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v
NASA CONTRACTORREPORT" NASA CR-111922
m\J
d_%
IlfllllllfJIlllllIFIQIrllllflltllIJflIfl/Ifllllll /lltl3 1176 00090 0192
CALCULATION OF INVISCID SURFACE
STREAMLINES AND HEAT TRANSFER
ON SHUTTLE TYPE CONFIGURATIONS
Part II. Description of Computer Program
by Fred R. DeJarnetteand
Michael H. Jones
Prepared by
MECHANICALAND AEROSPACEENGINEERINGNORTHCAROLINASTATEUNIVERSITY
RALEIGH,NORTHCAROLINA27607
for
DEPT.
NATIONAL
..:.., ':, - ;_,_-&-. 7LANGLEYRESEARCHCENTER
AERONAUTICSAND SPACEADMINISTRATIONRAMPTON, VIbRGINIA23365 "__' __
AUGUST,1971
https://ntrs.nasa.gov/search.jsp?R=19720004179 2018-07-16T09:24:15+00:00Z
NASA CR-I11922
CALCULATION OF INVISCID SURFACE STREAMLINES
AND HEAT TRANSFER ON SHUTTLE TYPE CDNFIGURATIONS
Part II. - Description of Computer Program
by Fred R. DeJarnette
and
Michael H. Jones
Distribution of this report is provided in the interest of
information exhhanEe. Responsibility for the contents
resides in the author or organization that prepared it.
Prepared under Contract No. NASI-I0277 by
Mechanical and Aerospace Engineering Dept.
NORTH CAROLINA STATE UNIVERSITY
Raleigh, North Carolina
for Langley Research Center
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
For sale by the Clearinghousefor Federal Scientific and Technical Information,
Sprlngfleld, Virginia 22151 - CFSTI price $3.00
FOREWORD
This research was supported under contract NASI-I0277 with the National
Aeronautics and Space Administration's Langley Research Center. The technical
monitor on the contract was H_ Harris H_milton II. The results are published
in two parts:
Part Io - Description of Basic Method (NASA CR-II1921)
Part Iio - Description bf Computer Program (NASA CR-II1922)
Part I contains a detailed description of the theoretical approach and basic
equations that were used to obtain a solution to the problem. The present
part (Part If) contains a detailed description of the computer program and
is intended to serve primarily as a user's manual.
ACKNOWLEDGMENTS
The authorswould like to express their appreciation to Mr_ H. Harris
Hamilton II of the Langley Research Center, NASA for his review of the
manuscript and valuable assistance during all phases of this work. In
addition, the helpful discussionsof Mr_ James C_ Dunavant of the Langley
Research Cen_er, NASA are gratefully acknowledged.
The authors would also like to thank the following people at the
North Carolina State University:
io Mrs. 3oyce Sorensen for typing the manuscript and assistin E with
_he printing.
2. Miss Eleanor Bridgers for her assistance in the printin E and
assembling of this report.
iii
CONTENTS
FOREWORD ili
ACKNOWLEDGMENTS
INTRODUCTION
lii
I
PROGRAM DESCRIPTION
Main Program
Simplified Flow Diagram
Function RUNGE
Subroutine STAGN
Subroutine NEWP
Subroutine STEMLN
Subroutine BGE_M
Subroutine SPLINE
Function FSPL
Function DFSPL
Function D2FPL
Subroutine STGPR
Subroutine PR@P
Subroutine SPCH
Subroutine PRESS
Subroutine PDER
Subroutine LAGRN
Subroutine TRANS
Subroutine BEGTR
Subroutine ENDTR ................
DESCRIPTION OF INPUT- - -
DESCRIPTION OF OUTPUT
DIAGNOSTIC MESSAGES ..........
TESTS CASES
Case Io - Blunted 15" Half-Angle Cone
Case II. - Blunted 70° Slab Delta Wing
Case III. - HL-10 Lifting Body
ACCURACY AND LIMITATIONS OF METHOD
3
8
9
9
ii
12
14
15
16
16
16
17
18
19
21
21
22
23
23
23
3O
32
33
34
38
42
47
v
CONCLUDINGm_um_s ..........
APPZND_XA - P_GRAM LIS_I_G
APPENDIX B - DETAILED FLOW DIAGRAM
APPENDIX _C - OUTPUT FOE CASE I ....
49
51
85
202
CALCULATION OF IN'VISCID SURFACE
STREAMLINES AND KEAT TRANSFER
ON SHUTTLE TYPE CONFIGURATIONS
Part II. - Description of Computer Program
by Fred R. DeJarnette
and
Michael H. Jones
North Carolina State University
INTRODUCTION
Part I of this report (NASA CR-II1921) presents a description of the basic
method used to calculate laminar, transitional, and turbulent heating rates on
arbitrary blunt-nosed three-dimensional bodies at angles of attack in hypersonic
flows. The present report (Part II) gives a description of the computer program
used in Part I. The main program and each subprogram are described by defining
the pertinent symbols involved, a detailed flow diagram, and a complete computer
program listing.
Input and output parameters are discussed in detail. Listingsare given
for the inputs used in Part I to compute heating rates on (i) a blunted 15 °
half-angle cone at 20 ° incidence and Math 10.6, (2) a blunted 70" slab d_ita
wing at i0 ° incidence and Mach 8, and (3) the HL-IO lifting body at 20"
incidence and Mach i0. In addition, the computer program output for two
streamlines on the blunted 15 ° half-angle cone is listed.
PROGRAM DESCRIPTION
Descriptions of the main program and each subprogram are given below along
with the pertinent symbols used in each program. Program inputs and outputs
are described in subsequent sections. The complete computer program is listed
in Appendix A, and a detailed flow diagram is given in Appendix B.
Main Program
The main program reads part of the input data, calculates the initial data
for each streamline on the _-circle around the stagnation point, and then cal-
culates and prints the heating rates and other pertinent data along each stream-
line independently of the other streamlines. A simple flow diagram for the
entire computer program is given in figure i, whereas a detailed flow diagram
is given in Appendix B. The program variables are listed below, and the corre-
sponding symbols used in Part I of this report are given if applicable.
ALP angle of attack, u (radians)
ALPD angle of attack, _ (degrees)
B ratio of principal velocity gradients at stagnation point
BBAR pressure gradient parameter, B
BBARS pressure gradient parameter at stagnation point, _s
BETA coordinate normal to streamline on body surface, _ (radians)
BETAD BETA in degrees
CAL cosine of ALP
CPSI cosine of PSI
CST constant to convert units of body coordinates to feet
CTHM constant used to calculate laminar momentum thickness
DDLBh _8
DDP _-_
DDX_x
DEPS radius of z-circle around stagnation point
FIGUREi, SIMPLIFIEDFLOWDIAGRAM
READ /
INPUT
DATA
I COMPUTE BODY GEOMo COEFF_ FOR SPLINE, LOCATESTAGNATION POINT, COMPUTE STAGNATION POINT PROPERTIES
_I-I
/_'! _o_L/ /-
i COMPUTE INITIAL VALUES ON e-CIRCLE
COMPUTE BODY ANGLES AND THEIR DERIVATIVES
ICOMPUTE PROPERTIES AT EDGE OF
BOUNDARY LAYER, CHECK FOR TRANSITION,
& COMPUTE HEAT-TRANSFER RATE
NO
INCREMENT DISTANCE
YES
ALONG STREAMLINE
DOB
DGP
DGS
DGX
DLPR
DPB
DPHB
DPS
DPSIS
DS
DSB
DSS
DTRB
DUEDPS
DUEDS
DXB
D2PB
EMEMIN
i _r
h _B
Lr
D_
DS
_r
@x
body angle 5¢
i
h _(P/Ps )
h _B
D(p/P s)
DS
D__DS
integration step size along a streamline
i _
h _B
D_
DS
i BB
h _8
DUe
D_
DU
DS
I _x
h _B
g _-f _f(p/ps)
Mach number a= edge of boundary layer, Me
freestream Maeh number, M
EMS
EM2
FF
FF_
F(1)
F(2)
F(3)
F(4)
F(5)
F(6)
F(7)
F(B)
F(9)
GM
HC
HE
HI
KMAX
HS
(E_ 2 , M 2e
same as EMS
body radius, f
body radius at stagnation point, fs
Dx
DS
DS
DhD8
U
_--_ h2Ps _
PeUs
DBm
D.D--"S-Dh,
gg'tS'g)
PeUe
DB
Ds
for KP = O,
for KP # 0
for KP = O,
fo_ K.P ,_ 0
for KP = O,
for KP # 0
.£oz K.P _ 0
for KP # 0DS
body angle F
compressible form factor, H
enthalpy at edge of boundary layer, he(ft2/sec 2)
incompressible form_ factor, H i
maximum integration step size
stagnation entha!py, _s(ft2/se_ 2)
HSP
RTR
HW
IMAX
ISTG
K
KBM
KG
KMAX
KP
KP_
KS
KTR
L
LI
L2
MR
NAM
NCS
NM/X
NR
NT
PE
PHID
PIN
P_P
P_Rv
QWS(6006)/(RS-HW), for a perfect gas RSP is the stagnation point
heattransfer coefficient (BTU/ft2-sec-°R)
transformed form factor, Sir
wall enthalpy, hw (ft2/sec 2)
see Description of Input'
indicator variable, see Diagnostic Messages
see function RUNGE subprogram, K=RUNGE
see Description of Input
counter for integration steps along a streamline
indicator variable, LI= i for laminar region, LI= 2 for
transitional and turbulent regions
indicator variable, L2 = i for laminar region, L2 = 2 for last
integration step before the beginning of transition, L2 = 3 for
interior of transition region, and L2 = 4 for last integration
step of transition region
see function RUNGE subprogram
see Description of Input
see subroutine SPC_
see Description of Input
number of differential equations to be integrated in RUNGE subprogram
indicator variable, NT = i for laminar region, NT = 2 fully turbulent
region, NT - 3 for laminar region over entire body
surface pressure, p(ib/ft 2)
circumferential angle, _(degrees)
freestream pressure, p= (lb/ft 2)
pressure ratio, P/Ps2
ps/(_sV_)
PNE_f
PS
PSI
Q_QS
Q_QSB
QWB
QWL
QWLB
QW5
QWSB
qwr
QWTB
RH_E
P_s
PaCs_
_NF
RNM
RNI
S
SAL
STR
TE
TKM-
TIN
.TRE
TRI
TW
UE
u_u
VlN
P_
(i -Ts)
snagnatZon pressura, Ps (ib/ft2)
angle
heating rate ratio, ___Jqw,s
Q_QS (i - ZW)/(_S5 + olSTE - ZW), for a perfect gas Q@QSB is the
ratio of the lo_al to the stagnation point heat transfer coefficient
(Q@QSB) (QWSB), (BTU-sec/ft 4)
lem_inar heating rat_, qw,lam (BTU/ft2"s_)
laminar value of QW]B
stagnation point heat-txansfer rate, _,s (BTU/ft2-se_)
QWS/(HS - HW)
turbulent heating rate, qw,turb (BTU/f_2-se=)
turbulent value _f QirB
density at edge of boundary layer, p (slug/ft 3)e
frees=ream density, p_ (slug/ft 3)
stagnation density, Ps (slug/ft3)
wall density at stagnation point, Pw,s (slug/ft3)
Reynolds number per foot, PeUe/_e (I/ft)
momentum thickness Reynolds number, Rn
integrated unit Reynolds number, Rn I
dissance along a streamline from stagnation point
sine (ALP)
STR=Y(1) for KTR=-I, STR=RNM for KTR=0, or STR=RNI for KTR=l
enthalpy ratio, te
inclination angle of invlscid surface streamline, 8
laminar momentum thickness, 8m,lam
freestream =_mperature, T=
}see Desuription _f Input
wall tem_perature, "R
velocity at edge of boundary layer, U e (ft/sec)
velocity ratio, Ue/V
freest:cam ve!oclt_, V (ft/se_)
V!SC
V!SS
VISW
WFN
x_
xY
ZW
ZZ
coefficient of viscosity at edge of boundary layer, _e (slug/ft-sec)
coefficient of viscosity at stagnation point, _s (slug/ft-sec)
wall coefficient of viscosity, U w (slug/ft-sec)
weighting function, wf
axial location of stagnation point, xs
y-coordinate
I s F(I)DS, where I = 1,2,...,NRo
wall enthalpy ratio, _w
z-_oordinate
F_nction RUNGE Subprogram
Function RUNGE is a subprogram which uses the fourth-order Runge-Kutta
method to integrate a system of NR first-order ordinary differential equations
with a variable step size. As a criterion for varying the computing interval,
the differential equations are integrated over an interval of step size H first
and then over the same interval with two step sizes of H/2o The two solutions
are then compared to give an estimate of the error for each variable. If any
error is larger than EPS = 10 -4 , these answers are discarded and the computing
interval H is halved_ If all of the error estimates are less than EPS, the
answers are allowed and the integration process continues. In addition, the
step size is either doubled or set equal to HMAX, whichever is the smaller,
for the next integration cycle_
The function RUNGE is used in the Main Program and has the form
K=RUNGE (Y, F, S, DS ,NR,HMAX ,MR)
where K can have one of three values:
K = 0 implies the completion of an integration cycle
K _ 1 implies the integration cycle has not been completed
K = 2 implies =he step size has been halved down to a value below 10 -8.
The dummy arguments used in the subprogram itself are
FITNCTI_N RUNGE (Y, F ,X, H, N ,HI_ ,MR)
8
where
Y
F
X
H integration step size
N number of differential equations
I_MAX maximum integration s_ep size
MR
array of dependent variables to be integrated
dY
array of first derivatives of _he dependent variables, F - d-_
independent variable (distance along a streamline)
indicator variable, if MR = i the previous integration interval is
recomputed _Tith a new step size H determined in the main program
Subroutine STAGN
Subroutine STAGN locates the Newtonian stagnation point and calculates the
two principal radii of curvature =here. A call to STAGN has the form
CALL STAGN(ALP ,IMAX,KMAX,X_, RPER,B ,ISTG)
where the arguments are
ALP angle of attack, _ (radians)
IM_X ]KMAX see Description of Input
X_ axial location of stagnation point
RPER principal radius of curvature in a plane perpendicular to plane
of symmetry a_ stagnation point,
B ratio of principal radii of curvature, _/RII
ISTG indicator variable, -1 for negative angle of attack, 0 for stagnation
point not found, or +i for stagnation point found
O_her program variables are
FF body radius, f
RPAR principal radius of curvature in plane of symmetry at stagnation
point, RII
Subroutine NE%_
This subroutine calculates the modifisd Newtonian pressure and its first
and second derivatives with respect to both the axial (x) and circumferential
9
(4) directions. A call to NEWP has the form
CALL NEWP (AA_F)
where the dummy arguments are p=
AA same as PNEWT in. main program, AA '= 1 - --Ps
F body radius, f
Other program variables are
CA cosine (ALP)
CP cos
FF body radius squared, f2
_2f
_x_¢
_3f_PP
_x_¢ 2
a3f
_x2a¢
a3fF3X
_x 3
PE ratio of static to stagnation pressure, P/Ps
(P/Ps)PP
PPP
pX ¸
PXP
PXX
SA
SP
a2(p/ps)
_x
B2(p/p s)
_xa¢
a2(p/p s)
_x 2
sine (ALP)
sin ¢
10
Variables in labeled COMMON /C_M 21 and /C#M 5/ are calculated in subrou-
tine STRMLN o
Subroutine STRMLN
FoY a given axial (x=XX) and circumferential (_=PPH) location on the
body, this subrou=ine calculates the body radius (f), body angles F and 8_'
and their derivatives in the axial and circumferential directions. A call
to subroutine STRMLN has the form
CALL STRMLN(XX,PPH,DLPH_DDP,DDX,GM,DGP,DGX,F,IMAX,KMAX)
where the arguments are
_LX axial position, x
PPH circumferential position,
DLPH body angle _
DDP -_
DDX _'__x
GM body angle r
DGP _F_
ALDGX_x
F body radius, f
KMAX see Description of Input
Other program variables are
FF body radius squared, f2
_f2FFP _
_f2FFX
_x
^2=2FFXP o __
ii
FFXPP
FFXX
B3f 2
_x_ 2
_2f2
_x 2
_3f2FFXXP
_2f2FF2P
_3f2FF3P
_fFP
_2fFPP
_2
_fFX
_x
_2fFXX
_x2
F2 2f
Subroutine BGE_M
Subroutine BGE_M reads the input data for the body geometry and calcu-
lates the coefficients for the two-dlmensional cubic spline function used
to represent the body geometry. A call to subroutine BGE_M has the form
CALL BGE_M(IMAX,KMAX)
where the arguments are defined in Description of Input° Other program
variables are
_3f2
AMD(I,K) (xi,_k)
AMDD(1) _temporary storage for elements of AMD(I,K)
AMnD (X) J
12
_2f2A_F(I,K) -- (xi,_k)
AMFF(I) %temporary storage for elements of A_F(I,K)
AMFF (K) !__22
(_j) corresponding =o inputAMFT (J) coordinates
a_
AMG(_,E)
AMGG(I) %temporary storage for elements of AMG(I,K)
AMGQ(K) h
22BMD(I,K) _ AMD(I,K)
_x 2
BMF (I ,K)
BMFF (I)
BMG(I,K)
temporary storage for elements of BMD(I,K)
3 2-- AMF(I,K)8x 2
temporary storage for elements of BMF(I,K)
3 2-- AMG (I,K)3x 2
BMGG(I) temporary storage for elements of BMG(I,K)
_f2
DLP (l,K) _-_ (xi,¢k)
DLPP(1) _=emporary storage for elements of DLP(I,K)
DLPP (K) A
FN(I,K) f2(xi,¢k)
FNN(1) _temporary storage for elements of FN(I,K)
FNN (K) J
FT(J) f2(_j) corresponding to input coordinates
3f 2
GAM(I,K) D--f-(xi'_k)
G_@/_A_'(1)_temporary storage for elements of GAM(I,K)
_P(K) _k - +k-1
H2T(J) Cj - _j-i
13
HX(I)
I
J
K
JM
J_m(1)
PH(K)
PHI (J)
x(_)
X_:(I,K)
XMDD(D
XMF(I,K)
XMFF(X)
XMG(I,K)
khMGG (I)
Y(J)
YY (I ,J)
z(J)
ZZ(l,J)
Note:
x i - xi_ 1
subscript corresponding to axial position
subscript corresponding to input circumferential position
subscript corresponding to rearranged circumferential position
number of input circumferential locations
number of input circumferential locations at x = X(1)
rearranged circumferential position, Sk
input circumferential position, Sj
input axial position, x i
_2-- DLP(I,K)8x 2
temporary storage for elements of XMD(I,K)
22-- FN(I,K)_x 2
temporary storage for elements of XMF(I,K)
82-- GAM(I,K)_x 2
temporary storage for elements of XMG(I,K)
input y-coordinate, see Description of Input
input y-coordlnate at x = X(1)
input z-coordinate, see Description of Input
input z-coordinate at x _ X(1)
The dimensions on the variables involving the subscripts I, J, and
K in the program listing are 20 for each subscript.
