passive nosetip technology (pant) program · interim report passive nosetip technology (pant)...

U.S. DEPARTMENT OF COMMERCENational Technical Information Service

AD-A026 613

PASSIVE NOSETIP TECHNOLOGY (PANT) PROGRAM

VOLUME XVII. COMPUTER USER'S MANUAL:

EROSION SHAPE (EROS) COMPUTER CODE

ACUREX CORPORATION

PREPARED FOR

SPACE AND MISSILE SYSTEMS ORGANIZATION

DECEMBER 1974

196077

SAMSO-TR-74-86Volume XVII

INTERIM REPORTPASSIVE NOSETIP TECHNOLOGY

(PANT) PROGRAM

Volume XVII. Computer User's Manual: ErosionShape (EROS) Computer Code

D. Rafinejad

C. Derbidge

Aerotherm Division/Acurex Corporation

SAMSO-TR-74-86

December 1974

AEROTHERM REPORT UM-74-57

Air Force Space and Missile 'C""' I"..Systems Organization

Los Angeles, California

Contract F04701-74-C-0069 .. ..(CDRL B002)

REPRODUCED BY

NA1iONAL TECHNICAL UINFORMATION SERVICE

U. S. DEPARTMENT OF COMMERCESPRINGFIELD. VA. 22161

A;"-. - - .. .

SAMSO-TR-74-86 C/N 7102.128Volume XVII

INTERIM REPORTPASSIVE NOSETIP TECHNOLOGY

(PANT) PROGRAM

Volume XVII. Computer User's Manual: ErosionShape (EROS) Computer Code

D. RafinejadC. Derbidge

DISRfL,-T .... :;-:.,7•'. A£

Approv.-c :D i,,: 1 : ';. .,

r DDC'\

•'j

-17

FOREWORD

This document is Volume XVII of tlhe Interim Report series for the Pas-

sive Nosetip Technology (PANT) program. A summary of the documents in this

Ir series prepared to date is as follows:

Volume I - Program Overview (U)

Volume II - Environment and Material Response Procedures for Nosetip

Design (U)

Volume III - Surface Roughness Data

Part I - Experimental Data

Part II - Roughness Augmented Heating Data Correlation

and Analysis (U)

Part III- Boundary Layer Transition Data Correlation and

Analysis (U.

Volume IV - Heat Transfer and Pressure Distributions on Ablated Shapes

Part I - Experimental Data

Part II - Data Correlation

Volume V - Definition of Shape Change Phenomenology from Low Tempera-

ture Ablator Experiments

Part I - Experimental Data, Series C (Preliminary Test

Series)

Part II - Experimental Data, Series D (Final Test Series)

Part II - Shape Change Data Correlation and Analysi.s

Volume VI - Graphite Ablation Data Correlation and Analysis (U)

Volume VII - Computer Us:er's Manual, Steady-State Analysis of Ablating

Nosetips (SAANT) Program

Volume VIII - Computer User's Manual, Passive Graphite Ablating Nosetip

(PAGAI) Program

Volume IX - Unsteady Flow on Ablated Nosetip Shapes - PANT Series G

Test and Analysis Report

Preceding page Inkiii

Volume X - Summary of Experimental and Analytical Results

Volume XI - Analysis and Review of the ABRES Combustion Test Facility

for High Pressure Hyperthermal Reentry Nosetip Systems

Tests

Volume XII - Nosetip Transition and Shape Change Tests in the AFFDL 50

MW RENT Arc - Data Report

Volume Xl'! - An Experimental Study to Evaluate Heat Transfer Rates to

Scalloped Surfaces - Data Report

Volume XIV - An Experimental Study to Evaluate the Irregular Nosetip

Shape Regime - Data Report

Volume XV - Roughness Induced Transition Experiments - Data Report

Volume XVI - Investigation of Erosion Mechanics on Reentry Mate-

rials jU]

Volume XVII - Computer User's Manual, Erosion Shape (EROS) Computer

ProgramVolume XVIII - Nosetip Analyses Using the EROS Cor.puter Program

This report was prepared by Aerotherm Division/Acurex Corporation under

Contract F04701-74-C-0069. Volumes I through IX covered PANT activities from

April 1971 through April 1973. Volumes X through XV represent contract efforts

from May 1973 to December 1974. Volume X summarizes the respecvive test pro-

grams and describes improvements in nosetip analysis capabilities. Volume X!

presents an evaluation of the ABRES test facility in terms of performing ther-

mostructural and reentry flight simulation testing. Voiumes XII through XV are

data reports which summarize the experiments performed for the purpose of de-

fining the irregular flight regime. The analysis of these data are presented

in Volume X. Volumes XVI through XVIII describe the background, de:elopment,

and check out of the PANT EROsion Shape (EROS) computer code. These volumes

jocuiment efforts performed unaer supplementary agreements to the Mintceman Na-

tural Hazards Assessment Program (Contract F04701-74-C-0069) between Aprl 1974

and March 1975.

This work was administered under the direction of the Space and Missile

Systems Organization with Lieutenant A. T. Hopkins and i.ieutenant E. G. Taylor

as Prcject Officers with Mr. W. Pcrtenier and Dr. R. L. Baker of the Aerospace

Corporation serving as principal technical monitors. Dr. Dariush Rafinejad

was principal Aerotherm investigator for the work described in this volume.

_v

This technical report has been reviewed and is approved.

E. G. Taylor, Lt., USAFProject OfficerAero and Materials DivisionDirectorate of Systems 7ngineeringDeputy for Reentry Systems

L,

vI

ABSTRACT

A computer program is developed to numerically model the in-depthtransient response and shape history of an ablating nosetip subjected to a

reentry environment. The generality of the input also allows the user to

conveniently analyze the boundary layer, shape change and in-depth responseof many materials in - variety of test facilities. The computer code iscapable of handling nosetips of shell or plug geometries. The boundary layerand heat transfer distribution are modeled for a variety of environmentsincluding hydrometer erosion. In addition, inviscid flow and heat transferdistributions for many types of blunt bodies in hypersonic flow can be readily

calculated.

Preceduig page blank

Vii

TTABLE OF CONTENTS

Section

1 INTRODUCTION 1-i

2 NUMERICAL MODELING AND COMPUTATIONAL PROCEDURES 2-12.1 Inviscid Flow Field 2-1

2.1.1 Shock Shape Calculation 2-12.1.2 Pressure Distribution 2-42.1.3 Boundary Layer Edge State 2-13

2.2 Boundary Layeg Heat and Mass Transport 2-152.2.1 Laminar Heat Transfer, Smooth Wall 2-162.2.2 Turbulent Heat Transfer, Smooth Wall 2-172.2.3 Transition Criteria 2-192.2.4 Surface Roughness Effects on Heat

Transfer 2-212.2.5 Transitional Boundary Layer Heat

Transfer .2-232.2.6 Hydrometer Boundary Layer Stirring

Effects 2-242.3 In-depth Conduction Calculations 2-25

2.3.1 Coordinate System, GoverningEquations

2.3.2 Finite-Difference Formulations 2-282.3.3 Time Step Control 2-29

2.4 Surface Ablation Response Calculations 2-30*2.4.1 Surface Energy Balance Formulation 2-302.4.2 Erosion Modeling 2-342.4.3 Use of Thermochemistry Codes to

Generate Input Data 2-352.4.4 Simpler Forms of the Surface Energy

Balance Equation 2-362.5 Surface Point Movement and Surface Smoothing 2-37

2.5.1 Shape Change Geometry 2-382.5.2 Surface Angle Definition 2-392.5.3 Apparent Nose Model 2-39

3 DESCRIPTIONS OF INPUT AND OUTPUT3.1 Input Instructions 3-1

3.1.1 Restart Information 3-1.3.1.2 Title and Heading Information 3-13.1.3 Table 01 - General Program Constants 3-23.1,4 Table 02 - Environment Table 3-2 23.1.5 Table 03 - Geometry 3-93.1.6 Table 04 - Transition Criteria 3-153.1.7 Table 05 - Weather Conditions 3-163.1.8 Table 06 - Material Properties 3-173.1.9 Table 09 - 3urface Thermochemistry 3-193.1.10 End of Input ,-23

3.2 Program Output 3-253.2.1 Output of Input 3-253.2.2 Calculation Output 3-26

4 SAMPLE PROBIEMS 4-1

REFERENCE3 R-1

APPENDIX A - DESCRIPTION OF UNLABELEDOUTPUT VARIABLES A-i

Nara n

ix PS I•

I}

LIST OF FIGURES

Figure Page

2-1 Nosetip shape change calculation procedure. 2-2

2-2 Shock shape evaluation technique. 2-3

2-3 Pressure distribution calculation nomenclature. 2-5

2-4 Schematic of pressure distribution calculations. 2-62-5 Flow chart of logic to determine nosetip sonic

point location (subroutine RUNLP). 2-11

2-6 Nosetip geometry...-,

3-1 Sketch of surface thrmochemistry table make-upfor a given material including leading constant cards. 3-24

x

LIST OF TABLES

Table Page

01 General. Program Constants 3-2

02 Envircr.rent Table 3-5

03 Geometry and Material Bound: riee 3-9

04 Transition Criteria 3-15

05 Weather Conditions 3-16

06 Material1 Properties 3-17

07 (Reserved for future use)

08 (Reserved for future use)

09 Surface Thermochemistry 3-19

x

LIST OF SYMBOLS

B' normalized ablation rate defined as /P eUeCM

CD drag coefficient

CH Stanton number for heat transfer (corrected (

for "blowing" if necessary

Stanton number for heat transfer not corrected

CHO for blowing or stagnation point Stanton number

CM Stanton number for mass transfer

Cospecific heat at constant pressure or pressure (Btu/lb*F)coefficient or(--

D diameter at start of aft cone (ft)

d minimum mesh size (ft)

d hydrometeor particle diameter (ft)p

F radiation view factor (---)

FK factor in Equation (2-27) (---)

FL ratio of local laminar heat transfer coefficient (---)

to stagnatior point coefficient

F ratio of local composite turbulent heat transfer (---)

FT coefficient to stagnation point coefficient

G erosion mass loss parameter (---)

H boundary layer shape factor (---)

HD dissociation energy (Btu/lb)

DI

xii

LIST OF SYMBOLS (Continued)

a 0 total enthalpy (Btu/lb)

Hr recovery enthalpy (Btu/lb)

h enthalpy (Btu/lb)

h material enthalpy (Btu/lb)cI

h enthalpy of species i at temperature Tw (Btu/lb)

h sensible enthalpy (Btu/lb)

hw enthalpy of gases adjacent to the wall (Btu/lb)

Eckert reference enthalpy (Btu/lb)

J interrna concucicr: :..:!e index C---)(X-dix -tionj

K interne! c.oncuct .,3de index (---)(N-dir,- agtcn)

mass f .i zpecies i(

Krough wall l;,mtn.• heating augmentation fact-.r

K-T rough wall turfnulent heating augmentationfactor

""KC rough wall comrpo.ite heating augmentation C---)factor

K1 material coefficient in Equation (2-46) (in -psi" ?7 7 )

k thermal conductivity or roughness height (Btu/ft-sec°F)or (ft)

xiii


k crater roughness height (ft)

ki intrinsic roughness of material in laminar (ft)flow

k effective sand grain roughness height (ft)

L internal conduction node index(ý-direction)

Le Lewis number A

M Mach number (---)

m net mass ablation rate per unit area (lb/ft'-sec)

me erosion mass removal rate per unit area (lb/ft 2 -sec)

min incoming hydrometer particle mass flux (lb/ft 2 -sec)

p individual hydrometer particle mass (lb)

thermochemical mass ablation rate per (Ib/ft 2-sec)

Pr Prandtl number (---)

p pressure (ate)

p PIPo --0

q heat flux (Btu/ft'sec)

q heat flux resulting from chemical energy (Btu/ft'sec)

""q heat flux conducted into solid material (Btu/ft'sec)at surface

q in heat flux radiated to surface (Btu/ft2 sec)

xiv

LIST OF SYYBCLS iContinuud)

qrad out heat flux radiated away from surface (Btu/ft sec)

sensible convective heat flux (Dtu/ft 2 sec)

Reff effective nose radius (ft)

RN geometric body radius of curvature at (ft)stagnation point

Re unit Reynolds number (ft 1)

roughness Reynolds number C---)

Ree momentum thickness Reynolds number

Re composite momentum thickness Reynolds number (---)based on reference conditions

RKL roughness - laminar heating parameter I---)

RKT roughness - turbulent heating parameter (---)

PL laminar Reynolds analogy factor 1---)

RT turbulent Reynolds analogy factor

r radius measured from body axis (ft)

rb iaternal conduction radius measured from (ft)body axis

s streamwise length (ft)

S transformed Z-coordinate, - Z/6 (--)

normal surface recession rate (ft/sec)

xv


T temperature (OR)

t tme (sec)

at time step size (sec)

u velonity (ft/sec)

X length measured from the axis along the (ft)i'nternal contour

Y distance measured normal to internal (ft)contour or fictitious interface

radial location of streamline (ft)

z axial length measured from original stagnation point (ft)

Z axial distance measured from the back of the body (ft)

zc axial distance from start of aft cone. (ft)

* modified diffusion driving force (see Refer-

ence 21, page 44 for definition)

Greek Symbols

material thermal diffusivity (ft 2 /sec)

Cw absorptivity of wall (

angle local tangent to the inner contour makes (deg)with the body axis

agnation point velocity gradient defined (sec-1 )as due/d2l

Svelocity gradient parameter

y specific heat ratio or plug shank inclination (--) (deg)angle with the axis

A distance from the inner.contour to the nosetip surfa,.e (ft)

xvi


a shock standoff distance (ft)0

distance from shank base to fictitious interface (ft)

6* boundary layer displacement thickness (ft)

11 transformed Y-coordinate, = Y/A

emissivity

s density ratio across shock (---)

momentum thickness (ft)f8 shock angle (deg)

A blowing reduction parameter (Equation (2-61)) (---)

A curvature of internal contour Cft-*)

Sviscosity (lb/ft-sec)

Sviscosity evaluated at Eckert reference enthalpy (lb/ft-sec)

density (lb/ft3 )

PC mass of hydrometer particles per unit (lb/ft')volume of air

surface material density (lb/ft3 )

p hydrometer particle density (lb/ft')

density evaluated at Eckert reference (lb/ft')enthalpy

a Stefan-Boltzmann constant (Btu/ft~sec*Rh)

Tw wall shear (lbf/ft2 )

xvii

- - - -- - - -


Sazimuthal angle (deg)

Subscripts

c cone condition

e boundary layer edge condition

i condition at start of cone or initial conditionor integration point index

L laminar

HN modified Newtonian

o stagnation point or total condition or notcorrected for blowing

R rough

s sensible

SP stagnation point

STIRRED modified to account for the effects hydrometerboundary layer stirring

T turbulent

TP tangent point

TR L TRANS transition point or transitional

tc thermal chemical only

xviii

I

LIST OF SYMBOLS (Concluded)

W condition at wall

I ci freestream condition

Superscripts

* sonic point condition

Ti value calculated at Ti

xix

SECTION 1

INTRODUCTION

The purpose of this document is to provide a description of the modeling

techniques and input requirements of the EROsion Shape (EROS) computer code

that combines environment modeling techniques developed by Aerotherm primarilyunder the PANT program (Reference 29) with in-depth transient conduction rou-

tines developed at the Aerospace Corporation (Reference 28).

