oo investigation of turbulent £ heat transfer at … › dtic › tr › fulltext › u2 ›...

AFFDL-TR-67-144 Volume III

oo INVESTIGATION OF TURBULENT

£ HEAT TRANSFER AT HYPERSONIC SPEEDS to

^f< Volume in. The Laminar-Turbulent pTnr Momentum Integral and Turbulent Nonsimilar Boundary Layer Computer Programs

R. T. Savage V^ » ±jm JäGCK

J. R. Mitchell

THE BOEING COMPANY

TECHNICAL REPORT AFFDL-TR-67-144, VOLUME III

December 1967

This document has been approved for public release and sale; its distribution is unlimited.

Air Force Flight Dynamics Laboratory Air Force Systems Command

Wright-Patterson Air Force Base, Ohio

Roproduced by fho CLEARINGHOUSE

for Fodorjil Scientific & Technical !"'ormstion Spnngfioid V<i 22551

r D D C

[ii Wl 1968 (ijl

C

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FURNISHED TO DTIC CONTAINED

A SIGNIFICANT NUMBER OF

PAGES WHICH DO NOT

REPRODUCE LEGIBLYo

INVESTIGATION OF TURBULENT

HEAT TRANSFER AT HYPERSONIC SPEEDS

Volume III. The Laminar-Turbulent PrPr Momentum Integral and Turbulent Nonsimilar Boundary Layer Computer Programs

R. T. Savage C. L. Jaeck J. R. Mitchell


NOTICES

When Government drawings, specifications, or other data are used for any purpose other than in connection with a definitely related Government procurement operation, the United States Government thereby incurs no responsibility nor any obligation whatsoever; and the fact that the Government may have formulated, furnished, or in any way supplied the said drawings, specifications, or other data, is not to be regarded by implication or otherwise as in any manner licensing the holder or any other person or corporation, or conveying any rights or permission to manufacture, use, or sell any patented invention that may in any way be related thereto.

MCESUMftr

tmt nm «im s ' NC mvama ntrnwa a jutTmnw

wsniMTioii/iwiuiiiin eooB

m. 1 AVAIL Ut/m VECMLj

/

Copies of this report should not be returned to the Air Force Flight Dynamics Laboratory unless return is required by security considerations, contractual obligation, or notice on a specific document.

AFFDL-TR-67-i44 D2-113531-3 Volume 111

; FOREWORD

This report was prepared by the Space Division, Aerospace Group of The I Boeing Company, Seattle, Washington, under direction of Messrs. A. L. Nagel j and V. Deriugln, program managers. The contract was initiated under BPSN

5(611366-62405334), Project 1366 Hypersonic Gas Dynamic Heating. Task 136607 Aerodynamics and Flight Mechanics. USAF Contract AF33(615)-2372, Investiga- tion of Turbulent Heat Transfer at Hypersonic Speeds. The work was adminis- tered by the Air Force Flight Dynamics Laboratory, Air Force Systems Command, Wright-Pattersoi. Air Force Base, Ohio. Mr. Richard D. Neumann (FDMG) was the Air Force project engineer.

Results obtained during this program are published in three volumes. Volume I, Analytical Methods; Volume II, Analysis of Heat Transfer and Pres- sure Data on a Flat Plate, Cone, Ogive, Cylindrical Leading Edge, Blunt Delta Wing, and X-15 Aircraft; and Volume III, The Laminar-Turbulent pivr Momentum Integral and Turbulent Nonslmllar Boundary Layer Computer Programs. Boeing document numbers assigned to these volumes are D2-113531-1, -2, and -3, respectively.

| This report covers work conducted between March 1963 and March 1967. The | report was submitted by the authors in May 1967.

I The authors acknowledge Barbara J. Safley for her exceptional effort in generating and editing the figures contained in this report.

This technical report has been reviewed and is approved.

PM» ffat^rfr. V. PHILIP^P. ANT0NAT0S Chief, Flight Mechanics Division Air Force Flight Dynamics Laboratory

11

ABSTRACT

This report presents a combined analytical and experimental investigation of turbulent heat transfer on basic and composite configurations at hypersonic speeds. The analytical results are presented in Volume I, the experimental results, including data-theory comparisons, are presented in Volume II, and computer programs incorporating the analytical methods described herein are presented in Volume in.

The two heat-transfer prediction methods programmed are the laminar-turbulent Pr^r momentum integral method and the turbulent nonsimilar boundary layer method. This volume of the report describes the numerical method and presents flow charts, program listings, input forms, and a description of the output lor each program. The programs are written in Fortran IV language for operation or. the IBM 7094.

iii

TABLE OF CONTENTS

Page

I INTRODUCTION 1

II THE LAMINAR-TURBULENT MOMENTUM INTEGRAL COMPUTER PROGRAM (Pr»ir) 3

1. DISCUSSION 3

3 9 9

2. PrPr PROGRAM DESCRIPTION 11

12 13

13 13 17 19 21 22 26

26 28 29 31 32 36 37 39 40 42 43 53 56 58 59 60 60 63 65 66 66 66

a. Method of Solution b. PrM r Program Schematic c. Nomenclature

pr Pr PROGRAM DESCRIPTION

a. PrM r Flow Chart b. Subroutines

1) ATMOS 2) AXI2D (Option D-l) 3) BEQI 4) CYLIND (Option D-3) 5) DBTP 6) DELTA (Option D-4) 7) DERV

8) DIVERG

9) DSIRCH 10) EBAR

H) FIND 12) FLOW (Program B) 13) HEMI (Option D-2) 14) INTIAL 15) IWALL 16) JAYELL 17) MUZERO 18) QTRANS (Program A) 19) REF 20) RORMUR 21) SIRCH 22) SOMEGA 23) SONENT 24) STACON 25) STREAM (Program C) 26) TABLE2 27) TABLE6 28) TABLE7 29) TABL14

s

TABLE OF CONTENTS (Continued)

Paiie e1

30) TABI18 67 31) TABJ.-19 67 32) TAEL20 68 33) TBLP 68

3. INPUT-OUTPUT DESCRIPTION 69

a. Data Input Preparation 69

1) Axisymmetric or two-dimensional bodies (Option D-l) 69

2) Hemisphere (Option D-2) 74 3) Swept infinite cylinder (Option D-3) 77 4) Sharp delta wing (Option D-4) 81 5) Streamline calculations (Program C) 87 6) Reference calculations (Program REF) 93 7) Boundary layer calculations (Program B) 95 8) Heat transfer calculations (Program A) 99

b. Output Description 104 c. Error Statements 108

4, PROGRAMMING INFORMATION 110

a. p ^ Program Listing 110 b. prLir Deck Setup 182

HI THE TURBULENT NONSIMILAR BOUNDARY LAYER PROGRAM (MSBL) 186

1. NOMENCLATURE 186

2. METHOD OF SOLUTION 189

a. Forward Integration 189 b. Numerical Method of Solution 191

1) Calc.ilation of tabular values 192 2) Values obtained from tables 192 3) Computation at initial x = x 192 4) Calculation of ÖT) J 193 5) Calculation of y-derivatives and v-profile 193 6) Cf.lculation of x derivatives 196 7) Calculation of 6*. 0, Q^, TW L, and Tw T at x 196 8) Calculation of u and I-profiles at x + Ax 196

Stagnation Region Calculation 197

VI

3.

TABLE OF CONTENTS (Concluded)

INPUT-OUTPUT DESCRIPTION

a. Data Input Preparation

1) Data card formats 2) Case dependence

b. Output Description c. NSBL Input Form

4. PROGRAMMING INFORMATION

a. b.

REFERENCES

NSBL Flow Chart NSBL Program Listing

Page

19H

19H

19H 203

204 205

206

206 207

222

Figure

1.

2,

3.

4,

LIST OF ILLUSTRATIONS

Pressure distributions on a hemisphere and unswept infinite cylinder

Boeing modified Newtonian hypersonic pressure coefficients

Streamline correlation for sharp delta wings

Correction factor for the velov ity gradient at a hemisphere stagnation point or cylinder stagnation line

Streamlines near the apex of a sharp delta wing

24

62

H2

VI i

SECTION I

INTRODUCTION

Two methods for the calculation of turbulent heating rates are discussed in Volume I of this report, the laminar-turbulent momentum integral method (prMr) and the nonsimilar method for turbulent boundary layers (NSBL).

This volume describes the computer program for each method. A description of the numerical method is given as well as flow charts, program listings, input preparation, and output description.

The first computer program was written incorporating the Pr^r heat transfer prediction method developed by R. A. Hanks of The Boeing Company. The method was developed in the course of the X-20 program and finalized under NASA Research Contract No. NAS 8-11321 (Reference 1). In addition to the PrHr method calculations, the program also contains subprograms for calculating pressures, gas properties, and streamline patterns for several special cases. The SRU 1108 version of the program was written during Boeing Company-funded research and converted to the IBM 7094 during this Air Force Flight Dynamics-funded study.

The program has four major sections. Program A contains the method per se. Given the external flow properties and wall condition, this section of the program computes laminar or turbulent heating rates. All required gas properties are computed by the program from the given pressure and edge velocity. The effect of transition is estimated by matching laminar and turbulent boundary layer momentum thicknesses at the transition point. The point of transition is determined by the program on the basis of a transition Reynolds number selected by the user. Program A can be applied to any geometry for which the external flow properties including nonisothermal wall effects can be defined by the user.

The second section, Program B, computes the local velocities and static tempera- tures along a streamline from a given pressure distribution.

If the streamline divergence is not known. Program C can be used to obtain this information. Program C calculates the path of two streamlines, the streamline of interest and one below, given a two-dimensional array of streamline angles. With two streamline paths known, the divergence parameter can be computed. This calculation, however, assumes zero divergence due to body geometry.

The fourth section of the computer program, called D, calculates or provides information for the three previous subprograms for four special cases; the axisymmetric and two-dimensional body, the hemisphere, the swept cylinder, and the sharp delta wing.

The second computer program, the turbulent nonsimilar boundary layer, inte- grates the boundary layer partial differential equations (described in Appendix C of Volume I) using finite difference methods. The program can calculate the turbulent boundary layer on unyawed sharp or blunt axisymmetric or two-dimensional bodies. Three-dimensional flow effects are calculated using the zero crossflow pressure gradient approximation which implies no rotation of the velocity vectors within the boundary layer. Tl.e program is also limited to attached flow in air. which is considered as ideal gas.

The program is capable of initiating its own boundary layer solutions, given only external flow properties, for the stagnation point of either sharp or blunt tip cones and plates.

Special inputs required are: pressure, wall enthalpy, a three-dimensional flow parameter and its derivative, the velocity gradient at the boundary layer edge and the normal velocity at the surface, all functions of the streamwise distance x. The user must also specify an initial and final value of x, the value of the x increments, and printout Information.

Output includes streamwise and normal velocity, temperature, total and static enthalpy, shear, heat transfer rate, a similarity parameter, and x derivatives of velocity and total enthalpy. These values are tabulated as functions of y (normal to surface) at each output position. Also printed In the output are the shear and heat transfer rate at the wall, and the boundary layer displacement and momentum thickness.

The Pr l^r progi'am requires about 0.5 to 1 minute of computer time per case, while the nonsimilar program requires 2 to 3 minutes for a flat plate case. The time estimate can vary and depends largely on the type of case.

SFCTIOX II

THE LAMINAR-TURBULENT MOMENTUM INTEGRAL COMPUTER PROGRAM (PrMr)

1. DISCUSSION

The computer pi-ogram described in this report is based on the Hanks PrHT

method of solution to the boundary layer momentum integral equation; and, as such, contains all of the assumptions pertaining to the method as discussed in References 1, 2, 3, and Volume I. The program calculates aerodynnmic heating on arbitrary bodies, with or without streamwise pressure gradients in laminar or turbulent flow. Three-dimensional flow effects are included in the form of streamlinf divergence due to body geometry (r) and crossflow or transverse pressure gradients (f)- The effect of a nonisothermal wall on aerodynamic heating is also calculated with modified methods of Ligbthill and Seban. This is discussed in further detail in Reference 1. The program described herein is applicable only to air in chemical equilibrium (References 4-7).

a. Method of Solution

The Pr^r integral program is actually four programs, any of which can be oper- ated separately. The first program, Program A, calculates heating rates for any body shape at any flight or ground-facility test condition given the external flow properties, gas properties at the wall, gas properties at the stagnation condition, total streamline divergence and the streamline divergence due to body geometry alone.

Program A has an additional capability of calculating transition effects based on a transition Reynolds number selected by the user. The program matches laminar and turbulent boundary layer momentum thicknesses at the transition point.

In addition, the program calculates reference bent transfer coefficients which are used to normalize the calculated local heat transfer coefficients. The laminar heat transfer coefficients are all normalized by the value of the hemisphere stagnation- point heat transfer coefficient evaluated at the same free stream conditions as the body of interest. The heat transfer coefficient on the stagnation line of a GO0 swept infinite cylinder is used to normalize all turbulent heat transfer coefficients. The hemisphere and cylinder radius are items of input.

Program B calculates the boundary edge velocity and temperature given the pressure distribution along the streamline or body.

Program C computes streamwise pressure and divergence parameters given a two-dimensional field of pressures and streamline angles. Since pressure data and streamline angles are often most easily available as spanwise plots at various stations, considerable crossplotting is required to provide specific values along the streamline passing over any specified point on the body. This work is performed by Program C, which begins at the point for which the user desires to calculate the heating rate, and traces out the streamline, proceedmg upstream to a specified boundary (e.g., the boundary layer origin). The streamline angle is determined by double interpolation of a two-dimensional array of angles which is input. The streamline angle is then used to determine the coordinates of a point on the streamline a distance dx upstream. This procedure is repeated until the upstream boundary is reached.

|,feet

It also interpolates as required to obtain initial values at the upstream boundary of the input array (4, 1) thus providing all information required for Program B.

Program D provides the pressure aad streamline arrays and initial values required by Programs A, B, and C for several specific bodies. These optional configurations are:

D-l Arbitrary two-dimensional bodies at an angle of attack and arbitrary axisymmetric bodies at zero angle of attack. The bodies may be sharp or blunt and composed of wedges and plates or cones and cylinders.

D-2 Hemisphere

D-3 Swept infinite cylinder (0 < A < 90°)

D-4 Sharp delta wing (60° < A < 80' and o > 0)

For options D-l and D-2. the user provides a table of body width or radius at various locations along the axis. For options D-2 and D-3. the cylinder or hemisphere diameter must be input and for options D-3 and D-4. the cylinder or delta wing sweep angle must be provided. For option D-4 the angle of attack must be stated. In all options, the user provides the free stream conditions.

Options D-2 and D-3 (hemisphere and swept infinite cylinder) use pressure distributions stored within the program. These distributions are shown in Figure 1. Options D-l and D-4 calculate pressures from the local flow deflection angle and a modified Newtonian pressure expression. For Ö > 0

2 sin" 6

1. or, 1.1025 (•"W sinö)'

1/2 , 0.. . 2^ 1.2<s sin ^

M 0.6

(1)

and for 6 « 0

sin" 6 (2)

This approximate analytical method was devised for predicting pressures from the Newtonian relationship in which K is allowed to vary.

C, P

K sin o (3)

The relationship for K was chosen to obtain the best agreement with the exact solutions for a wedge, cone, and blunt body stagnation point. A plot of the modified pressure coefficients is shown in Figure 2.

The D options also provide velocity and streamline angle distributions along a streamline. In option D-l, the velocity is computed from

1/2 if P

initial 328 P (4)

and

u = a' — e x

sonic point if P. .L. , > .528 P

iiutial o X < X

(5)

sonic point

1.0

.9

.8

.7

.6

.5

.4

.1

.09

.08

.07

.06

.05

.04 0

l^-"1 "^ 1 ___ J ^"""N

\ N, V

1 ■ II

I j I

-V i 1

1

1

^ \k 1 : ' \ A^T ^ 1 i

i j

1 ! \ V§ '

j , \ \ ^ i i i

i 1 \ V

1 1 L \ "* I 1 I 1

1 i ' \ \^

• 1 i

i \^ \ I \ w \ 1

i h - [ - i \ " \ L ..____

i ! y \l\ 1 \M i i -A »äJ

■ ■

1 10 20 30 40 50

6, degrees

60 70 80 90

Figure 1: PRESSURE DISTRIBUTION ON A HEMISPHERE AND UNSWtPT INFINITE CYLINDER (REF. 3)

o CM

00

's c

« JC

i 0 8 8: o

8 II '*

!

