Evvard, John C

Maslen, Stephen H

English

NASA Technical Reports Server

I,
I
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NATIONALADVISORYCOMMX12’E
FOR AERONAUTICS
TECHNICAL NOTE 2688
THREE-DIMENSIONAL SUPERSONIC NOZZLES AND INLETS OF
ARBITRARY EXIT CROSS SECTION
By John C. Eward and Stephen H. Maslen
Lewis Flight Propulsion Laboratory
Cleveland, Ohio
Washington
April 1952
.. —------ ——— ------- . . . . . . ... ... . ..—
https://ntrs.nasa.gov/search.jsp?R=19930083973 2020-06-17T17:37:53+00:00Z
lK IWI?IONAL ADVISORY COMMITTEE lHIRAEROIUD!I!IU3
TECHNICAL l!UT132688
THREE-DIMENSIONAL SOTERSONIC I?OZ!ZLESAND IXLETS OF
ARBITIVUIYEXIT CROSS SECTION
By John C. Evvard and Stephen H. Maslen
SuMyARY
A method is presented for obtaining
supersonic nozzles and inlets flromknown
three-dimensional
~tfic
axisymmetric fU3ws. Stream-
lines bounding the desired -t shape are traced through the Imown basic
flow solution to give the reqtired unsymmetric wall contours. Several
examples are given.
ImTRcnmmoN
. Because of the great complexity of the method of characteristics
in three dimensions (see, for ~ple, references 1 and 2), the accurate
theoretical design of geneml three-dimensional supersonic nozzles and
. diffusers has not been feasible. The designer iB thus limited to two-
&Lmensional and axisymmetric flows. In many cases this may be a serious
disadvantage. In the case, particularly, of a -sonic wind tunnel,
it would be very helpful if a three-dimensional expmsion could be
used to avoid the usual thin slit which would appear as the throat of
such a nozzle if it were two dimensional. Furthermore, such a nozzle
might have better secondary flow characteristicsthan the wrrespm3ing
,
two-dimensional nozzle in which transverse pressure gradients may cause
cross flow in, md 10cal build-up of, the boun&ry layer. However, the
usual three-dimensional nozzle is axially symmetric and thus precludes
the use of flat schlieren windows. In the design of supersonic inlets,
considerationsof space, as well as of permissible flow turning, make
the use of shapes other than two-dimensional or axisymmetric ones
desirable.
A simple method for obtaining three-tiensional nozzles from
axisymmetrical ones has been developed at the NAOA Lewis laboratory
.
and is presented
of the method of
.
herein. The procednre is exact within the limitations
CharacteristicCs.
.
4
———_-— -.— .._—. .———z— __. — _—— ____ —-—
ME!I!HOD
NACA TN 2688
In any intiscid fluid flow, the streamsheets,by definition, form
surfaces across which there is no flow and hence may be replaced by
solid lmundaries. Tbis fact will herein be applied to find unsymmetric
nozzles and Wets Mm axially symmetric flows. Two salLent features
of an axisymmetric flow are that the streamlines 13.ein planes through
the stresmwise axis and that the flow in any one such plane is the
same as that in my other.
In order to ftid the nozzle of a desired shape, the axisymmetric
nozzle having the desired length aud Mach nmiber characteristics is
computed and then, choosing the desired cross section at some station,
for exsmple, the exit, the streamEnes which pass through the periphery
of that section are found. The streamlines can be detezmd.nedeither
by constructing them from the known local flow directions or by the
better method of integrating the axisymmetric stresm function (appendix).
The streamsheet fomed by these streamklnes till constitute the walls
of the desired inlet or nozzle.
If the initial aud final flows are unif’om and -al, it can
readily be shown that the cross-section shapes at the inlet and exit
will be similar. For any arbitrary streamline, the ratio of the radii
at the initial and final stations is given by the one-dimensional
mntintity equation for isentropic flow:
y+l
(:y=*(:’+;)==, (1)
where M and r are Mach nuniberand radius, respectively, and the sub-
scripts i and f refer to the initial and final stations. This
ratio depends only on the initial.and final Mach numbers and hence is
independent of the initial streamline. Likewise each streamline con-
tinues in its own axial plane throughout the nozzle. Thus, the initial
and final shapes will be similar, all dimensions being changed by the
