On the instability of Taylor vortices

.1. Introduction

Taylor (1923) in his classic paper showed both theoretically and experimentally that the laminar circumferential flow (Couette flow) between concentrir rotating cylinders becomes unstable if the speed of the inner cylinder is increased beyond a certain critical value. He observed that the instability yielded a steady

second-2 Fluid Mech. Si

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17

Prinied in Great Britain

On the instabifity of Taylor vortices

By A. DAVEY,

Aerodynamics Division, National Physical Laboratory, Teddington, Middlesex,

R. C. DI PRIMA

Department of Mathematics,

Rensselaer Polytechnic Institute Troy New York 12181

AND L T STUART

Department of Mathematics, Imperial College, London, S.W. 7

(Received 14 August 1967)

It is known experimentally that laminar circular Couette flow between two

con-centric circular cylinders, the outer of which is fixed, becomes unstable when the speed of the inner cylinder is high enough. The flow is then replaced bya new

cir-cumferential flow with superimposed toroidal (or Taylor) vortices spaced periodic ally along the axis. At a higher speed stifi the new flow develops another insta-bihty, and is replaced by a flow in which the axially periodic vortices are simul-taneously periodic travelling waves in the azimuth.

In the present paper an attack is made on the problem of instability of the

Taylor-vortex flow against perturbations which are periodic both in the axial and

azimuthal co-ordinates and, moreover, travel with sOme phase velocity in the latter Subject to a number of assumptions and approximations, which are de-tailed in the paper, it is found that the Taylor-vortex flow is stable against

per-turbations with the same axial wavelength and phase, but unstable againstper

turbations diering in phase by ir After instability the new flow no longer has

planes separating neighbouring vortices, but has wavy surfaces travelling in the

azimuth. This feature is in accord with much (though not all) of the

experi-mental evidence.

The critical Taylor number (proportional to the square of the speed) at which the Taylor vortices become unstable is found theoretically to be about 8 % above

the value for which Taylor vortices first appear This must be compared with a value in the range 5-20 % for the experiments which our work models most

closely. The azimuthal wave-number given a slight preference by theory is 1, in

18 4. Davey, R. C. Di Prima and J. T. Stuart

ary motion in the form of toroidal vortices (Taylor vortices) spaced regularly

along the axis of the cylinders. Theoretically, the critical speed can be predicted by considering the linearized problem for the stabifity of Couette flow with

re-spectto axisyimetric disturbances. This leads to an eigenvalue problem for the

Taylor number T (based on the speed of the inner cylinder) which is a function of

the parameters a

_{= 2/1}

and = R1/R2 describing the basic velocity and geomtry up to scale factors, and the dimensionless axial wave-number A of the disturbance. Here and R1, R2 are the angular velocities and radii of the

inner and outer cylinders respectively.

Since Taylor's original work, there has developed a considerable body of

litera-ture dealing with the mathematical eigenvalue problem and with experimental

measurements of the transition boundary. Much of this work, including

generaliz-ations of the Taylor stability problem, is discussed by Chandrasekhar (1961); also see a brief survey paper by Di Prima (1963). It is probably safe to say that both experimentally and theoretically the transition boundary from Couette

flow to Taylor-vortex flow and the dependence of the critical Taylpr number on

the parameters u and is well understood. A note of caution is necessary, how-ever, as recently Krueger, Gross & Di Prima (1966) have shown

that for a

- 1

approximately, and, for a wide range of values of , non-axisymmetric disturb

ances become unstable at Taylor numbers slightly lowerthan the critical Taylor

number for axisymmetric disturbances. Apparently the instability leads to a weak spiral vortex motion as observed by Coles (1965, p. 399) and Snyder &

Karlsson (1965).

In the present paper we will be concerned with the development of finite

amplitude motions for T> 7,, where T,, is the critical value ofT at which Couette

flow becomes unstable. Further, we shall restrict our attention to the case in

which the cylinders (supposed infinitely long) rotate in the same direction or in which the outer cylinder is at rest (1a 0), and the gap between the cylinders is

small compared to a typiral radius ( 1). According to linear theory, when the

basic flow is unstable the disturbance grows exponentially with time for T> l. However, as Taylor observed, it is known that a definite equilibrium vortex

motion is attained. Moreover, the circulation in the vortices is a function of

T - 7. According to Taylor, 'A moderate increase in the speed of the apparatus merely increased the vigour of the circulation in the vortiëes without altering

appreciably their spacing or position, but a large increase caused the symmetrical

motion to break down into some kind of turbulent motion. .

In addition to Taylor's work, Coles (1960, 1965), Schwarz, Springett & Don-nelly (1964), Nissan, Nardacci & Ho (1963), and SchUltz-Grunow & Hein (1956) have made experimental observations of Taylor vortices for T increasing beyond for the casett s 0. Their observations appear to confirm Taylor's observations

for moderate values of T > On the other hand, for T sufficiently large, the

vortices assume a wavy form in the circumferential direction and have a certain

wave velooity in that direction. With increasing speed different wavy motions

develop, until at considerably higher speeds small irregularities begin to appear and the flow becomes turbulent.

19 one apparatus (with 0-88, p = 0) has been given by Coles (1965). While his

apparatus had a small length-to-gap ratio, being able to accommodate only 30

Taylor vortices compared with G. I. Taylor's 400, it is still pertinent to summar-ize briefly his observations. Let a given flow be denoted by rn/n where m is the number of Taylor vortices and n is the number of azimuthal waves. Then COles

found at a rising sequence of quite definite (and repeatable) speeds of the inner cylinders the sequence of states 28/0 (Taylor vortices), 28/4 (wavy vortices at about 1-51), 24/5, 22/5, 22/6,. - - In all cases for which n r 0, the boundaries

between neighbouring cells were wavy. The angular wave speed was about equal

to the average angular velocity between the cylinders at the first appearance of

the wavy vortices, but decreased with increasing speed to about 0-34 of the inner

cylinder's angular velocity. In addition, he observed that in the range of speeds

for which doubly-periodic flows were possible, different states of motion could be

attained at the same final speedthe state depending upon the manner in which the final speed was reached. In Coles' words, 'the experimental fact is that the steady Couette flow of a given fluid in a given apparatus is not uniquely deter-miiied by the speed of rotation. .

While Coles observed the transition from Taylor-vortex flow to wavy-vortex flow to be a 28/0 to 28/4 transition, Schwarz et al. (1964) using an apparatus which

could accommodate approximately 260 Taylor cells ₍

= 095, p = 0) have

apparently observed a transition to a non-axisymmetric mode with azimuthal wavenumber 1 at a Taylor number 3-8 % above critical. The mode appeared to be a subtle modification of the Taylor-vortex mode and moved with an angular

velocity nearly equal to the average angular velocity of the basic flow. In

addi-tion, the mode appeared to have a regular vortex spacing in the axial direction with planes, perpendicular to the axis and separating neighbouring vortices,_on which the axial component of velocity vanished. No stable modes of this type, exp [i(mO -wt)], for rn > 1 were observed. As T was increased the circulation

in the rn = 1 mode grew more vigorous and the axial form became more distorted with the vortex spacing sinusoidal in time. This evolution took place over a range

ofT but, as explained by Schwarz et al., at a T of about 20% above 'I appeared

to be complete, in the sense that it had a form which could definitely be described

as of wavy-vortex type

In recent years attempts have been made to compute the Taylor-vortex

mo-tion. An energy balance integral method has been used (Stuart 1958) which takes

account of the distortion of the mean motion by the disturbance, and gives a

finite non-zero equilibrium amplitude for the secondary circulation in the vortices.

Following this work an expansion procedure (Stuart 1960; Watson 1960) has been used (Davey 1962) to compute the amplitude and form of the

Tay1or-vortex motion for a range of speeds above critical. This analysis is valid to

second-order terms in the amplitude (though it involves consideration of third-second-order

terms), and takes account of the distortion of the mean motion, of the generation

of the first harmonic of the fundamental, and of the spatial distortion of the

fundamental. Good agreement with experimentally-measured torque data,

particularly for T near ?, is found in both analyses. It is indicated in the later

paper (Davey 1962) why the energy balance method isso effective at equihbnum

20 A. Davey, B. C. Di Prima and J. T. Stuart

In the above work it is assumed that the wavelength of the vortices in the axial

direction is the same as that predicted by linear theory. Even though it is found experiientally that the variation of wavelength with increasing T is not large

(Donnelly & Schwarz 1965), it has been suggested by Meyer (1966) that a suit-ably chosen variation of wavelength may lead to better agreement with

experi-ment. He has carried out an extensive numerical calculation of the

Taylor-vortex flow using a time-dependent finite difference procedure, and has found agreement with Davey's calculations for the fixed wave-number predicted by

lineaE theory. But by suitably varying the wave-number as T increased, he could

obtain agreement with the experimental torque data over a much wider range than obtained by Davey (1962) in the small-gap case. However, the required variation in wavelength was much larger than observed experimentally. Meyer also suggested that a possible mechanism for the transition from Taylor-vortex flow to wavy-vortex flow is a shear instability of the circumferential velocity

profile which has a large variation between neighbouring cells: This suggestion will be discussed in § 6. These regions of high shear in the circumferential velocity at the boundaries of neighbouring cells have also been noted experimentally by

Snyder & Lambert (1966). Finite difference procedures for computing the

Taylor-vortex motion have also been considered by Capriz, Ghelardorii & Lom-bardi (1964, 1966).