5

1

/ CM

/ o

I \

CO

iv / V

i, A 'O

> 1^ ,bxi

A CM

o

——

CO I—

UJ

o

O O UJ OH

on to UJ

a. o

O CO

c Q- •?. >-

«I O

o o

o

o CO

o ■

n oo

CN

o CM CN CN

O

CN

CO

Q. U CN

and

u = a* i e

6 1 ■ (kf .1/2

X > X sonic point

(6)

where a* is the velocity at the sonic point

Program options D-2 and D-3 obtain the edge velocity by integration of the pressure gradient ?long the streamline, while D-4 calculates ue from

u = u e «o

1 - 5600 (7)

The development of Equation (7) is presented in detail in Reference 3.

In addition, options D-3 and D-4 for the swept cylinder and delta wing also provide a two-dimensional array of streamline angles. The streamlines angles on the swept cylinder are obtained from

tan 0 = :—- e u_ sin A (8)

where the spanwise velocity ve is obtained by integration of the pressure gradient normal to the leading edge. Delta wing streamline angles are obtained from a correlation curve and

/d0\ R +

CL LE

(90 -A) m B9

CL (9)

where

tan R = (I)

tan /90 - A\ ' 57.3 /

PcL is the gradient of the streamline angle with

respect to the ray angle e

The streamline angle distribution across the wlug span is a curve fit obtained by matching the boundary conditions given by the angle and the gradient at the centerline and the angle at the leading edge. The calculations of program D are necessarily some' what approximate since exact flow field calculations are possible for only a few simple geometries. The approximations used in program D are those discussed in Reference 1 and are simple yet reasonably accurate.

8

Program D is easily modified and extended as more information becomes available, or as certain specific shapes become important.

b. Pr^r Prosra"1 Schematic

The relation of the four programs is illustrated on the following page: however, it should be understood that all four programs are coupled and function together as a single program when desired. The symbol OR indicates that the required data can come from either of the indicated sources.

c. Nomenclature

a* sonic velocity

Cp pressure coefficient

f streamline divergence due to crossflow pressure gradients

AI free stream Mach number 00

P local pressure

P stagnation point pressure o

P stagnation point or stagnation line pressure 2

streamline divergence due to body geometry

u boundary layer edge velocity

u free stream velocity

v spamvise velocity at the boundary layer edge

x distance along a streamline

6 local flow deflection angle measured from the free stream velocity vector

A total streamline divergence, includes body geometry and crossflow pressure gradients (rf)

e ray angle on a delta wing

9 angular location

INPUT

• Geometry • Test condition • Wall temperature • Transition Reynolds

number • Printout information

4 f PROGRAM D\

♦ D-l Axisymmetric or two-

dimensional D-2 Hemisphere D-3 Swept infinite cylinder D-4 Sharp d el to wing

<

1 Output from Pr ogram D OR I input to Program C

• Test condition • V/all temperature • Pressure • Streamline angle • Streamline divergence

due to body geometry • Transition Reynolds

number

i '

f PROGRAM C^)

i ' Output from Program C OR

input to Program B • Test condition • Wall temperature • Pressure • Streamline angle • Streamline divergence

due to body geometry « Streamline divergence

due to crossflow pressure gradients

• Transition Reynolds \ number

f PROGRAM B^)

\ Output from Program B OR

input to Program A • Wall temperature • Pressure • Streamline angle • Streamline divergence

due to body geometry • Streamline divergence

due to crossflow pressure gradients

• Transition Reynolds number

• _dge velocity • Edge enthalpy

| • Edge temperature • Edge viscosity

(^PROGRAM A^

Output from Program A • Heat transfer coefficient • Skin-friction coefficient • Heating rate • Shear stress

10

0 local streamline angle

A leading edge sweep angle

(|,T)) coordinate system

pr^r density-viscosity product evaluated at a reference condition

2. PrMr PROGRAM DESCRIPTION

Presented in the following section is a description of the program and subroutines. Equations, nomenclature input, and output from each subroutine are presented in alphabetical order by the subroutine title.

Before proceeding, a description of the program coordinate system is required. The program operates in a two-dimensional curvilinear coordinate system where the coordinate axes are designated as TJ and |. Restrictions on this coordinate svstem are:

lc t) and ^ are orthogonal.

2. Both T} and ^ must be parallel to the surface over which the boundary layer flow is being considered, and

3. i is taken to be in the general direction of the fluid flow.

Thus, no coordinate axis projects outward or normal from the surface area. In the case of a flat surface, T? and ^ will lie in the same plane.

CYLINDER

x, streamline distance

FLAT PL^"E

II

a. P w Program Flow Chart

INTIAL 1 r-SONENT * L-ATMOS

L-STAnON i

REF

-RORMUR

-SOMEGA—rTABLE 2 LTABL18

-TABLES IWALL- -SOMEGA—I-TABLE2

LTABL 18

T TABL 14 TABL 19 TABL20

-JAYELL

MUZERO—TABLE?

STACON SOMEGA

SOMEGA-r TABLE 2 LTABL 18

—r TABLE 2 LTABL 18

•-EBAR- -SOMEGA—T-TABLE2 LTABL18

DELTA K TBLP DBTP- -TBLP

CYLIND TBLP -FLOW i: SOMEGA—r TABLE 2

Li •TABL18 DERV

dsTREAMf—rFIND 1 J '-TBLP

•DSTP-

HEMI h-

Ü-r- Li

TBLP OIVERG

HAXI2D f i-TBLP c

h TBLP DBTP STREAM

"TBLP

TBLP -FIND -TBLP

DBTP TBLP

FLOW

SOMEGA-rTABLE 2 TABL IS T

h DERV SOMEGA-rTABLE 2

LTABL 18

^j BEOI [—pDERV

[OTRANS —pDERV -I WALL- -TABLE 6

-SOMEGA-T-TABLE 2 LTABL 18

-RORMURxTABL 14 kTABL 19 l-TABL 20

-iWALL TABLE 6 -SOMEGA-r-TABLE 2

•-TABL 18

-RORMUR-i-TABL 14 hTABL 19 l~TABL 20

EBAR —SOMEGA-rTABLE 2 LTABL 18

-EBAR- -SOMEGA—i-TABLE 2 L'

JAYELL-

'-MUZERO-

-SOMEGA-r-TABLE 2 L'

-TABLE?

TABL 18

TABLE 2 TABL18

12

b. Subroutines

The main program of the laminar-turbulent Pr^r momentum integral computer program —II (AS2419) —controls the input and the logic flow to the required mathe- matical subroutines. The input logic is arranged to minimize the modification required to change the input formats or the source of inputs (e. g., tape input or direct coupling to another program).

1) ATMOS(ALT, SONIC, PINF, RHO, TEMP)

Given the altitude, ATMOS calculates the free stream conditions. This subroutine is program AS1772, "Standard Atmosphere Properties, 1962".

t Input

ALT altitude, ft

Output

SONIC aÄ

PINF P«

RHO P«

TEMP T

free-stream velocity of sound, ft/sec

free-stream pressure, lb/ft

free-stream density, slug/ft^

free-stream temperature, "R

2) AXI2D (Option D-l)

AXI2D calculates the flow conditions for arbitrary axisymmetric and two- dimensional geometries using the following procedure:

(a) With the geometry specified by the coordinates (^j, yj), the slopes and pressures are assumed to occur midway between the input geometry points.

yi+l " yi

i+i M

i;;

Where y is the local body coordinate:

and

r O r

Xi+l/2 = Xi-l/2 + [(yi+l/2 " yi-l/2) + (5i+l/2 " ti-l/2)

1y 2 i 2 4-M sin 6 + 1

äin 0

where

C,

. 2X sm 6 = 0 for Ö £ 0

. 2. sm 6

= 1.05 + 1.1025 + 2 2

M^ sin 6 .

1/2 1.278 sin 6

M .6

for 6 > 0

The end points Xv and Xf and the pressures at the end points are found by linear extrapolation.

(b) If P at either end point is less than P^, set the respective P = ?„. If P at either end point is greater than P0, set the respective P = P0.

(c) Initialize x* = Xj and calculate the flow conditions at each increment dx along the streamline

x* = x* + dx i > 2 1 i-l

14

Linear interpolation is performed to find P/PSL(X*) at each x* using the tables of P and x calculated previously.

If P(x,) < . 528 P0

^»'"-['-(i-1)^ ll/2

If P(xi) > .528 P0 find the sonic location X* where P = .528 P0 from interpolation of the x and P tables. Then

a*x u (x*) = —— when x < X* e X*

-U-^ri! ue(x*) a i m\ 1/2

when x > X*

1/2 a* = (H/3) Sonic velocity

The remaining flow conditions are calculated from

A. i

(x*) =1.0

(x*) =1.0

Two-dimensional body

A(X*) = y(x*)

(x*) = y{x*)

Axisymmetric body

ie(x*)

w (x*) e

ZT (x*) e

H - I UeV)

f(P/PSL(x*). ie(x*)) j

f(p/?SL(x*), yx*)) [SOMEGA]

15

(d) Step (c) is repeated until the flow properties have been calculated for all points on the streamline.

Input

H H

XSI 4

Y yU)

XI xi

DX dx

POPSL Po/pSL

ACH M«

PINF P«

Output

II

XSTAR

POPSLX

UE

AIEE

ZTEX

OMEGAE

RORI

DODI

P/PSL(x^)

ue

(ZT)e(x*)

We(x*)

(r/rjHx*)

(A/Aj) (x*)

total enthalpy. ft2/sec2

coordinate in free-stream flow directic n, ft

geometry coordinate orthogonal to ^, ft

streamline coordinate at which calculations are to begin, ft

streamline coordinate increment, ft

stagnation pressure, atm

free-stream Mach number

free-stream pressure, lb/ft"

number of points calculated along streamline

streamline coordinate, ft

local pressure along streamline, atm

edge velocity, ft/sec

edge enthalpy, ft /sec

edge compressibility-temperature product, 0R

edge vircosity parameter, slug/ft-sec-cR

divergence parameter due to body-shock layer geometry

distance between streamlines, or total streamline divergence

16

3) BEQI

BEQI calculates the equivalent distance parameters at the initial streamline coordinate x». Following is the sequence of operations:

b b (a) Initialize -^^ = -^^ =1.0

(b) Divide the first streamline increment dx into 10 intervals, dx/10. Inter- polate to find the flow properties at each interval.

(c) Interpolate for i = f(T , P/POT) [IWALL] w w SL

ZT,

CJ,

= f(H, P/PSL) [SOMEGA]

ZT w

u w = f(iw. P/PSL) [SOMEGA]

(d) Calculate the reference terms

u; , o u , ZT , i , a [RORMUR] r frfr r r r

(e) Calculate d(A/A.)

dx [DERV]

(f) Calculate ie SL and EXPK from

N = x* d(A/£0 i_

(A/A.) dx

If N < .05; .99 < N < 1.01; EXPK = 0, ie SL = Je

--N(N-l) .05 < N < .99; EXPK = -.194e ' ^SL^N-^Ö,

-|(N-1) N.J

N > 1.01; EXPK = .194 e ' SL = ( N/

17

vP=UeCOSÖSL: VSL^'T

(g) Obtain EL and ET (EBAR)

(h) Calculate the equivalent distance parameter

b _eq

x 1

X* x* G(x*)

^ 'X, X,

where

and

GLMf^''r^(r/••l)2(f/t1)

2ELX,

A/A. f/f. = i r/r.

The integration is performed by applying Simpson's rule to the integrands GL and G-p to finu b L/X and b f/x, respectively.

(i) bg- /x is calculated for 10 inter/als. The 10th bg- /x is used as the new (b „ /x)v , the first interval is divided into 100 intervals, and the above eq AJ

calculations are repeated for the first 10 Intervals.

(j) The calculations are repeated for 5 iterations (1) where

dx i

dx. = —r and 1 101 (%) ] " [(%)

XI i ^ i-1

The final (b^/x)v is then used as the (bö /x)v for the general l CM

AII ii=5 ecl XI

calculation (.QTRANS).

18

Input

XSTAR x*

POPSLX P/PSL(x*)

TW Tw(x*)

UE ue(x*)

THETAS Öe(x*)

OMEGA E ü;e(x*)

DODI A/AjCx*)

RORI r/r^x*)

DX

Output

dx

BEQXLI (b r/x) XI

BEQXTI (b /x) eq, 1 x I

streamline coordinate at dx intervals, ft


vail temperature, 'R


streamline (edge) angle, degrees

edge viscosity parameter, slug/ft-sec-'R

dimensionless distance between streamlines



laminar equivalent distance paraneter at Xj

turbulent equivalent distance parameter at xj

4) CYLIND (Option D-3)

CYLIND calculates the coordinates 4 and TJ and the two-dimensional parameters Öe(l. ^)• p/po(^ > ^) and Tw^ ^• ^) for an infinite swept cylinder. The properties are calculated as a function of TJ at ^ - 0 and are assumed to be constant in the ^ direction.

Pressure:

where

p 0

P(0) pT

X

l"PT ' P

. o .

e = 57.3- T)

R CYL

P/PT (0) is found by linear interpolation on a built-in table (Figure 1).

19

T2

\7MN "'

2.5 Stagnation Line Pressure

M„, = M cos A

V'all temperature is found by linear interpolation on the input array.

Streamline angle:

9 (7?) - tan ;—j- e u^smÄ

where V(TJ) is the tangential velocity calculated in subroutine FLOW.

Then

P/P fM) = P/P (T?) for all | o' o

Input

RADIUS

SWEEP

ACH

VEL

ETAF

XSIF

TW

■yi.Tj) = Tw(i7) foralU

9 (M) = fl (T?) for all i e e

(u ) =u sin A e'xj

R, CYL

M„

w

cylinder radius, ft

sweep angle, degrees

free-stream Mach number

free-stream velocity, ft/sec

final T} coordinate for which streamline is to be calculated, ft

final ^ coordinate for which streamline is to be calculated, ft

wall temperature, "R

20

free stream pressure, lb/ft"


coordinate along axis of cylinder, ft

coordinate tangential to cylinder, ft

number of 4 values

number of TJ values

maximum 17 value, ft

streamline angle at Imax, degrees

pressure ratio array

wall temperature array, 0R

streamline angle array, degrees

DBTP is a linear double interpolation routine. Search time is reduced by starting the current search at the search result for the previous entry to this routine. DBTP uses TBLP to perform the interpolation.

Input

XX

YY

7.7.

X

Y

NX

PINT P«

POPSI. fVPSL

Output

XSI i

ETA V

M

N

ETA MAX w^) TH8MX %ax(0

PRESSR P/iy^n)

TWALL TwU,n)

THETAE eea,Ti)

5) DBTP

independent variable table


dependent variable table

independent variable


number of values in X table

NY number of values in Y table

NXS location in X table at which search is begun

NYS location in Y table at which search is begun

Output

Z dependent variable

6) DELTA (Option D-4)

DELTA calculates the body geometry, the streamline angles, and the two- dimensional pressure array P/P0(|, TJ) for a sharp delta wing. The following procedure is used:

{&) If R < 0 a sharp delta wing case is indicated and 6 = a (degrees) for all |.

(b) The pressure is calculated from

y 2 / CP \ 2 Pa,-n)=j P^M^ —2-) «in Ö + P«

\sin 6/

where

sin 6 = 1. 05 +

,1/2 1.1025 +

2 2 M„ sin 6 00

1. 278 sin 6

(c) The initial velocity is calculated from

<ue)xT = "ocl1" ö2/560o|

M .6

(d) The equations used to calculate 6e(4,J?) are:

6 + 1 ■"&mr am ■1 _1_

M.

