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Xuesong. Wu. Department of. Mathematics. Imperial. College. 180. Queens ..... right hand branchesof the neutral curve,it is straightforward to generaliseit to.

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On the Nonlinear Three-Dimensional instabiiity of Stokes Layers and Other Shear Layers to Pairs of Oblique WaVes __'_:_ - :== --

Imperial

College

London,

England

Sang

Ohio

Soo Lee

Sverdrup Brook

"

,_,_

-:- ....

::-

:-

--

Ohio

J. Cowley

Cambridge,

= .....

............

Cambridge England

and Insthute for Computa Lewis Research Center

tiona l Mechanic_

C!evela_nd,

_

Ohio

in Propulsion .... :

....

:

: : :=_

:= ,

--: ::&

December

:,.

..... :..................

...........

UniverSsity-of

_

,

Inc .......

and

Stephen

Propulsion :....... " .........................

..........

Technology, Park,

-

_

and Institute for COmputational Mechanics-in Lewis Researc__h Center " i_ -___- _........ "............. Cleveland,

=

:

7

1992 (_ASA-T_-I0591_) THREE OlMENSIO_AL STOKES LAYERS AND

I IASA

n

LAYERS (NASA)

TG PAIRS 51 p

ON THE NONLINEAR INSTABILIIY OF OTHER SHEAR OF

OSLIQUE

Nq3-15499

WAV_5

Unclas

G3/34

i

0139616

:

:

:



_war

ON

THE

NONLINEAR

THREE-DIMENSIONAL

OTHER

SHEAR

INSTABILITY

LAYERS

TO

Department

PAIRS

Xuesong of

OF

OF

STOKES

OBLIQUE

LAYERS

WAVES

Wu Mathematics

Imperial College 180 Queens Gate, London and

Institute

for

SW7

2BZ,

England

Computational Lewis Research

Cleveland,

Ohio

44135,

Stephen J. DAMTP, University Street, Cambridge

and

Institute

for

Cleveland,

Ohio

Inc. Group U.S.A.

Cowley of Cambridge CB3 9EW,

Computational Lewis Research

Propulsion

U.S.A.

Sang Soo Lee Sverdrup Technology, Lewis Research Center Brook Park, Ohio 44142,

Silver

in

Mechanics Center

England in

Mechanics Center 44135,

Propulsion

U.S.A.

Abstract The

nonlinear

evolution

hlgh-Reynolds-number when disturbances the

Reynolds

controlled

number.

effects

This

leads

axe included

amplitude

explosive

growth

the

importance

singularity, caying

and

between

flow

instability,

identified.

develop

are the

oblique

lead

to the

sufficiently equal, unsteady and

the

bursts

large the the

observed delays

viscosity

special solution

critical-layer wave-vortex

effects;

case

the may when

can evolve approach approach

in the

into

integro-

indicating

in experiments. lead

to the a periodic

to

that that

an such

Increasing

of the

streamwise

Vis-

solutions

we suggest

occurrence the

is then

1989).

too large,

singularity,

in a

e = O(R-1).

function

is not

by nonlinear

generally

scaling

kernel

viscosity

waves

& Choi,

distinguished of the

waves

is focused on times rates, where R is

of oblique

(Goldstein

a finite-time

can

For

the

When

be induced

exponentially.

link

studying

can

of viscosity

wavenumbers

linear

of a pair effects

modification

equation.

equation

growth

development

by

amplitude

explosive

of initially

critical-layer

to a complicated

differential the

The

by nonlinear

cous

of a pair

Stokes layer is studied. Attention of amplitude e have O(el/3R) growth

finite-time

disturbance and

de-

spanwise

oscillation.

A

to hlgh-Reynolds-number of Hall

& Smith

(1991),

is

AND

1

Introduction

The

flow generated

exact

unsteady

Stokes

layers

flows.

Although

has been

to the

linear

state.

Instead

seek

a sinusoidally of the

studied

the

disturbances

that

over

the

highest

a full

instabilities

& Davis which

period

the

of the

range

have

been

observed

paradox

has

been

partly

decay

a whole

_z Cowley

(1992)

unsteady,

(see

(see

also

Wu,

or non-equillbrium,

& Shukhman

(1988),

a linear development affect the evolution in a finite

time.

in Stokes

layers

1983).

they

(1977)

In this

This

approach

& Leib (1989)

paper

we extend

found

and

At

for which 1975).

