Spatial transonic gas flows in ducts

SPATIAL TRANSONIC GAS FLOWS IN DUCTS (0

PROSTRANSTVENNYKH

TRANSVUKOVYKH TECHENIIAKH

GAZA V KANALAKH) Vol.

PMM

23,

No. 4, 0. s.

1959,

781-

pp.

784

RYZHOV

(Moscow) (Received

14

April

1959)

and Falkovich [ 1,2 I have investigated that Frankl’ flow through a Lava1 nozzle where a gas accelerates supersonic

velocity

(throat). Such at the entrance on the the

other

hand,

pressure

both

ends

sonic

during

the

on the

of

the

regions

the

passage

flows take place and at the exit

near

the

at the-entrance does not then the stream will

walls

in works [ 3,~ 1.

the the

analogous walls of

the

arbitrary

In the

present

contained

to the

the equations and in axially note

spatial flows which contain a duct which has two planes constants

but

nozzle, adjacent

cross-section. A simple solution of describing such mixed flow in plane given

critical

side,

convergent-divergent

very

the

the

it

laminar to

cross-section

when the difference between of the nozzle is sufficiently

pressure

discharge

through

phase of the from subsonic

the pressures large. If

exceed very be subsonic

may contain

region

of

the

much at

supercritical

of the gas motion, symmetric nozzles,

solution

is

derived

was for

supersonic regions adjoining of symmetry. By proper choice

in the

solution

to be presented,

it

of is

possible to increase the regions of local supersonic flow and, as a result, to join them along the axis of the nozzle. In a sense such a flow is singular, because in this case the regime of the flow changes and the velocity Moreover, tained the

field the

in [5 1,

transition We shall

behind solution which

the critical presented describes

surface

from

investigate

the

cross-section here transforms the

subsonic flow

of

spatial to

becomes supersonic. into the solution

gas flows

supersonic

an ideal

differ only infinitesimally from the critical has two planes of symmetry. These two planes 1 ine, namely, the axis of the nozzle. Chosing

in the

region

obof

velocities.

gas with

velocities

that

velocity,

in a duct

intersect

along

which

a straight

the x-axis of the cylindrical coordinate system X, r, 8 to be coincident with the axis of the nozzle, we shall write the equation which determines the transonic gas flow in the form

1115

1116

O.S. Ryzhov

----I

where 4 is

apasp a:? axax2

the potential

of

,+$$+g+l

the

a?

a.

(1)

perturbations.

Also

ap

a

---L_--2,

Fx==“x*

v.s_l

I

a,

ar-

%+I

Y-g-7

r’

1

a?

*a

&+i =

(2)

and ~0 are the perturbations along x, r and e-axes of a where vx, V; equal in magnitude to the critical velocity (I, and reference velocity, directed along the axis of the nozzle, K is the POiSSOn adiabatic coefficient. Differentiating new function

relationship

a9 ax we obtain

the

(1)

with

c

u

u=-v

equation

respect

x+1 x

a,

to

x and introducing

a

)

(3)

i aw -- 2-@-+g+;?i-g+f&0 To describe

the

spatial

mixed

regions adjoining the walls of equation (4) in the form X+1 a. Substituting the constants

g(c)

of

V~ = u = 4 ‘;

flows

(4)

which

the nozzle,

contain

we shall

(1 + k2 + 2k cos 28) r2 + 4 $

supersonic for

a solution

g (c;)

< = cr + d (1 f k cos 29) r2

(5)

expression (5) into equation (4) it C, d, and k may be chosen arbitrarily

must satisfy

local look

the

ordinary

differential

is easily and that

seen that the function

equation (a2 = 1 + k”)

Hereafter

we shall

assume that

everywhere

c > 0 and d > 0.

Let us find now the remaining velocity components end it is simpler to make use of the equations which tion of irrotationality of the stream: I av, av, -_=_ r as ax p From the first two equations sideration formulas (5):

x+1

a.

avx

av,

ar=z-1 of

this

v,=-i6~kzrsln28-Bd~kg(~)rsin28+~X,(r,

x+1 -;;;-vu,=8$1+k2+2kcos28)zr+8d;(1+kcos29)g(~)r+Xl(r,

(6)

or and express

a wd

-=as ar system

we have,

~0.

the

To this condi-

au,

(7)

taking

into

con-

8) 8)

(6)

Spatial

Using

the

last

which connects determined:

The second

It

the

expressions

which

expressions

follows

that

components

gas

two functions

equation

substituting

the

of

the

transonic

the

vr and

8)

the

we obtain

the 8)

of

first

which

satisfy

equation

1117

ducts

and s(‘;

component

on the

in

functions

into

velocity ~0;

