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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
TECHNICAL MEMORANDUM NO. 1147
GENERAL CHARACTERTSTICS3 OF THE FLOIl THHOUCH
U03ZLE3 AT iEAR CRITICAL SPEEDS'
.3y R. Sauer
The characteristics of the position and form of
the transition surface tirou.gh the critical velocity
are computed for flow thr--UC.. fl:t and round nozzles
from subsonic to supersonic velocity. Corresponding con-
siderations were carried out for the flow about profiles
in the vicinity of sonic ,velocity.
'rhile useful retho.ds of calculation exist for pure
subsonic :ndl for pure sunersor.ic flows of comouressible
gasez difficultie arise in t-h i' athematical treatment
of mi:-edi flows with 3ubtonic and sup.ersonic raies.. Such
mixed flows exist: (1) in the tira!.sition from subsonic to
sutpersonic speeds in Lavai nozz .-s, (2) about rocfiles
in a flow at high su-bsonic speed ,:. the apple. ar:nce of
local sup-,r--onic ranges, (5) in fro-it of blunt bodies
in a flow &t supersonic 'velocit-' witih a locel subsonic
regicn behind the comnor--ssioa shcci: -in the vicinity of
the stagnation point.
The present report ta-es vui the firs; as the
simplest of the three problems nar.;ed. It is a fact that
for this case a rough view of flov conditions suffi-
cient for many problems cen be obtained by che methods
of hydraulics, that is, the nozzle is considered a
flow tube and the magnitude of the velocity is considered
"Allgemeine Eigenschaftan der Stromung durch
Disen in der Wfho der kritischen Geschwindigkeit,"
FB 1992, Zentrale fur wissenschaftliches Berichtswesen
der Luftfahrtforschung des Generalluftzu&-mnistere (ZWB)
Berlin Adlorshof, Sept. 25, 1944.
NACA TM No. 1147
constant in every cross-sectional plane. The critical
velocity (flow velocity: = sonic velocity) is then reached
exactly in the minimum cross section of the Laval
nozzle. In the strict two-dimensional treatment,
however, a curved line (critical curve) is obtained for
the passage of the velocity through the critical value
in the plane of the longitudinal section of the nozzle;
it begins at the nozzle wall in front of the minimum
cross section and cuts the nozzle axis downstream of
the minimum cross section. (See -fig. 1.)
In various reports, especially those of Th. Meyer (1),
G. J. Taylor (2), H. Gbrtler (3), and Kl. Oswatitsch and
H. Rothstein (4), expansions in power series for the
determination of the critical curves are given. In
what follows, general statements on the position and
curvature of the critical curve for a given nozzle are
obtained from such expansions in power series by
breaking off the series after the first two terms. The
investigations are concerned with flat and round nozzles
and provide sufficiently accurate results for practical
purposes for nozzle curvatures that are not too sharp.
Corresponding expansions in power series are
applied, in conclusion, to the flows through flat
nozzles with curved axes. In this way, information is
obtained on the variation of the critical curve for
profile flows with local supersonic regions.
II. POTENTIAL EQUATION
The nozzle axis is selected as the x-axis and the
origin of the coordinate system is placed at the point
on the axis at which the critical velocity is first
reached, (See fig. 1.) With the Lhyothesis of non-
vortical flow and perfect flow of a perfect gas with
constant specific heats c c ( = c /c ) the potential
F- (a2 2) + (2 2) 2 7 +.oa2v= 0 (1)
NACA TM No. 1147 3
holds. In this u, v are the x- and y-components of the
flow velocity and a, the local sonic velocity, which
is related to Li, v and the critical velocity a' through
a2 = a-- a- ku- + v (2)
S= 0 cha:-acterizes CL.e flat or two-dimensional flow,
0 = 1 the three-dimensional flow through stream tubes .of
of circular cros- section; x, y are cartesian for a =.0,
cylindrical coor'linates for a = 1i
Substitutin-. (2) i" equation (1) and introducing
the di:arensriories velocity co.oponents
l -- a', V = /a (3)
-the .~'~ ential eauatio-. (1) b-co,.-e;
U '1 ;2 + 1-l U2 V
x: + +
..Luv r 1-i _1 2+ v -.=0 (,)
k + 1 I:+ .v
Since the rrezent investigations w.as litrited to the
vicintty of the critical curve,
U = 1 + u, V = )
in which u an.I v are small quantities, and equation (.)