1/2

MXT = M^ sin £.

* ** 1 90 -A

tan I 6

'M 1 + N

M, M N (u/u«)

1 + - ü 6

1/2

22

y = 5($**)4

5(i**)4+ 1

e* = (90 - A)(i^***)r

N is found by linear interpolation on a built-in table (Figure 3).

If *** < 1.2 N = f(#**)

If ♦** > 1.2 N = f(A,***)

1 + ♦** _ NCL=—2 NCL LL M„+l CL

N

M 2

N

V = ^ tan (90 - A) max

6= (90-A)NCLR+ |Ö*-(90-A)NCL|R ^9

where

= _ tan^/g) tan (90 -A)

Then

0 =6* max

^W" T) = T). + A»;

i i-l

(e) The delta wing option D-4 assumes zero streamline divergence due to body geometry

— = 1.0 r.

i

CM

o

LU Q Q. cc: < »fcfcs

to 1^*1

< Or: o LX.

_ o o z o> o \ ^—

H- 00 * <

• ■Q' _J LU G£ Qi

«o o o LU z ■MM>

_-J

S -^■ <

LU Q: »— CO

■ •

CO a> k- =5 a»

CN

m

U |Z

24

Input

ACH M«

PINF P«

POPSL Po/PSL

VEL u«

ALPHA a

SWEEP A

RADIUS R

XSII «j

ETAI n,

XSIF lf

ETAF i7f

DELXSI A|

DELETA ATJ

Output

M

N

XSI

ETA

ETAMAX

TH8MX

THETAE

PRESSR

URATIO

max

max

e

P/Pc

uAu

free stream Mach number

free stream pressure, lb/ft'


free stream velocity, ft/sec

angle of attack, degrees

sweep angle, degrees

leading edge radius, ft

initial streamline 4 coordinate, ft

initial streamline ») coordinate, ft

final streamline ^ coordinate, ft

final streamline TJ coordinate, ft

increment used to calculate ^ coordinate, ft

increment used to calculate TJ coordinate, ft

number of { values

number of TJ values

geometry coordinate, ft


leading edge coordinate, ft

streamline angle at leading edge, degrees

streamline angle, degrees

pressure ratio

velocity i*atio

7) DERV (V, W, X, Y. Z. DBYDX, DX. ITYPE)

DERV calculates the derivative of a tabulated function f by Langrangian five-point formulas. Given a sequence of points x_2. x_2, x0, Xj, X2 separated by a common increment h, the corresponding derivative of f at each point is given by:

'^i'-Vi^'i^'-vy f2 = Ä(-fl + (2'

Input

V f-2 1

w f-l

X fo

Y fl

Z f2

ITYPE

DX h

Output

functional values

determines which derivative formula is to be used

function increment

DBYDX f function derivative

8) DIVERG

DIVERG calculates P/PQ. TW, 0e, r/r^ and A/Aj along a given streamline by the following method:

(a) Store the original streamline (1) data and calculate an auxiliary streamline 12) such that

n = T) ± An r p ^i max

26

(b) Calculate the following at each dx location on the streamline of interest

ß (x*) linear interpolation on 9 (x) e e

T (x*) double linear interpolation on T .(|,1)

P/P (x*) double linear interpolation on P/PJi.ri)

—{\*) =1.0 r.

i

cos fl (X*) A ^(x*) - n2(x*) e^

AT0*** _ V^lx*) - r,i2(x*) cos 0e (x*) i I

Where the subscripts 1, 2, and i indicate the original and auxiliary streamlines and the point at which calculations are begun, respectively. If the streamlines cross at any point, the above calculations are begun a distance dx downstream from the intersection point.

Input

XI XI

DX dx

FTAF "f

XSIF «f

M

X

XSI i

ETA »1

PRESSR P/PoU-

TWALL V^.T,)

XSIX 4(x>

ETAX rj(x)

initial streamline distance, ft


final streamline coordinate, ft

final streamline coordinate, ft

number of ^ input values

number of rj input values

streamline coordinate array, ft

streamline coordinate array, ft

pressure ratio array

wall temperature array, "P.

streamline coordinate, it


streamline coordiiwte, ft

n coordinate for geometry leading edge, ft

streamline angle, degreei


pressure ratio along streamline


dimensionless distance between streamlines

divergence parameter

wall temperature, 'R

final streamline x-coordinate, ft

number of points on streamline

9) DSIRCH(Z, X, Y, ZA, XA, YA, NZ, NX, NY, IX, IY)

DSIRCH uses DTAB to perform double interpolation on a table of the form

Z = f(x,y)

Input

ETAMAX max

THETAX *e(x)

Output

XSTAR X*

PRESRX P/P0(x*)

THETAS 0e(x*)

DODI (A/AjHx*)

RORI (r/rjHx*)

TW T

XF xf

II

X

Y

XA

YA

ZA

NX

NY

NZ

independent variables

table of X values

table of Y values

table of Z values

number of values in X table

number of values in Y table

number of values in Z table

28

IX order ot intenxjlacion in X direction

FV order of interpolation in Y direction

Output

/. dependent variable

10) KBAR (AVEW, AYESL. 11, SIGR. EXPK, POPSLX, OMEGE. OMEGS, GAMC, GAMO. EBARL, EBART)

K3AR calculates the crossflow pressure gradient parameters Ej^ and E-p and the pressure gradient correlation parameters rc and r using the following method:

(a) Interpolate to find

e, SL oL e, SL

(b) Calculate the local pressure in lb/ft^ and the Pß terms

P p =

r p i

SL (ps. ,

VT =

p.

R ^s

pc^e =

P

ir-e

(c) Calculate the mean enthalpies

'avg 2 (l\v ^ 'e. SL)

i = i + .2 (H - i or) a m,c avg e, SI, r

VT\JA

pe^e"/

(d) Interpolate to find

ZT = f(P/^T. ' ) [SOMEGA) m,c SL m,c

ZT = f(P/POTl i ) (SOMEGA) m.o SL avg

2i)

(e) Compute:

and

where

(I^-.294)«rr-355

FZC .4018

(£0-.294)<rr-355

FI0 = .4018

\ - ('*rpK

F _ ln„ \EXPK K

o = ("0)'

ZT 2 = m'c

c ZT e,SL

ZT 2 = m'0

0 ZTe,SL

(f) The pressure gradient correlation parameters are calculated from:

rc = -7m4FKc(\P\-1)

r0--71764FKo(VrT\-1)

(g) Compute the crossflow pressure gradient parameters:

ET = 1+ r L c

i+r \4

ET = (1+.76519 ro)_

30

Input

.WKW Ky

AYKSI. ^.SL

II H

SIGR ^r

POPSLX p/ps]

OMEGK "e

O.MEGS ws

Output

GAMC rc

GAMO r0

EBARL EL

EBART ET

Jl) FIND (ZA. Z. NTZ. XZS)

2 2 wall enthalpy, ft /sec

edge enthalpy at flow line of symmetry, ft /sec

total enthalpy, ffVsec

Prandtl number


edge viscosity parameter, slug/ft-sec-0R

stagnation viscosity parameter, slug/ft-sec-'R

pressure gradient correlation parameter

pressure gradient correlation parameter

laminar crossflow pressure gradient parameter

turbulent crossflow pressure gradient parameter

FIND looks up Z in an array ZA, dimensioned NZ, and stores the location of Z relative to ZA(1) in NZS. If Z is not in the range of ZA, the message "VARIABLE NOT IN RANGE OF TABLE" is printed, followed by the input values of Z, ZA(]), and ZA(NZ), respectively. NZS is then set to 1 if Z - ZA(1) and to NZ if Z ZA(NZ). Upon subsequent entries to FIND the search is begun at ZA(NZS) where NZS is now the location of Z from the previous entrv to FIND. For a normal exit NZS = i where ZAjiZ<ZA.+ 1.

input

ZA

Z

array to be searched

variable for which location is desired

."■1

NZ

NZS

Output

NZS

number of values in ZA

location of Z in ZA from previous call to FIND

location of Z in ZA

12) FLOW (Program B)

FLOW calculates the external flow parameters at the boundary layer edge as a function of streamline distance given the pressure and the stagnation conditions.

At the starting point, on the streamline, edge enthalpy is obtained from:

v2

("e)

Cel = « I

For the special case of the swept cylinder tangential velocity calculation

\ /x KsmAJ (0 = H - "I 2

Then

Let

and define

ZT = f(i , P/POT) (SOMEGA] e e öL

w - f(i , P/POT) (SOMEGA] e e oL

u - \**

K

2\u.

2 \u„

.21

X*,K

as the Kth iteration for u at the point x*.

32

Simüarly define |(ZTe) ^ i as the Kth iteration for ZTe at x*.

The iterative equation to be satisfied at each point x* on the streamline is

, , dxR x* „ x*-dx 2

K u„ (ZTeL

r_d_ /p dx ft I

avg

K \ o avg

where

d<p/IV _ i dx 2 lr<p/p°,i,.*ä1

<p/po'L. x*-dx

and

(P/Po) avg 2

P P — (x*) + —(x*-clx)

o o

The derivatives are calculated numerically using subroutine DERV.

For each iteration:

(i ) = !l-u 2\u >

K

For the swept cylinder tangential velocity

(U H - u " u 00 v* - 2 (u«si11 A

Then

x*

(-,

K

K

f Ce) P

' P. (X*)

K SL

SOMEGA]

;i:!

This va'ue of ZTe is then used in the above iterative equation to yield a new ux* These calculations are repeated at every x* until

K 'K-l

K

< €

where e = 10 . A maximum of ten iterations (K = 10) is fixed by the program. If the relative error criteria has not been satisfied after ten iterations at any point x*, an appropriate error comment will be printed. If convergence at x* has been estab- lished in the Ith step, then set u^ = {ux«} and

(Ue)x. - U-(2V)

(ZTe)x.= lK)x.!

KL-IKL

1/2

A special cass results when /ue) = 0. This corresponds to a stagnation point

where x, = 0 and d/dx(P/P0) = 0. The iterative equation for ue becomes

ue(x*) = ue(x* - dx) or due/dx = 0. Because this result is incorrect, the first step is calculated using the following:

The velocity gradient at the stagnation point xj is calculated from:

1/2

4fe) =f -*K) 4(f) \ «v J Xj »I Xj dx \ o /x

where the pressure derivative is obtained from the numerical derivative equation

2 £■(«!. i1)-2|-({2,T)) + |H|3,tj) d/P\o o 'o

A 2 \p dX \ Oy A«'

34

The use of this equation requires that the first three i input points be equidistant. Then

(Ual = u

I+dx

i /u d / e dx \ u

1 / 2

dx Jx,

I'M =H-^(U ) \ e' 2 \ e / I-dx Pdx

(ZTe) I-dx

K) I+dx

> = f Cel I+dx

SL I+dx

[SOMEGA]

The calculations continue, using the previously described procedure where the starting point is now Xj + dx.

Input

PR ESUX P/P0(x*)

II

ratio of local to stagnation pressure along streamline

number of values in pressure array

L"RATIO (ue/u«) velocity ratio at x

II H

VEL "*

DX dx

POPSL P0/

PSL

Output

total enthalpy, ft^/sec"

free stream velocity, it/sec



The following outputs are a function of streamline distance:

UE u,- edge velocity, ft/sec

POPSLX P/Pgjjx*) local pressure along streamline, atm

AIEE 'e

ZT EX ZT

OMEGAE we

XSTAR X*

13) HEMI (Option D-2)

edge enthalpy, ft"/sec

edge compressibility-temperature product, " R

edge viscosity temperature ratio, slug/ft-sec-'R


HEMI calculates the streamline and the pressure and divergence parameters along the streamline for a hemisphere geometry.

(a) Set x? = Xj

(b) Calculate x*, P/P^x*), r/r^x*), and A/A^x*) for all dx increments along the streamline until fl = 90°

e = 57.3 x* R HEMI

Interpolate to find P/PJx*) = f(e)

r/r.(x*) = sinj — HEMI

(c) URATIO - Up /u

Input

XI

A/A.(x*) = sinf —

RHEM RHEMI

XI XI

DX dx

Output

XSTAR X*

PRESRX P/P0(x*)

HEMI

hemisphere radius, ft

x-value at which calculations are to begin, ft

streamline increment, ft


ratio of local to stagnation pressure along streamline

3«

UORI r/rj(x*) divergence parameter due to tody-shock layer geometry

DOD1 A/Aj(x*) distance between streamlines, total streamline divergence

14) INTIAl. (LI, NTYPE)

INTIAL calculates the free stream conditions for wind tunnel and flight type inputs.

(a) Wind Tunnel Class i

2 u

H = 6000 T 2

Q« = 0.7 P^H.,2

P« - 1^/(1716 T^)

(b) Wind Tunnel Class II

2

i^= IT —r-

aao = Mi«) when 5.93 x 10(,< i«« 20.3 x 10° ffVsec2 [SONENT,

a^ = .6325 s/ i^ when !«< 5.93 x 10° It" sec"

u a

M. a«

,2 u„ \

T E \49M,

P« = P«,/(nif. T„)

P /P from [STACONl o oc

P-fe) SL\ P

o^

00 CC

q«> 2RT 00

(c) Flight

a^ = f (altitude)

P., = f (altitude)

T,, = f (altitude)

«« H = 6000 T,, + —-

u

00 a«,

q, = 0.7 P^M^2

UTMOSl

For wind tunnel I and flight

P /P = f(M , H, P^, u ) (STACONl

VPRT = U2- SL

Input

Wind Tunnel Class I, NTYPE = WTl

ACH M„ free stream Mach number

TEMP T.,. free stream temperature, "R

PiXF free stream pressure, Ib/ft^

38

Wind Tunnel Class 11, NTYPE - \VT2

VKL

II

POPSL

PINT

H

rJo/PSL

Flight, NTYPE = FLIGHT

ALT

VEL u.

Output


total enthalpy, ft /sec


free stream pressure, Ib/ft"

flight altitude, U


ACH M,

H H

PINT P„

POPiNF PQ/P^

POPSL Fo/pSL

QINF q,,

SONIC ^

V EL u

15) IWALL (T\V, POPSLX, AYEW)


total enthalpy, ft /sec^

free stream pressure, lb/ft

stagnation to free stream pressure ratio


free stream dynamic pressure, Ib/ft2

speed of sound, ft/sec


IWALL calculates the enthalpy at the wall as a function of the waii temperature and the local pressure.

(a) Set Tu ~ Tw/l.8 (conversion to 6K)

;;<)

(b) Calculate wal! enthalpy

i =- .432 T.. when. T . < .JOO'K

i -.432T.. ^ 3.82x 10 r>/T . -SOOT when 300 < T < l-iOO'K

iw = f(log.,0 p/psl. Tw ); |TABLE 6| when Tw > ISOO^K

2/c^2 (c) Convert iw to ft^/sec

Input

i /_iL\ . 2.,031 i (m wl 2 } w\ lb /

\ sec /

TW T 1w wall temperature, 0R

POPSLX P/PgL(x*) local pressure along streamline, atm

Output

wall enthalpy, ft^/sec Hv AYEW

16) JAYELL (AYEW, H, AYEE, ZTE, ZTS, SIGR, BETAS, POPSLX, XJL)

JAYELL calculates the laminar pressure gradient effect correlation parameter JL

from the following:

(a) Calculate the mean enthalpy

i = -(i + H) m.s 2 w

(b) Interpolate to find:

(c) Calculate

ZT = f(P/POT, i ) [SOMEGA] m,s SL m, s

(Z - .294) /i

Z .4018 s

( aw\ 355

40

where

ZT Z =

s m, s

ZT

i = H aw

i e — + a

H r 1/2 M)

(d) If /3 < 0 set j = -1

It ß > 0 set j = 1 s

See subroutine QTRÄNS *"or ß calculation.

