CALCULATION OF THE SUBSONIC FLOW OF A RADIATING GAS BY THE TIME DEPENDENT METHOD*

T. YA. GRUDNITSKAYA and A. V. SHIPILIN

Moscow

(Received 12 June 1972)

AN ITERATIVE process based on the use of the time dependent method is

proposed for calculating the subsonic flows of a radiating-absorbing gas. The

results of calculations of flows in a channel with coaxial walls are presented.

An iterative algorithm for the numerical calculation of the subsonic internal

axially symmetric flows of a radiating-absorbing gas is presented. As an

example the model problem of the equilibrium flow of radiating hydrogen between

two coaxial tubes of finite length is considered. The walls of both tubes are

rectilinear and are maintained at definite temperatures. Along some segment the

inner tube is strongly heated and, in radiating energy, affects the gas flow. The

absorption coefficient is considered to be a known function of the pressure and

temperature. It is assumed that the radiation is grey and is in local thermody-

namic equilibrium. The transfer equation is considered in the diffusion approxi-

mation. With these assumptions the problem of the flow of radiating hydrogen in

supersonic axially symmetry nozzles was formulated and solved [II. In a simi-

lar formulation, taking into account selective radiation, the subsonic flow was

calculated by the method of integral ratios (see, for example, [2-31).

(1.1)

1. The flow of a gas is described by the following system of equations:

ar pu

-+

ar pv

- = 0,

8X i)r

8U 8U 1 dP U~~v-~~~~~~

3X ar P ax

ilV 8V 1 JP U-+v-+--==o,

dx Or p dr

E’ = P(P, n, H = H(p, I’).

*Zh. u~c’chisl. Mat. mat. Pk., 13, 2, 510515, 1973.

318

Subsonic flow of a radiating gas by the time dependent method 319

Here x and r are the axial and radial coordinates, u and u are the corresponding

projections of the velocity w on the x and r axes, p is the pressure, T is the

temperature, p is the density, H is the enthalpy per unit mass, and & is the rate

of loss of energy per unit volume of the gas due to radiation. On the assumption

of local thermodynamic equilibrium and greyness of the radiation, we have

Q = x(4aP - U).

Here x(p, T) is the volume coefficient of absorption of the gas, c is the

Stefan-Boltzmann constant, and U is defined as the zeroth spherical harmonic

of the intensity of radiation, which apart from a constant factor is identical

with the radiation density. In the diffusion approximation the function U must

satisfy the equation

(1.2) L!-(&E) +;(-=J-w-40T")=0.

In order to calculate the flow in a finite segment of the channel a model

problem in which the boundary conditions are specified at a finite distance from

the hot segment of the wall is studied. Variation of this distance enables its

effect on the solution of the problem to be determined.

At the left boundary at x = 0 the constant values p = pot p = p,,, u = uO are

specified. At the right boundary for x = I(1 is the length of the channel) the

constant quantity p = pI is specified. The no-flow condition u = 0 is specified

on the walls of the tubes for r = r0 and r = r,. Here r,, is the radius of the inner

tube, and rl that of the outer. In deriving the boundary conditions for Eq. (1.2)

the following assumptions are used. At the left and right boundaries the flow

of radiant energy within the domain equals zero. On the walls of the tubes this

flow is determined by the temperature of the walls. It is assumed that each

wall emits radiant energy like a black body. In the diffusion approximation the

expression for the flow of radiant energy within the domain is of the form

where n is the outward normal.

We have the following boundary conditions:

2 dlJ u---_=o for x = 0,

3x 8x

2 au

“+izG-=” for x = 1,

320 T. Ya. Grudnitskaya and A. V. Shipilin

2 au U - - -- = 41sT,,~ (5)

3x ar for r = ro,

2 au U + z; = 4oTwz4 (5) for T = rl.

Here Tw r(x) is the temperature distribution along the inner tube, and TWz (x) is that along the outer tube.

2. The system obtained is solved by the time-dependent method. Thereby

the gas dynamic system of equations (1.1) is solved by the use of the finite-

difference scheme proposed in 1141 which was also used to calculate the internal

flows in i31. The transfer equation (1.2) is solved by using an explicit differ-

ence scheme, a non-stationary term dU/dt being artificially assigned to the

stationary equation. Let i be the number of the cell along the r-axis and j that

along the x-axis. Then equation (1.2) is approximated by the following difference

equation:

(p+u_ ($’

@.I) I1 (u:,:‘:l 7 -K

1 - + -J- - Ul’,“‘) _ %,j+l 1czj >

( 1 1

- -- +--- ) (rlt:’ - ul’,:ll) (C&2) -1 + Xzj XL,)-1 1

[( r,+i

+ z + 2_ (U[;l,j - Ul’,“‘)- “%I

Here r is the time step, h,, h, are the spatial steps, and the index n corresponds

to the preceding time layer, and n + 1 corresponds to the one being calculated.

It must be pointed out that the grid for the gas dynamic equations may differ

from the grid for the transfer equation. This is explained by the fact that for

radiati.on the characteristic distance is not the physical length h, or h,, but

the optical thickness & or xh,. The time step r for solving the system (1.1)

is chosen from Courant’s condition, and that for Eq. (2.1) from the condition

given in [6I on p. 215:

The following iterative algorithm is proposed for solving the problem. The

region considered in the xr-plane is subdivided into m layers along the x-axis

and n layers along the r-axis. In each cell at the initial instant some distribu-

tion of the flow parameters and of the function U is specified. Each iteration

consists of two parts. In the first part of the iteration for fixed values of the

Subsonic flow of a radiating gas by the time dependent method 321

li?yiL 0 4.2 84 I

FIG. 1. FIG. 2. z. = 0.135.

