a computer program for the aerodynamic design of axisymmetric and planar nozzles for supersonic and...
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AEDC-TR-.78-63
COPUE LEVELEI1,A COMPUTER PROGRAM FOR THE AERODYNAMIC
DESIGN OF AXISYMMETRIC AND PLANARONOZZLES FOR SUPERSONIC AND
SHYPERSONIC WIND TUNNELS
J. C. SivellsARO, Inc., a Sverdrup Corporation Company
VON KARMAN GAS DYNAMICS FACILITYARNOLD ENGINEERING DEVELOPMENT CENTER
AIR FORCE SYSTEMS COMMAND.ARNOLD AIR FORCE STATION, TENNESSEE 37389
LULDecember 1978
Final Report for Period December 1975 - October 1977
L Approved for public release; distribution unlimited.
Prepared for D D D
ARNOLD ENGINEERING DEVELOPMENT CENTERIDOTR JAN 8 1979ARNOLD AIR FORCE STATION, TENNESSEE 37389
D
79 0BEST AVAIL.ABLE -COPY
"4.
NOTICES
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Qualified users may obtain copies of this report from the Defense Documentation Center.
Reterences to named commerical products in this report are not to be considered in any senseas an indorsement of the product by the United States Air Force or the Government.
This report has been reviewed by the Information Office (01) and is releasable to the NationalTechnical Information Service (NTIS). At NTIS, it will be available to the general public,including foreign nations.
APPROVAL STATEMENTI LThis report has been reviewed and approved.
Project Manager, Research DivisionDirectorate of Test Engineedng
Approved for publication:
FOR THE COMMANDER
ROBERT W. CROSSLEY, Lt Colonel, USAFActing Director of Test EngineeringDeputy for Operations
UNCLASSIFIEDREPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM
(6 W. COMPUTER.PROGRAM FOR THE AERODYNAMIC nal epaDc 7
J. C./Sivellsl ARCO, Inc., a Sverdrup
Air Force Systems Command ogram Element 65807?
ArnldAiFrcSatonTenese_338__
16. AGERSLI N C S NAT MET Ao Ihi. OREpo(f dfeetfolotoln fi 1,SCRT LS,(fti eot
* IApproved for public release; distribution unlimited.
17, L3STRIOUTION STATEMENT (of tho abstract onteted ir Block 20, It different from Report)
IS. SUPPLEMENTARY NOTES
Available in DDC.
19. KLY WORDS (Contitnuo on reverse side It nce...ry and Identify by block "limber)
wind tunnel design boundary layerstransonic nozzles
* J supersonic nozzles* *..hypersonic nozzles
exhaust nozzle performance computer program2 0. ISTYAA CT (Con linue on .nre r.o .1 do I I nteo eeary and Identify by block numbe r)
A computer program is presented for the aerodyna-mic design ofaxisymmetric and planar nozzles for supersonic and hypersonic windtunnels. The program is the culmination of the effort expended atvarious times over a number of years to develop a method of de-signing a wind tunnel with an inviscid contour which has contin-uous curvature and which is corrected for the growth of theboundary layer in a manner such that uniform parallel flow can be
DD FOR 1473 EDITION OF I NOV 65 IS OBSOLET E
~~~ ~UNCLASS 7O 1 2 2IT %j 01 0ow 1
UNCLASSIFIED
20, ABSTRACT (Continued)
expected at the nozzle ex-t. The continuous curvature isachieved through specification of a centerline distribution
* of -'elocity (or Mach number) which has first and second deriva-tives that 1) are compatibl* -Lth a transonic solution nearthe throat and with radial flow near the inflection point and2. approach zero at the design Mach number. The boundary-layergrowtb ia calculated by solving a momentum integral equationby nuws'lcal iaitegration.
AILEV YEJi, W ii,oil- D D C
JAN 8 1979
IT ............... ........... .............
Slt. AVAIL
11B
UNCLASSIFPTD
-- ,,,.
AE DC-TR-78.63
PREFACE
The work reported herein was conducted by the Arnold Engineering
Development Center (AEDC), Air Force Systems Command (AFSC). The results
of the research were obtained by ARO, Inc., AEDC Division (a Sverdrup
Corporation Company), operating contractor for the AEDC, AFSC, Arnold
Air Force Station, Tennessee, under ARO Project Numbers V33A-A8A and
V32A-P1A. The Air Force project manager was Mr. Elton R. Thompson.
The manuscript was submitted for publication on September 12, 1978.
The author wishes to acknowledge the assistance of Messrs. W. C.
Moger and F. C. Loper, ARO, Inc., for providing thr. basic subroutines
for smoothing and spline fitting, respectively, which were adapted for
use with the subject program. Mr. F. L. Shope, ARO, Inc., provided
technical assistance in the preparation of this report. Prior to the
publication of this report, the author retired from ARO, Inc.
'I
AEOC-TR-73-63
*l CONTENTS
1.0 INTRODUCTION ................... ...................... 5
* 2.0 TRANSONIC SOLUTION ........... .................. ... 10
3.0 CENTERLINE DISTRIBUTION .................. ......... 154
4.0 INVISCID CONTOUR ........... ................. .... 20
5.0 BOUNDARY-LAYER CORRECTION ..... ............. ..... 24
6.0 DESCRIPTION OF PROGRAM ....... ................ ..... 34
7.0 SAMPLE NOZZLE DESIGN ........... .............. .... 38
8.0 SUMMARY .............................. ... 41
REFERENCES ............. ...................... .... 41
ILLUSTRATIONS
Figure
1. A Foelsch-Type Nozzle with Radial Flow at the
Inflection Point .............. .................. 6
2. Nozzle with Radial Flow and a Transition Region
to Produce Continuous Curvature ............ 7
3. Nozzle Illustrating Design Method of Ref. 13 , , , 8
4. Nozzle Throat Region ............ ............... 9
5. Relationships Obtained from Cubic Distribution
of Velocity from Sonic Point to Point E for
Axisymmetrin Nozzle ...... ................. .... 18
6. Limitations of Fourth-Degree Distribution of
Mach Number from Eq. (39) ..... .............. ... 19
7. Characteristics Near Throat of Nozzle with R = 1 . 23
8. Variation of Wake Parameter, A, w.,ith
Reynolds Number (Incompressible) ... .......... ... 28
9. Variation of Skin-Friction Coefficient with
Reynolds Number (Incompressible) ... ........ .... 29
10. Variation of Velocity Profile Exponent with
Reynolds Number Based on Boundary-Layer
Thickness ...... ......... .................. ... 30
3
h 4;
A EO C-TR-78-63
TABLE
Page
1. Input Cards for Sample Design ...... . . . . . . . 39
APPENDIXES
A. TRANSONIC EQUATIONS .................. . .. ... ... 45
0. CUBIC INTEGRATION FACTORS. .. . . . . . . . . . ... . 49
C. INPUT DATA CARDS . ....... . . . . . . . . . . . . .52D. COMPUTER PROGRAM................ . . . . . . . . . . 61
:"NOMENCLATURE ....................... 139
4
AE DC-TR-78-63
1.0 INTRODUCTION A
Supersonic and hypersonic wind tunnel nozzles can be placed in two
general categories, planar (also called two-dimensional) and axisymmetric.
Early supersonic nozzles (circa 1940) were planar for many reasons: the
state of the art was new with regard to both the design and the fabrica-
tion; the expansion of the air - the usual medium - was in one plane
only, thereby simplifying the calculations and requiring two contoured
walls for each test Mach number and two flat walls which could be usedfor all the Mach numbers; and the relatively low stagnation temperatureand pressure requirements did not create dimensional stability problems
in the throat region. Dimensional stability would in later years become
a primary factor in the development of axisymmetric nozzles.
Prandtl and Busemann, Ref. 1, laid the foundation for determining
the inviscid nozzle contours by the method of characteristics. Foelach,
Ref. 2, simplified the calculation of the contour by assuming that the
flow in the region of the inflection point was radial, as if the flow
came from a theoretical source as illustrated in Fig. 1. The downstream
boundary of the radial flow is the right-running characteristic AC from1',- the inflection point, A, to the point, C, on the axis of symmetry where
the design Mach number is first reached. The flow properties along thischaracteristic can be readily calculated; and inasmuch as all left-
running characteristics downstream of the radial flow region are straight
lines in planar flow, the entire downstream contour can be determined
analytically. Upstream of the inflection point, it was assumed that the
source flow could be produced by a contour which was a simple analytic
curve. In the Foelsch design the Mach number gradient on the axis is
discontinuous at the juncture of the radial flow region and the begin-
ning of the parallel flow region. This discontinuity produces a dis-
continuity in curvature of the contour at the inflection point and at
the theoretical exit of the nozzle.
5
.- .
AE DC-,TR-78-63
UniformrS~Flow
Figure 1. A Foelsch-type nozzle with radial flowat the inflection point.
As the state of the art progressed, it became desirable to cover a
range of Mach numbers without fabricating different nozzle blocks for
each Mach number. A limited range of Mach numbers could be covered by
using blocks with unsymmetrical contours which could be translated
relative to each other to vary the mean Mach number in the test section.
The widest range of Mach numbers with acceptably uniform flow in the
test section has been obtained in wind tunnels in which the contoured
walls consist of flexible plates supported by jacks which can be adjusted
to vary the contour to suit each Mach number. Inasmuch as the curvature
of a plate so supported must be continuous, methods of calculating
contours with continuous curvature were developed (Refs. 3, 4, and 5)
by introducing a transition region, A B C J, downstream of the radial
flow region (see Fig. 2). The shape of the wall between points A and J
was controlled to give continuous curvature. The contours used for the
von K~rmgn Gas Dynamics Facility 40- by 40-in. Supersonic Wind Tunnel
(A) at AEDC were obtained by the method of Ref. 5. Not only is a
continuous-curvature contour easier to match with a Jack-supported plate,
but it also satisfies the potential flow criterion for zero vorticity,,
dq/dn - Kq (1)
where q is the velocity measured along a streamline of curvature K and n
is the distance normal to the streamline. Inasmuch as the inviscid
contour is a streamline, this criterion implies that the flow will be
disturbed where a contour has a discontinuity in curvature.
6
AEDC-TR-78-63
D
B C
Figure 2. Nozzle with radial flow and a transitionregion to produce continuous curvature.
The usual wind tunnel criterion concerning temperature is that the
constituents of the gas should not liquefy during the expansion process
required to reach the test Mach number. For the usual pressure levels
involved, ambient stagnation temperatures can be used up to a Mach
number of about five. As the stagnation temperature is raised, dimen-
sional stability becomes more difficult to maintain in a planar nozzle.
Therefore, axisymmetric nozzles are used when elevated stagnation tem-
peratures are involved. Axisymmetric nozzles have also been used for
low-density tunnels (Ref. 6) because their boundary-layer growth is more
uniform than that of planar nozzles, which inherently have transverse
pressure gradients on the flat walls. The obvious disadvantage of
axisymmetric nozzles is that each one must be designed for a particularMach number. Moreover, disturbances created by imperfections in the
contour tend to be focused on the centerline.
Before the advent of high-speed digital computers, it was extremely
time consuming (Ref. 7) to calculate axisymmetric nozzle flow by the
method of characteristics (Ref. 8). Inasmuch as the assumption of
source flow saved time in designing a planar nozzle, it was logical to
use source flow as a starting point in the design of an axisymmetric
nozzle. In Ref. 9, Foelsch develops an approximate method of converting
the radial flow to uniform flow. Beckwith et al., Ref. 7, show that
Foelsch's approximations were quite inaccurate but utilized the idea of
7
./.
