calculation of the subsonic flow of a radiating gas by the time dependent method
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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
REFERENCES
1. ALEKSANDROV, V. V. The flow of radiating hydrogen in axially symmetric nozzles.
lzv. Akad. Nauk SSSR. Mekhan. zhidkosti i gaza, No. 4,9-28, 1967.
2. BELOTSERKOVSKII, 0. M. and FOMIN, V. N. Calculation of the flow of a radia-
ting gas in a shock layer. Zh. vchisl. Mat. mat. Fiz., 9, 2, 397-412, 1969.
3. LEBEDEV, V. I. and FOMIN, V. N. The flow of a hypersonic gas stream past
blunt bodies taking account of selective radiation and absorption of energy.
Zh. v?hisl. Mat. mat. Fiz., 9, 3. 655-663. 1969.
4. GODUNOV, S. K., ZABRODIN, A. V. and PROKOPOV, G. G. A computational
scheme for two-dimensional non-stationary problems of gas dynamics and the
calculation of the flow from a shock wave approaching a stationary state. Zh.
vychisl. Mat. mat. Fiz., 1, 6, 1021-1050, 1961.
5. IVANOV. M. YA. and KRAIKO, A. N. Numerical solution of the problem of a mixed
flow in nozzles. Izv. Akad. Nauk SSSR. Mekhan. zhidkosti i gaza, No. 5, 7743, 1969.
6. SAMARSKII, A. A. Introduction to the theory of difference schemes Wvedenie v
teoriyu raznostnykh skheml, “Nauka”, Moscow, 1971.
7. PROKOF’EV, V. A. The problem of the calculation of the radiation in the one-
dimensional stationary flow of a monatomic gas. Uch. zap. MGU. Mekhanika, No. 172.79-124, 1954.
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