Finite amplitude cellular convection

TLDR

When a fluid layer is heated from below and cooled from above, steady cellular convection arises once the Rayleigh number exceeds a critical value. The study introduces a method to determine the form and amplitude of this convection. The authors expand the nonlinear equations for motion and temperature into a sequence of inhomogeneous linear equations based on the linear stability solutions, and deduce a relative‑stability criterion that selects the realized solution with the maximum mean‑square temperature gradient. The analysis uncovers an infinite family of steady finite‑amplitude convection solutions with various plan‑forms; for high Prandtl numbers the amplitude is mainly set by mean‑temperature distortion, while for Pr<1 self‑distortion dominates; heat transport initially scales linearly with Rayleigh number and only slightly deviates at higher values; square plan‑forms are favored over hexagons in symmetric fluids, and the method applies broadly to shear flow or convection models with solvable stability problems.

Abstract

When a layer of fluid is heated uniformly from below and cooled from above, a cellular regime of steady convection is set up at values of the Rayleigh number exceeding a critical value. A method is presented here to determine the form and amplitude of this convection. The non-linear equations describing the fields of motion and temperature are expanded in a sequence of inhomogeneous linear equations dependent upon the solutions of the linear stability problem. We find that there are an infinite number of steady-state finite amplitude solutions (having different horizontal plan-forms) which formally satisfy these equations. A criterion for ‘relative stability’ is deduced which selects as the realized solution that one which has the maximum mean-square temperature gradient. Particular conclusions are that for a large Prandtl number the amplitude of the convection is determined primarily by the distortion of the distribution of mean temperature and only secondarily by the self-distortion of the disturbance, and that when the Prandtl number is less than unity self-distortion plays the dominant role in amplitude determination. The initial heat transport due to convection depends linearly on the Rayleigh number; the heat transport at higher Rayleigh numbers departs only slightly from this linear dependence. Square horizontal plan-forms are preferred to hexagonal plan-forms in ordinary fluids with symmetric boundary conditions. The proposed finite amplitude method is applicable to any model of shear flow or convection with a soluble stability problem.