Two-dimensional motion of a gas at large supersonic velocities


Material Information

Two-dimensional motion of a gas at large supersonic velocities
Physical Description:
24 p. : ; 28 cm.
Falkovich, S. V
United States -- National Advisory Committee for Aeronautics
National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


Subjects / Keywords:
Aerodynamics, Supersonic   ( lcsh )
Mach number   ( lcsh )
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


Abstract: The equation of motion of a gas is investigated for large supersonic velocities and a method is shown for the approximate integration, which gives sufficient accuracy for Mach numbers greater than 4.
Includes bibliographic references (p. 9).
Statement of Responsibility:
S.V. Falkovich.
General Note:
"Institut Mekhaniki Akademii Nauk Soiuza, SSR Prikladnaia Matematika i Mekhanika, Tom XI, 1947."
General Note:
"Report no. NACA TM 1239."
General Note:
"Report date October 1949."
General Note:
"Translated by Samuel Reiss, NACA."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003690375
oclc - 76884347
sobekcm - AA00006227_00001
System ID:

Full Text
LIAct4 iV-22,5




By S. V. Falkovich

A large number of papers have been devoted to the problem of
integration of equations of two-dimensional steady nonvortical
adiabatic motion of a gas. Most of these papers are based on the
application of the hodograph method of S. A. Chaplygin in which
the plane of the hodograph of the velocity is taken as the region
of variation of the independent variables in the equations of
motion; the equations become linear in this plane. The exadt
integration of these equations is, however, obtained in the form
of infinite series containing hypergeometric functions. The
obtaining of such solutions and their investigation involve exten-
sive computations. As a result, methods have been developed for
the approximate integration of the equations of motion first
transformed to a linear form. S. A. Chaplygin in'reference 1
first pointed out such an approximate method applicable to flows
in which the Mach number does not exceed 0.4.

S. A. Christianovich (reference 2), in solving the problem
of the flow with circulation about a wing in a supersonic stream,
gave as a first approximation a generalization of the method of
Chaplygin to the case where the region of variation of the velocity
in the hodograph plane lies within a sufficiently narrow ring
entirely inside the circle W pendently an analogous method was proposed by Tsien and von KArmAn
(references 3 and 4). These methods are not applicable for
Mach numbers near unity. In the papers by F. I. Frankly (refer-
ence 5) and S. V. Falkovich (reference 6), approximate equations
of motion are given suitable for the investigation of flows in
passing through the velocity of sound. In his recently published
paper, S. A. Christianovich (reference 7) showed that for Mach
numbers 1.05 < M < 3.5 the equations for the stream function and
the velocity potential may approximately be replaced by the equa-
tions of Darboux with integral coefficients and hence the general
integral obtained in finite form.

*"Institut Mekhaniki Akademii Nauk Souiza, SSR" Prikladnaia
Matematika i Mekhanika, Tom XI, 1947.

NACA TM 1239

The equation of motion of a gas is investigated herein for
large supersonic velocities and a method is shown for the approxi-
mate integration, which gives sufficient accuracy for Mach numbers
greater than 4.

1. Fundamental equation and its transformation. The equa-
tion determining the velocity potential of the two-dimensional
irrotational adiabatic motion of a gas has, as known, the form

(a2 ) 2uv + (a2 v2) 2 = 0 (1.1)
a2 -x by 4y2

To this equation the transformation of Legendre is applied.
As the independent variables in place of x and y are taken
ocp/ox = u and /cp/y = v and in place of the velocity poten-
tial cp(x,y), the Legendre potential is introduced.

@ (u,v) = ux + vy cp(x,y) (1.2)

Equation (1.1) becomes

(a2 v2) 2 + 2uv + (a2 u2) 2 0 (1.3)
uu2 &u yv r2

If the function c(u,v) is determined, then by differen-
tiation of equation (1.2), the coordinates x and y of the
corresponding point in the flow plane are determined.

x = M/ou y = cO/v (1.4)

Pass in the hodograph plane u,v to polar coordinates and
set u = W cos 8 and v = W sin 8; equations (1.3) and (1.4)
then become

a2 _- 2 o2( o2. a2 W2 o/
+ + -= 0 (1.5)
a2W2 e2 aW2 a2W aW

SC e s 6 ae + cos e (1
x = cos w y = sin e + 6

If only supersonic velocities are considered, it is more con-
venient in place of the independent variable W to introduce in
equation (1.5) the new variable

NACA TM 1239

z = 1/ 1 1


Thus z will be small for large Mach numbers and will increase
indefinitely as M->1.

