centrifugal instabilities of circumferential flows in finite cylinders: the wide gap problem

Centrifugal Instabilities of Circumferential Flows in Finite Cylinders: The Wide Gap ProblemAuthor(s): P. HallSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 384, No. 1787 (Dec. 8, 1982), pp. 359-379Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/2397228 .

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Proc. R. Soc. Lond. A 384, 359-379 (1982)

Printed in Great Britain

Centrifugal instabilities of circumferential flows in finite cylinders: the wide gap problem

By P. HALL

Department of Mathematics, Imperial College of Science and Technology, London SW7 2BU, U.K.

(Communicated by J. T. Stuart, F.R.S. - Received 18 February 1982)

The remarkable results obtained by Benjamin (1978), who investigated experimentally the nature of Taylor vortex flows in short cylinders have stimulated much theoretical work on the role of end effects in stability theory. Most of this work has been in connection with the Rayleigh- Benard problem in finite rectangular or circular containers where the side- wall conditions are simple enough for perturbation methods to be used. The experiments of Benjamin were performed in cylinders of variable length having end walls held fixed to the stationary outer cylinder. In this case the basic state set up when the inner cylinder rotates is never close to the purely circumferential flow of the corresponding infinite problem. Thus perturbation methods cannot be used directly to investigate Benjamin's problem but Schaeffer (I980) has proposed a model problem having simpler end-wall conditions. The conditions used by Schaeffer in fact correspond to porous end walls. Some predictions about Benjamin's problem can be made by perturbing the end-wall conditions of the model problem towards those of Benjamin's apparatus. Schaeffer's analysis is not valid for most of the available experimental results, which correspond to cylinders so short that only two or four Taylor cells occur. However, the bifurcation picture in the length-Reynolds number plane obtained by Benjamin for this configti- ration is probably similar to those appropriate to longer cylinders. Using qualitative methods and assuming certain numerical constants have the required behaviour, Schaeffer argues that Benjamin's results are plausible.

In this paper we investigate quantitatively the model problem proposed by Schaeffer using perturbation methods and determine explicit values for the numerical constants, which are so crucial in determining the possible equilibrium configurations. Moreover, the present formulation allows us to calculate the stability properties of these equilibrium configurations. Results are obtained for a wide range of possible ratios of cylinder diameters.

We also investigate the possibility of equilibrium configurations having an odd number of cells. The results obtained are compared with the observations of Benjamin.

INTRODUCTION

Our concern is with the influence of end effects in the Taylor vortex problem. It is now well known that in the Rayleigh-Benard problem imperfections associated with the application of realistic side-wall conditions cause the onset of convection

[ 359 ]



360 P. Hall

to occur smoothly rather than as a bifurcation from the motionless state. This result has been found experimentally for the convection problem by Ahlers (I975) while Benjamin (I978 a, b), hereafter referred to as B, has investigated the closely related Taylor vortex problem.

D

L B

Reynolds number, A FIGURE 1. The experimental results of Benjamin (I978a, b) for the

two-cell-four-cell interaction problem.

The experiments described by B were performed in cylinders of length Ld the same order as d, the gap width. The value of L was varied but particular attention was given to the case when only two- or four-cell flows could be set up in the apparatus. The apparatus used by B had end walls fixed to the stationary outer cylinder, and the primary flow set up when the speed of the inner cylinder was gradually increased from zero always had an even number of cells.

We now refer to figure 5 of Benjamin (I978 b), which for convenience is sketched here as figure 1. This figure indicates the types of flows that are possible for different values of the length and the Reynolds number. The three regimes of interest and their characteristic properties are:

(I) L > L2. The primary flow has four cells, but to the right of CD (and with L > L2) a two-cell secondary flow is possible.

(II) L < Ll. The primary flow has two cells, but to the right of BA (and with L < L,) a four-cell secondary flow is possible.

(III) L, < L < L2. The primary flow is not clearly a two-cell or a four-cell flow and loses its stability when CB is crossed with increasing Reynolds number. The resulting two-cell secondary flow is stable to the right of CD so that if when the flow is established the Reynolds number is decreased then the primary flow is re-established when CD is crossed.



Taylor vortices in finite cylinders 361

An analytical investigation of this problem is not possible because of the requirement that the no-slip condition be satisfied at the ends of the cylinders. Furthermore it appears that numerical investigations of this problem have been deterred by the size of the required computation. For these reasons Schaeffer (I980),

hereafter referred to as S, solved a model problem with artificial end-wall conditions simple enough for some analytical progress to be made. The conditions are

U. = (1-a) Out/On + aut = 0, (1.1 a, b) where u is the fluid velocity and u., ut denote the components of u normal and tangential to the end plates respectively. The parameter a is taken to be in the interval [0, 1]. If a = 1 the conditions (1.1) require that u should vanish at the ends of the cylinders. This is the situation appropriate to the apparatus used by B. If a= 0 the conditions require that the axial velocity component and the normal derivatives of the radial and circumferential velocity components vanish at the end walls. In this case some analytical progress can be made and by taking a, to be small and positive we can determine some information on how the possible equilibrium configurations depend on a.

It is hoped that the results of such a calculation do not differ dramatically from the solution of the problem with a = 1. Of course it is clear that the relation between the a = 1 and a < 1 results can only be verified by a detailed numerical investigation of the partial differential problem for different values of a.

