The Taylor-Couette Flow
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
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/r 3. At this new position, the centripetal pressure gradient is given by the local equilibrium with the centrifugal force f = L 2/r 3.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. Accordingly, the criterion for stability is given by: δf 1 dL2(r) 2V dV V = = + = φ(r) > 0. (8) δr r3 dr r dr r
Introducing now the effect of viscosity allows to formulate this criterion with the Taylor number as defined in equation (6). Indeed, the same mechanism ap- plies except that instability may occur only if the destabilizing force −δf over- 3 passes the viscous damping per unit mass fvisc ρνRv/ρR on a timescale R2 smaller than the viscous time τν ∼ ν which limits the displacement of the blob to δr τν v. As a result, the criterion for stability becomes:
νv ν2 δf = φ(r)δr − , that is : φ(r) − . (9) R2 R4 For the azimuthal profile V (r)=Ar + B/r, the above stability criterion 2 4 becomes 4AΩ(r) −ν /R , which in the most unstable situation (r = ri), and for a blob size scaling on the gap, is nothing but
Ta The mechanism we have described is based on the angular momentum stratification in the same way as the mechanism of Rayleigh-B´enard convec- tion is based on the vertical density stratification. Table 1 summarizes the other terms of the analogy. The driving force of Rayleigh-B´enard convection, the archimedian buyoancy, is replaced by the net balance of the centrifu- gal force and the centripetal pressure gradient. The temperature gradient is replaced by the velocity gradient and thermal diffusion is replaced by mo- mentum diffusion. The Taylor number is therefore analog to the Rayleigh number, the ratio of the archimedian force over the viscous dissipation force. Note that viscosity acts twice in the Taylor-Couette flow (so the dependence 6 A. Prigent, B. Dubrulle, O. Dauchot and I. Mutabazi to its square value of the Taylor number): it attenuates velocity gradients as the thermal diffusion attenuates temperature gradients and it dissipates the kinetic energy by friction. R.-B. convection T.-C. system coordinates z, (x, y) r, (θ, z) viscous dissipation µvd µvd driving force archimedian buyoancy centrifugal force d5 ∆T 3 d2 ρ0αg κ d v 4AΩρd ν v relaxation characteristic time d2/κ d2/ν 3 4 2 control parameter Ra = α∆T gd /νκ Ta = −4Aωid /ν instability threshold value Rac = 1708 Tac = 1708 critical wave number qc =3.12/d qc =3.12/d perturbation characteristic 0.063 0.041 time τ0 (Pr = 490) perturbation characteristic 0.385 0.260 length ξ0 (η =0.883,Γ = 70) Table 1. Analogy between Rayleigh-B´enard convection and Taylor-Couette flow. Let us point out that the stability criterion derived above is a local cri- terion, which for the moment we have turned into a global one by the most simple prescription concerning the localization of the instability. However, Esser and Grossmann [14], on the basis of the understanding of the basic mechanisms that we have recalled here, have recently shown that the sta- bility boundary can be much better approximated by an analytic expression when being more precise on the instability localization. As a result, they found a stability criterion of the form Ta = Tac provided one redefines the Taylor number as (Dubrulle et al, [15]): 4 2 1 2η(ξ(η) − 1) Ta= −Re (RΩ +1) RΩ +1− , ηξ2 (1 − η) with 1 − η dn ξ(η)=rp/ri =1+ ∆ a(η) , 2η d dn η 1 = − 1 , d 1 − η η(RΩ +1) −1 (1 + η)3 a(η)=(1− η) − η , (11) 2(1 + 3η) where ∆(x) is a function equal to x if x<1 and equal to 1 if x>1. Re,the Reynolds number built on the mean shear S¯,andRΩ, the rotation number The T.-C. flow: the hydrodynamic twin for R.-B. convection 7 built on the mean angular velocity Ω¯ are defined as: 2 Sd¯ 2 |ωo − ωi| Re = = rid (12) ν 1+η ν 2Ω¯ 1 − η ηωi + ωo RΩ = = (13) S¯ η ωo − ωi These control parameters, which at first sight may look artificial, are indeed based on dynamical propreties of the flow and therefore can be applied to any rotating shear flow. As a matter of fact, such parameters have already been used in rotating channel flows [16, 30]. They will be used when extending the analogy to the turbulent situation. 