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Produced by the NASA Center for Aerospace Information (CASI)
r^

(NASA-CN- 175338) SELF-Sl1STAINED N84- 161 31

USt;ILLAil')NS OF A SHUCK WAVE INIE&ACTING

WITH A IiOUNoAi^Y LAYLE UN A SUPERCRITICAL

AIRFOT'- Final i+epoct, Juu. 1981 - Dec. 1983 Uncias

(BosCoa Un1Y. 1 47 U HC A03/MF A) 1 CSCL O lA G3/01 00529

f

^^ Aw
Boston University

College of Engineering

Department of Aerospace & Mechenicaal Engineering

110 Cummington Street

Boston, MA 02215

NASA Grant No. NAG 2-179

Grant Title: Analytical Investigation of the Influence of

Transonic Buffeting on the noise and Unsteady

Loading of Helicopter Rotor Blades

Principal Investigators: C. S. Ventres (Boston University)

M. S. Howe (Bolt, Beranek & Newman, Inc.)

Final Report

.Tune 1982 - December 1983

Report Title:

SELF-SUSTAINED OSCILLATIONS OF

A SHOCK WAVE INTERACTING WITH A

BOUNDARY LAYER ON A SUPERCRITICAL

AIRFOIL

Authors: C. S. Ventres

Department of Aerospace & Mechanical Engineering

Boston University

110 Cumnington Street

Boston, MA 02215

M. S. Howe

Bolt, Beranek & Newman, Inc.

10 Moulton Street

Cambridge, MA 02238

Permanent address:

Faculty of Mathematical Studies

University of Southampton

Southampton, England S09 5NH

9
Acknowledgement

This work was t'ur,ded by the Ades Research Center of the

National Aeronautics & Space Administration under, NASA Grant No.

NAG 2-179. Dr. Charles Stith acted as technical monitor.

i
(i)

Summary

A theory is proposed of the self-sustaining oscillations

00 n weak shock on an airfoil in steady, transonic flow. The

interaction of the shock with the boundary layer on thn airfoil produces

displacement thickness fluctuations which convect downstream and generate

sound by interaction with the trailing edge. A feed-back loop is

established when this sound impinges on the shock wal7e, resulting in the

production of further fluctuations in the displacement thickness. The

details are worked out for an idealized mean boundary layer velocity

profile, but strong support for the basic hypotheses of the theory is

provided by a comparison with recent experiments involving the generation

of acoustic 'tor,ie bursts' by a supere itical airfoil secti=.
ORIGINAL PAGE 15

OF POOR QUALITY Table of Contents

Pages

P'

1, Introduction 1.1 - 1.3

2. Formulation of the model problem 2.1 - 2.10

Boundary conditons at the airfoil and wake 2.:t

Modelling the interaction of the shock and

boundary layer 2.4

Boundary value problem for 2.8

M

i 3. Solution of the boundary value problem 3.1 - 3.5

Boundary value problem for ty l 3.2

Boundary value problem for *2 3.3

y

The integral equation for T(A) 3.4

4. The characteristic frequencies of self-sustained

asCi1latxoas 4.1 - 4.6

5. Discussion of the Characteristic Equation 5.1 - 5.5 

Numerical Results 5.4

6. Conclusion 6.1

Appendices 1 and 2 Al - A4

References Rl - R2

Figures 1-6 F1 - F7

Figure Captions F8
ONIGINAL PAGE d.9

H. introduction ction OF POOR QUALITY

The production of sound by helicopter rotor blades having

transonic tip speeds is sitroaigly influenced by the presence of shock

waves near the blade tips (see, e.g., refs. [1-3]). Acoustic waves

are generated by (i) the interaction of the shock with inflow turbulence

and/or the trailing vortices of the other rotor blades, and (ii) the

intensification of the nonlinear quadrupole volume sources which occur

in the Lighthill theory of aerodynamic sound [4]. In this paper we

discuss a possible additional mechanism associated with an instability

of the shock caused by its interaction with the boundary layer on the

blade. This interaction is strongly dependent on the Reynolds number

when the flow is transonic [5-7], the position of the shock being

extremely sensitive to temporal variations in the properties of the

boundary layer.

gucci et al have reported [8] the occurence of intense, high

frequency and highly directional tone bursts during acoustic tests of a

scale model of a general aviation propeller operating at high subsonic tip 

speeds. The amplitude of the burst was of the same order as the propeller

noise, and observation strongly suggested that the tones were produced by

oscillating shock waves an the blades. The phenomenon was absent if flow

separation occured at the blade tips and if the boundary layers ahead of

the shocks Were made turbulent; this is presumably a particular manifestation of transonic buffeting [9]. The extreme sensitivity of the shock to

perturbations in the ambient flow is also illustrated by the experiments

of Tijdeman [ld] using an airfoil with an oscillating flap. These

demonstrate that under certain conditions the shock leaves the airfoil

and propogates as an intense acoustic wave to the far field, an effect

which is in qualitative agreement with theoretical work of Williams 1111.

Similarly, Magnus and Y'oshihara [12] argue that discrepancies between

1.1 
^^ v

4^^JQ

v

C7 O i

C) Gc,

I

O

^a

experiment and their numerical predictions of oration caused by a pitching

airfoil in transonic flow are probably a result of the strong interaction of

a shock wave with the boundary layer.

An idealized transonic flat problem is investigated below in

order to examine a mechanism of shock wave/boundary layer interaction

which is possibly responsible for the intenee acoustic fields observed by

Succi at al [8]. The analysis proceeds from the unsteady, transonic flaw

equations which are linearized about a steady mean flow with account taken

of the displacement of the shock, in the manner described by Williams [11].

The model problem is illustrated schematically in Figure 1. A twodimensional flat-plate airfoil is placed at zero angle of attack to a mean

" flow. Above the airfoil the mean flow is uniform and supersonic ahead of

^i

a weak shock wave which in the undisturbed state f , assumed to be normal

to the airfoil and of unlimited extent. Time harmonic oscillations of

the shock about its meen position are considere •:,^; At the root of the

shock the motion induc'; a fluctuations in the i;ispiacement thickness of the

boundary layer which propagate downstream as a surface wave on the airfoil.

At the trailing edge this wave feeds into an unsteady wake and is

responsible for the production of edge-generated sound which subsequently

interacts with the shock thereby inducing further fluctuations in the

ri boundary layer. Self-sustaining oscillations are possible provided the

returning sound waves are in an appropriate phase relation with the motion

of the shock, and are of sufficient amplitude. These conditions determine

`F a preferred wavenumber for the boundary layer waves, and our objective is

^h

'., to determine the frequency of the oscillations in terns of the characteristic

N

of the mean boundary layer.

