Nonlinear analysis of instability modes in the Taylor–Dean system

Nonlinear analysis of instability modes in the Taylor-Dean system Patrice Laure Institut Non Likaire de Nice, VMR 129 CNRS-Universitg de Nice, 1361 rte des Luciole$, Sophia-Antipolis, 06560 Valbonne, France

Innocent Mutabazi Groupe d’Energ&ique et Mkanique, Universitk du Havre, BJ? 540, 76058 Le Havre Cedex, France

(Received 2 July 1992; accepted 11 July 1994)

The linear and weakly nonlinear stability of flow in the Taylor-Dean system is investigated. The base flow far from the boundaries, is a superposition of circular Couette and curved channel Poiseuille flows. The computations provide for a finite gap system, critical values of Taylor numbers, wave numbers and wave speeds for the primary transitions. Moreover, comparisons are made with results obtained in the small gap approximation. It is shown that the occurrence of oscillatory nonaxisymmetric modes depends on the “anisotropy” coefficient in the dispersion relation, and that the critical Taylor number changes slightly with the azimuthal wave number for large absolute values of rotation ratio. The weakly nonlinear analysis is made in the framework of the Ginzburg-Landau equations for anisotropic systems. The primary bifurcation towards stationary or traveling rolls is supercritical when Poiseuille component of the base flow is produced by a partial filling. An external pumping can induce a subcritical bifurcation for a finite range of rotation ratio. Special attention is also given to the influence of anisotropy properties on the phase dynamics of bifurcated solution (Eckhaus and Benjamin-Feir conditions).

1. INTRODUCTION

In the study of the transition to chaos in systems far from equilibrium, the description of the critical or bifurcation points is a starting point to a better understanding of the fundamental properties of a system (critical parameters values, nature of the bifurcation, etc.). While for some systems such as magnetic fluids or electronic systems, a functional quantity minimization of which gives rise to the order parameter evolution equation near the onset of instabilities can be found, there is no such a quantity for many dissipative systems which can be derived from general physical princip1es.r However, it has been found that many known systems can be described near criticality by the complex Ginzburg-Landau equation which cannot be derived from any functional. Its predictive power due to its microscopic origin (order-disorder transition) allows a phenomenological description of the most experimental features observed near the onset of instability, despite their diverse physical nature, Much attention has already been given to the description of model systems such as the Rayleigh-Benard convective rolls or the Taylor-Couette vortex flow which have translational, rotational, and reflection symmetries.2-4 Because of the prac- tical importance of more complex systems, a recent attempt has been made to describe them, The flow in curved Poi- seuille channeL5 the electrohydrodynamic convection in liquid crystals,6 thermal convection in binary fluids,7 the print- ing instability8 taking place in a film between two eccentric cylinders and the Taylor-Dean system”” are mentioned as prototypes of such systems. Concerning the engineering ap- plications, the Taylor-Dean flow configurations are met in a rotating drum filter, in the paper and board-making industry,” in electrogalvanizing line in the steel-making industry which uses a roller-type cell to plate zinc onto the surface of a steel strip.ll

In the present work, we are concerned with the linear and weakly nonlinear analysis of the Taylor-Dean system which consists of the flow between two rotating cylinders in the horizontal position. As the space between the two cylinders is partially filled, the gravity to keep the liquid in place produces an azimuthal pressure gradient. In the core region, the base flow is the combination of a circular Couette flow induced by cylinder rotation and a curved Poiseuille flow due to the azimuthal pressure gradient.12 According to Rayleigh stability criterion for curved flows, the base flow velocity profile has potentially unstable layers alternating with stable layers, depending on the rotation ratio. Such a general velocity profile allows a global investigation of common properties of the Taylor and Dean vortex flows. The flow may exhibit at the instability onset, either Taylor-Couette or Dean instability modes, and also oscillatory modes which result from the competition between the two destabilization e mechanisms.13 In fact, recent theoretical and experimental studies’3-‘5 have shown that, the Taylor-Dean system is very rich in pattern forming and nonlinear phenomena. In particular, experimental studies have led to the observations of many patterns beyond the linear stability predictions: traveling patterns and coexistence of states with two different wavelengths,r4 spatiotemporal modulations close to the onset of the instability.‘” These observations are at the origin of the present linear and nonlinear investigations.

The full base flow in the Taylor-Dean system consists of an azimuthal flow in the core ac .’ a two-component flow near the free surfaces (the size of these regions is comparable with the gap between the cylinders). Recent numerical simulationsr7 and analytical calculationsr8 have particularly analyzed the two-component flow in the recirculation zone near the free surfaces. In particular Normand et al. l8 attribute the discrepancies between linear stability analysis of the core flow13 and experimental results14’16 to the intluence of recir-

3630 Phys. Fluids 6 (ll), November 1994 1070-6631/94/6(11)/3630/13/$6.bO Q 1994 American Institute of Physics

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culations eddies. However, we focus on the linear and nonlinear analysis of the core base flow and neglect the recirculation zones. Such an assumption seems valid for a small enough gap in comparison with the mean radius of the system. This condition is generally satisfied by most of experimental configurations. Moreover, linear analysis are made for both finite gap and small gap in order to check the valid- ity of previous computations.i3 In the first case, the numerical computations are mainly made for a radii ratio equal to 0.883 which corresponds to experimental setup. By using an efficient numerical procedure, we complete the previous linear rest&r3 which have missed an oscillatory branch. Let us finally note that with respect to the Taylor-Couette problem in which only integer wave numbers in the azimuthal direction are selected, noninteger azimuthal wave numbers are allowed in the Taylor-Dean system since it does not possess a rotational invariance. While our paper was submitted to the journal, Chen et uZ.“*‘~ published numerical results assuming an external pumping without a partial filling. Thus, they assume rotational invariance and they deal with disturbances having integer azimuthal wave number. Although to our best knowledge, there exists no experimental system with both simultaneous cylinders rotation and external pumping, such a configuration has the advantage of avoiding problems related to free surfaces. For example, the experimental apparatus described in Brewster et aLlo has an external pumping, but the gap between the two cylinders is yet partially filled up and only the inner cylinder can rotate.

