i(XoZ- fio)VJ,+ r.OoWd,= -i. + rn. OZ 2 '. (3.10a). (3.10b). (3.10c). (3.10d) with the necessary no-slip conditions ...... S.O. 1987 A nonlinear asymptotic investigation.
NASA
Contractor
ICASE
Report
181859
Report No.
89-34
ICASE ON THE INTERACTION OF STATIONARY CROSSFLOW VORTICES AND TOLLMIEN-SCHLICHTING WAVES IN THE BOUNDARY LAYER ON A ROTATING DISC
Andrew
P. Bassom
Philip Hall
Contract May
No.
NAS1-18605
1989
Institute NASA
for Computer Langley
Hampton, Operated
Research
V'trginia by the
Applications
in Science
Center
23665-5225
Universities
Space
Research
("/ASA-C:'-Idl_'=')) J_',_ TH" IrwTL_ACfI,_"t STATIONARY C_QUSFLL_W VjF._TIC'-:._ T:ILt_TEN-qCHLICHTT,_G ',4Art: _" I_"_ LAYFP, J'J A P,bTATING bTSC Cin._] ( I CA_._E )
£_-,, p
Association qb
C£cL
N/ A National Aeronaulics Space Administration
and Engineering
and
Langley Research Center Hampton, Virginia 23665-5225
NOO - Inun
OIA
r,;/o?
l
ON
THE
INTERACTION
AND
OF
STATIONARY
TOLLMIEN-SCHLICHTING
CROSSFLOW
WAVES
LAYER
ON
Andrew
IN
A ROTATING
P. Bassom
Department North
and
VORTICES
THE
BOUNDARY
DISC
Philip
Hall*
of Mathematics Park Road
University
of Exeter
Exeter,
Devon
EX4
4QE
ABSTRACT
There ferent the
many
can
interaction
upper-branch, been
the
Essentially, the
flow induced
to be linear stationary effects the
its own which
on
(1986). vortex that
the
right. admits
the and
of this
latter
whose
size
An
amplitude
solutions
vortex
This
problem stability
primary
nature
is chosen equation
corresponding
NASA
Langley
Research
Here
effect
is felt
vortex.
exhibit of this
for the to finite
the
evolution amplitude,
has
of investiTS wave.
TS wave
a linear
and and
is wholly
to
is taken
a nonlinear destabilizing
dependent
the
problem
this
mode
of the
TS
stable,
the
modification
the
stabilizing
that
and
structure
context
we examine
so as to ensure
we exam-
of a small
through
influence
Further,
Center,
in the
of both
Reynolds
waves
asymptotic
Initially,
both
large
(TS)
by dif-
we consider
In particular,
characteristics
* This research was supported by the National tration under NASA Contract No. NAS1-18605 at ICASE,
whose
(for the cases
instability.
disc.
is studied
of the can
be caused
(at an asymptotically
a rotating
on the presence
the
can
themselves.
Tollmien-Schlichting
crossflow
vortex
of transition
among
mode
flow above
and we show
that
TS wave
TS wave,
the
by the
in character vortex)
orientation
larger
of the
onset
lower-branch inviscid
by Hall it is found
mean
in the
stationary, effect
the compete
of instability
between
described
gating
occur
where which
of two types
which
the
fluid flows
mechanisms
interaction
number) ine
are
instability
upon with
is nonlinear wave
traveling
is derived waves.
Aeronautics and Space Adminiswhile the author was in residence
Hampton,
VA 23665.
a in
1.
Introduction
Many layers
studies
have
concerned
been
to turbulence
motivated
in fluid
description
of the
of the
wing;
which
wings.
which
describes
problem
the
and
the
an
of the
of the
flow).
triple
deck
Stewartson have
disc,
The
and
layers
the
china
the
which
is stationary predicted
approximately angle was
the
of flows
to an
over
a swept
of Laminar
Navier-Stokes
study
Flow
equations
of this
of the
first
at
the
form
than
such
same
rotating
disc
interaction
four
times
of 13 ° between
in excellent
at
that
for
of the
disc.
the
agreement
than
effective which
which
axes
of the
with
the
1
occur
would
Gregory
flow
the
et
al),
and
The
contains et al used
vortex
insta-
spiral
vortices
using
inviscid
in experiment
although the
by
a direction
Gregory
be observed
vortices
a small
3.
profile
of equiangular
seen,
and
when
velocity
disc
actually
experiments.
modes,
experimentally
appears
it vanishes.
(in
travelling
in three-dimensional
and
rotating
(1979),
in Section only
interactive
Messiter
thickness
basic
pattern
that
are
layer
mechanism
Stuart
of vortices
which
angular viscosity
by a classical by
in flows
the
kinematic
reviewed
waves,
occurs
upon
the
theoretically
the
spaced
to the
greater
are
can
both
location
and
is outlined
vortex
of a regularly
is based
boundary
structure
the
which
described
disturbances
that
to show
number
(which
TS
the
stationary
relative the
The
examined
is chosen
interaction
of the problem
(1982).
greater
The
technique
takes
of an
was
number
of which
instability
and
bility
theory,
layer
of the
of the
elements
vortex
point clay
boundary
flow is susceptible
analysis
are then
of these
disturbance
infiexion
lower-branch
in the
development
makes
TS waves
structure
et al (1955).
for the
key
Smith
much
crossflow
Gregory
disc
the
description
lengthscale
lower-branch
structure,
The
boundary
This
a theoretical
Reynolds
a typical
a wavelength
The
an
on the large
(1981)
wavespeed.
for
solution
asymptotic
and
waves)
particular
to the
of transition
TS waves.
asymptotically
velocity
flow.
vortices
boundary
relevance
disc
suitable
We will concentrate at
in the
is an exact
phenomenon
to as TS
This
boundary
a self-consistent
cross-flow
disc.
practical
rotating
with
referred
occurs
there
particularly
vortices
which
Further,
the
we are concerned
(hereafter
has
of three-dimensional
to understand
by a rotating
to that
a situation
instability
of stationary
waves
similar
Control
Here
interaction
flow induced
instability
the
by a desire
flows.
