receptivity _EJ kerschen.pdf

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Boundary layer receptiwity theory

E J Ke rschen

Department of Aerospace and Mechanical Engineering, The University of

Arizona, Tucson, AZ 85721

The receptivity mechanisms by which free-stream disturbances generate

instability waves in laminar boundary layers are discussed. Fre e-stre am

disturbances have wavelengths which are generally much longer than those

of instability wave s. He nce, the transfer of energy from the free -stream

disturbance to the instability wave requires a wavelength conversion

mechan ism. Recent analyses using asymptotic methods have shown that

the wavelength conversion takes place in regions of the boundary layer

whe re the mean flow adjusts on a short streamw ise length scale. This

paper reviews recent progress in the theoretical understanding of these

phenomena.

1.

I N T R O D U C T I O N

The important influence of free-stream distur

bances on boundary layer transition has been appre

ciated for many years. Exp erime nts clearly showed

that increased free-stream disturbance levels

enhanced the amplitudes of instability waves in the

bound ary layer, hastening transition. However, the

physical mechanism by which energy is transferred

from the long wavelength, free-stream disturbances

to the short wavelength, boundary layer instabilities

was not understoo d. This came to be known as the

receptivity problem.

In order for an external disturbance to generate an

instability wave, energy must be transferred to the

unsteady motion in the boundary layer at an appro

priate com bination of frequency and wavelength. To

simplify the discussion, it is useful to consider the

situation where the external disturbance is of small

enough amplitude that the unsteady motion can be

represented as a linear perturbation of the mean

flow. Atte ntion can then be restricted to a single

time harmonic, with the results for general time

depend ence obtained by superposition. This paper

focuses on linear, time harmonic unsteady flows.

The earliest theoretical analysis of instability wave

generation in a boundary layer was presented by

Gaster (1965). He considered the case of two -

dimensional Tollmien-Schlichting wave excitation in

a parallel boundary layer by a time harmonic distur

bance at the wall. Th e wall disturban ce was localized

in the streamwise direction, and hence the wave-

number spectrum of this disturbance was broad.*

Thus, the input disturbance contained energy at

the

appropriate frequency-wavelength combination t;..

directly excite an instability wave. <

The first attempts to predict receptivity to natur-

1

ally occurring free-stream disturbances (Rogler and

Reshotko, 1975; Tarn, 1981; Mack, 1975) were based

on a parallel flow formulation similar to that of;

Gas ter. How ever, naturally occurring free-stream}

disturbances (sound waves, turbulence, etc.) travel a j

much higher speeds than instability waves. Thus, the j

wavenumber spectrum of the free-stream distur-;

bance at a given temporal frequency is concentrated

at w aven um bers which are substantially differed j

from the wavenumber a

0

of the instability wave.

Hence, these parallel flow analyses succeeded only in

finding "particular solutions" which are unrelated to

the insta bility w aves. In order to transfe r energy j

from a naturally occuring free-stream disturbance to

an instability wave, a wavelength conversion process

:

-

is requ ired. A m ore com plete discussion of the j

differences between instability wave generation by

localized disturbances and by naturally occuring dis-

turbances can be found in Kerschen (1989). j

Ex per im en tal investigation s in the 1970s showed |

that rece ptiv ity can oc cur in the v icinity of the lead- j

ing edge and at localized dow nstream locations, j

Rev iews of these and more recent ex periments arc j

presen ted by Ka chan ov, Kozlov and Levchenko j

(1982) and Nishioka and Mo rkovin (1986). The;

experimental results stimulated theoretical investiga

-

tions,

w hich show ed that the wa velength conversion I

process takes place in regions of the boundary layer

Appl Mech Rev vol 43, no 5, Part 2, May 1990

S152 ® Cop yright 1990 Ame rican Society of Mechanical Engineers j

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Kerschen: Boundary layer receptivity theory S153

where the mean flow exhibits rapid changes in the

streamwise direction. This occurs (a) near the body

leading edge and (b) in any region farther down

stream where some local feature forces the boundary

layer to adjust on a short streamwise length scale.

The rapid streamwise adjustment requires that non-

parallel mean flow effects be included at leading

order, in contrast to the parallel flow assumption of

classical Orr-S om me rfeld stability theory. This paper

discusses recent progress in theories for receptivity

produced by small-amplitude acoustic and vortical

free-stream disturbances interacting with the leading

edge of a flat plate and with a localized mean flow

adjustment downstream of the leading edge.