(e) Calculate the streamwise pressure gradient function:

(1+2 CPES) \ß

where

(f) Calculate J,

Input

J. =

AYEW \v

H H

AYEE äe

ZTE ZT

ZTS ZT

s 2 CPES+ (■~-)/J

CPES = ZT /ZT

e s i /H e

1 + .7176 //T+ F. F 0 Z s s nf

wall enthalpy, ft2/sec

2 O total enthalpy, ft /sec

2 y 9 edge enthalpy, ft /sec'1

edge compressibility-temperature product, 0 R

stagnation compressibility-temperature product, "R

41

SIGK a Prandtl number r

BETAS H,. streamwise pressure gradient parameter

POPSLX P/Pgj (x*) local pressure along streamline, atm

Output

XJL JT laminar pressure gradient effect correlation parameter

17) MUZERO(OMEGR, ZTR, AVER, AYEE, H, POPSLX. XMUO, XL)

MUZERO computes the reference stagnation viscositj7 ix0 and the diffusion effect parameter £.

(a) Interpolate to find:

-p - f (log10 P/PSL, ie) (TABLE 71

If

/i < 0, set L/i = 0 iD/ie<o. set^

(b) Calculate fi0 and Z

v3/2 H = u (ZT ) ^

o r r'' (t)

e

ZT •*• 200 r

(f)zv 200

I = 1 ■m'i Input

OMEGR ^r

ZTR ZTr

AVER h

reference viscosity-temperature ifatio, slug/ ft-sec-8R

reference compressibility-temperature product, "R

reference enthalpy, ft /sec^

42

AYEE 'e

H H

POPSLX Py

Output

XMVO ^c

XL £

edge enthalpy, ft2/sec2

total enthalpy, ft2/sec2

P/Pgi (x*) local pressure along streamline, atm

reference stagnation viscosity, slug/ft-sec

parameter indicating diffusion effect on heat transfer

18) QTRANS (Program A)

QTRANS calculates heat transfer data for any body shape given the external flow properties, gas properties at the wall, gas properties at the stagnation conditions, and the streamline divergence parameters. The sequence of calculations is as follows (names in brackets are subroutines):

(a) Set up x-printout array

(b) Initialize 0 = 0.5

(c) Interpolate for

i = f(T , P/PCT) w w SL

[IWALL]

(d) Interpolate to find

ZT

w.

= f(H, P/PSL) [SOMEGA]

ZT w

u. w

= f(iw, P/PSL) (SOMEGA]

(e) Calculate

u , p u , ZT , i , a [RORMUR] r rrKr r r r

43

(f) Calculate

du d(Ä/A.) e i

dx ' dx [DERV]

(g) Calculate i and EXPK from

x* ^^^i) A dx

If NS.Ü5; ,99<N<1.01; EXPK=0, i e, SL 'e

.05<N<.99: EXPK= -.194e0 , fl = (N _ n ö SL e

T^'1) - /N-l\ N > 1.01; EXPK= .194e • 0OT = -vT— Ö

oL v N ' e

(h) Obtain EL and ET [EBAR]

(i) Calculate the equivalent distance parameters defined by:

1 x x* G(x*)

1

/b \ x* x. G(x.) i-^-j + I G(x)dx

\ / Y V X. X.

x* G(x*) x.G(x.)

eq .x*-2dx

i i \ x x. ^x.

i i

i G(x)dx+ f G(x)dx+ / G(x)dx x*-2dx

where the integrand G is calculated from

and

GL^pr^Ue T

/ \2E r V / f \ L

e 'i Vfi

G,, / r \5/4 / f \(5/4>ET

f A/Ai f. r/r. i \

44

Thus, for x* = Xj + (2K)dx (INTEGER x > 1) the integration is performed by applying Simpson's rule using Gj^ or G-j as the integrand to find (bg^/x), an^ (b^/x) , respectively. eq -p

eq _ i x /x* x*G(x*) XiG^-~

X; c x*-2dx G(x)dx

dx G(x* - 2dx) + 4G(x* - dx) + G(x*)

where

/ x. i

x*-2dx K-l G(x)dx = f D G [xj + (2j - 2)dx I + 4 [x. + (2j - l)dx j

G(x. + 2jdx)

This last sum is the result of successive application of the three point integration as the calculations move along the streamline. When x* - x, -t- (2K+ l)dx an extra point is required to compute (b^/x). The calculation is made by repeating the above sequence of operations (c) through (i) with x* incremented by dx.

(j) Calculate the streamwise pressure gradient parameter /3S from the following:

ZU**) = /Hx*-2dx) + F (dS - 0 ) s s dx\ s *'/x*-2dx

where

F = 71/(70+ l/cx)

G eq

L\ x

c = 1 x

x*-dx

. / eq 'ihr.

and

K) x*-2dx

H \/ ea

L J x*-2dx

u = 0 e

mu x* e dx'

Jx*-2dx u ^ 0 e

(k) Calculate the reference Reynolds number R „: calculate JT [JAYELL] and u £ (MUZEROl rM

then

eq (V*)L ; F

x,Q

■Li7/2ofe!

eq

'L

1/3

eq (b /x) y eg /.

- ; R x / .9/16 ' r.Q L

('

eq X ;

(b /x) V eq /.

4/10

p ^1 u (x /x ) r ^r e \ eq /

2 2 ^ F o o x.Q

11 Rr,Q a transition <input) a new value of (beq/x / is calculated using the following equations:

R p u u S T ^rrr e eq, L

L, S

R, 0 M u S ,„ MrKr e eq, T

T. S

46

e .664

(RL,S) 1/2 e e L

R .0(.2135R )

e e T B.T

^eKo^.S--407) 2.64

transition

fr) =(H-

./4

transition /

transition GTdx

TR GTx*

^ ^r Q < ^transition ^nPut)' a check is made at this point for x-printout values. If a printout point does not lie in the open interval (x* - 2dx, x* + 2dx), where x* is the current value of x, the sequence (c) through (k) is repeated until one is encountered.

(1) When a printout point lies in the above Interval and when the transition point is found the desired heat transfer data is calculated using the following equations:

Calculate JL and n0£ (JAYELL] and [MUZERO]

Then

eq M, L

c \

X/L

(b /x) \ eq /I

' 4/10"

-17

x.S

(b /x) T 17/4 \ «1 /-,

1/3

L /b /x\

x,Q

(h /x)

17/20 i^__TL /b /x\ [ eq JL

1/3

Reference Reynolds numbers:

p M u fx /x\ x* r r e \ eq /.

li r,Q

eci

~2~ 2" ^ F ^ o x,Q

p u u /S /x\ x r r e \ eq L

R r.S 2^ 2

M F 0 x.S

TT_ %Fx,SJL 3/2

/S 7x1 x* ' eq 'L

185 R r,S

!loS10(Rr,S+S000) 12.534

J.3/2 . F In \i/2 o x, S\ r, S/

— - . 332 -.——7—■ - u /S /x \ x* /SQ /x\ >

Skin friction coefficients:

L 1 C, =2

1, e, L u p u e e e

T 1 i, 0, T ' u p u

e e e

where P/P

Pe=1-232-ZT SL

48

Heat transfer coefficients based on enthalpy:

3/10 „ in \l/2

HL = L o x,Q\ r.Q/

.645 Tx /x\ x*

H, g-ej 3/1% F fe L Mo x.Q

T .645/ , t . a /x /x \ x* r ( eq L

.185 R r.Q

|1Og10(Rr,Q+3000) |2.584

H. H. =

L H

H„ H

T H ,. _ ref, T

Nonisothermal wall parameters:

n V-

local \ w, , w / ^ \ w. w. I S. ♦. , = (i - i + E f * - * F

local o/ i=l \ i

where

Fs; - ^ (SL). 1 L

Fs. = 1

5*\9/10

-(^T):

1/3

1-1/9

laminar

turbulent

and

L. 3 ;* _

(SL. " 3L. , ) ' 1 1-1/

local

Örp

G dx

N ; sn dx

local i 0

49

Adiahatic wall enthalpy:

i -iff (H - i ) aw, 1. e roe'

W^e^X-e) Nonisothermal heat transfer rates:

q-'Hii , - i + * \ L LI aw, L w L, . ;

> local/

"T'

Isothermal heat transfer rates:

q JL., lOU

(in) Print out heat transfer data.

L, iso L \ aw, L w/

T.iso ""Ti'aw.T ' w)

hi order for the sequence of calculations to reach this point, at least one printout point XpQ mutt lie in the interval (x* - 2clx), (x* + 2dx) where x* - K. + (2k)dx (INTKGKK k 2 0) is the current value of x. Answers of interest are now printed using linear interpolation for those printout points satisfying x* - 2<!'x < Xpn" x*. If a printout point satisfies the criterion x*< x <fx* + 2dx) ther, all quantities are saved for the current value of x to use for the interpolation scheme at x+ dx. Steps (a) through (m) are repeated until all printout points have been exhausted.

Input

Streamline Functions

XSTAR X*

POPSLX P/PSI(x*)

TW Tw(x*)

A IKK ie(x*)

streamline coordinate at dx intervals, ft


wall temperature^ 0 R

2/„'2 edge enthalpy, ft /sec

50

THETAS ee(x*)

DODI A/A^x*)

RORI r/r^x*)

UE ue(x*)

ZTEX ZTe(x*)

OMEGA E we(x*)

Initial and Reference Cond

H H

XI XI

RTRANS transition

SIGR <Tr

DX dx

XPO xPO

KF KPO

AYEvVO ^o

BEQXLI M BEQXTI UAI Output

XSTAR X*

POPSLX P/PSL(x*)

UE ue(x*)

A/Aj A/A^x*)

streamline edge angle, radians

divergence parameters


edge compressibility-temperature product, "R

edge viscosity temperature ratio, slug/ft-sec-'R

is


initial streamline value, ft

transition Reynolds number

partial Prandtl evaluated at ZTr


printout array, ft

number of values in printout array

reference wall enthalpy, ft2/sec2 (needed only if Tw ^ constant)

initial value of b /x (laminar and turbulent)




distance between streamlines

ROM r/rj(x*)

TW Tvv(x*)

ZTEX ZTe(x*)

OMEGAE we(x*)

AIEE ie(x*)

AYEW iw(x*)

BEQXL beq, L/x(x*)

BEQXT beq/r/x(x*)

AYEAWL

AYEAWT

HL

HT

QL1SO

QTISO

QDOTL

QDOTT

CFEL

CFET

RRQ

RRS

HL(x*)/H0

HT(x*)/Href i T

4iso,T<x*)

qL(x*)

qT(x*)

Cf)C>T<x*)

R r,Q

R r.S


wall temperature, "R

edge compressibility-temperature product, 0R

edge viscosity temperaturs ratio, slug/ft-sec-0R


wall enthalpy, f^/sec"

ratio of laminar equivalent distance parameter to distance x along streamline

ratio of turbulent equivalent distance parameter to distance x along streamline

2 / 9 laminar adiabatic wall enthalpy, ft /sec*

2 2 turbulent adiabatic wall enthalpy, ft /sec

laminar heat transfer coefficient ratio, lbm/ft2/sec

turbulent heat transfer coefficient ratio, lbm/ft2/sec

laminar isothermal heat transfer rate, Btu/ft2-sec

turbulent isothermal heat transfer rate, Btu/ft2-sec

laminar heat transfer rate, Btu/ft2-s8C

turbulent heat transfer rate, Btu/ft -sec

laminar skin friction coefficient

turbulent skin friction coefficient

heat transfer reference Reynolds number

skin friction reference Reynolds number

19) REF

REF cucut&tes the stagnation heat transfer coefficient based on a reference hemisphere of radius R*, and the reference heat transfer coefficient based on a 60° infinite swept cylinder given the reference cylinder radius. The following procedure is used to calculate the stagnation conditions:

(a) Initialize ie = H

(b) Obtain ^he following from the indicated subroutines:

iw [IWALL] o

ZT , ZT , ZT , o , a; . u) (SOMEGA) w e s w e s

w . p u , ZT i , <T (RORMUR] r r r r r r

J [JAYELL]

ti0, £. [MUZERO]

(c) Calculate the stagnation heat transfer coefficient and heat transfer rate:

1/2 _ .664g^

o .645 a r

■ RHEMI, ref .

\-\V-^ ) a

The reference conditions are calculated from the following:

(d) Reinitialize MN = 1/2 M^

(e) Obtain the following from the indicated subroutine:

P^T/P«' W' U^V, [STACON] ST " ST MN

Hv.ref tlWALLl

ZT , ZT , ZT , w , a) , w [SOMEGA] w e s w e s

,").')

UJ , p n , ZT . i , a r r r r r r

IROUMUR]

rC'ro

IF. BAH]

v^ (MUZEROl

(f) Calculate the following:

•86tiRCYL.rof

«I.L~ (i+ rc)u^N

x,Q

i+ r 1.6

1 + .7625 r O' -i

1/3 i+ r

i+ r

R r. Q, ref

.866 p u u x r r <*> eq, L

Mo x,Q

H .332gJ£ ''o x,Q

ref, L . 64o x a eq, L r

, " {Rr,Q,ref) 1/2

H 185 gX 'io x,Q

R r,Q, ref

ref.T .645 x a\ eq,L log10(Hr,Q,ref+3000)l

2.584

aw, ref, L = H

1/2 (■-^("-'e)

1/3' aw, ref, T

= >,-(!-,-)(„-g

ref, L ref, L aw, L w, ref

Vf, T Href, T (law. T " w, re^

.866 u a .645

ref, L

ref.T

z£ H ref.'

.866 u a oo r

.645

g^ ref.T

ref "f.C ref q^

Input

PINF P«

ACH Moo

VEL Uoo

VELP UMN

H H

POPSL Po/PSL

TWREF T w, ref

RHMREF RHEMI.ref

RCLREF RCYL, ref

Output

AYEWO Hv 0

HO Ho

QDOTO %

HREFL

HREFT

H ref, L

K ref.T

free stream pressure, lb/ft2



modified Newtonian velocity gradient, 1/sec



reference wall temperature, *R

reference hemisphere radius, ft

reference cylinder radius, ft

2 / 2 stagnation wall enthalpy, ft /sec

stagnation heat transfer coefficient, lbni/ft

2-sec

stagnation heat transfer rate, Rtu/ft -sec

laminar z'eference heat transfer coefficient, lbm/ft2-sec

turbulent reference heat transfer coefficient, lhm/ff-sec

AYAWUI. iaVv,reif,,

AYAW,lT W.ref/F

TAl'ilL Tref ^

TAURT Tref/r

QRKFL flref. L

QRKFT qret.T

CFREFL

CPREFT

%e, ref, 1,

^f, (:%ref,T

AYKREF iW(re{

laminnr reference adiariatic wall enthal|)y, ll-/sec~

turlnilent reference adiabatic wall enthalpy, ft2/sec2

laminar reference shear stress, lb/ft

turbulent reference shear stress, lb/ft

laminar reference heat transfer rate, lbm/ft -sec

turbulent reference heat transfer rate, lbm/ft -sec

laminar reference skin friction coefficient

turbulent reference skin friction coefficient

reference wall enthalpy, ft /sec

20) RollMUH

UOltMUR calculates the following reference properties:

(a) Viscosity parameter

ws K* =

1/14 1,005

W. .005+--

0J

K - (K¥)

13 3 ,_ -i

10 Ifi

U). F. - K

u).

a; -

vv

p7/8

ww F 1 1.2

1/10

.2-* F,

ÖC,

(b) DensHy-viscotity product

^ - fc ^

^r r r r R P„M_ = P

(c) Compressibility-temperature product ZTr

Change units of (jjr to slug/ft-sec-'K

u) - 1.8 w r,'K r,'R

For3.585 x 10"10 < w <8.03xl0'10

r

23213.13 x 10"20 T /, Ä„0„^ 1A18 2 \l/2"|2 ZTr,-K= 2 [l+(l -.478656x10 W ) J

Otherwise ZT »LT is obtained from interpolation

ZVK = f(lOg10P'Wrx) fTABL19]

ZTr "K is converted to "R by ZTr ,R = 1.8 ZTr .K

(d) Enthalpy ir

i r

RT o

f(log10P. ZT^^) [TABL201

/ I

i = 847559.8 ' r

r IRT \ 0)

(e) Prandtl number a.

a = f(ZT ) (TABL 14) r r

Input

POPSIX p/psi

OMKGS ws

OMEGE we

OMEGW ^w

Output

OMEGR ^r

ROMUR P M

ZTR ZTr

AYER V

SIGR ^^


stagnation viscosity-temperature ratio, slug-ft-sec-'R

edge viscosity-temperature ratio, slug/ft-sec-"R

wall viscosity-temperature ratio, slug/ft-sec-'R

reference viscosity-temperature ratio, slug/ft-sec-0R

2 M 4 reference density-viscosity product, slug /ft4-sec

reference compressibility-temperature product, "R

9 9 reference enthalpy, ft /sec

reference Prandtl number

21) SiRCH (Y, X, YA, XA, N, IO)

SIRCH uses TAB to perform single interpolation on a table of y = f (x).

