NASA Contractor Report 172249

ICASE NASA-CR- 172249

19840002433

\

THE EVOLUTION EQUATIONS FOR TAYLOR VORTICES

IN THE SMALL GAP LIMIT

Philip Hall

Contract No. NASI-17070

October 1983

INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING

NASA Langley Research Center, Hampton, Virginia 23665

Operated by the Universities Space Research Association

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THE EVOLUTION EQUATIONS FOR TAYLOR VORTICES IN THE SMALL GAP LIMIT

Philip Hall

Imperial College, England

Abstract

We consider the centrifugal instability of the viscous fluid flow between

concentric circular cylinders in the small gap limit. The amplitude of the

Taylor vortex is allowed to depend on a slow time variable, a slow axial

variable, and the polar angle 8. It is shown that the amplitude of the

vortex cannot in general be described by a single amplitude equation.

However, if the axial variations are periodic a single amplitude equation can

be derived. In the absence of any slow axial variations it is shown that a

Taylor vortex remains stable to wavy vortex perturbations. Furthermore, in

this situation, stable non-axisymmetric modes can occur but do not bifurcate

from the Taylor vortex state. The stability of these modes is shown to be

governed by a modified form of the Eckhaus criterion.

Research was supported by the National Aeronautics and Space Administration

under NASA Contract No. NASI-17070 while the author was in residence at the

Institute for Computer Applications in Science and Engineering, NASA Langley

Research Center, Hampton, VA 23665.

i

ivJ ' /

In two recent papers Tabeling [I] and Brand and Cross [2] have

independently proposed an amplitude equation which governs the slow azimuthal

and axial evolution of a Taylor vortex in the small gap limit. In this note

we show that this amplitude equation corresponds to a velocity field which

necessarily violates the no-slip condition at one of the cylinders. The

remedy for this difficulty is well-known in hydrodynamic stability theory

following the work of Davey, et al. [3] and requires the insertion of an

eigenfunction in the expansion of the disturbance pressure field. The

presence of this eigenfunction means that the evolution of a Taylor vortex

cannot be described by a single amplitude equation.

We shall see that if axial variations are ignored then it is possible to

describe the azimuthal evolution of a Taylor vortex by a single amplitude

equation. However, even this reduced equation differs from the reduced form

of the equation of Tabeling, Brandt and Cross. The appropriate amplitude

equation is discussed in some detail and it is shown that in the small gap

limit a Taylor vortex is stable to wavy vortex perturbations. Thus the

evolution equation approach to describe the azimuthal evaluation of a Taylor

vortex gives results which are not consistent with the classical results of

Davey, et al. [5] and the available experimental results. The implications of

this situation will be discussed later.

We consider then the stability of the flow of a viscous fluid of

kinematic viscosity v between cylinders of radii R1, R 1 + d. The outer

cylinder is held fixed whilst the inner one rotates with angular velocity

_l" We define the Reynolds number R and the parameter 6 by

U 0 d

R - u , (la)
d

6 - RI (ib)

In the limit 6 . 0 it is known that instability occurs when the Taylor

number

T = R2 6

is 0(60). Following Krueger, et al. [4] and Davey, et al. [5] it has been

customary in the small gap limit to consider disturbances with azimuthal

wavenumbers 0(6 -I_ ) even though all the available experimental results

suggest that only azimuthal wavenumbers of order 60 are important in the

transition from Taylor vortex flow to wavy vortex flow. Hence we shall take

a 0(60) a-o ~ hut the scalings of Davey, et al. [5] which were used by

Tabeling, Brand and Cross can be recovered at a later stage by considering a

further limiting process.

At this stage we restrict our attention to Taylor vortices of fixed axial

wavelength with amplitude dependent on the polar angle 0 and time. We

consider the limit 6 + 0 with

2 R 2 1 62 .] T = _ [To + 6TI + T2 + "" - 6 '

where TO = 3390 is the critical Taylor number in the small gap limit. The

velocity components in the radial, azimuthal and axial directions are scaled

on v/d, _i R1 and v/d, respectively whilst the pressure is scaled on

2

p v /d2 .

