2 sin20b(X)+~V-~ f cos 0b(0) dO, t%r~ #(fl)tx) # 1 f dO

(3b)

(3c)

y(°)(x, O) = r~(x) J u(°)(O)p(°)(x, O)' o

(3d)

vt°)( x, O) = u(o)(0) t3Yt°)(x, O)fl?x,

"i)

I

.(')(x, O) = u~ ( O ) + ~ ¢1

,......o.=

n,,,,/,+_, f ru,,, v2 , , r~b(x)j L Rb g,~o)

~

1

J p(°)(x, O) x,(¢,)

Of°)(x, O) dx,

Or(O) u(O)ly(O) y(O) c3x

Rb--'--~.+Jrlb

(3e)

~x

17

b COSOb(X)]JdO'

(3f)

where V is the free stream velocity, the subscripts s and b stand for the shock and body, the subscripts 2 and 1 refer to the condition immediately behind and ahead of the shock. Angle 0s and Ob are shown in Fig. 1. u(°)(0) and xs(0) are referred to the point of entry of

Radiating and absorbing steady flow over symmetric bodies

123

the streamline being considered. ~k~°), I//(1), U(2I) and pt2~)are, in turn, given by 1

0~°) - 1 +j plVr~

+j

'

(3g)

I//~1) = Pl VrJby~O) COS Ob,

(3h)

• . ~l[-'~-]¢,=,~o, cgu(°~l U{21)= --~ls - V s i n O b dy~ dx°) '

pt2t)=

(3i)

_ 0~1)(Opt2O)I [ dv{O) -I - ~ - i O = ¢ ~ o + p l V 2 2sinOb COS0b--d~-x ~ --sin 2 Obj. "

(3j)

For further detail, see CHERNYI. (2) Note that equations (3a, b) are identical to the expressions for the non-radiating case; (3c-f) are only formally equivalent because the density p(o~ is considerably affected by radiation. We shall proceed now to use the non-radiating solution of u t°~ and p(O} for obtaining T (°} and q~O}from the energy and transfer equations. Once T ~°} is known, p{O~can be obtained from the state equation. Then y(O}and vt°J can be determined and the zeroth order solutions are completed. With pto~ and y(O}known, one can go on to determine u (1) and pm and then use u m, pO) to start next order approximation.

C. Radiation transfer The main problem is how to treat the energy equation (ld) and the radiative transfer equation (le). We will consider the latter first. To begin, we introduce a radiation potential [COHEN" 1~] f~ Vfl = KO~,

(4a)

so that the vector equation (le) can be replaced by a scalar one, V2~ 1 (1) K2 -F~(V~). V = 3 f l - 4 a T 4.

(4b)

Proceeding now as in the cases of the continuity and momentum equations, we first write the left-hand terms in the usual boundary layer coordinates (x, y) V2a 1 [ 1 a2a 02a K 2 - K2(1 +y/Rb) l+-y/Rb -ex- 2 0+y 2 0(8 2)

0(1)

0(1/e 2)

y/R 2 ORb 1 ~D], (l+y/Rb) 2 ax +-~b Oy_] 0(0

(4c)

O(1/e)

(4d) O(e)

O(e)

O(1/e)

124

K . C . WANG

Next we transform from (x, y) to the von Mises' coordinates (x, q,) through •

0

(O~} = purJ(--~).

(4f)

To avoid the lengthy substitution of (4e, f) into (4c, d), we indicate at this point that, if we substitute (2a, b, c ...) into (4e, f), we find that

-~x y = -~x ~- EP(°~v~°)r~+ ev~°>r~(P~°>Y¢°>/Rb÷ p(1) +pW'v(~'/vm'+jpm'ym'(cosOo)/ro)+...(-~)

=

x,

o(1) ~ .

(4g)

(4h)

Clearly (O/Oy)x is always one order higher than (0/Ox)y. Based on the consideration that we wish to keep the leading term from each of the absorption (3f~), emission (-4aT4), and transmission (the left-hand terms) terms in (4b), we further expand T = T(°)+~T(X)+ -..,

(5a)

f~ = f~(o)+e_O(1)+ ...,

(Sb) (5c)

K = K(°)/e,+K(1)+ ... • Substitution of (4g, h) and (5a, b, c,) into (4b, c, d) yields for our zeroth approximation

1 O[ l Oa,°,l=

a(x, ~,) O-~b a(x,, ~,) d~k _i

3~(°)-4a(T(°))4'

(6a)

where

a(x, ~O) = K(°)/[p(°Ju(°)(~b )r~(x)],

(6b)

i.e. our original multi-dimensional transfer equation is now reduced to a locally onedimensional (y-direction) equation. Actually, the same idea becomes immediately obvious if we apply the usual boundary layer concept directly to (4c, d), where the order of magnitude estimate is indicated below each term. The preceding presentation however has the advantage of (a) being systematic with respect to the other flow equations and (b) allowing ready extension to higher approximation if so desired. Let dr/= a(x, ~,) d~,, then the solution of (6a) for f~o) is

[TW)(x, ~)]4 e-(43)(~-O d,-

f2(°)(x,~/)=~3{f q~ =

o

f

[T(°)(x,~)]ae-(43)(g-n)d~},

(6c)

q~ = ~(m° )

where we have employed the conditions that no radiation flux enter the gas layer from the

