the calculation of viscous gas flows in plane channels

THE CALCULATION OF VISCOUS GAS FLOWS IN PLANE CHANNELS *

A. P. BYRKIN and V. V. SHCHENNIKOV

Moscow

(Received 31 December, 197 1; revised 30 October 1972)

THE flow of a viscous thermally conducting gas in a plane widening channel

of finite length is discussed. The complete Navier-Stokes equations are used

to describe the flow. Numerical results are quoted for flows in two plane

channels of different configurations, with given characteristic values of the M and

Re numbers.

Introduction

The calculations so far published on the flow of a viscous thermally conduc-

ting gas in channels (nozzles) refer to cases with high Re numbers, when the

boundary layer equations can be used as the initial equations (see e.g. Cl, 21).

At the same time, cases which cannot be described within the framework of

Prandtl theory (low Re numbers, or flows with break-away etc.) are of considerable

practical interest. Such flows are described by the complete Navier-Stokes equa-

tions (under the assumption that the equations of the mechanics of continuous

media are valid).

In the context of interior gas flows, a change-over to the complete Navier-

Stokes equations implies more than just initial equations of greater complexity.

The formulation of the boundary conditions represents a vital step in the state-

ment of the problem. The difficulty lies in choosing the boundary conditions at

the ends of the channel. The progress in solving similar problems in the case

of incompressible fluid flows is explained to some extent by the existence of

the Hamel and Poiseuille exact solutions. The absence of such solutions for

interior flows of viscous thermally conducting gas makes it difficult to state

the relevant problems.

* Zh. vychisl. Mat. mat. Fiz., 13, 3, 728-736, 1973.

244

Calculation of viscous gas flows in plane channels 245

Exact solutions of the Navier-Stokes equations have been obtained in some recent papers 13-61, whereby the familiar solutions of Hamel, Poiseuille, Landau, Yatseev, and Squire have been extended to the case of compressible viscous gas flows. The solutions obtained have made it possible to pose correctly the problem of the interior flows of a viscous compressible gas.

In the present paper we state the problem and quote results for viscous gas flows in plane channels of different configurations, for given characteristic values of the Re and M numbers.

1. Statement of the problem

We shall consider the stationary flow of a viscous thermally conducting gas in a plane widening channel (Fig. 1). The shape of the channel is given

by the equation Y, = y,(x), where the sections BD and FH are straight, and DF is curved. Conditions for smooth matching of the function y,(x) (up to and including its second derivatives) are satisfied at the points D and F.

FIG. 1.

As the initial equations of the problem we shall use the non-stationary Navier-Stokes equations. The limiting solution of these equations as t -) M

(t is a time parameter) under the stationary boundary conditions will be interpreted, provided such a solution exists, as the stationary solution of our nroblem.

246 A. P. Byrkin and V. V. Shchennikov

All the quantities in these equations are dimensionless and are linked with

the dimensional variables by

t G&,X* --

x = dx*, y = ;iG,

U =-hi*, --

u = u/u*, p =p/ (F*i*3,

P = OFF*, -- -- --- -

h = h/uz+c=, P = Id&, Re = ~*u*x*/~*, -

where t is the time, x, y are the distances measured along the channel axis and

along the normal to the axis, u, y are the longitudinal and normal components of

the velocity, p is the pressure, p is the density, h is the enthalpy, F is the coe-

fficient of viscosity, Pr is Prandtl’s number, x is the adiabatic index, Re is

the constant Reynolds’ number, u - - - *, p*, h*, g* are the values of the respective

quantities on the channel axis at the initial cross-section (at the point A, see

Fig. I), and x* = OA is the characteristic length (see Fig. I).

In order to close the system (I. 1) we have to specify the dependence of the

coefficient of viscosity p on the enthalpy: ,u = (h)n.

To solve the above equations numerically it is desirable to transform the

flow region (with curved boundary) into a canonical region.

To this end we introduce the coordinate transformations

Calculation of lJiSCOUS gas flows in Phne channels 247

In 6, r~ coordinates the flow region becomes a strip of unit width.

When stating the boundary conditions at the input to the piece of channel we

use the exact solution of the Navier-Stokes equations 141. This solution describes

the flow of viscous thermally conducting gas in a widening plane channel with

straight adiabatic walls, The parameters 6, M, and Re of the exact solution

are taken equal to the corresponding parameters of the input piece of the channel,

i.e.p=&,,M=MA,Re=ReA7 where p,, is half the flare angle of the channel,

and M,, Re, are the Mach and Reynolds’ numbers, determined from the character-

istic flow parameters at the point A. As the boundary conditions for u, u, p, and

h in the cross-section AI3 we therefore take

u(1, 77) = $(I, “I), u(I, 71) =%(l, “t),

p(I, 77) = ?(I, 59, h(I, rl) = K(l, 50,

where z, c, F, ff is the exact solution corresponding to the parameters /$,, M, , and Re,.

