the linear stability of flat stokes layers

The Linear Stability of Flat Stokes LayersAuthor(s): P. HallSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 359, No. 1697 (Feb. 15, 1978), pp. 151-166Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/79535 .

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Proc. R. Soc. Lond. A. 359, 151-166 (1978)

Printed in Great Britain

The linear stability of flat Stokes layers

BY P. HALL

Mathematics Department and Physiological Flow Studies Unit, Imperial College, London, S.W 7, U.K.

(Communicated by J. T. Stuart, F.R.S. - Received 23 March 1977)

The linear stability of a flat Stokes layer is investigated. The results obtained show that, in the parameter range investigated, the flow is stable. It is shown that the Orr-Sommerfield equation for this flow has a continuous spectrum of damped eigenvalues at all values of the Reynolds number. In addition, a set of discrete eigenvalues exists for certain values of the Reynolds number. The eigenfunctions associated with this set are confined to the Stokes layer while those corresponding to the continuous spectrum persist outside the layer. The effect of introducing a second boundary a long way from the Stokes layer is also considered. It is shown that the least stable disturbance of this flow does not correspond to the least stable discrete eigenvalue of the infinite Stokes layer when this boundary tends to infinity.

1. INTRODUCTION

The stability of the flow between parallel plates when one of the plates oscillates in time has been investigated by Kerczek & Davis (I974). The flow was found to be stable for all Reynolds numbers less than 700 and wavenumbers greater than 0.3. The particular flow investigated by Kerczek & Davis (hereafter referred to as K. & D.) had a Stokes layer thickness equal to 8 of the distance between the plates.

It is perhaps useful at this stage to point out the most relevant results concerning the stability of periodic flows. A detailed review of this subject has recently been given by Davis (1976). The bulk of the work in this field has been concentrated on flows modulated about some non-zero mean. For example, Grosch & Salwen (I968) and Hall (Ig75 a) have considered the stability of modulated plane Poiseuille flow while Hall (I975b) and Riley & Lawrence (I976) have considered the stability of modulated circular Couette flow. The asymptotic results of Hall (I975b) and the numerical results of Riley & Lawrence (I976) suggest that modulation destabilizes circular Couette flow. It is not yet clear why these results differ from the conclusion reached by Donnelly (I964), who found experimentally that modulation stabilizes circular Couette flow.

The investigation of the stability of purely oscillatory flows is a more difficult problem. This is because the instability of flows modulated about a non-zero mean is usually associated with the mean flow. Thus the effect of modulation on the stability characteristics of the flow can usually be evaluated using perturbation methods.

[ 151 ]



152 P. Hall

This contrasts with the situation with purely oscillatory flows where numerical methods are usually required. For example, Riley & Lawrence (1976), following K. & D., used a Galerkin method to investigate the stability of a Stokes layer on a cylinder oscillating inside a fixed cylinder. The disadvantage of this approach is that in order to infer results for the infinite Stokes Layer the second boundary introduced to facilitate the use of Galerkin's method must then be allowed to tend to infinity. When this occurs the number of approximation functions required will increase rapidly because the disturbance will be confined to a smaller and smaller boundary layer.

In this paper we first obtain results for the infinite Stokes layer using the method of Seminara & Hall (1976). The stream function of the disturbance is first expanded as a Fourier series in time with coefficients dependent on the variable normal to the oscillating plate. The expansion is then substituted into the Orr-Sommerfield equation and like powers of eit are equated. This leads to an infinite series of coupled differential equations for the Fourier coefficients. The solution of these equations is then given as a double infinite sum of decaying exponential terms. It is clear that this procedure forces the disturbance to be confined to the Stokes layer. The results which we obtain show that when the Reynolds number increases the order of the dominant Fourier component increases so that for large Reynolds numbers the Fourier series must be truncated at relatively large order. This prevents the economi- cal use of the method at Reynolds numbers greater than about 400. We find that the infinite Stokes layer is stable in the range of parameters investigated. However, the damping rates which we obtain are significantly larger than those found by K. & D. Furthermore, we show that there also exists a continuous spectrum of eigenvalues associated with the flow in question. However, we note that the most unstable disturbances found by K. & D. had wavelengths of the same order as the separation of the boundaries.