Radiating and absorbing steady flow over symmetric bodies

125

body or from ahead of the shock. We now expand

~, =

~o~+~,+...,

(6d)

then (4a) becomes in the zcroth approximation

1

af~ ~°)

(6e)

h = h(°)+eh(~}+ . . . ,

(7a)

q~O~ = "Y a(x, ~k) O~k ' where ~/ry a ~°~ stands for the y-component of ~o)

D. Energy equation We make the expansion

The energy equation (ld) together with (4b) then becomes, in the zeroth approximation,

I

dx /t = plO~

J"

(7b)

Eliminating f~(o~between (6c) and (7b) results in a non-linear integro-differential equation for T ~°) (remember p~O)is known in (3b)). A rigorous solution of such an equation cannot be found except by numerical schemes. We shall simplify this by invoking the so-called "local temperature approximation" ; i.e. we replace T<°~(x, ~) by T~°)(x, ~1) in the integral of(6c), so that Oh(°) I = _ 2,[T'O'(x, ~k)]4a(x, ~k)r/(x)[exp{--(x/3)(~/,=,~o,--F/) } +exp{--(x/3)(~/--r/,=o)}], 8x I~, (7c) with #

/7 =

| a(x, ~b)d~b.

(7d)

J

The validity of this approximation is explained as follows: when the absorption is weak, (7c) recovers the correct thin limit because both exponential terms approach one. In the other limit, when the absorption becomes very strong, the exponential weighting functions are very sharply peaked and only the contributions near ~ = r/ matter. Physically, this simply means that when the gas is opaque, only nearby radiation can affect the local heat flux. One must be cautious, however, because even in the thick case, the approximation may fail if the temperature gradient is sufficiently large. This behavior occurs typically near a boundary (or a shock) and hence will not cause trouble with this approximation because it is confined to so small a region. In the middle of such a shock layer, as we shall see, the temperature gradients are similar to those for a non-radiating gas. It therefore follows that the local approximation reproduces the thin limit for V. ~, and is valid in most real cases for the thick limit. In the region between, i.e. of moderate opacity, there is no reason to expect any difficulty; but this cannot be proved. One must rely on comparisons with exact solutions for similar types of problems. Such comparisons are available, for example, in the works of W A N G , (1) Y O S m K A W A and CHAPMAN. (12) It should be emphasized that this local approximation is used only in the energy equation (7c) to obtain the temperature (i.e.,in V. ~,),but not in determining the heat

126

K.C. WANG

flux, ~,, itself. The reason is that in the thin limit, this approximation recovers the correct expression for V. ~, (= - 4 a T 4 ) , but not that for ~,. Our development so far is rather general. We have not invoked any particular form for the absorption coefficient K(p, T) or the state equation h(p, T). If one prefers, one can solve (7b) or (7c) numerically using real gas tables and opacity data. This would be much simpler than to attack the original system of equations (la) through (lh) even in terms of numerical schemes. For the purpose of obtaining simple analytical solutions for (7c), explicit forms of the state equation and the absorption coefficient are needed. The simplest state equations are, of course, those of perfect gases h = cpT, p = pRT, where % is constant. The corresponding expansions are h(O)+ eh(t) + . . . .

%(T~O)+ eT (1) + . . . ) ,

(8a)

and p(O)+ ep(1) + . . . .

R(ptO)/e + p(1) + . . . ) (T(O) + e T (1) +...).

(8b)

Since the gas constant R is of the order of e as ~ ~ 1, (8b) is self-consistent. For the absorption coefficient, we approximate K(p, T) by a power law K = K l p ~ T B,

(8c)

with constants K1, 0~,fl to be determined by fitting with opacity data. Expanding both sides of (8c) gives K(O)/e + K o) + . . . .

K~(p(O)/e + pO) +...)~(T(O) + eT(1) + . . . ~ .

(8d)

Substituting (8a, b, d) into (7c), we find that a simple solution is possible only when 1~- ~ + 1 = 0,

(8e)

so that t/is independent of T (°). Under this restriction, we have T(---~--I~) =

1+

a(x', ~b)rb(x ){exp[-(x/3)(r/#,=,T,-t/)]

[T{2°)(~b)] 3 xdqO

"1- 1/3

+ exp[- (x/3)(r/-r/#,=o)]}d x ' ]

,

(9)

with a(x, ~b) = Kl[p(°)(x, ~b)/R] ~- l/u(°)(~b)r~(x).

(9a)

Again T~°)(~,) and xs(~') are referred to the point of entry of the streamline being considered. The right side of equation (9) contains only known quantities. The condition/~- ~ + 1 = 0 for ~ = 1 is formally equivalent to first writing K = pK,, and then taking the mass absorption coefficient Km to be constant, say K"m.

