The Taylor-Couette flow: the hydrodynamic twin for Rayleigh-B´enard convection A. Prigent1, B. Dubrulle2, O. Dauchot2 and I. Mutabazi1 1 Laboratoire de M´ecanique, Physique et G´eosciences, Universit´e du Havre, 25 Rue Philippe Lebon, 76 600 Le Havre, France 2 Groupe Instabilit´es et Turbulence, DSM/SPEC, CEA Saclay, 91 191 Gif sur Yvette, France Summary. There is a strong analogy between Rayleigh-B´enard convection and Taylor-Couette system. This analogy is very well known, when dealing with the primary instability. It is essentially based on the existence of an unstable strati- fication in both systems. We show that the analogy can be extended beyond the linear stability analysis to the weakly non linear regime and even further to the fully turbulent one. 1 Introduction Early studies of the Taylor-Couette flow, the flow between two coaxial ro- tating cylinders, are contemporary of B´enard’s works on convection. Very soon, a strong analogy between both sytems has been observed [3]. In both cases the flow developping at onset of instability is made of vortices called respectively Taylor vortices and B´enard rolls (see figure 1). (a) (b) Fig. 1. Pictures and schematic views of (a) the Rayleigh-B´enard rolls (courtesy of F. Daviaud, Groupe Instabilit´es et Turbulence) and (b) the Taylor-Couette vortices. 2 A. Prigent, B. Dubrulle, O. Dauchot and I. Mutabazi The first studies of the Taylor-Couette flow consisted of viscosity mea- surements performed by Couette (1890, [1]) and by Mallock (1896, [2]). The instability of rotating flows was then considered by Rayleigh in a theoreti- cal paper [3] published the same year as his work on convection [4]. A few years later, Taylor [5], conducting both experimental and theoretical works, re-examined the whole problem. Using an apparatus where both cylinders could rotate independently, he predicted theoretically and verified experi- mentally the instability thresholds for both co- and counterrotating regimes. He showed that the flow, now called Taylor vortex flow, developing at onset of instability is axisymmetric, periodic in the axial direction and quantitatively similar to B´enard rolls of convection. This strong similarity is the signature of the analogy between the linear stability properties of rotating flows and the stability of stratified fluid first reported by Rayleigh and subsequentely analyzed by many authors [6, 7, 8, 9, 10]. In this paper, the analogy between the two systems is discussed beyond the linear instability. In the first section, we present the two flows and the physical mechanisms of their linear instability, which lead us to the detailed formulation of the analogy. In the next section, we consider the weakly non- linear situation and the secondary instability. Finally, in the last section, we discuss the implications of the generalisation of the analogy to the turbulent regime. More specifically, we show how it allows to predict precise scalings for the turbulent transport properties in the Taylor-Couette flow, on the basis of precise measurements conducted in the Rayleigh-B´enard case. 2 Basic flows and Instability mechanisms The Rayleigh-B´enard convection is obtained when a fluid layer of thickness d, is confined between two horizontal plates at different temperatures, the bot- tom plate being set to a higher temperature T0 +∆T than the top one. In the presence of the gravity field g, the temperature gradient, due to the difference of temperature between the two plates, induces a vertical density stratifica- tion. The flow is controlled by the Rayleigh number Ra = α∆T gd3/νκ,where α is the thermal expansion coefficient, ν is the kinematic viscosity, and κ is the thermal conductivity. For small Ra the basic flow is at rest, the tem- perature transport is purely diffusive and the density stratification remains stable. Then, there exists a critical value Rac above which instability occurs and Rayleigh B´enard rolls develop (see figure 1a). The Taylor-Couette flow is obtained in the gap between two independently 2 rotating coaxial cylinders. Choosing d = ro − ri as unit length and d /ν as unit time, the three natural control parameters are η = ri/ro and Ri,o = ri,oωi,od/ν,whereri,o and ωi,o are the radius and the angular velocity of the inner and outer cylinders. For a given geometry, η is constant and the linear instability threshold is given by a curve in the parameter space (Ri,Ro), which is usually described by: The T.-C. flow: the hydrodynamic twin for R.