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.

AD-767 280

THE INTERACTION OF A HYPERSONIC PLUME WITH AN EXTERNAL HYPERSONIC STREAM

John T. Kelly

Polytechnic Institute of Brooklyn

Prepared for:

Army Research Off ice-Durham Advanced Research Projects Agency

July 1973

DISTRIBUTED BY:

KHJ National Technical Information Service U. S. DEPARTMENT OF COMMERCE 5285 Port Royal Road, Springfield Va. 22151

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CONTRACT NO. DAHCD4-69-C-DG77 ARPA ORDER NO. 1442, AMENDMENT 2 PROGRAM CODE 9E3D

THE INTERACTION OF A HYPERSONIC PLUME

WITH AN EXTERNAL HYPERSONIC STREAM

by

JOHN T. KELLY

i«^^,

D D r .„,(■

r-ff

OCT 3 1975-

POLYTECHNIC INSTITUTE OF BROOKLYN

DEPARTMENT of

AEROSPACE ENGINEERING and

APPLIED MECHANICS

JULY 1973 I) APPROVED TOR PUBLIC RELEASE;

OISTRIBUTIOM UNLIMITED.

Reproduced by

NATIONAL TECHNICAL INFORMATION SERVICE

US Department of Commerce Springfield, VA. 22151

PIBAL HEPDHT No. 73-12

Jaüa ...1i.^^1.c,ljrm^l^.^.^iJSi-- —■ ^ -■ ^-■^■^■^ .. ^J.^-„..- --

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Unclassified ^SecurityClaBsification

DOCUMENT CONTROL DAT^ - R & D (Security r/...f<ic.Hon ol il,„. body ol ah.,„cl .nd lnd,,lnS .nnoHllon mual he ,„,.,.d»h»n the ov,..„ „po,, Is cla.Bllfd)

I ORIGINATING ACTIVITY fCo/po«/« «Ulhorj •J^~^^^" -~ '

Polytechnic Institute of Brooklyn Dept. of Aerospace Eng. & Applied Mechanics Route 110,, Farmingdale, New York 11735

2». FiePORT SECURITY CLASSIFICATION

Unclassified 2b. GROUP

3 REPORT TITLE ——— _

THE INTERACTION OF A HYPERSONIC PLUME WITH AN EXTERNAL HYPERSONIC STREAM

4. DESCRIPTIVE NOTES (Type ol report and Inclusive datta)

Research Report 5. AUTHORIS) (Flrsl name, middle initial, last name)

John T. Kelly

6 REPORT DATE

July 1973 7«. TOTAL N

8«. CONTRACT OR GRANT NO. . — - , „

DAHC04-69-C-0077 6. PROJEC T NO.

=■ ARPA Order No. 1442, Amendment 2

d Program Code 9E30

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PIBAL Report No. 73-12

ab. OTHER REPORT NOIS» (Any other numbers that may be aeelimed this report) '

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Approved for public release; distribution unlimited.

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12. SPONSORING MILITARY ACTIVITY

U.S. Army Research Office Box CM, Duke Station Durham, North Carolina 27706

A theoretical study of the gas dynamic interaction between a hyper- sonic plume and the opposed hypersonic external stream is presented. Steady, axisymmetric, inviscid, perfect gas flow is postulated for both the bow and far field regions. Limiting forms of solutions are obtained for the bow region by application of the Newton-Busemann approximation (i.e., ee,eio-0 M0Oe,M0Oio-<» such that M^e^M^ ^e^-O (1) ) to both

the exhaust plume and ambient air flow. Through asymptotic expansions and their matching, it is found that six regions are required to ade- quately describe the bow region. For the far field region, the hyper- sonic small-disturbance form of the Newton-Busemann approximation (i.e., Se-0» MODe-0, i5e-»0 such that M^ e 5^-0 (1)) is applied. From asymptotic'

expansions and their matching, it is found that "entropy wake" solutions are required to adequately describe the exhaust flow and the air flow near the contact surface. Analytical solutions are obtained which (i) define scaling parameters for the bow and far field flow; (ii)esti- mate the accuracy of the Newtonian impact theory in predicting bow region geometry and properties; (iii) establish the variation of bow and far field properties with variation in the primary system design parameters'.

DD FORM I NO V SS 1473 Unclassified

i«/ Security Classification

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Unclassified Security Classification

KEY K'ONOt HOLE WT nouE

Plume interaction Hypersonic Retro configuration

•b Unclassified Security Classification

fatfarii ^-^--■-v.::..v-->-^..^^.m^—^^it- nil l^lAüV1tt"lnli■iil^"^lli^ im - - ----1 mViTiiiirniiftinirn M .^^.^^^^.^.. L^m,^A;)-1<l^,i.ai-.-/J,v^.;-.J^,^lii.JH«i,-^>^j,.vc -.^.^.„^...^■^i.L^j^^j^^.^-.. ■..;.. ..: : -^ .:.■:..■, .L^.^-v^a;

•■ "■■»'»"'"■'"'"«"""'■•'•'■■"-'-"^^

THE INTERACTION OF A HYPERSONIC PLUME

WITH AN EXTERNAL HYPERSONIC STREAM

by

John T. Kelly

This research was sponsored by the Advanced Research Projects Agency of the Department of Defense and was monitored by the U.S. Army Research Office, Durham, North Carolina under Contract No. DAHC04-69-C-0077.

Reproduction in whole or in part is permitted for any purpose of the United States Government.

Polytechnic Institute of Brooklyn

Department

of

Aerospace Engineering and Appliec* Mechanics

July 1973

PIBAL Report No. 73-12

Approved for public release; distribution unlimited,

fC

hmni'i nr > — -■■■^-■--ii|-i>iiinniMifiirmniJriirr'c—-"-'-'- -""-■•—•■- ^...„.t..^^—^*^^**^^ .■^■^■..■■.■.w^.^....,... ..... ;.W^U-lJ,:ii.i^.;.Au:vit»lc«iv.ii^1il/.L:.-..;.. .,, ..... ,■■,.,....-^'^v.„iijJ.,;.ni1uJ,Aa

HI iaiiu*p..*.«iiuiin4iiii luni.juiiip.iiwwwfwww^mw .uiiiiiMI-iilMlimiMiJIiui -ni|i,iWilP^iJW»iiii||i.Mii*!ii,iWilllWiW«l« IIIIIUi-IJ|ilJWMIUIi«iUllWi|*l!*»t*W«HW*;l

THE INTERACTION OF A HYPERSONIC PLUME

WITH AN EXTERNAL HYPERSONIC STREAMt

by

John T. Kelly

Polytechnic Institute of Brooklyn Preston R. Bassett Research Laboratory

Fartningdale, New York

ABSTRACT

A theoretical study of the gas dynamic interaction between a

hypersonic plume and the opposed hypersonic external stream is

presented. Steady, axisymmetric, inviscid, perfect gas flow is

postulated for both the bow and far field regions. Limiting

forms of solutions are obtained for the bow region by application

of the Newton-Busemann approximation (i.e., e ,e. -0 M „M -oo 2 2 e 10 ooe» ooio

such that M ep»M e. -0(1)) to both the exhaust plume and UJ e •= oo IQ ■LU

ambient air flow. Through asymptotic expansions and their matching,

it is found that six regions are required to adequately describe

the bow region. For the far field region, the hypersonic small-

disturbance form of the Newton-Busemann approximation (i.e., e -0,

Mooe'*0,6e-'0 such that Moo 6eee~0(1)) is aPPlied- From asymptotic

This research was supported by the Advanced Research Projects Agency of the Department of Defense and was monitored by the U.S. Army Research Office, Box CM, Duke Station, North Carolina 27706, under Contract No. DAHC04-69-C-0077.

The author would like to thank Prof. S. G. Rubin, for his invaluable discussions during the course of this work.

Currently at Aerochem Research Labs., Princeton, New Jersey.

ib

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expansions and their matching, it is found that "entropy wake"

solutions are required to adequately describe the exhaust flow and

the air flow near the contact surface. Analytical solutions are

obtained which (i) define scaling parameters for the bow and far

field flow; (ii) estimate the accuracy of the Newtonian impact

theory in predicting bow region geometry and properties; (iii) estab'

lish the variation of bow and far field properties with variation

in the primary system design parameters.

11

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TABLE OF CONTENTS

«I

Section Pare

I Introduction 1

II Jet Exhaust Model 14

III Analysis of Bow Region 19

IV Discussion of Results 69

V Analysis of Corner Region 79

VI Far Field Region Analysis 91

VII Discussion of Results 119

VIII Conclusions 121

IX References ....; 124

Appendix (A) Bow Initializing Scheme 127

111

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LIST OF FIGURES

Page 1 Schematic of Jet and External Flow

Interaction 130

2 Flow Regimes ......... ^ 131

3 .Undisturbed Plume Model Exponent as a Function of Angle and Distance 132

4 Experimentally Determined Shock and Contact Surface Locations Referred t0 R»«, 133

5 Predicted Contact Surface Shapes 134

6 Comparison of Shock and Contact Surface Positions 135

7 Comparison of Contact Surface Pressure Distributions I36

3 Comparison of Calculated Shocks and Contact Surface Positions with Experi- mental Results of Zakkay I37

9 Contact Surface Pressure Distribution. 138

10 Variation of Bow Geometry with n 139

11 Variation of Bow Geometry with 9<p .... 140

12 Variation of Contact Surface Pressure with 6» 1^1

13 Vari ation of Bow Geometry with Xj ... 142

14 Variation" of Contact Surface Pressure with *) 143

15 Isobars and Streamlines for Bow Region 144

16 Density and Velocity Distributions for Bow Region X45

17 Isobars and Streamlines for Bow Region 1/f5

IV

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LIST OF FIGURES (continued)

10 Density and Velocity Distributions for Bow Region ;u 7

19 Corner Region 1/^

20 .Core Pressure to Ambient Pressure vs. Jet Exit Mach Number , 149

21 Far Field Shock and Contact Surface Position , ]_50

22 Far Field Shock and Contact Surface Position „ -|_c]_

23 Far Field Properties vs. Stream Function 2.52

24 Far Field Properties vs. Stream Function , ^ -jco

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ätiB

LIST OF SYMBOLS

t

c Plume model angle parameter

D Body diameter

F Deviation of internal shock from a sphere

Fx Axial force

H Ratio of radius o-J curvature of flow streamlines with- in layer to that of the shock

L Streamwise length scale

M Mach number

n Plume exponent

p Pressure

q Streamline velocity

r Distance measured away from axis

R Radius measured from jet in bow region or distance away from axis in far field region

T Temperature

u Velocity parallel to shock

U Free stream velocity

v Velocity normal to shock

x Distance measured along shock

y Bow region layer thickness

Y Far field layer thickness

ß Shock angle in corner and far field

5 Layer thickness

C Density ratio across shock

if Ratio of specific heats

lo Radius of curvature

VI

• ■ .*•«**

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n

ir

LIST OF SYMBOLS (continued)

Ratio of calculated n to analytical approximation

Viscosity

An£;le measured from jet centerline

Ratio of jet stagnation to free stream dynamic pressure

Streamfunction

Shock angle for bow region

Limiting angle for flow into a vacuum

0,1,2.

cs

e

<♦

o

je

s

00

SUBSCRIPTS

Denotes the order of the terra in the expansion

Contact surface

External flow properties

Internal flow properties

Point at which flow has crossed shock

At internal normal shock point

Layer

Shock surface

Free stream condition

Vll

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■ ■ ■■

INTRODUCTION

A theoretical study of the interaction of a highly underex-

panded jet issuing into a high Mach number free stream that

is opposed to the jet's expansion along its axis is present-

ed. At some distance, characterized by the lengtn R-ai«

the jet and free stream gases interact. The jet flow is there-

by deflected downstream by the external flow. The particular

case of interest to be examined in this study concerns flows

where the typical dimension of the body D is much less

than RsCo (see schematic (1)).

Schematic (1)

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For this case, analytical models of the upstream and downstream

gas dynamic interaction processes are formulated herein. The

following approximations are postulated:'

1. Steady Flow

2. Axisymmetric Flow

3. Continuum

4. Inviscid

5. Thermally and Calorically Perfect Gas

In further studies, assumption 4 can be partially relaxed by

applying boundary layer concepts along the shear surfaces for

high Reynolds» number flows. Also, assumption 5 can be re-

laxed quite easily by applying Mollier charts or similar models

for equilibrium gas dynamic properties.

A general schematic of the flow structure under these assump-

tions is given in figure 1, where three distinct flow regions

are discerned. The major characteristics of these regions

are as follows:

Bow Region ,

This region is characterized by nearly normal interior and

exterior shocks and relatively thin layers. The flow within

the layers is then necessarily subsonic and of high density.

Corner Region

The layers thicken markedly and turn in a downstream direc- t

tion. Both interior and exterior layers undergo transonic

expansions about the relatively motionless core region sur-

rounding the plume. The development of the plume ahead of

y.-.,!..:, ...,^:.J. ■■■ ,■■; ,,-.■ ■.>.■: i. ^*j.ii.^.:*-.Ki*^^.,to ■•i.C*,.-^-i*M*i&i*i.r>^ U*i,j«SÄ*t'itert'1ii.,.rt.i;./-rSi^.'. Jv.."r,.'t^:B»Ä .<.■:* .■■;■ . mMmmMH-i"-

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the internal shock resembles very closely that of a plume ex-

panding into i quiescent ambient.

Far Field Region

The exterior shocK and interface are at relatively small angles

with the freestream. The flow in all layers is supersonic and

nearly parallel to the axis.

Flow systems of these types are of considerable interest for the

practical applications of force vector control and attendant

surface thermal protection of re-entry vehicles. For force vec-

tor co'. rol, the altered pressure distribution on the body, as

well as the jet thrust, has to be considered in calculating the

total force on the body. For thermal protection, although the

hot external gases are blown free from the nose by the jet, it

is still necessary to consider the heat flux due to the poss-

ible reattachment of this separated gas flow on the body.

In addition to the effects already mentioned, which are mani-

fested near the body, we must also be concerned with the flow

field far downstream of the body. In this wake-like region,

the mixing of the jet gases may affect the chemical processes

occurring in the wake to such an extent that observable prop-

erties and hence detection or communication may be greatly fa-

cilitated or decreased. It is apparent that if such a sys-

tem were to be properly utilized, a detailed understanding of

the physical and chemical processes occurring within the dis-

turbed flow field must be obtained.

-■■ ... ...■

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vr "w ..,.>■.■■■ j........... .-.^.....i.!.^.-....,.^-«^.! i ni.mni^wB^uiiw ■■. ig»» IIII.HIII.III»I.I.IJ.HIJ.»IWI'IIH'MIII>J IHJIUUI i.. IUHII i wiw;ipiwwa^pi^Bi|;a>pw|WB|jwwWW<ipiW»piH||B| ,

Previous theoretical and experimental studies have added con-

siderably to our basic knowledge in this area. The first stud- 12 3

ies ' * were essentially experimental in nature, applying only

simplified modeling to correlate the data. A significant result 2 3

of these early investigations *J was that two modes of inter-

action v/ere observed to exist. The modes were dependent upon

the jet exit Mach number Mj , free stream Mach number Meat. ,

ratio of jet stagnation pressure to free stream dynamic pres-

sure IT , body size D , and shape. One mode is characterized

by large interaction distances with unsteady shocks and bound-

ing surfaces. This flow regime is designated as the unstable

case.

The second mode of interaction is characterized by a relatively

short interaction length and steady strong shocks. This flow

regime is the case investigated herein. The mechanism for

transition from the steady to unsteady flow was postulated by

Finley ^ to be a result of the development, for low Tt and

l^«0« or high Mj , of a multiple cell structure for the un-

disturbed plume before the" interaction region is reached. In

view of the subsonic flow existing behind the Mach disc sepa-

rating the cells, we have the possibility of upstream influ-

ence from the surrounding gases affecting the interaction re-

gion in a possibly unsteady manner.

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The boundary between stable and unstable flow,1»2»3for experi-

ments where the effect of booy size on the flow is negligible

is given by:

M 7.1 7.1 7.1 7.1 2.71 2.71

1.0 4.0 4.Ö5 5.30 3.10 3.90

Tf 2

3

15 20

4 Ö

Thus, to achieve stable flow, the jet stagnation pressure must

be much larger than the free stream dynamic pressure (i.e.,Tf

>^ 1 ) for moderate values of Mj . These are the same con-

ditions under which R5i0 becomes much larger than D (for D

approximately equal to jet exit diameter); therefore, the case

studied herein will always be of the stable type.

It should be noted that one investigator 2 observed a region

for largd values of Tf where the flow became unstable and

continued to be so for all higher values of TT . This result

has not been duplicated by other investigators, and indica-

tions of stable interactions /f'5»7 have been obtained from other

experiments at conditions that fall within the region of un-

stable flow found in Reference 2.

In addition, it has been observed 1»3»^ that the interaction

length RJCO for the stable condition is a function of TT, Vj

^•^Meo^jMj^eand that the contact surface separating the

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external stream from the jet gases is almost spherical in nat-

ure. These observations have been utilized in constructing

simplified models to describe the flow field associated wich

stable interactions.

Theoretical analyses of this jet interaction problem have ap-

peared only recently. In 1969, Laurmann 6,utilizing the

Newtonian impact theory, calculated quantitative results for

the upstream region. This paper presented general features of

the upstream bow interaction, but did not correctly treat such

areas as the corner region and the far field development of

the flow. In 1971, a time-dependent numerical technique 7 was

employed by Rudman and Vaglio-Laurin to calculate detailed

quantitative dynamic properties of the upstream flow interac-

tion. ■ In the calculations, the jet plume boundary layer in-

tersection with the shock layers was assumed to be a point in-

teraction and the core pressure was assumed to be that of the

freestream. The downstream flow was not considered in the

above report and only the bow region was evaluated.

The present study was undertaken for the primary task of cor-

recting the errors inherent in a Newtonian impact analysis, to

estimate the effect of the corner region on the total flow

field, and to calculate the far field interaction.

An analytic approach was considered since such a method pro-

vides for the' greatest insight into the physical processes and

■— ^ - — - M [1 1 ili^iW"-" —^..J-^-.-I-:.^—.. .^.^^M,^—^t..,,-. ^

.r.-~r^^SfrjWfJ^*^^r7V^^

also defines the accuracy associated with various approxima-

tions. Finally, this study also acts as a model for extend-

ing calculations-to more complicated configurations; e.g.,

where the body influences the flow or where the jet is at an

angle of incidence to the free stream.

We must now determine the operating conditions under which our

simplifying assumptions are satisfied. For a missile or ve-

hicle re-entering the earth's atmosphere, we have the follow-

ing range of conditions from approximately 400,000 ft. alti-

tude down to approximately 50,000 ft. altitude.

Ranze of Typical Operating Conditions for a Re-Entry Vehicle

Uooe (ft/sec) .15,000 23,000

Moo, 10 25

poo* (lbf/ft2) 6.92xl0~5 23.27

Over this range of operating conditions, we generally find

that rarefaction, viscous, chemical or physical effects may be

significant. This would violate our assumptions and therefore

we must examine the relevant parameters and their numerical

values in order to ascertain when our assumptions are fully

satisfied.

The first limitation on the parameters involves the assump-

tion of steady flow and requires that Tf» l,M<o«*> ij Mj s 0(i)

The reasons for this have been discussed previously. A second

limitation relates to rarefaction effects occurring within the

-~'-rwlr*?.7v*'.W'-.r.-m'>T*r* T'^^I^^-.-^-T-»-H^*"4--"fT^T'WT--w'W!ttrw:'-^!-''T.1 iir^.™T.pwT,v^'^^rw'.'vi.rr.^!'w^iww!W7-i..7J>«mLKl,-*'iwii. :i»■'i'i'.'^-w.rP 'M'fi.uiiww^.wJ.'i*,1 ''■■.!'>-Li-^-'i'üT^-ifl"MM'iipfl^-s-j-^>rÄ««c'.'!Ti"r,WP.JiwwvT^,wN^.w<^w^!fiiLp^^rv^^r^.^^wwi^TPw^ji

flowfield. For the upstream region, v/e have three areas where

rarefaction effects may invalidate the application of the ana-

lysis. The first area is the plume core itself where the large

degree of expansion may take the free plume flow into the tran-

sitional regime. From an analysis of a steady spherical source

flow expanding into a vacuum Kamel and V/illis derived an ex-

pression for the distance beyond which the source gar; becomes

collisionless. This is given by

R-t ^ (Re*) f*f

where f^ is the jet nozzle throat radius and Rene is the Rey-

nolds' number evaluated at the nozzle throat. This value is

only a function of the nozzle throat conditions and hence is

a constant for a particular set of chamber conditions. This

can also be simply related to the thrust of our jet

where I«* is the specific impulse, T is the thrust and R; is

the jet chamber pressure. In order that the plume flow for

our problem be a continuum, the interaction distance, Rsta ,

for the interior shock must be much less than the distance R-t«

The interaction distance Rst0 is determined by the condi-

tion of equal stagnation pressures along the axis. Therefore,

8

■^«mwwwwTOr^-^m^Hw™»-^^?^^

This expression is a result of assuming a source like behavior

for the undisturbed plume properties. The occurence of the

various quantities in the above expression will be detailed

in a later section relating to undisturbed plume properties.

The condition for continuum flow for the plume expansion is

then

In terms of the thrust of the jet, the above becomes

= L

Following Reference (9) we take the numerical value of fo ^ 10

to be safely within the continuum region. This bound is in-

dicated in Figure 2 for the representative conditions of

MeDe = lO , Pcj = 50 Otm. yyCUc = S-X »o"6 Ifef s.c/ft*J U* = 300ofps

^t =1«2S ^ X*p = 300 »ec.