Subroutine SPLINE
For an array of dependent variables, this subroutine solves the tri-
diagonal matrix system of algebraic equations to determine the coefficients
for the spline function. A call to SPL!NE has the form
CALL SPLINE(H,DE,K,AM,E,G,N)
where the dummy arguments are
14
H(J) array representln E the difference between the J+l independent
variable and the J independent variable
DE(J) array of dependent variables
K number of points used for the spline fit; J=I,2,._.,K
AM(J) array of coefficients for spline function
E,G derivative (slope) of the dependent variable at J=l and J=K, respectively
N indicator variable:
(1) N - - 2 indicates AM(2)=AM(1) and AM(K-1)=AM(K) are used in
place of specifying E and Go
(2) N - - 1 indicates AM(1)=0 and AM(K)=O are used in place of
specifying E and G_
(37 N = 0 indicates E is specified but AM(K-I)-AM(K) is used in
place of specifying G.
(47 N = 1 indicates both E and G are specified.
program variables areOther
A(J)
c(J)
D(J)
J
array of constants in uhe system of algebraic equations
subscript indicating position of dependent and independent variables;
J=I,2,o0.,K
P(J) 1Q(j) array of parameters used in the algorithm for solving the system of
U(J) j algebraic equations
Note: This subroutine calculates the coefficients for spline functions in
both the axial and circumferential directions. Therefore, the di-
mension on the subscripted variables must be the number of points used
in the axial or circumferential directions, whichever is greater.
The computer program listed in Appendix A uses a dimension of 20.
Function FSPL
The function FSPL calculates the value of uhe dependent variable
corresponding to a given value of the independent variable for any one of
15
the several spline functions. The function FSPLappears in the program in
the form
FSPL (AMI,AM2 ,XI,X2 ,YI ,Y2 ,X)
where the dummy arguments are
AM1 coefficient for spline function at X1
AM2 coefficient for spline function at X2
X value of independent variable, where X1 _< X < X2
X1 _two successive members of the array of independent variables, used to
X2 I determine the spline function, which bracket the current value of theindependent variable X.
Y1 dependent variable corresponding to X1
Y2 dependent variable correspondinE to X2
Function DFSPL
This function calculates the first derivative of the spline function
computed in function FSPL. It appears in the program as
DFSPL(AMI,AM2,XI,X2,YI,Y2,X)
where the dummy arguments are the same as those in function FSPL.
Function D2FPL
This function calculates the second derivative of the spline function
computed in function FSPL. It appears in the program as
D2FPL(AMI,AM2,XI,X2,YI,Y2,X)
and the dummy arguments are the same as those in function FSPL.
Subroutine STGPR
Subroutine STGPR computes stagnation properties for perfect gas or
equilibrium air° A call to subroutine STGPR has the form
CALL STGPR(PIN,TIN,VIN,RB@IN,EMIN,PS,RH#S,HS,_G)
where the arguments are
16
PIN freestraam pressure, p= (ib/ft 2)
TIN freestream temperature, T (°R)
VIN freestreamvelocity, V (ft/sec)
RH%IN freestream density, p_ (slug/ft 3)
EMIN freestreamMach number, M
PS stagnation pressure, Ps (_b/ft2)
RH_S s_agnation density, @s (slug/ft3)
HS stagnation enthalpy, H s (ft2/sec 2)
KG indicator variable; KG-0 for perfect gas or KG=I for equilibrium air
Other program variables are
HIN freestream enthalpy, h (ft2/sec 2)
H2 enthalpy aft of normal shock wave, h 2 (ft2/sec 2)
P2 pressure aft of normal shock wave, P2 (ib/ft2)
P/4_2 density aft of normal shock wave, P2 (slug/ft3)
RIN gas constant_for _r!R_Jo3_u45 x 32 1741 fu2/sec2-_R)
V2 velocity aft of normal shock wave, v 2 (ft/sec)
Subroutine PR_P
This subroutine' computes the local fluid properties at the edge of
the boundary layer from the stagnation properties and local pressure
for a perfect gas or equilibrium air. A call to subroutine PR_P has the
form
CALL pR@P(PS,HS,RB_S,PE,HE,R_@E,UE,VISC,EMS,KG)
where the dummy arguments are
PS stagnation pressure, Ps (Ib/ft2)
HS stagnation enthalpy, H s (ft2/sec 2)
RH_S stagnation d_nsity, Ps (slug/ft3)
PE local surface pressure, p (Ib/ft 2)
HE enthalpy at edge of boundary layer, h e (ft2/sec 2)
RH_E density at edge of boundary layer, _e (slug/f_3)
UE velocity at edge of boundary layer, U e (ft/sec)
VISC coefficieu_ of viscosity at edge of boundary layer, U e (slug/ft-sec)
EMS Mach number squared at edge of'boundary layer_ M 2e
17
KG indicator variable; KG-0 for perfect gas, KG-I for equilibrium air
Subroutine SPCH
Subroutine SPCH computes the turbulent skin friction coefficient from
the Spalding-Chi method based on momentum thickness. The corresponding
turbulent heat transfer rats is calculated from yon Karman's form of
Reynolds Analogy. A call to subroutine SPCH has the form
CALL SPCH(ZW,HE,RH_E,UE,HS,RNM,CF,QW,QWB,NCS)
where
ZW
HE
RH_E
UE
HS
RNM
CF
Qw
Qw_
NCS
Other
A@G
ACS
CFI
FC
FED
PR
RAF
REC
S_G
the arguments are
wall enthalpy ratio, _w
enthalpy at edge_f boundary layer, h e (ft2/sec 2)
density at edge of boundary layer, Pe (slug/ft3)
velocity at edge of boundary layer, U e (ft/sec)
stagnation enthalpy, H s (ft2/sec 2)
momentum thickness Reynolds number, Rnm
turbulent skin friction coefficient, Cf
turbulent heat-transfer rate, .__ (BTU/ft2-sec)
qw,turb/(haw-hw) , (BTU-sec/ft4) qw'tu_u
indicator variable; if NCS=0 the value of U from the previous step
is used in the Newton-Rhapson iteration scheme for U, if NCS=I the
initial value of U for the iteration scheme is calculated from an
approximate equation
program variables are
haw/h e
haw/h w
incompressible skin friction coefficient, Cf, i
F see note belowC'
FEB
Prandtl number, Pr
Reynolds Analogy Factor, 2NsT/Cf
recovery factor (0.89)
hw/h e
18
+0.4 U4
The equation used to compute FC in the program listing in Appendix
A is valid only fo_ a perfect gas. After the listing of the program
was printed, additions were made to subroutine SPCH to calcula=e FC
for equilibrium air as indicated in Part I - Description of Basic
Method.
Subroutine PRESS
Subroutine PRESS computes and prints modified N_wtonian pressures
corresponding to input body geometry coordina=es, and then a two-dimensional
spline fit to this data is ob=ained. If inpu= pressure da=a are supplied,
this pressure data is read-in and also printed in this subroutine, and a
new two-dimensional spllne fit to =his data is calculated to replace the
spline fit to =he modified Newtonian pressures.
has the form
CALL PRESS(KP,PNEWT)
where =he arguments are
KP see Description of Input
PNE_ (1 - pJps )
Other program variables are
_2p/p s
A_(I,K) (xi,¢k)
_2p/p s
Am,T(J) (¢j)_02
_2_P(I,K) -- A_m(X K)
_x2
BMPP(1) temporary storage for elements _f BMP(I,K)
FP body radius, f
m_2 (I) x2(I) - x2(I-1)
_(J) Oj - ¢j-i
A call =o subroutine PRESS
19
IM2X, I
J
JM
JMM(I)
K
KM2X
LL
P_i(J)
PH2 (K)
PN(I,K)
PNN(1)
PNN(K)
PeP
PR
PT (J)
PX(K)
PXE(K)
PxPP(K)
PXPPE(K)
XMP (I,K)
XMPP (I)
XT
X2(1)
YY(I,J)
2O
see Description of Input
subscript corresponding to input circumferential position
number of input circumferential positions
number of input circumferential positions at X2(1)
subscript corresponding to rearranged circumferential position
see Description of Input
indicator variable; LL-I for modified Newtonlan pressures at
input body geometry coordinates, and LL-2 for input pressure data
input circumferential positlo_, _j
rearranged circumferential position, Ck
P/Ps(xi'_k)
temporary storage for elements of PN(I_K)
pressure ratio, P/Ps
pressure ratio, P/Ps
P/Ps(_j)
BP/Ps (0, PR2 (K))
_P/Ps(x2(IM2X),PH2(K))
_2
_2PXE (K)
_¢2
_2p/ps
8x2 (xi' Ck)
temporary storage for elements of XMP(I,K)
axial location, x
x_
y-coordinate of _j
y-coordinate at xl,_j
z(J)
ZZ(I,J)
z-coordinate of _.j
z-coordinate at z i,_j
Subroutine PDER
For a given position on the body, this subroutine calculates the
pressure ratio (p/ps) and its derivatives from the two-dimensional spline
pressure function. If pressure data are input this function is a spline
fit to the input pressure data; otherwise, the function is a spline fit to
the modified Newtonian pressures corresponding to the input body geometry
coordinates. A call to subroutine PDERhas the form
CALL PDER(XX,PPR,PE,DPDP,DP2P,DPDX,DPXP,DPXX)
where the dungy arguments are
XX axial position on body, x
PPH circumferential position on body,
PE pressure ratio, P/Ps
(p/ps)DPDP
_2:(p/p s)DP2P
DPDX
DPXP
DPXX
_(p/ps)_x
_2 (p/p s)
_x 2
Other program variabies are defined in subroutine PRESS
Subroutine LAGPdq
This subroutine calculates the axial pressure derivatives _ith re-
spect to _x ) at the nose (x=0) and end of the body for _ = _k from
21
Lagrangian Interpolation functions° These pressure derivatives are used
as "end conditions" for pressure spline functions in the axial direction in
subroutine PRESS. A call to subroutine LAGRN has the form
CALL LAGRN (XX, X, FN, IMAX,M, N, F, FP)
where the dummy arguments are
XX axial location where derivative is to be evaluated, _x
X(I) array of independent variables (_x)
FN(J) array of dependent variables (p/ps)
IMAX number of axial stations
M points to the left of XX to be used in Lagrangian Interpolation
N points to the right of XX to be used in Lagrangian Interpolation
F P/Ps at x=XX and _'_k
(p/ps)FP at x=XX and _'_k
Subroutine TKANS
Subroutine TRANS is used only if transition is specified by geometric
location on the body (KTR=-I in input data)° It reads the body coordinates
specifying the beginning and end of transition, and it calculates the co-
efficients for a one-dimensional spline function for each set of data. A
call to subroutine TRANS has the form
IF (KTR
and the program variables are
_2
AMTB (L) -- XB (L)
_¢2
AMTE (M)
•EQ. -i) CALL TRANS
_2-- XE(M)
_¢2
subscript corresponding to a body coordinate used to specify
beginning of transition; L=I,2,.oo,LMAX
LMAX ) number of coordinates used to specify beginning of transition
(see Description of Input)
22
M
MMAX
PHB (L)
PHE (H)
XB(L)
YB(L)
ZB(L)
XE (M)
YE(M)
ZE(M)
subscript corresponding to a body coordinate used to specify end
of transition; M-I,2,...,MMAX
number of coordinates used to specify end of transition (see
Description of Input)
circumferential location of input coordinate for beginning of
transition, _(L)
circumferential location of input coordinate for end of tran_
sltion_ _(M)
Cartesian coordinates of position L specifying beginning of
transition
Cartesian coordinates of position M specifying end of transition
Subroutine BEGTR
This subroutine is used only when transition is specified by geometric
location on the body (KTR=-I in input data). For a given circumferential
location (PPH) it calculates the corresponding axial location (XTB) of the
beginning of transition. A call to subroutine BEGTR has the form
IF(KTR .EQ. -i) CALL BEGTR(PPH,XTB)
Subroutine ENDTR
This subroutine also is used only when transition is specified by
geometric location on the body (KTR=-I in input data). For a given cir-
cumferentia! location (PPH) it calculates the corresponding axial location
(XTE) of the end of transition. A call to subroutine ENDTR has the fo.-nn
IF(KTK ,EQ. -i) CALL ENDTK(PPH,XTE)
DESCRIPTION OF INPUT ....
The input data for thm computer program are listed below in the proper
sequence. Read statements for the input data appear in the Main Program
23
unless noted otherwise. In a subsequentsection the input data used in the
examplesin Part I of this report are listed, Thesteps which describe theinput are:
Step io
_aAD (i,i000) NAM
I000 F_RMAT (20A3)
NAM is the name used for the body designation_
Step 2o-
READ (i,!) IMAX_KM_
i FORMAT (215)
This read statement appears in subroutine BGE@M.
IMAX number of axial stations used to input body geometry data, including
the nose point (x=0) and the last axial station° Although IMAX in-
cludes the nose point (x=0), body geometry data are not input for
this point since the body radius f is zero for all circumferential
angles ¢. (See Sketch i on next page)
KMAX number of equally spaced circumferential planes to be used to spline
fit the body geometry in the circumferential direction, including
the windward (¢=0) and leeward (_=_) planes but only for the half
body 0_¢_zo These equally spaced circumferential planes are gener-
ally different from the circumferential angles corresponding to the
input coordinates described in step 3o Although the input body co-
ordinates are not generally equally spaced in the circumferential
direction and generally differ from one body cross section (axial
station) to another, a one-dimensional spline fit to the input data
at each cross section is used in the program to calculate rearranged
body points which are equally spaced in the circumferential direction.
This rearrangement of geometry data is needed to facilitate the spline
fit in the axial direction. (See Sketch i_)
Dimension statements in the program require that
IMAX < 20 and KMAX < 20.
24
x(:)=o x(2) x<3) x(4) f
I=i 1=4. I--IM_.X
iK=I_
K=I K=2
Y
Y Side View View from Rear of
Rearranged Circumferential
Planes (Equally Spaced)
Sketch i - Body geometry
Step 3.),_
D@ ii I=2,IMAX
READ (i, 3) JM,X (I)
3 F_.RMAT(15,F10.5)
_AD (l,4)((z(J),X(J)),J=I,JM)
4 F_RMAT (8F10.5)
ll C_NT INUE
The statements above appear in subroutine BGE_M, and they are the
read statements to input the Cartesian coordinates of the points used to
describe the body geometry°
I subscript corresponding to axial station number, where I=i is the
nose point (x=0) and I=IMAX is the last axial station number. As
noted in step 2, body geometry data are not input for the nose
point (I=l), and therefore the D@ loop listed above starts with I=2.
JM number of circumferential points input at axial station number I
(see Sketch 2 on next page°)
X(I) axial distance from the nose point (x=0) to axial station I. The
axial stations are not required to be equally spaced, but the :spline
fit in the axial direction is generally more accurate when the axial
stations are spaced approximately equally° <See Sketch 1 above_)
,
Z(J)
Y(J)
subscript corresponding to an input circumferential point, where
J=l is the windward plane and J=JM is the leeward plane.
z-coordinate for point J
y-coordinate for point J
oJ=l J=2
J=4
-_ z
Sketch 2.
¢Y
- Input coordinates of body cross section (viewed from rear).
The points J=1,2,3,... ,JM do not have to be equally spaced. However, it
is required that J=l corresponds to the windward plane, J=JM corresponds
to the leeward plane, and ¢(J+l)>¢(J) where ¢(J)=tan-l(z(J)/Y(J)). Also,
JM may vary from one axial station to another and is not related to KMAX
except the dimension statement in the program has JM<__20. The coordinates
X(I), Y(J), and Z(J) may be input with any consistent unit of length since
the constant CST in step 4 below is used to convert this unit to feet.
Step 4.
READ (i, 900) PIN, TIN, VIN, TRI, TEE, CST,HMAX, KG, KTR, KS, NMAX, KP_ _
900 F@RMAT(7E10o 0/5110)
PIN freestream static pressure, p= (ib/ft 2)
TIN freestream temperature, T (_R)
VIN freestream velocity, V (ft/sec)
26
TRI
TRE
CST
_AX
KG
KTR
KS
Nl_AX
KP¢
value of integrated unit Reynolds number Rn I or momentum thickness
Reynolds number Rnm, if either of these two parameters is used to
specify the beginning of transition. If transition is specified by
geometric location oz not specified at all, any number may be input
for TRI
same explanation as given above for TRI except TRE corresponds to
the end of transition
constant to con_ert units of input body geometry coordinates to feet
maximum step size along a streamline for inuegration by Runge-Kutta
method; HN.AX has the same units as the input body geometry coordinates
indicator variable; KG=O for perfect gas, KG=l for equilibrium air
indicator variable for transition; KTR=-I for beeinning and end of
transition based on geometric location, KTRm0 for transition based
on integrated unit Reynolds number Rnl, or KTR=I for transition based
on momentum thickness Reynolds number Rn . When transition is notm
specified, KTR may be any other integer as long as KS=0 is input
KS=O for laminar heating rates only, otherwise any other integer may
be input
maximum number of integration steps along a streamline
when the additional print out (see Description of Outputl is desired,
set K_I; otherwise any other integer may be input
Step 5.
