Transition to chaotic patterns in Rayleigh-Bénard convection in rotating cylinders

In rotating Rayleigh Bénard convection, Coriolis force stabilizes the conductive state, and the convective onset increases as rotation increases. The strength of the Coriolis force is represented by the dimensionless rotation rate Ω =2πfd2/ν where f is the rotation frequency. The conductive state becomes unstable to stationary convective parallel rolls as in Rayleigh Bénard convection, but overstability is also possible for small Prandtl numbers. In the nonlinear regime, one of the most interesting phenomena is the Küppers-Lortz instability. This instability has been the subject of many theoretical and experimental works mainly because the primary bifurcation is a direct transition from the conductive regime to a chaotic state (spatio-temporal chaos). Rigid boundary conditions have a strong effect on patterns, both in perturbing solutions of the infinite system and in selecting particular solutions. Besides the stationary and oscillatory rolls expected in infinite layers, experiments and linear stability analyses have shown that in a rotating cylindrical (finite) layer of fluid the conductive state may be also unstable to rotating travelling waves, side wall attached or spiral body modes. This side wall mode instability may set in at lower values of the Rayleigh number than the critical value corresponding to the parallel rolls expected for infinite rotating layers of fluids. The stability boundaries of travelling waves and bulk waves depend on the aspect ratio (Γ = radius-to-height ratio) and on the rotation rate of the cylinder (Goldstein et al. 1993). An asymptotic analysis in the limit of high rotation rate by Herrmann & Busse (1993) and by Kuo and Cross (1993) predicts that the onset of convection in the form of sidewall travelling walls grows as Ω and also that the wave propagates against the sense of rotation inside the cylindrical wall and with the sense of rotation if the fluid is outside the wall. We present here the numerical solutions of the equations based on a pseudo-spectral collocationChebyshev expansion in both non-homogeneous radial and axial directions (r, z) and based on the 2π periodicity of the solution in this configuration, a Fourier-Galerkin method is used in the azimuthal direction. The difficulty at the axis (r=0) has been avoided with a variable transformation by multiplying all the dependent variables by r (Serre & Pulicani 2001). The velocity-pressure coupling, has been overcome by the use of an improved projection scheme for time discretization (Serre & Pulicani 2001) using a pressure predictor computed at each time step. The time integration scheme is semi-implicit second-order accurate. It corresponds to a combination of the second-order Euler backward differentiation formula and the Adams-Bashforth scheme for the non-linear terms. The computations presented here were performed for confined layers of fluid with Prandtl number σ =5.3. The spatio-temporal behaviour of the convective flow has been studied in two geometrical configurations, rotating cylinders of circular cross section and of annular cross section. The values of the rotation rates are intermediate, Ω =60 and Ω =180. In the numerical results the first unstable mode observed in the circular cross section cavity is the slow body travelling wave, for both insulating and conducting sidewall thermal boundary conditions. The bulk travelling waves have not been obtained in the annular cross section cavities, since the bulk spiral patterns do not fit the annular cavity. In the transition to chaos we have found several nonlinear regimes depending on the geometry and on the thermal boundary conditions. Bulk and sidewall travelling waves The convective flow obtained in a circular cell of Γ =5, rotating at Ω =60 with insulating thermal boundary conditions at the sidewall is presented in Figure 1. The critical Rayleigh number Rac is in the interval 5300<Rac<5400.