Calculation of the shape of a two-dimensional supersonic nozzle in closed form


Material Information

Calculation of the shape of a two-dimensional supersonic nozzle in closed form
Series Title:
Physical Description:
29 p. : ill. ; 27 cm.
Cunsolo, Dante
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 )
technical report   ( marcgt )
non-fiction   ( marcgt )


The idea is advanced of making a supersonic nozzle by producing one, two, or three successive turns of the whole flow; with the result that the wall contour can be calculated exactly by means of the Prandtl-Meyer "Lost Solution."
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by Dante Cunsolo.
General Note:
"Report date January 1953."
General Note:
"Translation of "Sul Calcolo in Termini Finiti dell̕Effusore di una Galleria Bidimensionale Supersonica." L̕Aerotecnica, Vol XXXI, No. 4, 15 August 1951."

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University of Florida
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003778681
oclc - 86079810
sobekcm - AA00006163_00001
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Full Text

3r0 7qS /) ? 7?-7 T7





By Dante Cunsolo


The idea is advanced of making a supersonic nozzle by producing
one, two, or three successive turns of the whole flow; with the result
that the wall contour can be calculated exactly by means of the
Prandtl-Meyer "Lost Solution."


The subject matter of this paper is based on the artifice of not
letting the expansion waves emanating from one wall of the nozzle be
reflected from the opposite side, but of cancelling out their effect by
compensating compression waves emanating from this opposite wall. In
this way the difficulties attendant upon the intermeshing of the Mach
waves are avoided, and it is no longer necessary to integrate the differ-
ential equation of the hodograph by recasting it as a finite difference
equation, the solution of which is necessarily approximate. The tech-
nique illustrated here is nothing more than a quite direct application
of the Prandtl-Meyer relationship for flow around a sharp corner. On
the basis of this procedure, the calculation of the coordinates of any
point on the nozzle contour is independent of that for any other point;
that is to say, this way of handling the problem eliminates the lack of
precision usually associated with the tail end of the effusor in com-
parison with the better accuracy at the beginning sections.

In addition, what is of the utmost importance, is that it is not
necessary to determine the characteristics of the flow throughout the
interior of the nozzle, which leads to a tremendous saving in computa-
tional labor, with no deterioration in accuracy.

*"Sul Calcolo in Termini Finiti dell'Effusore di una Galleria
Bidimensionale Supersonica." L'Aerotecnica, Vol. XXXI, No. 4,
15 August 1951, pp. 225-230.

2 NACA TM 1358

Theoretical Background

Let us focus our attention now on the expansion which occurs when
a stream turns around a sharp corner (see fig. 1). In this figure 1
the following notation applies: is the Mach angle, is the com-
plement of the Mach angle, v is the angle through which the flow has
turned as it progresses through its turn around the corner, so that
related to this angle of the flow vector is the angle 9, which is the
angle between the normal to the wall and the radius vector originating
at the corner and lying along the points in the flow which have turned
through the angle v.

Then, upon making use of well-known theoretical relationships, it
follows that

S= v +

tan X1 = X tan

Y +

that is,

v = arctan tan X9 (1)

Now let us employ this Prandtl-Meyer relationship in order to
determine the streamlines of the flow. Let us denote a small lineal
element of a streamline by the symbol ds (see fig. 2), and then we
may take dp and pdO to represent the components of this element in
polar coordinates.

Thus one may write:

dp tan X9
--- = tan t = M
pdO X

dp tan X9
-- = -- d
p X

NACA TM 1358 3

and consequently

log p + log cos X3 = const.


p(cos xa) = const.

and upon selection of the value of 7 as 1.40, one obtains:

p cos6 -- = P (2)

One of the walls is thus the straight terminal side of the angle
of expansion while the other curved wall is defined by means of equa-
tion (2) as far out as the location where the angle characterizing the
radius vector has ultimately reached the value 'M, at which point the
desired Mach number will have been attained. From here on out this wall
is also straight and lies parallel to the terminal side of the corner
angle. As may be seen from reference to figure 3 the minimum length of
effusor which it is possible to have is one which terminates right at
the point where the wall stops curving, at which point the very last
expansion wave has just been included in the process of executing the
total turn. Of course, the depth of the "throat" or critical cross-
section is the quantity denoted by p4.

If one wishes to avoid reliance upon a perfectly sharp corner for
making the expansion, the two curved walls (see fig. 4) given by the

P1 cos -- = P


\ 6=
p2 cos = P2*

NACA TM 1358

may be employed. Of course, in this case the depth of the throat or
critical-cross-section is given by p* = p2* Pl*"

Naturally, for an effusor configuration such as this, the length
becomes longer than in the case of the sharp cornered type. The expan-
sion waves which originate from the number 1 wall are exactly counter-
acted or "swallowed up" when they meet the number 2 wall, without pro-
ducing any reflection. It is just for this reason, even in case one
wants to give some other arbitrary shape to the number 1 wall, that the
calculations may be carried out in parametric, but closed-form, provided
the Mach lines are maintained as straight lines.