S: + + T +
S- U k ( + u) v -
(k 1) K 't + v -
2(k u k + -i (1 j j (6)
NACA TM No. 1147
As x-O0, ->0, u,. and v approach zero- and also
on the basis of (6) 6v/,y and,consequently, the
quotient v/y approach zero, while the velocity
rise 6u/bx along the axis ought not.vanish at the
origin. Then if the small quantities u, v, 3 v/6y,
and v/y are considered linear only, the following
approximate relation is obtained from (6)
(k + 1) u u + 2v 0 (7)
(i Ys v -o o_ (7)
Since the nozzle flow is symmetrical with respect
to the x-axis, 6u/6y also approaches zero for x->O,
y-->0. Consequently the term 2v 6u/6y may be ignored,
by means of which (7) becomes further simplified to
("^ "l---0^ --8
b u 6v v
(k + 1) u 6- = 0o (8)
dith a = 0, (8) was already applied in another con-
nection by Kl. Oswatitsch and K. Wieghardt (5).
III. FLAT NOZZLES
Equation (8) furnishes general relations, for a = 0,
for the flow through flat nozzles in the vicinity of the
critical curve. Like Kl. Oswatitsch and W. Rothstein (4),
taking into account the symmetry around the x-axis,
= f0(x) + y2f2(x) + y f (x) +...
u = x= f'o(x) + y2f'2(x) + y4f'4(x) ... (9)
v = __ = 2yf2(x) + 4y5f(x) Ir('
NACA TM No. 1147
in which the primes signify derivatives with respect
to x, and, obtain from (8).
(k + I) (f' + y2f' + () (f"0
= 2f2 + 12y2f ... + 2 (f +
By arranging in order of powers of y
coefficients of the individual powers
system. of equations is obtained
+ 2 ^ 2
+ y2f"2 +..
2 -3 4,..
to zero this
2(1 + c) f = (k + 1) f' f"
2 0 0
2(6 + 2o*)f, = (k + 1) (p f" + fo f, )
... 0 -2 0 2 /
through which the functions f2(x),
f (x)... in order
can be expressed by the function f' (x) and its
derivatives. The function
f' = uox)
which characterizes the velocity distribution along the
nozzle axis, remains undeterrAined. Given u ( ) the
flova is completely established through (10) and can be
understood to be nozzle flow, if a pair of streamlines
that are mirror imamres of one another with respect to
the x-axis are assumed as nozzle walls. For further
f'o = uo(x) = a x (11)
k + 2x,
S (k + 1)2 5a
f a ,...'
Translator's note: The number 2 was omitted from the
German report. The correction here changes several
subsequent equations and several values in the table.
NACA TM No. 1147
follows immediately from (10) with 0 = 0. According
Sk + 1 2 2
u = ax+ 2 -2y2 + ...
v = (k + 1) a2xy + (k + 1) a3y5 + .
is obtained for the components of the flow velocity.
The dots represent terms of higher order of y. If (11)
is taken as the first term of a power series expansion
for uo(x), the expressicnsfor f2 and f4 in (12(a))
are also the first terms of power series expansions.
