Interaction between a high speed boundary layer flow and an anelastic deformable solid wall

"Wi.JOT,; ·~~v &.;;

Kr ~-_li,

:UYve:-\veg 1 - D:U=T

vonKARMAN

INSTITUTE

FOR FLUID DYNAMICS

TECHNICAL NOTE

86

INTERACTION BETWEEN A HIGH SPEED BOUNDARY LAYER FLOW AND AN

ANELASTIC DEFORMABLE SOLID WALL

H. W. STOCK

MAY, 1973

~1~

-~O~-

RHODE SAINT GENESE BELGIUM

~VW

TECHNICAL NOTE

86

INTERACTION BETWEEN A HIGH SPEED BOUNDARY LAYER FLOW AND AN

ANELASTIC DEFORMABLE SOLID WALL

H.W. STOCK

ABSTRACT

The hypothesis proposed by Probstein and Gold, that cross-hatching results from the differential deformation of an inelastic material. has been used in this analysis. Wall pressure and wall shear stress perturbations were calculated with a linearized small perturbation theory. The supersonic boundary layer was composed of an inviscid outer layer and a viscous sublayer. Both laminar and turbulent mean velocity and Mach number profiles were used in the calculation.

The theoretical results are in agreement with most of the experimentally observed features of crcss-hatching. For instance. for cross-hatching to occur the boundary layer must be supersonic and turbulent, whilst the inelastic body material must be of the Maxwell type. Theory and experiment also show that the pattern is characterized by a cant angle equal to the local Mach angle (transonic flow perpendicular to the direction of the grooves) and a wavelength inversely proportional to the static pressure, directly proportional to the viscosity of the material and independent of the downstream direction.

TABLE OF CONTENTS ABSTRACT

_•

'IABLE OF CONTENTS

_•

_•

_•

_•

LIST OF FIGURES

_•

_•

_•

LIST OF SYMBOLS

_{• •}

_{• •}

,

_•

• 1. INTRODUCTION •

_•

_•

_•

2. THEORETICAL ANALYSIS •

_•

_•

_•

2.1 General a s p e c t s . •

_•

2.2 Analysis of the inviscid outer layer •

_•

2.3 Analysis of the viscous sublàyer •

2.4 Wall pressure and shear perturbations and . their interaction with the viscous

deformable solid • • •

3.

CALCULATION PROCEDURE

_{• •}

4.

DI SCUSSI ON AND RESULTS • •

•

•

4.1

Influence of the amplification fact~r

ë

_I and the perturbation propagation speed cR' • • • 4.2 The wavy wall sweep angle

n

and the

influence of the wave number ~ •

_{• •}

4.3

The amplification factor Cl

4.4

Influence of the type of boundary layer

4.5

Influence of the Mach number MO _e

the Reynolds number Re _e

5.

COMPARISdN WITH EXPERIMENTS

6.

CONCLUSIONS REFEREHCBS • FIGURES APPENDICES

•

•

• •

and •

•

•

•

•

A - DERIVATION OF TllE THREE DIMENSIONAL, UNSTEADY PRESSURE PERTURBATION EQUATION IN THE INVISCID. NON HEAT CONDUCTING PART OF THE BOUNDARY LAYER B 1. THE QUASITWODIMENSIONAL PRESSURE

-PERTURBATION EQUATION IN THE INVISCID,NON HEAT CONDUCTING PART OF THE BOUNDARY LAYER 2. BOUNDARY CONDITIONS ~ ii iv v 1 3

4

6

8

10 13

16

17

18

19

19

20 21

23

24

EQUATION IN THE VISCOUS SUBLAYER

E - THE QUASI TWO-DIMENSIONAL PERTURBATION EQUATION OF THE VERTICAL VELOCITY FOR THE VISCOUS

SUBLAYER IN THE TRANSFORMED PLANE

F - SOLUTION OF THE QUASI TWO-DIMENSIONAL EQUATION FOR THE VERTICAL VELOCITY :PERTURBATION

G - CALCULATION OF THE FUNCTrON F(~) FOR COMPLEX VALUES OF I;

H - CALCULATION OF 'rHE EFFECTIVE THICKNESS

-r

_f OF THE VISCOUS SUBLAYER

I

-

CALCULATION OF THE SHEAR AND PRESSURE PERTURBATIONS AT THE WALL

1 2 3 4

5

6 7 8 9 LIST OF FIGURES

Cross-hatched patterns on wind tunnel models (Ref. 5) Typical flow and wavy wal 1 arrangement (schematic)

Mean velocity profiles

Distribution of the real part of the wall pressure and the imaginary part of the wall shear stress perturbation for which amplification (a) or damping (b) of the wavy wall amplitude e occurs

'E'1f_ . . . _

~ cl versus the ampl~f~cat~on factor cl 2G ~

ë

_r versus the amplification factor

ë

I

~~-

.

.

-r-

cr versus the propagat~on speed of the perturbat~ons cR 2~ ~

ë

_r versus the propagation speed of the perturbation cR ~~-

_K

_{cr versus th e I1ach numb er component Mo ex}

10 2tr ~

ë

_r versus the Mach number component Moe x

11 The real part of the wall pressure perturbation versus the

Mach number component Moe x

12 The imaginary part of the wall shear stress perturbation versus the Mach number component Moe

x 13

14

15 16 17

18

19 . 20 21 22: 23 'E'~_ -;:- cr

~~-7

Cr

and

2rr

~ cr versus the wave number ~ versus the wavelength À

bI!'"

-X-

versus the amplification factor

ë

_r 2tr ~ versus the amplification factor

ë

_r

'E'~_ .

K

cr versus the Hach number component i10ex for d~fferent types of boundary layers

2~ ~

ë

_r versus the Mach number component MOex for different types of boundary layers

cr versus the Mach number MOe 2G

T

ë

r versus the Hach number HOe

ET-T

cr versus the Reynolds number Ree

2G

T

ë

_r versus the Reynolds number Ree

Experimentally observed wave numbers

ä

versus the Mach number MOe

a e cl cR D eN eT E F G h f M mOe N p p' Po 'V p Re T T t u,v,w u' ,v'

,

w' U 0' W 0 'V 'V 'V U,V,W x,y,z LIST OF SYMBOLS v'eloei ty of sound

=

cR +

ier

amplification factor

perturbation propagation speed

variabIe defined in equ. (27) normal strain

tangential strain elastieity modulus

funetion defined ~n equ.

(16)

shear modulus

eharaeteristie he~ght of the viseous sublayer

Mach number

quantity defined ~n equ. (12) quantity in the exponent of the

power law velocity profiles statie pressure

statie pressure perturbation mean statie pressure

amplitude of the pressure perturbation

Reynolds number temperature

relaxation time of the material

time

velocity eomponents in the x,y,z direction

perturbations of the velocity eomponents

mean velocity eomponents

amplitudes of the perturbations of the velo city eomponents eoordinates in the physieal

plane Dimensionless Ë = E YPO

G

= G YPO h_f =

-

hf cS

p

=

L.