ratio K of equation {1). In most applications, the condition that
the flow be uniform and axial at the inlet and exit is desirable. In
the region between the two ends, the cross-section shape will generally
vary, although this variation has been found small in the flows computed
up to the present the. It may be expected that the variation of shape
increases with the rapidity of Mach nniber change between the inlet and
exit.
IRAMPms
Two -symnetric nozzles were designed by the method of character-
istics {see, for exam.le, reference 3). The solution was started by a
series expansion near the throat, using the method of reference 4.
---
. .
.
— ——— ..-.
.-,
l
.,
NACA TN 2688 3
These nozzles were designeii w expand from sonic velocity b a llach
number of 2.98 and 5.70, respectively, and to have unifomn tial inlet
and exit flaw. From these calculations, the resulting streamlines were
found by integrating along a characteristic h fm the stre~ *ction. .
The details of this computation are described in the appendix. In fig-
ures 1 and 2 are given the results of this calculation. With the use
of these results, nozzles or inlets of any shape are readily fozmed.
The &tails Zor the case of the Ma@ numiber2.98 nozzle of figure 3 wi12
be given. The exit section was selected as a square centered on the
axis and having a side of length 14.5. The stream function was then
found {at the exit) from equation (6) of the appendix for the periphery
of the square. Throughout the length of the nozzle on the wall.,~ will
be mnstant for constant values of the cylindrical angle. Thus, for
the example considered, at the corner of the section, $ = 13.10. Then,
from figure 1 at x = 30, for instance, for this value of *, r is
eqwl to 16.20. In this manner the shape of the nozzle may be found.
The result is shown in figure 3. This illustrates how little the
cross section varies from the initial shape for this ~icular basic
flow. In figure 4, there is shown a similar nozzle designed for
M= 5.70. In this case it is seen that the nozzle shape shows approxi-,
matel.ythe sane variations as in the case of the lower Mach number.
Of course, if -t cross sections with only two flat walls are required,
the other two could have circular-arc cross sections. Such a configura-
tion would probably be easier to fabricate than the square nozzles
already discussed but would still have flat side walls suitable for
schid.erenwindows.
If it is desired to have perfect flow at the @t, such as is
required in a wind tunnel nozzle, and still to have constant cross-
section shape, the method of reference 5 might be used to modify the
cross sections already found. ~S Possibility WaS pointed out by
Ferri at the February 1952 meeting of the American Physical Society.
On the other hand, if it is not essential to have perfect flow at the
exit, sufficient accuracy might be obtained by the use of constant cross-
section shape through the duct.
Another use of the method would be for the design of diffuser inlets.
Such a configuration is shown in figure 5. In this case the inlet and
exit sections are circles tangent to the tis of the basic flow. Since
it was designed to diffuse to M = 1, the imlet would have to be per-
forated in order for the shock to be swallowed, but not as much as in
the case of the usual axisymmetriw perforated inlet, because some
external compression is obtained by cutting the diffuser back along
the Mach cone A (fig. 5). For the same reason, if the inlet were
designed for only the I@ntrowitz contraction ratio [reference 6) so
that no perforations were needed, the permissible contraction ratio
based on the free-stream tube swallowed would be increased.
—. .—.—_ _____ —.
-.. — —..
4 IUK2ATN 2688
Tb illustrate this point, rough computationswere made for two
convergent-divergentdiffusers, one symmetric and the other not. A
conventional convergent-divergent diffuser designed for a Mach number
of 2.98 has a permissible contraction ratio of 1.39 and a pressure
recovery of 4A.9 percent when the shock occurs at the throat Mach number
of 2.63. The comparable diffuser designed by the present method would
be one which appears as in figure 5, except that the area ratio from
section B to the throat would be the IQntrowitz ratio corresponding to