The primary purpose of this code is to numerically model the in-depth

transient response and shape history of an ablating nosetip subject to a reentry

environment. The code calculates the inviscid flow and heat transfer distri-

bution for many types of blunt bodies in hypersonic flow. In addition, the

boundary layer and heat transfer distributions are modeled for a variety of

environments including hydrometer erosion. The in-depth thermal response is

capable of calculating the three-dimensional temperature field and surface

recession of noseti.ps at angle of attack. However, due to limitations of the

environment package to axi-symmetric geometries, the present code is restricted

to nosetips at zero angle of attack. A general thermochemistry model, including

kinetic effezts, is used in the surface energy balance formalation.

The generality of the input allows the user to conveniently analyze the

boundary layer, shape change and in-depth response of many materials in a

variety of test facilities, including wind tunnel, ballistic range, and archeater.

A description of the numerical modeling and calculation procedure is

given in Section 2." Input requirements and output are described in Section 3

and a sample problem is presented in Section 4.

SECTION 2

NUMERICAL MODELING AND COMPUTATIONAL PROCEDURES

The problem modeled by the computer code is that of determining theinstantaneous shape of an ablating axisymmetric nosetip reentering the atmos-phere at zero degrees dngle of attack, as well as the in-depthl heat transferand temperature rise. The requirement that the flow be parallel to the body

centerline reduces the problem to one of axisymmetric flow and recession. Thenosetip shape change events are modeled u3ing the cyclic calculation procedureoutlined in Figure 2-1.

The five computation elements illustrated in Figure 2-1 are describedin the following sections. Section 2.1 covers the inviscid flow solutions;Section 2.2 described the boundary layer heat and mass transfer calculations;Section 2.3 describes the details of the in-depth conduction calculation;Section 2.4 explains the surface ablation calculations; and Section 2.5 de-scribes the body movement.

The computational scheme is stable and accurate only if computationaltime steps are kept within certain limits. These limits are imposed by in-depth conduction and snape change and are described in Section 2.3 and 2.5,respectively.

2o. INVISCID FLOW FIELD

The inviscid flow field serves as a boundary condition for the boundarylayer solution. The actual boundary layer edge state is determined from theshock shape and the pressure distribution. The following t:,cee sections de-scribe the shock shape, pressure distribution, and boundary layer edge statecalculations, respectively. A complete description and justification of theinviscid flow field calculation technique is given in Reference 1.

2.1.1 Shock Shape Calculation

The bow shock geometry ahead of an axisymmetric ablated shape is computedby forming a piecewise Y tar curve with line segment slopes and lengths depen-dent on body point slopes and spacing. The technique for evaluating respectiveshock point locations is given in Figure 2-2. The procedure is to step along

2-1

Input EnvironmentHistory and lhiitial

Geometry

Irnviscid Flow Solution

1ioundary Layer Solutionand Heating Distribution

Nonablating option -

In-Depth Heat Conduct-IonCalculation te

Surface Ab-ation andRecessiLn RateDistribL L-on

Body Movement andNew Geometry -xA

NO Time to Call a

L2ý- New Environment?

YES

Figure 2-1. Nosetip shape change calculation procedure.

"2-2

--Wa4;f ~ ~ ~ •, At=e~te 7,,% &Vs ir.0/41

./ -ae -ep -

~05i Ee4Q~2/ 'A

ae 'V, r/ Z/'/// Y) J 19

Figure 2-2. Shock shape evaluation technique.

2-3

the shock by computing the next point based on the previous shock point and the

surface slope quantities. The following are needed to perform the calculation:

0 Body geometry quantities

0 Shock standoff distance on the stagnation line

a An expression which relates local surface slope to shock slope.

A correlation based on the results of Reference 30 is used to compute the

standoff distance (A ). The correlation, which includes dependencies on streamMach number, specific heat ratio and body bluntness ratio (i.e., r*/z*) isgiven by

A0 -(L) r* (2-Csphere

where (L0 (Y-l) A- + + 1 - 1 ,

sphere 4

r* is the sonic point ordinate and C is a bluntness ratio correction factorwhich is a function of r*/z*.

For subsonic flow in the shock layer the relationship between shock angle

and local body angle is obtained from Reference 3 and is given by

esi . 300+ -+ -+17 (2-2)

where e. and 8si are in degrees. The accuracy of this approach is discussed

in Reference 1.

For supersonic flow in the shock layer, the shock angle is determined by

the tangent cone ;pproximation. In this approach the shock angle is a functionof stream Mach number, ratio of specific heats, and body anle.

The procedure is used with all environment options except that for the

arc heater. Since hosetip shape change tests in arc heater environments aregenerally at a relatively low free stream Mach number, it is more accurate to

use a normal shock assumption for all inviscid flow calculations.

2.1.2 Pressure Distribution

The pressure distribution calculation is based on regional correlations

as indicated i.n Figure 2-3. Region I is defined as the subsonic portion of theflow forward of the limiting sonic flow charactetistic. Region II is the super-

sonic forebody. Region III is the flow over the aft conic surface of the nose-

tip and starts at the point where the body slope approaches the cone half angle.

2-4

44

00

L aL

II .

H,. 4

.ell

-* .2U

4,

I- 4,

I:l

1l.

2-,5

LA-

A

A flow chart identifying the various aspects of the pressure distribution cal-

culation is given in Figure 2-4. The procedures used to compute the pressurein the three regions are described in the following subsections.

M

r PCo:: it M IN a P50NIC PSTPEýMpu e PsTiuasow FOMRe 500~y

01S- PON1 TOZIUoNNO M1.1ATHC-H P •-l" A l}-I 4 E:: YES

INTEIZFOLAT 9ETWiN FINITCePOLATE TO SrbdRr oF cotieS~oNIC Pr AND Gs-7SO012 .L.CTION •1(.C

OP CONE To 6=:Zls

moc~iFiap N8E,"ToWJN -roMATCH POINT W1171 OE600

m CONE surc

4 M e-M .7 AFTBGE2Y

INERON AFT C ~ep~ C 02V \

Figure 2-4. Schematic of pressure distribution calculations.

2-6

2.1.2.1 Region 1, Stagnation Point to Sonic Point

The correlation of Reference 4 as improved in Reference 5 is used in

this region to more realistically represent the stagnation point velocity gra-dient and subsonic region pressure distribution on very blunt bodies. In addi-

tion, the correlati3on was extended to include low free stream Mach numbers.The correlation is an empirical extension and sjrnthesis of the modified

Newtonian ccrrelation, valid for spheres, but ircluding a correlation for flatfaced cylinders. It is expressed as follows:

LI '+=~ ((l--

1 SF- s/s)(l - j.)cos 28 + D sr [PiD- 1 + s/s*(l - p.)cos8

hee+ (2-3)where

P/po

Po - stagnation point pressure

RN - stagnation point radius of curvatureR =max (RN, R*)

R* =distance from sonic point to body axis, measured normal to the

surface at the sonic pointa= surface wetted length from stagnation point

e angle local tangent makes with body axis

sonic point

andJP'-= -/Po(2-4)

P - + (1 - %.)sin2 0 (2-5)

2 ky-lJ (2-6)

2-7

g~a h haeai - - e- a (2-7)

(flat faced cylinder pressure distribution)

with

S -n qn~*s (2-8)

The sonic point location is an important parameter in calculating thesurface pressure distribution, for it determines the surface length over whichthe Region I correlation is used in the 'subsonic" nose region. The importanceof the sonic point reflects the fact that geometry effects downstream of the

sonic point have no influence on the subsonic region flow field, and hence,

pressure distribution. In the code the sonic point is found using correlationswhich account for the effects of the following:

0 Free stream Mach nwuber (M)

0 Ratio of specific heats (y)

• Nose tip bluntness (r*/z*)

* Surface streamline recompression on biconic shapes (Bc)

The procedure is to estimate a sonic point location assuming modified Newtonian

flow and then correct the locat-ion for the effects noted above.

The modified Newtonian sonic point is the first point downstream of the

stagnation point (8 = 0), whilh has an angle (B) greater than the following:

= arc cos -( (2-9)

where p, and ' are defined in Equations (2-4) and (2-6), respectively, and 6*

is defined in Figure 2-3.

The nosetip geometry is then interrogated to determine the following:

"* Bluntness ratio at Newtonian sonic point

"* The existenne of a conic surface with 30* < c < 600 and the conicC

surface half angle 8c

The bluntness ratio, specific heat ratio, and free stream Mach number are usedto obtain a blunt body sonic point from correlations of exact numerical pre-

dictions.

2-8

y -1.4

2 < M<4

�* n080 - 3.495 J(r*/z')2 - a 2

*-49 + M - 2.0 (50.8- 49.9)

a -2.22 + i( - 2.2.0 (0 - 2.22)

4 < M<7

0* = 80* - 3.495 NJ(r*/z*)2 + b 2

Bo = 0. +M - 4.035o (51.3 - 50.8)

(2-10)b-oo+ N - 4.0 2-o

b = 0.0 +M - 4.0 (2.0 - 0.0)

M >___7

8 80* - 3.495 N 7 (r/Z*)2 + 4

0= 51.3

The uirves are hyperbolas. The expressions are written in a form to illustrate

as clearly as possible their interrelationships.

if a conic surface~ is formed in the vicinity of the sonic point, theminimum cone half angle for supersonic cone flow 4s also computed using the

folloiting correlations of exact solutions.

y 1.4

0 " 34.6 + 17.9e 3 7 " (M 2.0) (2-11)

2-9

- --.------ a ,.±--., ~ ~~o..-\- - - .-'2"--' -

O ., 26.0* + 23.60e-_ , (m _ 2.0)

y 1.1 (2-11)

Oc - 19.1 + 28.2ee0-314 (M - 2.0)

For other values of y, linear interpolation is used.

If the cone half Angle (ac) is greater than then the cone is super-

sonic and the sonic point i3 at the forward end.

The logic to decide whether the sonic point in controlled by cone flowor blunt body flow is shown in Figure 2-5.

2.1.2.2 Region II, Sonic Point to Match Point with Aft Body Correlations

In t!,e supersonic forebody of the nosetip (Region II) pressure distribu-

tions are computed either using the modified Newtonian expression (Equation

(2-5)) or, for biconic type configuritions, using a conic surface recompression

correlation. The cone recompression model is based on sphere/cone and

ellipsoid/cone exact solutions performed at various Mach numbers. The stream-wise length required to obtain the recompression is given by:

R)=,41.805(0 -e )2 -0.22 (2-12)0

where

00 = 1.047 radians

s = stream lengtb from stagnation point to the end of cone recompres-sion

RN = geometric stagnation point radius of curvature

Along a conic surface starting at si, the recompression pressures are given by

a linear function of stream distance; i.e.,

S - S.ii+SRc - Fi) for s < s < a (2-13)

" Pfor s > sSPC R

2-10

* AN4 15 PFUNSO AS

___________?_ LSOUNLPAR -Y'EZ

114TEGaRATiON MdiTAMS

VV1TMWINI I.E-4 RAViA~NS

6', rT ra, 6O

I ~C't.I -rAC exx E

YC-44

Figure 2-5. Flow chart of logic to determine nosetip sonic pointlocation (subroutine~ RUNLP).

where

PC Pc/P sharp cone pressure ratio

pi p/p = pressure ratio at start of cone, a

The pressure distribution computation in Region II of the nosetip mustalso blend together the results from the several correlations, including thefollowing:

0 Region I subsonic flow correlation

* Region II conic surface recompression, if any

0 Region III CD correlation, Prandtl-Meyer flow or modified Newtonian(see Section 2.1.2.3)

The smoothing is performed using a weighted average between an incremen-tal modified Newtonian expression and a linear aecay expression. For smoothingin the region einitial > a > 0fna

in iafinal

thTeineit + (1 >f Tnl d + a(O (Ff - i) (2-14)

f e

incremental linear interpolationmodified between end of mergingNewtonian region and sonic point

where a is the weighting function. Taking a linear weighting,

e - ei-" -(2-15)

gives

ef -2S+ e-e(1 -U({sin 2 - sin2 ei)

f i

÷ (2-16)

2-12

In a typical case, the smoothing expression might be used between thesonic point and the start of a forecone surface and between the end of the

forecone and the match point with the Region III correlations. In the casewhere concave shapes develop as in the sketch below, smoothing is performed

between the sonic point and the inflection point in the shape. Downstream ofthe inflection point, the modified Newtonian relation (Equation (2-5)) is used.

2.1.2.3 Region III - Aft Body

The correlation for aft cone pressures is one developed at Aerospace

Corporation (Reference 6). It has the form

=f 8F (2-17)ec~

where

8c = cone half angle

= axial distance from the start of the aft coneZc

D = diameter at start of aft cone

CD = drag coefficient of the forebodyCp = (p - p,.)/(1/2)poo• U2

The function f is determined by a series of polyaomial curve fits of exactnumerical solution for cones of varying bluntness, with cone half angle as a

parameter. The curves asymptotically approach the sharp cone pressure level.