YPO

T=

t=

_ó ü=

-.v-=

u' -UOe

tJ

_o= Uo

~

_e

,

x x =

&'

y v' -

w'

._ Wc

ü;-U , . Oe e Wo= =

1..

cS ' Uo

u;-

_e

-

z

z = -cS

X,Y,Z

~

Y f Cl ~ P p' Po T T '

'"

T o e w x z ST I R

.coordinates

in

the transformed plane independent variabIe defined

~n equ, (2F)

effective viscous sublayer, thickness in the transformed plaile

wave number

specific heat ratio

boundary layer thickness effective viscous sublayer

thickness

thickness of the layer of

X-Y'f

'6

f =

=

X

6'

Y f

-

6

material affected by deformation characteristic height of the

viscous sublayer in the transformed plane

wavy wall coordinate

amplitude of the wavy wall perturbation wavelength viscosity

variabIe defined in equ. (4F) density

density perturbation meán dens i ty

shear stress

shear stress perturbation amplitude of the shear stress

perturbation SUbscripts mean quantity e:

=

.§. ö

r

À =

_'6

T =

L-YPo

at the outer edge of the boundary layer at t he wa.ll

component in the x-direction component 1n the z-direction stagnation conditions

imaginary part of a complex quantity real part of a complex quantity

Y

= -Y

Z

=

-

Z

1. lNTRODUCTlON

One of the interesting phenomena in fluid mechanics in the last years has been that of "cross-hatching" 1-8. The name cross-hatching refers to highly ordered patterns which have been observed on the surface of ablating bodies exposed to a transitional or turbulent, supersonic boundary layer flow either in free flight or in wind tunnel tests. Cross-hatching consists of nearly straight grooves regularly spaeed and

running obliquely to the direction of the flow outside of the boundary layer, Fig. 1.

A considerable amount of experimental information on the development of such patterns is available 1-8 and many unsatisfactory attempts 9-14 were made to explain the physical mechanism which creates cross-hatching. The various theories

9-14 which were developed to explain the origin of cross-hatching made the common assumption that the pattern results from an interactioh between the bcundary layer and the ablation material.

Tobak 9 assumed that streamwise vortices situated in the boundary layer play an essential role. However. in reference

5

it was concluded that the presence of such vortices is not necessary for cross-hatching to exist.

roger 10 proposed that differential mass transfer rates create cross-hatching on the surface of ablating bodies. lnger calculated stationary perturbations of the ablation rate only on subliming bodies (other ablation modes were not con-sidered) using the rather restrictive assumption that the mass concentration of the vapor of the sublimed solid at its surface is equal to unity (i.e. very large mass transfer rates). lnger evaluated flow and material conditions under which the abla-tion rate in the bottom of the grooves is maximum, thus leading to a self-perpetuating "resonant" interaction between the

Nachtsheim 11 considered that the pattern resulted from the interaction between the boundary layer and a liquid film covering the body 8urface. Obviously this theory is

re-stricted .to melting materiaIs.

Probstein and Gold 12 proposed a model which is valid with or without ablative mass transfer, hence this theori is not restricted to a particular ablation mode. They assume that the material is inelastically deformable and that cross-hatching

is a consequence of wall shear stress perturbations leading

to differential deformations of the viscous solid. The turbulent boundary layer is approximated by a constant velocity profile and the wall shear stress perturbations are assumed to be pro-portional to the calculated pressure perturbations without phase shift. A physically unrealistic assumption was made, namely

that the velocity of the moving viscous solid material was

equal to the gas velocity. This boundary condition was corrëcted

in reference

13

by assuming that the velocity of the moving solid

was of the same order of magnitude as the perturbation terms.

Later Gold and Probstein 1~ did a more refined calculation by

using velocity and Mach number profiles of laminar and turbulent

boundary layers. The wal 1 shear stress perturbations were

eval-uated again from the wal 1 pressure perturbations. The ~atter

were obtained at an arbitrari~r · chosen height in the boundary

layer close to the wall as the inviscid pressure perturbation equatioD is singular at the wall. In their analysis. the phase

angle between the pressure and shear stress p~rturbations was

derived from the analysis of an incompressible flow over wavy surfaces 15

2. THEORETICAL ANALYSIS

Any theoretical approach to the phenomenon of cross-hatching should take into account the main experimental find-ings of the pattern development listed below :

1. The flow outside the boundary layer must be supersonic 1.

2. The boundary layer must be turbulent or at least transi-tional I, 2, 8

3. The Mach number component of the flow perpendicular to the direction of the grooves is close to unity 1-8.

4.

The streamwise length of a typical cellof the pattern is inversely proportional to the local statie pressure

3, 5, 6, 8.

5.

The cell s~ze increases slightly in the downstream

direc-t i on 2 ,5.

6.

The time needed to produce the pattern af ter exposure of the bodies to ablation conditions increases linearly with the cell size 8.

7.

The cell size increases if the viscosity of the solid ablation material increases 8

8.

Cross-hatching exists for all ablation modes 2.

9.

Cross-hatched patterns are observed in the absence of ablative mass transfer 8

The experimental facta

7, 8

and

9,

mentioned above. suggest that the phenomenon results most probably from an interaction between the boundary layer and the viscous deform-able body material, even in the absence of ablative mass

transfer.

Although the basic ideas of reference 12 are used for the development of the present analysis, the treatment of the problem differs considerably from that in references 12, 13 and 14. In the work or Probstein and Gold 12, 14 the pressure perturbations were calculated by an inviscid analysis. The wall

\

the pressure perturbations calculated at an arbitrarily chosen

height in 'the boundary layer close to the wall. Furthermore,

the differential deformation of the material was considered to be produced only by the wall shear perturbations.

In the present analysis which is a linearized small perturbation theory, the perturbations of both the wall shear and the wall pressure are calculated by taking into account the viscous sublayer. The height inside the boundary layer,

where the wall pressure perturbations are considered, is,

cal-culated and not arbitrarily chosen. Finally, the surface deform-ations are assumed to result from both the wall shear and pres-sure pert urb at i ons.

2.1 General aspects

Although cross-hatching is a three-dimensional phenom-enon, it can be treated as a quasi two-dimensional problem in which the pattern results from the superposition of two sinus-oidally wavy walIs, with waves symmetrically inclined with

respect to the direction of the flow outside the boundary layer.

A sinusoidally wavy wall introduces periodic sinus-oidal perturbations which are assumed to be linearly superposed to the mean flow quantities. Thus perturbations of the Tollmien-SChlichting type can be used in the analysis.

