Exact solutions of equations of gas dynamics

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Material Information

Title:
Exact solutions of equations of gas dynamics = Primer tochnogo reshenia ploskoi vikhrevoi zadachi gazovoi dinamiki
Portion of title:
Primer tochnogo reshenia ploskoi vikhrevoi zadachi gazovoi dinamiki
Physical Description:
12 p. : ; 28 cm.
Language:
English
Creator:
Kiebel, I. A
United States -- National Advisory Committee for Aeronautics
Publisher:
National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:

Subjects

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

Notes

Summary:
The equations of gas dynamics are written with the Cartesian coordinate x and the stream function psi as independent variables. An exact solution possessing both subsonic and supersonic velocities is described. A solution, analogous to that obtained for two-dimentional flow, can also be developed for axisymmetrical flow.
Bibliography:
Includes bibliographic references (p. 11).
Statement of Responsibility:
by I.A. Kiebel.
General Note:
"Technical memorandum 1260."
General Note:
"Report date June 1950."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003689933
oclc - 76833067
sobekcm - AA00006219_00001
System ID:
AA00006219:00001

Full Text
N A( M-1o







r *


.1
/ -


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1260


EXACT SOLUTIONS OF EQUATIONS OF GAS DYNAMICS*

By I. A. Kiebel


The equations of the two-dimensional stationary
dynamics are of the form


6u u

U --V + --=

dx by
U -- + V --- =


problem of gas


1 ap
P 6y
P y


+ 1 = 0
oy


u a I


1
p +


p = 0


where u and v are the components of the velocity along the
coordinate axes x and y, respectively, p is the pressure,
p the density, and K = cp/c_, the ratio of specific heats.
The equation of continuity permits the introduction of the stream
function from the equations


Pu = -
ay

pv =-
3x


The last of equations (1) gives


1

p = pS (M)


*"Primer Tochnogo Reshenia Ploskoi Vikhrevoi Zadachi Gazovoi
Dinamiki." Prikladnaya Matematika i Mekhanica. Vol. XI, 1947,
pp. 193-198.


7/pb


y P1
v
Ps






NACA TM 1260


where 6 is a certain function only of *. The first two of
equations (1) give the Bernoulli law


u2+v2
2


K-1
K K
+ K- p
K -1


= io (M)


where io is a function of I, and the equation for the vorticity
is


xv uy
ax ay "


odi
( dF


K-1 \
S KK d4
K-1 dI)


For the solution of the vortex problem in which at least one
of the derivatives dio/d* and d4/d* is different from zero, it
is convenient to pass from the variables x and y to the vari-
ables x* = x and *. Equations (2) and (5) then assume the form
(See, for example, reference 1.)


ox* u


ay 1
34- pu


Op v
5T 5x-*


The problem reduces to
u, v, p, p, and y of
and (6).


the determination of the five functions
x* and \ from equations (3), (4),


By using the last of equations (6), the function
duced with the aid of the equations

P -a


V _


X is intro-




(7)






NACA TM 1260


The first two equations of equation (6), on the basis of equa-
tions (3) and (4), then assume the forms


x i2 2K
i o 1
84' L2i- K-1


1


c-iL
axa


2K
21 -K
K -1


K-1 1

(x/, 1 2
x K

K-1 1
K 2 2
\x \a


where the asterisk on x has been dropped.


By differentiating equation (8) with respect to *
tion (9) with respect to x, y can be eliminated and a
equation for the function X obtained.


and equa-
single


In order to obtain an example of the exact solution of the
system of equations, io is set equal to a constant:
K-1
X = -H (') x 1 (10)

where H is a certain function of W to be determined. Equa-
tions (8) and (9) then give


K-1
6y K+12
H' x (2
OX


2
SK-1 ( 1


io -2
5-1


2o -
t F


2K K-l
K-1 \K+1


/K-1
\K +1


K -1

H) K


+ H'2


K-1
K


K1-
-2 -2 2L
K+1


(11)
1
2 +1
+ H'2 x K 1

(12)






NACA TM 1260


When equation (11) is differentiated with respect to and
equation (12) is differentiated with respect to x, terms with
the same degree of x are collected and after simple trans-
formations there is obtained


1 H"
K-1 H) K + H"
\K / 2 (13)

B' (H2 +KH H")' = (K-1) H" (H' 2+ H")

Successive integration of the second of equations (13) gives

H'2 + KH H" = ClH'1'1

K (14)

H = C2 (C_-H 1+) K


From the last equation, the relation between
be found with the aid of quadratures. It is more
ever, to introduce a numbering of the streamlines
the aid of H' and not 4.


* and H can
convenient, how-
directly with


Thus, equations (14) give H as a function of H' and, using
equations (13), permit finding t in terms of H' in the form


K-I C K-1/
2 W02
2K K+1


1-K
K


2M
K+1
H i1- K C, -H 1+ K


(15)


Substituting in equation (11) and integrating yields, for the
streamlines,


yK-1


L-1 1
-2 dr
CIH ,1-x + dx + F (H')


(16)


K







NACA TM 1260


Comparison with equation (12) shows that F = constant and
without loss of generality can be taken as F 0.

For the determination of u, v, p, and p the following
relations are written:


-r-
`2
S1-K 1+K
u2 =2io C1H'"Kx

K-1
v = H'x K+1

K 2K
Sc (CI +K 1+ K 1+K
p = C2 (Cx-E' I


2K+1 2
2K-i 2 (c1H1+) K+1 K-1 + K


(17)


(18)



(19)


For simplification it is convenient to introduce the nondimen-
sional variables x and y and in place of H' introduce T
with the aid of the following equations:

x = Lx


Y L (20)

H, =1K =1


where


i K+1 1
2 K-1 K-1
L = (?21o) C1


(21)


Equations (16) and (17) then assume the following simple form:






6 NACA TM 1260


1-s
P -2 1 --
m= m +1 -i d1 .
y m (22)


x = Tm () J


u 21o (1-m )
(23)
K-1 2
-I1 +1 1 i-l
v =21 -m =A/2i -m


If K= 1.4, the integral in equations (22) is evaluated and
gives


y 6 1 [1 + 2 ( m-) + ( -1)2] (24)


and the streamlines can therefore be easily constructed.