The anal, ^tical problem has similarities with that investigated

recently by Goldstein et al [13], concerning the instability of shocks

of arbitrary strength in cascades, although no account was taken of

L5

Y1 shock wave/boundary layer

1.2i

LL. 
is assumed to be sufficiently weak that the ,generation of vortici,ty and

r

entropy by its motion can be neglected. The unsteady flow in the main

e

^ stream behind the shock may accordingly be expressed in terms of a

^' 4

velocity potential At the outer edge of the boundary layer the

,,^ Ck normal derivative of equals the displacement velocity of the

Q^

unsteady boundary layer. Application of this condition is simplified

e

by means of an approximation analogous to that used in thin airfoil

theory [14], which permits It to be imposed on the upper surface of the

i

airfoil (c.f. [15-171).

The principal difficulty in formulating the problem is the

modelling of the production of the displacement thickness waves. The

I^

F approach adopted here is based on the hypothesis that the functional

4t y form for the transition in the boudary layer structure across the shock

is invariant in a frame of reference fixed with respect to the root of the

shock. This assumption of quasi-static behaviour at the root permits the

lids, unsteady motion in the boundary layer to be determined in a linearized

approximation provided the mean properties of the boundary layer are known.

r 

Results are given here only for the highly simplified case in which the

mean velocity profile is approximated by a step function, a procedure which

has been successfully exploited by Ffowca Williams and Purshouse [18] and

Goldstein [19] in analytical studies involving unsteady boundary layers.

Evidently this approach is also applicable to other shock wove /boundary 

r

r

layer interaction problems. For example, it provides an excellent starting

point for investigating the interaction of blade-tip generated shocks of a 

r^

ducted rotor with the boundary layer on the walls of the duct.

r'

The basis of the model problem is discussed, formulated analytically and

w'

wj solved in 562,3. In 64 the equation for the characteristic frequency of

the self-sustaining oscillations is obtained; numerical results given in

j ^ 15 aie examined in relation to the experiment of Succi at al CB^.

1.3'
ORIGINAL 

52. Formulation of the model problem OF POOR QUALITY

A two-dimensional rigid airfoil occupies the portion

- C < x l < p of the xl-axis if a rectangular coordinate system (xl,x2),

in the presence of a Dean flaw in the positive xh"direction. In the

i 

undisturbed state a normal shock wave extends from the upper surface of

the airfoil at xi W -f, to x2 w 4•• (Figure 2(a)). Attention is confined

to the weak-shock/transonic regime in which the speed U of the main flow

downstream of the shock is constant and

U M 02/ r (1)

where c is the speed of sound, and S ti rj__M_2" << 1 The RankineHugoniot relations applied to a weak shock imply that the main flow

Mach number M- , say, upstream of the shock is given by

f

Mr " l + 822 (2)

' Motion of the shock wave produces fluctuations in the boundary

layer and wake which in turn react back on the shock. It is assumed that 

all such back-reactions occur via acoustic paths intersecting the shock

from behind, i.e., possible interaction channels involving with the passage }

of round around the leading edge of the airf oil, after propagation through

the subsonic flow below the airfoil, are ignored. Similarly, the transj mission of acoustic disturbance into the supersonic region via the subsonic

boundary layer is neglected.

Consider a time-harmonic perturbation proportional to a iwt^

Let the instantaneous position of the shock be represented by

xl r 

t + z(x2)e"iwt

(3)

where z(x2 ) is assumed to be sufficiently small that the perturbation

equations may be linearized. For a weak shock the motion in the main

flow above the airfoil is described by a velocity potential m iwt 0

in terms of which the perturbation velocity v . 00 (the exponential time

2.1
ix II(xl) 1 2

2.2

^' I

G^

4

(8)

factor is bars and henceforth suppressed). In a linearized approximation

satisfies

2

ORIGINAL PA^'^ ^^ — ik0 + M W - - — ,^ 4 * 0 ^ (4) OF POOR QUALITY 1 ax12 ;x22

in the subsonic region x1 > - l + O(x2 ) # whereY.o ^ W/c .

In the supersorAic flow ahead of the shock the notion is steady, and

4

0 (x1 < - t + t(x2 )) (S) i

The potential must also satisfy the following conditions at

the undisturbed location xl - t of the shock:

+ 2iw a 0 :

(6) axl Oat

ca_

where it is understood that these conditions are to be satisfied by f and

ao/ax 1 as x1 - t from the downstream region. The derivatiaas of

these formulae are given by Williams 11,20].

In the ( shock free) main flow region below the airfoil the

motion is assumed to be subsonic everywhere and f is required to satisfy

equation (4).

:r Boundary conditions at the airfoil and wake

^i 4

Let vn (xl ) denote the normal (i.e. x2 -) component of velocity

V on the surface x2 0 6 lying just outside the boundary layer on the upper

surface of the airfoil, 6 being the boundary layer thickness; this will

be referred to as the boundary layer displacement velocity. On x2 n 6

y must satisfy
OF POOR QUALITY

In order to make the subsequent analysis tractable, this condition

• is imposed on x2 • + 0 , an approximation which is a4 acted to be

valid provided the length scale ti U/w of boundary layer disturbances

is large relative to 6. Boundary layer displacement fluctuations are

' assumed to be absent on the lower surface (x2 - 0) of the airfoil,

where it is accordingly required that

a.0 0 

<xl < 0' x2 • -0 (9)

2

We shall actually apply this condition over the semi-infinite interval

- » < x1 < 0 . This will ensure that acoustic disturbances cannot

impinge on the shock from the supersonic region, and avoids difficulties

(which cannot be incorpozated .'%,0o the prevent idealised model) arising

from the fact that the mean flow ahead of the shock must actually vary

with position

Similarly, if wh /U <c 1 , where h characterises the mean

thickness of the wake downstream of the trailing edge, the perturbation

pressure

p ' p0 (iw - U ax) e . (10)

1

00 being the mean density, taken to be constant), may be assumed to be

continuous across the wake. In the usual approximation of thin airfoil

theory [14], this condition may be imposed on the centre-line x2 a 00

X > 0 of the wake, and implies that

iwx /U

[#] 1

(xl > 0) .

where W - O(x l , + 0) - O(x l , - 0) defines the discontinuity in #

across the x l-axis. The value of the constant A is a measure of the

strength of vorticity shed from the trailing edge, and is determined by

the requirement that the perturbation pressure and velocity should remain

finite at the edge (Kutta condition).