Any perturbed quantity Q can be written for the

two-dimens i onal cas e

'V

Q(x,y.t)

=

Q(y) exp ia (x-ct)

'V

where Q(y) is the complex amplitude function, the real quantity

• ( 2'IT . ) .

a 1S the wave number a

=

~. À be1ng the wavelength and c 1S

a complex number. the real part of which is the propagation

velocity cR of the pertu~bations and the imaginary part of

amplification of the perturbations). Only the real part of Q has a physical meaning :

Real

~ ~

Q(x.y.t)

=

exp (cloaot)(QR(y)cosa(x-cRt)-QI(y)sina(x-cRt»)

( 2 )

The problem of explaining the or~g~n of cross-hatching is thus reduced to a stability analysis of the system formed by the boundary layer and the deformable material. The procedure

~s then to determine the conditions in the flow. in the wavy wal I characteristics and in the solid for which the deformation of the body material at its surface is amplified (ioe. an increase

~

of the wavy wall amplitude E. see Fig. 2. with time).

The flow conditions which were varied in this analysis are the Mach number at the outer edge of the boundary layer

MO • the nature of the boundary layer flow. laminar or

turbu-e

Vo opooo

_{e e} bulent" and the Reynolds number Re

e

=

where V

o

e is

the flow velocity outside the boundary layer. The wavy wall characteristics are the wavelength À. the wall amplitude and the angle of sweep

n.

of the waves with respect to the flow direction outside the boundary layer. Fig. 2. The anal-ysis takes into account the mechanical properties of the abla-tion material. the shear modulus.G. the elasticity modulus E. the viscosity ~. and the relaxation time T. as a function of cl and cR"

Only adiabatic flows are considered. This is justified by the experimental fa ct that the pattern is observable on

ablating materials even for very small heat transfer rates 5. 8

The boundary layer. whether laminar or tu~bulent.

was divided into an outer layer and a viscous sublayer. In the outer layer. the effects of viscosity and heat conductivity on the mean and ~he perturbed flow are neglected. but the com-pressibility effects are included. In the viscous sublayer.

the mean flow is compressible and viscous. but perturbations in density and viscosity are neglected. In the case of a tur-bulent boundary layer. the interaction between the random,

turbulent fluctuations and the perturbations in flow properties. due to the presence of the wavy wall. is neglected.

2.2 Analysis of the inviscid outer layer

Using the conservation equations in their inviscid. non heat-conducting forma applying the parallel flow assump-tion and linearizing. an expression can be derived (see Appen-dix A) which gives the pressure and the vertical velocity perturbations as a unique function of the mean velocity and Mach number profiles (i.e. independent of density and temper-ature perturbations).

This equation can be written ~n the three-dimensional form, equ. (2lA)

d 2

P

2

a

2

P

2

a

P

2 dM 0 x d P d 2

P

(l-M ox ) + - _{(l-M Oz )} + - - - • _ . - - 2Mo oMO

ax2 dZ2 dy2 Mox dy dY ' x z dXdZ

where P

and WO. ponents

(Ë

+ 2Uo

lf

+ 2Wo

II

+ 2 v'

_{- )}

dUO

at ax dZ Uo dy

,

=

~ is the dimensionless pressure perturbation. Uo

YPO

Mo

_x

and Mo are the mean velocity and Mach number

com-Z

of the flow in the x and z direction respectively. see Fig. 2.

The left hand side of equ. 3 ie the time independent part of the pressure perturbation, whereas for unsteady pertur-bations the complete equation (3) has to be considered. Equation (3)can be reduced for the wavy wall problem into a quasi

two-dimensional form, by choosing the coordinate system shown in figure 2, and applying the independence principle. In the Z -direction the wall produces no perturbations, thus

-ap

äZ

=

0

( 4 )

Hence equ. (3) can. be written for the quas~ two-dimensional case 2

-

, dU a a2

p

dMo Mox

(Ë

2 2 ap a ap (l-Mo .)

_{- -}

. - .-

x

₌

-

.-

+ 2Uo + 2

!...._)

x

_u

2

ox

2 _{Ma dy oy at at}

ox

_{Uo dy} x 0

It ~s shown in Appendix B that by coupling the pertur-bations in the vertical velocity v' and in the pressure

P

through the y-momentum equation and by a~plying equation

(1).

equation

(5)

can be written in the following dimensionless form : 'IJ 'IJ _dt-1 0 _{dUo - [} M2 (Üo-ö)

21

1

d2P dP

_{(2- .}

x 2

_{• -::-(_ C_>J -}

_P

_~

_(1-

Ox 0

- -

- +

-

=

dYL dy Mo dy Üo dy UO-c _ U2

x 0

(6 )

which ~s an ordinary. linear differential equation of second 'IJ

order with variabIe coefficients. where P is the complex am-plitude function of the pressure perturbation. Equation

(6)

can be solved numerically by specifying the boundary conditions

(see Appendix

B).

at

at

Outer boundary condition :

y

=

1

'IJ dP dy

Inner boundary condition

M

2

=

~.7;2 ~

(rf)

- 2

Uo

(8 )

where ~ is the dimensionless amplitude df the wavy wall. see equ. (15B) and

6

f is the viscous sublayer thickness. see Appen-die esF an d H.

One has a split boundary-value problem tor equ.

(6)

which can be difficult to solve especially for large values of

N

~ leading to a highly oscillating function p(y). However. the numerical integration can easily be done by a transformation of variables which leads to a more tractible initial value problem, see chapter 3. The mean velocity and Mach number pro-files used to solve equ. (6) with the boundary condi'tions (7)

and

(8)

are shown in Fig. 3. see also Appendix C.

2.3 Analysis of the v~scous sublayer

Using the afore mentioned assumptions. the three-dimensional continuity and Navier-Stokes equations in the x-direction can be rewritten using a standard compressibility transformation. see Appendix D.

Cont inuity - + - + -

aU'

av' aw'

=

0 ( 9 )

ax

aY

az

a

2

_u'

_a2

_u'

+.1.. au' dUo aw'

ataY + ua aXaY aY (Wo

ar)

- 'dY'"

äZ

=

x-Momentum

a

au'

~

•

p~w)J

'TI (är

+

ax

2 PO

(aw'

+

au'»)J

ax

aZ

(10 )

Considering again the quasi two-dimensional case and us~ng the arguments as in chapter 2.2. equation (10) can be re-written with the use of (1) and

(9)

in a much simpIer dimen-sionles s form. see Appendix E.

dUo 'U 4 'U

.-

_y-

c d2y mOe d4y

~a _(-=-) =

(11)

dY W _dÜo dYL _Ree dT'+

(-:::-) x

u a • P a

• Ó with _{Re ex}

₌

e e )Joe (12) 2

l.:1.-.

2

Ta

w

T

ST

and _mOe

₌

1 + ₂

Ma

_e = =

Ta

_e

Ta

_e

For the derivation of equ. (11). i t has been assumed that the wavelength À of the perturbation is much larger than the viscous sublayer thickness óf.i.e •• a.ó

f « 1, see Appendix E.