We now direct our attention to the internal and external shock

layers where rarefaction effects can result in the thickening

of the shocks and shear layers to such an extent that they

strongly influence the development of the layers. In a paper

by Bush it was illustrated that the thickening of the shocks

and growth of the viscous shear layers are related for the

shock layers; and, therefore, we need only determine the thick-

ness of the shock with respect to the layer thickness to de-

Mlli'illii ^■....■^--■^ ■.■-■■....■ .-^..^..-.-j.i,'. T „^ .■-...■.-.■■.. i,!^.-.1,.,. :--..■■•-■.iJ-vr..^.i,.-i...■»...!,^ ^.^.- ..,;■.,. .--...i.—-■■.:,. .■■.j.^---^--"-ij.ll>j| -iliiHliilMil M-'n'.ti. l-^-w,-wi.J.v^-.>-.*^.-a.,. -.-■■. ■ ■ ., ■...■,.-.. ■ ■ ,■ ........ .... . ■■...■. .. ■:-. ..L .- ... . «.^

p^-.-—,,^«,„.,»..,;.............-...,..-„ ~. , nn^vw.. . -. """""i' " '" 'vm*mmmmmmm*mm*^m*wmmmi!immmmmmmmm*mmmmm

termine where inviscid continuum theory is valid. It has been 11

shown that a continuum inviscid description of the flow is

appropriate when-

where in the above 5 denotes thickness, and X denotes layer,

5 snock and « , «- , external and internal layers respect-

ively. From Adams and Probstein 12 we have for the thickness

of the shock for either external or internal flow

Ss - V>*/C*

Where C* and V^ respectively are the speed of sound and vis-

cosity evaluated at M " 1 ,

After introducing the interaction length Rj;.!0 as a signifi-

cant length, this then reduces to 1^

Vfnere u) is the exponent for temperature in the power law

viscosity model, R« is the Reynolds' number arid M«, is the

Mach number, based on free stream properties. We now intro-

duce the first order thicknesses of the shock layers, which

will be derived in detail in a later section.

r\5tB Ksio

10

-. ....Ll ^.J.L-...- ' - ^ - —' "- —■* ^ ^ ■ -- - -

~T IM inwwiPiwieuMiiiiiiii. i. » '"■■• ■■."—»• •IIMHBHIU ■ in in -iivixii aifi^i ■wwuumiiJi'iijwnniiiiii.Mi u i i.i n iinuuiawwH* 11 n iLimjiii.i IUII I

V/here e«,^ are ths density ratios across the normal part of

the shocks along the axis of symmetry.

Therefore,

i^t — M-tt>* €« ROD, öS« -^«oe

iti- ~ ^gfeZ €t'.Va R 00 (.

where R« is based on the interaction distance Rsio . Using

the source model for the plume properties and the equality of

stagnation pressures to determine Rst'0 , we can reduce the

above ratios to functions of jet thrust,X , jet throat prop-

erties and external flow gas properties. The results of this

3.T*G *

Using values stated previously which are representative of

typical flight conditions, the above can be plotted as a func-

tion of altitude for a given thrust, above which rarefaction-

effects have to be accounted for. This has been carried out

in Figure 2 and it can be seen that for jets of 10 to 10^

pound force thrust, there is a large region of flow where the

continuum inviscid analysis will apply.

V/e must now examine the region where perfect gas behavior or

equilibrium gas behavior will apply. For the external shock

layer, there -are many analyses and data available which indi-

cate when nonequilibrium effects in such general flow fields

11

■ . - . . fcMttftj&Mäg a-—..-*-".. -■■ ^^^MifraaMaaaai^^ - --^---.■..■.^--.■- ■---:-*.-.^^■.t-;.-,-. . , ..,1 ..---■;. -V^ .i-J1 ^-U.. J ..--- ^. ■■■..- .f- ,.

*« t^m

5i~™™jwMifsmw4iw",iw»»reT *.-rnn%'nmwiww»»..iwiw, wnw )#jp.iiHJ,iisi^wi!»i»«iu«^ijJiW!> (■I.)üJI',»**I*IJ«II'I«! ^I>*«!^ä«WWWPWM!,TO,HII^

become important. For the plme and internal flow, this is not

the case. The reason for this is that rocket exhaust chemi-

cal composition,'especially those of the solid propellant va-

riety, varies greatly from application to application and in

each case the bounds of equilibrium are different. Therefore,

we cannot, define a single general condition when nonequili-

brium flow occurs. Therefore, we will set down the bound for

the external flow as the limit for the combined layers above

which the flow cannot be considered to be in equilibrium.

The external layer nonequilibrium bound is adapted from a

paper by Cheng14 and is based on the relaxation processes be-

hind a normal shock. When the distance for relaxation to equi-

librium conditions behind the shock becomes of the same order

of magnitude as the layer thickness, then nonequilibrium ef-

fects must be taken into account. Cheng's results gave a sin-

gle point at high altitude, which when combined with binary

scaling led to the result in Figure 2, above which nonequili- -

brium effects must be considered. Binary scaling derives from

the fact that at high altitude the probability for chemical

reactions to occur by three-body collisions is much less than

that for two-body reactions. Under these conditions, it can

be shown that if density and field dimension are held constant,

then the degree of nonequilibrium will be the same in each

case.

12

^T«™»!^?W"»w<^wF«r7TOr^^

Since the body dimension Rs,'6is related to jet thrust and den-

sity p«« is related to altitude, we can then establish a

relationship between thrust and altitude, which will maintain

the product of density and dimension constant, giving the

same degree of nonequilibrium. The binary scaling principle

only applies to high altitude, for moderate to low altitude

collisions should be prevalent enough that equilibrium flow

is maintained.

Referring to Figure 2, it is shown that for typical re-entry

conditions the extent of the equilibrium, inviscid, continuum

flow is about one half of the complete continuum, inviscid

region; however, there is still a considerable region where

all the assumptions stated previously are fully satisfied.

Now that we have determined our bounds of validity for an ana-

lysis based on several assumptions, we can proceed to outline

the analysis in detail.

13

gfiggg h - ■ -- - —»^-^.^.^..J. .«.-L^,—^".(iniiinii, - - „

■pr^ ' "Ty??1*W*Wl!l*!TOWJB™B<mW«»TO^

JET EXHAUST MODEL

The undisturbed plume which interacts with the external flow

to form the bow layer develops in an environment (the core

region) where there is little fluid motion and hence nearly-

constant pressure. Under these circumstances, the undisturbed

plume is analogous to the case of a jet exhausting into a

quiescent ambient which is at the core pressure. V/e can then

employ analyses developed for plumes exhausting into quies-

cent ambients to our retro plume case. To be consistent with

the analyses for the downstream and bow region, v/e will uti-

lize an analytical approach for the undisturbed plume. There

are many models available for describing the isentropic ex-

pansion of a jet far away from the nozzle exit. Many of them

have the form 15» l6» 17*

p oc co$nC9 (1)

Other properties can be determined from this expression by use

of the isentropic flow relations (No's. 30.2-30.3 from the Bow

Analysis).

*In this reference, an expression for Mach number is given which when substituted into the isentropic relationships for high Mach number gives a result identical to (1).

14

- - - - ^- - - - - -• - --1 ,,1, —— - - - - -■•---- - -■--- - - &iv^-^r^~L'--.:M^...i..':v,:-,.^^. t-^;-i-i .-:.■..» .. ■.,■■. . i-v-.... ....^-^ -, ■..■ ■■*:.... .^*.. ,■■,■{ ■■,,*&**'-*■-■•''■-■-

.MI,lWJ|l«!Ui>..UJJJi,^.i,U...J^

In (1) ÖD is the anp;le measured a.way from the axis of sym-

metry, r\ is radius measured from the nozzle exit and n

and C are constants for a given nozzle and jet gas compo-

sition. Various forms have been proposed for H and C as

functions of plume s^cific heat ratio 8j , exit Mach num-

ber r^j , and nozzle exit angle 0n . Using the form C =

TT/^ooo,where 0co is the sum of the Prandtl-Meyer limiting

turning angle into a vacuum and ©n^gives the correct theo-

retical result of zero density or infinite Mach number when

Op s ©oo . For this value of C , we can extract from a

15 Prandtl-Meyer analysis y near the nozzle lip the value forn.

n * »i-i

(2)

This result can also be obtained from aoplication of the snail 10

disturbance theory to hypersonic jets expanding into a vacuum .

As will be shown in the bow layer analysis section, these va-

lues for n and C give the physically realistic result in the

limit as 6 i0 -♦ O that the density and other flow vari-

ables remain of the same order as we go away from the axis.

For any other choice of C and n as a function of "J the

density would be finite on the axis and either zero or infi-

nite away from the axis. Even though the form of n as a

function of 8j is correct, its numerical value for a given #j

is still not clear. This is because comparisons with numerical

results give different values for Y\ , varying between the value

15

.. - - J.-—__ ^-^ . n . . . . . _..... „__-J___^.

pi mimwt\ mmm

given in (2) and one half that value 19. The difficulty is

that the form of equation (1) is not general enough to be

valid over a full range of R and ^ . This point is illus-

trated in an article by Boynton 15 where he shows for 9<.^

the proportionality constant is approximately one and for

9> -^ the numerical results lie closer to the curve for a

proportionality factor of two. Since the bow shock layer and

downstream layer depend critically on the undisturbed plume

flow, (as will be shown in the bow and dovmstream analyses

sections) then it was felt that a review of numerically cal-

culated undisturbed plumes to give "best" values for H was

justified. The expected value would probably lie somewhere

between the theoretical value S— anci 3 J J

Three sources of numerical plume calculation data were chosen

for examinition to determine "best" values for O . In re-

ference (20) numerical calculation of three plumes from super-

sonic nozzles expanding into a quiescent ambient were presented.

Two of the nozzles (Jfj = 1.15,1.18) are representative of

those used for launch vehicles, and the third ( */ * 1.2*0 is

representative of spacecraft nozzles. The method of charac-

teristics was used in this study to determine, for constant V. J , Mach number contours, and density and temperature dis-

tributions along and perpendicular to the axis of symmetry.

The second reference l6 contained th* calculation of the plume

resulting from the expansion of air through a sonic orifice.

ach number distributions perpendicular to the axis of sym- VI

16

>... -i.i! I'JM t^n^lii^^ne ■ ^JJJCJUL** ■ ...J^^.V .IJ-U.^- ...:...„.. fl ....~-....^LJr-..^ i^.^t^j^i^fotf^fyjfi^-rt-w^. ■ ■■.■,-■■■ i ■ y.i■ ■ . ■■:■.-..

.mi in iiiuii mupiiii. iiiiii.ii,u...,ii™.i»ufmi^ju jiunnn^^^WmiMllJHai ii.i.w^.i.in^i >->«P<I>II li^pja J ■■ Jl I HI »IHUWIMUIW ■■ »i i "I 1 \mim mmvmnmmmmi^mfimmm

metry were given. The third source used 15 contains results

of a numerical calculation, using a finite difference tech-

nique, of the flow from a nozzle representative of a launch

vehicle. Density is given as a function of angle from the

axis in this work. To determinenfrom the above sources for

points away from the axis, we simply use the isentropic re-

lationships and the assumed distribution (1) to determine

for a known value of the constant G« and jo at a given to

and R . However, near the axis this procedure does not

yield good results since large changes in n produce little

measured change in ^p or M in this region. In this case,

we then apply a derivative of the density distribution to ob-

tain a better relationship for determining n . For the first

and second references where properties are given as functions

of T , the distance perpendicular to the axis, the expres-

sion

(3)

establishes n from the density gradient perpendicular to the

axis. For the third reference, we use

" = -(^)^^£^l (4)

17

. Khiin i" ii ...^..^^-^^-._-....^i_^-.^.^.....,^....^-....,^^J..---..^aygflaiaajii .J...- .-.-...^■^■^..1,....^.^,._...■■: ^ -.^ ,^— ...... ....._

•'.^M^W-OT'.r.rr'^^^-'TO.U'-.Ju^^

Values of 7/ , the ratio of H calculated from 3 and 4, to

^/<*j-i) are plotted In figure (3) as a'function of J/ö^and

X/rj . From this graph, it is evident that n is not a con-

stant but varies both as a function of 9/Ö00 and X/r; . If

use is to be made of (1) in calculating retro plumes, then

for each change in X/Tj a new value of n must be determined

from figure (3).

The variation of h u±th<f/emis not as critical as the axial

variation since the bulk of the mass flow, which determines

the plume external stream interaction, lies near the axis and

therefore the behavior for J/Go^'S is not significant to the

interaction. Also, for 9/6« near zero the density is near

one for any value of h . Noting this, we then take the value

of n found at^/e^-.^b to be the "best" value for a given

^/Tj . Characteristic retro plume calculations have been

carried out with values of H determined from figure (3). Re-

sults of these calculations will be discussed in the sections

on bow and downstream analyses.

18

— -- ■- --- - — ■ - i-^

«"" —^~--^^^^^^»^^'^^—------^—^^-™^~^'"-^w^»»ww>—ww»^«»™™™-«-w-»^^^ " I' ""^i

ANALYSIS OF BOW REGION

For the problem studied here Moo« , Moo^ ^ 1 and ^e.A-^

The flow within the internal and external shock layers are

determined by a perturbation theory for strong shock waves

and small layer thicknesses. The procedure involves a limit*

of the governing equations and boundary conditions when

^ej€iö-*>0 Hooe,Noo:e-*oo and h«t€« ^oo^e^r Ofcjwhere

^e, €:C0 are the density ratios across the shocks at the

axis. To elucidate some detail of the flows within the la-

yers, the coordinates normal to the shocks are expanded in sd »Ti

powers of €e , €i0 where 4^ are to be determined.

The orders of magnitude of the various flow quantities in terms

of €ej €t6 and Mooe^-e ).M<oic6Le are derived from the

requirement that the shock relations and flow equations yield

a nontrivial system which includes all the physical effects

of interest as ee,^-> O . The coordinate systems are

shock oriented and are illustrated in schematic (2)

Tet

Schematic (2)

*This limit will be denoted the N-B limit after Newton-Bus emann

19

.■

i*^ ..,...■.■;.,.., ■...:,-..■■--...-. L^ iflmaaiMt ....... -: aal - - „^ .. ■. ^ ^...i^^-^^^L^atti^n^^ ^a»*^^^^^^^-»* v.^ ^.....,. ^J,, ,^_ y,^ ^„^ „.„..,, ..,„.^.JJJ.„., ..-

—■ i qn •...umn .p.ii.p.M. ..,...,.._, ,,, ,., ,n-mmmm^^mmi™™waiiimmi*u^^mmm*mmmmm**mm&nmmmimm\iH.iuiiiiuwmmmmmmimmm*iimvfi^mmimmm

Upon application of the. N-B limit mentioned above., it is found

that the flow field divides into several regions within which

unique asymptotic expansions for the dependent variables must

be determined. 'It will be shown in the following analysis

that four regions characterized by two major effects are re -

quired to describe adequately the external flow and two re -

gions characterized by two major effects are needed for the in-

ternal flow. In both the internal and external layers, a re-

gion characterized by constant density to first order is formed

near the axis of symmetry. Away from the axis, the density

and other properties vary both along and across the layers.

For the external layer, a region characterized by non-constant

stream velocity is formed near the contact surface in both the

near and away from axis regions. This contrasts with the char-

acter of the region near the shock where the velocity is con-

stant to first order.

The regions are numbered one through six in schematic (2)*,

Once the flow properties are established in each of» these re-

gions, then matching between the expanded properties will be

demonstrated to show their consistency and where necessary,

composite expansions will be formulated. The final step in the

solution to the bow region problem will consist of the numeri-

cal matching of contact surface pressure and position (i.e.,

flow deflection) between the external and internal shock layer

flows.

* These regions, of course, are not drawn to scale.

20

■--•--—"■^i- ^■-,-:*.,^~*i*^L-t . w.^-..■■■.. -»-•--.-■-- '■ ■~.~^- — -- lliiilinaiiri ■-- — -—-■■-■--■ -"-MI^.. »i i ,.^;. L^-A . - -■■, ^i.i .i.-ii-iiji'J.. ,„■ -..■ i '._

-— «uiiii.i niiüiii.iPi.i.ii-.ijiip.^^.-i.. imwuwammimmmm" m mmmmm " >' < <~»~*m***mmmmBmmmmi^mmmmim*mmil*mmmmimmm**,*i*'*

To achieve this result: (i) initial radii of curvatures and

positions of internal and external shocks will be found from

a scheme which uses solutions obtained for regions 1,3» and

5 (this scheme will be outlined in appendix (A)); (ii) shocks

will be extended away from the axis utilizing these radii of

curvature;(iii) external and internal contact surface pres-

sure and position will then be calculated, using solutions

from regions 2, 4 and 6. The pressures and positions are then

compared to determine if numerical matching is achieved; (iv)

if equality to a certain tolerance is not found, then the ra-

dii of curvatures are iterated until matching is achieved.

Having accomplished matching, the procedure starting from

step (ii) will be repeated using the newly found radii of cur-

vatures as the initial values.

A brief outline of the regions considered and their major

characteristics is now given.

For region two, which is of 0(6«) in thickness and O(l) in

length, the N-B limit gives the familiar hypersonic blunt body

result, which has been investigated in whole or in part by

10 21 22 many authors * ' . The orders of magnitude of the flow

quantities in this region are the same as those directly be-

hind the shock. A major characteristic of the solution found

is that the velocity along streamlines is constant in the

first order approximation. This result, adequate for points

near the shock, has been shown to be incorrect near the

21

'it rtMiinrn

■

..■■..^.■.w.....,..i...-...:_. ..i—■.,....

■ .

■—~'''","""''l"'""''",-"^'--'--™'''-,"--^~^^

contact surface in region 4; and, therefore, a new expansion

must be sought. In reference (10) the initial orders of mag-

nitude of the flow variables in region U were established as

well as the thickness of this region, which is OC€« /. To

first order when the N-B limit is applied, the constant stream-

line velocity result of the region 2 analysis must be replaced

by an expression for velocity which accounts for pressure gra-

dients along streamlines which are found to be the same for all

streamlines in this layer. After determining expressions for

the flow variables in region 4 matching with region 2 is then

demonstrated and a composite expansion is then formed which

is valid throughout regions 2 and 4. For general shock shapes

these expressions must be numerically integrated to yield va-

lues for the flow variables of interest. Having established

results for distances of 0(l) away from the axis, we now wish

to determine expansions valid near the axis in the N-B limit

for use in the initial radii of curvatures of the shocks

scheme.

In this study, unlike in past investigations,10' 21* 22 atten-

tion is focused on this near axis region because we are seek-

ing accurate and simple analytic expressions for the layer

thickness and contact surface pressure distribution as func-

tions of shock radius of curvature to be used in the initial-

izing scheme. It will be shown that the near axis region 1

22

.„^^^.i.^^:^:^-....-^.^.^.^'.^^^ j.j^^j.,!^.^«^^..-^-.-.. '■itinii*J,\l-miM,ua-M ■ ■- ■'...-■.-■----.■■^■.■■.v.>- .^w.^-:^-^ ■-t.-;,.....-. ■ - ■ ■ .r,■■:■..-.■.„<....\.-?-.^..-..^.< s. ..■:.■■-.■ .. . .,., ...„-..•■■„..mai

iinwmwjw'p«" < — • ■ „..,.,._,.. , j „ . IüUH. nun i ■ in i ■ ai mi iiiii«MMiPpaianr7n*^«wv» m^mmmmmm

will be ofO(€»)in thickness andOC^4jin length. This region is

characterized by constant density to first order. As in the

region away from the axis, the first order velocity along

streamlines is constant. Consequently, as in the away from

axis case in region 3 near the contact surface we must mod-

ify the expansions to include the effect of non-constancy of

velocity along streamlines. The dimensions of this region

are 0(€^) in thickness and Ofe) in length. The variables

found in this region are demonstrated to match with those of

region 1. A composite expansion for region 1 and region 3

is then formulated. Finally, matching between 1, 3 and 2, 4

is demonstrated for the specific case of a spherical exter-

nal shock.

For the internal layer analysis, more details will generally

be included than for the external flow since this work ap-

pears here for the first time. For region 6, which is 0(l) in

length, the N-B limit leads to the results that the layer thick-

ness and streamline velocity to first order, for non-trivial

results, must be of 0(€i"oA) . Also, the internal shock is

spherical in form and the first order pressure and density are

constant across the layer with an error 0(€t/*) These re-

sults will lead to interesting conclusions, which will be dis-

cussed at the end of the analysis. The solution found in 6

unlike the external flow case is valid throughout the shock

layer for distances of OU) . For general shock shapes, these

23

■■iMA&XMltr&tt rt^>^-4w^^i««ii»«wW^ ^ —• -■■■■ ~ '-■■■■■■--■■■-^- - ■■■■■ a ■■■:-..-^..—.^—■^- I • ^"■■nnt.iiiliüiMB.y ■diü

i '■■ ■•'" ■•■»»-■—•—p— ..l.... .1 .! i^=— PWII.I iHjii I-.IH.I . ■. .1 ma-ii ■«■■>i. J II.I ■^ai.nnpa.ii.u l i ■■iRwaOTjp>wM)«n<qi«aiBfinmaaMKini^^n^w^mvuni

equations must be numerically integrated away from the axis.

As in the external flow case, a region of 0(€v» ) ±n thick-

ness and of OC^i© ) in length is required near the axis to

be used in the initialising scheme. As for the external la-

yer, the near axis region 5 is found to have constant density

to first order. A solution technique is then applied which

makes use of this "characteristic to determine analytic expres-

sions for internal layer thickness and contact surface pres-

sure distribution near the axis to be used in the initializ-

ing scheme. Having found solutions for the variables in re-

gion 5i we then match these to the variables obtained in re-

gion 6 for the specific case of a spherical shock.