Tollmien-Schlichting layer
with
radius difference
as
his calculation vector
on
between
the the
experimental observationsand the results of the inviscid analysis has been shown to be due to viscouseffects. Malik (1986a) calculated the neutral curve for stationary disturbances and found a secondmode. Earlier, Federovet al (1976) had observed experimentally this secondmode, which, as in the inviscid case,appeared as a pattern of spiral vortices. The number of thesevortices was seento be between 14 and 16 and these had axes inclined at anglesof approximately 20° Hall
(1986)
by Gregory
has
et al (1955)
by Gajjar
(1989)
deck
structure
an
type effective
also
been
of the
Much
(1986b),
Bennett
over
examined
in this
the
was
no account wave
was
proportional grows also
like
the
examines
crucial
interaction ble
made
for
process,
at leading
the
order
been
found
extended a triple-
corresponds
This
to
problem
has
(1987,
1988).
of a non-stationary
See,
for example,
been
was
The
that
the
vortex.
of interaction
in both
of these
for a completely of the
in their
own
rights.
vortex
The
between problems,
Otherwise,
crossflow
the
involved the
wave
papers
adopted
of the
importance
vortex
be neutrally growth
was then Reed
of particular
approach
should
a TS
mode
TS
vortices
of
although
latter
of the
of
analysis
to force
aforementioned
description
modes
the
that
of the
in-
work
allowed
crossflow
(1988,
to the
A linear
amplitude
Malik
therein.
exponentially
rate
as-
& Smith references
found
was
growth
instability (1981),
we refer
to grow
of the
of crossflow-TS
TS waves.
In the
rational
instability
the
type
and
in the
permitted
of an exponential.
question
Hall
and
to the
interaction
(an
Nayfeh
(1988),
(1988)
paid
problem
vortices
in particular,
effects.
of the
al.
& Hall
growth
such
amplitude
each
GSrtler
et.
this
vortex
important
with
although,
nonlinear
exponential
that
the
has
paper,
wavelength
However, fact
mode
by MacKerrell
to
vector.
elucidated
wall.
in relation
exponential'
and
the
the
properties
Bassom
investigated
used
to the
wavelengths. the
who
of appropriate
at
has
which
the
waves
attention
to a 'double
problem
also
examined
Bennett
et al (1988),
teraction
lead
(1986)
approach
surfaces).
(1988),
little
vortices
have
of TS
& Hall
1985)
work
vortex
stress
nonlinear
curved
relatively
(1984,
Hall
inviscid
This
of stationary shear
performed
However,
Reed
of the
radius
instability. been
b), Daudpota
effects.
zero
(1988)
interaction
flows
account numbers.
type
a weakly
& Gajjar
of the with
1989a,
with
crossflow
sociated
nonlinear
profile
has
asymptotic Reynolds
second
using
work
interaction
for large
for the
velocity
Bassom
version
a linear,
to examine
studied
Further,
given
to the
due
ignores of the stato any
interaction processis no larger than the growth experiencedin its' absenceand the importance of the role of the interaction is impossible to assess. For this reason, together with the neglect of nonlinear terms in her subsequentanalysis, it must be concluded that the relevanceof Reed's work to practical situations is at best doubtful. The primary motivation for the presentinvestigation stemsfrom the question of how the presenceof an inv}scid stationary of the with
TS
wave.
We
calculations
consider
the
of the
vortex
of the
orientation increase
to the
small
neutral
stability
point
nonlinear is felt
the
vortex.
through
of the
problem
and
we examine
the
vortex.
TS wave
draw
with
some
the
in the
We impose
problem
are
of the
correction
TS
in character.
it and
wave
is obtained.
wave TS
vicinity
upon
the
depending
wave
right
effect We
upon
is then
in its own
the
allowed
and
to
amplitude
obtained. between
It is concluded
effect
TS
structure
in the
is linear
we
that
vortex
to the
a stronger, for both
on the
effective
nonlinear the
stability
mean
vortex
linear
and
of the
the
TS wave
flow induced
by
the
vortex.
procedure
The
nonlinear
of the interaction
the
layer
of the
of the
ourselves Firstly,
structure
superimposed
the
concern
an asymptotic
layer
critical
amplitude
TS wave
is considered.
problems
this
not
characteristics
instabilities. has
a critical
wave
it becomes
we do
which
properties
The
of the
problem
TS wave
primarily
The late
the
vortex
presence
evolution
TS
the stability
respective
or destabilise
mode.
at which
has
flow and
stabilise
latter
for the
a smalt
can
basic
stage
vortex
vortex
a very
In addition, and
The
affects
this
of the
crossflow
has
vortex
at
rates
of the
of the
equations
(1986).
on the
the
growth
point
then
vortex that
the
that
of a small
in Hall
inflexion
The
find
for
problem
as described
emphasize
vortex
indicate a small
neutral
a larger,
conclusions
remainder the TS
stability is increased now
of the basic
in Section
on
characteristics in size
6.
is follows.
disturbance
disturbance
nonlinear,
paper
structure this
for the
configuration
of this
to become
vortex
In Section
wave
nonlinear
is considered
2 we formu-
linear
crossflow
in Section
3 and
presence
of the
in the
in Section
in Section
4 and
the
5. Finally,
we
2.