With this in mind we also investigate the stability of a thin Stokes layer between parallel plates to disturbances with wavelengths of the same order as the separation of the plates. Suppose that we follow K. & D. and let ,8 denote the ratio of the separation of the plates to the Stokes layer thickness. Now consider a disturbance with wavelength 2X/a scaled on the separation of the plates. In the absence of the Stokes layer, the damping rate A of the least stable disturbance is given by the even Dolph & Lewis (1958) eigenvalues. Hence we have that

A = iu(a)/2,81, (1. 1)

where It is determined by the transcendental equation,

aIV(a -a2) --tan {!V(a-a2)} coth (la). (1.2)

Using the method of matched asymptotic expansions when,8 -- 00 we show that if the Stokes layer is present then (1.1) is modified to give

A = 2(a) {+ I + O(3-4)4, (1.3)



The linear stability of flat Stokes layers 153

where y is a positive constant proportional to the square of the Reynolds number based on Stokes layer thickness. We see that the presence of the Stokes layer further stabilizes the flow. Moreover, since (1.3) shows that for larged,, A ,f-2, we conclude that, when 8 ->-oo, the least stable disturbance of the finite Stokes layer does not reduce to that of the infinite Stokes layer if the latter flow is stable. This appears to be the reason for the discrepancy between our results and those of K. & D. Indeed, the behaviour suggested by (1.3) agrees qualitatively with the numerical result of K. &D.

The procedure adopted in the rest of this paper is as follows. In ? 2 we formulate the differential system governing the stability of the flow induced by a rigid wall oscillating transversely in an infinite viscous fluid. In ? 3 we obtain a series solution for this system, while in ? 4 we consider the effect of introducing a second boundary. In ? 5 we discuss the results of ?? 3 and 4.

2. FORMULATION OF THE PROBLEM

Suppose an infinite rigid plate oscillates with speed (UO cos wt, 0, 0) in an infinite viscous fluid of viscosity v. This oscillation generates a Stokes layer of thickness (2v/wo)l at the boundary and if we choose dimensionless variables X, Y, Z, and r by writing (X i

(X, Y, Z) v( Y )

r (=o (2.1)

then it follows that the corresponding velocity field is given by

(u, v, w) = Uo(cos (r- Y) e-7, 0, O). (2.2)

If this flow is perturbed two dimensionally by writing

(u, v, w) = UO(cos (r - Y)e:r + Y, T) (2.3)

then, by substituting into the momentum equations and eliminating the pressure perturbation corresponding to #/(X, Y, r), we can show that i/ is determined by

_ 1 8a 82 82) 82; 1 82 82 2 24 +i)Re ajV (Y2 - ReX2+ aY2}V (2.4)

where ii is given by v= c -Y) e-, (2.5)

and Re is the Reynolds number based on Stokes layer thickness defined by

Re = UOJ(2/vwo).

If we now look for a disturbance of the form

? = exp {ia(X - cT)} f( Y, T) + complex conjugate, (2.6)



154 P. Hall

then we can show that ? is determined by,

L 2c 2 8 2; 1 ( Reix+ eT}- 9 NT-a y2=i , N2TJ (2.7)

~Re iaRe arjT ix Re

where N is given by, N _ aM/a y2 (2.8)

On the basis of Floquet theory we can assume that QT is a periodic function of T since any exponential growth of i/ can be absorbed into the complex wavespeed c. The boundary conditions required to completely specify the problem are

s'=? [/8Y =?0, Y=0- J (2.9)

W, a/a Y, 0, y oo,J so that the no-slip condition holds at the boundary and the disturbance is confined to the boundary layer. For given (real) values of a# and Re, (2.7) and (2.9) constitute an eigenrelation for the complex wavespeed c. The flow will be unstable or stable depending on whether or not ci is positive or negative respectively. It is also worth mentioning that, if the Stokes layer is driven by a pressure gradient then, by making the transformation used by Hall (I 975b) the appropriate eigenvalue problem can be reduced to that specified by (2.7) and (2.9).

