Blunt body flow of a viscous radiating gas

BLUNT BODY FLOW OF A VISCOUS RADIATING GAS *

I.M. BREEV, YU. P. GOLOVACHEV, YU. P. LUN'KIN and F.D. POPOV Leningrad (Received

6 February

1970)

Introduction THE results of a number of papers D-41 show that in the calculation of the blunt body flow of a hypersonic air stream taking radiation into account, it is necessary to take into consideration the radiation cooling absorption of energy and the dependence of the absorption on the temperature and density, but also on the frequency. may lead to the appearance

of a significant

part of the shock wave, which throws shock

layer

is absorption

into a region the radiative

of non-viscous transfer

temperature

of the shock layer, the coefficient not only The radiation cooling gradient

some doubt on possibility flow and a boundary

of energy

leads

in the main of dividing

layer.

to interaction

the

When there

between

the

It is extremely difficult to take account of main and boundary flow regions. consideration of these regions of the shock layer, especially if we take into account the complex nature of the dependence of the absorption coefficient of air on the frequency. Therefore, in studying the blunt flow of radiating air taking into account radiation cooling and absorption, it is necessary at the same time to consider the entire flow between the shock wave and the frontal surface of the body. This problem was considered in Fll using the parabolic system of equations of a thin compressed layer. In calculating the radiative term of the energy equation, the radiation of the air was taken into account only in the continuous spectrum. The solution was obtained by the combined finitedifference integral method. In this paper the flow between the body is studied within *Zh.

themselves v$hisl.

by means completely

Mat. mat. Fiz.,

the shock wave and the frontal

of the simplified

NavierStokes

the terms of the gas-dynamic 10, 5, 1228-1237,

200

1970.

surface

equations, viscous

of containing

flow equations

Blunt body flow of a viscous

radiating

gas

201

and of the boundary layer equations. To approxima~ the spectral decadence of the absorption coefficient of air two models were used, one of which takes into account radiation only in the continuous spectrum, and the other both in the continuous spectrum and in the spectral lines. The initial system of equations is solved by the method of finite differences. 1. Formulation

of the problem

1. THE INITIAL EQUATIONS AND THE GAS DYNAMICAL BOUNDARY CONDITIONS To describe the flow in the shock layer we use equations generalized to the case of a radiating gas in thermodynamic equilibrium [51. We write these equations in dimensionless form, using the following notation:

where x is a coordinate measured from the critical point along the surface of the body; y is a coordinate measured from tlm surface of the body along the normal to it; t is the radius of blunting; u, u are the velocity components tangential and normal to the surface of the body; h is the enthalpy; p is the density; p is the pressure; p is the coefficient of dynamic viscosity; X is the total thermal conductivity taking into account particle collisions and heat transfer resulting from diffusion and reactions occuring in the establishment of chemical equilibrium at each point; T is the temperature; m is the molecular weight; R is the universal gas constant; q, is the radiation heat flow; Re is Reynolds number; ~1, and X, are the values of p and X on the axis of symmetry immediately behind the shock wave; U, is defined by the relation h, f U,’ I 2 = U,’ i 2. The subscript m denotes values of the parameters in the oncoming flow. For the case of the flow round a sphere, omitting the primes on the dimensionless variables, we obtain

-&(PU)-t (1 -t- Y>&

(pV) + 2pv i_ pr.4ct,g e

= 0,

202

I. M. Breev,

Yu. P. Golouachev,

Yu. P. Lun’kin

and F, D. Popov

Equations (1.2) contain entirely within themselves the terms of the corresponding equations of non-viscous flow and of the equations of the boundary layer. To them must be added the equation of radiative transfer SJVf” = xv (I”, - IV).

(1.3)

Here Iv is the spectral intensity of the radiation, I,, is the equilibrium spectral intensity of the radiation, xv is the absorption coefficient taking into account stimulated emission, and 52 is the unit vector in the direction of propagation of the radiation. The radiation heat flow q, is defined in terms of the intensity radiation as follows:

of the

fn ~rformin~ calculations the approxima~ anal.ytic represen~tions of the thermodynamic functions of equilibrium air, proposed in I.61, are used. The values of the viscosity and of the total thermal conductivity are adopted from 173. The flow region considered is bounded by the shock wave, the axis of symmetry, the surface of the body and the ray 8 = &. = const. The ray 8 = & is chosen in such a way that the flow downstream from it exerts no significant