AE DC.TR-78-63
a region of radial flow followed immediately on the axis by uniform
flow, as in Fig. 1. As in the case of planar flow, the discontinuity in
Mach number gradient on the axis produces a discontinuity in curvature
on the contour (Ref. 10). Such discontinuities have been eliminated by
the design methods of Refs. 10, 11, and 12; here, an axial distribution
of Mach number (or velocity) between points B and C (Fig. 2) introduces
a transition region between the radial and parallel flow regions, thus
gradually reducing the gradient and/or second derivative to zero from the
radial flow values at the beginning of the parallel flow. As shown in
Fig. 3, the upstream boundary of the radial flow region is a left-running
characteristic from the inflection point, G, to the axis at point E. The
flow angle is the same at points G and A. Both are shown to illustrate
a general nozzle design. As described in Ref. 12, the contour upstream
of the inflection point can be calculated for an axial distribution of
velocity in the region between points I and E, which makes the transition
from sonic values to radial flow values. On the axis, the sonic values
of first and second derivatives of velocity with respect to axial distancewere calculated by an adaptation of the transonic theory of Hall, Ref.
13, or Kliegel and Levine, Ref. 14. The upstream limit of these cal-
culations was the left-running characteristic from the sonic point on
the axis.
Inflection Reglon
M * 6°
SI F -
Figure 3. Nozzle illustrating design method ofRef. 13,
8
-: • .'*. ..'. U,,.J-
AEDC-TR-78-83
This characteristic is also called a branch line. Between the theo-
retical location of the throat and the intersection of the branch line
with the contour was a region which was not calculated but which in-
creased in size as the throat curvature increased. This gap in the
contour has been eliminated by the method described herein which util-
izes a right-running characteristic originating at the throat as shown
in Fig. 4 (where point I has been moved from the sonic line to the
throat characteristic). With this latest improvement upon the method of
Ref. 12, contours can be designed which have throat radii of curvature
of the same order of magnitude as the throat radii although such an
extreme curvature would not normally be recommended from other stand-
points. A recent (1975) design of a Mach 6 nozzle utilized this method
with a throat radius of curvature of about 5.5 times the throat radius.
Sonic Line Branch Line
YO! Throat Characteristic
Figure 4. Nozzle throat region.
After the design method was developed for axisymmetric nozzles, it
was adapted for planar nozzles having a prescribed centerline dis-
tribution of Mach number (or velocity). This approach to such a design
is considerably different from that of Ref. 5. The current design
method is incorporated into the computer program included herein. As an
option in the program, a complete centerline Mach number distribution
9
AEDC-TR-78-63
can be used which does not include a radial flow region. Parts oZ the
computer program are subroutines for computing the boundary-layer correction
to the inviscid contour, for smoothing the contour, and for interpolat-
ing points at even axial positions by means of a cubic spline fit of the
contour.
2.0 TRANSONIC SOLUTION
In many early nozzle designs, it was assumed that the flow at the
throat was uniform (M - 1) and parallel. This assumption impliey that
the wall curvature is zero and that the acceleration of the fl, Is
zero (i.et, the acceleration starts from zero at the beginning .-. the
contraction, reaches a maximum in the contraction but is reduced to zero
again at the throat, and must be increased again in the beglaning of the
supersonic contour and reduced to zero at the nozzle exit). A nozzle so
designed therefore becomes considerably longer than one in which the
flow reaches its maximum acceleration in the vicinity of the throat,
where it is approximately proportional to the reciprocal of the square
root of the radius of curvature. The above argument indicates the
fallacy of some so-called "minimum length" nozzles, although some
designers have combined a contraction having a relatively high throat
curvature with the supersonic section having zero throat curvature.
For a throat with a finite radius of curvature there have been many
transonic solutions. Hall, Ref. 13, developed a small perturbation
transonic solution for irrotational, perfect gas flow, in both two-
dimensional and axisymmetric nozzles, by means of expansions in inverse
powers of R, the ratio of the throat radius of curvature to the throat
half-height, or radius. His solution gives the normalized (with the
velocity at the sonic point) axial and normal velocity components in the
form
(y,Z) + . . .. + + .Y. . . (2)R R2 R• 3
10
10
AEDC-TR -78-63
2
- + 12 Lv (y,z) v+(y,z) + v,(y,) V(L,( 1 2 + L. .( 3 )
where
and x and y are coordinates normalized with the throat half-height or
radius, yo. The value of a is zero for two-dimensional flow and one for
axisymmetric flow. Kliegel and Levine in Ref. 14 extended the applica-
bility of Hall's axisymmetric solution to lower values of R essentially by
making the substitution
- S-1 + 9-2 + S- . .(5)
where S - R + 1, into Eqs. (2) and (3). In the method used herein, the
same substitution is made in Eq. (4) for two-dimensional flow as well as
for axisymmetric flow and therefore becomes a special case of the general
transonic solution described in Ref. 15. The complete general equations
in terms of S are given in Appendix A.
At the throat, x - 0, y - yo' v 0 0, for planar flow,
n= 1+ _1_ - (14y-7 5) + (274Y2
- 861y,+ 4464)3S 270S
2 1701003
d u . -(32y2 + 87y 561) +
dx/y. S 540S2
and, for axisymmetric flow,
+ 1-4y -57) + (2364y 2 - 3915y + 14337) + .. . (8)4S 288S 2 82944S3
Sd__+__ = _ -(64y2 + I 17y - 1026) + (9)dX/yu as 1152S
2
;!: 11
AEDC-TR-78-63
where the derivatives are with respect to x nondimensionalized by the
throat half-height or radius, respectively, and
2
A _____(10)
On the axis, y - 0, v - 0, for planar flow,
u = 1 - _i + Y-5 782y 2 + 3 5 07y - 7767 +6S 270, 2
2721 60S'
+ + 34y2+ 429y_ 12+].4Y" 4320S 2 1
,Y," 6 36S
(,& -z' (2y 2 - 33y + 9)/72 + . (1)
and, for axisymmetric flow,
S1 - _. + IOy-. 1L5 _ 2708y2 + 2079y + 2 1 1 5+4S 288S 2 82944S 2
+& -1 -t I + 92y2 + 180X - 9 ++Y \ 8S 115282
()2 (.2~ + +
(--A)' (4Y2 - 57y + 27)/144 + (12)Yr,
Because the sonic line is curved for finite values of R, the mass
flow through the throat is reduced by the factor CD (discharge coef-ficient), which is the rat 4 .o of accual mass flow to that which couldflow if R were infinite and the sonic line were straight. For planar
flow,
CD I- [i - 4y24 + 334•2 - 457y + 4353 + (] 13)
I12
i 12
AEDC-TR -78-63
and, for axisymmetric flow,
C[ 1 - y [1 - - .754y2 -757y +36U. + (14)96 S' 24S 28EIOS'
The flow which passes through the throat also passes through the
sonic area of the source flow which is at a distance r from the source.
In planar flow, 0(
•! or
oY0/r1 = ,/C1 (16)
where the inflection angle, n, is in radians.
In axisymmetric flow,
7y. = ry~o CD =21 r• 1 -cs (17)
or I
yo/r- 2 sin (1/2)/cD (18)
In the calculation of the throat characteristic used herein, thevalue at x - 0, y - yo Eq. (6), is the starting point. The half-height'•~
or radius, y0, is divided into 240 equally spaced values of y. Inasmuch
as the characteristic is right running, its slope at each point is
dy/dx tan( (19)
wheresin I = 1/M (20)
Also
W = M + K-l M2) (21)y+1 Y+-
sin • = v/W (22)
and
drdo + d- o '""a d• (23)
y
13
- -
AEIDC-TR-78-63
Sde dx/cos(o - t)- dy/sin(o - 1) (24)
'The term i is the Prandtl-Meyer angle in two-dimensional flow,
y an. (2I - - tan-1 (I2 -1) (25)Y-1 y+l
Equations (19) and (23) are the characteristic equations and are solvedby finite differences. If all values are known at point 1, the valuesat point 2 are found (y is known a t both points) by
S + 2(y-y) (26)x2 - 1 tan( l-A, + tan
A (Y2 -Y)2 + (x2! - x1 )2 (27)
V02 = 01 + 01 - 0 2 + VI + - V 2 (28)
At the starting point W is the value of u because v - 0. Values of v2I
are calculated at each point (x 2 , y2 ) from the transonic solution,
and Eqs. (26) to (28) are iterated until convergence is reached. For
evaluating the term in brackets in Eq. (28), the ratio v/y is defined by 4the transonic solution even on the axis where both v and y are zero.This fact eliminates thbi general problem ia axisym-etric characteristics
solutions of evaluating the indeterminate sin */y in Eq. (23) on the
axis of symmetry.
It may be noted that the value of W as calculated from the character,-
istic value from Eq. (21) differs from the value (u2 + V 2)I12 calculated
from the transoutic equations, but the difference decreases wT:Uh in-
creasing R. For the final point of the throat characteristic which 1,us
on the axis, the value of d 3 u/dx3 from the transonic solutioa for the
axial distribution is "corrected" to make u - W for the axisymmetric
case fur values of R less than 12. The correction is abvut 16 perccnt
for R I and decreases rapidly as R increases. This correction is made
14
•
F • ' " i I i ' i I ii i i i ln n. . " . . ..
AE :)C-TR -78-63
so that v,'alues of du/dx and d udx 2 can be calculated from the transonic
solution for later application. The correction is believed to be JuatifiedI1•'•much as the accuracy of the transonic solution -s limited, particularly
for low values of R, because the series expression for u is truncated
after the x• term.
3.0 CEN'TERLMN DISTRIBUTION
In the radial flow region, the distance r, measured from the sourne,
is related to the locAl. Mach number by
Lr)+ _ (_L + h.-y+ I+ M 2)qy (29) i 'r Y+1 ++
or
)1+a W-(Y+' I W2-- (30)
First, second, and third derivatives of W -it M with respect to r/r can
be obtained as described in Ref. 12. Along the axis x - r when x is
measured from the source. Inasmuch as all, coordinates ma.st be normalized
by the same factor, r,, the transonic equation in terms of A/yo and y/yo
can be transormed by Eqs. (16) and (18), after which the distance froom
the source to the throat station must be taken I.-ntc account. This
latter distance is generally unknoun until after the distance from point
I to point E is determined.
In radial flow, the term on the right-hand side of Eq. (23) can beevaluated dimply. Inasmuch as sin y/r and d& - dr/cos v,
H ik9! S21 tan fy r
but
. -tan = (M2
-I ) 2
"15
AEDoC-TR-78-63
and, from Eq. (29) 'for a - 1,
. dr (2- 4M
r 2(1 +2 E: M2) Mi2
Thus,tnan s dr = (M 2 -1)
2 dM
r 2(1+-Z2 MN2 ) M2
I
From Eq. (25), do M(I +Y-1 M2 ) M
2
"therefore, Eq. (23), in radial flow, becomes
dot/ + do4 - 2- dot (31)21
which applies for characteristic AB or GF. Similarly, for the left-
running chaivacteristic EG,
do - do - ad (32)Thc.:efore, 2
13 - A + -G (33)
and G - OF, = (a + 1) (34)
and, from the design values n and MB (and/or ), MA, M, M' WE, and
the necessary derivatives can be calculated.
