On the theory of the Laval nozzle


Material Information

On the theory of the Laval nozzle
Series Title:
Physical Description:
16 p. : ill. ; 27 cm.
Falkovich, S. V
United States -- National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


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


Presents an investigation of a plane streamline flow of a gas in a Laval nozzle in the vicinity of the transition line (sound line) between the subsonic and supersonic velocities. The investigation rests on the basic equations of motion. An analysis of the mathematical solution leads to the conclusion that the point of intersection of the axis of symmetry of the nozzle and the sound line is a singular point. The Frankl results presented are thus obtained by a simpler method.
Includes bibliographic references (p. 15).
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by S.V. Falkovich.
General Note:
"Report date April 1949."
General Note:
"Translation of "K teorii sopla lavala." Prikladnaya Matematika i Mekhanika. Vol. 10, no. 4, 1946."

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University of Florida
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All applicable rights reserved by the source institution and holding location.
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aleph - 003804227
oclc - 123502974
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Full Text

'I2 2 1i t7 3.c/.7




By S. V. Falkovich

In the present paper, the motion of a gas in a plane-parallel
Laval nozzle in the neighborhood of the transition from subsonic to
supersonic velocities is studied. This problem was first consid-
ered by Meyer (reference 1) who sought to obtain the velocity poten-
tial in the form of a power series in the coordinates x,y of the
flow plane. The case of the nozzle with plane surface of transi-
tion from subsonic to supersonic velocities was further considered
in a paper by S. A. Christianovich and his coworkers (reference 2).
For computing the supersonic part adjoining the transition line,
Christianovich expanded the angle of inclination of the velocity
and a specific function of the modulus of the velocity in the power
Series, using the velocity potential and the stream function as the
unknown variables. In a recently published paper, F. I. Frankly
(reference 3), applying the hodograph method of Chaplygin, under-
took a detailed investigation of the character of the flow near the
line of transition from subsonic to supersonic velocities. From
Sthe results of Tricomi's investigation on the theory of differ-
ential equations of the mixed elliptic-hyperbolic type, Frankl
introduced as one of the independent variables in place of the
modulus of the velocity, a certain specially chosen function of
this modulus. He thereby succeeded in explaining the character of
the flow at the point of intersection of the transition line and
the axis of symmetry (center of the nozzle) and in studying the
behavior of the stream function in the neighborhood of this point
S by separating out the principal term having, together with its
derivatives, the maximum value as compared with the corresponding
corrections. This principal term is represented in Frankl's paper
in' the form of a linear combination of two hypergeometric func-
tions. In order to find this linear combination, it is necessary
to solve a number of boundary problems, which results in a complex

In the investigation of the flow with which this paper is
concerned, a second method is applied. This method is based on
S the transformation of the equations of motion to a form that may
be called canonical for the system of differential equations of

*"K Teorii Sopla Lavala." Prikladnaya Matematika i Mekhanika.
Vol. 10, no. 4, 1946, pp. 503-512.

NACA TM No. 1212

the mixed elliptic-hyperbolic type to which the system of equations
of.the motion of an ideal compressible fluid refers. By studying
the behavior of the integrals of this system in the neighborhood of
the parabolic line, the principal term of the solution is easily
separated out in the form of a polynomial of the third degree. As
a result, the computation of the transitional part of the nozzle is
considerably simplified.

1. Fundamental equations. The equations of the two-dimensional,
steady, nonvortical motion of an ideal gas in the absence of friction
and heat conductivity have the form

T y- = =o (1.1)

W2 + P (1.2)
2 X-1 P X-1 Po

where u and v are the components of the velocity along the x
and y axes, p is the density, p is the pressure, W = /u2 + v2
is the magnitude of the velocity, X = c-/c PO and po are
the density and pressure of the gas at rest.