In our analysis of the growth of Taylor vortices and their instability, we shall

consider the interaction of two axisymmetric disturbances of differential axial

phases with two non-axisymmetric disturbances of different axial phases.

Follow-a method described elsewhere (StuFollow-art 1961), we derive Follow-a systemof four non-linear equations for the amplitudes of the fundamental disturbances as functions

of time. The coefficients in the amplitude equations are functions of the

para-meters of the problem, namely (i) 1a and , which describe the laminar velocity and geometry; (ii) the axial and circumferential wave-numbers of the disturb-ance; and (iii) the Taylor number. These coefficients can be determined in a systematic manner by solving a set of linear ordinary differential equations. Possible equilibrium states of our mathematical model (the non-linear

ampli-tude equations), the stabifity of the equilibrium states, and the transition

from one equilibrium state to another as the Taylor number increases, will be

discussed.

The present model is sufficiently general to admit equilibrium solutions corre-sponding to the mode observed by Schwarz et al. (1964), and also the more com-plex wavy-vortex modes, and.to give results concerning, their stabifity. Finally,

we mention that this work illustrates the conceptof successive instabilities sug-gested by Landau (1944) in the sense that the model includes (i) a stabffity

ana-lysis of the Taylor-vortex flow with respect to non-axisymmetric disturbances

and (ii) the possible states of motion that might result from an instabifity of the Taylor-vortex flow.

In the. next section the small-gap disturbance equations are derived. In § 3 a Fourier analysis of the disturbance and suitable expansions of the Fourier corn-ponents in powers of the amplitudes of the four fundamental disturbances are

21 equations and their stability. Finally, in § 6 we discuss the relevance of the

theor-etical results to experimental observations; our conclusions are summarized in § 7..

2. The disturbance equations

Let r, e, z denote cylindrical-polar co-ordinates, and let 'ui, u0, u denote the components of velocity in the r, 0 and z directions respectively. Consider two

infinitely-long concentric circular cylinders with the z-axis as their common axis,

with radii 14 and 14 (>14), and rotating with angular velocities _{and Q2}

re-spectively. The equations of motion for a viscous, incompressible fluid admit the

exact steady solution

u=u=O, u8= V(r)=Ar±(B/r),

(2.1)

where A and B are constants chosen so that (2.1) satisfies the boundary condi-tions at r = 14 and r = R2. In order to study the stabifity of this flow we

super-impose a general disturbance on this basic solution and write, for example

u0 = V(r)+v'(r,O,z,t). (2.2)

Substituting in the Navier-Stokes equations of motion and in the equation of continuity (e.g. Whitham 1963), we obtain a system of four non-linear partial differential equations for v' and for the perturbations u', w' and p' in

u, u, and

the pressure respectively.

In the present analysis we shall restrict our attention to the 'small-gap'

case, in which the gap d = 14-14 is so small compared to the mean radius

14 = (R1 + R2) that terms O(d/R0) can be neglected. The derivation of the small-gap equations is essentially the same as for the classical Taylor problem except

that now we must also consider terms involving differentiation with respect to

the circumferential co-ordinate 0. The proper scaling for 0 has been discussed by Krueger (1962), by Bisshopp (1963) and by Krueger et al. (1966) in their analyses

of the linear stability problem for non-axisymmetric disturbances. Briefly, the

reasoning is as follows. Consider the second momentum equation

u5 u8 au0 I2u9

\

(2.3)

where v is the kinematic viscosity. Letting denote a reference angular velocity, and scaling t in units d2/v and u0 in units of 14 , we see that the second term in (2.3) is of apparent scale (L d2/v) as compared with the other terms in the

equa-tion. Recalling that, for the classical Taylor problem in the limit dIR0

_{- 0, the}

dimensionless combination (R0d/v) (d/R0) is kept flxed,t we see that,, if the second term in equation (2.3) is to be retained, then jae must introducea factor (R0/d)l. In that case (0d2/v)a/a0 has a scale (0R0d/v) (d/R0), and the termmust be retained.

For the linear stabifity problem it is often convenient to obtain a single sixth-order equation for the circumferential perturbation velocity v', by using the

22 A. Davey, B. C. Di Prima and J. T. Stuart

tinuity equation, and the 0 and z momentum equations to eliminate w', p' and u' respectively. For the non-linear equations this is no longer possible. It is still

convenient, however, to follow a similar procedure to obtain an equation whose linear part has the above form. At the same time we obtain subsidiary equations

which relate u' and w' to v' (the pressure need not concern us here). Letting

dIR0 with R0d3/v2 and v0/0d2 fixed, we have

32v 1 32p1 32 1

LM+a

1P2

LMLvTc(x)

= --

_-

_3x3çb

Lvu=P2,

(24)

a

I 3u 3v 3w )

Here we have chosen =

2+

) =

l(1 +) where a

= 2Il1, and have

introduce4 the dimensionless variables and operators

d2 d2 4ALlOd4)

r=R0+dx, z=Cd, 0=

°

ç, t=T,

T-v v V(r) =R0Q0CA(x),

(x)= (1ocx),

cz= 2,

(2.5)

M=-2+k2, L=Mcl(x).

₎

The parameter T is often called the Taylor number. Moreover

3u 3u

P1=

u--av+waTv2,

P2=4av+4,

(2.6)

3w 3w 3w

P3= u--av+W.

If the non-linear terms F1, P2 and P3 are neglected, then equations (2.4) reduce,

with slightly different notation, to the linearized equations for stabifity with respect to non-axisymmetric disturbances considered by Krueger et al. (1966).

The choice of = C ( 1 + a) is a convenient scale if the cylinders rotate in the same direction; however, if the cylinders rotate in the opposite direction, a more appropriate scale for fl(r) is In this case the only changes that are necessary

in equations (2.4), (2.5) and (2.6) are the replacement of T, i(x) and a by

- 4AQ1d4fv2,

(1 +p)ax and 1ia respectively. The choice of scale for

'a' and

w' relative to v' is the natural choice from the equations of motion as indicated by Davey (1962). It should be emphasized that equations (2.5) are small-gap equations, obtained from the full equations by letting diR0 with the

inde-pendent variables x, q5, z, r, the deinde-pendent variables 'a, v, w, and the parameters 4a and T held fixed.

V V

23

Finally, in deriving equations (2.4) it has been assumed that the components of the disturbance depend upon z. For harmonic components which are

inde-pendent of z, those equations are inconvenient and do not determine w. However, the equation for w separates from those for u and v and the system of equations (2.4) is conveniently replaced by the equivalent system

ía

a a2

\

a2P

_L-_aa) U

₌

Lw = (1/cc)P3, (2.7)

au

a,

= 0,

with the understanding that aa is to be set equal to zero wherever it appears in

the operator L and the expressions for P2 and P3.

The requirement of no slip at the boundaries leads to the conditions

u=v=w=0

at x = ±

. Furthermore, equations (2.8) imply for the system (2.4) that

au

?v

a(Lv)

a2

ax

at x = ± ..; while for the system (2.7) we obtain only the first of equations (2.9).

3. An expansion procedure

For a given value of p < 1 the velocity distribution (2.1) is unstablet accord-ing to linear theory for Taylor numbers greater than a critical value which depends upon p. As mentioned earlier, forp 0 and T slightly greater than 7, the instability leads to a new motion composed of toroidal vortices spaced regularly

in the axial directions and superimposed on a circumferential motion. With increasing T this laminar motion becomes unstable, the second instabifity ap-parently leading to a 'wavy' vortex motion.

Consider, first, the linearized problem for the stability of Couette flow. The

critical Taylor number occurs for an axisymmetric disturbance, so we look for a solution of the linearized equations corresponding to equations (2.4) of the form

v(x,r,.C) =f(x)cosA.eaT. This leads to an eigenvalue problem for a(T,u, A). The critical value of T and the corresponding critióal value of A, A, are deter-mined by the requirement that be the minimum value ofT over all positive A

for which there exists solutions with a = 0 but not a > 0. For p = 0 it is known (Davey 1962, p. 363), that

169495, A 313. (3.1)

For T>

linear theory predicts that the Taylor-vortex disturbance will

grow exponentially. In fact, however, as a disturbance of axial wave number A

t For a> 1 it has been rigorously shown that Couette flow is stable to axisyrnmetric

disturbances (see (Jhandrasekhar 1961, § 7O) A corresponding proof for non-axisyinmetric

disturbances has not yet been given.