For our Taylor-Dean configuration, the bifurcated struc- tures are either stationary axisymmetric modes or oscillatory modes. Assuming the small gap assumption, the large scale effects in both azimuthal and axial directions can be charac- terized at the onset by partial Ginzburg-Landau equations, the coefficients of which have been computed. We analyze the supercritical and subcritical nature of the bifurcation depending on the rotation ratio and conditions imposed on the flow rate. We also derive the linear part of the phase equation in each case. This allows us to obtain a criterion similar to Benjamin-Feir-Newell instability criterion for the anisotropic systems. Note finally that if the amplitude equation is only valid at the onset, the phase equation concept is still applicable far above the onset. The phase equation can provide a description of the slow dynamics of the structure as long as the gradient of the wave number remains small. However, far from criticality its calculation is more complicated since it requires the numerical computation of the structure for higher values of bifurcation parameter.

The paper is organized as follows: in the next section, we will give the governing equations and main parameters of the problem. Section III describes and discusses the results from linear stability analysis. The weakly nonlinear analysis is addressed in Sets. IV and V where the amplitude and the phase equations for stationary and oscillatory modes are analyzed. Section VI is concerned with the nonlinear behavior near codimension two points and the last section is the conclusion. To make this paper clear, the equations handled in the small gap approximation are given in an Appendix.

FIG. 1. Geometry of the Taylor-Dean system.

II. GOVERNING EQUATIONS

We consider the flow between two horizontal coaxial cylinders with a partially filled gap (with a filling angle 0,>. The inner and outer cylinders of radii R and R + d rotate with angular velocities fl and ,X s1 respectively (see Fig. 1). In the following, space coordinates, time, velocity components, and pressure are scaled by:

7 pvRfl I”=& u*=RQ; t*+ n*c,.

In cylindrical coordinates (r, 0,~) the dimensionless velocity field u =(ur,uO,u,) is described by the Navier-Stokes equations

2 due an $=A+-$ -7de -dr

-.Re

2 du, ~=Au~-$+~ -g

(1)

and the incompressibility condition

1 &ru,) 1 duo au, ---+;,e+~=O r dr c-9

with the Laplacian, in the cylindrical coordinates, given by

1 d d 1 d2 d2 A=;% rap +-;~aijz+z.

( 1 If we consider infinite long cylinders in the z-direction,

the control parameters are the radii ratio v=R/(R + d), the cylinder rotation ratio p and the Reynolds number Re=ROd/u. ‘We assume in the sequel that the two free surfaces at angle B= 0 and 0, are flat and that their positions do not change with the time. The former is usually true in the small gap approximation (v-+1) and the latter for a suffi-

Phys. Fluids, Vol. 6, No. 11, November 1994 P, Laure and 1. Mutabazi 3631


ciently slow flow motion (low values of Rer’). For example, numerical computations carried out by Chen and Changi7 assuming flat free surfaces agree very well with their experimental results obtained at moderate Reynolds number and ~=0.879. Therefore, the boundary conditions and mass conservation condition in the dimensionless form become

vr=vZ=O, v@=l at r=rl=Rld,

vr=vZ=O, vO= 5 at r=r2=(R+d)/d,

~=vs=v,=O at B=O and 6= of,

L “2 I J v B drdz= 0. 0 r1

A. Base flow solution

If we neglect the free surfaces and the recirculation zone effects, the base flow in the core region is purely azimuthal and depends on the only radial coordinate r,

v,=v,=o;

It consists of the superposition of a Poiseuille tlow (maintained by a pressure gradient assumed to be constant) and a Couette flow [due to the rotation of the two cylinders). Due to the gravity, the azimuthal pressure gradient is generated as soon as one of cylinders moves and it is proportional to the constant C. In the formula (3), the subscript p and c refer respectively to the Poiseuille and the Couette flow. With the above notations, we have

A= ld 17)

- P l-$ A,= v&$>

- v2 ln( 17) cl-P)?1 BP==(prl)Z(+$); B’=(p?l”)(l-rl)’

I

c=-2 (1-17)(~7~-77’--2 ln(7j7)7j2+p(2 ln(17)#-#+1))

771(1~-22z+774-~217 Mv))2)

These coefficients are determined by using the nonslip boundary condition on the two cylinders and assuming zero azimuthal mean flow. The latter condition comes from the two free surfaces since at moderate Reynolds number the fluid does not go through the upper region. The three coefficients A, ,B, ,C depend linearly on the rotation ratio ,u. Hence, the azimuthal pressure gradient is also a function of the rotation ratio. On this point, the present experimental arrangement is rather different from those proposed by Brewster et a1.r’ or ‘Chen et aLI1 where a constant azimuthal pressure gradient is maintained by an external pumping. Nevertheless with this latter configuration, it is also possible to cancel the azimuthal mean flow for a suitable value of pumping.

The Couette flow alone can be obtained for a specific value of the cylinder rotation ratio ,q,

For ~=0.883, we obtain that ,I.,+= -0.849 and ,u~--+ - 1 as v-+ 1 . These particular values will allow us to validate our numerical procedure by comparison with available results on the Taylor-Couette problem.“0

Due to the nonaxisymmetric boundary conditions near the free surfaces, the base flow in the Taylor-Dean system does not possess the rotational symmetry as in the Taylor- Couette system. This property renders the flow system similar to the Poiseuille tlow in a curved channel where the en- trance and exit zones make it nonaxisymmetric.

B. Perturbation equations

In the following, we will neglect the end effects and write the perturbation equations around the core flow delim- ited in azimuthal direction by @= 0 and 0= Sf (the recirculation zone is of order S= d/R). The perturbation U to the purely azimuthal base flow satisfies the following equations:

f&J-VIl I 27&O)

-Re v(O); -Re

i

--ug

(&O)lj. vw

r r 0

v- u=o

together with the boundary conditions

ur=ug=u,=O at r=rl=R/d and r=rz=(R+d)/d,

u B drdz = 0; where L is the length of cylinders, (5)

u,r drdB = 0.

