A class of de Laval nozzles


Material Information

A class of de Laval nozzles
Physical Description:
15 p. : ill. ; 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:
Aeronautics   ( lcsh )
Supersonic nozzles   ( lcsh )
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


A study is made of irrotational adiabatic motion of a gas in transition from subsonic to supersonic velocities. A shape of de Laval nozzle is given, which transforms homogeneous plane-parallel flow at large subsonic velocity into a supersonic flow without any shock beyond the transition line from subsonic to supersonic regions of flow.
Includes bibliographic references (p. 13).
Statement of Responsibility:
by S.V. Falkovich.
General Note:
"Technical memorandum 1236."
General Note:
"Report date October 1949."
General Note:
"Translation Institut Mekhaniki Akademii Nauk Siuza, SSR Prikladnaia Matematika i Mekhanika, Tom XI, 1947."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003692425
oclc - 77073704
sobekcm - AA00006226_00001
System ID:

Full Text
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By S. V. Falkovich

A study is made herein of the irrotational adiabatic motion of
a gas in the transition from subsonic to supersonic velocities. A
shape of the de Laval nozzle is given, which transforms a homoge-
neous plane-parallel flow at large subsonic velocity into a super-
sonic flow without any shock waves beyond the transition line from
the subsonic to the supersonic regions of flow. The method of
solution is based on integration near the transition line of the
gas equations of motion in the form investigated by
S. A. Christianovich (reference 1).

1. Fundamental equations. A plane, steady, irrotational,
adiabatic flow of a gas is considered. In this case, the equations
of motion, as is known, have the form

OPU ov 0
+ =

2 p

av au

K-1 PO


p density of gas

u,v components of velocity along x- and y-axes

p pressure

W absolute value of velocity

K adiabatic exponent

Subscript 0 denotes condition of gas at rest.



*"Institut Mekhaniki Akademli Nauk Siuza, SSR" Prikladnala
Matematika I Mekhanika, Tom XI, 1947.

NACA TM 1236

From equations (1.1) there exist two functions, the velocity
potential cp(x,y) and the stream function 4(x,y), determined
by the equations

dP = u dx + v dy

If u = W cos 8 and

d* = p- (-v dx + u dy)

v = W sin 8,


between the velocity vector and the x-axis,
equations (1.3) and are solved for dx and

Sco dP! sin d
w P W



ae 6 is the angle
are substituted in

0 cos 0
p w


The concepts x and y and
variables W and 0; then

4p and are functions of the

dq = dW + d

dI = dW 4

a* d

If these expressions are substituted
tions (1.4),

dx = C aP dW +
\w VT vW aw

dy = sin e a
\w aw

p cos 0
PO v


for d0P and d4I in equa-

cos acP O sin 0 8
\ w p W 80/

(sin 0


cos 0



In order for dx and dy, determined from equations (1.5), to be
total differentials, it is necessary and sufficient that the fol-
lowing qualities hold:


NACA TM 1236

6P_ P0 sin a\ /COB arp 0 sin e
wp w w \w p W

6C PO cos e 6a
w + p W X)

S/ine a P cos e
= \ w e p W



From equation (1.2) and the condition of adiabatic flow,

d /PO\ PO W
W C P a2


a velocity of sound

If the differentiation In equations (1.6) is carried out,

6cP PO


-= (1-M2)1
W i w Te

where M = W/a is the Mach number.

In equations (1.7), as the independent variable, in place of
the velocity W a new variable a (reference 1) is introduced,
which is connected with W by the relation

- a

-W2 dW

Equations (1.7) then assume the form

- =- ao


K(s) =- 2


inasmuch as equation (1.8) is a function of s.

a (cos s

- (ine
ae W w

a U a0'




NACA TM 1236

The system of equations (1.9) in the regions where the flow
velocity is subsonic, W < a, is of the elliptic types Hence,
any solution qC= cp(e,s) and = *(e,s) of equations (1.9) rep-
resents two functions analytic in the variables s and e up to
the line of transition to the region of supersonic velocities.