The crucial simplification of the a = 0 problem is that it leads to a basic circumferential flow identical to that of the corresponding infinite problem. If a is perturbed away from a = 0 the basic state at low Reynolds numbers consists of a dominant order a? circumferential flow together with a weak vortex flow of order a. The latter circulatory flow becomes stronger when the order-a? flow is linearly unstable and nonlinear effects must be considered. It is the development of this order-a flow that we expect to be similar to the development of the circulatory flow for a = 1.

The analysis given by S does not apply to the two-cell-four-cell interaction problem and Hall (I980) used the artificial end-wall conditions proposed by S to investigate that problem. It was found that the curves AB, BC of figure 1 were predicted by the model problem, but the phenomenon associated with CD was not detected. The degeneracy of the two-cell-four-cell problem is discussed in detail by Hall (I980).

Using abstract methods S had argued that the interaction of two modes of amplitude X and Y could be described by the solution of two coupled cubic equations in X and Y of exactly the same form as those found in the convection problem by Hall & Walton (I979). The approach used by S did not allow for time-dependent amplitudes so that the stability properties of these solutions could not be discussed. It was found by S that only for certain values of the coefficients in the cubic equations would it be possible to produce results similar to figure 1.

Thus, influenced by the experimental results, S assumed that the coefficients had the required properties, but such an assumption is open to criticism since his

'3 Vol. 384. A



362 P. Hall

analysis fails for the two-cell-four-cell problem. Here we determine whether the assumption made by S is ever justified and actually make precise predictions about possible Taylor vortex flows in finite cylinders. In this paper we follow the approach of Stuart & Di Prima (I980) and use a perturbation expansion that is able to handle time-dependent perturbations and give expressions for the constants that determine possible equilibrium flows. We also outline how we have calculated these constants numerically. We are thus able to predict the types of flows that should be observed in cylinders in which more than two cells can be accommodated. Of course these predictions assume that the results for a = 1 and ac < 1 are similar. More precisely we predict that an 'upside-down' version of figure 1 is relevant to the four-cell-six-cell interaction problem. Since this paper was originally submitted for publication this result has been found experimentally by Mullin et al. (1981). In addition to symmetric flows we also investigate whether asymmetric flows should be expected in the finite Taylor problem. Our results suggest that, as is found experimentally, such flows are probably not accessible by slowly increasing the speed of the inner cylinder.

The procedure adopted in the rest of this paper is as follows. In ? 2 we formulate the partial differential system to be solved in ? 3 for modes having 2m and 2m + 2 cells where m > 2. The amplitude equation obtained in ? 3 determines such flows and they are solved in ? 4, while symmetric-asymmetric interactions are investigated in ? 5. Finally in ? 6 we draw some conclusions.

2. FORMULATION OF THE PROBLEM

We consider the flow of a viscous fluid of kinematic viscosity v and density p between concentric circular cylinders of length Ld and radii R1 and R1 + d. The flow is driven by the inner cylinder, which rotates with angular velocity Q while the outer cylinder is held fixed. We take cylindrical polar coordinates r, 0, z scaled on the gap width d and the corresponding velocity vector U is scaled on QR1. If the cylinders are of infinite length then the basic flow is

U = (0, V(r), 0), (2.1)

where V =-rr,/(l + 2r1) +rl(l +r1)2/r(1 + 2r,). (2.2)

If we now write U= (0, V, 0) +u (2.3)

and scale the pressure perturbation p associated with u on pvQR1 d-1 and time on d2v-1, where p is the fluid density, then we can show from the momentum and continuity equations that U (u, v, w, p)T satisfies

P U/t + RU+ AMAIU+ CU = AQ(U) U. (2.4)

Here A is the Reynolds number defined by

A = QRldr-1, (2.5)



Taylor vortices i nfinite cylimders 363

1 0 0 00 -2V/r 0 0 01 0 2a0 0 0 (.ab while P= 0 0 1 0 M ? (2.a,b)

10 0 0 0 L0 0 0 0

C= | o a/az r R=(82arar+a2P o

L/ar+ i/r 0 //zz 0 Jr0 0 00 (2.6c, d)

- u alar - w a/az v/r 0 V/r - u/Wr - w J/?z 0 0 (.e Q(U) = 0 -u a/ar-w a/az o

L 0 ~~~0 0 O

where 2a = dV/dr + V/r. The above equations are to be solved such that the total fluid velocity l? satisfies

O& (0) 1, 0), r =rl,

p(, 0, 0), r= r +1, (2.7)

= ?, (1- a)a0t/an+a t = O, z= +L,

where qi& and 9t are the normal and tangential components of 9/respectively. Thus the no-slip condition is to be satisfied at the cylinders r = r1 and r = r1 + 1, while the parameter a, on which the end-wall conditions depend, is assumed to be in the interval [0, 1].

We shall concentrate on the case 0 < a < 1 and we note that in terms of u we can rewrite (2.7) in the form

u=0, r=rl, r1+1, (2.8)

Un = 0, (1- a) aut/an +ct =a-ta(0, V, 0). (2.9)

These boundary conditions together with (2.4) completely specify a partial differential system for U.