Tt.' transition between Regions II and III is effected at the point wherethe pressure distribution curves for the two regions intersect. That point isdetermined iteratively since CD is a function of its location.

The calculation procedures used in Region III (aft of shoulder) arebased on hypersonic considerations. They are used for % > 5. To better modelthe flow for M, < 4, the modified Newtonian calculation procedure for Region IIis extended to Region III. For 4 < M. < 5, linear interpolation is used between

= 4 and M. = 5 predictions.

Alternate procedures are used for cylindrical afterbodies.

2.2.3 Boundary Layer Edge State

The actual boundary layer edge thermodynamic state is determined by alook-up on pressure and entropy in a real gas Mollier air table. Pressure isknown from the inviscid flow solution, and entropy is calculated from considerations

2-13

of bow shock shape and boundary layer mass flux. The sketch shown on the

following page illustrates the path of a streamline passing through the. shock

layer. At the point where the streamline, originating at y, enters the bound-

ary layer, the mass flux cai be expressed as follows:

Boundary layer edge

~e e

y _ _e_

For the laminar boundary layer

2 4.52 rue Re, (2-18)

For the composite m=-el of the turbulent boundary layer, which is described in

Section 2.2.2, the free stream location of the boundary layer edge streamline

is computed from

100 + 25ReP--i 10+ie 45re 8T (2-19)

The turbulent Reynolds number, Ree , is computed using a roughness augmentedT _

momentum thickness. This expression for YT passes smoothly from the laminar

value at !W = 0 to the previously used expression for turbulent flow at large

le.The entropy used to compute the edge conditions when the streamline

enters the boundary layer is the entropy existing at the radial coordinate YL

or- T just behind the shock. This entropy is evaluated from the free stream

conditions, using oblique shock relations (see Reference 6).

The boundary layer edge velocity over most of the body is computed from

energy conservation along an effective inviscid flow stream tube as follows.

Ue - (Ho - e) (2-20)

2

Since the boundary layer edge state is determined accounting for entropy

swallowing, the edge velocity also is affected. This is the only mechanism

by which swallowing influences the boundary layer solution.

In the vicinity of the stagnation point the velocity is assumed to be

linear for pe/Pt2 = 1.0 to 0.999. The velority at the 0.999 poiat is calcu-

lated accounting for entropy swallowing using a first guess obt&inei by assum-

ing Pormal shock entropy. Therefore, the velocity gradient, due/dsl , is oval-uated directly from the pressure distribution including entropy layer effects.

The edge viscosity is determined from the following correlation taken

from Reference 6.

S3.0 x 98. T < 2000OR

= 1.9 x 10-0 T > 2000OR (2-21)

T in *R, V in lb/ft-sec units

The Prandtl number is assumed constant at 0.7.

2.2 BOUNDARY LAYER HEAT AND MASS TRANSPORT

The boundary layer heat and mass transport events are modeled using a

film coefficient approach. The momentum integral equation is solved assuming

that zero pressure gradient relations between skin friction and momentum thick-

ness apply in the presence of pressure gradients. Reynolds analogy and compres-

sibility corrections are applied to obtain the nonblown heat transfer coefficient

distribution. Effects of blowing are accounted for as a function of local abla-

tion rate, and the mass transfer coefficient is taken as a constant ratio of

heat coefficients.

Details of the solution procedure for laminar flow are given in Section

2.2.1. The turbulent solution procedure is *discussed in Section 2.2.2. Tran-

sition criteria options in the code are given in Section 2.2.3. Techniques

used to compute the roughness effects on laminar and turbulent heating are

reviewed in Section 2.2.4, and relations used to compute heat transfer in re-

gions of transitional boundary layer flow are discussed in Section 2.2.5. The

effects of hydrometer boundary layer stirring are covered in Section 2.2.6.

2-15

2.2.1 Laminar Heat Transfer, Smooth Wall

The stagnation point heat transfer coefficient calculation is discussed

in Section 2.2.1.1, and the laminar distribution evaluation technique is de-

scribed in Section 2.2.1.2.

2.2.1.1 Stagnation Point Heat Transfer Coefficient

At high altitude or low Reynolds number conditions, the energy flux to

the surface is limited by the total energy content of the free stream. Thecorresponding heat transfer coefficient is, therefore,

qimit p"UHo (2-22)

PeueCHOl limit =p.u

For other conditions, the stagnation point heat transfer coefficient is

computed using the relation of Fay and Riddell (Reference 7) with Pr 0.7.

pUC 0.944(o o .o)°'s [.0+ (Lees 5 2 - 1.0) (2-23)

where, as suggested in Reference 8,

0o • To 5000ORH/Ho= (2-24)1 - 0.308(To/Ho), To > 50009R.

The Lewis number used is the average between the Lewis numbers evaluated at

the wall and edge temperatures. These are computed from the approximation

c1.2 , T < 5400ORLe -(2-25)1.2 - 5.5 x 10- (T - 54000R), T > 5400R(

2.2.1.2 Laminar Heating Distribution

The method of Reference 9 as simplified in Reference 10 is used to obtain

the laminar heating distribution. The correlation is expressed as the local

heat transfer coefficient divided by the stagnation coefficient, i.e.,

2-16

e PGUeCHL (2-26)

L POUeCH,O

p.

pL 0 (2-27)

where

S.10 -0.1625 + 0.0625 O )21

(.uJJ'1.116 + 0.411 "' 0 062()

du

2ý- 2(_ (fe(!e)u r2 ds

The corresponding laminar momentum thickness Reynolds number is obtairnd byapplying Lees transformation to the Blasius incompressible flat-plate skin fric-

tion relation. The resulting equation is

Re 0.664 ueue re _ 2 ds (2-28)[ L eir [Pe7o p0 ePO

This Reynolds number and the associated momentum thickness are used to obtainboundary layer thickness parameters for use with transition criteria, transi-tional heating correlations, and turbulent boundary layer starting conditions.

2.2.2 Turbulent Heat Transfer, Smooth Wall

The compresiiible boundary layer momentum integral equation is solved to

evaluate the fully turbulent heat transfer coefficient distribution. The im-

portant assumptions included in the solution are as follows:

0 Blowing effects may be decoupled from the boundary layer solution(computes nonblown transfer coefficient).

* Boundary layer shape factor H - 6*/B is taken as -1.

2-17

0 A modified Reynolds analogy (explained more completely below) isused to relate heat transfer to skin friction.

0 The Crowell incompressible composite skin friction expression (Ref-

erence 11) modified for compressibility, is used.

The integral momentum equation may be written as

d TW Ur + l due 1 6(+u)e) =e (2-29)en~ %o

Using properties (C, 5) evaluated at the Eckert reference enthalpy (Reference 12)

S=O0.5 hw + 0.3 he + 0.2 Ho (2-30)

The Crowell composite skin friction expression modified for compressibility is

0.0128 Pun.222 (2-31)ce Re '

where

A =and Re100 + Re,

This expression substituted into the integral momentum equation (which is then

integrated) determines 9(s). Trial calculations for sphere cone geometries were

carried out assuming -1.0 < H < 0.5. Although H - 0.5 is probably most realistic

for conditions of interest, the skin friction was found to be relatively insen-sitive to variations in H. The closest approximatio;. (within 10 percent) be-tween 0 (0) from the composite model and the laminar calculations is obtained forH = -1. This value., often assumed because it simplifies the momentum equation,was adopted here.

In using Reynolds analogy to determine the Stanton number from the skin

friction, separate factors are used to multiply the laminar and turbulent con-tributions to the skin friction. The laminar Reynolds analogy factor (taken to

be independent of s) is determined from the requirement that the composite modelyields the correct heat transfer at the stagnation point, that is,

2-18

e e-.H)lamS,u "e(2-32)

0.278 (• .

The turbulent Reynolds analogy factor, FT, is currently taken to be unity over

the entire body. There is, however, some evidence from turbulent BLIMP solutionsthat T is a function of pressure gradient, being about 0.95 on the nose and

about 1.15 on the cone, th.erefore, the factor, R, was chosen to be

RT 1 (2-33)

This is close to the suggestion of Kays' (Reference 14) in which the exponent onthe Prandtl number is -0.4. The composite expression for the turbulent heattransfer coefficient becomes

•e 0.0128 PUeAPeueCH,T 0.278 - RL + 01 - (R (1 X) + RT) (2-34)

The turbulent heating factor is defined as

PeueCt~rFT -eUeCH,2-35)

2.2.3 Transition Criteria

Built into the code are several optional techniques for determining the

conditions for boundary layer transition. In summary these are:

"* Momentum thickness Reynolds number versus boundary layer edge Machnumber.

"* Run length Reynolds number versus boundary layer edge Mach number.

"* Axial distance from stagnation po nt versus vehicle altitude.

"* Roughwall transition criterion based on momentum thickness and wall

cooling ratio.

"* Roughwall transition criterion based on displac-ament thickness andrun length.

"* Fully turbulent flow from the stagnation point (composite nodel).

The appropriate boundary layer quantities are computed and compared to critical

values input in tabular form by the user. For the two roughwall criteria,

2-19

critical values of the appropriate parameters from Referenco 15 are built into

the code. For the transition correlating parameter involving the momentum

thickneso, critical values are

Re1 k 255, onset1, BI Pe 1215, location) (2-36)0+ + 4

For the transition correlating parameter involvirg the displacement thickness,critical values are

Rek•. /,_ . 2300, onset I(-7_•

{ 2000, location

where ki is the intrinsic roughness of the surface appropriate for laminar

flow conditions as specified by the user and Rek = Uek

Th'. onset conditions are determined first and, if s-ticfied, the pointof transition is found from the location condition. Equation (2--37) indirectiy

accounts for the effects of surface temperature on trarsition through the dis-placement thickness, 6*. The displacement thickness is computed from the momen-

tum thickness, 0, and wall to edge temperature ratio, (Tw/Te) as follows:

0HO WALL 0 OLD WA [1.104 - 0.348(T/I•o)]

(8COLD WALL from Eq. (2-28)) (2-38)

[2-8-o(T.oT)- 0l.0] i6* 0HOT WALL [280uT/e .640]

2.2.4 Surface Roughness Effects on Heat Transfer

Correlations from PANT wind tunnel data (Reference 16) are included toaccount for the effects of roughiiess on laminar and turbulent heat transfer;

in addition, surface roughness modeling accounting for intrinsi - roughness,

scallop roughness, and crater roughness are included. Roughnesti effects on

lminar and turbulent heating are discussed in Sections 2.2.4.1 and 2.2.4.2

and surface rcughness modeling is covered in Section 2.2.4.3.

2-20

'-S

2.2.4.1 Roughness Effects on Laminar Heating

The effects of rouohness on laminar heating are correlated with theparameter

2 HOT WALL )o %HOT WALL

The effect is accounted for with the multiplicative factor KL or. the smoothwall laminar Stanton number, ClL, where

S10 RKL < 50

L CHL 1.307 Ln(OE) + 23.09 -if- 6 > 50

The correlation is applied to all laminar flow locations.

2.2.4.2 Roughness Effects on Turbulent Heating

The effects of roughness on turbulent heating are corr-lated in Refer-ence 16 using the following parameter:

( ) Peuekt (Te 0.5

=Re e CHT"H,T (2-41)

As with laminar flow, the effect of roughness on turbulent heating is accountedfor with a multiplicative factor KT on the smooth wall Stanton number, CHT.The correlation equation is as follows:

1.0 , •-r• < 10

CH-- R = Oglo(RT). , l0 < T-- < 10' (2-42)

CH, T

3.0 ,WK > 10

In the expression for the composite turbuls--t heat transfer cofficient, thelaminar and turbulent contributions are au.nuented individually, so that cn arough wall,

2-21

1'e 0.0128 "u ePeueC H,T,R =0.278 0 RLKL-- (RLKY. (I - X) + RTK T) (2-43)•e

For the purpose of output a composite turbulent augmentation factor is definedas

We TR (2-44)"TiC PeUeCH,T

2.2.4.3 Surface Roughness Modeling

Three types of surface roughness are modeledI Intrinsic roughness, ki

* Turbulent or scaliop roughness, kt

0 Crater roughness, kc

Intrinsic roughness (ki) is that associated with the basic material granularity

and is input as a constant for ea(:h material. The intrinsic roughness is used

in the transition and laminar heating correlations, unless there are particle

impacts as described below.

The turbulent ro3ughness (kt) is the effective sand grain roughness that

results from turbulent ablation and is used in the turbulent heating correla-

tions. The roughness height, k,, is specified in one of two ways. A uxiiform

value of kt for all turbulent regions may be input by the user; or a value may

be obtained by using the scallop dimension correlation from Reference 17. From

the correlation, the effective turbalent region scallop depth Js computed as

follows:

kt = lX (2-45)

where

kt the effective sand grain roughness height suitable for use in

Equation (2-41)

K1 a material dependent property determined from experimental data

and input by the user (a nom.nal value for graphite is0 77

N1 0.93 in-psia )

2-22

pa - instantaneous local edge pressure

Crater roughness (k ) results from the impact of hydrometer particles.

Presently, the assumption of hemispherical craters is used in conjunction with

the mass loss parameter, G, described in Section 2.4.2. Therefore, the crater

depth is derived from

kc 1 d/ (2-46)

where

Cp =hydrometer particle densityp = surface material densitym

dp hydrometer particle diameterG - me/min = mass loss parameter

kc - crater depth

Since crater roughness (kc) occurs over the entire body, the local roughness

(either intrinsic (ki) or turbulent (kT)) is compared to the crater roughenss

(k c) and the larger is used.

2.2.5 Transitional Boundary Layer Heat Transfer

Transitional heating is computed using a modified version of the corre-

lation of Reference 18. The correlation used is expressed as follows:

H,TRANS,R CH,T,R (2-4)

The values of ATR and n are computed differently depending on the approach to

fully turbulent heating; that is £Zr

1 > CHTR " H,TRANSR > 0.4- (CH,T,R -CH,L,R)

then

n = 0.8S

and

TR(cHR CHLR (2-48)

2-23

|--~--- - - -

-W -Y-.. . - --••r ,

where TR denotes the values at the point of transition. For

0.4 > CH TOR - CHTRANSR

(TCHTRCHLRTR

then

n - 2.0

and

ATR [Re2 (CH - CHTRASR)] (2-49)

where the nubscript, 0.4, denotes the point where ATR is reevaluated.