Formulation
of the
We consider velocity
fl.
the
case
Relative
and in which lengthscale
problem
in which
the
linear
the
disc
rotates
to cylindrical
r and
z have
been
l, the continuity
and
in the region
and
polar
made
axes
,cortex
about
(r,R,z)
dimensionless
Navier-Stokes
structure
the
which with
z-axis
with
rotate
with
respect
equations
angular the
to some
reference
for an incompressible
fluid
z __ 0 are
V.u = 0, On
vqt + (U'V)
Here the
u is the velocity fluid
pressure
flow is given
the
r is the
u the
by R = fll2/u
sections
axes
u + 2 (f] A U) + aA vector,
and
Anticipating lowing
disc
it is found
rotate
with
and is taken
disc,
vector,
viscosity.
The
to be large
which
convenient
the
(fl A r) -- --1VP P
coordinate
kinematic
the structures
(2.1a)
govern
the
to define
the
basic
p is the
Reynolds
flow
fluid
density,
p is
R for the
the following.
interactions small
(2.1b)
number
throughout
the
steady
+//V2u"
described
parameter
is given
by
in the fol-
e = R-_. the
As
Von-K£rn_n
solution
u = uB : m where
z--
_12_
and
p=
u, v, w satisfy
the
p/2n2_24p(_),
equations
II
I
+2_:0, Here
primes
conditions
denote
p +_W
differentiation
with
respect
=0.
to _ and
on
---*0, We shall
perturbed
--_
II
(2.3c,
the
appropriate
d)
boundary
are fi=_=t_=0
to now
(2.2)
derive
be concerned the
equations
with
_=0,
(2.3e)
_----*--I
as
_---,oo.
perturbations
to the
basic
governing
these
disturbances.
flow and If the
it is convenient
steady
solution
is
by writing
U
-"
U B
-{- l_"_ (V,
V_
4
W),
p
._
p/2_-_2
(_
7t" p),
(2.4)
where are
U, V, W, P are small
substituted
we then
obtain
three-dimensional
into
the
governing
the
following
disturbances
equations
(2.1)
perturbation U
OU
-r +
and
for the
if expressions
(2.4)
flow in the rotating
frame,
equations: 10V
cOW
+ --+ _cOz = o, r O0
-_r
(2.5a)
"_ O + r_ _r 0 + v-_ O + e12_ -_z O / U + eU + rW-_z dfi - 2 (I + _) V - "7" V 2
(2.5b)
/
(
O V O W O U_r +-- r O0 + _z
+
-_
+ r_r
+ O-_
)
OP Or
U-
+ e'zff_
(
1 + -R
U r2
L(U)
V + _V + rW-_z
2 r 2 -_
'
+ 2(1 + O)U + --r
(2.5_) +
U
_+---+Vr
c3 O0
W "_z o )
0
0
y -
rlOPO0
+-R I
(L(v)_
+ _--_)W+
0
"_+r"_O-'O" V 2 0U)
'
_nW_z (2.5d)
+
vo --r a8 +
u_+ 0
_
w-
Oz +
L(W).
Here, 1
6 2
L-
2.1
The
linear
Since which,
we will
in the
finitesimal vortex
first
consider
instance,
is absent.
quantities
EWe shall some
point
restrict
our
a
expand
-" a 0 +
effect to be need
Following
the
on the r and
£40q
"_'- • • • ,
size
work
boundary
_
TS
wave
so as to have
an in-
the
of the
of Gregory layer
structure
et al (1955),
thickness
given
by E,
and defined
+ O,(e)}] . disturbances
fl in the
on a small
to consider
0 dependence
to examining a and
vortex
of such
we first
scaled with
of the
exp [e-_{ fra(r,e)dr
attention
r = rn and
the
be taken
vortex,
wavelengths
disturbance
with
will
on the
has
62
structure
the TS wave
vortex
1 0
- -+ r _ + _-V_" Or 2 + r_ 00--z
be concerned
influence
when
inviscid
vortex
02
the so we by
(2.6a)
in a neighbourhood
of
forms
=
flO
"_ 64fll
+
....
(2.6b)
_2
where
"702_
satisfies the
ao2 + _-
a Rayleigh
point
boundary
and
say at ( = _. This
_o.