Equation (11) ~s an ordinary, linear differential equation of the fourth order with variable coefficients, where

'V

V is the complex amplitude function of the vertical velocity perturbation.

Using the following transformations

-1=Y---~c~

(dU

a )

dY w and

e

=

l-t. f

(13)

(14)

where t.

f is a measure of the visCOUG sublayer thickness in the transformed plane

X, Y

[

- J-1/3

Ree dUO t. f

=

ä.

4 x (~)

ma

dY w e

equation (11) can be reduced to

=

0

(16)

as may be seen in Appendix F. lts solution is, following Holstein 18 and Lighthill 19

'"

V

=

x where and dU 0 ( - )

dI

~Crf) [~-E;"

+ F ( E; ) w F'(E; ) -w F(E;) = dF F'

=

dE; E;E;

II

00 co 1/2 (1) E; oH1!3 F(E;)

j

F' (E;w) 2 3/2

(3'

(i E; ) ) dE; dE;

(18)

The function F(E;) and its derivatives have been tabulated by Holstein 18 for real values of E;. vlhen ë

I

1:

0. E; is complex (see equs. (13) and (14)) and F(E;) can be determined by the method described in Appendix G. for lëII « 1.

The first term on the right hand side of equ.

(17)

is the inviscid part of the solution which satisfies the tangency condition on a wavy wall placed inside the viscous sublayer at a distance y above the mean surface 19

The second term represents the viscous part of the vertical velocity perturbation. which vanishes for E; > 1 as shown by LighthilI 19. The calculation of the thickness

(6

f ) of the effective viscous sublayer is given in Appendix H.

2.4 Wall pressure and ~hear perturbations and their interaction with the viscous deformable solid

It is shown in Appendix I that the variation of the amplitude and phase of the pressure perturbation across the viscous sublayer ~s neglig~bly smalle Hence

where P(6

f ) is calculated from equs (6), (7), and (8).

The wall shear perturbation is given by the following expression, see Appendix _{I "}

1 -2 2 2 [

Tl

IV U

o

Moe mo F(~ ) F"(F; ) IV ('rf) x e F;-F i (

~

,

)

:

W P (20) T =

_;r

Ree F'{F; ) w ÜOC6:f)6f . w w °x

Jf

w

There are basicly ,two Iways: of descriping I the visco-elastic

behaviour of materials 21

Kelvin body

Normal stress: T·-deN _dt =

_T

Pw

-

_eN (21)

de'l' T

Tangential stress: T w (22)

~ =

2G

-

eT

Maxwell body

Normal stress: T deN T dpw + Pw (23)

d t

= Ë

d't

E

de

T ." d'r T

Tangential stress: T .Jo w

+ w (24)

ëit

=

2G

ëit

2G

where Pw and Tw' eN and eT are the normal and tangential stresses

E

and strain respectively. E is the elasticity modulus and G

=

2

the shear modulus of the material (Poisson coefficient v - 0) and T its relaxatiun time. The mechanical properties G, E and T of the ablation material are assumed to be constant in this analysis. A more detailed study should include variations of these quantities inside the material not only in the direction normal to its surface but also along i t because of the existing temperature variations in these directions.

It is shown ~n Appendix J that a Kelvin body behaves in a stabIe manner. whereas a large amplification of the wavy wall amplitude

~

exists for Maxwell bodies. Therefore, in the

following analysis. only Maxwell bodies will be considered. For the latter.one can write with the use of equ. (1) in a

dimen-sionless form. see Appendix

J.

'"

Normal stress: Pw-

lr

6

=

-~€ . " ' - - -a c T

'"

Tangential stress: T W

2'"

-= a € c T

(26)

where

!

is the dimensionless thickness of that part of the solid which. near the body surface. has a sufficiently high temperature and thereby a low elasticity modulus and viscosity to be affected by the normal stress es.

3. CALCULATION PROCEDURE

To start the calculation. the Mach number Mo and the e

Reynolds number Re based on conditions at the outer edge of e

the boundary layer must be given. The nature of the boundary layer must also be specified (laminar or turbulent). so that the velocity and Mach number profiles of the mean. anperturbed flow can be chosen. The parameters characterizing the geometry of the wavy wall must also be selected. ieee the wave number

ä

and the sweep angle

n

from which Moe _x and Ree _x can be derived.

see Fig. 2, Finally. the propagation velocity ë_R and the ampli-fiaation or damping factor

ë

I of the perturbation must be fixed. The first etep ~s to calculate by an iteration proce-dure. as indicated in Appendix H, the effective thickness of the viscous sublayer I f ' Here the values F(~ ) and F'(~ ) are

w w

obtained by a standard interpolation procedure and using the

1 8

-tabulation of Holstein • If cl

:f

0, ~w is complex and Appendix G describes a method to calculate the F-function and its deriva-tives for lëII « 1.

The second step ~s to solve the pressure perturbation equation

(6).

in the inviscid outer layer. Equation

(6).

together with boundary conditions

(7)

and

(8).

form a split boundary· value probler.,

A simple transformation of the dependent variabIe. similar to that in reference 14. changes (6), (7) and (8) into an initial value problem. Defining the new cariable

'U dP D

₌

~

_'U P transforms (6) into dD 2 dM Ox + D2 - D(--- --- M O -J.y x dy

Equation (28) is a first order non linear Ricatti equation in D •. which forms, together with

(7) :

At

y

=

1 'V dP

...&....

._(

2 D

=

'V = -~a Moe x p

an initial value problem.

_

)1/2

(1_c)2 - 1

Equations (28) and (29) can be decomposed into their real and imaginary parts and integrated simultaneously by a standard Runge-Kutta methode The calculation is started at

y

= 1 (outer edge of the boundary layer) and stopped at y :0: 61'

(outer edge of the viscous sublayer). The calculated value

D(6

_f ) gives, together with equation _{'V _}

(8),

the pressure

perturba~ion amplitude P at of;

or with (19) 'V P w

=

_2 9 2 Mo x

rrr

o

(30)

The third step is to calculate the wall shear pertur-bation from equations (20) and (31).

'V Lw 'V e: 2 mOe ~ 2 F"(f; ) w U"o(6 1' )6f Derf)

The last step is to evaluate the material p~operties of the Maxwell body from equations (25) and (26)

'"

Normal stress

--

E-T = (~)R

rw

-

1

6. E CL

_cl

'"

Ë-T" p 1 = -(..:!)

-

_6. '" _E I _CL

_cR

(34)

'"

T 1 2'G-T" = E-T =

("''W)R

-2~ E _CL

_cR

Tangential stress

'"

T 1 2'G-T" =

Ë-T"

= (~'W)I 2 -E _CL

Cr

with 'G-î-

=

~. where ~ ~s the di~ensionless viscosity of the solid_

4.

