an asymptotic investigation of the stationary modes of instability of the boundary layer on a...

An Asymptotic Investigation of the Stationary Modes of Instability of the Boundary Layeron a Rotating DiscAuthor(s): P. HallSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 406, No. 1830 (Jul. 8, 1986), pp. 93-106Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/2397811 .

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Proc. R. Soc. Lond. A 406, 93-106 (1986)

Printed in Great Britain

An asymptotic investigation of the stationary modes of instability of the boundary layer on a rotating disc

BY P. HALL

Mathematics Department, University of Exeter, North Park Road, Exeter EX4 4QE, U.K.

(Communicated by F. T. Smith, F.R.S. - Received 6 January 1986)

The paper investigates high-Reynolds-number stationary instabilities in the boundary layer on a rotating disc. The investigation demonstrates that, in addition to the inviscid mode found by Gregory, Stuart & Walker (Phil. Trans. R. Soc. Lond. A 248, 155 (I955)) at high Reynolds numbers, there is a stationary short-wavelength mode. This mode has its structure fixed by a balance between viscous and Coriolis forces and cannot be described by an inviscid theory. The asymptotic structure of the wavenumber and orientation of this mode is obtained, and a similar analysis is given for the inviscid mode. The expansion procedure provides the capacity of taking non-parallel effects into account in a self-consistent manner. The inviscid solution of Gregory et al. is modified to take account of viscous effects. The expansion procedure used is again capable of taking non-parallel effects into account. The results obtained suggest why the inviscid approach of Gregory et al. should give a good approximation to the experimentally measured orientation of the vortices. The results also explain partly why the inviscid analysis should not give such a good approximation to the wavenumber of the vortices. The asymptotic analysis of both modes provides a starting point for the corresponding nonlinear problems.

1. INTRODUCTION

In recent years there has been much interest in the manner in which three- dimensional boundary layers become unstable (see, for example, Reed (I985), Hall (I985), Malik (I986) and Lakin & Hussaini (1984)). Much of this work has been motivated by the need to understand the instability mechanisms that are operational in the boundary layer on a a swept wing. This research has been directed towards the development of laminar flow aerofoils, and the possible instability of the flow to Gortler vortices, crossflow vortices, and Tollmien- Schlichting waves must be taken into account. Thus, for example, Hall (I985) considered the Gortler vortex instability in a weakly three-dimensional boundary layer and found that an asymptotically small spanwise velocity component is sufficient to prevent the G6rtler mechanism occurring at finite G6rtler numbers.

The crossflow vortex instability mechanism has also been the subject of many investigations (see, for example, Gregory, Stuart & Walker (I955), Cebeci & Stewartson (1 980), Malik et al. (1 98 I ) and Reed (1 985 )). This instability mechanism occurs when the effective velocity profile in a parallel flow approximation to the

[ 93 ]



94 P. Hall

linear instability equations has an effective velocity field with an inflection point where the velocity field vanishes. The importance of such a profile was explained by the inviscid analysis of Gregory, Stuart & Walker (hereafter referred to as G.S.W.) in the context of the rotating-disc problem.

The noteworthy feature of this profile is that it can support a stationary vortex pattern relative to the disc. G.S.W. showed that the normal to the vortex boundaries made an angle qS of about 130 to the radius vector. This was found to be in excellent agreement with their experimental observations, but the number of vortices predicted by the theory was found to be too large by a factor of about 4. The latter discrepancy has been attributed to viscous effects, but the reason why the angle qS should not also be significantly altered by such effects is not clear. The asymptotic investigation of the G.S.W. mode that we shall give in ?3 will shed light on this question.

A recent parallel-flow numerical investigation by Malik (I986) found that the point at infinity of G.S.W. in the wavenumber-Reynolds-number plane is connected to a curve corresponding to stationary modes at finite Reynolds number. However, 0 varies along the curve and the critical Reynolds number corresponds to qS 0 110, and there is also a lower branch on which 0 asymptotes to about 390 when the Reynolds number is larger.

