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The thermal convection equations for a thin layer of fluid are solved numerically as an initial value problem. The calculations include only those nonlinear terms which have the form of an interaction of a fluctuation in the velocity and temperature with the mean temperature field. In the present calculations, the velocity and temperature fluctuations have one horizontal wave number, and satisfy free boundary conditions on two conducting horizontal surfaces. The computed steady state velocity and temperature amplitudes show many of the observed qualitative features. In particular, the experimentally observed boundary layering of the mean temperature field is correctly reproduced, and, at large Rayleigh number, the total heat transport is found to be proportional to the cube root of the Rayleigh number, provided the fluctuating temperature and velocity amplitudes have that horizontal wave number which maximizes the total heat transport. However, the heat transport found here for free boundaries is about three times the experimental value for rigid boundaries. The mean temperature gradient can become negative near the boundaries for large Rayleigh numbers and large horizontal scale motions. The linear stability of the system is also investigated, and it is concluded that the stable solutions for all Rayleigh numbers investigated (R < 108) have horizontal wave numbers which very nearly maximize the total heat transport. The stability study also indicates regions in which two or more horizontal wave numbers are required to support convection.
Jackson R. Herring (Mon,) studied this question.