Reviews of receptivity by other authors are pre

sented in Kozlov and Rizhov (1987) and Goldstein

and Hu ltgren (19 89). A discussion of the role of

receptivity in transition prediction can be found in

Heinrich, Choudhari and Kerschen (1988).

2. L E A D I N G E D G E R E C E P T I V I T Y

In this section, we examine receptivity in the lead

ing edge region of the Blasius boundary layer.

Results are presented for both free-stream acoustic

waves and conv ected g usts. The analysis conside rs

low Mach num be r, two-d imen sional flow. Th e flow

far from the plate consists of a small amplitude, har

monic disturbance superposed on a uniform mean

flow U

0

. The Reynolds number is assumed large,

and hence the outer problem for the unsteady flow

corresponds to the inviscid interaction of the small

amplitude free-stream disturbance with the semi-

infinite flat plate. This outer solution provides the

distributions of pressure and slip velocity which

drive the unsteady motion in the boundary layer on

the plate. Since the free-stream disturban ces are

assumed to be of small amplitude, linear superposi

tion is valid. He nce , the inciden t disturba nce

velocity can be separated into components parallel

and perp end icular to the plate surface. The parallel

component contributes directly to the slip velocity on

the plate surface, while the perpendicular component

contributes to the slip velocity via its scattering by

the plate surface. Near the leading edge, this sca t

tered component has a square root singularity corres

ponding to inviscid flow around the sharp edge,

while far downstream the scattered component takes

a simple form app ropria te to a doubly infinite p late.

Goldstein's (1983) analysis of the unsteady flow in

the boundary layer shows that the asymptotic solu

tion for e « 1 («T

6

= Ug/wv) contains two distinct

streamwise regions. The receptivity occurs in the

first region n ear the leading ed ge, w here

X= x'w/U

0

= O (l) and the motion satisfies the lin

earized unsteady boundary layer equation (LUBLE).

Farther downstream where x = e

2

X = O( l) , the un

steady flow satisfies the classical large Reynolds

number, small wavenumber approximation to the

Orr-Somme rfeld equation (OSE). Goldstein exam

ines the LUBLE for X » 1 and the OSE for x « 1,

and shows that the first asymptotic eigenfunction of

the LUBLE, with amplitude C

l 5

matches onto the

T-S wave which eventually exhibits growth farther

downstream. Hence the amp litude of the T-S wave

is linearly proportional to C

l 5

which we call the

"receptivity coefficient." To find C

1 ;

a numerical

solution of the LUB LE is required . Unfortunately,

for X » 1 the first eigen function makes only an

expon entially small con tributio n to the solution. In

order to extract C^ from the numerical calculation,

the equation is analytically continued into the com

plex plane and solved along a ray where the first

eigenfunction dominates the solution. Further details

may be found in Heinrich (1989).

Figu re 1 is a plot of the rece ptivity co efficient for

a convected gust as a function of disturbance orien

tation. The velocity and frequenc y of the free-

stream disturbance are held constant as the orienta

tion is varied. Fre e-stre am co nvected gusts can be

interpreted as a two-dimensional model of weak

free-stream turbulence. The convected gust is vorti

cal,

has no pressure fluctuations, and the velocity

fluctuations are perpendicular to the wavevector.

The maximum and minimum levels of receptivity are

found for free-stream velocity perturbations parallel

(9

= 90°) and perpe ndic ular to the plate surface,

respectively. The weak recep tivity levels produced

by the perpendicular component of free-stream

velocity are rather surprising, since this component

produces both a singular flow field near the leading

edge and a significant slip velocity in the doubly in

finite plate limit far dow nstrea m. Essentially, 0(1 )

receptivity levels are produced by the individual

contributions (such as the singular flow near the

leading edge) to the slip velocity, but for the con

vected gust the amplitudes and phases of these vari

ous contributions lead to a high degree of cancella

tion. Hen ce, the receptivity for convected gusts is

produced mainly by the parallel component of the

free-stream disturbance velocity.

The receptivity produced by free-stream acoustic

waves is illustrated in Fig. 2, for two values of the

Mach num ber, M = 0.01 and 0 .1. Here, 0 = 0° cor

responds to incident acoustic waves propagating

downstream parallel to the plate surface, and was the

case considered by Goldstein, Sockol, and Sanz

(1983).

