On the interaction of stationary crossflow vortices and Tollmien

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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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