Suppose now that we introduce a second rigid boundary a distance ,? times the Stokes layer thickness away from oscillating plate. It is an easy matter to show that the appropriate differential system governing the stability of the flow becomes

K 2c 2 a 1 2f 1 T Re+iaRe aT]

V a y2 iaRe N

(2.10)

VF=a=0)y=0,,8 ay

where N is as defined by (2.8) and uR is given by

= T

sinh (I h+ i,Y) + complex conjugate. (2.11)

It is clear that for large values of/8 and Y of 0(1) we can approximate ui in (2.11) by u given by (2.5).

3. THE INFINITE STOKES LAYER

Since q is a periodic function of r we can expand in the form 00

F = z ,nr inn(y). (3.1) n=- oo

If we substitute for I from (3.1) into (2.7) and equate like powers of eiT we can show that

{M - 2in + 2iac} MT. = 2 {e-Y('+i) [M - 2i] Wn-l + e-YU1- [M + 2i] Vfn+1

[fn = d9[n'/dY _ 0 Y-=O0




where M =_ d2/dY2- a2.

Following Seminara & Hall (I976) we now obtain a solution of (3.2) in the form of a double series expansion.

If we neglect the terms on the right hand side of (3.2) we see that the solution V* having the required behaviour at infinity is

W.* = ane 0ay + lne-O'y 33 where an, ,n are constants to be determined and o-n is defined by

-n = V(a2 + 2in-2iac). (3.4)

Here the branch of the square root with positive real part is taken. This ensures that, the disturbance is confined to the Stokes layer. If a second boundary were introduced some distance away from the Stokes layer we would have to retain the exponentially growing solutions in (3.3). It is clear that for a given value of n the coupling torms in (3.2) will cause I9n to have contributions arising from I for all values of m. For example, W* will generate a term equal to

iaRe(, an(-2i) exp (-aaY-(1 +i) Y) 2 [(a + ( 1+ i))2 _ a2] [(a + ( 1+ i))2- a2 _ 2i(n + 1 ) + 2iocc]J

ia Re f ln(on -a2- 2i) exp (- o-n Y - (l+i) Y) 2 [(On + (1 +i))2-a2] [( + (1 +i))2-a2 - 2i(n + 1) + 2iac]

in Vn+j and a similar term in V'n-l with (n + 1) replaced by (n - 1) and (1 + i) replaced by ( 1-i). These terms then interact to generate more terms in VJn and initial terms in Wn+2. The cascade continues indefinitely and we find that (3.3) leads to a contribution to Vfm of the form

a)

vfm = an E {Anmj exp-{a + (m -n) i-j} Y} j=O

00

+fln {Knmj exp-{o. + (m -n) i-j} Y} (3.5) j=O

where Anmj and Onmj are given by

AnmO OnmO =nm' (3.6) and for j > 1

Anmj = lia Re {Anm-ij-1 [(- a -(m-1-n) i -j + 1)2 - a2 _ 2i] + Anm+lj-l [(- a - (m + 1 - n) i-j + 1)2 - a2+ 2i]/

[(a - (m + 1n) j)2 - a2} {(- a -(m + 1) j)2 -a2 - 2i(m + 1) -2iocc}], (3.7)

and Onmj is given by the same recurrence relation with a replaced by on inside the round brackets. The general solution for E1m is then obtained by summing over n to give

00 00 VIM E an E Anmj exP -{a + (m - n) i +j} Y

n.= -_0 j=n



156 P. Hall

If we now apply the boundary conditions at Y = 0 we obtain the following infinite set of equations for aw&, ,8

00 00 oo

E (XZn E )lnmj + f3n E Onmj _ O, n= -co j=O j=O

00 00 00

E jn E Anmj(a + (m - n) i +j) +dn Ei 0nmj(n + (m - n) i+j) (3.9) n-=-coo j=O j=O

m = ?, 1, + 2, etc. If this system is to have a non-trivial solution, the determinant of the matrix of coefficients must vanish. This condition determines an eigenrelation of the form

c = c(x, Re).

The nature of this eigenrelation will be discussed in ?5.