(11)

^ a
q ^Ca^ Qq^ A non-zero value of • A is associated with the presence of

an asymmetric ( sinuous) disturbance in the wake. Symmimetric "breathing"

modes are alr ► possib"lj^x # however. When mh/U is small the pressure

is continuous for such modes * but the normal velocity exhibits a simple

discontinuity across x2 0, x l > 0 In this case the phase velocity

is equal to the minimum mean velocity Urn , say, in the wake [21]. and

if Oct w Umsin /w , the breathing node satieffes

Ox ` Be 

Usxl (xi > 0) ^ (12)

2

Where B is a constant. The value of B is obtained by requiring that

there be no net flux of fluid from the Wake and boundary layer. This

condition is derived from the equation of coatin-dity on the basis that the

characteristic acoustic wavelength is large relative to the thickness of

both the wake and the boundary layer. Thus in the absence of displacement

velocity fluctuations on the lower surface of the airfoil, B is related

to the displacement velocity vn by

(^ 

1 vn(X

r iK x

l)dxl t J Be s 1 dxl 0 (13)

-C 0

Convergence of the second of these integrals is assured by assigning to

W - UK S a small positive imaginary component which is subsequently allowed

I

to vanish. The validity of this procedure is a consequence of the causality

condition which requires all field variables to be regular for sufficiently

large and positive Im w The disturbances in the Crake arise as a result

of the displacement velocity fluctuations in the boundary layer and of the

motion of the shock. Hence

B iKS J n(xi )dxi

(14'

-C

2.4
ORIGINAL PAGE 13

OF POOR QUALITY Modelling the interaction of the shock and bound&;y

1xyRr

Oscillatory motion of the shock a frequency w generates

displacement velocity fluctuations whose specification according to linear

theory depends on & ^4nowledge of the mean boundary layer velocity profile.

M

We consider here only the highly idealised velocity profile illustrated in

Figure 2(b), in which the Dean shear in the outer region of the

boundary layer is concentrated into a vortex sheet at a stand-off

'

I

distance 6 from the airfoil. If the flow in the inner region

A < x2 < 6 has speed V in the x,-direction, the vortex sheet can

iK x

a support displacement velocity waves proportional to e 1 , where

t,

i U - 1, w6 }

'^,( K^ ^^ 1 t V 1 V

l

(15)

L

7 u2 9 - V

n- ^+ - u)

L

Zl

I

J

D

^

I

lri:.vided w6/V is small, i.e., that the wavelength is large relative to 6

(c. f. ref. [16] , 53) .

yMM

In the undisturbed state there will exist a length scale d

determining the distance in the xl-direction over which the mean properties

of the boundary layer change across the shock. When oscillations occur

at frequency w it is anticipated that at distances exceeding d downstream

of the shock the unsteady motion in the boundary layer consists of a linear

combination of the two displacement velocity waves determined by the

w

Y

dispersion equation ( 15). This notion leads to the quasi-static

K

representation of the interaction of the boundary layer and shock described

below.

In the steady state the boundary layer thickness 6 increases

rapidly with x across the shock., and we write

6 0 60F(xl + 1) (16)

2.5
ORIGINAL PAGE IS

' OF POOR QUALITY

and assume that

F(x1+b) ^, for sl + f + - d

r %1+61/60 for x,+f;d

so that d - 60 , 60 + 6; respectively in tho aygions upstream and

dowastresse of the shock. The generally such slower variations of 6096o

with x due to the natural growth of the boundary layer are ignored.

In generalising equat ion (16) to unsteady notion, observe that in

practice " fanning" of the shock soot occur in the vicinity of the

boundary layer, and therefore that the shock - location equation (3)

becomes ill-defined as x2 * 0 This fanning will be neglected

to the extent that when x2 -+ 0 it will be assumed that a representative

value of to - s(0) can be defined to determine the instantaneous

position of the root of the shock just outside the boundary layer.

last C(x l ,t) denote the unsteady boundary layer thickness,

and introduce the representation

ik(xl+t)

C fa 60 F (x+1) - 

s0 r kZf)e k-u k-K* ^

 I I

where KI , KZ respectively denote the wavenumber K t defined by equation

(15) when the -, + sign is taken. The path of integration in (18) runs

below both of the poles at k - K I , K* (this can always be ensured by

taking w to lie in a suitable region of the upper half-plane). The

function f(kd) is assumed to be independent of w , to be regular on

the real k-axis, and to vary significantly only for variations in kd

which are 0 (1),. In practice d and 6 0 are of comparable magnitudes,

I,

4

(17)

2.6
OF POOR QUALITY

and both are here required to be small relative to the wavelength

til/Re (KI ) of the displacement velocity craves. This fmplies that the

singularities of f(kd) in the complex plane lit further from the real

axis than the simple poles at k • KI ,K* Accordingly, when

x + I » d > 0 the value of the integral in (18) is dtainated by the

residue contributions from these poles, and the motion thereby described

is entirely associated with the corresponding displacement velocity waves.

When W is real, K I,K* respectively determine boundary layer disturbances

* which grow and decay exponentially with increasing zl .

In the opposite extreme in which (xl+il ! 0 ( d) , the

t displacement velocity poles make an insignificant contribution to the

r inutgral, and (18) reduces to

ik (z +J)

1

f.

C 60 F(z+R) - a0 f(kd)e dk Cl9)

r

The hypothesis that the structure of the boundary layer in the neighbourhood

w.

of the shock is invariant in a reference from fixed relative to the shock

' requires that f (kd) be interpreted as the Fourier transform of

F'(z1 ) dF/dxl , into,

f(kd) i1cz1

F^ (z1)e dz1 . (20)

s

in whi ch case ` (19) becomes

^G 60 ! 

r 

F(xh^t l

-z0) + 0 (z2) (21)

In the following discussion the functional form of F(z+L) is assumed to

be known.

f

F

2.7 

_. ._. ..._..._., __... _ ^.r 
ORIGINAL PAGE If'

OF POOR QUALI'T'Y

The displecAtmant velocity %(xl )' any now br, expressed in terms

of the displacemeri4, 

s0 

of the root of the shock by substituting the

rspresentation ( 18) into the linear theory formula

vn + U

i, G 

0 (xl > (22)

giving

ik(x +t)

v (x) i z 6 (2 3) k2(or-Uk)f(kd)e 1 dk . n 1 ^ o o ^ (k-K 

I 

)(k-K*)