Within the accuracy of Eqs. (11) and (12), the second derivative of
velocity ratio at the sonic point is negative for values of R loss than
11.767 for planar flow and 10.525 for axcisyrmetric flow. The second
derivative of Mach number at the sonic point is positive for all values
of R. Inasmuch as the second derivative of either W or M is negative
for source flow, it seems better to use a velocity distribution rather
thau a Mach number distribution between points I and E. On the other
hand, a Mach number distribution between points B and C is preferable
16-IT
A E DC-T R -78-63
because the velocity ratio approaches the constant value of [(Y +
1)/Y _ 1)] 1/2 as the Mach number increases to infinity; therefore, the
change in velocity between points B and C becomes small relative to the
change in Mach number.
I.
The velocities and their first and second derivatives at points I
and E are used to determine the coefficients of the general fifth degreepolynomial
.,W C1 + C2 X + C3 X2 + C4 X3 + C'5X4 + C6X5 (35)where
X - (x - xl)/(XF - X[) (36)
Similarly, the Mach numbers and their first and second derivatives at
points B and C are used to determine the coefficients of the polynomial
M = 1) + D2 X + D3 X2 + 0)4 X + D5 X4 + D6X" (37)
where, in this case,X .: (x - XI)/(X. - xv) (38)
and the first and second derivatives at point C are usually set equal to
zero.
In these equations, the lengths (xE-xI) and (xC-xB) must be specified,
but can be determined by the conditions that C6 and D6 equal zero,
thereby reducing the polynomials to fourth-degree ones. If the velocity
at point E is determined by iteration, the third derivative at point I
or E can be included as a criterion for the fourth-degree polynomial;or, by setting C5 - 0, one can find a third-degree polynomial with a con-
stant third derivative. In either case, the Mach number at point B is
found from Eqs. (33) and (34) after the value at point E is found. All
"of these options are included in the program, but unless there are other
factors involved, the preferred options are the cubic between points I
and E and the quartic between points B and C.
17
- .'i- T-n i n
I
AEDC-TR-78-63
* For the cubic distribution for axisymmctric flow, the Mach number
at point E is related to the radius ratio as shown in Fig. 5 for y
1.4 for various values of inflection angle. Cross plotted are lines
of constant values of the ratio * E/n. Such values for most axisymmetric
nozzles lie in the range covered in this figure, and inasmuch as *F/n
h + 4, values of MF can also be obtained.
2,8
2.41,4 "-2-2, 22
1, 8 - 12
.11.6
1,2 4
0 4 8 12 16 20 24Radius Raik (R)
Figure 5. Relationships obtained from cubicdistribution of velocity from tonicpoint to point E for axisymmetricnozzle.
j18- - - - -' ,. -
AEDC-TR-78-63
In determining the length of the segment between points B and C,
using the fourth-degree polynomial distribution, there is a minimum
value of the Mach number at point B for the design Mach number at point
C. As given in Ref. 12,M= ,; + 0.715 M " /M (39)
where the primes indicate derivatives with respect to r/rI. This
relationship is shown in Fig. 6. For an axisymmetric nozzle designed
for a Mach number greater than about 3.4, the minimum Mach number at
20
+ ~18 /
16
14
,, 12
10 Axisymmetric
Planar
0 I . , I
0 2 4 6 8 10 12 14Minimum Mlach Number at Point B
Figure 6. Limitations of fourth-degree distribution
of Mach number from Eq. ?39).
point B is about two-thirds of the d2sign Mach number. Using such a
value visually causes the length to be •xcessive, and more realistic
19
zzzizI,7,.: ,. 71
* AEDC-TR-78-63
values of B are 75 to 80 percent of M It is important, however, as
illustrated in Ref. 16, that the distance between points B and C be
sufficient to allow for accurate machining of the contour between
points A and J, which lie on the characteristics through points B and C,
respectively.
4.0 INVISCID CONTOUR
The flow properties are determined at a desired number of points
along the key characteristics (i.e., the throat characteristic, TI, as
described earlier (a sub-multiple of 240 is used for subsequent calculations),
the characteristics EG and AB bounding the radial flow region by Eqs.
(33) and (34) for equal increments in n, and the final characteristic CDalong which the Mach number is constant and the flow angle is zero).
The flow properties are also determined at axial points from Eqs. (35)
and (37). The network of characLeristics is then calculated in the
region TIEG starting at point E and progressing upstream and in theregion ABCD starting at point B and progressing downstream.
The equations for a right-running characteristic were given previously.
dy/dx = tan(o - p,) (19)
+ ,. d (23)
wherede - dx/cos( -It) dy/sin(r/ - p) (24)
For a left-running characteristic, the equations are
dy/dx tan (0 + p) (40)
do- =b '!2s_0nL,uiJ d4 (41)y
where
Aldo dx/cos (0 + p) = dy/sin(0 + p) (42):, Also
do - .~J- dM acot I dW (43)(1 + 1.i M2) M W
2
[I 20
A__ __
AEDC-TR-78-63
Values of x, y, *,and M are known at the general point 1 on theright-running characteristic, C, and at the general point 2 on the left-
running characteristic, C. The characteristics intersect at the general
point 3 where the values are calculated by numerical integration of Eqs.
(23) and (41) along the respective characteristics.
S(si 6,mn A + , 02. n P~2 ) (44)
whereA4 c -( " 2) sec 1 (45)
andY3 Y2 tan~ 0 tan (0 3 + jA3) + 1. tan (02 + A2 (46)
3 222
-3 011 + (3- 01) = =
4, (Min0mi&AMn +. sinl 6minp (47
where (3- x1) sec. a (48)
and
=tan a Itan ((3 13 +1L tan(~ (49)X -X 2
Adding, substracting, and rearranging gives
03 (0 + - 0 + S6 + P2 + )(50)2
03 -(01 2 + 0,+ 02 + --(51)2
In planar flow, P1 p2 0 because a -0 and Eqs. (50) and (31)
can be solved directly, 14 is obtained from *j by t~he inverse applica-14-1
tion of Eq. (25), and vNsin (1/M ). In axisymnietric flow, the equa-
tions must be solved by iteration. A useful first approximation for P1
and P2 is the radial flow values, P1 -(p- 1)/2 and P2 ) ~ 3 /2.
21
.-- . .. ..
AEDC.TR-7S-63
At all points except on the axis in axisymmetric flow, Eqs. (44)
and (47) are defined because Y2 and y, are nonzero. On the axis, the
terms sin /Y2 and sin 0,/y, are indeterminate with the form zero/zero.
These indeterminates can be evaluated by assuming that the general
points 1 and 2 on the axis are very close together and that U f P s 13
and W OW W2 * W3 . Equation (41) can be written
Cot g L do + 'in 6 sin I dx (52)W y cos ( + -)
and Eq. 23 can be written
cot A dW= -d9 + sin 0 idnp dx (53)W y cos( S -)
as 964 9S-Psino, O±IA-.±ts
and tan 3 = i__i..-- =. 33 2 1 3
In finite-difference form,
co int an• IA.. (X3 -x2t a (W3 - W2) = 3 + . 3 3 2)
0 tan X sin 0. tan P (X3 - X2)
Y3 Y3
-- 2 sin 03 tan M3 (x3 - x2)/y 3 (55)
Similarly
S(Wl - W3) =3 + sin 0, tan /I3 (x1 - x3)/y 3 (56)Wa
-* 2 sin 03 tan p 3 (xl - x 3)/y 3 (57)
Adding Eqs. (55) and (57) and rearranging,
lia • =i °~ __ w (58)y4O Y 2 W dx
and s"' 0'2 Min (M1 -)Y (d' (59)
2 2
. , ,. . - . , ,, . , ./ '. , . . . , 2 2
AEDC-TR-78-63
for use in Eq. (44) when point 2 is on the axis, and
HilIl H st 410Yl 2W W \I (60)
for use in Eq. (47) when point I is on the axis.
In starting the calculation of the network of characteristics in
the region TIEG, point E becomes point 1 and the first axis point up-
steam of point E becomes point 2. The complete left-running character-
istic approximately parallel to EG is calculated, and the point on the
contour is determined from mass flow considerations as described in Ref.
17. The flow properties along this characteristic are then used to
calculate the next left-running characteristic, again starting on the
axis. This process is repeated until point I is reached, after which
the starting point for each left-running characteristic is a point on
the throat characteristic as illustrated in Fig. 7. The process in
region ABCD is similar except that right-running characteristics are
calculated for each point on the contour.1, 4
UI
1.0
0.8 -
Y0.6 "-
0.4 -
0.201
0 0.2 0.4 0.6 0.8 1,0 1.2 1,4 1,6 1.8 2.0x
Figure 7. Characteristics near throat of nozzlewith R = 1.
23
-. 'IL,
AEDC-TR-78-63
5.0 BOUNDARY-LAYER CORRECTION
To each ordinate of the inviscid contour must be added a correction
for the boundary-layer growth to obtain the viscid or physical contour
of the nozzle. Except for very low stagnation pressures, the boundary
layer is assumed to be turbulent. Generally, the boundary-layer cor-
rection will be made for one design condition of stagnation pressure and
temperature although it is theoretically possible to reshape a flexible-
plate type of planar nozzle to account for different boundary-layer
thicknesses corresponding to different stagnation conditions. The
correction for a planar nozzle is usually applied to the contoured walls
only, but the correction also allows for the growth of the boundary
layer on the parallel walls in order to maintain a constant Mach number
along the test section centerline. Therefore, the correction applied is
greater than the displacement thickness on the contoured walls, and the
flow in the test section is diverging in the longitudinal plane normal
to the contoured walls. In the longitudinal plane normal to the parallel
walls, the flow is converging because of the boundary-layer growth;
moreover, there is a tendency for the boundary layer to be thicker on
the wall centerline because of the transverse pressure geadients present
on the parallel walls. Although these physical effects make a true
correction impossible for a planar nozzle, the calculations described
herein are made as if the cross section were circular, with the cir-
cumference at each station equal to the periphery of the actual rec-
tangular cross section.
The method of calculating the boundary-layer growth is based on
obtaining a solution to the von Kfrmfn momentum equation written for
axisymmetric flow.
0I M 2 -M 2 +H dM + I dr C (61)dx M El + (y.-_) M2 /2 dx rw dx 2
The term [(I/rw)(drw/dx)1 becomes an effective one for planar flow as
just described. For either type of nozzle, the inviscid value is used
24
• , . ,.k
AEDC.TR -78-63
as a first approximation. The entire solution is iterated several times
with new values of rw and dr w/dx - tan 0w obtained each time by adding
vectorially the displacement thickness to the inviscid contour.
The value of mcmentum thickness used in Eq. (61) is defined by
PI -__d__(62)
0___ P'I '( 1-- q )d (2
where z is measured normal to the wall.
Also5
iiO. (,_ dz (63)
The quantities P* and 0 may be considered to be the displacement and
momentum thicknesses when the boundary-layer thickness is small with
respect to the radius, r . These values are related to tota2 valuesw
a* and 0, obtained from mass-defect and momentum-defect considerations
by12
- (64)
and -- .• 62 (65)
Because rw-6 coo Cw + y, where y is the inviscid radius, Eq. (64) mayw a~ wbe rearranged to give
4-- w I Y 2 SC2 2 - y sec 9W (66)
For the final correction, the value *a sec w is added to the in-viscid
radius in order that no correction be made to the longitudinal location.
The integrations of Eqs. (62) and (63) are performed numerically
using Gauss' 16-point formula, with the assumption of the power-law
velocity distribution
q/q, (67)
25 1
.-. ....... '..