(2.8)

24 A. Davey, R. C. Di Prima and J. T.Stuart

grows, non-linear effects become important, altering the exponential growth so that an equilibrium state is attained. As described in the introduction, this non-linear problem has been studied by Stuart (1958) and Davey (1962) fOr T slightly

greater than .

Here, we shall extend this earlier work on the growth of the toroidal vortices

by studying their instabifities and the form of motions consequent upon the

instabifities. Thus we consider the interaction of a Taylor-vortex disturbance which is periodic in the axial direction only, with a non-axisymmetric

disturb-ance, which is periodic in both the axial and the dircumferential directions. Both

disturbances are assumed to have the same axial wavelength 2ir/A (the variation of A with T is slight in experiment for some range ofT> 7, as shown by Donnelly & Schwarz 1965); however, to allow for phase shifts in the axial direction as the

motions develop, we consider the interaction of modes proportional to cos AC

and to sin AC t Through the non linear terms they mteract and give rise to a mean velocity and to higher harmonics.

The general Fourier series representation for a functiOn which is periodic of

period 2niA in and 21T/k in ç!) is

v(x, C ç5, T)

=q=

{voq(x r)₊ Vcnq(X, r) cos nAC + vsnq(X, 7) sinnAC}

where k is related to m of § 1 by ic = m)0d2Jv. Here the c or s subscript denotes whether the function is the coefficient of a coine or sine; the first (n) and second (q) number suffixes refer to the harmonics of the axial and azimuthal wavelengths

respeotively; an4 the coefficients corresponding to q negative are the complex

conjugates of the corresponding coefficients for q positive

₄

The terms

v10(x, r) cos AC, v810(x, r) sinAC, (3.3)

which represent Taylor vortex-modes, and

v11(x, r) cos AC. v811(x, T) sin AC. e9, (3.4)

which represent non-axisymmetric modes, are the four fundamentals whose

interaction we wish to consider. The additional terms in (3.2) represent harmonics

and the mean-motion corrections, and arise from the interaction.

When we substitute from equation (3.2) for v, together with similar expressions for it and w, in the systems of equations (2.4) and (2.7) and separate out the

vari-ous harmonics we obtain a system of infinitely-many partial differential equa-tions (coupled non-linearly) for the funcequa-tions v10(x, r), u10(x, r), w10(x, r). By using an expansion procedure devised by Watson (1960) and Stuart (1961), this system of coupled partial differential equations can be reduced to a system of linear ordinary differential equations which can be solved in succession.

For the Taylor.vortex motion u and v are symmetric functions of while w, by the continuity equation, is anti-symmetric; consequently for that motion it was sufficient to

use cosine and sine Fourier-series respectively (see Davey 1962).

Thus v0(x, r), v,,0(x, r) and v00(x, r) are necessarily real, while the other terms in the series may be expected to be complex-valued.

25

Briefly, the idea is as follows. Associated with the fundamental v10(x, r) cos AC is an amplitude A(r). According to linear theory if we substitute

v10(x,r)

we find A(r)

_{exp {a0r]; hence, dAn/dr=}

_{a0A where a0, the amplification}

rate, is positive for T>

_{Note that a}

depends upon 1u, A and T. With increasing A this linear relation will cease to hold, the right-hand side of the equation for dAn/dr being replaced by aoA plus higher order terms. Similarly

with v810, v11, v311 we associated the amplitudes A3, B, B8, respectively. WhileA

and A8 are real-valued functions of r, B and B3 will in general be

complex-valued functions. When the fundamentals interact they give rise to the first

harmonics of the fundamentals and to a mean-motion correction. These effects are represented (to first order) by quadratic terms inA, 48, B, B8. In turn, these terms react with the fundamentals and lead to distortions of the spatial form of the fundamentals and, moreover, force higher harmonics; such effects are

presented by cubic terms in the A, A8, B, B8. The process cascades to higher

amplitudes but, in a sense which will be discussed later, we may consider a termination of the series at this stage.

Thus we expand the velocities in suitable powers and products of the

ampli-tudes A('r), A8(r), B(r), B3(r), the coefficients being functions of x.

Correspond-ingly, the amplitudes satisr a system of four non-linear first-order ordinary

differential equations. It is not difficult, though it is rather lengthy, to show that the correct expansions are as follows.

The four fundamentals.

v10(x,r) =

_Af0

+

Af1 ± AAf2 + AJBI2f3

_+AIB8J2f4

w811(x,T) ₌

_{Bh30+ ....}

₎

Here the f's and h's are functions of x alone, and a tilde denotesa complex con-jugate. The expansions for v310, u810, w10 and v811, u11, w011 respectively, are the

same as those just given with A and A3 and B and B8 interchanged, and with the

f's replaced by g's and the h's replaced by i's. Notice that the leading _{terms in} the expansions are first order in the amplitudes, and that the corrections _are third order.

The mean motion.

v00(x,r) =

AF1+AF--

_{IBI2F3+IB8J2F±...,}

w00(x,r) = 00

(

The F's and G's are functions of x alone. The fact that u00(x, r) 0 follows from

the continuity equation and the boundary conditions.

u10(x, T)=

Af20 +...,

+A8B.j5A8$B8f6±

(3.5)

w310(x, T) _-4cf30+.

v011(x, r) ₌

_{Bh0 +BIBI2h1 ±BJB8I2h2+BAh3}

26 A. Davey, R. C. Di Prima and J. T. Stuart

The first harmonics.

v20(x,r) = Am1+Am2+ IBl2m3+ 1B312m4,

v820(x,'r) = AA3n1+B3n2+B0B3ns,

v22(x,r) =Bp1+Bp2, v822(x,r)

BB3q1,

v21(x,r) = ABri+A3B3r2,

v821(x,r) = AB3s1+A3Bs2.

The sries have been truncated at quadratic terms, and the functions rn, n, p' q,

r, 5 depend on x only. The expansions for u20, w820 are the same as that for v,20

with rn1,. ., m4replaced by m21,..., m24 and m31,. .., m34, respectively. The expan-sions for u320, w20,..., u321, w21 are determined similarly. In addition, for the

har-momic components which are independent of ,

v01(x,r)

ABt1+A3B3t2,

u01(x, T) = A0Bt21 ±A8B8t22,

w01(x,r) = AB8z1+A3B0z2, v02(x, r) = B Yi + B Yz' } u02(x,r) = By21+By22, w02(x, r) = BB3z3, (3.8) (3.9)

to second order in amplitude. The functions t, y, z depend on x only.

The amplitude equations. For the above expansions to be consistent with

equa-tions (2.4) and (2.7), the amplitude funcequa-tions A(r), A3(r), B(r) and B3(r) must

satisfy a system of ordinary non-linear differential equations of the form

The equation for A3 is similar to the equation for A with the as's replaced by a8's

and A and A3 and B and B3 interchanged; similarly the equation for B8 is the same as that for B with the b0's replaced by b3's and A and A3 and B and B8

interchanged. The parameters a,. . . , a, a,..., a36, b0,..., b, b,..., b

are func-tions ofa, A, k, T. They, as well as the funcfunc-tionsf0(x),. .., z3(x), can be determined in a systematic manner, which we now discuss.

Substituting the series expansions (3.2) and (3.5)-(3.9) for v10, u10, w810,..., w02(x,T) in equatiOns (2.4) and (2.7), using equations (3.10), and equating

co-efficients of A, A, AA8, etc., we obtain equations for the functions f0(x),

f20(x), f30(x), f1(x),.., z3(x). While it is not practicable tO write out all of these equations,it is helpful to record a few of them in order to ifiustrate the method of

solution. First, however, we define the following operators:

N(A,a,k) = D2-A2-a0-ikfl1(k),

M(A, a, k, T) = N(A, a, k) (D2 - A2)N(A, a, k) + A2T(x).