We can see from (2.4), (2.8) and (2.9) with a = 0 that U = 0 is a possible solution. Thus when ac = 0 the velocity field of the corresponding infinite problem is a solution of the model problem. Suppose that, having set a = 0, we perturb this flow such that

U = (U(r) cos kz, V(r) cos kz, W(r) sin (kz)/k, P(r) cos kz), (2.10)

then we can show from (2.4), (2.8) and (2.9) that the eigenrelation A = A(L) corresponding to such modes is determined by

RU+AMU+?CU =0, (2.11 a)

U= V= W=0, r=rl,rl+1, (2.11b)

W=0, U/18z=V/18z=0, z=+L. (2.11c) 13-2



364 P. Hall

The axial wavenumber k is thus determined in terms of A by the usual eigenvalue problem

(DD*- k2)2 U = 2A VVVk2/r, (DD*- k2)V = 2AaU,

W =-D*U, (2.12)

U = V = W = O, r=rl,rl+t,

where D = d/dr and D* = D + r-1. The ordinary differential system (2.12) must be solved numerically for different values of r1 and the neutral curves k = k(A) are of the form shown in figure 2. If A is chosen to be greater than some critical value depending on rl, then there are two possible values of k. Such modes satisfy the end-wall condition if kL = mic, for m = 1, 2, .... The flow fields associated with such modes are symmetric about z = 0 and have 2m cells. The eigenrelation A = A(L) for these modes is obtained from the eigenrelation k = k(A) of the infinite problem by replacing k by mir/L. In figure 3 we have sketched the eigencurves in the L, A

100_

90

80-

70 l I I 0 2 4 6

k FIGURE 2. The neutral curve of the linear stability problem

for the infinite case with r, = 1.597.

m=1 2 3 4

L FIGuRE 3. The eigencurves in the A, L plane for symmetric modes with 2m cells.



Taylor vortices in f nite cylinders 365

plane corresponding to these disturbances. The minimum points on each of the curves correspo.id to the minimum point of figure 2. We see that the number of cells of the most dangerous linear mode depends on L, the non-dimensional length of the cylinders. However, for some values of L, the eigencurves cross and the question of which is the preferred mode in the neighbourhood of these points can only be determined by nonlinear theory. We shall restrict our attention in ? 3 to points of intersection corresponding to modes having 2m and 2m + 2 cells for m > 2. The case m = 1 is degenerate and has been considered by Hall (I980). In ?5 we shall investigate asymmetric modes.

3. FINITE-AMPLITUDE SYMMETRIC MODES FOR a < 1 Consider the point where the eigencurves of the 2m- and 2m + 2-cell modes inter-

sect. We now take a to be small and positive and confine our attention to a neighbourhood within ac' of this point. Thus if the coordinates of this point are (L*(m), A*(m)) we expand L and A in the form

L = L* + ciL1+..., (3.1a)

A = A* + aiA1 + . . ., (3.1 b)

where L1, Al are O(o?). We denote the wavenumbers corresponding to A = A* by k, and k2 and without any loss of generality we take k, < k2 so that

klL* = mit, k2L* = (m+ 1)nr, (3.2a, b)

for some positive integer m > 1. To investigate the stability of the finite-amplitude solutions that we shall construct it is necessary for us to define a slow time variable -by -r = cct. (3.3)

We then expand U in the form 00

U= ac3n Un, (3.4) n=1

and substitute into (2.4) after replacing a/lt by acla/ar and A by (3.1 b). We then equate terms of order aC, ai, etc., in turn and solve the resulting partial differential equations subject to (2.8), (2.9). The order-ac partial differential problem to be solved is of course just the linear system (2.1 1) with A = A* and L = L*, which we can solve to give U1 cos k1z 1 U2 cos k2z

U1 = 2X(r) [Wio k, z + 2Y(r) [ V2cosi 2z (3.5a) Wlk-'sinklz W2k1 ~sin kz

P1 cos k1z J P2cos k2Z ] where X(r) and Y(r) are amplitude functions to be determined at higher order while (U1, V1, W1) and (U2, V2, W2) are the velocity eigenfunctions satisfying (2.12) with A = A*, and P1, P2 are the corresponding pressure distributions. The eigenfunctions are normalized such that V'(rl) = V'(r1) =



366 P. Hall

At order ac we find that the partial differential system to determine U2 is RU2+ A*MU2+CU2

[F1cos 2k3Lz G1 cos 2k2z [H cos (k1 + k2) Z

= 2X2 F2cos 2k1z + 2y2 G2cos2kl2z XY H2cos (k1 + k2) z F3sin2k,z G3sin2k2Z H3sin (kO+k2)Z

0 j L 0~~ 1 0 ] I1 cos (ki-k2) z ]

+2XY [I2cos (k-k2) Z

+X2 ki + Y2 K1 (3.5b) I3sin (k - k2) Z [0]

where Fl, F2, etc.,_ are functions of r depending on the first-order eigenfunctions, whose precise form is not given here. At this order the boundary conditions to be satisfied are identical to those appropriate to the linear problem discussed in ? 2. Thur the solution of (3.5b) can be written in the form

r Ull1 cos 2k1z - U222 cos 2k2Z

= 2272 4Vll cos 2k1z 2y2 V222 cos 2k2Z W111(2k,)-l sin 2k1z + W222(2k2)-lsin 2k2z

'L A cos 2k,z L P222 cos 2k2Z

U2l.jcos (k1-k2) Z

+2XY V211 cos (kL-k2) z W2L(,- k2) 1sin (k1 - k2) Z]

P21 cos (k1 - k2) Z

U212 cos(k1 + k2) 1 o 0vo

+2XY V212 cos (k i + k2) Z + X2 VM + Y2 VM2. (3.6) W212(k, + k2)-l sin (k, + k2) Z o o