AEDC-TR-78-63
ai and
pl/pý " TT (68)
whereT = T (Taw - Tw) q/q, + [Te - a - Tw) - Tw] (q/q.) 2 (69)
which is Crocco's quadratic temperature distribution if a - 1. 'However,
as shown in Ref. 12, a value of a - 0 gives a parabolic distribution
which agrees better wit1' data obtained in hypersonic wind tunnels with
water-cooled walls. The same distribution is obtained if T - Taw'w &
which is likely to be the case for planar, flexible-plate nozzles.
Before using the Gaussian integration, one must replace the values of z
and dz with 6(q/q dN and N6 (q/qe)N-1 d(q/qe), respectively, in order to
avoid the infinite slope, dq/dz, when q and z equal zero.
The value of the compressible skin friction coefficient, Cf, in
Eq. (61) is assumed to be related to an incompressible value, Cfi
by a factor F c, introduced by Spalding and Chi, Ref. 18,
Fc Cf =Cr f (70)
and Cf is related to an incompressible Reynolds number, R, which isfI
relateý to the compressible value, Re , by a factor F,c
FIa RO, RO (71)
The factor Fc, also used by van Driest, Ref. 19, is given by
L I/(p/pe)2 d (q/qe)] (72)
which uses Eqs, (68) and (69), In Refs. 18 and 19, a value of a ' 1
was implied, but Eq. (72) is used herein with a - 0 also, to give
a "modified" value of V . The factor Fc may be considered to be thei ~cratio of a reference temperature to the free-stream temperature. The
factor FR , as used by van Driest, isV F68 = ,/JAW (73)
26
-. Ni
Zi
A E DC-TR-78.63
The compressible momentum thickness, 6 upon which R is based is
the flat-plate valueC
li P% q P q" 'z 74
because the values of Fc and FR were developed to correlate flat-
plate data.
The equation used herein for incompressible skin-friction coef-
ficient is that of Ref, 20,
If (log H0 + 4.56]) (log t0o - 0.546)
This equation is believed to agree with experimental data slightly
better than the von Kgrm~n-Schoenherr equation,
C (0,242)2, (fI (log 110 + 1.1696) (log lie + 0.3010) (76)
at high Reynolds numbers. Also as shown in Ref. 20, Eq. (75) agrees
with the equation, Ref. 21, based on Coles' law of the wall and law of
the wake,S~L(2/C1 )2 fn 15 + 0.5 Fn (C /2) + YC + 211 (77)
if 1I varies as shown in Fig. 8 from about 0.41 at R6 - 400 to a maximum
of 0.5885 at Re - 50,000 and then decreases to about 0.49 at R.7i107. in order for Eq. (76) to agree with Eq. (77), I1 must continually
increase with increasing R0 as shown in Fig. 8. The data shown in Fig.
8 were computed by Coles iniRef. 21 from Wieghardt's flat plate data,
Ref. 22. A comparison of friction coefficients from Eqs. (75) and (76)
is shown in Fig. 9 together with Wieghardt's values as recomputed by
Coles. The constants K and C are 0.41 and 5.0, respectively. The
relationship between e and 6 is obtained from the logarithmic velocity
profile by neglecting the laminar sublayer, representing the wake function2
by a sine distribution, and integrating to obtain
____ (78)
j27
A ED C-TH -78-63
and
_8* C= i(~,. 2 179 11 + 1.5 n2) (Y9)
8 8 2K
•,'• I"0.6 ' 0 0
0.50
U From Eas. (76) and (77)'-Iata Tabulated In Ref. 21;
Identifled as Wloghardt Flat Plate Flow
0.3
0.2I~ l ll J I I l lll ] I I ~ ll I I I I I f ll I I I J I ll
103 104 105 106 107
Figure 8. Variation of wake parameter, n, withReynolds number (incompressible).
The value of N in Eq. (67) is assumed to be a function of Reynolds
number based on the actual boundary thickness, not corrected by FR
and is evaluated through the use of the kinematic momentum thickneis
q I- dz (80)from which
Ok/6 N/(N2 + 3N + 2) (81)
or
N -a+ - 6\ + 1I (82)
28
I
A EDC-TR -78-63
0 . 00 6 r I I I I [1 I T l l 1 TrTiJ I l f l I ! I I 1 1 1 1
0,003 5
O.004 Wleqhardt's Flat Plate DataTabulated In Ref, 21Ct,
0.003
0.002 Eq .(75)
0. L0 Eq. (7610
0 1 1 11 tl 1 1 f 111 ilj ,J I J iIlll ,I I I I ILill'
104 1 5 106 Ia7Re,
Figure 9. Variation of skin-friction coefficient withReynolds number (incompressible).
The value of 0k/ 6 is obtained from Eq. (79), where the value of nI isevaluated from Eqs. (75) and (77) with ek used instead of 0i. The re-sulting variation of N with Rd is shown in Fig. 10.
Two options contained in the program subroutine for the boundary
layer utilize Coles' law of corresponding stations (Ref. 23),
(83)
if Cf /Cf . F is calculated from Eq. (72) for a - 0 or a 1 1, then onei c
option gives
F 1 1 8 ~ ~/I~0''~ ~(84)
The second option divdes Eq. (83) into the two parts,
Cfl/( =f I' /T,, P11 (85)
29
AEDC,.TR-78-63
and
= !S/!L.(86)
where pis evaluated at the temperature
'T', . + 1?.2(Cf/2)2 a(Ta - T) 305(Cf,/2) [a(Taw - Tw) + Tw T.](87)
10
.1 8
N
6
104 106R6
Figure 10. Variation of velocity profile exponent with Reynoldsnumber based on boundary-layer thickness.
Still another option defines the incompress~ible skin-friction
coefficient as
Cf 0,0888(log 118 + 4.6221) (log 118 1.4402) (88)
where
CTWAW (89)
and F Cis calculated from Eq. (72).
30
V t.
AEDC-TFR-78.63
The wall temperature in the above equations can be the adiabatic
wall temperature or can be allowed to vary between a throat wall tem-
perature, T WT, and a nozzle-exit wall temperature, T WD, both of which
are input to the program. Two options are available for the variation
of wall temperature,
I- (A i/A*)'- I A/A* (90)
where m can be 1/2 or 1, A/A* is the area ratio corresponding to local
Mach number, and Ac/A* is the area ratio corresponding to the design Mach
number at the nozzle exit. Equation (90) is used in lieu of more
accurate values and approximates the way the heat transfer decreases as
the Mach number increases from 1 at the throat to the design value at
the exit. For a water-cooled throat, the value of T can also becalculated by the program, T
T ' + 00 I('l' l)-1 )'I; T= (91)
h. + 0
where h is the airside heat-transfer coefficient at the throat asacalculated by Reynolds analogy from the throat skin-friction coef-
ficientSp p C 2/ c2 (92)
with a constant specific heat based on the thermochemical BTU
T (y -. i) 777,64885, (93)
and Q is an input which is a function of the properties of the throat
material, the cooling water, and the geometry and would be a constant if
the properties were constant. The assumption is made that the bulktemperature of the water is 15*F less than T and that p2 /3 is the
square of the recovery factor used to obtain Vhe adiabatic wall tempera-
ture, T •
S31
AEDC.TH ,78.63
For "he integration of Eq. '61), the values of x, y, dy/dx, M,
and d&/dy .. : obtained from the iviscid contour at unevenly spaced
points as a result of the characterio:t:cs solution. With the inputs of
stagnation pressure and temperature, ga,• constant, and recovery factor,
the unit Reynolds number and static and adiabatic wall temperatures can
be calculated at the same points as functions of Mach number withSutherland's equation used for viscosity. With the inputs of T andw_
TWD, the wall temperatures can also be calculated as functions of Mach
number, although T may need to be obtained by interation if thewToption to input a value of Q is exercised. Sutherlandts equation is
also used with wall temperatures to obtain the viscodities at the wall.For any static temperature below the Sutherland temperature, 198.72*R asused herein, the viscosity variation with temperature is assumed to be
linear.
The integration of Eq. (61) is started at the throat where it is
assumed that dOidx w 0 in order to obtain a value of e. Iteration is
involved at each point because C f is a function of Reynolds number basedupon 8, and the relations e/6 and 8*/8 depend upon the value of N,
which is a function of Reynolds number based upon 6. After all itera-tions converge within specified tolerances, the value of P is calculated
afrom the value of 6*, and the values of B and dO/dx are used in the
calculation at subsequent points. The values of dO/dx are integratednumerically to obtain the increment in 0 to be added to a previouslydetermined value of 8. The trapezoidal rule is used to determine the
second point, the parabolic rule for the third point, and cubic integra-tion for the fourth and subsequent pointa.
For convenience, Eq. (61) may be written 8' + 8P - Q. The general
integration for the nth point is
On =On3 + Gu_3 1 n 3 + Gn-2 0 n-2 +I Gn 1+ Gn 0. (94)
32
S.... .. . : ': : .... . . .-- • " • -- • ' .... .. • -.. . . . ..• " + + + "+" , i ................. ..... ..........
AF DC.TR-78-63
where th' C's are functiw~s of the spacings s, t, and u between thepoints and are given in Appendix B. Except tor e and ef, the other
n fnvalues in Eq. (94) are known from previous calculations. Inasmuch as
0 Q P 0 (95) 1Eq. (92) can be rearranged to g(ve
on. (011_,1 111 0.,, : 3 1. "_• + G t + G; 0' 4.- G Q. .(6
(0 + Gri )
After convergence of the iterations, Eq. (95) is used to obtain de/dx.
Inasmuch as Eq. (94) depends upon the knowledge ot en-3, the value of
a2 is calculated by
on-2 = On-. 3 + "' n• - + + "'n-1 0,1-1 + p, lon (97)
which becomes the Gn3 for the next point to be calculated. The valuesof the Fls are also given in Appendix B. The values of 02 and e3 obtained
from Eq. (95) are used in the calculation of P* and 6P instead of thea
initial values obtained by the trapezoidal or parabolic integration.
The success of the above type of integration depends upon thespacing of the points. The values of the increments s, t, and u must
be of the same order of magnitude, although t is usually larger than s
and smaller than u if the parameters itnvolved in the characteristicssolution are selected with care.
After the values of 6* sec w are calculated, the values ofa w
d(6* sec 0 )/dx are obtained by parabolic differentiation and added toathe inviscid values of dy/dx to obtain dr /dx. This procedure is believed
wto be more accurate than differentiating the value (* sac w + y)
.a wbecause dy/dx is obtained directly from the characteristics solution andnot by differentiating y with respect to x.
In general, the boundary-layer correction at the throat will have
a gradient such that the viscid throat will be slightly upstream of the
33
AEOC-TR-78,63
inviscid throat. This displacement and the value of the viscid curve-
ture at the throat are calculated using the assumption that both the
inviscid throat and the boundary-layer correction are parabolic in
shape.
6.0 DESCRIPTION OF PROGRAM
The computer program is written in Fortran 1V for use with the IBM
370/165 Computer. The program consists of a main section, three functions,
and 16 subroutines arranged so that the program can be overla4,d toconserve computer storage. The four overlays consist of AXIAL, CONIC,
SORCE, and TORIC; PERFC; BOUND and HEAT; SPLIND and XYZ. The inputdata cards are described in Appendix C, and a listing of the program is
given in Appendix D.
Program MAIN. MAIN calls for the various overlays. The title card is
read in with the designation as to whether the nozzle is planar or
axisymmetric. A card defining the gas properties and a few pertinent
dimensions is then read in. The first subroutine called is AXIAL, in
which the upstream axial distribution is defined. PERVC Is called to
calculate the upstream contour. AXIAL is recalled to define the downstream
distribution, and PERFC is recalled to calculate the downstream contour.