3 Weakly nonlinear description and secondary instabilities Both Rayleigh-B´enard rolls and Taylor vortices dynamics can be represented by a field Ψ = A(t, z)eiqcx where the amplitude A(t, z) satisfies, near the onset of instability, the Ginzburg-Landau equation : 2 ∂A 2 ∂ A 2 τ0 = A + ξ0 − g|A| A (14) ∂t ∂z2 where =(Ta− Tac)/T ac or (Ra − Rac)/Rac is the relative distance from the onset of instability. The real coefficients τ0 and ξ0 depend only on the lin- ear terms in the flow equations and represent respectively the characteristic time and the coherence length of perturbations, g is the Landau saturation constant. The Ginzburg-Landau equation depends on the boundary condi- tions which differ from a system to another. Its validity for Rayleigh-B´enard convection has been confirmed in experiments by Wesfreid et al [17]. For Taylor-Couette system, it has been derived from Navier-Stokes equations by many authors [18, 19, 20, 21]. In the Taylor-Couette system, the effect of boundary conditions have been investigated by Dominguez-Lerma et al [22] and Ahlers [23]. They showed that for cylinders with finite length L,the critical Taylor number depends on the aspect ratio Γ = L/d as follows: 2 2 Tac (Γ ) 2π ξ0 2 − 2 =1+ 2 +2ξ0 (q qc) (15) Tac (∞) Γ where qc and Tac (∞) are calculated values in the case of infinitely long cylinders. The spatial variation of the pattern for Rayleigh-B´enard or Taylor- Couette flow can be represented by the real amplitude A0(t, z)andthe iφ(t,z) phase φ as follows: A(t, z)=A0(t, z)e . The real amplitude satisfies the Ginzburg-Landau equation (14) and the phase satisfies the following equation which is the simplest phase equation valid only for <<1: 8 A. Prigent, B. Dubrulle, O. Dauchot and I. Mutabazi 2 2 2 ∂φ ∂ φ − 3ξ0 (q − qc) = D 2 with D = D0 2 2 . (16) ∂t ∂z − ξ0 (q − qc) The phase dynamics of a stationary pattern was theroretically investigated by Pomeau and Manneville [24] who have established the phase equation from general principle. When the diffusion coefficient vanishes at the Eck- 2 2 haus boundary =3ξ0 (q − qc) , the stationary pattern pertains a modula- tional instability called Eckhaus instability. The experimental investigation of the phase dynamics in Rayleigh-B´enard was performed by Wesfreid and Croquette [25], they found D0 =2.35 for a silicon oil with Pr = 490. In the Taylor-Couette system, the phase dynamics was investigated theoretically by Paap and Riecke [26] who have delimited the Eckhaus instability domains. Experiments were performed by different groups [23, 27, 28, 29]. The lat- ter found the value of phase coefficient for Taylor vortex flow to be about D0 =1.643 for a system with a radius ratio η =0.883 and an aspect ratio Γ = 70. Considering the analogy between the two systems when dealing with the secondary instabilities, there are still important similarities. In both cases, (a) (b) Fig. 3. Pictures and schematic views of the secondary flows in (a) the Rayleigh- B´enard convection (courtesy of F. Daviaud, Groupe Instabilit´es et Turbulence) and (b) the Taylor-Couette system. the transition to turbulence occurs via a cascade of successive bifurcations in a globally supercritical scenario. Also, the visual inspection of secondary flows calls for a formal analogy. Figure 3 displays modulated patterns obtained in both systems. They look strickingly similar. For details on the observed flows, The T.-C. flow: the hydrodynamic twin for R.-B. convection 9 one may refer to [31] for Rayleigh-B´enard convection and to [32] and [33] for the Taylor-Couette system. Unfortunately, no systematic study of the analogy has been proposed. 