The data in this step are input only if the beginning and end of
transition is specified by geometric location (KTR=-I). Otherwisep skip
this step and go to step 6.
The statements below appear in subroutine TRANSj and they are the
read statements for the Cartesian coordinates of points along the body used
to specify the beginning and end of transition by geometric location.
READ (I, i) LMAX
F_T(15)
READ (1,31 ((XB (L) ,YB (L), ZB (LI) ,L=I, L_L&XI
FO_ (SE10o01
READ (1, I)MMAX
27
The variables are
total number of input points used to specify beginning of transition,
including one in the windward plane and one in the leeward plane
(o2_2_)
L subscript corresponding uo input point to specify beginning of tran-
sition; where l<L<LMAX, L-I must be the windward plane (_=0) and
L-LMAX must be the leeward plane (_=_)
XB(L)Ix, y, and z coordinates, respectively, for input points used to!
YB(L)>specify beginning of transition by geometric locations The units!
ZB(L)I are the same as those used for specifying the body geometry coordi-
nates. Also it is required that _(L+I)>_(L) where
+(L)-tan-I(zB(L)/YB(L)) and +(i)=0, #(LMAX)-_
The variables listed below pertain to the end of transition, and the same
general comments given above for the variables pertaining to the beginning
of transition apply to these variables also.
MMAX total number of input points used to specify end of transition
M subscript co:responding to input point to specify end of transition,
l<M<MMAX
xE(M •YE(M)_X, y, and z coordinates, respectively, for input points used to
-fZE(M)_specify end of transition
READ(I,2)ALPD,ZW,KBM,KP
2 FORMAT(2FI0ob,215)
ALPD angle of attack, _ (degrees)
ZW ratio of wall to freestream stagnation enthalpy, _w
KBM total number of streamlines to compute for the half body (0_)
KP indicator variable; I_P=0 for simplified streamlines (see Part I for
explanation), KP=I for streamlines computed from modified Newtonian
pressures, K9=2 for streamlines computed from input pressure distri-
butlon
Step 7o i
The data in this step are input only if surface pressures from some
other source are supplied (KP=2) o Otherwise, skip this step and go to step
28
8. The pressure data are read-in as the ratio of surface pressure to stag-
nation pressure around the circumference of the Body at several axial
sta1_ions, in a manner similar to the Body geometry coordinates described
above. The statements below appear in subroutine PRESS.
READ(I,15)PN(1,1),IM2X,KM2X
15 F_T(FI0_Bi215)
D#.II I=2,1M2X!
IF(LLoNE.I)READ(I,3)JM,XT
3 [email protected](I5,FI0.5)
IF(LL.NE. I)READ (i, 4) ((Z (J) ,Y(J) ,PT (J) ,J=l ,JM)
4 F@IhMAT(6FI0oB)|
ll CdNTINUE
The program variables are:
PN(I,1) ratio of pressure at the nose point (x-0)
IM2X
KM2X
JM
XT
PT(j)
J
to stagnation pressure
number of axial stations where pressure da_a are supplied, in-
eluding the nose point (x=0) and the last station. Note that
these axial stations are not required to coincide with the axial
stations used above for the input Body geometry coordinates
number of equally spaced circumferential planes to Be used to
spline fit the rearranged pressure data in the circumferential
direction, including the windward (¢'0) and leeward (_=_) pla_
Although the value of KM2X generally differs from KMAX, =he same
comments given for KMAX above apply here also_
axial station number, where I_i is the nose point (x=0) and
I=IM2X is the last station number
number of circumferential positions where pressure data are input
axial distance from the nose to the local axial station, in the
same units as input body geometry coordinates
y and z-coordinates, respectively, of position J where pressure
ratio is input (units are same as input Body geometry coordi-
nates). Note that _(J+l)>¢(J), where ¢(J)=tan-l(z(J)/Y(J))
ratio of surface to stagnation pressure at position J
subscript correspondlng to a circumferential position where pres-
sure ratio is input, where Jml is the windward ($=0) plan_ and
J=JM is the leeward (¢=_) plans.
29
Note that JMmayvary from one axial station to another and the points
J=l,2,o.o ,JM do not have to be equally spaced_ However, the dimension
statement in the program limit JM, IM2X and KM2X to 20.
Step 8°
KB= 1
3 READ (I, 4) BETAD
4 F_T (F10 o5)
19 IF(KBoGEoKBM)G_ T_ i0
KB=KB+I
G_ T¢ 3
The statements above read-in a value of BETAD for each streamline to be
computed, where
KB streamline nu_nber, I<KB<KBM
KBM total number of streamlines for half-body 0<_¢<__
BETAD angular orientation (B) of each streamline on the _-circle (see
page 36 of Part I), measured in degrees with BETAD=0° corresponding
to windward streamline, BETAD - 180. corresponding to leeward stream-
line, and O.<_BETAD<__I80. A total of KBM values of BETAD must be input.i
DESCRIPTION OF OUTPUT
----_Ah example of the output obtained from the computer program for twok_ !
st_-ines _(B=l Q and 45 _) on a blunted 15" half-angle cone at 20" inci-
dence and M m 10.6 is given in Appendix C. The format for the output was
changed slightly in Appendix C to facilitate printing it in this report.
The output data printed are:
l. Body designation specified by NAM in the input data
2o If transition is specified by geometric location (KTR=-I), the
Cartesian coordinates of the input points for the beginning of tran-
sition (XB,YB,ZB) and end of transition (XE,YE,ZE) are printed. The
total number of points used to specify the beginning of transition
is given by LMAX, and the number for the end of transition is MMAX.
3. Freestream properties: pressure (pc), temperature (To) , density
(pc), velocity (V), and Mach number (M=)o
Stagnation properties: pressure (ps), density (ps) , and enthalpy (Hs).4_
3O
Q
6.
1
8
o
lO.
Input parameters TRI,TRE,CST,KG,KTR,KP,HMAX,NMAX,KP_,KS,IMAX, and KMAX.
The input body geometry coordinates are printed, and the modified
Newtonian pressure(ratioed to stagnation pressure) is printed next
to the coordinates of each point. Th_ modified Newtonian pressure
ratio corresponding to each input body coordinate is printed even
when input pressure data from some other source is supplied°
If input pressures (ratioed to stagnation pressure) are supplied, this
data is printed along with the corresponding body coordinates.
ANG. OF ATTACK - angle of attack, u(degrees)
ZETA(WALL) - wall enthalpy ratio, _w
NO. OF STREAMLINES TO BE CALCULATE - KBM
B - ratio of principal radii of curvature at stagnation point, _/R_I
TW- wall temperature, Tw (QK)
QWS - stagnation-point heat-transfer rate, qw,s(BTU/ft2-sec)
HSP - QWS(6OO6)/(Hs-hw) , (BTU/ft2-sec-QR) where the factor 6006 is
the specific heat at constant pressure for a perfect gas (air),
H s is the stagnation enthalpy, and hw is the wall enthalpy.
Note that for a perfect gas HSP is the stagnation-point heat-
transfer coefficient.
For each position along a streamline,
following data:
X - x-location from nose
Y - y-location same units as
Z - z-location input bodygeometry data
H - scale factor or "equivalent radius", h
S - distance along the streamline, measured
from the stagnation point
BBAR - pressure gradient parameter, 8
P/PS - ratio of local _o stagnation pressure_ P/Ps
QW/QWS - ratio of local to stagnation-point heat-transfer rate, _/_s
If KP_=I is input, then the following additional data are printed at
each position along a streamline:
ire - velocity at edge of boundary layer, U (ft/sec)e
HE - enthalpy at edge of boundary layer, h e (ft2/sec 2)
the normal print-out gives the_._--
31
RH@E - density at edge of boundary layer, Pe "[slug/ft3)
MLrE - coefficient of viscosity at edge of boundary layer, _e
(slug/f t-sen)
MACH NO -Mach number at edge of boundary layer, Me
RN/FT - Reynolds number per foot, PeUe/_e (ft -1)
MOM THK - momentum thickness, 8m (ft)
RNM - momentum thickness Reynolds number, Rnm
P_I - integrated unit Reynolds number, Rn 1
R - body radius, f (in units of input body data)
PHI - angular position on body, ¢ (degrees)
DR/DPHI - aff (same units as R)a¢
_2fD2R/DPHI2 (same units as R)
_2
_f
DR/DX - _-_
D2R/DX2 - _2___f (units of R -I)
_x 2
QBIQSB - (QW/QWS)*(Hs-hw)/(haw-h w) where H s is freestream stagnation
enthalpy, haw is adiabatic wall enthalpy, and hw is wall
enthalpy0
DIAGNOSTIC MESSAGES
Listed below are descriptions of diagnostic messages printed in the
output:
i= NEGATIVE ANGLE OF ATTACK ENCOUNTERED, STAG, POINT SET TO 0.0, PROGRAM
COUNTINUING. If a negative angle of attack a is input to the program,
then the program places the stagnation point at the nose (x=0) in
subroutine STAGN, assigns the value -I to the indicator variable ISTG,
prints the message above and continues the computations. For those
examples where a negative angle of attack is desired, the body geome-
try data should be input upside do%m and then a positive angle of
attack can be used in the input.
32
2. STAGNATION POINT NOT FOUND, RUN ABORTED. An iterative technique is
used to locate the Newtonian stagnation point in subroutine STAGN.
If the stagnation point should not be located, the program assigns
ISTG-0, prints the message above, and terminates the calculations.
3. CALCULATIONS FOR BETA - }{AS TERMINATED AFTER INTEGRATIONS_
If the computations along a particular streamline should reach the
end of the body before the number of integration steps (L) has reached
the maximum allowed (I_MAX in the list of input), then the message
above is printed at the end of the computations for that streamline°
4. *** SUBSCRIPT OUT OF RANGE **_, CALCULATIONS FOR STRE_4MLINE NO.
DISCONTINUED_ When the number of integration steps (L) along a
streamline has reached the max±mum allowed (NMAX) before the end of
the body is reached, the message above is printed.
5. INTEGRATION OF STREAMLINE DISCONTINUED DUE TO STEP SIZE,
*** B =***o When the integration step size [B) in function
RUNGE is decreased to a value less than l0 -8, the message above is
printed and computations for _his streamline are discontinued.
60 SCALE FACTOR (_) GOING NEGo, ,ee H = ST_REAMLINE DISCONTINUED.
The message above is printed if the scale factor or "equivalent
radius" (H) becomes less than l0 -4, and the computations for that
streamline are discontinued.
7. REY NO (RNM) GOING NEG, *** RNM -, STRE_LINEDISCONTINUEDo
If the momentum thickness Reynolds number (Rnm) becomes less than
lO -4, the message above is printed, and the computations for that
streamline are discontinued°
8. **** BEGINNING OFTRANSITION ****o This message is written just before
the print-out at _he beginning of the transition region.
9. e*** END OF TRANSITION ,e**_ This message is written just after the
prlnt-out at the end of the transition region°
TEST CASES
Test cases were calculated for: Case I. - A blunted 15 _ half-angle
cone at s - 20 ° and M - 10.6, Case IIo - A blunted 70 ° slab delta wing
33
at _ = i0 ° and M - 8, and Case III. - The HL-10 Lifting Body at _ = 20 _
and M = i0. Inputs for these cases are given below, and output data for
two streamlines on the blunted 15 _ half-angle cone are given in Appendix Co
Results for all three cases are presented graphically in Part I of this
report, and the results for the bl_nted 15 = half-angle cone are also sho_m
to compare favorably with experimental data.
Case Io - Blunted 15 ° Half-Angle Cone
Input data for a blunted 15 ° half-angle cone at a = 20 _ and M = 10o6
are presented on the pages which follow. Written beside each section of
input data is the step number which describes that data in Description of
Input° Step 1 corresponds to the body designation, and for Step 2 the Body
geometry was specified by input coordinates at 19 axial stations (IMAX=I9)
and 20 planes around the half body for the rearranged circumferential points
(KMAX=20). As noted in Part I of this report, the body geometry could be
represented by simple analytical expressions, But the coordinates of 20
points around the circumferenoe of the half Body at 19 axial stations were
used to generate the body geometry in order to test the accuracy of the
doubly cubic splice function° This data corresponds to Step 3 of the in-
put, and the coordinates are input in inches°
Freestream properties are shown in Step 4, where p= = 2.6614 1b/ft2_
T= = 89°971 =R, and V - 4928_i ft/seCo Since only laminar hea_ing rates
were calculated here, TRI=O and TRE=0_ In order to convert the input
coordinates from inches to feet, a value of CST=0_0833333 was used_ The
maximum step size was set at HMAX=0o4, and a perfect gas (KG=0) was
specified. Again, since only laminar heating rates were calculated, KTR
could be any integer (KTR=0 was used here) and KS=0. The maximt_n number
of integration steps along a streamline was input as NMAX=!00, and since
the additional print-out along each streamline was desired, KP_=!.
Step 5 of the input data was not included because transition was nDt
_equiredo For step 6, an angle of _ttack (ALPD) of 20 = , wall enthalpy
ratio (ZW) of 0.251, number of streamlines (KBM) of 2, and streamline com-
putation by modified Newtonian pressures (KP=l) were input_ Step 7 of
_4
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35
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_._ =._ _,. ,,_ _. -_ _ f_'_ _ _ _ _1_
C'; 0 C_O. c_ O-
e _ @ $ 11
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37
the input data was omitted because modified Newtonlan pressures were used
in lieu of an input pressure distribution.
The input data given here only have two streamlines to be calculated
for Step 8 - BETAD of 1 degree and 45 degrees. Note, however, that the
data presented in Part I of this report used BETAD values of I*, 10*j 15 e,
and 45".
A complete listing of the output data for this case is givenin Appen,
dix C. For an explanation ofthls data, see Description of Output. From
the data listed in Appendix C, it is observed that the integration step
size along a streamline is small near the stagnation polntD but it in-
creases rapidly downstream due to the variable step-size Runge-Kutta
integration routine. The streamlines move from the wlndwardiside of the
body and approach the leeward plane downstream. The scale factor or
"equivalent radius" increases all along the BETAD - i" streamline, but
it decreases along the BETAD- 45" streamline after it has "essentially"
reached the leeward plane (PHI - 180.). It should be noted, however, that
the streamlines are continued into the "shadowed" region of the body where
the surface pressure is assumed to be freestream static pressure. It is
possible that the flow separates from the body somewhere in this region_
and therefore the calculated properties are questionable there. Along
each streamline the heating-rate ratio QW/QNS decreases and them gradually
levels out. The computational time (CPU) for this case on an IBM 360/75
computer was 1:47 minutes.
Case II. - Blunted 70* Slab Delta Wing
Input data are given on the following pa_es for a 70 @ swept delta wing
with a cylindrical leading edge which is tangent to a flat slab on the
upper and lower surfaces. The blunt nose is a spherical cap, and the
radius of both the spherical cap and cylindrical leading edge is one foot.
Again, the step number corresponding to that in the Description of Input
is written beside each section of the input data.
Step 1 corresponds to the body designation, and in Step 2 the body
geometry is specified by input coordinates at 19 axial stations (IMAX-19)
and 19 planes around the half body for the rearranged circumferential points
38
HI-I
41
mr,3
14o
.i.I
i-i
0
Z
..j_J
J
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I I
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0 (DI
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e e e
¢M
04 _
| I I | I | I ! |
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I I I _ _ I I I !
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f_O _" '_"_ "_ _ ',C .'_,
e _ o • g _ e e
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39
._d"
t% C I/_ C 0"- kt_ t'.-, _-J
t'_ t_ 0 '_ _ ,",J _ _'_ ,_
_'_', D'. _. L"_, ,,,I" C-,J C,# 0_o r'.- _ _ ',_ I'--- t,,; t-Jt_ r_j cth o,_ co D,, _-,o P,- _'_ .,',Job'=,, on,_
°g. °0o go
0
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_ _ _ _ _ _ _ _ 0 _ _ _, _
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0
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g o o • og _ o o e e _ _ _ •
_ot _te 0ee eli _eg000 000 OOO _00 000
i I I i !
. . . . . .0 . . . . . • . . .
_ i 0 I I Q I I 0 I I 0 I I 0 I I
40
(EMAX-19). The coordinates ell3 points around the circumference of the
half body at 19 axial stations were used to generate the body geometry in
Step 3. All body coordinates were input in feet.
In Step 4 the freestream properties are p_ - 103 ib/ft 2, T_ - 416"R,
and V= - 8000 ft/sec. For this case transition was assumed to begin at a
momentum thickness Reynolds number of i03 (TRI-103) and end at 2 x 103
(TEE=2 x 103) for eachinviscid surface streamline. Since the input body
coordinates have units of feet, CST - I. 0. A maximum integration step
size of HMAX - 0.05 was input, and a perfect gas (KG-O) was specified.
Since transition wasbased on momentum thickness Reynolds number, KTR-0
and KS-I. The maximumnumber of integration steps allowed along a stream-
line was NMAX=IO0, and KP_=l was input to obtain the additional print-out
data along each streamline.
Input data corresponding to Step 5 were omitted because transition was
not specified by geometric location. In Step 6 the angle of attack was
ALPD-10 °, the wall enthalpy ratio was set at ZW-0.4,:the number of stream-
lines to be calculated was KEM-8, and the streamlines were calculated from
modified Newtonlan pressures (KP-I).