DISCUSSION AND RESULTS

It was shown in Appendix J that large amplification rates of the deformation occur only for the Maxwell type of material. Accordn.ng to equs. (33·) and (36) omly the real part

of the wall pressure and the imaginary part of the wall shear stress perturbation are responsible for the damping or the amplification df the perturbations. i.e.

- -

_{E T}

-r

1

(i'2

'U the wall amplitude E.

(38)

To gain a better physical insight. the real part of the wall pressure and the imaginary part of the wall shear stress per-turbation have been sketched in Fig.

4

for both cases

ë

I > 0 and

ë

_r

< O.

The following discussion will only be concerned with cases for which amplification of the perturbation occurs. ~.e.

ë

_I > O. The influence ~f the flow and wavy wall parameters on the terms on the left hand side of equs.

(37)

and (38) will be examined. These expressions represent essentially the dimension-less viscosity ; of the material

(Ë

T

=

20

T

=

2;)

and the

amplification factor

_{ë r}

calculated from either the wall pressure. equ. (37). or the wall shear stress perturbation. equ. (38).

Unless otherwise stated. the results are obtained f~r turbulent boundary layer profiles (N

=

7).

with a free stream Mach number of 2.0 and a free-stream Reynolds number based on the boundary layer thlckness Re

=

2.104 •

4.1

rnfluence of the amplification factor

ë

_r

and the perturbation propagation speed

ë

_R

Figures 5 and 6 show that the influence of cr on and 2 ~ ~

ë

_r is negligibly emaIl for values of

ë

_r < 10-4 ,

especially for increasing values of the wave number ~. Although the sweep angle

n

of the wavy wall was chosen such that the com-ponent of the Mach number perpendicular to the direction of the waves was unity, the results of Fig. 5 and 6 apply equally to other flow conditions and sweep angles.

To demonstrate,that ë_r

=

10-4 gives a considerably large amplification rate, the expression for the wavy wall coor-dinate will be used. From

(1)

or

m

Assuming a ~ I, UOe ~ 10 2 s and ê ~ 10- 3 m. the expression ineide the brackets of exp becomes for

ë

_r = 10- 4

exp (lOt)

with t in seconds.

8 ~"f-

.,.;-~-Figures

7

and show that ~ cr and 2u ~ cr are nearly

/ ) . .

.

.

constant for small values of the perturbat10n propagat10n veloc1ty

(ë

R < 10-4 ). rt was shown 1n Ref.

8

that the downstream propa-gation speed of the cross-hatched pattern was of the order of 1 mm/s, which g;ves, together with UOe ~ 10 2 ~, a value of CR ~ 10- 5 • Hence, one can conclude that this propagation speed

1S toa small to affect the theoretical results.

Based on the arguments mentioned above. the calcula-tions were carried out for sufficiently small values of cr and

ë

_R (~R

=

ë _r < 10- 4). Thus,it is also ensured that

ë

R «

Ü

o

(8

f ),

i .e. that the critical layer

(ë

_R

=

rro)

lies inside the ViSCOU5 Sublayer.

4.2 The wavy wall sweep angle

n

and the influence of the wave number a

-

_{E T -}

-In Fig.

9

and 10. ~ cl and 2G T cl are plotted versus the Mach number component MOe • which is the important

x

parameter changing with

n.

The Reynolds number Ree , which con-_x tains UOe. is only of minor importance.

n

was altered in such a way that Mo ex covered both the subsonic and supersonic regimes. As may be seen. for wave numbers

ä

=

0.1 - 2.0, both quantities

Ërr--r

cl and

iG'

T

ë

_r

have at

Ho

_ex = 1 their maximum values, which decrease with increaeing

ä.

For values of

ä

larger than 2.0, the maximum values remain approximately constant, but shift to higher Mach numbers ~Oe slightly larger than unity.

x

Figures 11 and 12, give the real part of the wall pressure and the imBginary part of the wall shear stress pertu~bation . ~s a function of the Mach number MOe •

x

ET

In Fig. 13 - - -

ë

_r

and

2rr

~

ë

_r

are plotted versus the F

wave number

ä

for Moe_x

=

1. Both quantities are almost inversely proportional to

ä

or nearly directly proportional to the wave-length À as shown 1n Fig. 14.

Although, strictly speaking , the linearized theory used in this analysis fails in the transonic range, the results are qualitatively valid,as shown below. For an irrotational flow the problem is singular for Moe

x

=

1.0 using a linearized theory. i.e. the wall pressure perturbation wou1d be infinite. But

applying boundary layer velocity and Mach number profiles, i.e. assuming rotational flows, the wall pressure perturbation calcu-lated at the outer edge of the viscous sublayer stays finite. The fact that its amplitude is maximum, but finite in the present analysis when Moe

=

1, is not due to the linearized theory.

x

This is shown in Ref. 22, where non linear terms are included. The amplitude of the longitudinal velocity perturbation, which is related to the pressure perturbation. is doubled for an in-crease of l>1o_{ex from 0.9 to 1.0 22}

The phase of the pressure perturbation with "respect to the wavy wall can be predicted with the present analysis. its amplitude 1S certainly overestimated in the transonic Mach

number range. Inger 10 has compared his theoretical results

" "

(linearized theory) with experiments. Inger measured the wall pressure perturbation on a wavy wall (

f

=

0.03) in the transonic regime (MOe

=

0.95-1.8). He concluded that the phase of the

x

pressure perturbation was predicted with reasonable accuracy (error at most 20°). but its amplitude was overestimated by the theory by a factor two.

4.3

The amplification factor cl

For constant values of a. ~oex' MOe and Ree one can write for ~I < 10-4 :

ËT'

_'V 1

1:

_cl and

(40)

2G

T

' V _ 1 cl

Equation (40) indicates that neutral stabi1ity

(ë

_r = 0) occurs for ~ ~ and

G T

equal to infinity. Hence. one can conclude that perfectly elastic

(T

=

m) and rigid materials

(Ë

=

G

=

m) are neutrally stabIe. On the other hand. for increasing amplification rates cl'

Ë

Tand

G

T get smaller. which may be physically ob-vious. see also Figs. 15 and 16.

4.4

Influence of the type of boundary layer

Figures 17 and 18 show the influence of the type of

.

ET-

-boundary layer on the express10ns ~ cr and 2G T cr" For the

N

=

7 power law velocity profile (turbulent profile) both quan-tities are the largest in the subsonic to slightly supersonic velocity range.

For the N

=

3 velocity profiles, the curves are similar to those for N

=

7,

they have their maximum at MOe _x

=

1 for

ä

< 2. For the laminar profile (N

=

1) both quantities in-crease linearly with MOe and they,are about 20 times smaller

x

than for the turbulent profile for Mo_ex

=

l.O.

4.5

Influence of the Mach number MO _e and the Reynolds number Ree

The influence of the Mach number outside of the

boun-ËT-

- - - .

