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 ThreeDimensional 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 .... :
....
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:= ,
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December
:,.
..... :..................
...........
UniverSsityof
_
,
Inc .......
and
Stephen
Propulsion :....... " .........................
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Technology, Park,

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and Institute for COmputational Mechanicsin Lewis Researc__h Center " i_ ___ _........ "............. Cleveland,
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7
1992 (_ASAT_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
Nq315499
WAV_5
Unclas
G3/34
i
0139616
:
:
:
•
_war
ON
THE
NONLINEAR
THREEDIMENSIONAL
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
hlghReynoldsnumber 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
criticallayer wavevortex
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(R1).
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 finitetime
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
criticallayer
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
finitetime
disturbance and
de
spanwise
oscillation.
A
to hlghReynoldsnumber 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 nonequillbrium,
& 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 threedimensional
to
investigated.
numbers
& Thomann,
for twodimensional
criticallayer
Goldstein
Merkli
basic
theory
the rapid growth of small highfrequency 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
normalmode
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
NavierStokes
flow is unidirectional,
stability
linear
This
above solutions
following
layers could to :blowup' and
transition
of a pair
of oblique
waves. The
importance
instance,
in the
turbances
are
iments,
e.g.
of threedimensionality case
Schubauer
tant
Kachanov
For the
twodlmensional
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,
twodimensional
Floquet
Reynolds
instability and
best
layers,
(1975).
downstream,
appears
For
controlled
et al.
Kachanov
controlled
available. a linear
critical
of the
quasisteady
to the
of boundary
early
of introducing,
threedimensional
(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
threedimensionality 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
twodimensional
disturbances
threedimensional (1984).
TollmienSchlichting
predominantly
twodimensional 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
lefthand
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 lefthand disturbance
the
mode
neutral
A number
gives
of the
modes
a wavenumber
fast.
an amplitude
constant
of Goldstein this
modes
rate
Ar
= rr0
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 highfrequency 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 leadingorder of the unsteady, u_, and (2.30)),
shows
that
in the viscous,
the
basic
flow will be evaluated
we wish to force viscous
criticallayer 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 threedimensional 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
righthand
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 finitetime 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 wavevortex 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
Stokeslayer
to the w I,
i.e.
solution
is
(v, v, w) : (u, v, w)  (cos(r We denote
y* is normal
We nondimensionalise
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,
2R1
+ 00uOz + v COl] coy

(2.1)
COP coz '
(2.2)
2R10_
+ O.0v cOx
cOw 2R10rv+ 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
multlpletlme 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_v2t
...
,
(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
40, fil has
the
y_ is the
following
jth
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
ear/+_pjT] The
function
+...
v2 takes
the
,
and
following
v2 = V2(y,tl)Ecos_z where
a relatively
included
large
in order
longitudinalvortex
to match
to the
of the eigenfunction
from
Rayleigh
equation
The
behaviour
2+
....
+ c.c + ...
component,
inner
solution
its neutral
(see
state,
oU_YY 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 "Uc 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
=
Vyy2
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)
basicflow
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
OyThis
has
the
U..
+
=
UC
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
spanwisedependent 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).
leadingorder
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 criticallayer 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(21_CosOb_:
fil
_
(ia)
lsin
z01
_
_1
sin Ob_]
singularity
corresponding
results
in the faster
twodlmensional
(2.26)

become "_''"
in ul is a simple
of a twodimensional that
(0
20b_711
(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]Ay1Ecosfiz
p
.._
e ial(f_cosOAbfEcosflz+c.c+
2.2
1 sin'
+ b_sj
Solvability
OAb]Y1Ecosflz
[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_
(oc) 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
jth
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
ymomentum
equation
(3.7)
(b +  b_) is zero. gives
01"1 0, OY and
so the
appropriate
solution
(3.8)
is
P1 = ialOu 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
leadingorder
+ _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
12 = {(_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
righthand
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
blowup
1992). the singularity
less,
singularity
finitetime
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
blowup 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
twodimensional to be valid.
an explosive
(e.g.
sign of _ vj/3_ _ , it is
a finitetime
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 finitetime
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 spanwisedependent
mean
flow is
disturbance.
of the
wave
is that
can
could
an 'unsteady'
small
A feature
there
amplitude
a disturbance
consisting
or 'nonequilibrium'
This
is of course
involving
the over
nonlinear
neutral
addressing
of a pair
criticallayer
related
Rayleigh
range
which
of validity the
from
(1.1)
and
as specified
the
of (4.7).
growthrate
(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 wavevortexinteractions
interaction?
a wavevortex
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 wavevortex
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