Combined Spectral-Perturbation Approach for Systematic Mode Selection in Thermal Convection

Abstract

A nonlinear spectral approach is proposed to simulate the post critical convective state for thermogravitational instability in a Newtonian fluid layer heated from below. The spectral methodology consists of expanding the flow and temperature fields periodically along the layer, and using orthonormal shape functions in the transverse direction. The Galerkin projection is then implemented to generate the equations for the expansion coefficients. Since most of the interesting bifurcation picture is close to criticality, a perturbation approach is developed to solve the nonlinear spectral system in the weakly post critical range. To leading order, the Lorenz model is recovered. The problem is also solved using amplitude equations for comparison. The similarity and difference among the three models are emphasized.

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

“…In this case, the deviation from the critical Rayleigh number is defined as e ¼ Ra À Ra cr =Ra, which is smaller than 1 for any Rayleigh number. This is in contrast to e 0 ¼ Ra À Ra cr =Ra cr , the deviation from the critical Rayleigh number used by Ahmed et al (2013), which makes their solution limited to a very narrow Rayleigh number region above criticality. Using the expansion parameter ε, one is able to identify the required modes (to different order) to accurately capture the convection even far from the threshold.…”

“…The convective state just above the critical Rayleigh number is described by simply giving the amplitude A, corresponding to the projection of the motion onto this mode with, possibly, correction for small adjustment in the shape of rolls." However, from a theoretical point of view, the original amplitude equation method is valid only close to the critical Rayleigh number found from linear stability analysis (Dauby et al 2001;Dondlinger et al 2007;Ahmed et al, 2013).…”

“…However, the amplitude of the motion, the preferred pattern and the size of convective cells can only be predicted by considering nonlinear effects. Common nonlinear approaches to predict the convective flow pattern are the low-order dynamical system approach such as the Lorenz (1963) model, the method of amplitude equations (Newell and Whitehead, 1969;Segel, 1969;Cross and Hohenberg, 1993;Parmentier et al, 1996;Parmentier et al, 2000), and the perturbation approach (Malkus and Veronis, 1958;Albaalbaki and Khayat, 2012;Ahmed et al, 2013).…”

“…Perturbation expansion is another approach to study finite amplitude convection, where nonlinear inertial and thermal convection terms are considered as perturbations of the linear convection problem (Malkus and Veronis, 1958;Ahmed et al, 2013). Existing nonlinear methods to study natural convection are mostly based on arbitrary mode selection (Curry, 1978;McLaughlin, 1976;Aceves et al, 1986).…”

“…Existing nonlinear methods to study natural convection are mostly based on arbitrary mode selection (Curry, 1978;McLaughlin, 1976;Aceves et al, 1986). Recently, Ahmed et al (2013) used a spectral approach to identify the required modes to accurately capture the convection, when the deviation from the critical Rayleigh number is small. However, their results are limited to the very narrow Rayleigh number region above criticality.…”