4. THE FINITE STOKES LAYER

In this section we obtain an asymptotic solution of (2.10) in the limit J-+ 0oo. We focus our attention on disturbances having wavelengths the same order as the separation of the boundaries. Thus we rescale a by writing

a= af-i, (4.1)

where a is now taken to be 0(1). We assume that the Stokes layer plays a passive role so that the stability characteristics are, to first order in fi, those of a motionless fluid between parallel plates. These disturbances are just the well-known Dolph- Lewis eigenfunctions and the corresponding values of the wavespeed are purely imaginary. However, the interaction of such a disturbance in the Stokes layer with the basic flow produces a contribution to the disturbance synchronous with the basic flow. This contribution will itself interact with the basic flow to produce further mean terms and a first harmonic in time. This cascade process continues indefinitely to produce all the harmonics. However, we are able to evaluate as many of these terms as we require by a suitable expansion procedure. The situation is not unlike that discussed by Hall (I975 b) who investigated the stability of plane Poiseuille flow modulated at high frequencies.

In the Stokes layer we can approximate uR in (2.10) by vi. Thus we wish to obtain a solution of

{ a2 a2 2a) ( a2 a2 -;j + 2icac - ~- _ jj- -: [

tay2-22A-T taY2- fl

ia Re 2e1a-(1?i) Y a2 ] + /1 i) a a2 1 12i] - 2flj yaY2

- i f -ir-e"-I" Ly2- 2iIP




We further require that as Y-->oo the solution matches the solution of (2.10) satisfying the boundary conditions at Y = ,. Since IF is periodic in T we write

00 =fi Wn( Y) ein (4.3)

n = -oo

In the absence of the Stokes layer the least stable solution of (2.10) is given by

y =Icosh (aY a\ - cos (V(It-a2) (Y-,#) cosh la Yf = cosh o~V(a~2) =xYII6) (4.4)

where iac = 4u(a)/2fl2, (4.5) and ,u(a) satisfies the eigenrelation (1.2).

In the Stokes layer we can show from (4.4) that IF Y2//82 + 0(/8-3). Thus we expand [ in the form

g[I. = -2{([I?! 1+O(f-2)} * (4.6)

The nonlinear interaction terms in (4.2) then suggest that we expand Pif? in the form

=/f=r83(9J810?+jf+O(/f)}' l (4.7)

=l f=r{-3 10+ IJ11. + O(1f-2)}J

The above terms will then interact with the basic flow to generate the first terms of Yff of order 8-4 and a contribution to To of similar order in /. This cascade process continues indefinitely and we see that the appropriate expansion of VJn is

- f21(nlOJo+ A nl[ +0(/f32)} (n= 0, + 1 + 2, etc.). (4.8)

In view of (4.5) we further expand iasc in the form

2ixc = /o-2 + l+O(/f2)}* (4.9)

If we substitute from (4.3), (4.8) and (4.9) into (4.2) and equate steady terms of order if-2 and f-3 we can show that

Ioo = AO Y2+B BY3), (4.10) Ihol = A1Y2+B1 1Y3,

where AO, Bo, etc., are constants to be determined. However, if this solution is to match with the Dolph-Lewis eigenfunction at the edge of the Stokes layer, we must choose Be = 0. Having determined fo we can equate terms proportional to e?ir of order f-3 to show that

VI1O = PO{e-Y('+i) -1 + Y(1 +i)}-iaReAOF(Y),

-1o = Qo{e-('-i)- 1 + Y(1 -i)}-ia Re AO iP(Y), (4.11)



158 P. Hall where

F(Y I -Y(+1)iY3+5(I i) 2+13 Y + 13 13__4_12 F(Y) 4(1 -[ei) 3+ (+i)Y+ 4 +4( 1+i) 4(1+i)]

and PO, QO are constants to be determined. The important feature of (4.11) is that, unlike the basic flow, T + 1 do not decay exponentially to zero at the edge of the Stokes layer. Thus we must solve for f + t in the region outside the boundary layer. Indeed, following Hall (Ig75 a), this region must itself be split up into two further regions since TJ+ 1 will have Stokes type boundary layers at the upper boundary where the disturbance velocity is reduced to zero. This layer will clearly be of the same thickness as the lower Stokes layer.