In order to carry out this transformation, from the equation
of Bernoulli the expression of W is found in terms of z. Thus

W2 a2 h2~e
2 K-I 2

(h2 +l 6)

1 1
2 -(-1) M

is replaced by z, from equation (1.7)

2 h2aV2 (l+z2)
+2 2
1+h z


By making use of this expression, pass in equations (1.5) and
(1.6) from the variable W to the variable z; then

z) 2 (h2-1)2 a2)
az2 (1+z2)2 (1+h2z2)2 ae2

X =

y =-

(+z2) (l+h2z2)

(.1z2) (l+h2z2)

3h2z4 + 2h2z2 + h2 2 o
(1+z2) (l+h2z2)z )z

cos 8 + sin 8
s z

sin e -- cos 8




The characteristics of equation (1.9) may be taken in the form

Tf M2



NACA TM 1239

/ f(h2-1)dz e+ (h-l)dz

0 (1+z2) (1+h2z2) = (1+z2) (l+h2z2)

The line of maximum velocity z = 0 in the hodograph plane
will correspond in the plane of the characteristics AX to the
line p A = 0. If the integration is carried out,

A = 8 (h arc tg hz arc tg z) g = e8 (h arc tg hz arc tg z)


2. Investigation of equation (1.9). If equation (1.9) is
referred to the characteristic coordinates XA,

o20 h2 -- 2 2z2 h2z4 (4 ao\ ( ,2.
'ON -- 4(h2 l)z0

From equations (1.11),

v A = 2(h arc tg hz arc tg z) (2.2)

From equation (2.2) it follows that the coefficient of the
equation (2.1)

h2 2 2z2 h2z4
L(p A) = (2.3)
4(h2 l)z

is a function of the difference p A. Equations (2.2) and (2.3)
give this dependence in parametric form.

From equation (2.3) it follows that the function L(V A)
is negative for z2 > (h2 2)/h2 and positive for
z2 < (h2 2)/h2. For z2 = (h2 2)/h2 the function L(p A)
becomes zero. According to equation (1.7) this function corre-
sponds to the Mach number

M = 2/ f13/E = 1.565

S. A. Christianovich showed (reference 8) that for a given
value of the Mach number there is a change in the direction of
curvature of the characteristics in the flow plane. The graph of
the function L(C A) is shown in figure 1.

NACA TM 1239 5

If the right side of equation (2.2) is expanded in a series
in powers of z,

S- = (h2 1)z h z3 + (2.4)
2 3

from which

S-h2 + 1 r )3
2(h2 1) 24(h2 1)3

If this series is substituted in equation (2.3),

h2 2
L(- A) = 2(g 2)

(h2 + 1) (h2 2) + 6 ( ) + .
24(h2 1)2

Set h2 = 6. Then

L(n ) 0.057 ( ) + .

For M > 4, assume

L(g 2)

Equation (2.1) then becomes

_20- 2 (<0 o _0j


Equation (2.5) is the equation of Darboux with integral
coefficient. The general integral of this equation has the form

SX() Y() X'(A) + Y'(p) 2 X(A) Y(p)
X al 5 p P (A3- p)2 (N- 4)3


where X(A) and Y(p) are
The expression (2.6) is the
equation (2.1) for z --0,

arbitrary functions of their arguments.
asymptotic integral of the exact
that is, for M- -o.

6 NACA TM 1239

Equations (1.10) for the coordinates x and y, after
transformation to the characteristics A) becomes on the
basis of equation (1.11)

x -- 1

y +

+ si + -

+ cos -
dk/ 2 z Vd




From equation (2.2) for small values of z,

z= P -A = -
2(h2 1) 10

If this value of z is substituted in equation (2.7) and the
arbitrary functions $(A, g) eliminated from equation (2.7),
with the aid of equation (2.6) the final solution is obtained.

