NASA Contractor Report 182037

ICASE Report No. 90-32

ICASE

CONCERNING THE INTERAC_ON OF NONSTATIONARY CROSS-FLOW VOR_CES IN A

THREE-DIMENSIONAL BOUNDARY LAYER

Andrew P. Bassom

Philip Hall

Contract No. NAS1-18605

May 1990

Institute for Computer Applications in Science and Engineering

NASA Langley Research Center

Hampton, Virginia 23665-5225

Operated by the Universities Space Research Association

National Aeronautics and

Space Adminislralion

Langley Research Center

Hampton, Virginia 23665-5225

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CONCERNING THE INTERACTION OF NON-STATIONARY

CROSS-FLOW VORTICES IN A THREE-DIMENSIONAL

BOUNDARY LAYER

Andrew P. Bassomland Philip HalP

Department of Mathematics

North Park Road

University of Exeter

Exeter, Devon

EX4 4QE

UNITED KINGDOM

ABSTRACT

Recently there has been much work devoted to considering some of the many and varied

interaction mechanisms which may be operative in three-dimensional boundary layer flows.

Here we are concerned with resonant triads of crossflow vortices. In contrast to much of the

previous work we examine the effects of interactions upon resonant triads where each member

of the triad has the property of being linearly neutrally stable; then the importance of the

interplay between modes can be relatively easily assessed. We concentrate on investigating

modes within the boundary layer flow above a rotating disc; this choice is motivated by

the similarity between this disc flow and many important practical flows and, secondly, our

selected flow is an exact solution of the Navier-Stokes equations which makes its theoretical

analysis especially attractive. Firstly we demonstrate that the desired triads of linearly

neutrally stable modes can exist within the chosen boundary layer flow and then subsequently

obtain evolution equations to describe the development of the amplitudes of these modes once

the interaction mechanism is accounted for. It is found that the coefficients of the interaction

terms within the evolution equations are, in general, given by quite intricate expressions

although some elementary numerical work shows that the evaluation of these coei_cients

is practicable. The basis of our work lends itself to generalisation to more complicated

boundary layers and effects of detuning or non-parallelism could be provided for within the

asymptotic framework.

tResearch was supported by the National Aeronautics and Space Administration under NASA Contract

No. NAS1-18605 while the authors were in residence at the Institute for Computer Applications in Science

and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665.

§1. Introduction.

The desire to understand the phenomenon of transition to turbulence in fluid flows

has motivated a wealth of studies concerned with the instability of three-dimensional

boundary layer flows. Here we address the problem of providing a self-consistent

asymptotic description of the interactions within a resonant triad of travelling crossflow vortices occurring in a three-dimensional boundary layer at a large Reynolds

number. In particular, we relate our analysis to vortices in the boundary layer of the

flow induced by a rotating disc. This flow is susceptible to instabilities similar to those

which occur in flows over a swept wing; a situation which has practical relevance to

the development of Laminar Flow Control wings. Further, there is an exact solution

of the Navier Stokes equations which describes the rotating disc flow which makes

this particular flow especially suitable for a theoretical analysis of the interaction of

travelling cross-flow vortices.

The resonant interaction of a triad of waves is one of a number of possible mechanisms that play important roles in describing the nonlinear evolution of disturbances.

Such an interacting triple is possible whenever three waves with wavenumbers k i and

frequencies _j (j -- 1,2,3) such that kl = k2 + k3 and _1 = _2 + _3 can coexist.

This type of mechanism is the subject of our current investigation, but an excellent

account of the variety of interaction phenomena which are possible within the scenario

of fluid mechanics has been given by Craik (1).

The cross flow vortex instability structure considered here can only occur in threedimensional boundary layers and was first examined both theoretically and experimentally by Gregory, Stuart & Walker (2) in the context of rotating disc flow. They showed

with the china-clay technique that this instability takes the form of a regularly spaced

pattern of equiangular spiral vortices which are stationary relative to the disc. In this

paper GSW showed, using inviscid theory, that the stationary vortex mechanism is associated with an inflexional profile in which the inflexion point coincides with a point

of zero velocity somewhere in the flow. Their calculation of the number of vortices

expected to be seen in the flow above a rotating disc gave a number approximately

four times greater than the observed value of about 30, but the angle of 13 ° between

the axes of the vortices and the radius vector predicted was in good agreement with

that seen in experiments. Brown (3), and later Cebeci & Stewartson (4) solved the

Orr-Sommerfeld equation numerically and obtained values for the critical Reynolds
number for transition much lower than that observed by Gregory et. al. and used

by Stuart in his calculation for the number of vortices. Malik (5) found the neutral stability curve for these stationary disturbances and he showed the existence of

a second stationary mode (this one however being viscous) which corresponds to zero

wall shear stress of the crossflow velocity profile. The asymptotic structures (at high

Reynolds numbers) of both this viscous mode and of the inviscid mode of GSW were

obtained by Hall (6). The linear analysis presented in that paper has been extended

by MacKerrell (7) and Gajjar (8) to include nonlinear effects.

The aforementioned investigations have concentrated on stationary modes. However, Fuller & Kaylor (9) and Lilly (10), in their studies of periodic perturbations

imposed on Ekman boundary layer flows, mentioned the existence of an instability

with non-zero phase speed. Federov et. al. (11) showed that in rotating disc flow

a similar instability occurs much earlier than the stationary mode observed in the

experiments of Gregory et. al.. This instability also appeared as a pattern of spiral

vortices, but which was moving relative to the disc. The number of these vortices was

observed to be between 14 and 16, and had axes inclined at about 20 ° to the radius

vector. Recently, Bassom & Gajjar (12) have conducted a theoretical analysis of some

aspects of non-stationary vortices in rotating disc flow and, in particular, investigated

the properties of neutrally stable nonlinear modes. Our present concern is with the

time dependent version of the high Reynolds number inviscid crossflow vortex whose

flow structure was elucidated by Hall (6). The modifications required to his analysis to incorporate the effects of unsteadiness are relatively straightforward and are

considered in the following section.

There has been much work in relation to the important problem of the interaction

of Tollmien-Schlichting disturbances with GSrtler vortices (an instability associated

with flows over curved surfaces). See, for example, (13-17) and the references therein.

Also, some study has been made of the interaction between ToUmien-Schlichting waves

and steady crossflow Vortices, (18-20_), and of the triad interaction between three unsteady crossflow vortices under consideration here. In particular, we refer to the work

of Reed (18, 19) and E1-Hady (21), who found that the growth of each of the instability modes could be substantially influenced by their mutual interaction mechanisms.

Additionally, non-parallel effects were included and it was concluded that the evolution of the vortices was critically dependent upon the initial amplitudes of the modes.
However,theseaccounts ignored the crucial fact that for a completely rational description of the importance of the interaction process in the development of the disturbances

each of the instability modes involved should be neutrally stable at leading order in

their own rights. Otherwise, the vortex growth due to the interaction mechanism is

no larger than would be experienced by the modes in the absence of the interaction,

and the true importance of the mechanism is extremely difficult, if not impossible, to

accurately assess. For this reason, our present aim is to first illustrate that a suitable

triad of travelling crossflow vortices can exist in a three-dimensional boundary layer;

crucially, these modes are chosen so as to ensure that each is individually neutrally

stable at leading order.