Cowley

idea

was

(1987)

developed

disturbances

of Hickernell

and others.

They

our

analysis

found

consisting

Wu

using

the

Churilov

that

of instability

disturbances

by

(1984),

stage, nonlinear interactions inside the critical of disturbances sufficiently to cause the amplitude

to three-dimensional

to

investigated.

numbers

& Thomann,

for two-dimensional

critical-layer

Goldstein

Merkli

basic

theory

the rapid growth of small high-frequency to nonlinear effects preventing the linear

Hall,

1991)

of the

However,

number

by Tromans

approach

Floquet

Reynolds

(e.g.

resolved numbers can lead also

used

period.

included

(periodic)

unsteadiness

at all Reynolds

this

of such

of unsteady

(1978)

a complete

simplest

instability

normal-mode

to the

Hall

experimentally

that at high Reynolds over part of a period period

due

and

stable

studied,

who argued disturbances over

over

flow was

end

The

of the instability

(1974)

grow

wall is one of the

a conventional

flow is not possible

Kerczek

plane equations.

as a paradigm

of the

yon

oscillating

Navier-Stokes

flow is unidirectional,

stability

linear

This

above solutions

following

layers could to :blow-up' and

transition

of a pair

of oblique

waves. The

importance

instance,

in the

turbances

are

iments,

e.g.

of three-dimensionality case

Schubauer

tant

Kachanov

For the

two-dlmensional

on Stokes to analyse

number

of theoretical

has

been

yet

expansion

the

waves a graph

frequency are

from

of the

neutral

stages

of our

instability Since

theory

number

in the

to be even then

curves

waves. Rayleigh's

of a Stokes

Under

these

equation layer

plotted

then with

Klebanoff & Thomas

more

observing,

imporartificial

there

are

as yet

disturbances.

of Stokes critical

analysis,

layers,

there

Reynolds a weakly

spirit

we assume that the Reynolds of the Stokes layer, w, is much satisfy

Saric

knowledge

no finite

Even e.g.

(1987),

and

dis-

exper-

of transition,

two-dimensional

Floquet

Reynolds

instability and

best

layers,

(1975).

downstream,

appears

For

controlled

et al.

Kachanov

controlled

available. a linear

critical

of the

quasi-steady

to the

of boundary

early

of introducing,

three-dimensional

(1977) is not possible. Instead is large so that the frequency O(wR)

with

approaches

identified

about

indeed layers

the

(1984),

in transition.

in carefully

to significance

three-dimensionality difficulty

disturbances;

no experiments In order

layer,

apparent

realised

Nishioka

only in the

growing

& Levchenko

Stokes

of the

only

(1947),

dominate

disturbances

(1962), because

& Skramstad

been

instability

two-dimensional

disturbances

three-dimensional (1984).

Tollmien-Schlichting

predominantly

two-dimensional et al.

of the

has long

nonlinear

of Seminara number, smaller

conditions, (Tromans

are a

number & Hall

R, of the flow than a typical linear

1977).

as parametric

instability Figure functions

1 is of

time.

This

is for a flow where

(z*,y*,z*)

and

the velocity

t* are dimensional

of the boundary

Cartesian

y" = 0 is (U0 cos wt*, 0, 0),

coordinates

and

1

the

Reynolds

cosity.

The

number

streamwise

5 °-1 are denoted exist, times

and

and

Stokes

and

by a and

(1987)

for further

thickness

Stokes

assume into

the

duced branch

layer,

that flow.

into

i.e. the

as a result Stokes

including

of one of the

this mode

receptivity

curves

will begin

with

how

time

suppose

that and

a simple wave

where those

linear

course layer

theory

of oblique

& Choi

in an unsteady scaling

exists,

modes

are

other

mean

authors

and

modes

are

by the

left-hand

profile

slowly

introevolves

then

arise:

is such that nonlinear branch of either curve is introduced at the

r, where

growth

in

introduced

of possibilities

e0. At a time

crossing excited

are

given

amplitude left-hand disturbance

the

mode

neutral

A number

gives

of the

modes

a wavenumber

fast.

an amplitude

constant

of Goldstein this

modes

rate

Ar

= r-r0

as /r0RAr

e_ (Wu, involved

although

Even

structure

dynamics

1992a).