Input

X

YA

XA

N

IO

Output


dependent array

independent array

number of values in XA and YA arrays

order of interpolation

dependent variable

22) 30MEGA (ENTH, 0OPSLX, ZT, OMEGA)

SOMEGA calculates the viscosity-temperature ratio u; and the compressibility- temperature product ZT as functions of enthalpy and pressure.

(a) Convert enthalpy to Btu/lb m

i(Btu/lb ) = 25301 rf/sec2) x m'

(b) Calculate ZT

For i < 180

For i > 180

ZT = i/.432

ZT = f(log10 P/PSL, i) [TABLE2]

Calculate u

ZT < 111 CD = 14.454 x 10 ■10

m<ZTs20„„ w=(i-T)^(_i|«4__)M.45x 10 -10

ZT > 2000 w - f(log10 P/PSL, ZT) (TABL18]

(c) Convert ZT and w to "R

ZTtR = 1.8 ZT.K

u slug w ft-sec-'R 1.8

Input

ENTH

POPSLX P/P SL

enthalpy


.)!.•

( )UtiJUt

ZT ZT

OMEGA w

23) SONKN'T (EXTH, SONIC)

compressibility-temperature product, 'R

viscosity-temperature ratio, slug/ft-sec- "R

SOXENT calculates the speed of sound as a function of free stream enthalpy. The result is obtained by linear interpolation on tabular arrays constructed from the equation:

f>RT„ ,1/2

where

T« = f(U/RTrt)

T«, = MT«,)

Input

ENTH

Output

2 / 2 free stream enthalpy, ft /sec

SONIC a^ speed of sound, ft/sec

24) STACON (PSTPNF, AYEST, VELP, ACH, ACHN, H, PINE, VEL)

STACON calculates the stagnation-free stream pressure ratio, the stagnation enthalpy, and the modified Newtonian pressure gradient.

(a) Stagnation enthalpy

iorr = H - -— ST 2

'M 1 -

N M

(b) Pressure ratio P0/P

P

(H^1-2^3'5^- 2.5

MN21

no

3.5

("ST)

M. 1 +

X

o SL MN < 1

P2 GMN P 2

1 -MN ' 5

Interpolate for

(ZTST) - f(PST ' W) ^ 'o v o /

fSOMEGA]

(psT)r^:+K ^ \ \pi PSL/L

6006 (ZTST)

ST 2(P2/P1)- 1

Interpolate for

Krj/^srW [SOMECAl

(p-)2=fej k - ^ 2(P2/P1)

6006 ZTST

'ST 2(P2/P1) - 1

p (PST) o _ ' 'o P (P /P )

(c) Modified Newtonian pressure gradient

MN "?'- ' P

"KVC'V

6006 IZT (ZTST

ST

1/2

Interpolate to find the empirical correction factor to the modified Newtonian velocity gradientas a function of M^cos A (Swept cylinder) or M«, (Hemisphere).

UMN

= KIN) ' (CORRECTION FACTOR)^

For the correction factor jee Figure 4.

61

c v E

I x 0)

X

1.14

1.12

1.10

c

I 1 -08 v Z

'x ^ 1.06

1.04

1.02

1.00

—

!

\ ! \ !

0.5 1.0 1.5 2.0 2.5

M^ or M^, cos A

Figure 4: CORRECTION FACTOR FOR THE VELOCITY GRADIENT AT A HEMISPHERE STAGNATION POINT OR CYLINDER STAGNATION LINE

(i2

Input

ACH M«,

ACHN MN

H H

PINT PcC

VEL U=o

Output

PSTPNF PoK AYEST 'ST

VELP ui«v


normal Mach number

total enthalpy

free stream pressure, Ib/ft^


stagnation to free stream pressure ratio

stagnation enthalpy, ft /sec4'

modified Newtonian velocity gradient, l/sec

25) STREAM (Program C)

STREAM calculates the streamline coordinate x as a function of the geometry coordinates ^ and TJ . A complete description of the geometry and streamline pattern are provided by the coordinates ^ and TJ, 0e(£,»)), ^maxd) and ^max^*

Öe(^,Tj) must be provided for the grid of ^ and TJ that lies within the geometry boundaries. With ömax(4) known along the upper TJ geometry boundary the complete streamline pattern on the body is defined. V^^^) specifies the upper n boundary thus defining the geometry.

The coordinates (^f, Tf) determine a point on a particular streamline. The streamline calculation begins at (If, Hf) and proceeds upstream until the geometry boundary is reached. The calculation of the streamline is done by the following procedure:

Let (4X)1 - Ij, (Hx) - l^ and (x)] = 0. Linear interpolation is performed on 6e{i,V) and ömax(4) if necessary, to determine (ox).. Then

(d\)1 = dxsin{ex)1

(^K)1 = dxcos(9x}1

G3

Then the coordinates of the new point are:

Continuing this procedure, the general equations are.

H.rf[MK»Kj /d 4 \ = dx sin /eJ \ ^ Xk-1 [ Xk-1

(dr) ] = dxcos (9 ) \ X/K-I \ X/K-I

and

K-l

(^)K = v - s {^ K-l

(\)K ' "F - S (<■.,). v 'K i=l 'i

K-l (X)K= Z d(Xx).

1=1 1

The streamline calculation is completed by rearranging the 4Xi ^x and x arrays such that 4 x' Iv and x are measured from the upstream end of the streamline.

The subroutine STREAM assumes zero streamline divergence due to body geometry

I- =1.0 r.

1

64

Input

M

N

XSIF «f

ETAF "f

DX dx

XSI 1

ETA T

ETA MAX :'r

TH8MX «r

THETAE 9e

Output

II

XS1X

ET AX

X

THETAX

XF

max

max

^x

ex

numl)er of 4 coordinates

number of r? coordinates

final streamline ^ coordinate, ft

final streamline of n coordinate, ft

incremental streamline distance, ft



body edge coordinate, ft

streamline angle at body edge, degrees


number of points calculated along streamline





final x-coordinate on streamline, ft

26) TABLE2 (AYES, PEDGFL, TS, CHECK)

TABLE2 obtains by table interpolation the compressibility-temperature product as a function of pressure and enthalpy.

Input

AYES i

PPLOG log10 P/PSL

enthalpy, Btu/lb m

log of ratio of pressure to sea level pressure, pressure in atm

Output

TS ZT

CHECK

compressibility-temperature product, 'R

program check to determine success of interpolation (if the interpolation is not successful, sn error statement is printed)

27) TABLE« (TVV, PPLOG. AYEW)

TABLE6 interpolates to find wall enthalpy as a function of pressure and wall temperature tables.

Input

TW

PPLOG log10 P/PSL

wall temperature, 'K


Output

AYEW Aw wall enthalpy, Btu/lb m

28) TABLE? (AYEE; PPLOG, AYEAYE)

TABLET interpolates on a table of pressure and edge enthalpy to find the dissociation on reaction enthalpy ratio io/ig.

Input

AYEE

PPLOG log10 P/PSL

Output

AYEAYE iD/i



dissociation reaction enthalpy ratio

29) TABL14 (ZTW, SIGT)

TABL14 obtains by table interpolation the partial Prandtl number as a function of the compressibility-temperature product.

fifi

Input

ZTW ZT compressibility-temperature product. °\l

Output

SIGT Prandtl number

30) TABL18

TABLls determines by table interpolation the viscosity-temperature ratio as a function of the pressure and compressibility-temperature product.

Input

ZT ZT

PPLOG logJ0 P/PSI

compressibility-temperature product. 'K

loi? of ratio of pressure to sea level pressure, pressure in atm

Output

OMKGA

CHECK

viscosity-temperature ratio, slug/ft-sec-'K

program check to determine success of interpolation (if interpolation is unsuccessful, an error comment is printed)

31) TABL19

TARU9 performs table interpolation to determine the compressibilitv-temperature product as a function of pressure and viscosity-temperature ratio. The table is composed of data in TABLED and TABUS.

Input

OMEGA u)

fPLOG log10 P/PsL

viscosity-temperature ratio, siun/ft-sec - "K

log of ratio of pressure to sea love! pressure, pressure in atm

Output

ZT ZT compressibility-temperature product, "K

(IT

32) TABL20

TABL20 performs table interpolation to find i/RT0 as a function of pressure and compressibility-temperature product.

Input

ZT ZT

PPLOG log10 P/PSL

compressibility-temperature product, 0 K


Output

dimensionless enthalpy AYERTO i/RT0

33) TBLP (XT, YT, X, NTAB, N)

TBLP is a linear single interpolation routine. To save search time in finding the location of the independent x value in the x table, the search is begun at the previously loca.ed x value.

Input

XT

YT

X

NTAB

N


dependent variable table


number of values in the x table

location in x table at which the search is begun. This value should be set = 1 upon the first entry to this routine. Thereafter, it stores the location of the previous search.

08

s

3. INPUT-OUTPUT DESCRIPTION


Data are input to this program via punched cards. The purpose of this section is to define the required input and the form it takes on the cards. Input sheets for each option or section of the program are shown. Use of this type of form greatly reduces the amount of effort recjuired to obtain results from the program. Data may be key punched directly from the forms.

1) AXISYMMETRIC OR TWO-DIMENSIONAL BODIES (Option D-l)

Card Columns Format Description

| i 1-6 A6 Type of case: FLIGHT, WTl. or \VT2. I This card indicates the type of free-stream | condi'ions to be input.

f 1 7-10 Leave blank.

1 11-72 10AG Title or some description of case.

Cai'd -2 will consist of only one of the following types oi input: Flight. Wind Tunnel 1 (WTl), or Wind Tunnel 2 (WT2). Wind Tunnel 1 case should be used for tunnels operating in the ideal gas region, while Wind Tunnel 2 case should be used tin- high energy tunnels such as shock tubes.

FLIGHT

2 1-10 F10 Flight altitude, ft.

2 11-20 CIO Flight velocity, ft/see.

WTl

2 1-10 F10 Free-stream M;ich number.

2 11-20 F10 Free-stream static temperature. H.

2 21-;50 F10 Free-stream static pressure. lb/!'t-.

WT2

2 1-10 F10 Free-stream velocity. It/see.

2 11-20 F10 Total enthalpy, ir,.secJ.

(I!)

2

3

3

4

21-30

41-50

1-10

11-20

21-30

1-14

F10 Ratio of stagnation pressure to sea level pressure.

2 F10 Free-stream d^amic pressure, lb/ft .

« F10 Radius of 60° swept infinite cylinder for

the turbulent reference heating rate, ft.

F10 Radius of hemisphere for the laminar reference heating rate, ft.

F10 Reference wall temperature, " R.

A6 Contains the word AXISYMMETRIC or TWODIMENSIONAL.

1-5

6

7

15 M is the number of values in the geometry tables y(4) and § . 3 < M < 50.

8F10 This table contains M values of ^ , ft.

8F10 This table contains M values of y(^) corresponding to 5 in the preceding table, ft.

Note: A minimum of three points must be input into the preceding two tables. If the body is very slender, more points should be used.

1-10 F10

8 11-20 F10

8 21-30 F10

9 1-5 15

Initial x location, point at which calculation begins, ft.

Increment in x, ft.dx > (x. - x )/750, i o

Edge velocity at x., ft/sec.

ITC is the number of values in the wall temperature versus surface distance tables. 2^ITC ^50. If ITC 2*3 then the wall temperature must be nonisothermal.

70

10 8F10 This table contains ITC values of surface distance, ft.

11 8F10 This table contains ITC values of wall or surface temperature corresponding to the preceding distance table, R. If T = constant then enter two values, w

12 1-5 15 KpQ is the number of printout locations desired. KpQ < 500.

12 11-20 F10 Transition Reynolds number, see definition of Rr)Q in Program A, labeled QTRANS in section 2. b, 18.

13 8F10 This table contains KpQ locations at which printout is desired.

71

s

n (/) Z o H

z. o

o 7

1- o o

8 z

K Ul (O U.

£ _, _.

L LU 0. > I-

8 UJ a: a. U P < ft

a.'

u

d z i u < 5

—

8 HI o: 0. o i < z >- □

8 Ul

0. o

<

_l

a.0

>

UJ

>- t-

Ö O -J

>

1

I

ü u.

s H

5 ai X

ir

_i

> u

UJ z O h O

i oc u

<0

-I

< o

Z o

c y- M 0. UJ Z z ? Ul

S S > n CO

X < 1

u—— -.„■■,.- _«

72

o 00

oT

x V

-

in Ul D -i < > O

UJ _i CD

< t-

ä

< > o o.

I

2) HEMISPHERE CALCULATION (Option D-2)

Card Columns Format

2

3

4

10

Same as Option D-l

1-10

1-10

1-5

1-5

11-20

A6

F10

15

Description

Contains title of program - HEMISPHERE.

Hemisphere radius, ft.

ITC is the number of points in the Tw

versus x tables. 2<ITC£50.

If ITC > 3 then the wall temperature must be nonisothermal.

8F10 This is a table of ITC values of x where T... are input, ft.

8F10 This is a table of ITC values of Tw

which correspond to locations given in the preceding x table, or if Tw = constant, enter two values of Tw-

15 KpQ is the number of printout locations desired. KpQ < 500.

F10 Transition Reynolds number, see definition of Rr,Q in Program A, labeled QTRANS in section 2. b. 18.

8F10 This table contains Kpg surface distance locations around the hemisphere at which printout is desired.

74

§

c

f-

5

81

^s

.1 UJ D K

<

to 2 5 <r a

i a.

UJ I

5 a o

u h-

-j L '■ 5

tu I

rrt~i

j i

L_ L

s

to UJ

-I < > 2

in -i ca <

J o a.

y

76

3) SWEPT INFINITE CYLINDER (Option D-3)

Card Cuiumns Format Description

Same as Option 0-1

1-8 A6 This card contains the word CYLINDER to allow the program to select the D-;5 options.

1-10 F10 Cylinder radius, ft.

11-20 F10 Cylinder sweep angle, measured Irom a line normal to the flow, 0 < A < 90 .

1-10 F10 |r is the abscissa at which the streamline calculation begins, ft. £j- must be large enough so that the streamline has very nearly zero slope at the sta^naticn line. Cases have been run by the authors using ^f ^ 10 and 20 ft for A 10D and GO .

11-20 F10 Tjj- is the ordi ate at which the streamline calculation begins, ft.

21-30 F10 dx is the increment in distance along the streamline used in the boundary layer calculations (Program A), ft.

o ■> 1 ''2 dx>(|f- t nf) 'f;oo

1-5 15 ITC is the number of values in Tv, and ij tables. 2 < ITC < 50. 11 ITC > .'! the wail temperature is nonisolhcrmal.

8F10 This table contains ITC values of 77. where T) is the distance around the cylinder, ft.

8F10 This table contains ITC values of Tw

corresponding to the -q table/H.

10 1-5 15 Kpo is the number of x printouts desired along the streamline. KpQ < 500.

10 11-20 F10 Transition Reynolds number, see definition of Rr,Q in Program A, labeled QTRANS in section 2. b. 18.

10 21-30 F10 Arjf is an increment used in Program C to locate the starting location of second streamline, ft.