Once the existence of the desired triad is established we move on to consider the

problem when all three modes are simultaneously present. In the case of a single,

infinitesimally small mode the asymptotic flow structure consists of a main layer (coincident with the majority of the boundary layer) augmented by a thin, viscous wall

layer next to the surface of the disc and by at least one linear critical layer present

somewhere in the flow. With all modes of the triad present, the corresponding flow

structure becomes much more complicated, for now we have at least three critical layers (one for each mode) and sometimes more. It transpires that in order to obtain a

rational description of the evolution of the triad each of the critical layers needs to be

carefully analysed, for each contributes non-zero terms to each of the final, coupled

amplitude equations for the three modes.

An important characteristic of our analysis is that each vortex is taken to be sufficiently small and to develop on appropriately slow length and timescales such that all

the critical layers remain linear in character. We emphasize that although the equations obtained in this paper are specifically oriented towards the rotating disc flow, for

the cases of other basic velocity profiles the derivation procedure would be essentially

identical. The method adopted also allows nonparallel effects to be accounted for without formal difficulty. The linked triad of evolution equations, somewhat unfortunately,

suffer from the handicap that they are very complicated and require much delicate nontrivial numerical work to establish the interaction coefficients for any particular flow.

Here we perform some of the calculations for the rotating disc situation to demonstrate

that the derivation of the evolution equations is practicable. The equations so derived

have complex-valued coefficients and are not dissimilar in form to those derived by

3
Smith & Stewart (22) who were concerned Wi_th:lllgh-frequency resonant triads within

boundary layers. We consider that the principal result of the current work is that in

three-dimensional boundary layers resonant triads of non-stationary crossflow vortices

do exist. Further, given such a triad of modes which are neutrally stable at leading order, we show that we can develop a fully rational asymptotic account of the interaction

processes.

The procedure for the remainder of the paper is as follows. In section 2 we

briefly consider the asymptotic structure of a linear, travelling crossflow vortex in a

three-dimensional boundary layer, making specific reference to the case of rotating

disc flow as an example. Numerical work in section 3 establishes that the desired form

of a resonant triad of modes exists for this special basic flow and, armed with this

information, we make a careful study of the interaction mechanism in section 4 to

enable us to derive a coupled triple of evolution equations for the modes. Finally, we

perform some numerical work to establish the interaction coefficients, discuss a few

solutions of the relevant evolution equations and we conclude with some discussion

and suggestions for further work.

§2. The basic equations and flow structure for a linear crossflow vortex.

We have already stated that we intend to make an investigation of the interaction

processes which take place between the three members of a resonant triad of crossflow

vortices and a natural first step is to consider the disturbance structure for each individual mode in isolation. Here we concentrate on the rotating disc flow problem for

definiteness, although the work can be trivially modified to account for other threedimensional basic flows; for example the asymptotic suction profile Or the attachment

line flow, see Hall & MacKerrell (23). In the particular problem at hand, we Consider

the case in which the disc rotates about the z-axis with angular velocity f_. Relative

to cylindrical polar co-ordinates (r, 8, z) which rotate with the disc and in which r and

z have been made dimensionless with some reference lengthscale L, the continuity and

Navier-Stokes equations for an incompressible fluid in the region z > 0 are

V.u 0,

0U

0-7+ (u.V) u + 2(a Au) + ft ^ ^ r) = + V2u.

P

4

L
Here u is the velocity vector, r is the co-ordinate vector, p is the fluid density, p is the

fluid pressure and _ the kinematic viscosity. The Reynolds number R for the flow is

given by R = _L2/v and is taken to be large throughout the following.

It is found convenient to define the small parameter e = R- }. As the axes rotate

with the disc, the basic flow is given by the Von-K_rm£n solution

u : u,_ = LG (r_(_),ro(_),e3tv(_)) , p _- pL2_2e6lD(_), (2.2)

where z = e3_ and _, v, zv satisfy the equations

fi2--(1-i-V)2+fl'_----fl '', 2fl(1-t-I))+I)'ti)--V", (2.3a, b)

zD' -t- 2_ = 0. (2.3c)

Here primes denote differentiations with respect to _ and the appropriate boundary

conditions are

= v=w=0 on _=0; fi_0, 0---+-1 as _------eoo.

The structure of an infinitesimal unsteady crossflow vortex imposed on the above

basic flow follows closely that given by Hall (6) for the steady case. The mode has

wavelengths scaled on the boundary layer thickness and an O(1) wavespeed and so we

consider perturbation quantities with r and _ dependence given by E, defined by

E-_exp[_-5(/'e_(r,e)dr-t-813(e)-'_(e))] •

(2.4a)

We shall restrict our attention to examining disturbances in a neighbourhood of

some point r = r,_ and expand the wavenumbers and frequency in the forms

O_ -- S 0 -t- COL1 -{- ..., /_ -- /30 + e/_l + ..., _ = _0 + e_/1 + .... (2.4b)

In general these quantities will be complex but here we shall be concentrating on

neutrally stable disturbances (at least to leading order) and so we take s0,/30 and _0

to be real. The disturbance structure in the z-direction is sketched in figure (1). There

is an inviscid layer (I) of thickness O(e 3) (the same depth as the boundary layer), and

to satisfy no-slip conditions at the wall a viscous layer (II) of thickness O (e 9/2) must

5
be present. If the basic flow is perturbed so that the total non-dimensional flow field

is given by

(u,_,_,p) = (_,rv, _3,_,_6p)+ _[(U,V,W,P)E + _._.]+..., (2.5a)

where _ _< 1, then in zone (I) the disturbance quantities U, V, W, P (which are functions of _ alone) expand in the forms

(U, V, W,P) = (Uo + eU1 + ..., Vo + eV1 + ... , Wo + eW1 + ...,Po + eP1 +...).

(2.5b)

Substituting expressions (2.5) in the continuity and the linearised Navier-Stokes

equations yields at leading orders

(oo o+ + rn

and

If we define

=0, (2.6a)

i (nor. _ + flov - _o) Uo + r.a'Wo = -iaoPo,

i (aor._ + _o_ - _o) Vo + r. v'Wo = - i_°po,

rn

I

i (nor. a + _ov - So) Wo = -Po.

Ui =aJr-_+flJV-GJ,

then we obtain the classical Rayleigh equation

00(w0'- o Wo)=00Wo,

where _/_ - a2o + _-_• The required boundary conditions on (2.7a) are, firstly,

IWo(oo)l< _, Wo(0+) = 0,

(2.6d)

(2.7a)

(2.7b, c)

to avoid exponential growth at the edge of the boundary layer and to give the typical

inviscid condition of tangential flow as the wall is approached. Technically, to reduce

this slip velocity to zero at the surface of the disc, analysis of the viscous layer (II) is

required but such analysis does not affect the leading order quantities no, fl0 and g/0

which are of the primary interest here.