The detailed analyses for each of the six regions will now be

given.

24

- - - — ■-■ ■■- I I I IM • - - J.—^—^-^ ^..^..^.^■^... . „...., ., , ^ , . . . . Lj^fcÄ

*q*in*^*W'9WmW*~~irrr~~*mirw*~~wrT-~l~-~7~r^*^Tr*TWWr^

External Layer - Region 2 Analysis

Following previous investigations,21 we obtain the orders of

magnitude of the variables inside the layer from the values

obtained from the shock relations. These relations are:

V = U«oe £" S^n^•

(1.1)

(1.2)

(1.3)

(1.4)

Non dimensionalizing the pressure byjo^U«^ velocity by

U«t and density by ^)eot , we obtain from (1.1-1.4)

y^eK *• «oe -*■(>-£) singer

«^ = COS O"

Taking the NB limit, we find from (2.1-2.4)

(2.1)

(2.2)

(2.3)

(2.4)

z «^ sin cr

U s ooscr

o GeSIKlO'

J> 'S V€e

(3.1)

(3.2)

(3.3)

(3.4)

25

^^iiiBiiiMBiiBiiii^ mm Hmma* -'■• ■'-■'-'■ -]-j" -

v^.i,!J.^jiuiiv^-^,^-^wvw'y^'.'i^.v,-?^^"-.wf..'*- ■yu-**~wx^.'rnzr7rrS!*xrWp^^

Using the orders of magnitude established by (3.1-3.4), we

can now write the following expansions:

(4.1)

(4.2)

f> ■= p«, H- e« px + • • • (4.3)

(4.4)

In order that the flow variables described previously lead

to a meaningful description of the flow, we must stretch the

normal coordinate y. From the continuity equation we find

for y

•

Since rt:o(f<SdlJ which is set as the unit length scale, then

jr ^Ofee). Therefore, to obtain non-trivial results, we must

stretchy byO(€.). The expansions of the variables and the

stretched coordinate y should now be applied to the govern-

ing equations

^^F + ^^i = 0 H^x-fc^ tj— " w • ■ * ^ (5.!) U^C + HV1| ^ KUV + > |g = 0 ^.2)

^T -^ ^ (5.4)

26

"■:-'- ""'' ^"■-^^■■^- ■■- ^-wu^-u ^Wj^^^uLtJ^t»^,.^^^.^^»..,^......, .. r ■■ _ . - -

ffy*PWiw»^vr^v7T'ji^<.jii»«jmi^^^ Di. .1 .UiJiWlUBpi

Before substituting in the expansions of the variables, it

will be advantageous for final integration of the equations

to change the above system to one employing the stream func-

tion as an independent variable.

Define "

1£ = Hrpv

The above system under the transformation

(6.1)

(6.2)

then becomes

H bx (7.1)

(7.2)

(7.3)

(7.4)

The distance across the layer in terms of V is obtained from

(7.5)

. ..--..a^- "■■--■^r"-ftrmnil-imii.rmi-»ni-*ü.«rtni

27

. . ..„-^..o.. ■- - -—- -■'—■"— -■— —■- - 1 a -• -■■■■- ■

■■*?^*vis*?i*^-**F*jvvFwr*y^

In this system we then must add

r = To ■+ ^eTa. + • • ■ (3tl)

H = 1 - K.oJj.cee t .- • • (Ö.2)

Expanded boundary conditions at the shock are:

u = cos^ (9.2)

V « £«Cl+cot2£.) Sin^- (9.3) Mt»««-^

P ^ l/(€e(l+ coüf )) (9.4)

This then gives the result for first and second order boun-

dary conditions:

po « smxff (10.1)

Uo - COS^ (10.2)

Cfe - (l ^ c6i'c/Ml^6e)s»^o: (10.3)

^ - l/Ci+ Cd-t^/fl^e) (10.4)

pt a VM«fee - (A + «t*(Sf/Maa€eOsm,ö: (10.5)

v. = o (10-7)

^ = o (10-ö)

Substituting (4.1-4.4) into (5.1-5.4) and then collecting

terms of like order in €e we obtain

28

"'--—---"-:-- -^"■■^"■^-"'i^^ --■' ^....^:.,...^..^,..v-,..,..„..,.-..--:...........J... ...■.,-^;..^.-....,-,..,.-,.......,....J .^M

First Order Problem

Second Order Problem

V

(11.1)

(11.2) ■

(11.3) |

(11.4)

(11.5) i'i

Pi = Pl- fttSsUi -O-^i + tiAUaCfe^re+^coSfl-)]^ (12.1)

Ui = - Pa-»- I n pfc/pb} (12.3)

(12.4)

(12.5)

/»oUoro

Equations (11.1)to (12.5) describe all the properties of in-

terest near the shock, with an error 0(c« j.

External La\er - Region 4 Analysis

Bush has determined from these results that equations

(11,1) to (12.5) are rot valid at the contact surface due to

the change in order of magnitude of streamline velocity and

hence another expansion is required. Following his results,

we derive the system of equations' valid in the contact sur-

face region with an error Ov€e /,

29

■

^mm

• • ■

The expansions for the variables in this region are;

^ = p; + e«. pi -^ * •' "

I- = To H- €eVa r^. +• • • ■

H ^ 1 + Kofr €eVa + • •

. - < VatTT

(13.1)

(13.2)

(13.3)

(13.4)

(13.5)

(13.6)

(13.7)

(13.8)

These will be substituted into the governing equations (5.1)

to (5.4). The resulting systems of equations are:

First order system of equations

* i*W=0 1| = o

Second order system of equations

(14.1)

(14.2)

(14.3)

(14.4)

W = O

•bsr Sx ^tr sjf

o* Po*

30

(15.1)

(15.2)

(15.3)

■ ■- 1— —i —- ■..-.. -. -,. ...,._ - i..

We can extract from (14.1) to (15.4)

First order results

Uo = Uo* - 2po*/fi* In Ro/f^c

Second order results*

(16.1)

(16.2)

(16.3)

pi * flex) (17>1)

2.u.at = ^Qu - ^ (. ^ - (17.2)

^^^ T.^?{ (17.3) Po« ffe«

These body layer results must be matched to the external la-

yer region 2 results in an intermediate region where both ex-

pansions are valid.

* Vp is not considered since it does not affect the calcu- lation of P*^ to error greater than ofee^and is itself very small of OCc«*)?

31

■■—-' ~ — ^■■■„L-.-^.——..* ^.-J—, >...„^——. -■ -— a ■

We now demonstrate matching for the variables p»P and U.

Matching occurs within the intermediate region defined by 77 .

In this region

oo

-^ *1 In terms of the variables ^^and^^with^ ^ of 0(l)we have

from the expression for layer thickness, (7.5),

The order of ?[ must then fall in the range

OCÄ ^ < 0(L)

23 Following Cole -% matching will be achieved if for each power

of €c we have

W C? - f) ^ o

where T is the variable of interest to be matched. Taking

the matching of the pressure first

i.™ (pel) - pcf;9*^o)) ^ o

32

- — -■- - - -.-—■- -~ ■ ■ -: ^-.J..^J.;..^-^..-u.^J^J.. ,.;—-. : ,. .~~M

Therefore, the pressure in the body layer is equal to the

pressure at the inner boundary of the external layer. A com-

posite expansion valid in both regions can be determined by

subtracting the common term to both expansions which is the

pressure at the contact surface. The correct pressure dis-

tribution will then be that derived from the external flow.

For matching of the density, we must have

u (f cf,9s-?po -£(§>%■*ncp*J}$o

From (16.3) and (17.3) we have

^%*

Since p* and pi are just functions of ^1 then PX®) ; there-

fore

The density within the body layer is then equal to that at

the inner boundary of the external layer. As in the case of

the pressure distribution, the uniformly valid density distri-

bution is simply given by that derived for the external layer.

The velocity now has to be matched. Once again, as for p and

p , we must find

33

.-., MWMMOUMttU • .

■ -— L..^....,.,..^..■■..^■..,.^,.,.J, ..^J1..M.,^^„^...J : ., -.. .^ 1 ..- ....... . '

l^^-^-^rrw-Fim^nTWBT-iwi^^^

For the body layer, we have

Ufo« = *§* ^.^ ■= fa* * 1 U^ = O

f 1« = o

Under the above conditions U, from (16.2) and (17.2) becomes

~ - ''a, / •A' « - 2. In ^„Cf ^

The near shock expression for the velocity is from (11.3) and

(12.3)

u = e* - ^ ^ In fVp^

For matching in the intermediate region f\ we transform the

independent variables such that

34

- !— - —-■ ■ ,..^.,A^^. im-imifcaiirrt 1, - 1 -

^■•-■-^^S.W.Mf'(W.i«F-»^i^^

The matching condition then becomes

©»7,71

The above can be seen to match to error 0\6e ) . A compo-

site expansion valid in both regions is then

£c(^Vh,JnP0/Po» - I^F^o)) (e*2 - ^€e In poC9;0)) »/2.

(18.1)

The layer thickness can now be calculated from

Once the radius of curvature and position of the shock are

known, then all the layer properties may be calculated.

35

j

»■- . —^— — — ... ._ ..

:^L."iw^-..\j^^R'.n-|r'«'*w?n^'*i?-i:t'fjp .i! .v.. y\i'ivwfliujuiwx^^li!^m^^^^rHm>^m^^^w^wm!v^''m-m^'smimiwiw«ui^wiwaffWWIWW'IMWJ^W^W^W'W.WIJIiiiiiwiM^^>JWPi!J|'W«^*fl»*H«w^B5iW!w^w!!.|iivu^giiii^wpiii»^AHH^Iiipiii

Now that we have the results for the external flow away from

the axis, we will now determine if they apply in the region

near the axis. 'From the external flow, the velocity parallel

to the shock is given by

As we approachthe axis both f and ^ , as well as 6^ be-

come smaller than 0(1) . For this situation, the expansions

based on the above quantities being of order one are then no

longer valid and new expansions must be obtained. It is of

importance to develop an accurate solution in this near axis

region as the final matching of the external stream layer flow

and jet layer flow depend on knowing the initial locations

and radii of curvature of the shock surfaces near the axis of

symmetry. 1

External Layer - Region 1 Analysis

To establish these initial values for the external flow, it

is convenient to determine the shock layer thickness and con-

tact surface pressure distribution in an analytic form. This

leads us to describe the external shock as an expansion in

even powers of 9 for the region near the axis. The expan-

sion is

R - 1 + CU^ + be ^ +

36

- •- - - —~^——^J.—^^^-^.—^—w - _ ■■ - - ■*«

rj.,-,,.,,.VB),^H1T(,W),WjliWHW(„r_1,l ,.., , ,„,.,,, s^v, Kfr„,rrmnrrwr~r?rrr*fw-n*.twt*K**■»'!-*WWLU^i^v-^.T^.V¥WVSW«W;wj^r^T"s^w;v»jwf 17'•(.iwi 11 ,«?J ^. U M,V»HP, »vw j «P»w ,,..1 ..^-r>i i^■,.«- ^...."J.IWM ^IWu..^...^ .^-.W,.-,VIwWP^-^^»«««'I'iwp'.w'i'n'.w .? mmftH

It is sufficient to retain only the first two of the above

terras for accuracy consistent with the number of terms in the

dependent variable expansions. The expansions for the de-

pendent variables which give the relevant physical behavior

in the near axis case and are consistent with the shock re-

lations are

P f U.

V

pö + ee. pi + • • •

jV/sc + pi' + ••••

^ - e^

Inserting tho above in equations (7.1) - (7.5), we obtain for

the first order results

U, / „

s Uoj(f

jf o' - f iai!

For the second order, we obtain

^-C^+fe'--fe)^4

(19.1)

(19.2)

(19.3)

(19.4)

(19.5)

(20.1)

(20.2)

37

■ ■

^^v%Tu,...^^^^^,-.^.--^R»|,.,ifiiSP^'HlWfl'fSWH^WWW iw«..i-i»i...^. i^j r*~urwm?Vwlwfy*rrr.w-.W'.*'•Wmwr.Wmm*>-wwwmrriW™rwy™*V*"^VVV*Wi'K I,",-WJi«"«il^4IP^WW"«•!■'; »■ 1^7 !W'n».:WfW'l^f.tVTi'lJ^H^.ifMKI^I

ÄU.'Ul' -.V^* = ZU^UU+V<£ -*-(&- ^) (20.3)

u.' = r^^ (fi.' + ^ + rV) (20-4) a ■J0/

;>.'u<>'r«' -f1' "•' r»'

The boundary conditions obtained from (1.1) - (1,4) are

pt; = - c i +• 9« ^i - ;ia•)^ - V M-*. e. )

For R»l + ae9' (^.l to 19.5) and (20.1 to 20.4) give for

first order

Uo = Cl - X-ae ) 9«

V,' = (9*/^')a

,' =_L (1 - xa»)

For second orderi we have

0- - &/$')

(21.1)

(21.2)

(21.3)

(21.4)

• f22 1^

?; = u-^>,a + c^'-r^-i«.)4 - (22.3)

38

If ~J .....■.■,..J.^.^.^.,.^.>,.^.t^,.^.....,..^.^-,.-^.......,..^ ^^^...^^^^^.^^.^^^.JlM|m :.. i^ .^J,......a ,J.^J^,.^^...^^i.,

ffT1^ * '' ^^.?^.:^^!^Vi^fWI?■-VW»:W[T7r^ir? T;t^ra^rw^JK«7r»T-7W'^-TT"?r^^v-5^"i (>, >.■ i«,. ..iflj»^ «r^'iT^iTTTTTnW^i^ns^^

External Layer - Region 3 Analysis

As in the case away from the axis, we must develop a body

layer solution near the contact surface. The relevant ex-

pansions for the variables in this region are

p = po" + €e pa." +

a = £e Uo" +

X = 6eX" u. = €*'»-"

V = ee'Vo" +

a-= ^~z Substituting the above in the equations (7.1) to (7.5), and

collecting terms of like power in 6« , we obtain for first

order results

**&' (23.1)

RoVfc'' = P**/po* (23.2)

For second order results

lEi' -o W (24.1) Uo"1 = U."^ + zCPxI -Pi") (2t.2)

p.;' ^Po" p,;/ Rf The boundary conditions which must be applied are

39

r—— i ""i"' !■—-"—w« ■»Jim ; in laimn^^vmiiimii i i i. Miiiii.in i ■■ 11 J i «mini» ■ i ■■■■>■■ M i in iinwiijuiu i.iin mpppgppippigaMapp^^pw^HI

pi» - VMAde -1

The final results which must be matched to the near axis outer

region.case are then

po - po CX)

pi" = PI'CK)

fa." =/t"Cx)

(25.X)

(25.2)

(25.3)

(25.4)

(25.5)

Following the same procedure as in the away from the axis

case, matching requires that

po'cx) ^ po' = 1-

JD0"CX) = po' = 1 (26.1)

(26.2)

(26.3)

(26.4)

The values obtained for (26.3) and (26.4) are simply those de-

rived from the near shock region expressions evaluated at the

contact surface. The value of Uo'is then

40

*.-.■.-. M.--; „%■.■--■--■■. MUMiniVTH'J'liii'l ' ' i ' 1 '■ ii iV •|,"i 'i ■<' ■■■■... -■■-....,. ■ ..t.

f i ..•pa.>.l>..u.< 1.^'j.,. . .1 •>•- .1.11. .omi [„m mil 1..1 WJR J-I ui imtmxiH.ii i i iniiiiiii u i ii II>I«I«>II>>JI'" u ' i.m mwwmmi'm^^mimmmmimimmmmmminmim'^Hmiinim

Matching to outer layer results, we require that

l\^ ( u' - u" ) ^ o

, (- f ti-^ - ci-^rcri - <Pnsr/p ')/3 -

^ ' ^,(1-20«) g^^ 3/ /

Expanding the square root

** From boundary conditions derived on the basis of mass flow through body layer

41

^«*-'t-'">—"■•'•- ■ ■■■-■ ■ ■ ■■■ ' —■ '■•■ ■ -■-■ - -■ ■■■■■ ■ ■ ■■ ■ ■■'■■■—

^:^;i;uf^M^^fe^*<^^-^)irt^^

— «< ijipvijiiKxi^Kiiiiiia-xuii-j' ijpi>iii..».>pi IU.^I.»««» i . i,lli|,uniiljinia.pliiB i ..jiji P1.1.1U m-t ■iinijwi-iuiupaum'ppnifCTOTiii.iiLniiiiiii l"1 ,l""l«l"'«"l".H»llH"0«WB«PPWr3W><,?WW«WI"^l^""-»«lulW».i

It is seen that matching is assured. A uniformly valid re-

sult for velocity in both regions is then 1/2

^(Cl- vufqC* ^eeC^,Cl-2A.^ i)f(27.i)

Using the uniformly valid results, we calculate the layer

thickness from (22.2) (27.1) and (7.5), which gives

The contact surface pressure distribution is given by (21.1)

and (22.1)

Pcs - 1 - ^~(l-*ae)*£.9'le + -IS— (28.2)

Expressions (28.1) and (28.2) are then used in a scheme, out-

lined in Appendix A, which determines the external shock ra-

dius of curvature and position along the axis of symmetry,

for given external flow and jet conditions. These values are

then used as initial conditions for determining flow proper-

ties away from the axis.

42

^'^u^..^,,..'. . ..^^-l.^.--.-^ ^..^-^r .^- > ^..^t^ ^ a.^., ^.■■..„■„^ J^.... .„■■..;... ^.. • .^J.-.^ * -iiL-^■.-l.—^. .A^- ■ .-^ ■-»'.: ■'■- . W^ .■,■.. .^I. .,-■ ,■■1,- w... "•^"^■-tjtfliitiUidinililiHii ' t ^ i HIIAI- ( ~l--* . »■■^■^ -^^.i-- .^.^ ^- ■ ■- --:- .,— ,.-,.■:.->■-...■■ L ^.., -.:,.,. ^. V.> ■:../..-,^.!...J^^...J ^..J.^.^-y^i..,-:-^ ... .■-. ■..:.■.--. .Li^JI>^

«Sf^traro^B ■ ----T-TTTI v-JuM*!mW4^^vyi*^,.CT''-w«iw?.iPW-W'^■ U»«JWfvw*«rff^iy^JW'sw^1.■»''!^l^'MP^jwP«Jiy*(>wjj*w»w^!^J < ..^ ■ J ,sw' «^ifif-wi,™!u. -itPviiv.ppjNJw.vw^w^urw, -.i-,^?«^111: -..■'■"»' i- J1 Mi JWpi

The solutions found for regions 1 and 3 as the variable

(^ tends to infinity must match the solution found for re-

gions 2 and 4 as <p vanishes.If matching is achieved, then

we have proof that the expansions for the region near the

axis was the correct one.

For this analysis, a composite expansion for the near axis

and away from axis regions is not sought. Therefore a match-

ing of these regions for the special case of a spherical shock

will demonstrate the consistency of the expansions and will in-

volve a much smaller calculational effort than if the general

shock case were attempted.

The away from axis results, equations (12.1), (12.2), (10,1),

for a spherical shock and small U? are

u» = 9,* - yj/z + *6e(f*% -9*3/5f - 9»a)

- i - Z 3r 3f 12.7 figel

3 m* 2

9" O 'g^

2

43

. .. . •■ .-^„..■.^.«.«««^—«iiw^^ ,.,■•. .^ -..«»iiÄ-iu «nu^^AMMiiij^^,

—-. ■■i.-ipiii.- ■....PPWIM.JI. i.n mmmmmmmi iMiiiiiiu mmm^mmmmmm*mmm*m wmmmmmm

The near axis results for a spherical shock, CU = 0 » from

equations (27.1), (21.1), (22.1) and (22.2) are given by

u»= €e 9;x + ae^C-^ -xC-f'V ^zSt^r)

- (l -(J»^')'*))

Following the same matching procedure as was outlined for re-

gions 1 and 3 and 2 and 4, we define a variable 7? which des-

cribes an intermediate region where both the away from axis

and near axis expansions are valid. For this case,0(fe VKOQ)

and therefore for U matching

lim €e^O

Matching is assured for U'

44

Main mm-in ii in inn — — ---' --" - - •-■ -— "- ^ i J^—■ ■■ .:■ „tt^... ^.^. L;. ■.,.■..... -■■ •-•--■--.«lit^Maltr ■ -r ..^^^^-■i^vm^atffnTir1-1-1-^"-""--'^---^!- im .1 i! i [HnHlMM

■ -. ■■.•-„r.^jtmi fVMJvwpwi^ -..I ...^vMw^^wiy^.ww-s^i'^! H^!^«^™viJ' f-■"■"■' '> J•!■'-»■WT?W■- '• ^V5■"WT^-'»r™^M■ f wmu-tw ■ ■ i.j-! i -i. i»>-;» ,'. uji'. ■' ■ i-.', ,^■l9ll!W^5fl|^CT.,i.P-WI^'AlVV»Pp■,W',,' J■-'r-Pf*';!, W*'W'>*w-i.™.J.™mvfWrv**mXUV ■■ * ■."■'.'«'' I' n n T-"TW':^.'WJnT-rr.w^

For p we have

Therefore, matching of p to this order is assured.

For p we have

Therefore, matching p to this order is assured.