First-order terms. In the equations for the coefficients of cos AC, sin AC, exp

[ikcJ cos AC, and exp [ikç] sin AC, we have respectively

.M(A,a0,0)f0 = 0,. f20 = .N(A,a0,0)f0, /30 = -A1Df20; (3.11)

A3: M(A,a,0)g0 = 0, g20 = N(A,a,0)g0, g3 = 1c'Dg20; (3.12)

B: M(A,b,k)h0 = 0,

h20 = N(A,b,k)h0,

lc'(-Dh20+ikah0);

(3.13)

B3: .M(A, b30, k)10 = 0, 120

N(A,b,k)l,

l3 = A-'(D120-ikal0). (3.14)

dAn/dr = coAc+aciAac2Ac+c34cIBcI2+ac4AcIBsl2

'I

dB/dT = bC4BCA

310

From equations (3.6) and (3.13), with some simplifications, the boundary

con-ditions associated with the equation for h0 are

h0=D2h0_N(A,b,k)Dh0=O at x= ±.

(3.15)

The boundary conditions for l, g and f are the same as (3.15) except for

0 replace b0 by b80, for g0 replace b0 by a and set k = 0, for f0 replace b0 by a and set k = 0.

The homogeneous linear differential equation. for f with the associated

homo-geneous boundary conditions determines an eigenvalue problem for a0 as a function of A, Au and T, namely the linear stability problem for

v(x, r, ) = f0(x) exp [a0 rJ cos A.

The parameter a0 is clearly the amplification rate which is equal to or greater than zero for A = A and T or T > respectively. The eigenvalue problem

for a0 can be solved numerically for given values of p, A and T and the cor

responding eigenfunction f0(x) tabulated, as has been done by Davey (1962) and

Krueger et al. (1966). Similarly, equations (3.12), (3.13) and (3.14) with the

appropñate boundary conditions determine eigenvalues problems for a30(A, Au T), b(A, AU k, T) and b(A, Au k, T). These eigenvalues and the corresponding eigen-functions can be computed numerically. For-a given disturbance, i.e. given values of A and ic, and for fixed Au and.T, we are interested in the most highly amplified

(or least damped) modes, that is, in the solutions of (3.11 )(3. 14) corresponding

to the largest eigenvalues a and a, and the eigenvalues b0 and b80 with the largest real partt respectively. It is clear from equations (3.11)(3.14) that the s

and c eigenvalues and eigenfunctions (appropñately normalized) are related by a80 = a0 = a0, b80 = b0 =

g0 =fo g20 =f20, g30 = f30' (3.16)

= h 120 = h20, 3O =

Second-order terms. In addition to corrections to the- mean flow, second-order terms arise from the following:

cos 2A, sin 2A, exp [i2lcçS] cos 2A, exp [i2kç5] sin 2A,

exp [ikçb] cos 2A, exp [ikçb] sin 2A, exp [ikç.], exp [i2kçbJ

- and their complex conjugates. Typically, the expansion of the mean motion

v00(x, r) gives ₁ - A: (D2-2a0)F1

---D(f0f20),

) (3.17)

IBI2: D2 (b0+0)F3 = D[h200+20h0],

with

F1=F3=0 at x= ±j.

(3.18)

The corresponding equations for F and F4, together with the conditions (3.16), subject to the reasonable assumption that 2a0 and (b0 ± ) are not eigenvalues of the homogeneous forms of equations (3.17), yield

F2=F1, F4F3.

- (-3.19)

28 A. Davey, R. C. Di Prima and J. T. Stuart

Once the first-order problems for a0, b0 and the functions 10, fo h0 and h20 have been solved, equations (3.17) can be integrated.

As one other example, consider the expansion for v22(x, r), the coefficient of exp [i2kçf] cos 2A. The terms proportional to B yield

M(2A, 2b0, 2k)p1 = (2A2/a)D(h0 + h0) - {(1/)N(2A, 2b0, 2k) (D2 - 4A2) D + i2k(D2 + 4A2)} (h20 h0) + (A/ci4 (D2 + 4A2) (h20 h30) - {A2T +2k2 aD

- ikN(2A , 2b, 2k) (D2 - 4A2)}h2

- A{(1/a)N(2A, 2b0, 2k) (D2 - 4A2) + i4kD} (h0h30). (3.20)

The boundary conditions are

=!2z-i

N(2A, 2b0, 2k)Dp1 = 0

at

x = ±

. (3.21)

Again, once h0, h20 and h30 have been determined there is no difficulty in

integrat-ing equation (3.20) for p1, provided that 2b0 is not an eigenvalue of the corre-sponding homogeneous problem. This assumption is certainly valid for A =

and T near 7. Further, the terms proportional to B in the expansion of v

and proportional to B .88 in the expansion of v822 lead to equations similar to

equations (3.20) and (3.21); from which it can be shown that

P2

-p1,

q1 = 2p1. (3.22)

Proceeding in the same manner, we find that the leading terms in the expan-sions for the mean motion and the first harmonics can be computed once the

first-order eigenvalue and eigenfunction problems have been solved. In addition, upon using (3.16), we find the following relations:

v,,20, u20, w820, v820, u820, w20 m3, m3, m33. = 2P21'

=

-Z3 = 0. (3.23) (3.24) (3.25) (3.26) (3.27) m2 = - m1, m22 = - m21, m32 = - m31; m4 = m24 = -m23, m34 = -rn33, n1 = 2m1, n21 = 2m21, n31 = -2m31; n2 = n3 = n22 = n23 = m23, fl32 = n33 = -V2, u22, U)822, V822, U822, W22

P2

Pi'

P22 = -P21' P32 P31;

q1 = 2p'

q21 q31 c2l, uc2i' u8 V8 _{821' '"c21}

r2= -r1,

r22 = -r21, r32 = -r31;

91=82=r1, s21=s22=r21, 831=832=-r31.

= .F1, F= F3.

G3.=

G= 0,

02 is purely imaginary. v01, ; v02, u3, W01, W02

29

Third-order terms. Finally, we consider several typical third-order terms in the expansions of the fundamentals, v10, v810, v,11 and v811. The coefficient of A in the expansion of v10(x, r) yields

M(A, 3a0, O)f1 2a1(D2 - A2) (D2 - A2 - 2a0)f0

+

{- D+_ (D2 +

A2)}(f20m31 f30m21) - A2T(2f0F +f0m1)

- (1/z)N(A, 3a0, 0) (D2 A2) [D(f20 F1 + f2O m1 + m21)

± A(f30F1 + U0m31 - f30m1)], (3.28)

with _f1

D2f = N(A,3a0,0)Df1aiDf0 = 0 at

x = ±..

(3.29)

Except for theparameter a1, the non-homogeneous terms in equations (3.28) and (3.29) are known.t The determination of;1 has been discussed elsewhere (Davey

1962; Reynolds & Potter 1967) and does not involve the non-axisymmetric

components of the flow; the argument is the same as that for determining b4 which

is discussed later in this section. Once a1 is determined, f1 can be computed

numerically.

Consider now the equation for f2(x) corresponding to the term ACA in the

expansion for v010(x, r). Making use of the relations. (3.16) and (3.22)(3.27), we find that the equation and boundary conditions forf2 are identIcal with those for

f with (f1, ;1) replaced by (f2, a2). Thus a2 = a1, f2 = f1. A similar procedure for the remaining third-order terms in the expansion for v10 and for the third

order terms in the expansion for v810 yields the relations: a81 = a82. = a2 = a1 = a1,

_{g = g2 = f2 = f;}

a83 = a3 = a3,

a84=a4=a4,

a85 = a5 = a5,

a=a=5,

Here r used as a subscript denotes the real part. The c and s subscripts are no longer necessary. Later we use i to denote the imaginary part.

Similarly, for the third-order terms in the expansions for v11 and v811, we find

b85 = b5 = b4, l = h5 = h3 - h4;

b = b6 = b1 - b2, 16 = h8 = h1 - h

Again, the 8 and c subscripts may now be dropped.

t The term involving a81 arises from the term a81A in the expansion of dA8/dr. b81 = b1 = b1, b82 b2 = b2, b83 = b3 = b3, b84 = b4 = b4, 11= h1; 2 = h2; 13 = 14 = (3.31)

g3 = f;

g4=f4,

a5 =(a3a4),

q6=f6=f5.

g5 =f5, f5T (3.30)

30 A. Davey, B. C. Di Prima and J. T. Stuart

To ifiustrate how the parameters a1,..., a5, b1,..., b4 are determined, consider the equation for h4 =14 which involves. b4, since this parameter, as well as a1, plays an important role in the subsequent analysis. The equation for h4 is

M(A, b0 + 2a0, k)h4 2b4{N(A, b0 + a0, Ic) (D2 -A2) + ilcaD}h0 + fr, (3.32)

where

= (A2/a)D(f20r21 - h2m21 -f30r31 - 2f30z.1 + Ic30 m31)

- {(

1/a)N(A, b0 ± 2a0, Ic) (D2 - A2)D + ik(D2 + A2)} x U2ori+2h2oFi-h2omi-hom2i+for2d

- (A/2) (D2 + A2) [-f20 r3 +2f20 z1 + Ic20 m31 - Ic30 m21 +f30 r21]

-{A2T + k2aD - ilcN(A, b0 + 2a0, k)(D2 - A2)}

x [2h0F1 +f0r1 - h0m1] ± A{( 1/cc)N(A, b0 + 2a0, Ic)

x (D2_A2)+i2kD}[_ 2h30F1-f0r31+ 2f0z1+h0m31-h30m1 +f30r1].j

The boundary conditions are

h4=0, D2h4=0, N(A,b0+2a0,k)Dh4=b4Dh0 at x= ±. (3.34)

Again, except for the parameter b4 the non-homogeneous terms in equations

(3.32)and (3.34) are known.

