docAc-/Kr14

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

TECHNICAL MEMORANDUM 1295

GAS FLOW WITH STRAIGHT TRANSITION LINE*

By L. V. Ovsiannikov

On the basis of the solutions obtained by S. A. Chaplygin

(reference 1), an investigation was made of the limiting case of

a gas flow when the constant pressure in the surrounding medium

is exactly equal to the critical pressure for the given initial

state of the gas; the results are presented herein. For a jet

flowing out of the opening in a vessel with plane walls, it is

shown that equalization of the flow in the jet is attained at a

finite distance from the start of the free jet, the line of transi-

tion being a straight line.

1. According to Chaplygin (reference 1), every problem on the

determination of the subsonic flow that is satisfied by some condi-

ticns reduces to the solution of the system

2 'r 'a

(1-7)

= (1-T)

or the equivalent system

S^ 2T oe

(1.2)

ST 2aT(1-T) )P+l e

0 9 CL-T C*,

for the corresponding boundary conditions.

In these equations, W = W(e,T) and P = P(0,7) are the stream

function and the velocity potential respectively; = V2/V2max, where

*"Ob Odnom Gazovom Techenii s Priamoi Liniei Perekhoda."

Prikladnaya Matematika i Mekhanika. Vol. XIII, 1949, pp. 537-542.

NACA TM 1295

V is the magnitude of the velocity vector and Vmax is the maximum

value of V that satisfies the given initial state of the gas; and

e is the angle between the velocity vector and the x-axis. Further-

more, P = 1/(K-l) is the adiabatic power of the function of the

density on the temperature, where for brevity,

1 K-1

2P+1 K+l

so that the value T = a corresponds to the critical velocity

Ver = (+1Vmax

Chaplygin (reference 1) gives the solution of the problem of

the flow of a gas jet out of a vessel with plane walls forming an

angle of 1800 into a medium with pressure PO = constant in the form

n sin 2n (1.3)

n ln Zn0

n=l

( = C + 1 dT +- 1 + 1 -n xn cos 2n (1.4)

Q 2 T(1- -)) n=l L n zn0

where this solution satisfies the following boundary conditions

(fig. 1):

S= Q on ABC

I= 2 Q on A'B'C'

In equations (1.3) and (1.4), Q is the relative quantity of

flow in the gas jet and C is an arbitrary constant depending on

the choice of the origin from which the values of ( are computed.

The magnitudes zn, ZnO, and xn are defined by the equations

NACA TM 1295

zn = Zn(T) = Tn yn(T)

Zn0 = n(T 0)

7 Yn'(7)

x. = xn(T) = 1 + T YT)

n yn TFT

-T n'(T)

n Znf7

(n = 1,2, .)

where yn(T)

is that solution of the hypergeometric equation

T(i-T) Yn" +[2n + 1 (p-2n-l)T ]yn' + 3n (2n+l) y = 0

which is obtained for T = 0; To denotes the value

responding to po.

of T cor-

In Chaplygin's investigation, it was assumed that T ~<0Ta. In

this case, equations (1.3) and (1.4) converge everywhere in the region

of flow and give the proposed solution of the problem. The case

where TO = a is considered in the following development:

It is recalled that the functions zn(T) and xn(T)

0 Tg:a satisfy the following inequality of Chaplygin:

T(1-T)2p n ; zn;O

1T Zn

0(1_-T)2 Zno '

a(l-7)

for

(1.6)

q+ CAXn -

nl/3 (1-7) ca(l-T)

(1.7)

q =c\/2B2 (1+2) = constant

2. It shall be shown that the potential of the velocity

by equation (1.4) for T--TO

( given

(a) increases without limit if T0sa

(b) remains finite if T0 = %

S (1.5)

NACA TM 1295

It shall be assumed that the limit T --T0 is effected by

moving along some streamline. The following notation is introduced:

ST(1--T)2

To(1-To)21

P(e,T;T) = 1 zn xn cos 2ne

n ZnT

n=l

It is readily seen that for O- r- T0zO

For the potential C at the points of the x-axis (e=0), the

following expression is obtained from equation (1.4):

C = c + + -1 + P(O,T;-0O (2.1)

Q 2 T(1- T) 1)

On the basis of the inequalities of equations (1.6) and (1.7),

for the magnitude of P(0,T;T0),

n-7 +n jn -P(O ,T ) Zn

(1-T) J n 1 +3 n 1 pcL(-) 1 n ZnO

n=l n=l n=l

(2.2)

Assertion (a) now follows in an obvious manner from the second

of inequalities (2.2) because Zn/znO--l for T-*T0 and the sum

of the series with the general term z/nzn0 therefore increases

without limit; whereas the coefficient preceding the term, because

TO / a, approaches a positive limit.