At the channel output (cross-section GH) we specify the conditions for the

flow in a channel with straight walls to have similarity properties; in the

variables 6, 71 these conditions are

au a, ah -= _= __zc 0. at at at

On the line AG we pose the conditions for flow symmetry

au ah ap -=-= 0, v =o,

a? all a?l

and on the channel wall (BH) the conditions for fluid adhesion

u=?,?=o.

The wall is assumed to be thermally insulated (which is the same as the condi-

tion for the flow to have similarity properties in a channel with straight walls),

i.e. ah/an = 0, where n is the normal to the wall.

In the variables [, ‘1 this condition becomes


The density at the wall and at the output of the channel will be found from

the non-stationary equation of continuity in the context of the boundary condi-

tions. The pressure is eliminated from the system of equations by using the

equation of state.

As the initial condition (t = 0) we take the exact solution, corresponding to

the parameters of the input piece of the channel:

ZJ (f, 77> = X, $1 u ([, 4 = 369 $9

P (5, 77) = #X9 4, h(f, 59 = CC 4.

2. Numerical method of solution

As the initial condition (t = 0) we choose the exact solution, and make use

of the two-step method of establishment [71 with a symmetric difference approxima-

tion of the space derivatives, which provides an approximation error of order

O(h2), where h is the maximum step of the space mesh, as we move towards the

stationary solution.

The time step r was selected by starting from the stability conditions for

the difference scheme, the conditions being obtained by considering the system

of linearized equations. The system (l.l), linearized in the light of the specific

features of the present problem, takes the form

au au (2.1) ___=-A _

at ax

where

/I ; - B = 1) T/pxW 0

0 P 0 V 0 0

V l/xM2 ’

I 0 0 T(x-1) v

au d’U a2u -B_+C__ +D__,

dY ax2 aY2

A=

U P 0 0

TlpxlW u 0 I/?&l”

0 0 U 0

0 T(x-1) 0 u


0 0 0 0 0

C=o 4/3p 0 Re 0 0

i/pRe 0 y 0 0 0 x/p Re Pr

0 0 0 0

0 D=o l/pRe 0

0 0

413~ Re 0 * 0 0 0 x/p Re Pr

When the system (2.1) has constant coefficients, the matrix representing a transi-

tion over the complete time step is

G=I-QZ-iQ-(I-iQ)S,

where I is the unit matrix, and Q = Q(A, B), S = S(C, D), these latter functions

being linear.

If we now require that the spectral radius of the matrix GG* be bounded by

unity, then perform simple but rather laborious calculations we find that the

sufficient condition for stability of the linearized difference scheme is

(2.2) T f ,u, y;, + * , TG O*5h2gPxRePr ,

where h is the minimum step of the space mesh, and

6=max{2p,2T(x--1), (T+p)/(pxM’)}.

In flows without stagnation zones, concrete computations show that the first

of conditions (2.2) is the definitive one. With the appearance of developed

stagnation zones, in which the density may prove to be a small quantity, the

estimate r = O(hZ) becomes definitive (see [8I). In these cases there is a large

expenditure of computer time if the method of [71 is used. It is easily shown that,

if the system (2.1) is modified by writing it as

au au au cm d=u (2.3) F_=-A_-B-+C__+D__,

at ax JY dx2 JY’

where F is a real positive-definite, symmetric matrix with the norm I(F\/ = O(I +

r/ hZ), the stability condition allows the computation to be performed with r =

O(h) [9, 101.


FIG. 2.

In this case the system (2.3) loses its physical meaning though its limiting

solution as t -t m, if it exists, can be interpreted as before as the stationary solu-

tion of system (2.1).

It must be mentioned that the difference scheme employed in the present

paper loses its stability if a disturbance of the density p occurs in a region where

ZJ and u are small (e.g., if there is a strong intake through a crack in the channel

wall). For, it is easily shown by examining the expression U*GU, where u* =

(fOOO), that

JU*GUI >, IU*U/.

In this case we have to modify the approximation of the equation of continuity,

by introducing an extra “viscous” term.

3. Some computational results

Our somputations related to gas flow in two plane curved channels. For one

channel, the wall angle at the output was less than at the input (case l), and for

the other, it was greater (case 2).