P212 cos (kI + k2) Z PM1 PM2

We now turn to the solution of the system obtained by equating terms of order x in (2.4). The nonlinear interaction of the solutions of orders ac and x, reproduces the two fundamental modes with wavenumbers ki and k2 together with further harmonic functions with wavenumbers + (k1 ? 2k2), ? (k2 ? 2k1). For convenience we denote by U3 the part of the order-x solution that depends on the two fundamental modes. The partial differential system that determines U3 can be written in the form

RL3+ A*MCT3+ C&3 N11 Cos k1z N21 cos k1z]

N1sin k, z IN2Y N2sin k z1 =-PaU1/8-A1 M U1+ 2X3 |N12cs1 [2XY2 22cos1 2

Q12cosk2l Q21cosk2Z] +2Y3 QlQcosk2 Z +2YX2 Q22cosk2z I (3.7)

IQ13sin k2 +2Y Q23sin k2z Z L 0] 10]




U3 = V3 = 3 = 0, r = rl, r = r+1, t23/Dn = 2(-1)mLlk2XU2+2(- )m+lL1k YU2, z = +L*

aP1/1n = Lk2 2, z = +L*, (3.8) J' = 2(- )m+lL W1X+2(-1)mL1W2 Y, z = +L*,

JJ = 2(-1)mL1JIX+2(-1))m+lLW2 Y, z =-L*.

Here the functions Nll, N22, etc., depend on the solutions of the order-xt, -x* systems. The homogeneous form of (3.7), (3.8) has the non-trivial solution given by (3.5a) so that we require a solvability condition in order that this system should have a solution. After Stuart & Di Prima (I980) this is obtained by first deriving the solution of the linear partial differential system adjoint to (2.1 1). If we denote the adjoint by IF = ( 1, 2' 3, y 4)T then, by multiplying (2.1 1 a) by VpT and integrating over the region A defined by r, < r < r1+ 1, -L* < z < L*, we can show that the adjoint functions corresponding to k, and k2 are I r11cos k z 1 r

21 coss l2Z

= -f12 cos ki Z I -fi22 cos k2Z

-f13k'1 sin k z , 2 i23k-'sink2z L V14 cos kz L 24 cos k2Z J

where (fl, O121 V131 V14) and (p21 p22' p23, p24) are determined by solving the sixth- order ordinary differential system adjoint to (2.12) with k = k1 and k = k2 respectively. If we now multiply (3.7) by PT, 2T in turn and integrate over the region A then we find that X and Y satisfy

dX/dr = (Al 1 + L1 81) X + X(J1 X2 + y1 y2) + 1jl, (3.9) d Y/dT = (A1l 2 + L1 82) Y + Y( Y2 + Y2X2) + 02J

Here the constants o1, 8i, etc., are defined by

1 = r (V V Y11 - 2aU1 VlY2) dr/c,,

61= L* c U1 1r1 +kj 1V V12 + W1V13 + W1V114) dr/cl,

r1= f+ (Nk 1 + N12 V12 + N13 V13 1c 7) dr/el,

Y= (N211311 + N22 Vr12 + N23 V23 ic 1)dr/,

-)m+l rr,+1 (01 L*

JrL V12V dr/el,

where c1 = f+ (V11 U1 + Vt 12V1+ Vl 3' 1 i 2) dr,

and o2, &2, etc., satisfy similar equations with V1 replaced by V2, etc.



368 P. Hall

I a co

C> ****>

- c- r co

t 104 _4 a t- ,: .,

*

_ _ Or ce o4 Cs O

1 0 "It 0

Co o r o

o 0- cea n+e

o. 10 t t 1

- - =

t- e - C co

Cb n4 n- n4 0 *

to _ e0

O0'I 1 > I-* 0 C >1

'It C> t C> co C>

_. R n nn -

cq 0 - t- I~~

* a a

b C 00 '0 00 00

\4 C6 r 6 cesCe

~~ - 0~0 C4

b q " -0

C6 C6 C

m ro a) t o r co

za 1 - 00 00

C* *I l * *i *i *

c- co n 0 *

- 0 R * 0o r~ - r~ o 00 '




co coa)

sN N x r- M co r

co o - r t- a) r

0 aq o GQ

c: 10 -4 U

co~~~~~~~~1 coe se - c 0u O|C oo 00 CZ s oc U*nese

C- 10 >10 qo0 O -1 0cmx -t

,-icq 0 q 1 0 - oc XO OOOOOO tOO OOO

1 14 1to 1t t 1 1t1 1 1q 1- 1q 10 '10 0 'It -0 '.0r '. 10'0 10 10OD 1 10 10 '0 '0'0 '0 '0 '0

oo 0 I- o 0 o o t o o 0 o 0 10 10 00 aq t- 44 a oo lo o- o o o0 o0 o0 o o oo

*O 10 16 4 CY cy Cn Cn cy cy cl 16 4 4 4 4 C+ C+ C4 cy cy C< C< a nn

*

^ )o o Ot s o N oo a) m -- m 10 a C> c o a4 0 "4 00 N - t- C aq es s

_l c - C> c- m c - aq c: V- _I t- oo --I cr c4s c:

O :e c c sr I co

r 4 Oq

s ce 8 o cec C> -I = "J C OD co to c = oo oo oo COCOC

' 1 1 1 CI I n n n * * * *i *i 1I1 II I I I I *I *

to a o M s a ) t- 0 e es q t o - t-o CD N

t- -, oeo co l)o C9 c c a eo t-n m ce o aq C> m n> a

ce C o oo -4 ce ce es di es es e

a) OD 00 m 10 m cq CO co ese s s

I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

* co

= * to CD .- mo CDo aq r

e oo l5 o "r N o ce t- N = m -r o =o o s

s~~~~~~~~~C ci 1. c:6 O6 C C6 C6 C C6 C6 C6 1q 16 O6 rC6 eno onOrc sr

~~~ 1 0 -~~~~~~~~~~~'

o ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~OV41 01- O

CD 0 - 0 ~ - ----0 V- -t - 4 - - --t - t- "d

b 10 oo o 10 10 10o X t N 4 0 e r + cq -I * oo e "t n

O '.1 4 O aq b t- C> O- co d4 ce C Q O

O 0 = oo lo cq C= m nq o- o

lo x x c r- --I l l l lo m,- x x c N m -t 0 c m x t- =

lo 0 0 = ce

r c ce e t- t- N N CZ os*o*Xoc )\

b rc aq c> 0 es io io c o- s so a oD m ceo = 00 aq aq a)4 a

k lo lo l ,. -d -t -t -. m m - l l l l llt c o lo lo lo lo lo l )c ssro l) o t o t s o t -

r )ci *i * *i *i i * c *i (- *i ar ci cli o i ei ol rl i *i Oi *i e o

c r a c ca o, o- o -- 0 N t- r c e 1 a + coe

m~~~~~~~~~~~~~ _ -4 o o 0 x -t N

60 - C> N 0 t- 0 l' 10 0 ca 10 0 ur 10 0 r0 es o

N to co r 10 to t:o to 10 a) a)ll~

9~~~~~* ainn+cee ec c6 c i c6 r.s o6 sC;l es <6 e6 ee4 ca4 CQ4 C csii c6 ca i6 c6 t: eo c;

n~~~~~~~~~~~~- esoornn ec 4z; -c sla



370 P. Hall

The foregoing constants must be calculated numerically before we can discuss the possible solutions of (3.9) and their stability properties. This was done by first solving the ordinary differential systems appearing in this section by a fifth-order Runge-Kutta scheme with 100 steps on the interval [rl, r, + 1]. The constants o1, 41, etc. were then calculated by using Simpsons rule. The results obtained are shown in table I for r1 = 1.597, which is the value corresponding to Benjamin's experimental configuration. In table 2 we have shown the corresponding results for other values of rl. We have computed results only for 2 < m < 8 because, as we shall see in ? 6, the relevance of the model in the limit L -? oo is limited.

4. THE SOLUTION OF THE AMPLITUDE EQUATIONS

FOR SYMMETRIC MODES

We now describe the equilibrium solutions of (3.9) appropriate to the numerical values of the coefficients appearing in these equations. The discussion appearing in this section is in parts similar to that given by Schaeffer but is included here for completeness.

The equilibrium solutions of (3.9) are found by setting dX/dr = dY/dr = 0 and then eliminating X or Y from the resulting equations. The equation obtained is of degree nine and, before discussing the possible solutions of this equation, we consider the perfect problem obtained by setting O1 = 02 = 0. For definiteness we assume L1 < 0 so that X is the most dangerous linear mode. The four types of solution of (3.9) are

(a) the zero solution with X = Y = 0, which exists for all A1 and L1 and is stable for A1 <- .1L1/o-l and unstable for A, > - 61 -;

(b) the 2m-cell mode with X = + [(A1 oc + L1 4'1) (- fi)-]I, Y = 0, which exists for A1 > - 1Ll/o-l and is stable;

(c) The 2m + 2-cell mode with X = 0, Y = ? [(A1 2+1462)(- i2)1]i, which exists for A1 > - 62L1/o2 and is unstable until it suffers a secondary bifurcation to (d) after which it is stable;

(d) the mixed-mode solution with

X = +{[A1(ajfi2- .2Y1)+L1(61/h2- 2Y1)]/(Y1Y2-/l42)}I,

y = ? {[A1(o1Y2-o2/) + L1(Q1Y2-6'2/31)]/(fi1/32-Y1Y2)}i,

which bifurcates from the pure Y mode when A1 = Ll(41/#32 - 62Y1)/(f2 1 - 0a1f2)

and is unstable. These solutions are illustrated in figure 4 and we note that if L1 > 0 the picture is

essentially identical except that X and Y change roles. Thus for L1 > 0 the pure Y mode bifurcates first from the zero solution whereas the pure X mode is initially unstable until it suffers a secondary bifurcation to the mixed mode. We stress that the foregoing description of the solutions of (3.9) corresponds to the value of the coefficients 61, etc., shown in tables 1 and 2. This bifurcation picture is similar to that




found by Hall & Walton (I979) in the context of Benard convection and that proposed by S.

We now describe how the solutions of the amplitude equation are changed when w, and t2 are non-zero. In the limit A, -? - oo we can show that there is only one solution of the amplitude equation and its asymptotic structure is

X - -(ll/Al ?-11 y ^W- (2/Al '2'

We shall refer to this solution, which exists for A, -+ - oo, as the primary flow. The normalization of the eigenfunctions that we have used in determining the constants in tables 1 and 2 is such that the radial velocity component is positive in [rl, r1 + 1]. This means that in the limit A. -> oo both modes lead to a radial inflow at the ends of the cylinders. This result is consistent with the observations of Benjamin.

(b)Y(c

A /2__\( A

\L1(&12-&2 Y1) (d) 027y701 fl2

FIGURE 4. The finite-amplitude solutions of (3.9) of types (b)-(d) for ) =C)2 = 0 and L1 < 0. , Stable solutions; - - -, unstable solutions.

