Investigation of Problems in Thermal Convection: Rigid Boundaries

Abstract

An investigation of thermal convection in a thin layer of fluid has recently been reported (Herring, 1963). The calculation included only those nonlinearities having the form of an interaction of a fluctuating quantity with the mean temperature field. In addition, free boundary conditions were employed and the fluctuating velocity and temperature fields were composed of one horizontal wave number, α. In the present paper, the calculation is extended to include the effects associated with rigid boundaries and many horizontal wave numbers. The results of the multi-α study indicate that the stable steady state solution contains only one α, provided the Rayleigh number, R, is less than ?106. Above R?106, the stable solution contains at least two &alpha's. The stable single-α solutions have a somewhat different value of α than either that predicted by the maximum heat flux principle of Malkus (1954) or that predicted by the relative stability criterion of Malkus and Veronis (1958). At present, we are not able to characterize the stability of the system by postulating an extremal for some simple property of the flow. The value of the Nusselt number found here for rigid boundaries is N=0.115R½, for large R. This value for N is within ?20 per cent of the experimental value for large Prandtl number fluids. The rms values of the velocity and temperature fluctuation fields computed here appear to have the form expected for large Prandtl number fluids. The lack of accurate experimental data prevents us from drawing definite conclusions as to the numerical accuracy for these quantities. The computed mean temperature profile is qualitatively correct, but develops a thin stabilizing region with a stable temperature gradient just exterior to the thermal boundary layer. It is concluded that the stabilizing region represents a self-adjustment in the flow which compensates for the omission of the effects of eddy processes on the equations of motion.