From (15(a)) the critical curve is obtained from the
(1 + u) v = 1
from which in (13(a)) the parabola
x k + 1 ayK2 (14(a))
is obtained, therefore, with u = 0 as a first
approximation. It cuts the nozzle axis, normally, at
the point x = y = 0 and its curvature there is
1/pK = (k + 1) a (15(a))
The vertices, therefore the points, of the streamlines,
adjacent to the nozzle axis, are given by v = 0 and,
according to (13(a)), lie on the curve
Sk 1 ay2 (16(a))
Considering, now, the streamlines at a given vertex
distance yS (fig. 1) from the nozzle wall, then
E =k + 1 2 (17(a))
E = "xS "S(1 Ys
NACA TM No. 1147 7
is the distance from the center point of. the narrowest
cros-s section downstream to the intersection point of
the naozzle axis and the critical curve.
The junction point of the critical curve with the
nozzle wall is upstream and is obtained, as a first
approximation from (14(a)) with yK = yS, which yields
S- = a2 = 2y (18(a))
for the distance r from 1the junction point to the
narrowest cross section.
Finally the ca L.:ulIs.tion of the curvature at the vertex
of the nozzle -wall is to o-' i-..:dc. In moving outward
from thi vertex. to the wall by an element of
curve: dI = dx, the tangent turns through the angle
(.d v 1 6v 1 .iv
+ u v = ds
1 + u 1 + + u oX
Therefore the curvature at the vertex is
1 d 1 v
PS s~ 1 + u oX
and taking into account (15(a)) ani- (15(a))
1/pS = (k + 1) a2y = l/PKaS (19(a))
Up to now the velocity distributiDn uo(x) had been
considered as Liven by equation (11) and the nozzle
walls computed for it. Practically, tie reverse is
the rrobl ir:., u-nely to asc'-rtain for a particular
nozzle, that is f'or assigned valu.S f y,, PS, the
flo: in the norrowust cross s-ct: n, ther.fore the
NACA TM No. 1147
values a, c, 1T, and pi. On the basis of the equations
that have preceded it Ts possible to write down the
solution to the problem, immediately, namely
t (k + 1)pSyS
PK = k + 1 (20(a))
n = 26 = (k + 1)
If the flow is in the positive direction of the x-axis,
as in figure 1, then the subsonic region is to the left
and the supersonic region to the right and, accord-
ingly, a> 0.
In practice it is recommended that all distances be
expressed, dimensionless, in terms of yS, and
accordingly set yS = 1 throughout (20(a)).
IV. ROUND NOZZLES
Corresponding relations for round nozzles come
from (9) and (10) with a = 1. retainingg (11) for
the velocity distribution u (x) along the nozzle
axis, the following relations are obtained in place
of (12(a)) through (20(a)):
f2 = 1 -2x
f2 = u x,
(k + 1)2
f = 64 a
INACA TM No. 1147
k + 1 2 (- + 1) ..
v = -+2- + 0 ,
k +1 2
xK = a4 K
x = -
'" + 1
- +1 -
: "(;;= : ) = (': + 1) 7 T2 : c
1 k+1+ 1
p 2 a' .= y
NACA TM No. 1147
= (k + l)p y
1 PS s
A 2(k + 1)--
To compare a flat nozzle with a round one of the same
longitudinal section, formulas (20(a)) and (20(b))
are compared. (See fig. 2.)
For the flat nozzle T = 2E, for the round nozzle T) =.
E round 3 -
C flat .\2 ~ 1.06
rn ound 3 0.5
'0 flat \ 0
a round i---
The velocity increase is, therefore, around 40 percent
larger for the round nozzle than for the flat nozzle.
All of these results are fully confirmed by the nozzle
flows calculated by Kl. Oswatitsch and "'. Rothstein, as
the enclosed table on page 18 shows.
k + 1
NACA TM No. 1147 1:
V. C,(1 i.T:0 T FLO'i-T'U .-.'..: T..PC
Th- sutho2'.3 rC .s ilt e -.-. :, ..e r.,;"--.O w i .t.
thr f o1'0 '- b'.b? ar,.:.' oxi *- .[.l" 9 ,;.,";,." .. ,- .;,.'.?o: e
t1 I? t nZ 'I
in veloc t.; ct icr ,t ." ( -" )) .: (
by he f-oe f low-- b eci .~'s r -.=. .. ro. ,
Sden- ity lon the ton ,'1 a-:i in U.c vrc i.-. o 'nLic
narrov,/e L& cros- s cto ... l ::c: r o"l'. "s
e = -. ij
= CO. t.,
~ co,-I t.