3632 Phys. Fluids, Vol. 6, No. 11, Wvember 1994 P. Laure and I. Mutabazi


The two latter conditions of zero mean flow (flow rate conservation) in the axial and azimuthal directions are imposed by physical arguments. They enable one to take into account the two extremities of the cylinders on the one hand and the two free surfaces in the L‘recirculation ” zone on the other hand. Such a condition is commonly used when rigid walls are replaced by periodic boundary conditions.21 In the azimuthal direction, it only expresses the mass conservation as the liquid does not go through the two free surfaces. More- over, these two conditions are automatically satisfied by critical modes as soon as they are z-periodic ( qc # 0 in the sequel). However, they are particularly important for the nonlinear analysis as they directly act on the mean flow of the perturbation. As explained in Refs. 21 and 22, the only consequence from a mathematical point of view is that only the pressure gradient is periodic in z and 9 directions. In other words, the mean part of the perturbation modifies the pressure gradient’in these two directions by constant terms.

Note that an azimuthal external pumping modifies the condition (5) as it cancels the mean value of the pressure gradient in the azimuthal direction. Consequently, it yields the following condition,

-rdrde= 0. (6)

Due to periodicity in 8, we get the same condition for the Couette-Taylor problem in the small gap approximation.Z So, even if we adjust external parameters in order to get the same base flow v(O) for these two different configurations, the difference between the conditions (5) and (6) induces two different azimuthal mean llows for the perturbation. In the sequel, we show that this modification can change the behavior of bifurcated solutions.

Under all these approximations (the two free and rigid surfaces replaced by periodic boundary conditions and the base flow assumed purely azimuthal)’ the perturbation problem is now invariant under azimuthal and axial translation. Thus’ the perturbation equations (4) with the boundary conditions are invariant under SO(2) X0(2) symmetry group (the translations along the axial and azimuthal directions and the reflection .24-z). This latter symmetry is associated to the operator SZ given by

SzvCt,r, 0;~)

={vAt,r,B,-zj,vB(t,r,O,-t),-vJt,r,e, -2)).

III. LINEAR STABILITY ANALYSIS

Let us linearize the system (4) and look for eigenmodes of the form

,TJ= fi(r).$4z+P~), II= f~(rje~(V+PB). (7)

associated with some eigenvalue (T, 4, and p being the axial and azimuthal wave numbers. The form of the linearized operator from Eq. (4) and the reflection symmetry ZH -Y-Z imply the following properties of the eigenvalues:

cr(-q,p,Re,~)=a(q,p,Re,~),

a(q,-p,Re,~j=a(q,p,Re,~u) if pf0.

These properties can also be summarized in the formal ex- pression cr= a(q2,ip).

In the following, we introduce the Taylor number Ta=Refi and a new azimuthal wave number p* =pfi which depend on s=d/R=(l - 17)/o (&=0.364 for ~=0.883). As expIained, in the Appendix, these variables are commonly used in the “small gap approximation” (which means 84 0 or v-+ 1): As their critical values depend weakly on the radii ratio 7, a direct comparison with previous results is easier. l3 ‘The numerical procedure is the same as that used in Ref. 23: the linearized equations from (4) are discretized in the radial direction using the Chebyshev-Tau method. The generalized eigenvalues of the resulting matrix problem are then calculated by means of a library routine (EIGZC of IMSL). The critical value of the control parameter Ta, is obtained by minimization with respect to wave numbers (4, p) by iterative procedure of Newton-Raphson type. The most unstable mode is given by the eigenvalue o. with the largest real part. We have

(9) and the critical Taylor number Ta, is given by

Ta,=min Ta,(q,p,p) ca,P

(lo)

with Tao the marginal solution (i.e., for which the real part of o. is equal to zero):

Real(~o(~,p,~ao,~u)j=O.

Note again that p is not necessary an integer since the%ystem does not have the rotational symmetry. In the -Taylor- Couette problem p is an integer, even though the smaI1 approximation allows the existence of noninteger p*.” The critical values Ta, , o, = Im( ao) ,qc ,p,* (computed numeri- cally) are plotted versus the rotation ratio p in Figs. 2 and 3 for ~=@883. First, we find the same results as those of Mutabazi et al.: l3 the stationary solution Ta; (ao=O=pj corresponds to two different branches which intersect near ~=0.38. As’noted in Ref. 13, these two different stationary branches are associated to distinct -physical instability mechanisms: the left branch corresponds to the Taylor modes and results from the destabilization of potentially unstable Couette zone and the right branch describes the Dean modes from the destabilization of potentially unstable Poiseuille zone. Second, there exists an oscillatory axisymmetric mode (p=O) connecting these two stationary branches but this mode is never the most unstable. Indeed, Ta, is reached for a complex eigenvalue cre = iw corresponding to the nonaxisymmetric mode p#O. As shown in Figs. 2 and 3 for v= 0.883, we also have two different branches of oscillatory critical modes; for o= 0.883 the first one occurs in the range - 5 G J.J,G pi = 0.307 and the second from pi to pFL* = 0.585. This first oscillatory branch was missed by previous authors apparently because of a simplified numerical procedure. For ,G=,u* the most critical Taylor number is equal to Tat corresponding to the stationary axisymmetric mode. The two

