Abstract
The problem of thermal convection between rotating rigid plates under the influence of gravity is treated numerically. The approach uses solenoidal basis functions and their duals which are divergence free. The representation in terms of the solenoidal bases provides ease in the implementation by a reduction in the number of dependent variables and equations. A Galerkin procedure onto the dual solenoidal bases is utilized in order to reduce the governing system of partial differential equations to a system of ordinary differential equations for subsequent parametric study. The Galerkin procedure results in the elimination of the pressure and is facilitated by the use of Fourier-Legendre spectral representation. Numerical experiments on the linear stability of rotating thermal convection and nonlinear simulations are performed and satisfactorily compared with the literature.
Highlights
Thermal convection in a fluid layer has been a cradle of nonlinear hydrodynamic stability studies
The classical Rayleigh-Benard problem of thermal convection in a horizontal layer which is heated from below is the most studied problem of the convective flows. This is due to its stability behavior exhibiting a sequence of discrete steps from steady regime to periodic, quasiperiodic regimes and eventually to chaotic regime as well as the simplicity of its geometry. This geometry of infinite fluid layer confined between rigid plates has been approximated by a periodic horizontal extent in the numerical studies and by large-aspect-ratio containers in the experiments
The expansion of the velocity field in terms of the solenoidal bases and the subsequent projection onto the solenoidal dual space provide the automatic satisfaction of the divergence-free condition and the elimination of the pressure. Removal of these common algorithmic difficulties helps to focus on the extraction of the dynamics hidden within the model dynamical system equations
Summary
Thermal convection in a fluid layer has been a cradle of nonlinear hydrodynamic stability studies. A unique feature is observed to occur that steady finite amplitude convection can exist for Rayleigh number which is lower than the critical Rayleigh number for a limited range of rotation Theoretical studies include those of Kuppers and Lortz [10] that investigated the stability behavior for the case of infinite Prandtl number and free-free boundaries. Puigjaner et al [23] studied numerically stability and bifurcation in convective flow in air in a cubical cavity heated from below They used a divergence-free Galerkin spectral method to discretize the system and a parameter continuation method to determine the different branches of solution. The dual space is spanned by dual bases that are required to satisfy the solenoidal condition in a form that incorporates the associated weight arising from the inner product in the spectral representation This is required in order to eliminate the pressure term in the projection procedure. The use of Legendre polynomials significantly simplifies the construction of the solenoidal dual bases due to associated unity weight
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