Equations (1.1) represent the condition of the absence of vor-
tices and the equation of continuity. Equation (1.2) is Bernoulli's
equation for adiabatic motion for which

Po P (1.3)

For the velocity of sound a

a2 = X (1.4)

From equations (1.2), (1.3), and (1.4), the following equation
is derived:


P =po 2,2.) (1.5)

NACA TM No. 1212

(a02 = Xp/p0 is the velocity of sound in the gas at rest) from

d PO W
" T P a


From equation (1.1), it follows that there exist two functions:
the velocity potential cp(x,y) and the stream function (x,y),
which are determined by the equations

dap = udx + vdy

In place of the velocity components
coordinates, setting u = W cos 0 and v
the angle between the velocity vector and
stituted. Equations (1.7) are solved for

d = -- (- vdx + udy)


u and v, the polar
= W sin 8, where e is
the x axis, are sub-
dx and dy, thus

cos 0 P0 sin 8
d- W pW dW
dx = dcp-- d

sin 0 PO cos
dy- dcp+ p-


If x and y as well as W and 0 are considered as func-
tions of the variables y and *J, then dx and dy must be
total differentials, so that the following equations must hold:

S/cos 9\ o0 sin 6
Tp P

S/sin )( oa cos e

In carrying out the differentiation,in taking account of the
fact that according to equation (1.5) in which p0/p depends
only on the magnitude of the velocity W, and in making use of
equation (1.6), the following equations are obtained:

b0 cos 0 W 0 o s e 0 sin 8 :2 w
sin e + = + cos e "

o sin 0 w PO sn e PO sin 8 (
cos sin T' w W ap 1 2 a
bJI~~~ k ~/ a~-P ac

NACA TM No. 1212

By solving these equations for the derivatives 90/~qc and

-e P W =
TC- Pow F*

oe PO a2-W2 _
+p iP O= 0

This system of differential equations will be of the elliptical
type if the magnitude of the velocity W is less than the velocity
of sound and will be of the hyperbolical type for supersonic velocities.

The new function iT is considered instead of velocity
is related to W in the following manner (reference 3):

W and


S= aW dW /3

Equations (1.9) then assume the form

+ b 0
aJ+b(rl) b o

i b( q =
,1 -b----- =o0


P V a a2
b(ij) =~2 a


as a result of (1.10), is a function of the variable T.

Equations (1.11) are the fundamental equations for the inves-
tigation of two-dimensional, nonvortical motion of a gas when the
velocity of the flow passes from subsonic to supersonic velocity.
In some cases,it is more convenient in these equations to sub-
stitute 8 and n as the independent variables and take cp and
I as the required functions. After this transformation, equa-
tions (1.11) assume the form

T + b T) = o

i- b(l) O = 0

2. Investigation of variable T. The variable T deter-
mined by equation (1.10) is considered in more detail. For



NACA TM No. 1212

computing the integral entering this equation, the square of the
velocity of sound is

2 k+1 2 k-1 2
a =-2- a


In substituting the preceding equation in (1.10)

(e1 ii r 1- h i2 /3
The integration results i

The integration results in

W 2
a X = h

l+h h2-X2
-- 2
1-h h
V 2-?

1 =

By expanding equation (2.3) in a series

T1 = (h(h2-1) /3



From equation (2.3), it follows,

that 1 > 0

for A< 1

i- < 0 for X > 1, that is, in the plane of the variables e and
T, the region lying in the upper half-plane will correspond to the
region of subsonic velocities and the region lying in the lower
half-plane will correspond to the supersonic velocities. The line
of transition from subsonic to supersonic velocity will correspond
to the line T = 0, that is, the axis of abscissas. From equa-
tion (1.10), the value of the velocity W = 0 in the plane e,r
corresponds to an infinitely distant point. For A > 1, equa-
tion (2.3) assumes the form

_= (2/3

(h arc

g 2
tg F-

arc tg h 2
h -I/

characteristics in the plane of the hodograph of the
for two-dimensional, nonvortical motion of the gas are
epicycloids (fig. 1), the equations of which are (refer-




known as
ence 4).


NACA T3 No. 1212

S= C & h arc tg h-2_ arc tg h
Vh-? Vh2 X2

Because for a point transformation characteristics go over
into characteristics, the following equations of the character-
istics in the plane of the variables 8 and n are found by using
equation (2.5):

e = (- + C (2.6)

from which it follows that the characteristics assume the form of
semicubical parabolas with the cusps on the axis of abscissas
(fig. 2).