45

. - ■

■^■^._..... .......M^.^.^^^^^^ MtttWKMRUlMA

lftfliniln'fii'd«'«1iüi1ii-"^>-1-t-J—

^WTWWWWWWUU^' ■ ".^ i' irv-v-rvviwviwm 'i^rH^r-P??w^P!^TO5^pBi(>iw.^^^

Internal Layer - Region 6 Analysis

We start the analysis from the shock relations, which are:

(29.1)

r " F9*1 /e (29.2)

U « Ufloi CöS>ö (29.3)

V » € Uooi Sin.ö (29.4)

Unlike the external uniform stream case Jw,fu^ pfc,; are not

constants but functions of the undisturbed plume flow. Their

functional relationships are determined by the assumed source

distribution

n ^ jpw 'JW-jprH,, (30.1)

p»i - fwQW^».-.)*' (30-2)

^ -c^(fiuni+^^)-ri(3o-3)

where n = b/(^t- l)

We now expand the boundary conditions for€4-»0/V*<» such that

CioMc^;-» O(l) . This will then indicate the orders

of magnitude of the variables within the layer near the shock.

46

.^.^..^■^.^.^.■^.^.-.^.^^^ , .v-.^,..^.... .^W~...-..-T...', ...,.„....^., w>.,^.^ ■—■■■: . :,. ... - .

^^T^^j^^^wB^Hif^^i.^i^iin^T.«i.,jji!i i.,.i.,v.|^M<^i,q<i^.MW?L.,.>.?HU^v>;iK!!i.v^JniK-tH"'!Mf>n■'.l■"w'JF-W"T-w'.i....vHi,.1.7.^*¥!-.«y!-.,,,if,,^..1,,.*.,^<f^«^T.nrKfi^Jn^^M.'Wr^-^fHW.^«.^V'VMiiiff-^A^.-j 1.^IV.*I|U.I. -^UiN,.,. «^iWJWRTW.W'S^y.'ÄiMlWWW,WWlWSMPW.Ai-'..!.^ ...,.L 1 -Vljl.iq|

It is hoped that the orders of magnitude found for the vari-

ables will be valid through to the contact surface. If this

is not the case, then the solution based on the orders of mag-

nitude near the shock will break down and a second expansion

valid near the contact surface will be required. Expanding

the undisturbed plume parameters, we find that

When applied to (30.1),(30.2) and (30.3), we obtain

f«; V-'o-flllL ~ "^ (3i.i) £»i » f<6Ä> *§£!$„ ~ od) (31.2)

The first order expressions, (31.1), (31.2), and (31.3), are

then introduced into the expanded form of the Rankine-Hugoniot

relations, (29.1), (29.2), (29.3), (29.O and (29.5), which

then give the order of magnitude results

ü - oCe) Ucoc v ~~ ofeco) Uoot

where 9 is the angle the free stream makes with the direction

normal to the shock.

47

■- - - *^~~s~^u :. „...^ ._.. ..^....^ _.a .....^....... „. :-■.-...

a^rB^^wrTOr!^r»P7rr?T^T?^^

At the axis of symmetry, 6co is the density ratio across the

. Up to shock and is given by 610 = 'JJizJs. •+- vj ^"

this point, the magnitude of the velocity U parallel to

the shock has not been determined in terms of ^Jo ; however,

its lower bound can be determined from the condition that the

shock layer is of zero thickness in the limit as €co"*0 and,

therefore, the flow must be turned 90 degrees upon crossing

tht shock. The deflection angle is given by

-tan S = JM&L&nlÄ - l)co-tS

where ^ - 1L - © for near normal shock 0<<* 1M001 ^ ^■

"tan § = C6<;ttft ha^e = 6 OO

This then yields

«IV* e 1 ^ OÖ

V/e can conclude that G ^ OC^i'o) and is obviously less than

OO«) • This result can be written as

u = oieio*1) 1 > hn > o

The dependent variables can now be expanded in terms of 6LO »

48

— - ■- "" ——'■'—"•*- .«■ >.-,.^fc-^.1-^uJ.^J^H^.....J^»-lJ. ■■ • ■ • ■■■■■■■ ■-■-'-' ' ■■■; ,-■■■ ■

■■.. . ■.... 3....,.^.l..i ■..-.-- ■/.■- ; ■■■ ■ ■■- -■■^^!■ ■ :-- ■ u

.I.J»IIIIWPI.IWIMW'J»>«"—«»••»«»".■••■-.■•.»"I..I' "■■ ■■.••MI III«MJ>P«<>I » "■■'■ •!- ■■'..'. - ■■i,ij.m> i I.IUJI »i.nii, ijiiiLiiiiiu.ii.iiuj iimnip*nm««Pinm<nnMit*4BBii|ipnmBnpnip>imiM|

following the first order magnitudes found from the shock re-

lations. The non-dimensionalized expansions will then be:

^/foOiM^i = Po + «EtV^pi +■ 6i0 pa + XHl.X

to /x+ ...

+ St'o^Ui + €t'03^ Ua +

+ e^V, ■»- € a,hi+i

(,« ^ +

It is assumed that X will be of order one, from geometry,and

«^ will depend on €cö to some power n' The unknown pow-

ers in and n will be determined by requiring that the govern-

ing equations, when written in terms of ec* , yield a physic-

ally reasonable non-trivial system of equations. The govern-

ing equations are:

^ un oX continuity

X momentum

u. momentum

energy

•bx ^-

where IC is the radii of curvature of the shock surface and

H=l-K.^ is ehe ratio of the radius of curvature of a constant

y. surface, with respect to the shock curvature at a given

value of X.

49

.. ■ . •

■■-■■ -• ■ ■■ ■ - ■. ■ -■•■■■■ ■--■ ,>.j^v..i.., L-, ,:.:*ml^.<L.-iäi^. mtflaaaaaa ■■-.-—: -.- .- ■ ■ .— ■'«■-M;,.. L-,AÄlU*Mfc*.»».<

' ■" ■IIIW«.^IIWU." -i -i " ..p......pJvvi.i"j.« nil jimiHiiipimnwnmpaipiiHiiijiwiOT<iininwivijiuiiiijiiiin^«wi^«m*«pm*npv^mmtM^m^mmmmmmammiminmimmmi'mmmmmmmmm

Substituting the expansions into the governing equations, we

obtain the following results relating the unknowns m and n:

^X " v \ x momentum

4U~' = oieu1'") £ momentum

continuity-

Applying the source condition l^at: 0(l}we find that m = i. ox

It then follows that n = I from the other two relationships.

Sinceö^H-yö is now found to be proportional to €io^ t

then in the limit as €to->e , ß must equal Wz . The

shock surface is then a spherical shell which is centered at

the jet exit. The magnitude of the perturbation from this

spherical shell will then be derived b] the use of the known

value of 6-0(6io ). The radius can be written as R=i-«-GCx)

From this, the value of 6 can be calculated R'Cx) = G'O<) = 6^9

• The magnitude of (Sex) is then 0(6c*Va) which then leads to

the result RCx) * 1 + FCx)> 6^ . This will be the

form used for ROO . With the values of m and n'established

the expansions are

f - fo/€Co + fx/€C0Vz + Pz+ "•

Ü = €t6V2 Uo + ^co Ui -•- 6^* t(l -t- • • •

50

-■'■■- j ■- - ■■■■■■-■■■■■ ■■■ ■■ '■■-■' ■ ■ ■-■■ ■■■■ -■■■■-■■ -.■■^-■. ■ ■■■: ^ ..-■ . ^^ .. ■-, -. ■,,.,:.^..^,.,^^-^^^^*::^.^..■:■....., ,.,.:.,- i... ..,.,,. . . ; , t-J..m^._^ - ■- - : : , , ■ ■ - ■ ■■ ; . ; ^ ; i ■ ■■■ - ..- ■-■ ■ ■■ .--.'■■.. - -

........ ,, „..-i.., , r^—^~, , -,———.......„,,„.,„,.,,, „ i i liai! ii j|. iiiiui>ui.uiiiij«iuiii>wwuii«npvninniiiijijii .inii «QwanwniHip^'^

The flow equations become considerably simplified for inte-

gration across the layer if they are written in terms of the

Mises variables X,, V .

The distance downstream is X and S^ is the streamfunction.

The flow equations in this coordinate system are

K^lE. = K.U -1- "^ (32.1)0. momen- ö^ a?c turn equation

VLi+y-* +-^a. - CM (32.2) inte- « -f grated X mo-

mentum

p/p^ = bC^) (32.3) energy equation

"by,, - - _J (32.Zf)stream- ^^7 ~ jau»- function defi-

nitions which satisfies con-

H ~ 1 - K^ tinuity direct-

After substitution of these expansions and introducing an ex-

pansion for ^ and K in terms of X and 4^ and collecting

terms of like order, we then develop the following system of

equations.

First Order Expressions

Momentum equation across layer

"^ft. = O (33.1)

51

■ ■-■■-■-■

' ■■' ■ ■ ■—i—■ -.^..i.—iiwipiii ia.i.iiiiMiiwiimiu w i.ii -....i • .... ■IUIIII [■iiiii.iiHM.inwHui iiuii «m. iji juin^«*n«paim,iiua«iwnii i ii iii«i . mi i i JW

Integrated momentum equation

(33.2)

Integrated energy equation

po - /b (33.3)

Definitions of streamfunction equations

Definition of streamfunction

(33.4)

"bus. _ Vo iJJ ""■ 77"

(33.5)

For the second order results, we obtain for the momentum

equation across layer

(34.1)

Integrated momentum equation

ßo* Po (34.2)

52

MB '^.-..■^-■...,, ^Ju^^a.iiiu.t. .-.:,...^-.:.:,:.i.,...:*i..x .. .Jj.^l-^Wu»---^^--i.--"-'-'' •■■-ii|^^f|Trtllf|t-ll|-|>|i|r-l'"-^*^-'--''^*^;--*---"-■■■-■ ■ - ■■ -■ ■■■■■ ■■ . '-■■ ■ - ■- .-.■■■-■....- L.--... .., , , :. ...i .. .... . „,■:■ , - .■.,-...-.^ . -,...u.;-..-..,. :..... ;_■■ . ■ ..... . :- -.rHUfl

'■w.—»^»-IWIW^IW-WWH ,m* .,.H.,W.,«.^..,.,«,L H,« •■■^■r>'uii|BV,NiuiiajpnnpJiiVViiiMW««-uvji>-MwiiiHip>.i ■iUnH«ji iRjiuuiJMBijainfn^pnpni>i|ii.wiiw|ui'|tn." JJIJMII« «ijBwiiiiL^iuiiiiHPPiwiiuiiiuqjwmcnvnvwipp'Rpv^fvw^^ ■ ■■■■■n

Integrated energy equation

pi - f 1 (34.3)

Definition of streamfunction equations

>>^i - 1 r0 \ J^o Uo r© / (34.4)

Definition of streamfunction

(34.5)

For the third order results, we obtain momentum equation

across layer

-)

Integrated momentum equation

Ui* + 2.UzUQ + Vo1, = 2.Ma*Uo^ + MxÄ* + (35.2)

/t»* ^* /fc* ^*

53

... ■.q^^.^ww.-i-t,.:,.

...

-^•^rrew^^fK-^p^ir"^^

Integrated energy equation

Definitions of streaaifunction

We must apply the expanded shock boundary conditions to the

outer edge of the layer. In the limit as Ct'o"*© , Moo^-^oo

such that €-00 ^eo^0 -* Oi^-J ^Q obtain

First order results

fo - COS (36.1 )

jp. - ^"m (36.2)

; Uo = F'C$>) (36.3

Vo ^ 1 C36-4)

^e =. O (36.5)

Second order results

px = '^F7"^^ . (37.1)

^ ^ -^F cos" USi (37.2)

54

I .. .,.,.„...^^.::..ll.J,...,-.,.:.n:^-.w-,.j-..<c^,.i..-J....-i .■^,;,^.:,,..^.^I.^1>^^.^^JI^^.^ .. ^.. ..... .,^..^.. . ,v.,.,.. .,..; ., w. ^. . ...^ - ,-.., .■.. . L-^ .:: - ■ ^^__ ■ ■■■■■ ■■ ^'■ ■ ■-■ ■■ .:■■■■ - .■ __ka±,

' "■—'--''■■■'""'»--•■'"'■•■"--^-"'-"^^'--"-■"^-'-■"^^

Ui = - F'F (37.3)

CU - O (37.4)

Third .order results

?a..^n|ft(3F^i.F- + Tt_:ft) (38.13

b^ÖTc-iy^C^ - F/a) (3Ö.2)

CJt= FF' - F'V* (3a.3) u;2 = o (30.4)

Before we integrate the equations, we introduce another

transformation of the independent variable T . This vari-

able describes the strearafunction in terms of the position

at which the streamline has crossed the shock. It is de-

fined from the source conditions as:

H7 ={ cos* Eton sm9* 49* (39.1)

A point within the layer will then be described by the co-

ordinates ( 9 > T* ) • Substituting (36.1 to 39.1) in the

equations (33.^ to 35.4) and integrating across the layer

we obtain

55

r- ,,m„w i.u.n , ,■„„—.w^.-™ , „mmii*-. IIUJ..... P..- H n I. I.I. ... I .>•.■».■» LMM IB iiULawiiiii in i^n m | I wrmHCM|gr£K>PMIPn«m^

First order system

p. -

Us

Vo

s. COS

= co& 2©oo

•= CF - Z\yy Po/Po*)

9«

Second order system

p! - -2Fco5W SU^

Third order system

(40.1)

(40.2)

(40.3)

(40.4)

(40.5)

(41.1)

(41.2)

(41.3)

(41.4)

(41.5)

pa = COS CSF1-!- F" (42.1)

56

'■ ■-■^■■i ■-■■ ■- ' ..--•■.,.i-i ■■ ....... ..■■■■ ...■■...-.-■..■..—....-■-..■..-... .■■ ........„■.^.---. v. ....i. i.-^. ... ^^^.Vtl.! .>-.V ,.. ■■■--■ ■ 1^ I ■■ --■ • ■■ - ■■■—"**

(p»*/po* - ?**/fGm ^ ^ln^)) (42.2)

ro J3«»^ /oUo UoTej ^yr t[^

f) (42.4)

Examining the above integrals, we note that the calculated

variables remain finite and non-zero from the vicinity of

the shock through to the contact surface. This indicates

that the solution is uniformly valid. V/e now must exam-

ine the behavior of the solution as v/e approach the axis

of symmetry. This is important since in order to evaluate

the matching of the shock layers we must determine the ini-

tial radius of curvature of the shock surface, as well as

57

iMna MMH - - - --'■"--' -■'-.■-^ —

its position in space. This requires that we know both la-

yer thickness and contact surface pressure in an analytic

form so that tho matching of contact surface pressure and

position can be determined. For the case when ROO * i.O

the variables can be expanded in even powers of (D near the

axis, and then integrated across the layer. These results

will then be used to illustrate the change in order of mag-

nitude of some of the dependent variables near the axis of

symmetry.

From the source conditions, we can write for small ^, $

f. - 5 - i - ^r which then leads to the first order results

Vo = (9../f )*

The second order results then become

58

Ül*r~. ■l-.^.^^J^. --riiimlllilil-l^--'-'-^^M^t..^^..-^.^^Uw.^.»^^.^-■..r....w 1 ■ ^ v..^^.:^..^v-H^.■„^.i.^.i;■ ^-J ^^s^.■.^..>„■.-..;.■.,.rw,.,^..^.^,-...,.^ij^.^.^ .^^,., ..;.^,-,-,.._ ^.,,..^.^^..^■.J;,^..r...: ,:._ ,.,_..■.,■„.^.......... i -... . ■:..■■. ^.......■-:L ....,^^.-.^

It can be seen that as we approach the axis <$*>§-*■ ofeX*)

and

U ^ € J^Uo ^ 0(6,0

v ^ euVo - O^eco)

The change in orders of magnitude of the variables then ne-

cessitates that a new expansion procedure be applied. To

first order p,^ and V do not change; however, U and ,*

are altered and the new expansions must reflect this.

59

■^■i.!:.--^-;^ „ :. ^^^.^L^..r^i.,.-.r--......^^..■.^-.-.,.J..^^,1L..^ -.■ -■ ^-.^■.^. -.,. J^^^^...^. ....■-. .„■^:-...^.-^.---:^^t..-u .;■...■:;..^ ^k.k^,:^„h^.^.:.^,^,^:1 ...■.:■:... ■.■■..■.......^ . . , ^ ■ --^^-^..■TV/.vattW^a^^g,^^^^^

Interual Layer - Region 5 Analysis

The shock position in this region is described by an expan-

sion in terras of even powers of y away from the axis;i.e.,

R = i + a,6c'o94-f ai.£uz<pH+ •

Based once again on the shock relations, we can expand the

dependent variables in the form

U - dtoUo + €U*Ui, 4- - •

p = Po + €copt + • ' " f> ~ fo/e to + p* + • * '

to

Substituting the above in the governing equations, we obtain

the first order system

These are of the same form as the constant density flow equa-

tions.

60

.......-..„....■..■■■...^^.^...■.■v. ..■.^^.^.^...>1.^<^^,^^,^,. ,^^,„-^:,...^„..1.„.^„. .,^.,.J.;^QJ^...-,1 J-..^.;..^t.^,..,^.^,-^'--^-'--'-':-.^...^J.,_, ■„....■.■■ .:

We therefore apply a constant density flow solution method

to this first order system. A streamfunction is defined

which \^ill satisfy the continuity equation identically. This

is

"^ = -Foüo "äiE = r.vfc

If ^f is known throughout the layer, then Uo and Vo can

be calculated. The equation for ^ to this order of approxi-

mation is

V/here S is the vorticity within the layer. For an incom-

pressible flow, the circulation must remain constant through-

out the flow region and this leads to the constancy of the

product of vorticity and distance away from the axis. Not-

ing this, we can then develop the vorticity inside the layer

in terms of its value at the shock and its radial position

within the flow. The result is

The equation that must be solved is then

61

Substituting in a strearafunction of the form

(44.1)

we find from (43.1) that

From the definition of the streamfunction we have

From (45.1) we find

-^(jL^+HaAcp^o* -f- ZCQ + ^.69^0 = ToVo

(46.1)

At UQ =• O which is the shock surface, we have Vo = L

which then gives from (46,1) C - V^L

From (45.2) we find - («^^^a»)^ ^ D + G?*- -n,Uo

at the shock Jo = O Uo « ^ai<f and from (47.1)

we obtain

5 = O G = - -2.at

62

IMMMi-il*iii*imMriii>iiiin^»imilllMMi«IMitlilti^ii-*lnii«tiiiiaiaMiin>^ _. .^.

These results then give us the first order velocity compo-

nents in this region.

vo = i - Haty. - OÖ^T+^i) V (43.1)

and p© = p© - J-

The normal coordinate £- to first order as a function of <£>*

and O can now be calculated from

IXs. =- 4^8- C49.1) V<5 Ue

Substituting in Vo and Uo in (49.1) and integrating, we

obtain after applying the shock boundary conditions

or

u0. ,^ W^^HC^V^I^I; )

The pressure field can now be calculated from the known ve-

locity field given by (48.1) and (48.2). We must integrate

the streamwise momentum equation to obtain the variation of

pressure along the streamlines with velocity. This result is

üo^Vc1 - UoJ ^ Vo: - ^CPI - Pi*) ^^

63

-■■'- *—'• —.*.■..,■,—^■■■,. ^--.. ^.—^ ,-J—-~— -niMi i -■ - ■ ■ ■- i ■ m* i i i i t^iMH

From the shock relations

Applying this to' (51.1), we then obtain

^i. /

- (i -^^o-C^^Wj^)^ (52.!)

From the conservation of entropy along streamlines relation-

ship, we also have

fi~fi~ fx* +f^ (53.1)

Making use of thejalue of pt-p^and noting from the shock

relations that pL^ " C^^^j)^ we can then write

for (53.1)

■^ *

%

The complete expansions for the dependent variables in the

near axis region can now be written

v = euvc + OUc*) (55.2) (55.1)

64

■^■■„..J..;-,...^... ^.^.^.. .., ^■■....w^^^,...^....,.. .,..J,-..-...J...^^^.u..^.,^.,,^.^^,,.......,....^^^M.i^m..Mj.,...... ,...,.., J,..J.,.-......;.^.1...^.,.. ,■.......,,.„„..

(^-(gj^^Of** ^ OCete) (55.4)

These variables in the limit as ^^ tend to infinity must

match the variables obtained from the equations away from

the axis as the variablesf,^ vanish. If matching is ac-

hieved, then we have proof that the expansion for the region

near the axis was the correct one. Since in this analysis

a composite expansion for the near and far away from the axis

regions is not sought, then a matching of the solutions for

a special case will be adequate to indicate the correctness

of the expansions and will be easier to perform than a gen-

eral matching procedure. The special case chosen is that

for a spherical shock. For this case ai" - 0 and lthe inner

expansions can be written as

P-1 - «£ - ^i^^Ci-^VV ?&)^ 56. D

-P-V #-^-^T)HW#-^^ (56.2) u = ei.nfe.yjy. + o(et'.')

1 ' (56.3)

65

■■■■■■■ "r- r Xamt*.

■-nn«iMiir ^^^.-i^-rf^i.^ ....- — ■- 1 .■.., ._ .- .— . .

For the away from axis region, we have the following equa- ^

9,9« tions for small

P = L-*k{%£$' ^O<€t0 (57.1) f =C1 " Sfe)z9a>co + ofeJ2) (57.2) U = €^ ^(g^^C1 -^^r^ 0^) (57.3) V = €to(3Wf)a +- 0(6^) (57-4)

Following the same matching procedure as that for the exter-

nal layer flow, we define a variable^ which describes an in-

termediate region ^vhere both the away from axis and near axis

expansions are valid.