For definiteness we shall now concentrate on the four-cell-six-cell interaction problem corresponding to r1 = 1.5-97. Thus in the following discussion X and Y denote the amplitudes of the four- and six-cell modes respectively. For a given value of L,, then, depending on A,, there can be one, three, five, seven or nine equilibrium flows. To make predictions that can be checked experimentally we describe how these equilibrium flows change when L, is varied. In figure 5 we show the dependence of the equilibrium solutions on A, for L, = 0. In this case linear stability theory in the absence of any forcing from the end walls predicts that the four- and six-cell modes are equally likely. However, we see in figure 5 that when A, increases the primary flow becomes asymptotic to the stable solution of the perfect problem with X #A 0 and Y = 0. Thus if the Reynolds number is increased slowly with L, = 0 then a four-cell flow with radial inflow at the ends develops smoothly and is always stable. In addition to this stable primary flow there are three further stable secondary flows, which, when A, -? oo, are asymptotic to the remaining three stable solutions of the perfect problem. One of these secondary flows will have a definite four-cell structure when A, -? oo, while the other two have a definite six- cell structure in that limit. Moreover, we note that two of the secondary modes



372 P. Hall

(labelled VI, VIII in figure 5) lead to a radial outflow at the ends and so using the classification of Benjamin we refer to these as 'abnormal' secondary flows. The abnormal secondary modes play no role in the mechanism by which the primary flow becomes a six-cell flow when L1 is increased. We note that all three secondary modes are such that, having been set up by some means, they ultimately lose stability and disappear when A1 is decreased. We denote by Al, the value of A1 above which the normal secondary flow with six cells is possible.

2 VIII

IX

-2 _

XIIII

B VII

V~~~~i

-2 VI FIGURE 5. The finite-amplitude solutions of (3.9) for L1 = 0, m = 2. , Stable solutions;

- - , unstable solutions. (Note that X, Y denote the amplitudes of the four- and six- cell flows respectively.)

Suppose now that L1 is varied so that, dIepending on whether L1 > 0 or L1 < 0, the six- or four-cell mode is the most dangerous on the basis of linear theory. In fact if L1 is decreased from zero the bifurcation picture is similar to figure 5. However A1, the value of A1 at which the normal six-cell secondary flow appears, increases when L1 decreases. If L1 is increased from zero then the minimum distance between branches I and II of figure 5 decreases until the branches touch when L1 = 0.123 74. Thus for 11 < 0.12374 the smoothly developing primary flow is always stable and ultimately has four cells, while a normal six-cell solution is possible for A1 > Al(L1).




If L1 is increased slightly beyond L1 = 0.12374 then the equilibrium solutions corresponding to I, II and III in figure 5 are as shown in figure 6. In this case the primary flow loses stability at A1 = Au and the flow -adjusts to that corresponding to the normal six-cell secondary mode possible for A1 > A1. In addition a normal four-cell secondary mode is possible for A1 > A'. If L1 is further increased the S-shaped part of the primary flow loop disappears and As and Au coalesce when LI = 0.12395. For L1 > 0.12395 the primary flow is always stable and develops smoothly into a six-cell mode, while a normal four-cell secondary mode is possible for A1 > Av.

A1

4.36 4.3 A4.7 4.9

I I -0.7 _ I

xI -0.8 - I

-0.9 _

0.9 _

0.8 -

0.7 _ '

0.6l I l I 4.3 A A1 A 4.7 4.9

A1

FIGURE 6. The finite-amplitude solutions of (3.9) for L1 0.12375, m = 2. , Stable solutions; - - - -, unstable solutions. (Note that X, Y denote the amplitudes of the four---and six-cell flows respectively.)

In figure 7 we have plotted A1, Au and Av as functions of L1 in a part of the region where these quantities are defined. We see that this figure is an 'upside-down' version of figure 1, which we recall was found experimentally by Benjamin for the two-cell-four-cell interaction problem. This result has now been found experimentally by Mullin et al. (I98I) although details were not available to the author. The depth of the cusp found by Mullin is significantly larger than that found here. Nevertheless the extrapolation of our results to the case a = 1 appears to give qualitative agreement with the experimental observations.

Suppose now that we apply our analysis to the case a = 1, which is perhaps beyond the range where our perturbation expansion is valid. Then we find that the region of figure 7 where hysteresis occurs corresponds to a non-dimensional length



374 P. Hall

interval AL 0.0002 and Reynolds number interval AA 0.3. Thus it would be extremely difficult to determine experimentally the structure of figure 7. It is also pertinent to question whether such subtleties could be found by numerical methods with grid sizes large compared with AL.

An investigation of the solutions of the amplitude equations corresponding to the other coefficients given in tables 1 and 2 leads to curves similar to either figure 1 or figure 7. We do not give numerical results for these cases but we note that the lengths of the intervals AL, AA over which hysteresis occurs are again found to be extremely small.

0.1240 (7) A

C

0.1239 (8)

0.1238 -

30 Al 60

Alx B

0.1237 -d D

4.4 4.6 4.8 A 11

FIGURE 7. The dependence of Al, AU and A' on L1 for = 2. FIGURE 8. The regions in the A1, L plane where secondary modes are possible: curves a, b

normal and abnormal four-cell flows respectively; curves c, d, normal and abnormal six-cell flows respectively.