we

in the

In addition

are unsteady

and

viscousin nature. We arethus interestedin when the

nonlinear modes

GC, Wu close to

effects

become (1991)

the evolution

become

important

unstable

and Wu

of a pair

or just

near

before

_ Cowley

a neutral

they

(1992),

of high-frequency curve,

stabilise.

oblique

i.e. either

As explained

it is appropriate

modes

soon

after

in detail

to concentrate

by

on times

1

T = r0 + eivl for some

suitable

Therefore

rl = 0(1),

we introduce

i.e.

times

the time

,

at which

the

linear

growth

rate

is O(e_R).

scales 1 , tl = _ciRr,

(1.1)

and

t= RT to account carrier

for the

wave

The

frequency

basic

sufficient

_slow' nonlinear

flow

of the

to express

growth/decay

disturbance,

U" evolves

on the

its profile

(1.2) disturbance,

and

the

'fast'

respectively.

very

at time

of the

slow

time

r as a Taylor

scale

T, and

series

about

it turns the

out

neutral

to be

time

T0:

0(y,T) = 0(y,To) + A&(y, T0)T:+ .... Hereafter otherwise

all quantities stated.

associated

with

In order to maintain

maximum

generality,

to appear at leading-order of the unsteady, u_, and (2.30)),

shows

that

in the viscous,

the

basic

flow will be evaluated

we wish to force viscous

critical-layer u_, terms

at T0 unless diffusion

equations. An elementary in the critical layer of width

the

be assumed

balance e] (see

we require R -1 = Ae ,

where

terms

parameter

A is introduced

to be of order

to reflect

one in §2 and

(1.3) the

§3. The

importance

highly

of viscosity.

viscous

case

It will

corresponding

to )_ being asymptotically The overall evolution

large will be discussed in §4.2 and §4.4. of a three-dimensional disturbance is summarised

in figure

2 for the

flow

of curve

case

when

A. As illustrated, its

growth

interactions that

there

the

the

rate inside

nonlinear

disturbance

decreases the

are four

goes

is initially

to

O(e_R)

when

critical

layers

control

times

scales

illustrated

slow time

scale,

r, over

1. The

very

2. The

slow time

3. The

faster

4. The

fast

time time

scale, scale, scale,

near

rl, over tl, over t, over

=

the

which

the

the

which

which

linear r:

in this

which

the

the

and

grows

O(1).

At

evolution.

branch exponentially

this

stage,

We wish

figure: Stokes

growth

the

right-hand

rate

disturbance

disturbance

layer

evolves;

evolves; grows; oscillates.

until nonlinear

to emphasise

We note that although our analysisis basedon being closeto either the left or right hand branchesof the neutral curve, it is straightforward to generaliseit to wavenumberscloseto the apex of curve A in figure 1 (cf. Hickernell, 1984). The paper is organisedas follows. In §2 we construct asymptotic perturbation expansionsin the %uter' region awayfrom the critical layers. The limiting forms of these solutions near the critical layersare then determined; as usual these contain unknown _jumps'acrossthe critical layers. A solvability condition is alsodeducedfor an inhomogeneousRayleigh equation. In §3, we analysethe unsteady,viscousand weakly nonlinear flow within the critical layers. By matching the inner and outer solutions the unknown jumps are evaluated. Then by combining the solvability condition with these jumps, we derive the amplitude equation which is a main result of this paper. The amplitude equation is studied in §4, both analytically and numerically. In particular, a finite-time singularity structure is identified as in GC, and confirmed by numerical solution. In addition, exponentially decayingsolutions are found under certain conditions. The viscous limit is discussedand a llnk is establishedwith the wave-vortex interaction work of Hall & Smith (1991). Finally, in §5, we summariseour main results, and discussthe implications of this study. In appendix B, we deducethe amplitude equation for free shearlayers by combining the presentresults with those of GC.