If Arjf is small, in some cases the resolution between streamlines near the streamlines origin or apex may be very poor. The poor resolution leads to heat transfer distributions that are not smooth. In the event this occurs Arif should be increased, ft.

11 8F10 XpQ is a table containing locations along the streamline at which printout is desired.

78

Hi

5

J i-

s 5

8 cr

o a

0 I

8

j

I a

,8

Ui O

P <

5

1 I

a:

-) >- o

a: za

u

H

<c P*

g

i

Ul

-I < > O

ul -I o < 1- i

t-

<

I1 3 CD <

I1

I il

80

4) SHARP DELTA WING (Option D-4)

Cards Columns Format Description

Same as Option D-l

4 1-6 A-6 Contains the word DELTA.

5 1-5 15 ITC is an indicator for wall temperature gradient. If the wall is at constant temperature input a 1 and one value of wall temperature in table.

6 1-10 FIG Angle of attack, degrees.

6 11-20 F10 Wing sweep angle measured from a line normal to the flow, degrees.

6 21-30 F10 4 f is the final or end coordinate along the streamline at which heating data is desired, ft.

6 31-40 F10 T]J is the final or end coordinate along the streamline at which heating data is desired, ft.

6 41-50 F10 xj is the point along the streamline at which the calculations begin and may have to be greater than zero as indicated in Figure 5. The resulting change in streamline curva- ture, due to interpolation, near the apex produces a discontinuity in the heating calculations, ft.

7 1-10 F10 A^ is the increment in | which the program uses to setup an (£.i?) array of streamline angles, velocity, and pressures, ft.

^f

si

■Region of interest

.0125

Tj, feet

■Points in o two-dimensional array of streamline angles

090

^ , feet

Centerline - Streamline I

.135

Figure 5: STREAMLINES NEAR THE APEX OF A SHARP DELTA WING

82

7 11-20 F10 ATJ is the increment in n the program uses to setup an (£,T)) array of streamline angles, velocity and pressures, ft.

Ari > - 50

7 21-30 F10 dx is the increment in distance along the streamline at which calculations will be performed, ft. dx > (Xj- - xj)/750.

Note: If l\v is a constant value, enter ope value into card 11, and delete cards S, 9, and 10.

8 1-5 15 M is the number of points in the 4 array. 2 < M < CO.

8 6-10 15 N is the number of points in the n array. 2 < N < 50.

9 8F10 This is a table of M values of i where wall temperature will be input, if ITC > 1.

10 8F10 This is a table of X values of TJ where wall temperature will be input if ITC > 1.

11 8F10 This is a table containing (X) • (M) values of wall temperature, input so i varies with each T) .

12 1-5 15 KpQ is the number of x printouts desired along the streamline. Kp()< 500.

12 11-20 F10 Transition Reynolds number, svv definition of Rr.Q In Program A, labeled QTRANS in section 2. b. IS.

12 21-30 F10 See card 10, svept infinite cylinder. option D-o.

13 8F10 Xpo is a table of Kpo locations along the streamline where printout is desired.

o as

I UJ a. >

8 a.

s 5

—

8 a

8 a.

eT o a.

0 I

8 3

i

I o -I

8 3

IU Q 3 H H _l <

tu i

o GC

u in 1

< H -1 UI O D K

84

t3

^

My 0

■

1 ! 1 j

i ! i |

! | |

—i—i

! i

1

- to UJ 3

5

I

{

i -1 EC

UJ 1

> |

UJ!

CD !

< 1

:l I

o to

0)

< > O a.

UJ _l a) <

PJ

<1

I O a.

86

5) STREAMUNE CALCULATIONS (PROGRAM C)


Same as Option D-l

1-6

1-5

A6

15

Must contain the word STREAM.

M is the number of points in the array £. 2 < M < 50.

6

6

6

6-10

11-1;

1-10

11-20

21-30

31-40

41-50

15

15

F10

F10

F10

F10

F10

8F10

N is the number of points in the array n 2< X <50.

ITC is an indicator used to test whether the wall temperature is a constant or a variable. If T^. is a constant ITC = 1 and enter one value of T into card 16 and delete cards 13. 14. and 15.

4f is 5 at the final or end point, ft.

n£ is T) at the final or end point, ft.

dx is the increment in distance along the streamline, ft.

1/2

dx >

/. 2 'i iv ^ V2)

750

Xr is the point along the streamline at which the calculations will begin, ft,

ue xT is the edge velocity at the initial or start of calculations, ft/sec.

This is a table of M values of 4 2 < M < 50. 9e, P/P0 and T^, ui input as functions of ^ and r\ . ft.

87

8 8F10 This is a table of N values of TJ

2 < M < 50. Ge, P/P0 and Tw will be input as functions of | and TJ , ft.

9 8F10 This is a table of M values of Tlmax as a function of {. rtmSül defines the upper or outer boundary of the body, ft.

10 8F10 This is a table of M values of the streamline angle 0max along the boundary as a function of ^, deg.

11 8F10 This is a table of (M) . (N) values of the local streamline angle 6e and should be input along lines of constant TJ and { . deg.

12 8F10 This is a table of (M) • (N) values the local pressure divided by the stagnation pressure and should be input along lines of constant TJ and varying {•

13 1-5 15 MT is the number of | values entered into the table Tw = f(4,Tj). Also see ITC above.

13 6-10 15 NT is the number of TJ values entered into the table Tw = f({ ,TJ). Also see ITC above.

14 8F10 This is a table of MT values of £ at which Tw will be input. Also see ITC above.

15 8F10 This is a table of NT values of n at which Tw will be input. Also see ITC above.

16 8F10 This is a table of (MT) • (NT) values of Tw as a function £ and TJ , 0 R. Tw

should be input as a function of 4 at constant TJ .

17 1-5 15 KpQ is the number of x locations along the streamline at which printout is desired. Kp0 < 500.

88


17 11-20 F10 Transition Reynolds numbfr. see definition of Hr.Q in Program A. labeled QTHANS in section 2. b. 18.

17 21-;i0 F10 Sec card 10. Swept infinite cylimler. option D-;}.

18 1-10 F10 xpo is a table of Kpo locations along the streamline at which printout is desired.

M)

I <o g

0-

8 a

8 a.

-J

■— o a.

o I

8

I o -I u.

6

LU Q D h- h-

<

_J

>- ()

(C 5

oc o

< LU

a: a. z

in

2

in

5

• >

•-• X

X ■D

tr

—

«/>

in ut 3

< >

-I

<

90

o oo

Ü3

< >

OJ

< >

«5 UJ

-I < >

3 < >

<

Ei

< I I

CD

< CO

< H a.0

a.

2

!»I

s

•Si UJ 3 < > 5 j -i CO

<

05 Ui D -I < > Z UJ _l OQ

 o

o

X

6) REFERENCE CALCULATIONS (PROGRAM REF)

Card Columns Format Descriptor.

Same as Option D-l

1-10 A6 Must contain the word REFERENCE.

o CD

I

8 a.

8 5

—

8

a?

o 0.

I

8 3

I- I ?

1-

i UJ I

a:

_i >- u

K

N LU () a 7

ü UJ cc (T a. UJ 2 u.

Ul tr

!)-l

7) BOUNDARY LAYER CALCULATIONS (PROGRAM B)


Same as Option D-l

I 4 1-6 A6 Must contain the word FLOW.

; 5 1-5 15 IS is the number of values in the P/PQ,

f ee, &/&\ and r/rj tables as a function of | streamline distance. 2 < IS < 50.

6 8F10 This is a table containing IS number of x locations, ft, along the streamline at which P/P0, öe, A/Aj and r/rj will be input.

7 8F10 This is a table containing IS values of local pressure each normalized by the model or vehicle stagnation pressure. Tho P/P0 correspond to the above x table.

8 8F10 This table contains IS values of the local streamline angle relative to the axis of symmetry, degrees. The values of 9e

correspond to the above x table, degrees.

9 8F10 This table contains IS values of the streamline divergence parameter A/A^. The values of A/A^ correspond to the above x table.

10 — 8F10 This table contains IS values of the body shock geometry parameter r/q . The values of r/r^ correspond to the above x table.

11 1-10 F10 xj is point at which the calculation will begin, ft.

tjf)

Card Column Format Description

11 11-20 F10 dx is the increment in distance alon^ the streamline, ft

XF!NAL"XI dx > -

750

11 21-.10 F10 uex is the edge velocity a ft sec.

12 1-5 15 ITC is the number of values in the Tw

versus x table. 2 < ITC < 50.

If Tw = constant then ITC = 2.

13 8F10 This is a table of xlocations (ITC) along the streamline where Tw will be input.

14 8F10 This is a table of ITC values of Tw along the streamline corresponding to the x table. If ITC = 2 enter only two values of TW, 'R.

lo 1-5 15 Kpy is the number of x locations along the streamline at which printout is desired. Kp() < 500.

15 11-20 F10 Transition Reynolds number, see definition of Rr Q in Program A. labeled QTRANS in section 2. b. 18.

16 8F1Ü This is a table of KpC) locations along the streamline where printout is desired.

Of)

g

I UJ a.

s 5

1—

8 a

8 a.

_i

0 a.

I

8 3

1

I i

i UJ I

_1 >- o

a: !D

* if) r~i 0 o rr 1 0. -1

u. CO

V) UJ D -I < > CO

UJ -1 m <

< a. OL

s

<]

X ■o

in Ul D -1 < >

Ul J to <

c

JJ

< > O a.

I .°l

98

8) HEAT TRANSFER CALCULATIONS (PROGRAM A)


Same as Option D-l

1-6 A6 Must contain the word QTRAXS.

1-5 15 IS is the number of values in each of the following tables: x. P/PgL' ue- "c A/Aj and r/rj. 2 < IS < 50.

1-10 F10 Xj is the initial x location a^ong the streamline at which the calculations will start, ft.

6 11-20 FlO dx is the increment in distance along the streamline, (ft.)

XFINAL'

XI

dx> — IOO

6 21-30 F10 ue.xT is the boundary layer edge velocity

atxj, ft/sec.

7 8F10 This is a table containing x locations (IS) along the streamline at which P/Pg^. ue . 8e , A/A, and r/q will be specified, ft.

8 8F10 This is a table containing IS values of the local pressure normalized with the sea level pressure, and at locations corresponding to the x table.

9 8F10 This is a table containing IS values of the boundary layer edge velocity corresponding to locations specifiec in the x table, ft/sec.

10 8F10 This is a table containing IS values of the local streamline angle 6C relative to the axis of symmetry. The angles correspond to x locations specified in the x table, deg.

99

11 8F10 This is a table containing IS values of the streamline divergence parameter A/A.- and correspond to x locations specified in the x table.

12 8F10 This is a table containing IS values of the body-shock parameter r/r. and correspond to x locations specified in the x table.

13 1-5 15 ITC is the number of values in the T... versus x table.

2 < ITC < 50 If T is a constant, ITC =• 2. Input two values of Tw into card 15.

14 8F10 This is a table containing ITC values of x along the streamline, ft.

15 8F10 This is a table of ITC values of Tw along the streamline and corresponding to the x locations in the preceding table. (° R)

16

16

1-5

11-20

15 ^po ^s ^e num^er 0f x printout locations desired along the streamline. KpQ < 500.

F10 Transition Reynolds number, see definition of Rr,Q in Program A, labeled QTRANS in section 2. b. 18.

16 21-30 F10 BEQL is the ratio of the laminar equivalent distance parameter to the distance along the streamline. If xj > 0 then a value of BEQL is required. If xj = 0 input BEQL=0,

16 31-40 F10 BEQT is the turbulent equivalent distance parameter. See BEQL above.

17 8F10 This is a table of Kpo locations along the streamline where printout is desired.

100

8

J ID 0. > (- u

I O

s 3

UJ C D

_1 <

2 tu X

a:

-j >- u (0 v> v. o < o X

tr H o. o £

101

o CO

■

—_

102

o 00

o I-

< >

to 111 D _) < > o

tu _J CO <

W -1 CD <

K

—T~~\—

I

J

>

c 0 M c

!■

1 in

o Q.

103

b. Output Description

INITIAL CONDITIONS

VELOCITY

MACH

ENTHALPY

TINF

PINF

QINF

PO/PSL

PO/PINF



2 , 2 free stream enthalpy, ft /sec

free stream static temperature, 0R

2 free stream static pressure, lb/ft

free stream dynamic pressure, lb/ft

(model total pressure)/(sea level static pressure)

(model total pressure)/(free stream static prepsure)

REFERENCE CONDITIONS

HO

.24HO

QDOTO

HREF.L

RHEM,REF

RCYL, REF

SWEEP

.24HREF,L

QDOT.REF.I,

^k/âw'Sv) = ^c/cp where ô is the hemisphere stagnation point heat transfer coefficient, lbm/fr-sec

. 24 ho/Cp , Btu/ft2-sec-0 F.

2 hemisphere stagnation point heating rate, Btu/ft -sec

^REF.l/âvv.L" Hv,REF) = hREF,L/CP where hREFsL is the laminar heat transfer coefficient based on enthalpy at the stagnation line of a GO" swept infinite cylinder, lbm/ fr-sec

hemisphere radius, ft

cylinder radius, ft

sweep angle of the swept cylinder used for the turbulent reference condition, degrees

•24 bREF,l/CP

laminar heating rate at the stagnation line of a 60° swept infinite cylinder, Btu/fr-sec

104

HREF.T

.24HREF.T

QDOT.REF.T

LAW, REF, L

IAW,REF,T

I\V, REF

TAUREF, L

CFINF.REF.L

IW.O

TAUREF, T

CFINF,REF,T

TW.REF

^REF.TAiaw.T " V.REF) = hREF,T/cP where hREF, T is

the turbulent heat transfer coefficient at the stagnation line of a 60° swept infinite cylinder based on enthalpy, lbm/fr-sec

.24 hREF)T/Cp

turbulent heating rate at the stagnation line of a 60; swept infinite cylinder, Btu/fr-sec-c R

laminar adiabatic wall enthalpy on a 60° swept infinite cylinder, ft^/sec^

turbulent adiabatic wall enthalpy on a 60° swept infinite cylinder, ft^/sec2

reference wall enthalpy, fr/sec

laminar shear stress on a 60c swept infinite cylinder, Ibj/ft2

2Tref l/(P'ou'*)' laininar skin friction coefficient based on free stream conditions

stagnation wall enthalpy, fr/sec2

turbulent shear stress on a 60° swept infinite cylinder. Ibj/ft2

2Tref T^^«11«'" k^bulent skin friction coefficient based on free stream conditions

wall temperature for 60" swept infinite cylinder heating calculations, 0R

GEOMETRIC PARAMETERS

SWEEP CYL cylinder sweep angle, degrees

SWEEP DELTA WING delta wing sweep angle, degrees

ALPHA angle of attack, degrees

RCYL cylinder radius, ft

RHEIvII hemisphere radius, ft

10.1

STREAM UXE

XSI

ETA

THETAE

coordinate, ft

coordinate, ft


BOUNDARY LAYER CALCULATIONS

X

ETA

XSI

BEQXL

BEQX~

POPSL

AYEAWL

A YE AWT

UE

HL

HT

DODI

N

QLISO

QTISO

RORI

?treamline distance, ft

ordinate of coordinate system, ft

abscissa of coordinate system, ft

(b /x)^. laminar equivalent distance parameter

(beq/x)^ turbulent equivalent distance parameter

local pressure in atmospheres

2 2 iaw L, adiabatic wall enthalpy in laminar flow, ft /sec

2 9

law j, adiabatic wall enthalpy in turbulent flow, ft /sec"


laminar heat transfer coefficient divided by the reference hemisphere stagnation point heat transfer coefficient

turbulent heat transfer coefficient divided by the reference turbulent heat transfer coefficient at the stagnation line of a 60° swept infinite cylinder

A/A^, total streamline divergence

x/^A/Ai) (d(A/Ai)/dx), streamline divergence parameter

laminar isothermal heat transfer rate, Btu/ftr-sec-0R

turbulent isothermal heat transfer rate, Btu/ftr-sec-" R

r/r., streamline divergence due to body-shock geometry

10()

QDOTL

QDOTT

TW

CFEL

CFET

ZTEX

RRQ

RRS

OMEGAE

AIEE

AYEW

THETA

iMUO

ZTR

EBARL

EBART

JL

XEQL

FXQ

FXS

OMEGAR

laminar heating rate, Btu/ft^-sec- R

9 turbulent heating rate, Btu/ff-sec-^ R

wall temperature. R

laminar skin friction coefficient \ based on dynamic > pressure at boundary

turbulent skin friction coefficient ; layer edge

ZT , compressibility factor times the edge temperature, "R

heat transfer reference Reynolds number

skin friction reference Reynolds number

2 edge viscosity divided by ZT . lbf-sec/ft -' R

2 2 i . edge enthalpy, it"/sec

0 2 1, wall enthalpy, ff/sec" w

Q . local streamline angle, degrees

ß , absolute viscosity at the stagnation reference condition. lbf-sec/fr

compressibility factor-temperature product evaluated at the reference enthalpy, l H

laminar erossflow pressure gradient parameter

turbulent erossflow pressure gradient parameter

laminar strcamwisc pressure gradient profile parameter

laminar heat transfer equivalent distance divided by the local distance along the streamline

see definition of F ^ in QTRANS

see definition of F in QTRANS

reference \iscosity divided by reference tcmperatiire. lbrsee/ft2- H

107

THETAL laminar momentum thickness, ft

THETAT turbulent momentum thickness, ft

c. Error Statements

ERROR IN CONTROL CARD (NTYPE - name)

ERROR IN CONTR il, CARD (NPROG - name)

The above errors are caused by ari input error of the code words NTYPE or NPROG. "Name" is the control word that was input. This error will terminate the run.