Blunt

body flow

of a viscous

radiating

203

gas

effect on the parameters of the gas in the region 8 < &. The shock wave is considered to be a surface of discontinuity, and the values of the gas-dynamic parameters behind it are determined by means of the Rankine-Hugoniot relations On the line 8 = 0 the symmetry conditions are used. At the surface of the body the adhesion (u = u = 0) are assumed, the temperature of the surface of the body is taken as constant (T, = 2000°K). 2. THE RADIATION TERM OF THE ENERGY EQUATION In calculating the divergence of the radiant heat flow the shock layer is considered to be locally onedimensional, and in each section it is represented as an infinite plane layer with thickness equal to the local value of the exit of the shock wave. Approximations of the locally onedimensional layer are given to a sufficient accuracy in [81, and the possibility of replacing the geometry of a shock layer by the geometry of an infinite plane layer is demonstrated, for example, in (9, 101. It is considered that the radiation in the shock layer from the flow situated ahead of the shock wave, does not advance, and that the surface of the body absorbs all the energy supplied to it, and radiates a quantity of energy which is negligibly small in comparison with the radiation of the shock layer. With the approximation of the locally one-dimensional plane-parallel layer, and with the stated boundary conditions for the radiation field, the solution of the radiative transfer equation (1.3) leads to the following expression for the divergence of the radiant heat flow:

divq,

= 2n 1 xv [ 21,, - ‘f I,,& 0

(1zy -

t, I)&] dv,

(1.5)

0 u

where

E1 (z) -- 5 w-le-w’dw 1

is the exponential

%,dy

integral function, ~~ = s 0

is the optical coordinate, and -cvE(~) is the optical coordinate of the shock wave. In view of the complexity of the spectral response of the absorption coefficient of air, a model enabling an absorption coefficient avrying strongly with frequency to be represented by step functions of frequency is widely used. In this paper we use the step models proposed in ill1 and [121 to approximate the spectral response of the absorption coefficient of air. The first of them (called model A below) enables radiation to be taken intD account both in the continuous spectrum and in the spectral lines, and the second (model B) takes account of radiation in the continuous spectrum only.

204

1. M. Breeu,

Yu. P. Golouachev,

2.

Yu. P. Lun’kin and F. 0. Popov

method of solution

Using relation (1.51 and the boundary conditions indicated in section 1, part 1, the initial equations (1.2) are solved by the method of finite differences, which best enablesac~i~a~ account to be taken of the complex form of the profiles of the gasdynamic par~e~rs, characteristic for the flow of a viscous, hea~onducting, r~iating gas in the shock layer. After ~ansf~ming the normal coordinate by the formula

where E=u/s(B), s(0) is the exit of the shock wave, d is a numerical factor by the variation of which the computing net can be shifted, and the region considers becomes rectangular, and the lines 6 = const are compressed to the surface of the body in such a way that in the boundary region, charac~ri~ by si~ificant gradients of the gas-dynamic parameters, there is a sufficient number of calcula~d points. In the variables 8, z the initial equations and the boundary conditions for the crowns u, u, T, p are written in difference form. If the position of the shock wave of the system is not known beforehand, the difference equations are open. An additional closing relation is obtained by writing at the surface of the body (z = 0) the projection of the momentum equation on the z-axis The system of ~uations is solved by Newton’s meth~. 3.

Discussion

of the results

Calculations were performed for the following flight conditions: 25 < Mn, < 42; 0.~1 atm < p, < 0.001 atm; 0.15 m < r < 3.5 m. The profiles of the gasdynamic parameters are obtained in the entire shock layer. The shape and position of the shock wave are de~rmined. The vaiues of convective and radiative heat flows at the critical point and on the frontal surface of the body are found. The dist~bution of the heat exchange and friction parameters over the body is studied. By using two spectral models of the absorption coefficient, the effect of different radiation mechanisms on the behaviour of the gas-dynamic parameters in the shock layer, the heat flows and the friction coefficient is analyzed. Some results of the calculations are presented below and an analysis of them is given. Figures 1 and 2 show the profiles of the gasdyn~ic parameters along the axis of symmetry and the ray 8 = 0.225 for M, = 40, p, =j0.~2 atm, r = 1.5 m, TV = 240.6OK. The continuous curves show the results obtained by using the spectral model A, the dashed curves those by using model B. As is seen from Figs. 1 and 2, the use of the different models of the absorption c~fficient leads

Blunt

body flow of a visc~ous radiating

FIG.

FIG. 2

gus

1

FIG.

3

to a considerable difference in the enthalpy and density profiles, The pressure values are practically the same for both variants of the calculation. The velocity profiles differ negligibly. The results given in Fig. 1 show that the output of

206

f. 161.Breev, Yu. P. Golwuchev,

Yu. P. Lun’kin and F. D.