P. Laure and 1. Mufabazi 3633 Phys. Fluids, Vol. 6, No. 11, November 1994


Ta

160

80

0.8

0.6

-5 -3 -1 1 3 5 -5 4 -3 -2 -1 0 W P (4

T

2 m-+-4 : ; I ; : ; ,+

O-4 -’ -3 -1 1 a 5

P

p=o lJ*

“\ -60 d--+++-+--+iA,

(d) -s 4 -3 -2 -1 0 1

P FIG. 2. Critical parameters as function of the rotation ratio p: (a) critical Taylor number Ta, versus p; (bj critical wave number qc in the axial direction versus y; (c) critical wave number pT = S”‘pin the azimuthal direction versus ,u; (d) critical frequency o, versus /.L. In addition, we give the frequency of the axisymmetric solution (p=O) connecting the two stationary solutions. For rotation ratio p * (p: = 0). Focus of curves (a) and (b) are shown in Fig. 3.

the critical modes become stationary and axisymmetric

different oscillatory branches intersect at the point pi= 0.307, the critical Taylor is Ta,= 175.7 and valnes of the other critical parameters are p * = 0.57; y = 4.08; w = 20.84 and p *=1.12; q=5.52; w=- 66.61 for the first and the second branches respectively. This corresponds to a codimension two point with two different nonvanishing frequen- cies. At the second codimension two point (,u = p* = 0.585, Ta,= 126.5) where the oscillatory and stationary branches intersect, the frequency w and azimuthal wave number p vanish continuously.

The occurrence of these oscillatory modes can be under- stood using the properties (8) and the Taylor expansion of a0 in the neighborhood of (q= qs ,p = O,Ta=Tas) which yields

-ia5P(4-4s) (11)

where the coefficients Ui are real (their physical meaning will be given in the next section). In this way, the critical Taylor number for a small value of p is given by

Tar=TaS,+ zp’+... . (12)

3634 Phys. Fluids, Vol. 6, No. 11, November 1994

So, if the ratio ax/u1 is negative, the most critical Taylor number is reached for a nonzero azimuthal wavelength p and we verify that the frequency w is of order -+J. This ratio, plotted in Fig. 4, illustrates two facts. First, for negative values of p, down to ,u-- 1, this ratio is rather small; so the critical Taylor number Ta, is very close to the stationary critical Taylor number Tai . This behavior is also pointed out in Fig. 5, where the functions

Tal’(p*,pu)=min Tao(q,p*,p) 4

(13)

for three values of rotation ratio ( ,X = - 5, 0, and 5) are plotted. For ,u= -5, we show that the curve is very flat (Ta,-Tag-O.025 ; pr=O.36, oC= -5.4) and the absolute value of the frequency w increases quickly with p * . There- fore, the oscillatory modes in the Taylor-Dean system are due to the anisotropy without parity invariance (pf0). Sec- ond, the transition between oscillatory and stationary critical modes for ,~=p,* is due to the change of sign of coefficient a3. The description of the neighborhood of the point p,=p,.* is postponed to Sec. VI.

Finally, the results are compared with those obtained in the small gap approximation (see the Appendix). First, we

P. Laure and 1. Mutabazi


o=i o;p=O

Ta ‘O”

180

160

140

120

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 (a) P

8-r

6.5

5

(b) I-c FIG. 3. Zoom of Figs. 2(a) and 2(b) for OGp~0.8. At velocity ratio ,u* the critical modes are stationary and axisymmetric (p: = 0). Other solutions l%, of the marginal stability are represented by dashed lines.

verify that the critical values obtained under this approximation are well reached when the aspect ratio 77 tends to 1. This limit process is made for three values of rotation ratio p and is plotted in Fig. 6. In Table I, we report the different critical values in order to highlight the inaccuracy of this assumption. We have also compared the previous results

a3 --- a1

L ‘_,.,.,\L.+ -5 -3 -I 1 3 5

I-L

FIG. 4. The “anisotropy” coefficient a,/~, versus /L. The two coefficients are defined in Eq. (11).

“cdl” e

0.05

0

-0.05

(4

-0.1 0 0.3 0.6 0.9 1.2 1.5

P’

30 I

co t pz.5

-70L--&.LLL-.--LLI 0 0.2 0.4 0.6 0.8 1

@) P” FIG. 5. Normalized critical Taylor number (Ta; -Ta;)/Ta; and frequency w plotted against p* for p= - 5, 0, and 5.

(v= 0.883) with those obtained under the small gap approximation, and we find a relative error of 9% for the critical Taylor number. Additional computations show that, the critical Taylor number decreases with 7 in the range -0.5<,u<O.4, and increases outside this interval.

IV. ENVELOPE EQUATIONS

In this section, we present the envelope (or Ginzburg- Landau) equations which describe the spatiotemporal evolution of the amplitude A at criticality for stationary and oscillatory modes. As this analysis deals with slow variations with respect to the wave number in both azimuthal and axial directions, it applies to systems having long cylinders and a small gap. The flow velocity is expanded into a series of the form

U=AUo+AAU11+A2U20+~.c.+... (141 where A is the complex amplitudes slowly varying with the coordinates t) and z, U. is the critical mode of the linearized problem, Ull is the second order mean flow contribution, U, is the second harmonic correction, C.C. stands for complex conjugate. The general form of the Ginzburg-Landau equation is obtained by means of invariance properties of the system, and the main coefficients entering the equation are obtained by numerical computations.20 Of course, the linear

Phys. Fluids, Vol. 6, No. 11, November 1994 P. Laure and I. Mutabazi 3635


200

Ta 160

0 0.5 0.6 0.7 0.8 0.9 1 “.5 0.6 0.7 0.8 0.9 1

(4 q (c) rl

6- -5 -

g 0

. - -,5 = I’ -1 5 --

j

\

--------

-25

4 --

------ -35

/--p=-1 3 -k------p=1

_--- .I&= o------

-

2 I * ----t---+-t t---w o’5 0.6 0.7 0.8 0.9 1 (b) 0.5 II.6 0.7 0.8 (4 0.9 1

q rl

FIG. 6. Influence of the radii ratio r] on the critical values for various values of rotation ratio p: (a) critical Taylor number Ta versus 7; (b) critical axial wave number 4 versus 7; (c) critical azimuthal wave number p” versus 7; (d) critical frequency w versus 7.

part may be obtained from the dispersion relation cr(Ta,q,p) near the critical point. In our case, those computations have been made for ~=0.883 (Navier-Stokes equation) and for p= 1 (“small gap” equation), but we only report in the tables the results for the aspect ratio p=O.883 since those from the small gap approximation do not bear any qualitative significant difference. The normalization of the critical eigenfunction UO=(~Or,~Oe,~Oz)(~) Xexp(ip,ti+q,z) is chosen in such a way that Re IC,+&,,+ SU~&,~+U~~U~~= 1 in the middle of the cavity (Y = (1 + 77)/(2( 1 - 7))). In fact, this normalization uses all

TABLE I. Critical parameters for various values of rotation ratio p and radii ratio 7, where p*=p ((l- r/)/~)~“.