6

=.
The Rayleigh equation (2.7a) additionally has singularities at any positions where

90 = 0, and, if this occurs, the presence of an internal critical layer Cat that position)

is suggested. The details of the asymptotic analysis of such a (linear) critical layer

follow well-known lines (see for example Drazin & Reid (24)) and so we merely sketch

a few of the relevant results here. If in the vicinity of the critical layer, at _ = _ say,

we have the Taylor expansion

Oj --_ 7. (_-_-_ , (2.8a)

r =1

where we have defined B_ - (a_r,_ (k) + _jv (k)) ]e_--, then the solution of (2.7a) as

_ _+ is given by

Wo= g,

T-----O

where g0, gl are constants upon which the remaining coefficients g, (n = 2,3,...),

G, i n = 1,2,...) depend. In particular, we have G1 = goBo2/Bol. The solution

(2.8b) implies that U0 and V0 in (2.5) have logarithmic singularities as _ _ _+ and

to smooth these we need to invoke viscous effects in a thin layer (III in figure 1)

surrounding _ = _. Balancing inertial and diffusive terms suggests that this zone is

of thickness O(e 4) and if we define _ = _ + e¢ here, where ¢ = O(1), the disturbance

quantities develop as

U = e-l_0 + lneU1 + _r2 + eln e_r3 + e_4 + ...,

(2.9a)

V = e-l%+ Ine_'l+ _2+ eln e_z3+ eV4 + ...,

W = W0 + eln eI2V1 + eW2 + e21neW3+ e2W4 -b "'',

P- P0 -b elnePl + eP2 + e21neP3 -b e2P4-b ....

(2.9b)

(29 )

(2.9d)

Briefly, we find on substituting (2.9) into the linearised Navier Stokes equations

that

_'o=

IB-oll c_0

' 2 oo

BOl /

exp[_irsgn(Bo1)<_b+_oO1)]exp( 3,_01,) dr,

(2.10 )
where Po = Po(().

(2.6) as _ _ $. Across the critical layer we also find that

B01 '

and

d212_/2

d¢ 2

This solution, as [¢[ _ co, matches with the solution for U0 in

B o] g0B02 (¢ + (2.10c)

iBo2go

-- 'BoI' /0 exp [-irsgn(Bol)(¢+ B10]]exPB01 / ( 3]_-01[ ) dz. (2.10d)

If we integrate (2.10d) once and then apply the asymptotic results given by Haberman (25) concerning the behaviour of integrals of the general type

fO °°M(v)(e-i'#r -1') d'r'

for large argument ¢, we may show that the derivative dl?V2/d¢ suffers a discontinuity

across the critical layer and in particular that

dl_2] '_-'-'_ __ iTrBo2go (2.11)

dr J - IB011

The implication of the solutions (2.10), (2.11) is that the Frobenius solution (2.8b)

for the eigenfunction W0 as ¢ 4--4 $+ continues to hold in the limit ¢ _ $- so

long as the replacement of ln(¢- $) by (ln(_-¢)- iTrsgn(Bol)) in (2.8b) is made.

Equivalently, we have that as ¢ _ (-,

Wo = g, (,_ - _-)" + _ G, (_ - _3" [ln($- _) - ilrsgn(Bol)], (2.12)

,=0 ,=I

just below the critical layer. This is the well known phase shift of 'it' across a linear

critical layer.

We now have a complete specification of the problem to determine the leading

order neutral wavenumbers a0, fl0 at some given frequency F_0. The Rayleigh equation

(2.7a) needs to be solved subject to the boundary conditions (2.7b, c) and where the

replacement given by (2.12) in the local series solution (2.8b) across the critical layers

is applied. The analysis presented here is specific to the case of rotating disc flow,

but may easily be modified to account for a variety of other three-dimensional basic

velocity profiles.

8
§3. The isolation of resonant triads.

At the outset our aim was to verify that resonant triads (with the property that

each member of the triad is neutrally stable at leading order) are possible within

particular three-dimensional boundary layer flows. In the context of rotating disc

flow, we therefore wish to seek ordered triples {(a05,/30i, _0i ) ;J = 1,2, 3} such that

o_01 "- a02 + oL03, _01 :/302 -4-/_03, _01 -- _02 + _03, (3.1)

and with the characteristic that each triple is an eigenvalue set of the system (2.7),

(2.8) & (2.12). To this end a numerical approach was adopted.

A first concern is to find the location(s) of the critical layer(s). These occur

wherever _'0 = 0; i.e. wherever

aor,, _ + _0_ = _0, (32a)

or, if we define the leading order wavespeed co = glo/r,_sO; wherever

U, --- _ + #i_- co, (3.2b)

with # = flo/(r,,ao). Hence, the 'effective' basic velocity profile U. is a function of p

and this flow is sketched in figure (2). It is known that a' (0) = 0.510, V' (0) = -0.616

and then as the parameter # varies the effective basic velocity acquires a number of

distinct possible forms. The boundary conditions in (2.3) suggest that Uo ----* -# as

----, oo and for # > /.tc (_ 0.8284) the profile is a monotonic decreasing function.

For 0 < # < #c, U, vanishes somewhere other than at the origin, and indeed when

# _ 0.235, the point of inflexion of the basic flow Uo coincides with the point at

which it vanishes. In this case the boundary layer becomes susceptible to the inviscid

stationary vortex found by GSW. For # < 0, U. is always a positive function and for

# <_, -0.6 it is monotonic as well. We see from figure (2) that, depending upon the

values of # and co in (3.2b), we may have either one or two critical layers.

The numerical scheme used in practice requires that a value of # be preselected

and then s0,/_0 and _0 are determined such that # =/30/(r,_a0). It is easily shown

from (2.7a) that as _ _ oo the eigenfunction W0 is proportional to exp(-"/0_).

A suitably large value of _, say _oo, was chosen and the asymptotic behaviour of

the eigensolution was assumed to hold there. With these starting conditions, the

9
Rayleigh equation (2.7a) was integrated with a fourth order Runge-Kutta scheme to

a point a small distance, say A, above the first critical layer encountered. There the

series solution (2.8b) was assumed to hold and go and gl were chosen so that the

necessary matching conditions held. Then, using the replacement suggested by (2.12),

the solution just below the critical layer was established which could then be used to

integrate (2.7a) until either the next critical layer position was reached (in which case

the process described above was repeated) or until ( = 0. In general, the predicted

value of W0(_ = 0) is complex and non-zero and so a Newton iteration on the real

values a0 and co was employed to ensure that the required condition (2.7c) was indeed

met. Given the eigenvalues a0 and co for some chosen value # it is then a simple

task to retrieve the leading order azimuthal wavenumber fl0 and the frequency f_0.

This numerical work was checked for independence from the chosen value _o,, the step

length used in the Runge-Kutta integration and of the distance A from the critical

layer(s) at which the series solutions were applied.