Popou

FIG 5

radiation may lead to the appearance of a considerable enthalpy gradient in the main part of the shock layer, and this indicates the need to take into account the the~al conductivity of the gas in the entire shock layer. In Fig. 3 the continuous curve for one of the calc~ations ti%&= 33.7% p, = 0.0002 atm, I = 1.5 m, T, E 240.6OK, the spectral model B) shows the shape and

Blunt body flow of a

viscous

radiating

gas

207

FIG. 6

FIG. 7

position of the shock wave relative to the surface of the sphere. The dashed curves show the lines of flow. The dashdot curve corresponds to the sound lines. Figures 4 and 5 show the dependence of the heat flows at the critical point on the radius of the sphere for M, = 38.85, p, = 0.000437 atm, T, = 257.2”K. The continuous curve in Fig. 4 shows the results of calculating the convective flow obtained by using the spectral model A. The dashed curve shows the results obtained by using model B. The dash-dot curve shows the values of q, calculated by the formula of Fay and Riddell [131 with the nsual assumptions:

208

1. M. Brecu,

Yu. P. Golouachev.

I,

Le = 1, h, > L (pop, / p8pp)o.i= subscript

s denotes

Comparison radiative

of the results transfer

decreases

qe -

values

the convective

as the radius

I/

‘)‘r,

layer ignoring

where Le is Lute’s

of the gas immediately

number,

behind

proposed

radiation.

of the convective

than follows

in [131, which is based The use of different

flow differing

is taken

heat flow at the critical

increases,

of the

point,

from the theoretical on a study of the

spectral

by more than 10%.

and the

the shock wave.

of Fig. 4 shows that when account

of energy,

more rapidly

dependence boundary

parameters

Yu. P. Lun’kin and P. D. Popov

models

Figure

leads

to

5 shows the

results of calculating the total heat flow 9, + 9, and its components, obtained by using model B for the absorption coefficient. The short-wave component of the radiation flow (X < 1100 1) is denoted by Q~,, and the long-wave component 9,, decreases (h > 1100 1, by 9,.2. It is seen that when the radius is increased after passing through a maximum; for small radii qr2 increases practically linearly. As the radius increases the qrl(r) relation deviates more and more from linear, this being explained by the increasing part played by radiation cooling of the shock layer. The character of the qT,(r) and qr,(r) relations indicates the strong absorption

in the shock

layer of the short-wave

radiation

(h < 1100 1) and the

almost complete transparency of the shock layer for long-wave radiation (X > 1100 A). For r, < 0.8m the main contribution to the radiation heating is made by shortwave radiation, and for r > 0.8 m by long-wave radiation. For small radii the total

heating

flow.

at the stagnation

Beginning

point is determined

with r = 0.4 m, the convective

mainly

by the convective

flow becomes

less

heat

radiative,

and

radiation plays the principle part in heating the surface of the body. The total heat flow at the stagnation point has a minimum for r = 0.7 m and as the radius increases Table

further

depends

1 gives

weakly

a comparison

point (model B) obtained

on r. of the values

of the heat flows at the critical

in this paper with the results TABLE

I

obtained

1

U,, km/see

Ii, km

r, m

Yc, k&cm'

9,.kdcm’l

12.5

55

0.305

1.49

0.775

2.265

1.24

2.68

1.44 I

in [ll.

:$;A4

/

Source Reference This

paper

The discrepancy in the values of the radiation flow is expanded by the different methods of allowing for the spectral dependence of the absorption coefficient.

An additional

source

of discrepancy

in the values

hI

of the radiation

Blunt body flow of a viscous radiating gas

FIG. 9

FIG. 8

flow may be the possible

209

non-coincidence

of the values

of the shock wave exit.

The variation of the heat flow ever the surface of a sphere for M, = 40, P, 0.0002 atm, r = 1.5 m T, = 240.6”K is shown in Fig. 6. The continuous curves show the results obtained with model A, the dashed curves show the results obtained

with model A, the dashed

when account

is taken of radiation

curves

those

with model B. It is obvious

in the spectral

lines

of the radiation flux at the critical point are approximatelg when account is taken only of radiation in the continuous 8 increases

the radiation

when model A is used. difference

flux to the surface The convective

of the total and radiation

of the absorption

coefficient

that

by model A the values 2.5 times greater than spectrum (model B). .4s

of the body decreases

more quickly

heat flow, which can be found as the

fluxes,

depends

only slightly

on the choice

model.