It is convenient to rescale the region away from the lower boundary layer by writing z = Y,8-I so that away from the lower Stokes layer, f is determined by

{82a+a2[+ / -. 212f aa {a82 _a2 O. (4.13) We now expand VI in the form

-c

=E *f (z) einr, (4.14)

and note that, since the basic flow is exponentially small in this region, when we substitute from (4.14) into (4.13) and equate like powers of eir the resulting equations for O4n(z) will not be coupled. In this region we expand V/n(z) in the form

*e { *oo>I*ol +0(/h-2))

Rn = {fno+ >bfnl + O(j8-2)} bn (j), (4.15)

where bn(/3), n = + 1, + 2, etc., are to be determined. If we substitute for ?bf from (4.15) into (4.13) and equate terms of order b1(fl) we can show that ?lo can be written in the form

VIfo:= Cocosha (z-1) +Dosinha(z -1), (4.16)

where Co, Do are further constants to be determined. It is clear that we cannot choose C0, Do in (4.16) so that fro = '= 0 at z = 1. Thus, as stated above, we must introduce a new region near z = 1 to remedy this situation. We define a variable s for this layer by writing

s =,8(1-Z) (4.17)

and it follows that in this layer T is determined by

S[1-o + -2 --(a- = O. (4.18)

We again expand L in the form

fo + E Sn(s) Oinr + _n e-in r (4.19)




We note that it is not necessary to distinguish between the steady part of 1 in the upper two regions because, being steady, i'% does not have a boundary layer charac- ter near z = 1. In the uppermost layer we write

Tln = Cn(I) (0no+ Onl]+O(/f )}, (4.20)

where cn(fl) are to be determined. Substituting from (4.20) into (4.18) and equating terms proportional to ejr of order cl(/3) we find that the solution satisfying the required boundary conditions at s = 0 is

01o =

Go{e-s(l+i)+s(1 +i)-1}. (4.21)

Here the constant Go together with c1(fl) are determined by the condition that y2 and 0 match at the edge of the uppermost layer. We find that

o (fi) - ,8-'b1(,),

Go = (1+i)' (4.22)

co0= 0.

The constants Do P0 and b1 are then determined by matching [' and ?/f, at the edge of the lower Stokes layer. We obtain

PO= 0,

bl = t3-2, (4.23)

Do = 13ia Re/(64a cosh a).

Similarly, by considering the terms proportional to e-it we can show that Q0 = 0,

b-,= fl-2.

Having determined P0, Q0 we now return to the lower Stokes layer and determine

VJ02. Substituting for YJ' from (4.6) into (4.2) and equating terms of order l-4 and

using (4. 1 1) we find that

02a= 60 Y4 -Uo Y4AO+a2AORe2 6 12

(8( +i [12Y +3 Y2(1 3i) + 16 (32 + 17i) + -1 (65 + 55i)] 13i * a2Re2AO{12.~ * (.4

+ -28 e-y1-i) + complex conjugatej+ 256 120-90Y} (4.24)

As stated above, it is sufficient to consider the two upper regions simultaneously when evaluating the steady part of fbo. If we substitute for VFfrom (4.14) into (4.13), equate steady terms, and then equate like powers of /5 after substituting for 3bo from (4.15) we can show that

S3N = 0, )) (4.25)

Woo= d/fooldz = 0, (z = 1)J

YVfo]L - 1,.(d2/dz2 - a2) Vfoo, (4.26)

V,ol =dVolldz = O (z = 1),)



160 P. Hall

Id2 /2 = /k2I ---2 Vfoo -Ii.*-a1/ Yvf02 =-2(dz2 a OO] rdz2,-Ia2r (4.27)

V"02= dVf02/dz 0 (z = 1),

d2 ~ ~ d2\ d2 \ -Tv 03 =-t3 \dZ2- ) 200 -2 ( Z2da2 l dZ2-a2 ) 02, . (4.28)

#03 =d/rO3/dz =0 (z = 1),

where -- -a2 +o o a2)

We further require that ifo and YfJ should match at the edge of the lower Stokes layer. We can show from (4.10) and (4.24) that the matching condition requires that, for small z, Ioo matches on to the series solution of (4.25) satisfying the conditions.