The recep tivity for this case is found to be

only about 1/4 that produced by a convected slip

velocity parallel to the plate surface (Fig. 1,0 = 90°).



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S154 MECHA NICS USA 1990

Appl Mech Rev 1990 Supplement

4 . 0 -

180k0

Fig. 1. Lead ing edge receptivity coefficient for a

convected gust.

The explanation for this may be the slower phase

speed of the convected gust compared to the acoustic

wave. Ho wev er, the receptivity to acoustic waves

rises rapidly as the incidence angle increases, reach

ing a maximum value at

6

180°, which corresponds

to an acoustic wave propagating from downstream

infinity tow ard the leading edge. The high recep tiv

ity for oblique acoustic waves is caused by the per

pendicular component of the free-stream disturbance

velocity. This component produces a flow around

the leading edge whose strength increases as M"

1

/

2

as

the Mach number decreases and the acoustic wave

length becomes large compared to the hydrodynamic

length scale of the problem.

180.0

Fig. 2. Lea ding edge receptivity coefficient for a

plane acoustic wave.

The above result applies only for cases where the

acoustic wavelength is short compared to both the

plate length and the distance to any surrounding sur

faces.

To address the effect of nearby surfaces, we

have analyzed the receptivity to upstream travelling

waves for the case of a semi-infinite flat plate cen

tered within a channel. The slip velocity was calcu

lated using a Wiener-Hopf analysis and confirmed

for low frequencies by a matched asymptotic expan

sion analysis (MA E). Figure 3 contains results as a

function of channel width H for a free-stream Mach

num ber of O.J. The receptiv ity coefficient grows

rapidly with increasing H and reaches a maximum

C

x

= 35 at uH/c =

TT

For higher values of wH/c, Cj

exhibits a regular pattern of weakly damped oscilla

tions,

gradually approaching the isolated plate result

C j = 20 (see Fig. 2). Th e exp lanation for this be

havior can be found in the alternate cut-on of up

stream and downstream traveling acoustic modes.

4 0 - .

saSU ™

Wiener - Hopf

Fig. 3. Effects of finite chann el width on the lead

ing edge receptivity coefficient for an upstream

traveling acoustic wave.

3. LO CA LIZE D REC EPT IVITY M ECHANISMS ,,

In this section, we examine receptivity mechanisms

which occur when some local feature downstream of j

the leading edge causes the boundary layer to adjus j

on a sho rt streamw ise leng th scale. Examples of local j

features are short scale wall humps or suction strips i

or shock - bound ary layer interac tion. The asymp- \

totic descriptions of the mean and unsteady flow \

com pon ents in the local region involve triple deck j

structu res. For sufficiently weak mean flow distur-1

bances (small hump heigh t, for exam ple), the mean j

flow triple deck equations can be linearized and \

solved in closed form. Th e unsteady flow can then

\

be solved via Fourie r trans form s, and the instability j

wave is given by a residue in the complex wave-1

number plane.

For localized recep tivity me chan isms, the wave- ;

length conversion process is produced by the un- [

steady flow conforming to the local short-seal';

geom etry. Essentially, the long w avelen gth, unstead; j



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Kerschen: Bounda ry layer receptivity theory S157

H

(jss4

J

04

legend

So = 0, 44

- 90 . 0 - 45 . 0

0.0 4S.0 90 .0

vac (degrees)

Fig.

tion

6. Norm alized receptivity coefficient as a func -

of acoustic wave propagation angle •

| A CK N O W LED G EM EN TS

; Finanacial sup port for this research was provid ed

by the Nation al Science Fou ndation unde r gran t

ME A- 8351929 and by the McDo nnell Douglas

Research La bora tory. The author would like to

thank Dr. M.E. Goldstein for a number of stimulat

ing discussions.

REFERENCES

Bodonyi, R.J., Welch, W.J.C., Duck, P.W., and

Tadjfar, M. (1989), A num erical study of the in

teraction between unsteady free-stream distur

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Choudhari, M. (1990), Boundary layer receptivity in

laminar flow control applications, Ph.D. Disserta

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Choudhari, M. and Kerschen, E.J. (1989), Receptiv

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