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NASA CONTRACTORREPORT" NASA CR-111922

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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 F@.RMAT(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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e

c_

ii

_J3",

o

,,:t

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

Z

L,,'3

-.dO0

P,- ,P4

I I

f_ O" r_J

gdg

NOt,4f_O_

a 0 t00_

I t

gdd

,,1_ ,,,_ __,

0 (DI

_. C'. ¢3

e e e

¢M

04 _

| I I | I | I ! |

ddd ggd

I I I _ _ I I I !

_' 0 _ L,_ ,-_ _ _, C, Lt_

f_O _" '_"_ "_ _ ',C .'_,

e _ o • g _ e e

_o , *To I io

o

_,: _,_.

_.

',- _ 0 C::, ,mi I 0

e _ e e

3--

_.+ _ _'. ,_,.._ _

_" 0 0 ,:0 (0 0

./

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

o ,_ • . .g o o , _ , . .g oII I! II It Ii

_ _ _ _ _ _ _ _ 0 _ _ _, _

... go, og

0

I I I I I I

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

|

i::',

I

I

I

41

(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

, u

i4

i

i

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0

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i.J

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-44

:f', C,_ ,,t"0 O._q

_., .o

"._r__'" F'" D"

I @ • •

0

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,,Z_ ,-'-j

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45

@

L_

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0

p,.

co

p_

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ef_

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c_J

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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

51

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53

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,,,,"-'-'='_ _ Za',.IJIJJ_,.I.j _ .C

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r_IQ -,_ _.:

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_-'_ _ .,_, _.,,_'

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0

0

x

,-_0._.,_._'%!

_E_Q,

_L

_- t

•_ I

_t_.L _.

It ,,,J I,_ ,,_

_Z._. ._.,..__',_

L_

.T

i

!:

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

JI

!

!,

< sT_T)

NAM

I ,,

- N ITIN zVTN,TI_,TP, E

CST,9MAX,KG, /KTR,KS, NMAX,

NO

87I

.

•/ 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

9O

, w_T_I

,_,K,, I

YES

YES

YES

C_NNECTSTI_PAGE 91

KBM- /

Cg_INECTS

TEl PAGE i07

:i

I_ :

i

!

J

[

ii

m_MPAGE 90

c_wz

BB_

BEXP,G7,AS,BS,

Q_I,Z_NK,QWS,

owsB[nS?

WR_ETW,QWS,HSP

¢91

GS,P_RV,DEPS

'- !., ,"

92

READ

BETAD

C_MPUTE

BETAI L1

L2_MR,

STR,NT,NE

YES

YES

PAGE 106

!

i ,ii:'I i

,I

I

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!:

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

94

N_

N_

YES

YES

Y(2)

I!

= ! C_PUTE

Ix(3) ,Y(4)

"Lill

.

YES

95

N¢ YES

Y(5)

Y(6),L,EXET

YES

C_WESXET" 'I'

i ii:

i

!

AS I=SBE**2

. !i

96

i

YES

BSI

YES

CCM_UTE

BSI

q s_=o. I

97

AMQ,SUM

1

YES

YES

SUM

N¢__I ¸

91 " C_MPUTE

S

Ce_PUTE

Y (NR),K

I

i

ii

98

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

-q

N¢

C_MPUTE

SGM,CGM,

CPSI,SPSI,

STH,F(1),CTH

YES

YES

C_[PUTE

THE

YES

C_JNNECTST¢ PAGE i01

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

,;

:i

'I

FROM

PAGE 111

CCNN_CTST_ PAGE iii

162 C_MPUTE

TE

L

YES

C_NNECTS,

T_ PAGE 116

103

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'

YES

CONNECTS,

T_ PAGE 102

i:

,i

i

i

T_ PAGE 110

S • NO

FROM

PAGE 107

117

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

i

i

I

i

I

,i

I

IiJ

PAGE 122

I 4 J-O

II

F_M

PAGE 118

FP_MPAGE 118

121

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

N_

Jr0

..... i

J=J+l I

ICQNPUTE

Y2(J),F(J) ,Y(J)

l 56

YES

I

] CCm_EcTsT_ PAGE 124

123

J=0 l

IJ=J+l ]

YI(J)

c¢_s

YES

LeaP,RUNGE ,M

F_M

PAGE 125

124

RETURN > F_0MPAC-E 123

)

_ YES

C_/NECTS

T6 PAGE 126

125

126

RETURN

I 80 J-0

YES

>

I

81 c_uTsY(J),

F(J)I ,Y2(J)

X,H

CCNNECTST¢ pAGE 127

YES

FP,CMPAGE 125

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

L

'I

i

_t K..K÷I ]

YES

IC_MPUTE

YES_

XMPZa, R_ER,

_,_ i

i

v

129

13CALPi

c_I,r_E

SGI

II,,1 I

I

I

li I'Ii

CmeUTE

SG2

YESw

130

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

N_

SG,CG,

DX,_

YES

f

N_ YES

C_R_NECTS

T@ PAGE 133

x_

YES

132

CRn_ECTST_ PAGE 131

, YES >

ISTG-O 1

l

7 _STAG=I ]_

J

ll,:aI]I

133

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

, K-I ["