Since the boundary layer is transitional, the momentum thickness Reynolds

number (Re9 ) in Equations (2-47) and (2-49) is computed by integrating the re-

duced form (i.e., H = -1) of the momentum integral equation, Equation (2-29),

assuming unity Reynolds analogy factor and using the rough wall Stanton number

(C H,TRANS,R) from the previous boundary layer integration station, i.e.,

p erRe eli-1 + (peUerH,TRANS,R)i_l (si- Si-I)Re = ri (2-50)

It should be noted that use of rough wall Stanton numbers, C andH,L,R

CH,T,R, in the transitional heating correlation provides for a reasonable

transformation from the laminar to the turbulent roughness effects models

described in Section 2.2.4.

2.2.6 Hydrometer Boundary Layer Stirring Effects

Experiments indicate that in regions of laminar flow hydrometer parti-

cle impaction and subsequent erosion can cause significant augmentation to

the undisturbed laminar heat transfer rate. An option is provided in the

code to model this laminar stirring augmentation. The correlation is in terms

of the ratio of the disturbed (stirred) heat transfer coefficient to the undis-

turbed coefficient. The correlation is of the form

C C[D - (1 + G) sin2e (2-51)~STIRRED T~

2-24 j

2•'1

where

pp = particle density

p. = freestream air density

G = erosion mass loss parameter (described in Secticn 2.4.2)

0 = local body angle (8 = 90* at the stagnation point)

The constants (C and c) are presumed to be a function of the surface material.

Reference 19 indicates that graphite data are best correlated by

C = 0.098

c 0.317

The implementation of the stirring augmentatiun logic is flagged by the

JROUGH flag described in Section 3.1.8. When the stirring augmentation corre-lation is employed the augmentation factor calculated from Equation (2-51) is

compared with the factor calculated from Equation (2-40) and the larger is used.

2.3 IN-DEPTH CONDUCTION CALCULATIONS

This section briefly describes the numerical technique used to solve the

heat conduction equation inside the nosetip and the coupling between the sur-

face energy balance relations (Section 2.4) and the in-depth conduction solu-

tion. The details of the conduction package are described fully by Crowell

(Reference 28). In this section only a brief review of the procedure is

presented.

Section 2.3.1 describes the coordinate systems and governing equations.

The finite-difference formulations of the differential equations and their solu-

tions are explained in Section 2.3.2. The conduction time step control is dis-

cussed in Section 2.3.3.

2.3.1 Coordinate System, Governing Equations

The ir-depth coordinate systems for shell and plug geometries are

illustrated in Figure 2-6. For the shell geometry a body orientcd coordinate

system which is located on the internal contour is used (x,y,#). For the plug,

the geometry is split into two sections, separated by a fictitious boundary

(shown as a dashed line in the figure). The location of the interface between

the two s,.Lions is chosen such that the geometry of section I is exactly that

of the 5hell configuration. Thus, the coordinate system for region I is also

body oriented and located on the fictitious boundary. The shape of the interface

2-25

-- ~ A __ - -

.7. .4 9.ZzX

K-~ . ~4~ (~/L7~Aio*-

~ S~&. ~/O,

x2~~El&-aN

AC~~/A0Y4 ~1 AMA~I\ I

Figure 2-6. Nosetip geometry.

2-26

is taken to be spherical for convenience. Cylinderical coordinates are used

in region II (the shank portion of the plug).

In the body oriented coordinate system (region I) the heat conduction

equation for temperature dependent properties and isotropic conductivities

may be written as

PmCprb(a+AY) T = 7X] + [k r r(+AY) Al

at 3X l+AYY a+ a 11+m (2-52)

where A is the curvature of the internal contour or the fictitious interface.

For the cylinderical coordinates (region II) the conduction equation

takes the following form

p-(aT a rb a + k ) +- (ka-T +) (2-53)

It should be noted that due to the axisymnetric assumption, there ib. -o 'tem-

perature variation in the #-direction, although the conduction package iscapable of handling full three-dimensional problems.

The boundary conditions on all surfaces consist of specified heat flux.

For most nosetip problems ali the boundaries except the receding surface are

insulated and these fluxes are zero. In case of the plug geometry the ficti-

tious interface between regions I and II is not a boundary of specified flux

or temperature. The temperature distribution along this boundary is computed

by requiring that the temperature and heat flux in the regions I and II be

identical at the interface. At the recedin.g surface the boundary condition is

-k T = qond (t, (2-54)an T, n)1w

where n denotes the direction normal to the nosetip surface and the functional

form of q is determined from the surface energy balance formulation

(Section 2.4.1).

In order to solve the moving boundary conduction problem over a fixed

domain, the surface movement is incorporated into the heat conduction equation

through the use of the following transformations

2-27

Region I:

Region II:

Tin rb rt

The differential equations (2-52) and (2-53) and the boundary conditions

(2-55) are transformed into the new coordinate system and then solved by a

finite-difference technique. The description of this finite-difference pro-

cedure is given in the following section.

2.3.2 Finite-Difference Formulations

The finite-difference scheme which has been adopted is the Dufort-

Frankel method that is an unconditionally stable explicit technique. This

method uses a central time difference and therefore, requires storage at two

time levels. The Dufort-Frankel method does not require the restrictive Limestep limitation of standard explicit technique (At < A-s), but for consistencypurposes it requires that At goes to zero faster than Ax.

Variable mesh spacing is used throughbut. First order derivatives are

written in second order central or one sided difference forms and for the

second order terms the Dufort-Frankel form is used. For the details of the

finite-difference formulation the reader is referred to Reference 28.

"When the equations are differenced, the left hand side will contain the

temperature of a node (jkl) at the n+l time level and the right hand side will

contain the temperatures of the neighboring nodes at n-l and n time levels andTn+lknown geometric parameters. The difference equation is then solved for jkl"

In order to start the calculations both the n-l and n time levels are set equal

to the initial temperature distribution.

In region 1, the domain over which the temperatures are obtained from

the differential equations runs from J = 2, JMAX-l in the X-direction, K = 2 to

KMAX-l in the Y-direction and L = 1 to LMAX in the #-direction. In region II,

the temperatures are calculated from the differential equations for JP - 2 to

JPMAX-l, KP = 2 to KPMAX-l and L = 1 to LMAX in X, Z and *-directions respec-

tively. In the present axisymmetric code, the computations are only performed

in the L - 1 plane. The boundary temperatures are calculated from the finite-

difference forms of the boundary conditions.

2-28

Following the calculations of the interior point temperatures from thedifference equations, the centerline values (X-0, J-1 and JP-l) are obtained

by back extrapolation from the known values of J=2 and J-3 nodes. For axi-

symmetric nosetips the following condition must be satisfied along the

centerline:

3T = 0 (2-55)

These derivatives are written in one-sided for.iard difference forms and solved

for centerlir~e temperatures including the stagnation point temperature.

The surface temperatures and recession rates are determined from simul-taneous solution of tht difference form of Equation (2-55) with the surface

energy balance relations.

2.3.3 Time Step Control

The computational time steps are controlled by a comprehei~sive technique

to achieve numerical stability, economy and output versatility. The code has,basically, two kinds of time steps: a conduction time step and an environmenttume step. The print time step is currently set equal to the environment time

step.

2.3.3.1 Conduction Time Step

The time step of conduction calculations is t- minimum of the followingvalues:

0 Explicit finite-difference stability limit: d2/4a where d is theminimum mesh size and a is the thermal diffusivity of the nosetipmaterial. This is not currently in use because the Dufort-Frankel

scheme is unconditionally stable.

* Surface temperature rise control: Atold(STRD/STRM) where STRD isthe input desired surface temperature rise in one time step, andSTRM is the maximum surface temperature rise achieved during the

previous time step.

* Surface heat flux rise control: Ato(qo/qn CrF, where q isthe maximum surface heat flux and CTF is the desired growth factor.

A recommended expression for CTF in terms of the desired maximum

surface temperature rise is the following:

CTF - 1.5 + (STRD - 140)/300 (2-56)

2-29

SSurface recession control: 6 /Smax, where S is the maxixum valueof surface recession rate and 6 is the smallest distance between the

first and second nodes in the Y-direction.

At the first conduction step when a majority of the above quantities cannot becalculated, the following time step is also used.

(&tc - SED .(2-57)

2.3.4.2 Environment Time Step

The environment time step determines the frequency with which the invis-

cid and viscid solutions are to be updated and is equal to the user specified

value in the absence of time step stability criteria. In the presence of sta-

bility criteria, the environment is redefined whenever either one or both of

the following conditions are satisfied.

* If the local surface temperature changes by a factor greater than1.4.

* If the tangent of the local body angle changes by a factor of twoor more.

The reference condition for the above two tests is the last environment defini-tion.

The computation time step, DTH, is the minimum of the conduction and theenvironment time steps. Furthermore, the computation is automatically termin-

ated if the compvted time step is less than the user specified minimum, DLTMIN.

2.4 SURFACE ABLATION RESPONSE CALCULATIONS

The formulation of the surface energy balance technique used to computethe surface ablation response is discussed in Section 2.4.1; modeling of erosiondue to hydrometer impacts is described in Section 2.4.2; computer codes to pro-vide the necessary input data are described in Section 2.4.3; and simplified

means of the surface energy balance equation are presented in Section 2.4.4.

2.4.1 Surface Energy Balance Formulation

The ablation rate and surface temperature at points on the nosetip aredetermined by accounting for energy, mass, and species conservation at theablating surface. The sketch below illustrates the ablating surface control

2-30

qrad qrad

qsen qche m i out m ehc

L L._L __....L_,.

; hc qcondC

Sketch of Surface Energy Balance Control Volumeand Energy Flux Terms

volume and the energy fluxes of interest. The surface energy balance equation

employed is of the convective transfer coefficient type. In the program, this

energy balance equation takes the following form:

PeUeCH(Hr -hew) + PeUeCM L (Ze - ziw)hi - Btchw + mtchc cond

qsen qchem

+ q - FawT w= 0

qrad qrad

in out

(2-58)

Before commencing a term by term discussion of Equation (2-58), however,

it will be useful to describe the general nature of this transfer coefficientexpression. Like all such expression, Equation (2-58) is an approximation, the

usefulness of which depends mainly on the validity of the transfer coefficientapproach. A discussion of this subject is far beyond the scope of the present

document. It may be observed here that transfer coefficients have successfullycorrelated both data and "exact" solutions in simple heat or mass transfer prob-

lems, and in combined heat and mass transfer problems for unity (or near unity)

Lewis number. Equation (2-58) attempts to extend the transfer coefficientapproach to both nonunity Lewis number and unequal mass diffusion coefficient

problems, still allowing for chemical reactions and net mass transfer effects.This approach was suggested in Reference 20. Its validity is discussed in

References 21 and 22.

In Equation (2-58), the term q represents the "sensible convectiveheat flux." Physically, this is the convective het flux which would occur

for a frozen boundary layer and a noncatalytic wall in the absence of mass

2-31

transfer;* it excludes all che.iical energy contributions. The term qsen isperhaps more Lnually written in the form

q PeUeCH (H - hs) (2-59)

but, since generally it is more convenient for the user to input Hr rather than

Hs" qsen in Equation (2-58) has been written in a modified form in which Hrappears. This form has the additional advantage that the driving force forenergy transfer involves only edge gas states. The derivation of the modifiedform from Equation (2-58) is given in Reference 20 and Reference 21.

The quantity hew in the qsen term is part of the input thermochemical

data discussed below. The transfer coefficient peu eCH and the recovery enthalpyHr are time dependent variables computed in the program for each analysis loca-

tion. The transfer coefficient is automatically modified from the nonblownvalue to implicitly account for the effect of the computed ablation rates. The

following relation is used:

CH Ln (1 + 2 XB~cL= H (2-60)

H,O 2I•tc

where

B• implicitly determined normalized thermochemical ablation rate(tc/ PeUeCM)

I= an input number discussed belowCH

- = ratio of blown to nonblown Stanton numberCH, 0

Specified values of X allow the user to fit blowing correction curvesof CH/CH,0 versus B'c to account for special effects in the few cases wherethese are known with confidence, such as molecular weight effects or variableproperty effects. In view of the uncertainties, it is generally recommendedthat X = 0.5 be used for laminar flow. A value X = 0.4 appears to correlate

constant properties for turbulent data somewhat better. For graphite in air,studies have indicated that a value of 0.7 for both laminar and turbulent

flow is most appropriate.

More generally in the presence of chemical reaction it is the diffusive heatflux from the gas to the wall even in the presence of net mass transfer, pro-vided the boundary layer is frozen and the wall is catalytic.

2-32

The term qcond in Equation (2-58) is obtained from the in-depth coraduc-tion analysis as a function of Tw and S (Equation (2-54)).

The term qchem in Equation (2-58) represents the net amount of chemicalenergy fluxes at the surface. The z*-difference term represents transport ofcnemical energy associated with chemical reactions at the wall and :..n the

boundary layer, it is the ciiemical energy parallel to the sensible convectiveheat flux term. The z* driving forces for diffusive mass transfer include theeffects of unequal diffusion cbefficients; for equal diffusion coefficients thez*'s reduce to the familiar mass fractions Ki. The Btchw term represents energyleaving the surface in the gross motion (blowing) of the gas adjacent to thesurface. The mass transfer coefficient (eUeCM) is obtained from the blown heatcoefficient (peueCH) using a user specified factor, CM/CH. Remaining quantities

S*T T C/Hnot yet discussed are Bc' he ,Zhw, Ezi hl, and T does not appeartc e iei Iw i w w2xplicitly but is necessary to Jluate the temperature dependent values ofvarious quantities. The quantitles TwI ,LFZwhi', and hw are input by the useras the dependent variables in a table with three i-idependent variables: p,ae eCM, and BE (if no chemical kinetic effects ;Are considered only two inde-

pendent variables p and Bt are required).B,

Similarly, the quantities EZeh." and h are input as the dependentie A. h aew ipta h eednvariables in a table with p and T as independent variables. These tables aretypically generated by the ther'iochemistry codes described below. Furtherdiscussion of these tables is given in Section 3.1.9.