-B. convection 3 Ta(Ri,Ro,η)=Tac(η). (1) For Ta<Tac, the basic flow, called circular Couette flow, is purely azimuthal and depends only on the radial coordinate r. The velocity and the pressure 2 are given by uθ = V (r)=rΩ(r)=Ar + B/r and P (r)=P0 + ρV (r)/rdr, where the constants A and B are determined by the no-slip boundary condi- tions uθ(r = ri,o)=ri,oωi,o: 2 2 ωi µ − η ωir (1 − µ) A = and B = i , (2) 1 − η2 1 − η2 where µ = ωo/ωi.ForTa > Tac, instability occurs and the Taylor vortices appear (see figure 1b). One can already note at this stage that the circular Couette flow is inde- pendent of the fluid viscosity in the same way that the resting base state in Rayleigh-B´enard convection is independent of the thermal diffusivity. Also, as first shown by Jeffreys in 1928 [6], the analogy between both flows goes far beyond the similarity evidenced on figure 1. In order to explicit this analogy, we now turn to the linear instability analysis of the basic states, focusing on the Taylor-Couette case and refering to the paper of Manneville in the present volume for the Rayleigh-B´enard case. Let us consider an infinitesimal axisymmetric perturbation of the form u (r, z)=(ur,uθ,uz)andp (r, z). The linearized Navier-Stokes and continuity equations read: ∇.u =0 ∂ (3) ∂t + M u = −∇p + ν∆u where 2V 0 − r 0 M dV V = dr + r 00 (4) 000 is the inertial operator for axysymmetric inviscid perturbations. As a result one obtains that the flow is linearly stable when the Rayleigh discriminant 2V dV V φ(r)= r dr + r is positive [11, 12]. The above equations are indeed sim- ilar in structure to the linearized Boussinesq equations for Rayleigh-B´enard convection [13]. When expanding the perturbationfieldintonormalmodes, these equations, in the small gap approximation for the marginal mode, v(ξ)exp(ikz) reduce to: 3 d2 − q2 v(ξ)=−Ta[1 − (1 − µ)ξ] q2v(ξ)(5) dξ2 where ξ =(r − ri) /d, q = kd and Ta is the Taylor number defined by 4 −4Aωid 1 − η Ro Ta= =4 Ri Ri − . (6) ν2 1+η η 4 A. Prigent, B. Dubrulle, O. Dauchot and I. Mutabazi Apart from the radial dependence of the r.h.s coefficient, equation (5) is formally identical to the equation for the temperature perturbation Θ in Rayleigh-B´enard convection as soon as one replaces the z-derivative by the ξ-derivative and the Rayleigh number by the Taylor number Ta [7]: 3 d2 − q2 Θ(z)=−Ra q2Θ(z). (7) dz2 In the limit µ → 1, the Taylor number for which instability occurs is Tac = 1708 and the corresponding wave number is qc =3.12 [7]. These values are precisely those obtained in the Rayleigh-B´enard convection with rigid-rigid boundaries. Outside of this limit, the dependence of the insta- bility threshold on the radial coordinate calls for a physical prescription to determine the position at which the instability first takes place. Studying the mechanism of the instability, will now provide us with a physical ground of the analogy and thereby with the desired prescription. Let us consider a spherical fluid particle with radius R rotating near the inner cylinder in the Couette flow. We introduce a small perturbation in the form of a radial shift of this blob with velocity v (see figure 2a). This is the analog of a temperature perturbation in the form of a vertical shift in the fluid at rest between the two differentially heated plates in the Rayleigh-B’enard case (see figure 2b). If this blob experiences forces which tend to reinforce its movement, the flow is unstable. In contrast, if forces tend to bring it back to its initial position, the flow is stable. We first consider the inviscid (a) (b) Fig. 2. Displaced spherical fluid particle in the laminar circular Couette flow (a) and in the fluid at rest in a Rayleigh-B´enard cell (b). 2 uθ case. In the Taylor-Couette system, the centrifugal force per unit mass r The T.-C. flow: the hydrodynamic twin for R.-B. convection 5 acting in the radial direction on a fluid element at position r is balanced by the centripetal radial pressure gradient at this position. Now, consider an outward displacement of the blob, initially at r,towardsr = r+δr.Owingto axisymmetry, the angular momentum per unit mass L = ruθ is a conserved quantity. At its new position, the particle is submitted to a centrifugal force per unit mass f = L2/r3. At this new position, the centripetal pressure gradient is given by the local equilibrium with the centrifugal force f = L2/r3.Ifδf = f − f>0, the pressure gradient pushes back the element to its initial position. On the contrary, if δf < 0, the fluid continues to move outward until it reaches the outer cylinder. There, it is forced to come back and so follows helical paths that constitute the Taylor vortices.