YES

F_PAGE 135

136

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

s_I_ BGE_M (IMAX,D_.X)

START

C_MP_EIM2X, KM2X,

R_,PI,AXM

I

138

2 C_M_V_E

FN (I,K),AMF(I,K),

¢

139

J_J+l

I

41 c_E _ "YY(I,J) ,ZZ(I,J)

YES

FT(1),rr(dM),JMM1

FT(J)

140

IZ

'II

N¢ YES

A _. _

Psz(J)

..__--._

c_z

PHz(J)

141

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

_Ir[ J-J+l I

YES

q

()

jIjM_YES

., , C

..... FN(I,K)

N_ YES

I

p,-

143

145

I K-o. l

l,,

K-K+I ....L

I

K-O

YES ,.._

144

C_NNECTST_ PAGE 144

d

CCNNECTS

T_ PAGE 139

N¢

i0 C_UTE ;A_(I,K)

CCm'UTE !

YES

• K-O

PAGE 147

145

l I-0 l

i

-/

, PAGE 147

12 C_MPUTE

AMFF(1),

F_m(1)

YES

SPLINE

I

SPLINE

II, _-°,1

I, I , ,, ,

146

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

fF_M

• PAGE 149

DLP(I,I),

DIP (I,KMAX)r

148

I K-0 I

IL

--.<_

c_t_E

D_(I,K)

PAGE 149

PAGE 151

fls°c_,"' GAM('T,K)

" YES

[

I

I=I+i. 1

_ ]

K-O ]

K_K+I __e.

C@NNECTST_ PAGE 148

I,

149

YES

I

11SPLINEI

I _.oI

AI_ (I,K),AMG (I,K)

C_NNECTS

T_PAGE 149

FROMPAGE 151

150

f_

i:

i,

i!!

i

i

!]

PAGE 153.

C_CTST_ PAGE 150

151 -_I

YES

SPLINE

I

II I'lCALL

SPLINE

I

ilCALLsi,LINE

!! 11SPLINE

C_NNECTST_ PAGE 151

152

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

SUBP,C_E SPLINE<R,DE,_<,_,E,G,N)

START

, I .......

NO

COMPUTE

A(1) ,B(1),

c(i)

NO __///___ YES

154

156

l

C_MPUTEC(K) ,D(K)

__ N_ _< YES

I D(K),,0.

Q(1),u(1),P(a.)

C_UTEB(j),c(j),A(J) ,D(J)

I

P(J),Q(J)

2O

o(J)

tF

N_

157

AM(K),EM1

J-0 1

J"J+l I_

I

.122 c_EJR,AM(JR)

YES _

158

FUNCTI@N FSPL (AM1,AM2 ,XI,X2 ,YI,Y2 ,X)

START>

RETURN

159

FUNCTION DFSPL (AM1 _AM2 _XI iX2 ,YI _Y2 _X)

c_HDT1,T2_T3,T4,DFSPL

160

i_IN_I_N D2FPL (AMI,AM2 ,XI.X2 .YI ,Y2 .X)

TI,T2,D2FPL

I

161

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

¢

.i

c_w_.P2,H2,PR,

HE,DEN

Rl_2 ,V2

NO YES

HR21PR2 _TT D

T31T4pPSI

PSEXIHSEXI

YES

163

/P2,u2,_2,//v2,Ps,Hs,//. _'J /

I

LI EM2-EMIN**2 [

COMPUTE

T3,T4,PS,

m_s

164

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.

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_EDUES,UE,EMS

166

ili

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SUBR¢UTI_Z SPC_(ZW,P_,_S,UZ,HS, PaM,CI',QW,QWS,_CS)

SPECIFY

PR,REC,RA

c¢_nz

CFI, U

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+l o-_oo.I

167

COMPUTEU2,U3,U4,USI

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C@NNECTS

T_ PAGE 169

168

CCNNECTST# PAGE 168

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PAGE 168

169

SUBP_t_NE PRESS (EP,PNEWT)

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C_MPUTE 1PN(I,1)

PN(I, i) ,

IM2X,

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_[ 20AKM'KM2X-I _I

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PAGE 184

FROM

PAGE 171

PAGE180

C_NNECTST_ PAGE 170

/.,_ J_IXT_i

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WRITE

x2(_)

z(J) ,Y(J)

°

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172

yES,_. N_'

N_ YES

173

N_

I :!PH_(1) ,PHz(_c)

• Jl_l

I

I, , [ ....

PH_(J)

YES

PAGE 175

174

c_E

PHi(J)

_ I5 COMPUTE

HPT(J)

1

1C,e_WE

_TCa_)

CONNECTSTO PAGE 174

, , 1 ,, ,

JuO ,,,l C_NNECTST@ PAGE 176

175

i

rlCALL

STRMLN

IcCte_z

CPH,SPH

iCALL

NEWP

22 C_MPUTE

PT(J)

-= YES __////

176

l J'J+l

!