-‘I 0.883 62.32 0.87 3.50 - 18.81 0.98 64.94 0.91 3.63 -21.42 la 65.43 0.92 3.64 -21.71

0 0.883 136.4 0.983 4.78 -45.22 0.98 130.6 0.993 4.72 -44.27 la 129.6 0.994 4.71 -44.08

1 0.883 82.66 0 3.61 0 0.98 87.10 0 3.81 0 la 87.92 0 3.85 0

Ta P* 4 w

%e case v=l corresponds to the small gap approximation.

dA dA d2A tPA d2A dt+a2dB=alEA+a3~+aqdZ2+iag-

f?ec?z

-cdA12A 05)

where e=(Ta-Ta,)/Ta, is the critical distance from the onset. The coefficients ai(i= 1 ,. . .,5) are real and reported in Table II, they have the following physical meaning: alis the linear growth rate of the perturbation, u2 is the group velocity in the azimuthal direction, u3=u1&, a4=u,&. The quantities co, &, represent the curvatures of the marginal stability surface Tao=Ta,-,(q,p) near the critical point

3636 Phys. Fluids, Vol. 6, No. 11, November 1994 P. Laure and I. Mutabazi

the components in order to avoid divergence problems due to the cancellation of the velocity profile along the azimuthal direction according to destabilization mechanism. On the contrary, in the Couette-Taylor problem, the normalization can be imposed only on the azimuthal velocity in the middle of the cavity.

A. Stationary modes

The critical stationary modes occur in the range ,ua,u* and are described by the amplitude A U,,(r)exp i(qg) and its complex conjugate. Their amplitude obeys the following equation and it is based on the expansion of the growth rate go of the stationary modes [see Eq. (ll)]:


TABLE II. Numerical values of coefficients of the Ginzburg-Landau equation for stationary modes.

F at a2 a3 a4 a5 b

0.6 a52 -8.745 0.097 2.091 - 0.364 -0.408 1. 35.60 -5.809 0.448 1.815 -0.736 -0.149 2. 23.88 -4.425 0.413 1.802 -0.710 -0.053 . 3 . 21.99 -4.338 0.336 1.867 -0.665 -0.041 4. 21.49 -4.384 0.285 1.896 -0.650 -0.036 5. 21.32 -4.440 0.250 1.906 -0.644 -0.034

(qc ,pc= 0,TaJ; a5 gives the correction to the group velocity in the azimuthal direction due to the axial wavelength modulation, g is the nonlinear saturation coefficient called Landau constant. The total group velocity due to the long- wavelength modulation in both axial and azimuthal directions u,=[a2+a5(~-~J] is added to the base flow and therefore its effect cannot be easily detected. The nonlinear saturation constant g is negative for all values of p, imply- ing that the bifurcation to stationary modes in the Taylor- Dean system is supercritical.

The obtained equation contains the one-dimensional Ginzburg-Landau equation describing the axisymmetric b = 0) Dean and Taylor-Couette vortex flow.‘,20

B. Oscillatory modes

The oscillatory modes are found in the intervals -5 Gpspi and Iui~~~~L”. In general, there are four critical modes

Uo=U(r)exp i(p,0-q,z+ o,t),

Ui=iT(r)exp i(p,8+q,z+ w&) W)

and their complex conjugates Uo,U1

where $ is the image of U under the reflection symmetry SZ. Note that qc and w, are assumed positive in order to associate the mode U. to the right traveling wave. So, the critical azimuthal wavelength pc would be either positive or negative depending on the oscillatory branches. The expansion of the eigenvalue in the neighborhood of the critical point gives:

TABLE III. Numerical values of coefficients in Eq. (18) for the two oscillatory branches.

x(q-q,)2-iaS(l+iC5)(P-P,)(q-~,) (17)

where the coefficients ui are real and have the same signifi- cance as for the stationary case, b2 is the axial group velocity. The quantities ci in the expansion coefficients are related to the dispersive properties of the oscillatory modes. Let A and B denote the amplitudes of the perturbation associated to critical modes U. and U1 respectively, they satisfy the following coupled Ginzburg-Landau equations:

=u,(l+i cr)EA+ua(l+i C&2A

+a4(l+i c&$A+iu5(l+i c&A

+g(l+i ~a)IA]~A+d(l+i c6)IB12A,

d,B+u,d$-b24B

=ul(l+i cr)eB+u,(l+i &B

+u4(1+i c&B-i+(lfi Cg)&B

+g(l+i ~~)lB1~B+d(l+i c6)1A12B. (18)

where all the coefficients are listed in Table III for both the oscillatory branches.

Computations show that in the parameter space (p,Ta,j, the azimuthal group velocity u2 is positive for the branch starting from negative /J and is negative for the other one. Next, the axial velocity group b2 vanishes at ,x- - 0.1 and 0.34. This case should correspond to wave packets of oblique rolls propagating only in the azimuthal direction. Fi- nally, the nonlinear saturation constant g is negative for all values of ,u, which means that the oscillatory modes in the Taylor-Dean system occurs via a supercritical Hopf bifurcation.