Radiating and absorbing steady flow over symmetric bodies

127

In the transparent limit, (le) becomes V. ~r = - 4trK T 4, and (7b) can be integrated without the restriction (8e), to give T(°)(x, ~b) T(2O)(~b) = .( 1 + 4( 4 + fl - a )trK x R ~ - ~' [T(2°)(~b)]4+a-"

cpu'°~(q~)

[p(O)(x' ~k)],- 1 dx~] - 1/(4+/1-~) .

f

(9b)

9

xs(~,)

Once T ~°~ is known, the streamlines y~°~(x,~), radiation flux Try .~o~, velocity components v~°~,u"~, and pressure p"~ can be determined from (3c) through (3f) and (6c, e). Representative examples of both pointed and blunted body flows will be given first for the transparent limit in Section III and then for arbitrary opacity in Section IV. E. R a d i a t i o n p a r a m e t e r s

Before turning to the discussion of special examples, let us comment briefly on some confusion which may arise in the presentation of radiation effects. This is caused by the fact that in the general case of arbitrary opacity, there are two governing parameters for equation (4b). One is the absorption parameter (10a)

% = K o L o,

where z is the optical thickness, K volumetric absorption coefficient, and L length; subscript zero stands for some "reference state." z0 is also referred to as the "intrinsic radiation parameter" (GoULARD (1~)) or, in the literature merely as the reference optical thickness. The other parameter is the radiation-convection parameter, which may take any one of three different forms F = trT'~/%TopxV,

Fn

---- I " Z 0 ,

I"k = F / z o ,

(10b)

where F n and F k are often called the thin and thick radiation parameter (because in those limits, each is the only radiation parameter appearing). The choice among these three depends on the problem and the point of greatest interest. F o r the wedge and cone flow of Section IVA, since there is no fixed reference length, we shall choose F, so that z 0 can always appear together with x, the coordinate along the wedge or cone surface. F o r the sphere case of Section IVB, we shall choose Fn, so that the absorption effects can be better displayed near the thin limit. In any case, care must be taken in interpreting limits of the results. III. SOLUTIONS OF TRANSPARENT

LIMIT

We present here a separate detailed treatment in the transparent limit for several reasons : (a) the absorption coefficient is not restricted to any particular value of ~ or fl, (b) higher order approximations to the pressure and velocities are obtained in simple closed form from which general conclusions can be made ; and (c) the transparent solutions themselves are often good in practical problems.

128

K . C . WAN6

The transparent solution becomes particularly simple if a is taken to be unity. This is generally valid for air at a temperature below 15 000°K. A. Cone case

For a cone, Rb = oo, 0b = constant = 0c where 0, is the half cone angle. Equations (3a, b) and (9b) with a = 1 become

u(°)(x, ¢,)

(lla)

= u(2°'(~) = V c o s 0~,

(11b)

p(°)(X, ~b) = p(2°)(x) = Pl V2 sin20c, T(°)(x, ~b)/T(2°)(~) = [1 + F,.x(1 - x~(de,)/x)]- 1/(3+p),

(11c)

where F~ = 4(3 +/~)rJLo, .....

=4(~+/s)

aptKl[T~°)(¢)] '*+p ~ YlU~2 t'p~t 2 ~W]

.

Lo is some arbitrarily chosen reference length. In fact, Lo need not be specified here because Fc is always associated with x. The streamline equation is tanOc

3+fl

(1 + Fr.x)t2B+ 5)/(3+~)}

3,}].

(lld)

(1 +F,~)(3 +fl)~r 1

I

xrt,],~,! (15+2)1(15+3)

tk +r ll- lJ

- ( I + F ~ ) ~+2'1~+

The shock shape y~O)is given by setting xs(~) = x in (lld). For F~x ~ 1, y~O)~ (x tan 0~)/2. This result is familiar from the non-radiating constant density solution. The radiation flux to the cone surface (q~°))b (or across the shock (q~Oj)s)is (n(Oh •l,y Jb

½pl(VsinOc) 3

=

? 1 l - - [ 1 + (fl + 2)(1 + F,.x)- (3 +fl)(1 + Fcx)ta + 2)/ta+ 3)]. r + l f l + 2 F,.x

(lie)

With yOJ(x, ~) given by (1 ld) and using the relation dy~°)(x)/dx = tan 0c + [dy(°)(x, x~(~,))/~x],,,(,)= x,

one can readily obtain v(°), u "~, p") in algebraic form (WANO")). We shall, however, omit these expressions but present instead some calculated results from which general conclusions can be drawn regarding the radiation effects on pressure and velocities. Figure 2(a) shows that with radiation, - u " ) increases from the shock toward the cone surface, implying that the total u-velocity (u = u ~°~+ eu "~) decreases across the shock layer. Figure 2(b) shows that without radiation v(°~ increases linearly from the cone surface; with radiation the profile for v~°' is slightly curved near the shock. Figures 2(c) and 2(d) show that with radiation p") (hence the total pressure p = p¢O)+ tpO)) is decreased slightly along the cone surface and across the shock layer. Results presented in Figs. 2(a) to 2(d) clearly show that the radiation effects on the velocities and pressure are small. This agrees with previous numerical results (BIRD,~14) and WANG(4)).

Radiating and absorbing steady flow over symmetric bodies

129

1-0

Nonradiating

0"8

y(s~O I

FcX = 2 0 ~

0.6

_rcx = I0

0"4 0"2 I

I

0"40

0"30

O' 20

"

0.50

-ull)/u10)tan2 0 c

(a)

lo I 0"8

u (~) distribution

across the shock layer.