We shall now make further predictions about the normal and abnormal secondary modes possible for sufficiently large Reynolds numbers. In figure 5 we see that a normal six-cell secondary mode exists for A1 > Al, while abnormal six- and four- cell secondary modes exist for A1 > AB and A1 > AA respectively. Similarly for L1 > 0.12395 a normal four-cell secondary flow exists for A1 > Av. If we now vary Ll we can plot the curves A1 = A1(L1), A1 = Av(L1), A_ = AA(L1) and A1 = A1B(L) in the A1, L1 plane to produce figure 8. If L1 is held fixed and A1 increased from A1 =-oo then the secondary modes indicated exist to the right of the curves shown. We note that the region where the curves appropriate to the normal four- and six- cell secondary modes meet is shown in figure 7.




5. ASYMMETRIC MODES

We assumed in the previous sections that Taylor vortex flows having an odd number of cells do not occur. Such modes are not driven by the end walls, so it is reasonable to assume that the symmetric modes discussed previously are more preferred. We recall that the symmetric modes satisfy the end-wall conditions if kL = mit when m is a positive integer, and the flow then has 2m cells in [- L, L]. The asymmetric modes have axial velocity component proportional to cos kz and satisfy the end-wall conditions if kL = 1(2m - 1) i, and the flow then has 2m -1 cells in [-L,L].

We now consider the interaction of a 2m-cell symmetric mode and a 2m - 1-cell asymmetric mode in a neighbourhood within ac of the point in the A, L plane where they are equally unstable on the basis of linear theory.

If kI and k2 (k1 < k2) again denote the two real positive wavenumbers of (2.12) then the eigencurves of the symmetric and asymmetric modes are determined by

klcL=-(2m-1)it, kl2L= mi,

and we again exclude the degenerate case m = 1. The eigencurves meet at (A*, L*), where kj/k2 = (2m- 1)/2m and L* = (2m- 1)it/2k1. Thus, A*, the value of the Reynolds number at which the eigencurves of the 2m - 1- and 2m-cell modes inter- sect is the same as that appropriate to the point of intersection of the eigencurves of the symmetric 4m -2- and 4m-cell modes. In the neighbourhood of (A*, L*) we expand A, L as in (3.1) and we retain the expansion (3.4). The order-a, terms in the expansion have U consisting of an asymmetric mode of amplitude X and a symmetric mode of amplitude Y. Following the procedure of ? 3 we find at order ac that X, Y satisfy

dX/dT = (lAL+ 1L L)X+X(fl1X2+ y Y2), (5.1a)

dY/dr-= (o2Ak+62L4) Y+ Y(/32Y2+Y2X2)-w2. (5.1b) Here the coefficients ,, CJ2, 41, 2, fh I2h yi and v2 are exactly as defined in ? 3 but with L* now equal to (2m- 1) i/2k1.

For definiteness we now take r1 = 1-597 and let m = 2 so that X and Y are the amplitudes of three- and four-cell modes respectively. We note that one set of equilibriuin solutions of (5.1) can be obtained by setting X = 0, in which case Y is determined ly

0 = (of2Ll+52Al) Y+fl2 Y3-w2.

These solutions are symmetric four-cell flows. For A1 -o - co we can show that there is only one such mode and this disturbance leads to a radial inflow near the ends of the cylinders. This solution develops smoothly as a single-valued function of A1 but its stability properties depend on L1. In addition to this solution a pair of solutions is possible for A1 > A+ where A+ is the value of A1 at which dAl/d Y = 0. The stability properties of these solutions also depend on L1 and we postpone a discussion of this point until we have investigated further equilibrium solutions of (5.1).



376 P. Hall

If X is not zero then we can divide (5.1 a) by X and then substitute for X2 into (5.1 b) to give

o = [72L1 +42L1-A 2(aAj + 4Li)] L,Y + [2 ,B Y

Y3- )2

This cubic equation always has at least one real solution, and three real solutions for A1 > A++ where A++ is the value of A1 at which the solution curves of the equation have dAl/dY = 0. However, these solutions must be such that after substituting back into (5.1 a) for Y the resulting equation for X2 gives real solutions for X. This condition reduces further the range over which these mixed-mode solutions can exist. Furthermore we note that, where solutions do exist, if (X, Y) is a solution

2 - 2 -

-Y 20 Ae II 4 Iks G -;; - S y 40f 2 X 0 A A,

-2L -2 II

FIGURE 9. The solutions of (5.3) for L1 = 1, m = 2. Stable solutions; - - -, unstable solutions.

0.8 -

0.4 0.8 - C :'

0.6 -

I ~ ~ ~ ~ ~~~1 3 5 0.4 1/ A1 02 I

0.2 A- 1 3 AX, 5 -0.4

A1

-0.8 _ FIGURE 10. The solutions of (5.3) for L1 --1.025, m = 2.

, Stable solutions; - - -, unstable solutions.




then so is (- X, Y). Thus the bifurcation picture in the A1, X plane is symmetric about the X axis.

In figures 9-11 we have shown the equilibrium solutions of (5.1) for m = 2, ri = 1.597 at three different values of L1. In figure 9, for L = 1, the four-cell mode is the most dangerous linear disturbance and when A1 increases a four-cell flow, I, is forced that remains stable for all A1. This motion is normal in the sense that it has a radial inflow near the ends of the cylinders. An abnormal four-cell secondary mode is possible for. sufficiently large values of A1. Two further stable secondary modes, III, which become dominantly asymmetric when A1 oo, are also possible solutions

1.4 - .1.44-

A1=A0 1 5 A, -01.0 V

-y

0.6 - ~~~~0.6 - \ 0.2 ~~~~~~0.2A7

x ~~~~~~~~~~~Al A=A~~~ 1 5A -0.2 1 A19

-0.6-

-1.0

-1.4 FIGURE 11. The solutions of (5.3) for L1 =-1.1, m = 2.

Stable solutions; - - -, unstable solutions.

A'1 -3 0 3 6

-1.00 -

-1.05

-1.10

FIGURE 12. The dependence of A', Au and A' on L1 for m = 2.



378 P. Hall