At finite Reynolds numbers the parallel flow approximation of Malik cannot be justified. At high Reynolds numbers Malik's equations can be thought of as the final step in a successive approximation solution to the full equations. Here we develop formal asymptotic solutions to the full equations based on a large Reynolds number assumption. For both types of modes the expansions developed can take care of non-parallel effects in a self-consistent manner. Moreover the structures of the expansions are appropriate to more general three-dimensional boundary layers and so our analysis is relevant to flows of practical interest. Thus, with only minor modifications, our expansions could be used to predict the high Reynolds number stationary disturbances that occur in, for example, the boundary layer on a swept wing. The consistency of our asymptotic results and Malik's calculations suggests that such predictions will be reasonably good over a significant range of Reynolds numbers. The procedure adopted in the rest of the paper is as follows. In ?2 we formulate the partial differential equations that govern the instability of the flow on a rotating disc.

In ? 3 we give an asymptotic description of the upper branch mode. It is shown that the inviscid analysis of G.S.W. corresponds to the zero-order approximation to the instability equations. It is found that viscous effects are required to satisfy the no-slip condition on the disturbance velocity field at the disc. The orientation of the vortices where they are neutral is found to be dependent on the Reynolds number. The zero-order approximation to the orientation is that found by G.S.W. but the next-order correction is a viscous effect not given by them. The sign of the correction term possibly explains why the inviscid analysis of G.S.W. gives good agreement with experiments at finite Reynolds numbers. It appears that the nonlinear critical layer structure of the G.S.W. mode has not been investigated; the calculation given in ?3 must be done before such an investigation can be undertaken.

In ?4 the lower branch modes are shown to be governed by a triple-deck



Three-dimensional boundary-layer instability 95

structure. The effective velocity profile for the disturbances has no shear stress so that in the lower deck the eigenfunctions are determined by parabolic cylinder functions rather than Airy functions. The upper and main deck structures of the eigenfunctions are essentially those found by Smith (I979) for Blasius flow. The first two terms in the expansion of the angle determining the orientation of the vortices are found. However, unlike the situation with the upper branch modes, only the first term in the wavenumber expansion is calculated.

Finally in ?5 we draw some conclusions.

2. FORMULATION OF THE PROBLEM

We consider the flow of a viscous fluid of kinematic viscosity v in the region z > 0. The motion of the fluid is induced by the steady rotation with angular velocity Q of the plane z = 0 about the z axis. We take cylindrical polar coordinates (r, 0, z) with r and z having been made dimensionless with respect to some reference length 1. The Reynolds number R for the flow is defined by

R = Ql2/1, (2.1)

and if the axes rotate with the plane, then the basic steady velocity field is

U = ub = lQ?{r(Riz), rvi(R2z), R-ii(R'z)}. (2.2)

Here the functions u-, v and w are determined by

u2_(V 1) t-U u= 0, (2.3a)

2u(v+1)+v'-v" -= 0, (2.3b)

2u-+w'/ = O, (2.3c)

where the prime denotes differentiation with respect to z. The appropriate boundary conditions are

u =O, = =O, w =0, z=O, (2.4)

We now perturb the above flow by writing

u = Ub+ Ql{U(r, 0, z), V(r, 0, z), W(r, 0, z)}, (2.5)

where U, V, and W are small and steady. The expression (2.5) is then substituted into the Navier-Stokes equations in the rotating frame and linearized to give

U_a 8+f a + w ad U+-U-2{P+ 1) V+rWou- 7+ + { IY'U- 2 OV

- U

(2.6a)

{rU_ +va +R- 'w-aa V+iUV+2{v+1}U+rW-=?-r +! YV+ 2 W0 2 OJr Oz0o r 60 R r20rJ (2.6b)

{rua +va +R-1 } W+R-2 W8 = W

R ff (2.6c)



96 P. Hall

a2 1 a2 la 2

where -+-2 -+--+ where S-a~~~~~~~r2 +r2 002 +r Or + Z2

and P is the non-dimensional pressure perturbation. The equation of continuity then becomes

au+ U 1V av+w = W . (2.7) Or r r a0 Oz

Finally, we must solve (2.6) and (2.7) subject to the no-slip condition at the wall, although sufficiently far away from the wall we insist that the disturbance decays to zero. However, we shall see below that the length scale for this decay to zero will depend on the type of disturbance under consideration.