Nonradiating~

0'b y(s~ °)

Radiating

0"4 0"2 I

0

0"2

f

0"4 -v(O)/v sin 8C

t

()'6

0"8

(b) v(°) distribution across the shock layer.

0.3I

Nonradiating

q

0.2 _(1) Radiating 0.1

I 4

I 8

1

[-,×

12

i

16

c

(c) p(l) distribution along the cone surface.

I

20

K . C . WANG

130

1"00"8-]"Cx = 1 0 ~

nradiating

0"6-0"4 0-2 I -0-2

-0.4

0

I 0"4

0-2

p(1)/p(O) (d) p(l) distribution across the shock layer. FIG. 2 . ~ =

l, fl= 5.

B. Sphere case For convenience we shall choose 0(= ~/2- Oh(x)) and ~b(= 1r/2- Ob(~)) as independent variables. Since only the zeroth approximation is intended to be carried out here, the subscripts b and s are not needed. For a sphere we have

u(°)(x, ~) = u(2°)(d/)= Vsin ~b, pt°)(x, ~b) = p~ V2(sin 30+sin s ~b)/3 sin 0.

(12a) (12b)

The temperature along a streamline (~b is fixed, 0 > ~b) is

T(°)(x'qJ) = T(2°)(~b)

[

l+(3+fl)4F.

(0--~)(C0S~)2(3+~)]-1/(3+fl) s~n~

'

(12c)

where F n =

aKlpt[T~°)(¢,, = O)]4+#Rb p, V%T¢2°)($ = O)

Across a shock layer (0 is fixed, 0 _ ~b _ 0) the temperature distribution is T(°)(xAb) T'°)(x, ~k) cos 2 q~ T~°)(~ = T~°)(~k) cos 2 0'

(12d)

where T~°)(x) is the temperature behind the shock at x. The streamline equation is

3+0sin 3 ~[ l+(3+fl)4Fn(O-~)(c°s~-)213+#)]-l/(3+#)d~'sin ~ ] sin 3~c°s3

Yt°)(x' I b R~k)-

(12e)

The shock shape y~°)(x) is obtained by letting q~ = 0 in (12e). For small 0 and small F c, y~O) ~ Rb as was shown by FREEMAN. (3)The radiation flux into the sphere t,,(o)x I,"lryIb or across the shock (q,r)s (o) is o

e 4 rno f(cos4,) lplV3 _ ~:-F1 slnvd

[

.........

(cos

s-~

J

d~b.

0

(12f)

Radiating and absorbing steady flowover symmetricbodies

131

For small 0, (12f) becomes 1

( q ~vO3 -) ) -b ? f4r. I 1 +(3 +fl)4F. ~1 ½p

]

~-1 -(4+#)/(3 +#)

d~.

(12g)

0

Again the continuous cooling along a streamline and the decrease of shock distance are obvious radiation effects from equations (12c) and (12e). The singularity at 0 = 60 ° in equation (12e) is a consequence of the zero surface velocity which, although not a good approximation, is nevertheless consistent with the transparency assumption. In the absence of absorption a particle continues to radiate energy so that the surface temperature is theoretically zero. A particle with zero temperature certainly cannot accelerate around the body, i.e., the velocity remains zero as at the stagnation point. The temperature distribution across a shock layer is shown in Fig. 3(a) for a = 1, fl = 5, R b = 5 ft. Near the stagnation region, the temperature decreases continuously behind the shock. Down stream, it first increases due to the shock curvature and then decreases to zero at the surface, due to the radiation cooling. In spite of many differences in the calculation details, a comparison with WILSONand HOSHIZAKI'Stl s) results obtained by an integral method is also shown for 0 = 30 ° and Tt2°)(x= 0) = 15 000°K. This value of temperature corresponds approximately to a flight velocity of 50 000 fps at an altitude of 19 000 It, the conditions used in their calculations. The present result falls in between their two limiting curves calculated with the upper and lower estimates of air emissivity. Figure 3(b) gives (q,r)b/½Pl co) 1 V a with ? = 1"1. The higher the stagnation temperature immediately behind the shock (T~2m)x=o, the smaller is the ratio (q,v)b/-~P ~o) 1 1V 3 ; i.e., the smaller is the portion of the total particle energy being converted into radiative heating to the body. Figure 3(c) presents the radiation flux to the body normalized to its stagnation point value. The fact that all three curves are so close to each other suggests that such distribution is not sensitive to the change of flight conditions. Similar results were also obtained by WILSON and HOSHIZAKI~ls) as represented by circled dots in Fig. 3(c). IV. SOLUTIONS

OF

ARBITRARY

OPACITY

A. Wedge and cone cases (1) Solution. u~°) and prO) as

given by equations (1 la, b) still hold. The temperature, T ~°), from equation (9) now becomes

T{°)(x'~k)

1

[1

--

ff

r,xl#l+J-#'+Jl-]

F

+- ,,xj1OX L-r l

lJ

Iz

,ZT)-

d~'

where = x,(0)/x,

F, = (,/3)[K,p](T(z°') ~] t a n Oh, =

2,/3 V

1

(13a)

132

K . C . WANG 1"0

Wilson and H o s h i z a k i

zff/A ~

.