for A1 > Au,. These modes ultimately have flow in opposite directions at the end walls. If L, is decreased then the three-cell mode becomes the most dangerous linear mode when L1 < 0. However, the bifurcation picture described remains essentially unchanged until L1 = - 0.995 when the branches I and II touch. The situation for L1 slightly less than this critical value is shown in figure 10 and we now see that the primary flow becomes unstable when A1 = Au and the flow must then change discontinuously when A1 is increased further. In fact the flow will adjust to either of the stable solutions, which are possible secondary flows for A1 > Al. However, if A is decreased, then the flow will not jump back to the primary state until A, = Al. We further note that if A1 is slowly increased then for large A1 an asymmetric flow field with three cells will be set up..In addition a stable four-cell secondary flow is possible for A1 > Av and if L1 is increased further then eventually the situation is as shown in figure 11. In this case Al = Au and the flow no longer changes discontinuously when A1 is increased through A1 = Au. However, a stable four-cell secondary flow is still possible for A1 > Al. In figure 12 we have shown the curve obtained by plotting Al, Au and Av as functions of L1. The curve is similar to the corresponding one obtained for the symmetric-mode interactions except for the following principal differences.

(i) The curve can be drawn for A1 less than the critical value at which Al = A1, and the resulting locus gives the position where the symmetric flow bifurcates to one of two possible mixed-mode solutions, each of which becomes dominantly asymmetric when A1 -o co.

(ii) The value of L1 at which a symmetric mode first becomes unstable to asymmetric disturbance is relatively large. This suggests that in practice asymmetric flows might never be set up by slowly increasing the Reynolds number.

The foregoing discussion also applies to all the solutions of (5.1), which can be determined from the results of tables 1 and 2. Similarly, we can consider the interaction of modes having 2m and 2m + 1 cells. Essentially the outcome is as already described but now the even mode is the most dangerous linear mode for L < L* = mt/k1. A direct consequence of this result is that, for these modes, the curves corresponding to figure 12 are upside-down versions of figure 12.

The reader is referred to the paper by Benjamin & Mullin (I98I) for a detailed discussion of the experimental properties of the abnormal modes.

6. CONCLUSION

Our numerical calculations have shown that for the model problem proposed by S it is indeed possible to predict the form of figure 1 found by B. However, we recognize that the model is valid only for a < 1 whereas the experimental situation has a = 1. Moreover, the results that we have obtained are for interactions involving more than two cells whereas figure 1 refers to the two-cell-four-cell interaction problem. Our numerical calculations have predicted that an 'upside-down' version of figure 1 applies to the four-cell-six-cell interaction problem. This result has now been found




experimentally. Thus it would appear that the model problem has some relevance to the experimental observations.

We have also shown that a modified version of figure 1 applies to interactions between symmetric and asymmetric disturbances. However, we found that, since only the symmetric modes are forced, there is a strong preference for flows having an even number of cells. More precisely we see in figure 12 that a smoothly developing four-cell flow will not become unstable to an asymmetric three-cell flow until L= - 0. 995 even though the latter is the most unstable on the basis of linear theory for L1 < 0. Moreover, it is possible that at this stage the four-cell flow might have been replaced by a two-cell flow either as a result of a bifurcation or because the primary flow for L so much less than L*, is in fact a two-cell flow. Such considerations apply to the other interaction problems between symmetric and asymmetric modes and suggest that, as reported by B, experimentally asymmetric modes might never bifurcate from the primary flow. This strong preference for symmetric flows could of course be destroyed by applying non-symmetric boundary conditions, which would cause all possible modes to develop smoothly. Experimentally this can be done by removing the upper boundary at z = L and thus having a free surface at one end and a rigid stationary plate at the other end. Some experimental results for such a configuration have been reported by Cole (1976).

This work was begun while the author was a visitor at Rensselaer Polytechnic Institute, Troy, New York 12181, and was partially supported by the U.S. Army Research Office.

REFERENCES

Ahlers, G. 1975 Fluctuations, instabilities and phase transitions (ed. T. Riste). New York: Plenum.

Benjamin, T. B. I978a Proc. R. Soc. Lond. A 359, 1-26. Benjamin, T. B. I978b Proc. R. Soc. Lond. A 359, 27-43. Benjamin, T. B. & Mullin, T. I98I Proc. R. Soc. Lond. A 377, 221. Cole, J. A. 1976 J. Fluid Mech. 75, 1-15. Hall, P. I980 J. Fluid Mech. 99, 575. Hall, P. & Walton, I. C. I979 J. Fluid Mech. 90, 377. Mullin, T. et al. 1981 (Submitted for publication.) Schaeffer, D. I980 Proc. Camb. phil. Soc. 87, 307, 338. Stuart, J. T. & DiPrima, R. C. I980 Proc. R. Soc. Lond. A 372, 357.



centrifugal instabilities of circumferential flows in finite cylinders: the wide gap problem

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