Since 9 = OCl) and f - O^^) then 0(^ ^ < ^^

In this region, we have

then hi^ ^ -♦ O and liyyi g> -* 00

Applying the above limiting procedure in the region of common

validity, we must have

€10 -* o

V/here fo and fc denotes the various expansions for the vari-

ables for the outer and inner region respectively.

66

■---J- ■-- a ... --^-~-.-^—f—" maaflMy—.^.„w....... „.^...■..■*.„...:,.■.. ■■..:-,.i...,..-■. '■.»»...■i - .■..■;-^.-J^.^......—...........—..,_..,...-...,.,....^.^.....^„■„.■i^v,,.J.J.,., .,~-.i,*tik

Matching V

In the-intermediate region, it is known that

Therefore the above is zero in the limit, assuring matching.

Matching U we can write

Again, making use of the expression for 'jL we obtain match-

ing for U .

Matching p

+ (1-M(^)^) - t^Wf. ) Introducing (j0 again, we then achieve matching for P

The above matching process has demonstrated the correctness

of the near axis expansion. V/e now use the more general ex-

pansions, (55.1to 55.4) to determine the layer thickness and

contact surface pressure distribution near the axis. The

results are •

67

aa i i— — ■■ —^..■~....^^^.——1..^.....-.,...-;-.. ■■ — -.,..-,.,. -..■,.■.,..,...;.. . .... „,-, „.^..^w.. - ; ^ tj.^-.-- i.,.^.-....,;.,.. ....L-MI

These expressions ( y. and p ) will then be used in

Appendix (A) to derive matching conditions for the internal

and external layer near the axis of symmetry. From this pro-

cedure, we can extract internal shock and external yshock ra-

dii of curvature and the positions of the shocks in space.

This information will be sufficient starting conditions for

an integration across the layer of the away from the axis

equations. By this procedure, the full bow region, from the

axis to the corner region, may be calculated. ■

68

^ .. .■ - .>■-,..-.■^.^..■^.-,.JJw....,...-^.-..^...■...■.„.■ ■ ..^^ ...j Jjl.-..t?J..^,..Jj..^.:...,...A:t.-;..l.--;^^Jf^..,. ^.u^..-.:..,^-,..-.^.0...■.■.. ^-.^ ■- ■ . ■ . . ■■ .. .. _!, , „^j U , __„. lifau, *l*IX&i>L*äi*

DISCUSSTOn OF RESULTS

From the analytical bow region results scaling parameters are

defined which can be used in experimental simulations and in

extensions of calculated results to other conditions. They

can be "obtained by examining the first order results of the

bow analysis. For the internal flow, we find to the first ap-

proximation that the geometry of the layer (i.e., shock posi-

tion and layer thickness) depends on the parameters h , S;0

öeaCMj^ön, ^j) and Rsi0 (Tt,8oo , n , *,') . To this list,

we must add external layer parameters €e and Rse . How-

ever, from the matching of contact surface position, it is

determined that R9e is a function of Rs».^»^^^ and there-

fore only €e is added, giving as the significant defining pa-

rameters of the bow region 6»^ n^ 6i0 ^ ^e , Ksdo • These

may be further reduced if we consider very high Mach number

flow for which €zit and €e reduce to &j' and Xe . The geo-

metry of the flow is now independent of stream Mach number.

For a correct experimental simulation, all of the above para-

meters should be matched between the actual system and the ex-

perimental setup.

Also, for a given system with fixed ambient and exhaust gas

composition (i.e., fixed ^jj ^t ) and nozzle conditions (i.e.,

fixed ©»»n ) the bow geometry will scale with Rstoor ^ -or

all flight conditions (i.a«, altitude, velocity) and system

69

. .-..^^ ^^-^■^...^■■^. ^-_>^^—^- . - ■^,^..—...^—n,,,,. , ^ ..._.. ^ ._■— ^.^

thrusts (i.e.jFoj). This last effect is illustrated in figure

4 which compares in a coordinate system which is reduced by

the scale ixsio experimentally determined ^»7,24 bow region

geometries obtained at different values of TT . Also shown

in figure (4) is a comparison between experimental results

carried out at nearly the same TT but with different exter-

nal flow Mach numbers. As can be seen, the agreement in

positions both for varying IT and Mooe is very good, indi-

cating that the scaling parameters are correct, and also that

for high Mach numbers the geometry is effectively independ-

ent of stream Mach numbers.

The correctness of the scaling is further substantiated by ex-

amining figure (6) in Charwat and Faulmann . In this figure,

many experimental values of axial bow layer thickness divided

by r<Stoare plotted versus Tt . For both 1^»«= 2.75 and 7.1

it is shown that the above ratio becomes constant as Tf becomes

large, indicating the correctness of the scaling parameter

Rsio . It should also be mentioned that the value of the

ratio reached forTY»! and M«je = 7.1 is in agreement with

the value obtained from figure (4).

Having established that the geometries of the shocks and con-

tact surface are fixed in a reduced coordinate system, we then

note that to first approximation all the nondimensionalized

flow variables will also scale since they are only functions

of <p> R/R-sio , ^©oo^j^ev/hich are fixed quantities for any

70

--•— -- ■*—— ■ -■■- -- - ■ ■■ - —. ■ - . _ .. „-, , ,

t™™aHP!w«p»TfTWf'r^--'™»^^

given point in the flow if the nozzle conditions and exhaust

and ambient gas composition are fixed.

Another result of the analytic analysis of the bow region is

that it defines the accuracy of the Newtonian impact approxi-

mation which was applied by Laurmann6 to this problem. In

his technique, the contact surface position is determined by

the balancing of Newtonian impact pressure along the contact

surface.

This approximation has been shown to yield good results for

surface pressure for the external bow region flow 25.

For the internal bow region layer flow, we must examine our

analytic solution to establish the accuracy of the Newtonian

impact theory in describing the surface pressure. From the

first order results, the layer thickness is a constant. Also,

to first order pressure is constant across the layer. The

actual contact surface pressure will then be determined by the

shock shape, which for constant layer thickness is ythe same

as the contact surface. Therefore, we would expect the New-

tonian impact analysis to give reasonably good results for

the internal as well as the external flow contact surface pres-

sure. The contact surface shape in the bow region,as deter-

mined by the matching of the impact pressure across it, should

then be reasonably well predicted by Laurmann1s6 analysis.

71

„ , -^ .,._. -■^...^■■■^ ^ -... trz^n^^ i-^^i:-- .;-.-.. .-»wia

> iwmmmmmi^m^^mKmmmmmmimmmmmmmm mc^^m^tmi^^m^mm^mmmmmmmmimmmmmmm

'I

To test this hypothesis the contact surface position predict-

ed by Newtonian impact analysis is compared in figure (5) with

the predictions made in this study and in reference (7) for a

characteristic bow interaction. The agreement shown is very

good, indicating the usefulness of this approximation to quick-

ly obtain bow region geometry. The success of the technique

in this axisymmetric case gives one confidence in extending

this simple technique to asymmetric bow region flows where

the angle of attack of the jet to the free stream is not large.

For the transonic corner and supersonic far field regions,

Laurmann's ' technique must necessarily fail due to the tnick-

ening of the layers and the acceleration of the flow in these

regions.

A number of calculations of the bow region flow have been car- ■

ried out, utilizing the analyses outlined in the preceding

section and in Appendix A. These calculations have attempted

to: (i) compare predicted bow interactions with results from

an existing numerical technique; (ii) compare predictions

with an experimental result by Zakkay7; (iii) determine the

variation of bow region geometries and pressure distributions

with variations in 6«. and *j (primary jet flow parameters);

(iv) give detailed predicted properties for two cases which

are characteristic of "cold" jet and actual jet operation.

72

Ba-.i.v.w;.. . . ■ .. .. U«*aUfMUü<iMCM«tk4MtiAäs( ■■.. lüÄfc tlXSAäÜuk* r-w-Urtac^vI'^:..--;.. ............

-- «.p. F*I>—•"F ■—'" ■ i i ■ I" ii i in ii i 11 i ■ M i i ii ii i i i i| i I»PI !■ t\m ■ "IW^

In the comparison of the present predictions with those of

Rudman and Vaglio-Laurin^ identical stream and undisturbed

plume parameters were utilized. Predictions, which included

terms with errors 0(€to) and o(€S%ere made of flow field

geometry and pressure, as well as the other variables of in-

terest. As can be seen from figure (6), the present results

and those calculated by the nuraeriikl technique of reference

(7) do not agree very well in position, although shapes are

similar. Also, the comparison of pressure distributions in

figure (7) do not agree very weH. There are two possible

sources of error which could cause the poor agreement shown.

The first possibility is that the example calculated, with

^•^j = 1.4 is just too difficult a test for the present the-

ory, which is based on having »e^y t: 1.0. Since the internal

flow solution proceeds in half powers of €i0 then for the ex-

ample calculated terms higher than the first are really not

very small. This causes poor convergence of the solution,

which is indicated by diminishing oscillations of the vari-

ables about their correct values as higher order terms are

added to the solution. This type of behavior is observed in

figure (7). However, figure (6) does not show this effect;

and, therefore, this cannot be the total explanation for the

lack of agreement.

A possible source of error in the technique of reference (7)

is the imposition of a point interaction (i.e., non penetra-

73

;

-. ^ _^.„^,._J^^>^>^__^MM^_^_^^^^„ - - , ••.—i

»^•«^•■WWi^lMK^SWWlWII^^WIW^

tion) for the corner region in the calculations of reference

(7). As will be explained further in the section on the cor-

ner region, the point interaction assunrption puts a constraint

on the bow region calculation, which could possibly over-em-

phasize the corner region flows upstream influence on the bow

layer properties. This could result in the large difference in

internal layer thickness which is observed in figure (6). Tak-

ing the above two effects together, it is possible, within the

bounds of error of the present results, to show agreement bet-

ween geometry and pressure distribution.

It should be mentioned that the results of reference (7) have

been comparea favorably with the experimental results of Zak-

kay . However, the comparison is not definitive since the

analytical plume flow model used had an exponent of 2.5, which

has been shown in the section on jet exhaust models to be in

larga error for the case calculated.

Good agreement between the prediction of bow region geometry

and the experimental results of Zakkay7 is shown in figure

(Ö). No linear scale was given in the experimental schlieren

photograph; and, therefore, it was necessary to arbitrarily

scale the experimental and predicted results by setting the

jet to internal shock distances equal to each other. The

contact surface pressure distribution for this same case is

74

It ; '"■-■ .:...■..,..~*^^«-.:.:, I . .^^^^^^t^^^jJ^^j^^^^^^^^^^^J^^^^^,, ,.,...,„. .,„„.. .„.^ ..... ^^ ■...,■-,.„,, ......_.,. .,. ■,..-, ■ - ■ , .„.....,. ; ....^ .^^

» .,J..I.«.»«.PI..PP-.. i » .._.... •liinii-nva u^jinpimn;»!'■" . .1 JI iinwnn i i njiJJ u ii> 1.1,» mmm .11 iij iiuiiiw^n^w ii« 1 mtmmmmtm^jufggfg^mummfn

shown in figure (9). Two other calculations were made for

these same conditions with the only difference being that

in the analytic plume modelj exponents H = 5.0 and 2.5 were

used instead of the more correct 4.17 value. In figure (10),

these predictions show a marked sensitivity of bow region geo-

metry (i.e., shock and contact surface position) to plume

model exponent. The variation of the bow geometry with plume

exponent is larger than the error bounds of the solution in-

dicating that if accurate predictions of bow region properties

are to be obtained then the plume model used must also be very

accurate.

In noting the good agreement between the predicted and experi-

mental geometry, one is tempted to state that the calculated

flow properties must also be in good agreement with experi-

mental values. However, this is risky in that experience with

co-flowing plumes and the good agreement of the calculated

results of reference (7), using an incorrect plume model, with

experimental results show that the coarse geometry of the

interaction (i.e., shocks and contact surface) can be pre-

dicted relatively easily but in all probability the detailed

experimental flow properties are not so easily matched by cal-

culations.

Figures (11), (12), (13) and (14) show the effect of varying

©oo and Qj on bow geometry and contact surface pressure

distribution.

75

.■.■...■^-■. ..,.-... ... .- ^ —... ... _..... ■*-"—----w^w^MHwwuwifc

H«IPJ»P«II.IIIJWIUI]1IH»I.»I™.I»"I"WIM<I«..»I.I.I«. . i .■jiijMiiLiHmnmpwnjiiniiiaiauiii.iiii.Niaii .iiiiuwillWl J^wmni«aCI|POTRIMmnWMmn9lilHI|PH|HmPnWiH«ipi

From figure (11) it can be seen that increasing ©oo from 125.5

degrees to 135.0 degrees, which could be accomplished by eith-

er increasing Ön or decreasing Mj , results in a definite

increase in layer thickness both for the internal and external

flow. ■ Since external layer thickness is directly proportion-

al to contact surface radius of curvature, then an increase of

layer thickness indicates an increase in radius of curvature.

Consistent with the increase in contact surface radius of cur-

vature, we note from figure (12) an increase in contact sur-

face pressure with an increase in ©«. In general increases

in 0«D with all other parameters fixed give increases in

contact surface radius of curvature, yielding a more blunt bow

region interaction. In figures (13) and (14) we can see the

effect of increasing Yj with all other parameters fixed. In-

creasing Xj from 1.2 to 1.4 in figure (13) causes a definite

thickening of the internal and external layers. Also, from

the contact surface pressure curves in figure (14) we note

that increasing tfj results in an increased pressure level.

Both of these effects are primarily the result of an increase

in radius of curvature of the contact surface. In conclusion,

an increase in either Gaoor ^j for all other parameters fixed

causes an increase in radius of curvature of the contact sur-

face resulting in a more blunt bow interaction. In this case,

pressure, density and temperature within the layers do not

decay as rapidly as in the less blunt interaction case.

76

^^-..^-.^■■^....^... .v.-.-.., ■;:, . :...-. ..^-^ .■^.■J...U,;.^.^- ..■■-^J.J^..,.,..■/-,■■ ■■■ .^^.^:.^r..^^.^-^^...^:^^.,^,: -.....■. .^-.^, ^,.. j^U. ^.-.^^J ^^.V.^vw^..,::^ w-^ . W^l.-..^^

■■-*=--^'^w'7r!,"^'"'wr«T^^ ■■"^-*WJV .W-^WK.,**"W-.PJU.^Htf.^u.,..'-^^.1 ^npnfffiinQiVipgppBiiRfvm'! ■■ ^ ^'^«vwffliwwHW« ^ivnvnim<V9n!pVfW'.»-^w^^w*r»ww.?'^.-»^.'U..!HWTOWP

In figures (15-13) detailed distributions of flow properties

for a "cold" jet and actual jet operation are given. The cold

jet case, shown ■ in figures (15,16) would be representative of

a wind tunnel simulation where the ratio of jet to freestream

stagnation temperature is 2.0 and air is used as the jet gas.

The actual jet case shown in figures (17,10) is representative

of a real system that is operating at 150,000 ft. altitude,

whose jet gases are composed primarily of H20 and C02 which

is modeled by setting ^j = 1.24 . The ratio of jet to free-

stream stagnation temperature is .75, which is representative

of the actual system.

As can be seen from figures (15) and (17), the flow geometry,

streamlines and isobars for the two cases are very similar.

However, from figures (16) and (IS) we note wide differences

between the two cases. In figure (16) it can be L:een that

the velocity discontinuity between the internal and external

flow is very small at the contact surface and consequently,

mixing effects along it will be small. ':Je also note a mode-

rate jump in density as we go from the external to internal

flow. Since pressure is equal across this surface, then the

above result indicates that we have a less dense and hotter

external layer flowing past a more dense and cooler internal

layer. In figure (10) we note some significant differences

from the "cold" jet case of figure (16). Here, we find that

there is a large velocity discontinuity which leads to strong

77

*-■-■ -■ - -- — — — i ■"-'*■—- ——■—---— - - —

' ^^^^^-^•Twr-^n«^»^-!rnr,m^^^

mixing effects. Also, the very large discontinuity in den-

sity indicates that a very hot and low density external layer

is flowing past a much cooler and denser internal layer.

If heat transfer were allowed, there would be a heating up of

the cooler internal layer by the hot external layer. It can

be concluded that in the actual case mixing and heat trans-

fer effects will be much larger than the experimental simula-

tion case.

78

imiiiiiiiTf iniini 11 iiniTiftiiiiiiiii"' ii inMiiiiii«ii t in iriininiiaMlilllMlMMi—miililfciMtmiimiriiiir' -- - ■■ — - - m

V^^W^^rrjr^w^^^-^rr^^rrrr^^^

ANALYSIS 0? CORNER REGION

For an exact treatment of this region, the inviscid external

and internal layer bow flows, as well as the relatively mo-

tionless cor*- and the viscous mixing region separating the

core and bow flows, must be simultaneously calculated. The

reasons for utilizing this coupled approach have been discus-

sed previously in references (2) and (3). Such a complex

treatment is beyond the scope of the present study and sim-

plified analyses are applied. Unlike the bow region and down-

stream region analysis (to be discussed), the corner region

flow is not amenable to analytic treatment because of its

transonic nature and complicated coupling with the core flow

and the viscous layer separating these regions.

An approach is then taken which will provide an estimate of

some of the flow characteristics within the corner region and

will also assess the significance of the corner region flow

on the bow and downstream region solutions. In brief, this

section will outline: (i) the significance of the corner

region flow on the bow region and downstream flow properties

as calculated by their respective analyses; (ii) an engineer-

ing estimate of the pressure level in the core region; (iii)

the location of the plume boundary based on this core pres-

sure; (iv) the qualitative penetration of the intercepting

shock layer mass flow into the internal layer, and the loca-

tion of the internal layer sonic line.

79

- ■ - - - ■ um- MI ■— - i - -- „..i -■--■ - , .. » , i,«___

Wr-'--^.'m*T***f*ri^i-mP^^

Before establishing the significance of the corner region flow

to the bow and downstream flow solutions, a brief description

of the flow in the corner region will be given. Referring to

figure (19), the jet gases upon exiting the nozzle expand in-

to the' low, nearly constant pressure core flow. An intercept-

ing shock and shock layer form which are identical to those

for a plurae in a quiescent ambient of the same pressure. This

intercepting shock and layer eventually intersect the strong

internal shock, resulting in a triple point shock configuration

similar to that found at the Mach disc for plumes in a quies-

cent or co-flowing ambient. A complex system of shocks and

expansions are then generated within the internal and inter-

cepting shock layer, which accelerates and turns the flow in

the downstream direction. The viscous layer separating the

core and internal flow reattaches at a downstream location,

recompressing the flow to a pressure greater than that of the

core. All of the above flow processes interact, establish-

ing a unique geometry for the corner region. '

Having defined the corner interaction an order of magnitude

assessment of its effect on the bow and doi^mstream flow so-

lutions will now be given.

30

aHMaUMUUMM^IMI iiiiMHimiHir""'-- : ■-'-■■- ■iHBlMMttilMWMllUMaUIH^Hk - ■ -■ -

^VP^#M»MW«WW.W!WH'.<i'*fli -,-1 o-WilWflTl-W,'!": »'W* TV^'M(«!C^«W^li.W !J|JWJH|IIAV?*IWMIWU|8»I li'U!! J1 I IJ.i'l. i .■ ,l Jmm. 'I «m fUlftJl^iy WiWWAlW.rfL^^l^lW?!'WW»-J■^W.',l*ftlW^W,tp?,l,»•■'!.'" !'"^W».l «SillI»WW^'J l.li li, I, .*■ J«.J l.,.l S: t,H| WW jnOpppiiqVpminPnnpniPqni

Significance of the Cornor P.eräon en the Dow and Down st r e an Solution """"" ' —

Since the bow region layers are thin and bounded by strong

shocks, then the flow properties within the layers are strong-

ly influenced by the local shock conditions and only weakly

influenced by the conditions in the corner region near the

sonic point. V/e can therefore conclude that the bow layer

properties can be calculated independently of the corner re-

gion without incurring serious error. Since, as is discus-

sed in the next section, the downstream flow depends on the

balancing of jet and external flow interaction forces along the

contact surface, then the significance of the corner region

flow on the downstream solution is assessed by determining the

axial force balance that would occur if the corner region flow

is included or neglected. For the case when it is neglected,

the jet flow is assumed to expand to its vacuum limiting angle,

Qto •

The particular example studied herein is that given in a report

by Rudman and Vaglio-Laurin . The axial force created by the

jet flow is given by

Fx = - jf % C^.- nx) ^ A - ( p n» c< A

Applying this to the cases with and without the corner region

and taking their difference divided by the total force of the

jet, we obtain ^ e<B

**tM f cc>s*msMycos$Ay O

81

^--'— - m -^—. _______ J, _„. . ... __ ._ ^jj

lm„r.. ^,.^,,^^l...^.,wmm„A,. T^~^-., jMiuiiwaniiui.ii iii.imii.i jiiiiiii uii ii mi 111 IL.IMI uiii LI ^wpn^mjiuiiiu •• i 4 m-umi n in jnm^w^^^i i i .mn imm^^m

where ' ± < f < 1 and 0 < V < 9^

Substituting in the numerical values 0»= 130.45, (f^ = 55 de_

grees, H = 4.17jco8>>= * We obtain

'Jl — = .0425 or 4 percent rxtot«J

This error is well below that created by using the approximate

analytical source flow expression (1) which in some cases can

be in considerable error 1^.

Since the details of the corner region flow are not signifi-

cant to the calculation of bow or downstream properties, then

engineering estimates of core pressure, plume boundary and

penetration will be sufficient to approximate the corner re-

gion flow for this study.