. .

dary layer M Oe on

-=-

cl and 2G

'r

cr ~6 shown ~n F~g. 19 and 20

6. •

for MOe

=

1. Assum~ng constant stagnat~on pressure and

temper-x P e"V 0 e"Ó

ature. the dependenee of the Reynolds number Re

=

on

e lJ

e

MO was taken into account. As may be seen from Fig. 19 and 20 e

an increase ~n MO has the main effect of lowering the quantity

ËT

e

ë

_r but i t increases the quantity

2G T

ë

_r • For increasing

ËT-values of Re , for otherwise constant conditions. ~ cr and

e ~

2G

T

ë

_r decrease, the latter at a larger rate, as shown in

5.

COMPARISON WITH EXPERIMENTS

It was shown that large amplification rates can occur only for the Maxwell type of inelastic material. This is in agreement with the expe~imental findings of Ref.

8.

Ablation materials such as wax and camphor belong to this category 8

The calculat~ons

.

show that the quant~t~es ~

. .

ËT-

cl and

6.

2~ ~

ë

_I reach their maximum value for wave numbers

ä

< 2.0 when

the Mach number component perpendicular to the direction of the

w~ves of the wall is equal to unity. This confirms the

experi-mental observations that 1. the Mach number at the outer edge of the boundary layer must be superson~c and 2. the cant angle ~

of the cross.-hatched pattern is nearly equal to the Mach angle

( i.

e. the flow_ p~rpendicular to the grooves is transonic). The experiments of Ref.

5

and

8

indicate that indeed

ä

< 2.0. In

Fig. 23, the wave number

ä

=

aö is plotted versus the Mach number MO , obtained from the experiments 5,8. As may be seen,

ä

~

0.5

e

for the whole Mach number range which was tested. The boundary layer thickness ö was calculated at the location xl (x being the distance from the apex of the cones) where the cross-hatched

• 2'IT •

pattern started. a 1.S equal to

T'

"rhere À 1.S the wavelength

of the pattern taken normally to the direction of the groovee. Although the boundary layer thickness ö grows in the downstream

- 2'ITÓ

direction x, i t is s t i l l believed that a ~ aö

=

-r-

~ 2.0 for the total body region which is covered with cross-hatching. The distance from xl to the base of the cones is not large (~ xl). furthermore, À grows slightly in downstream direction 2,5. The change in the ·Reynolds number Re due to the increase of ó has

Ë

T

e

only a minor effect on cl and

2G

T

ë

_I , see Chapter

4.5.

t:

E T -

-It was shown that both quantities cl and 2G T cl

6.

are about 20 times larger for turbulent than for laminar boundary layers for MOe

=

l.O. Although the theory predicts instability

x

for laminar boundary layers, the amplification rates are very smalle Thus, the experimental observation that the boundary layer must be turbulent for cross-hatching to exist is evidenced.

Figure 14 demonstrates that one can write approximately for Moe_x

=

1.0

~'T'--

cr 'V _À 6 ₍₄₁₎ 2G _{T cr} 'V _À or with cr cr =

0;-e Ë = E ypo

tr

= G yPO T" T.U oe = Ö À =

ll.

ö Equation (41 ) gives E T cr 'V À Po

-

₆ (42) 2G T cr 'V _À Po AS3uming stant thickness

-

6,

a constant amplification factor cr and a con-equations (42) indicate that the wavelength À

increases in proportion ~o the viscósitY\~fthe material and is inversely proportional to the static p~essure pO. Both findings have been proved experimentally 3,5,8. Furthermore, equations (42) show that À is independent of the boundary layer thickness ó, i.e.

À is independent of the downstream direction x. The experiments indicate a slight increase of À with x 5,8.

6.

CONCLUSIONS

The following theoretical results are in agreement with the experimental findings :

1. The boundary layer must be supersonic.

2. The boundary la~er must be turbulent or at least transitional. 3. Cross-hatched patterns develop more readily on body materials

of the Maxwell type.

4.

The Mach number component perpendicular to the direction of the grooves is unity.

5.

The pattern wavelength or cell size is inversely proportional to the statie pressure.

6.

Theoretically the pattern wavelength does not change with the downstream direction. the experiments show only a slight

1ncrease.

1.

The waveleneth of the cross-hatched pattern increases in direct proportion to the viscosity of the material.

Although the agreement between the results of the

present theory and the experiments is considerabl~. the analysis fails to predict the waveleneth À.

REFERENCES

1. LARSON. H.K., MATEER. G.G.: Cross-hatching - A coupling of gas dynamics with the ablation process.

AIAA Paper 68-670.

2. LAGAUELLI, A.L •• NESTLER, D.E.: Surface ablation patterns A phenomenology study.

AIAA Paper 68-671.

3. WILLIAMS, E.P.: Experimental studies of ablation surface patterns and resulting roll torque.

AIAA Paper 69-180.

4. McDEVITT,

J.B.:

An exploratory study of the roll behaviour of ablating cones.

J.

Spacecraft and Rockets, Vol. 8, No 2, Feb? 1971, pp 161.

5. STOCK, H.W., GINOUX,

J.J.:

Hypersonic low temperature ablation-An experimental study of cross-hatched surface patterns VKI TN 64. July 1971.

6. STOCK. H.W., GINOUX,

J.J.:

Experimental results on cross-hatched ablation patterns.

A I AA

J.,

Vol. 9, No

5,

M ay 1971. p. 97 1 •

7. STOCK. H.W., GINOUX,

J.J.:

Further experimental studies of cross-hatching.

AIAA J. Vol. 10, No 4, April 1972, p. 557.

8. STOCK. H.W., GODARD, M.: Cross-hatching - A comparison between the behaviour of liquefying and subliming materiaIs. VKI TN 8;" 1973.

9. TOBAK, M.: Hypothesis for the origin of cross-hatching. A I AA

J.,

Vol. 8. No 2. Feb. 19 70, p. 3 30 •

10. INGER, G.R.: Compressible boundary layer flow past a swept wavy wal1 with heat transfer and ablation.

Astronautica Acta, Vol. 16, No 6, 1971, p. 325. 11. NACHTSHEIM. P.R.: Stability of cross-hatched wave patterns

in thin liquid films adjacent to supersonic streams. The Physics of Fluids. Vol. 13, No 10, Octo~ 1970, p. 2432.

12. PROBSTEIN, ~.F., GOLD, H.: Cross-hatching: A material response phenomena.

A I AA

J.,

Vol. 8, No 2, F eb. 1970, p. 364.

13. STOCK, H.W.: Role of the anelastic behaviour of the ablation material on cross-hatching.

14. GOLD, H., PROBSTEIN, R.F., SCULLEN, R.: Inelastic deformation and cross-hatching.

AIAA Paper 70-168.

15. BENJAMIN, T.B.: Shearing flow over a wavy boundary.

J. Fluid Mechanics, Vol. 6, Part 2, Aug. 1959, p. 161. 16. LIGHTHILL, M.J.: Reflection at a laminar boundary layer of

a weak steady disturbance to a supersonic stream, neglecting viscosity and heat conduction.