BOUND is called to calculate the boundary-layer growth. SPLIND is
called to determine the coefficients of cubic equations to fit the
unevenly spaced points along the contour, and XYZ uses these coeffici-
ents to obtain ordinates at evenly spaced points along the axis or, in
the case of the planar nozzle, at discrete points along the surface ofthe flexible plate at which the supporting jacks are located.
Subroutine AXIAL. In this subroutine, cards are read in with theparameters used to define the axial distributions of velocity and/or
Mach ,Lumber and with integers which define the number and spacing of thepoints on the axis and on the key characteristics and the sequence of
34
AEDC-TR-78-63
subsequent calculations. If the throat characteristic is called for,
the upstream end of the upstream distribution starts at the intersection
of the throat characteristic and the axis. An option can be exercised
to not use the throat characteristic and thereby start the distribution
at the point where M - 1. This option would normally be used for a
nozzle with a large throat radius of curvature, e.g. a planar nozzle, or
if it were desired to repeat a calculation as in Ref. 13. Another option
is to avoid a radial flow section altogether by using a polynomial dis-
tribution from the throat to the beginning of the test cone or rhombus.
Other options will be described in Appendix C when the input cards are
discussed.
Subroutine BOUND. This subroutine is used to calculate the turbulent
boundary-layer correction to the inviscid contour. The stagnation
conditions are input, as are the parameters to describe the wall tem-
perature distribution, the temperature distribution in the boundary
layer, and the factors relating the compressible skin-friction coefficients
to incompressible values.
Subroutine CONIC. This subroutine is used within AXIAL to give the
derivatives of Mach number with respect to r/r 1 in radial flow from Eq.
(29).
Function CUBIC. This subroutine is used to obtain the smallest positive
root of a cubic equation.
Function FMV. This subroutine determines the Mach number for a given
Prandtl-Meyer angle.
Subroutine FVDGE, This subroutine is used within PERFC in conjunction
with NEO Lo smooth the inviscid coordinates as desired.
35
, .-
AEDC-TR-78-63
Subroutine HEMT. This subroutine is a dummy called by BOUND but is
included so that with a more elaborate subroutine a heat balance can be
made to determine the wall temperature if the material conductivity is
specified and the cooling water passage geometry and quantity of flow
are specified.
Subroutine NEO, This subroutine is used with PERFC in conjunction with
FVDGE to smooth the inviscid coordinates as desired by modifying the
ordinate such that the second derivative is more nearly linear after
smoothing than beforehand,
Subroutine OFELD. This subroutine is used within PERFC to calculate the
properties at the intersection of a left- and a right-running char-
acteristic.
Subroutine OREZ. This subroutine is used to make all values of an array
equal to zero prior to a new calculation.
Subroutine PERFC. In this subroutine, the properties along the key
characteristics are first calculated to go with those along the axis.
The intermediate characteristics are then calculated and the contour
points obtained by integrating the mass flow crossing each character-
istic. If desired, certain designated intermediate characteristics may
be printed out. If smoothing of the ordinates is desired, the inputs
associated with the smoothing are read and the smoothing applied.
Inasmuch as the wall angle is interpolated from mass-flow considera-
tions, independently of the coordinates, the wall slopes are integrated
from the inflection point toward the throat for comparison with the
"interpolated ordinates. Parabolic integration is used for this purpose
as well as for the mass flow. Also calculated for comparisorn are the
ordinates of a parabola and a hyperbola which have the same radius
ratio, R, inasmuch as the transonic solution should be equally applic-
able to these shapes for the number of terms retained in the series,
36U ,,
AEDC-TR-78-63
Eqs. (2) and (3). Finally, the scale factor, the value of rI in inches,
is applied to obtain the inviscid coordinates in inches, and the abscis-
sas are also shifted as desired.
Subroutine PLATE. This subroutine is also a dummy to allow additional
calculations to be made for a flexible plate contour after the coordinates
at each Jack location have been interpolated by SPLIND and XYZU
Subroutine SCOND. This subroutine is used in BOUND, NEO, and PERFC for
parabolic differentiation of coordinates to obtain the slopes, or of
slopes and abscissas to obtain second derivatives. Three points at a
time are used to establish the parabola, and the slope is obtained at
the center point. The slopes at the first and last point are also obtained,
but with less accuracy.
Subroutine SORCE. This subroutine is used within AXIAL to give the
derivatives of velocity ratio, W, with respect to r/rI in radial flow
from Eq. (30).
Subroutine SPLIND. This subroutine computes the coefficients of cubic
equations that fit the unevenly spaced points obtained from the char-
acteristics solution. The initial and final slopes are used together
with the coordinates to determine the cubic coefficients.
Function TORIC. If the velocity gradient is known at the axial point
where M - 1, this function gives the value of radius ratio, R, which
would produce such a gradient from the transonic theory used. This
function is used in AXIAL if the option is exercised of specifying the
Mach number at point F but not specifying the value of R. It is also
used to determine the value of R for calculating streamlines other than
the contour itself.
37
A E DC-TR-78-63
Subroutine TRANS. This subroutine calculates the throat characteristic
from the transonic theory. In AXIAL, at the point where the throat
characteristic intersects the axis, the derivatives of velocity and
Mach number are used to determine the coefficients of the polynomial
describing the axial distribution. In PERFC, the flow properties along
this key characteristic are used at the number of points specified as
one plus a submultiple of 240.
Subroutine TWIXT. This subroutine is used in PERFC and BOUND to inter-
polate the ordinate and other properties at a specified point. A four-
point Lagrangian interpolation is used with two points on either side of
the specified point.
Subroutine XYZ. This subroutine uses the cubic coefficients obtained in
SPLIND for calculating the ordinate, slope, and second derivative at
specified values of the abscissa read as inputs in the MAIN section of
the program. The points may be at even intervals in the abscissa or at
arbitrary uneven intervals. The points may be the same points as those
input to SPLIND if a comparison is desired between the derivatives so
determined and those obtained el8ewhere in the program.
7.0 SAMPLE NOZZLE DESIGN
The design of a Mach 4 axisymmetric nozzle is selected to illus-
trate use of the computer drogram. The input cards for the sample
design are given in Table 1. An axiiymmetric nozzle is specified by
leaving JD blank (JD - 0) on Card 1. Leaving SFOA blank ou Card 2
specifies that the upstream axial velocity distribution is not a fifth-
degree polynomial. Leaving FMACH blank cn Card 3 specifies that the
value of FMACH will be computed by the program, and leaving IX blank on
Card 4 specifies a cubic distribution. The computed value of FMACH is
3.0821543, which is greater than the value of BMACH specified on Card 3;
t38
4
A ED C-TF-78.63
therefore, BMACH also becomes 3.0821543. The negative value of SF
means that the inviscid exit radius of the nozzle is 12.25 in. The
value of PP means that the inflection point will be 60 in. downstream of
an arbitrary point. Leaving XC blank specifies the downstream axial
distribution will be a fourth-degree polynomial, and the positive value
of IN on Card 4 specifies a Mach number distribution. The values of MT,
NT, MD, ND, NF, and LR determine the number of points on the key char-
acteristics and are all odd numbers because each includes both end
points of each distribution which is divided into an even number of
increments. The negative value of NF specifies the contour points to be
smoothed according to Card 5, and the negative value of LR specifies
that the transonic distribution be printed as the first page of the
sample output. The NX value of 13 specifies the spacing of the axial
points between points I and E to be close together near Point I with the
last increment about 3.17 times as large as the first increment,
(201.3 191.3). The JC value of 10 specifies that every 10th left-
running characteristic will be printed for the upstream contour together
with the right-running characteristic through Point E. The smoothing
incegers on Card 5 are used to control the emoothing subroutine.
Table I. Input Cards for Sample Design
CARD IITLE JU
MACH4
CARD 2GAM AN ZO RO VISC VISM SFOA XAL144 1716s563 1. 0,896 2.26968E-8 198,7a 10000
CARO 38.67 60 3. 4. -12.25 60.
FTAD RC FMACH BMACH CNC SF PP MC
CARD 4MT NT Ix IN IG MO NO NF MP MG dA Jx JC IT LR NX41 21 10 41 49 -61 1 10 -21 13
CARD 9NOUP NPCT NODO
5S 85 S,)
CARO 6PPQ TO TwT IWAT GFUN ALPH IHT IR I1 LV200. 16J8* 900. 540. .36 1 5
CARD 7XSt XLOW XENO XINC Hi XMID XINC2 CN1000. 46. 172. 2.
39
-.
A EDC-TR-78-63
For the boundary-layer calculations for stagnation conditions of
200 psia and 1638R, the value of QFUN of 0.38 overrides the specified
throat temperature of 900R and produces the throat temperature of 866R
as indicated on the output. Leaving ALPH blank causes the temperature
distribution in the boundary to be parabolic for both the calculation of
the boundary-layer parameters and the calculation of the reference
temperature. Leaving IHT blank causes the longitudinal distribution of
wall temperature to vary as a square-root function of the area ratio
* corresponding to the local Mach number; m a 1/2 In Eq. (90). ILeavi.ng IR
:' blank causes the transformation from incompressible to compressible
values of skin friction coefficient to be calculated using a modified
Spalding-Chi reference temperature and a Van Driest reference Reyuolds
number. Specifying ID - I takes into account that the boundary-layer
thickness is not negligible relative to the radius of the inviscid core,
* and its positive value causes the boundary-layer calculations to be
printed for the first and last iteration; the number of iterations is
specified by the absolute value of LV (LV m 5 fcr the example).
For the final coordinates, interpolated at even intervals, speci-
fying XST - 1,000 (the same value as XBL on Card 2) keeps the X-coordinates consistent with the location of the inviscid inflection
point at 60 in. downstream of an arbitrary point.
The main parameters selected for the sample problem were the inflec-
tion angle, the curvature ratio, and the Mach number at the point B.
The selected values of 8.67 deg, 6, and 3.0821543 (computed), respectively,
are not necessarily optimum but result in a nozzle with an upstream
length of about 14 in. from the throat to the inflection point, a
length of about 31 in. from the inflection point to point 3 (see Fig. 3),
and nearly 120 in. from the inflection point to the theoretical end
of the nozzle. Such dcwnstream lengths are probably conservative and
could be reduced to some degree although experience with Mach 4 axisym-
metric nozzles is very limited.40
.:•,.40
"AEDC-TR-78-63
The number of points used on the key characteristics should be con-
sistent with the number of points used in the axial distributions in
order that the individual nets in the characteristics network should not
become too elongated (e.g., see Fig. 7). The spacing of the points on
the final contour should also progress in an orderly manner. Several
trials may be necessary to optimize the various inputs to the program.
8.0 SUMMARY
A method and computer program have been presented for the aero-
dynamic design of planar and axisymmetric supersonic wind tunnel noz-
zles., The method uses the well-known analytical solution for radial
source flow and connects this radial flow region to the throat and test
section regions via the method of characteristics. Continuous curvature
over the entire contour is attained by specifying polynomial distribut-
ions of the centerline velocity or Mach number and matching various
derivatives of these polynomials at the extremities of the radial flow
region, the test section, and a throit characteristic. The inviscid
contour is obtained by initiating characteristics outward from the
centerline and then integrating the mass flux along these character-
istics to compute the inviscid nozzle boundary. The final wall contour
is then obtained by adding to the inviscid coordinates a boundary-
layer correction based on displacement thickness computed by integrating
the von KArmAn momentum equation. To illustrate the method, a sample
design calculation was presented along with the associated input and
output data. A listing of the computer program and an input descrip-
tion are included.
