On linearized supersonic flows in laval nozzles

ON LINEARIZED SUPERSONIC FL.OWSIN LAVAL NOZZLES (0 LINEARIZOVANYKA SVERKBZVUKOVYKB TECBENIIAKA V SUPLAKR LAVALIA) PMM

Vol.27,

No.4, M.P.

(Received

pp.714-718

1963,

RIABOKON’ (MOSCOW) March

2,

1962)

In this paper we investigate the supersonic gas flows near the SXiS of Lava1 nozzles, which are described by linearized hyperbolic equations of the second order. An expression will be obtained for the determination of the form of the stream lines as a function of the distribution of Mach numbers along the axis of the nozzle. The effect of discontinuities in the initial values on the behavior of the stream lines will be investigated. These solutions velocity of sound. 1. The equations metric irrotational

do not apply in the region

of transition

through the

of the characteristics of supersonic axially symflows (Fig. 1) in the characteristic coordinates (q, c) have the following form:

where p is the Mach angle, M is the Mach number, v is the critical velocity and 8 is the anile between the velocity vector The characteristic coordinates are expressed coordinates by the following relationships:

1091

in terms of Cartesian

M.P.

To linearize the equations clination of the stream lines

R iabakon’

let it be assumed that the angles are small

of in-

(1.3)

e = 0 (1)

and in the equations we shall neglect small terms of higher order. This condition applies in a certain neighborhood of the axis of the nozzle. Equations (1.1) show that in any characteristic triangle, a - oe and 0 are quantities of the same order of smallness (here oe is the value at a given point of the triangle). Hence it follows that within the linear approximation (3 = ae = const and p = ~(a) = const. In linearized

form equations

and the relationships

(1,l)

become

between the coordinates

are

The characteristics of both families are straight lines having conAfter elimination of CTfrom stant angles p and +_I with the x-axis. of 0 (1.4), we obtain the differential equation for the determination in Cartesian coordinates

i!?+_E$-c$.!&o, To SOlW? equation (1.6) it is necessary The second initial condition, derived from is efx, 0) = 0. Usually in calculations of tion of M numbers (or o(M)) along the axis connection between (aO/a,ty=e and a(%. 0) equations (1.4). We have

that %/ay be given at y = 0. the condition of symmetry, Lava1 nozzles the distribuof the nozzle is given. The may be obtained from any of

(1.7) 2. Substitution

To obtain

of

solutions

u = e/y

leads

to the Darboux equation

which are ~ontinuons

at y = 0 it

is necessary

to

Linearized

assume,

in order

flows

supersonic

to account

for

in Lava1

the singularity

1093

nozzles

in the second

term

(au I a?/),=e = 0 This equation is one of the initial conditions. Therefore, second of the initial conditions may be chosen arbitrarily u (2, 0) = cp(2)

or

cp(2) =

T

w I ay),,=,

u b, 0) =

3. We shall first find the solution the simple initial condition

U of equation

(2.2)

(2.1).

satisfying

for s> %I for x
1I

(x, zo) =

only the

0

we shall look for the soluEquation (2.1) is homogeneous; therefore, Substitution of this function tion U as a function of z = (X - x0)/y. into (2.1) leads to the ordinary differential equation

(9 The boundary conditions

u and the solution

The continuous

(3.2)

I

for

x -

z, = cy

for

x-

2,~

or

-cy

z = or

c

(3.3)

2=--c

-r = 0 for

function

for

=:I

T(X) may be represented

x,, > x according

of

w (5, 4 dso sdcp

(4 = cpb - cy) +

to the increase

value

U

z>c

may be used for the construction of a solution but continuous initial values of (p(x).

T

The full

0

fJ have the form

forI<-c,

4. This function terms of arbitrary,

leads

/ dz =

is

U=O

because value

for

zdU

0

1

=

9) ddU / dza +

z

X-W to (3.1).

in the function

II, obviously,

in the form (4.1)

dxo

An increase

at the point

is represented

in

(x,

in the initial

y)

by equation

1094

M.P.

Riabokon’

x+CU

u=

‘p

gyJ(y)dx.

\

b --cY) +

(4.2)

r-cy

because

u = 0 for

taking

into

x,, > x t

account

XSCY

UZ- 1 Y

the to

CY. Hence,

values

s x--cy

of

dU cp(20) dz

when integrating

II on the

by parts

characteristics

(3.3).

and we obtain

dxo

It may be shown that equation (4.3) applies to all instances where distribution of M numbers along the nozzle axis is continuous, i.e. Note that at points where the flows without strong discontinuities.

function &/ay 5. tion

6(x)

and its derivative dq/dx 0 is not satisfied.

= 0 for

have

discontinuities

condition

y -

Returning

to

the

function

0 = uy.

we obtain

as a solution

of

equa-

(1.6)

s

%+CU

cp(4

6=

$

(5.1)

dxo

2-W Consider

the

function

8 for

which (5.2)

Let tinuity

us assume the of order n.