4 Turbulent regime So far, we only considered the analogy in the regimes where perturbations are small, resulting in linear or weakly non-linear equations of motions. Dubrulle and Hersant [34], have shown that this analogy can be extended into the tur- bulent regime. Indeed, at small scales, turbulent motions are mainly slaved to the large scales. Their dynamics can then be approximated by a linearized equation of motions, in the spirit of the equations of the Rapid Distorsion Theory. The same remark can be applied to Rayleigh-B´enard convection, since an effective shear is generated by large scales motions. This extension of the analogy into the turbulent regime yields a number of interesting con- sequences on the turbulent transport and profiles. 4.1 Turbulent transport A classical quantity in convection is the non-dimensional heat transfer Nu = Hd/κ∆T, called Nusselt number, where H is the heat transfer. The analog of this in the Taylor-Couette experiment is a non-dimensional angular momentum transfer (with respect to the laminar value). This can be writ- ten using the non-dimensional torque G = T/ρν2L,whereL is the cylinder length and T the torque: G Nu∗ ≡ , Glaminar G (1 − η)2 = . (17) Re 2πη The normalization by Glaminar ensures that in the laminar case, Nu∗ =1, like in the convective analog. Near threshold, theoretical [35] and experimental [36] studies of convec- tion lead to identification of two regimes just above the critical Rayleigh number. For =(Ra − Rac)/Rac ≤ 1, one has a linear regime in which Ra (Nu− 1) = K1. (18) Rac where the constant K1 depends on the Prandtl number. For Pr =1,itis K1 ≈ 1/0.7=1.43 [35]. For larger , a scaling regime appears in which [36] Ra 1.23 (Nu− 1) = K2 . (19) Rac 10 A. Prigent, B. Dubrulle, O. Dauchot and I. Mutabazi where, K2 is a constant which is not predicted by the theory. In the Taylor-Couette flow, the analogy yields the same scaling laws with Ta and Nu∗ instead of Ra and Nu respectively. Figure 4 shows how the results of Wendt obtained for η =0.935 near the instability threshold com- pare with these two predictions. One sees that the linear regime is indeed obtained for ≤ 10, while the scaling regime is obtained for larger values of 10 <<100. 1000 100 c 10 ( Nu*-1) Ta / 1 0.1 0.1 1 10 100 ε = ( Ta-Ta ) / Ta c c Fig. 4. Comparison of the theoretical near instability onset behaviour with the data of Wendt [39]. The symbols are the experimental measurements. The two lines are the theoretical formula predicted by analogy with convection for <1 1.23 ((Nu∗ − 1)Ta/Tac =1.43 and for >1((Nu∗ − 1)Ta/Tac ∼ .) In the latter case, the proportionality constant is not constrained by the analogy, and needs to be adjusted for a best fit. Further from the threshold, one needs to compare with the turbulent the- ories of convection. In the Taylor-Couette flow, transport properties in the turbulent regime have been investigated in a number of work. In the case of rotating inner cylinder and resting outer one, one observes a tendency of thetorquetobehavelikesomepoweroftheReynoldsnumberG ∼ Reα. There is no clear consensus about this dependence yet: marginal stability computation of King et al [37] or Barcilon and Brindley [38] predict that the non-dimensional torque should vary like G ∼ Re5/3.Oldexperimen- tal data indicated the existence of two scaling regimes, one for Re > 104 where the exponent is 1.5, and another for larger Reynolds number, where The T.-C. flow: the hydrodynamic twin for R.-B. convection 11 the exponent switches towards 1.7 − 1.8 [39, 40, 41]. Recent high precision experimental data yielded no region of constant exponent, and measured a “local”exponents d ln(G)/d ln(Re) which varies continuously from 1.2to1.9, with a transition at Re ∼ 1.5×104 (for η =0.7246). This transition was later found to correspond to a modification of coherent structures in the flow [42]. This situation is reminiscent of turbulent convection where different scaling regime for Nu as a function of Ra are observed. In the classical theory of convection, one usually considers three regimes: a first one, la- beled as “soft turbulence ”, in which Nu ∼ Ra1/3 [43]; this regime is followed by a “hard turbulence ”regimes in