J,Z(J),J,/

Y(J) ,J,/PT(J) /

ilCALL

SPLINE

lC_L_E

KMXM1

i _-_ [I

IFROM

PAGE 178

177 _ _

YES

a

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178

PN(I,i),PN (I,_,_2X)

.... j

l _.o I

!9 CCm_UTE

Pm_(K)

SPLINE

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[ K'° 1........ I

PAGE 180

179

C_NNECTST_ PAGE 171

N_

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C_NNECTS

T_ PAGE '179

FROM

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IK=K+ 1

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180

!

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181

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FROM

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182

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183

CONNECTS

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184

!i "_UBP_UT iNE PDER (XX _PPH, PE, DPDP, DP 2P _DPDX, DPXP, DPXX)

START

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T(_ PAGE 186

185

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PAGE 185

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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--

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/YES, , _"/__, Np

r

C¢_CTSx_ PAcX 188

187 _..

C_NNECTST_ PAGE 189

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PAGE 187

,_ C_NNECTST_ PAGE: 189

188

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PAGE 188

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189

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C_NECTSTJ PAGE 193

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C_NNECTST_ PACE 191

PAGE 193

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191

cCMP_z

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C_NNECTST@ PAGE 191

C_ECTST_ PAGE 190

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PAGE 190

193

194

SUBP4_I/rINE TEANS

, START

i

WRITE

LMAX

I

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()

i_.

196

PEI_(i),LMI

J

7 c_mt_E

PHB(L),HPB(L)

C_UT_PH:B(LMAX),

l

II,

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JI

f

WRITE /MMAX

![ _.oI

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r

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197

WRITE

_,XE (M) ,M,

YE (M) ,M,

ZE(M)

C%m_ECTS

T@ PAGE 197

YESM-MMAX

PKE(1) ,MMI

FP4_PAGE 199

198

YES

c_PHE (_LMAX),RPE(m_X)

N_

CALL

SPLINE

II

199

SUBROUTINE BEGTR(PPH,XTB)

START >

., . , I

_f--I, L=L+I I

YES

N_

XTB

2OO

k,I

!

201

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

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X

_ Z

o

IIII II II

0

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O

_uOOOO Oo

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IIII x

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Z

l.U

'e

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U_

_0_u

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

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_ 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

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0042947E-01. 0.16849E_00 -. 0o44452E-0I 0_62665E-0I

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0.:1,350_E 02 -0.39062E 01. -0®47638_03 0.3812_E O0O. 15537E 02 Oo 15003E-06 0=6892_E-02 O, 26758E-02

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0.15937E 02. .Oo53792E-06 0,6892_E-02 0;25329E-02o,_3873E 06 0o30594E 07 0=30_47E-05 O*36861E-06

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0.1_566E 02 -0._1907E 01 -0.50893E-03 0.32218E00:0.16737E 02: 0.76550E-06 0.68924E-02_ 0.22501E-02

0,_3873E 04 0,30594E 07 0,3046TE-05 0,36861E-06

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0.39660E 0_ 0,36239E 05 " 0,12888E-0_I 0.66703E 03

0.6_275E 05 0.62856E Ol O.18001E 03: -0.68415E-05-0.16020E-03 0,26795E O0 0.11127E-05 0.26880E-02

0,_5274E 02 -0._3804E Ol -0.53196E-03 0.28282E O00.1_537E 02 0.0 0.6892_E-02 0.19704E-02

0,_3873E 0_. 0.30594E 07 0,30_47E-05 0.3686IE-060.39660E 01_ 0.36239E 05 0.13800E-0I 0.500IOE 03

0.65_83E 05 0.43806E OI 0.18001E 03 -0.9_807E-05-0.11275E-03 0.26795E 00_ 0;12383E-05 0,23235E-02

0.15628E 02 -0.44752E 01 -0.54348E-03 0®2631_E 00:0.17937E 02 _ 0®0 0.6892qE-02 O.18315E-OZ

0._3873E O_ 0.30594E 07: .0.30_67E-05 0,36861E-060.39660E 01 0.36239E 05 0.14867E-01 0,53802E 03

0.66691E 05 0,44752E 01 0.18001E 03 0.68809E-05-0.86860E-06 0,26795E'00 0. I3652E-05 0.21597E-02

0.15982E 02 -0.4570IE 01 -0.55500E-03

0.18337E 02 0.0 0.6892qE-020._3873E 04 0.30594E 07_ 0.30447E-05

0,39660E Ol: 0,36239E_05 0,16060E-01

0.24346E 00:0.16931E-02

0.36861E-060.58200E 03

250

Oo67899E 05-0.63853E-04

0.4STOLE 0[:0o26795E_00:

0.1800[E 030.13_34E-05

-O.[O090E-O_.0_19965E-02

_lt. 5UBSCR|PT_OUTOFRANGE *_* 'CALCULATIONS FOR STREAHLINE_NO. 2DISCONT[NUED

THE CALCULATIONS FOR 2 STREAMLINES HAS BEEN COMPLETED

PROGRAH_TER_INATING

251