Quart. J. Mech. Appl.Mech. l I l , 1950, p. 302. 17. DUNN, D.W., LIN, C.C.: On the stability of the laminar

boundary layer in a compressible fluid.

J. Aero.Sci., Vol. 22, No 7, July 1955, p. 455. 18. HOLSTEIN, H.: Uber die äU3sere und innere Reibungsschicht

bei Störungen laminarer Strömungen.

ZAMM, Vol. 30, No 1/2, Jan./Feb. 1950, p. 25.

19. LIGHTHILL, M.J.: On boundary layers and upstream influence 11. Supersonic flows without separation.

Proc. Royal Soc., Vol. l27A, 1953, p. 478.

20. SCHLICHTING, H.: Zur Entstehung der Turbulenz bei der

Plattenströmung.

Nachr. Ges.Wiss.Gött. Math.-Phys., Kl. 2, 1933, p. 181.

21. FREUDENTHAL, A.M.: The inelastic behaviour of engineering materials and structures.

Wiley, New York, 1950.

22. SICHEL,

M.,

YIN Y.K.: Viscous transonic flow past a wavy wall. Fluid Dynamic Transact. Vol. 4, pp 403-414.

Institute of Fundamental Technical Research, Polish A cademy 0 f Sci., Wars ·aw.

T5T ,. '07 oK '_{T5T = 397} TST = 399

PST • 32.0 kgf/cm2 PST = 30.' k9:f/cm2 PST ,. 30.~ k9_f/cm2

t = 13 "sec t ,. 11 se-c t ,. 12 sec

ex: =00 (ANGLE OF ATTACK) ex: ,. 0° (ANGLE OF ATT ACKI ex: = 10° (ANGLE OF ATT ACK)

e

= 10° (TOtAl CONE ANGLE)

a

" 28° (TOTAL CONE ANGLE)

e" = 24° (TOTAL FLARE ANGL,E)

Q

Pattern sweep angle

(Q

=

90

-4> )

y

MOe

Ma

(y)

~-1~

.

(/)

w

0 - l I:::>

u..

0

~

a::

0 .~

-

CL

E

11 oZ

.

>-

_...

-.J I.!)

.

U 0

9

W

>

--IZ

...

Z I~

...,

«

w

IJ ~

l:f

M

.

Cl

.

-0 lL. 0

-

I~ 1.0

_d

0

(PW)R

E

(PW)R

Ë

(a) cl

>

0

(unstabIe)

(1w)

I

€

(tw)

I

Ë

(PW)R

Ë

(PW)R(1W)I

€

Ë

(fSw)

R

E

(b)

ë

I

<

0

(stabIe)

(1W) I

r

Fig.4 DISTRIBUTION OF THE REAl PART OF THE WAll PRESSURE AND THE IMAGINARY PART OF

THE WAll SHEAR STRESS PERTURBATION FOR WHICH AMPLIFICATION (a) OR DAMPING (b)

ET

,c

I

II

5.0

o

10-

8

Fig.5

,

_{MOe = 2.0}

, ,

r

~

ä=0.3

Moex = 1.0

i

I

I

I

/

!

K

Ree

= 2xl04

. - ~.-_.

!

/

--- -- ----T

_~

_N

₌

₇ :

I

/

.

~

i ! ,

,

i

~\

I I

----

-t-

---

---

--

I , _{/ I I} , ! !

I

i

i

:

I ,

I

I

'

~

I

/

I

;

:

/1.0

I

:

i - - -

-M

I

I

----,

i _, I /

I

/3.0

! _i ,

I

I

10-

6

_10-

4

Cl

10-2

ET

'CI

VERSUS THE AMPLIFICATION FACTOR

Cl .

- - - -- ---- ---" ..

_ -

Moex = 1.0

2GTë

_I

Ree = 2xl04

N

₌

7

I 1

0.2

--- ----_ .. _-

,

< - - - -,_ .. _---

L

_. - .- _.- - -_.-_. - -

_

..• I I

I

I

I

I ---t- -I

0.1

i 3.0

o

10-

8

_10-

6

_10-

4

Cl

10-

2

Fig.6 2GTë

ET

--==-·c

_I

~

5.0

o

10-

8

I

I

~

_~

I

I I

I

I

10-

6

_10-

4

I

_I MOe

=

2.0

i

Moex

=

1.0

!

I

Re

=

2xl04

~

!

N = 7

~=0.3

I

~

i

.

i

~

!

~

.~

:

/3.0

~

I

-CR

10-

2

Fig.7

ET

.ë

I

VERSUS THE PROPAGATION SPEED OF THE PERTURBATIONS

CR.

2GTë

I

0.2

-~

I _{MOe = 2.0}

I

! I

i

Moex

=

1. 0

+-

---t---

-..

._._

-<X.

=

0.3 _{Ree = 2x104}

I

I

_I

I

I

_{N = 7}

I

I

I

.

I

I

I

I

I

i

i I I i , !

!

I

I

I I

I

I

I

i

i

I

t

I

t---·_-

I . I I

I

!

I

J

1\

i

J ! I J

-

i

I

i I I

---0---+

--_.- -

- ---

_.

r

'

---

-

---

_I i

I

I I : I

I

I

I

I

I

i I I I ! I i ,

I

I

i

i ! _I

/1.0

I

i 1 ._. I _-

-I

0.1

I

I

I

/3.0

o

10-

8

10-

6

10-

4

CR

10-

2

ET . cl

11

20.0

10.0

I

I

t/n:o.l

/0.2

V

/0.3

/ 0.5

!J

~

J

2.0

~

... '-~

-.

MOe :

2.0

Ree : 2xlO'

N

=

7

--" - -- ---f---- --- - - --- ----_._--- --- f--~~-- _-

---o

0.8

1.0

1.2

1.4

_MOex

Fig.9

~

T .

cl

_{VERSUS THE MACH NUMBER COMPONENT Moex '}

~

-

~ 0 0 N

-

) ( t"--N 11 11 11

z

0

-~ ei 11 CD Id ei

-

-t-0 0 0 N ('I") ~

' t

-.

o

UJ

::>

Z ~ Z

o

u

Ol Ol

u...

2GTcI

1.0

0.5

o _

0.8

!

i

,---

-

-r I - .-

----+

--

--t-1-- -

-i

I

I

I I I r ! i

t

--

_._._-._- -_._ -+-I VO.2 . 1-•

J

VO.3 .

~

VO. S

k::,1.0

2

.

0

1.0

1.2

I . -- -.

--1.-

--

_- -. --

---_ . i MOe

₌

2.0

i

Ree

=

2xl0 4

N

7

!