Vkoo = d?/foo/dz = 0 (z = 0). (4.29) Thus it follows that

Woo-X(Z), 1a 1a(c), A0 =()

where x and It are as defined by (4.4) and (1.2) respectively. The presence of the Stokes layer first effects the matching condition at order f-3. The functions o11 and /02 are again required to match on to the small z series solutions of (4.26), (4.27)

respectively which satisfy the no-slip conditions at z = 0. Thus it follows that

=1- 2 =0 and V0fo j02 are then constant multiples of the function X(z). At order f-3 we find that the matching requires that Vf03 matches onto the series solution of (4.28) satisfying the conditions

O3- ?= ~, dWfr3/dz =-1?a2Re2X"(0), z = 0. (4.30) If the system specified by (4.28), (4.30) is to have a solution, an orthogonality condition must be satisfied. This condition is found to reduce to

J x(z) d2X_a2X dz = 45a6Re2(X (0))2 (4.31)

Using (4.4) we can show that this condition can be further reduced to give

45a2Re2 cos2(1V(1uO a2)) ,u 128(1 + sinV(,uo - a2)//(,uo - a2))*

We see immediately from (4.32) that t3 and I0 are of the same sign. Since the motionless state is clearly stable (ao > 0) it follows that the presence of the Stokes layer further stabilizes the flow. This is consistent with the results of K. & D.

5. RESULTS AND DISCUSSIONS

(a) The infinite Stokes layer In order to obtain the eigenrelation satisfied by a, Re, and c the determinant

associated with the system of equations (3.9) must be evaluated. The elements of this determinant can be obtained to any accuracy by summing the series which define them, by using a computer. The infinite determinant was then approximated




by restricting m in (3.9) to be less than M, say. The determinant is then of order 4M + 2 and was evaluated using the NAG subroutines supplied by the Imperial College Computer Centre. The eigenvalues (iccc) for which the determinant vanished were found using the iteration procedure due to Mueller (1956). In figure 1 we have shown the dependence of iacc on M for a = 0.15 and Re = 140. We found in general that the required value of M to obtain sufficient accuracy increased with a Re. In figure 2 we have shown acci and accc as functions of Re at a fixed value of a. No eigenvalues were found near the discontinuous points on the curves. We believe that for

1.95 -

- cci_

1.75 -

1.55

2 4O 6 8 10 12 14 M

-0.15. CCr

-0.25 FIGuRE 1. The behaviour of the growth rate as a function of M.

some value of the Reynolds number in these regions no solution to the problem representing a disturbance confined to the Stokes layer exists. This is suggested by the fact that c, approaches zero as these regions are approached. We can see from (3.4) that o- then becomes purely imaginary and the solution no longer decays to infinity. Thus no solution confined to the boundary layer exists. A similar situation arises at zero Reynolds numbers where [ is determined by

(N -2a/at) NW> = O, w = avflaY = (Y = )r (5.1 W, a V/a Y o (Y oo).

It is an easy matter to show that this system has no solution of the form

TF= ?b(Y)expiac(x-cr) (5.2)

which decays to zero at infinity. However, if we relax the condition at infinity by merely insisting that W is finite at infinity then, for a given wavenumber a, the spectrum of disturbances becomes continuous and J is given by

' = expiia(x- cr) (e-ay - cos J( + 2iac - a2) Y + a(2iac - 2) sin \/(2iac - a2) Yj



162 P. Hall

and c is purely imaginary and can take all values satisfying 2ioc > z2. In fact, it is easily shown that for finite Reynolds numbers the same continuous spectrum of eigenvalues exists. These disturbances are all synchronous with the basic flow. We can see from (3.4) that for the range of values of c mentioned above o-O is purely imaginary. Thus, the term in (3.3) proportional to ,0 will not decay exponentially to zero at the edge of the Stokes layer. However, if we again merely stipulate that VI is finite at infinity, then we see that

uO* = aox e-aY +/5, sin V(2izc -_ 2) Y + Co cos -(2ia _ a2) y

satisfies the required condition at infinity and contains three unknown constants ao eo ,6l. The other solutions T'*, (n =A 0), are unaltered and decay at infinity. We can then follow the interactions as previously and we see that the terms generated by

0.4

OtCr

0.2

C , \ 80 160 240 320 Re

-2

cxci

-3 FIGuRE 2. The behaviour of ac as a function of the Reynolds number for a = 0.15.