In the small gap limit the basic flow (0, _I RI v, O) has the

asymptotic form

v = I - x + 0(6)
where x is a radial variable scaled on d. The equations governing the

stability of this basic flow can be written in the form

_p

Lu = - _-_+ QI + T_ v + 0(6) (2a)

Lv = - _ 63/2 _-O+ 8p Q2 + u _-_+ d-_ 0(6) (2b)

Lw = - _--Pz + Q3 + 0(6) (2c)

_u 1⁄2 _v _w

_--_+ W 2 6 _-_+ _Tz= 0(6), (2d)

where z and t have been scaled on d and _/d 2 respectively. The

nonlinear functions QI' Q2 and Q3 are 0(60 ) whilst the operator L is

defined by

i_ ./'__ _ 22 22

(3)

L _ _-_+ 6 ' 2 v De _x2 _z2

Since the Taylor number differs from its critical value by 0(6) we

expect a finite amplitude motion of 0(6 1/2) and therefore expand u =

(U,V,W) in the form

, _ = 61_ _0 + 6_I + 63/2 _2 + "''' (4)

together with a similar expansion for the pressure. We then define the slow

1/2 --

time scales T = 6 t, t = 6t. The details of such an expansion procedure

follow closely those of Tabeling, Brand and Cross and at order 6 I_ it is

found that

_0 = A(0'T'_)eiaZ 2 (U0(x),Vo(X),Wo(X)) + COMPLEX CONJUGATE
where a is the critical axial wavenumber whilst (U0, V0, W0) is the

velocity eigenfunction corresponding to the critical point on the neutral

curve. At order _ it is found that A must satisfy the equation

s0_+_-= 3A 3A 0

and Tabeling has calculated sO numerically and found that sO = .5261. It

follows from the above equation that A = A(i,_) where

= @ _ _0 So T.

2

At order _ the first harmonic and mean flow correction are determined and it

is at this stage that the difficulty overlooked by Tabeling, Brand, and Cross

arises. The mean flow correction at this order is in the O-direction and we

denote it by vM. However, we see from the equation of continuity that this

mean flow drives a radial mean velocity field of 0(6 3/2 ) which we denote

by uM. The equation which determines uM is

3UM _ //T-o3vM

3x 2 30 '

and this equation must 'be integrated to satisfy uM = 0 at x = 0,I. This

cannot be achieved unless vM contains some arbitrary function of O and

t. It is for this reason that the solution given by Tabeling, Brand and Cross

does not satisfy the no-slip condition everywhere. The remedy is to allow for

a pressure eigenfunction in the manner discussed by Davey, et al. [3] and

DiPrima and Stuart [6]. Thus the perturbation pressure must be expanded in

the form
5

1/2 _3/2 -1/2 -- --

p = 6 PO + 6Pl + Pl + "'" + 6 po(0,t,r) + .-- ,

i

where the relatively large size of the induced mean pressure field is, of

course, a lubrication effect. The equation for vM now becomes

_2

VM _ _ _P0 I IAI2 d

_: (U0 V0) , (Sa)

_x2 TO 3e 2

which can be integrated subject to vM = 0 at x = 0,I. We can then

substitute for vM into the equation of continuity to find uM. The

condition tht uM should vanish at both x = 0 and x = I gives

22 PO 6 _ IAI2

_T_e2 - 3-O QO (5b)

where

i

Q0 = f F0(x)dx 0

with

x 1

Fo(X) = f U0 V0 dx - x f U0 V0 dx. 0 0

The equation for po is now integrated once and the arbitrary constant which

_0

appears in the resulting expression for De is fixed by insisting that PO

be periodic in e. The function vM is then completely determined and we

find

= - i [AI (x) +

VM 2 2Fo {1⁄2 o0 - 3Qo [A[m}{x 2 x} (6)

QO 2_

where o0 = 3 --_-f IAI2 de. The amplitude equation found by Tabeling,

0

Brand and Cross corresponds to vM = - _ IAI2 Fo(x) so that the radial mean
6

flow induced at higher order in their expansions cannot satisfy the no-slip

condition at both cylinders. At order 63/2 we find that A satisfies the

equation

_A - T{ A + c3 --- 32A c4 AIAI 2 + c5{1⁄2 o0 3Q0 IAI2}A, (7)

_ _2

where T[ - cO

2T0 [TI - TI] with T I the order 6 correction to the

axisymmetric critical Taylor number. The constants Co, c3, c4 are given by

Tabeling as

co = 26.16, c3 = 2.609, c4 = 40.2 with Vo(i/2) = i

whilst c5 has been calculated by DiPrima and Stuart [5] who give

c5 = -4.76.