3. THE INVISCID MODES

From the inviscid theory of G.S.W. we expect these modes to have wavelengths scaled on the boundary layer thickness. Thus we must consider modes with a lengthscale of order R-2 in the r and 0 directions. It is convenient to define the small parameter e by

6 = R-16

and then write U = u(z) exp (i/e3) {fa(r, e) dr+t/3(e)} (3.1)

together with similar expressions for V, W and the pressure perturbation P. The wavenumber a will, in general, be complex and is determined in terms of e and ,/. However, we shall restrict our attention to neutral disturbances and find a and /3 such that the flow is neutrally stable at the position r. We now expand a, /3 as

a = ao+CaJ+5 . . . 5 (3.2a)

I# = 1#0 + 61#1 + 5 - - - - (3.2b)

The disturbance structure in the z direction is then fixed by the following considerations. Firstly, from the results of G.S.W. we expect there to be an inviscid zone of depth 0(63). To satisfy the no-slip condition on the velocity at the wall, a viscous layer must exist. The thickness of this layer is then found to be 0(64) by balancing the convection and diffusion terms in the disturbance equations.

In the inviscid zone we expand u, v, w, and p in the form

U = U0(0 + cui(6) + **1 ...(3.3a)

V = V0() + CVr) + 5 ...' (3.3b)

w = WO(O) + CWA() +, ...(3.3c)

p = Po(6) + CP1() + 5 ...5 (3.3d)

where Z = ze3. The above expansions are then substituted into (2.6) and (2.7) with




O/Or -replaced by a/ar + (i/C3) {aXO + ea1 +, ... ,} and with 0/80 replaced by (i/63) {flo + c/i1 +, ... ,}. If we equate terms of order c-4, we obtain

iiuo + rwO U' = -iaopo, (3.4a)

iivo + rwo v' = -(i3o/r) P (3.4b)

iuw =-Po (3.4c)

iac uo + (i/ivo/r) + w' = 0, (3.4d)

where u = ao ur + Ih v. If uOI vo and the pressure from the above equations are eliminated, we find that wo satisfies

u[wO-yowo]-u wo = 0, (3.5)

with uii now acting as the 'effective' or 'equivalent' two-dimensional velocity profile, and y2 = a2 + fl2/r2 is the effective wavenumber. Thus, wo satisfies Rayleigh's equation and y0 is determined as an eigenvalue when (3.5) is solved subject to

w0 = 0, = 0, O. (3.6)

We further note that (x/flo is chosen such that '= and v=" vanish at the same non-zero value of C = ~; in this case (3.5) has no singularity at C = .

If ~-+4 in (3.4a, b) it is found that uo, v0 behave like 1/(-) there; however, the combination oo uo + /lo vo/r has no such singularity. Since the calculation that follows depends only on the latter combination it is not necessary for us to determine the critical layer structure of uo and vo. The eigenvalue problem was solved by using central differences; we obtained

Yo= 1.16, (3.7a) ao/fo = 4.26/r, (3.7b)

= 1.46.

The eigenfunction w0 normalized with w' = 1 at C = 0 is shown in figure 1.

0.24 -

0.16

0.08t\

0 2 4 6

FIGURE 1. The inviscid motion eigenfunction.

4 Vol. 406. A



98 P. Hall

Having calculated wo we can use (2.8), (2.9) to solve for uo, vo, and po; however, it suffices here to note that when C-*0

i{c0 uo + fl0 vo/r}--> w(0). (3.8)

Before proceeding to the next order in the inviscid zone, we calculate the zero-order solution in the wall layer. If we write

6 =C4z,

then, in the wall layer, u- expands as

u = Muobr+..*

and v-, w- are expanded in a similar manner. The disturbance velocity and pressure now are written as u = UO(6)+cu1(6)+, ...,

V = Vo (6) + e V(6)+'n

w = sWO(6) + 62Wl(6), ...,(3.9)

p= Po(6)+62Pl()+.+I

After substituting the above expansions into the disturbance equations, equating the dominant terms and performing some manipulations, we find that

cO UO + flo Vo/r satisfies

[co UO+/30 V0/r]"'t-i6[cx0i0r +flO P0] [azo UO + o V0/r]' = 0 (3.10)

and the solution of this equation that satisfies UO + /Jo VO/oxo r= 0 at r = 0 and (3.8) is

V w(0)J Ai(ys) ds [

? ? r ] ' ~~~~~~~~~(3.11 ) J Ai(ys) ds

where y = {i[x0O i0r +0 V-0}3. (3.12)