I

(o)

I/ 0"5

Ys

0=5

o

]

o :

I,

o

/

/

0"5

: 30 °/I

1"0

I

50 °

I '

1"5

2"0

(a)

1.0

Wilson and Hoshizaki

0"20 I

0.8

O

0"15

0.6

(i/2)p~V~

O

(q~?)). (0) [(qr,)b]x=O

(~(0)~ tIr~ Jb

O. i0

0.4

0"05 0.2

0

I 10

I I 20 30 0 (deg)

(b)

] 40

I 50

0~---~i 0 (c)

20 30 0 (deg)

FIG. 3. a ~- l, f l = 5 , ) , = 1"1. -

-

(T(2°))~ffio = 1 2 0 0 0 ° K .

(T(2°))x=o = 14 000°K. (T[°))x:o = t5 000°K.

(a) Temperature distribution across the shock layer. (b) Radiative heat transfer normalized to fluid particles energy. (c) Radiative heat transfer normalized to stagnation point value.

40

50

Radiating and absorbing steadyflowover symmetricbodies

133

0b is the constant half wedge (or cone) angle, j is zero for wedge and one for cone. F1 and F2 are essentially xo/Lo and F defined in equation (10b), but are grouped with other constants for convenience. For a wedge (j = 0), equation (13a) can be integrated explicitly to yield

T(°)(x, ~k) T(2O)

=

(l+r2[rax(1-#)e-r'x~+(1-e-r'x"-"))]} -a/3.

(13b)

This is completely analogous to the result for the non-similar piston problem (WANG(a)). In other words, in the present approximation of our thin shock layer expansion method, the classical analog between a two-dimensional steady wedge flow and an unsteady onedimensional piston problem remains true. Conceptually, the validity of such an analog in a radiating flow is somewhat unexpected because of the directional dependence of the thermal radiation. In the transparent limit, (13b) becomes

T(°)(x, q/)/T~°) = [1 +2F2Fax(1 - # ) ] - 1/3. In the thick limit, (13b) becomes [1 + F 2 ( 1 - e- r'x"-~))] -a/a, r(°)(x, qt) _ T(°)

[1 + F2(1 + Fax

e-r,~,)]-

p-} 1

1/3,

( l + F 2 ) -a/a

/.~ "-~ 0

0
We notice that the proper parameter here is F2Fa = F~ for, say, the transparent limit and F2 for the thick limit, and that, in the middle portion of the shock layer, the temperature is constant. For a cone,j = 1, (13a) can only be partially integrated to yield

T(°)(x, ~b) T(2°)

= [1 +

2r2{½ra Itx. xs(g,)) + [½Fax e - t~'x''*)"3~ - ½r~x~(g,) e- t~,x.,,)J/~]

+ (½r ax,(~))e[E,(½r ax,(~))_ Ea(½rax,(~)~)]}]- a/~,

(13c)

where 1

FlI(x, xs(~b)) =

rlxf

exp[-½Ftx#'(1 -/~2/#,2)] dK,

#

and E a is the exponential integral of order one, defined by Ea(x ) =

fe

-t

dt. t

x

With T (°) given by equations (13b, c), the streamlines (3c) are determined by

J , yt°)(x, q/) = x tan 0b f r(°)(x,#') T--~2o) # ,~"o#. 0

(13d)

134

K . C . WANG

The shock shape is obtained by letting /~ = 1 in (13d~ In the non-radiating limit, the temperature is constant (to the zeroth approximation), hence, /o)

(x tan 0b)/(1 +j),

i.e., the shock layer thickness for a wedge is twice as that for a cone. This result is familiar from the constant density solution. The y-component of the heat flux q~O)is given by ,y q,0,

_ 2 F x x ~~[ T (°)(x, # ) ,] ~ 4

Jo[

JY

]- F x / ~ + j _ # , l + ~ ) l }

1

2r~x t" T'°'(x,/) '* ___[-

F,x

g_,l+j

_X

d/

+j%]~.otj

(13e)

//

Notice that the local temperature approximation is not used here. The radiation flux into the body (q,y)b (o) or across the shock (q~O))~is given by letting/~ = 0 and one in (13e). This implies that q~O)and q(0) are calculated here by approximating the local shock layer as an infinite plane slab. (2) Calculated results and discussions. Sample calculated results are plotted in Figs. 4(a)-4(e) for wedge and cone flows. Figure 4(a) gives the temperature variation along streamlines with F2 = 2. For the non-radiating case, the temperature is constant along any streamline. In the transparent limit, all particles are cooled down exactly the same way and there is no difference between the wedge and cone cases. With absorption, differences arise between one streamline and another and also between cone and wedge flow. The corresponding streamline is cooler in the cone flow than, in the wedge flow. Nonradiating (wedge and cone) - ~

, Cone Wedge

......

I'C

O.

IF1Xs(O)=

2

2

O"

r2 : 2

~

a

n

~

O. I

[

2

4

2 r l(x - x

I

6 s(*))

I

I

8

I0

FIG. 4a. Temperature distribution along streamlines.

Radiating and absorbing steady flow over symmetric bodies Nonradiating----~

- ......