Th_ amplitude equation given by Tabeling, Brandt, and Cross corresponds to

setting c5 = 0 in (7). The linearized form of (7) shows that the nonaxisymmetric mode with wavenumber M is linearly unstable for

T I > TIc = c3 M 2 (8)

and the finite amplitude axisymmetric mode which bifurcates from T_c is

A = A e = _IT{ - c4c 3 M 2) eiMi (9)

and of course only integer values of M have any physical relevance. The
first mode to bifurcate is the Taylor vortex solution which has M-- 0. In

order to investigate the stability of (9) we write

A=Ae+b

and the linearized equation satisfied by b is

2

8b_ c3 8 b

8_ 8_ 2 (T{ - c3 M2)(b + be2iM!)(l + g) + c3 M 2 b

iMi 2_

e g . _

+--T[TI c3M2 ] f (beiM_+ b_iM_)d_ (I0)

0

where

3Q0 c 5

g -

c4

If we set M = 0 in (I0) we can study the stability of a Taylor vortex to

dependent perturbations. We can then see from (I0) that the growth rate of a

disturbance proportional to icos Mi or isin Mi is -c3M2 so that, in the

small gap limit, there is no bifurcation from a Taylor vortex to a wavy vortex

solution. The non-axisymmetric modes with M # 0 are susceptible to the

Eckhaus-Benjamin-Feir sideband instability mechanism. Following Stuart and

DiPrima [7] it can be shown from (I0) that the non-axisymmetric mode is

unstable to sidebands with integer wavenumbers yM,(2 - y)M for

-i < y < 3, _ # I. The non-axisymmetric mode with wavenumber M which

bifurcates from T[c is found to be unstable to such a sideband for

. (Y+I)(3-y)l .

Tic < T{ < [I + 2(l+g) JTlc
which reduces to Eckhaus" result T{ < 3Tic in the limit T . I with g = 0.

We see then that the Eckhaus criterion is altered if _ # 0 so that the

pressure eigenfunction decreases the unstable regime. In the present problem

only integer values of yM are physically acceptable so that the nonaxisymmetric mode is unstable for

TIc" < T{ < [I + [42(I_) - I/M 2]Tic.

It is of course possible that stable wavy vortex solutions to (7) exist

but do not bifurcate from Taylor vortex flow. In order to investigate such a

possibility (7) was integrated numerically using a fully implicit finite

difference scheme. Several runs were made using difference initial

distributions for A at different supercritical values of T{. At

sufficiently large values of t the numerical solutions approached one of the

equilibrium solutions (9) for some value of M, we found no steady state

solution other than those given by (9).

In previous descriptions of non-axisymmetric motion in the small gap

limit it has become customary to take 2R2 6 = T and 8____ _-I/2 in the 8e

equations of motion. This procedure was followed by Tabeling, Brand, and

Cross who then allowed T to differ from its critical axisymmetric value by

0(g) and considered azimuthal variations with 8___~ 80 0(_) I_ >> 1 The

8

procedure outlined in this paper has _-@-~ 0(I) but, by taking _ = _ in the

work of Tabeling, Brand and Cross and inserting the required pressure

eigenfunction, we can recover (7). Alternatively by taking the limit TI + =

~ T_ I/2 A ~ T{ I/2 in (7) we recover the corrected form of the

8

with _

equation of Tabeling, Brandt, and Cross. Thus the expansion procedures are

equivalent and it is not necessary in the small gap limit to ignore the

8

available experimental results and take _-@->> i.
We shall now derive a generalized form of (7) which takes account of slow

variations of the vortex amplitude in both the axial and azimuthal

directions. Such an equation has been given by Tabeling, Brand, and Cross but

the velocity field associated with that equation does not satisfy the no-slip

condition at one of the cylinders. We again assume that _--0-~0(I) and now

choose to look for axial variations on the same length scale as those in the

azimuthal direction. We therefore define

= 61/2z

and retain the expansion (4). However we now write

_0 = A(O,_,T,t) 2 [U0(x) ' V0(x ) ' W0(x)]eiaZ + COMPLEX CONJUGATE,

and note that in the absence of any pressure eigenfunction the radial mean

flow at order 63/2 cannot satisfy the no-slip condition at both cylinders.