ALTITUDE INPUT TO ATMOS EXCEEDED RANGE - ASSIGN VALUE OF 2,300.000 FT.

ALTITUDE INPUT TO ATMOS BELOW ROUTINE RANGE - ASSIGN VALUE OF 0 FT.

Input altitude out of range of ATMOS subroutine. FLIGHT case only.

STREAMLINE COORDINATE DIMENSION (750) EXCEEDED IN SUBROUTINE AXI2D

The input value of dx is too small for an axisymmetric or two-dimensional ease. Also printed with this comment is the x-value at which the array dimension was exceeded, the maximum x defined by the geometry input, and the iiiput dx. The run is terminated by this error.

DBTP ERROR

The independent variable is outside the range of the program tables. The independent variable and the table limits are printed. The program will use the table limit and proceed.

STREAMLINE CROSS AT XSI =

When the twe divergence calculation streamlines cross, the above comment is printed with the value of the i coordinate at that point. The program moves downstream, by dx increments, until the streamlines are not crossed and begins calculations at that point.

10«

CONVERGENCE NOT ESTABUSHED IN VELOCITY CALCULATION

The velocity iteration routine in subroutine FLOW failed to converge on a solution in 10 iterations. The program proceeds using the final calculated velocity.

SIRCH ERROR

Table interpolation error in the general interpolation routine SIRCH-

ERROR IN TABLE2 LOOK-UP

ERROR IN TABLI>5 LOOK-UP

An error occurred during table interpolation in subroutine TABLE2 or TABL18.

FREE STREAM ENTHALPY EXCEEDS TABLE VALUES IN SONENT

The enthalpy value used for table interpolation in subroutine SONENT is outside the table l>mits.

ERROR IN STREAMLINE CALCULATION

This comment is printed when a calculated streamline point lies outside the (9e(£ .T)) grid in subroutine STREAM.

109

4. PROGRAMMING INFORMATION

a, PrPr Program listing

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181

b, PrMr Deck Setup

The following diagrams indicate the deck setup and the core storage requirements for the Pr^r program. Each link is composed of the subroutine decks (symbolic or object) listed in the allocation diagram. All control cards are indicated by a $ in column 1 and must be punched exactly as shown in the deck setup diagram. All items in parenthesis indicate subroutine or data card decks.

182

OOöOö

LP1

LP2

LP3

Notes:

1. Storage locations are on actual numbers.

2. Load points (LP) are indicated as follows:

LP1 - 04076

LP2 - 60005

LP3 - 65623

SCHEMATIC DIAGRAM OF COMPUTER STORAGE ALLOCATION

183

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184

The '•ontpnts of each link are:

Link 0 AS2419 (main program) TAB DTAB DBTP TBLP XTAB LOC XLOC

Link 1 DIVERG UND STREAM

Link 3 DELTA

Link 4 CYUXD FLOW

Link 5 AXI2D HEMI

Link 6 QTRAN BEQI

Link 2 WALLT 1 WALLT 2 BLKDTA DERV EBAR IWALL JAYELL MUZERO RORMUR SOMEGA TABLE 2 TABLE 6 TABLE 7 TABLE 14 TABLE 18 TABLE 19 TABLE 20

Link? INTIAL REF SONENT STACON ATMOS

tn;

SECTION ni

THE TURBULENT NONSIMILAR BOUNDARY LAYER PROGRAM (NSBL)

1. NOMENCLATURE

Program Symbol

CP

Math Symbol

* DET.FTA 6(rj /p)

DELSTR 6*

* DHDX 3I/8X

♦ DHDY ai/ay

' DMUDY öti/dy

DQ dq/dy

DRDX 'di/bx

DS AS

* DUDX 3u/ax

* DUDY 9u/9y

* DUDYL (au/a.y)

* DVDY dv/dy

DX Ax

Description

sj^cific heat of air at constant pressure, 60Ö6

boundary layer thickness

increment in boundary layer parameter

boundary layer displacement thickness

derivative of total enthalpy

derivative of total enthalpy

derivative of viscosity

derivative of the heating rate

derivative of three-dimensional flow parameter

increment for stagnation region problems

derivative of velocity

derivative of velocity

velocity derivative at boundary layer edge

derivative of velocity normal to surface

incre^rnt in x-direction

Units

2 2 ft /sec -0R

ft

ft3/slug

ft

ft/sec

ft/sec

lb-sec/ft

3 Btu/ft' -sec

ft

1/sec

1/sec

1/sec

1/sec

ft

* Those symbols marked with an asterisk are arrays (tables) in the program. Throughout this document, a subscript i on either the program symbol or the equivalent math symbol indicates that reference is being made to the i^1 position within the array.

1%

Program Math symbol symbol

DY Ay

* D2HDY a2i/ay2

* D2UDY a u/ay

ENDX xf

* ENTH i

EPS e

EPSY a«:/ay

* ETA n

* H i

* PR a

* PRESSURE P

* QP q

QW %

QQW QQW

R r

E

* RHO P

SI

STARTX

* T

*TAU

TAULP

Description

increment in y-direction

second derivative of total enthalpy

second derivative of velocity

final value of x

static enthalpy

eddy viscosity

derivative of eddy viscosity

similarity parameter

total enthalpy

Prandtl number

pressure

heating rate

heating rate at the wall

heating rate at the wall

thiee-diroensional flow parameter

universal gas constant, 1716

density

initial value of S for the calcuiation of 17

initial value of x

temperature

shear force

local laminar shear force

Units

ft

2

1/sec

i /sec-ft

ft

,2. 2 ft /sec

lb-sec/ft

lb-sec/ft"

u2/ 2 ft /sec

lb/ft"

/ 2 Btu/ft -sec

2 Btu/ft -sec

2 Btu/ft -sec

ft

ft7sec2-0R

2 4 lb-sec /ft

ft

ft

0R

lb/ft2

lb/ft2

187

Program Math symbol symbol

TAULW Tw.L

TAULY 3Tl/ay

TAUTP rT

TAUTW TK.T

TAUTY dTT/dy

TAÜW \

THETA e

TURBS 3TT/3y

arL/ay

* U u

* V

X

XI

*XMU

*!

M

Description Units

2 laminar shear force at the wall lb/ft

3 derivative of the laminar shear force lb/ft

2 local turbulent shear force lb/ft

2 turhuient shear force at the wall lb/ft

3 derivative of the turbulent shear force lb/ft

2 shear force at the wall lb/ft

boundary layer momentum thickness ft

ratio of the turbulent to the laminar shear force derivative

velocity in the x-direction ft/sec

velocity, v" in Volume I (Appendix C) ft/sec

coordinate tangent to the surface ft

initial value of x for the calculation of 17 ft

2 absolute viscosity uf air lb-sec/ft

coordinate normal to the surface ft

Subscript notation:

Subscript

e

i

I

L

o

T

w

evaluated at the boundary layer edge

evaluated at the i^ position within an array

initial value

laminar

initial value

turbulent

evaluated at the surface

188

2. METHOD OF SOLUTION

a. Forward Integration

In the following discussion, the equation numbers in parenthesis refer to an appropriate equation in Appendix C of Volume I. The equations solved by this computer program are the equation of state (C-3),

p - P/(RT)

Continuity (C-17),

1

au. 2u. 1 1^

ITy ' Ay

2 /I ap 1 dri iu i +

i \p ax r axi ap^ _a_ / au i

" ax ^ ay v ay;

aTn

a>

ip ay (a ayj

dih du { du \ dp + IM * T + U —- ay ay \M8y Tj dx

1 2"i vi-i /av\

x-momentum (C-6),

au = _1_ ax pu

ap a / au \ 1 ^H. u av

and Energy (C-7):

9I = -L A dx pu ay

M ai • / 8u v ai_ u av

In addition, the relationship for turbulent shear stress, Equation (C-39), is required for the computation.

9y I P. 1 ^mi^u^ w 1 m

e di S =ä äy

s!l

Since v is expressed as a function of input data it can be determined explicitly at each point in the boundary layer at the initial or start position.

The y derivatives in the above expressions are calculated using three-point central differences at all positions except at y = Ay, Since the turbulent velocity and enthalpy profiles change very rapidly near the wall, an off-center method of calculation of derivatives is used at y = Ay.

au\ aV2

1 6 Ay (9u3 -2U4

-6u2)

8 2/ 1

2<V -3u3. -9u2) GAy

With v defined and with u, I and y derivatives, da/dx and dl/dx can now be determined. With the 9u/3x and 8l/8x derivatives determined the u and I profiles at the next station (x + Ax) can be obtained by forward intt ation using:

u = u + x+Ax x

1 ^A =! + x+Ax x

The y derivatives, v, du/dx, and 8I/9x calculations move out from the wall along a line normal to the wall until a specified limit is reached. The profiles are then stepped forward and the calculations repeated.

At each point in the boundary layer, a similarity parameter TJ is calculated as indicated below in subsection b., item (4). A value of V max is an item of input used to limit the calculation in the y-directions, which are to be printed out.

Also calculated at each station x are the displacement and momentum thickness, heating rate, and the laminar and turbulent shear stress at the wall using:

6,

Ü >■■ i i'-Ä»

/ o - i6 L£L./i-iL

0 KUel V dy

1 The notation x0 will be used throughout this section to denote the initial value of x.

190

i fö\ ai si \ =-778 y [pU87+PV37-UT dy

6' _?J1 üü ^L M a 2 + "57 3y ■X dy

The heating rate and shear at the wall are calculated by an integration over the boundary layer rather than from the definitions because of the greater accuracy which can be obtained with the equation.

b. Numerical Method of Solution

The purpose of this section is to give the procedure used in solving the partial differential equations defined previously including the required numerical approximations.

The following variables and tables are input. Tables are indicated as functions of some independent variable, e.g. P = f(x) indicates the table of pressure as a function of x.

u = f(y) at x

I - f(y) at x^

ar ax

au 9y

P - f(x)

r = f(x) w

= f(x)

= f(x) e

In addition, the following quantities must be specified:

Ax Ay T) max f

191

(1) Calculation of tabular values

x. - x. , x. , - x.

l+l l X. . - X, l l-l X. - X. i+1 i i i-U x. , - x. ,

i+1 i-l

i = 2, 3 (NP-1)

= 0 ap ax

i=l

(2) Values obtained from tables (linear interpolation)

P = f(x) (|^) =f(x)

9P 3x

f(x)

r - f(x)

v^f.:)

i =f(x) w

_ar ax -f(x)

(3) Computation at initial x = x

y. = (i - 1) Ay i = 1, 2, 3, .... 200

u. = u._1 + (-|ü) Ay i = (LIST + 1), .... 200

I ^IST

LIST = number of input values in u and I tables.

T. = (I " u /2)/Cp

-8 1/2 Ji. - (2.272 x 10 T. )/(T. + 198. 6)

p. = P./(1716T.) ii r

C = 6006 ft2/sec-0R

i -2, 3 200.

192

(4) CalcUi. tion cl 6^

S fl\ , A 3/5.4/5 [p) ~- <ue r Ay "e )/S

S = SL^pu ur x- initially. At x + Ax

= o u u r Ax x+Ax e e^e

(5) Calculation of y-derivatives and v-profile

3v

9y

•m Ay

3^

9y

Mi+i - Vi 2 Ay

du i 1 T- =7T-(9uo -2u^ " 6uo) oy 6Ay v 3 4 2'

A ay

il 9y

ay

6Ay 2("4-3u3-9u2)

6Ay <9I3-2I4-6I2-Il)

2 6Ay' 2^4-3I3-SI2 + 5V

j =2

8u Vi - ui-i 2Ay

ai 'S?

^1 - ^-l 2Ay

a2 o u u. + u. , - 2u.

i+l i-l i

v 2

^y

193

82I

I. +1. -21. i+l i-l i

ay = M.

32U

'»y2

IE 9y

3u

: 9y

.833 . 12

= .054

L 'L

?) ©ft)©, Ay 2

T' = T T T 2

^rVi^^i^i-i)«^)-?

i = 2, 3; . . ., MIN (j, 199)

The edge of the boundary layer is defined by j = i at which the following criteria is satisfied:

Ay 9u aui

9y" 9y| eJ < .00001 —%?- < .00005 I 9y

The following are calculated for the range - < i < j

T = r - r T. T . T . i ii lj

T =7' - T' L. L. L.

i 'i lj

T - T - T i L. T.

i i

T = - T' - T' 1 T . L .

'J 'J

194

t. - TT /(9U/9y).

v.833 , 12 "#)(Mm-'0 (3u/ay).

dy a2i

9y ay ^y i

2Ay (Ui+l Vl " "i-l Vl'

• • /dq qi = ^-l + W

i-l dy ) 2

T,.™ , 2/I d- 1 dP\ PART 1. ="- u. (- ^- + r=r -T-)

i \ r dx P dx /

= _1_/^L| + ^T _9P| \ i p.u I 3v 3y . OX . J

PART 2

1 dq I 1 PART 3. = hr1 + (u- , T. - u. , T. ,)hTT-

i P.u. dy . v i+1 i+l i-l i-l' 2Ay

PART 6. = u. i i

2v- i A i-l 9v

i-U

v. = i 3u 2ui

^y

PART 1. + u. PART 2. i t i

(I - "j /2)

-— (PART 3.

- u. PART 2.) - PART 6. i i i

9y = 7^^ -v^J -

3v Av v i i-l' #)

i-l

195

(6) Calculation of x derivatives

ax „. „ i 3u

= PART 2. ä- 1 U. oy

l I J

31 ax = PART 3. —i^-

i u. ay i i

(7) Calculation of ö*. fl, Qw, TW L, TWJT at x

J 6* = Ay ^

2 1 -

p.u. + P. , u ri i i-l i-l 2 p.u.