Notice that the erosion mass loss rate (me) does not appear in Equation(2-58). This is because the eroded material is assumed to leave the surfacewith the enthalpy of the solid (hc) and, hence, cancels with the net incomingmass rate (m) to give

inhC -mehc =itchc (2-611

where tc is only the thermochemical portion of the total mass rate (m). Thetotal mass rate (6) (and hence i ) is required, however, in the conduction equa-etion (Equation (2-56)) to compute the net recession rate (S). The calculationof the erosion mass loss rate (le) is described in Section 2.4.2.

The surface energy balance solution procedure may be summarized as fol-lows:

1. Obtain Hr Pe ueCH,OD and p from environment definition routines.

2. Reduce the in-depth conduction matrix to calculate the constants inthe equation for qcond (Equation (2-54)).

2-33

3. Calculate erosion mass loss rate (me)

4. Correct or adjust PeUeCHo for blowing effects

5. Compute eeC = (CM/CH) (PeUeCH)

6. Assume B'c

7. With p, PeUeCM, and Btc, look up in iniput surface thermochemistry

tables values of Tw, Ez*w hT%". hw

8. With p and %, look up in input edge gas thermochemistry table

values of Fziei, hew

9. Construct Equation (2-58), noting departure from zero, if any

10. Adjust Btc quess to reduce departure from zero (Newton-Raphsoncorrection)

11. Go to Step 4 and continue

This procedure converges on a new B'I value in very few iterations. The sametcprocedure may be used wih Th as the independent variable and B' as a depen-dent variable.

2.4.2 Erosion Modeling

Surface erosion due to hydrometer particle impacts is modeled by two

different types of correlations depending on the surface material. For gra-phite type brittle materials the erosion mass loss correlation is of the form

e bmc d-=G =Aum sin 8 (2-62)* lwpMin

where

me= erosion mass loss fluxe

in= Pcusin 8 = incoming particle mass flux

PC = mass of particles per unit volume of air

u = vehicle velocity

m = wd3p /G = individual particle massp p p

d - particle diameterp8 local body angle relative to the axis (8 = 900 at the stagnation

point)

2-34

The co•stants (A1 , b, c and d) in Equa 4 ion (2-62) are determined by correlationof ground test data for each given material. For materials which char (i.e.,carbon phenolic) a different set of constants is required for the charred andvirgin plastic materials. The two erosion rates predicted for a fully charredand virgin surfacc are "bridged' together based on the relative rates of sur-face erosion and in-depth char generation. Additionally the body ancrle (6)dependence in Equation (2-62) does not fully collapse all carbon phenolic dataand, hence, lo' zad high angle erosion correlation constants are required aswell as a "bridging" function. Currently specific erosion correlation constantsand bridging functions are built into the code fer carbon phenolic. Refer-ence 19 discusses the details of this modeling and Reference 23 covers thegraphite erosion models. The input section (Section 3.1) describes how thesespecific correlations may be evoked.

For malleable type metal materials the erosion mass loss correlationis of the form

.~U2

e G (2-63)

where

CN = damage coefficient

The damage coefficient (C,) is determined from experiment and is primarily afunction of surface temperature (Tw). Typically the damage coefficient decreasesas the surface temperature approaches the melt tempe-ature. Built-in valuesof CM versus Tw are available for tungsten. Reference 19 gives the details

of the tungsten correlation and the input section tells how it may be evoked.

The input required for erosion calculations is the cloud profile, whichis a table of:

* Mass concentration (p

* Particle diameter (dp)p

0 Particle specific gravity (ys)

versus altitude.

2.4.3 Use of Thermochemistry Codes to Generate Input Data

Section 2.4.1 above makes it clear that some complex tabular thermo-chemical input is required if the surface energy balance boundary condition isto be used. These tables are generated by any one of a number of separatecomputer codes. The most recent such code is designated General Nonequilibrium

2-35

AI

Ablation Thermochemistry Code (GNAT). 7t is a general open and closed system

thermochemical nonequilibrium code spe':ifically constructed for this purpose

(Reference 24). Other Aerotherm thermochenistry codes which treat only equi-

librium thermochemistry are in existence. The most recent c, these is the

Equilibrium Surface Thermochemistry Code, Version 3 (EST3), which is described

in Reference 25. A generally similar code which differs from EST3 only in added

detail is designated ACE and is described in Reference 26. An older version of

EST3 was designated EST2 and is described in Reference 27. To obtain the nec-

essary data tables for input, the user selects sets of values for the pressure

(p), transfer coefficient (PeueCM), and nondimensional thermochemical ablation

rate (B't). (Note, if chemical kinetics are not considered the only parameters

required are pressure (p) and ablation rate (B'c).)

The user specifies the elemental composition of the environment gas andthe ablating material, and supplies some general species thermochemical data

for all molecules to be considered in the system. Finally, the user specifies

the unequal diffusion coefficients if they are important. The thermochemistry

code then computes all the dependent quantities of interest at each table point•.,Twin the p xB matrix of independent variable values, namely, Tw, EZi h. , and

Lc 1W 1 Thw, and punches this information on cards. Similarly, the tables of )Ziehiw

and hew values are prepared as functions of p and T, and punched out on cards.

All these cards form part of the Table 09 card input deck (see Section 3.1.9).

2.4.4 Simpler Forms of the Surface Energy Balance Equation

As noted in Section 2.5.2 above, for equal diffusion coefficients the

z' driving forces reduce to the simple mass fractions K,. If in addition toT

equal diffusion the user specifies that PeueCM = eU eCH, then sinre KehW = hew

and Ki = hw by definition, Equation (2-58) simplifies to the more fami)iar

form

Su CH(r- (1 + Bc)hw)+ mtbc - q + a - FacT4 = 0 (2-64)

In this expression hew and • Zhiw do not appear, hence the correspondingl~e 1

table is not necessary and need not be included in the input (see Section 3.1.9

below).

A steady state ablation option is also available. If this option is

specified the q term in Equation (2-58) is calculated by taking an energy

balance on a control volume extending from below the ablating surface down to

the thermally unaffected material (see the following sketch).

2-36

T T W

S" I --'• cond' d/ "'c IL I E/dr J T - Tj x 530OR

; Ttmhc .

IC Sketch of Control Volume Around Thermally Effected Material

Energy conservation on the above control volume gives

•Ti dEmdhc - q mh - dE (2-65)

where Tih - the enthalpy of the thermally uneffected material before

exposure (Ti is assumed to be 530°R)

dE- ". rate of energy storage in control volume

The steady state assumption implies that dE/dt is zero and corresponds to thephysical situation when the temperature profile relative to the moving surface

is invwriant with time. The assumption is accurate for low conductivity abla-

tors and for high ablation rate situations. By considering dE/dt = 0, qmaybe calculated from Equation (2-65) as

cond" ( hc - h) (2-66)

Notice that for the steady state assumption, q is independent of materialthermal properties and response history.

2.5 SURFACE POINT MOVEMENT AND SURFACE SMOOT-v.-,G

The surface energy balance determines the recession rate normal to .he

surface at each body calculat'... pint. Based on the time step size (At) these

points are then moved the e.s'.; esp',3c ng distance to define new body points at

the end of the time step ':- :,.. cdy points are then used to define new sur-

face inclination angiE-. "i e body points. In a typical nosetip shape change

problem the size of important geometric features in the stagnation region de-

creases in turbulent flow, and eventually becomes smaller than can be efficientlymodeled with typical body point spacing. When a numerically limited nose radiusis reached logic is applied to define an apparent nose radius.

2-37

The numerics of shape change are described in the following sections.

[ The shape change geometry for body point movement is described in Section 2.5.1;

surface angle definition techniques are presented in Section 2.5.2; and the

apparent nose logic is discussed in Section 2.5.3.

2.5.1 Shape Change Geometry

With reference to Figure 2-6, the location and shape of the surface is

completely defined by A(X,*,t). The rate of change of A with time can be re-

lated to the surface normal recession rate, S by the following equation:

.. ~ [ _____ ~ .\fl /2

S+ -AA k +R+Acosa ,%2-67)

where A, R and B are the geometric parameters of the internal contotr defined,

respectively, as: curvature, radial distance from the nosetip axis end angle

of inclination with respect to the nosetip axis.

The Equation (2-67) is written in an explicit finite-difference form

and solved for the values of A at the n+l time step. Along the centerline

where R = 0, Equation (2-67) is not applied and the condition 3A/3X = 0 is

used to determine the new position of the stagnation point.

In the steady state conduction option the body points are moved along

lines of constant radius as indicated by the sketch below.

r

The relation for the amount of axial movement is

Snormal AtAz 1nl2-681sin 8

2-38

When using the steady state energy balance, conduction considerations do

not limit time step size; hence, the only consideration limiting time step sizeis shape change. In other words, the calculated recession rate distribution

cannot be applied over such a time span that the body shape (and, hence, reces-

sion rate distribution) changes in a drastic manner. The criteria applied is

that the tangent of the local body angle may not change by more than a factor

of two.

2.5.2 Surface Angle Definition

Numerical shape change calculation techniques are strongly sensitive to

the method used to define the local surface angle since this angle strongly

influences the surface pressure, heat transfer, ablation, and erosion calcula-tions. Circular curve fits and straight line interpolation techniques are

The circular curve-fit method involves basically fitting a circular arc

through the point of interest and the points on either side (i.e., three points

define a circle). The body angle is then defined as the tangent to the circle

at that point. Exceptions are taken to this definition if the radius of curva-ture is negative in order to avoid unrealistic concave shapes.

2.5.3 Apparent Nose Model

As shape change proceeds on nosetips for which transition is near the

nose, the stagnation point radius of curvature becomes too small to model withthe typical body point spacing. The code has internal logic to determine when

this numerccally limited nose radius is reached, and at that time an effectivespherical nose radius is computed. This specification controls anly the detail

at the stagnation point and does not limit or redefine the overall shape ofthe nosetip.

The apparent nose radius logic used with the circular curve fits involves

basically fitting a tangent sphere into the *cone" formed by the second and

third body points, and is primarily based on geometrical considerations.

Neither of the apparent nose radius or body angle definition techniquesdescribed above is ccmpletely successful in predicting all observed nosetip

shape change regimes. Hence, both must still be considered to be in the de-

velopmental stage.

2-39

SECTIM 3

DESCRIPTICNS OF INPUT AND OUTPUT

This section provides detailed user oriented instructions for code inputand a description of the output. The input instructions are described in Sec-tion 3.1 and output features are covered in Section 3.2.

3.1 INPUT INSTRUCTIONS

The input to the code can be read either from data cards for an initialrun or from magnetic tape or disk for a restart run. The details of the inputfor each of these types of runs are described below. The basic input for an

initial run eonsists of:

"* One restart information card.

"* Three title cards.

"* Nine input tables.

Not all nine of the input tables are required for every run. Each table ispreceded by a single card containing the identifying table number.

For a restart run only the single restart information card is required,the rest of the information is read from magnetic tape or disk.

The following sections describe the restart information, title cards,

and nine input tables, respectively.

3.1.1 Restart Information

The "restart' card is the first card in the data deck. If the run is arestart it is the only card required, and tells the code the iteration fromwhich to begin restart. For an initial run the restart card tells the code how

often to write restart files. All restart reading and/or writing is done onLogical Unit 11 and it should be assigned accordingly. The restart card for-

mat is as follows:

3-1

Card

No. Columns Format Data Units

1 1-3 13 ISKI.2 - number of environmental calls

between restart file writes.

4-6 13 IRESRT - restart flag

0 - first run no restart

>0 - transient restart - read theIRESRTth set of data on Unit 11to start the run. No other in-put is required

<0 - steady state restart

Note that if both ISKIP and IRESRT are equal to zero no data will be read or

written on Unit 11 and it need not be assigned.

3.1.2 Title and Heading Information

The second set of input data are three title cards. They are used totransmit title and heading information to the output. The first 72 columns ofeach of these cards may be used for the title, the alphameric information in

.olumns 61 through 72 of the third card being used as a page heading on all

pages after the first.

3.1.3 Table 01 - Genitral Program Constants

These cards supply the code with computation tame information and program

flags ovhich indicate options to be subsequently read.

Card

No. Columns Format DatA Units

1 1-2 12 Table No.

2 1-12 E12.5 Initial value of problem time sec

13-24 E12.5 Final value of p:oblem time sec

25-36 E12.5 First output tine increment. This in-terval represents the time incrementfor output. Provision for changingthis time increment within a run isprovided by tha NTIC flag describedbelow.

37-48 E12.5 DLTMIN - time step stability flag sec<0 - '..i, stability

-0 set to 10-S sec>0 - stability criteria used

3-2

II

Table 01 (continued)

Card

No. Columns Format Data Units

49-60 E12.5 CTF - defined by Equation (2-56).

If CTF is less than 1.2 or greaterthan 1.7 it is set to 1.3.

61-72 E12.5 STRD - maximunm desired surface tem-perature rise in one conduction time ORstep. If STRD is less than 49 orgreater than 201 it is set to 750R.STRD is not required for the steadystate option.

3 1-3 13 TC - Flag denoting type of transi-tion criterion to be subsequentlyinput in Table 04. TC < 0 denotestransitional heating is used in thesurface energy balance and TC > 0denotes abrupt transition is used.

ABS (TC)

0 - all laminar flow (Table 04 isnot needed)

1 - momentum thickness ReynoldsNo. vs. edge Mach No.

2 - run length Reynolds No. vs.edge Mach No.