The
that
Wlo
solution
wavenumber
we obtain
of the effective
conditions
for ao and
effective
equation
of inflexion
flow vanishes,
is the
basic
stationary
together
with with
0 as _ ----* 0 and
of this
eigenproblem
= 1.46,
P30 satisfy
the
zone
that
the
Wlo
by demanding
(2.9)
and
that
at which
this
its appropriate
oo yields
the
values
gives
9o = 1.16,
it is also found
Thus
the point
as _ _
ao _ ,'_0
In this inviscid
disturbance.
modes
flow coincides
criterion,
_
of the
mean
4.26
(2.10)
'rn
flow quantities
U30,1730,
W30,
equations I
2U3o
+ W3o
- uI
II U3o -
_U3oI
-
2fiU3o
V3o"
tDV3o '
-
2'aV3o -- 9'
-
W
=
(2.11a)
O,
30 + 2(1 + 0)173o = 0,
W3o- 2(1 + _)U3o= o,
(2.11b)
(2.11c)
I
P_o= -2 (iW3ol') To determine
(2.11d)
these correction terms we need to consider explicitlyboth the
wall layer and the criticallayer structure (at _ = _). In the wall layer where
we
defne the 0(1) co-ordinate ,_ by _ = e4,_, the total velocity and pressure fields assume
the forms
,.,:,-,,I_=o_'_ +...+,,[iU_o +...l_+c._.] +(5 2 [(U:o
+ ...)E2
'
v=r_(D)_=oe
w = _(_")_=oP°2
+ c.c. +rn_-S(U;o
'-
Z +...
+ ...)]
[,,.:o
+ _5 (
+ ...)E
+ ...,
+ c.c.
(2.12a)
]
_ +...
+ ,5[] +....
(2.12_)
(2.12d)
In general, sider
neutral
turbance There
the
wavenumbers
disturbances
structure is an
to satisfy present.
no-slip
inviscid
the
(the wall
to the earlier by the
problems zone
of O(e12), at the
to be
velocity
in Hall
layer
pressure
fields
sections.
assume
the
con-
The
dis-
(1989).
layer),
the
a crucial
mean role
We find
v = rr_O(_)+_
[(Vlo
+ e4Vll
flow
that
in the
forms
C2.7a)
2 + c.c. + rne -12 (U3o +...)]
+ ...)
be
in the
= r._(e)+_ [(U,o+ E'ull +...) E + c.c.] + 62 [(U2o + ...)E
and
O(e 16) must
consider plays
only
Gajjar
boundary
of thickness
correction
in subsequent
real.
(1986),
as the
to explicitly
this
we shall
to be
depth
we need
because
described
and
same
a viscous
work
vortex
but
ao,al,...,_o,j31..,
is as described
conditions
induced
interaction
so take
quantities
z-direction
layer
In contrast
correction
and
in the
inviscid
will be complex
+ O(63),
Z + c.c.]
(2.7b)
+ ,9 [(V_o+ ...) E_+ c.c.+ r,,_-'_ (V_o+...)] + 0(_), w = _'_(_)+_ [(w_o+ _w,_ + ...) E + _._.] + 6_[(W_o+...) E_+ c._.+ (W_o+...)] + O(6_), P'_- e24p(_)+
_ [(Plo
+ e4pll
+ ...)
C_.7c)
E + c.c.]
(2.7d)
+ _ [(p_o+ ...)E _+ c.c.+ (P_o+ ...)] + o(_). Here
6 0. This
be of either
and
effect
other
that
may
-16.50
the
presence
for Re(T3)
# > 0.297
we can
by the
equivalently
between
of the TS wave
particularly,
is stabilised
Re(T3)
for
direction)
More
be drawn
that
that
angles
making
basis
may
of _ (or
we find
on the stability
T3.
TS wave
We notice
making
waves
the
conclusion
in Figure
vortex
the coefficient
< 0 then
opposite
of the
nonlinear orders (1989), first
remainder
small
vortex that
structure
of the
to considering TS
wave
that
of nonlinearity flow structure
the
(of size
is unaffected
we will be concerned
who showed
effects
revert
problem
A as defined
by the
with.
We can
size reaches
are
encountered
in the
26
linear.
of the
use the
the vortex
It is this
interaction
in (3.2)).
presence
when
remains
of the
TS wave,
results
O(R-_')
critical sized
layer vortex
In this at
described (=O(eS)), although which
we
consider
here
is now
(1981)
due
and
following and
we note
O(e 4) which
is necessary
?
and
is larger to
the
ideas
temporal
the
than
mean
the
properties
Haberman
the
that
(1972).
dependence
-[5i
induces
3, we expect
Jr O_(e)
-
by
mean
an
the
vortex
flow correction
layer,
the
F1, defined
&(r,e)dr
large
critical
vortex
function
generated
This
nonlinear
the
in Section
exp
correction
itself.
of the
as in the
F1 -
vortex
Since
presented
flow
see
Stewartson
O(_ 4) correction
TS
wave
to have
and spatial
by
}]
_(e)t
,
(5.1a)
where,
a =&o,
=
= Here,
as in (4.3),
_00 and
+...) + 4( oo +...),
_0o denote
frequency
of the
TS
wave
amplitude
of the
TS wave
C5.1b)
+ ...) + 5( 0o + ...).
due
the
to the
to evolve
changes
presence
in the of the
on an rl
azimuthal
vortex.
lengthscale,
wavenumber
We permit
the
and scaled
where
r = r,_ + e4rl,
replaces
the
derive
definition
an evolution
hand
side
of the
TS
shown
to that
linear
of the
that
C4.14)
Indeed,
the
for with
following
stability
performed
will satisfy
theory
type
absent.
already
disturbance
The
equation equation
3 & 4 it is easily identical
(4.2).
of the
and
an equation
(5.2)
that
Or1 -
TS
wave
the
nonlinear
the
work
TS wave the
of the
the
will term
us to
on the
right
contained
is governed
linear
allow
evolution
in Sections by an analysis
equation
of the
form
Q200d
+ 3,,,
where and
A is the QI,Q2
correction
scaled
& Q3 are parameters
Further,
as
in the
contains
the
mean
to give
functions which
of the of the
follow
discussion
Finally,
concerning
TS
from
induced the
effect
Q3 is given
by the
27
wave,
the
of the
analysis
(4.14),
TS
of the
it is only
by the
vortex
and
of the
vortex
on
same
1^
/_ood --- r_'_oo,
orientation
following
flow velocities
information
perturbation.
amplitude
formula
l_lOOd = r_-_i2oo
and
certain
nonlinear the
layer.