REFERENCES
1. Prandtl, L., and Busemann, A. "Nahrungsverfahren zur zeichnerischen
Ermittlung von ebenen Stromungen mit uberschall Geschwindigkeit."
Stodola Festschrift. Zurich: Orell Susli, 1921.
41
rAEDC-TR-78-63
2. Foel2ch, K. "A New Method of Designing Two Dimensional Laval
Nozzles for a Parallel and Uniform Jet." Report NA-46-235-1,
North American Aviation, Inc., Downey, California, March 1946.
3. Riise, Harold N, "Flexible-Plate Nozzle Lesign for Two,.Dimensional
Supersonic Wind Tunnels." Jet Propulsion Laboratory Report"No. 20-74, California Institute of Technology, June 1954.
4. Kenney, J. T. and Webb, L. M. "A Summary of the Techniques of
Variable Mach Number Supersonic Wind Tunnel Nozzle Design."
AGARDograph 3, October 1954.
5. Sivells, J. C. "Analytic Determinatiton of Two-Dimensional Super-
sonic Nozzle Contours Having Continuous Curvature."
AEDC-TR-56-11 (AD-88606), July 1956.
6. Owen, J. M. and Sherman, F. S. Fluid Flow and Heat Transfer at
Low Pressures and Temperatures: "Design and Testing of a
Mach 4 Axially Symmetric Nozzle for Rarefied Gas Flows."
Rept. HE-150-104, July 1952, University of California,
Institute of Engineering Research, Berkeley, California.
7. Beckwith, I. E., Ridyard, H. W., and Cromer, N. "The Aerodynamic
Design of High Mach Number Nozzles Utilizing Axisymmetric Flow
with Application to a Nozzle of Square Test Section."
NACA TN 2711, June 1952.
8. Cronvich, L. L. "A Numerical-Graphical Method of Characteristics
for Axially Symmetric Isentropic Flow." Journal of the Aero-
nautical. Sciences, Vol. 15, No. 3, March 1948, pp. 155-162.
9. Foelach, K. "The Analytical Design of an Axially Symmetric
Laval Nozzle for a Parallel and Uniform Jet." Journal of
the Aeronautical Sciences, Vol. 16, No. 3, March 1949, pp.
161-166, 188.
42"i•' ' 4
AEDC-TR -78-83
,* 10. Yu, Y. N. "A Summary of Design Techniques for Axisymmetric
*• Hypersonic Wind Tunnels." AGARDograph 35, November 1958.
11. Cresci, R. J. "Tabulation of Coordinates for Hypersonic Axisym-
nmetric Nozzles Part I - Analysis and Coordinates for Test
Section Mach Numbers of 8, 12, and 20." WADD-TN-58-300,
Wright Air Development Center, Dayton, Ohio, October 1958.
12. Sivells, J. C. "Aerodynamic Design of Axisymmetric Hypersonic
Wind-Tunnel Nozzles." Journal of Spacecraft and Rockets,
Vol. 7, No. 11, Nov. 1970, pp, 1292-1299.
13. Hall, I. M. "Transonic Flow in Two-Dimensional and Axially-
* Symmetric Nozzles." The Quarterly Journal of Mechanics
and Applied Mathematics, Vol. 15, Pt. 4, November 1962,
pp. 487-508.
14. Kliegel, J. R. and Levine, J. N. "Transonic Flow in Small
Throat Radius of Curvature Nozzles." AIAA Journal, Vol. 7,
No. 7, July 1969, pp. 1375-1378.
15. May, R. J., Thompson, H. D., and Hoffman, J. D. "Comparison
of Transonic Flow Solutions in C-D Nozzles." AFAPL-TR-
74-110, October 1974.
16. Edenfield, E. E. "Contoured Nozzle Design and Evaluation for
Hotshot Wind Tunnels." AIAA Paper 68-369, San Francisco,
California, April 1968.
17. Moger, W. C. and Ramsay, D. B. "Supersonic Axisymmetric Nozzle
Design by Mass Flow Techniques Utilizing a Digital Computer."
AEDC-TDR-64-110 (AD-601589), June 1964.
AEOC.TR 78.63
18. Spalding, D. B. and Chi, S. W. "The Drag of a Compressible Turbulent
Boundary Layer on a Smooth Flat Plate With and Without Heat
Transfer." Journal of Fluid Mechanics, Vol. 18, Part 1,
January 1964, pp. 117-143.
19. Van Driest, E. R. "The Problem of Aerodynamic Heating."
Aeronautical Engineering Review, Vol. 15, No. 10, October
1956, pp. 26-41.
20. Sivells, J. C. "Calculation of the Boundary-Layer Growth in a
Ludwieg Tube." AEDC-TR-75-118 (AD-A018630), December 1975.
21. Coles, D. E. "The Young Person's Guide to the Data." Proceedings
AFOSR-IFP-Stanford 1968 Conference on Turbulent Boundary Layer
Prediction. Vol. II, Edited by D. E. Coles and E. A. Hirst.
22. Wieghardt, K. and Tillmann, W. Zur Turbulenten Reibungsschicht
bei Druckanstieg. Z.W.B., K.W.I., U&M6617, 1944, translated
as "On the Turbulent Friction Layer for Rising Pressure."
NACA-TM-1314, 1951.
23. Coles, D. E. "The Turbulent Boundary Layer in a Compressible
Fluid." RAND Corporation Report R-403-PR, September 1962.
AEDC-TH-78-63
APPENDIX ATRANSONIC EQUATIONS
When Eq. (5) is substituted into Eqs. (2),(3) and (4), Eq. (2)
can be written as:1 GR GS
u i 3-- cy)S -S2 S3 "S GT
+ x(i -aF + GT +Gv'' x,8S 2 /
+2X2 -2 V + )- + 33 K ++,-/-(1 - -• .- ...
2 S 3y2 y4 U Y2 U y6 U y4 +U 2 y
2+ 2+ 4 + 4 ... U2 + 63 . 43 + 23
2S _2__3
S y42(2+ UxP2 Y U0 PO )
222
+ 3 - (10 - 30)y\+2 \ 4S + A i
where the coefficients are written in the terminology of the program
and x and y are normalized with respect to yo. For planar flow,
GR - (15 - y)/270 (A-2)
US (782 2 + 3507 y + 7767)/272160 (A-3)
GT - (134 y2 + 429 y + 123)/4320 (A-4)
av - 5 y/18 (A-5)
GK - (2y 2 - 33y + 9)/24 (A-6)
U4 2 - (y + 6)118 (A-7)
U2 2 - y/9 (A-8)
45
- ,j .1,, i
AEDC-TR-78.63
63 = (362 y2 + 1449 ,y + 3177)/12960 (A-9)U43 y
43 ' (194 y + 549 y - 63)/2592 (A-10)
U2 3 (854 yL + 807 y + 279)/12960 (A-11)
" Up2 (26 y + 27 y + 237)/288 (A-12)
2LU0 = (26 y + 51 y - 27)/144 (A-13)
For axisymmetric flow,
GR - (15 - 10 y)/288 (A-14)
GS - (2708 y 2 + 2079 y + 2115)/82944 (A-15)
GT w (92 y2 + 180 y - 9)/1152 (A-16)
GV - (y + 0)/8 (A-17)
* GK - (4 y2 - 57 y + 27)/48 (A-18)
U42 (2 y + 9)/24 (A-19)
U2 2 (4 y + 3)/24 (A-20)
U6 3 - (556 y + 1737 y + 3069)/10368 (A-21)
U4 3 -(388 y2 + 777 y + 153)/2304 (A-22)
U Y2U2 3 (304 y + 255 y - 54)/1728 (A-23)
P2 - (52 + 51 y + 327)/384 (A-24)
Up0 - (52 y2 + 75 y - 9)/192 (A-25)
The first part of Eq. (A-i), which is independent of y, can be recognized
as Eq. (11) for planar flow or Eq. (12) for axisymmetric flow inasmuch
as x and y are normalized here with the value of yo.
46
own'!
--- ... :-• •ig lo o ") i:- ,, _ T3•:I• r"l~:IIr•I ]1'
AEDC-TR-78-63
In a similar manner, Eq. (3) can be written as
4 2
2-) 42 V22 +V 0 2NS• 2(3 •)s + s2
6 v y4 +V 2+ V6 3 - 4 3 +v 2 3 y -V 0 33'
2+ x I (2y +- 12 - 3+)y 2y 1 .5q
(9 - 3o)S
6U6 3 y4 -4 U4 3 + 2 U2 3 +,
S2
( 4 U2 '
X12x2 (2 Up PO+
2 S -2
+ x3 "4 y .. )/ (A-26)
For planar flow,
V4 2 - (22 y + 75)/360 (A-27)
V22 - (10 y + 15)/108 (A-28)
V0 2 - (34 y - 75)/1080 (A-29)
V6 3 - (6574 y + 26481 y + 40C59)/181440 (A..30)
" V4 3 - (2254 y2 + 6153 y + 2979)/25920 (A-,31)
2
V2 3 - (5026 y2 + 7551 y - 4923)/77760 (A-32) '
-03 (7570 y + 3087 Y + 23157)/544320 (A-33)
47 ,
44'i
IIIi: t A EDC-TR.78.•03
L, , t
, For axisymmetric flow,
v4 2 " (Y + 3)/9
v,22 (20 y + 27)/96(A-35)
V0 2 " (28 y - 15)/288 (A-36)
V6 3 - (6836 y2 + 23031 Y + 30627)/82944 (A-37)
V4 3 " (3380 y2 + 7551 Y + 3771)/13824 (A-38)
V2 3 - (3424 y2 + 4071 y - 972)/13824 (A-39)
"0O3 - (7100 Y2 + 2151 Y + 2169)/82944 (A-40)
48
* I , ,
48i
A EDC-TR -78-63
APPENDIX :CUBIC INTEGRATION FACTORS
If a curve through four points with ordinates a, b, c, and d,spaced at uneven increments in abscissa, s, t, and u, is definedby a cubic equation, the area under each section of the curve canbe found in the following manner:
Area b F a+ b-+ Fc +dd (B-i)a-b as Fbs cs Fds
Areab-c at a + Fbt b + F c + d (B-2)F Aresc-d "F a + Fbu b + F c +F d (B-3)
c- u u cu Fdu -3li Itota G a Gb + G c + Gd d (B-4)
where
2S(s + 4t (+-5)
•",2 I Zs + 4t + 2u
iij a + 2SFbs +--)(B-6)bs 2 12 t(t + u)
a 3 (s + 2t +(2B)cs T2tu(a + t) 's- 7)
I:: s_(L±+ 2t,)Fds 12 (s + t + u)(t + u)u (B-B)
t 3t+2u (-9)at 12s(s + t)(s + t + u)F - 2+ t + 2u
(bt 2 12s(t + u)
t 2 (2s + t -2B1)F " ct 2-u(s + t) (B-11)
49
_ _....__ _ _ _ _ _ __._ _ %.-- - .,.