Similarly, since F'(x) vanishes for z + I - d , upstream of

the shock, equation (14) becomes

ikk

E 0 i ti060Ka F kk-K )ek^^ dl4c (24)

I -.. I I

q,

ti

s

Boundary value problem for +

.Collecting together the principal strands of the above discussion,

the boundary value problem for the potential # may be stated thus:

For (xl > - lt, x2 > o) and (- . < x < •, x2 < 0) , find

admissible solutions of the equation

2

^- ik0 + 

M ^z , - 02 - 02 0 (25)

1 Sxi Ix2

which satisfy that radiation condition at large distances from the airfoil

together with the following:

2.8

4e
ORIGINAL PAGE19 I. Conditions on the Airfoil:

OF POOR QUALITY

• i z 6 (•

0 0 J

ik(s

k7 (W-Uk)f(kd)^t 1 

ft)

k-xI'dk )(k--%*,)

2

Ii. Conditions in the wake ( s2 - 09 al # 0)

i aXl N

^ -

Cam i

KS% B e ..

(26a)

(26b)

(27a,b)

III. Conditions at the *bock:

+

2i2` . 0 (sly - t, 0 < s2 < •) ; (28a)

1 U8

z0

- -^ (x - to s2 • 0) (28b)

C8

The coefficient A is to be determined by application of the Tutta

condition at the trailing edge of the e'-foil, and B is given in terms

of z0 by equation (24).

When used in conjunction with the dispersion equation ( 15), the

solution of this boundary value problem provides the characteristic

equation for the admissible frequencies w for which self-sustaining

oscillations of the shock gave are possible.

2.9
k^

F

E>

t

s

t.^

13. Solution of the boundary value problem

ORIGINAL PAGE 19

The Pr&idtl-Glauert transformation OF POOR QUALITY

0 • ` eiKMX

K • k0/B

X ® x1/0

reduces equation (25) to the standard form

2

+ 3 2 +K2 ^^0

aX2

ax2

and the boundary conditions (26) - (28) becosae:

I. Conditions on the airfoil:

i

: t Q v2 ( W-Uv)f(vd)ei (oX+VL1

ax2

0 0 j v-K1) (v-K* Qv

(- L<X<0,

x2 •

+0) .

where L t/S a • vO + KH

ax 2

II. Conditions in the wake (x2 0p x > 0)

C^^ • A 

eiKX

(K • W/SU)

iKax -

Hay-2

B e (K • K S + KM)

•

3.1

(29)

(30)

(31a) 

(alb)

(32a

(32b)
III. Conditions at the shock: ORIGINAL PAGE IS

OF POOR QUALITY

ax+ iK* « 0 (X x2 ). 0)

z a- , 1. * einm (_ .• - y , ZOO). 2 

0 c02

Squation (33a) has been simplified by making use of the condition

42 « 1 .

Since the problem is linear we can set

(33a)

(33b)

(34)

0 ' *1 + *2 •

where ^1 represents the flow induced by the boundary layer displacement

{ velocity ,fluctuations, and* 2

is that produced by the oscillating

motion of the shock. The functions * 10 *2 must satisfy reduced forms

of conditions ( 31), (32), and their sum, *, must satisfy (33).

botwidary value problem for 1

The perturbation due to the displacement velocity fluctuations

is determined by ignoring the presence of the shock, i.e., by assuming 

the flow to be subsonic everywhere and by imposing (3la) over the semiinfinite interval (- « < X < 0) This procedure is Permissible

provided that the composite potential satisfies conditions ( 33) an

the shock.

A Fourier integral representation for *1 is obtained in Appendix

1 by the Wiener-Hopf technique [22]. In x2 > 0 we find

«

i z06 0 v2 (ur-Uv) f (vd) j 1 1

4T j v-K (v-Ki 1 —7(k) k - a + i0

FZF+l r-K--10]

i (vL+kX+Y (k) X2 }

X e dkdv (35)

3.2
• in which the notation t id indicates that the contour of integration

in the k-plane passes above the pole at k - a and below the poles at

k - Ka , +c The function y(k) is defined by

Y(k) - Of:P. /F; t" I-J U K (36)

where branch cuts for the radicals on the right-hand side are taken

respectively in the upper and lower halves of the k-plane,, such that

when K is real and positive, y(k) is positive on the real axis for

JK1 < K and positive imaginary for jk) > K .

Boundary value problem for *2

Motion of the shock causes sound to be radiated in the downstream

direction. Since the influence of the displacement velocity is included

in 01 , the shock associated sound must satisfy

a^

a 2 

-0 ( <Z <0 '

x2 - 0) (37)

rather than ( 31), where again the condition is imposed on the half-line

( < R < 0) . In the wake there is no need to account for breathing

modes generated by the displacement effect upstream of the edge, and 2

is therefore required to satisfy conditions ( 32) with B - 0 .

In the absence of the edge an appropriate representation of the

motion in x2 > 0 would be

*2 f V (a)cos ( ax2 ) e`Y(A)(R+L) dA (38)

0

If

*2 - *12 *2 . (39)

^2 is the diffracted field produced by the edge and wake, and may again

be determined by the Wiener-Hopi procedure. In Appendix 2 it is shown

that, for x2 > 0

3.3
Apo -, ^"da Y 1)) 1

2

2w2 

ORIGINAL

OF pOOR QU,ALIV l i(kx+Y(k)x2+'7(A)L)

} - k-y A x -10 a dk (4i?)

i

Equations (38)-(40) determine *2 in terms of the as yet unknown

function VM •

The integral equation for Y(a)

Condition (33a), to be imposed at the undisturbed location of

the shock, provides an integral equation which determines Y(A) in terms

of the displacement z0 of the root of the shock. To obtain this

! equation we introduce the Fourier cosine transforms =(u) of a function

g(x2) defined by the reciprocal relations

g(U) g(x2)cos (ux2)dx2

0 (4la,b)

g(k2 ) g(u)cos(ux2)du

0

(see, e . g. Erdelyi et al [23]) .