3 - axial distance vs. altitude

4 - rough wall transition basedon

S2300, onsetRek ' =k (*) (2000, location

(Table 04 is not needed)5 - rough wall transition based on

Ree11o7+ + B' Pe) 15, onset

4 +1 215, location

(Table 04 is not needed)

6 - fully turbulent flow (Table 04is not needed

4-6 13 ENV - flag denoting environment optionE-be subsequently read in Table 02

"1 - flight option

2 - wind tunnel

3-3


CardNo. Columns Format Data Units

3 - ballistic range4 - general

5 - arc heater

7-9 13 CF - flag controlling curve fit andapparent nose option

0 - circular curve fits and noapparent nose logic

2 - circular curve fits and appar-ent nose

10-12 13 SO - special output flag

0 - boundary layer solution only(no ablation or shape change).

1 - general problem with ablationand shape change. Shape pro-files written on file 15

13-15 13 NTIC - number of time interval changes(not number of time intervals)NTICmax = 10. A non-zero entry in thiscolumn causes sets of time intervalchanges to be read from the next card.

16-18 13 ISS - conduction option flag

0 - transient conduction option.Sphere-cone initial geometr;with geometric progression dis-tributions of surtace andin-depth grids.

1 - steady state conduction option.Initial geometry and surfacepoint distributions same as abcve.

2 - steady state conduction option.General initial geometry andsurface points distribution. Thedetails of this input are de-scribed in Section 3.1.5.

19-21 13 IPRNT - flag which determines the amountoT-environmental output at print times.Six output tables are available and thecontents of each is described in Sec-tion 3.2.2.1. IPRNT < 0 denotes outputfor each integration point and IPRNT > 0denotes output for body points only.

3-4

Table 01 (concluded)


ABS (IPMT)

0 or 1 - print Table 12 - print Tables 1 and 23 - print Tables 1, 3, 4 and 5

4 - print Tables 1, 3, 4, 5and 6.

22-24 13 LPRNT - flag similar to IPiUNT which de- --

I-erines the amount of output at inter-mediate computation times (i.e., whencomputation time not equal to print time).LPRNT < 0 denotes output for each inte-gration point and LPRNT > 0 denotes out-put for body points only.

ABS (LPRNT)

0 - no output

1 - abbreviated output of envir'n-ment and recession only

2 - print Tables 1 and 2

3 - print Tables 1, 3, 4 and 5

4 - print Tables 1, 3, 4, 5and 6.

4 This card supplies information for changes in the output time inter-val and is read only if NTIC > 0.

1-12 E12.8 Second output time interval sec

13-24 E12.8 Time for change to second output time secinterval

25-36 E12.8 Third output time interval sec

37-48 E12.8 Time for change to third output time sec

interval

49-60 E12.8 Fourth output time interval sec

61-72 E12.8 Time for change to fourth output time secinterval

5 Same for NTIC output time interval changes. Three sets per card, to(etc) a maximum of ten sets.

3.1.4 Table 02 - Environment TableA

The basic environment information required by the code is the freestream

state (pressure and density' and vehicle/gas relative velocity. Given this

3-5


information the code performs real gas calculations for air to find the stag-nation conditions. To aid the user in performing calculations for variouscommon flight and ground test facility environments, five environment input

options are provided. They are:

1. Flight environment - input altitude and velocity as a function of

time; free stream conditions a: e found from a built-in ARDC standardatmosphere table.

2. Wind tunnel environment - input supply pressure and temperature as a

function of time as well as free stream Mach No. and ratio of specificheats (y); free stream conditions are found based on assumed airidentropic expansion at constant y.

3. Ballistic range environment - input projectile velocity and rangepressure as a function of time; air assumed to be at 75tF to obtainfree stream density.

4. General environment - input free stream pressure, density, and veloc-ity as a function of time.

5. Arc heater environment - input consists of a quantitative descriptionof the arc heater flow field and of the start-up transient which themodel experiences when injected into this flow field.

For the flight, wind tunnel, ballistic range, and general environmentoptions, Table 02 is basically a time table of envirorment conditions. Forthe arc heater option the environment is more complex and in this case Table 02provides input of both th- spacial and tetporal variation of environment.

There are a ma-dmum of 50 entries in this table; at least two entriesare required.

3.1.4.1 Input for Flight Option, ENV = 1


1 1-2 12 Table No. 02

2 1-2 12 Must be blank --

3-14 E12.8 Time sec

15-26 E12.8 Altitude ft

27-38 E12.8 Velocity ft/sec

3 Same as Card No. 2 for increasing time.(etc)

3-6


3.1.4.2 Input for Wind Tunnel Option, ENV - 2


1 1-2 12 Table No. 02


3-14 E12.8 Time sec

15-26 E12.8 Supply pressure psia

27-38 E12.8 Supply temperature

39-50 E12.8 Froe stream ratio of specific heats --(Card No. 2 only)

51-62 E12.8 Free stream Mach No. (Card No. 2 only) --


3-14 E12.8 Time sec

15-26 E12.8 Supply pressure psia

27-38 E12.8 Supply temperature OF

4 Same as Card No.. 3 for increasing time.tetc)

3.1.4.3 Input for Ballistic Range Option, ENV = 3


1 1-2 12 Table No. 02 --

2 1-2 12 Must be blank

3-14 E12.8 Time sec

15-26 E12.8 Range static pressure atm

27-38 E12.8 Projectile velocity ft/sec

3 Same as Card nu. 2 for increasing time.(etc)

3.1.4.4 Input for General Envirornment Option, ENV = 4


1 1-2 12 Tabia No. 02 --

1-2 12 Must be blank --

3-7



3-14 E12.8 Time sec

15 26 E12.8 Free stream static pressure atm

27-38 E12.8 Free stream static density lb/fts

39-50 E12.8 Free stream velocity ft/sec

51-62 E12.8 Ratio of specific heats (Card No. 2only)


3-14 Z12.8 Time sec

15-26 E12.8 Free stream static pressure atm

27-38 E12.8 Free stream static density lb/ft'

39-50 E12.8 Free stream velocity ft/sec

4 Same as Card No. 3 for increasing time, maximum of 50 entries.(etc)

3.1.4.5 Input for Arc Heater Fnvironment

The free stream environoent produced by an arc plasma generator, such asthe AMPDL 50 Wf RENT facility, has characteristics distinctly different from theenvironment options described above (ENV = 1-4). In general, the pressure levelis constant with time but varies with distance from the nozzle exit because of

nonparallel flow streamlines. Input consists of one card containing the steadyoperating conditions (including model location) and a table of normal shock totalpressure ratio as a function of distance from the nozzle exit.

An option also exists for specifying a free stream pressure variati.onduring a start-up transient. The ratio of instantaneous total free stream pres-sure to the steady value for two or more times are input. The option is flagged

by reading in a nonzero entry for the length of the start-up transient (DTIME)which is read from the second card. The values of total pressure ratio versustime are input using the same read statement which specifies the spacial varia-

tion of pressure ratio, by adding 100 to each of the times. Again a maximum ofSO entries is allowed.

3-8

---- --- -- - -- ------ -- ---

Table 02 (concluded)Card

NO. Columns Format Data Units

1 12 Table No. 02 --


3-14 E12.8 Ptstady, total free stream pressure atm

15-26 E12.8 HO, total enthalpy Btu/lb

27-38 E12.8 X0 , axial distance from nozzle exit in.plane to stagnation point (assumesmodel advance with recession)

39-50 E12.8 K1 , multirplying constant in the Fay &Riddell stagnation point heating rela-tion (if K1 - 0, it is set to 1.0)

51-62 E12.8 DTIME - Duration of start-up transient sec

63-74 E12.8 y, specific heat ratio (if y = 0, it --is set to 1.2)

3 1-2 12 Must be blaalk

3-14 E12.8 X, distance from nozzle exit plane in.

15-26 E12.8 pressure ratio -e

4 Same as Card No. 3 for increasing X values.(etc)

n 1-2 12 Must be blank

3-14 E12.8 Time + 100 sec

15-26 E12.8 Pt . /Ptt , pressure ratio --

steady'n+l Same as Card No. n for increasing time, maximum of 50 entries includ-ing Card No. 3. There must be at least two x-values and two time

values when this option is used. Time variant environment logic isnot debugged.

3.1.5 Table 03 - Geometry

In the first part of this section initial geometry and in-depth grid art.described and in the second part the input format of Table 03 is explained.

3.1.5.1 Initlai :;eometry

The initial body geometry is input as a table of coordinates for severalpoints on the body. This input is different for steady state and transientconduction options. The differences are described below.

3-9


1. Transie.t conduction option - a sphere cone initial geometry and

geometric progression dietribution of body points are assumed.

Input sphere radius, cone half angle, number of body points and

the common ratio of the gecmetric progression. The initial geom-

etry may be of a shell or plug category. For a shell, the internal

contour is assumed to be a sphere cone whose sphere radius and cone

half angle are input. For a plug, the shark geometry is ALrput.

The conduction option flag in this case is ISS 0. The details of

these inputs are described in Section 3.1.5.4.

2. Steady state option - under this option, three types of input are

possible for initial geometry and surface point distributions:

0 Sphere cone geometry and geometric progression distribution ofsurface points. This input is the same as the transient option

input. The conduction option flag in this case is ISS = 1.

0 Sphere cone geometry with uniform surface point distribution.

Input sphere radius, cone half angle, axial length and number

of body points desired on sphere and cone.

0 General body - input a table of up to 30 body coordinates (r,z;.

The conduction option flag in the above two cases is ISS = 2.

3.1.5.2 In-Depth Grid

The input of the in-depth grid applies only to the transient conduction

option of the code. The in-depth grid system is shown on Figure 2-6. Since

the code is presently limited to axisymmetric geometries, we will only consider

the grid distributions in the X- and n-directions.* Geometric progression dis-

tributions of grids in both X- and r-directions are assumed. Therefore it is

only required to input the number of the grid points and the common ratios in

X- and n-directicns.

3.1.5.3 Surface Temperature

The code assigns one input value of surface temperature to all bodypoints for all input options except the general body. If no surface tem-

perature is specified, a default of 530*R is utilized. For the general

shape option a surface temperature distribution can be input.

n defined on Page 2-28.

3-10


3.1.5.4 Table 03 Input Format


1 1-2 12 Table No. 03 --

Read The Follcwing Cards Only if ISS = 0 or 1. If ISS = 2 go to page 3-14.

2 1-4 4A PS - "PLUG" or "SHEL",pecifying the nosetip geometry

9-13 4A SB - ignored if PS = "PLUG", must5'e "SOLI" if a solid body isdesired. The solid body is a shellwith no internal contour.

3 1-5 15 ABIATE - integer variable definingsu-' e movement.

= 0 surface movement is specifiedin subroutine TRANS

1 surface movement is calculateQin subroutine ABL8

6-10 15 MOVE - integer variable defining=su--ace movement

= 0 no surface movement= 1 allow surface movement

If ABLATE = 1, MOVE must = I

11-15 15 KAPFLG - has meaning for shellgeometry only

zero curvature of the inter-nal contour and BETA =/2,ZJ = XJ, ZB = 0 and KAPPA = 0

= 0 finite curvature of theinternal contour

16-20 i5 IPHI -. flag for extrapolation in=,- -'anes. Applies only to three-

dimensional in-depth computations.

0 all back extrapolations tobe done for half the 0-planes;the other half are definedsymmetrically

1 the back extrapolation to bedone for L=l only; the otherplanes are set to the L=lvalues

3-11



21-25 15 NOSYM - flag for symmetry in O-direction --Applies only to three-dimensional shapechange calculations.

0 assumes a symmetry planeimplies no symmetry

r 26-30 15 IZERO -flag for angle of attack --

= 1 zero angle of attack;T7FLG and T22FLG will beturned on, NOSYM set tozero and LMAX=3.

In the present axisymmetric calculations,a value of 1 should b( entered for IZERO.

4 1-5 i5 JMAX - number of X steps +1 (maximum-o -7-3)

6-10 15 KNAX - number of ij steps +1 (maximum=-6)

11-15 15 LMAX - number of * steps +1 (maximum

5 1-10 F10.0 AE - controls the mesh spacing in then-direction

11-20 F10.0 AX - controls the mesh spacing in ther--directionAE and AX are the common ratios of thegeometric progressions for grid dis-tributions in n- and X-directions.For uniform mesh, set AE=AX=1. For afine body point distribution near thestagnation point enter AX>l, and for afine in-depth grid near the nosetipsurface enter AE<I.

geometric Parameters Relative To The Internal Contour(For a plug these variables are calculated internahlyand the input values ignored.)

6 1-10 F10.0 RN - nose radius ft

11-20 F10.0 THC - cone half angle deg

21-30 F10.0 XLEN - maximum X-distance. In the ft,SOLID case, XLEN must be an angle radin radians.

3-12



31-40 F10.0 BSUBB - square of the ratio of the --major to minor axis of an ellipse.Enter a value of 1 for sphericalcontour.

7 1-10 F10.0 T2FLG- flag for X-variation

= 1 no X-variation, TERM2 andT2 are internally set to 0

= 0 allow X-variation

11-20 F10.0 T7FLG - flag for #-variation

W 1 no #-variation, TERM7 andT2 are internally set to 0

= 0 allow %-variation

In the present code enter a valueof 1 for this flag.

Cards 8, 9 and 10 Are To Be Input For The Plug Geometry Only

8 1-10 F10.0 GAMMA - angle from the horizontal degU--W-defines the inclination of theplug shank

11-20 F10.0 RLEN- radius of the plug shank (r 1 ) ft

21-30 F10.0 ZLEN - length of the plug shank (Z1 ) ft

9 1-5 Is KPMAX -number of Z steps+I --

10 1-10 F10.0 T22FLG - flag for #-variation in --th-79ank

= 1 no #-variation; T22 is set

equal to zero

= 0 allow *-variation

11 1-10 F10.0 TINITL - initial temperature of the OR

Geometric Parameters Relative To The External Contour

12 1-10 F10.0 RN2 - nose radius ft

11-20 F10.0 THETA2 - cone halt angle deg

21-30 F10.0 ZL - nosetip overhang for SHEL; total ftaixial length for PLUG or SOLID (seeFigure 2-6)

13 1-5 15 NNMAT - material index assigned toal-lsurface and in-depth grid points

3-13

- .. , . " " . o r.. r 1' 7s'r 7 . -r7" -



Read The Following Cards Only if ISS = 2

2 1-5 15 NS - number of points on the body7irface (maximum 30 points)

>0 sphere-cone geometry

40 general shape

6-10 Is NPN - number of points on the nose;a'•licable only to sphere-cone option(NS>O)

11-20 F10.5 RSTAGI - initial nose radius ft

21-30 F10.5 ZM&X - maximum axial length ft=-p ere-cone option only)

31-40 F10.5 ANGLI - initial cone half angle degsphe're-cone option only)

41-50 F10.5 TS - initial body temperatures ORIT entered zero, it is set to 530*R

General Shape Option Only - (read only if NSO)

3 1-2 12 NC- flag to read the coordinates of --

We body points

=0 keep reading

S 0 stop reading. This indicatesthat the card is the last ofits kind.

