calculation of subsonic viscous gas flows in plane channels and wakes

264 A. P. Byrkin and V. V. Shchennikov

REFERENCES

1. VLADIMIROV, V. S. Mathematical problems of the single-velocity theory of particle transfer. Tr. MIA.WSSSR, No. 61, 1961.

2. RUMYANTSEV, C. Ya. Boundary conditions in the method of spherical harmonics. Afomnaya energi?~a, 10, 1, 26-34, 1961.

3. SMELOV, V. V. Lecfures on the theon, ofneutron transport (Lektsii po teorii perenosa neitronov), Atomizdat. Moscow, 1978.

4. S.ZIELOV, V. V. The notation of the transport equation and its representation by the method of spherical harmonics in arbitrary curvilinear orthogonal coordinates. In: Some problems of numerical and applied marhemofics (Nekotorye probl. vjYchis1. i. prikl. matem.), 172- 183, “Nauka”, Novosibirsk, 1975.

U.S.S.R. Comput. Maths Math. P/JJ,s. Vol. 19, pp. 264- 271 0 Pergamon Press Ltd. 1980. Printed in Great Britain.

CALCULATIONOFSUBSONICVISCOUSGASFLOWS INPLANECHANNELSANDWAKES*

A. P. BYRKIN and V. V. SHCHENNIKOV

Moscow

(Received 7 April 1977)

THE PROBLEM of the laminar flow of a viscous gas in plane channels of constant cross-section and

finite length, formed by a lattice of plates, and in the wakes behind the plates, is considered, using

the complete Navier-Stokes equations. The case of the subsonic flows of a gas is investigated

when a direct condensation jump occurs at the channel input. The calculations are carried out in

the range of Reynolds numbers Re = 20 to 200. The wall temperature is assumed to be equal

to the stagnation temperature. The data of the calculations confirm the features of the flow of a

viscous gas in long channels revealed by the approximate treatment of the problem.

The overwhelming majority of papers devoted to the study of the flows of a viscous

compressible gas in channels (nozzles) published up to the present time are based on the use of a

well-known model of viscous flows: the separation of the flow into a boundary layer and an

inviscid kernel.

As theoretical and experimental investigations have shown, under certain conditions this

model quite satisfactorily describes the actual flow.

However, in recent years aerodynamic equipment has appeared in which flows corresponding

to fairly small Re numbers (Re = 300 to 500) are realized. The creation of such equipment

necessarily poses the problem of the numerical solution of the complete Navier-Stokes equations

over a fairly wide range of variation of the Mach number. As was pointed out in [ I] , the

fundamental difficulty in the numerical solution of internal problems of the dynamics of a

viscous compressible gas, consists of the correction of the statement of the boundary conditions at

the ends of the channel segment considered.

*Zh. v@hisl. Mar. mat. Fiz., 19, 1, 252-259, 1979.

Short communications 265

In [2], with reference to the subsonic flows in a channel of constant cross-section.

extrapolation-type conditions are used at the channel output. It must be pointed out that the

formulation of such conditions is justified only in the case where they are laid down in the

neighbourhood of a closing cross-section. It is obvious that it is then necessary either to provide

for a sufficiently accurate isolation of the singularity in the solution, connected with the

phenomenon of blocking of the flow, or (in the case of a “through” calculation) to justify the

accuracy of the results obtained.

The results of a calculation of the subsonic ilow in a nozzle are given in [3],

1. We will consider the laminar steady- flow of a viscous gas in a plane channel of constant

cross-section of finite length (Fig. 1 .a).

r FIG. 1

The initial equations are of the following dimensionless form:

where

266 A. P. Byrkin and K Z’. Shchennikou

The dimensionless quantities are connected with the dimensional quantities by the

following relations

X0 Y0 u” VO e” J-=-, II=-, VZT-

Y,‘” !,=-,

?J ” UfO 11,O , c=-,

(a,“)2

p’po, p= po p2, Re = PlO~lOYWO Pi” p,“(ul@)y ’ PI0 P*O

where x0 and J.O are distances measured along the longitudinal axis and along the normal to the

plane of symmetry of the channel; u O, p” are the longitudinal and normal components of the

velocity: e”, p”, p”, p” are respectively? the internal energy’, density: pressure, and dynamic

viscosity; UI 0. pi 0, ~10 are the values of the corresponding quantities in the plane of symmetry

of the channel in the initial cross-section x = 0; the subscript it’ denotes the channel wall. The

quantity K is the adiabatic index and Pr is Prandtl’s number.