1 T/X" + Y"
x = _

ofir Y- ,
10 (r )3

w 11)2

10 / y"

- 2 X
( )3 /


- 6 X + 12 --- Ycos
(X-H)4 (.-))c/

0 -Y') T A+ 4
- 2 ) cos A +
23 2

X' Y' x-
-6 X + 12 x-YN
(X -)4 (--))5

sin _-_+

Consider the equations for the velocity potential op(W,0)
and the stream function i(W,8):

SPo ( 1) l1 i
aw = F

a p w ow
210 P aw

NACA TM 1239

If the variable z is introduced in place of the modulus of the
velocity W according to equation (1.7) and equation (1.8) used,
these equations are transformed to the form

= (1 + h22) 1- (h2- l)1-" (1

- (l+ hzl) (1+ z2)-1 (h2

For small z,
the equations

2 1-K z*
+ z2) K z

- 1)1 ej

equations (2.8) may be replaced approximately by

K K+1
=-[ 1- K 1- K a*
= W h-1 z -

(h2 1)


If the velocity potential qp is eliminated, the equation for the
stream function is

(h2 1)2 e2

h2 i*
z Oz
z aZ


the characteristics of which have the form

S= (h2 1)z

g = 0 + (h2 l)z

If equation (2.9) is referred to the characteristics, it is trans-
formed into the form

S3e h2 2 =
a + 2(A 2 ) o =o

Setting h2 = 6 gives


+ 0-- -
X g id7< =n


NACA TM 1239

The'equation of Darboux with integral coefficient, which is anal-
ogous to equation.(2.5), is integrated in finite form.

3. Criterions of similarity. By examining equation (1.9)
for Mach numbers near 1 (z-oo) and also for large Mach numbers
(z-->0), certain criterions of similarity can be established
that may be useful in evaluating experimental data obtained in
wind-tunnel tests. For large values of z, equation (1.9) can
be replaced by the approximate equation

o20 (h2 1)2 20 3 3
2 h4z8 e2 z = 0 (3.1)

and for small values of z by the approximate equation

2 (h2 1)2 2h + -2 l = 0 (3.2)
U -2 2O (3.T)

A thin slightly cambered airfoil at small angle of attack is
now considered in a plane-parallel nonvertical gas flow with Mach
number M0 at a large distance from the airfoil. The profile
chord is denoted by 1 and its maximum thickness is denoted by 6.
If it is assumed that MO is near unity, in equation (3.1)

9 = 9*8/B z = z*z0 z0 = 1/M2 1)

and equation (3.1) becomes

2 h2 1 02, 3 0
K -- + 2 = o (3.3)
1 721 h4z"*8 "*2


K = (M2 )3 (3.4)

The number K1 may be called the criterion of similarity in
the sense that two flows having different Mach numbers about two
airfoils of different thicknesses 5 and different chords I but
for which the values of K1 are the same will be determined by
the same "nondimensional" equation (3.3); hence, for the drag
coefficient Cx

C = f(Kl)

NACA TM 1239

In an entirely analogous manner starting from equation (3.2),
a second criterion of similarity valid for large Mach numbers is

K /M02 1
K2 = 1 MO

This criterion of similarity has recently been obtained by Tsien
by a different considerably more complicated method.

Translated by Samuel Reiss
National Advisory Committee
for Aeronautics


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

2. Christianovich, S. A.: Flow of a Gas about a Body at High
Subsonic Velocities. Rep. No. 481, CABI.

3. Tsien, Hsue-Shen: Two-Dimensional Subsonic Flow of Com-
pressible Fluids. Jour. Aero. Sci., vol. 6, no. 10,
Aug. 1939, pp. 399-407.

4. von Karman, Th.: Compressibility Effects in Aerodynamics.
Jour. Aero. Sci., vol. 8, no. 9, July 1941, pp. 337-356.

5. Frankly F. I.: On the Theory of the Laval Nozzle. Izvestia
Akademii Nauk SSSR, Ser. Matematicheskaya, vol. IX, 1945.

6. Falkovich, S. V.: On the Theory of the Laval Nozzle. NACA
TM 1212, 1949.

7. Christianovich, S. A.: Approximate Integration of Equations
of the Supersonic Gas Flow. Trans. File X-121, Ballistic
Res. Labs. (Aberdeen Proving Ground, Md.), March 11, 1948.

8. Christianovich, S. A.: On Supersonic Gas Flows. Rep. No.
543, CAHI, 1941.

NACA TM 1239

Figure 1.

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