According teristics

to

function

(5.1)

the

x = f cy has

T(X)

at the

expression

the

for

point

x = 0 to

8 in the

region

have

a discon-

between

charac-

form x+ev 0=

s

axon ‘;

dx,

(1.3)

the

(5.3)

0

As a consequence y = const; of

therefore

the

angle

6 along

Let

us introduce

of

linearizing

relation

(5.3)

a stream

line

stream

lines

at y = y,, represents

correspond the

y = yu.

the new variable (5.4)

x=x+cy This substitution x coordinate to the and the

line

y = ye.

to

variation

is equivalent to translation point of intersection of the

of the origin characteristic

of the x = -cy

Linearized

The expansion

of

supersonic

dU/dz

into

in

flows

a series

Lava1

in terms

1095

nozzles

of

(X -

xe)/yu

has the

form dU

2

dz

xc

-=-

Let

us also

For n > with respect

. ..k(A~--o)k+. YOk

Jrgyz++.

expand

0 into

a series

in terms

of

(5.5)

. .]

X/y,

1 all the coefficients C. have meaning. When integrating 1 to X we obtain an expansion of the equation of the stream

1 ines &p+2.5

-+.

co

.

n + 2.5

q(x)

Relation (5.7) of order n,

propagates are absent line the

is

shows that for a discontinuity

greater

than

1.5.

x1= Expansion

of

into

dU/dz

the

Xl

x9

a series

for

Let N be a positive as follows:

for

e=ylP+l

(

n #-

/lo+.

8 is

(5.7)

function line

of

the

8 in

change

of

front

in the

or zero.

(5.8)

x0 +

Xl

of

x1/y,,

has the

following

Then (5.10)

of

variables

form

. +ak$+...)

transformed

integer

=

in terms

~=-&f$~~(if.. The expression

behavior

x = CY we introduce

cyo-

. .

x = -cy. When strong discontinuities of the discontinuity in the stream

To investigate

characteristic

+.

QXk (n + k + 2.5) yak

a discontinuity in the initial of order n + 2.5 in the stream

along the characteristic on the x-axis the order

reflected

. +

(5.9) manner:

may be represented

0.5 + fV . .+

A,%+. . . +x’ YOk

>

n+1.5

Yo”.5

(

Bo_t...

+l?,X1L+.. . (5.11) > Y$

1096

M.P.

for

Riabokon’

n = 0.5 + N

CO+.

(5.12)

4+.;.+o,X’k+...

..+&+...

Yak

YOk

When differentiating the latter relation we see that the curvature of the stream lines k = %/a~ for n = - 0.5 on the reflected characteristic has a logarithmic singularity. An analogous result was obtained in [31 for the derivatives of the velocity for this particular case. Behind the reflected characteristic x = cy we have for 8

where

This expression transforms into a polynomial of order real number or zero, otherwise it has the form analogous (5.12).

n. when n is a to (5.11) or

With the help of farmulas (5.3) and (5.1) and also by means of integrating and differentiating them we obtain the following relations for the angle 6. the displacement of the stream line y - ye and for the curvature .of the stream line k which are valid in the entire region:

0 = Yon+lF1

(f)

,

Y-

yo =

YO”+~Fa

(f ) ,

k = y,nF,

(;)

(5.15)

6. As an example of the application of equation (5.1) let us Calcuwhich transforms a radial flow into late the shape of the Lava1 nozzle, parallel flow. The distance from the “source. to the point on the x-axis of transltion from radial to parallel flow shall be denoted by R and the outlet r/R = 6, is the angle of the wall radius of the nozzle by r. Obviously, of the nozzle in the radial region. Let us locate the origin of the xaxis at the point of contact of the two regions. Obviously

cp(4 = { I/ (R o+ 4 The quantity

l/(R

for 2<0 for z>O

+ x0) may be represented

as follows:

Linearized

supersonic

flows

in Laual

1097

nozzles

term I/(R + x) in this exSince y/(R = x) = Q(8) only the first Using (5.1) we obtain the expression for pansion needs to be retained. 8 0 P 1 -dUdx 6E: (6.2)

5R +x0

dz

-w

To calculate the equation of the stream line y = r approximately equation must be integrated with y = r. Here we can assume

That equation

this

has the form

T. When eliminating the function coordinates x and y, we obtain

8 from equation

(1.4)

by using

the

(7-i) From the

initial

conditions (7.2)

the

solution

of equation

(7.1)

has the

form (7.3)

8. For COmpariSOn with the relationships obtained above we gfVe (without derivation) some results for the case of plane flow. equations for 8 and 0 have the form

(8.1) snd the solution

q

(D’A1embert.s~

(x) =

(3

(05 oh

of these

e (2, 01 = 0,

equations

(aa / dy)-

is

= 0

When the values of Q are continuous on the axis, which is the case in the absence of strong discontinuities on the axis, the stream lines have discontinuities of an order higher than the first, i.e. they do not have corner points.

M.P.

1098

R iabokon

The order of the discontinuities of stream lines, propagating along the characteristics is greater by two than the order of the discontinuin the case of the simplest initial conities of q(x). In particular, ditions (3.1) the stream lines in the region -cy dx - x0
RIRLIGGRAPRY 1.

Chen Yu Why, Supersonic metry.

Commun,

Pure

flow through nozzles with and AppE. Math., Vol. 5. NO.

rOtatiOna

SYB-

1, 1952.

2.

Chen Yu Why. Flow through nozzles and related problems of cylindrical and spherical waves. Commun. Pure and Appl . Math., Vol. 6, No. 2. 1953.

3.

Meyer, R.E., The method of characteristics pressible flow involving two independent and Appl. Math., 1, 1948.

for problems of comvariables. Quart. J. hfech.

Translated

by

L.R.W.