Qs
which
is the
stability
as determined
flow
critical
function
hence the
mean
only of the
T3 save
term TS that
the mean flow derivative terms presenton the right hand side of (4.16) should be replaced by the equivalent terms for the (larger) vortex-induced mean flow. To summarise the above,we concludethat the evolution equation for the TS wavefollows immediately oncewe havedetermined the meanflow correction due to the nonlinear vortex. We consequentlyconsider this problem, in a manner similar to that used in Section 2 and below we determine the governing equations for the mean flow.
5.1
The
nonlinear
We
follow
nonlinear
Gajjar
critical
z = e 12 f, the
vortex
(1989)
layer
implied
and
use
calculation velocity
scalings
to be
and
fields
We reiterate m) and
are
that
azimuthal
solutions defined
take
the
the
inviscid
to the
zone,
forms
(5.4b)
+ e12_(_)
+ e16t_,_
_,_
are the mean
of _ and directions
rl
mean
flow terms
induced
expansions
(terms
in these
and
that
the
mean
larger
than
the
fundamental
functions
may
(5.4d)
+ ...,
order
flow terms
are
latter
(2.6)
leading
in terms
be rewritten
(5.4c)
+ ...,
of the
flow
corrections
crossflow
vortex variable
by the with
vortex.
subscript
in the terms.
We
x, where
that
there
expansions
is a solution
for the
(5.4)
into
vortex
radial seek x is
as
E -- exp(ix).
Substituting
where
v = _0(_) + _4_,0m+ _8(Vlo+ oral) +...,
for these so that
In
appropriate
(5.4a)
all the
functions
vortex
u = r,_(_) + _4r_m + _8(Ul0+ _ml) + ...,
p = eSplo _m, Om and
the
addressed.
pressure
w = eSWlo
where
for
(2.5) terms
(2.6')
and of the
comparing
suitable
terms
yields
form
(Ulo, Vlo, Wlo, Plo) =
14(,,o_ (,-,,,_)cosx, ,,ol(r,,,_)cos:_,wo,(,,_,_)sin_;,Po,(",,,_)cosx),
28
(5.5)
where A
is the
scaled
size
J_0
of the
vortex
r.
VOl, WOl, pol)
0_
--UoBVO1
+
satisfies
i
=0,
--UoBUOl
+ rnf_ wol
-----o¢0POl, (5.6)
'
that
(uol,
G_W01
_XoU01 + --vol
We recall
and
rnv
Wol
J_0 --P01, rrt
=
UoB -- r,_aofi+/_oV
and
UOBWO1
note
that
t --Pol"
=
Wox satisfies
the
Rayleigh
equation
(2.9). Additionally, to (2.11),
and
we find
that
the
mean
flow terms
satisfy
equations
very
similar
in particular d_m 2tim + d2 V_m
_ df_m w d_
d_ 2 d2 fjrn
_ df_rn
wall layer
2 for the the
linear
induced
structure
vortex
mean
and
the
most
treatment
to become derivative by that
of the
across
_ -- _ + e4z, z in the
critical
critical
=
layer
and the
_m,_,_
to that
boundary
_0
the
work
zone.
For
now
to compute
As before,
O(1),
is analogous
we need
between
layer
in order
_ = _.
that
f=0,
change
appropriate
vortex
(5.7c) in Section
conditions
for
form
on
dramatic
(5.7b)
2_m - v wm- 20 + _)_m= 0.
so we conclude
flow of the
(5.7a)
fi'z_m + 2(1 + O)Vrn = 0,
for this nonlinear
fim=_,_=t_m=0
The
-
O,
_' _
w d_
d_ 2 The
2_fim
-
d_
then relevant
the
as
here
and
that
we expect
jumps
in the
we suppose
that
the
critical
the
of system
(5.6)
analysis
expansions
u=ro_+e4(_o+_,_-_
+e8
(5.8)
given
a nonlinear mean
as _ _
is
analysis
flow
layer
earlier
and
its'
is described _ implies
are
_+
= _S_o+ _,_(_, + _z) + ..., p = eS_bo + el2/_ I + ....
29
f---,oo.
+...,
(5.9,)
(5.9c) (5.9d)
Using these expansionsin (2.5) revealsthat the first order unknowns tbo,/3oin (5.9) are given by @o= @.sin x
where
Poo = Pol(()
clear
that
these
in the
the
continuity
This,
when
gives
that
as defined
solutions
If we define
by
match
unknowns
equation
matching
(5.6)
fzjk :
the
and
_0
the jump
that
layer. After
that
1
straightforward
_3o satisfies
where/31o
-
We recall
(5.6)
+_y_)k,
layer
_m
-- rnayfi(k)(f--
(5.10),
solutions
=
) + /gy0(k)(0.
outside then
(5.10)
the
critical
we find upon
that
It is
layer. substitution
fioo = Coo (a constant).
surrounding
•
the
_0_r
critical
layer,
n
that
in Vm across
(Bol,_+/310)--_-
and
on using
r._'o o
'_i.e.