3. Differential equations of motion of a gas in neighborhood
of transition line. The flow in a Laval nozzle near the line of
transition from the subsonic to the supersonic velocities is con-
sidered. This line is hereinafter designated the sound line.

If a straight line perpendicular to the axis of symmetry of
the nozzle is directed away from the axis, it will intersect the
streamlines with constantly increasing curvatures and will there-
fore encounter particles of the gas having constantly increasing
velocity. The sound line will therefore be a curve that is con-
vex toward the supersonic velocities1 with vertex on the axis of
symmetry (fig. 3). The point of intersection of the sound line
with the axis of symmetry is, according to Frankl, denoted as the
center of the nozzle.

In the plane of the variables cp and 4, the region of flow
is transformed into a strip the width of which is determined by
the amount of gas flowing through the nozzle (fig. 4).

The point of origin of coordinates in the cp,i plane cor-
responds to the center of the nozzle in the flow plane.

The determination of the flow reduces to finding two functions
T=--T(p,i) and 9=0()cp,) that satisfy equations (1.11). Because
the flow is to be symmetrical with respect to the streamline
* = 0, it is necessary that the required functions satisfy the

1When the streamlines have points of zero curvature, the sound
line will be a straight line perpendicular to the axis of symmetry;
this case was considered by S. A. Christianovich (reference 2).

NACA TM No. 1212

n(cp,4,) = T(cpq-*) e(cp,I) = 0e(cp-) I(o,o) = 0

which are based on equations (2.2).

The solution of equation (1.11), in the form of a power series
in the variables cp and 4, takes into account equations (3.1)
2 2 3 2
S= 1cp + a2p + a3i i- acp + a5c .

6 = bfpy 3 b2cpW + b3 3+ b4C9p+ .. (3.2)

from which it follows that if the flow in the neighborhood of the
origin of coordinates is considered, that is, if cp and are
assumed to be small magnitudes, the following equations may be
obtained from equation (3.2):

I = 0(c) e = (cpW) = O0()

e= 0() = 0(1) 0(cp) (3.3)

With the use of equation (2.1) and the notations introduced in
equations (2.2), equation (1.12) for the function b(T) may be
reduced to the form

0o /h(1-12)
b(n) = P- \h )
P 'V(h2-^2)T

In accordance with equation (2.4), the following equation is derived:

Po 1/3 k k-1 1/3
b(0) ) (xl)3 (k)l)

In taking account of the order of smallness of all terms
entering equations (1.11), it is concluded that near the origin
of coordinates the system of equations (1.11) may be replaced by
the following equations:

NACA TM No. 1212

Sb(o) 0-o

By setting

b(0)W =W, the final result is

n e 0
two 0

p be
T1 C-n 0

where, for simplicity, the bar over has been dropped.

4. Investigation of flow in neighborhood of center of nozzle. -
It is evident that the functions

6 =A2p*- A3 %3

T = Acp -2 2

where A is an arbitrary constant, are integral of the system of
equations (3.4), and satisfy conditions (3.1).

The significance of the constant A will be explained. From
the second equation (4.1), ) = Acp along the axis of symmetry of
the nozzle ( 0 = 0). Differentiation results in

A = dr = d1 dW dx
dp dW dx dp

Furthermore, by using successively equations (1.10) and (2.1)

S-1 a2-W2 _ha 1-?_
dW aW 7 -(h2)_2

Moreover, along the line 4 = 0

dx 1
6=p w

dW =u
dx ox

Hence,for A the following relation is obtained:

1-7?2 ux
1th2 ) x )y =0

- S#0 T6



A( h-
A= W2

(x 2l)



NACA TM No. 1212

where to obtain the last result, equation (2.4) was used.

The value of A is thus proportional to the value of the
derivative of the velocity at the center of the nozzle.

It is assumed that Zu/ox > 0 so that A will be a negative

Along the sound line, T = 0. Hence, according to the second
equation (4.1), the following equation of the sound line is

A= 2 (4.3)

that is, in the plane cp,4', the sound line will be a parabola.