Figures 7 and 8 show the distribution over the frontal surface of the sphere of the heat flows, referred to their values at the critical point (Mm = 38.85, P, = 0.000437 atm, r = 0.305 m, T, = 257.2”K, spectral model B). The dashed curves show the results of [ll. It is obvious from Figs; 7, 8 that in proportion to the distance from the critical point the radiation heat flow to the surface of the body descreases much more rapidly than the convective heat flow. Figure distribution

22

9 shows for Mm = 40, p, = 0.0002 atm, r = 1.5 m T, = 240.6”K the over the surface of the sphere of the convective flow, relative to its

210

1, M. Brew,

Yu. P. Golovachev,

Yu. P. Lun’kin

and F. D. Popou

FIG, 10

value at the critical point, for the different versions of ~~c5~~t~~~ for the radiation tcontinuons curve - model A, dashed eruve - model ES), The dash~ot curve on this graph shows the distribution of the convective flow caIcuIa~d by the formula rrC(@) _ 0.55-j- 0.45cos28,

Ycw

(3.1)

proposed in El43 for the case of the flow round a sphere of equilibria dissociated nonradiating air. It is obvious from Fig. 9 that the variation of qc (0) f QE(0) differs n~gligibI~ for the diff~~nt ways of taking account of the radiation. It is also obvious that in the flight conditions considered, when the ionization of the gas and radiation are eonsiderabIe, there is a discrepancy between the variations considered here and the variation given by (3.1). of %P) MO) The variation of the coefficient of friction over the snrfaee of the sphere for Mo. = 40, r = 1.5 m, T, = 240.6”K and for different values of the pressure in the oncoming flow are ~p~sen~ in Fig. 10. The ~ontinuuus curves show the results obtained by using model A for the absorption c~f~cient, the dashed curves show the results fur model B, The coefficient of friction was defined by the formula

As is obvious from Fig, 10, the use of different spectral models for the absorption c~fficient does not lead to a si~i~cant di~e~nc~ in the values of the coefficient of friction. Tr~~s~~~e~ by J. Berry

Blunt

body flow of a viscous

radiating

gas

211

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2.

AFIMOV. N. A. and SHARI, V. P. Solution of the system of equations of the motion of a selectively radiating gas in the shock layer. IZU. Akad. Nauk SSSR. Mekh Zhidkosti i gaza, 3, 1%25, 1968.

3,

LEBEDEV, V. 1. and FOMIN, V. N. Streamlining of blunt bodies by the hypersonic flow of a gas taking selective radiation and absorption of energy into account. Zh. u$chisl. Mat. mat. Fiz., 9, 3. 655-663, 1969.

4.

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5.

TOLSTYKH, A. I. Numerical computation of the supersonic flow of a viscous gas round a blunt body. Zk. uj%kisl. Mat. mat. Fiz., 6. 1, 113-l%, 1966.

6.

MIKHAILOV, V. V. Approximate analytic representation of the thermodynamic functions of air. Inzkere~yi sb., 31, a)6-216, 1961.

7.

HANSEN. C. F, Approximations for the thermodynamic and transport properties of Hugh-tempe~ture air. NASA TR, R-50, 1959,

8.

GOULARD, R. BUGNER. R. E., BURNS, R K. and NELSON. G. F. The flow of a radiative gas in the conditions of entry into the atmosphere of a planet. Teptofiz. uysokikk temperatur, 7, 3, 542-565, 1969.

9.

KENNET, H. and STRUCK, S. L. 31, 3, 3701372, 1961.

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10. CHISNELL, R. F. Radiant heat transfer in a spherically symmetiic medium. AIAA Journal, 5, 7, 1389-1391, 1868. 11.

CALLIS, I,. B. Time asymptotic solutions of blunt body stagnation region flows with non-gray emission and absorption of radiation. AIAA Paper, 68-663, 1968.

12

ANDERSON, J. D. Jr., Heat transfer from a viscous non-gray radiating shock layer. AIAA

Journal,

6, 8, P570-1573,

1968.

13. FAY. J. A. and RIDDELL. F. ‘l%eoretical analysis of the heat exchange at the leading critical point around which flows dissociated air. In: Gas dynamics and heat exchange with chemical reactions. (Gazodinamika i teploobmen pri naiichii khim. reaktsii). Izd-vo in lit., Moscow, 1962, 190-224 14. MURZINOV, I. N. The laminar boundary layer on a sphere in the hypersonic flow of equilib~~ dissociated air. Izu. Akad. Nauk SSSR. Mekh. okidkosti i gctzu, 2, 184-188. ,1966.