Byk
rnotjftk
inviscid
[O_0 _rn-q-"
/30 = Apoo cos x,
and
on to (5.5)
(2.5a)
with
,LI_+
or, equivalently,
-- AT,,POv,,_ sin x, Bol
the
the
analysis
layer
is proportional
(see Gajjar
(1989)
to the jump
for further
in urn across
details)
it is found
equation
afro A_I2poo x + --
Bo,
Blo
critical
-'
. cgfJo smx--_= Apoo
#0
rn76v_
1
sinx
B01 ] 2-
I
+
02_0
(5.11)
+ Coo.
that
Bol
< 0 and
this
enables
us to write
(5.11)
in a canonical
form.
If we define
= go(Y - V*), Bo,C
C.Y-
Bxo Bo
3 = ,_-1,
BoxC, +_,
(> 0)
Ko=
C*= -
"_----_
_o I
_
r,,'yhv_'_ BOl2-
e
]'
A "I_ poo , B_, (5.12)
we can
reduce
(5.11)
to the
form
-0.
3O
(5.13)
We note that conditions
C, < 0 so _ _
to ensure
a match
+oo
with
v" -. Y as Y _
4-00 where
The
system
well
known,
then
we obtain
inviscid
_, ly
(1972),
apply
applying
the
the
standard
boundary
appealing
03¢
was
the
1972)
numerical (1982))
that,
and
(5.15b)
and
whereas (5.16)
V*
are
= 02¢/i)Y
2
04¢ -
Ac c3y---_,
(5.15a)
as
Y---++oo.
(5.15b)
using
very
large
for small
data
K0
of (5.15)
if we define
kindly
¢ _ Ac, ¢ _
then
(5.15a)
with
respect
integrating
over
a period
integration
with
respect
to Y, in x.
to Y we find
(5.12),
2;;;,.
of ¢ on the
and
a further
on using
-
solution
dependence
plotted
becomes
characteristics
If we define
of integrating
f0
2_r
is well the
= 1 f2,_ ?1" Jo then
the solution
(1981).
2 + sinXcoy3
conditions
& Bodonyi
(5.14)
(Vm) respectively.
Stewartson
technique
to periodicity
Haberman
The
the forms
Ko
We may
(see
now take
'
r_"Om'TY+lnlYlcosx+B_(x)+...,r_-
2
the boundary
system
2
After
solutions
one and
03¢
OY
zone
lim_____
is a classical
Y oxoy
a_
to Y ---* q:oo and
r"(_m)_ + +O K0 -7-
& (5.14)
see Haberman the
the
(_m) _: denotes
(5.13)
corresponds
( B+
parameter
supplied
_ which
documented
phase
shift
_ Bo)
(Haberman
(-6)
corresponds
0. Using
the
-
2r,_%B01
(1972),
Smith
by
sinzdz,
A, is shown by Dr.
(5.16)
(B0 +- Bo)sin d,:.
J.
Gajjar.) to
transformations
the
(5.17)
in Figure
(6).
It is known result (5.10,
for
(This that
a linear
5.12)
figure as Ac vortex,
we find
that
becomes
(5.18a)
[_({)]_+ _
31
_._o
These are the We can (from
easily
mean
show
(5.10),
flow jump
that
(5.12))
conditions
for Ac >> 1 (and
and
the
jump
which
hence
criteria
are
¢ _
the
counterparts
_r), the
approach
those
size
of (2.21).
of ,_,
of (2.21)
diminishes
for the
linear
vortex. We need flow
across
terms
the
derive
equation
the
critical
in expansions To
reduce
the
boundary
@-'-*-61
and
necessary
this
the shift
necessitates
information
Vl.
The
so we do problem
in the
derivative
consideration
we can
governing
not
repeat
for vl
0a_ Y OxOy with
to consider
of the
of the
mean
next
order
(5.9).
the
(1989)
above
layer
to determine
by Gajjar can
to extend
equations
them
to the
proceed for
here.
canonical
to obtain these
Instead,
a differential
unknowns
we merely
are
state
given
that
we
form
O3g_ 04_ 2 + sin x-O-_- _ - Ac Oy 4 -
KoW_ C, '
X(X, Y)
(5.19a)
conditions
[I+rr'B°IC*(
2 0'm) _ ]y3'r'_C'-r"_
[Bol(gml)
_ - B,o(Om)T-]y2+O(YlnlYI), '
(5.19b) as
IYI
tively,
oo. where
is a very
these
(0m)T,(Om,)
mean
complicated
find
upon
that
the
critical
Here
layer
(5.19a) X(x,
and,
flow terms
expression
integrating function
_ denote
Y)
are defined involving
with
plays
no part
on reapplying
in the
tb (the
respect
respec-
..[02(_)]_----_$_='[9'n'(¢)]¢---'_ expansions
solution
to Y and
the
(5.12),
Also X(x, Y)
of (5.15)).
integrating
in determining
transformations
(5.4).
However,
over jump
we obtain
a period
in x,
in _:n across the
we also
find
by the (5.8)
we now have
nonlinear and
jump
vortex;
namely
conditions
studied
in Section
lutions
(Ua0, Va0,W3o)
0o)
that
___
Consequently,
the
result
= t0 ( )ll + Similarly,
we
(5.18)
2 and
a complete the
defining
& (5.20).
in practice
found
=
+. determination equations This
system
(tim, 0m, t0m)
earlier.