From assertion (b), it is seen that

on

S1 =- log(l-) = log T-(1-)2

n -0(1-70)2P"

n=1

NACA TM 1295 5

by virtue of which, for T7 = a as T--a in equation (2.2) on the

left, the first component approaches zero and the second component

approaches a finite magnitude equal to

oo

q n- 4/5

The same reasoning, together with the first of inequalities

(2.2) is repeated for P(e,T;a) for e O0, so that the assertion

is completely proven.

3. From the previously proven boundness of the velocity potential

it follows that at a finite distance from the opening BB', the jet

is intersected by a certain line L along which T = a, so that the

velocity is equal to the velocity of sound. It will now be shown that

at all points of L, e = 0.

First, it will be observed that along any fixed streamline,

e varies ronotonically, as is true of the boundary streamline,-and

the transformation (e,T)--(9p,*) is a single sheet transformation

at every interior point of the flow region. Next, by fixing some

value T = for which 1 1 Q, the limit is approached as

2

T--w,. and eC = lim 8 is set for T-~a and 4 = 4; this limit

exists because of the finiteness of e ana the monoticity of its

variation.

Approaching this. limit in equation (1.3) for 4 = 4 yields

S- if e0i>0

0 2/ 2

= o sin 2n90 =

n=l 29 +

0 : if 902

that is, in all cases for o0 / 0, Ii = Q, which contradicts the

assumption that I71c Q. Hence, the equality 0 = 0 must hold.

It will now be shown that the line L is straight Along L,

q = constant; this result is readily obtained if the previously con-

sidered transition to the limit is carried out in equation (1.4).

NACA TM 1295

It is sufficient to note that every displacement in the plane xy

is connected with the corresponding displacement in the plane cp by

the relation

dx = cos sin e (d.

V V(1-T)

from which it follows that in a displacement along L in the

xy-plane, dx = 0, because in a displacement along L in the

cp J-plane, according to what was previously proven, e = 0 and

dp = 0. The line L in the xy-plane is thus a straight line

perpendicular to the x-axis.

The equalization of the jet occurs along the line L. Behind

this line the jet becomes uniform, flowing with constant velocity

everywhere equal to the velocity of sound.

The distance of the line L from the edge of the opening will

be computed. Along the boundary of the jet, i = constant and

T = a = constant, so that at the points of this boundary, equa-

tion (3.1) assumes the form

dx = cos e de

Vcr e

Substituting the expression for p (equation (1.4)) taken

for T = a yields

dx = 5 2Xn(a) sin 2ne cos e de (3.2)

]V (1-M)P 0E

cr n=l

Inasmuch as

2 sin 2n6 cos 8 = sin (2n+l)e + sin (2n-1)e

integrating equation (3.2) from x = S = it, to x = xL, = 0

yields

Q 4n Xn(a)

L B Vcr(l. )- 4n2-1

NIACA TM 1295

If h denotes the width of the jet where it is uniform, then

Q = Vcr(1-a)P h, so that the required distance is finally obtained

in the form

OD

XL-XB = 4n

h = 2: 4n2-1 xn(a) (3.3)

4. If any two streamlines of the obtained flow are taken as the

walls of a certain nozzle, the subsonic flow within this nozzle will

be determined by equations (1.3) and (1.4). This flow becomes uniform

at a certain distance and has a straight transition line. This fact

corresponds entirely to the result obtained by S. A. Christianovich

in his investigation (reference 2), where a general device is given

for the construction of a Laval nozzle with straight transition line.

The first derivatives of 0,T with respect to CP,l are

estimated near the line L inasmuch as the behavior determines the

possibility of continuing the obtained flow across L as a super-

sonic flow. The investigation of this complicated problem has been

started and, it is hoped, will be presented in a forthcoming report.

Remarks herein will be restricted to the following:

In the first place, from the previously noted properties of the

line L, it follows that the derivatives 0/0T, 8 T/6J, and 80e/cp

are equal to zero on L.

In the second place, the derivative 6T/~c is evaluated at the

points of the x-axis. It is sufficient to consider the derivative

0j /, inasmuch as, on the x-axis, because e = 0 and r = constant,

_- 1

--- 1 (4.1)

Substituting equation (1.3) for 1 in the second of equa-

tions (1.1) for T = a and setting zna = zn(a) yield

z]

S(1 +2)-- cos 2nG

2na1-T)+1 zna

L n=l

NACA T 1295

or, on the x-axis where 8 = 0

= Q(a-T) 1 n

Sn=l

(4.2)

For estimating the value of the expression in brackets, which

is denoted by S, the last of relations (1.5) and the inequality of

Chaplygin (equation (1.7)) yield

+ .A n2/3 ^ log zn(T) n ,-T

1-T dy a:(l-7)

(4.3)

