A combined spectral-amplitude-perturbation approach for systematic mode selection in thermal convection

Abstract

Purpose – The post-critical convective state for Rayleigh-Benard (RB) convection is studied using a nonlinear spectral-amplitude-perturbation approach in a fluid layer heated from below. The paper aims to discuss these issues. Design/methodology/approach – In the spectral method the flow and temperature fields are expanded periodically along the layer and orthonormal shape functions are used in the transverse direction. A combined amplitude-perturbation approach is developed to solve the nonlinear spectral system in the post-critical range, even far from the linear stability threshold. Also, to leading order, the Lorenz model is recovered. Findings – It is found that very small Prandtl numbers (Pr < 0.1) can change the Nusselt number, when terms to O(ε5/2) and higher are considered. However, to lower orders the Prandtl number does not affect the results. Variation of the Nusselt number to different orders is found to be highly consistent. Comparison with experimental results is made and a very good qualitative agreement is observed, even far from the linear threshold. Originality/value – Unlike existing nonlinear formulations for RB thermal convection, the present combined spectral-perturbation approach provides a systematic method for mode selection. The number and type of modes to be included are directly related to the post-critical Rayleigh number. The method is not limited to the weakly nonlinear range.