2

Outer

We take where plate

the

Expansion flow to be described

z* is parallel and

z* is the

to the

direction

spanwise

r = wt*, and

write

for flow over

an oscillating

the

by Cartesian of oscillation

direction.

velocity

coordinates

plate,

Then

the

time

basic

the

perturbed

with

Stokes-layer

to the w -I,

i.e.

solution

is

(v, v, w) : (u, v, w) -- (cos(r We denote

y* is normal

We non-dimensionalise

as Uo(U, V, W).

plate

of the

(x*, y*, z*) -- S*(x, y, z),

0,0).

flow by

(u,v,w)=(O+u,v,w). 2.1 Outside

Asymptotic the

is governed,

critical to the

Solutions layers, order

the

Near unsteady

of approximation

Critical flow

Levels

is basically

required

in this

linear study,

and

inviscld.

by the

It

inviscid

equations

Ou

av

Ow

CO--_ + Oyy + cO---_ = 0,

2R-1

+ 00uOz + v COl] coy

-

(2.1)

COP coz '

(2.2)

2R-10_

+ O.0v cOx

cOw 2R-10-r-v+ The

elimination

of pressure

_

O. cOw _-

Op cOy '

(2.3)

cOp Oz"

(2.4)

yields:

(2R-'

- cO cOy + U_)(-g;z

cOw N )

cOO" cOw OyO_

- O,

(2.5)

and (2R_,O+ On introducing

the

to be transformed

multlple-tlme according cO

velocity

scales

referred

to above,

__1 _ CO +

ot

(u, v, w) and

- _--_x=0. cOy U_

-

(2.6) the

time

derivative

needs

to _R 0_+_

o,The

O.O)V,v

,

2R_ot,

the pressure

p of the

CO +

cO

c_-g_n

o_ "

disturbance

are expanded

as follows:

4

u

=

eul + eiu2

+...

v

=

' evl + e_v2-t-

...

,

(2.7)

,

(2.8)

4

w

=

ewl + eiw2

+...

4

p The form

'early

time'

linear

=

the

function

A(tl)

is just

a normal

amplitude

,

(2.10)

mode,

A(tl)5,(y)cos_zE

is the

(2.9)

epl + e_p2 + e_pa...

solution

V 1 :

where

, 5

so we seek

solutions

of the

+ c.c.,

of the

E = exp(io_x

-

(2.11)

disturbance,

(2.12)

i0(t)),

and dO dt For simplicity principle the

we have

case. and

Note

that

in the

that

and

+ 1)t_e½f_(r0) 2 the

to extend

amplitudes

as in Wu & Cowley

r_ is parametric

Rayleigh's

assumed

it is straightforward

asymmetry

lc_c(ro)

+ ....

two oblique the

analysis

complicates (1992),

waves

the the

will not be written

(2.13) are of equal amplitude.

to unequal algebra,

dependence

amplitudes;

especially on the

out explicitly.

The

however

in the

slow time

function

In

viscous scales

Vx satisfies

equation (O. _ c)(D 2 _ (_2)_,

_

O.yy'Vl

where

a = (_' + _)½ 8

_-

O,

(2.14)

r

The boundary conditions are that _1= 0 on y We let r/= r/_

V - Y_, where

-4-0, fil has

the

y_ is the

following

j-th

asymptotic

= 0, and

critical

_1 _

level

0 as y _

at which

0

o0. = c. Then

as

behaviour

vl "_ aj+ Ca + bf[¢b

+ pie,

log ]_I],

(2.15)

where 1

2

ea--r/+_pjT] The

function

+...

v2 takes

the

,

and

following

v2 = V2(y,tl)Ecos_z where

a relatively

included

large

in order

longitudinal-vortex

to match

to the

of the eigenfunction

from

Rayleigh

equation

The

behaviour

2+

....

+ c.c + ...

component,

inner

solution

its neutral

(see

state,

oU_Y-Y c )]v2 = (ia)-l{[-dA-(iaU'vl)A]dtl

asymptotic

l+qjT]

form + _0,2)cos2flz

deviation

[n'-(a2+

eb=

and

,

v_(0.2) 2 cos 2flz, §3).

The

satisfies

(U:2_f_ - c)

(2.16) has

had

function

to be

O_ is the

the inhomogeneous

_- zc_V_Yr "U-c vi A }@1

(2.17)

of _2 as y ---* y_ is

1_71 + (_=r_ + bJ=_A_log I_1+... +c_¢, + d_[¢b + pjCalog I_1],

,--, -bfrj

log

(2.18)

where PJ

=

--_0_ ,

qj

:

2(_2+2

(2.19)

Up

1 _fyyy

Vy

_]uu

=

Vyy-2

dA

-5[[

(2.20)

---_

(iaU.r_)A]

(2.21)

A.Cr. +(_dA dt, Recall

that

all the

level y_. The layers in §3. From

the

iaU_.rxA)

basic-flow

quantities

(a + -

a_-), etc.,

jumps continuity

equation,

are

UuUuYu U3 evaluated

will be determined

we can write

wx = A_lEslnflz

+ c.c. ,

Cr. -U2uu } (2.22) at time

r0 and

by analyzing

at the

critical

the

critical

where _1 satisfiesthe equation

O---yThis

has

the

U..