From equation (3.4), the differential equation of the char-
acteristics has the form

\dIJ -

By substituting the value of i from equation (4.1)
()2 A22 Ap

In the integration of this equation, set
Acp= 2 A* = x

The equation then assumes the form

(1-22-2e xy 2 2 or 1-2y2-2y = y

After separating the variables and integrating, the following
equations of the characteristics are obtained:
2/3 1/3 2/3 1/3
x(y+l) (2y-1) = C x(y-l) (2y+l) = C

NACA TM No. 1212

In order to obtain the characteristics passing through the
origin of coordinates, set C = 0. Thus the variables qp and *

A.2 A(4)
CP = 2 44)

Hence, the characteristics passing through the origin of coor-
dinates in the cp,$ plane are parabolas tangent to each other at
this point and tangent to the sound line (fig. 4). The origin of
coordinates will therefore be a singular point of the integral of
equations (3.4) determining the flow in the nozzle.

In considering the character of this singularity, it is evi-
dent from figure 4 that the characteristics and the sound line
divide the neighborhood of the center of the nozzle into six
regions. It shall be investigated how the neighborhood of the
center of the nozzle is transformed in the plane of the variables 0
and TI by the integral of equation (4.1). By eliminating from
equation (4.1) the variable cp, the following cubical parabola is
used in determining the stream function:

A33 + 3An 30 = 0 (4.5)

This equation has one real root if its discriminant
8 = 902/4 + T3 > 0 and three real roots if 5 < 0. Because the
point (cp = 0, = 0) corresponds in equation (4.1) to the
point (9 = 0, T = 0), the equations of the characteristics cor-
responding to equations (4.1) are in accordance with equation (2.6).

S02 3 = 0

Thus regions I, II, and III of the plane are transformed into the
same region of the plane 0,6T lying between the characteristics


Furthermore, the streamlines ( = & q correspond, as seen
from equation (4.5), to the straight lines in the 0,,T plane.

2A 2

NACA TM No. 1212

The transformation of the neighborhood of the nozzle in the
G,I plane will thus have the form of the folded surface shown in
figure 5. The corresponding regions in figures 4 and 5 are denoted
by the same numbers.

In order to compute the streamlines in the flow plane, equa-
tions (1.8) are used in which dW is set equal to 0, after which
they assume the form

dx = d cp

dy sin

By substituting for
magnitude of the velocity
function of the variable

dx = w cos (A A2q dc

its value from equations (4.1), the
is, according to equation (1.10), a
Thus along the streamlines I = q

dy= sin
dy = sin (2K-q

A2q) dp

Integration results in

r 1
x= = Cos


k 3

- AZqcpdcp

0 W1m sin +3 A2 dcp


where H is the width of the nozzle at the critical section.

In equations (4.6), set according to equations (4.1)

A2 2
T = Acp q

The computation of the integrals in equation (4.6) reduces,
evidently, to the computation of the two integrals of the type


12 =

sin A2qp
W(Tl) d-


cos A2q d

12 NACA TM No. 1212

with the aid of which x and y are expressed as follows:

33 33 A 333
x = Il cos A -- + Ia sin A 3 y = 2 sin "A I1 cos A,


5. Nozzle with surface of weak discontinuity. The case in
which weak discontinuities are formed along the Mach lines issuing
from the center of the nozzle is here considered. For this inves-
tigation, it is necessary and sufficient that the derivative
(ou/ax)y=0 possess a discontinuity at the center of the nozzle
(reference 3). It is assumed that both values [(uox) x and
[(ou/ox)y=-O are positive.

From equation (4.1), it is evident that the magnitude A will
have the value A = A1 in the regions VI, V, and IV (fig. 4) and
the value A = A2 in the region III where, according td equa-
tion (4.2), A1 < 0 and A2 < 0.