"~

135 Cone wedge

1'0

0"8 50

T(0~

0" 6

0"4

0.21 f "

I

0

I

0'2

I

0"4

I

0'6 (0).

Y

I

0'8

1'0

(0)

/Ys

Fro. 4b. Temperature distribution across the shock layer.

Nonradiating 1'0'

Nonradlating

x = 1

1"01 x = 5

L

/

'1 °5

j

T (0) (x, 0)

0"

0"1

o

(0),

Y 1.0

o

(0) ( o ) Y /Ys

(0)

/Ys

Nonradiating

1.01 x = 20 Nonradzatmg I

r1

T (°) ~x, ~1

°

5

| B

1

T2(0) (x) 0"I

%

1!o (o), (o)

Y

/Ys

i!o

% (o). (o)

Y

/Ys

FIG. 4C. Temperature distribution across the wedge shock layer at different x stations.

136

K.C. WANG 20

15

r " (0) tan 0 b

5

o~

I

5

I

I

10

15 F1

I

I

20

25

x

FIG. 4d. S h o c k shape. 0.8

0.6

(0)

qry

~ ( T 2 ( 0 ~ 0"4

•f

~

Body

.....

Wedge

0.2

0

" ~ ' ~ ' ~ - -~~. 0

5

i I0

Xl"'-Sh°ck 15

20

25

FIo. 4e. Radiation flux across the s h o c k a n d body.

Figure 4(b) gives the temperature profile across the shock layer for different values of Fix. For small Fix, the gas layer is optically transparent, and the temperature shows a typical continuous decay behind the shock in the absence of absorption. As Fix increases, the profile has an inflection point. For large Fix, the gas layer is optically thick; and the profile is flat in the middle part, with a boundary-layer-like behavior near the ends. Increase of F2 (= 20) further reduces the temperature. Figure 4(b) demonstrates trends similar to those shown in Fig. (11) of the work of YOSmKAWAand CHAPMAN.{12> On the other hand, if we choose F~ instead of F2 in our presentation, the temperature distributions appear quite differently. Since F1 does not always appear together with x,

Radiating and absorbing steady flow over symmetric bodies

137

we have to calculate each x-station individually. Figure 4(c) shows such temperature distributions at four x-stations along the wedge surface. In particular, it is noticed that as F 1 increases, the temperature goes up instead of down and approaches the non-radiating value in the thick limit instead of in the transparent limit. These facts merely point up the warning in Section IIE that care must be taken in interpreting results of using different parameters. Figure 4(d) gives the shock shape comparison. Without radiation the shock over both the wedge and cone are straight. With radiation the shock distance is considerably reduced, but its curvature is hardly noticeable in the present plots. The shock curves toward the body if the body is cool and absorbs heat from the gas. Figure 4(e) gives the radiation flux across the shock (q,y)s (0) and into the b o d y (q~O))b. It exhibits a pattern similar to those first obtained by YOSI-nKAWA and CI-IAPMAN"~) and later obtained by the author (I) for a non-similar piston problem. For small Fix, (q,y)s (0) is nearly equal to ~,yt"(O)~b.For large Fxx, (only regions close to the shock and the body are really concerned in these radiation flux calculations) (q,y)~ (o) is much higher than t,,(o)~ ~ry Ib, because the temperature near the shock is much higher than that near the body. At the apex of a wedge or cone the radiation is theoretically zero.

B. Sphere case (1) Solution. In the case of a sphere, j = 1, u t°) and prO) are the same as in equations (12a, b). Since p(O) is no longer constant, we consider a = 1, so that a(x, ~0) and hence t/in equations (9a) and (7d) will appear in simple form. This process amounts to taking the mass absorption coefficient Km to be constant, say K"m. The temperature from equation (9) then becomes along a streamline. 0

T(°)(x' ~)

[ 1 ' "'-" c°s6 ~b/"

sin

4, sin ~

- 1/3

(14a)

,

where B

Zo = plKmRb, r . = pIKmRbGET~0'(¢, = 0)]4/%T~°'(¢, = 0)pIV.

Near the stagnation region, 0 is small, equation (14a) is simplified to

Y(°)(x, ~,) ... T(°)(x, d/) ..~ ~1 + 6 V"O [ 1 - e -(43)~:°(1-'~/°)] [1 + e -(43)~°q'/°] T(2O)(~) -- T(2O)(x) = ( (x/3)Zo~b

- 1/3

(14b)

The streamlines (3c) are given by

y,O, ~ rw, O,(x,~]] gb -- ; L T ( ~ )

3 cos ~b' d~b' ]*=*' (sin 30+sin s ~b')"

(14,:)

0

The shock shape y~O)is obtained by letting ~b = 0 in the integration limit of equation (14c).

138

K.C.

WANG

The y-component of radiation flux becomes ~(0) fi(O) _

-,ry

¢'lry

½plV 3 cosSOf~[T(°)(x,~l')'] 4

....

[

....