When computing either case, the specified profiles of the gas-dynamic

parameters in the initial cross-section AB correspond to similarity flow in a

channel with straight walls and p,, = 23.5O(tg&, = 0.433), Re = 300, and M = 3.

The gas was assumed to be diatomic (x = 1.4), Pr = 0.71, and n = 1 in the

expression for the coefficient of viscosity ~1.

FIG. 3.

The computations were performed for a relative length AG = 6, the shape of

the channel being specified as follows:

a) ZA G z G XC, Y,,,(Z) = tg pox;

b) xc < x < xB,

c) XE<XGX6, !/w(x) = Ylo(Xzc,> + !h’(XE) (x - x,).

In either case the computations were performed in a fixed mesh A4 = 0.15,

A7 = 0.1 (the number of computational points was 41 x 11 = 451) for the values

xc = 1.45 and 3cE = 3.7.

Case 1 related to the data a = 0.15, tg/3 (x,) - 0.095, p(xE) = 5.5’, and


0.4 0.6

FIG. 4.

case 2 to a = -0.20, tg/3(+) q 0.883, P(Q) = 41.5”.

In Figs. 2-4 we show the computed profiles of the dimensionless quantities

u, u, h and.p/p, in the channel cross-sections (the subscript 0 refers to values

on the axis). Each curve shows how the relevant quantity varies along the

length of the channel.

The continuous curves refer to case I, and the broken curves to case 2.

In either case the numbers of the curves and the values of x are connected as

follows: corresponding to curves 1, 2, 3, 4, 5 we have x = 1, 2.5, 4. 5.5, and 6,7.

In case 1 the Mach number decreases monotonically along the axis to M = 2

at the output.

The profiles of the transverse velocity v can be seen to have non-monotonic

properties; the same is true of the p/p0 profiles; at the end of the channel the

Calculation of viscous gas flows in plane chwanels 253

pressure becomes virtually constant over the cross-section.

The close similarity between the profiles of all the quantities in the last

two cross-sections suggests that the rightwards flow changes into a new simi-

larity mode for a channel with straight walls.

In case 2 the Mach number increases continuously along the axis and

reaches M = 4 at the output.

As in case 1, the pressure profiles are deformed non-monotonically, though

in this case we get a pressure gradient over the final cross-section.

Notice finally that, in order to obtain a change-over into a new similarity

mode rightwards, the length of the straight section PH needs to become greater,

the greater the Re number.

Translated by D. E. Brown

REFERENCES

1. BYRKIN, A. P. and SHCHENNIKOV, V. V. A numerical method for computing laminar

boundary layers. Zh. v%hisl. Mat. mat. Fiz., 10, 1, 124-131, 1970.

2. BYRKIN, A. P. and MEZHIROV, I. I. Computation of gas flow in a hypersonic nozzle.

allowing for the influence of viscosity fthe direct problem), Uck. zap. TsAGf, 2, 1, 33.41, 1971.

3. WILLIAMS, J. C. Conical nozzle flow with velocity slip and temperature jump, AIAA Journal, 5, 12, 2128-2134, 1967.

4. BYRKIN, A. P. An exact solution of the Navier-Stokes equations for a compressible gas, Prikl. Mat. i M&k. 33, 1, 152-157, 1969.

5. SHCHENNIKOV, V. V. A class of exact solutions of the Navier-Stokes equations for the case of a compressible thermally conducting gas, Prikl. Mat. i Mekk. 33, 3, 582.584. 1969.

6. BYRKIN, A. P. Exact solutions of the Navier-Stokes equations for compressible gas flow in channels, Uck. ZQP. TsAGI, 1, 6, 15-21, 1970.

7. BRAILOVSKAYA, I. YU. A difference scheme for the numerical solution of the two- dimensional non-stationary Navier-Stokes equations for a compressible gas, Dokl. Akad. Nat& SSSR, 160, 5, 1042-1045, 1965.

8. BRAILOVSKAYA, I. YU. Explixit difference methods for computing break-away flows of a viscous compressible gas, in: Some applications of the mesh method in gas dynamics (Nekotorye primeneniya metoda setok v gazovoi dinamike), No. 4, VTs MGU, Moscow, 6-85. 1971.


9. CHENG. S. J. Numerical integration of Navier-Stokes equations, AIAA Paper, No. 70-2. 1970.

10. CROCCO, L. A suggestion for numerical solution of the steady Navier-Stokes equations, AIAA Journal, 3, 10, 1824-1832. 1965.

the calculation of viscous gas flows in plane channels

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