International Conference on Computational Heat and Mass Transfer Proceedings of 4 th ICCHMT May 17 – 20 , 2005 , Paris-Cachan , FRANCE 146 NUMERICAL INVESTIGATION OF THE TRANSITION TO SPATIO-TEMPORAL CHAOS IN ROTATING RAYLEIGH-BENARD CONVECTION

The Küppers-Lortz instability (KL) is a paradigmatic example of spatio-temporal chaos which appears in a rotating Rayleigh-Bénard layer. Surprisingly and contrary to this proved scenario, Bajaj et al. [1] observed experimentally in a cylinder, square patterns in the range of parameters where KL instability was expected. In this work we present the square patterns obtained in a numerical study performed taking account of realistic boundary conditions. The three dimensional flow in a finite cylinder has been obtained by solving the NavierStokes and heat transport equations in the OberbeckBoussinesq approximation. The numerical method is pseudo-spectral and second order accurate in time. The critical Rayleigh number and wavelength agree with the expected linear theory results for the onset of convection in cylinders. The rotation velocity of the square pattern increases linearly with the control parameter ε = Ra/Rac 1, as in the experiment of Bajaj et al. [1]. Furthermore, it was observed that this velocity decreases linearly when the aspect ratio of the cylinder increases. These results indicate that the square pattern appears when the flow is laterally confined. The range of Ra for which this pattern is stable tends to vanish for more extended layers. INTRODUCTION Convection in a fluid layer heated from below and rotating about a vertical axis has long been a favoured subject in nonlinear fluid dynamics because of the onset of convection in the form of spatiotemporally chaotic motions. In contrast to the direct transition from a laminar into a turbulent state observed in shear flows, the chaotic state of a rotating convection layer is induced by the supercritical Küppers-Lortz (KL) instability [2] in the limit of vanishing convection velocity. Typically, convection rolls become unstable to disturbances in the form of rolls oriented with their horizontal axis at an angle of about 60° in the sense of rotation with respect to the original rolls. As the disturbance rolls grow and replace the original pattern they themselves become unstable to disturbance rolls with angle of 120° relative to the orientation of the original rolls. As this process occurs in an uncorrelated fashion in patches throughout the convection layer, it involves a spatio-temporally chaotic state. Starting with the early experiment of Heikes and Busse [3], (see also Ref. [4, 5]), the (KL)-chaos has been investigated in numerous laboratory studies as in Ref. [5] as well as in numerical simulations as in Ref. [6, 7]. Surprisingly, and contrary to theoretical predictions (Clever and Busse, Ref. [8]), Bajaj et al. [1] observed within a Γ= 4.8 circular layer (Γ = R/d, with R the cavity radius and d the height of the layer) that for Ω=2πfd2/ν≥70 (f is the rotation frequency and ν the kinematic viscosity) the nature of the 4 International Conference on Computational Heat and Mass Transfer pattern near onset changed. Instead of typical KLpatterns, square patterns were found to be stable although the bifurcation remained supercritical. This pattern slowly rotates in the prograde direction relative to the rotating cylindrical box and qualitatively its properties are the same for argon at Prandtl number σ = 0.69 and water at σ = 5. Thus, we are faced with a major disagreement with theoretical predictions in a parameter range where one might have expected the theory to be reliable. A possible explanation for these disagreements was suggested by Hu et al. [5] and is related to the side wall, which generates defects travelling into the bulk. Since the defects travel at a constant speed rather than diffusively, they could in influence the system interior. These authors conjectured that the defects break up the KL-domain and thus alter the characteristic length and time-scales. In contrast to the non-rotating case, sidewalls exert a destabilizing effect in case of a rotating layer. For theoretical studies of the side wall modes we refer to Goldstein et al. [9], Herrmann and Busse [10] and Kuo and Cross [11]. Experimental observations of the onset of sidewall convection [12] have been reported by Liu and Ecke [13, 14], Ecke and Liu [15], and Bajaj et al. [1], and its dynamic has been studied numerically using a Ginzburg-Landau equation in Ref. [16]. These convection waves propagate along the side wall in the retrograde direction with a much faster phase velocity. The main goal of this numerical study is to investigate thoroughly the properties of the square pattern and its dependence on relevant parameters. After describing the basic equations of the problem and the numerical methods employed for their solution, we discuss the main results obtained numerically. In the choice of the parameters we are guided by experimental studies of Bajaj et al. [1] and Ecke and Liu [15]. In general, a good agreement between numerical simulations and experimental measurements is found. MATHEMATICAL AND NUMERICAL MODELS Rayleigh-Bénard convection is described by the Boussinesq equations written for the velocity vector V r and the heat equation for the temperature T relative to the rotating frame of reference as follows: 2 2 V Ra V V P z V V Tz t σ ∂ + ⋅∇ =−∇ − Ω × + ∇ + ∂ r r r r r r r r ,