AEDC-TR-78-63
t 3 (2s + t) (B-12)Fdt -- 12u(t + u)(s + t + U)
F u u3(2t + u)F = 32 )(B-1 3)
au 12s(s + M)s + t + U) '3iu3(2s + 2t +_y (B14
rbu - 12st(t + u) (B-14)
u (2s + 4t + u) (-5cu 2 12t(s + t)
..R u2 (2s + 4t + 3u) (B-16)du 2 12(t + u)(s + t + u)
G -F + Fat + F (B-17)
Gb bs + Fbt + Fbu (1-18)
Gc F + + F u (B-19)
G -F + F + F (B-20)d ds dt du
If all increments are equal, then
s t - u h (B-21)
Fds -F at F -dt -Fau -h/24 (B-22)
F -F-5h/24 (B-23)Co •bu
50
AEDC-TR-78-63
F = F - 19h/24 (B-24)
Fas Fdu -9h/24 (B-25)
F bt Fct - 13h/24 (B-26)
Gt 3h/
d 3h/8 (B-27)
G - c G 9h/8 (B-28)
The values of G's in Eq. (96) correspond to those in Eq. (B-4).The value of F's in Eq. (97) correspond to those in Eq. (B-i).
* '5 1
2 2$WA,,
j'""
AEDC-TR-78 -3
APPENDIX CINPUT DATA CARDS
Input Columns
Card 1
ITLE 2-12 Title
JD 14-15 Blank (0) for axisymmetric contour,-1 for planar.
Lard 2
GAM 1-10 Specific heat ratio.
AR 11-20 Gas constant, ft2/sec2 R.
ZO 21-30 Compressibility factor for an axisym-metric nozzle, constant for entirecontour. Or, for a planar nozzle, ZOis half the distance (in.) between theparallel walls, and the compressibilityfactor is one.
RO 31-40 Turbulent boundary-layer recovery factor.
VISC 41-50 Constant in viscosity law.
V1SM 51-60 Constant in viscosity law. If VISM isequal to or less than one,
VISC* T lb-sec/ft 2
If VISM is greater than one,
1.5VISC*T 2"
11 VI- T- lb-sec/ft2 . IfS"T + VISM
T is greater than VISM,
VISC* T ; T <VISM.2 VISM1/
2 -
SFOA 61-70 Used for nozzle with radial flow regionif 5th-deg axial velocity distributionis desired. If positive, the distance,in inches, from the throat to Point A
52
'- .~-
AEDC.TR-78-63
on the characteristic diagram. If nega-tive, absolute value is distance from thethroat to Point G. If Blank, 3rd- or 4th-deg distribution iu used depending on valueof IX on Card 4.
XBL 71-80 Station (in.) where interpolation isdesired (e.g., the end of a truncatednozzle). If XBL1000., the spline fit
subroutines are used to obtain values atincrements evenly spaced in length.
Card 3
ETAD 1-10 Inflection angle in degrees if radial flowregion is desired. Two characteristicsolutions are obtained, one upstream andone downstream of Point A. If ETAD = 60.,the entire centerline velocity distributionis specified and only one solution isobtained and the inflection point must beinterpolated. If ETAD - 60., IQ 1 1, IX - 0,on Card 4.
RC 11-20 Ratio of throat radius of curvature tothroat radius. Must be given if ETAD - 60.or FMACH - 0. If FMACH is given, RC iscalculated. If LR - u, IX = 0 givws third-deg equation betweev Mach 1 and EMACH,matching first and second derivations ateach end. If LR 0 0, the value of RC foundfor LR = 0 is used with given value of FMACHto define a fourth-deg equation. If IX = ±1and FMACH is given, RC is calculated todefine a fourth-deg equation. If LR ý 0,a new value of FMACH is found, compatiblewith the value of RC calculated for LR = 0.
FMACH 21-30 Mach number at Point F if ETAD # 60. Nega-.tive value specifies Prandtl-Meyee angleat Point F as IFMACHI *ETAD (usually around-7). If PMACH and RC are given, IX = 0and 4th-deg distribution is used. IfFMACH - 0 and IX = 0, a 3rd-deg distribu-tion is used. If FMACH = 0, and IX = ±i,a 4th-deg distribution is used. FMACH is
calculated if not given. If ETAD = 60.,Point F is not defined.
53
. . . .
•'•'• •'• • '••'• .... .. ' • •±• ,"• •P, --- • "-- --'; '• " • .. ... "• . .............. .. ........ ............. ..
AEDOCTR-78-63
BMACH 31-40 Mach No. at Point B if ETAD • 60.
(MC 41-50 Absolute value is design Mach No. at PointC. If ETAD 0 60, positive CMC giyes d 2M/dx 2
-0, and negative CMC gives d2 M/dx' 0 0. If jETAD - 60., CMC ts positive.
SF 51-60 Scale factor by which nondimension coordi-nates are multiplied to give dimensionsin inches. If SF - 0, nozzle will have aninviscid throat radius (or half-height) ofI in. If negative, nozzle will have aninviscid exit radius (or half-height) ofISF1 in.
PP 61-70 Station (in.) at Point A. PP - 0 givescoordinates relative to geometric throat.Negative PP gives coordinates relative tosource or radial flow (ETAD 0 60.).
XC 71-80 Nondimensional distance from source toPoint C. XC 1 1. requires centerline MachNo. distribution from Point B to Point Cto be read in as input data on Unit 9.Otherwise, positive XC gives 5th.-dog dis-tribution if CMC positive and 4th-deg if CMCnegative. XC - 0 gives 4th-deg dietributionif CMC positive and 3rd-deg if CMC negative.Negative XC and IN gives 3rd-deg distributionwith d2W/dx 2 not matching source flow atPoint B. If ETAD - 60. and XC > 1, XC is ratioof length, from throat to Point C, to throatheight. Negative XC gives 3rd-deg distribu-tion in M; XC - 0 gives 4th-deg distribution;XC > 1 gives 5th-deg distribution. XC - 1.requires centerline Mach No. distribution tobe read in as input data on Unit 9.
Card4
MT 1-5 Number of points on charact',ristis EG ifETAD ' 60. or CD if ETAD - 60. Maximumvalue about 125. Use odd nuwjer. A zero ornegative value stops calculation after tonter-line distribution is calculated if NT positive.
54
=l . . .. .. i I I III . •ll lN.• . . . . . f , ,
AED-TR - -63AEOC T-78*8
NT 6-10 Number of points on axis IE. Maximum valueis 149-LR. Use odd number. A zero or nega-tive value stops calculation before center-line distribution is calculated but afterparameters and coefficients of distributionare calculated.
IX 11-15 Determines if third derivative of velocitydistribution is matched. IX - 1 matchesthird derivative with transonic solution.IX w -1 matches third derivative with sourceflow value. IX - 0 does not match third deriva-tive but gives constant third derivative ifRC - 0 or FMACH - 0.
IN 16-20 Determines type of distribution from Point Bto Point C, positive for Mach No. distribution,negative for velocity distribution. IN - 0 forthroat only. If XC is greater than 1., thedownstream value of the second derivative atPoint B is 0,1* JINJ times the radial flowvalue. Similarly, If ETAD - 60., the secondderivative at Point I is 0.1i*IN times the"transonic value.
IQ 21-25 Zero for a complete contour if ETAD 0 60., 1 forthroat only or if ETAD - 60., -1 for downstreamonly.
MD 26-30 Number of points on characteris'ic AB. Maxi-mum value about 125. Use odd number. A zeroor negative value stops calculation similarlyto MW.
ND 31-35 Number of points on axis BC. M~aximum value is150. A zero or negative value acts like NT.
NY 36-40 Absolute value is number of points on character-istic CD for ETAD , 60. Maximum value is 149or 200 - ND - MP - jMQj - number of points onupstreem contour. Negative value calls forsmoothing subroutine.
MP 41-45 Number of points on conical section GA ifFMACH 0 BMACH. Use lialue to give desired in-crements in contour -, usually not known forinitial calculation.
55
* . |-- . ............. . . . . . .
A AEDC-TA-78-63
* I MQ 46-50 Number of points downstream of Point D ifparallel inviscid contour desired. A nega-tive value can be used to eliminate theinviscid printout.
rB 51-55 Positive number if boundary-layer calcula-tion ir desired before spline fit. Nega-tive number transfers control of program toJX. Absolute values greater than one areused to approximately halve the number ofpoints on the upstream contour even thoughLR + NT - I points are calculated from char-
* Iacteristic network if LR > 2, or (NT + 1)points if LR a 0.
JX 56-60 Positive number calls for calculation of stream-lines, zero calls for repeat of inviscid calcula-tions requiring new cards 3 and 4, or, ifXBL 1 1000., for spline fit after inviscid calcu-lation, negative number calls for repeat of cal-culations requiring new cards 1, 2, 3, and 4.
JC 61-65 If not zero, calls for printout of intermediatecharacteristics within upstream contour if JCis positive and downstream contour if JC isnegative. Characteristics are (NT - 1)/JC or(ND - 1)/(-JC). Opposite running characteristicthrough Point E (or B) is also printed.
IT 66-70 Number of points at which spline fit is desiredif points are not evenly spaced, such as Jacklocations for a flexible plate. Used only fora planar nozzle, inasmuch as a nonzero valuecalculates distance along curved plate surface.Positive value of IT tequires additional cardsto be read in (8 points per card) after boundarylayer is calculated.
LR 71-75 Absolute value is number of points on throatcharacteristic used in characteristics solution.Negative values give printout of transonic solu-tion. LR - 0 gives M - 1 at Point I.
NX 76-80 Number from 10 to 20 determines spacing of pcintson axis for upstream contour. NX w 10 giveslinear spacing. NX > 10 gives closer spacing ofpoints at upstream end than at downstream end.NX - 0 same as NX 20. Ratio of downstream
56
* , I I
A E DC-TR-78-63
increment to upstream increment is (NT - 1 )NX/10 -
-NX/1O(NT - 2) . Optimum values, usually 13 to 15,determined by trial and error for specific con-tour desired. Negative NX used with negative LRlimits printout to transonic solution.
NOTE: A zero value of MT, NT, MD, or ND will allow a repeat of cal-culations for parameters specified by new cards Nos. 3 and 4.A negative value will allow a repeat of calculations for newcards Nos. 1, 2, 3, and 4.
Card 5
NOUP 1-5 If smoothing is desired, negative NF. Number oftimes upstream contour is smoothed.
NPCT 6-10 Smoothing factor in percent. Smoothing factor- NPCT/100.
NODO 11-15 Number of times doimstream contour is smoothed.
Card 5 If boundary-layer calculation is desired usingorinviscd points calculated from characteristicsor solution. (No smoothing). '
Card 6 If boundary-layer calculation is desired usingevenly spaced points interpolated from spline
or fit of points from characteristics solution.
Card 7 If boundary-layer calculation is desired usingevenly spaced points interpolated from splinefit of smoothed points.
4 PPQ 1-10 Stagnation pressure (psia).
TO 11-20 Stagnation temperature, Rankine.
TWT 21-30 Throat wall temperature, Rankine, if QFUN - 0.If TWT - 0, the wall temperature is assumed tobe the adiabatic value.
TWAT 31-40 Wall temperature, Rankine, at Point D. Forwater-cooled wall, the bulk water temperaturei.s assumed to be 150 lower than specifiedTWAT. The cooled wall temperature distribu-tiou is assumed to be
57
t=,.. .. ...... . !. j. • "u•' ..........
I
AEDC-TR-78-63
(TWT C WT) "' A* 1)TWTWAT +
where A/A* is the area ratio corresponding tolocal value of Mach number and Ac refevs toPoint C.
For negative IHT
TW - TWAT + ,TWT-IA x A A*
QFUN 41-50 Heat-transfer function at the throat.