If the Jacobian of the transformation of the region in the
plane e,8 on the plane qP,%

does not become zero over a certain segment of the transition line
and therefore also in a certain region on the supersonic side, the
functions ((e,s) and 1(6,s) may be analytically continued across
this segment into the supersonic region of the flow. Hence, the
flow in the subsonic region determines the flow in the supersonic
region near the transition line. Equations (1.9) retain their
meaning also for supersonic velocities. For, if W > a, s will
be purely imaginary and K negative. .Setting s = ii and
K = -K in equations (1.9) gives

Thus, if in the solution P(e,s) and L(K,s) of equations (1.9)
determining the flow in the subsonic region, s is set equal to iT
and the real parts of the expressions thus obtained are taken, the
solution of equations (1.11) determining the flow of the gas in the
supersonic region near the transition line is obtained.

2. Investigation of equations (1.9) near the transonic line
V = aE. From equation (1.8), it follows that in the plane of the
variables 6,s the upper half-plane s > 0 will correspond to
the region of subsonic velocity and the axis of the abscissa
s = 0 to the line of sound velocity. It is therefore necessary
to consider the behavior of the function K(s) for small values of
the variable a.
Equation (1.2) is represented in the form

W2 (+l)a*2 M42 (2.1)
2+(6-1) M2

NACA TM 1236

Substituting this value of W in equation (1.8) gives

(h2-l)t2 dt

h2 2 +1)

Computing the integral gives

5 = log

-[/(h l


If the equation is expanded in a power series in t,

h 2-1 3

h4-1 t5

h6-1 t7

t ala1/3 + a3e/3 + age5/3 + .


al h 1

Further, from equation (1.2) and the adiabatic
simple transformations,

PO0 P 1-
P 0. (

condition after


h2 K -

Substituting this value in expression (1.10) gives for K(s)



= rt
0 I



+ .

(t = 1-2 ,

NACA TM 1236

Substituting in equation (2.5) the series for t (equation 2.3)

bsl/ b 81/ 3 + b953s/3 + 8. +(b = f h )
1 (a) 13 5 5/5 1 1

It follows from equation (2.6) that for a velocity near that of
sound, that is, for small s, consideration may be restricted in
series (2.6) to the first term by setting 4TK blsl/3. The
fundamental equations (1.9) then assume the form

= b~1/s a C b 81/3 s (2.7)

whence for the functions *(6,s) and p(e,s) there is obtained

+- +0 L =a 0O (2.8)
2 2 39 a ae2 2 39 se

Having determined from equations (2.8) the velocity potential
q(9,s) and the stream function 4(0,s), by equations (1.5) the
coordinates x and y are found in the flow plane. It is easily
seen that equations (1.5) can be represented in the form

dz = 1 sc e o 6 sin e ds + cos sin d
w as P as C ae P Lae j

SP 9 de be (si a ]
dy = sin + cos e ds + in e + -- cose 6 de
as P a ae P ae/

If, however, we pass from the exact equations (1.9) to the approxi-
mate equations (2.7), the expressions (2.9) cease to be exact dif-
ferentials. Hence, simultaneously with the passing from equa-
tions (1.9) to equations (2.7) it is necessary to introduce

NACA TM 1236

instead of the relation (1.8) between a and W a new relation
such that the expressions (2.9) remain exact differentials. In
order to obtain this relation in equations (2.9) are substituted the
values of the derivatives of the function P from equations (2.7)
after which the condition that dx from equations (2.9) is a
total differential assumes the form

a 6- 1/3co e 0 a
ble rcos e sin 1
*e 6_\1e 0 p s

= le /cos 9 c+ in

Carrying out the differentiation and making use of the first of
equations (2.8) satisfies this condition if the following equations
are satisfied (the expression for dy likewise then becomes a
total differential):

P 1/3 d 1/3 .1 d
PW 1 d \W da

2 3/2
Setting = 3/2 gives

P= ()1/ b b) 1 d 1/3b d P (2.10)

Thus, the equation determining 1/W is obtained, namely,

d-2 1 0 (2.11)