Core Pressure

Qualitatively, the external and internal flows do not bend im-

mediately downstream after passing the intercepting shock la-

yer boundary due to the considerable lateral momentum impart-

ed to the layers in the bow region. Instead, the external

shock moves continuously outward along with the bulk of the

external and internal layer mass flow, leaving behind in its

"wake" a low pressure (with respect to the shock value) flow

adjacent to the core region. On this basis, we would expect

the core pressure to be low and of the same order as ambient

pressure.

82

■..^^.at.^^.:^^..^-^.^^^a^Jt^LtI-^jJ^J^^^.^^W.A..^.V^--.W......^.^^^w;;iv^.^^i^-*.^i^^.^^ .- „,-,.„.,,■.,■■, n 1 .^v^„.^,.„.■..^1..,..„.--„^..^......^-^■w^.;. ^^^^^.^^^■^.■^.^.^.^...■.- iM nmftIM

I,l, ■",,, " "'■ " ' ' uj—iiii. ■•uniiiiwiu. mrau ■»(^•i ,HIII.UIIIMIIII.I»I«IM.I mi jinn^mwOTnp«ia||inMHpp!p||HPn(IMWi*«H«mp«il«mf||||MMIIVi

In determinins the core pressure, the experimental results of

Finley ', Gharwat and Faulmann^, and Jarvinen and Adams 26 v/ill

be utilized. From figure (7) of Finley, it can be seen that

as TT is increased, the core pressure drops to a nearly con-

stant value. This behavior is also consistent with figure (11)

in Jarvinen and Adams and figure (15) in Charwat and Faulmann.

In our analysis, which is the limiting case when Tt ^> L , the

core pressure will then be at a constant level for all values

of Tf .

These results are for exit Ilach numbers of 1.0, 2,6, 3.1, 3.9,

and 4.3 and freestream Ilach numbers of 2.5, 2.75, .6 and 2.0

V/e therefore conclude that this behavior with TT does not de-

pend on exit or freestream Mach number.

Noting the lack of dependency of the core pressure on Tt for '

Tl»l we can now correlate all experimental results to estab-

lish the best value for P core. From figure (20), it is seen

that the most general result, especially for high external

Mach number, would be Pcore = 1. This result is reason-

able considering the wide variety of Meo.,Mjuncier which

tjflrR has been obtained and also in consideration of the roo«

nonuniformity in pressure throughout the core region.

83

■ ■-■-1 li hMlliL-iT-rt"!!-!-1'-■•■■■"--■- ■ -'■ '• '■ •■-■ ■^---.■--^--■-'---^—^^^■-■■■^■■■■■■■^ ■.^■A^J.^,iJ.^^.1 j.iJ^.ir....l,^-..M.-:-,--..-.-J^-J.>;,...-J/... -.■:.. .... 1..,.■.■..... .. .■^....-■....^.Cl^^.J:..^^.^ _ ■■■. ....

Plume Boundary

As was mentioned previously, the plume develops in a rec ■-

of nearly constant pressure and, therefore, it is analogous

to the case of an underexpanded jet into a still ambient. Va-

rious analytical approaches describing the location of the

boundary of the plume have been developed . (for eg. see re-

ference 27 and reference 25). Host of them give reasonable

results near the nozzle exit but diverge considerably from

the actual boundary auay from the exit. For this study, an

20 analytical technique ' is used which assumes a fixed plume

shape for all nozzles when the coordinates are nondimension-

alized with respect to plume boundary maximum radius and its

axial coordinate. The advantage of this method is that it

gives the correct boundary location not just near the nozzle

exit but far from it as well. The method relies on setting

up momentum and force balances between the exhaust and core

region gases. The shape of the plume when nondimensionalizcd

is represented well by

J

v/here j =2.5 gives the "best" fit with numerical data. The

balance of lateral and axial momentum give respectively ,7t iv»

Pore TTT^ = Poj A* fz (X^/r«)

84

ii imMniiiwiiiin '^ i .II»J» , i inuuunii IIIUUI wmm u i i> ■ i i i ■<wWüRimnmmiii

where Ja, and J*. are functions of the nozzle geometry and ex-

haust gas conditions. From the above expressions X^/r* and

grrrt/r* can be found for any jet exit and ambient condition,

thereby giving the plume boundary. This technique was uti-

lized in locating the plume boundary for the example given

m Ptudman and Vaglio-Laurin's study. Good agreement is

found with experimental results as can be seen in figure (S).

Since we are dealing with highly underexpanded plumes (i.e.,

Ti>^l ) where the intercepting shock layer is thin, then this

boundary location is also assumed to be a good approximation

for the intercepting shock location.

Penetration of the Internal Layer

Referring to figure (19), we can establish a qualitative des-

cription of the corner region flow. The reflected shock ema-

nating from the triple point, at a, crosses the intercepting

shock layer and intersects the plume boundary at c. Since

the boundary must be at constant core pressure, then the shock

must be reversed and reflected as an expansion c-b. Crossing

the layer, this expansion then intersects the slip stream sur-

face at b, being once again reflected and partially trans-

mitted as an expansion into the slightly supersonic internal

layer flow d. The surface b-a remains nearly straight since

only a small drop to sonic pressure over this region is ex-

pected, which requires very little turning along b-a of the

supersonic flow in region a-b-c. The internal flow sonic

point occurs at b because if it were sonic before this point

85

i.,(,A"it.VKi.-i-:h>ii*-~it,.-KO.;.i»..i-.v,...-r--v., ■■.,■.. ...-^■^■..■i,;,,,,..-j->. •.j-,-- .'^■.v..,,.^,^,...;,.;...,,,,., w!-- *'■■■-..■:- ■

™?rw?»w»»fp8wmww3s^p^|B*«^^ ^^.«Bi^i^9^flmi^*^^HPi^^Hn!^^

the non divergence of the streamlines in this region would

unrealistically prevent the flow from accelerating further.

Also, the sonic point cannot lie beyond b because the di-

verging nature of the flow would not allow sonic velocity

to be achieved from thfe initial subsonic state.

Within the slightly supersonic flow region, d, the trans-

mitted expansions b-c are reflected off the nearly constant

pressure sonic line surface as compressions. These return

to the intercepting shock layer and are transmitted to the

core region. The continual proc?ss of reflection, transmis-

sion and interaction of these waves results in the rapid turn-

ing of the flow in the dovmstreara direction as is illustrated

in figure (19). Some idea of the shape of the jonic line and

its limiting characteristics can be obtained by applying the

ideas developed by Hayes and Probstein 30 for blunt body flows.

Taking u) as the angle that the contact or slip stream sur-

face makes with the sonic line at these respective surfaces,

we can write '

\^>S/b|e

v/here /-. v

and ty^e is streamline velocity, f^ vorticity and S and n

are the coordinates along and perpendicular to the slip stream

and contact surface. At point b we have nearly a straight

86

■-■-■■—'- ■ - -•■^--■■^■■-•■■■•"imhrmMfci.iiiiliiiif iiiirwi^--'----^ ll' jriyaaiiijiiiaa**^ ■^.-.-—»-i. •.■,..,■■ ..■,... ■■ ■ ........... „■,. ■■■.■-.... ■■.■L...,,.......,....,I..L, ■..-....,.. ,.:.■

r -- . -ii'r>.<-^..yFvei^TJW^^»^-lfn-'^m«9ffr?'f>ru'^"'nn''^™*,.«WJPS"!-1^'liHWW!?WWA«»1 i1.?,ffPf III.^WW(W.-wnpii.v i...,..t,»..i.f.i..,1-,.. 1,.. JI. p,«;^ 1 wmmf\v-.*yV'-rM>wl*h-»-'.f.UJ.««.:]..4-.P.4l^n»«'i*! WWV,*!*?.WWlll.p.Ul'-M-WW' WW'Ü'JJJ-WiW»!?'"^'-''^»«'"«^,»!W*W:.*; _>.,[ujljiiqi

slip stream b-a and therefore the second term in /^>Qr\ is

negligible. Also, the shock is nearly spherical and in the

vicinity of point a the vorticity generated is small. Under

these conditions, ^^>St j d O and therefore uJ ^ ^V^. at

b. At the contact surface, point e , different conditions

prevail. Using the first order expressions derived in the

bow region analysis section and assuming for simplicity that

the internal shock and contact surface are spherical in nat-

ure, then we can write in terms of nondimensionalized variables,

7 Using Rudman's exa-nple, wu can write

also, we have

(Hi - ^ S-'- ^ The ratio of these expressions gives

tan cO = l.g5 which for Q =55° gives tO = 62 degrees

Now that the sonic line angles are estimated at both bound-

aries, we can then sketch its location throughout the layc -

as shown in figure (19). The initial turning of the sonic

line in the downstream direction from point bjdue to centri-

fugal pressure gradient effects', is typical of the rapid tran-

sonic expansion occurring in d and gives the sonic line its

87

- - - - ■ — - UM L..-.^^..^^—^.. ^ _ _. . . ..„ ,. _ ... . , ^

i wiiipiiiiiaMpn«iMi"L<»i.< ■•<•■■> ■»•wiNuw 11 in.uiimiHUlMHii nv .ijiwBOTimiwMmnuMmvx^vKOT^vnmH [wmmnmimummmmim^Kmiu uii». i ip t "• ■■

characteristic s shape. The laßt limiting characteristic

lines which originate in regions d and f are also shown in

figure (19). Having obtained a qualitative estimate of the

penetration of the intercepting shock layer into the internal

layer, we can now make an estimate of the actual penetration

for the example calculated by Rudman and Vaglio-Laurin7. The

first step is to estimate the intercepting shock layer thick-

ness which can be determined on the basis of mass flow con-

siderations. Subscripting layer properties by £ and utiliz-

ing standard notation for undisturbed plumc^ quantities, we

can write ©00

es. Substituting in the analytic expressions for jo«^ and Uvi

which can be derived from the analytical plume flow model,

and, letting

we obtain Ä

For 600= 130.45° ©A = 55° and ^ = 1

have Sä, = .0627 Rsu,

This value is characteristic of plumes whereTf»l . The se_

cond step involves determining the reflected shock position

from the triple point solution. Using the initial conditions

ntol = 9.2, ß = 06° for the strong shock and/5= 20°

we

88

•*..nJ\ir,*i\^.-w.i*.-*.. ;'...." ........ •-rlriirt-Vi'i-.iiT-riii ^iif[iiii-THHJ^Irt^fV-itf.iii'tiWn i^-.inT.n-|f'»i.lJitih^^.'...^^^t.)^i^j^-M^t^-^^a.J.^iB^-^^^^ .. . '......",,".. .-'<.. ..■ .\i*\'.<*wAMb&irt4*>'.--*~*---*-.-*-,'--.- •■■. ■■ ■ ... -v mtftiirtTitffriiliftftfr1* H^fTr '" ÜUfM

"j..wv,WW)iWiiWiimi™..u i"'»!™^

for the intercepting shock, we obtain the reflected shock po-

31 sitton from D'Attorre which is shown in figure (8).

Assuming that the Mach number downstre'am from the reflected

shock is constant, we can then determine the initial reflected

expansion wave's position and intersection with the slip stream

surface. The position of this wave as shown in figure (Ö)

defines the extent of the penetration into the internal layer.

Unlike the point interaction or zero penetration model pro- 7

posed by Rudman and Vaglio-Laurin , we note a considerable pe-

netration of the internal layer by the intercepting shock la-

yer flow. This fact is borne out experimentally in a study ^

on the similar problem of a jet impinging normally and at an

angle to a flat plate. The constraints on the numerical so- 7

lution of Rudman and Vaglio-Laurin imposed by their assump-

tioii of a point interaction (i.e., zero penetration) for the

corner region could be a source of considerable error in their

calculation of the bow region.

It should be mentioned that the large penetration indicated in

this study does not necessarily mean that the bow region flow

is strongly influenced by the corner region penetration. This

is because, in the absence of the corner region flow, the inter-

nal flow streamlines would penetrate the layer much like the

slip stream separating the internal and intercepting shock

layer flows. This can be seen by comparing streamlines ob-

tained from the bow analysis in figure (17) with the slip

stream position shown in figure (8).

89

P^^^T«^W™IWW!I»WITH^ W»^IWW?^"»W™!^»ä«^ Kimim>'!>mumhw^"i^*M»v'm***,Mw**¥*v*!mi ..i IUWUHI

From the results presented a scaling criteria for the corner

region is now outlined. The major characteristic to be scaled

is the undisturbed plume boundary, which for a given nozzle

and exhaust gas composition depends on Poj/p«»e . For an

exact simulation of the plume boundary, this ratio must be

matched. If, for a given system, it can be shown that the

plume boundary coordinates are proportional to Bs^ or Tt'^

then the plume boundary will scale like the bow region. The

technique used to locate the plume boundary normalizes the

geometry by »"^ a^ XM the plume maximum radius and its

axial location. When these are plotted as a function of

^j/poDft it is found that

• « Hk

Since V is proportional to P^j/poo.M«,^ then

Thus, if Moo, is fixed, then the plume boundary will scale

with Rs.'o . The scaling of the bow region and plume boundary

then requires that ^ , ^e , 9«, H , ^ 1^ be fixed over

the entire flight range. Since M«c does not vary greatly

over a typical system trajectory, then the above scaling re-

quirements are satisfied for some systems of interest.

This scaling result is substantiated by some experimental'dl^a 24

in which the plume boundary triple point distance away from

the axis, when divided by the scale length Rs^ls nearly con-

stant for values of Tf from 30 to 627.

90

t*'- ■■ -■---. .....-■..- -. ..■■■:..^. -.„■■.. ^ ..,,,...., ...-: ,.:.-.■ fa ,.,.1 — ..-. p. ,_.-;. .J.... .,.,.-».■....; -...■...■. - . J,. ..r.i, .*..,..,: J... 1. ■ 1 ■■>.-.... ...... ...... - .^.O. ..-..a.,.-. .._.

j,-,,.W-JV-rv .1» l-rw *t.*X-*fim*'--m* PS.^W7T*4'lf«i-"'««!-«^»«»« P«W^W K.' -'^ ....>. !l,l. „J^iy lljill^lll ||- ^WW^***>H-».:_«;(P^pww^J«.rW-'tf«,M,.?WW|W-1-1 rfWHflV^-^ ?'.W>^T^Wv^m«**^««,!!»^!«!1 WW^»,-M^.»« FAJt^'^WT'^M?1«^WI?PVTM?«p*'';' 1 I-I'.I UiCÄHllllF^

FAR FIELD REGI0:i ANALYSIS

The dowmstream flov/ region between the external shock and axis

of syniiTietry consists of supersonic layers bounded by mixing

regions and crossed by a wake recompression shock as is illus-

trated in schematic (3)

.mixing region

mixing region

Schematic (3) This general problem cannot be handled by approximate analy-

tical techniques; therefore, we will seek to solve a reduced

problem within the framework of the assumptions listed in the

introduction. This problem will involve:

1) the neglect of all mixing regions (wake

and shear surface)

2) the neglect of the recompression shock

Assumption 1) will be valid at moderate altitudes and 2) will

be reasonable since the recompression shock is weak compared

to the external shock. A schematic of the flow and notation

used is then:

Schematic (1+)

91

■■ '■ i ■ -iiirri'iiriVn.mü>»Hia'^iri-irir'i i aaaBBit t ,.... » ■...**

»i . iioiii.!...! ii ................... , ..„..PIIII iiii.an.i ,i mi ii.jiuii.i i .n jji.i i MI ■■iinnniuiiiii II i ii ii i i ii iii..iiu.ii.ijwini«iii«(ia .•■UIIIIUL i WLIIIIU.HJIIIIU

The numerical matching criteria between the internal and exter-

nal flow requires that static pressure and flow deflection must

be equal at the contact surface. Also, the stagnation pres-

sures will be equal due to the upstream matching condition.

The particular region of interest in this downstream analysis

will be at a point characterized by the distance L , where

L» Rs^ts. The ratio of R9to L will be a significant

parameter for the downstream problem and we denote it by the

symbol 6« a Rs/L . Once again, as in the bow region, we will

be considering the case when M^M»,»! a €i:0j€e«L. ^n ap-

proximate analytical method of solution is available for the

external layer flow near the shock surface in region 1. This

method was first applied by Cole 33 and involves taking the

limits in the inviscid flow equations as the perturbation quan-

tities.

n ooe CO •e O , §

Ma^e-^oo Mo^Sc'-ee

O

oci)

The solution found by this procedure is valid to a distance of

OQCeSeL) measured inward from the shock surface. For attache

shocks, this solution would be valid throughout the external

layer. However, if the shock has a blunt nearly normal for-

ward region, the external layer becomes thicker 'than the above

order of magnitude and the solution breaks down away from the

shock toward the contact surface. This is due to the viola-

92

j.^AfcjiuJiteM^^i^.wlLf,..^.,*.« —- ---^- - - ~ .- - -~ -JJ.._ ::...■^■....-..■.t.—j.^,■r.f.-.y i^.■ ^,.rr ,.,^1. .,,..i>.■■ ^,■ j-i/t-MmlTiffjfialifinaiil^ilii i-imfir' •

™",'Tp!w«fwiwff?*?wwwM!^rKwn^^

tion in the blunt region of the assumption that the shock

makes small angles with the free stream. As was pointed out

by Cheng and Kirsch ^ in the equivalent unsteady case, a

region of thickness o(SeL/ , denoted 2 in schematic (4), is

then found near the contact surface which requires a differ-

ent expansion in order to develop a valid solution in this re-

gion. The solution found in this "entropy wake" matches with

the outer layer, and the composite expansions for the entire

external layer will be given.

For the internal layer, we have two conditions imposed by the

external flow at the contact surface, which can be used to es-

tablish the orders of magnitude of the flow variables within

region 3» the interior layer. They are:

Based on these conditions, we can then prescribe expansions

for the flow variables in terms of perturbation parameters

that will lead to a system of analytical solutions for the

inner layer.

Like the external "entropy wake" the internal layer is of thick-

ness OfSeLj . Utilizing the orders of magnitude found from

the contact surface and upstream bow region conditions, we

93

- .^—— --iM-t i, n Ä _ |, I, if miri .ir-iiii^ .. ...-.^^

" » pwiju|»uillMi"IH.Nl mill, ill ^MH>| ii ill mil •mniai.n m«nMwnm*w^Bmmmmmm*.*n!&tt*m^mmm^mmmm*^mmmmmmmmm^***

then develop expansions which will be solved giving expres-

sions for layer thickness and other variables of interest.

Numerically matching these expressions with those of the ex-

ternal flow, along the contact surface, we then achieve ana-

lytic solutions in the far downstream region. These results

are found to depend on the upstream bow interaction between

the plume and external stream. Taking results from the bow

region consistent with the accuracy of the downstream region,

we then find the complete solution for the downstream flow.

In the following discussion for the external flow, frequent

reference will be made to the work of Cheng and Kirsch ^.

They solved an unsteady problem that in many ways is anala-

gous to our external layer problem by the application of the

hypersonic equivalence principle. For details of the exter-

nal layer analysis, the Cheng and Kirsch paper should be con-

sulted.

94

(^w-wüu'JiwwBi'ywuiuViii«* Mii*asutiazaix*i,.^*&*-*:..^ ■ ,,-. •.■.:.-iWöia..^M»"*i4bft^ii»i^^^

• -.—".WI.-.1II1. ■ p-w'iiMPiiiiuii.ii.i. ii iiiiip *»™.III» -.».i. i .. i npiu.i. i i >-■'^■■v^KWTCa^raptmwRnmmimümnvfBnwnnniraiiPtill

External Layer - Region 1 Analysis

Utilizing the notation illustrated in schematic (4), the fol-

lowing general set of equations and boundary conditions for

the downstream problem in terms of von Mises variables

are •

^Xe Ue (1.1) stream- line slope

m ^ V^l "fTuTRe (I-2* continu-

t^r?- + \^T = 0 (1.3) momentum normal to streamline

(1.4) Bernoulli integral

}>*/$>?* ' f^e) (1.5) entropy integral

Boundary conditions at external shock

U, = U«. C1 - ^.CMooe* sm1^ - 1}.4:^)H^ ) (1.3)

Ve = U^e ( * CMof« »««»A - l) cot^e/fve^a) M«? (1 *9 ^

Boundary conditions at contact surface

Pecs * Pics (1.10)

<lfiec& = aRcas (1.11)

95

?-^7rmwHr*rvnr**rrrp**iri>w^^

Velocity parallel and normal to the shock are:

V = £e' Ufto« s\ytßc

(1.12)

(1.13)

The expanded Rankine-Kugoniot relations lead to the follow-

ing orders of magnitude for the flow parameters near the

shock:

(2.1)

(2.2)

(2.3)

(2.4)

where €& is evaluated at the point where >5e s TT/^. . These

conditions will be the upper boundary values for the external

layer. From the orders of the variables near the shock, the

dimension normal to the shock can be inferred from continuity

considerations.

p. ~- o(f«o» LU\ *<?)

ß' — OCp«a/€e) <** Ue — <3 (u eoe /

- o u t ^e Lleo« )

^ - 0(6e^L-) (2.5)

The dimensions of the streamwise coordinate are easily se^n to

be

x - O(L.) (2.6)

96

■ - mum —-'—■■ ..^^t******^*******,^^ -., , _...-.,.,..._. ;. .,... m— ^..,^..^^..,.^. a

These orders of magnitude, (2.1 to 2.6), are then used to form

asymptotic expansions which are

Pa = J'ooejaoe/Cft +" '*'

Ue - Uaoc •♦" SeaUooe Uoe + • • '

Ve' - Uooe ^e€G Voe + •• •

^e = LSe Gc ^ot -h • • •

Substituting these expansions into the equations of motion,

(1.1 to 1.5), and boundary conditions, (1.6 to 1.13), we can

develop a sequence of equations which can be solved and then

summed to yield the solution to any desired accuracy in terms

of £«,S») Mooe .