QFUN - ha(Taw - TWT)TWT - TWAT + 15
where ha has dimensions of BTU/sec/sq ft/R andis obtained by Reynolds analogy from the skin-friction coefficient. If QFUth is specified,input value of TWT is ignored and TWT is calcu-lated from QFUN.
ALPH 51-60 Parameter specifying temperature distribution inboundary layer. ALPH - I. uses quadratic dis-tribution both in the calculazion of the refer-ence temperature TP and the calculation ofboundary-layer shape parameters. ALPH w 0 usasparabolic distribution in boti. calculatiovs.ALPH -u -1. uses quadratic distribution for TPand parabolic in the calculation of boundary-layer shape parameters. Within bcundary layer,
T - Tw + a(Taw- TW) (U/11)
+ Tes- (Taw- Tw) - Twj (U/Ue) 2
where a 1 for quadratic dist.
a -0 for parabolic dist.
IHT 61-65 Integer which determines temperature distribu-tion (see TWAT). If nonzero, IRiT determineshow often subroutine HFAT is called. An absolutevalue of IHT greater than KO, the number of points"on the upstream contour, will prevent HEAT frombeing called but will allow the choice of tempera-ture distribution to be made.
58
11 p 1 Ml 1 11.01
AE DC.TR -78-63
NOTE: HEAT is a special purpose subroutine for determ.ningheat-transfer values for the upstream contour. Thesubroutine HEAT incorporated in this program is adummy.
66-70 Integer, parameter specifying transformationfrom incompressible to compressible values.If YR w 2, Coles' transformation is used forCf and Re If IR - 1, TP is calculated bya modification of the Spalding-Chi (Van Driest)method. If IR - 0, the Van Driest value ofRe6 is used, but if IR- -1, Colas' lav: of
corresponding stations is used.
Cf Cf * TE/TP, Re - RD*Ree,i f
ID 71-75 Integer. If ID - ±1, axisyminetric effectsare included in momentum equation and in cal-culation of boundary-layer parameters (6 nornegligible relative to coordinate normal toaxis). If ID - 0, these effects are omitted.Negative ID suppresses the printout of theboundary-layev calculations.
LV 76-80 Integer. Absolute value, usually 5, deter-mines number of rimes boundary-layer solutionis iLerated so that radxus terms in momentumequation refer to viscid radius instead of Jinviscid radius. Value of 0 or absolute val-eof 1 uses inviscid radius, Positive LV repeatsboundary-layer calculations for new set ofparameters on a new card if XBL j 1000.
Card 5 If streariflines are desired, JX positive, (Nosmnoo thing.)
METAD 1-10 Inflection angle in degrees for streamlinedesired if ETAD 0 60. for Card 3. If ETkD -60. on Card 3, use ETAD - 60 on this card.
* 11-20 Fract ion of r'.ottour desir.-ed if ETAD - 60.Otherwise, QON - lTAD (in Card 5 divided byETAD oa Card 3.
X,1 21-30 Value to upd•ite JX for subr.eq4oent calcula-tion, JX XJ.
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AZ-DC.TR-78.63
Card 5 If SPLIND used after inviscid calculation* (JX zero or negative and JB zero or nega-
or tive). (No smoothing.)
Card 6 If SPLIND usad after viscid contouir (,7B posi-tive and LV zero or negative). No v'.oothingof inviscid contour. Or, if inviscid contour
or is smoothed before SPLIND is used.
Card 7 If inviscid contour is smoothed, boundary layeris added and SPLINE Ls desired.
XST 1-10 Station (in,) for throat value of X. IfXST a 1000., program uses value previouslydetermined by specifying PP on Card 3. Other-Wise, value of XST is used to shift contourpoints by desired increments for arbitrmryStation 0.
XLOW 11-20 Starting value for interpolation. Second valueof interpolated X a XLOW + XINC.
XEND 21-30 End value for interpolation. If zeto, SPLINDis used to calculate slcpe and d2y/dx2 at samepoints as previously defined.
XINC 31-40 Increment in X for interpolation. If zero, andBJ > 10, contour is divided into BJ increments.
BJ 41-50 Value to update JB for subsequent calculation.JB - BJ. If negative and XEND - 0, interpola-tion is made at diecreate points read in on sub-sequent cards similar to case when IT > 0.
XMID 51-60 Intermediate value for interpolaticn. Distance(XMID-XLOW) is divided into increments definedby XINC, and distance (XEND-XMID) is dividedinto increments defined by XINC2.
XINC2 61-70 Inctements in X between XMID and XEND if differ-ent than XINC.
CN /1-80 Number of copies desired of final tabulation ofcoordinates if more than one copy is desired.
ý,j
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AEDC.TR-78-3
NOMENCLATURE
A Area
AC Exit area, inviscid contour
A* Sonic area
a* Sonic speed
C Factor in logarithmic skin friction law,Eq. (77)
C1, 2 ,3,4,5, 6 Coefficients, Eq. (35)
* C. Ratio of actual mass flow to that if R were
* infinite
Cf Skin friction coefficient, compressible
1Cf Skin friction coefficient, incompressible
C Specific heat at constant pressure
p
D1,2,3,4,5, 6 Coefficients, Eq. (37) J
Ratio, Cfi/Cf
I Fn Multiplying factors, Eq, (97)
Ratio, R /RFR i c
Gn Multiplying factors, Eqs. (94) and (96)
139
APEDC-TR-78.63
Ratio, 5*/1
h Heat-transfer coefficient
K Streamline curvature
In Natural logarithm (base e)
log Common logarithm (Base 10)
M Mach number
rm Exponent in Eq. (90)
N Velocity profile exponent
n Distance normal to streamline
P1,2 Factors in axisymmetric characteristicsequations
Pn Coefficient of 6 at nth point on contourin
Pr Prandtl number
Q Factor related to heat transfer, Eq. (91)
Qn Coefficient in momentum equation
q Velocity along streamline or, in boundary-
layer equations, velocity within boundary
layer
qe Velocity at edge of boundary layer
140
-. • w ~-.- T -- ...- .~, - T-- T ' -.. .. . .... . . ..
AEOC-TR-78-63
R Ratio of throat radius of curvature to
throat radius (half height, a - 0)
2 2Rg Gas constant, ft /sec R
,R Reynolds number based on 6, compressible
R1 Incompressible Reynolds number
R0 Reynolds nuitiber beed on 0c, compressiblec
Re Incompreauible Reynolds numberi
Sr Distance from source
r Distance from source where M 1, used tonon-dimensionalize distances for inviscid
calculations
rV Radius of viscid contour
I
SR+ I
,9 s,t,u Cubic integration increments, Appendix B
T Temperature within boundary layer
T Adiabatic wall temperatureaw
Tc Reference temperature, Eq. (87)
T Free-stream temperature at edge ofe
inviscid contour
141
•I.. ... ..... . ,
AEDC-TR-78-63
T Wall temperaturew
T Wall temperature at nozzle exitwD
TWT Wall temperature at nozzle throat
u Axial component of velocity, normalized
by a*
v Normal component of velocity, normalized by
a*
W Velocity along streamline, normalized by a*
X Ratio in Eq. (36) or (38)
x Axial distance, normalized by yo in transonic
equations, normalized by rl in .inviscid
calculations, not normalized in boundary-
layer calculations
y Normal distance, normalized same as x
y Y Throat half height, used to normalize
x and y in transonic calculations
y* Theoretical throat height if R is infinite
z 1Function of x in transonic equations, or
distance normal to contour in boundary-layer
calculations
142L •,= •!• •: :•:• : & •:|!J • W •- s~l i .• '•.• ; ,,F • .• ';.•4.,.2iL.•.. , ..... ;,.. ,:L•..,•.,;,'•,al,.1;,.;,,,ili,• ;',' •'•"" ::' ' i . ., ' . , .,, I,,; ,
AEDC-TR-7843
a Mean angle of right-running characteristic,
or factor in temperature distribution in
boundary layer
Mean angle of left-running characteristic
A Prefix to indicate increment in value
Specific heat ratio
S~Boundary-layer thickness
6* Displacement thickness in boundary layer
a* Displacement thickness when boundary layer
is large relative to rw
6t Incompressible displacement thickness in boundary
layer
Distance along left-running nharacteris.ic
n Inflection angle, radians
e ,Momentum thIckness in boundary layer
6 Momen'tum thickness whon boundary layer
is large relative to r
c Compressible 0 for flat platec
0 1 Incompressible value of 0
0 k Kinematic rementum thicknesa
143
-I- ~ -. -..
AEDC-TR-78-63
0 Value of 0 at nth point on contourn
K Constant in logarithmic skin-triction law
I2(L y + I)S
Mach angle, sin (1/M)
PC Viscosity at value of T c
Viscosity at value of TI PW Viscosity at value of Tw
e e
Viscosity at value of T
Distance along right-running characteristic
Wake variable in logarithmic skin-friction law
p Density within boundary layer
Pe Density at edge of boundary layer
a Zero for planar flow, 1 for axisymmetric flow
4) ,Flow angle
•w Flow angle of viscid contour
Prandtl-Meyer angle
'44
AE2DC-T R-78.83
SUBSCRIPTS
1 Values at point 1 on right-running
characteristic
2 Values at point 2 ona left-running
characteristic
3 Values at intersection of characteristics
A,B,C,DOE, Variables evaluated at points on Figs.
F,G,I,J,T 1 through 4F
second-, and third-order approximations,
respectively
OTHER NOTATION
4/ dx:
OUTPUT NOMENCLATURE
BETA Pressure gradient parameter
2 8* d ci d
YM2 P~Ci
C(Y) Coefficient of third-degree term if throat
contour is a cubic
C(YI) Coefficient of third-degree term if integrated
throat contour is a cubic
145
....
AEDC-Tn-78-63
C(YP) Coefficient of third-degree term determined
from slope of contour
D2A/DX2 Second derivative of boundary-layer correction
evaluated at the throat
D2R/DX2 Second derivative of corrected contour
evaluated at the throat
DA/DX Slope of boundary-layer correction
DELR(IN) Boundary-layer correction to inviscid contour
DELTA* 6* from Eq. (66)a
DELTA* - 1 6* from Eq. (63)
FMY Bracketed term in Eq. (61)
HYP/YO Value of hyperbola with same throat curvature
ratio
IcY 10' [C(YI) - C(Y)] for Point 2
INT.Y/YO Value of Y/YO obtained by integrating contour
slopes starting at inflection point
KCF 1000 Cf
KCFI 1000 Cf
KCFS KCF sec cw
146
. ............ x
AEOC-TR-7843
lO 1000 dO/dx
MSS Result of mass integration along characteristic
EG or AB (measure of accuracy of numerical
integration)
PAR/TO Value of parabola with same throat curvature
ratio
PE/PO Ratio of static to stagnation pressure
R(IU) Ordinate of viscid contour
R•ASS C1/0+0)
RTHI Incompressible Reynolds number based on
momentum thickness
SKPP Second derivative of Mach number in source
flow evaluated for BMACH
SMPPP? Third derivative of Mach number in source flowevaluated for BMACH
THETA - 1 0 from Eq. (62) used in Eq. (61)
WE Velocity ratio at Point E (Fig. 3)
W1 Velocity ratio at Point I (Fig. 3)
WO Velocity ratio on axis at throat
SWoFP Third derivative of throat velocity distribution
147
BEST AVAILABL[ COPY
ANDC-TR.7-43
WRPPP Third derivative of velocity ratio in source
flow evaluated at WE
w0 Velocity ratio on vail at throat
fV
148BEST AVAILABLE COPY'