We consider the flow of a viscous gas in a channel, when at the channel input there is

a normal pressure shock. In this case we have succeeded in correctly, formulating the conditions

at the channel input.

In order to avoid the difficulty, connected with the formulation of the boundary

conditions at the channel output, we will consider the problem of the flow of a viscous gas in a

lattice of channels and in the wake behind it, as shown in Fig. 1, b. The channel walls are assumed

to be absolutely thin. We also assume that the length of each channel is less than the critical

length for which blockage of the flow occurs.

Because of the periodicity of the flow and its symmetry about x the domain of integration

may be regarded as a strip of unit height. The boundary conditions will have the following form:

Short communications 267

ii= I. l.Zil, e=(,. p=p$ for 1.=1-l. II<j,<l. 1i=o, v=o. e=E,. p=p1 fQr s=o, y=l. all de

-=

8Y 0, r=o, - = 0 , ?=O for r>n,

i,Y SY fJ=o,

,I = 0, !-=(I. (= y_ for fI(crc,r,, ii= I.

i,!i -=(I. _, --=fl for [.E_n “’ .r B I ,\. ii= 1: <I ‘, “!I

u u ue 0. L.=o, -=O for I” \.

hr ficy<i.

2.7

(1.1)

where e,=l/x~~-l~M,~, p,=l/z!G,‘, Ml is the Mach number at ~1 = 0 in the initial

cross-section, and e,, corresponds to the specified wail temperature.

Conditions (1 .I) follow from the physical picture of the flow in the wake at great distances

from the trailing edges of the channels and are equivalent to the following:

U-U,. l‘+ 0, e-e, for I-x. Il<y<l.

where urn coo are the values of the corresponding quantities at infinity, determined by the integral

conservation laws and by the solution of the problem as a whole.

The nature of the asymptotic behaviour of the laminar flow of a viscous gas in the wake

behind a lattice as x + 03 was investigated in [4] . and it was shown that the perturbations of

velocity (U - u,) and temperature (e - e,) as x -+ m decrease exponentially. The results in [4]

can be represented as follows: for .Y = SL, 0 <,I, < 1

ou,~o.r ,x’ V(‘, L’.i sJ -=-- -=--I II - ii a Re, ’ e-f=, He, Pr

(1.2) al,

1

.‘I 1 --1 ,x2 - = -- ii.r

r’+-siri(.-rYl 2[P[J,0)-cm] ---$- Re, e, ( PI. )I Re, ’

where R~‘,=!I,“I~,“,,.“~I_I~‘. and _yL is the distance at which the asymptotic conditions are

considered.

The use of conditions (1.2) to obtain a numerical solution makes it possible to reduce the

computed field because of the choice of the quantity .yL commensurate in order with I ,> (J-! = ?.rl, I,

The values of uoR coo are found by an iterative method. However. we note that the proposed

procedure for obtaining a numerical solution requires an “excellent” initial approximation.

Such data can be obtained. for example, from calculations of the flow of a viscous gas

in a channel using the “shortened” Navier-Stokes equations, valid for long channels and identical

with the boundary layer equations [S, 61.

2. Calculations were carried out using the splitting method proposed in [7]. The temperature

of the wall of the channels was assumed to be equal to the stagnation temperature of the incident

flow in the initial cross-section x = 0, and the Mach number in the initial cross-section was assumed

to be Ml = 0.4236.

268 A. P. Byrkin and V. V. Shchennikov

The gas was considered to be diatomic (K = 1,4), Pr = 0.71, and the exponent in the formula

for the viscosity as a function of temperature @ - 7n) was chosen to be n = 0.76.

The chief characteristics of the cases computed are shown in the table, where K and L

are respectively, the number of nodes of the mesh in the longitudinal direction on the channel

segment and on the entire segment of the domain considered.

The calculations of almost all cases were performed on a mesh with 18 X 101 nodes.

To determine approximately the critical length of the channel segment and the values of

urn, eo3 in the wake calculations were carried out of the flow in the channel using the “shortened”

Navier-Stokes equations [6] . In connection with the simplified treatment, the profiles of the

reduced velocity ub) and reduced pressure p/p1 when Re varies are the same for values of x

satisfying the condition x/Re = const. The latter holds for futed profiles of the reduced velocity,

temperature and Ml number at the channel inlet.