Figure 1 shows the streaml.nes for q 1, 2, 3, 19.

This motion possesses both supersonic and subsonic velocities,
for the line of transition (shown dotted in fig. 1) is obtained if

2 K-1
u + v 2 1Ki

that is,

-2 K-1 2 K-1
-2 -2
i+1 1 K+1 K-1
i -m +-m -
2 K+1
S(25)
or

2 K-- K+1 1
m K+1
2,







NACA TM 1260


It is possible, without difficulty, to construct the character-
istics in the x,y- and u,v-planes. Instead of the equation of
the epicycloid in the u,v-plane, in the case considered a more
complicated equation arises.

In the vorticity problem, along the characteristic there
occurs


ctg a sin a cos a
d F dw = d log
v K-i


(26)


where a is the Mach angle, v the magnitude
0 the angle of inclination of the velocity to
tions (23) yield


2 21o-u


2io-w2 cos2

w2 sin2 B


of the velocity, and
the x-axis. Equa-


(27)


But 0 depends only on H', that is, on
equation (15), the following is obtained:


K- 1


According to


2K
- ( 2-1)K+l
TI


C K-1 2K
2 2 (v2 sn' 0) K'+l (2zio.-W)v2
21-w cos O

(28)


Substituting this value of 0 in the right side of equa-
tion (26) yields the differential equation for the characteristics
in the O,w-plane.

Finally, the question arises whether it is possible to obtain
such vortex motion by transition through a surface of strong dis-
continuity of some other kind but with irrotational motion. This


1-K

K-1 E~-


1-K
K-1 -I-l r'
h K (2 K+1






8 NACA TM 1260


problem may be answered in the affirmative. For on a surface of
discontinuity there must, among others, be satisfied the relations


K-i 2 K P
- + ++
+1 +e+


(29)


p-1
P.


2 (
K+l


K P+ 9 P+
P +06;+P)


where p., p0, and 0+ are t]
of propagation of the surface o:
surface and p_ and p_ are t]
side. The magnitudes p+ and
motion. The magnitude e9 is
cribing the motion and in terms
of discontinuity. Finally, p_

relation p_ = diP -_ where
degree of arbitrariness) magnitl
discontinuity the motion is irr


he pressure, density, and velocity
f discontinuity on one side of this
he pressure and density on the other
oP may be taken from the vortex
found in terms of the elements des-
of the inclination of the surface
and p can be connected by the

t1 is a constant (up to a certain
ude. (At the left of the surface of
otational.)


Inasmuch as


1

P". K + 19
P+-) 0 b


(30)


then


1

K-l -


= GTC (n2-1)


Cl 2

C = Kl GZK "

2
M=-
p,


I-1
2(Al


Inasmuch as


- 0 = u cos 6 + V sin 6
+ + +


where


2K
K+1


(31)


(32)







NACA M4 1260


where 6 is the angle between the normal to the surface of dis-
continuity and the x-axis (the normal is directed toward the
"positive" region); M may be expressed in terms of known magni-
tudes and 6.

By expressing tan 8 in terms of the derivative of x
with respect to r along the surface of discontinuity and using
equations (22), (18), and (19), the differential equation for
determining the surface of discontinuity (K = 1.4) is obtained
after simple transformations:


m A .A/ 71+1e2 -)
S3 M6 (1q2-1)

(33)

where M is expressed in terms of 12 from the transcendental
equation (31) in which the constant G is, to a great extent, arbi-
trary. The velocity in the "negative" region will be determined
first on the surface of discontinuity and then extended on the
negative region by the usual graphical method of Busemann. A model
of the motion about a contour having an angle is obtained.

The surface of discontinuity extends out from the angle and
on passing through the angle the motion reconverts to the rotational
motion herein considered.

A solution analogous to that developed can also be obtained
for the problem with axial symmetry. Taking for the independent
variable the distance r* = r from the axis of symmetry z and
the stream function yields the relations

z Vz

r (34)

z_ 1
or* r*pvr

where vr and vz are the velocity components. As before,

1






NACA TM 1260


Bernoulli's law will have the form


K-1

1 ( 2zV 2) + 2- Kp
2 ('r ^z K-1 p


= i


The equations of Euler give

ovr* r*p
z' rwp
are at


The function X(r*,W)
conditions


can therefore be introduced from the


v = -ax

i W
p r
r* or*


(35)


Equations (34) and (35) now permit writing (the
are dropped)


asterisks on r


Oz aX I
0*- lJ vr


(36)


1
K+1 -

r r/


Vr= [2io


- 2 1


K-1

ic


1
, 2 2

kcw/


By eliminating z from equations (36), a single equation for
the determination of the function X is obtained. Particular


where







NACA TM 1260


solutions, analogous to the solutions in the first case, can be
constructed by seeking X in the form

2K-1
-2
X= r + H()


Translated by S. Reiss
National Advisory Committee
for Aeronautics.


REFERENCE


1. Kochin, N. E., Kiebel, I. A., and Rose, N. V.:
Hydromechanics. Pt. II, 1941, p. 80.


Theoretical






NACA TM 1260


13
12 ---

11 ------ 0 .

10
/ Supersonic



1.2 1.4 1.6 1.8 800
7 00 -








2 00 -Subsonic

1"0 '0 zone

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19





Figure 1.


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