More precisely, Eq. (18) possesses two types of non- trivial solutions, either right (A # 0; B =0) and left (A =O; B#O) traveling waves or standing waves (IAl = ]B]). The former are stable with homogeneous perturbation when g<O and d-g<O, the latter when g-d<0 and g+ d <O. Since the constant d is negative for all calcu-

EL

-5.0 -4.0 -3.0 -2.0 -1.0

0.0 0.3 0.3 0.4 0.5 0.575

al Cl

23.76 0.27 24.75 0.35 26.16 0.43 28.63 0.52 35.51 0.64 65.61 0.76 89.08 0.83 69.04 0.16 63.38 0.21 58.36 0.18 54.12 0.03

a2 bz a3 =3 a4 C4 a5 8 C2 d C6

5.342 0.584 0.087 3.24 1.92 0.036 0.624 -0.41 -0.026 0.45 -0.046 -0.22 5.310 0.785 0.145 2.45 1.95 0.057 0.642 -0.48 -0.027 0.55 -0.045 -0.27 5.337 0.968 0.248 1.64 1.98 0.088 0.669 -0.50 -0.028 0.62 -0.044 -0.32 5.595 1.096 0.370 1.22 1.97 0.136 0.715 -0.40 -0.029 0.69 -0.043 -0.43 6.789 0.980 0.631 0.85 1.83 0.210 0.840 -0.09 -0.031 0.83 -0.043 -0.81

11.13 -0.187 1.602 0.06 1.43 0.127 1.241 -0.06 -0.055 1.65 -0.045 -4.82 13.83 -0.073 1.966 0.01 1.27 0.144 1.434 -0.10 -0.100 2.40 -0.051 -1.22

- 10.11 -0.271 1.016 -0.712 2.12 -0.078 0.662 0.48 -0.407 -0.21 -0.804 -0.08 -5.52 0.229 0.877 0.117 2.13 -0.074 0.524 0.00 -0.434 -0.18 -0.836 -0.05 -9.17 0.374 0.520 0.810 2.14 -0.065 0.455 -0.18 - 0.444 -0.14 -0.855 -0.04 -9.10 0.073 0.005 2.10 2.10 -0.013 0.321 -0.03 -0.434 -0.02 -0.867 -0.01

Phys. Fluids, Vol. 6, No. 11, November 1994 P. Laure and 1. Mutabazi 3637


lated values of p., the coupling between left and right traveling waves has a stabilizing effect. In fact, the effective linear growth rate coefficients for the amplitudes A and B become c~,=a~-~d~~B~~ and c~~‘b=a~--[d~IA~~ respectively, reducing this way the degree of linear instability. The stability criterion of the traveling wave is fulfilled in the intervals ,U E [-5,-0.1351 and p E [~~,p*], while for ,X E [ - O.l35,pJ, the traveling waves become unstable with respect to standing waves. Note that the transition from traveling to standing waves has been observed in the Taylor- Dean system for ,z=O when the inner cylinder rotation is modulated.24

Additional computations are made assuming an azimuthal pressure gradient created by an external pumping [boundary conditions (6)]. With the same base flow (3), we find that the coefficient g which characterizes the sub- or supercritical behavior of the transition, changes its sign with the rotation ratio p. This coefficient becomes positive for -0.7<p<pi (~~0.883) and -1*2<,u<,Ui (17-l). A similar result is obtained in the classical Couette-Taylor system as the spirals are subcritical for p< - 0.78 ( v-- 1) .22 In particular, this point confirms our computation for the specific value ,u= - 1. For subcritical bifurcation, higher order terms must be added to Eq. (18) in order to saturate the instability. A detailed analysis in which the nonlinear Landau constant changes sign with the control parameter has been performed by Eckhaus and IOOSS.~

Before concluding this section, it is worthwhile mention- ing that in the presence of an axial mean flow due for example to Ekman rolls at the boundaries, the group velocities and the other coefficients in the amplitude equations will be different for rolls traveling to the left and to the right respectively. This imperfection will break the reflection invariance assumed in the above equations.

These two coupled Ginzburg-Landau equations are obtained from the core region far away from the Ekman vortices and recirculation flows. They can be used to model the amplitude evolution for the entire system by addition of ap- propriate boundary conditions. This method has been used in the Rayleigh-Bhnard convection26 and the Taylor-Couette problem.” Nevertheless, as noted by Finlay and Nandakuma? such an approach does not predict the spatial variation of vortex amplitude for curved channel because of the active Ekman rolls.

C. Absolute and convective instability

In the case of oscillatory modes, the concept of absolute/ convective instability28 can be addressed for the Taylor- Dean system. In fact, the linear G&burg-Landau equation contains the necessary ingredients to address that question. Following Deissler,” a set of traveling waves exp i(-yg+w,t) will be convectively unstable if its axial group velocity satisfies the following condition:

Ib21>2{lala4~l(l+~,2)}1/2, and al>O.

Above the onset and for sufficiently small E, the system is convectively unstable, but there exists a second value E, beyond which the flow becomes absolutely unstable. From the last relation, we obtain

(19)

From the value listed in Table III, one can verify that the transition between convective and absolute instability occurs at E~s~X~O-~, i.e., very close to the onset of instability. There also exist two points where the instability is absolute at the onset as the coefficient b2 is equal to zero at ,U = - 0.1 and - 0.34. It is also shown in the Taylor-Couette problem3’ that the onset of absolute instability becomes close to the convective one as the radii ratio tends towards 1.

V. PHASE DYNAMICS

In real systems, critical wavelengths are never precisely selected because of boundary constraints, but there exist small variations which should induce phase modulations. In this section, we give the phase equation and the stability conditions of the bifurcated solution with respect to a general perturbation. Calculations leading to the phase equation are very tedious and will not be included in this paper, but the methodology and technique are rather usual.‘1-34 Hence, we will mostly focus on the influence of cross derivative term (coming from coefficients a5,c5 in the equations below) which is encountered in all anisotropic system without reflection symmetry. We exhibit criterion similar to Eckhaus and Benjamin-Feir stability conditions for the stationary and oscillatory modes.