We therefore expand the pressure in the form

p = 6 1/2 P0 + 6Pl + "'" + 6- 1/2 Po(O,_,r,_) + pl(O,_,r,t)

+ 61/2 +-.. , (11)

where P0' PI' _2' etc. are pressure eigenfunctions introduced in order that

the radial mean flow satisfies the required boundary conditions. However, it

60

follows from (2c) that TO will drive an axial mean flow

of order if

-- _ O. Thus we set - 0 and - 0 in order that the azimuthal and
I0

axial mean flows induced by the disturbance should be comparable. If we

denote the mean part of W I by wM it follows from (2c) that

a2 WM aP--2

ax2 a_ ,

so that

i aP2

WM 2 a_ (x2 - x), (12)

whilst VM, the order 6 azimuthal mean flow again satisfies (5a) so that

i [A[2 aP0 (x2 - x)

VM = - 2 F0(x) - r 0 a@ 2 (13)

The radial mean flow at order 63/2 is now driven by both vM and wM and

satisfies the no-slip condition at x = 0,i if

a2 -- a2

Po P2 a IAI 2

a°2 + a_2 - 6-_-_ qo /--_ , (14)

a

where Q0 is as defined earlier. We see that if --= 0 then the pressure ae -

eigenfunctions P0 and P2 both vanish whilst if -- E 0 we recover (5b) a_

It is of interest to note that in the corresponding pressure equation in

parallel flow stability theory P0 = P2"

The axial mean flow of order _ interacts with the fundamental terms of

order 6 I_

to reproduce the fundamental so that the solvability condition at

ap2

order 63/2 will depend on _- • After some analysis we find that the

appropriate solvability condition is
II

- T{ A + c3 a2 A a2 A a2

aA A 2

a_ _-_-+ c6 ---_+ a_ icy _--_-_- c4 AIAI

ap0 ap2

- c5 _--_--A- ic8 _--_-A, (15)

where c3, c4, and c5 are as defined previously whilst Tabeling has

calculated c6, c7 and gives

c6 = .9837, c7 = .3948.

The constant c8 satisfies

I

a f [I Vo(X2 _ x)V0 _ _I U;(x2 _ x)(U0" - a2U0) + U0 U;]dx 0

c8 = I , (16)

[U;(U_ - a2 U0) - V+0 V0]dx

0

where (U_, V_) is the function pair adjoint to (U0, V0). We have not

calculated c8 which is purely real but there is no reason to suppose that it

is zero. The equations (14) and (15) are coupled so that in general it is not

possible to describe the axial and azimuthal evolution of a Taylor vortex by a

single amplitude equation.

In order to demonstrate how (14) and (15) can he solved let us suppose

that all the disturbance quantities are periodic in _ with wavelength 2L.

We can integrate (14) once to give

-- a2 p0

ap2 _ a _ IAI2 d_" - (_ + L) --+ F(8,_), (17)

a_ - 6Qo 2 ae -L ae2

where F is an unknown function of e and t. However, if P2 is periodic
12

L

in _ _2 TO _ 3Q0 _ 3 f IAI2 ", (18)

3B2 L 2 _0 _n

so that

8p2 _ - 6Q0 __0 f_ {IAI2 _ 12L fL [AI2 d_'}d_ . + F(0,_), (19)

-L -L

and if we integrate (19) once more with respect to _ and impose the

condition that po be periodic in _ we obtain

F(O,_) - -3Q0 _ 8§0 fL f_ {[Aim --_ I fL ]AI2 d_'}d_" d_ (20)

-L -L -L

The solution for P2 obtained by this procedure is unique only up to an

arbitrary function of e and t. This nonuniqueness of can be seen

-- 32 3P2 P2

from (14) and (15) directly, since only 3_ and appear in these

3_2

equations.