From equations (4.1), it is concluded that in the regions VI,
V, and IV

2 A13 3 A2 2
eA = A, = Al 2 (5.1)

and for region III

SA23 3 A22 2
9 = A222 =2 (5.2)

According to equations (4.4), the equations of the character-
istics separating the regions IV and V from regions I and II and
the equations of the characteristics separating regions I and II
from region III have the forms

Al 2 A2\2
P=- 4 CP=-- (5.3)

Substituting the first of these equations in equations (5.1)
and the second in equations (5.2)

NACA TM No. 1212


e =+ A213
*~ Tr

TI = -

rp = +

A2 2

In order that the flow in the nozzle has no discontinuities,
it is necessary to determine 8 and q in regions I and II from
equations (3.4) in such a manner that the characteristics condi-
tions, equations (5.4) and (5.5), are satisfied. In order to
integrate the system, equations (3.4), set

1 = (f 92



where f and g are functions to be determined.

For this substitution, equations (3.4) are transformed into
a system of ordinary differential equations with the independent
variable t = cp/2

2f-2tf'-g' = 0

ff'r3g-2tg' = 0


By the elimination of G'

g = [4tf (f + 4t2) f']


By differentiating equation (5.8) and substituting the result in
the first equation (5.7), a differential equation of the second
order for determining f is obtained

(4t2 f) f" f'2 2tf' + 2f = 0 (5.9)

From equations (5.6), (5.4), and (5.5), it follows that the
boundary conditions for the function f will be

f = for

t A1

f = A for t =


In order to integrate equation (5.9), it is written in the form



NACA TM No. 1212

f'+t- = 0

(The solutions 2tf' f = 0, that is, f = cq/ which do not
satisfy equations (5.10) are eliminated.) In carrying out the

f'+2t 1 or1 2
or f- f
2tf'-f 2c1 2(t-cl) t-cl

that is, the integration of the linear equation results in

f = 4Clt 8c12 + C2 1 (5.11)

The boundary conditions, equations (5.10), which the obtained
solution equation (5.11) must satisfy, have the form: f = fl for
t = t1 and f = f2 for t = t2 where it is easily seen from
equation (5.10) that the points (tl, fl) and (t2, f2) lie on the
parabola f = -4t2 and that t < 0 and t2 > 0. Hence, in order
to satisfy the boundary conditions, it is necessary from the family
of parabolas equation (5.11) to choose the parabola passing through
(tl, fl) and (t2, f2). Upon satisfying these conditions

tl2_tlt2t22 2 16(tl-t2)2 (tl+2t2)2 (2t t2)2
cl = 3(t+ t2) 2 =

It is necessary that along a streamline the velocity in the
flow direction should increase monotonically, tat is,that T
should decrease monotonically. Because T = f according to
equation (5.6), f' < 0 must be in the range tI < t order to obtain this result, it is necessary that c2 < 0. This
condition is possible only for 2t1 + t2 < 0 and tl + 2t2 > 0
when in accordance with equations (5.10), the following condition
is obtained
Al I A2 --

for which a flow without discontinuity is possible.

Translated by S. Reiss
National Advisory Committee
for Aeronautics.

NACA TM No. 1212 15

1. Meyer: Uber zweidimensionale Bevegungsvorgange in einem Gas das
mit bberschallgeschwindigkeit stromt. Forschungshefte, Nr. 62,

2. Levin, Astrov, Pavlov, and Christianovich: On the Computation
of Laval Nozzles. Prikladnaya Matematika i Mekhanika, vol. VII,
no.'1, 1943.

3. Frankl, F. I.: On the Theory of Laval Nozzles. Izvestia Akademii
Nauk SSSR, Ser. Matematika, vol. IX, 1945.

4. Kochin, Kibel, and Rose: Theoretical Hydrodynamics. 1941.

NACA TM No. 1212

Figure 2.

I /
Figure 4.
Figure 4.

Figure 1.

Figure 3.

Figure 5.

0 m

o m0

SI .--I
O r

0* A o
to 2 .

o *5
o m
0 a !

g I




9 Q

to 11 0l

04 0 0
Ol r4

0 1 0

M14M P 943
+ a a
0> 0

1- oD flJ
6 4> +

4 5

0 0a

0ri 4P
0W R

0 0 u

S rd

0 0i I d
5 5 ai