Gas flow with straight transition line

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Material Information

Title:
Gas flow with straight transition line
Series Title:
NACA TM
Physical Description:
13 p. : ill ; 27 cm.
Language:
English
Creator:
Ovsi︠a︡nnikov, L. V ( Lev Vasilʹevich )
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Aerodynamics   ( lcsh )
Gas flow   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
An investigation was conducted on 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.
Bibliography:
Includes bibliographic references (p. 12).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by L.V. Ovsiannikov.
General Note:
"Report date May 1951."
General Note:
"Translation of "Ob odnom gazovom techenii s priamoi liniei perekhoda." Prikladnaya Matematika i Mekhanika, Vol. XIII, 1949."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003871226
oclc - 156856542
sobekcm - AA00006213_00001
System ID:
AA00006213:00001

Full Text

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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