The solutions of the eigenproblem for various values of the parameter # (which

is tan -1¢, where ¢ is the angle which the considered disturbances make with the

outward radial direction on the disc) are shown in figure (3), where we illustrate the

dependence of the leading order radial wavenumber a0 and the frequency f_0 on the

wavenumber/_0. There are several noteworthy features of the solutions. Firstly, there

are two distinct branches in solution space which are independent of each other and do

not coalesce. On one branch (labelled I on figure (3)) we see that there is a non-trivial

solution of the eigensystem with zero frequency and this corresponds to the inviscid

stationary mode of (2). Here a0 ,-_ 1.1, fl0 "_ 0.26r,, and # _ 0.235, and then the basic

profile U. has a point of inflexion coincident with the point at which it vanishes. As

the parameter # increases we find that the frequency becomes negative and using the

sketches of the basic flow in figure (2) we see that we are reduced to an eigenproblem

with a single critical point. Appealing to the ideas of Stuart in (2), in the case of

only one critical layer the neutral wavespeed co is determined by the condition that

the function U. in (3.2b) must have a point of inflexion where Uo = co.

For a waveangle parameter # less than the inviscid stationary mode value of 0.235

we now have two critical layers in the flow and the neutral frequency is positive. As

# _ 0 the wavenumbers and frequency become small. Finally for this branch, as

# _ 0 or # _ oe then the wavenumbers in the radial and azimuthal directions

10
becomeof different orders of magnitude and the symmetry of the governing equations

(2.6) is lost. The structures which would govern these limiting cases are then not

covered by the work of section 2 and it would be interesting to identify and analyse

these structures.

On the second branch of solutions (labelled II on figure 3) it is immediately seen

that only positive frequencies are found. In particular, we remark that as # increases

and approaches #_ _ 0.187 (corresponding to a disturbance waveangle of 10.6 °) the

wavenumbers and frequency become small. This suggests the existence of a longer

wavelength disturbance with a small wavespeed and this structure corresponds to that

of the 'upper-branch' mode investigated in (12). In this paper, small perturbations

with wavenumbers O(R,_) and wavespeeds O(R-_) were shown to be possible at

waveangles between approximately 10.6 ° and 39.6 °. As # decreases from p: in our

current problem and becomes negative the convergence of the numerical scheme employed becomes increasingly difficult to achieve and only the neutral solutions which

can be reliably located by our method have been shown in figure (3). Indeed, for

sufficiently negative # the basic profile is monotonically increasing and has no point

of inflexion. At these values of parameter, no neutral solutions of the sought type can

exist and so it suggests the existence of some p, < 0 at which our solutions cut out. It

is seen from figure (2) that as # decreases from zero the neutral disturbances have two

critical layers up to the point where the basic velocity is no longer non-monotonic.

Then the two critical layers must merge, cease to remain asymptotically distinct and

a fresh analysis is required. Since the principal thrust of the present work is to study

resonant triads we do not explore this aspect any further here. However, we remark

that these considerations show that neutrally stable linearised disturbances are possible at propagation angles > tan-l(#,). There is therefore, a lower bound for this

angle but no formal upper bound since from figure (3) we can see that neutral modes

exist for p, < p < oo. This contrasts with the results of (12) where it was found that

for linear disturbances the possible waveangles associated with upper branch neutral

modes were bounded both above and below.

Given the eigenvalues of the problem specified by (2.7), (2.8) & (2.12) we sought

resonant triad triples (a0j, ;30j, _0j' ) of waves which are each neutrally stable at leading

11
l

order and which satisfy requirements (3.1). We found two such sets, in particular

c_Ol =0.8514,

a02 =0.7343,

a03 =0.1171,

801 = 0.2648rn,

802 = 0.0731r,,

803 = 0.1917r,,,

1201= --3.137 x I0-2,

_02 -- 1.12 x 10-2 ,

1203= -4.25 x 10-2,

(3.3a)

and

a01 =0.5847,

a02:0.3605,

a03:0.2242,

fl01 - 0.2764rn,

802 = 0.2699r,_,

803 = 6.46 x 10-3r_,

_01 = -6.29 x 10-2 ,

1_02 = --7.16 x 10 -2, (3.3b)

_o3 = 8.76 x i0-3.

We do not claim that these are the only possible triads but the important result

here is that resonant triples with the sought neutral stability properties do indeed exist

_within a three-dimensional boundary layer. The type of analytical and numerical work

presented here can be applied to other basicveloclty proffies. In(iced Very similar

results as to the forms of the neutral stability curves in a0/80 and D0/80 spaces

appear for the attachment line profile and, of course, given the existence of triads

. _ : . : _ ......... : ....... , :==

for a simple rotating disc flow we have no reason to doubt that a similar conclusion

holds for other much more complicated and more physically realistic three-dimensional

boundary layers.

............ . : : ......... _.

This verification of the existence of resonant triads leads us on to an integral

part of the current work- that of considering the disturbance structure of the three

members of the triad When theyare permitted to interact and evolve on a suitably

slow timescale.

_ _{-:

...... _,. , = =

_4. An asymptotic description of the interaction mechanisms in a resonant

triad of crossflow vort_s.

We have illustrated that within the context of three-dimensional boundary layer

flows resonani triads comprising of crossflow vortices can arise. We now investigate

an asymptotic description of how the members of the triad may evolve in time and

_--_i272::2:_=::2 7 ? i: : : 2! _ : ! _ ±_ 2:i 2: 2 :

space once the interaction processes are accounted for. To ensure that the structures

of the criticM-iayers in the flow remain as siml_e aS possible for this study we assume

12
that the amplitude of each vortex is 0(6) where _ << 1 is sufficiently small such that

all the the critical layers are linear in character. Then, through the nonlinear terms

in the Navier Stokes equations, it is found that each pair of modes combine to drive

the third mode via terms of size 0(5 2 ) and this in turn suggests that the interaction

mechanisms force the vortices to evolve on slow length and time scales of size O(_f).

If again we relate our working to the rotating disc problem as a concrete example,

suppose that we wish to describe the interaction of three linear vortices with leading

order wavenumbers a0j, f_0j and of frequencies _0j (j = 1, 2, 3) such that the resonance

conditions (3.1) hold. Then we define "

Ej-exp [;_ (f"_(r,_)d,-+0_j(_)-tnj(_) )], (4.1a)

Tw,

where

aj =a0j +ealj +..., flj =fl0_ +efll_ +..., Ftj =_0j +eFt14 +..., (4.15)

and E1 = E2.E3. We allow the disturbance vortices to develop on slow lengthscales

rl = 5 (r -r,,), 01 = 50 and timescale tl = $t, so that in the governing Navier Stokes

equations we formally use the methods of multiple scales to replace the r, 0 and t

derivatives according to the transformations

0

_T

0 0 0 0 0

0 0 0

ot , g+ e-_.