Step 7 was omitted since surface pressures from some other source were
not used. For the 8 streamlines to be calculated, the values of BETAD in
Step 8 are 0, 1, 10, 38, 60, 90, 170, and 179 degrees.
The computational time (CPU) for this case was 3:02 minutes on an IBM
360/75 computer° Surface streamlines and heating rate ratios are presented
graphically in Part I of this report.
Case III. - HL-IO Lifting Body
The third, and last, test case computed was for the HL-10 lifting body,
which is described in Part I of this report. Input data for this case are
listed on the following pages, where again the Step numbers corresponding
to the steps in the Description of Input are written beside each section
of inpu_ data. Step i gives the body designation, and Step 2 lists the
number of axial body stations (IMAX-20) and rearranged circumferential
planes (EMAX=20) used to describe the body geometry. In Step 3 the
42
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v
I-Ii-Ii,-,i
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-44
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coordinates of the input body points are listed at 20 axial locations and
19 circumferential positions, where all coordinates are input in inches°
The freestream properties are given in Step 4, where p= = 103 IB/ft 2,
T= = 416 °R, and V - I04 ft/sec. In this case transition was specified
by geometric location; therefore TRI and TRE could be any numbers (values
of TRI-IO. and TRE=20. were arbitrarily input). Since the input body co-
ordinates were in inches, a value of CST = 0_08333 was used. The maximum
integration step size was choosen to be HMAX - 18_1235, which is one-tenth
of the distance from the nose tO the last axial body station_ Equilibrium
air (KG=I) was used for this case, and KTR=-I was input Because transition
is specified bygeometric location. For laminar, transitional, and turbu-
lent heating, any non-zero value of KS may be used, and here KS=I was input.
The number of integration steps along a streamline was limited to NMAX-100,
and only the normal print-out along each streamline was desired (KP@=0).
In Step 5 data was input to specify beginning and end of uransition
by geometric location. Three pointswere used to specify the beginning
(IMAX-3) and end (MMAXm3) of transition. In each case theaxial location
was arbitrarily chosen to be x = 33.073" for beginning and x = 54.239" for
end of transition.
For Step 6, the angle of attack was ALPD = 20", wall enthalpy ratio
ZW = 0.i, number of streamlines KBM - 5, and the streamlines were calcu-
lated by the "simplified method" discussed in Part I (KP=O). Step7 was
omitted. For the five streamlines tO be calculated, values of BETAD in
Step 8 are O, 45, 90, 135, and 180 degrees. '
Heating-rate ratios for this case are presented graphically in Part I
of this report for the windwardstreamline. The computational time for
this case (CPU) was 1:48 minutes on the IBM 360/75 computer.
ACCURACY AND LIMITATIONS. OF METHOD.
The method used in the computer progr.am described herein is applicable
to a large class of configurations subject to the following restrictions:
i. The body must have at least one plans of symmetry, and the free-
stream velocity vector must be parallel to this plane of symmetry.
47
2. The body radius r = f(x,$) must be single valved.
3. The nose of the body must be blunted, and at x=O f(O,$)=O and
-xf(O,,) * ..
A two-dimensiona!cubic spline function is utilized to represent both
the body geometry and the input pressure distribution. The computer program
listed in Appendix A has dimension statements which limit the number of
axial and circumferential positions to 20; however, these dimensions could
be changed to accommodate as many positions as desired. As indicated in
Part I of this report, the spline function does not smooth out data points
(althoughit does smooth derivatives), and therefore data points used in the
spline function should be reasonably accurate. When pressure data from some
other source are used, they should be smoothed before using them in the com-
puter program.
The accuracy of the spline function is affected by both the number and
location of data points used. More accurate results are generally obtained
when the data points are nearly evenly spaced in the axial and circumferen-
tial directions. In some cases a large number of data points are needed to
make the spacing between data points nearly even and also have a reasonable
number of points in regions where the data is highly varying. As an example,
consider the cross sectional geometry of a blunted slab delta wing far down-
stream of the nose. At this axial station the thickness of the wing is small
compared to the span, and a large number of body coordinates are needed to
make the data evenly spaced in the circumferential direction and still have
a reasonable number of points near the tip of the wing.
Input freestream properties for this method are limited to supersonic
speeds, and when modified Newtonian pressures are used the speed should be
hypersonic. Although the angle of attack is not limited to small angles,
heating rates on the lee side of the body are questionable at large angles
of attack. The approximate methods used to calculate heating rates are
more accurate for cold walls (_w<<l) than hot walls.
In Part I of this report the present method was found to predict
laminar heating rates very well on the blunted 15 ° half-angle cone at 20 _
incidence and M = 10.6. Additional comparisons of the present theory with
48
experimental data on other body shapes and for transitional and turbulent
heating are needed to assess the accuracy of the theory more thoroughly°
CONCLUDING REMARKS
The computer program described in this report was developed to calcu-
late inviscld surface streamlines, and laminar, transitional, and turbulent
heating rates on shuttle-type configurations° Approximate methods were
employed to keep the computational time and storage down to relatively
small values but provide reasonably accurate results. Input and output
data were programed to be easily used by the ordinary engineer who as not
trained as a computer specialist. Although the computer program was devel-
oped on an IBM 360/75 computer, it is also compatible with the IBM 370/165,
CDC 6400, and CDC 6600 computers.
49
APPENDIXA
PROGRAMLISTINGfrom IBM 360175 Compu=er
Conte_ts
Main Program
Function RUNGE
Subroutine STAGN
Subroutine NEWP
Subroutine STRMLN
Subroutine BGE_M
Subroutine SPLINE
Function FSPL
Function DFSPL
Function D2FPL
Subroutine STGPR
Subroutine
Subroutine
Subroutine
Subroutine
Subroutine
Subroutine
Subroutine
Subroutine
PR@P
SPCR
PRESS
PDER
LAGEN .........
TRANS
BEGTR
ENDTR
52
62
64
65
66
69
72
73
73
74
74
75
76
77
80
81
82
84
84
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It ,,,J I,_ ,,_
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L_
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i
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j :
i
APPENDIX B
DETAILED FLOW DIAGRAM
Symbols used in the flow diagram conform to the l_ternational Orgauizatlou
for Standarization (_S0) Recommendation on Flowchart Symbols for Information
Processlngp and are consistent with the fewer symbols .adopted by the American
National Standards Institute, Inc. (ANSI).
CONTENTS
Main Program ..........
Function RUNGE
Subroutine STAGN
Subroutine NEWP
Subroutine STRMLN
Subroutine BGE_M
Subroutine SPLZNE
Function FSPL
Function DFSPL ......................
Function D2FPL
Subroutine STGPR
Subroutine PR_P
Subroutine
Subroutine
Subroutine
Subroutine
Subroutine
Subroutine
Subroutine
SPCH .... - ........... - - _ ....... - - -
PRESS -
- - -LAG_ ................
BEGTR
ENDTR.
87
118
128
134
135
138
154
159
160
161
162
165
167
170
185
187
194
200
201
' 85
•/ PaAD /ALPD,ZW,
/ "KBM,lIP
1ALP ,SAL,
CAL
I
l
"IfIICALL
STGPR
IPIN,TIN, /
_iN,VIN, /
EMIN /
L
/ /PS, RR_S,HS
88
,i
[ TRI,TKE,CST,KG, i
/ m_¢,Ks, /
CmmUTE
PNE_T
Ji
I
PRESS
I I
i
ttW)_)VlSW, RH_SW
I
li:::l'lt
DUED?S,
CTHM1,CTHM
J
89
!
i ,ii:'I i
,I
I
i
_t
!:
i,
[
, ti
NCS,DS
KB,
BETAD
I ,
/ (C_L.l _GS.)
(ADD'L
C_L..
_DGS.)
ICBE, SBE,
-I CB4 _CBS,Y(1) ,ZZ _YY
93
i
]
:!
;i !
i:
i
I
i
!
I!
!i
i
I
C_NNECTS
T_ PAGE 108
FROM
PAGE i05
C_NNECTST_ PAGE 106
C_NNECTS
T_ PAGE 102
N_ YES
YES
S C_MPUTEK
YES . N_
III'STRMLN I_
PAGE 102
99
I!"
t
:i
i!'
i _!I'
i
if'
m_aPAGE I00
THE-PI+THE I
_C2)
YES
U
c_n_vzE.
EM, DGS ,DSS, DP SIS,U_U, P_R, PUI'I,BIFF,
DPS, DUEDS, F(4) ,F(5),
RNF, I_ 1 ,TI4M,RNM
¢101
102
C_ECTSm_ PAGE 99
C_NNECTS
T_ PAGE 107
YES N_ PAGE 99
YES/ PAGE 117
_-- C_)NNECTS -
_kv/J T_ PAGE 116
YES N_
Y(1),YY,ZZ,/Y(3),S,_BAR,/
P_2_P,Q_QS /
Nfl _ YEs
FP4BM
PAGE 115
FROM
PAGE 116
104
N_
/=,_E'mOE,WSC,EM,/
/ FF,PRID,_,_P,// FX, FXX, Q_QSB /
@ YES
C_NNECTS
TZ PAGE 107
i
i]'i
, !
'i
J
z i
P
F
!!
i!l,i i
CCNNECTST_ PAGE 99
YES
IS
:IMP)
0
FP4_.
PAGE i
_CCmrSCTS
V T¢ PAGE I06
YES
I
/"N¢ _
C_NNECTS
T¢ PAGE 106
FR_MPAGE 115
F_MPAGE 116
FR_MPAGE 113
C_NECTST¢ PAGE i07
PAGE 106
PAGE 107
FP_M
PAGE 107
103
v I
FP_MPAGE 105
F_MPAGE 105
KB=KB+l
.
WRITE /
KB ,,/
C_R_NECTST@ PAGE 9 2
T_ PAGE 105
106
FR#MPAGE 99
'C_NNECTST_ PAGE i05 /310 /
WRITE
KB, DS
I
T_ PAGE 117
_F_MPAGE 108
FROM
PAGE 102
¢
'i
'I
,I
i
i,
i
i,
i' '
i
i
FP_MPAGE 102
C_TNECTST@ PAGE 105
_, FP_M
PAGE 104
f T
- 314 /
WRITE /RNB,KB
C_NNECTS
T_ PAGE 105
F_MPAGE iii
/30/WRITE
/". •.RUN
/ ABORTED"
_ C%NNECTST@ PAGE 10S
YES
i
FRCHPAGE ii0
_170 C_MPUTE
C%NNECTST_ PAGE 117
107
108
CCNNECTST¢ PAGE106
YES
N_
I
CALL
STRMLN
}C_MPUTE
THE, SPH, CPH,
SGM, CGM, STH,
CTH, SDLP, CDLP,F(1) .F(2)
YES.t--
C_NNECTST_ PAGE llO
I _oo.I :ii
,I
i
i i_i i: _
I
'i!
F(3)-Y(6)
ICALL
PDER
,,,,,
PE=PS*P_P
CALL
PR_P
hT_
N¢
C_MPUTE
EH, U_U,PUH,
F(4) ,P4_R,DSS,
DXB, DPHB, DPB,UESQ,F(5)
YES
109
C_MPUTE
HTI, HT2, DSB, DTHB,
DPS ,DGB ,DDLB,HT3,
HT4 ,HTS,HT6 ,HTT,
D2PB,F (6),F(7) ,_F,
RNi,THM, RNM, DUEDS
YES
C_NNECTST@ PAGE 116
PAGE 117
ii0
YES NO
YES
YES
C@NNECTST@ PAGE i07
C_NNECTS
T_ PA_E 106
I:
[
i
iiI!
l
i
i,
i_ ill
I
, i
i,
NO
L-L+1 I
!
YES
C_NNECTST_ PAGE 112
STRL-STR
Ic_E
STR',TRIL,
TREL
i J
C_TNECTST_ PAGE 107
F_PAGE i03
111
[_ CCN_ECTST0 PAGE 114
.F_MPAGE iii
N_ YSS
YES _
NO _ _YES
YES
N_
I12
C_ECTST_ PAGE 104
Cj2JNNECTS
T_ PAGE 113
CONNECTS
Tg PAGE 115
: Ii
I
i
Ii I
L
FROM
PAGE 112
R_
CCNNECTST@ PAGE 114
DS ,MR,
L2
NR-NR+2
IMR,LI,L2,
Y (NR-I))HC,
HTR, HI ,Y (NR)
WRITE /"BEG. _F
TRANS."
J/_ _E /
(I),_,zz,/(3),s.m_m_J
P_P,Q_QS /
@ YES
113
PAGE113
FP_M
PAGE 112
,HE,RH_E,VISC, /
,RNF,THMF ,RNM, /
I,FF ,PHID,FP ,/
,i_'_(,FXX, Q_QS B/
I
K,DS
YES N_
1i4
ENDTR
@ YES
+
c_NPU_E
NCS,XI)EXW)
WFN,QW,QWB,
Q@QS)Q@QSB
_ C_NNECTST_ PAGE i16
_ C_kTNECTST_ PAGE 115
I
!
i
ii!
i
i'
:i
.!
F_PAGE 114
CCm_CTST¢ PAGE 105
85 C_UTEDS,MR,L2
I8_ C_MPUTE
MR,NT,Q_QS,
Q@QSB,QW
Y(1),n,zz,/(3),s,B_aJ
P_P,Q_)QS /
N_ _ YES=_
/' WR_ITE :
/UE,HE,RH_E,VISC, //RNI,FF,PHID,FP,.! FPP.FX.FXX,O_0SB
I/ _ /' "E_D CF.
TRANS."/
__ C_NECTST_ PAGE 104
PAGE 112
115
116
FROM
PAGE 103
C_NNECTS
T_ PAGE 1047 6 C40MPUTE
QW,Q@QS,
q_QsB
YES
8o c_PUZZ t
II !'1CALL
SPCH
CCMPUTEHC,F (NR-I) ,WI,W2,
W3,W4 ,W5 ,W6,
WT,F(NR)
F_M
PAGE ii0
F_MPAGE 102
F_MPAGE ll4
ii
,ii_' ii'
FI_TNCTI_N RUNGE (Y,F,X,H,N ,HMAX,MI_)
_/_ START _)\ i
INTEGER
RIPNGE
i!
i DATA
I M,L_P,EPS/
i -04/
M,,M+I 1
N_
I
i
YES
YES
r_
C_NNECTS
T_ PAGE 12].
C_NNECTST_ PAGE 121
I18
! ;
[
!
!i
PAEE 118 YES
zzs.
C_NECTS
T_ PAGE 120
cCz_zc_sT_ PAGE 127
>/_ YES
I
J j-o ]
, I .....
JuJ+l LI ,
PAGE 120
119
y_ (a),_z.P
YES
X_=X
I
21 J=O
YES
• l
I_22 C¢_ZZ
SAWY (J),
PHI(J) ,Y(J)
C_R_NECTS
TO PAGE llg_
FRZM
PAGE 127
i
, i
120
C_E.
FT Ii'ii
C_NNECTST_ PAGE-121
144 _EPU_(J)j (J)
YES
cem_rE I
X DRUNGE
I
I
i '
Y(J)
' _ -
YES :
FROM
PAGE 119
i:
,!
i
ii
J,i
122
PAGE 126
FROMPAGE 119
I'
L_P ,M.
I
I
92 cCzPuzzZ(J) ,F(J)
L
C_[PUTE
X,H
I
YES
C_qNECTST_ PAGE 120
cCs_zczsT_ PAGE 120
127
SUBP4_UTINESTAGN(ALP,IMAX,KMAX,X_,RPER,B,ISTG)
START
I
IISTG=-I
12x_-°.°I
I
_i0 C_MI=VZEn,_2 I
0
C_NZCTSTJO PAGE 129
128
C_NNECTST_ PAGE 130
FROMPAGE 133
i'S°I"S°2¸I
I
I,
w[
IJ-O
IJ-J+l
r,, , l
CALL
STRMLN
FP_MPAGE 130
131
SUB_TIN_ ND_ (AA,F)
START >
C_UTE
FF,F3X,FXP,FXPP,FXXP,TD,TDP,TDPP,TDX,
TDPX,TDXX,T,TX,TXX,TXP,TP,TPP,D,DX,
DXX,DXP,DP,DPP,DPSI2,PX,PXX,PP,PXP,PPP,PE
i
134i
i[
i',[
SUBROUTINE STm_N (XX)PH,DLPH, DDP,DDX )GH)DGP)DGX,F,IMAX,EMAX)
START >
YES
YES ,v
YES ._r
_ C_NNECTST_ PAGE 136
135
AMF21FI_F2_FF_FFPIFF2PDAMFXI_AMFX2_FXI_FX2_FFX_FFXP_AMFXX1,
AMFXX2,FXXI_FXX2,FFXX,TDLP,DLPH,CDLP,SDLPIT2jDDP,DDX_TGM,
GM, CGM,T3,DGX,DGP,F,AMDI_AMD2_FFPI,FFP2,AMGI_AMG2,F2XIIF2X2.,
FFXX,FFXPP,FFXXP,FF3P,AMGXI,AMGX2,F3XI,F3X2_FF3X,F2,FX,FXX,_'P,PPP
P,ZTUI_ >
137
C#_ECTST_ PAGE 140
N_
FROM
PAGE 143
142
5 C%MPUTE
troT(j)
YES
,
_T (JS),
, _T(Z)
I
SPLINE
iI_-_-_ I
. J-1 I
ii --
,!
:_ C_NNECTST_ PAGE 146
.: . / J
YES
13 c_E 1
B_F(I,K)II!x_(I,K)|
@ tYES _.
N¢
PAGE 148,
L' _i .