'1 + 1 \ )" -
+ 1 22-1
(1 + T)-- 1 .
(1 + u) 2uL -1
- conas. (1 u.)
Taking the velocit' d is.tri ';,-1 i.n (].) 1 a -,.s.sf, for
the ozze r -,.oz e w- s .in bt.-. -..1 t 1-. ,'.rro...-t
cros2 section tlihe 1.'II ;.n '.. oj'c.i.r i n ..l c? o'
a flat nozTlt
2/ ). 2= -1
7 1k u = c0,1 -= V- = i'
.r L -'
= 1 + o:
and. in the c-.s of r'. rof ;-'. n i ziz'. z.
-' u = c on--:. = 7 V .= -.1.
Fi'otil thi.s, th curvature at the vc.'t.x -1.
1/p, = .-* "
1/p., = a2:'
NACA TM No. 1147
and in place of the first equations of (20(a))
a -0 -
The comparison of (20(a),20(b)) and (21(a),21(b)) shows
that the velocity increase a in the flow-tube approxi-
mation for both flat and round nozzles is too large by
a factor k + 1, that is for k = 1.! about 10 percent.
The values for the flow-tube aproximation are added
to the table, pnge 18, in square brackets.
VI. FLOV ABOUT PROFILES
Through similar power-series expansions the flow
past a profile in the vicinity of transit through the
critical velocity can be discussed. If a streamline
adjacent to the profile is thought of a rigid wall,
the flow -;*- also be explained as flow through a
nozzle with a curved axis. The origin of the coordinate
system is located at that point of the profile at
which the critical velocity is just reached, the
x-axis downstream on the profile tangent and the y-axis
normal to the profile, outward. (See fig. 3.)
Since the flow is no longer symmetrical to the x-axis,
instead of (8) the rather complicated equation (7)
which must be specified with G = 0, by all means, is
taken as a basis and in place of (9)
S= f0(x) + yfl(x) + y2f2(x) + y3f (x) +...
u = x= f'o(x) + yf' (x) + y22(x) + yf3(x) +... (22)
v Q fl(x) + 2yf2(x) + 5y2fj x) +...
6y 2 2
NACA TM No. 1147 13
3,y putting (22) in (7)
(k + l) (f + yf'+... (f0 + yf ..) = (2 + ,f2 +..
+ yf2+...), + 2',, )
and by comparing coefficients
if + ____
f = ( + 1) (f f + r" f' + ( f + f'
0 0 1f
while )nly the f unction f'o(x) haJ renmainel arbitrary
in the n-)zzle flow, here there are tv..o arbitrary
functions f'0(-x), fl(x). :.ith f'o(x) and fl(x)
the remaining coe'fficient tf~nctio)ns f2(x), f(x),...
in order may be c'omp.uted fron1 ..he systemi of equations (25),
by which the flow 1i completely .established.
Similar to (11) sot
f0 = ax, fl = -PX (24)
and obtain fror (25)
f2 = a + 7-
k + 1 a. 4P 3 (+ 1 2 +p2x
S6 3 2
and from (22)
NACA TM No. 1147
u = ax Py + a + 2...
v = x + k + 1)a2 + 2 'xy (25)
+- r (' + 1)1 2 22 x !
Here a signifies tile velocity increase T- along the profile
on transit through the critical velocity; it is therefore
positive at the starting point A and negative at the
terminal point B (fig. 4) of a local supersonic region.
ITe profile curvature is given with p > O at the
point A or B as
P... p x=y=0 (26)
For the slope of the critical curve with respect to
the profile tangent with
(1 + u)2 + v2 2u = 0
from (25) the relation
tan ~ p
I ~ ___~~I
'Translator's Note: This 2 was omitted through
error in the original German report. The correction
applied here changes values of the constant in the
succeeding equations and some values in the table.