We now integrate (18) to give

P0 3Q0 1 27

_0 - L {IAI2 - _ f IAI2 d0}d_, (21)

-L 0

where the unknown functions of _ which results from integrating (18) has

3P2 3P0

been fixed so that P0 is periodic in e. The functions 3_ and 36 can

then be substituted into (15) to give an equation depending only on A which

must then be integrated numericallly subject to the periodicity of A in e

and _. In the more general case where the perturbation is not periodic in

such a simplification of (14) and (15) is not possible.
13

CONCLUSION

We have given a self-consistent derivation of the equations governing the

evolution of a Taylor vortex flow in the small gap limit. The equations which

we have derived differ from those derived previously for the same problem.

Our results for the situation where there is no slow axial variation suggest

that, unlike the finite gap case, there is no bifurcation to a wavy vortex

flow from a Taylor vortex flow. Furthermore we have found that nonaxisymmetric modes are possible stable states in the small gap limit. This

latter result is again different from the classical results of Davey et al.

[5] for the finite (but small) gap problem. The expansion procedure used in

the latter paper allowed the axisymmetric and non-axisymmetric eigenvalues to

be split apart by taking _->> 1 but results were obtained by subsequently

taking _--_~ 0(1). The expansion procedure used here has 8--_~ 0(1)

throughout and at first order in our expansions we have a multiple eigenvalue

so that a self-consistent expansion can be developed. It is possible that the

procedure we have used is valid only for very small values of _ whilst that

of Davey et al. [5], though not a formally self-consistent gives good

agreement with experiment at moderately small values of 6.
14

References

[i] P. Tabeling, J. de. Physique Letters, 44, 16, 665, (1983).

[2] H. Brand, and M. C. Cross, Physical Review A., 27, 2, 1237, (1983).

[3] A. Davey, L. M. Hocking, and K. Stewartson, J. Fluid Mech., 63, 529,

(1974).

[4] E. R. Krueger, A. Gross, and R. C. DiPrima, J. Fluid Mech., 24, 52,

(1966).

[5] A. Davey, R. C. DiPrima, and J. T. Stuart, J. Fluid Mech., 31, 17,

(1968).

[6] R. C. DiPrlma and J. T. Stuart, J. Fluid Mech., 67, 85, (1975).

[7] J. T. Stuart and R. C. DiPrima, PRS(A), 362, 27, (1978).



1. Report No. 2. GovernmentAccessionNo. 3. Recipient'sCatalogNo.

NASA CR-172249

4. Title and Subtitle 5. ReportDate

The Evolution Equations for Taylor Vortices in the Small October 1983

Gap Limit 6. PerformingOrganizationCode

7. Author(s) 8. PerformingOrganizationReportNo.

Philip Hall 83-55

10. Work Unit No.

9. PerformingOrganizationNameandAddress

Institute for Computer Applications in Science

and Engineering 11. Contractor Grant No.

Mail Stop 132C, NASA Langley Research Center NASI-17070

Hampton, VA 23665 13. Type of Reportand PeriodCovered

12. SponsoringAgencyNameand Address Contractor Report

National Aeronautics and Space Administration

Washington, D.C. 20546 14. SponsoringAgencyCode

15. Supplementary Notes

Langley Technical Monitor: Robert H. Tolson

Final Report

16. Abstract

We consider the centrifugal instability of the viscous fluid flow between

concentric circular cylinders in the small gap limit. The amplitude of the Taylor

vortex is allowed to depend on a slow time variable, a slow axial variable, and the

polar angle e. It is shown that the amplitude of the vortex cannot in general be

described by a single amplitude equation. However, if the axial variations are

periodic a single amplitude equation can be derived. In the absence of any slow axial

variations it is shown that a Taylor vortex remains stable to wavy vortex perturbations.

Furthermore, in this situation, stable non-axisymmetric modes can occur but do not

bifurcate from the Taylor vortex state. The stability of these modes is shown to be

governed by a modified form of the Eckhaus criterion.

{

17. Key Words(Sugg_tedby Author(s)) 18. DistributionStatement

centrifugal instabilities 34 Fluid Mechanics and Heat Transfer

Unlimited-Unclassified

19. S_urity Oassif.(ofthisreport} 20. S_urity Classif.(of this _ge) 21. No. of Pages 22. Dice

Unclassified Unclassified 16 A02

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