To determine b4 we proceed as follows. The homogeneous boundary-value

problem corresponding to equations (3.32) and (3.34) is M(A, b0 + 2a0, k)h4 =C)

with the boundary conditions Ic4=D2Ic4=N(A, b0 + 2a0, k)Dh4 =

0 at x

= ± .

But in the limit T

T,, a0 -0, and we obtain precisely the same equation and

boundary conditions as for the linear eigenvalue problem for b0 and Ic0, equations

(3.13) and (3.15). Since this homogeneous boundary-value problem has a non-trivial solution (by the choice of b0), we can anticipate a smgular behaviour of

the solution of the non-homogeneous boundary-value problem (3.32) and (3.34)

as T

7 unless b4 is properly chosen. Indeed the solution will behave as a'

as a0 0. To remove this singularity, it is necessary and sufftcientt to set a0= 0

m (3 32)-(3 34) and to require that the mtegral from - to

of Ist times the righthand side of (3.32) is equal to b4[Dh0D2hfl*. Here h, the adjoint of Ic0, is

the solution of the adjoint eigenvalue problem:

M(A,b0,k)ht=0,

.

=Dh = (D2 - 2A2 - b0 - ikcl1)D2h- 0

at

x = ±

This procedure yields the condition.

I1'l

b4 ₌ h (x)fr dx / [Ict N(A, b0, Ic) (D2 - A2)h0 + Ic0 N(A, b0, ic) (D2 - A2)hfldx,

(3.36)

where it is understood that the calculation is carried out at a0 0.

The parameters a1, a3, a4, a5, b1, b2, b3 and b4 can be determined in a similar manlier. The arguments of Watson (1960) have been applied by Davey (1962) to

show that for a given T (+ 7) within a supposed range of convergence of the See Ince (1956, § 9.34).

31

expansions, there is no 'better' choice of a1 than that given by the formula

analogous to (3.36) when working to cubic order in amplitude; this is presumably

true at higher order in amplitude. A corresponding statement applies to b4 as

given by (3.36).

4. The mathematical model

Using relations (3.16), (3.30) and (3.31), the amplitude equations (3.10) for A, A5, B, and B5 to third order in amplitude take the form

dAn/dr =

+a5A5Bfl3+â5A8.B5,

dA8/dr =

+5A8B,

dBJdr = boB+b1BIBI2 + b2B!B8I 2 + b3BA±b4BA

+ (b3b4)B5AA8+ (b1b2)B,

dB5Jdr = b0B5+b1B5IB8I2+b2B8JBI2+b3B3A+b4B5A

+ (b3 - b4)BCA SAC+(b1 - b2)fl5B, where the a and b coefficients, which depend on the parameters of the problem,

are determined by the procedure described.in §3.

We now discuss how the possible solutionsof equations (4.1) and their stability characteristics as functions of the parameters A, k, p, and T provide a model for the transition with increasing T from COuette flow, through Taylor-vortex flow, to wavy-vortex flow. In particular we will consider the case 1a = 0. For assigned values of the wave-numbers A and ic, there is a critical value ofT, (A, k) such

that for T> 1(A, ic) this particular disturbance grows according to linear theory.

The minimum of 7(A, k) over all positive A and k determines 7 such that for

T < 'I

all disturbances are damped, again according to linear theory. For dU = 0,

occurs for k = 0 and A = A as given in equations (3.1). For much of the

dis-cussion A will be fixed equal to A, but it will be necessary to consider the varia-tion with ic of 7(A, k) > 0) = 7. We shall normally use the notation J,(k),

with especially (0) = However, the notation /,(m), m being an integer, is also used. For the instability of the Taylor-vortex flow, according to non-linear

theory, we denote the critical Taylor number, as a function of m, by T' (m). The calculations of Krueger et al. (1966) and Roberts (1965) indicate that, for values of k of physical interest, 7(k) increases monotonically with k, at least for

the range of k of the calculations. Though the change in (k) is only a few per cent, it follows thatfor A AC and an assigned value of k + 0, a0> 0 for T> 7,

but b01. < 0 up to (k) > This will be discussed in more detail in the next

section.

The calculation of the parameters a1,..., b4 has not been carried out in full

except in a certain limiting case to be discussed in § 5. However, it is stifi useful to consider the general system (4.1). At the end Of this section we will give some

justification for truncatmg our expansions at third order, but first consider the

signiThance of the possible equilibrium solutions of equations (4,1).

32 A. Davey, R. C. Di Prima and J. T.Stuart

I(0).=Laminar Conette flow. A possible solution of equations (4.1) is

A A3₌B B8

=

0,

i.e. the periodic perturbation is zero. This solution, which gives the basic

Couette flow, is stable to small perturbations m A, A3, B and B3 provided

that a0 and b < 0. This is the case for T <7, and Couette flow is stable within

the model for T <

1(i).Taylor-vortex flow. It is easy to show that a second class of solutions of

equations (4.1) is

B =B3= 0;

A3 =.CA, A+A =Ka0eaaon/(1_Ka1e?aoT),

(4.2)

where K and C are arbitrary real constants. It is known that a1< 0 and a0 0

as T

l. Hence for T>

(A+A)-(a0/a1)

A

as r-'.

(4.3)

This equilibrium solutiOn represents a Taylor-vortex flow, the paramöter C giving, the C-phase. Without loss of generality C can be taken equal to zero. The azimuthal velocity perturbation from laminar Couette flow takes the form

v(x, _,r) v00(x, 'r) + v10(x, r) cos AC+ v(x,r) cos 2AC +

[AF ± O(A)] + [Af0 + Af1 ±O(A)] cos AC

±[4m1+O(A)]cos2AC±...,

(4.4)

and asT-+ we obtain the equilibrium motion

v =A8f0(x) cos AC+ AF1(x) + m1(x) cos 2Afl + O(A). (4.5)

For details of (4.4) and (4.5), the reader may consult the paper by Davey (1962),

where the theoretically predicted torque and the experimentally measured

torque are shown to be in excellent agreement for a small range of T above T.

Consider now small perturbations from the Taylor-vortex flow. We write

A(r) =

Ae+X

Linearization of (4.1) for small values of_x A8, B and B3 gives.

d dA3

=-2a0, -;-=O

dB

-

_'b

b3

_\

B_c' dB3

-

"b

0o)

b4

\

B₈

Since a0> 0 for T> 7 (where the Taylor-vortex flow exists),

_x

decays. The

result for dA3/drreflects the presence of the class of solutions (43) with an

arbi-trariness in the axial phase of the Taylor-vortex motion (see Segel (1965) for

a similar situation in the thermal-convection problem). The real parts of

b0-- b3a0/ a1 and b0 - b4a0 fa1 determine the growth of decay of B and B8

respec-tively, and hence the instability or stabifity of the Taylorvortex motion with

rospect to in-phase and outofphase non-axisymmetric disturbances respectively.

The calculation of the parameters a0, a1, b3 and b4 and the answer to these

important questions will be discussed in § 5.

A natural question which might be raised is the relevance of the stability

equations (4.6) to a 'straightforward' discussion of the stability of steady

Taylor.vortex flow, the azimuthal component of which is given to second order in amplitude by (2.1) plus (4.5). Then and w components of velocity are given by formulae similar to (4.5). Suppose this Taylor-vortex flow is perturbed by a small

non-axisymmetric disturbance of such a form that its azimuthal velocity is

v'(x, ) exp {qr + ikç]. Upon expanding v' and q as series in the small amplitude A6 = (- a0/a1), we can show that to second order in amplitude, q = b0 - b3a0/a1 if v' has the same phase in as the Taylor-vortex mode, but that q = b0 - b4 a0/a1

if v' differs in phase by

n. These formulae confirm the stability coefficients

given by (4.6) through terms O(A). In § 6 we shall discuss the possible influence

of terms O(A) on the preóise location of the zeros of the stability coefficients

(4.6), that is of q above.

Other solutions of equations (4.1) are tabulated below. Stability criteria are

given for those cases where the result can be stated simply; this is the case for the 'simple' modes, which have only one of A, A8, B, B3 nonzero, and for the spiral

mode B8 = ± iB. In these solutions r8 denotes an arbitrary time phase.