J J

A J P.". / "A P- , u- i / u- i\

2 ^ p.u. ^ u./ p.u. V "j /

% 778 ^ « 9I

i i ax ai

+ pi-i Vi T* i-i

ai + P.V. -5- ri i ay

ai + ^i-i vi-i ^

i-i - (U.T. -U. , T. )-r— v i i i-l i-l' Ay

QQ = ^Sv 778 '

1 Ay

ova

M1 2

Tw, L " TLj

T = - T' w,T TTj

(8) Calculation of u and I-proflles at x + Ax

ai Wx-Vax Ax

d"\ A U A = U + -r— Ax x+Ax x 9x

'x

} i = 1,2 j

196

ui =UJ

1.1 = I, ■x+Ax 'x+Ax

i = j+ 1, j + 2 200

c. Stagnation Region Calculation

A slight modification to the previous discussed procedure is used to obtain stagnation point profiles. Input profiles are corrected by integrating the u and I profiles, but x (not equal to zero) is not increased during the integration. The profiles are assumed correct when

W. x < •1 "LIST for a11 i

This convergence criteria is obtained with the velocity similarity stated below.

/au\ u a / e \ _ _e 9u \ dx)~ u ax

91 e

8x = 0

Experience has indicated that an enthalpy profile convergence criterion is not necessary.

For the stagnation region, the equations for the velocity and enthalpy profiles are changed to

I =1 +^ i,S i,S-AS ax

AS

i,S au

u + i,S-AS ax

AS x + AS

o

If ![ or uj become negative for some i, AS is halved and the case is restarted. This may be repeated a maximum of two times for any combination of Ij or uj < 0.

Additional input quantities are AS and MITR.

MITR is the maximum number of iterations that the program will make in trying to converge on a solution to the stagnation problem.

197

AS is a pseudo length similar to Ax. The program also uses AS as a test to determine whether the stagnation point calculati i is to be performed or bypassed. For a flat plate case AS = 0,

Calculations for a hemisphere may be performed near the stagnation point and also along the surface. This is accomplished by the input of AS > 0 and Ax > 0.

3. INPUT-OUTPUT DESCRIPTION


Data are input through the medium of IBM cards. The purpose of this section is to define the required inputs and the form they take on the cards.

A sample data sheet may be found at the end of this section. Use of this type of form greatly reduces the amount of effort required to obtain results from the nonsimilar boundary layer program. The data may be keypunched directly from the form.

All cards except cards 1 and 4 must be keypunched with decimal points and power of ten indicators as follows: if only a decimal point is needed, the number may be punched anywhere within the respective ten column field; however, if the power of ten indicator E is used, the exponent must be right-adjusted in the field.

The information on card 1 is alphanumeric and may be keypunched as desired.

The data on card 4 are all integers and are keypunched as fixed point numbers (i. e., no decimal or power of ten indicator E is to be used). Each number on card 4 must be right-adjusted in its respective five column field. For example, if JOT1 = 1 then a one must be keypunched in column five.

1) Data Card Formats


1 1-48 8A6 Any information may be given on this card. It is used to identify the case.

2 1-10 7F10 Ax, the increment in the x direction. See the discussion on numerical stability in Appendix C (Volume I) for the criteria to be used in choosing a value.

2 11-20 7F10 Ay, the increment in the y direction. Should be chosen to provide at least ten points in the boundary layer for accurate results.

198

2 31-40 7F10

2 41-50 7F10

2 51-60 7F10


2 21-30 7F10 ^max' the limiting value on the number of calculations in the y direction.

XQ, the inicial value of x. x0 > 0.

xf, the final value of x.

a, the turbulent Prandtl number.

2 61-70 7F10 CHECK, case dependence criteria. If CHECK = 0 then the current case is not dependent on previous calculations. If CHECK = 1 then the current case is dependent on the previous cade.

3 1-10 4F10 AS, Calculations may be made for a bluLt body. This involves calculating profiles near the stagnation point as well as along the body. This type of problem is indicated by AS > 0 and MITR > 0 (card 4). For nonstagnation point calculations, AS = 0.

3 11-20 4F10 Turbulent Prandtl number - must be equal to Prandtl number on card 2.

3 21-30 4F10 Sj, value of S used to initialize

S = ST + p u v^ x lee I

3 41-50 4F10 Xj, value of x used to initialize

2 S ^ S, + p u r xT lee I

The data on card 4 control the number of values to be input to each of the iuc- ceeding tables. The values must be greater than zero. There is an upper limit on the number of values that may be input to each table. This limit is given with the description of the table. The limit may be changed by altering DIMENSION statements in the program.

4 1-5 1415 JOT1, the number of tabular values in the DJOT and XJOTT tables. 0 < JOTl < 10.

4 6-10 1415 LIST, the number of tabular values in the u and I profile tables. 2 < LIST < 50.

199

11-15 1415 NR, the number of tabular values in the three- dimensional flow parameter table (r), the table of its x-derivatives, and the tables of their corresponding values of x. 2 < NR < 50.

16-20 1415 NP, the number of tabular values in the pressure table and the table of corresponding values of x. 2 < NP < 50.

21-25 1415 NW, the number of tabular values in the wall enthalpy table and the table of corresponding values of x. 2 < NW < CO.

26-30 1415 NDU, the number of tabular values in the (9u/9y)e table and the table of corresponding values of x. 2 < NDU < 50.

31-35 1415 NV1, the number of tabular values of velocity normal to the surface and the table of corresponding values of x. 2 < NV1 < 50.

36-40 1^15 MITR, the maximum number of iterations allowed to calculate the profiles for the stagnation region.

Cards 5 and 6 are used to specify different frequencies of output and the corresponding ranges. The maximum number of values in these tables is 10.

7F10 Table DJOT, assigns value to first, second, and successive printout intervals,

7F10 Table XJOTT, assigns value of x up to which each interval of table DJOT is used.

2 In the case of tables with more than seven input values, the card number refers tJ a set of caris (e. g., if ten entries are made in the DJOT table, then card 5 actually refers to two cards).

200

Example: XQ = 0.5 and Xf = 1.0, Printout is desired every . 05 (value of x) until x = .25, every .01 until x = .3, and every . 1 until x = Xf. The required input would be:

Card Column Value

m 1-5

5 I

1-10 11-20 21-30

.05

.01

.1

6 1-10 11-20 21-30

.25

.3 1.0

(right-adjusted)

Columns Format

7F10

7

7

7

8

1-10

11-20

21-30

7F10

;FIO

7F10

7F10

Description

Table u-profile = f(y). The purpose of this table is to provide the program with an initial velocity profile at x = x0. Normally, this table must be input and contain at least three values. Exception is made when a case is dependent on a previous case (see Section 2).

This table contains LIST tabular values. The maximum number of values in this table is 50.

Velocity at wall.

Velocity at Ay from wall.

Velocity at 2Ay from wall.

Table I-profile = f(y). The purpose of this table is to provide the program with an initial total enthalpy profile at x = x0. Normally, this table must be input and contain at least three values. Exception is made when a case is dependent on a previous case (see Section 2).

This table must contain the same number of values (LIST) as the u-profile table. The maximum number of values in this table is 50, Entries are assumed to be consistent with Ay.

1-10 7F10 Total enthalpy at the wall.

201

Card Columns Format

8 li-20 7F10

8 21-30 7F10

9 7F10

Description

Total enthalpy at ay from the wall.

Total enthalpy at 2Ay from the wall.

Table r = f(x). This table contains NR tabular values of r, the three-dimensional flow parameter (streamline divergence due to body geometry). The maximum number of values in this table is 50,

10 7F10 Table (ar/ax) = f(x). This table contains NR tabular values of {dr/dx). The maximum number of values in this table is 50,

11 7F10 Table x. This table contains NR tabular values of x corresponding to entries made in the r and (dr/dx) tables (cards 9 and 10),

12 7F10 Table pressure = f(x). This table contains NP tabular values of pressure. This table is numerically differentiated for (9P/ax). The first value of (dP/dx) is set to zero. The maximum number of values in this table is 50.

13 7F10 Table x. This table contains NP tabular values of x corresponding to the entries made in the P table (card 12).

14 7F10 Table wall enthalpy = f(x). This table f ntains NW tabular values of the total enthalpy ^ i.he wall. This table is used for nonisothermal wall problems. The maximum number of values in this table is 50,

15 . 7FIO Table x. This table contains NW tabular values of x corresponding to the entries made in the wall enthalpy table (card 14).

16 7F10 Table (9u/ay)e = f(x). This table contains NDU tabular values of (du/dx) evaluated at the edge of the boundary layer. For all problems not involving vorticity (9u/ax) - 0, The maximum number of values in this table is 50,

202


17 7F10 Table x. This table contains NDl tabular values of x corresponding to the entries made in the (au/3y)e table (card l(i).

IS 7F10 Table Vw = f(x). This table contains NV1 tabular values of the velocity normal to the wall. This table is used for mass injection or leakage problems. The maximum number of values in this table is 50.

19 7F:o Table x. This table contains NT 1 tabular values of x corresponding to the entries made in the Vw table (card 18).

20 NOTE: In all cases where the values in the above tables are zero, a ?ero must be entered on the input sheet.

E>ample: the normal velocity at the wall is zero:

V table (card 14) 0,0 0.0 0.0

x table (card 15) 0.0 1.0 2.0

2) Case Dependence

The NSBL program has a criterion which allows a case to be c'ependent on the success or failure of the previous case. This involves the input CHECK and LIST and the test for ERROR.

If: Then:

LIST CHECK ERROR

0 0 — Not dependent on the previous case, program continues.

0 1 0 Dependent on the previous case which worked, program continues, using final profile from previous case.

0 1 1 Dependent on the previous case which tailed, so present case stops and program continues to next case,

0 — 0 Dependent on the previous case which worked, program continues, using final profile from previous case.

0 — 1 Dependent on the previous case which failed, so present case stops and program continues to next case,

20;'.

b. Output Description

The printed output consists of the input data and at each printout interval the following are printed as a function of y:

y u q dv/dy v 9u/9x

T T I T) ai/ax e

The following are printed as a function of the x at the end of each printout interval:

K Tw,L

Tw,T

Ö*

9

P 1

ap/ax

QQW

S

204

c. NSBL Input Form

CARD TITLE CARD FORMAT 8A6 ALPHANUMERIC ANY SPACE 1-48 1

Ax Ay Vma* *o xf Prandtl no. CHECK

AS Prandtl no. «I XI 50 60 70

JOT1 LIST NR NP NW NDU NV1 MITR

mnr 5 10 IS io 25 30 & 40

XJOTT

1 1 1 1 1 1 1 1 7 u-profile

8 I'Protil6

9 L

10

11

x

I I I I I I 1 x table

12 Pfwsure

13 x table

14 Wall anthalpy

,K xtabte

1 1 1 1 1 1« Wu/dy)e

! 1 1 I 1 1 17 x table

i 1 ! ^ 1 1 v

Ifi w

1 1 1 1 1 1 iq x table

1 1 1 1 I 11 21 31 41 51 61 71

20Ü

PROGRAMMING INFORMATION

a. Program Flow Chart

INPUT

X PRINT INPUT j

LOOK UP p 9? cfc 3ul P 5x,r'8x,8y|e

,vw

AS FUNCTIONS OF x

CALCULATE

. dH du 31 d2u a2l aTL 9TT T. T. „

FOR ALL POINTS IN THE PROFILE (»I < TJ^x)

I CALCULATE

de \, TT. T. C.-p-, AO, Q", PART 1, PART 2, PA.TT 3

PART 6, v, |^ , li FOR ALL POINTS IN PROFILE 9v 8x

CALCULATE qw, Tw, 6*, 9

X PRINT OUTPUT

| INCREASE x tYAx j

Pi* 'ax'

LOOK L"?

ox ' w ' 9vl AS FUNCTION OF x+Ax

——" 1 -L

CALCULATE u AND 1 PROFILE ji.T, p AT x+Ax

206

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REFERENCES

1. Hanks, R. A.: and Savage, R. T,: Thermal Design Methods for Recoverable Launch Vehicles with Consideration of Arbitrary Wall Temperature and Surface Conditions. NASA CR-74714, August 1965.

2. Nagel, A. L.: Fit/simmons, H. D.; and Doyle, L. B.: Analysis of Hypersonic Pressure and Heat Transfer Tests on Delta Wings with Laminar and Turbulent Boundary Layers, NASA CR-535, August 1966.

3. Thomas, A. C.: Perlbachs, A.; and Nagel, A. L.: Advanced Reentry Systems Heat Transfer Manual for Hypersonic Flight, AFFDL-TR-65-195, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio, January 1966.

4. Hansen, C. F.: Approximations for the Thermodynamic and Transport Properties of High Temperature Air, NACA TN-4150, March 1958.

5. Hilsenrath, J.: and Beckett, C. W.: Thermodynamic Properties of Argon-Free Air (0.78S47 N^ 0-21153 O2) to 15,000 °K, NBS Report No. 3991, July 1955.

6. Feldman, S.: Hypersonic Gas Dynamic Charts for Equilibrium Air, AVCO Research Report No. 40, January 1957.

7. Logan, J.; and Treanor, C: Tables of Thermodynamic Properties of Air from 3,000 to 10,000 "K. Cornell Aeronautical Laboratory, Inc., January 1957.

8. Ames Research Staff: Equations, Tables, and Charts for Compressible Flow, NACA Report 1135, 1953. " ~

222

JJNCTASRIFTF.n riU ( 1.. vsij i, .ili-'H

DOCUMENT CONTROL DATA - R&D tirttv l.i-.-ilu .,tv.n -I ml.-, U.tiv .'! .ihslrHt t .«n.J phliJunK .n.ij.taüon muvt *>«■ rnlrr-d vvlu-n t!i>- i>v<*r.Jl rf^ort is < lusst(i»,'i>.

1. ORIGIN*! IMG ACTIVITY l(\,rIN.r.Hi- ..ulii.irP

The Boeing Company P. O. Box SF^S Aerospace Group Seattle, Washington Space Division—Kent Facility 98124

2.i REPOIT SECURITY CLASSIFICATION

UNCLASSIFIED 2h. GROUP

3. REPC- : riTLE

INVESTIGATION OF TURBULENT HEAT TRANSFER AT HYPERSONIC SPEEDS Volume III. The Luminar-Turbulent prMr Momentum Integral and Turbulent Ncnsimilar Boundary Layer Computer Programs

4. DESCRIPTIVE NOTES iTvr'-> "' '<T"'t ami imlusivc dm.-M

Final Report. March 1965 to March 1967 5. AUTHORS iKirsr ram-. raiH.ll.- inilial, I.1-.1 nann-

Savage. Ricru rd T. Jaeck, Carl L. Mitchell. James R.

8. REPORT DATE

December 1967 7 i TOTAL NO. OF PAGES

222 71.. NO. OF REFS

8a. CONTRACT OR GRANT NO.

AF33(615)-2372

Project No. 1366

Task No. 136607 ^BPSN 5(61136-62405334)

9^.. OR1 JINATOR'S REPORT NUMBERS

D2-113531-3

9li. OTHER REPORT NO(Sl (Am othrr numbers iVuit may hi- assign.-.I

AFFDL-TR-67-i44! Volume III 10. DISTRIBUTION STATEMENT


11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTI "T-

Air Force Flight Dynamics Laboratory A;r Force Systems Command Wright-Pattp.rsnn Air Fornp. foifiP, Ohin 4fi4a?

13. ABSTRACT

This report presents a combined analytical and experimental investigation of turbulent heat transfer on basic and composite configurations at hypersonic speeds. The analytical results are presented in Volume I, the experimental results, including data-theory comparisons, are presented in Volume II, and computer programs incorporating the analytical methods described herein are presented in Volume III of this report.

The two heat-transfer prediction methods programmed are the laminar-turbulept Pj.^,. momentum integral method and the turbulent nonsimilar method. This volume of the report describes the numerical method and presents flow charts, program listings, input forms, and a description of the output for each program. The programs are written in Fortran IV language for operation on the IBM 7094.

FORM DD 1 NOV es 1473

U3 4802 1030 REV 7/66

UNCLASSIFIED S. . urit-, fUissili

UNCLASSIFIED

*£> *CSDS L'^r £

N'onsimilar Boundary Layer

Momentun Integral

Aerodynamic Heating

Laminar Flow

Turbulent Flow

DDINOV651473 U 3 4802 1030 REV 1 ftft

P A R T 2 O F i

JINCLASSIZIEIL SecunlA Cliissifu .ill

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