3-14 E12.8 ZSP - body point axial length measured ft'rom the stagnation point

15-26 E12.8 RSP - body point radial length ft

27-38 E12.8 ATS - body point temperature. If ORe-ntered zero, it will be set to TS

39-40 12 IMAT - body point material index

4 Same as Card No. 3 for the rest of the body points.(etc)

3-14

- - - -v -X -

3.1.6 Table 04 - Transition Criteria

This table communicates criterion for specifying when boundary layertransition occurs. Of the six transition criteria available three require tab-ular information which is transmitted through this table; they are momentum

thickness Reynolds No. (Re0 ) vs local Mach No. (Me), run length Reynolds No.

(Res) vs local Mach No. (Me) and axial transition location vs altitude. TheTC flag read with the general constants denotes which criterion applies. This

table must contain at least two, but no more than 30 entries.

3.1.6.1 Input for Re0 versus Me (ABS(TC) = 1)


1 1-2 12 Table No. 04 --


3-14 E12.8 Local edge Mach No. (Me) --Me)

15-26 E12.8 Transitional moment-m thickness --Reynolds No. (Ree)

3 Same as Card No. 2 for increasing Me.(etc)

3.1.6.2 Input for ese versus Me (ABS(TC) = 2)


1 1-2 12 Table No. 04


3-14 E12.8 Local edge Mach No. (Me) --

15-16 E12.8 Transitional run-length Reynolds No. --(Res)

3 Same as Card No. 2 for increasing Me-(etc)

3.1.6.3 Input for Axial Transition Location vs Altitude (ABS(TC) = 3)


1 1-2 12 Table No. 04


3-15



3-14 E12.8 Altitude ft

15-26 E12.8 Axial transition location in.

3 Same as Card No. 2 for increasing altitude.(etc)

3.1.7 Table 05 - Weather Conditions

This table inputs the hydrometer environmental conditions for use inperforming erosion calculations. The input consists of certain erosion calcu-lation flags and a cloud altitude profile. This table is not required forclear air calculations.


1 1-2 12 Table No. 05

2 1-5 15 NCL - number of cloud entries. IfNCL < 0, cititudes are in meters

6-15 E10.2 Maximum altitude of cloud ft or m

16-25 E10.2 Minimum alaitude of cloud ft or m

38-39 12 NOSLO - particle shock layer slow-aio f lag

0 - no slowdown

1 - particles are slowed down asthey impinge on the shock wave

2 - particle mechanical breakupmodel employed

40-41 12 NOHEAL - crater roughness healing

0 - no healing of craters

1 - crater healing by ablation ismodeled

3 1-10 F10.5 Altitude (independent variable). The ft or munits arb dictated by the meter flag.

(2-62)) (2-62) (2-58)

3-16



11-20 F10.5 Particle mass concentration gm/mr3

21-30 F10.5 Particle diameter mxl0-6(micron)

31-40 F10.5 Particle specific gravity. If enteredas zero it is set to 1.0.

This table must contain at least two but no more than 20 entries and the alti-

tudes must be entered in an increasing order.

3.1.8 Table 06 - Material Properties

Table 06 is used to input material surface roughness and thermal proper-ties. The material index number (MAT) assigned to a given material need (in

general) only be consistent with the material indices used in the nosetip con-

figuration input (Table 03). If, however, hydrometer erosion effects are to

be included the following assignments must be followed due to built-in values

for certain of the erosion correlations.

0 Carbon phenolic MAT = 2

0 Tungsten MAT = 3

Two types of surface roughness are input:

* Laminar or intrinsic roughness (ki) which is used for rough wall

transition criteria and in calculating roughness augmentation tolaminar heating.

* Scallop or turbulent roughness (kt) which is used in calculating theroughness augmentation to turbulent heating.

Turbulent surface roughness may be input either as a constant or calculated--. 77

according to k -Kl_0'77 where Kiand a maximum roughness height are input.

Three roughness heating augmentation options are allowed for; they are

h No roughness heating augmentation.

- Laminar and turbulent heating augmentation according to the modelsdescribed in Section 2.2.4 - but n hydrometer stirring effects.

SSame as above -but including hydrometer stirring effects.

3-17


The material thermal properties required include certain constants (i.e.,

density, heat of formation, etc.) plus tabular values of quantities which are a

function of temperature (i.e., specific heat, conductivity., and emissivity).

Notice that for the steady state conduction option the specific heat and con-

ductivity are not required and may be entered as dummy values (e.g., 1.0).

Cardo__. Columns Format Data Units

1 1-2 12 Table No. 06 -

2 1-2 12 MAT - material index. If erosion --effrects are included use the follow-

"ing assignments.

1 - graphite

2 - carbon phenolic3 - tungsten

3-10 E8 RHO - material density lbm/ft 3

11-20 El0.5 TFO - datum temperature for heat of ORTrofmation. For JANNAF data,TFO = 536*R

21-30 E10.5 HFO- heat of formation Btu/lbm

31-40 E10.5 TBRPL - laminar blowing rate reduction --

parameter. A in Equation (2-60).

41-50 E10.5 TBRPT - turbulent blowing rate reduc-tion parameter. A in Equation (2-60).

3 1-2 12 NERODE - erosion law number

=i generalized input(read the next card)

=2 carbon phenolic erosionmodel

=3 tungsten erosion model

-----.Next card input the constants of erosion law (Equation (2-62)) -----(for NERODE = 1 only)

4 1-10 F10.4 A1

11-21 F10.4 bConstants in Equation (2-62)

21-30 F10.4 c

31-40 F10.4 d

3-18



5 1-2 12 JROUGH - roughness heating augmenta- --Fin--flag.

0 - no augmentation

1 - roughness augmentation, but nostirring augmentation

2 - roughness and stirring augmenta-tion

3-14 E12.8 RUFL - intrinsic roughness height, ki in.

15-26 E12.8 RUFMAX - turbulent roughness height in.

>0 - constant turbulent roughnessequal to RUFMAX

<0 - calculate turbulent roughnessaccording to

kt = -

with ktmax = ABS (RUFMAX)27-38 E12.8 K1- turbulent roughness height at in.

e= 1 psia. Read only if RUFmAX < 0.

6 1-2 12 NC - flag, nominally zero, +1 marksterminal card of last material propertytable, -1 marks terminal card of otherintermediate material property tables.

3-10 FI0.5 Temperature (independent variable) OR

11-20 P10.5 Specific Heat Btu/lb-OR

21-30 F10.5 Thermal conductivity Btu/ft-sec-oR

31-40 F10.5 Emissivity

This table must contain at least two but no more than J0 entries.

Currently Tables 07 and 08 are reserved for future use.

3.1.9 Table 09 - Surface Thermochemistry

Table 09 consists of the parameters necessary to utilize the surface

energy balance formulation described in Section 2.4. The following paragraphsdescribe the input for a given material.

3-19


A single lead card specifying the material index is read first, followedby a card containing control integers, followed by a set of tabular thermo-chemxstry data cards.

3.1.9.1 Table No. and Constants 3


1 1-2 12 Table No. 09

2 1-5 15 MAT- material index no. --

3 1-5 15 NBPF - flag controlling reading of --mechanical fail quantity (for usewith blowing correction)

NBPF

0 - no mechanical removal1 - mechanical removal tern read

and used to reduce blowing rate

6-15 FI0.5 CMH - ratio of mass to heat transfercoefficients (typically 1.0).

3.1.9.2 Edge I.ithalpy Data

Equation (2-58) of Section 2.4 indicates that if diffusion coefficientsare not equal or if the ratio Cm/CH is not unity, then the surface energy balancerequires data about the edge gases of the boundary layer. These data are pro-

vided in special "edge tables" which precede each pressure section of the sur-face tables (the various sections of the surface tables are described in Sec-

tion.3.1.9.3 below). The independent variables for an edge table are pressure

and temperature. Dependent variables are h and the sum Tz* h~w.ew ~le 1


4 1-8 F8.5 Pressure atm

9-16 F8.5 Blank

17-24 8X Blank

25-33 F9.4 Temperatu.-e (*R if nega-tive in whichcase enthal-pies beloware Btu/lb)

34-38 F3.3 Unequal diffusi.n exponent y

3-20



Tw39-47 F9.3 Summation ziehni cal/gr

(Btu/Ib iftemperatureis enteredwith minussign)

48-56 F9.3 Enthalpy of edge gases hew cal/gr(Btu/lb iftemperatureis enteredwith minussign)

57-58 12 -1 (flag signifying that thiscard is part of the edge gastable)

59-60 2X Blank

61-66 A6 Unused

67-78 2A6 Leave Blank

5 Same as No. 4 fo- remaining entries in "edge table' for this pressure,(etc) maximum of 12 temperatures for each pressure.

Note that although the thermochemistry codes described in Section 2.4.3will provide data decks using *K and cal/gr, in those rare cases in which a

user wishes to supply his own deck and prefers to work in OR and Btu/lb, he maydo so simply by introducing a minus sign as a flag in front of the temperatureentries.

The table length is limited to 5 pressure sets (it may have only 1 pres-sure set) with not more than 12 nor less than 3 temperature entries in each set.The series of temperature values may be different for the edge table at each

pressure set. The table is organized as a series of sections, eacN represent-ing one pressure and each preceding the corresponding pressure group of the sur-face thermochemistry deck as described below. The temperature entries withineai- " ction must be ordered, either ascending or descending. Similarly, thepresb.,res must be ordered either ascending or descending. Decks generated by

the thermochemistry programs will have been automatically ordered properly.

3.1.9.3 Surface Thermochemistry Tables

3.1.9.3.1 Description of Surface Thermochemical Tables

This table comprises a series of sections. Each section represents onepressure and one transfer coefficient value. More than one transfer coefficient

3-21


may be necessary if the effects of kinetics on the surface response are con-

sidered. Nondimensional ablation rate, BtL, forms the third independent vari-able within a given section. The table has three dependent variables: 2Z! hiw,

v 1whw,w-nThe thermochemistry programs generate separate groups for each pressure,

one at a time. All these groups together make up the surface thermochemistry

deck. Within each pressure group the transfer coefficient values will beordered. Within each transfer coefficient section, ablation rate entries neeAnot be ordered in any particular way on the ablation rates; any necessary order-ing Js made automatically by the code as it reads in the data.

Users providing their own thermochemistry decks must ensure that thetrensfer coefficients are ordered, but the ordering may be either ascending or

descending in each case. The surface thermovhemistry cards are identified bya unity flag in column 58, as described in the form,-- specification below.

The number of pressure groups may not exceed 5 (and may be only 1); the

number of transfer coefficient values in each pressure group may not exceed 5

but may be only 1. If no kinetics effects are to be considered a transfercoefficient of zero is acceptable. The sequence of transfer coefficient values

need not be the same in the different pressure sections. Within each transfercoefficient section the number of ablation rate entries may not exceed 25 ea-dmay not be less than 2. The series of ablation values, Btc may be unique oreach section.

The *R-Btu/lb option described for the edge tables in Section 3.9.3.2may be used for these tables also.

3.1.9.3.2 Card Formats


n 1-8 P8.5 Pressure atm

9-16 F8.5 Transfer coefficient* lb/ft2 sec

17-24 F8.5 Nondimensional ablation rate -i/PeUeCM = B'

25-33 F9.4 Surface temperature OK(*R if riega-tive in whichcase enthal-pie3 beloware Btu/lb)

Not provided by most Aerotherm thermochemistry codes.

3-22



34-38 F5.3 Unequal diffusion expunent y

39-47 FS 3 Summation •z h~w cal/grSw 2. (Btu/Ib iftemperatureis enteredwith minussign)

48-56 F9.3 Enthalpy of wall gases h. cal/gr(Btu/Ib iftemperatureis entered

with minussign)

57-58 12 Flag indicating surface thermo-chemistry table entry

59-60 2X Blank

61-66 A6 Chemical symbol of surface species.(ACE and GASKET programs print suchsymbols arranged alphabetically anOtruncated from right end if necessary).

67-78 E12.3 Nondimensional mechanical failrate = ma*/peUeCM = Bail

remaininn+l Same as Card No. n for remaining entries in th-is section; maximum of

25 entries in each section.

3.1.9.4 Assembled Thermochemical Deck

Figure 3-1 shows a picture of an assembled thermochemical data deck

for several pressures. The deck corresponds to repeating the input describedin Sections 3.1.9.2 and 3.1.9.3 for each pressure.

The surface equilibrium data deck must be terminated by two blank cards.Output decks of the thermochemistry programs do not have such cards, and the

user must supply it.

3.1.10 End of Input

The end of the input data deck is signaled by a single card with a -1

punch in columns 1 and 2.

3-23

SAj

"\Blank Cards

(Un to 5 Pressures)

Surface Table forSecond Pressure

Edge Table forSecond Pressure

'--Surface Table forFirst Pressure

Edge Table forFirst Pressure

Control Integers

Material Index Card

Figure 3-1, Sketch of surface thermochemistry table make-up for a givenmaterial including leading constant cards.

3-24

3.2 PROGRAM OUTPUT

The program output consists of output of the input data for check and

verification purposes, and output of actual calculations. The output %f theinput is covered in Section 3.2.1 and the calculation output is described in

Section 3.2.2.

3.2.1 Output of Input

Program output begins with an output of the restart information, and

follows with tUe contents of input Tables 01 through 09. Most of this output

is fully 1le,'ee and is printed exactly as input by the user. For those fewoutput items r,.t fully labeled, Appendix A provides a description.