For simplicity and without loss of generality, we consider the complex Ginzburg-Landau equation describing the evolution of amplitude A. Thus, we analyze stability of both right traveling waves and stationary vortices with respect to the perturbations of infinitely small wave number. The ad- vective terms may be eliminated by a transformation to a moving frame of reference at velocity (0,a ?, b2). As only the sign and the weight of each coefficient with respect to others are important, we can make the following change of variables and scales which preserve the bifurcation parameter in the amplitude equation:

t-z1 t, z--+zl~/~, t7~y=8l&iJ&

A+A(lbl/a,)V2.

The resulting equation reads

$=(l+. 2 2

z c,)EA+(l+i C4)$f(lfi c3,$

fia,(lfi cg)- d$z -(l+i c2)IAj2A4. (20)

Note that if the coefficients ci are zero, we recover the stationary case [similar to Eq. (15)]. Equation (20) has wave- like solution which describes for example a right inclined traveling wave propagating in the (0,~) plane with wave numbers slightly different from the critical ones:

A=Q exp i(S,13+ S4z+ 3,tj, w

Q2= cz- $- a;+ CY~C~~~~PS~, 02)



s,=(-cz+CI)E+(CZ-Cg)~3.(C2-C4)~

--a5 (lfcac,)S,S*.

The Eckhaus instability will take place in both axial and azimuthal directions if

(23) e<3&+3&

The linear analysis around this solution yields two eigenvalues. The first eigenvalue is always negative for positive 6 and is related to the amplitude perturbation. The second one comes from the existence of a marginal mode due to the phase invariance of Eq. (20). The linear part of this second eigenvalue gives the linear orders of the phase equation (the highest order terms are not considered in the sequel),

Y Y

Such a condition is represented by a paraboloid centered on aq4= S,= 0. This condition is a generalization of the one- dimensional Eckhaus criterion to two-dimensional anisotropic systems.

The anisotropic terms cys ,cs introduce more complicated dependence in high order derivatives and in nonlinearities, These terms are not considered in the present work, but they

34 a4 a24 a24 $+-a~~+~z~=D,~+D~z~ +D f?tt would be important in order to make a local analysis in the z a,z2 neighborhood of points where the homogeneous solution

(24) (21) becomes unstable with respect to phase perturbation.36

where the group velocities a0 and a,, VI. CODIMENSION TWO BIFURCATIONS

aB=2(c3-c2)Sp+~s (1+c2cs)aq4; In this section, we briefly address the nonlinear analysis

of the codimension two points {that means bifurcation de-

which usually depend on the longitudinal wave number modulation, ‘are presently modified by the tranverse modulation because of the anisotropy factors CQ,C~. The same behavior is observed on the diffusion coefficients D B and D, ,

1sc; Dz=l+C4C2- F(28q-ff5C5S,)2

while the constant part of the cross diffusion constant D oz,

resulting from these anisotropy terms is the signature of the nonparity invariance of the problem. Consequently, the criterion giving the possibility of Benjamin-Feir instability34Y35 for a right traveling wave (a,= Sq= 0) is more complicated and will occur if either

scribed by two independent parameters). From the linear stability analysis, we have detected two cases when the two oscillatory branches intersect at the point (pi=O.307,Tak= 175.7) and when the right oscillatory branch smoothly becomes a stationary branch at the point (,x * = 0.585,TaT = 126.5). The neighborhood of these codimension two points requires a specific study as states with different spatial and temporal properties can rise up.

The first codimension two point corresponds to the coexistence of oscillatory modes of incommensurate wave numbers. From a theoretical point of view, this case corresponds to Hopf-Hopf interaction with O(2) symmetry. A general description of spatiotemporal properties can be found in Refs. 4 and 37. The complete description in the case of the Taylor-Dean system is underway and will be devoted to a forthcoming work.

The second codimension two point occurs when the second derivative of cro with respect to azimuthal wave number p vanishes. The relevant Taylor expansion of the eigenvalue u. near the codimension two point is

cro=a-ia2p+pp2-a4(q-qc)2-iasp(q-q,)-a~p4 05)

or

or both D, ,D iZ are negative. From Table III, it follows that the perfect traveling waves (S,= Sp= 0) are never unstable

DSZ

with respect to the phase perturbation.

Dkz=De- 40,

For stationary modes, all the dispersive coefficients ci

4(1+c~c4)(l+c~c3)-a~(c?_-c5)2

are zero and the coefficients entering the phase equation are

= DZ

(0

where all the coefficients are real and the two small bifurca-

other case. The critical azimuthal wave number p and the

tion parameters a and p are defined by

critical eigenvalue we are always zero (for an axial wave number qc- 4.4). After a suitable scaling, the corresponding

a=ai(Ta-Tar)+a;(,u-p*),

Ginzburg-Landau equation in the moving frame (0,a2 ,O)

P=b,(Ta-Ta:)+b;(p-p*) (35)

reads:

where p is negative for JJ, greater than ,u* and positive in the

simpler:

ae= cysSq;a,= ffgSp,

D.=,-23; a2

D8=l-24; D,=-4s”s”. Q Q

dA a”A a4A a2A yg=aA--P---y+ T+ cj)t2+icy5 de de $$ -1~1~~

iW where LY and /3 are two small parameters defined in Eqs. (26).



We have the same basic “homogeneous” solutions defined in (21), but the modulus of the perturbation is slightly modified

Q2=a- s”,+ps;- s; (28)

while the frequency becomes . b,= ffgSpSq. (29)

The traveling waves correspond to a solution with Sp # 0, while stationary modes correspond to a solution with Sp = 0. The phase equation is the same as Eq. (24), but with different coefficients Do and DBz

Do== - 4spsq(p-2c$)

Q2 ’ The new conditions for the stability of the “homogeneous” solution with respect to perturbation of infinitely small wave number are

3p2+ 146 - 15ps; -p+6$ <a* (30)

Then, for ,L?> 0 (that means ,x<,Y,*), the axisymmetric stationary solutions (8, = 0) are all unstable, while a set of traveling waves is stable to long wavelength perturbations.