FIG. 2

Short communications 269

1.0

0.8

06

-04

- 0.2

D

h”@ ‘2

.’ 8

FIG. 3

In Figs:2 and 3 the dashed lines give the results of the approximate calculations for the quantities u(g, 0) and p(E)/~l, where l= x/Re is the reduced channel length.

From these results we may draw the conclusion that blockage of the flow for the conditions considered at the inlet and on the wall of the channel will occur for .$cr = 0.175.

The data about the reduced critical channel length thus obtained was used to specify the values of xK in cases l-5 computed, for which numerical integration of the Navier-Stokes equations was carried out.

In performing the calculations the initial data were specified differently. For example, as the initial data in the whole of the domain computed, apart from the channel wall, a homogeneous flow was specified, corresponding to the conditions at the channel inlet. Then as the initial

approximation for u_, coo in the wake their values obtained from the approximate treatment of the problem were used.

270 A. P. Byrkin and V. V. Shchennikov

However. for the purpose of reducing the computer time required to obtain a solution close

to the steady state solution, the initial distributions in the computed domain were specified

taking into account knowledge of the approximate picture of the flow.

In the calculations of each case the time step 7 was varied from 0.01 to 0.1. The number of

time steps did not then exceed 500.

In Figs . 2 and 3 the continuous curves give the results of calculations of the quantities

u(t, 0) and p(l, 0)/p] for the cases considered.

The results for cases l-4, Re > 50, testify to the validity of the laws of similitude for the

flow of a viscous gas in long channels [5]. Only in case 5 for Re = 20 is a deviation from these

laws observed, this being most noticeable in Fig. 2.

FIG. 4

In Figs. 4-6 for case 1 the profiles of the velocities U, Y and of the temperature e/e1 are

given in various cross-sections of the channel and wake. Each cross-section is labelled with the value

ofN(x =NAx).

In Fig. 4 it is seen that in the cross-sections situated in the immediate vicinity of the inlet

the variation of the quantity u as a function of y is not monotonic. The transverse.velocity

(Fig. 5) close to the inlet crosssection is directed from the wall to the plane of symmetry of the

channel, and on approaching the right end changes sign. Along the entire length, with the

exception of the neighbourhoods of the inlet and outlet cross-sections, the pressure is practically

constant over the channel cross-section.

Short communications 271

’ c . _

FIG. 6

For this case Fig. 3 gives the distribution of the number MO along the channel axis as a

function of the length, from which it is obvious that at the cutoff of the channel the number MO

attains the supersonic value 1.03.

On average the computation of one version occupied 1 hour on the BESM-6 computer.

Trarlslarcll b>, J. Berq.

REFERENCES

1. BYRKIN, A. P. and SHCHENNIKOV, V. V. The calculation of viscous gas flows in plane channels. Zh. vjbhisl. Mat. mat. Fiz., 13, 3, 728-736, 1973.

2. BORISOV, A. V. and KOVENYA, V. M. Application of an implicit difference scheme for computing the internal flows of a viscous gas. In: ,Vumerical methods of the mechanics of a continuous medium (Chisl. metody mekhan. sploshnoi sredy), Vol. 7, No. 4, 36-47, “Nauka”, Novosibirsk, 1976.

3. CLINE, M. C. Computation of two-dimensional viscous nozzle flow. AIAA J., 14, 3, 295-296, 1976

4. BYRKIN, A. P. and MEZHIROV, I. I. On the damping of the perturbations in the isobaric wake behind a body flowed round by a viscous compressible gas. Uch. zap. TsAGI, 8, 3, 49-55, 1971.

5. BYRKIN, A. P. and MEZHIROV, I. 1. Computation of the flow of a viscous gas in a channel. Izr. .4kad. Nauk SSSR. Mekhan. thidkosti igaza, No. 6, 156-162, 1967.

6. BYRKIN, A. P. Numerical calculation of the laminar flows of a viscous gas in channels. UC/?. zap Tstl GI, 4,17-24,1913.

7. BEREZIN, Yu. A., KOVENYA, V. M. and YANENKO, N. N. On an implicit scheme for computing the flow of a viscous heat-conducting gas. In: Numerical methods of the mechanics of a continuous medium (Chisl. metody mekhan. sploshnoi sredy), Vol. -3, No. 4, 3-18, “Nauka”, Novosibirsk, 1972.