Substituting 0 - *l + *2 into (33a), using the representations

(35), (38), (40), and taking the cosine transform, we find that T

satisfies

(K-Y

,z+Y u T (u 0

X . _ 1 0iL(Y(u)+V(A))da

(Y(A)+Y(P) u7)

^) K (2x ♦ iKJ

1 . (42)

7C^-L

3.4

TO) 227r
where the cosine transform ^1 is given by

i (VL+ax) dv ORIGINAL PAG'^ jU v2(^Uv)f(vd)e OF POOR '

.' QUALITY 1 0 0 v-KI v-K e K-uz-c2)

v2 ( ^r-Uv)ff(vd) + ib z0 ^-K (V=K* Y (U) Y(u} +aa Y(u} +c

+ , (u^ [-y^ f(u)x)dv =11j+ ei(v'-

tr u ^

. The various terms in equations ( 42), (43) have the following

interpretation:

VW is the contribution to (33a) from V2

the direct

field general d by the shock; the integral involving T(X) on the

left of (42) represents the contribution 'l2 of sound waves generated

at the trailing edge by diffraction of the pz-ir; field V2

:ie

f^

first term on the right of equation (43) denotes the influence of the

boundary layer dispLmcement velocity fluctuations when the edge and

wake are ignored; the second term represents the effect of sound

generated at the edge due to its interaction with the displacement

velocity wave.

(43)

3.5
64. The characteristic frequencies of self

sustained oscillations

A second relation between so and VW can be obtained from

condition (33b) imposed at the root of the shock, vist

ORIGINAL PACE- 10 aiM f

OF POOR ;Q^AUT

zo 

s - 0 1,^1(—L°d) +(—L'O) 

+ Ti ('^1+ 00)} (k4) CO2

The elimination of s 0 , TM between (42), (44) provides ra equation

relating the frequency w and the conjugate displacement velocity

wavenumbexs KI ,Ki . The roots of this equation define those admissible

values of K,,K* for which self-sustained oscillations are possible, and

when substituted into the dispersion equation (15) yield the characteristic

equation determining w in terms of the boundary layer thickness

d - 60 ♦ d0 downstream of the shock.

Self-sustained oscillations are envisaged to occur as a result

of the following feed-back loop of mechanical processes (see Figure 3):

1. The displacement z 0

 of the root of the shock generates

? displacement velocity waves in the boundary layer;

2. Sound waves are produced by the subsequent interaction of

F the displacement velocity waves with the trailing edge;

3. The impingement of the sound on the shock closes the

k

loop by generating further disturbances in the boundary

7.

w

layer.

These observations lead to the following simplifications of equations

(42), (43) and (44):

In equation ( 42) the second term an the left hand side is

discarded. This expresses the influence on the shock of sound waveM

r+

orignally generated by the shock ( ^r2) and subsequently diffracted at

^. the trailing edge. The amplitude of these waves is expected to be small

4.1
^^4 % con hared with the sound produced by the boundary layer/edge interaction

because of the exponential growth of the displacement velocity waves

d as they propagate towards the edge.

When the second integral on the right of equation (43) is

inserted into the right hand side of (42), it describes the effect on

the shock of sound generated by the boundary layer/edge interaction.

This interaction Will be dominated by the exponentially growing

displacement velocity wave (proportional to eiKIxl,6 ,whose effect is

determined by the residue contribution to the integral from the pole

at V W K T . This will be taken to represent the principal component

of the edge generated sound.

Taking account of these approximations we accordingly reduce

equation (42) is reduced to

V(U) " ZO(Xl (U) + X2 ( U) ) r (45)

where X1 X2 , respentively denote the local and edge-diffracted 

,. influences on the shock of the boundary layer motion, and have the

explicit representations:

-iKMI.

X10)

-rape v^,(w^-Uv) (o+K)f (yd)dv (46)

J

[K +Y(U)] (V K;)(V—Ki) (K2-y2-02)

2 i(KIL+Y(p)L)

ita K (w-UK )f(K d)e

` X (u) O K -YO) I

I

I

2 2 KK

(K -K*)

x

KSIKI

- t + 1 1

Y() K Y μ +3 r[^o^ Y(^+K Y U ^I

+ S I

where

vi = KIS + KM . (48

i.

4.2
o„Qr,C 10R (^up^.1TY

+pp Turn attention now to equation (44). The first term in the

brace brackets on the right hand side accounts for the backreaction

of the bound_:; layer on the motion at the root of the Oock. As above,

this decomposes into two components, the first describing the local influence

of the boundary layer (when edge effects are ignored), and the second the

eriect of acoustic waves produced by the boundary layer /edge interaction.

Both effects are contained in the double integral representation (35).

When X 4 - L on x2 + 0 the integration contour in the k-plane may

be displaced to - is . In so doing the integral along the real axis is

transformed into the sum of two terms 200 1 , z002 , say, respectively

equal to the residue of the simple pole at k w a (which characterises

the local effect of the continuum of boundary layer displacement velocity waves

specified by equation (23P and an integral arouni a contour enclosing the

branch cut of ► i+k , which extends from k w - X to - K - i- . The i

latter represents the edge generated sound.

Thus we find:

-iK v2 (w-Uv) f (vd) dv 0 1 doe J

7(Q) (v-K I ) (v-K* (49) I^ r

iwl the branch-cut integral can be reduced to the form.

02 - I dv I VV d& , (50)

0

f

where 3'(v 1 is regular on the & - axis and ti 0(1/&) as + .

Now 0 2 « 1 , so that KL = k0R /S 2 is a large parameter which

characterises the foreshortening of the wavelengths of sound waves generated

at the trailing edge and propagating upstream against a near sonic mean flow.

To leading order, we therefore have

4.3
6 00 ^ $(v,0)dv ^ ^^ dE

2 0 ORIGINAL. VIRGO ri;

OF POOR QUAL11°1f'

KL J (v.0) dv (51)

The remaining integral with respect to v is approximated by the residue

s

contribution from the pole at v . K (see equation (35)) or: fiche basis,

discussed above, that the dominant edge -generated sound is produced by

the exponentially graving displacement velocity wave proportional to

aiKIX1 In this way equation ( 51) is found to have the explicit form:

(T_ ! KI (w-UKI K IK

02 is - d0

^K _Ka I f(Kd) sI

I I (K+KS)

..1 + 

K (j+K

i CKIZ+KL-^r/4]

R+oI A+oI^ a (52)

I

This and equation (49) determine

*1 (~L.0) " 9O{el+e

2} .