The functions satisfying this equation are called Airy functions.
Tables of these functions have been computed by V. A. Fock
(reference 5). Thus

-= C1 u(j) + C2 v(I) (2.12)

NACA TM 1236

where u(Ti) and v(rT) are two linearly independent tabulated
integral of the equation (2.11). The constants of integration are
determined from the conditions

(ii) 1
0 =j*



PO a7
blp*a* 4

where the latter relation is obtained from equations
(2.6). After computation,

1 = [yv(O) v'(O)

(2.3) ld

(2.10) and

2 [u',(o) yu(0)]


where from reference 5,

u(0) + iv(0) =

u'(0) + iv'(0) =

2 Ahf in
exp --
34/3 (2/3)

2 41e -in
4/3 exp
3 r(4/3)

From the first of equations (2.10) and (2.12),



c u'(T() + C v' ()
bl + C2
ClU(r) + C2v(?)

The expressions (2.12) and (2.14) thus found for
must be substituted in equations (2.9).


1/W and P/P

3. Determination of flow in feed part of de Laval nozzle. -
The shape of the de Laval nozzle was determined such that the
distribution of the velocity over the section tended, with increasing
distance from the critical section upstream of the flow, to a uni-
form flow with a certain subsonic velocity WO near the velocity

NACA TM 1236

of sound. It was necessary that the
x-) have a horizontal asymptote
nitude H being determined from the
velocity W0.

walls of the nozzle for
y =B*H (fig. 1), the mag-
amount of gas flow and the

In the plane of the variables 8,s, there corresponds to the
infinitely distant section in which the velocity is constant and
equal to WO, the point A with coordinates 8 = 0, a = s0. To
the lines of flow there correspond in the e,s plane a bundle of
curves issuing from the point A. In order to obtain a solution
of equations (2.8) having the stated properties, in the upper half-
plane of e,s bipolar coordinates are introduced (fig. 2).

2 2
9 +(e+so)
= o 82+( 0o)2

Thus the lines
with centers on the
a family of circles
point A (fig. 2).
are, respectively,

S= are tg e22
2 +2-80 2


a = constant constitute a family of circles
a-axis and the lines 0 = constant constitute
with centers on the 8-axis passing through the
The equations of the families of these circles

e2+(s+s0 cth

a)2 0= 2

(e-s0 ctg p)2 + s2 802
sin2 P

The first of equations (2.8) transformed into bipolar coordinates
has the form

2 _+ 1 (l+ch a cos P)L + sh a sin =
2 2 3 sh a(ch a+cos B) B +


W= (ch a + cos p) /6 (ap) (3.3

Equation (3.2) is reduced to the form


S. cth aC 1
C)2 3 i! +- 6 X




in which the variables are separable.

NACA TM 1236

In seeking a solution of equation (3.4) of the form
X= X(a) Y(B), the ordinary equations for X(a) and Y(P)
are obtained:
d2Y 2
d2y + n2Y = 0

d2X cth a dX +
d + 3 d+ -36
da2 3 a. 36

n2)X = 0

where n is an arbitrary constant. If n is set equal to

d2X cth a
2 3

dX 1+ x=
da 31=
do. 36

The first of equations (3.5) gives Y = C1 + C20 and the second,
by the substitution t = ch a, reduces to the hypergeometric


t(l-t) 1d t) dX 1 X = 0
2 2 6 dt 144

the general integral of which can be represented in the form

X t-1/12 F 7 1 1
i 12' i, + C4F ( 72 1 ,

If C1 = C4 = 0 and considering equation (3.3), the required solu-
tion of equation (3.2) is in the form

S ch a. + cos
60 = ch a

F 1 1
7F 12 12 mh2
V" i" ch" a/

Returning to the initial variables e and s according to
equations (3.1), after simple transformations

d2Y 0



NACA TM 1236

S/ 2800 6 /, e2 +(-sF 0 2+(s+so)22
"; 2 1 2 l\2 / 12 l (e2.82+802)2

tan-'l 29--0


This solution corresponds to the flow of gas in a nozzle having the
shape shown in figure 1. Such flow cannot, however, be continued
into the region of supersonic velocities. In order that a certain
subsonic flow having a straight streamline may be continued into
the supersonic region, it is necessary and sufficient, as has been
shown by F. I. Frankly (reference 2) (see also S. A. Christianovich,
reference 3, ch. V.), that the stream function 1(8,s) have on
the transition line the form

*(e,o) = A1e/3 + A9 3/3

+ Aso5/3
c A5B

- .