Once it is decided what the shape of the number 1 wall is to be,
one may find a value of v which corresponds to the value of B
defining the direction of the Mach wave with reference to the y-axis.
This value of v locates a point with coordinates xl and yl on the
number 1 wall. Consequently, the point lying on the number 2 wall which
is marked out by this h-ray will have the coordinates (see fig. 5):

x2 = x1 + p sin 6 (

Y2 = Yl P cos 6


p = p* (cos 1

Obviously it is necessary to have xl and yl given as functions of 8.

Let us follow through on the details in the case where the number 1
wall is taken to be circular, with radius = po. Then the value of 9,
used as the independent parameter, locates a point with coordinates
(see fig. 6):

xl = PO sin V

Yl = P* + PO(l cos v)

NACA TM 1358

wherein the value of v is given by:

v = -8 arctan (6 tan -

and the point (xlyl) lies on this 0-ray.

Likewise let us examine more fully the case where the number 1 wall
is parabolic in shape at the start, with a radius of curvature equal to
PO at the apex of the parabola (see fig. 7). Taking the equation of
this parabola to be

y = p, + -

then its slope is given by

tan v= dy = x (6)
dx PO

Consequently, since B is taken as the independent parameter, one

v = arctan 6 tan -

xl = PO tan v (7)

Yl= P* tan v

In those cases where the slope dy/dx has more complicated formula-
tions (these would be cases devoid of any practical interest for that
matter, in view of the extreme simplicity of construction exemplified by

NACA TM 1358

the contour of the number 1 wall for the configurations just examined
in detail) one may still solve the problem, to whatever degree of
accuracy is desired, by working with the isocline system:

v = arctan ( tan --

dy = tan v (8)

y = f(x)

which gives the slope of the contour applying at each assigned value of
the parameter 3. Then, after carrying out this construction, one may
find the corresponding values of the coordinates x2 and y2, by means
of equations (4).

If, however, one insists on having the working section not offset
from the axis of the throat, it is necessary to apportion the total
angle through which the stream is turned into three pieces. To be more
precise, let this total deviation of the flow, corresponding to the
ultimate Mach number to be attained, be denoted by vM. Then the three
partial turnings of the flow would be, respectively, an upward turn of
amount VI', a downward turn of amount -- = v', and once again an
upward turn of amount V M = V". With this arrangement one
now has an effusor made up of three expansion regions separated by two
sections of uniform flow, wherein the Mach numbers attain the values M'
and M", respectively.

Now let v be the net angle through which the flow has been turned
up until the time it has reached a certain location in the effusor; and
let 9 be the angle linked to this value of v by means of the Prandtl-
Meyer relationship, equation (1). In addition, let a be the angle
swept out by the Mach wave with respect to the y-axis. Then, in the
first expansion region, we shall have that a = (see figs. 8 and 9).
The first Mach wave beginning the second region of expansion will be
characterized by the angle a = 2*' Y', while within the second region
it will be true that a = (29' ') + ( 5') = 8 2v'. The last
Mach wave ending the second region will be characterized by the

NACA TM 1358

angle a = 9" 2V', and the first wave beginning the third expansion
region is then given by a = 24" (6" 2v') = 23" 2v" 0" + 2v' = 8" vM.
For like reasons we have that, within the third region, the Mach angle
is given by a = (6" vM) + (b t") = b VM. In summary, we have that,
in the three expansion regions, the following hold, respectively


a = 2 2v

a = 3 VM

Let us consider what the appropriate relationships are, in the case
of a tunnel design such as illustrated in figure 8, wherein the sections
of tunnel wall effecting the expansions, that is to say, the convex por-
tions of the wall, consist of circular arcs. For this example the value
of v' must turn out to be calculated in such a way as to make the pro-
jection of the broken line A B C D E F G H L upon the y-axis have the

LM y 2

where oM and ao are the widths of the effusor at the very end and at
the throat. Writing this condition in explicit form, we obtain the

Scos P2 cos v' +
cos6 -

P2 cos ( -

3 1 cos

v + -- cos (I" 2V') -
cos --


Pl(1 cos v') -

V = -m


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where v' and v" are the net angles through which
turned at the end of the first and at the end of the
regions; i.e.,

tan Xi'
v' = -3' arctan

V"= V' +- =

tan Xd"
- arctan

the flow has been
second expansion


In fact, one has in particular that:

(AB)y = P1

(BC)y = -pl cos V'