/sin 4'-sin tk')l

= -2r",+l sin0 Jok

sin0

d~b'

0

4,

The radiation flux into the body ,,(o) u,b or across the shock u,~(°)is again given by (q~°y))b or (,(o)~ "lry ; $ , i.e., by letting 4~ = 0 and 0 in (14d). Despite the fact that along the stagnation line (0 0) the present formulation using the von Mises variables is singular, all our solutions (14a)-(14d) are well behaved in that region. (2) Calculated results and discussions. In the present case, the absorption parameter z 0 does not ,depend on T~°) (or V); the cooling (or radiation-convection) parameter F. does not depend on Pl- Therefore, we can fix one and var.y the other just by fixing pl and varying Vor vice versa. Figures 5(a)-5(c) display the temperature profiles across the shock layer at different angles 0 with F. fixed. Figures 3(d)-3(f) display the same thing with % fixed. Near the stagnation region, the temperature field in Figs. 5(a) behaves as that over a flat wall shown in Fig. 4(c). Downstream, the curvature effect of the body's shape gradually enters and the temperature :pattern across a shock layer progressively changes, as in Figs. 5(b)-5(c). In addition to those features shown in Fig. 3(a) for the transparent limit, we notice that as absorption increases, the temperature increases over the value for the =

l-0

O =5 ° 6 Fn = 5 " 3 6

P l' 0'

("

= 0"

0 1

=v(x" ~J)~

2"5 5 10

Ys 'rransparent~

0"

~'---- Nonradiating °

0

Fro.

0"2

0'4

0"6

0'8

i'0

5a. Temperature distribution across the shock layer over a sphere (F. fixed).

Radiating and absorbing steady flow o v e r s y m m e t r i c b o d i e s

139

0 = 30 °

1" 0 6 Fn = 5"36

o= 10)

r o___Jil/l\ \ 1

o

•

)2.5-----#V

/

/

I

Nonradiating.

~

/

~

10

0"4

o. 0

/ 0"2

0"4

0"6

0"8

I'0

I

1"2

I

1"4

T (0) (x t ¢')

T(20)Ix)

FIG. 5b. Temperature distribution across the shock layer o v e r a s p h e r e (F. fixed). 1.0 O = 50 °

6

5-36

~

~

/-~Nonradiating

~

o

O" f~- 0 = 2"5

0'

0.

0"

0-2

0"4

0"6

0"8

1"0

1-2

1"4

1-6

1"8

2-0

2'2

T (°) (x, ~) T (0) (x) FIG. 5c.

Temperature distribution across the shock layer o v e r a s p h e r e ( F . fixed).

transparent case and approaches that for the non-radiating case. Figures 5(d)-5(f) show that at fixed Zo the larger F. is, the larger the radiation cooling effect becomes ; hence, the lower the temperature. In any case, the temperature is always zero on the body surface. This is due to the nature of a stagnation flow in the transparent limit and to the u¢°~-approximation of equation (12a) in the arbitrary opacity case. The latter can be readily taken care of, if the method

2"4

K.C. WANG

140 1"0

0 =5 ° J'3-r 0 : 5

0-8

6rn= (o)

0.6

y~x~°-ro. 4;

0.~

0

0"2

I 0"4

I 0-6

I 0-8

I 1-0

T (°) (x, ~) T2

(0) (x)

FIG. 5d. Temperature distribution across the shock layer over a sphere (z o fixed).

1.0

0=30* /

//11

0.8

6 r -j 0-6

~

to o i2o--

//

/ I

\\1

\y/ I /I II

0-6

0"8

0.4

0'2

0

0-2

0-4

1.0

T (0) (x~ ~) T2

(0) (x)

FIG. 5e. Temperature distribution across the shock layer over a sphere (Zo fixed).

suggested by MASLENtlG) is used. As will be seen later, such a local failure affects the calculation of radiation flux into the body in the thick limit. Figures 5(g)-5(h) give the corresponding results of the shock shape calculations. As z 0 at fixed F. increases the shock distance increases from that for the transparent case to its non-radiating value. As the increasing absorption reduces radiation losses, the gas is less compressed and the shock distance is naturally larger. With increasing F. at fixed Zo,

Radiating and absorbing steady flow over symmetric bodies

141

the radiation cooling becomes stronger; hence the gas becomes more compressed. As a result, the shock moves closer to the body. The shock shape has a singularity at 0 = 60 ° inherited from the corresponding non-radiating solution. 1" 0

0 = 50 ° qF3",r 0

=

5

/ I \I 1

or.o

0.8

V /

0.6 _ (o)

0.4

0.2

o

-

I

0"2

0"4

0"6

I

0"8

I

1"0

1"2

I

1"4

T (°) (x, ¢)

T 2(0) (x) FIG. 5f.

Temperature distribution across the shock layer over a sphere (To fixed).

40 F Nonradiatin

(o

6 Fn = 5-36

(0) 3.0]Ys 2.0-

j3"'r 0 :

5" 25'

l ' 0 h ~

O 0

'

FIG. 5g.

10

~

_

_ J 20 30 0 (deg)

40

,5O

Shock distance from a sphere (F. fixed).