(4.2)

As previously, we choose to examine vortices in the neighbourhood of some point

r = r,, and then, in the main inviscid region (I) where z = e3_, we are led to consider

perturbation quantities of the forms

v=_E(_oo_+...)A_zj+_.c. +6 2 V_o_E_+... +..., (4.3_) [

w-_,_(Woo;+...).4;E;+_._. +52,_[2WlojEj+... +..., (4.3_)

[j=l

13
P=&Lj_=I(POoj+'..)AjEj+ f' c.c :i+6 2 [ P10jEj+... I+ .... ; (4.3d)

In (4.3) the scaled amplitudes Aj are functions of the slow scales rl, 81 and tl

and U00f, V00j,..., U10f,... are functions of _ alone. Substituting (4.2) and (4.3) into

the Navier Stokes equations and comparing coefficients of Ef we find, as in section 2,

that at leading orders Woof satisfies the Rayleigh equation

O0, (Wo'o,- 72, W00,) -O'o'hWo0;, =0, k= 1,2,3, (4.4)

where 0ih = aih r, a+/?f k V_gtih and .y_k = a2 h +(fl_/r,). 2 2 The functions Uooi and Vooj

may be found in terms of WOOs by suitable manipulations of the governing equations,

which are precisely (2.6) with no, rio, Qo replaced by aoi, fl0i, QOj and Uo, Vo, Wo, PO

by U00_, Voos, Woof, Po0i.

LFrom here onward we concentrate on deriving the evolution equation for the

vortex proportional to El. Given the obvious symmetry in the problem, together

with the generalised way in which the" amplitude equations will be derived, once the

governing equation has been obtained for one mode it is formally straightforward

= :

to obtain the corresponding equations for the other two modes. At O(& 2) in the

Navier Stokes equations we find that the function WI01 in (4.3c) satisfies the nonhomogeneous Rayleigh equation

II II

oo ( _ 0ol / _ aOl _-rl +

where

]_01 OA1

r. -_1 / + I.A2A3,

(4.5_)

L = i [(Zo3aol +

Uoo3Vo02 _o2Zol)

rn

+ (fio2_ol + -o3Zol)

Uoo2Vo03 2

+ GO1UOO2UOO3+

_--_0_1VO02 VO03 ],[ _oo_(oo_oo_,_O_oo2),_oo_(oo_oo_÷_

rn . rn 7"n

-_ [_(oo_oo_oo_, oo_oo_oo_, _o__oo_oo_ __O_oo_oo_)

r, r,

+Woo3Wo02, + Woo2Wo03,] ,

(4.5b)

14

I

=

=

%-_

=_

=
and

-- -b r,, fl ÷ e . (4.5c)

The appropriate boundary conditions on the function W01 are very similar to

those for W001 in (4.4). In particular, to avoid unbounded growth as _ _ c_ requires

that ]W101(oo)] < oo and for no distmrbance at the surface of the disc, W10I(0+) =

0. Additionally, it is clear that appropriate jump conditions need to be imposed at

the position of the critical layer(s). For ease of notation, suppose that the mode

proportional to Ej has associated critical layers located at 4_" and note that there

could well be more than one critical point for each mode. Inspection of (4.5) suggests

that singularities in the solution W101(_) may be expected at all the critical layers 4_

for j = 1, 2, 3, not just at the positions 41.

The form of the solution of (4.5a) in the vicinity of the critical layers at 4_ may

be one of two distinct types. Close to the position _1 we can easily show that the

right-hand side of (4.5a) is asymptotically proportional to 1/(_ - 41) and from this we

can deduce that for 0 < _ - 41 << 1,

w101= :_0log(_-41)+ o(1) + ..., (4.6a)

where X0 is a constant which may be found analytically. Close to the other critical

layers, 4_' (J = 2, 3), the behaviour of W101 is more complicated. Then we obtain, from

(2.6) & (2.8), that

and then

Xo_ XI_ X2_

x. ~ (_-- _)3 + (_-_)2 + (__ _) +..., (4.6b)

D0_ ln(_ _.)+ D2_. + D3;(_ _.)ln(_ _.)+ (4.6c) w_0_~ (__ 4j) + D1; - - - ...,

where the constants Xkj., D,j. are given in the Appendix. The solutions (4.6) are valid

for _ > 4j and clearly need to be modified for the region _ < 4_'. This requires careful

scrutiny of the critical layer properties in the vicinity of( = 45 in much the same way as

was outlined in §2. The main difference now is that the algebraic manipulation needed

within the critical layer region becomes much more complex due to the presence of

all three modes. Rather than give details of the inevitably lengthy (but conceptually

fairly straightforward) critical layer al/alysis, we elect to merely state that the result

15
of this work is as is to be expected from the classical theory: that is that we obtain a

'i_-' phase shift in the logarithmic terms as characterised by (2.12). This then enables

us to write down the behaviour of W101 for 0 < _j - _ << 1 and thence to continue the

solution W101 into the region _ < _#.

To derive the evolution equation for the amplitude A1 we follow the usual process

of multiplying (4.5a) by W001 and integrating over the range [0, c_), paying particular

attention to the behaviour of the integrals close to the critical layers. Recalling the

solutions (4.6) and the phase jumps across the critical layers, combining the integration

with the boundary conditions imposed at _ = 0 and at _ = _ leads to the desired

amplitude equation. After simplification, and substituting the definition of l: given by

(4.5c), we obtain

IBolllBotlr,.,_ Wool (W001) finite part _B-'0_]I_

-r._ ( Bo212 Bo13] i'2_._Bol2Wdol2 2

IBol_l_'_ _ W_ol B_ll ",-,

+i "_'01 W(J01 - t d_ + 2i aolr,,W_old_i

_rBol2r._/'W_O 1

]BolllBoll

+

fo" ( "'' °°°"w_ ]

+

+

(

]B011]B011 _Bo_ _,Woo_(Woo0 ,finite part I_-;7o_1W_o_ 7r_" +

z_" VB012W{_01 )

IBonl \B'_on B--'_-lnz W°2°t B_ll ¢,

+ 2i fo®_olW2oold_] Oal

' 2

7rBol2VW6ol

]Boll]Boll

( _.

B011

• 2 2 2

Bo13_ W201 zlr Bol2Wdol)

B-_n,' B_n ¢--,

_o_ Oo'_,_,__] o,4_

]

A2A3 = O.

16

(4.;)

|

v_
In this amplitude equation we have defined

B0;_ = (r.ao_(_)(_) + _o_O(_)(_)) ,

and have supposed that

(4.s)

oo

Woo =

/_--0

close to the critical layers _ (j = 2, 3) at which this function does not have singularities.

Otherwise, the notation is as previously defined.

In (4.7) the expression _t denotes the Hadamard (finite) part of the corresponding

integral. Some of the integrals which are contained in the coefficients of OA1/Orl,

OA1/O01 and aA1/Otl are singular at _1: the integral in the coefficient of A2A3 is

singular at the other critical points as well.