(7),

x(r,

these

(8)

flows

equation

have to

may be found motion

of

these

equations

is

t)

does not determine uniquely Eo every solution U(X, r, 8)

contrary,

derived

from the

x1 = r4 [1 sin 48 ~2 = 4r3 [-

'jp

1 cos 48 f

by

(1):

of equation (4) there corresponds an infinite number of solutions linear partial differential equations (9) and (10). The particular tion

be

of two solu-

formulas

*is d3cwYk (1 $ k2) sin 231

‘is d3c-3k

(1 + k2) cos 23 + d3c-3 (1 + k2)]

(11)

The remaining solutions of the system (9) and (10) may be expressed, using the principle of superposition, in terms of harmonic functions. To describe axes cause the

the

of

flows

in the

symmetry,

then function

the IJO

it

is

functions will

nozzles,

the

sufficient v

T

cross-section

to make use

and

vr

will

be even

of

of

the

with

which

has two

solution respect

(111,

be odd.

Now let us investigate end we integrate equation

the (6)

function g(t) in greater and obtain as a result

detail.

To this

g -$- = g + a2E The constant

of

be-

to 8 while

integration

here

is

(12)

included

in 4,

since

equations

(1)

and (4) are Invariant with respect to a displacement along the x-axis. The differential equation of the first order (12) has one singular point [ = g = 0, which for any value of a is a saddle point. On the curve g=-

of

the derivative Therefore, the

a25

finite.

equation

(12)

will

becomes qualitative be the

zero, for g = 0 this derivative configuration of the integral

same for

Is shown in Fig. 1. To obtain an exact Introduce the change of variables

any values solution

of of

the this

parameter equation

is incurves a,

as

we shall

(13)

From the

resulting

equation

gq

(61 we have

dg = qd,o

q” +

now ($2

1118

O.S.

is

(14}

Equation

easily

Ryshov

integrable: 1

where tities

e is q1

an arbitrary constant which may be real or complex, the and g2 are expressed in terms of a2 by means of equations

Eliminating formula

which

dg/&

= Q from relationships

defines

the

function

g(r)

(12) in

and (15)

implicit

quan-

we obtain

the

form: (17)

(g + a”E/ qP (g -I- a25/ qdeql = e For k = I, i.e. solution obtained symmetry

expression (17) becomes the in the case laminar flows, in [ 3 ] ; equation (17) describes the flows with axial

investigated

in [ 4

1.

For the description of the mixed flows we have to choose from the four families of curves, shown in Fig. 1, the curves of the family A. Indeed, in the tives

solutions of

which

the velocity

correspond components

points g = 0; if we chose the velocity field in the local

supersonic

appears

zone

in formula

velocities

is

increased

the the

section in Fig.

because

and for

the velocity

becomes supersonic. 1 by the straight

ordinates the author

t = r cos

8,

B and B’,

become

on the magnitude

When e is

decreased

e = 0,

the

In this limiting the axis of the nozzle. g(c) is represented by the broken curve changes,

branches stream

branch A1 for the desired duct will be supersonic.

depends

(17).

to of

field

the

@ is

given

the

constant

region

of

of v

the

zone

graph

Then the

aob.

downstream

the

of

type

is the of

critical

solution g(t) in terms of

by’the

derivaat the

solution g(f), The dimensions

supersonic

case

The corresponding line aoc; function

y = r sin

of the

the

infinite

formula

then of the

e which supersonic extended

to

function the

flow

crossis represented Cartesian co-

obtained

by

in [ 5 1 :

(18) [3 + 2k2 +r/l +2 $13 The form of the function equations (8) and (11). The derived

+ 49

+ 2ke F VI v r and

ve

+ (5 ‘f

v’1 + 4~29kf z*+

+ 4az--(5 F t/l

is

easily

f 4ae) kJ y*

obtainable

when using

solutions as given by formulas (5) and (8) may be used to as it is varied systematically by increasing the the gas motion, of pressure between the entrance and the exit of the nozzle. near Also the extent of the supersonic zone, which is formed initially the walls of the channel in the region of the critical cross-section, is describe difference

Spatial

increased the

gradually.

surface

of

transonic

gas

the

region

of

the

in

of

pressure

When the difference

transition

from subsonic

Fig. described

flows

critical

by formula

to

e zz o-

is

supersonic

further

flow

The gas flow which is given

(g = -

This tions,

surface, is

according

to

characteristic

for

dc-’

the

a25 1 ql,a)

is

(19) not an which

(1 + k cos 29) r2

theory

equation

of

(4).

partial

(20) differential

A detailed

equa-

investigation

solutions with second derivatives of the velocity potential, having continuities on the particular characteristic surfaces, was carried in [6

in

in this region by the equation

flow with a local supersonic zone is which describes a singular analytical solution. The derivatives of the velocity components, a discontinuity on the surface: correspond to it, contain x = -

increased,

develops

1.

cross-section. The solution

(18).