(53)

The second term in the brace brackets of equation (44) is the

contribution of the primary field due to the moti;,6 of the shock, and

it follows immediately from ( 36) that

^►I(-L,O) • 2 J T(A)da (54)

0

i•

The final term in (44) is discarded for reasons discussed

previously, since it represents the inf lueme of sound produced by the

f

diffraction of *2 at the trailing edge. Hence substituting from (53),

f

I

(54), equation ( 44) may finally be cast into the app roximate form

s

4.4
a{

,I

*c82

()

X 0 w 0 (^S)

CO2 ORIGINAL VAUI;

ar POOR QUALITY

When this expression for x0 is inserted into equation (45),

J the following integral equation is obtaineds

12a iKML/WCO2 

(Xl(lj)+X2(V))

{i + (el+e2) ._.cam.., o

(0 < u < (56)

The characteristic equation relating admissible values of w, K1 , Ki

is now deduced by integrating this result over 0 < N < »

iKML i + y=! -

^rc6 2

2 Vet+02) +

 I"-

 (X l (u) + X2 (N))du a 0 . (57)

0 j

♦ more convenient form of this equation is obtained as follows.

De f ine

A (KI.Ki.w) w 2,

iKML

iKmL 

we, 

+ f Xl(N)duj

WCO2 0

26 0 1 j v2(u-Uv) f(vd)dv

WC 02 2 T y(Cr)v-KZ V-Ki

i^du T V2 ( w-Uv)(o+K)fNOdv (58)

0 

(V-KI) (V-K*) (K2-u2-o2 ) (K+Y(U))

where use has been made of equations (46), (49) .

The presence of the factor e iy(u)L in the representation

(47) of X2 (u) indicates that, when KL - k0 l

/02 is large, as assumed

here, the value of

T1

X2Wdu may be approximated by the method of

0

4.S 
1 + A (KI.KI.W) + 8 (KI.Ki.W) eie - 0 .

ORIGINAL PAGE 15

OF POOR QUALITY

F

t

4.6

stationary phase [22] . Taking account of this remark we define

iKML r8 *9 - 2..e..._._ 2 8 (KI KI•w)e CO2 –r + X2(u)du

0

J where (from (47) anA (52)) in a leading approximation

60K2 Ki(,^-vK

cs2 KL 

(;K

+& K —K a f (Kid) I I

x —s./ I — w + K^. 1 _ 1

r

►'►^

s

K+ol ^ f 1C+ic IC+p I, '

I

and

e * K + k01 — "

I 1—M Wi

Th`e characteristic equation (`) may aceardingly be lit in the

s

(59)

(60a)

(60b)

form
5.1

15. Discussion of the Characteristic Equation OF DOOR QUALITY

The characteristic equation (61) gives the condition which

must be fulfilled if self-sustaining oscillations are to occur at

frequency w for given values of the displacement velocity wavenumbers

KI , Ka The first two terms on the lefthand side represent the

influence of the moticm at the root of the shock when diffraction at

the trailing edge is ignored. The term in B gives the backreaction

at the root of sound produced during the passage of the unstable

boundary layer wave over the trailing edge. This interpretation is

evident from the structure of

Re 8 a R1: (K ID ) + 1-M - 4 (62)

wherein (i) Re(K IL) is the phase change associated with the convection

of a boundary layer disturbance from the root of the shock to the

trailing edge, and (ii) k n

L /1-M) - R/4 is the phase change experienced

by sound waves radiating upstream from the edge to the root together With

a correction (- r/4) due to the cylindrical spreading of the waves.

In order to derive quantitative predictions from equation (61)

it.is necessary to introduce an explicit representation of the function

f(vd) given by equation (20). Consider first the value of f(KId)

which occurs in the definition (60a) of B By hypothesis, both the

transition length d and the boundary layer thickness 6 - d o + 6'

are small relative to the wavelengths of the displacement velocity waves.

Since F'(x1 ) is significantly different from zero only for Ix l j < 0(d),

it follows from (17) and (20) that, to leading order,

/^ 6'

f(K I d) M 2e J V (xI )dxl ' 2w63)(6

10

and this estimate will be employed in calculating B .
4

To estimate the corresponding values of the coefficient A

from its definition (58), it will be assumed for simplicity that

F I N +R) is an even function with respect to the undisturbed location

a

xl - f of the root of the shock. When KId << 3 the leadins order

j asymptotic approximation to equation (58) reduces to

A

(KI 'KI 'bj)

d f (71)

r- i 2tn f - In rl

X (64)

+01)]

s S l ) t

Numerical results are presented below for the came

(6oibo)

f (A)

2w ( 1+X2)

for which,

.,

A (K j _eK* .^) • - ZM i °^ 2 Ln f Fcn, - tn_ + ri (6.6) j 1 l

This and ether expressions appearing in the characteristic

equation ( 61) will be expressed in non-dimensional form in terms of

the reduced frequency S and wavenumber W defined by

S k t _ wR

o c

(67a,b

WWR+iWI ki

0

f.^t

where WR,WI are the real and imaginary parts of W If the mean

a velocity in the wake is assumed to relax rapidly to that of the mean stream M

} U , the wavenumber K a of the breathing mode in the were is equal to

w/U , and this value is used below.

r Equation ( 61) may now be expressed in the form

8 (WR + 11M, - z + arg G 2n^r ,

t^^^09^^1^,L PUALI'^''/ ` (68a,

OF POOR QUALITY • to IG I ,

5.2

(65)
where

d — ORIGINAL FACE" IS

r

o

0 60 0 MIS 

0W) _ T+ A OF ?OUR QUALITY

6 o S ;

M2

W_ 1 1 - W - iW^

1+ M M 1+M Mx

r82W- 1+M^ (1+M+8 W) 4

}

 ^+M+ 2- (1+ 8

• (

M 60

2 OT (d^ 2 An t 282

1 1t8

S

17 ^t , - a f n L,-- - 1

' and n is an integer. Equation (68s) is the feedback loop condition that

the total phase change around the loop should be a multiple of 2n .

` The term erg (0 ) is the change in phase introduced into the loop

during the production of displacement velocity and sound waves; the

remaining terms on the left of (68c) account for the effects of propagation

discussed above. Equation ( 68b) determines the growth rate of the

instability wave in the boundary layer, wi ch must be such as to ensure

that the amplitudes of the sound waves returning to the shock are sufficiently

great to sustain the loop. 

w

Equations (68) are to be solved simultaneously with equation (15)

relating the displacement velocity wavennmber to the characteristics of the

assumed boundary layer velocity profile. In terms of dimensionless

variables equation (13) gives for the exponentially growing wave

l^ 1 -

W 1 w

w (70)

I - MG V [1-M2 ( 1-V)2] }

where

ii O V/U

is the ratio of the inri^, ,- velocity of the model boundary layer (Figure 2)

to the main flow velocity and 6 - 6 0 + 6^ is the boundary layer thickness.