(CD) = CDi cos = -p' cos =

(DE)y = -p2 coB v'

(EF)y = p2 cos (EF,y) = P2 COB

2 -

(FG)y = p" cos (FG,y) =


cos (-g" 2v')

( v- v

(GH)y = P3 cos

p cos '
co6 -

NACA TM 1358

In this case where such circular sections of wall are considered
to effect the expansions, it is necessary to utilize equations (5)
and (4) (when suitably modified for treatment of the flow in the second
and third expansion regions) for getting the solution, but if this
circular wall contour is replaced by the curve given by means of equa-
tions (3), then this case is handled by setting

Pl* cos -
c6 d'
Cos --

cos -

cos (i' 2V') +
cos6 -

cos (B" 2V') +


PI* o}M 0*
S cos (-M VM) =
cos6 M 2

provided it is assumed that

IDE| = BCJ = pl = *
C6 O 5
cos -

GHI = IEF = P =

c 6.


10 NACA TM 1358

In fact, one has in particular this time that

(BD)y = cos '
COB6 6
cos6 _'

angle between ED and y = (ED,y) = -' 2v'

(EG)y = cs (E,y)
6 '"

angle between EG and y = (EG,y) = S" 2V'

angle between GH and y = (GH,y) = o" VM


(HL)y= l*
cos6 M

wherein the symbol *M stands for 'M VM.

Numerical Applications

For illustrative purposes let us suppose that one wishes to design
a tunnel which will reach a Mach number of

M = 2.5

by means of an effusor such as depicted in figure 6, or else with one
with three kinks as in figure 8.

NACA-TM 1358

Setting down the numerical values available to us, we get the
following listing:

M = arcos = 660 25.3'

M = 6 arctan

tan M = 6 arctan M2 = 1050 327'
f6 46

vM = 390 7.4'

_M 1 ( =+ M2
OM o f_5= = 6.5919
cos 6_ 6

M 1 /5 +M2 3 = 2.6368
a* M R 6

Let us compute the value of v which applies to the type of
circular arc expansion incorporated into the figure 8 design. In this
case if we let

a = p = 1

P1 = 0.5

02 = 03 = 1

M 0*
= o.8184.

then, from equation (10),

NACA TM 1358

Let the function A(V') represent the left hand side of equa-
tion (10) (which is legitimate because ', v", and B" are all func-
tions of a'), and we then propose to find the value of 9' which makes

A(') = 0.8184 (13)

In order to effect the solution of equation (13), the author made
use of the method of approximation by secants (see fig. 10). Let us
assume for example that 91' and B2' are two values of the independent
variable which when substituted into the expression A(V') renders a
result which is too small in the first case and one which is too large
in the second case in an attempt to find a value of V' which makes
equation (13) hold true. In addition, let 3' be a value of the
independent variable which is much more accurate, and which can be
obtained by linear interpolation. Then the two secants P1 P3 and
P2 P3 will cut the horizontal line A(4') = 0.8184 in two points,
which may be labelled S1 and S2. A better approximation to the
exact solution of equation (13) is then obtained by selecting V' as
equal to a value '4 which falls within the interval defined by the
end-points S1 and S2. After discarding one of the more distant
points on the curve (in the case illustrated by fig. 10, the point '1i
would be dropped) the whole process is repeated by working with the
points P2, P3, and P4 to start with again.

We shall try out this procedure by first selecting the value of 9'
as = 600. Then one finds that:

B' = 600

V' = 11.90

v" = 31.50

-" = 94.30

A = 0.59 <0.8184

NACA TM 1358

The value of 6" listed here is obtained by use of the relationship

V" = j" -.arctan 6

tan -"

through use of the method of approximation based
a tangent. According to this method, if a first
which is too large is determined and denoted by
accurate approximation to the exact value of -3"

*2 = '

V" V" V V"
- 1 1
dv" 5

on interpolation along
approximate solution
i then a more
is given by

6 + cot2 1
2 if6)

where the value
given above for

of V1l is obtained by an analogous formula to the one
v"; i.e., here one uses the relation

Vi = *l" arctan (V

tan )"

-' = 600
the value

the above derived numerical result it is seen.that the guess
is a too small solution for equation (13). Consequently if
' = 700 is now tried, it turns out that:

3' = 700

v' = 16.90

v" = 36.5

8" = 101.70

A = 1.47 > 0.8184

14 NACA TM 1358

Thus the trial value B' = 700 is a too large solution for equa-
tion (13). Linear interpolation then results in the value for a'
of 62.60.