The solution (3.8) does not, however, satisfy this condition inasmuch
as on the transition line (a = 0) it has the form

(eo0) =- ( 1/6

F 1,, 1 t -n 21 so

In order to continue the flow with the stream function of equa-
tion (3.8) into the supersonic region, it is necessary to add to
equation (3.8) the solution of the first of equations (2.8) satis-

fying the condition *i(e,0) = e /3. Such a solution, as is shown
in reference 4, has the form

2 = s 2/3

A3*5 + 3AT1 3 = 0

whence setting A = 3/3, we obtain
whence setting A = 3 we obtain

(1 -/ ( 6 2 -+ 2 + (2)


NACA TM 1236

Hence, in order to construct the flow in the de Laval nozzle
having the shape shown in figure 1, it is sufficient for the stream
function 4(e,s) to assume the form

(e0,s) = AO 0 + A11 + A33 (13 = 0) (3.11)

where AO, Al, and A3 are arbitrary constants and the functions
0 and *1 are given by equations (3.8) and (3.10).
Equation (3.11) for *(e,s) is obtained if in the expansion
(3.9) the first two terms are retained. With the values of the
constants A0, A1, and A., a nozzle may be constructed sufficiently
near the given nozzle of the shape under consideration.
The equations determining the functions 40 and *1 in the
supersonic region will be determined. In the expression (3.8) for
a = 0, the argument of the hypergeometric function attains the
value unity and for the supersonic velocities, that is, for imagi-
nary a, although remaining real, becomes greater than unity.

The formula giving the analytic continuation of the hyper-
geometric series (reference 6) is used

F (a,b,c,t) = r(c) r(c-a-b) F (a,b,a+b-c+l, l-t) +
r(c-a) r(c-b)

r(c) r(a+b-c) (l-t)c-a-b F (c-b, c-a, c-a-b+l, 1-t)
r(a) r(b)

which in the case considered has the form

1 7 ( 1 7 2 ,
S 12' = (11/12) (5/12) 2 12' 3

r(-l/3) 1/3 /5 11 4 t)
r-pl7) y 71zy(l-t) F -, -, 1-t
(1/12 /12) 12 3

and the characteristic coordinates e= 9 is and p = e + is
are determined. After computations, the following equations are

NACA TM 1236

/ 2 /6
2sP7o /
"-4 s~

(11/12) r(5/12)

r ri/3 r so(W-X) 2/3 F
r r(-/1) X-V92 )
r(1/12) 7/12) 2
Ap4e0 /

5 11 4
12 l2 S


(A1+82) 2

are tg 2s(+-)

l (=,7) 1(i /3

(3qTx+,g+ q4X-F)



These expressions determine the flow in the regions 1 and 2 between
the transition line and the characteristic passing through the center
of the nozzle and directed upstream of the flow (fig. 1). The fur-
ther computation of the supersonic part of the nozzle can be carried
out by the method of characteristics.

Translated by Samuel Reiss
National Advisory Committee
for Aeronautics.


1. Christianovich, S. A.:
Subsonic Velocities.

Flow of a Gas about a Body for Large
Rep. No. 481, CAHI, 1940.

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

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

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

5. Fock, V. A.: Tables of Airy Functions. Nil, No. 108, NKEP, 1946.

6. Whittaker, E. T., and Watson, G. N.: Modern Analysis. The
Macmillan Co. (New York), 1943.

1 7 2
12' 12'3

(- -0-- o )
2 (P )2
~o (

NACA TM 1256

Figure 1.

NACA TM 1236

0 = constant o

Figure 2.

MACA-Langley 10-5-49 950


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