32
This
(5.20b) of the mean (5.7),
multiplication
boundary
is almost
is merely
flow induced
identical
a multiple
factor
conditions
arises
to that of the
due
so-
to the
difference in the jump conditions (5.18a) and (2.21a). This observation leads us to some immediate conclusionsconcerning the effect of the vortex on the stability of an infinitesimal TS wave. For a given orientation of the TS wave, the function appearing
in (5.3)
quently, mode
the
nonlinear
makes
¢ < -16.5
°, and
the
forms
size
is proportional
tions
(5.12),
know
that
small.
and
,kc _
the
of the
on the
direction
weakly
across
to the
jump
across
(5.10)
and
(5.18a),
the
(4.14).
TS wave
Conse-
if the
latter
if 0 ° < ¢ < 16.5 ° or if
the
critical
critical
it is seen
layer.
layer. that
jump
to returning
to the
corresponds
to a large
across
linear
vortex),
on
the
the
the
TS
transforma-
in tim across
the
(7) illustrates
Firstly,
as A ----* 0 we
critical
layer
of Section
2.
the
theory
from
effect
From
arise.
we con-
We know
of )_c. Figure
naturally
r so the
problem
the jump
is a function
cases
vortex
or destabilising
the
to CA 2, which
¢ _
nonlinear
stabilising
two asymptotic
,kc (which
radial
equation
otherwise.
that
c_ and
evolution
influence
outward
(4.16)
corresponds
for small
a stabilising
flow jump
is proportional
This
ally,
mean
solutions
dependence,
of Tz in the
investigation
and
wave
this
¢ with
this of the
layer
has
is destabilising
of (5.18a)
critical
multiple
vortex
an angle
To conclude sider
is a real
Q3
we have
from
becomes Addition-
(5.12)
that
in this
limit
2
the
scaled _
amplitude
,kcC (1), where
of the
vortex
A ,_ O(A_-_).
¼C(1)
=- 1.379
(Smith
It is also
& Bodonyi
known
1982).
that
Hence
the
mean
flow
!
jump
is O(,k_ -_)
be shown layer
changes
strongly
earlier the
with
critical
thus
the
TS
main
layer.
of our
to study
problem
in this
difference
following
case
from
that
similar
the
of a strongly discussion
33
_e _-),
the
linear
influence
Of interest
limit
of the
vortex; Ac _
it can critical
mean of the would the
for
a
flow is required.
of a nonlinear
changed
nonlinear
(1989)
et al (1983)
of the
of the
of the
nature
Bodonyi
effect
the
of the
of Gajjar
analysis
the
TS wave.
work
by
to that
from
nature
of the
the
given
a modified
section
arising
orientation the
thus
is very
the
A --_ O(e-
to that
and
As previously,
by the
this
layer
wave
Following
amplitude similar
shown
of the
large.
scaled
critical
determined work
the
to a structure
have
stability
so becomes
when
nonlinear
We the
that
and
vortex
vortex
on
considered
flow jump
across
vortex
is largely
be the
extension
flow structure
0 given
above.
for
6.
Conclusions
In this
paper
the problem
disc
in the
to the
we have
and
of the
wave
wave
in the
inside
the
solutions
pass
the
that
is felt
through
entirely
analysing the
this
TS
particular and
the
practical
destabilise
linear
are
above
this
adapted
to study
presence
is not extension
larger
wave
crossflow
° with
the
role
by the
nonlinear
of the
behaviour
vortex
which
the
leading
order
TS
In addition,
we have
of the travelling
TS wave
of the
on
makes
the
vortex.
On
orientation
of
that that
effect-
an angle
direction.
it is possible
is TS
or a destabilising
radial
and
as the
presence
latter
flow
is of
As far
depending
in a three-dimensional
this
flow.
and
stability
a stabilising
that
of the
unaffected.
if the
and
was obtained
problem
of the
the outward
namely
structure
as far
as
vortex,
vortex
being
the
last
This
when
the
effect
The
although
of the
analysis
for the
in
between work
these
crossflow
0°
has
two
an
types
vortex
by the
5 and be
34
that
described
to the study
with
are
vortex
on
the
linear
for
work
problem
considered
may
the
extended paragraph
for the stability
of Section which
also been
in the
is concerned.
extendable
study
has
described
the implications
is currently
Further,
-16.5
least
of a linear
so readily
either
the
layer
that
the TS wave
findings
at
TS
found
wing
and on weakly
layer
on the
been
flow is susceptible
vortex
critical
region
has
can
wave.
The
change,
infinitesimal
have
stabilises
been
flow generated
we have
interaction
vortex.
layer
of
wings.
linear
a passive
of the vortex
can
present
plays
critical
mean
has
boundary
layer
implication,
the TS
The
the
the
vortex
vortex
work
this
crossflow
on both
of the
of the
the effect
the
of instability
by
the
16.5 ° or less than
obvious
part
phenomenon,
wave,
in our
critical
through
demonstrated
vortex
neighbourhood
main
is concerned,
of a linear
interaction
of a swept
Control
analysis
in the flow induced
The
because
layer
Flow
asymptotic
vortex
wave.
flow
boundary
of this
interest
TS
basic
structure
the effect
a rational crossflow
branch
of Laminar
the asymptotic
Of particular
situated
in the
development
TS waves. TS
lower
of this particular occur
investigated
to provide a stationary
a classical
which
At first,
between
context
to instabilities relevance
attempted
of interaction
by a rotating studied
we have
of a non-
largely the
unaltered
stability
vortex
was
of a weakly
nonlinear
for the case
of a nonlinear
this
larger
TS
of an easily
TS wave
in
vortex
amplitude.