Consequently, using this value it results that:

4' = 620 36'

v' = 130 5'

v" = 320 39'

a" = 960 4'

A = 0.762 < 0.8184

Now continuing the computation to obtain a better approximation by
means of the method outlined above (see fig. 10), one gets that

a' = 630 20'

v' = 130 26.7'

v" = 330 0.4'

S" = 960 36.9'

A = 0.8183

Consequently the solution of equation (13) is v' = 630 20' with
an error which is less than 1 minute.

In carrying out the actual construction of the effusor contour it
is necessary to point out that, if 0 is taken to denote the direction
of the flow with respect to the x-axis, the following sign relationships

dP = dv in the first (I) and last (III) expansion regions

dB = -dv in the central (II) expansion region

NACA TM 1358 15

Now perform the integration indicated by equations (14), substi-
tuting the proper limits, and one obtains

3 =v

0 = 2 v (15)

p = v VM

in the regions I, II, and III, respectively.

If the coordinates of the points B, E, and H are assumed known,
then the equations (15) afford the means of determining the coordinates
of the point (xl,yi) which lies on the convex portion of the wall contour
and which is related to an arbitrarily selected fixed value of the param-
eter 0. In addition, let the former equations (4) be modified to read

X2 = xl + p sin a

Y2 Yl T p cos a

p = p* cos

where the + sign in the second equation holds just for the central II
region of expansion. These expressions, together with the equations (9),
allow one to compute the coordinates (x2,Y2) of the corresponding point
lying on the concave portions of the wall.

Working with the illustrative case depicted in figure 9, let the
starting data be selected, for example, as

P* =1 =

Pl* = 0.3

P2* = P1* + P. = 1.3

16 NACA TM 1358

Upon substitution of these values into equation (12), one will
find that

B(1*) = 0.8184


Continuing the computation by an entirely
used previously, it is found that

analogous procedure as

-' = 630 5.7'

v = 130 19.6 '

v" = 320 53.3'

9"= 960 26.2'

In order to carry out the actual drawing of the effusor shape (see
fig. 9) the relationships given as equations (9) are again employed and
the results applying to the three convex portions of the wall will be
given by, respectively:

x1 = XB + P1 sin a

yl = YB P1 cos a

xI = XE + P1 sin a

Yl = YE + Pi cos a

xl = xH + Pl sin a

l = YH P1 cos a


P =

NACA TM 1358

The determination of the concave portions of the wall is carried
out in an analogous way, except for the mere replacement of the sub-
script 1 by the subscript 2.


The advantages accruing from the methods expounded here for design
of the various kinds of effusor lie essentially in the great degree of
precision with which it is possible to draw in the wall contour. A
blemish of the method suggested for design of the effusors possessing
three successive expansion regions (see figs. 8 and 9), which must be
acknowledged, is that the length is longer than what would be found
necessary if the effusor had been calculated according to the usual
hodograph method, although the latter would be less exact (see fig. 11).
The lengths of effusor, when non-dimensionalized by reference to the
throat depth, have the following magnitudes in the cases exemplified
by figures 8 and 9, respectively:

= 12.2

= 14.2

The same ratio in the case of the effusor represented by figure 11
has the value 9.8, but it will be cut down to only 4.9 if the effusor
shown in figure 11 is considered to be merely one half of a complete
symmetric tunnel.

This defect in the effusors described here ceases to exist in the
case of the skewed effusor designs first mentioned in this paper. These
off-set designs can be fruitfully employed nevertheless where there is
restricted room for the setup, because it should not entail a very great
deal of trouble to incorporate a suitable compensating kink in the sub-
sonic part of the tunnel.

A promising compromise design can be obtained by use of an effusor
having two expansion regions (with equal and opposite amounts of turning).
With this configuration it is clear that the working section will be
lined up parallel with the axis of the throat section, except that it

NACA TM 1358

will be offset laterally (see fig. 12). The length of the effusor and
its lateral displacement will have the following sizes, respectively

1 9.7


As is evident, this displacement s/p. is not of formidable size,
while on the other hand the length of the effusor has been somewhat
reduced, in comparison with the cases illustrated in figures 8 and 9.

Despite all that has been said, it is still worthy of note that in
the case of effusors designed for Mach numbers which are only slightly
greater than unity; that is, for effusors whose lengths do not exceed
that of the working section, the effusor designed with three expansion
regions can still be employed to good purpose. Such an effusor designed
to produce a Mach number of M = 1.2 is illustrated in figure 13.

Translated by R. H. Cramer
Cornell Aeronautical Laboratory, Inc.,
Buffalo, New York

NACA TM 1358

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20 NACA TM 1358

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NACA TM 1358 21



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22 NACA TM 1358


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NACA TM 1358


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24 NACA TM 1358

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NACA TM 1358

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26 NACA TM 1358

NACA TM 1358

A (0) = 0.8184

Figure 10


NACA TM 1358


NACA TM 1358

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