AEDC-TR-.78-63 COPUE LEVELEI1, A COMPUTER PROGRAM FOR THE AERODYNAMIC DESIGN OF AXISYMMETRIC AND PLANAR ONOZZLES FOR SUPERSONIC AND S WIND TUNNELS J. C. Sivells ARO, Inc., a Sverdrup Corporation Company VON KARMAN GAS DYNAMICS FACILITY ARNOLD ENGINEERING DEVELOPMENT CENTER AIR FORCE SYSTEMS COMMAND .ARNOLD AIR FORCE STATION, TENNESSEE 37389 LUL December 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 1979 ARNOLD AIR FORCE STATION, TENNESSEE 37389 D 79 0 BEST AVAIL.ABLE -COPY
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A Computer Program for the Aerodynamic Design of Axisymmetric and Planar Nozzles for Supersonic and Hypersonic Wind Tunnels
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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
When U. S. Governiment drawings, specifications, or other data are used for any purpose otherthan a definitely related Government procurement operation, the Government thereby incurs noresponsibility nor any obligation whatsoever, and the fact that the Government may haveformulated, furnished, or in any way supplied the said drawings, specifications, or other data, isnot to be regarded by implication or otherwise, or in any manner licensing the holder or anyother person or corporation, or conveying any rights or permission to manufacture, use, or sellany patented invention that may in any way be related thereto.
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)
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.
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
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
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,
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)
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.
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.
(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.
,, 2 !*.;.a a'11d598Y '
I,
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.
- ~ f 0SI IAIA. US &- 0K-: M- 0- OI K 'I0N0 IV 1010-M 04 1 I - O M 3 U I .1
- II- S~A IA 0I 01 II O A0 11 ** -.. .E .- U~8544
AEDC-TR-78-63
'asp
ix a 20 6 0 06.
'v @A a ,, $- 1 z 4 . zW
cc u a
-jW La a0 0- 0 -'! '. W x
a, z It 'vo
W Z-ý-j 00 - -N U. T r U. w 0W0zWu0_jx 41 lux
0 0 OZ IWO 42 0- al .0ty v T U. 0
-9.jW 0 ; 2 ; 4 .. 0.z W W z k. -i a -M a
W'.. u un W 'd *a"HAZO Wo rw
mmmýw #W -j LL ... 'a czo-ONO = . N.ým U. 0 -j
IZJAWJTOZO;zx UWO ;am X,.Wz Z% -2;: X-z a OzW 2 Cal 4 .. -9 cr . W .. Olt ltt Lt Ucc W to- tj D in W (Y.J a Ito 0 In 0 a
bjý - &L 9- Czw to ýO K -L.) CP W n a24 0 W 19 Z OXO U, a 't,
Wýý W i! on
IL I -i Ic.9 uzkvt a z tA " 2.910 - - i P.-ka Uj 4A -Cýj tAzw t' C; OD M XV -09V u
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