Large-amplitude Bénard convection

Abstract

Calculations are presented for two-dimensional Bénard convection between free bounding surfaces for ranges of Rayleigh and Prandtl numbers. The variables are expanded in a series consisting of the eigenfunctions of the stability problem and the system is truncated to take into account only a limited number of terms. The amplitudes of the eigenfunctions are evaluated by numerical integration of the resulting non-linear equations. In all cases considered, the system achieves a steady state with the motion consisting of a single large cell. Results for Nusselt number vs. Rayleigh number are given for a range of Prandtl number varying between 0·01 and 100 and show that heat flux increases slightly with decreasing Prandtl number. The calculations agree with those of Kuo where the ranges of Rayleigh number overlap. A simple heuristic argument based on the assumption that turbulent boundary layers exist is also given and the conclusions of the latter indicate that heat flux should decrease with decreasing Prandtl number. Thus the behaviour is qualitatively different from that of the calculations. The reason appears to be associated with the fact that the single large cell in the computed cases enables the fluid to accelerate through repeated cycles until it achieves a steady state with the amplitude of the motion much larger than could be acquired by a single turbulent blob free-falling in the gravitational field.