The first order system from the above procedure is

l&c ^-J- ^ ^« Rs» ^x

Sc-

1_ CP>e/poe) = O

op. =. r H.

e ^ ^

(3.1)

(3-2)

(3.3)

(3.4)

The boundary conditions on the above at the shock surface are

^ = i^ (4.1)

97

...,■■-

-- ^MMUMIMMi '.*K* ,

7oe. ~ L »'

(4.2)

(4.3)

Integrating (3.1 to 3.4) and applying the boundary conditions

(4.1 to 4.3) > we obtain:

^-m+k.^-^) (5.i; ) (5.2)

(5.3)

(5.4)

Making the substitution

^e* = Ri^/^ (5.5)

in the above, we then obtain results identical to those of

Cole 33

^^r^-irl-a-^) (6,; foe « poe/^i^)4

As previously mentioned, these results are expected to hold

)

(6.2)

(6.3)

(6.4)

98

- — — ■ --—'

——. ..^—. . —■,...,,.... ■...■

near the shock surface; however, their validity near the con-

tact surface has to be examined. From the bow region solution

we know that a blunt interaction occurs. Consequently, as

we follow the shock surface upstream to the vicinity of the bow

region, we note that the shock must become nearly normal in

form. Therefore, it invalidates our assumption of small free

stream to shock angle, initially made in our above analysis.

Under these circumstances, one would not expect our expansion

procedure to be valid in such a region;and this turns out to

be t..j case, as was found by Cheng and Kirsch ^ for the un-

steady problem. What we must do is alter our expansion pro-

cedure so as to be consistent with the strong part of the shock.

External Layer-Rej?:ion 2 Analysis

Following Cher- and Xirsch 3/f we assume that initially/ due

to the strong bow interaction of the jet and external flow,

the external shock moves out in a blast wave manner. This

requires that the external shock take the form:

as Xe-^O where A is a function of the upstream bow inter-

action. A requirement which must be met by the contact sur-

face in the vicinity of the blunt interaction is that:

99

^- ^---^-L^^^^--^-.;.».^:..,.. ...■■.......■...- ^..^■■.-*A^-..„^.,11,>............_, ^..■ji.^^.... ;■....alJ^,,JJ-^^;:„....,;..:;....... ,.. ■ . ..-.. ... ;.■.-■. .... . ,_-...KJ..-- -

This states that initially the contact surface must move out-

ward slower than the external shock, thus guaranteeing the

dominance of the blast wave result for the upstream interac-

tion. . Up to this point, we have paralleled the unsteady-

problem quite closely. The equivalence principle is most ac-

curate near the shock with the streamwise velocity given by:

The error involved in assuming K« "^ Uooeis seen to be small,

of error O^M.ootOej . However, as we approach the contact

surface we have for the streamwise layer velocity, from the

Bernoulli Integral.

U - Uooe + OCUooeS^^^)

The error involved in taking U~Uo)Ä in order to apply the

equivalence principle is seen to be large for Jfe. — J_ of

In order to utilize the equivalence principle, we then must

stipulate that ^c cannot equal one but can be near one as

long as öe is small enough so that the term de *"• does

not become of order one in the downstream region of interest.

If these conditions are met, then we can apply the equival- 34

ence principle and, therefore, make use of Cheng and Kirsch's

100

, hüft ■■■■ .-■- .'..■.-■...-■■..■... ..,J.^.^^,.i_.. ,^.^..1......:-.^-^yi.--^. .^^-l-i... ,.:.. ^.- ^■. .-w^-,^.^,,,^.,:...^^,^^...^.^-: :i,.. .............. „..,.. .,..■. ■ ....... J..,...,...,.^i. ..,..., .. . ^.....-.,^ ■^;^.^.;^^-J.^.L,-^-J--'-.^-...,-.„.,'.._.;.J.

results v;ithin the entropy layer.

Assuming this form of initial behavior, we have then speci-

fied the type of breakdown of the solution due to the near-

ly normal character of the shock in the vicinity of the nose.

It was found that the expansion for u^ based on flow near

the shock for initial blast wave behavior would be of the

form:

^ ~ oO* Yc.) + o OXlnV.,)*)

v/hich would then become infinite and invalid as an expansion

as Yo^O , or as we approach the contact surface. This

form indicates that the expansions cannot be rnade uniformly

valid by simply altering the scale of Yo^ by a power of Gc •

In his analysis, it was determined that the scale normal to

the shock should be in terms of No* to some power which is a

function of €e . V'e can then determine the orders of magni-

tude of the other variables by applying entropy and energy

conservation principles.

Following Cheng and Kirsch ^ we then establish a new set of

variables, fulfilling all the necessary conditions within the

region adjacent to the contact surface. These are:

Ve = Ucoc ^e: Voe -H • • • (74)

101

BMM ■-• .■-■•-^ ■ -—^--■■--- "-■ ■-• --' ■ ■■ ■-■■■■ .■,..-...■ --^-...^-..■v .-. .„^ws..;..,-.--... -.--.■■.-.^ 1 ,,, .J.,.. .,-,...■ _ ..■■■^■..._..^^iJ„^_..^J.-^-......-^--.^H.^--'^.^--------^-.-~-'-

Ue = Ufloe ■»"••

P* = foo* Uo?; 5a* p"oe + •

fe - ^ooc Re,,, p^e/(ge +■

^ = Se L ^o -H • • •

X - Lx 1 = Re^Vo-o2^

(7.2)

(7.3)

(7.4)

(7.5)

(7.6)

(7.7)

Substitutins these variables into the equations of motion,

(1.1 to 1.5), we obtain the following set of first order re-

sults:

- O (3.1)

(Ö.2)

(Ö.3)

(S.4)

The carrying out of the integration in (S.l) and

(Ö.2) gives

got - ^oe'U^O) -^ O-0^/p0e (9.2)

Matching of the inner to the outer solution is carried out by

determining the behavior of the solutions in an intermediate

region where they are both valid. The results

procedure are:

34 of thii

102

l—[Mini«nMI«iiMMl—mil i—lil ■^a^—-^—..^——-.K.^.-!. .^—-^-^-.^ ... .. tm

Equations (10.1) and (10.2) are nonlinear second order dif-

ferential equations for the pressure distribution and shod:

position. If vie can obtain an inner layer solution which

gives r\cs as a function of p0e , then we can combine those

results into the above to determine "Rs«- and Pics •

Internal Layer - Region 3 Analysis

Referring once a^ain to our flow diagram to establish the

system of coordinates, we 'have the following set of equations

and boundary conditions for the downstream internal layer

problem in terms of von Mises variables.

= O

(11.1) stream line slope

(11.2) contin- uity

(11.3) momentum normal to stream- line

(11.4) Bernoulli Integral

(11.5) Entropy Integral

103

:.■...-,. »KMAWMMM ■....-.., .......^uj.

■■■-■-■■.--•-'■■> ---'-.■.- ■^^p^^H»^^T«l^,^w^iJiiii*.., -^r-r^j^^r^-^-^M..', ^.I..P,I^S ^L^^L.f..-. .^.IMW"^ ■wr^. »WI-WJ «(•? ^■.'.T^rr?"1?r.''^T-Tw-«!.!'(.'.»«v ■ ^ i-'j^^q^

Boundary conditions at the internal upstream bow shock are

PC -f»i -^ ÄiCÜ. ^ocCMoJsi^^i ~ 1) (11.6)

Vi = Uooi C 2-CHeAc'si^t - l)«)^/^^!) Roo* ) (11.9)

Boundary conditions at the contact surface are:

Ptc» * Pecs (11.10)

aRics •= aR«cs (n.ii)

At the axis , Re = O , the syraietry conditions are applied.

Also, in the bow region solution, we note the condition that;

P l «K^nabo* - P« «WMA.'KO^ (11.1?)

Fron the external layer results, v;e can infer the order of

magnitude of some inner layer quantities. From the contact

surface condition, (11.10), vie have

P^s ^ Pec» = 0(fe0eUoo*V)

In addition, from the stagnation point condition, (11.12),

^ooi'oUooJ — O C^oocUooe)

where P<K>iQ iG value of density along axis in front of in-

104

.J..^.-... .. - ^—......-.^ ,....,..-^ .- -.....„ ,^... , ......: ., ...„,.....,. ,

■'*™rr**TTrrvrrm**™rrKf^^

ternal shock, v;e find

P L cs = O tj>coiti U^ ^e2) (12.1)

Since entropy is conserved alon^ streamlines, we can es-

tablish the order of magnitude of density in the internal

layer near the contact surface from (11.5)

For Mooi^l , s\y\x Bi ~ 1 the shock conditions, (11.7)

and (11.6),for the streamline which wets the contact sur-

face become

V/here subscript o denotes normal shock location in bow

region. '.Then substituted into the entropy equation, (11.5),

the result is: \

fUs = oCfcoCo ^^Veco) (12.2)

From the second contact surface condition, (11.11), we find

105

• ■ . -.-...■■--.wntiJ-ibi*

■ - • - - -

i—^, ,,„„,.n~ M, .iw iV| nn mi i in mm uii.niai .« ■! • ILI .in mi'ui i Miiiuiiuiii^Ki^p^oiii. ■piuiiiiuiiKiiiiiiia.uiiiin.

This also requires that

Vt/u^ = oCSe.) (12.3)

near the contact surface, v/hich is an upper bound on this

ratio. Another expression v/hich can be utilized is Bernoulli's

equation, (11.4), alon^ a streamline. Substituting (12.1 to

12.3) into this equation and assuming Ui of the form

U t - Uooi -«- Uc"

we obtain

Uco* 4- ZüniUi" + o(S'e*Uoi + zSe'Uo.iUc")

Therefore, we find that

Ui" V (12.4)

Therefore, we can conclude that for U,'to be constant in the

first approximation in the above flow system, it is neces-

sary that Vc," j* 1 and Jj» be small enough so that the term

Oe v. »i y << oCl) . These are the same conditions which

are necessary for the external layer solution to be valid

near the contact surface; therefore, we assume that this will

be the case 'in the present problem. This leads to the result

U't •= O(UOOL) (12.5)

106

- - - ■ 1 •—- m -- - -- --- -- ■

-^..^•■.u.,.^.! MIUI.,11 i, .■m-m..*j,,,wm'.wti-m ill .lllip|,nMaipRWnWl!>BI .»WÜMmipif fii ■.■.■iuilillB!ii«.MJimi>W!Uim.rilin;iW'HJNIIIIl lUimH 4IJW«!W»Wip.p>WI|llll|ipillJTOWI""U,M*B^W!^

Fron the contact surface condition, (12.3), "e find

Vt = O (Uflot oe) (12.6)

V/e have nov; established the orders of magnitude of all inte-

rior variables of interest near the contact surface. V.Te now

postulate expansions for the variables based on these orders

and then substitute them into the equations of motion to yield

a sequence of solutions, which can be summed to give the solu-

tion to any order of accuracy for the internal layer flov:.

The expansions are for £ ^ -^ O 9 Oe -♦ O 5 ^toU ~* co

Pi •= p<*U Se^ol/^to +■ * ' "

Vi - Uoot Se Vot ^ * * '

Ut - Uoot +• • • •

The first order set of equations is

c-_^ R.9V

(13.1)

(13.2)

(13.3)

(13.4)

107

..-...■■^.■.■.i..,:.,- ---■»^' - '-.■^J^.U,-.-:;*^..- ■"-;-■ ■■ ^--

■ ■ - ■ .. ,

fAHftgiiin i^iiJiAlHuAU.ufeu ■■■■^---■"--■- UKUi . ,.-^

^*uwi!|*^■fF.fcWi.»,*!«? *!;",'-■ M.'' ^-"-r v .., WWV. ■,',^*MJ iJ^'iR'-^.JWrwv^n^^VHlJTO^^JMIB.IW^- ?i«»rPW>^-r-rTii^?^rT?!TTT^^^^ ^i^WTflwrrirwTSrw^TFTT^T!^?^^

Integrating, v;e then have

pot - poCCx)

1 Jö foi

Combining (14.1, 14.2 and 14.3), v/e obtain

Res1 - 2. B^4 Poc

(14.1)

(14.2)

(14.3)

(14-4)

Now that we have results for the internal and external flow,

.ve can combine then to determine the location of the shock

and contact surface, as well as find the values of the pro-

perties across the layers.

From the external flow, we have the results:

Jse |d ^R Se (10.2)

(10.1)

Also from the internal flow, v/e have the result

pDlCX^ = Tjex.

R.CS -^O /^i*- (14.4)

Since footU(Ä = J^aotoUc?^ from thG stagnation point con-

dition, then poets = -po^Cx.) • •■'e can now combine

108

"■'"-'—- --—-^-■^- ■...-.-^^.^■^^^..■-^.■.^^■■^■^^^.^^■^s.x... aayjaa—.:..■■...■...-.,.::,.,..■■..-. ■M.;..^.^^t^-,1(V'tl'^MliMMii,ii''»nimfiiir""'^i^~'-'L-: ■-- Ül

ii i ii jMiiiiiw«mM«qmw>iMiiiuuJuiiiiiujwiiiuiiiiii i ■ ■(■■■■HWWH-"» iii.iiiiiiuij-iiiipiBH9WiM"m"w)"w»»iiM i,» HI iinm, i ■•«» ntmmmmim lui^iWLaiiiiiaiii i iiini.

these relations (10.1, 10.2 and 14.4) from which property-

variables can be determined.

Taking (10.1) and (14.4) we have

Poees or

Combining (15.1) v/ith (10.2), v;e then obtain for the shock

shape a second order non-linear differential equation

f(4jLy ^ i^L^jRu = ^ H- ^l^01^ C15.2)

For "X/R.stj, >V i , which is in the downstream region of

interest, (15.2) admits a solution of the form

Rs« = C x Vz (15.3)

When (15.3) is substituted in (15.2), we find the value of

C to be given by

VH (15.4)

109

. . ■. ■ ■

- -. • ■ 1 iiliiWririiiiiT'ii'fiftMriliiViriilViiiA^iii.irrtfrtfttfti AMiimn^ .^.^ ^-.—;..,.^.^...^^^^^-..u... .^-u .„■„ -....■ .L.--: :. :.,.■..,.:.,. ■ -... ■ ......... .._,„... :.,^::/,virt.Ä^.,.;..^«f,.^,-..j. .,-: _.. - .

4u(>uKwmmfcvy^MW'l)M>m

Using (15.4) in (10.2) to find Po^Cx) (14.4) then be-

comes

fPc^J^t (15.5) 8fvc* Re»*" *o J^tV

Combining (15.4) and (15.5), we tnen obtain the following

expression for the contact surface shape in terms of the shock

shape

RC8 = ^e/CÄ^c + 1}Va (15.6)

V/e now have determined the values of Rs« > fves in terms

of the functions Ca and J -^J^. «^t . These

expressions are functions of the blunt interaction between

the external and internal layers at the extreme upstream

position in the bow region and are a result of the matching

of pressures across the upstream contact surface.

We, therefore, must postulate an upstream interaction based

on our solution for the bow region flow which is consistent

with the approximations made in this downstream region.

For the downstream analysis, we require €^,€e-*Q MODJ.', M»--»®.

These limits must also be applied to our bow region matching

conditions which are the equality of pressure and flow deflec-

tion along the contact surface. Since our bow region internal

no

■ . -^ ....■.-- —-■i>..». i-ihM-.iJ.. ,■ .. , .^—. ■.,i.._-i.._:i.kiJ-.—jfc^-,- '■ n n^ i .mi i -in »f.i--jii ■■ inif. '■'-• •-—'--*^*i—-—*. ••■■—■■■-■ ■ ■-. ^-n--. .tJj*kai

~T>^WfT^nw™i<B?»W!?5(^^ -TI^TTW^^T^T^^IW^

flov; is a perturbation of the case where the shock and body-

surf ace are a spherical shell, for Gco - 0 , then v.re

make the approximation that the upstream interaction region

is spherical in nature. For the external flov/, it is a well

35 established fact that the downstream shock surface does not

depend on the details of the pressure distribution along the

contact surface,but only on the total integrated pressure dis-

tribution (i.e., drag) in the axial direction, adding further

validity to the use of the above assumption. Along this sur-

face, the pressure induced by the internal flow must match

that of the external flow. To first order, which is the de-

gree of accuracy for the downstream region, the pressure change

across the shock layer is zero and, therefore, we can calcu-

late the pressure on the contact surface by simply finding

the pressure level behind a spherical shock surrounding the jet

flow source. For the case Heo,;»^ 1 ^ s*^ y&c ä 1 ,

the expression for pressure behind the shock reduces to

Poo to Non-dimensionalized with respect to the external conditions,

this relation becomes

From the equality of stagnation pressures, this then reduces to

111

.....^..1.wJ-^-.-..:-:■.. .■■.-:r:;..-..-,--...^..... •^-.- '■■■ njtBHMj ..,n^..,,^.....^J^^.,.„--,^.,....^^.,^.c..^.... '-•■■• ■^■w^MtuW.Mm,i,,, ^ ..^..^^^.^^^^.^ ■■.^■.. ,... ....: -.:,.. ■-.......■... ,..—..^..-. .......^-^ _. . ' - M

lyijlij.DWiinnnni i u.-...«..^«.!^» <•• u ... M ■ i>iiiiaMiHi<>w>iii. i "' ■ ■ p^ -ii.n.n. I»II.»I ■.. L.I.IW—, i«< •!•'•■' > .L i.iki uu iiijiui. JII u ii IIIJHI mmRininBnmcwmniimq

From the source conditions, we have

fee « <^hm 2.01 co

V/ith the pressure level and contact surface shape determined,

we have satisfied our matching criteria and our next step is

to calculate the functions cr0 and 1 Ffei« cl^t Jo Pot*

Evaluating the integral from the internal layer solution, we

find

/ cos^iaL 61^9 d^

In this result, the upper integration limit requires some dis-

cussion. Strictly speaking, in order to calculate the pres-

sure and density distributions over the full range of CP

one would have to take account of the effect of the plume

boundary "nd the mass flowing in the plume boundary layer on

the internal shock. However, for highly underexpanded jets,

there is relatively little mass flowing in this layer and,

therefore, the integration of the above to the vacuum limit,

<J ■= ©eo creates only a small error in the downstream

analysis.

Ue now seek to determine the value of cr< o • It can be shown

that for ^/Ks«'« <* 1 our postulated initial blast wave be- 36

havior for the shock shape can be written as

112

.;.■ -■■■ ■-■■■.■ ■ . .- ■■■ ■..;,.v.,i.■.-;■.',i ,.;.i\>—Cu-jv^iiH-ij- -|-;-. ' -r ■ -.'i i iVTr-yi^rt^^rVirariv-'-*---''''-'--"'■■■-■ '■■ •■■--^i-^-.^-i.;v^--i-w....;.. .■■■-. ^. ;■■■.,..-..■ ...... ■,.,.,-.■.■>—i.....-^,f^^,:,^..~.-^S"-y-'&M^i\.i-':.-y'->\-<'. ..V..-.;.-<L'>V—-IM^-V.-:.. ■..,,..-...... ■■-,..■

^W^raHWW^w^OP^nrwflW^TWw^^ iMLiu-^TMW^i

V/here lb to our desreo of approximation can be written as

lb = fc^- fre* ^1

In the above CON is the none dra^ coefficient of a hy^jth--

tical blunt nosed slender bod-/, which takes on the shape of

the contact surface. C t>N is directly related to the

blunt interaction and is written as

Since we stipulated that ^ csj/Rs "* O

then equations(10.1) and (10.2) give

as *i /R. CO

c^r^-^i-)^^^ X/RSl.0 ^ O

from which o-« ■= lb

'..Te now calculate CbN based on our postulated upstream in-

teraction. For a spherical shell interaction with the pres-

sure distribution given by

ien

anc

2.000

n • m x ■= cor. (^

dA - 2.Tr l^ste1 stncj c|(p

• '.ich leads to

i i j

i i'i.iji™iu]i»uij(«impiiu»wiium.iHi»i...(iiiijiJuijüijji1iiii lim. pp . m.iip »n i iibii ■ ipiiwunitiuii «WHWW ■H^CMwnMmnRmpanwmnmwiu nua

C^N = H r COS II$!!LSIH9* ^S^K» ^9*

As in a previous inteGration, we use Goo as the upper limit of

integration instead of the exact result. In this case, v/e art

further substantiated in this approximation by the fact that

near öco^^tl he contributions from the above integrand t o •bN

are near zero due to the cos 9* factor in the above.

V/ith the previously calculated values, the shock shape can

now be written as

Geo wQD vVV 1/2

Also from (15.6) we have

Res = S, /fi^lif-''"^^ . ^

V-'e are now in a position to calculate layer properties based

on these results. For the external flow wa have

V.'ith ^ v \Ks,/ /

This distribution will bo valid throughout the external layer

The density distribution is given by

114

. ■■---.;-.-,.- ^-^ ._ ■■. ^ _ ^-..^— --.. . ^- .^ .- - ^--- >.-- . .. a.^^»-,.^. :*,.=.* ^^,..—^■^.. ■ ^^^■,. ■-..■. ^„„^ ^^^^^w^ ^A, ^^^^-'i- ^' ■ ^ -i--. ■ '^ ■■■ .. - v...^ .■■.■„... .^wi^^vi^j^täiitÜM

^»^eW"IM«?™!WP!l««WW«IIW>W?«mP»WWW^»>™w»JpBiPll^^

Q0 = (l + ^*/?>)%/2.?'