Integrating equation (4.3) from T7:;7r 0 to T = a yields

apa

n J dT + on2/5 dTL >log Zna

a(-T) 1-T Zn

UT J T

np

UTJ

1 a-T dT

T a.a(l-T)

Raising the upper and lowering the lower limits and carrying

out the integration give

2n

371 Wa-Ta)

(a-T)3/2 + qn2/3(ma ) log Znna 2n (a-_)3/2

1-a. Zn 3aW

(4.4)

If a-T = z(OcZz a--Tl), where z--0 as T--a and

2

T = q 0

q = -- 07. X)

5 2

equation (4.4) assumes the form

y nz3/2 + qIn2/3z>-log ZLn,.bnz3/2

zn

r a(l-T)

NIACA 'TH 1295

T-en the inequalities

exp(-rnz S/2 -qn2/3z) zn Zeexp(-5nz3/2)

ZncL

are I:'b.ained.

Because of equation (4.5), the following inequalities are

obtained for S:

'?1 exp(-ynz3/2 -qn2/3z)}S4l +

n=1 n=l

The sui S2 is readily computed;

1 1 I

2 2 exp(bz3/2)-_ 2

In order to estimate the sum Sl,

un = ex(-qn2/3z)

vn = exp(-Ynz3/2)

S' = Vk

k=O

exp(-6nz3/2) = S2

+ 1

5z/2

unVn =

n=O

exp [-qn2/3z-rnz3/2]

Applying the transformation of Abel to am yields

m-1

aU = (n-un+l) Sn' + um-m'

n=O

(4.5)

(4.6)

(4.7)

m

m n=

n=O

(4.8)

IACA T:: 1295

For m-2/3 cwzw(m+l)-2/3 a, where o = T-2/3(10. 2)2/3,

for the sum Sn'

n

Sn' = exp(-ykz3/3)>n1 (4.9)

k=0

The differences un-Un+1 are evaluated by the forrnular of finite

increments. Therefore

un un+1 = exp -ln2/3z2 ep [- 1(n+ )2/3

S z (n+a)-1/ exp[- ql(n+I')2/3 z (0
3

Replacing. ( by 1 yields the inequality

un- un+ l z (n+l)-1/3 exp[-ql(n+1)2/3 z] (4.10)

Combining equations (4.9), (4.10), an.' (4.11) yields

m-1 m

m > ql(n+1)2/3 exp -q(n+l)2/3 z = qn2/3 z exp -qn2/3

n=0 n=l

The function

f(x,y) = xy2/3 exp(-xy2/3) (4.11)

is considered for ytO, x-0. The derivative with respect to y

varies as

>0 for x-

Of 2 -1/3 p( 2/3)( 2/3) 0 for-3/2

xy exp(-x )(-xy ) = 0 for y = x

L 0 for y x"3/2

For fixed x:O, the function f(x,y) increases at first from

0 to e-l, then decreases and, for y-, approaches zero. Hence,

for any x>c, the inequality

H'ACA TM1 1293

1 + n2/3 x exp(-n2/3x);

n=l 0

xy2/3 exp(-xy2/3) dy = g

(4.12)

is obtained.

The value of the integral g is obtained by applying the

s.s'.bsit.-.titln .x2/ = t. Then

-=

- 2.-53/2

t3/2 e-t dt

(X=m2/3x)

(4.13)

Jo

Substituting equation (4.15) in equation (4.12), setting

x = c-,Z, and noting that m2/3z-w yield

t2/3 e-t d z-3/2 1

hence

S+ + 1 t2/3 e-t

Sm ~~2 3/2 t2L/ e- dt z-3/2 (4.14)

On the tasis of inequalities (4.6), (4.7), and (4.14), it may

be concluded that there are two positive constants 61 and T1 such

that the inequalities

1 13

813/2 71z3/2

(4.15)

hold over the entire interval O0-zga T7. Returning

variable T and comparing equation (4.2) with equation

the following result: Two positive constants 82 and

exist such that for any T in the interval TlTa.S,,

1 < 1

-2 a--IT-- --

to the

(4.15) gives

2 2(4.22)

(4.16)

n?

n=l

NACA TM 1295

From equations (4.1) and (4.16) there follows finally

2Va T> >r24 (a7) (4.17)

which is proven for the points of the x-axis.

In a similar manner, it may be shown that on the x-axis the

second derivative 82/6 2 remains finite as T--oa.

Translated by S. Reiss

National Advisory Committee

for Aeronautics

REFERENCES

1. Chaplygin, S. A.: Gas Jets. NACA TM 1063. 1944.

2. Astrov, Levin, Pavlov, and Christianovich: On the Computation of

Laval Nozzles. Prik. Mat. i Mek., vol. VII, no. 1, 1943.

NACA TM 1295 13

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UNIVERSITY OF FLORIDA

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