+

=

U--C

solution

_1 = _-1

sin 8 (_0_

c_1 - _l,v )

(2.23)



where sin 0 = fl/_ The

velocity

ul has Ul

where

----

fil is obtained

that

in order

spanwise-dependent order wise

(see sllp

also

GC).

inside

that

the

Similarly,

"u_°'2)(y,

-1

{[_U---_Y

c'Vl

ul to be able flow,

the

critical

critical

layer.

magnitude

dynamics. represented

we write

the

cOS

--

later,

with this

mean itself

vl,u}

flow is driven is generated

pressure

it has

be viewed

perturbation

p_ = A(tl)_Ecos_z

(see

§3),

a

at leading by a stream-

by

flow is large

waves,

Its dependence on tl can by v_0,2) and _0,2).

leading-order

solution

to be included

mean

fundamental

(2.25)



inner

has

which the

+

the

2_z,

Although

as the

(2.24)

3 I- c.c.,

sin 2 e

_dl,y]

tl)cos

layer,

2_z

as

to match

_°'2)(y,

the

_1)

equation

As will be shown

same

on the critical-layer longitudinal vortex

+

continuity

--(io0

across

interaction

cos/Yz

the

mean

velocity

it has

=

for

form

A_IE

from

"Ul

Note

the

.

a nonlinear in the

sense

back

effect

no

as forcing

the

as

+ c.c. ,

where

pl As y _

y¢a, the

the

asymptotic

Note

that

point

characteristic

is this difference with

the

--1

ZO_

COS

solutions

--

of pl, ul,wl

Pl

""

i(2-1_CosOb_:

fil

_

-(ia)

-lsin

z01

_

_-1

sin Ob_]

singularity

corresponding

results

in the faster

two-dlmensional

(2.26)

--

become "_-''"

in ul is a simple

of a two-dimensional that

(0

20b_71-1

(2.27)

+...

,

(2.28) (2.29)

-1 -4- ....

pole (singular)

nonlinear case. 10

,

rather

than

disturbance. evolutionary

the

logarithmic

branch

As GC observed, time

scale

compared

it

We now introduce

an inner

variable: y-

The

outer v

expansions .._

written

in terms

eb_AEcos_z

7/1

--

(2.30)

of this

inner

+ e{loge](-b]rj

+ b_sj

are then:

+ b_pjAY)Ecosflz

+ e,! [(-bj + rj log ]Y[ + df) + A(a_Y + e_ log e][(a]rj

variable

+ b_pjYlog

+ d_pj)Y

[Y[ )]Ecosflz

+ 1Apjb_Y']Ecosflz Z

+ e_[c_Y

+ (a_r_

u

.._

e_(-ia)

w

_

e]a -1 sinOb]Ay-1Ecosfiz

p

.._

e ia-l(f_cosOAbfEcosflz+c.c+

2.2

-1 sin'

+ b_sj

Solvability

OAb]Y-1Ecosflz

[Y]Ecosflz

+ c.c. + ... + c.c + ...

both

to y, and

using

the

condition

for (2.17):

the

sides

of (2.17)

asymptotic

+ J2r_A

following

sum

is over

integrals

=

-_

(2.15)

-

J

layers,

(2.34)

from

and

(2.18),

b-_c-_)-

they

After equation plitude equation,

integrals

should

and

0 to +oo

with

we obtain

a solvability

-

a-_)

-(ajdj + +-a-_d;)}

J1 and

J2 are

constants

=

f0 +°°

(0

are

+

0_0.,.

(or_

singular;

be interpreted

through

we only

to the jumps.

, defined

U_ _[dy, - _)_

(2.35) by the

the

in the

analysis

sense

(2.36) Oy_

(o-c) of the

]_dy.

(2.37)

critical

layers

This

need

the jumps; to consider

consideration

thus

for the

those

purpose

parts

of the

the

algebra

simplifies

11

in §3 shows

of Hadamard.

the jumps (a+-ay), etc., are determined in the next section, can be derived from (2.35). The nonlinearity is introduced equation

respect

respectively

dO

that

(2.32) (2.33)

by vl, integrating

= f. +oo [these

,

....

solutions

all critical

J_

Notably

, (2.31)

,

-pj(b+d+-b;d-_) where

+ c.c +...