Equation (10) characterises a disturbance which grows exponentially as it

5.3
u

propogates at phase velocity V downstream from the shock.,

oRIGII4AL PAGS Ea•

Numerical Results OF pOOR QUALI'T'Y

Equations (68), (70) are used to determine the dependence

of the reduced frequency S of the self-sustained oscillations on the

( boundary layer thickness ratio 6/1 The calculation is performed

r

for fixed values of 60/1, d/8 1

0

 , K and V The values of S and

8/1 determined by this procedure are actually discrete, and parameterized by

the integer n (S increasing with n) . Their variations with n turns out to b

i

sufficiently smooth, however,for it to be convenient to present the

results graphically as continuous plots in the (S,8/1)-plane. The

results illustrated in Figures 4,5 correspond to the case 80/1 - 0.0005

and M - 0.99. The range of variation of 811 is set by the constraints

(i) that 6 ; 6 0

>o,the boundary layer thickness upstream of the shock,

and (ii) the upper limits of validity of the approximate dispersion

equation (70) (wherein the second term in the brace brackets is required

to be small relative to unity). The solid and dashed curves are for

w

d/6o - 1 and 10 respectively; V is equal to 0.2 in Figure 4 and 0.6

in Figure 5. The lm er of these velocities is intended to model the

order of magnitude of the phase velocity of the long wavelength

disturbances in the boundary layer [24] . Prominent, large amplitude

boundary layer structures tend to convect at about 60% of the main flow

velocity, and this is the situation modelled in Figure S. In both cases

it is evident that the predicted variation of S with 611 is not

significantly dependent on the transition width d/6 I.

In the experiments reported by Burdges [7] the increase in the

displacement thickness d* of the boundary layer across the shock is

typically of order 0.0011 If, following Ffowcs Williams and Purshouse

[18], we identify the boundary layer thickness 6 of the vortex sheet

h model with the displacement thickness of the real boundary layer,

5.4 
ORIGINAL PAGE 13`

OF POOR QUALITY

i.e., we take 6o/f - 0.001, the results depicted in Figure 6 are obtained

when V - 0.2. It is of interest to consider this case in relation to

i

the experimental findings of Succi at al [8] . In their experiment

Z S 0.6 ems and acoustic tone bursts were observed at a blade referenced

frequency of 12.6 KHz. This is consistent with the present theory in as

much as in Figure 6 the reso _ance frequency fR - SOW varies between

10.2 - 21.6 KHz when d/do - 1 , and between 8.1 - 16.3 KHz at the

higher value d/6o - 10 .
56. Conclusion

A linearized theoretical model of the interaction of a normal

shock wave with the boundary layer on ,a supercritical airfoil has been

discussed. The system can oscillate at certain discrete resonance frequencies provided an appropriate feedback-loop condition is fulfilled.

This is dependent on the convection velocity and growth rate of dieturbances generated in the boundary layer by motion of the shock and on

the subsequent interaction of those disturbances with the trailing edge

of the airfoil. Resonance frequency predictions are consistent with

measured "tone burst" frequencies observed in the study of the noise

produced by a transonic propeller.

The analysis of the coupling between the shock and the boundary

layer is based on the hypothesis that the motion in the neighbourhood of

the root of the shock is steady in a reference frame moving with the root.

There are obviously many differences between the model and real flows: 

shocks are not usually weak nor of infinite extent, are often accompanied

by a separation bubble and never enter the boundary layer in one front.

In addition, disturbances generated downstream of the shock leak through

the boundary layer and can also propagate around the leading edge of the

airfoil to modify conditions ahead of the shock. However, the theory

provides a useful first step in the understanding of unsteady shock/

boundary layer interactions, and would also be applicable to other aeroacoustic problems, such as the interaction of blade-tip shocks of a

ducted, transonic rotor with wall boundary layers.

i 

6.1
ao

TX 

-eiax

X<0 0 x2 -+0.

2

(a- Sv+KM) .

Having obtained the solution 0 (v), say, in this case,the complete

solution is given by

(A2)

(A5)

APPENDIX 1

k ORIG1NAt. QUA

►-^^ OF p(^OR To solve equation (30) subject to conditions (31) (with

(3la) extended to the interval - « < X 4 0), (32) and the Kutta

condition at the trailing edge. In order that the solution be

causal, i.e., that the displacement velocity waves arise as a consequence

i

of the motion of the shock ( and are therefore present only for X > - 1.)

it is assumed initially that w is appropriately situated in the upper

half-plane to ensure that KI , Ki both have positive imaginary parts.

' 

+ The final result is obtained by analytically continuing the solution on to

the real w-axis.

Set

i(kX±y(k)x2)

C # (k) a dlc , (Al)

f according 'as x2 0 Wiener-Hopf functional aquations for C± (k)

are obtained by the usual procedure [22, Chapter 23. To simplify the

argument condition (31a) can be replaced by

- _ z . V2(w-UV) f(vd) La p (v) eiv^dv

0 1 o u v-KI v-KIF o

Conditions (A2), (31b) lead to the functional equations

(A3)

7( k ) C+ (k) - 2^r k o+i0) a L1 (k) . (A4)
where Ll0 L2 are regular in In k < 0 and vanish as k ♦ - i- .

Conditions (32) respectively yield

ORIGINAL PAGE 1:4

OF POOR QUALITY

+ (k) - C_(k) - 2xi(k -i0 Ul ( k) (AG)

K' /V

Y(k) CC+ (k) ♦ C_ (k + 2w(k-KS-iO) - U2 (k). W)

where Ul, U2 are regular in In k > 0 and vanish as k + + i- .

Using the method described by Noble [22] to eliminate

L1 ,L20 U19 U2 one finds ultimately that

Y (k) C (k) 1 — a+ — - Ks v/

+ 4^r-o+10 -Ke-io k

+ 47P 1 _ iA-:'-K (A9) 4n(k-oai0) k iO

The perturbation velocity and pressure will remain finite at the trailing

edge (Rutta condition) provided y(k)C+ (k) vanishes at least as fast as

1/k as k - This will be so if

A -i (A9)

Inserting this into ( A8) 9 substituting into ( Al) and applying the integral

operator ( A3) gives the result ( 35) of the main text.

(A2)
OPENDIX 2 ORIGINAL PAGE 19

OF POOR QUALITY

To solve equation (30) subject to condition (32) (with

a-o v

^tf^

a*

^ 8xx • 0 for X < 0 „ s2

0

2

(A10)

and the Kutta condition at the trailing edge.