the average Mach nuder at section B ffig. 5). To get some numbers,
this average Mach nwiber may be assumed to be 2.70. The permissible
contmction ratio for this Mach nmiber is 1.35 giving an over-all
contraction ratio for the diffuser of 1.76 and a pressure rewvery of
54.8 percent when the shock occurs at the throat Mach number of 2.38.
These * inlets have the same mass flow and entrance area, but the
unsymmetric one has about 10 percent higher theoretical pressure recovery
than the usual convergent-divergentdiffuser.
.
The configuration shown in figure 5 could also represent a convergent-
Uvergent exit jet. Such a jet might be useful in fairing the exit to
the tall structure or aft ~rtion of an airplane.
COWLUDING REMA13W
A method has been presented for obtaining theoretical wall.contours
of supersonic nozzles snd inlets of arbitrary cross-section shape from
lmovn axisymmetric or two-dmensional flows. If the basic fldw is
axisymmetric and uniform at the throat and the -t, the desired nozzle
will have similar cress sections at these stations. For the examples
computed, the wall contour cross sections between the throat and exit
Md not deviate drastically from similarity with the initial shape.
Lewis FkLght Propulsion I&mratov
National Adviso~ Committee for Aeronautics
Cleveland, Ohio, February 5, 1952
.
—-
NACA TN 2688
flwlmmx
5
.
.
.
.
,.
D~ON OF STMMKDWS IN AXCSYMMEUU C FLOWS FROM
KNOWN VELOCITIES ON A CHARACTERIS!EICNET
The continuity equation in sxl.allysymmetric flow may be written as
where p is the density, W the speed, and the remaining quantities
are defined by the sketch.
B
*racteristic
Hence, there exists a stream function, $(x,r), such that
where PO is the stagnation density. Then, along the ~-characteristic
or, from the sketch
%=* M
aT b
cos(~+f3)+~sin(B+e) (3)
If equations (2] are substi~ted into (3), there results, after some
manipulation,
From the sketch,
dq
. .—. . ——. —c —— —. .-
.—-—
—
NACA TN 26886
Therefore,
If the interval AB is small, say the distance between two points
on the characteristicnet, then it may be assumed that the quantity
pw takes its average. Experience shows that this will be
p# sin (p + e)
a particularly ~od appro-tion along characteristicsproceeding
downstream away from the axis. Then
[
pw
)(
pw
VB-$A=
)]4p# sin (~ + El)A+ 4p@ sin (f3+ 6) B
(rB2 - rA2) (5}
At the exit, e = O and M = constant. In this case, from the second
of equations (2), if the axis is taken as the ~ = O streamline,
Equations [5) and (6) are as accurate as the characteristic diagram.
In the applications of this re~rt, W has been made nondimensional
by dividing by the maximum speed.
1.
2.
3.
4.
(6)
IW?ERENCES
Moeckel, W. E. : Use of Characteristic Surfaces for Unsymmetrical
Supersonic Flow Problems. NACA TN 1849, 1949.
Thornhill, C. K.: The Numerical Method of Characteristics for
Hyperbolic Problems in Three Independent Variables. ARE Rep.
No. 29/48, British M. O. S., Sept. 1948.
Meyer, R. E.: The Method of Characteristicsfor FYoblems-of Compress-
ible Flow Involving Two Independent Variables. Part I. Quart.
Jour. Mech. and Appl. Math., vol. 1, 1948, p. 196.
Hooker, S. G.:
of the Throat
1429, British
The Flow of a Compressible Fluid in the Neighborhood
of a Constriction in a Circular Wwd Channel. R & M
A. R. C., my 1931.
q
l ,
——
I?ACATN 2688
m
5. Ferri, Antonio: The linearized
cation to
1951.
6. Kfndnmwitz,
gation of
‘
Practical I?onMnear
7
CharacteristicsMethod and Its Appli.-
8upersotic Problems. NACA TN 2515,
Arthur, and ~naldson, Coleman duY.: Welidnary Investi-
Supersonic Diffusers. NACA A(!RL5D20, 1945.
—— --— _ ——. — -. ..—. — ———— —. .-.--—
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NACA TN
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2688 11
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Figure 4. - Square mzfie for M* ~ Of 5.70-
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——.-.—— ~—. _ —— ._ .._ —.—._ _ . ....._ __ ————
1.2 NACA TM 2688
5
Axial. dlsbnce
x
4 I
—
- ~x
m 5. - tkwDm3Mc dmular Inlet or mzzle for Mach nmhsr of 2.9S.
.
.
4’
.
—.—
NACA - I.ant?ley Field, Va.


Three-dimensional supersonic nozzles and inlets of arbitrary exit cross section

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