T_ PAGE i_5
147
ii •
I_I+i
I
20 C_M_UTE ]
XMD(I,K) ;XMG(I,K) ,
BMD(I,K),
BMG(I,K),,
v
C_NNECTS
T_PAGE 151
.e YES
i
153
SUBP4_UTINE STGPR (PIN, TIN ,VIN,PSI#IN ,EMIN ,P$ ,RB_S ,HS ,KG)
CCM_E iRIN,HIN, RI_IN,
HS, EMIN
cCm'uTE
AI,A2
WRITE /
HIN, RH@ IN/
Iv¢-o.o 1
YES
CCNNECTS
T_ PAGE 164
162
J-] J,,J+l I
¢
SUBR¢_IN_. PR¢_ (PS,HS, _¢S,PE,HE,p_E, UE,VISC, _S,KG)
N¢YES
C¢_CTST¢ PAGE !_6
UE
I
C_M_UTEUE.
J
C_MPVIE IAEM2
16S _
V " ' •
!
I C_UTE
EMS ,HER2 ,AS D
PEE2 iA6 _V_$C
t '
I
|
C_MPUTE
HE,TE_V_SC _
_EDUES,UE,EMS
166
ili
i
:I
SUBR¢UTI_Z SPC_(ZW,P_,_S,UZ,HS, PaM,CI',QW,QWS,_CS)
SPECIFY
PR,REC,RA
c¢_nz
CFI, U
N_ _ YES
+" ' i
+l o-_oo.I
167
SUBP_t_NE PRESS (EP,PNEWT)
STAKT
I
I C_I¢OUTEP I ,X2, LL
YES
C_MPUTE 1PN(I,1)
PN(I, i) ,
IM2X,
EM2X
I
_[ 20AKM'KM2X-I _I
I,
PAGE 184
FROM
PAGE 171
c_E
PHi(J)
_ I5 COMPUTE
HPT(J)
1
1C,e_WE
_TCa_)
CONNECTSTO PAGE 174
, , 1 ,, ,
JuO ,,,l C_NNECTST@ PAGE 176
175
YES N¢
N¢ YES
CCNNECTS
T_ PAGE 180
FROM
PAGE 184
SPLINE
I
SPLINE
II _-o )
I
t _-° 1
CJNNECTS
T¢ PAGE 181
182
!i "_UBP_UT iNE PDER (XX _PPH, PE, DPDP, DP 2P _DPDX, DPXP, DPXX)
START
I
i _ I
I I=I+l
YES /_
_ N_
T(_ PAGE 186
185
YES N¢ :
FRCM
PAGE 185
"_125
K=K÷I
1c_e,_E
_fP1
AMP2,PI,P2,PE,
DPDP,DP2P,AMPXI,
AMPX2,PXItPX2,
DPDX,DPXP,AMPXXI,b-MPXX2,PXXI,PXX2,
DPXX
CCNNECTS
T_ PAGE 185
186
SUBROUTINE LAGRN (XX_X_FN_,X_M_N,F_p)
i
START
i ,
. 1
i I=I+i 1--
v
/YES, , _"/__, Np
r
C¢_CTSx_ PAcX 188
187 _..
II_-°..... I
!_i K-K+I I-I
__[__
D ,AN ,ANP ,J_K
C_NECTSTJ PAGE 193
I a-a_ I
li
YE S-- N
C_NPIr_E
TR,JK
'I I
C_NNECTST_ PACE 191
PAGE 193
i
o
o
I-I
I.uz0
I,U0!
¢1I.U
Z
.I
202
U..0,,_tuu,),.__m
,JNtn_,,w,,,d
t-.O
tnZl.N
tn C=tu
wn,O
o. No
tu O
b.. tu_.
t_ X .,,' tuOit. t=-
Or
_N
_Ov3uJ
¢L
NO
eC3WZU_
_0
UU__O
eO
_U
_O
u_w"_ :O
u_o
u_
ZuJ
uu_4
u_ y.
O _O
_;felZ _U_
Z_U
u,_O
c_
uJc0
_o
_u
O
fq
0
I! IIII
o0
0
0
N
X
_ Z
o
IIII II II
0
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Z
l.U
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U_
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Ov)uJ
¢=
_OC_
_3U
U.k-
_u_u_ _100
II II 11 II I! II
q== w
O. Q..I_, 0._ 0.
O. O. I_. I_. 0.. '_.
_0_ oo_0
4) 0 0 • @ e
000000
II It tl It 11 II
_-_1..>. )..>->..
oo00oo
II II II' II II tl
Z( 7'|= 0.18094. Y( 71=Z( 8)= 0°-19793 YI 81=Z( ,9)= 0020952 Y( 91--2110)= 0.21539 Y(IO)=
Z(II) = 0.21539 Y(II) =
Z(12}= 0.20952 Y( 12l---Z|13)= 0.19793 Y(13)=Z(14)= 0.. 1B094 ' Y( 14)=Z (15)=_ 0o.15901 Y( 15)=
Z(16)= 0.13275 Y_ 16}=
Z( 17)= 0-I0287 , Y_17)=
l(18)= 0.07018 Y(18| =Z(19_= 0.03557 Y_ 19)=
Z|20)= 0=0 Y_20)=
MODoNENT=PRESS_AT
0.118210.086820.053060,01785
-0=01785-0=05306-0.08682-001182.1-0=14638
-0.17056-0=19008-0;20442-0°21318-0.21613
INPUT COORDo
FOR JN:= 20AND X( 3)= 0.13710
THE Z_Y ANOP COORDINAT'ES ARE:
Z( 1}= 0=0 Y( l;=Z( 2;= 0.04771 Y( 2l =Z( _310 00094121 Y( 3)=Z( 4)= 0°13796 ¥( 4}=
Z( 5)= 0°17804 Y( 510Z( 610 0021327 V{ 6)=
Z( 7)_ 0.24267 Y( 7)=Z( 8)= 0=26546 Y( 8)=
Z! 9)= 0.2a100 Y( 9)=
Z( t0|= 0=28888 Y(.10)=Zttl)= 0.28888 Y( 11)=Z(12)= 0o2_100 Y( 12)=1(13|= 0®26546 V( 13)=Z(14)= 0°24267 Y(14)=Z(15)= 0=21327 Y(15)=
ZII6)= 0.17804 Y(I6)=
Z(171= 0013796 Y(I7;=Z(18)= 0,09412 Y(18)=
0®2B9870.285920°274170.254940_22875
00196330°158550®1164_0=07116
0.02394-0.02394
-0.0.7116-0.11644-0=15855-0319633-0.22875
-0°25494-0.27417
PIPS( 7l=PIPS( 8)=PIPS(.9)=P/PS(10)=
P/PS(I1)=P/PS(12)=P/PS(13)=P/PS(14)=PIPS(15)=
P/PS|16}=PIPS(17)=PIPS{18|=PIPS(19)=PIPS(20|=
P/PS(I)=PIPS(_2)=PIPS( 3)=PIPS( 4)=PIPS( 51=
P/PS( 6) =PIPS( 7)=PIPS( 8)=PIPS( 9)=
PIPS(IOI=PIPS(11}=
P/PSi12)=P/PS(131=PIPS(14)=P/PS(15)=P/PS(16)=PfPS(17)=
P/PS(181=
0°76857007195500668620°61757
00567990,521060_477970.439630°40687
0037929
0.357980o3427100333540°33046
0°7421700.736040o7178600688670=65001
0.6035600551630.49664
0°440690.386110°334390°287040=245170.209240°179520°155930,13810
0°12572
203
_:H0OoN E WT ,_PR.ES S ,,AT
_.FOR J_ = 20 AND X(
V{l?1= ,-0=28592.v{zoi=-o.28e87
INPUT'COORDo
41= 0.,2056_
T._IE
:,"Z_'•L )=
.i{ ,B_=., ZL_ .4_=
Z ( -6.;=Z ! :,T)=Z_ 8J'=Z( 9t=Z(_O)=Z(ll)=
Z ( 12 D.='Z{t3)=
,Z ._151=Z(l:6I=
.. CrZ:( 1 "_ ) _
.__:.I.( 1.8 ) =.:Z 119)=-
_, Z{2.0)=
Z,Y AND.;.P:CQORDINATES ARE:
0,0. - ,0 ;0-55'07O; 1086,_
• 0.159240,20550
• : 0,246160;280.I00.;306400,324340._3344
O; 33344O. 32_!34
0;306.400 ,; 280t 00 .;2_&:l 60.205500;159240;10864 ..
• 0,;05507.0.0 "
Y( _1)=Y! 2)=,Y-{. ,;3)=Y! 41=Y( 5)_
Y( 6)=Y( _7):
Y( 8 )=Y(. 9_=Y{ 101=
Y( 11)=¥( I2)=Y( E.3)=
Y(14)=Y( 15|=
. 0.3._58 _
,. 0;33002
0,31645.0,29_250;26403 _
0_22660
0.t8300_0,134_010;08213.0_02763
-0;02¥63-0;08213-0;13440-0.18300
-0_22660
Y |16 | = .- -0.26403Y(1¥|= --0_29425Y|181=, -0_3t6_5Y(19)= . -0,;33002Y( 201=: • -0.33456
...:; :..; :
:_OD,_:_T.PRESS,,AT INPUT COORDo
/C .... :
: :_HE Z._V _ND P CQQRDE_ATES _RE:
;.:Z:{'. 2' _= 0,,059'_5 Y_ _2)= 0=35627.
:_Z'_ 3_ = O.IIT2B Y_ ,3_= 0_3_163
'i•:J.•
PIPS( 19l =
PIPS(20|=
p/Ps(il}=pips( ,2•|•=
PtPS(:._)=P/PS(_5)=PIPS(6|=
PIPS( T|=P/PS(S):P/PS(gt_.PIPS(LO.t=P/PS(II_=P/PS(12I=
P/PSI13|=PIPS.(I_)=PIPS[15).=PIPS|16|=
P/PS(I_)=P/PS(_8)=PIPS(I?)=
P/PS(20|=
P/PS_ 2)=
P(PS_ _i =
0.536300,530280.51260
0;_8_16
0.40308
0.355010*304600*255190.20810
o.z_ps30.128660.0977b0.073200.05_27
'0;0_057
0.031270.025430.022220.02i:2!
204
Z ( 4)=Z( _5)=Z! 61-Z( 7)--Z( ,81=
Z I[ 9)=ZItO|=ZIII)=
Z 112 )-
ZI13)=
Z(14)=
ZI15;=
Z|16)=
Z I 17)=
Z|18|=
Zll9)=
Zl20l=
031:_71910;22185
-0.265740.302380.33077
0,350140*359960.35996
0.350140. 330770.302380.265740.:221850.17191
0,11728
0.059450.0
YI 4)=' YI :5)--
YI +6I=Y¢ 7)-YI. +81=Y[ 9)=YI lO)=
Y111)=Y( 12):Y( 13 )=
Yi 14)=
Y(15)=Y( 16 I=,Y|IT)=
Y(1B)=Y( 19)=Y( 20)=
0.317660.28503
_0324463
0,197560,;145090.08867
0302983-0.02983-0.08867-0.14509-0319756
-0,24463-0.28503-0.31766
-0.3_163-0.35627-0.36120
NOO.NEtWT.PRESS..AT: INPUT COORD.
FOR JM = 20 AND Xl 6I= ,0;28169THE Z_Y ANO!P COOROINATES ARE:
Z( I)=ZI2)=
Zl:3)=ZI 4)=ZI 5)=ZI6_=
Zl,7)=ZI 81=Z! 9}_ZllOI=ZIIII=Z(121=
Z(13|=Zl14)=Zli5)=
0,0 YI 1)=0,05978 Y! 2)=0.11793 YI 3I=O; 17287 YI _4)=
0;22309 . YI 5)=0.26722 Y( 6l=
0,; 30406 y( 7)=0,33261: YI 8)=0335209 Y! 9)=0.36197 Y110)=0.;36197 Y( 1I)=0.35209 Y(12)=
0333261 Y( 13)=0.30406 Y114) =0.26722 Y( 15)=
0.363210.358250.3_3530.3t9_30;28662
-032_599
0,198650;145900,089160.02999
-0Q02999
-0,;08916
-0_14590-0,19865-0,24599
P/PSI:4_=P/PSI:5)=P/PSI 6l=P/PS(_71=P/PSI_8)=
PtPSI_91=P/PSI101=P/PSI111 =P/PS(12I=P/PS|131=
P/PSI14)=P/PSI15)=PtPS(16I=
PIPS117)=P/PSI18)=P/PSI19)=P/PSI20)=
• P/PSi 11=P/PSI,2)=PIPS( 3)=P/PSI 4)=PtPS(51=,P/PSI6/=
PIPS( 71=P/PSI 8)=P/PSI 9)=P/PS(_OI=PIPS(Ill=
P/PSI121=P/PS(;3)=P/PS(14I_=P/PS(15;=
0.296590,26581
0o2292I0o19003
00152010;11558
0_083470_:0563.80303522
0;020580,011580°007590.006890°006890°006890_006890°00689
0_330390.32499
0_3102'00.28718
0.256850-220710o182190.145030.10943
0.078300_05215
0.031980,018330.010270.00715
2O5
Z116|= -0.2Z309 Y( 16l= --0_28662ZILT_--_ O.1TZS? • YIt71= -0.319'_3Z{I8|-- 0.:_IZ93 Y( 181=: '-0o34353
Z{19}: 0,05978 Y(.t9I=- -0.35825Z_20|= O_:Q ¥|20) = _ -0.;36321
MOD.NE_¥,PRESS.AT iNPUT_COORO*
FOR JH = 20::ANO X( _71=, 0o50000 _
THE Z,V ANOi-P:COOROENATES ARE: •
Z[_t)= 0,0 Y(: li=ZI 2)= 0*0694L Y(:2|=Z{ :3_= 0,13693 Y( /31=Z( _)= 0.20071 Y( 41'=Zl.5)= 0025901 Y(. 5)=,Z( 6_= 0_31026 Y( 61 =
Z( _?|= 0;35303 Y| _T)=Z_ 8_= 0;38618 Y| 8)=Z( 9}= 0;_0880 YI 9)=Z E_0)= 0.42026 Y( 10)=.
Z(l:l)= 0,_2G26 YEll)=Z|_2]= 0.4_880 Y(12I =Z{i3)= 0_38618 Y(:I3)_Z(l:4|'= 0o_5303 Y|14)=Z{_5}= 0.3i026 Yl115) =
1(16_= 0_25901 _ y( 16)=Z_ i¥_= _.ZOO71 Y( 171=Z{_SJ= 0.E_693 Y(.18 |=_
Z{ZO)= 0.0 YI20I=
• 0.421T00;41595
0,39885- 0_37088
0_332T8,0-28561
0o230650_169_00,10352
0_03_82-0;0348Z-0;10352-0o169_0-0;23065
-0_28561-0_33278-0o370B8
-0_39885-0o41595-0_4_1T0
NOD=NEWT®PRESS.AT ENPUTCOORDo
P/PS(]6t:
P/PSI17)=PtPS(15)=
P/PSI19)=PtPS(20|;=
P/PS(_I|=P/PSI-2|_P/PSI3)=PIPS( 4|=P/PSI 51 =
P/PSI :6) =P/PSI _T)=P/PSI8)=p/PSI • 9)=PIPS(tO|=
P/PSI11)=PIPS(12|=PIPS(13|=P/PSII_=
P/PS(151=P/PSI16)=PIPS(-_T)=
PtPS(18_=P/PSI19)=P/PS(ZO)=
0;00_89
0.00689
0o00_890,00689
0o00689
0.33.4390.329360._1636
0.290470.Z59600;22_12
0oI86120014780
0,_1t.81
0°079850.053450o0332_0;019190.010_!
OoOOT30
0o0068_0_00689
0.006890o00_890_00689
FOR JM = 20 _ND X( 81= 0=75000
THE Z_V AND-P COORDINATESARE:
2O6
Zi :1}= 0o0 ?I 11=Z! .21= 0.08044 Y( 2)=Z( 31= 0.15868 ¥0 3)=
Z( 4)= 0.23259 Y( 4)=Z( 5)= 0,30016 Y( 5)=
Z( _6l= 0.35954 Y{ 6)=Z( 7)= 0.40911 Y( 7)=Z( 8l= 0,44?53 Y( 8)=Z( .91= 0.47373 Y( 9)=ZIZOI= 0.48702 Y(101=
Z(II)= 0.48702 Y(111=ZIl2)= 0.47373 YI 121=Zi131= 0,44753 Y1131=
1014|= 0.40911 YI 141=ZI151=. 0,35954 Y( 151=ZI161= ,0.30016 Y(161=Z(LT)= 0,23259 Yi 17_:
Z(181= 0.15868 Y( 18)=Z(191= 0,08044 ] Y(19)=Z(201= 0,0 Y( 201=
0e488690,482020,46221
00429790.385640.330980,26729
0.19630
0.119970,04036
-0.04036-0olt997
-0.19630-0°26729-0,33098-0.38564-0,42979
-0.46221-0.48202-0,48869
ROD.NEWT,PRESS.AT INPUT_COORO,
FBR JR = 20 AND X! 9_= leO0000
THE Z_Y ANO P COOROINATES ARE:
Z_ 11= 0.0ZI 2)= 0.09146Z(;31= 0918043Z| 4|= 0_26_47
Z( 5)= 0.34130Z(b|= 0.40882Z( _7)=. 0,46519Z( 81=, 0_50887Z( .91= 0,53867
ZII01= 0.55378ELL1|= 0.55378Z(12) = - 0.53867
Y( 11=Y( :2 )=
Y( .31:y( 4)=
y( 5)=
Y( 6)=Vt., _71=
Y( 8)=Y( 91=Y(.IOI=Y(11)=Y(12)=
0°555680_548100.525570.48870
0.438510.376350,30393
0.223210.136410.04589
-0.04589
-0,13641
P/PSI:l)=PIPS( 21=
P/PSi3)=PIPS( 41=P/PSI5) =P/PS( 6) =
P/PSI 7)=PIPS( _8)=P/PS(:�)=PIPS(IO)=PIPS(/I)=PIPS|121=
P/PSi13)=P/PS(14)=P/PSi15|=PIPS(161=
P/PS(I?)=PIPS(18)=P/PSi19)=P/PSi20|=
P/PSi 11 =PIPS( 2)=P/PSI 3| =
P/PSi 4)=PIPS( 51=P/PSI 6)_ =P/PSI 71=P/PS(_81=
PIPS( 9)=P/PSllO|=
PIPS(Ill=PIPS(121=
0_333440032830
0.313360o_289640_25895
0.223270_185170_147090_1_11220.0?9500°05313
0o032930.018980o010_40.:00726
0_006890o006890©006890o006890;00689
OQ333&90_328590o3136I
0o289880o25913
0°223500,18542
0o14726
0_11400.079590.053210003301
207
i
/
/"
Z(13)= 0,50887 Y(13}=Z(I_)= 0.46519 Y( 14)=Z{15)= 0.40882 Y115}=
Z()_6)= 0=34130 Y{16} `=
Z( _.T)= 0.Z6447 Y(17)=
Z{18|= 0.;18043 Y{18) =Z(199= 0.091_6 ¥(19) =
Z(20_ = 0,0 Y(20)=
MOOoNEWT.PRESS_AT
-0.22321-0.30393-0.37635--0.43851-0._8870-0.52557-0,54810
-0.55568
_PUT COORD.