1119

ducts

of disout

1.

We shall

introduce

2 -+3i Then the

the +4a”)

expressions

+

notations =A,

(18) vx = A,

-1

will

4

5 ‘F 7/l + 4a2 k==h 1/ 1 + 4aa

3 + 2k2 $

have the

+ ‘42 (l/g -

(21)

form [ 5,6 ] :

n) 23+ A2 (l/4 + n) y2

l

From formulas the

degree

of

(20)

deviation

and (22) of

the

it

follows

form of

the

that

quantity

particular

k characterizes

characteristic

sur-

face from a surface of revolution,and parameter n characterizes the degree of deviation of the stream velocity field from axial symmetry. It is interesting to point out that one of the two solutions, given by the equation c = 0, while describing the gas motion with a supersonic velocity

1120

O.S.

field

downstream

of

the

critical

Ryshov

cross-section

account for all possible flows of correspond values of n of absolute presented veloped

by formula in this

(18).

paper

This

that type. magnitude

is the

in accordance

of

nozzle

does not to which

Only the flows not exceeding 5/16

case,

with

the

because

in the

relationship

are

re-

solutions

(20),

the

de-

charac-

teristic surfaces same time in [ 5 ]

always pass through the center of the nozzle. At the it has been shown that similar surfaces exist only for The absence of particular interval - 5/16 4 n< 5/16.

s lying in the characteristic surfaces in the solutions for 1 n/ > 5/16 that such solutions cannot be obtained from formula (5) limiting point to

when the magnitude of constant e is approaching process, the instability of the corresponding mixed gas flows.

d

e Fig. In those does

not

fore, which

and also the fact by means of a

cases

differ

when values

zero,

e

2.

e are near

zero,

the

graph

of

function

g(t)

greatly

from the broken curve sob shown in Fig. 1. Thereit is easy to obtain the shape of the curves using formula (181. are formed by the intersection of the sonic surface with the mutual-

ly perpendicular planes y = 0 and z = 0. It may be shown that for l/3 in both of these planes, curves are obtained which have the

1k 1 <

usual are

form;

they

obtained

for

are

represented l/3

1k 1 >

in Fig.

in either

Za.

The curves

the plane

y = 0 or

shown in Fig. z = 0.

Zb,c

The first

of these figures corresponds to the case l/3 < \ k ] < 1 and the second to - \ k] > 1. The form of the intersection of the sonic surface with the second plane qualitatively coincides I wish attention

E = 0 or y = 0, respectively the same as for 1 k 1 < l/3.

with to

one of

express

to the

the

axes

y or

my sincere

results

z depending

gratitude

given

for these values of k are In Figs. 2a, b, c the abscissa

to

upon the

1.5.

sign

of

k.

Nemchinov

who drew my

theory

Lava1 nO!ZZleS).

in [ 3 1.

BIBLIOGRAPHY 1.

Frank]‘, Izv.

F. I., Akad.

K teorii

sopel

Nauk SSSR,

ser.

Lavalia Moth.,

(On the Vol.

9.

1945.

of

Spatial

transonic

gas flows

sopel Lavalia S. V., K teorii PMM Vol. 10, No. 4, 1946.

2.

Fal’ kovich, nozzle).

3.

Studies Tomotika, S. and Tamada, K., flows of compressible fluids. Pt. No. 4,

4.

1121

in ducts

(On the

theory

of

the

Lava1

on two-dimensional transonic 1. Quart. Appl. Math., Vol. 7,

1950.

Tomotika,

S.

ible

fluid

Vol.

29,

and Hasimoto, through

No.

2,

2..

an axially

transonic

symmetrical

flow

nozzle.

of J.

a compressMath.

Phys.,

1950.

5.

Ryzhov, flows

0. S. , 0 gazovykh in Lava1 nozzles).

6.

Ryzhov, O.S. ot tsentra

and Cherniavskii, sopla (Concerning

from the

center).

nozzle

On the

techeniiakh PMM Vol.

v soplakh 22, No. 3.

Lavalia 1958.

(On the

gas

Ob strazhenii slabykh razryvov S. Iu., the reflection of weak discontinuities

PMM Vol.

23,

No.

1,

1959. Translated

by J.R.W.