which Nu ∼ Ra2/7 [44]; fi- nally at very large Rayleigh numbers a ultra-hard turbulence regime oc- curs in which Nu ∼ Ra1/2 [45, 46]. More recently, Dubrulle [47, 48] used a quasi-linear turbulence model to predict slightly different dependence: at low Rayleigh number, the dissipation is dominated by the mean flow, and 1/4 −1/12 Nu = K1Ra Pr ; at larger Rayleigh number, the kinetic energy dissi- pation starts being dominated by velocity fluctuations, and the heat trans- 1/9 1/3 2/3 2/3 port becomes NuPr = K2Ra / ln(RaPr /20) . Finally at very large Rayleigh number, the heat dissipation becomes also dominated by (heat) fluc- 1/2 3/2 tuations, and Nu = K3Ra / ln(Ra/Rac) . This classification summarized in table 1 shows that it is unlikely that the three regimes coexist in Taylor- Couette experiment. Indeed, since the temperature analog is related to the velocity, it might be impossible to excite velocity fluctuations without ex- citing pseudo-temperature fluctuations. This would mean a direct transition from regime 1 (mean flow dominated) to regime 3 (fluctuation dominated). Re −→ R.-B. dissipation mean flow velocity fluctuations velocity and heat fluctuations T.-C. dissipation mean flow velocity fluctuations Table 2. Dissipation mechanism in both flows when increasing Re. The theory predicts two regimes for the torque behaviour: – a “low Reynolds number”regime, where mean flow (Taylor vortices) dom- inates, in which (3 + η)1/4(ηRe)3/2 G = K4 . (20) (1 − η)7/4(1 + η)1/2 – a second regime in which (3 + η)1/2 (ηRe)2 G = K7 3/2 3/2 , (1 − η) (1 + η) (ln[(K(η)(ηRe)2]) (1 − η)(3 + η) K(η)=K8 . (21) (1 + η)2 12 A. Prigent, B. Dubrulle, O. Dauchot and I. Mutabazi The value of the unknown coefficients is found by the analogy or by best fit of the data. The analogy with convection predicts that K4 =2πK1.The small aspect ratio convective experiment extrapolated at large aspect ratio gives K1 =0.75 × 0.31, which translates into K4 =1.46. This is in very good agreement with the prefactor measured by Wendt [39] who found the same exact dependence in η and Re for 400 1010 109 G 108 7 10 η = 0.935 6 10 η = 0.85 105 η = 0.68 104 4 5 100 1000Re 10 10 Fig. 5. Torque vs Reynolds in Taylor-Couette experiments for different gap widths η =0.68, η =0.85 and η =0.935. The symbols are the data of [39]. The lines are the theoretical formula obtained in the soft and ultra-hard turbulence regimes and computed using the analogy with convection. Soft turbulence eq. (20) (full line); ultra-hard turbulence eq. (21) (dotted line). There is no adjustable parameter in this comparison, all the constants being fixed either by the analogy with convection, or by the comparison with the data of [42]. 4.2 Velocity profile Mean profiles have been measured recently for different Reynolds number by Lewis and Swinney [42] in the case with outer cylinder at rest. They The T.-C. flow: the hydrodynamic twin for R.-B. convection 13 observed that the mean angular momentum L¯ = r 2 2 2 2 Ωi − Ωo η Ωiη (1 − 2η )+Ωo(2 − η ) u∞(r)= + . (22) 4r 1 − η2 2(1 − η2) In contrast, the same technique applied to convection predicts isothermal temperature profiles at large Rayleigh numbers, even though the maximizing principle differs only slightly from the maximizing Taylor-Couette problem. According to Busse, the difference originates in different ratios between the dissipation of the mean profiles and the fluctuating components of the ex- tremalizing vector fields. While this ratio approaches unity in the case of the extramilizing solution for the Couette problem, it reaches twice this value in the asymptotic case of convection. This may be due to the fact that the real convection is tri-dimensional (and not bi-dimensional as required by the analogy), resulting in more degrees of freedom. 4.3 Fluctuations The analogy provides perhaps more successes when dealing with fluctuations. 