For large 6 it is found that 00

WO 111 w' 6+ w'(0) A'(0) /y Ai (s) ds,

so that w1, the order e inviscid zone normal velocity component, must satisfy

w?+ w'(0) A'()/vf) Ai(s) ds, C-*0. (3.13)

We now turn to the next-order problem in the inviscid zone. Thus we now equate terms 0(6-3) in the inviscid zone disturbance equations to obtain a set of equations similar to (3.4) with (uO, vo, wo, po) replaced by (u1, v1, w1, p1) but now having inhomogeneous terms. If manipulations carried out on (3.4) to get (3.5) are repeated it is found that

v[1- y0 w1] -v W = 2u~ {zO z + flO ll/r2} w0 + {1-/h x0o/3o} r{u" - '/ w- . (3.14)




The second term on the right-hand side of (3.14) causes w1 to have a logarithmic singularity at C = '; this can be removed in the usual way by introducing a critical layer at ( = ~. The solution of (3.14) that satisfies wl(oo) = 0 is

w = 2G + 2 }w(() g- w J w2(0) dO

+ {a l l _ rl } rwo d 2(O) [|"(0) [ (O) ( "(O)iv(O)] dO, (3.15) 1 ,8~~ J0J~W w(y) J 0 [i~

where C is such that C> g. A routine viscous critical layer calculation shows that the above solution is valid for C < C if the path of integration is deformed appropriately into the complex plane near C= . It can then be shown from (3.15) that wl(0) = 2 r + w0(0) } w0(0) '2' (3.16)

where I = fw2(0)dO (3.17 a)

and '2=/of w2[ U$2 dO, (3.17b)

where the path of integration in (3.17 b) is deformed below the singularity at C =

to be consistent with a viscous critical-layer calculation. The matching condition (3.13) produces the eigenrelation

i 0 2 IX L2{L + fl' flo}11 +{' -0 l

r1}2I (3.18)

vfAi(8) ds l

Our calculations showed that

Ii = 0.094, 2 = 0.058-0.029i,

and using the well known values for Ao) f0 Ai(s) ds, we obtained

acO CXi + 2 1=-14r Yo0,

2_ r = 29r3.

The above equations can be solved for al, /31; however, it is more useful to evaluate

v (x2 +/2/r2) = y+x0cxj + /o 30l/r 2] 6/yo +,

= 1.16-( 14.4/R1) r-+, ..., (3.19)

which we interpret as the 'effective' wavenumber of the disturbance. The wave angle 0 then expands as

tan [2E-0] = A ?o+r {-ci2f?}c+, ..., = 4.26+29r-/R6+, . (3.20)

4-2



100 P. Hall

Thus we have calculated the first correction terms to the classical results of G.S.W. The sign of the correction term in (3.20) has some important consequences, which we shall discuss in ?5.

4. THE WALL MODES

We have seen in the previous section that the 'effective' velocity profile for a three-dimensional disturbance with wavenumbers a and ,8 in the r and 0 directions is auir +,8 f. The inviscid modes are such that acur +/hJ and au""r +,/i' vanish simultaneously. It can be shown that lower branch disturbances having a triple deck structure of the type discussed by Smith (I 979) for Blasius flow can also exist. However, such modes are necessarily time-dependent with a, ,3 real if the effective wall shear au'r+,/f' does not vanish. We therefore choose to look for stationary modes for which the effective wall shear vanishes at zero order.

It can be shown that the appropriate triple-deck structure is based on the small parameter c now defined by

6 = R-ll6, (4.1)

and the lower, main and upper decks are of thickness 69, 68, and 64 respectively. The disturbances structure in the main and upper decks is essentially the same as that found by Smith (I979), who investigated lower branch disturbances to Blasius flow. The wavenumbers in the r and 0 directions are now 0(6-4); we therefore write

U = U(z) exp (i/64) { (r, e) dr +0,()

together with similar expressions for V, W and P. We define 6, , Z by

6= Z/69, = Z/68, Z = Z/64. (4.2 a, b, c)

The wavenumbers then expand as

a = a0+62aj+63 a2 +, .............. ,(4.3a)

fl = flo + 6218S1 +63 fl2+, (4.3b)