PIPS(t3)=PIPS|14}=PIPS(!5)=PIP'(16)=PtPS|iT_=
P/PS_=
0.01903
0.01079
0,00727
0=00689
O.O0_B9
O_OO_B9
FOR JM = ZO AN_ _(lO)= 2_00000
THE Z_V ANO_P £OOR_INATES _RE:
Z( I)= 0.0 Y(1) =
ZI 2)= 0.13556 Y( ,2|=
Z( 3)= 0.267_3 Y( _-3l=
Z( 4)= 0_39200 V( 6)=
Z( 5)= 0.50588 Y(5) =
Z(6) = 0J60596 Y(6) =
Z( T)= 0.b8_51 Y| ,7)=Z( 8)= 0_75425 Y( 8)=Z_ 9)= 0.?_42 Y( 9i=
Z(_O)= 0_82081 Y(lO|=Z(lll= 0,8208[ Y(II}=
ZI12)= 0-79842 Y(12)=Z(t3)= 0®75425 Y(13]=Zll_)= 0.68951 Yllk)=
Z(_5)= C060596 Y(15)=!
Z(16}= 0-50588 ¥(161=Z(IT)= 0.39200_ Y(1T)=
Z(18)= 0.26743 Y(18) =
Z(19)= 0,_3556 Y(19)=
Z(.20_= 0.0 V(20)=
0.823630.81239
0.779000.724360.6_79604557830,_50_80,33085
0-202190.06801
-0.06801-0°20219-0.33085
-0,650_8--0_55783
-0064996-0,72636
-0.77900
--0°81239-0.82363
MOO_NE_T.PRESS.AT INPUT COORDo
PIPS( _I)=
PIPS( 2)=PIPS( 3)=PIPS( #)=P/PS(5)=P/PS( _)=PIPS( ,T)=P/PS( 8)=
PIPS( 9)=PtPS(IO)=PtPS(11|_P/PS(12)=P/PS(;3)=
PIPS(1_)=P/PSI15)=P/PS{16)=PIPS(IT)=PIPS(iS)=
PIPS(19)=
0°33369
0_3t3530o28982
0.259070,223430.!853_
0_14722
0°07955
0.05319
0.03299
0301902
0o010770.007270°006890.006890.0068_
0_00689
0.0068g
I08
FOR JM = 20,1AND X(11)= 3.00000•THE ZeY AND P COORDINATES ARE:
Z! , 1)=ZI 2)-ZI _3)=Zi 41=Z( 5)=-ZI 61=
Zl TI:Z( , 8)=ZI ,91=
Z 11-0])=Z[ll)=Z(1210Z(13}=
Z ( 14)=Z(15}=Z( 16)=Z(IT)=
Z(18)=Z(19}=Z(20}=
0.0O, 17967O, 35443
0.5191530.670460-803100,913830.99963
1.05817
1 =087851 .;087851.058170.99963
0,913830.80310
0.670460,51953
O. 354_,30.;179670.,0.
Y( 1)= 1.09157Y( 2l= 1,;07669
Y( ;3)=. 1003293Y( 4)= 0,96001Y( 5)= O. 8614l
Y( 6}= 0.73930Y( 7)= 0.59703Y( 8) = 0.63868Y( 9)=:. 0,26797
Y( 10)= 0,_09014Y(11)= -0,09014Y( 12)= --0.26797Y(13)= -0,43848
Y( 14)= -0.59703
Y(15)= -0.73930Y( 16)= -0. 86141Y(IT)= _ -0,,9600I
Y( 18)= -I.032/,3Y( 19)= -1.07669Y(20)= - 100_9157
HOD°NENT,PRESSoAT INPUT COORD.
P/PSI 1)=P/PSI:2)=
P/PSI 3)=PIPS( 4)=P/PSI5|=
PIPS(6) =P/PSI,7)=P/PSI 8)=PIPS( 9)=
P/PS(IO)=P/PS(11)=PiPS(IT)=
PIPS(13)=PIPS(14)=PIPS(15)=P/PS(_6)=P/PS(I7)=
P/PSI18)=P/PSI19)=P/PS(20)=
0=333610°32851
0.313560.289830.25908
.0.223430.185350.16723
00_II350.079560,053190003299
0=0190200010770=007270°,006890o006890°006890°00689
0_00689
FOR JM = 20:AND X(12)= 4,00000THE:Z_Y ANDP COORDINATES ARE=
Z( 1)= 0,0 V(, 1)= 1o35952Z( 2)= 0,22377 Y( 2)= 1034098ZI _3)= 0°44144 Y( :3|= 1.28586Z( 4)= 0.66706 Y( 4)= 1®19567
Z( 5)= 0.83.506 Y( 5)= 1.07286
Z( 6}= 1.00023 Y( 6)= 0.92078Z( .7)=_ 1o13815 Y( :7|= 0.74359Z( 8)= 1.26502 V(8)= .. 0e54611
Z( 9)= 1.31792 Y( 9)= 0.33376
P/PSI,I)=P/PS(_2)=
P/PSI;3)=PIPS(4)=
PIPS( 5)=P/PSI 6)=
PIPS( 7)=PIPS( 8)=
P/PSI ._9)=
0o333620032849
0=1313560.28983002590Z0°223.440018535
0=147230°21134
209
Z(IO|= 1,,35488 Y(IO)=Z{ll)= 1.,35488 Y(I1) =Z(12}= 1.31792 Y( 12)=Z( 13)= 1.24502 Y( 13)=
Z(14)= 1o13815 ¥(14)=
Z(15|= 1=00023 Y(.15)=Z(16_= 0,;83504 Y(16)=Z( 17_-= 0.6H706 Y(.17)=Z_I8_= 0o_414z_ Y(18)=
Z(19_=_ 0=Z237T Y( 19)=Z_20)= 0=0 Y(20)=
MC_D. NE_JT. PRESS. AT
0.11227-0.11227
-0, 33376-0,54611-0.74359
-0,;92078-1.07286-1.19567
-1=28586-1.34098- 1. 35952
INPUT COORD.
P/PS(IO|=PIPS(Ill=
P.IPS(121=P/PS(131=P/PSi141=PIPS(15)=P/PS(16J=P/PS(17)=P/PS(18)=
P/PS(191=PtPS(20}I=
0.079550.05319
0°032990.01902
0.010770°007270°00689
0.006890.006890.006890®00689
FOR Jt_ = 20 AND X;13)= 5=50000
THE Z_¥ .ANO P COOP, DINATES ARE:.
Z( 1)=
Z(.21=ZI_3)=Z( 4l=Z( 5)=Z(6I=
Z(_7)=Z_ _81=Z(9}=
Z¢IO;=Z(II)=
Z(12)=
Z(13)=Z(14;=Z_IS)=
Z|_6_=
Z(i7)=
Z(19)=
Z¢20)=
0.01
0.289920.57194
0=83836I_0!819G
Eo29594
1._7462!.6i309
1.70755
1.755431,755431.707551=61309
1=47_4621=29594
I.G8190
0o83836
0,57194
0.289920.0
Y( 1 )=
Y( 2)=Y( 3)=y( 4)=Y( 5)=Y( 6)=
Y{ 7|=YI 8)=Y( _91=Y! 10)=Y( 11)=
Y(12)=Y( 131=_Y( 14)=
Y(15)=Y{16)=
Y(IT)=Y(18)=Y(19)=Y( 201=
1.76145
-I._37H2
1.66601
I_54915
1.39003
1.19300
0=963620.707570°632410.14546
-0=14546
-0.63241-0.70757-0°96362-1.19300-1.39003-I=54915
--1.66601-1-73742-1.76145
P/PS(1)=P/PS{_2)=
P/PS( 3)=PIPS( 4t =PIPS(.5)=P/PS(6)=
PIPS( 7l=P/PS(_8)=PIPS( 9)=P/PS(IO|=P/PS(II)=
P/PS(12) =PIPS(13)=P/PS(IH)=P/PSI15) =PIPSI16)=
PIPS(17)=PIPS(18)=
PIPS(19)=P/PS(20)=
0.333610=32850
0.313560o289820°259070.223H4
O. 18535
O= 147240o_I1340.079550.05320
0.032990.01902
0.010770.00727
-0.006890.006890.006890.00689
0,00689
210
MODoNENII',, PRESS. AT _INPUT COOROo
FOR JN = 20 .AND X(l.kl= 7,;00000
THE ZtY AND P COORDINATES ARE:
Z(, 1)= 0,0 _ Y(,.,I,)=Z( .2)= 0.35608 " Y( .23=
Z(:3) = 0,70245 Y( 33=ZI 4)= 1.02965 Y( 4)=
Z( 53= 1.32877 Y[ 51 =
ZI -63= 1.59164 YI 6l=Z( .73= 1.81I!0 Y( ..7)=
Z( 8)= 1.98116.. Y( 83=Z( 9|= 2..097[7 Y( 91=Z(103= 2.15598 Y(|,O) =Z(I_]= , 2.15598 Y(113:
Z ( 123= 2.09717 Y{ 123 =
Z{13) = 1.98116 YI 13)=Z(141= 1.81110 i Y( 14)=
Z (15)= 1..59164 Y{ 153=
Z(163= 1.32877 Y( 163=Z(17)= 1.02965 Y( 17}=Z( 1.8)= 0,70245 Y( 18)=
Z(193= 0,35608 ' Y(19) =
Z{201= 0.0 • Y(20)=
2.163372. 13387
2 °1046151.902631. 707201. 4652 I1.18325
0._ 869020.5:3108
Oo 17865-0.17865
-0.53108-0.;86902- I.i 18325
-1.46521- I - 70720- I. 90263-2_04615-2,D 13387-2.16337
MOO.NEWT,PRESS.AT INPUT COORDo
FOR JM = 20 AND X(153= 10,00000THE Z,Y ANDP COORDINATES ARE:.
Z[ 11= 0.0 Y[ 11= 2.96722Z( 21= 0.48839 Y( 2l= 2,926T5Z( _3| = 0.96345 Y(_31= 2,80645
Z( 43= 1.41224 Y( 43= 2.60959Z( .5)= 1.82250 . Y( 53 =: 2,3!4155Z( 631= 2.16305 Y( 6)= 2.00964
PIPS(,1|=P/PSI2|=P/PS[_33 =
P/PSl_k)=P/PSI 5)_P/PSI b|=PIPS(73=PIPS( 8)=
PtPSIg)=P/PS[10)=P/PS(II|=
P/PSI12)=P/prS(13) =P/PS(_'_)=PIPS[t5)=P/PSI1:63 =P/PSI173=
PIPS(18)=P/PS(193=P/PS(203=
PIPS( 13 =P/PS[_23 :P/PS(_33=
PIPS( 43=P/PS(_53_P/PS(;6I=
0.333610o32850
0.313540°289820_259070,223440;18535
0.147230oLII340_079550_053200°032990=0_9020,010770°007270.00689
0o006890°006890.006890.00689
o.333620.328500.31354
0ol28982
0.259070.22344
211
Z( 7)= 2.48606 Y( 7)=Z( 8)= 2.72730: Y( 8)=Z( ,9)= 2°87662 V(_9) =Z([Ot _ 2.95708 Y(IO)=Z([I|= 2.95708 Y(II|=
Z(12)= 2.87662 Y(12)=Z(_3|= 2°71730 Y(13)=Z(14)= 2°68606 Y(14)=
Z(15)= 2=I_305 Y(15;=
Z(16)= 1.82250 Y(16)=Z(17)= 1_122_ Y(17)=
Z_18_= 0_96345 Y(18}=
Z{19)= 0._8839 Y(19.)=Z{20|= 0,0: Y(20)=
HOD.NE_T.PRESS_AT
1;622911.191920,72861
0326503-0,26503-0.72841-1.1_9192
-1.62291-2.00966
-2.:36155-2.60959-2°80645
-2,92&75-2°96722
_NPUT :CODRD,.
PIPS( 7)=
PIPS( 8)=PIPS(9}=
PtPS(IO)=PtPS(I1)=PIPS{12)=P/P$(Z3)=PtP$(14_=P/PS{15_=
P/P_(16)=
PIPS(17}=Pt_l_)=
P/P_(I_)=PtPS(2_=
0.185350.147230,1,1136
0.079550.053200.03299
0.01902
0.010770.I00727
0o006_90o00689
0.0D689
0o006890®00689
FOR JM = 20 A_E) 7,C!6}= 13.00000 :
THE Z_Y _ND, P CO0_OINATES ARE::
Z( 11= 0.0
Z( 2)= 0-62070
Z( 4)= 1.79483
Z_ 5)= 2'31624
Z( 6_= 2=77_66Z( _7)= _.15¥01
Z( 8)= 3®:45344
Z( 9)= 3®:65567
Z{lO_= 3.:_5819Z(II_= 3_758i9Z(1Z}=. 3o:_556TZ(13)= 3=_53_6Z{t4)= 3.'15701
Z(15}= 2.7T_46Z(I_)= 2.31636Z(t7)= 1.79483Z_18)= [-:22_66
Y( :1)=
Y( 2)=Y( ,3l=Y( 4)=Y( 5)=:Y( 6)=Y( 7;=Y( 8)=
Y( 9l=Y(lOl=
Y(ll)=
Y(12)=
Y(13)=Y(14)=Y(15}=Y( 161=Y(17)=Y(18) =
3._71073°719633.566743o31_552.975902'55_07
2.062581_514820.92574
0.31161-0.3114I--0.92574
-1.51482-2.06258-2.55607-2;97590-3_31655-3.56674
PIPS( l}=PIPS( 2}=PIPS{ 3}=PIPS( 6)=PtPS(5}=
PIPS{ :6)=PIPS(7) =PIPS( 8)=P/PS( 9)=
P/PS(IOI=PtPS(11)_
P/PS(12)=PIPS(13)=PIPS(14)=P/PS(151=PIPS(16)=P/PS(I_|=
PIPS(18)=
0°333610;328500.313560.289830.:25907
_0;223440.18535
O. 147230. _II36O. 079550;05320
0,032990.019020.010770.00727
_0,006890.00689
0=0068g
212
Z(19)= 0.62070 Yll9}= -3.71963
Z(20)= 0.0 Y(.20)= • -3.77107
MOO,NEUT..PRESS, AT INPUT_COORO,
FORJH =.20 AND X(17)= 16.-00000
THE ZtY AND:.P COORDINATES ARE:
Z(:[)= 0,0Z( 2)= OQ7530Z,
Z( :'31= 1.48547:Zl 4)= 2.[7742.Z( 51= 2.80997
Z( 6_ 3.36587Z_ 7)= 3.82996
Z¢, 8)= 4.18958Z( 9)= 4_43472ZilO)= &_55929Z(ll|_ 4°55929
ZI12_= 4.43472Zil3)= 4.18958
Z(14)= 3.82996Z(_5)_ 3.36587
Zil6)= 2°80997.Z(17)= 2.17742_ZI1B)= 1.48547
Z|L9)= 0o7530_Z(201= 0.0_
Y( ,1)= 4_5749[
Y( 52)= 4.51252Y( _3)= 4.32703Y( 4)= 4°02352Y( 5,)= 3,61025
Y{ 6_= 3°09850Yi_o7)=! 2,50224
Y'( 8)= 1_83772YI 9)= 1.12307
Y(IO)= 0.37?79Yllt_=_ -0.37_79Y(12}= -1..12307V(13)= -1.83772Y(14)= . -2,5022_Y(15)= -3-09850
Y( 16]= -3.6"1025Y_ 17)= -4.;02352Y{ 18)= -4.;32?03Y( 19)= -4_51252Yi20)= -4,,57491
ROD-NEHT,PRESSoAT:[NPUT COORD.