The sensible enthalpy term output from the material properties informa-tion (Table 06) is defined as

T

hc= HFO + C dT (3-2)

TFO

where

hc = sensible enthalpy

HFO = heat of formation (input by user)

C = specific neatp

T = temperature

The integration is performed numerically by simming over the table entries.

The surf.-e thermochemistry table is output zeordered with increasingablation rates in each section. For e,--h entry in the thermochemistry tables

tha program computes an'" outputs the quaz.tity

"TCHEM hw + e w)hiw - B+hw - hw (3-3)

Notice that this term combines all of the tabular input surface thermochemis ry

terms and %hen combuined with the general surface energy balance equation fEqua-tion (2-59)) gives

PeueCH (Hr + TCHEM) - aowqr - w = 0 (3-4)

3-25

The purpose for the creation of the TCHEM term is to reduce computer storagerequirements. For the simpler case of equal diffusion and heat and mass trans-

fer coefficients (i.e., no edge gas tables and CM - Ca) the TCHEM term is re-defined to be

and Equation (3-4) is sLill applicable.

3.2.2 Calculation Output

Two types of calculation results are output, they are the environment

calculation and the surface energy balance/in-depth calculations. The environ-

ment output is jiscussed in Section 3.2.2.1 and the surface energy baiance and

in-depth cutptut is covered in Section 3.2.2.2.

3.2.2.1 Environment Output

The er.vironment output consists of six tables which may be called

according to the procedure given in Section 3.1. The following paragraphs

give a brief description of each of these tableb and the sample problem (Sec-tion 4) shows typical output.

Table 1 - Summary Information

Table 1 contains a summary of current geometry, trajectory and statevariable values. This table is especially useful as a quick check on the cur-rent nose radius, stagnation point state, and stagnation point recession.

Also inclu.3 d are the transition and sonic point locations, free stream Machnumber, and number of calls to the environment and conduction packages (theseare equal for the ateady state option). For the flight option, the current

value of altitude is given.

Table 2 - Summary Distribution

Table 2 is essentially a selective condensation of Tables 3, 4, 5, and6. Table 2 contains current body geometry, edge pressure ratio, and Mach num-ber distributions. Also included are heat transfer coeffic?-nt distributionsand the momentum thickness Reynolds number distribution. The LAM flag outputin Table 2 indicates the boundary layer flow regime and has the following threevalues:

3-26

* 1 - laminar flow

0 0 - transitional flow

* -1 - fully turbulent flow

Table 2 finds its major application in cases where it is desired to keep the

amount of printout to a minimum. This is particularly useful when a largenumber of output intervals is expected, and minimwmi output is sufficient.

Table 3 - Entropy Swallowing Information

Table 3 contains the current body geometry, shock shape, and the entropyswallowing distribution.

Table 4 - Boundary Conditions

Table 4 shows the computed distributions along the body of boundary layer

edge properties and recovery and wall conditions. Included in Table 4 (and in

Tables 5 and 6) is the NTB or transition flag. This flag has four values indi-cating the flow regime at a given integration point. These are:

* -1 - laminar flow

* 0 - onset of transition

0 1- transitional

* 2 - fully turbulent "low

Table 5 - Heat Transfer and Boundary Layer Quantities

Table 5 displays distributions of heat transfer coefficiernts, heating

parameters and heat flux. Moientum and displacement thickness and laminar momen-

tum Reynolds number distributions are also tabulated.

Table 6 - Roughness Heating Quantities

Table 6 gives distributions of laminar, turbulent (augmented and smooth),and transitional Stanton numbers. Other useful entries include distributions

1/3of surface roughness height, both transition parameters (Rek(s/6*) and

ie

the turbulent .-ating augmentation parameter, and the net (laminar and turbu-

lent) heating augmentation factor.

3-27

3.2.2.2 Surface Energy Balance and In-Depth Output

Following the environment output comes the in-depth and surface energybalance output in that order. The in-depth output consists of time stepinformation, current nosetip material configuration, and temperature distribu-tion. The time step information gives the size of the various controlling timesteps (see Section 2.3.4), the current conduction iteration, and problem time.

Since, typically, more conduction time steps are required than environ-

meant time steps the temperature arrays are output only for the last conduc-tion time step preceding an environment call. The time step information isoutput for every conduction time step. For the steady-state conduction optionno in-depth output is required or given.

The surface energy balance and new body point location output is thefinal output prior to the next environment call. The output consists of:

[ New body point locations.

a Surface temperature.

* Recession rate (both thermochemical and erosion).

. Nondimensional ablation rate (B').

* Crater roughness height.

* Surface heat flux.

0 Heat transfer coefficient (both blown and nonblown).

For the transient conduction option these outputs represent the results of thelast conduction time step preceding the environment call. For the steady stateconduction option the output applies over the entire time step between environ-ment calls.

3-28

SECTION 4

SAMPLE PROBLEMS

4-1

Sample Problem No. 1

Sample Problem No. 1 is a steady state wind tunnel prediction

of an 80 sphere cone camphor model with a 1.5-inch nose radius.

This problewr is typical of a low temperature ablator (LTA) test.

It demonstrates the generality of the material and thermochemistry in-

put, as well as the variable time step output and environment input

operations. The initial zhape is code-generated by implementing the

sphere-cone option. Also incorporated is the stability controlled

time step option.

4-2

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Sample Problem No. 2 is a steady state weather flight prediction bof a 90 ATJ-S graphite sphere cone nosetip with a 0.65-inch nose radius.

This problem demonstrates the use of the flight environment

option. In addition, the weather option is utilized. Also, the sphere-

cone input option is repeated.

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Sample Problem No. 3 is a steady state clear air flight prediction

of a 70 ATJ-S graphite sphere cone nosetip with a 0.65-inch nose radius.

This problem repeats the flight environment option, however, this

time employing a clear air condition. Again, the sphere-cone input

option is used. Also, a short output option is demonstrated.

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REFERENCES

1. Abbett, M. J. and Davis, J. E., "Passive Nosetip Technology (PANT) Program,Interim Report, Vol. IV - Heat Transfer and Pressure Distributions onAblated Shapes, Part II - Data Correlation and Analysis," Aerotherm Report74-90, Aerotherm Division, Acurex Corporation, January 1974.

2. Foiles, S. R., "Shock Configuration Around Spherically Blunted Cones,"Lockheed Missiles & Space Company, TIAD/831, LMSC Internal Report,Sunnyvale, Calif., May 1966.

3. Lomax, H. and Inouye, M., "Numerical Analysis of Flow Properties AboutBlunt Bodies Moving at Supersonic Speeds in an Equilibrium Gas," NationalAeronautics and Space Administration, NASA TR R-204, July 1964.

4. Love, E. S., et al., "Some Topics in Hypersonic Body Shaping," AIAA papei.69-181, 1969.

S. Moyer, C. B., et al., " A Coupled Computer Code for the Transient ThermalResponse and Ablation of Non-Charring Heat Shields and Nosetips," AerothermCorp., Mountain View, California, NASA-CR-1630, October 1970.

"6. "Hardening Technology Studies - III, Vol. I, Combined Thermochemistry andThermomechanical Response in Nosetips," SAMSO-TR-69-181, March 1969.

7. Fay, J. A. and Riddell, F. R., "Theory of Stagnation Point Heat Transferin Dissociated Air," J. Aero. Sci., Vol. 25, 73-85, 121 (1958).

8. Lockheed Missiles & Space Company, "Advanced Composites (RESEP II), Nose-tip Response and Thermal Buckling Analysis," Final Report, Change OrderNo. P001, LMSC-D052386, Sept. 6, 1969.

9. Cohen, H. B., "Boundary-Layer Similar Solutions and Correlation Equationsfor Laminar Heat Transfer Distribution in Equilibrium Air at Velocitiesup to 41,100 Feet Per Second," National Aeronautics and Space Administra-tion, NASA TR-118, 1961.

10. Hearne, L. F., Chin, J. H., and Woodruff, L. W., "Study of Aerothermodynamic*Phenomena Associated with Reentry of Manned Spacecraft," Final Report,

Lockheed Missiles & Space Company, LMSC Y-78-66-1, May 1966, ContractNAS 9-3531.

11. Crowell, P., "Turbulent Heating Near a Stagnation Point," InterofficeCorrespondence, The Aerospace Corporation, March 1973.

12. Eckert, E. R. G., "Survey on Heat Transfer at High Speeds," Report ofUniv. of Minnesota, Minneapolis, Minn., 1961.

13. Aerotherm Corp., Mt. View, California, "User's Manual, Boundary Layer Inte-gral Matrix Procedure, version C (BLIMPC)," Aerotherm Report UM-70-20,June 1970.

141. Kay., W. M., "Convective Heat and Mass Trnasfer," McGraw-Hill Co., N.Y.,(1966).

15. Anderson, A. D., "Passive Nosetip Technology (PANT) Program, InterimReport, Vol. III - Surface Roughness Effects, Part III - Boundary LayerTransition Data Correlation and Analysis," Aerotherm Report 74-90, Aero-therm Division, Acurex Corp., January 1974. (Confidential)

R-1

-- t- - - - - - -- - - - - - - --

16. Powars, C. A., "Passive Nosetip Technology (PANT) Program, Interim Report,Vol. III - Surface Roughness Effects, Part Il - Roughness Augmented HeatingData Correlation and Analysis," Aerotherm Report 74-90, Aerotherm Division,Acurex Corp., January 1974. (Confidential)

17. White, C. A. and Grabow, R. M., "Influence of Scallop Roughness on NosetipShape Change Behavior," SAMSO TR 73-88, January 1973.

18. Persh, Jr., "A. Procedure for Calculating the Boundary Layer Developmert inthe Region of Transition from Laminar to Turbulent Flow," NAVORD Report4438, March 1957.

19. "Minuteman Natural Hazards Program," Aerotherm Final Report,* Contract No.PO 4701-74-C-0069, to be published.

20. Kendall, R. M., Rind'l, R. A., and Bartlett, E. P., "A MulticomponentBoundary Layer Chem' ýlly Coupled to an Ablating Surface," AIAA Journal,vol. 5, no. 6, June 1967, pp. 1063-1071.

21. Moyer, C. B. and Rindal, R. A., "Finite Difference Solution for the In-Depth Response of Charring Materials Considering Surface Chemical andEnergy Balances," Aerotherm Corporation Final Report 66-7, Part II,March 14, 1967 (also NASA CR-1061, June 1968).

22. Bartlett, E. P. and Grose, R. D., "An Evaluation of a Transfer CoefficientApproach for Unequal Diffusion Coefficients," Aerotherm Corporation,Mt. View, California, Report 69-50, June 30, 1969.

23. "Erosion Studies Program," Aerotherm Final Report, C{ontract No. FO 4701-74-C-0069, Supplemental Task 4.3, to be published.

24. Kelly, J. T., "User's Manual General Nonequilibrium Ablation Thermochemistry(GNAT) Conputer Program," Aerotherm Report UM-74-54, October 1974.

25. Aerotherm Corporation, Mountain View, California, "User's Manual, AerothermEquilibrium Surface Thermochemistry Program, Version 3," Aerotherm ReportUM-70-13, May 1970.

26. Powars, C. A. and Kendall, R. M., "User's Manual, Aerotherm Chemical Equi-librium (ACE) Computer Program, Aerotherm Corporation, Mountain View,

'California, May 1969.

27. Aerotherm Corporation, Mountain View, California, "User's Manual, AerothermEquilibrium Snrface Thermochemistry Program, Version 2," June 1966.

28. Crowell, P. G., "Three Dimenzsional Heat Conduction and Surface ShapeChange of an Ablating Nosetip at Angle of Attack." Technical ReportAerospace Corporation - to be published.

29. Interim Report, Passive Nosetip Technology (PANT) Program, SAMSO-TR-74-86June 1974.

30. Guy, T. B., *A Note on the Shock Standoff Distance for a Spherical Bodyin Supersonic Flow," AIAA Vol. 12, No. 3, 1974.

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APPENDIX A

DESCRIPTION OF UNLABELLMD OUTPUT VARIABLES

A-1

- - - -- - - -- -

Name Description Units

BPRIM - Nondimensional ablation rate --

BeUeCM

CH - Blown heat transfer coefficient (i.e. effect lb/ft ' -secof mass addition abcounted for)

CHZ - Non-blow heat transfer coefficient lb/ft 2 -sec

CM - Mass transfer coefficient lb/ft 2 -sec

DLTC - Explicit finite-difference stability seclimit time step

d 2DLTC -I

DLTCl Surface heat flux rise control time step sec

)LTCl = At (!ne1d CTFAti qnew)

DLTC2 - Surface temperature rise control time step sec

STRDDLTC2 =Atold ST

DLTE - Time to next specified environment call sec

DLTIS" - Conduction time step for first calculation secEtep

DLTIS = STRD q 2DLTS - Surface recession control time step sec

6DLTS = 6

Smaxc

where 6 is the distance between the surface and theadjacent in-depth nodes

DTH - Conduction time step finally chosen from secall of the above limits

HCH - Sensible enthalpy of the solid material Btu/lb

HE - Entholpy of boundary layer edge gases at Btu/lbthe wall temperature, hew

HFO - Heat of formation Btu/lb

A-2

Name Description Unmts

TwHZ SUMation izh Btu/lb

HZW - Summation z*iwhiw Btu/lb

RHO - Material density lb/fts

RIOUT - Radial distance from the body centerline into the material interface boundary

ROu,' - Initial surface point radius in

SLOP - Initial body angle slope with respect to degthe centerline

SfRM - Maximum surface temperature rise during ORthe previous time step

TBRPL - LNminar blowing rate reduction parameterA in Equation (2-61)

TBRPT - Turbulent blowing rate reduction parameterA in Equation (2-61)

T-DIST - Explicit grid temperature array OR

TT-DIST - Implicit grid temperature array OR

TCHEM - Ablation parameter defined by Equation Btu/lb13-3) or (3-5!

TEMP - Temperature OR

TFO - Datum temperature for the heat of formation OR

TSEN - Enthalpy of wall gases, hw Btu/iblOUT - Initial surface point axial coordinate in

A-3 :

Datum.r...-5

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