Vii. CONCLUSION

In the Taylor-Dean system, the destabilizing mechanisms of the Couette and Poiseuille potentially unstable zones compete and generate branches of different nature (oscillatory or stationary critical modes) in the parameter space. First, we have improved the linear stability analysis made in Ref. 13, for the radii ratios v= 0.883 and v- 1 and we have detected two oscillatory branches for ,uu(O.31 and 0.31<G<O.585 (pcO.33 and 0.33<,~<0.66 if 7-1). These new results are now compatible with Chen’s results.lg Moreover, we have shown that the occurrence of nonaxisymmetric oscillatory modes depends on the sign of “anisotropic” coefficient in azimuthal direction. We have observed a weak dependence of critical Taylor number with the azimuthal wave number for large rotation ratio (1~1 > 1). In this case, similar critical Taylor numbers can correspond to different azimuthal wave numbers [see Fig. 5(a)]. Thus, the experimental azimuthal wave number will depend strongly on boundary conditions which have been neglected in our theoretical model. The two end effects can be introduced by using Normand et al’s results.18 They have shown that the transition from almost symmetric whole base flow (i.e., in- cluding end recirculations) to asymmetric whole flow occurs at p= -0.38. Therefore, if the circulations at the both ends are rather similar (two damped eddies on both sides), the azimuthal wavelength is null and stationary rolls are observed. On the other hand if the two end circulations are different (boundary layer regime on one side and eddies on the other side), the critical azimuthal wave number is nonzero, hence the appearance of inclined traveling waves. In

this way, it is possible to understand the discrepancy with the experimental results for large positive and negative values of ,x. So, further work must be performed to include the recirculation zones in the linear analysis. On the other hand, when the influence of azimuthal wave number p on the critical Taylor number is obvious, a qualitatively good agreement between linear and experimental results are observed and both approaches predict oscillatory rolls for ,X E [ - 0.3,0.58] and stationary rolls for p E [0.58,1.2].

The linear analysis is also complemented by a weakly nonlinear analysis of the transition towards the spirals or the stationary rolls. We have found that this transition is always supercritical if the Poiseuille-Couette flow is induced by a partial filling [boundary condition (5)]. However, an external pumping [boundary condition (6)] induces a subcritical bifurcation for specific values of rotation ratio p. Therefore these two configurations are not always equivalent either for linear analysis (non-null azimuthal wave number selected by recirculation flows) or for nonlinear analysis (sub- or supercritical bifurcation). Finally, the phase dynamics of the stationary and oscillatory modes has been presented, and the anisotropic terms have been included in the generalization of the Eckhaus and Benjamin-Feir instability conditions.

Then, the Taylor-Dean system can be considered as an example of anisotropic system (due to the preferred direction of the base flow) and this work unifies and generalizes the nonlinear description of the centrifugally driven instabilities near their threshold. It might be applied to rotational driven instability such as that observed in a straight channel rotating about its own axisa As the amplitudes equations are mainly obtained by using symmetry arguments, the present weakly nonlinear analysis can be applied to other systems with the same symmetry properties. Moreover there are good candi- dates to numerical investigation of different two-dimensional patterns (defects, interaction between counterpropagating waves, etc.) for anisotropic systems without parity invariance. Such a work has already been performed for isotropic systems and anisotropic, parity-invariant systems.34

ACKNOWLEDGMENTS

We would like to thank G. Iooss, J.E. Wesfreid, C. Nor- mand, C.D. Andereck, and C. Bertin for their material support and for stimulating discussions during this work. This project has benefited from NATO support and I.M. also ac- knowledges financial support from the DRET (Contract No. 921447/AOOO/DRET/DS/SR).

APPENDIX: THE TAYLOR-DEAN PROBLEM IN THE SMALL GAP APPROXIMATION

The equations studied in the small gap approximation (d/R 4 1) are deduced from the Navier-Stokes equations after the change of variables

r=z(l+&); 8=51i2 y with S=d/R.



/ 2 Ta v;%,, +u; \ In addition, we define the scales of the velocity components in such a way that incompressibility condition preserves its form in the new Cartesian coordinates

vi=Re ur, ui=Re $” ve, vi=Re u,.

Substituting the dimensionless variables in Eq. (4), and neglecting terms vanishing for 6= 0, we obtain a set of equations in the new Cartesian coordinates (where the primes have been omitted) which describes the flow motion in the core of the cavity:

dv, - -$e&-g - ( Qg+v s+uzg -v; , y dY 1

!$&,-f(y)- (

dV u,~+uy~+7J:-g ,

i

(Al)

where A is the usual Laplace operator in the two coordinates x and z, and, the azimuthal pressure gradient is replaced in the second equation by an arbitrary function f which only depends on y, This additional arbitrary term is usually for- gotten but it is of great importance for the computation of coefficients of the amplitude equati0n.l’

The control parameter is now the Taylor number Ta= Refi which appears directly in the boundary conditions

ux=uz=O, u,=Ta at x=0,

vx=uz=O, vy=,u Ta at x=1.

Assuming a constant azimuthal pressure gradient [e.g., f(y) is constant], there exists a purely azimuthal solution of Eqs. (Al) which reads

v,= VI= 0;

u,=Ta(3(1+,u)x”-2(2+p)x+l)=Ta via)(x). (A3)

This solution is also the superposition of the linear Couette profile and the parabolic Poiseuille profile. We can also get this solution by means of a Taylor expansion of the base velocity (3) with respect to S

,(o)=uvJ) s (~)-&(1+3~+3x(2~+1)-2(i+~)x~)

13~+8 2 x - ~ x 3

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dU c x=AU-VII-Ta VP):+ [ -Ta u~~x+fIy~]

-(U*V)U, v.u=o. bw where the tilde over the differential operators indicates that they act only on variables x and z.

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Nonlinear analysis of instability modes in the Taylor–Dean system

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