VfO* in V. (n 4 0) will all decay exponentially to zero as Y->oo. Furthermore, the terms of To arising from interactions involving P. (n 7& 0) all decay to zero at Y -oo. Thus, the only Fourier component which is finite as Y -> oo is the steady one so that as Y -* oo

P -* cocos 1(2iacc - 2) Y + ,80 sin V(2izc - a2) Y.

Furthermore, if the Fourier series is truncated at m = - say then the boundary conditions at Y = 0 determine acc, /3n (n = 0, ? 1, etc.) together with ec. In fact, there are now 4M + 2 equations to be solved for 4M + 3 unknowns. Thus, we can always find a solution for Ca, fBn (n , 0 + 1, etc.) in terms of ec so that we again have a continuous spectrum of eigenvalues.




Mack (1976) has recently suggested that the Blasius boundary layer also has a continuous spectrum of eigenvalues. It is interesting to note that the spectrum suggested by Mack is identical to that occurring in this paper. This is not surprising since the spectrum arises in both problems because of the lack of an upper boundary. The phase of the eigenfunction outside the boundary layer will, however, be sensi- tive to the precise nature of the boundary layer. This is because, for example in our problem, the constants f0, eo must be chosen to satisfy the lower boundary conditions. Thus, it follows that, though the continuous spectrum of eigenvalues is identical in both problems, the phase of the corresponding eigenfunctions outside the boundary layers will be distinct.

OCCr

0.1

0 0.13 0.17 a 0.21

-1.2

-1.4

otcj

-1.6

FIGuRE 3. The behaviour of ac as a function of a for a fixed value of Re = 68.0.

We also notice that in figure 2 the damping rates are directly proportional to the Reynolds number on any of the sections of the curves. However, the slope of these curves decreases in going from any one section to another. The damping rates on the fourth section in figure 2 are always less than the maximum damping rate on the third section. Thus it is possible that at higher Reynolds numbers the flow is unstable. However the present method is not efficient enough to settle this issue.

Calculations were also made for different wavenumbers at fixed values of the Reynolds numbers. We found that the damping rate increases as the wavenumber increases. In figure 3 we have shown oc as a function of as for Re = 68.0. We again find



164 P. Hall

that when the frequency of the disturbance goes to zero the disturbance is no longer confined to the boundary layer and only the continuous spectrum of eigenvalues exists. If a is increased further we find that the eigenvalue reappears with some non- zero frequency for a finite interval and then vanishes again. This behaviour is repeated as a increases again. The behaviour found at different Reynolds numbers was similar to that found at Re = 68.0.

We have seen that the discrete spectrum appears and disappears from the continuous spectrum when either ax or the Reynolds number is varied. The wavespeed cr is found to be zero at such points. We can show from the Orr-Sommerfield equation that if c is an eigenvalue, then so is - -. Thus, in addition to the eigenvalues given in figure 2, there is a set with the same value of ci but with wavespeed being the negative of that shown in figure 2. It follows that the points where the discrete spectrum branches off from the continuous spectrum are points where the eigenvalues c, -

coalesce. The eigenvalues c and - c correspond to waves propagating to the right or left damped at the same rate. Clearly, both such waves must exist because there is no preferential direction associated with the Stokes layer velocity profile.

We assume that for fixed values of a and Re then, if Re is sufficiently large, there will exist several eigenvalues c(x, Re). However, at the values of the Reynolds numbers which we considered none other than those shown in figure 2 were located. At higher values of the Reynolds number we expect that other eigenvalues will appear.