Condition

By multiplying

ia-'J_-_

+ d]pj)Ylog

of deriving inner

solutions

to a certain

the amplitude into the amthe

amplitude

contributing extent.

3

Inner

Equations layer

Expansion

(2.31)-(2.34)

take

the

suggest

following

that

the

inner

expansions

within

the

j-th

critical

form u

=

e]U1 + e]U2 + e_U3+...

,

v

=

e]Va + e_V2 + e_Va +...

w

=

_]W1 + e_W, + e_W3 +... '

(3.1)

,

(3.2)

,

(3.3)

It

p

where

O(e n log el)

as deriving

the

they

onto

match

match,

these

experiment The

e]P1

have

not

amplitude the

i.e. matching

However, and

terms

=

equation

outer

+

been

solutions

must

+

e'_Pa

+...

explicitly

they

are

automatically

whenever

does

any

be included

(3.4)

,

included.

is concerned,

at O(e" log el)

terms

e]e2

not yield

if a quantitative

This

is because

passive

in the

the

solutions

additional comparison

jump

as far

sense

that

at O(d _) conditions.

between

theory

is to be made.

function

Va satisfies

the

equation

0_Vl Lo Oy 2 -

0 ,

(3.5)

where

Lo = Ot---_ + (O_Y + (Lrl) The

solution

which

matches

the

outer

expansion

¼ = A(tl)Ecos_z where

A = bjA,

The

expansion

and

- )_-5_2 " is

+ c.c.,

b+ = bf = bj, i.e. the jump

of the

y-momentum

equation

(3.7)

(b + - b_-) is zero. gives

01"1 --0, OY and

so the

appropriate

solution

(3.8)

is

P1 = ia-lOu cos OAE cosflz + c.c.. The

function

(3.9)

W1 satisfies

OP1

(3.10)

LoW1= --5-; We let

(3.6)

Wx = I)dlE sin/3z

+ c.c.,

then

W

satisfies

L¢01)#,= ion,sino coscA,

(3.11)

where L(o,,)

0 Ot_ + nia(_f_Y 12

+ O.rl)-

02 A Oy 2 •

(3.12)

Equation (3.11) can be solvedusing Fourier transforms to yield the solution

_V_ = _Gsin0cos0_Vo (°),

(3.13)

where

¢¢(o") = fo +_ C A(tl-_)e-"_'-_nt

(3.14)

d_

and fl = a(Gy Similarly,

the

leading-order

+ _r.r,)

streamwise

1 2-_ s, = _Aa U,_ •

, velocity

U1 can

be written

(3.15) as

UI = _IE cos_z + c.c.. It follows from the continuity equation that

& = -G sin _0_Vo (°).

(3.16)

At O(e]), V_ satisfies

L oV2 ,y y = L i V1 + _

0 [0Sll _

O&,]

+ -_z

(3.17)

'

where il and

Sn

Sn,

and

1-2 = -{(_U_,_Y $31 are Reynolds

Sal can Sil

&, After

+ (I_.riY

some

1 _x + -_fJ._rr_)

stresses Sn

-

SaI

-

be rewritten

defined

ou_

ogl v,

+ --

Ox

ou1w, 8x

(3.18)

+ _ _Oz0

by +

OY

o_w,

+ _

OY

ou1w1 Oz

,

(3.19)

ow?

+ --

(3.20)

Oz

as

=

¢(o,o) _) COS2/3Z + q(2,o)E2 '_11 -4- o(o 011' _'11 _

=

¢(2,2)_2 s?,")sin2_z+ _, ._ sin

calculation,

+ A _rlO:_}-_

+ _(2,2)._2 b'n /_ cos 2/3z + c.c.

2/3z + c.c.