Set

r

^2 ' ^Z ♦ *2ll)(A

where^Z is given by equation ( 36). *2 the edge diffracted field is

when displacement velocity waves are ignored. First solve the reduced

problem for

X20 say, for which 0

2

is replaced by

X20 * cos(Ax2)eiY(I)X (Al2)

} The total diffracted field is then given by

*2 r X20( 1rMe'y(')"da

.

J

(A13)

0

Define

i(kY.±y(k)x }

e(k)e 2

Y*

dk .

X20

t (A14)

"' `s r according as x2

0 This form automatically satisfies (32b)

M

when 8.0 On x2 

0 , X < 0 equation (A10) implies that

t

Y (k)C (k) ^- L (k) . (Ale)

where L(k) is regular in Ink < 0 and vanishes as k — i• .

^; F u c

i

r

(A3) 
_t.

Condition (32a) gives ORIGINAL PAGE 19

OF POOR QUALITY

2C(k) +

A

A - T .. W k, M U(k) r

where 0(k) is regular in Ink > 0 and vanishes u k + + i-,.

Solving (A1S)r (A16) by th* Wiener-Hopf procedure and choosing

the value of A to satisfy the rutta condition, we find

C (k) a 1 ,. 1

4ea k"IC W (k-y(K)-i0

Substitution into (A14) and application of the integral operator (A13)

yields the result (40) of the stain text.

(A1

(A

^.-i

rj

rs

^' S

V

4

^i r

1'.

(A4)
r.

IJ

REFERENCES ORIGINAL. PAGE 19

OF POOR QUALITY

1. A.R. George, 1978, AIAA J. 15, 707-715. Helicopter Noises

State-of-the-Art.

2. C. Kitaplioglu and A.R. George, 1977. AIAA Paper 77-1360.

A study of the far-field sound due to unsteady shocks on rotors.

3. F.H. Schmitz and YoH. Yu, 1980. Paper No.22 presented at Sixth

European Rotorcraft and Powered Lift Aircraft Forum, University

of Bristol, September 1980. Transonic Rotor Noise-Theoretical

and Experimental Comparisons.

4. M.J. Lighthill, 1952. Proc. R. Soc. Load. A211, 564-587.

On sound generated aerodynamically. I. General Theory.

5. G.L. Loving, 1966. NASA TND 3580. Wind-tunnel-flight correlation

of shock induced separated flaw.

6. T.C. Adamson and A.F. Messiter 1980. Ann.Rev.Fluid.Mech. 12, 103-138.

Analysis of two-dimensional interactions between shock waves and

boundary layers.

7. K.P. Burdges, 1981. J. Aircraft 16, 431-437. Experimental measurements

of shock/boundary-layer interaction on a supercritical airfoil.

S. G.P. Succi, D.H. Munro and K.U. Ingard, 1982. To appear in J. Sound

Vib. Propeller Tone Bursts.

9. C.J. Wood, 1960. J. Roy.Aero. Soc. 64, 683-686. Transonic buffeting.

10. H. Tijdeman, 1975. NLR Report TR-75038U. On the motion of shock waves

on an airfoil with oscillating flap in two dimensional

transonic flaw.

11. M.B. Williams, 1980. AIAA J. 18, 615-624. Unsteady thin airfoil

theory for transonic flows with embedded shocks.

12. R. Magnus and H. Yoshih ara, 1975. AXAA J. 13, 1622-1628. Unsteady

transonic flows over an airfoil.

13. M.E. Goldstein, W. Braun and J.J. Adamczyk, 1977. J. Fluid Mech.

83, 569-604. Unsteady flow in a supersonic cascade with strong

in--passage shocks.

NO
k.

ORIGINAL PAGE 1$

OF POOR QUALM Y

14. H. Ashley and M. Landahl, 1965. Aerodynamics of Wings and

Bodies. Reading, MA: Addison-Wesley.

15. M.S. Howe 1981, J. Sound Vib. 75, 239-250, The displacement•

thickness theory of trailing edge noise.

16 M.S. Howe, 1981. Proc. R. Soc. Lond. A374, 543-568. The role

of displacement thickness fluctuations in hydroacoustics,

and the jet-drive mechanism of the flue organ pipe.

17. M.S. Howe. 1981. Phil. Trans. Ray. Sac. Load, A303, 151-180.

On the theory of unsteady shearing flow over a slot.

18. J.E. Ffowcs Williams and M. Purshouse, 1981. J. Fluid Mech.

113 0 187-220. A vortex sheet modelling of boundary layer noise.

19. M.E. Goldstein, 1981. J. Fluid Mech. 104, 217-246. The coupling

betvaen flow instabilities and incident disturbances at a

leading edge.

20. M.H. Williams. 197'9. AIAA J. 17, 394- 397. Linearization of unsteady 

transonic flows containing shocks.

21. R. Betchov and W . O. Criminale, 1967. Stability of Parallel Flows.

New York: Academic Press.

22. R. Noble, 1958. Methods based an the Wiener -Hopf technique. Oxford:

Pergamon Press.

23. A. Erdelyi, ed. 1954. Tables of Integral Transforms. (Bateman

Manuscript Project). Vol.l. New York: McGraw-Hill.

24. C.J. Gedney, 1979. Rept. No. 84618-1 of Acoustics and Vibration

Laboratory, Mass. Inst. Tech. Wall pressure fluctuations during

transition on a flat plate.
IW

AW

a

a

w

W

0

a

0

u

w

a

13

.04

u

ucn

r+

c^

w

ORIGINAL PACE IS

OF POOR QUALITY

R

V

40

R}I

a ^ 4

C 'C

^ C

A

d in

CT ,v

W V

Y

C a r f

E 3

u

O ^ ^

v

u

N'1 >

^Aw

C ^

O A

^ LL.

F.1
w..u

0

a^

.,. to

C > d d

E 3

41U

d

vi 0

N

x

v

N

t

11

x

ORIinINAL PAGE IS

OF POOR QUALITY

N

^

N

AM

F.2

u

t

Zn

tV ^N

Co.

I

AA

•

a

u

0

vS

u

rs

c^

14

..4V

w

0

C

0

y

u

0

0w 

rA.

>

d

.M

u

My 

" 

C114

+

W—
ORIGINAL PAGE 19

OF POoR QUALITY

i

4

•.1

W

a

I

4

I

U

O

7

I I.a

rA

+l

.G

ii N I

•^M^ 

d

0-4 

.i1

f N

Pee

P.3
OF Poo;i% QUALITY

40

i

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