Here we have anticipated that the order e terms are zero, and we again seek;zx, /,b, etc., such that the flow is neutrally stable at the location r. In the upper deck u = 0, v -1, and U expands as

U = 63Uo(Z) + 64U1(Z) +,

and V, W and P have similar expansions. We found that the zero-order equations to be solved in the upper deck are

30 UO = aO Po, /30 VO = o Po/r, il3o Wo = dPo/dz,

iaco Uo + i/o Vo/r + d W0/dZ = 0,

and the solution of this system that decays to zero when z -x 00 is

PO = CeYoZ, U0 = (xo/fi0) C e-Yo Z, (4.4a, b)

VO = (C/r) e-yoZ, WO = (iyo/fo)Ce- Yoz, (4.4c, d)




where Yo = V(ax +/3/r2)

and C is an unknown function of r. In the main deck the disturbance expands as

U-=(I1 /6) UO(O) + UA() +I ..*.I*

V = (I /6) Vo(O + V() +)-

W = 63WO(C) + 64W1(C) +,

p = 3C+C4p1(C) +, -...

where we have anticipated that P is independent of C to order 6 and therefore equal to C. Substituting into (2.6) and (2.7) we find that uo, vo, wo satisfy

iaO ru-uo + i/o Vu+ru wo = 0,

iao ru-vo + i/o 0vvo + rv'wo = 0,

iaO uo + (i/?o/r) vo + dwo/dC = 0,

and a solution of this system that matches with the upper deck solution is

uO = Cryo 0`//3f (4.5a)

vo = Cryo V-'/fl8 (4.5b)

w = - (iCyol//8) (atO ru- +30 v-). (4.5 c)

We note from (4.5c) that wo in fact satisfies the no-slip condition when C-- 0; however, unless u-' and v' both vanish at o =0 the other velocity components are non-zero there. If we choose ao and ,80 such that

aO 0'(0) + (lo3/r) `-(0) = 0, (4.6)

which gives aO r//?o = 1.207, then aO uo + i8 vo/r - 0 when C--> 0. It is the imposition of the constraint (4.6) on the effective velocity profile that enables us to find stationary disturbances. If we expand u-, v- for small C and write 4 = /C, we have

U-= CUo-6+62U-162+3U-263+,..., (4.7a) u Cuo u U2 ~~~~~~~~~(4.7ba) -

= V6+62Vl62+03V 63+, ..,(4.7 b)

when u = H(0)/j!, r_ = vP(0)/j!; forj = 1, 2, .... To match with the solution (4.5), written in terms of 6 using (4.7), we therefore expand the lower-deck disturbance in the form

U ,2C L8o + 2eitl + . . .1+ ()+ Uo(6) + CUl (6) +, .. ., (4.8 a)

V= 2 [o+2e6+ + -v()+cv1(6)+1 ...' (4.8b) c/3o

W=_ i?'> e C [(Xo 81 r + A0 fl) 62 + ....] + 6 (4.8 c)

P =,63P1(6)+ ............(4.8d)



102 P. Hall

We must now substitute the above expansions into the disturbance equations and solve for (QU1, V_1), (U0, V0), (U1, V1, W1, P1), etc. From the continuity equation we obtain immediately that

V1 = -(aO r/flo) U_1, (4.9a)

VO = -(aO r//3o) Uo, (4.9b)

where a0//30 satisfies (4.6). From the radial momentum equation we obtain

-i[rto -1. + /lo -1] 62 U 1 + d2U /d62 = 0, (4.10a)

-i[raO U? +/O P1] 2U0 + d2 UO/d2 = -rUO W1 + [raO U2 +,80 v2] 63U1, (4.10b)

which must be solved subject to

U_1 =-ryo Co//32 Uo =, 6 = ,) (4.11)

U_1, UO-*O, 6 -o.