NACA TM No. 1147 15
For the slope of the streamline at a distance y
fror the profile
1 1 iv ~ r ,2L 2
y -1k + 1_1 a2 + 2P yI
S sL J
Z [(k + 1) a2 + P2] y
is obtained and taking (26) into consideration
S -i(k + 1) a2+ -+ y
PS PD I PP 2
p: P + Y 1 + (k + 1) a2 21
PS P P
according to (27) thc- critic tl curve at A (a > 0)
srd B (a < 0) is steeper the larger the velocity
increase or decre.c. anud the less the profile is
curved?. By (231 in the limiting case of a = 0 the
circle of curva'tur.e at S is concentric with the
circle of cur-vatu'e of' the profilee and, as is already
known from the flow-tube approximation, becomes still
flatter with increasing a.
Approximation formulas were developed for the
position and curvature of the critical curve for
transition through the critical velocity in the
neighborhood of the narrowest cross section of flat
and round Laval nozzles. In comparison with
16 NACA TM.No. 1147
nozzle flows calculated by Kl. Oswatitsch and
W. Rothstein (it) they showed satisfactory agreement.
In addition, corresponding approximation formulas were
deduced for the flow at profiles with local super-
Translated by Dave Feingold
National Advisory Committee
NACA TM No. 1147
1. Meyer, Th.: Uber zweidimensionale Bewegungsvorgdnge
eines Gases, das mit Uberschallgeschwindigkeit
strLmt, Diss. Gbttingen 1907.
2. Taylor,G.J.: The flow of air at high speeds past
curved surfaces, Aeron. Res. Comm. Rep.
a.Mem.N.1381, London 1930.
3. G'rtlerH.: Zum tjbergang von Unterschall-zu
Uberschallgeschwindigkeiten in Disen,
Z.angew.Math.Mech.19 (1939), pp.325-337.
4. Oswatitsch,Kl. and Rothstein,V.: Das Stromungsfeld
in einer Lavaldise, Jahrbuch 1942 der deutschen
Luftfahrtforschung, I 91-102.
5. OswatitschKl. and Vieghardt, K.: Theoretische
Untersuchungen iber stationare Potentialstr6mungen
bei hohen Geschwindigkeiten (noch nicht
A6 N 7?- ... .5-,D ,1
18 TABLE ACA TM No. 1147
COMPARISON OF APPROXIMATION FORMULA. (20(a)) AiD (20(h))
WITH THE NOZZLE FLOWS COA;';'UT. 37 EL. COS,'ATIYTCH
AND W. r 'OT '..1 .
a flat I 0.20 (0.2"')
in fl-.. t1 I !O. 2-
L Iute _
a ru.mCd G.29 (C.28)
Ia ro ui I i' -"
Tubj I '.3l
C flat c.o8 (0.03)
E round 0.09 (0.10)
fI flat 0.16 (0.16)
rn round 0.09 (O.CS)
; ( .,7) 0.o 5 ((.
41 (0.37) V c.530 (o.
1_ --- ----4- -
The unbracketed numbers have been computed with approxi-
mation formulas (20(a)) and (20(b)), the numbers in
curved brackets taken from figures 7 anrd 10 of thJ
paper by Kl. Oswatitsch and W. Rothstoin ()4). The
square bracketed numbers are from formulas (21(a))
and (21(b)) for the flow-tube approxima.tion.
___ ~ _I~ _I
NACA TM No. 1147
- criLical curve
Figure 1. Illustration of terms.
....... round nozzle
Figure 2. Comparison of a flat and a round nozzle.
- critical curve
Figure 3. Profile flow at transit through the critical velocity.
Figure 4. Profile with supersonic region.
NACA TM No. 1147
UNIVtH5I Y UP- 2LUHIUA
3 1262 08106 299 3
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