FOR JN = 20 ANOX(18)= 19,00000THE: Z,Y ANDP_COORDINATES ARE:
Z( 1)= 0.0 Y( :1)= 5.37876
Z( 2)= 0.88532 Y| -2)=. 5.30540Z( ,:3)= 1o74648 Y( ,3)= 5.08733
P/PSI 19} =
PIPS(201=
P/PSI 1)=PIPS( 21=
P/PS(3)=PIPS(4).=P/PS{ 5)=PIPS| 6)_
PIPS(:7)=PIPS( 8|=PIPS( :9)=P/PS{IOi_=P/PSfll]_ ,
PtPSi12)=PIPS(t3)=
PIPS(I_|=P/PS(15)_
P/PS([6|=P/PSI[T)=P/PS(|8)=P/PSi19)= "
P/PSi20)=
P/PS(I)=P/PS(2I=P/PS(,3)=
0°00689
0_00689
0-333610;32850
0_313540.289830.25907
0.18535
Oo 14723Oo Fl134
Oo 079550_053190°03299
0.01902O_OlOTTOoOOTZT
0_006890;00689
0_006890=:006890000689
0.333620,328500.31354
213
Z| 4|= 2.56001 Y( 4)=ZI-5)= 3.30370 Y( 51 =Z{ _6}= 3.95728 Y( 6)=Z( Ti= 4;50292 Y( 7|=Z( 81= 4.92573 Y{ B}=Z( 9)= 5.21417 Y( 9}=
Z(IO|= 5=36039 Y(IO)=Z_EI_= 5.36039 Y(1I) =Z{t2)= 5.Z1417 Y(121=
Z(13_= 4.92573 Y(13)=Z_14}= 4.50292 Y(14)=Z(iS_= 3._5728 Y(15)=Z_16]= 3.30370 Y(16I=
Z_171= 2.56001 Y(17|=Z_18_= _=74648: Y(18)=Z(19)= 0.88532 Y(19)=Z_20)= 0.0: Y(201=
_.730484°244603.642962=941902.16062
1.320410°44417
-0.44417
-I,320_I
-2,16062
-2,94190-3.64294
-4.2_J_60-4.73048-5.08733-5.30540
--5.37876
MOD.NE_T_PRES$oA_INPUT COORD.
PIPS{ 4)=P/PS(SI=P/PSi 6}=PIPS(7)=
PIPS{ 8)=P/PS(.,9_ =P/PS(_OJ=
PIP${121=p/ps(I3_=P/PS(14J =PIPS(15_=PtPS{16_=
P/PS(17_ =P/P$(18}=P/PS{_9}=PIPS(Z0)=
0°289820.259070.223.440.18535
0._147230.1113_0.079550.053200,032990.019020.01077
0®00727
0,006890.006890.006890°00689
0.00689
FOR JN= 20AND _191= 22.00000
THE Z,Y _NO P CO0_OINATES ARE:
Z{ I)= 0,0 Y(I) =
Z( 21= 1.01762 Y(, 2;=
Z{ 31= 2.00749 Y(,3}=
Z{ 4)= 2._4260 YI 4)=
Z{ 5)= B.T9744 y( 5)=
Z_ 61= 4.5_869 Y( 6| =Z( _7}= 5.17587 Y{ 7)=:Z( 8}= 5.66!87 Y( 8} =
Z{ 9)= 5.99342 Y( 9}=Z(iO)= 6.|,_i_9 Y(IO}=
Z{il) = 6.1_!49 Y(ll} =
Z{121= 5.99342 Y(12} =
Z(13}= 5.6b187 Y{131=Z(t4j= 5.17587 Y(141 =Z_IS)= 4=54869 Y(151=
6®182616.098295=84762
5.437444=878954®187373°381572=483531.5t7740.51056
-0=51056-1.51774
-2.48353-3.38157-4.18737
PIPS( 11=PIPS( 2)=PIPS{ 3| =PIPS{ ;1=
P/PSI 5}=PtPS{_6|=PIPS{ 7)=PIPS{ 8)=PIPS( 9)=PIPS{_O}=PtPS{_I)=P/PS{I21=
PIPS(13t=PIPS(14;=P/PS(15|=
0.333610;32850
0.313540.289830.259070.223_40=185350.147230=_II34
0°079550=053200.032990,0_9020.0107T0®00727
214
/
/ 1/'
ZI|61= 3e79744 Y(I6|= --6®87895Z(17)= 2.94260 YI[7I= -5.43746Z(18)= 2.00749 Y([8)= -5=8¢762Z([9|= 1.01762 YiI?)= -6=09829
ZI20|=- 0.0 Y(20}= -6.18261
STREAMLINE CALCULATIONS
INPUT_DATA:
ANG, OF'ATTACK= O.20000E 02 DEG®ZETA(WALL)= O._IOOE O0
NO, OF STREARLINES ,TO BE CALCULATED =
P/PS(161=P/PS(17}=
PIPS(18|=PIPS(19)=P/PS(20)=
0®O06890.006890.00689
0=006890,00689
8 = 0,99335E O0
STAGNATION POINT,XO = 0=22615E-01
TW = 0.53006TE 03 OEG. R QHS =HSP = 0.227223E-01 BTU/SQ FT-SEC_OEG R
0.359411E 02 BTU/SQ FT-SE
FOR STREAMLINE NO. I_WHERE BETA = OolO000E 01DEG.THE CALCULATIONS ARE AS FOLLONS:
X yS BBAR
UE HEMACH NO_ RN/FT
RNt RD2RIDPH[2. OR/DX
0=37092E-0I, 0.1626tE,00
0.37256E-01 0.68442E O0
0;24302E 03 O,t2654E 08OotO802E 00. O.2TO36E 05
ZPIPS
RHOE
MOM THK iPH!02R/DX2
0.65007E-03 _
0,99188E_00_0.10593E-03.
0.16879E-030,93173E-02
-0.9_113E-060-16261E: 00
0.20TBOE 01O.ZZgOSE 00_
-0.32710E 02
HQW/QWS
HUERNM
DR/DPHI-QBIQSB
0.37250E'0I
0.91683E 000=95220E-06
0.40228E 01-0.35879E-05
0.91726E O0
215
\
\
0=38723E-01 G.!6596E.O0 0.71488E-03 0._0984E-01O=409EiE-Ot 3._2739E O0 0.98977E 00_ 0o942_7E O0
0.27279E 03 O=kZ646E 08 0.10577E-03 0.95186E-06O®_Z129E O0 O.30313E 05 0.1_407E-03 0.43671E 010=_127E Ol 0o1659bE O0 0=24681E O0 -0=37898E-05
-0=93940E-04 0.20262E Ol -0,30774E 02 0.9_303E O0
O=42066E-G_ C_kTZ60E GO 0.84333E-03 O_8_T7E--Ol
0o464_E-01 _.57318E OO O.9B_gBE O0 _='9_554_ DO
O._30_IE 03 0=t2629E 08 0,105_1E-03 0._109E-060=t_7_9E OO D.36663E. 05 0.13969E-_3 _,51216E O1
0,2_?_5E 02 0o_7260E 00; 0°27994E O0 -0=_2541E-0_-O_B87_SE-'-O_ 0._19287E Ol -0=27364E 02 0_96737E O_
0=49205E-01 O._B569E O0 0.10972E-02 0=_2974E-010_633_iE-01 O_5411iE O0 0.97310E O0 0=97492_ O0
0.4_3_9E 03 _=i2585E 08 0=I0450E-03 0=9_9[_E-06
0®I_775£ O0 0.4,8848E:05 O..13TSgE-03 0=672IIE O_0=82_61E _2 0.t8559E 00' 0,33854E O0 -0.48620E-0_-O=T4779E--,O_ 0.!7545E 01_ -0.21995E 02 0o976_3E OC
0o5683_-Oi _19848E O0 0o13_81E-02 0.77392E-010=782_._-_ _53646E 00_ 0®95817E O0 0.96854E O0
0.554_0E O_ _12530E 08 _0. I0335E-03 0.94670E-06
0=2_7_E O0 _60568E_05 0.13774E-03 O_83425E Ol
0_083E OB 0.19849E OO_ 0.38916E O0 -O.SZTBTE-05-0=_29E-04 0=16029E Ol -0.18031E 02 0.97090E 03
0®64973_-0_ 0=21096E O0 0.1596_E-02 0o91_50E-01O=g_I_E-_! 0_5_053E O0 0=94023E O0 0.95743E O00=65_E OB 0_i2462E 08 0=1'0196E-03 0.9_370E-060.2_305E OO _.71!B99E 05 0.13851E-03 0.99588E Ol
O.2_3llE 03 0.21097E O0 0.43355E O0 -0.51271E-05-0=312iGE-04 G®I_b96E OI -0,15039E 02 0=96079E O0
0_269_E-01 0=23490E O0 0.20838E-02 O=llgb4E O0
_®I2Z_BE O0 G=55605E O0 0..89566E O0 O.g2_20E;O0O=BB_6BE 03 0_1229!E 08 0.98_88E-0_ O.gB603E-06
216
0o39990E O0 0o93295E 05 0°14205E-03 _ 0,,13160E OZ0.43.863E 03 0,23491E 00, 0.,50826E O0 -0.35900E-05
0o33884E-0_ 0o12443E O1 -0,1092IE 02: 0.95198E O0
0,10227E 00. 0.25736E O0 • 0o25574E-02 0_14684E O00;15273E O0 0=5670_E. 00: O,,84LSTE 00 0,88306E 00'
0;11045E 06 0.12074E 08 _ 0o96202E-04 0.9262TE-060.50258E: _ O0 0o1 _12_32E 06 0o16508E.-03 0o16296E OZ
0.6965tE 03 OoZ5737E 00, 0.56933E _ O0 -O._400Z'/'E-06
0.10582E-03 0o10596E Ol. -0,82818E. Ol 0o89165E O0
0.:1_6_4E O0 _ O.29727E 00. 0.34555E-02 0,198_1E 00:0o21.233E O0. 0..61310E O0 0.71573E O0 ; 0.:77_53E O0 _'
0,15205E 04 0.11528E- 08 0=839IOE-0_ 0,90125E-060,70808E O0 O=L_4156E 06 0.15741E-03 0o22283E 02
0.13301E 0_+ 0.29729E 001 0.66599E O0 0.65593E-050o15928E_03 0o76852E O0 -0,.5t.671E 0I 0o78893E O0
0.1963¥E 00. 0.32969E O0 0o42718E-02 _ 0.24528E, O00o_7193E O0 0.85924E O0 . 0.56690E' O0 : 0.65621E O0 :
0*1948¥E 04 OolOT85E 08 O.TIO39E-O_ _ 0=86613E-06
0.93823E 00. 0.15983E 06 0.;17505E-03 0-27978E'020;20827E 04- 0.32972E 00: 0.7_23_,E O0 0.70036E-05
0=11879E-03 0.542_bE O0 -O=_1477E 01 :: 0o676_9E O0
0,20960E O0, 0.33653E _ O0 i 0o44595E-02 _ 0.:25606E O0
0.28683E O0 0.97469E 00 _ Oo52278E_.O0/ _0.62243E O00_20715E 04 0,10538E _08 0®670_5E-0_ 0.85417E-06
0;10089E 01_ 0=I6259E_06 0o18096E-03 0=29422E 020=22830E O_ 0;33656E O0 0_75920E 00: 0.76_.88E-05
0=15989E-03 0.49264E O0 :, -0,41644E O1 0.66425E O0
0.23683E O0 0,34862E O0 : O.,/_8116E-OZ .: 0=2762¥E O00o31663E O0 0.I0888E_ 0! O._31tgE O0 _ 0o54209E O0 :
0.23280E 0,_.. 0.99738E. 07 0.58_27E-04 0.;8262_.E-06
0, I1655E 01 0.I6462E 06 .0..19650E-03 0.323_9E 020..26703E 04 _ 0=34866E O0 _ 0.79074E O0 _ 0.4180_E-05
O.;ZT_ZTE-O_ 0o39713E 00_ -0,3-flOSE 0t 0.50632E 010
217
0o2649_E 00: 0_35848E O0 0.5132_E-02 0_29469E O00_346_3E 00: 0o88841E O0 0.35680E O0 0.45950E 001
0,25_37E 04 0=94485E OT 0.51035E-0_ 0=799_3E-06O_8_E 01_ 0°16238E 06 0.2[602E-03 0.35077E 02
0_0969E 04: 0®35851E 00,. 0=82027E 00: -0°61819E-050_!3740E-03 0.30505E 00: -0.23727E Ol 0._8423E O0
0,27209E O0 0=3_058E, 00 0.52080E-02 _29902E 000.35388_. 00_ ©;75869E 00; 0®3_369E O0 0=4%087E 00_
0.25829E 04 O.g3480E 07 0o_9689E-04 0.79422E-060.13357E. 01 0._6159E 06 Oo22077E-G3 0o35675E 02
_.3!975E O_ 0.3606[E_00. 0.82750E O0 -0.10934E-0_ :
-O_E910_E-04 0.28238E00 -O, LS?57E 0I 0.46538E 0_3
0.2738_E O0 0=36108E 001 0o52267E-021 0.30009E 00
0.355_E 001 _720[3E 00_ 0o36085E 00: 0.43638E O00=25_4E 04 0.93258E 07: 0._9395E-04 0,79306E-0_
0._4_7E O_ OoL6140E 06 O.Z2IglE-O3 0.35817_ 020_2226E 04 0o_6I_2E'00 0.8293_E O0 -0.1_2130E-0_
-0.I_31E-04 0.27672E 00: -0;[7674E 0_ 0._608[E O_
0_2_74_E OO: _=36205E_00 0o52637E-02 0=3022IE 00._o359_E 001 G=&4877E O0 0=33569E 00. 0o42796E O0
_,26070E 04 0.92853E 07 0,48860E-04 0.7909_E-06
0._28_ GoI_IOSE 06 0o22_10E-03 0;36090E 020=_3726E O_ 0o36209E O0 0.83293E O0 -0.1_613E-0_-D,_B398E-04 0.26775_ 00: -0;_5420E Ol 0=45223E O0
0=2810_ 00: 0_3630!E 0.0 0o53005E-02 0=30433E O00,3631_E _, 0o600_5E O0 0.33109E 00. 0._2068E00:
0.26_10E O_ 0_9248BE 07, 0.48380E-04 0o78903E-060®13627E Oi 0_15071E 06 0o22619E-03 0.36352E 02
C_3322_ O_ 0_36305E 00_ 0=83655E O0 -0.17388E-04
-O.t8377E-G3 0.26428E00 -0=14015E O1 0.44480E O0
O_Z8288E_DO 0=363,49E O0 0.53189E-02 0-30538E O00=36506E OO 0=58350E 00_ 0.32893E 00: 0._174tE 00_
O-Z6276E O_ 0o92315E 07 O-Z_8155E-04 _ 0.78813E-060-1367_E O1 O.t6055E_O& 0o2272IE-03 0®36_78E:02
2._.8
0o33_5E O_-0°22556E-03
0.36352E O0 0o83836E O0 -0o18345E-040o26432E O0 -0.13586E 0I 0.44148E O0
0o28648E00' 0°3644_E_00 0.:53559E-02 0.3075IE 00_0.36878E O0 0°55080E O0 0.32_87E O0 0o_1115E 00;
0.26_00E 06: 0_91988E 07 0o47727E-04 0o7864lE-06
0.13763E Ol _ 0.16023E 06 Oo2291gE-03 0.36724E 02:0,339¥3E0_ 0.3644BE_00 O.rB_lggE O0 -0.22_85E-04-0.30523E-03 0_26454E O0 -0.12767E OI 0.43509E OO
OoZ9368E.'O0 _ 0.3663_E O0 0.54305E-02 O,3_ll?SE O0
0.37623E O0 0o48516E 00_ 0o31765E 00: 0.39954E 001
O.26622E 06 0°91399E 07 0o46969E-06 0o78331.E-06O.13923E.OI 0.15963E 06 0,23294E-03 0o37186E 02
0.3_966E O_ 0.36638E00 0,86927E O0 -0,26281E-0_!-0,_936E-03 0=26497E 00 -0o11191E Ol _ 0_4232ZE 00
0.30809E_00 0.37016E. 00 0.55820E-02 0,32047E O0 .