The function U_1 is given by

U~1 iuoyoCr U(O, V\2Ai6) (4.12) -1 A2 U(O, O)

where A = i{faOrUl+/3o v} (4.13)

and U(0, V\2Ai6) is a parabolic cylinder function. The functions U0, V0 cannot be determined until W1 is calculated. The latter function can be found by considering the next order approximation to the radial and azimuthal momentum equations. If we multiply these equations by iao and i/80/y respectively and add them we obtain

1d2U1+3 d2 ___

i ?X d62 +r d62J + Yo P+2icxO V1 r2i/30 U ljjxo

r~~~~~~~iloV = i{co rH1 + flo p} {iczo u1 + A/31 62 + 2i6{cx0 ru1 + 3o V} W1

-2{a, rU- + , v- - {aO U+r) (4.14)

and the z momentum and continuity equations give

dP1/d6= 0,

a{O UJ+l+_ _

+ dW y = i U- [ fl1oP0] i-ial U- fl- V-1 (4.15) iJ0 +floj2 + 1 r r (IS

so that P1 = C. It is important to point out at this stage that the terms proportional to UJ1, VW1 in (4.14) are due to Coriolis effects; thus the structure of the neutral curve for stationary small-wavenumber disturbances depends both on viscous and Coriolis effects. We can eliminate U1, V1 from (4.14) and (4.15) to give




d3W1 W d___- i{x0 rui + /? v1} 62 d + 2i6{x rH1 + W Pl} W1

= C+r{lC+ 2 r- + o - + W 1

+ , {+r2U U( O) , (4.16a)

where s=z1i6. (4.16b)

We write the solution of (4.16) in the form

W1-i{t -r+/A V} C + A- CF(s) +u2] f (0,0)

(4.17) where k1 is a constant and F1 satisfies

FY'-s2F+ 2sF1 = 1, F1( )= F1 (o) =O,

and F2 satisfies a similar equation with the right-hand side of the differential replaced by U(0, V2s). In fact, it is possible to express F F in terms

equation 1~~~~~_ lo io 0 (' 2) (.1a

of integrals involving parabolic cylinder functions. It remains for us to satisfy Uw = Ve = 0 at 8 = 0; from (4.15) and (4.17) we can show that this condition leads to the eigenrelation:

YOI3[ 1+ Y2 = o2iyo C (4.18)

Here the integrals 13, 14 are given by

2 _U(0,0) T U2U(0, 6O

F 8 F' U2(0, 0) = 0 ,

and (4.18) can be solved to give

{ti/%Lzo} = 2yo(+i5o/fiuo) - 2.312 r-i. (4.20)

We see at this stage that it is still not possible to find ex and F independently; however, it follows from above that 1, the angle between the radius vector and the normal to the vortices, is given by

tan [2 -qS] = 1.207 + 2.312 e2r+ , . (4.21)



104 P. Hall

and the total wavenumber (1/64) V(x2 +/32/r2) is given by

( I /64) v\ (X2 + I2/r2) = 1.224 r-1/64 +, .... (4.22)

The above expansion procedure can be continued in principle to any order and can take non-parallel effects into account as in Smith (I979).

We stress that (4.21) and (4.22) have been obtained by taking the Coriolis effect into account; an Orr-Sommerfeld approximation to the full equations gives incorrect values for the second term in (4.21) and the first term in (4.22). The sensitivity of the structure of the lower-branch modes to a combination of viscous and Coriolis forces means that, unlike the upper-branch modes, for a more general three-dimensional boundary layer this class of modes might not even exist. Finally, we note that time-dependent modes with a sufficiently long timescale are also possible and introduce a frequency into the eigenrelation (4.18).

5. CONCLUSION

The Reynolds number RA, based on the boundary layer thickness and the local azimuthal velocity of the disc, is given by

RA = Rr2.

The inviscid modes have local wavenumber kA defined by

ka= =\/(a2+fl2/r2)

where the appropriate lengthscale is the boundary layer thickness. On the lower branch the local wavenumber ka is defined by

kA = R-V (X2 + 2/r2),

so that (3.19) and (4.22) are equivalent to

kA = 1.16--14.4 Ra3+,..., (5.1)

and ka = 1.22Ra+,..., (5.2)

respectively. Similarly (3.20) and (4.21) become

tan [217- ]= 4.26+29R-3+ ... (5.3)

and tan [12-] = 1.21 + 2.31 R-4 +, (5.4)

Thus, if the neutral values are expressed in terms of RA, then ka and 5i have no explicit dependence on the radial variable r.

In figures 2 and 3 we have compared the above asymptotic predictions with the numerical results of Malik (I986), who solved the parallel flow approximation to (2.6) obtained by setting a/ar _ iac, a/la = i,J. Such an approximation is valid only for R -* oo but to the order shown in (5.1)-(5.4) our asymptotic results apply to the system solved by Malik.