We notice that if the various integrals and summations were to be evaluated numerically and the formulae (4.7) rearranged appropriately, we could obtain an evolution

equation for the E1 mode which assumes the form;

OA1 Otl + ),11 _ + ,_12_- . OA11 + A13A2A3 = 0, (4.9a)

where the coefficients ,_11, _12, _13 are some constants. By repeating the analysis outlined above we can, in an identical way, derive evolution equations for the other two

OA2 + OA2 OA2 ,

_21_Trl + _22-_1 + _23A1A 3 = 0, (4.95)

OA._..._3 OA3 OA3 .

+ A31"W-'-+ A32_ + A33AIA 2 = 0, (4.9c)

Orl o_1

modes of the types

where an asterisk on a quantity denotes the complex conjugate of that quantity.

Clearly, the numerical work required to actually evaluate the coefficients in (4.9)

for any particular basic flow is nontrivial. However, to illustrate the type of evolution

system which may be obtained we considered the computational procedure relating to

the resonant triple given by (3.3a). For each member of the triad we computed the

eigenfunction W00j' by solving the Rayleigh equation (2.7a) subject to boundary conditions (2.7b, c) and jump conditions (2.12). The corresponding radial and azimuthal

velocity fields U00_, V00i may then be determined in terms of the functions /70_. and

W00j by eliminating the pressure from (2.6b, c). Since the problem (2.7) is linear we

17
neededto fix some normalising constraint and this was chosen such that as _ ----, or,

Wo0j was a real, positive function and that max{0 < _ < co : IW00j I = 1}.

At this stage the coefficients )'sl and )_2 in (4.9) were evaluated. The integrals

involved in finding these unknowns are singular only at _i and the analytic nature

of these singularities may be deduced in a routine manner. To find the finite parts

of the integrals across the positions _j, we subtracted the singular behaviour of the

integrand in the neighbourhood of _i and integrated the remaining (necessarily well

behaved) function across _j. By utihsing the formula given by Lighthill in (26) it

was possible to deduce the Hadamard part of the subtracted singularity function and

thence determine the value of the finite part of the original integral. The method

worked well in practice and the corresponding results are given below.

We turn now to address the much more complicated task of determining the

interaction coefficients _3 and again, as a concrete example, concentrate on the determination of _13 for the method for calculating the other interaction coefficients

follows in a very similar way. Then the function I. is as defined in (4.5b) and was

computed by using the information previously derived on UOOs, V00_ and Wo0_' and

detailed above. It can be shown by straightforward analysis and using (4.6b) that the

integrand (I, WOOl/O01) has-singularities at each critical layer and to calculate this

integral a technique was employed which is very similar to that already described. In

the vicinity of each critical layer the singular part of the integrand was removed as

before, except now we found that the remaining function was very erratic near the

critical point. This is due to the fact that the integrand can only be determined numerically at the outset and the process involves the estimation of several derivatives of

Uooj, Voo_ which is a probable:source of numerical error. On making the usual checks

for numerical stability, the determination Of the finite part of the integral was found

to be strongly influenced by the choice of the intervals around the critical points in

which the singular behaviour was removed. This is obviously unreasonable and a more

satisfactory procedure was found in which the integrand was written as the sum of

the singular part and an interpolated polynomial in the vicinity of the critical points.

Even so, in this case there was found to be some variation in the determined value

of the coefficient _13, but it has to be remembered that this coefficient depends on a

calculation in which four separate critical layers are traversed with a strong singularity

needing to be accounted for at each critical layer. Consequently, the difficulties en18
countered in obtaining accurate values for the interaction coefficients in the evolution

equations (4.9) are not surprising.

In this paper we have managed to obtain approximate values for the interaction

coefficients. What we have shown is that determination of the interaction coefficients

for the flow structure considered here is possible in practice, and, with the use of more

sophisticated numerical methods, there is no reason to doubt that the accuracy of the

coefficients could be markedly improved.

We now describe some properties of the solutions of the evolution equations and

conclude with some discussion.

19
§5. Results and Discussion.

The numerical work described above leads to evolution equations which assume

the forms

__ _3,0A1 OA1

OA1 = r, (-0.114, 4.09 x 10 )'_rl + (0"305'0"111)'_1 + (2.0,0.6)A2A3, (5.1a) 0tl

OA2 _ r,(7.52 × 10-2,-7.74 x 10 -2 ) _1 + (0.265,0.117)-_1 . OA2 + (0.15,-0.15)A1A3, ,

¢9t1 (5.15)

. cOA3 . OA3

OA3 _ r,,(-2.4 x 10-2,-0.14)--_-rl + (0.14,0.13)-_- 1 + (0.3,0.4)AIA_, (5.1c) Otl

for the resonant triple given by (3.3a) and (4.3). As already mentioned, the effect

of having four critical layers within the flow makes the calculation of the interaction

coefficients very sensitive and we believe that the values of these quantities quoted in

(5.1) are correct to within about ten per cent.

Naturally, the evolution of the wave amplitudes A1, A2 and A3 is very dependent

upon the initial conditions and the behaviours of these quantities may, in general, only

be determined by a numerical approach. We do not study this facet of the problem here

but, instead, draw an analogy between the forms of (5.1) and those derived by Smith &

Stewart (22) in their investigation of resonant triad nonlinear interaction in boundary

layer transition. The evolution equations derived by these authors were a simpler form

of (5.1), (effectively with 0/0rl and 0/081 replaced by constants) with the interaction

coefficients purely imaginary quantities. Smith & Stewart integrated their equations

using a predictor-corrector formulation to march forward in time and the resulting

evolution of the wave amplitudes took a very complicated form. These amphtudes

generally increased with time although this trend was modified by the superposition

of a very short period, large oscillaticm which had the overall effect of giving the

function IA_ (tl)] a very spiky appearance. Similar types of evolution equations have

been derived by Avis (27), who was concerned with resonant triads in axisymmetric

flow on a cylindrical geometry, and by Bowles (28), who investigated triads within

transonic flow. These authors obtained numerical solutions for the development of the

mode amplitudes, and both found behaviour qualitative to that described in (22).

Very little is known about particular solutions of (5.1), although some work on

related equations has been studied by Craik (1, 29, 30). No general analytic solutions

for these equations have been found. However, an important property of the system

2O
(5.1) is under what conditions these equations admit singularities in the solutions Aj,

(j = 1,2,3), either in a finite or an infinite time. Wang (31) has derived bounds for

the solutions of three-wave interaction systems with arbitrary coupling coefficients.