The result is not strictly valid since the density is not

zero at the contact surface.

V/hen referenced to values at the shock, the results become

p./p= - Pe/fe

Te/fe

** (16.1)

= V* C^/^)(l + W4/Cp) do.2)

= HVy* ** (16.3)

For the internal flow, we have the following results:

pct = Ca/8X (l6.4)

foi = CVSX (16.5)

VCH r ^/C/^V^^^^V (l6-6) c ex>

V/hen referenced to values at the shock, the above become

pi/pe - V^ (16:7)

fl6.8)

(16.9)

»«.

** Not valid at contact surface

-— - — "— -■-— -^ -^^—"■^—^—^•-^^-—-^ .-. ^— — —.— —

7-.*MVHUiHwmiiiiiii « ■•■ P» mmti .* ■»■"•■P-^III . iu.ui tmmimm ti IIVMI /■ mw ■«■ I<<.MIWIMWI»V>II w•»•,IIJWH»»W"<«IP.« ■■■■> — ■ i -■ ■ > i*"<•■■. M >wif«-<ai «««wi^iBiHipv^i^npinBnq^nmmfOTmnpvw

The first order solution obtained thusfar from the hypersonic

small disturbance form of the Newton-Busemann approximation,

is adequate to describe layer thicknesses, pressures, and den-

sities in the far field region near the contact surface. How-

ever, due to the assumption of ^"»Ithe temperature to the

first approximation is in considerable error for "x/p .>>1.

This is because the expression for temperature, 3=" -(■&) **

gives the unrealistic result that for all ^o , ^L+las ^->1

This result is acceptable for small X/^, , where p/p is of

order one, however, for X/^l & is smaii and even for ^

near one X. << l. Ts

To obtain a more accurate value for temperature near the con-

tact surface, the unexpanded forms of the variables in terms

of iS will be retained. Referring to Cheng and Kirsch34, the

unexpanded for S' results for the 'entropy wake* are

(17.1)

J=>e= fe'Vs*^ (17.2)

The retention of the exponent, /*., in a?.1), (17.2) is justified

because the relative errors are smaller than any integral po-

wer of 6e . Applying the same reasoning to the internal la-

yer expressions, (14.3), (14.4), we obtain

V ptW

116

«iit'„v...,.r'.V:ivi, , ..- ■ ■■■' ■ '■■■■■ ■' ■■- ■■■■■-■■--■ .-.•..■■-,•.-.■ ...■■■..- ^ ,....■■.,:■. --,..,:.-^,- v ..■..--..: ■.- -..- .- ^--'t.-b-^^.:M.t.|U^ft],.t^|ll-,iV|||iM|rtini,t^'>t,irrii • f i - -■..■■■.■■.- - ' y- n t ■ ■■, ■■:.,■■ ■ ■ ...... '.■ -... :.....-.. ..,/.-./i.„ti^»J^4iUF^«AijAL'k.'i^<-<>&>vi,vV>rf.>',-',

^■^W'VTWBJPBWWWWSW^^^

Matching (17.1) to the near shock layer results (6.1) and

then combining v;ith (17.4) we obtain

plA'(fU -R«) - o-/^ This is the same form as expression (10.1) with Po replaced

by p« . To the accuracy needed for X/fe,• ^> 1 it is ade-

quate to use the formal result (10.1) which utilises po in- '/fee

stead of pc . However, for the distribution of temperature,

we use

ti

^ - fe)/ (17.5)

for both the external near the contact surface and internal

flow. This expression will give accurate values of -3=" for Is

small P/ps as well as for p/p5 near one. To be consistent,

we also use the density expression

instead of p = ps p/p^

Making use of (17.5) and (17.6) and the expression for pres-

sure (16.1) the temperature and density distributions near

the contact surface for the external flow and within the in-

ternal layer are

T = Ts (^_\

117

:

■ ■ ■ ■ ■ ■ ■ ■ - ■ ■ ■ ■ ■....-..,.,

V,T^-,l.l-'!-'".>-T^«'in.-'—"1—^T IIIMIIUWU^^ MM II W W-^u».<, .Jtflwn; n W-«. y-i.-i ^(»■^■iw.irifHf.iiB-T .fi^..

The above expressions, as well as those previously found, are

used to predict geometry and flow properties in the far field

region for typical systems of interest.

118

—^-..^--.^.■^—,- ..^ . -— .-.- ^ ....-i.-^*——.— - - ■-^■•■■^•—-^-^'-- ■■•' " r-.-..-f-.w.^l..ii,..-

!.■■■ .i.«....! -,.,,._,^.»1^,^_ ' in " iiiiiiwm™mi>^mm«mmnmrmmmmmm*mm*mmimninHtmm^mmimm*mmmmim[tmf*mwnw\liu>vi*^wiKfmi*mmit

DISCUSSION OF RESULTS

Examining the far field analytic solution, we find that an

experimental simulation of an actual flow requires matching

Sj , ife , 0», n and Rsto f the same scaling parameters used

in the.bow region simulation. If, in addition to the above,

roj/Pooe i5 matched then the bow, corner and far field geo-

metry will be simulated. For a given system with fixed am-

bient and exhaust gas composition (i.e. ^e,^' fixed) and noz-

zle conditions (i.e.,6co,n ) the flow geometry will scale as

does the bow region with Rst'0 or Tf /*. If Moo« is also as-

sumed to be fixed, then the bow, corner and far field regions

will all scale with Rs^ . Calculations of far field flow

geometry and properties have been carried out and the results

appear in figures (21) to (24). As in the bow analysis, the

far field geometry is sensitive to the exponent used in the

exhaust plume model. From figures (21) and (22) it can be ob-

served that the internal layer occupies a large fraction of the

shock layer flow. This is due to the greater amount of str-

ongly shocked, and hence lower density, gas in the internal

flow compared to that in the external flow.

As shown in figures (21) and (22), increasing 0» or *j re-

sults in the moving outward of the shock and contact surfaces.

This behavior is consistent with the results from the bow re-

gion. Also, near the contact surface, temperature, pressure,

and densityfor the internal and external flow increase for

increases in Goo or ^1

119

M --•■-——•■■——-—^ .iWwMt««„^....„ ^nmmmmm -

' —' ■■—-.w^w i mm ■»-...PM^. mmawiNuu MIUI«II i p«jvv^7pwi«i UIIIMIIIVW;« . m P«MJ nwn« ■!■«■■ «W «»-..«...-....I—*-I^I^.UP,|.IH^III IW«>I ^w^^nw^matl WIl.PliiUllpJIIJl! U MIinW^pqmpiV^R«

The effect of the internal layer flow on the external flow

pressure, density, and temperature is to maintain them at a

higher level than that created by just the bow interaction.

This can be seen from expression 15.4 in the far field analy-

sis section where the coefficient for the pressure decay

expression consists of the normal bow interaction term plus

the effect of inner layer thickness.

In figures (23) and {2U.) the detailed property distributions

across the layers are given for the same "cold" jet and actual

jet cases that were analyzed in the bow section.

Consistent with the bow region results, we find that the act-

ual jet case shown in figure (24) has a hot, low density ex-

ternal layer, which flows over a cooler and more dense inner

layer. If allowed, considerable heat transfer and mixing bet-

ween the external and internal layers would occur. The "cold"

jet case does .not exhibit as sharp a change in flow proper-

ties across the contact surface as that of the actual case.

This indicates that "cold" jet experiments must be interpreted

carefully in light of the greater mixing and heat trans-

fer effects for the actual system versus the "cold" jet simu-

lation.

120

^^.■.■■.-^uj..ij^v\L.^ttj^^fc^J<^j:<.k-i4d>!ibJ^,.^....f...J.■,-,.■_■■......ii .^.....:.,-,...., ,c.,. rjivrffrii'fä^awmiTilim'iaitBlhiiHn'liT'ii'iji

'mr*-Vt*rr**'*ir?v™rPW7rr^^ ,rvw^.»^^ ^-«.^lIMÄMIi«.!^!..! W^

CONCLUSIONS

Analytical techniques for predicting flow properties in the bow

and far field regions of an opposed hypersonic plume in a hy-

personic stream have been developed. It has been shown that

they are valid for a wide range of altitude and jet thrusts

for typical systems. The influence of the corner region flow

on the bow and far field flow predictions has been shown to be

negligible. Consequently, only a qualitative outline of the

flow processes in this region are given.

Some of the major results of the analyses and calculations

are:

1. Experimental simulation depends on the matching of

^e > *l ^0O ^ n ^ ^*io ) Fkj / pooe

2. For a given system with fixed ambient and exhaust

composition, external Mach number and nozzle con-

ditions (i.e., fixed ye,Vj,6eD,n,Mooe) the entire flow

geometry will scale with Rsieor «T

3. This analysis confirms the good accuracy of,the

Newtonian impact analysis when applied to the axi-

symmetric bow region, and suggests that the extension

of this simple technique to predict asymmetric bow

geometry might be successful.

4. The bow and far field solutions are sensitive to the

exhaust plume model exponent used and consequently

for accurate predictions of flow properties an accu-

rate plume model must be utilized.

121

... . ■ ..■.....■ ...

■.■.,.--.^,..-..-^^.,L.,,..,-J.1J....--.....-.'—----■.■^.■.^.■.■..1-.-... ^..„..-.■^■„■;.-^-.^.lJJ^.^..^^-.. A. 1iatB|ja^Ya||iiMt^|-j|^ ■;,, ,f.>........-J..,-.,.„j;...,i.,.M^:^K..; : ^■-l. --^^

rwwwPTEws^wwP'»^CT'T,r!^^ ^»•-TTiTTT^^r-^r^.^^-KT'Ti'TTTTOT^^

5. Penetration of the intercepting shock layer flow into

the internal layer is considerable. However, the

influence on the bow and far field properties is

small.

6. Good agreement is found between a calculated bow re-

gion geometry and an experimental result.

7. Calculations show that mixing and heat transfer ef-

fects between the hot external and cooler internal

layers are more pronounced in the actual case than in

the experimental "cold" jet simulation.

Ö. Increasing the primary system design parameters &oo

and &j with all other parameters fixed results in

a. The thickening of internal and external layers

in the bow and far field regions

b. An increase in radius of curvature of the con-

tact surface and thereby an increase in bluntness

for the bow region

c. An increase in the angle between the far field

external shock and free stream direction

d. A higher level of pressure density and tem-

perature both across and along the internal and

external layer flows.

■^ 0a;, is related to noaalc exit Mach number anä angle and exhaust gas conposition. For a fixed composition, increasing nozzle exit angle or decreasing exit Mach number gives in- creases in öco . Vj is related to exhaust gas composition which in general will decrease as the degrees of freedom or the complexity of the exhaust molecules increase.

122

*- ■ ■ - ■„.... .; ■....,- : , ._*_,_!__*.,**»_^—„——llrfl

— uiiinwiiKIUHIIi wmmmnwmwjumtiuu I «xipxan^nmmimi^np^a I1IJ m UIU II IPIJI.UIIIII WPWWmBOTmpa^nn^üK"""-' "»Um I W^P Uli 111 ■ 11«1I.1IHH»W»»!WWW^I

9. The effect of the internal layer on the far field

region properties is to maintain the pressure, den-

sity, and temperature of the flow at a higher level

than that which would be created by the bow region

interaction with the far field internal layer flow

absent. i

For high Reynolds' number flows the present technique may be

extended to treat mixing effects along the contact surface

by the use of boundary layer methods. Also, equilibrium chem-

istry effects may be easily incorporated into the present mo-

del through use of Ilollier charts.

Further extension of the bow layer technique to include vis-

cous chemically reacting and merged layer effects requires

considerably more effort than the above extensions. In this

regard, an advantage of the present technique over the numer-

ical technique of reference (7) is that a complete solution of

the bow region requires approximately 13 seconds on an IBII

360-65, whereas the numerical technique requires 120 seconds

on the much faster CDC 6600. This economy may prove to be

significant in extending the above techniques to be able to

predict viscous chemically reacting or merged layer bow pro-

perties where an order of magnitude increase in computational

times is expected. This makes the technique of reference (?)

uneconomical for use in parametric calculations whereas the

present technique would still fall within the practical time

limitation for these calculations.

123

■.. .■■

W^W«WIW»*J™W>J»»ä^^

REF5RE:ICES

1. Romeo, D.J. and Sterrett, J.R., "Exploratory Investiga- tion of the Effect of a Forward-Facing Jet on the Bow Shock of a Blunt Body in a Mach 6 Free Stream", NASA TN - D-1605 (1963)

2. Char-vat, A.F. and Faulmann, D., "Investigation of the Flow and Drag Due to Control Jets Discharging Upstream into a Supersonic Flow", Proceedings of the XVth Inter- national Congress, V/arszawa 1964» Vol. Ill pp. 85-114

3. Finley, P.J., "The Flow of a Jet from a Body Opposing a Supersonic Free Stream", J. Fluid Ilech., 26, pp. 337- 36Ö, (1966)

4. Cassanova, R.A. and V/u, Ying-Chu Lin, "Flow Field of a Sonic Jet Exhausting Counter to a Low-Density Supersonic Airstream", The Physics of Fluids, 12, pp. 2511-2514 (1969)

5. Smithson, H.K., Price, L.L. and VJhitfield, D.L. , "v/ind Tunnel Testin-1: of Interactions of High Altitude Rocket Plumes with the Free Stream", AEDC-TR-71-llbv, July 1971

6. Laurmann, J.A., "Elementary Mewtonian Theory for the In- teraction of Two Opposing Hypersonic Streams", Institute for Defense Analysis, Research Paper P-509, July 1969

7. Rudman, 3. and Vaglio-Laurin, R., "Flow Pafterns and Re- gimes for Retro-Plumes", Advanced Technology Laboratories, Inc., Report ATL-TR-154, January 1971

8. Hamel, B.B. and V/illis, D.R., "Kinetic Theory of Source Flow Expansion with Apolicaticn to Free Jet", The Physics of Fluids, 9, pp. 829-341, (1966)

9. Simon, G.A., "Rarefaction Effects in High-Altitude Rocket Plumes", A.I.A.A. J., 10, pp. 296-300 (1972)

10. Bush, V/.B., "On the Viscous Hypersonic Blunt Body Problem", J. Fluid Llech., 20, pp. 353-367 (1964)

11. Probstein, R.F., "Shock 7,7ave and Flow Field, Development in Hypersonic Re-entry", ARS J., 31, pp. 185-194 (1961)

12. Adams, M. and Probstein, R., "Jet Propulsion", Vol. 28, pp. 86-39, 1958

124

- ■ ■ - - — - - - .—.^..M»^«—.^ ^..^-^^-faM. -^.L».-., .v .......... . ,. ^.^ ^_ „.„..., -m nitti^iilur' — - -- --■-- iir ■■■■!

RSFEREMCSS (continued)

13. Van Dyke, Milton, "Second-Order Compressible Boundary Layer Theory with Application to Blunt Bodies in Hyoersonic Flow", Hypersonic Flow Research , Progress in Astronautics and Rocketry, Vol. '/, ed. F. R. Riddell, Academic Press, N.Y., 1962 . ' '

14. Cheng, H.K., "Recent Advances in Hypersonic Flow Research", A.I.A.A. J., 1, pp. 295-310 (1963)

15. Boynton, F.P., "Highly Underexpanded Jet Structure: Exact and Approximate Calculations", A.I.A.A. J. 5, pp. 1703-1704, (1967)

16. Ashkenas, H., and Sherman, F.S., "The Structure and Utiliza- tion of Supersonic Free Jets in Low Density V/ind Tunnels", Rarefied Gas Dynamics, Vol. II, ed. J. H. de Leeuw, Acade- mic Press, N.Y., 1966

17. Albini, F.A., "Approximate Computation of Underexpanded Jet Structure", A.I.A.A. J. 3, pp. 1535-1537, (1965)

10. Hirels, H. and Mullen, J.F., "Expansion of Gas Clouds and Hypersonic Jets Bounded by a Vacuum", A.I.A.A. J., 1, pp. 596-602, (1963)

19. Hill, J.A.F. and Habert', R.H., "Gas Dynamics of High Al- titude Missile Trails", MITHRAS Inc., Report No. MC61-13- Rl, 1963

20. Andrews, E.H., Jr., Vick, A.R. and Craidon, C.B., "Theo- retical Boundaries and Internal Characteristics of Exhaust Plumes from Three Different Supersonic Nozzles", TN j-2650, March 1965, NASA Washington, D.C.

21. Chester, ',7., "Supersonic Flow Past a Bluff Body with a De- tached Shock, Part II, Axisymmetric Body, J. Fluid Mech.. 1, pp. 490-497, (1956)

22. Freeman, N.C., "On the Theory of Hypersonic Flow Past Plane and Axially Symmetric Bluff Bodies", J. Fluid Mech., 1, pp. 366-337, (1956)

23. Cole, J.D., Perturbation Methods in Applied Mathematics. Blaisdell Publishing Co., Mass., 1968

24. Romeo, D.J. and Sterrett, J.R., "Flow Field for Sonic Jet Exhausting Counter to a Hypersonic Mainstream", A.I.A.A. J., 3, pp. 544-546, (1965)

125

---■•

^.^.—..-., —■...-.. ,

REFSREMCES (continued)

25. Chernyi, G.G., Introduction to Hypersonic Flow,Academic Press, N.Y,, 19^1

26. Jarvinen, P.O. and AdaraS; R.H., "The Effects of Retroroc- kets on the Aerodynaaic Characteristics of Conical Aero- shell Planetary Entry Vehicles", A.I.A.A. Öth Aerospace Sciences Ileetinr-,, *Iev; York, N.Y., A.I.A.A. Paper No. 70- 219, 19-21, January 1970

27. Charv.rat, A.F., "Boundary of Underexpanded Axisymmetric Jets Issuing Into Still Air", A.I.A.A. J. 2, pp. I6I-I63, (1964)

23. Latvala, E.K. and Anderson,T.P., "Studies of the Spreading of Rocket Exhaust Jets at High Altitudes", Planetary and Space Science, 4, PP' 77-91, (1961)

29. Luce, R.VJ. and Jarvinen, P.O., "An Approximate Method for Predicting Plume Sizes for Nozzle Flow into Still Air", A.I.A.A. J., 6, pp. 182-184 (1968)

30.

31.

32.

33.

34.

Hayes, V/.D. and Probstein, R.F., Hypersonic Flow Theory, Academic Press, New York, N.Y. , l^p^ ""

D'Attorre, L., "Experimental and Theoretical Studies of Underexpanded Jets Near the Mach Disc, - Appendix", Gener- al Dynamics Astronautics Report GDA-DBE-64-OO8, Febru- ary 1964

Sterrett, J.R. and Barber, J.B., "A Theoretical and Experi- mental Investigation of Secondary Jets in a Mach 6 Free Stream with Emphasis on the Structure of the Jet and Se- paration Ahead of the Jet", Separated Flows Part 2, AGARD Conference Proceedings, No. 4, May 1966

Colev J.D., "Newtonian Flow Theory for Slender Bodies", J. Aero. Science, 24, pp. 448-455, (1957)

Cheng, H.K. and Kirsch, J.VM, "On the Gas Dynamics of an Intense Explosion with an Expanding Contact Surface", J. Fluid Mech. 39, pp. 289-305, (1969)

35. Cox, H.H. and Crabtree, L.F., Elements of Hyoersonic Aero- dynamicst Academic Press, New York, N.Y., 1 -jo5, page 135

36. Mirels, H,, "Hypersonic Flow over Slender Bodies Associated with Power-Law Shocks", Advances in Aopliej MGchanics, Vol. VII, Academic Press, New Xor.c, N.Y. 19C2

126

■ - — -....- ,.,„.—-..—-..T.^..-^—u - -iiiimimi 1 ,.;...-.,..■..■ ■JJ.^..,.,^..,.-...i...J „,.;-; .■„,.■..„...

Appendix (A)

Bow Initializing Scheme

To integrate the bow region equations for points away from

the axis, we must first obtain the initial radii of curva-

tures and positions of the shocks at the axis. Writing equa-

tions (2Ö.1), (28.2), (5Ö.1) and (58.2) in the form

^ *.

ceo + bCaO ft* (1)

(2)

(3)

(4)

we develop expressions for <£t/^ and shock radii of curva-

tures R5io/f?st . The geometry which relates the internal

flow parameters to those of the external flow is illustrated

in schematic (5)

Schematic (5) i-r-A

127

— - ■UM^tMMMMiMIM --'-- ■ ..^--^-^ "-■"-^""tuaiilriaiiKilftiin'ir-rrlii

■

In the above illustration

Re - Rst Cl + Q^-ÄC^^- BCqc^e) (6)

Referring to schematic (5) we can write for

Angle matching

^t - $e + 2.(.Q.C + B,-2(ac'A)9c/Cl + ^ ■+

Pressure Matching

-0/2.

9e/^ 'C||)

(7)

(Ö)

V/e now have two equations with three unknovms 9e/$ J > ^«i ^t

Consequently, we must relax one of the parameters a;, Qe

in order to obtain a unique solution. Choosing Qi = 0 the

expressions for A (0.0, B ( <aj), C ( Oi) and D ( di) simplify

considerably.

Substituting (8) into (7) we obtain q« which also gives <pe/&l .

From these results the ratio of shock radii of curvatures at

the axis, Ks^/f^ , and the relative position of their cen-

ters must be obtained. Referring once again to schematic (5)

we have from geometry

sth frr- "(fri) - gin o^. = sioC^t z$kl Re RC J/

For small values of the angles we can write

128

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their relative positions at the axis, we then use these quan-

tities as input data to the computer program, which predicts

bow properties for regions away from the axis of symmetry.

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