0.39113E O0 0_35520E O0 0o30651E_00. 0.37961E O0 _
0,2696.8E_0_ O=90_TZE 07 O=45T87E-O_ 0.77840E-060o14176E OI 0.I5863E 06 0_23955E-03. 0.38000E 02
0o36941E 04 0.37020E 00.. O.B6395E O0 -0.33537E-04-Oro68059E-03 0=26575E_00 -0_82829E O0 0,_0273E.00
9_33688E.00 0.37783E O0 0o58956E-02 0.338_4E OO
0o42093E O0 0_11840E 00_ 0.29500E 00_ 0.36963E O00.27331E 04_ 0.89488E 07 0.46552E-04 0.7731TE-06
0,14446E O_ O._ST49E 06 O.Z_gZ2E-03, O;39Z_8E 020=40866E 06 0.37788EO0 0.89396E 00_ -0.41011E-04"-0.9453TE-03 0o26705E O0 -0.34105E O0 .O.3TI54E O0
0,39445E O0 0,39325E 00: Oob5726E-02 0.3772_E 00:
0o_8053E O0 0.0 0.29956E_00 0.32835E00_O,ZTI86E 0_' 0.89881E 07. 0,45043E-04 0o77526E-06
0.14338E_01 0,15795E 06 0o25792E-03 _ 0._0740E 020o48686E 04 _ 0.39330E 00_ 0.95754E 001 -0.3_560E-04-0,9073_E-03 0.26868E 00_ O.Z8IBOE O0 0.34869E O0
0.50955E O0 0.424tgE O0 0.8L556E-02 0._6790E00
219
0.59973E GO. 0®0 0.33643E O0 0.33207E O0
0.26048E 04 0.92911E 07 0.48936E-04 0.79125E-060oI_512E Ol 0.¢6110E 08 0.2645lE-0_ 0._2613E 02
0.6454BE 04 0.42427E O0 O. IIOI4E Ol 0,91712E-05.-0=94494E-04 O.26872E O0 0o30629E O0 0.35087E OG
0_62468E O0 0®45503E O0 O. IO0_IE-Ol 0o57592E O0
0.7_893E O0 0=37715E-01 0.34203E O0 0.32348E DO
0=25879E 04" 0.93351E 07 0.49517E-04 O.79354E-D_
0. I.3392E 01 0.16148E 06 0.27736E-0_ 0.44789E 02
0=80586E 04 0.45514E O0 0.12641E O_ 0°;9088_E-050.41i39E'04 0.26779E O0 -0.64252E-01 0,34154E O0
0=73982E O0 0._858LE O0 0.12218E-01 0=70069E O0
0._38_4E O0 0.48316E-0I 0.33388E O0 0®30696E O0
0=26i35E lO_ 0.92710E O? 0.48672E-04 0.79019E--06
0.i3_65E _t 0°t6092E 06 O.29089E-03 0.46809E 02O=96&OOE 04 0.48596E O0 0.]_407E 01 -0_16227E-0_
-G=_713E-03 ,0.26772E O0 -0,98914E-01 0.32444E GO
O._7007E O0 0.5473BE O0 0.17552E-01 9. I0'061E 01
0.10_65E O_ 0=0 0.33291E O0 0.28306E O00=28_5_E0_ 0.92633E 07 0._8571E-04 O.78979E-06
0=1358¥E _Z 0o16085E 06 0.30929E-03 0.49748E 02
0. Z28-54£ 05 0.54766E 001 0=18366E 01 -0.13036E-04
.-O_Z9794E-O_. O.26BO4E O0 0.35037E-01 0.29921E O0
0®I1532E O_ 0,59898E O0 0.2315tE-01 OoZ3266E 01
O.ZZT6_E Ol 0_0 0o33542E O0 0.2T210E O0
O=26O_BE _4 0.92832E 07 0.48B32E-04 0.79083E-06OoX3_33E Ol 0.16103E 06 O=32251E-Q3 0.51933E 02
0=_5537E 05 0.599#2E O0 O.ZZI34E O[ -0.22466E-05
_2g_4E-C3 0o26799E O0 0.15065E-0_ 0=28754E O0
O.Z5495E Ol OoTOIgOE O0 0o37787E.01 0.21650E Oi
O. i6765E Ol 0.51126E-02 0o33513E O0 0.25260E O0
0_26087E O_ O.g2809E 07 0°4880tE-04 0.79071E-060.13540E 01 O.L6101E 06 0.34897E-03 0.56186E'020.20905E 05 0.70292E O0 0.30816E Ol -0.29927E-04
220
-0.7_868E-04 0°26793E O0 -O.6_Z66E-02 •0-26694E O0
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0=43785E :04 0.30980E 07 0.31417E-05 0_37223E-05
0o39333E 01_ O.36q55E_ 05 :O.52978E-OZ 0°19578E030=1986_E 06 0_48256E 01 _ 0°13307E 03 0°_0143E-04
0o25678E_--03 0o26795E00 0058670E-06 0_69891E-02
0017248E02 -0o35221E O1 003420_E 01 0'37278E03.
O°LgBYbE 02 0059_21E 001 0o69358E-02 OoS_916E-OZ0_43861E O_ 0o30649E 07 0o30584E-05 0,36913E-06
0°39613E O1 0°363_0E,05 0_53987E-02 0o196I9E 03
0o19986E 06 0°49094E Ol 0o13586E 03 0°49655E-05O=t5818E-03 0©26795E_00 0.5342IE-06 0=6_750E-02
0.17563E 02 -0=37415E OI 0_3307_E 01 0037684E 03
_2027&E 02 O=4BOIBE--01 Oo68631E'02 0®51098E.-020=43882E O_ 0.30557E 07 0_30355E-05 0.36827E-06
0_39692E Ol 0o36170E 05 0.54_53E-02 0o19587E 03OoZOiO6E 06 0°49937E 01 0o13852E 03 -0o15922E-05
0=5787[E-04 0.26795E O0 0.64_54E-06 0=60259E-02
0_17881E 02 -0.39535E 01 0.3[881E OI 0o38046E 030=20676E 02 O°O 0.6880_E-02. 0o50406E-02
0o438¥?E 04 0.30579E 07 0.30_09E-05 0.36847E-060o39673E 0I_ 0_362,IOE 05 0.53918E-02 O_lq52_E 03
0o20227E 06 O°50788E:D_ 0o14I_2E 03 -0o22789E-050=37167E-04 O=2&795E O0 0.76284E-06 0o59440E-02
0=17960E 02 -0=40053E O[ 0_31574E Ol 0o38135E 030=20776E 02 0.0 0o68883E-02 0o50_92E'02
0°43874E 04 0-30589E 07 0o30434E-05 0o36857E--060.3966_E O[ _ 0.36229E 05 0.53845E-02 0o[9507E 03
0=20257E 06 0o51002E OI 0o14175E 03 0.10371E-0_0;52652E-04 0o26795E 00: 0®78886E-06 0.59540E-02
0-_¢80_-_. -0°4031[E Ol 0o31418E_ 0I 0_3!8[¥9E 030=20826E 02__ 0=01 0.68919E-02 0o50532E-02
0=3686IE-06
0=19_99E 03O.1z&ogE-06
0o43873E O_ .....'¢._3.05..93E O-d' 0,,30_6E--05
o:z9660E Oo3 O,S S OE-02 "06
0o64[07E-04 ' 0o26795E OU *._0_7697[E-06 0=59587E-02
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0.18080E 02 -0._0823E 01 0.31106E Ol 0®38268E030.20926E 02 0o0: 0®68981E-02 0.50606E-02
0._3871E 04 0.3060IE 07: 0.30465E-05 0.36868E-060.39654E 01 0.36252E 05 0.53745E-02 0.19_84E 03
0.20302E 06 0.51323E 01 0.14269E 03 0.:16032E-0_0_92_[C_c--0_ 0o.26795E O0 0.78392E-06 0.59674E-02
0o182_1E 02 -_1832E 01 0.30_70E OI 0_38449E 030.2._126E 02 0o01 0.69062E-02 0=50726E-02
0o_3869E O_ 0.3061IE 07 0,30491E-05 0o36875E_0_0_396_5E 0i 0.36271E 05 0;53637E-02 0.19455E G3
0.20363E 05 0.51753E 01 0.14393E 03 0.20887E-0_0=15526_-03 0o26795E O0 0.72558E-06 0.59813E-02
_.18563E 02 -0._3799E Ol 0o29160E 01 O_388L8E 030®21526E 02 _._134E-01 0.6909&E-02 0=51262E-02
•_o_3868E O_ G.30616E 07 0.30501E-05 0®368.82E-06
0;3_6_E O_ 0.36279E 05 0.53483E-02 0o19403E 030.2048_E 06 0.52618Ei01 0.14635E 03_ 0.28419E-0_
0'25596E-0_ 0,26795E O0 0o50975E-06 0.60_22E-0_
0._888_E 02 -0o_5695E OI 0.27804E OI G_39!93E 030=2192_E 02 _33701E-01 0.69032E-02 0.51728E-02
0.43_?0E O_ 0.30608E 07: 0.30_8[E-05 0.3687_E-06O.3_BEO1 _.3626_E 05 0.53379E-02 0.19357E 03
0=20605E 06 0,53_89E 0I 0.14868E 03 0.35014_-040,24351E-03 0.26795E O0 0.24515E-06 O.60995E-G_
0_192_6E _2 -0._522E O[ 0o26_05E Ol 0._9570E 03
0.22326E C2 0=30IllE-0[ 0.68944E-02 0.51739E-020,43872E O_ 0.30596E 07 0.30_54E-05 0.3686_E-06
0.3_658E OI . 0.36243E 05 0.53291E-02 0.1931_E 030_20725E 06 0;5_365E O_ 0,15094E 03 0.3_16_E-0_
0.6626_E-04 0o26795E O0 -0.38373E-0_ 0.610[0E-02
0_9380E 02 -0._8_1[E 01 0,25692E 01 _=39757E 030.2ZSZ6E 02 0.20653E-01 0.68911E-02 0®5161gE-02
0._3873E O_ 0;30592E 07 0.30_3E-05 0.36860E-06
230
0.39661E 01, Oo36Z36E 05 0=532_4E-02 0_17293E 03O_20TSbE 06 0=54806E 0I 0.15204E 03 0_30874E-04-0_79056E_06 O=26795EOO-O=16857E-06 0o60869E-02
0o19710E02 -0=_0139E 0I 0=24239E 0I 0.¢0131E 03
• 002292.6 E 02 0,50047E-02 O0.68879E-O2 Oo51371E-OZ0o43874E 04 0=30588E.07_ 0o30;33E-05 0,36856E-06
0o39665E 01o. 0.36228E 05 0o53142E,02 0o19252E 030_2090¥E 06 0,55690E 01 0=15420E 03 0.39408E--04
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0.20042E. 02 -0.5!803E O[ 0.22755E_01 0o40502E-03O=Z3326E 02 0=0, ,,- -.0_68879E-02 O.51ZbSE-OZ
0o43874E 04 0,30588E 07 0=30433E-05 0.36856E-06
0_39665E, 01 0.36228E:05 0=53032E-02 0o1921.2E 030=2t027E 06: 0.56580E O1 0o15629E. 03 Oo3ZB78E_O_-0.37869E-03 0o26795E-00 -0o10.113E-05 0,60651E-02
0o2037bE02 -0=53405E Ot 0.212._4E 01 0.40871.E 03
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0.39663E Ol 0.36232E 05 0.52922E--02 O=l_175E 03
0=_1148E 06 0o57475E 01 0=15831E 03- 0=19_63E-04-0=33067E-03 0©26795E O0 -0°14882E-05 0_60582E-02
0o20712E 02. -0,54947E 01 O. tgTOSE 01 0®4124lE 03
0o24126E 02 0.0 0068917E,.02 0.51483E-02O0438T3E 04 0.30593E 07 0o30_45E-05 0.36861E-06
0_3966[E 01 0°36237E 05 0_52816E.02 Ool?139E 03
0;21269E 06 0.58374E 0I 0.16027E 03 0.17064E-05-0_19673E-03 0=26795E!00 -0°17563E'05 0.60708E-02
Oo21049E 02 -0.56;30E 01 O. LB153E 01 0=616IIE 03
0o26526E 02 0=0 0o68931E-02 0=51581E'020°63873E 04 0®30595E107 0.30469E-05 0=36862E-06
0.39659E Ol 0o36240E05 O.SZ719E-O2 0o1:9105E 030=21390E 06 0=59278E 01 0®16217E 03: -0.986_6E-05
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0.21388E 02 -0.57858E_0I 0. I6578E Ol O_L981E 030.24926E 02 0.0 0.68936E-02 O.51668E-OZ
0.43873E 04 0.30595E 07 0.30451E-05 0.36863E-060°39659E Ol 0°36241E 05 O°5263IE-OZ O.I90T4E 03
O.Z151OE 06 0.60186E 01. 0.1640IE03 -0.42_362E-050.36107E-05 0.26795E O0 -0.1_835E-05 0.60926E-02
O°2IT28E 02 -0.59231E OI 0.14989E OI 0.42352E 030°25325E 02 0.97009E-03 0°68935E-02 0°51817E-02
0_43873E O_ 0.30595E 07 0.3045IE-05 0.36863E--060.39659E 01 _ 0o3624IE 05 0.52550E-02 0.19045E 03
0.2t631E 06 0°61098E01_ 0.16580EG3 -0oI1520E-0_-D.9076IE-05 0.26795E00 -0.38535E-06 O._:lIOZE'O_
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O.43873E 04 0.30595E 07 0.30_49E-05 0.36862E-060.39659E Ol 0°36240E 05 0.52477E-02 0.19018E 03
0.2t¥52E 06 0.62014E OI 0.16753E 03 0.64796E-05-D,65084E-O_ 0.26795E 00: 0.50460E-06 0.61225E-02
_ CALCULATIONS FOR BETA =O.IO000E OI HAS TERMINATED
AFTER 93 INTEGRATIONS t_
FOR-STREAMLInE NO, 2 WHERE BETA _= 0.45000E 02 DEG°
THE CALCULATIONS ARE AS FOLLDWS:
X Y
S BBAR
UE HE
MACH NO RNIFTRNI R
D2RIDPH_2 DRIDX
0.33368E-010.41158E-01
0.15237E 00_0.72033E O0
Z H
PIPS QWIQWS
RHOE MUE
MOM;THK_ RNMPHI- DRIDPHI
D2R/DX2 QB/QSB
0.26340E-0I 0°37250E'-010.99219E O0 0o89825E O0 :
_2
0023824E 03
0o105.89E 00:0®93798E-02
0.45506E-0_
0o12655E 08 0©10596E-030_26509E 05 O015225E-03
0=15463E O0 0©98078E 0t0_22093E O! -0o38022E 02
0o95225E-06O_60361E 0I-0=46050E-05
0o89865E O0
0o34.631E-01. 0_15470E O0 0o28954E-0I 0.40954E-010_44883E-01 0_,65260E 00, 0_990IIE O0 0.92819E O0
0_26816E_03 0,12648E 08 0oI0580E-03 0o95192E-060o11922E O0 0o29804E 05 0o14656E-03 0_3682E Ol
0=87518E 01 Oo15739E O0 OolO601E 02 0..53337E-050_6¥907E-0_ "- 0o21625E 01 -0036060E 02 O092872E O0
0o37265E-01 0o15935E O0 0_3_150E-01 O=48269E-O[.0o52333E-01 0_58865E 00 0°98537E 00 0°95743E O0
0o32647E 03 0=12630E 08 0oI0544E-03 0_95115E-060_14525E O0 0_36190E 05 0_1_I20E-03 0o51101E O1,0_27249E 02 ,0o16296E O0 0=12096E 02 0o19980E-04
0.10478E-03 0o20724E Ol -0o3249ZE 02 0_95824E O0
0042947E-01. 0.16849E_00 -. 0o44452E-0I 0_62665E-0I
0=67233E-01 0.54975E O0 0o97356E 00. 0_96995E 000=43988E._03 0=12587E 08' 0ol0453E-03 0_:94723E_06
0_19604E GO 0o48441E 05 0;13841E-03 0o67048E OI0_81850E 02 0.17425E O0 0_147¥9E 02 0_32529E-04
0o14706E-03 0o19056E Ol -0©26595E 02 0,9_144E O0
00_9162E-01 0o17738E. 00 0o54661E-0I 0=76856E-010o82133E-01 0o5!4158E O0 0o95870E O0 0=96478E O0
0o55125E 03 0.12532E 08 0_10339E-03 0o9_678E-060o24622E O0 0_60198E;.05 O013B3bE-03 0°83287E Ol
0;14934E 03 0_18561E O0 0°1_127E 02 0.28250E-040=14610E-03 0o17555E 01 -0o22023E 02 0o96710E O0
0o55892E-01 0_18601E 00 0=64778E-0I 0o90875E-010.97033E-01 0o54277E O0 0°:94089E O0 0®95382E O0
O_&6168E:03 0=12465E 08 0o10202E-03 0_9_381E-060_29633E O0 0_71520E ,05 0o[3909E-03 0.99478E_01
0_23116E 03 0019696E O0 0=19201E OZ 0..13069E-040=98928E-04 0=16202E Ol -0°18452E 02 0=95713E O0
233
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0.88168E_03 0.12295E 08 _ 0o98577E-04 0,93623E-060.39758E O0 0.92833E 05 0oI_I67E-03 0,13152E 02
0.43562E 03 0.21737E 00: 0.22705E 02 -0o5422IE-05-0.84918E-04 0.13866E O1 -0.[3395E 02 0.9285gE O0
D=87755E-O1 0,217_8E 00: O.[0400E 00_ 0_:1_487E 00,0.15663E O0 0.57968E O0 0.84245E O0 0o88160E 00
0.11012E 04 0.12077E 08 0.94272E-04 0.92643E-06
0=50102E O0 OoLI206E 06 0,1_551E-03 0.16306E 02G=_9052E 03 0o24107E 00_ 0.25558E 02 -0o12364E-0_-0.20395E-03 0.11917E 01. -0o10108E,02 0.89012E O0
0.10647E O0 0®23124E O0 0.12266E O00o186_3E O0 0=60494E 001 0.78021E 00,
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0.17027E 00.
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O.L52@BE;04 0o11514E-08 0=83653E-0_ 0.90059E-060.71284E O0 O=I42IOE 06 0.15768E-03 0=22406E 02
0.13267E O_ 0.28118E O0 0.29990E 02 -0,42208E'040°2I_21E-03 0.88226E O0 -0.62505E O1 0.78800E OC
0®13768E O0 0.24909E O0 0-1_918E_00 0.20599E O00=23113E CO 0=65595E O0 0o67767E O0 0.74288E O0
0=16337E O_ 0.11349E 08 0®80698E-04 0°89292E-060.76677E 00: 0.1_765E 06 0=[_172E-03 0o23877E 02
0.15067E 04 0.29035E O0 0.30717E 02 -0o40671E-040.299_9E-03 0.8171IE O0 -0=56040E 01 0.75887E O0
0=16037E O0 0.25899E O0 0.16575E O0 _.22811E O00.26093E O0 0.72884E O0 0o60573E O0 0.68070E O00.18399E 04 0,10991E 08 0.7_82E-04 0.87600E-06
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0=20569E 04 0.10572Ei08 0°67589E-0_ 0.85583E-060_99928E_00 0,16229E06 0_18226E-03 O.29579EOZ
0=22812E 04 0=32294E 00_ 0.3:_t58E/02 0.15782E-0_
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0.1968TE O0 0o27072E 00 0=18868E_00 0o25874E 00_
0=30563E O0 0=97949E O0 0°4874_E O0 0,58282E O00.21699E06 0o10329E 08 _ 00637_5E-0_ 0.8_394E-06
0_10675E OI 0.1639TE 06 0,18901E-03 0.30993E 020.24839E 04 0.32999E 00. 0o36875E 02 0°19524E-04-0653730E-04 0.54031E O0 -0=41455E 01 0o60532E O0
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0=26312E' OO: 0.2818TE 001 O°2ZO6OE OO. 0_30159E O0
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