1(u). Non-axisymmetric simple mode.

A=A8=B3O, BcfleeT3),

Pe = (b0,./b1,.)1,

_{0) = b0b1b/61.J}

( This solution exists if b./b17> 0, and it is stable if a0 --. a3 b1,,./b,, a0 - a4 be./bj,,

- b0,. and b,(1 - b2/b1) are all less than zero. A generalization of this motion is = A8 = 0, B = fiexp [i0)(rr8)], B3 = fl83exp [i0)(r-73)] with fiL+fl6 = fl.

11(i). Wavy-vortex flow.

A3 = B = 0,

A = A

aobb.a4bor B8 =

a4 4ral lr

/32

aiboraob4r

_{= b01+b11/1+b4A.}

a4 ,.a1 lr

This solution, which represents the interaction of a Taylor-vortex mode with

an out-of-phase non-axisymmetric mode, is of particular interest. It exists

when a1 b0,. - a0 b4,., a4 b4,. - a1 b11. and a0 b1 - a4 b1 all have the same sign. If only

A and B3 modes are allowed, it is stable if a1A + b1j < 0 and a1 ba,. a4b4> 0. Note that if the latter condition holds, then the existence of the wavy-vortex

flow requires a1ba0b41, < 0, which in turn implies that the Taylor-vortex

flow I (i) (with a1 < 0, a0> 0) is unstable according to (4.6) to a B3 perturbation! For A8 and B perturbations the corresponding statement is much more complex.

The form of the azimuthal component of the disturbance velocity (3.2), correct through terms O(A) and O(fl), is

v(x, r, ç5, )

+ {AF1(x) +flF3(x)} + {Am1(x) +flm3(x)) cos 2AC

+ 2/{[ p1(x) cos 2AC+ y1(x)] exp (i2{kçb + 0)(r

-+ 2A6/36{r1(x) sin 2Aexp (i[kØ -+ cu(r - r8)] )). (4.9)

This result should be compared with the Taylor-vortex flow (4.5).

3 Fluid Meóh. 31

(4.8)

-34 A. Davey, B. C. Di Prima and J. T. Stuart

11(11). Non-axisymmetricvortex flow.

This solution exists if the numerators and denominators of A and fi respectively

are of the same sign. We do not consider the question of stabffity here. The

terminology 'non-axisymmetric vortex flow' is used to indicate that while the

motion depends on ç5, the A and B disturbances are in phase m C and the vortex motiOn is bounded by planes of constant .

11(iü). Spiral mode.

A8

=

0, B

= B8=

_{fleexP[(T-7s)±T],}

( 11)

fl = _b0,./2b2,., 0) = b0±2b21fi. ₅

This solution exists if - b/2b2r> 0. It is stable if b,,> 0, be/b,,> 1, and

a0 - (a3 + a4)b/2b < 0. Because of the phase difference of 7T in B and B3 the terms in the velocity distribution combine together to give a wave travelling in both the and ç5 directions Alternatively, it may be regarded as a spiral pattern which rotates with a certain angular velocity.

III

and IV Unless certain relations exist between the a and b coefficients, there are no triple modes. Quadruple modes are either generalizations of the

double modes A , B3 and A ,B or (in principle) completely new modes The latter

possibility has not been investigated.

In concluding this section we return to the questions raised earlier of truncating

the expansions at thir4 order in the amplitudes. Assuming that a1, a3, a4, a5,

b1, b2, b3 and b4 are 0(1), we see from equations (4.3), that possible equilibrium solutions of (4 1) have amplitudes which are 0(at) and/or O(bj) For a given value of the Taylor number b/a0 is fixed, so that the amplitudes are 0(a) where a0 is a small number if T is close to Whereas the linear and cubic terms on the

right-handiside of equations (4.1) are then both 0(a), the terms omitted which are of qumtic or higher order in amplitude are 0(a)

To give some justification for neglecting such terms, we will discuss global

stabifity of the system (4.1) hi order to assess whether all solutions of amplitude

0(a) :retain that order, or whether there are solutions which become unbounded

as r

If any solutions had such behaviour, it would be invalid to truncate

the set of ordinary differential equations (4.1) at cubic terms. Happily, however, it is possible to obtain conditions on the coefficients a, b1, etc , which ensure that

solutions within some bounded domain O(at) cannot escape from that domain;

so that, consequently, solutions of equations (4 1) retain amplitudes 0(4)

This statement of global stabifity can be proved by constructing a Liapunov

function (Minorsky 1962) of the form

L _{acA+z3A+flcIBcI2+fi8IB8l2,} (4.12)

where a, fl, /13 are real and positive. It is found that dL!dr is negative if L is

greater than some value of order a0, provided certain conditions hold on a1, b1,

A8 e

0_' B A

a1b,,.- aobsr

A A

aobir-ab

B0₌fi8 exp (i&(r

-(4.10)

-C -e 7.

a3v38a1u1T

c. = b + b1 fi + baL A. a3b3-a1b1,.

etc. This result ensures that, for amplitudes larger than O(a) the trajectories are heading 'inwards'; consequently, the system (4.1) is globally stable. The

con-ditions on a1, b1, etc., are satisfied (Stuart 1964) for the 'simplified 'mathematical

model proposed in the next section. We note, in passing, that if the cubic terms in (4.1) were ignored (as in linear theory), the system would not be globally

stable when a0> 0 and/or b> 0, since then there would be exponentially

increasing solutions.

Before leaving the topic of the truncation of the amplitude equations at cubic terms, we reiterate a second point mentioned earlier in this section. At values of

the parameters near those at which the 'local' stabffity of an equilibrium solu-tion changes character, the quintic terms may be important in a specificasolu-tion of the precise location of that change in character. This will be discussed in detail

in6.

5 A SimIified mathematical model

In order to complete our discussion of the possible equilibrium solutions of

equations (4.1) and their stability, we need to compute numerical values for the coefficients a0, a1, ..., a5, b0, ..., b4. Of particular interest are the combinations

b,._Za0 and b_4!a0,

(5.1)

a1 a1

which, as indicated by (4.6), determine the stability of the Taylor-vortex flow.

Since a0 = 0 at T = 7(0), each equals b and hence is negative. Assuming,

how-ever, that our theoryis a good model of the physical problei, we may anticipate

that for a fixed value ofA(=A) and within a range of T > T there will be a

mini-mum value of T, say T', greater than with a corresponding critical value of k

for which one of (5,1) will become positive. Above that value of T the Taylor.

vortex motion will be unstable, and the corresponding value of k will determine

the azimuthal wave numbert of the critical non-axisymmetric disturbance. A calculation of the a's and b's as functions of k and T for fixed 4u( =0) and

= A) is aformidable task, and it is natural to consider possible simplifications. One such simplification is suggested by the work of Gross (1964), as reported in part by Krueger et al. (1966) which was mentioned earlier. Gross showed that, for = 0 and a given value of A, both the critical Taylor numbers and the amplifica-tion rates depend only slightly on ic, even for (quite typical) values of k of order

10. For the case of small-gap, his results are displayed in table 1. Note that while

1(m) is monotonic increasing, the amplification rate as indicated by (db/dT) evaluated at T(m) is monotoñic decreasing. The actual critical values of A and

T for the assigned values of K are given in the last two columns of table 1; they show that there are on1y slight changes if A is replaced by A(m). Gross showed

further that while results from the small-gap equations are changed by a few

We recall (from below (3.21)), that although k = mfl0d2/v is a continuous parameter,

it is only necessary to consider those values which correspond to integers, m, for the physical wave.number. From the deiiitions of T and a in (2.5) with dIR0 - 0 in T, it

follows that for u = 0, k = rn(T8)/2 = K(4T) where K= m(-6) and 8 = diR0; thus for

a given geometry, values ofK = m(lô) are assigned

36 A. Davey, .1?. 0. Di Primaand

J. T. Stuart

percent if the exact linearized equations are used, these errors occur uniformly.

For example, using the exact equations with p

= 0,

i =

095 (i.e. 8

0.05),

A=

3127, he found that

= 1755, a =0.00746 for m

=

0 and

= 1763,

a

= 000742 for m 1. A comparison of these results with those in the first two

lines of table 1 shows that the correction for gap size is essentially the same for

m = 1 as form = 0; it may also be helpful to refer to figure 4 on p. 535 of KPueger etal. (1966).

t The values of dbo?/dT at T(m) have been inferred from Gross's data Tnis 1. Data from linearized theory (6'= 005, u = 0)

Assuming jbr the moment that the coefficients of the non-linear terms in

equa-tions (4.1) are also only slightly dependent upon k (orK),we consider the form

taken by the a's and b's as k -0. Even for the case m =4, 8= for which the

change in l(m) from 7(0) is about 6%, it is hoped that results in this limit will at least give qualitative information.