Figure 5(i) gives the radiation flux into the body ,~rvt~(°)~Jband across the shock t~.to)x,~is. ,y The specific heat ratio 7 is taken to be 1.1. As expected, (q,y)b -(o) and (q,y)s -(o) decreases around the body from a maximum at the stagnation region. For fixed F., (q,y-t°~)band (~/~°))s usually decrease as the absorption parameter *o increases. Referring to equation (14d), we notice that an increase Of,o decreases the exponential damping term but increases the temperature. Hence, whenever the later increase outweighs the former decrease, the net result is an

142

K . C . WANG

increase of the integrand. This is why (?/~o)), for (~/3)% = 1 is larger than (,(o)~,yj~ for z 0 = 0 (i.e., it is transparent). For sufficiently large absorption (~o2 > 1), only the temperature near the body (or the shock) is really important in determining (?/~o))~(or (q~Oj)~).But the temperature is lowest near the body for the steady flows considered here (note that in the piston problem, the temperature is higher near the piston than near the shock if the piston velocity v v is oc t" and n < 0), hence (q,y)b -(o) is much smaller than .=(o ~q,rh,," Figure 5(j) gives the radiation flux for fixed (~/3)% = 5. Optically this is a nearly thick case. Increase of F, means increase of (?/~°r))~ but has little effect upon (g/~°r))b.Of course ifr o is fixed at some smaller value, X~ry tz(o)~]b will also increase as F, increases. 2-0

/ Vr3~O = 5

/]

1-5 6 F =) (0) Ys 1-0 Rb

10

3o--~£

L

0"5.

)1 0

I

I

10

I

20

30

I

40

I

50

0 (deg) FIG. 5h. S h o c k distance f r o m a sphere (% fixod).

6 rn : 5-36 (fixed) 0" 1 2 5 ~ 1 m c

0" 1 0 ~ , , ~

k

I~

~__'r0 = 0 (Transparent case,

s.oek a.d hod,,

0

10

20 30 0 (deg)

40

50

FIG. 5i. Radiation flux across the s h o c k and body (F, fixed).

Radiating and absorbing steady flow over symmetric bodies 0'25~.....,~

143

¢~"r 0 : 5 (fixed}

0.0L_

\

I

Shook;

\

//

L 7///

_

1 v3 2Pl

0-10

0"05 Body; 6 Fn = 5"36, 10, 20, 30 0

I

tO

I

I

20

30

I

40

I

50

0 (deg) FIG. 5j. Radiation flux across the shock and body (% fixed).

In other words, the present results show that at fixed flight speed (i.e., fixed F,), increase of ambient density (i.e., increase of z0) generally decreases the dimensionless radiation flux (C/~°))band ,.~,ytzt°h,s,"but increase of flight speed at fixed ambient density increases (?/,y)b and t=.to)x -w) is somewhat ~tley Is" In both Figs. 5(j) and 5(h), it should be pointed out that (qry)b underestimated for z 2 >> 1, because the temperature near the body is not accurately determined. Acknowledgements--The author wishes to express his gratitude to Dr. S. H. Maslen for his valuable suggestions and his review of the manuscript, to Dr. S. C. Traugott for several discussions on the choice of radiation parameters, and to Miss B. Snyder for calculating the numerical results.

REFERENCES 1. K. C. WANG,Martin Company Research Report RR-67 (1965), also Phys. Fluids 9, 1922 (1966). 2. G. G. CHERNYI, Introduction to Hypersonic Flow, (English Translation by R. F. I~OBS~IN). New York, Academic Press (1961). 3. N. C. F ~ M A N , J. Fluid Mech. 1, 366 (1956). 4. K. C. WANO, J. FluidMech. 20, 447 (1964). 5. R. GOULARD, and S. C. TRAUGOTT, Proc. 1 lth Int. Congress of Applied Mech., Munich (1964). 6. D. B. OLFE, AIAA J. 2, 1928 (1964). 7. P. Cn~,~G, AIAA-Paper No. 65-81 (1965). 8. S. C. TRAUC_,OTr,Proc. Heat Transf. Fluid Mech. Inst. 3, (1963). 9. P. ChaNG, AIAA J. 2, 1662 (1964). 10. K. C. WANG, Martin Company Research Report RR-73 (1965). 11. I. M. COHt~, AIAA J. 3, 981 (1965). 12. K. K. YOSHIKAWA, and D. R. CHAPMAN, NASA TN-D-142A (1962). 13. R. GOULARD, A&ES 62-8, Purdue University Report (1962)~. 14. G. A. Bmo, J. Aerospace Sci. 27, 713 (1960). 15. K. H. WmSON, and H. HOSHIZAKI, AIAA J. 3, 67 (1965). 16. S. H. MASLEN, AIAA J. 2, 1055 (1964).

144

K . C . WANG DISCUSSION

CLAIR CHAPIN, Purdue: I would like to ask a question on your "local temperature" approximation. The reason you can use it is that the optical thickness z is large and that the factor e-" essentially cuts off the sourcefunction. What if the optical thickness is neither small nor large? Dr. WANG: The "'local" approximation used here is rigorously justified in both the thin and thick limits. In the region between, i.e. neither thin nor thick, there is no reason to expect any difficulty; but this has not been proved. Evidences in support o f this claim are available from comparisons with more accurate solutions o f similar type of problems, see for example, my own Martin Company Research Report RR-67 ct) and the work of YOSHIKAWAand CHAPMAN.(12)