=

i

- - t - - - - -- --

-

-

---

-

-i i i r I

I

I

1.4

_MOex

Fig.l0 2GTcI VERSUS THE MACH NUMBER COMPONENT Moex ·

-I

I

~ ~ 0 0 N x t'--N 11 11 11

...,.

·

-Z

~----

-

I

! N

·

-I I

I

I

_~

-

__

__

+_

____

_

J

:

~

i

w

ï-

-

-~

-o

I

d

0 0

·

- 0

W

I

! I I I I

::::>

z

t=

--+-

--I

z

0 U

I

0

-LO

-

-

0 COOl

-

Ln

diL

0 0 0 IU

.

I~ 0 Ó

K!>

N

I

MOe

=

2.0

I Ree

=

2xl0

4

(Pw)

R

I - - -....

--Ë

2.0

1.0

I 7 ' IA

o

0.8

N = 7

!

L

-.---.----.

I

I

I , .. - _. -+----. __ . I -'or.

1.0

1.2

1.4

_Moex

1.6

Fig.l1 THE REAL PART OF THE WALL PRESSURE PERTURBATION VERSUS

-,

--

-

-r-

11

- ,

_

-\t-\J\.-

_L

1

; ltS

I

I

~---<-I

I

- - - J . -' -

I

-tt----IL- I N

.

-~ 0 0 N )( c--. N 0 11 11 11

-

.

Z 0

w

:::>

z

...

z

0

u

--

.

a::

0

-

_IlllW

Ó

co

,Ol 0

-

Lri

Ó

u..

(Ew)

I

E

~ _ _ _ _ _ +-_ _ _ -j!~_+-___ '-= _----1 __ _

0.041

J n ...

"

.0

MOe Ree

N

=

2.0

=

2xl0 4

=

7

~---~I ~ ~--

-0.02

1 1 / I 11

j

-~~--~~-

----o

0.8

1.

0

1.2

1.4

Mo

_ex

1.6

Fig. 12

THE IMAGINARY PART OF THE WALL SHEAR STRESS PERTURBATION

-i

t----

-~-~

~--\---\-

~

I

~---+---+

---ET

.ë

l

~

,

-- . -- --- -

~

----

-

-­

i

I

_I

I

I

.

r-I

I

-

t-

I I I MOe

=

2

.

0

Moex

=

1.0 Ree =

2xt0

4

N

= 7

- ---11

2

G

T

cl

5.0 I

\{

f"....:

10.25

o

I

I

.

I

I

I

0

o

0.5

1.0

1.5

a.

2.0

Fig. 13

ET

.c

I

AND 2GTë

I

VERSUS THE WAVE NUMBER

a..

ET

eI

II

- + -- - 7' , I

2GT~I

5.0 1

I

/ : / " =

10.25

MOe

=

2.0

I I 7' I Moex

=

1.0 Ree

=

2x104 N

=

7 Ol~

I

I

I

I

I

10

o

10

20

30

40

À

50

60

Fig. 14

ET.

cI

AND 2GTë

I

VERSUS THE WAVELENGTH

À.

-~ I - - - _ . ₀

-_ N ( ' t ' )

.

_.

_.

_{0 0} 0 0 0 ('t')

I~

l

L

I

-~ _,

. ' ,

Iu

"

"

'"

' . '

ct:

~~

₀

t -U

~

Z

0

~

U ~

u..

0 0 0 N ) ( t"--(0 --l N I ~ 11 11 11 11 0 ~

-)( <{ QI QI QI Z 0 0 QI ~ ~

ex:

_W I t -(f)

::::>

(f)

ct:

W

>

It-II<J

IW

Ln

-

.

00 I

en

0 Î.L.

-ct) ~ 0 0

II-II<J

0 0

-

IUJ

-

-MOe

=

2.0 Moex

=

1.0

2GT

0.2

0

.

3

l.:.: -::::0:"""" -::::0:"""" -::::0:"""" ~ /

4

"""'" '""""" ---= --...: Ree

=

2xl0

N

=

7

10

4

I

-::::0:"""" ... -::::000""", ~-...:::-- 7~

=

=: ...

=:::~=t==~~::::::~~;:::~~~:-.::::-~---...:~

_{...: T""'--...:}

10.0

10

0 , ! !

10-

8

_{1 0 -}

6

_{1 0-}

4

ë

l

Fig.16

2GT VERSUS THE AMPLIFICATION FACTOR

Cl

ET

.c

I

!!l

6.0

4.0

2.0

o

MOe Ree

a

= 2.0

= 2x 104

-~' I - --- -_._.- - ---I

= 0.3

~

N=7

-~ I I

!

/1

0.5

0.6

0.7

0.8

0

.

9

1.0

Mo

_ex

1.1

1.2

Fig.17 .tL'C

I

VERSUS THE MACH NUMBER COMPONENT Moex FOR DIFFERENT

·._----~ 0 N IU 0

.

-It-I _{IW l<l}

-0') Ó

co

Ó __ -+-~ 11 I:' Ó (,0 Ó Lr) Ó 0

0

w

:::::>

z

I -Z

0

U t"

-Cl

.-u..

0

.

IU--

lt)

.

0

.

It-I

IUJ 1<3

--

-H~

__

lo

-_ \--_ _ 11 0')

ó

0 CX)

d

I:' 0 CD 0 lt)

ó

.

Cl

w

:::::>

Z

i=

Z

0

u

I:'

-.21

U.

--- ---

---i-I

MOe

:: 2

.

0

I

0.2

Ree -=

2xl0'

I

_ä:

_{:: 0.3}

I

0.1

/

V

N

::

7

3

.

~

~

/~

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

\

~

o

0.5

0.6

0.7

0.8

0.9

1.0

_{1.1 Moex}

1.2

-IU I~ I~ N ! I I 1

I

, I

...s

o

.

o

t'--11 ('f')

~

>---

---..

~

0 N 11 41 0 ~ -4' 0

-

q ) ( .-N 11 11 :'IÖ ~ ('I')

o

_.

o

\

N

o

.

o

J

/v

V

/

.-'"

\

1

\

\

\

,

~

o

.

o

o

-o

-C1> Ó ex:> Ó

o

w

::::>

Z

<.0

r-Ó

Z

o

u

LO

d

ex)

-

_Ol

u..

~ 0 0 N >< q N ('I'l 11 11 11 41 : I Ö 0 ~ et::

-o

.

-0')

ó

o

w

:::>

Z

(Q ...

o Z

o

u

---l~~---I~~~~~-L---~~---~~~L---~U) ~

_Ó

_{IC> 0 Ó IL} N

C?

o

o

Ëf

,c

I

I:l

5.0

o

2.0

ei:

0.3 - ---_.-.-. - - - -

-3.0

3.0

4.0

Fig. 19

ET

'C

I

VERSUS THE MACH NUMBER MOe.

I:l

Moex :

1.0

Ree : 2xlO' (FOR MOe: 2.0) N :

7

5.0

_MOe

6.0