For the first-order terms, from (3.11) to (3.14),

b0=a0;

h0=f0,

h20_f20, h30=f30 (5.2)

in the limit k -0. Using these results in the second-order equations, as typified by (3.17) and (3.20), and letting k0, we find

.F3-2F1,

_02=0;

m3= 2m1, m23=2m21, m33= 2m31;

Pi =m1, P21 =m21, Psi=m31; (5.3)

= 2m1, r21 =2m21, r31 =2m31; Y21

=

0, Yi =

P;

t21 =0, t1= 2F1.

Finally, from the third-order equations, as typified by (3.28), the same limit gives a3=

6a1, f

= 6f1; a4=2a1,

/4=2/1;

a5=2a1, fs=2f1;1

b1= 3a1, h1 = 3f1; b2=2a1,

h2=2f1;

. (5.4)

b3= 3a1, h3 b4=a1,

h4=f1.

_J

(1964, table 9

The implication of these results for the stability of Taylor-vortex flow is

-

a0

= -

2a0, be,, - a 0 as

k -0.

a1 a1 ). (5.5) Ic=

m()

m T(m) for A = 3127 a(m) = (db0/dT)T(m)t fOr A = 3127 A(m) T(m) for A = A(m) 0 0 1695 000772 3127 1695 O1581 1 1 1701 000770 3131 1701 031623 2 1720 O00758 3143 1720 047434 3 1753 O00740 3163 1752 063246 4 1800 000720 3190 1799 079057 5 1866 000690 3225 1863

On the instability of Taylor vortises

Thus for a discussion of the stability of the Taylor-vortex mode (As) with

respect to in-phase (Be) non-axisymmetric disturbances, it is probably valid to

set k= 0. Moreover, since a0> 0 for T> the Taylor-vortex mode is stable to

in-phase non-axisymmetric disturbances. On the other hand (4.6) and (5.5) show

that stability with respect to out-of-phase (B3) non-axisyinmetric disturbances depends upon the correction terms in k, precisely because b0 a0 and b4 a1

as k -+0. To examine the behaviour of [b7 - (b./a1)a0J more carefully as k -- 0 with A=

A and T - 1

small, we note that it can be shown that (with

2k(T6)4)

a0(T) =

a(0)[TI(0)]+O[TT(0)]2,

(5.6)

and T(m) =

/(0)+k211+...,

(5.7)

b(m, T)

= cs(m) [T 7,(m)] + O{T

-=

= a(0) [T - 7(0)] + k2@1[T - (0)] a(0)} + 0(k4) + O[T

-(5.8) b4,.(m)= aj+k2b4ri+O(k4).t (5.9)

Neglecting terms0[T - 7(m)]2, O[T -1(0)]2and 0(k4),

b -

a = k2 b4l? a(0)] [T

-

'I(0)] -

a(0Yli}. (5.10)

Thus, the stability or instabifity of the Taylor-vortex flow to an out-of-phase non-axisymmetric disturbance depends to first-order upon the k2 correction

factor (5.10).

AtT= l(0) it is clear that, since a(0) > 0 and 7 > 0, (5.10) must be negative. If, at some critical value ofT, (5.10) changes sign, the Taylor-vortex flow will be

unstable for higher values of T provided the terms neglectedin (5.10) are truly

unimportant. Moreover, since the azimuthal wave-number m appears only

through k, any critical value ofT derived from (5.10) must be independent of m; terms 0(k4) would have to be added to (5.10) to obtain such a dependence on m.

We shall return to this matter in our discussion of the results in §6.

The preceding argument suggests that a simplified model may be considered

with a3, a4, a5, b1, b2 and b3 of (4.1) replaced by their limiting values as k-0 as given in equations (5.4), but with a0(T), b0(m, T) and b4(m) taking exact values (for given u, A). As it affects the question of instability of Taylor-vortex flow with respect to B3 disturbances, the simplified model has the same accuracy as (4.1), since the B8 equation of (4.6) is exact; but its accuracy is less in the determination

of the equilibrium state arising from that instability. In our study of the

simpli-fied model, we shall find it convenient to write

(X, Y, Z, V) (- a1)1(A, A3, B, B3), (5.11) (e, o, ya1) = (a0, b0, b4), (5.12)

38 A. Davey, B. C. Di Prima and J. T. Stuart

where y, which is a complex number,.approaches unity as k -0. Then equations

(4.1) become

dX/dr =CX_X8_XY2_6X1Z12_2X1V12_2Y(ZV+2V), dY/dr =

eY y3_ Y12_6Y1V12_2Y1Z12_2X(ZV+.ZV),

dZ/dr=

oZ_3Z1Z12_2Z1 V2_ 3ZX2_yZY2_(3_y)VXY_2V2,

(5.13)

dV/dr =

TV_3V V2_2 VZS_3VY2_yVX2(3y)ZXY_ VZ2

Corresponding to the possible equilibrium solutions of equations (4.1), we have

the following possible equilibrium solutions of the simplified system (5.13). In

stating the solutions, the facts that e> o,. and a1

<

0 have been used, and a

suffix e denotes the equilibrium value. Note that generalizatiOns of the solutions

can be effected by alteringthe and r phases, as was pointed out in the previous

section.

1(0). Laminar Couetteflo

XYZV0

(5.14)

exists for all e and is stable fore<

0 (T <7), and unstable fore> 0 (T> 7).

1(i). Taylor-vortex flow.

X=e, YZV0

(5.15)

exists for e> 0 (T> 1) and is stable or unstable as 0r Yr

1(u). Non-axi8ymmetric 8imple mode.

Ze

= (0r)exp(ioiT),

(X Y V 0) (5.16)

exists for o,> 0 (T> 1,(m)), but is unstable.

11(i). Wavy-vortex flow.

V=(Tt6)e17

_(YEZ0),

(5.17)

where =

o - y X. It exists if -y

<

1 and o7> y e. Whenever it exists, it is

stable. Note that if oS,.

> ye,

the Taylor-vortex flow is unstable.

11(u). Non-axisymmetric vortex flow.

g2

2oe

_Ze

(_Or/3)exp(i.jT),

_{(Y V 0) (5.18)}

exists for e < oS,.< 3c, which is expected for some T> 1(m), but is unstable.

11(h). Spiral mode.

= o3exp (io1r), Ve

=

± iZ,

(X Y 0) (5.19)

exists for o,.> 0 (T> l(m)), and is stable or unstable as o e.

There are no triple modes since y 4 0, as we shall shortly see, and the quad-ruplé modes are simply generalizations of the double modes 11(i) and 11(h).

.39

The most important observations (assuming there exists a T>

such that

ye) are:

e <0 Lamihar Couette flow exists and is stable.

e> 0, o < yT e. Laminar Couette flow is unstable, and Taylor vortex flow

exists and is stable.

o-> Yr6 Taylor-vortex flow is unstable, and wavy-vortex flow exists and is stable.

Perhaps of lesser, though notable, importance is the existence and stability

of the spiral mode II(iii) for o-> c.

Turning now to the computation of e, o and y (or, equivalently, a0, b0, a1 and b4) we note, as mentioned earlier, that a0 and a1 have been computed by Davey (1962) and a0 and b0 by Gross (1964). It remains only to compute b. However; since, as part of the calculation of b4, it is necessary to recompute a0 and b0 and since, moreover, b4 = a1 at k = 0, there is no practical advantage in our using the earlier results. Thus for given values of 1u, A, ic and T we first solve (numerically)

TABLE 2. Non-linear coefficients (tI = 005, p = 0, A = 312657, T = 1694.95).

Scaling is f0(0) = 1; h0 scaling is irrelevant for b4, a1.

the linear stability problems for axisymmetric and non-axisymmetric disturb

ances to determine a0, f, f2o' f3o f and b0, h0, h20, h30, ht respectively, wheref

and h0+ are the adjoints of f and h0 respectively; then the second-order terms m1, m21, m31, r1, r21, r31, z1 and F1 are evaluated. Finally, b4 and a1 (by setting k 0)

are determined by the condition (3.36). The computational methods and

pro-cedure are broadly similar to those describe4 in an earlier paper on

Taylor-vortex flows (Davey 1962). Here, however, there is the added complexity

brought about by the complex arithmetic and the vast number of functions and

their derivatives which are involved, as a glance at (3.32) and (3.33) will show.

The critical Taylor number, 7 = T(0), given by the calculations is

l(0) = 169495 at

A = 3-12657

with p = 0,

ic = 0. (5.20)

For p = 0 and the value of.A given in (5.20), a1 and b4(m) have been evaluated at as discussed in § 3, and for integer values of m for & = 0.05 with k = m(T8)t/2. The results are given in table 2.