He showed that if

dao dal da2

d"-_ = iwoao + #oala2, d"-i- = iwlal + I._laoa_, d-"t" = iw2a2 + #2aoa*l, (5.2)

with initial data a,(0) = a,o, i = 1, 2, 3: then explosive instabilities (instabilities that

become unbounded eventually, possibly after infinite time) cannot exist if

< 0 and ll_(o)ll < (5.3a)

where

a = min{ max{$o0, ,k12,),21}, max{A01, _,10,_22},

max{$oo, _01, $22}, max{A01, _12, _21},

max(_02, )_10,_21}, max{)_02, All, X20}),

(5.3b)

1

m (Ip, I}, and A,¢ I ;I- 2Im (co,). (5.3c)

Here Ilall- Jl(ao, al, a2) J] _ (_2 ,=0 a,a[ ) 1/2

If we allow the disturbances proportional to (4.1) to be neutrally stable in space

co-ordinates rl and 81 we can substitute

-- and i_l, (5.4)

r.

for 0/Or1 and 0/0el in (5.1) with crl and/31 real. Then using the results (5.3) we

can find the region in (ai,_l) space for which explosive instabilities of the system

(5.1) are impossible, at least for sufficiently small initial disturbance amplitudes. This

information is illustrated in Figure (4) where we indicate the minimum size of the initial

amplitude required in order to ensure'that the possibility of explosive breakdown is

maintained. Broadly speaking, we can conclude from figure (4) that once we are

located in that part of (al,/31) space at which a finite initial amplitude is required to

enable explosive breakdown then by decreasing the radial wavenumber al and/or by

increasing the azimuthal wavenumber _1 there needs to be a corresponding rise in the

initial disturbance amplitude if the breakdown mechanism is to remain possible.

21
It should be noted that the point corresponding to the choice o_1 _- _1 -- 0

lies in the zone for which the possibility of explosive instability cannot be discounted

irrespective of the initial conditions. Then the governing equations reduce to the forms

dA1 (2.0, 0.6)A2A3, dA2 (6.15, -0.15)A1A_, dA3 (0.3, 0.4)A1A_.

dtl dtt dtl

(5.5)

These are, of course, the evolution equations obtained by allowing the disturbances

to evolve temporally whilst suppressing spatial dependence. It is a straightforward task

to write bl = A_, b2 --" A2, b3 -= Aa and to normalise so that the system (5.5) may be

rewritten as

db_

-_1 = 8,b;b_, (i,j,k = 1,2,3 cyclically), (5.6)

where the interaction coefficients s_ - exp(i¢_). Craik (29, 30) has shown that 'phaselocked' solutions of systems analogous to (5.6) exist where O = 01 -t- 02 + 03 is constant

and 0_ = phase(b_). Then ® must be a root of

tan(¢i -- ®) = 0.

In this case the existence of roots of this equation is guaranteed and then, for appropriate initial conditions, the solutions b_ can exhibit a finite time singularity. It

has to be remembered that in practice once the disturbance amplitudes become too

large however, then the assumptions concerning the smallness of the amplitude within

definitions (4.3) cease to remain valid and a revised asymptotic analysis is required.

Apart from the results relating to explosive instabilities described above, very little

is known about solutions of system (5.2) for general parameter values. In the course

of an investigation into interaction processes between crossflow vortices in practically

important cases one would have to resort to extensive numerical calculations to provide

a full description of the characteristics of the solution of the evolution equations. This

provides scope for extending the elementary ideas presented here to the potentially

more important interactions in experimentally generated boundary layers.

The derivation of the evolution equations relating to the second resonant triad as

given by (3.3b) was also considered. However, for this triad, the mode proportional to

E3 has a critical layer very close to the Wall. In practice, it was found that the numerical

scheme described in §4 was too crude to accurately model the singular behaviour close

22
to the wall to provide satisfactory convergence. This discovery points to the need to

develop improved numerical procedures to compute the interaction coefficients.

The asymptotic arguments behind the derivation of the amplitude equations for

modes with spatial and temporal depe.ndences as in (4.1), (4.2) relied on the assumption that the amplitudes of the disturbances were O(*) across the majority of the

boundary layer, where $ << 1. This condition was introduced primarily to ensure that

the viscous critical layers within the flow assumed their simplest possible forms: i.e.

linear critical layers encompassing the usual 'it' phase shift. We can formally relax

this requirement by making the following observations. Recall that

-e p [, (/" )], (4.Za)

and suppose that, according to linear theory, the mode proportional to E i is neutrally

stable when

¢zj =o¢0i + EOLlj + ...... + _k&h i + • ..,

_ =_0_ + e_Z_ + ...... + d',b,,_ +..., (5.7)

_2_ =_20_ + efZz5 + ...... + ck (Zk_ + ....

Suppose we allow the disturbance to evolve on slow space and time scales given by

(4.2) when we replace

m0 by d' (o ) (5.8) 8r1 _rl + i&hj ,

and make similar replacements for _(0/0_1) and _(O/0tl). Then the analysis of the

interaction mechanism will proceed in a very similar way as before, so long as the

critical layers remain linear in form. Following the work of (8, 20), we know that

the critical layers appearing in this type of disturbance structure remain linear in

character until the disturbance amplitude reaches the size O(e 2) at which point the

critical layer configuration is modified due to the presence of (relatively) large mean

flow corrections and the calculation of the phase shift across the layer becomes a

numerical task. However, we can see that the derivation of the evolution triad of

equations (4.7) may be trivially modified to account for the interaction between three

modes of finite amplitudes, so long as these amplitudes remain less than O(e2).

To conclude, we believe that the major finding of our work here is that triads

of linearly neutrally stable crossflow vortices can exist in three-dimensional boundary

23
layers; further these triads are such that we have a resonant set of modes at leading orders in wavenumber/ wavespeed expansions based on the asymptotically large

Reynolds number. Further, we have developed a fully rational description of the interaction processes which promises to re_'eal the degree to which these mechanisms

can alter the stability characteristics of the flow. We have concentrated on a special

three'dimensional flow as an iUustrative example but the results obtained here Suggest

that the interaction process will be operative in more practically oriented :boundary

layer flows. Our work can be regarded as a first stage in a very complicated field:

obvious generalisations could be tackled to explain situations in which, for example,

we have critical layers of other than the simple linear kind; or in which the modes

have slight detuning. In this context we note that Avis (27) has made a careful study

of the effects of detuning on resonant triads in his axisymmetric flows. The analysis

adopted here allows nonparallel effects to be accounted for without formal difficultiesthese effects can play significant roles in practical applications. The largest drawback

of our approach is that the computation of the interaction coefficients appears to be

not straightforWard, especially when the modes have critical points close together or

when such critical points are near the bounding surface. We feel that to be able to use

our approach to its best advantage further study needs to be devoted to developing

more powerful and sophisticated techniques for resolution of the numerical problem

as opposed to the fairly rudimentary ideas employed here. Overall however, we have

here the first asymptotic description by which the importance of interaction mechanisms upon a resonant triad of linear neutrally stable crossflow vortices may be truly

assessed.

Acknowledgements.

The research of A.P.B. was supported by SERC under Contract No. XG-10176.

We thank the referees for their useful suggestions which have led to improvements in

the paper and we also thank Mr. J.W. Cole for providing us with Figure (1).

24
Appendix.

Here we briefly describe the derivation of the constants {Xkj, Dkj } appearing in

(4.6). We shall, for convenience, determine these quantities when j = 3 for then we

may easily modify the results for j = 2.