We see in figures 2 and 3 that there is satisfactory agreement between the asymptotic theory and Malik's results. Thus the asymptotic approach could be a useful tool in finding the structure of the possible stationary modes in other three-dimensional boundary layers rather than having to solve the full parallel-flow




100 - -

inviscid mode

10-2

kA lower- branch modes

. I I IiI 102 104 106

Ra

FIGURE 2. Comparison between the results of Malik (I986) and the asymptotic wavenumber predictions.

40 -lower - branch mode

20-

0 I I 102 t10

RA

FIGURE 3. Comparison between the results of Malik (I986) and the asymptotic wave angle predictions.

equations numerically. Similarly, the asymptotic theory could be used to identify the stationary modes that are likely to be important in a Navier-Stokes investigation of this problem.

It is interesting to question why the lower-branch modes have not been investigated earlier. The reason appears to be that in most experimental investigations of the disc problem, only the modes with 5i 0 130 were observed. However, there is some discussion of modes with 0 - 200 in the paper by Federov et al. (I976). These modes were found to exist closer to the centre of the disc than the G.S.W. modes and have a different vertical structure. Thus, it is possible that the lower



106 P. Hall

branch modes may bifurcate suberitically and therefore do not persist into the region where the G.S.W. modes occur. Obviously, a weakly nonlinear theory at least could settle this issue. However, it is interesting to note that S. Allen and J. T. Stuart (I985, personal communication) have pointed out the possible existence of a subcritical mode with azimuthal wavenumber n = 2.

The upper-branch asymptotic results are again consistent with the results of Malik. The positive sign associated with the first correction term in (5.3) has important consequences. Thus if (3.3) and (5.4) are to be connected at some finite Reynolds number the higher-order terms in (5.3) must be negative. This means that for some range of RA, the value of 0 along the upper-branch modes will stay close to the infinite value of about 13?. This is exactly what Malik found numerically and, even at the critical Reynolds number, 0 is still close to 130. The wavenumber, however, changes much more along the upper branch; this could explain why G.S.W. predicted 0, but not the number of waves, so well.

Finally, we turn to the relevance of the lower-branch modes in other boundary layer flows. At first sight one might think that our analysis is directly applicable to a two-dimensional boundary layer having zero shear stress at some position along the boundary. However, it can be shown that the structure given in ?3 is only applicable to boundary layers having a non-zero third normal derivative of the streamwise velocity component. This constraint effectively means that there are no neutral modes of the type found in ?4 for spatially varying two-dimensional boundary layers. The Stokes layer velocity profile is another matter; at high Reynolds numbers the flow varies slowly in time, and the modes discussed in ?4 are relevant to the times in a cycle when the shear stress instantaneously vanishes at the oscillating wall during the fluid motion.

For three-dimensional boundary layers we expect that the lower-branch modes are directly relevant. Moreover, it is of course possible that in such flows nonlinear effects might cause them to be more important than the G.S.W. modes in the development of crossflow vortices. This matter can, of course, only be resolved by further calculations.

This research was supported by the National Aeronautics and Space Admini- stration under NASA contract no. NAS1-17070 while the author was in residence at the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, Virginia 23665, U.S.A.

REFERENCES

Cebeci, T. & Stewartson, K. I980 AIAA J. 18, 1485. Federov, B. I., Plavnik, G. Z., Prokhorov, I. V. & Zhukhovitskii, L. G. I976 J. engng Phys.

31, 1448. Gregory, N., Stuart, J. T. & Walker, W. S. 1955 Phil. Tran8. R. Soc. Lond. A 248, 155-199. Hall, P. I985 Proc. R. Soc. Lond. A 399, 135-152. Lakin, W. & Hussaini, Y. I984 Proc. R. Soc. Lond. A 393, 101-116. Malik, M. I986 J. Fluid Mech. (In the press.) Malik, M., Wilkinson, S. P. & Orszag, S. A. I98I AIAA J. 19, 1131. Reed, H. I985 AIAA Pap. no. 84-1678. Smith, F. T. 1979 Proc. R. Soc. Lond. A 366, 91-109.



an asymptotic investigation of the stationary modes of instability of the boundary layer on a...

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