The amplification rates and frequencies, given by the real (b) and imaginary

(b0) parts of b0, are given in table 3 for several values of m(&

₌

) and a range of values of T, but with A and p given by (5.20).

The stability coefficient bth. - b47 a0/a1 0r - Yr6 as a function of T is shown in table 4. It can be seen that, for each value of m, the stability coefficient changes

sign at some T'(m), the value being about 1821 for m = 1, 1824 form = 2, and

1834 form 4. The meaning of these results, the effects of the approximations,

m b4,. b42 _b4/a1

0

10035

0 1

1

95063

072035 094731

-2 7-9569 15238 079291

40 A. Davey, R. C. Di Prima and J. T. Stuart

and the relevance of the theory to experiment, are discussed in § 6. For the

moment we note that the values of b01.- ba0/a1 form = 2 and m = 4 are, to a rough approximation, 4 times and 16 times the values for m = 1 respectively.

Tsia

3. Amplification ratesof lineartheory (8 = 0-05, 1u = 0, A = 3-12657)

T&mx 4. The stability coefficient b0 - b4rao/ai (8 = 0-05, u = 0, A 3-12657) This is in accordance with equation (5.10). Indeed the data in table 4 can be used to justify the following approximate formulae, valid for & = dIR0

b01. - b4,a0fa1 = k2{1-8264.x 10-5[T - 1(0)] - 0-22848 x 102}

- (0086202 x 10-5)k4, (5.21)

where k2 = 21-1875m2, and

T'(m) = T(0)+125+m2.

(5.22)

The k4 effect, which leads to m2 in (5.22), is seen to be relatively small. While the values of T'(m) for m 1, 2,4 are close together, they are quite separate

from (0).

6. Discussion of theory and comparison with experiment

It has been argued in the previous section, and confirmed by calculation, that

instaFifity o the Taylor-vortex flow is connected with subtle changes in the

stability coefficient (be,.- ba0/a1). Indeed, we have found that these changes do

T 0 .1 2 4 ( a0 \

b0

r b0 -'

b0

be.

1694-95 0-00000 0-04791 4-8441 0-19170 9-6901 0-76789 19-396 1715 0-15444 +0-10611 4-8738 0-03895 9-7495 0-62023 19-515 1735 0-30771 0-25896 4-9032 +0-11263 98O84 0-47371 19-633 1755 0-46021 0-41104 4-9325 0-26345 9-8670 0-32794 19-750-1775 0-61195 0-56235 4-9616 0-41351 9-9252 0-18292 19-867 1795 0-76293 071292 4-9906 0-56282 9-9832 0-03862 19-983 1815 0-91317 0-86275 5-0194 0-71140 10-041 +0-10496 20-090-1835 1-0627 1-0119 5-0480 0-85926 10-098 0-24784 20-214 1855 12115 1-1602 5-0765 1-0064 10-155 0-39001 20-32S 1865 1-2856 1-2342 5-0907 1-0797 10-184 0-46084 20-385 T 1 2 4 1694-95 0-04791

019170

0-76789 1715 0-04020 0-16141 0-65511 1735 0-03254 0-13136 0-54320 1755 0-02492 0-10146 0-43188 1775 0-01735 0-07171 0-32113 1795 0-00981 0-04211 0-21093 1815 0-00231 0-01266 0-10128 1835 + 0-00515 + 0-01664 + 0-00783 1855 0-01258 0-04581 0-11640 1865 0-01629 0-06034 0-17048

produce a reversal of sign of the above coefficient at some value of T of order 8_% above the Taylor number at which the Taylor-vortex flow itself appears. It may

be concluded, therefore, that if the approximations we have made are valid, the mathematical model does yield an instability of the Taylor-vortex flows; later in this paper we shall compare the consequences of this theoretically derived instability with experiment. In the meantime, however, it is desirable to assess the magnitude of known approximations in the mathematical model, especially insofar as they affect the precise value of the Taylor number at which the

co-efficient (b. -, b4a0/a1) changes sign.

The approximations are of four main kinds: (i) the amplification rates a0(=e) and b01.

(=

o1.) are regarded as small enough for the amplitude equations (4.1)

to be truncated at cubic terms; (ii) the small-gap approximation (dIR0 -0)

has been used both in the derivation of the basic partial differential equations (2.4) and, by implication, in the limit k used in much of the analysis of the simplified mathematical model of§5; (iii) the number of basic modes has been

restricted to four, all of which have the same axial wave-number whereas two of

them have an azimuthal wave-number of zero and two have some other given azimuthal wave-number; (iv) the cylinders are infinitely long.

Let us consider now the question of possible instabifities of the Taylor-vortex flow (as opposed to the question of the nature of the motion which developsfrom

any instabilities). Clearly, in the present work, attention is restricted to

insta-bffities associated with perturbations of the form of the four basic modes, whose wave-numbers are subject to the restrictions described under (iii) above. Although

it would be of value to have information concerning the stability or instability

of Taylor-vortex flow against other perturbations, the four modes were originally selected because of their especial relevance to several experiments. In this sense the choice is felt to have been a sensible one. Nothing more can be said about the

more complete problem at this stage, but the generalization from (iii) needs to

be borne in mind

The small-gap approximation (ii) is known to be a good approximation in

linearized theory (Krueger et al. 1966), and further the error is uniform in k. The

accuracy of the approximation for the non-linear problem cannot, however, be

assessed until calculations have been done from the full equations (without

dIR0 -. 0). It is hoped that, as in linearized theory, the results are accurate to

within 5_%or so (and if uniform ink would not change the critical azimuthal wave-number), at least with reference to the instability arising from the B8 mode. The additional approximation k -0 is also a good approximation in linearized theory,

and has some validity for the non-linear problem in the sense that for m = 1,

which corresponds to the smallest value of k used, the value of b/a1 differs by only about 5_%from unity (see table 3). Other aspects of these approximations,

especially a reconsideration of the result that B perturbations decay, will be

discussed later in this section.

We now come to what is, in principle, a major approximation in the

calcula-tion of the critical value, T'(m), for instability of the Taylor-vortex flow; this

approximation, namely (i) above, has already been hinted at in _§4, but we must

42 A. Davey, R. C. Di Prima and J. T. Stuart

condition, we have balanced out the two parts of (be,. - ba0/a1), each of those parts being proportional to a 'small' parameter. (Although, as seen in table 3, a0 and b0,. are of order 1 in the range of interest, this value is small compared

with A2 10; the case for this criterion of 'smallness' has been argued elsewhere by Davey (1962, p. 346).) The neglected terms, due to truncation of the amplitude

equations (4.1) at cubic terms, may well be smaller than each of the terms b0 and b4a0/aj individually, but may be very important in the neighbourhood of the zero of that coefficient. Indeed, it is conceivable that such terms could even prevent the occurrence of a zero in the range of validity of the present type of

theory. It is for this reason that we must give detailed attention to this question. The first and second parts of (b),. -, b, a0/a1) arise respectively from linear and

non-linear (cubic) terms of the relevant equations of (4.1). Let us consider the

form which those equations would take if quintic terms were included. But

rather than deal with the complete set (4.1), we restrict our attention to the pair

of equations for the Taylor-vortex mode (An) subject to linearized B8 perturba-tions. It is this procedure that led to equations (4.6) at cubic order, and therefore to the stabifity coefficient (bth. - b,.a0/a1). If quintic terms are included, (4.1) in a generalized form yields

(6.1)

8 ₌

_{b0B8+b4B8A+b5B8A+...,}

_(6.2)

wherO a2 and b5 are new parameters, unrelated to thea2 andb5 in (4.1), which must be determined.

Equation (6 1) has an equilibrium solution, which is descriptive of Taylor

vortex flow; the first three terms of (6.1) are each of order a, whereas the quintic

term is of order a. It is thi8 feature which implies that a cubic truncation (as in (4.1)) is uniformly valid for a calculation of the equilibrium state (as 0),

though not necessarily, we hasten to add, for a discussion of stability or instabifity of that state. With a0 regarded as a small parameter, the equilibrium solution of (6.1) tO order a is

A =

(aO/a1)aa2/a

(6.3)

Substitution of this expression in (6.2) yields

B8{b - b4a0/a1 + [b5 -- ba2/a1]a/4}, (6.4)

so to O(a), the stability or instability of the Taylor-vortex flow against B8 per-turbations depends on the sign of the expression

bj,.b4rao/ai±

(6.5)

In order to assess the importance of the term O(a) in (6.5), it is necessary to

know the magnitude of b5 and a2. Reynolds & Potter (1967) have calculated the

Taylor-vortex flow to sixth order in amplitude, and from their data we have