FIG. 3. u, =0.675.

FIG. 4. u,, =0.135.

FIG. 5. ,,=0.675.

&ID u 4.2 B.4 f

FIG. 6. FIG. 7.

322 T. Ya, Crudni’tskaya and A. V. Shipilin

FIG. 8.

gas-dynamic parameters the transfer equation (2.1) is considered up to the

steady state. In the second part the system (1. I) is calculated for the values

of U found. Each part of the iteration is continued until the maximum error of

the corresponding stationary equations becomes less in modulus than a speci-

fied quantity. If after the ending of the calculation of the gas-dynamic equations

the stationary equation (1.2) is satisfied immediately without further iterations,

the calculation is ended.

3. Calculations of the flow of hydrogen for two different values of the

longitudinal velocity at the input were carried out as examples. The following

parameters were specified: r,, = 1 cm, r, = 2.68 cm, 1 = 8.4 cm, TW2 = 300°K

for all values of x;

i

5000°K for 0 4 x < l/3 and 21/3 ,( x 6 1,

T,r= (1 + “/z sin[(3n/Z) (x - Z/3)1) 5OOO’K for l/‘/3 < x < W3,

p,, = p1 = 5 atm; T, = 4000°K.

In the calculations a simulated absorption coefficient whose temperature

variation at p = 5 atm is shown in Fig. 1 was used. Here II is measured in

Cili_‘. Since the calculations carried out showed that, to the accuracy of the

calculations, the pressure remains constant, the variation of the radiation

coefficient with pressure is ignored. The equation of state, the variation of

the enthalpy with p and T, and formula for the isentropic speed of sound were

taken for ionised hydrogen. These formulas are given in [71.

‘Il!~- -7rculations were carried out for two values of u0 at the input, equal

to 1000 and 5000 m/set. These correspond to the Mach numbers 0.135 and

0.675 respectively. In the calculations dimensionless variables were used,

defined by the formulas

Subsonic flow of a radiating gas by the time dependent method 323

Here a, is the isentropic speed of sound at the input, pO is the density, and

R is the gas constant.

In the solution of the gas-dynamic equations the domain considered was

subdivided by the grid into 320 cells (8 intervals along the r-axis and 40 along

the w-axis). The spatial steps hx and h, were taken the same, equal to 0.21.

In the solution of the transfer equation a spatial step one third of this was

taken (24 intervals along the r-axis and 120 along the x-axis).

Here the maximum optical thickness of a cell of the grid is equal to 0.392.

Linear interpolation was used in calculating the temperature occurring in Eq.

(2.1). The value of the function U occurring in the energy equation of the

system (1.1) was taken as the arithmetic mean calculated from the values in the

corresponding nine cells.

As the initial approximation for the gas-dynamic parameters a uniform flow

satisfying the boundary conditions was taken. The values of U in the initial

approximation were calculated by the relation U = 407”. The calculations were

performed with an accuracy of up to 3%. For the calculation of one version 4 5

iterations are required.

Figures 2-8 show some of the results of the calculations. All the quantities

are shown in dimensionless form. The variation of the gas temperature with x

in the sections r = const are shown for both versions in Fig. 2 and 3. We - mention that the value T = 0.6 corresponds to a value of the temperature equal

to 4000°K. Figures 4 and 5 show the isotherms in the lcr-plane for these versions.

The effect of the quantity u0 on the formation of the temperature profile is

obvious from a consideration of the graphs. A gas, moving with a lower velocity

succeeds in being heated to a higher temperature and then cools because of

luminescence. Figures 6 and 7 show the dependence of the longitudinal

velocity component on x in the sections r = const. In this problem the radiation

has a weak effect on the kinematic motion. The pressure remains constant within

the limits of accuracy of the calculations. For both versions the transverse

velocity component v does not exceed the value 0.02 in the entire region of the

flow. The main effect of the radiation shows itself in the variation of the

temperature, and accordingly, in the density of the flow. Figure 8 for the - - version with u0 = 0.135 shows the curves of constant Q, the rate of energy loss

due to radiation; here the value of Q is increased by a factor of 10’. The

dashed curves correspond to Q = 0 and separate the region of absorption of

radiant energy by the gas from the region of emission.

In order to check the effect of the boundary conditions chosen, an additional

324 T. Ya. Grudnitskaya and A. V. Shipilin

calculation was performed for x = 0 and x = 1. For & = 0.675 the flow in the

shortened domain x, ,( x ,( x, was calculated, where x, = 0.84 cm, x2 = 7.56 cm.

The boundary conditions at x = 0 and x = 1 were transferred to the straight

lines x = x, and x = x, respectively. The temperature distribution along the

wall of the inner tube was unchanged by this. A comparison shows that the

difference between the flow and radiant energy density parameters is within the

limits of accuracy of the calculations.

It is obvious from the calculations made that the proposed iterative process

is convergent. The calculation of one version requires a considerable time

(3 hours on the BESM-6 computer). The computing time can be reduced by the

use of an implicit difference scheme for the transfer equation, but most of the

time is spent on solving the gas-dynamic equations at high temperatures. In

this case the value of the speed of sound is large and the time step, determined

by Courant’s stability condition, is found to be small.

Translated by J. Berry

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