This
authors. of Gajjar
(lg80)
by asymptotic
points means.
to an This
even again
provides
scope
Finally, action
we should
between
is likely the
for future
other
relative looking
vortex
mode stress
at the
developed
Reynolds
numbers
TS wave
numerical
and
be present
of other
possible
effective here
at lower
we have
crossflow
(1986)
can the
deal
importance
Reynolds
numbers
to be able
of the
crossflow
at
the
stationary
only be resolved
zero
we note
on
the
by pursuing
wall
that
asymptotically
vortex
it
to classify
by having
In addition,
interactions
flows
In particular, and
is characterised
with
of inter-
In realistic
TS waves
profile.
can
problem
mechanisms.
between
velocity
only
the
it is desirable
interaction
which
crossfiow
considered
TS instabilities.
and
of interaction
by Hall
and
here
could
problem
described
approach
the
modes
for the
that
of specific
importance
we are
shear
remark
a pair
that
work.
our large
stability
of
extensive
calculations.
Acknowledgements
The
research
10176,
and
under
NASA
the
data
authors
that
used were
Engineering,
of A.P.B. of P.H.
Contract for
by
No. Figure
in residence Hampton,
was
supported
the
National
NASI-18107. (6).
Part
at the VA 23665,
by
SERC
Aeronautics We thank
of this
Institute USA.
35
under and
Dr.
research
for Computer
Space
J.S.B. was
Contract
Gajjar
carried
Applications
No.
XG-
Administration for supplying out
whilst in Science
the &
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A.P. & Gajjar, boundary-layer
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A handbook
1988 Non-stationary Proc. R. Soc. Lond.
Bassom, A.P. & Hall, P. 1988 On the generation of GSrtler vortices and Tollmien-Schlichting waves Rep. No. 88-43. To appear in Stud. Appl. Math.
of mathematical
functions,
cross-flow vortices A 417, 179-212.
of mean in curved
in three-
flows by the interaction channel flows. ICASE
Bennett, J. & Hall, P. 1988 On the secondary instability vortices to Tollmien-Schlichting waves in fully developed flows. 445-469.
of TaylorGSrtler J. Fluid Mech. 186
Bennett, J., Hall, P. & Smith, F.T. 1988 The strong nonlinear interaction of Tollmien-Schlichting waves and Taylor-G6rtler vortices in curved channel flow. ICASE Rep. No. 88-_5. To appear in J. Fluid Mech. Bodonyi, R.J., Smith, F.T. & Gajjar, J. 1983 Amplitude of boundary layer flow with a strongly nonlinear critical layer. 30, 1-19.
dependent stability IMA J. Appl. Math.
Daudpota, Q.I., Hall, P. & Zang, T.A. 1988 On the nonhnear GSrtler vortices and Tollmien-Schlichting waves in curved channel Reynolds numbers. J. Fluid Mech. 193,569-595. Federov, B.I., Plavnik, G.Z., Prokhorov, sitional flow conditions on a rotating disc.
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Gajjar, private
J.S.B. 1989 Nonlinear communication.
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I.V. & Zhukhovitskii, J. engng. Phys. 31, layers
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Figure Captions
Fig (1). The dependenceof (2.7)
and
nate
_.
satisfying
Fig (2). the
TS
wave
Fig tions
(3).
to the
of the
neutral
defined
The neutral
(2.13),
mean (2.14),
(2.21))
non-dimensional
by (3.1)
functions
expressed
(i) M1,M2
wavenumbers
flow terms upon
the
wavenumbers as functions
and
and
Uao, V3o and
frequency
boundary
&d,/_d and of the
(ii) Rx,R2
(defined layer
due
I2d for
¢ = tan -1/z.
determine
TS wave
by
coordi-
frequency
waveangle
which
of the
W3o
the
to the
correcpresence
vortex.
Fig
(4).
evolution
The
equation
Fig which
The
(2.11),
the
(5).
The
determines
real (4.14)
real the
Fig
(6).
The
Fig
(7).
Dependence
is defined
by
part
coefficient
expressed
part
of the
coefficient crossflow
q_ = ¢()_c)
of the (5.17)
and
7"4 of the
as a function
of the
effect
function
(5.15),
of the
vortex
given
39
term
evolution
equation
7"3 in the vortex
by
in the
TS
of #.
on the
(5.15)
quantity
A is the
nonlinear
scaled
stability
and
TS wave.
(5.17).
CA 2 upon vortex
of the
(4.14)
the
amplitude
parameter size.
)_c. Here
(.D
°m T,-
ii
C_
J
r_D |
0 C_J
40
41
,-,.
--I Lib
I |
Tm
U')
Tm
X
C_ C_j
42
m,.
LL
I
I
n
A
_
Tm
!
!
43
44
C_ o_ UI
I
L_ _lo
I
I
_1I
c_
I
_
I
Tul
C_
Tm |
|
C_J |
45
o-
IL II
I
C_
46
LL
I
I
C_ Tm
c_l
47
iln |
I.I.
I
C_I
_" I
48
U_
I"
.__ LI.
I__
l
1
I
__I
I
49
_D
-L_
m
,ira
I
0 (_1
0
SO
_
I
L
I
.1.
L
.L
d