,

(3.21) (3.22)

we find that 1

-2

(3.23)

_'nq(°'°) =

_iaU_

sin 2 O.4"FV (i)

11 s(o,2)

liao:

sin2 O_.Wo(I)

,

(3.24)

_'11¢(='°)--

1 -2 _ic_U_ sin = 0[fiA_o O) + 2sin 2 0 fiz(°hEd°)l,,o ,,o , ,

(3.25)

11 S(2,=)

_iaO_

(3.26)

"31q(0'2) =

_/3U_1 -2

S(2,2)

i - 2 &_F_(1) _f_u_ cos'

_

:

sin2 Oii I_V(I) , COS2

0[A*_,r(l)

13

_11_ 2sin

2 0],_0(0)1)¢'O(0)]

(3.27) (3.28)

From inspection V2 has a solution

of (3.21), (3.22) of the form V2=

The L1V_ Wu

II2 0 is a constant. This is significantly different where viscosity causes the disturbance to saturate

Although

rather

shear

how

Wavenumber

gr is now negative. As can be seen, viscosity delays the time to singularity if 3_ is not too large. However, once A exceeds a critical value 3 (between

and

start

for these

here

results

case

for 0 = 60 ° and

.0r is again

noting

that

the

disturbance

show that

viscosity

values

oscillations

decays

both

& = 0.8 on the and

at moderate

However,

the

negative,

when

A and the

of as in

rather

as A increases,

large.

sign of .0r determine

role

can induce

disappear

A is sufficiently

branch

a similar

of ),, viscosity

gradually

to (3.99). This conclusion is supported coefficients which we do not report

right-hand

plays

The

calculations

the terminal

form

of the

by other numerical calculations using here. However, it is worth observing 4

that

because

always

.0r depends

possible

can occur.

on 0, for any given

to find some

In this sense,

0 such

blow-up

1992). the singularity

less,

singularity

finite-time

nonlinear

effects,

bursting

phenomena

et al., 1976). the

amplitude

and

Moreover, of the

as GC

that argue,

disturbance

this the

A as an example.

theory

present order

theory one,

case 0 = 45 °, again

3No attempt has been made to determine intensive to integrate the amplitude equation

This

growth

blow-up Merkli

may

does

taking

break

point

the

by

to the

1975;

ttino

down

until

flow will be

_ = 1.2 on the is special

the critical value precisely because for ,k close to its critical value.

27

Neverthe-

be related

not

angle

case

is induced

& Thomann,

at which

propagation

singularity

two-dimensional to be valid.

an explosive

(e.g.

sign of _ vj/3_ _ , it is

a finite-time

in the ceases

nonlinear

becomes

hand

than

that

in experiments

equations. the special

of curve

our

does indicate

governed by the Euler Finally, we examine branch

common

occurs,

we suggest observed

_ and

.qr > 0, i.e. such that

is more

(cf. Wu & Cowley, Of course, once the

that

wavenumber

rightbecause

it is very CPU

in both the

inviscid

amplitude

equation

figure

vanishes.

9, we depict

to develop

viscous

This

evolution

a finite-time

significantly. into that

()_ = 0) and

does

curves

not

for four

singularity.

As shown

()_ = +oo) occur, values

However,

in figure

limits,

the

however,

nonlinear when

of ,_. The large

the

)_, the

clearly

We note

transient

state

becomes

The

A feature

interaction

of Ha_

modify the mean is a link between also involve

first

layers.

evolve

(1991)

Are the stable?

scale

further.

Interactions surprisingly

large

mean

flows

a spanwise-dependent

mean

flow is

disturbance.

of the

wave-

is that

can

could

an 'unsteady'

small

A feature

there

amplitude

a disturbance

consisting

or 'non-equilibrium'

This

is of course

involving

the over

nonlinear

neutral

addressing

of a pair

critical-layer

related

Rayleigh

range

which

of validity the

from

(1.1)

and

as specified

the

of (4.7).

growth-rate

(A.7)

waves

at least

r-

evolves,

transient

disturbances

interaction

weakly

partially

to determine

while

'chaotic'

'chaotic'

one amount. It seems natural to ask whether especially since Rayleigh wave-vortexinteractions

interaction?

a wave-vortex

analysis

a rather

This

evolve

to demonstrate at two different

as $ is increased

that

fundamental

In particular

through

fact

change

can

waves

to the

stage

questions:

be established

in the

the

first

a wave-vortex

of these

From

(1.3)

questions

and

evolves,

as specified

ro~O(_IR_)

,

time

by tl = O(1),

in such

scale

over

(4.6)

interaction

it is instructive it follows

by _1 = O(1),

that

theory if

assumes

that

these

(4.16)

which

the

disturbance

amplitude

is

two

,-- R-_ since

both

time R-_

scales

are then