With I. as defined in (4.5b), we suppose that close to the critical layer _3 the

eigenfunction W003 develops according to

W003 _ g_3 + O ((_ - _3) In [_ - _3[). (A1)

Further, in this region the mode proportional to E2 has a regular Taylor expansion

and so

(Uoo2, Voo2, Wo02) " (Q,,02, Q,,02, Q_02) + (_ -_3) ( Q',,02, Q:o2, Q',_02) + ..., (A2)

f){*} where the quantities "_{.02} are constants. Finally, we know that the behaviours of

U003 and V003 can be obtained by suitable manipulations of the basic equations (2.6).

Substituting this information into (4.5b) yields the behaviour

X03 XI3 X23 (A3)

where

X13 --

and

[Bo31[J

_o3Bo31

2iQ,_ o21Bo31lJ (aoi/3o3 - ao3/_oi),

X03 - ?r_O3B031

(a03_01 -- a01fl03)[(_01 + _03) Q,02 + (aol + a03) Q,,02]

iglaBo32BollQ,_02 iJ[B031]

B_31 _'_o3Bo31

I

(_olZo3- _o3Zol)q,_o2,

(A4)

(A5)

ig_3Bo32Bo12Q'_02 iQ_°2"/21[B°311 (_02_03 - ao3f102) J223 -- B231 Irf103B031

+ + -

B_31

(A6)

Here

j_-

7ra03B031gI3 _= gI3

_23[Bo311 IBo311 "

25
Then we can show that

Do3

w101~ (_-_3) + D131n(_-6) + D32+ D33(_-_3) ln(_-_3) +..., (A7)

for _ > _3 with

and

Xo3

Do3 = 2Bolo'

2Bon X13

Di3 - Do3

Bolo Bolo '

X33

B°l----_lD13 + _021D03 + _.

D33 = B010 BOIO

The quantities D13, D33 appear in the evolution equation (4.7) for the mode proportional to El. Of course, we need to obtain quantities equivalent to those given

above in order to traverse the critical layer(s) _2, but these can be deduced merely by

interchanging the subscripts '2' and '3' in the relevant formulae above.

tr

r

26
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28. R.I. Bowles, Ph.D. Thesis, University of London (1990).

29. A.D.D. Craik, Proc. R. Soc. Lond. A 406 (1986) 1-12.

30. A.D.D. Craik, Proc. R. Soc. Edin. 106A (1987) 205-207.

31. P.K.C. Wang, J. Math. Phys." 13 (1972) 943-947.

28
F -i

¢ t- ................. I

m nnl m u m n m m m m m m m mUll m n

o(tj_) I I

¢

o(__)

Figure (1).

Flow structure for the Rayleigh modes described in §2. The inviscid layer I is of

thickness 0(_ 3) (the same depth as the boundary layer and of the same size as the

horizontal scale for the structure). In addition, we have a viscous wall layer (II) of

depth O(e 9/2) and a critical layer (III) of thickness O(e 4) at those positions where

the 'effective' basic velocity profile 90 (see (2.7)) vanishes.

29
0.00

-I.00.

-1.50J

=

F

Figure (2).

a) Basic velocity profile U. - _ + #0 as a function of the boundary layer coordinate _. (i) /._ = --1.2; (ii) /_ = --0.8; (iii) /_ = --0.4; (iv) # = --0.0; (v) # = 0.3;

(vi) # = 0.55; (vii)/_ =/_, = 0.8284 and (viii) p = 1.3.

3O
0.50.

0.25

0.00

Figure (2).

b) mustration that for 0 < /J < #c there can be either one or two points where

U, = co for some given co. Hence there ca_ be either one or two critical layers in the

flOW structure.

31
1 .20_

1 ,00_

0.80.

0.60.

0.4'0.

0.20.

0.00

0.00 I v I" 1

"0.50 I .00 I .50 2.00 2.50

XtO-1

I

3

Figure (3).

Solutions of the eigenproblem (2.7) for various values of the waveangle parameter _.

a) Radial wavenumber _0 as a function of the azimuthal wavenumber 80.

32
0.20_

- .20.

- ._0.

- .GO.

- .80.

1'.oo ,'.5o 2'.oo zT_c 3

Figure (3).

Solutions of the eigenproblem (2.7) for various values of the waveangle parameter #.

b) Frequency _10 of mode as a function of/_o.

33
\ \

o

.41,, "_

v _

v

o

I]1 _"

0

=

=

r

L

F

34


__ Report Documentation Page

T.Report No. 2. Government AccessionNo. 3. Recip_ent'sCatalog No.

NASA CR- 182037

ICASE Report No. 90-32

4. Title and Subtitle

CONCERNING THE INTERACTION OF NON-STATIONARY

CROSS-FLOW VORTICES IN A THREE-DIMENSIONAL

BOUNDARY LAYER

7. Aurar(s)

Andrew P, Bassom

Philip Hall

9. Performing Orgenizat_n Na_ and Addre_

Institute for Computer Applications in Science

and Engineering

Mall Stop 132C, NASA Langley Research Center

Hampton_ VA 23665-5225

12. S_nsoring Agency Na_ and Addre_

_ational Aeronautics and Space Administration

Langley Research Center

Hampton, VA 23665-5225

5, Report Date

May 1990

61 Performing Organization Code

8, Perfo_i_ Organization Re_rt No.

90-32

10. Work Unit No.

505-90-21-01

11. Contract or Grant NO,

NASI-18605

13_Type of Report and Period Covered

Contractor Report

14. SponsoringAgency Code

15. Su_n_ W Not_

Langley Technical Monitor:

Richard W. Barnwell To appear in Quar. Jo Appl.

Math.

Final Report

_16. Ab_ract

Recently there has been much work devoted to censidertng some of the many and

varied interaction mechanSsms which may be operative in three-dimensional boundary

layer flows. _ere we are'concerned with resonant triads of crossflow vortices.

In contrast to much of the previous work we examine the effects of interactions

upon resonant triads where each member of the triad has the property of being

linearly neutrally stable; then the importance of the interplay between modes can

be relatively easily assessed. We concentrate on investigatln B modes within the

boundary layer flow above a rotating dlscj, this choice Is motivated by the similarity between this disc flow and many important practical flows and, secondly,"

our selected flow Is an exact solu=ion of the Navier-Stokes equations which makes

its theoretical analysis especially attractive. Firstly we demonstrate that the

desired triads of linearly neutrally stable modes can exist within the chosen

boundary layer flow and then subsequently obtain Evolution equatlonsto describe

the development of the amplitudes of these modes once the interaction mechanism is

accounted for. It is found that the coefficients _the interaction terms within

the evolution equations are, in general, given by quite intricate expressions although some elementary numerical work shows that the evaluation of these coefficients is practicable. The basis of_our work lends itself to generali_ation to

more complicated boundary layers and effects of detunlng or non-paralle_llsm could

he provided for within the asymptotic framework.

17. K_ Words (Suggested by Au_or(s))

triad, crossflow, instability

19. Security'Classif. (of this report)

Unclassified

18. D_tribut_n Statement

02 - Aerodynamics

Unclassified - Unlimited

_. Securiw Clef. Iof th_ N_)

Unclassified

21. No. of _s _. Pri_

36 A03

NASA FORM 1(_6 OCT 86

NASA-Langley, 1990