The apparent inconsistency between our results and those of K. & D. is easily explained. The results which we obtained in ? 4 show that the most dangerous modes of the finite Stokes layer are not localized in the layer and all have zero wavespeed. This is precisely the result found by K. & D. and the eigenrelation obtained in ? 4 agrees qualitatively with that found numerically by K. & D. When the separation of the boundaries tends to infinity it seems likely that these eigenvalues 'merge' into the continuous spectrum of the infinite Stokes layer, which we recall corres- ponded to disturbances having zero wavespeed. Thus it appears that K. & D. correctly calculated modes of the finite Stokes layer but, as the separation of the boundaries tends to infinity, these modes have progressively less dependence on the Stokes layer. In contrast to the latter behaviour, the discrete modes found in ? 3 depend entirely on the motion in the layer and are only slightly perturbed by the introduction of a distant boundary.

(b) The finite Stokes layer

In figures 4, 5 we have shown /to and a3 as functions of a. Since t3 is always positive it follows that the presence of the Stokes layer further stabilizes the flow. We see that, for any fixed value of Re, the damping rates (on the scale o-1) tend to those of the Dolph-Lewis eigonfunctions as 8l-? oo. This is similar to the results shown in figure 7 of K. & D. However, we cannot compare our results directly because those given by K. & D. correspond to a fixed value of a/fl whereas we have kept a fixed.




100 It3

80-

60

41 -

40-

39 20-

37 1 11 0 2 4 a 0 1 2 a 3

FIGURE 4. The behaviour of IAO as a function of a.

FIGURE 5. The behaviour of ,3 as a function of a.

500 6 - 400

300 A ~~200

100

Re-0

4

2-

0 10 20 18 30

FIGuRE 6. The behaviour of the damping rate A = o/2I2 + /t312fl5 as a function of f for a

6 Vol. 359. A.



166 P. Hall

We further notice that the damping rates tend to zero as ,8-oo. Thus, in any region of the (cx, Re) plane where the infinite Stokes layer is stable, it follows that the least stable disturbance of the finite Stokes layer does not tend to that of the infinite Stokes layer as the second boundary tends to infinity. Thus, unless there is a cross- over of modes at finite Reynolds numbers, it seems that the disturbances given by K. & D. have no connection with those of the infinite Stokes layer.

In figuro 6 we have shown the damping rate A correct to order fl-6 at different values of the Reynolds number and a fixed and equal to unity. The dotted line denotes the damping rate at zero Reynolds number. We see that the range of validity of the expansion increases as the Reynolds number decreases.

It is relevant at this stage to mention the results of Riley & Lawrence (I976) and those of Seminara & Hall (1976). The former authors used the method of K. & D. and discussed the stability of a Stokes layer on an oscillating circular cylinder inside a stationary cylinder. The numerical results obtained in the limit of the outer cylinder tending to infinity agreed well with those of the latter authors who tackled the infinite problem directly using the method of ? 3. It is because the latter flow is unstable that the least stable mode obtained by Riley & Lawrence (1976) tended to the correct mode of the infinite problem when the outer cylinder tended to infinity.

It was found by the above authors that a Stokes layer on a cylinder of radius b becomes unstable when the Reynolds number Re based on Stokes layer thickness a = (2v/wo)1 is such that

Re = (b-1232.5)i

Thus it is likely that even for very small curvatures, centrifugal instability will arise at quite small Reynolds numbers.

REFERENCES

Davis, S. 1976 A . Rev. Fluid Mech. 8, 57. Dolph, C. L. & Lewis, D. C. 1958 Q. appl. Math. 26, 97. Donnelly, R. J. I964 Proc. R. Soc. Lond. A 281, 177. Glosch, C. E. & Salwen, H. I968 J. Fluid Mech. 34, 177. Hall, P. 1975 a Proc. R. Soc. Lond. A 344, 453. Hall, P. I975 b J. Fluid Mech. 67, 29. Kerzeek, C. & Davis, S. I974 J. Fluid Mech. 62, 753. Mack, L. 1976 J. Fluid Mech. 73, 497. Mueller, D. 1956 Math. Tabl. natn. Res. Coun. Wash. 10, 208. Riley, P. & Lawrence, R. L. 1976 J. Fluid Mech. 75, 625. Seminara, G. & Hall, P. 1976 Proc. R. Soc. Lond. A 350, 299.



the linear stability of flat stokes layers

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