2 In [42], the azimuthal turbulent intensity iθ = −0.125 iθ =0.10Re . (23) Using the analogy, this intensity is related to the temperature fluctuations at mid-gap, in the ultra hard turbulent regime (regime 3). The total analog temperature fluctuation in fact also includes vertical velocity fluctuations (see Table 1). In an axisymmetric turbulence, one could therefore expect that the turbulent intensity measured by Lewis and Swinney is proportional to the temperature analog. Recent measurements of this quantity at Rayleigh num- ber up to Ra =1015 have been performed by [51] in a low aspect ratio Helium experiment. They found θ/∆T =0.37Ra−0.145, but this was obtained in a 14 A. Prigent, B. Dubrulle, O. Dauchot and I. Mutabazi regime where the Nusselt number varies like in regime 2 (velocity fluctuation dominate but NOT temperature fluctuation). Using the analogy, this would −0.29 translate into a regime where iθ ∼ Re , in clear contradiction with the data of Lewis and Swinney (see figure 6). This might therefore be another proof of the absence of the regime 2 in Taylor-Couette experiment. Unfortunately, we are not aware of laboratory temperature measurements in convective turbulence in the ultra-hard regime. In previous analysis of tem- perature fluctuations in the atmospheric boundary layer (a large Rayleigh number medium, presumably being in the ultra-hard convective regime), Deardoff and Willis [52] showed that temperature fluctuations follow the free convection regime θ Nu ∝ , (24) ∆T (PrRaNu)1/3 where the proportionality constant is of the order 1. Using the analogy, this canbetranslatedinto: K11 iθ = . (25) ln(Re/Rec) Figure 6 shows the application of this scaling to the data of Lewis and Swin- ney. The best agreement with the experimental fit of Lewis and Swinney is obtained for a prefactor K11 =0.18 and Rec =44.4. 0.035 θ i 0.03 0.025 0.02 0.015 104 105 106 Re Fig. 6. Azimuthal velocity fluctuations in Taylor-Couette flow. The circles are the power-law fits of experimental measurements by [42]. The triangles are the power- law fit to the temperature fluctuations (analog of azimuthal velocities) in Helium by [51]. The line is the prediction obtained with the analog of the free-convective regime [52]. The T.-C. flow: the hydrodynamic twin for R.-B. convection 15 5Conclusion We have detailed the analogy existing between the Taylor-Couette system and Rayleigh-B´enard convection. Despite of different symmetry properties, since in Rayleigh-B´enard convection both transverse directions are equivalent while z and θ directions are not equivalent in the Taylor-Couette system, the instability mechanism appears to be the same in both experiments. One is based on density stratification and the second on angular momentum stratifi- cation. The patterns appearing at onset of the primary and secondary insta- bilities are also similar. As suggested by Rayleigh [3] and already performed by Jeffreys [6], when studying the linear stability of the Couette flow in the small gap limit, one obtains the same equation of motion. Consequently the same threshold value and critical wavenumber are found. This analogy has also been used to develop stability criteria in other systems [9, 12, 60, 12, 61]. Above, but not too far, the instability threshold, both Taylor vortices and B´enard rolls may be described in the framework of Ginzburg-Landau equa- tions. The phases obey the same simple equation and may become unstable against Eckhaus instability. Further above the threshold, the secondary in- stabilities still present strong similarities but ask for a detailed comparison which, as far as we know, has not been conducted yet. Finally, we have discussed the extension of the analogy to the turbulent regime when the mean shear, generated by large scales, is taken into account into the Rayleigh-B´enard system. Then, analogy allows theoretical predic- tions for the angular momentum transfer and velocity profiles. In turbulent convection, three regimes with different modes of dissipation are identified. At low Ra, the dissipation is dominated by mean flow. 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