Internally Heated Convection and Rayleigh-Bénard Convection

Six canonical models of convection are described: three configurations of internally heated (IH) convection driven by constant and uniform volumetric heating, and three configurations of Rayleigh–Bénard (RB) convection driven by the boundary conditions. The IH models are distinguished by differing pairs of thermal boundary conditions: top and bottom boundaries of equal temperature, an insulating bottom with heat flux fixed at the top, and an insulating bottom with temperature fixed at the top. The RB models too are distinguished by whether temperatures or heat fluxes are fixed at the top and bottom boundaries. Integral quantities important to heat transport are examined, including the mean fluid temperature, the mean temperature difference between the boundaries, and the mean convective heat flux. Integral relations and bounds are presented, and further bounds are conjectured for the IH cases. Similarities and differences between the six configurations are emphasized.

Three configurations of internally heated convection and three configurations of Rayleigh–Bénard convection are analyzed mathematically. The standard methods of determining the linear and energy stability thresholds of the static states are explained. These analyses yield Rayleigh numbers above which the static states are linearly unstable and Rayleigh numbers below which they are globally stable. The resulting thresholds are reported with high precision for various boundary conditions on the velocity. Exact analytical values, calculated by long-wavelength asymptotic, are given for configurations with heat fluxes fixed at both boundaries. We explain how the background method is used to prove lower bounds on the mean fluid temperature in all three internally heated configurations, including one for which no bound has been reported previously. In every configuration, these bounds guarantee that the mean temperature of the fluid grows with the rate of volumetric heating, H, no slower than H ^2∕3. The bounds are compared with upper bounds on the Nusselt number in RB convection.

Laboratory experiments and numerical simulations of internally heated convection are reviewed. The emphasis is on quantitative results, especially integral quantities important to heat transport and their dependence on the Rayleigh number, which is proportional to the heating rate. For all experiments and three-dimensional simulations, the various measures of mean temperature can be fit to powers of the rate of volumetric heating. The exponents of these fits range from 0. 75 to 0. 77 when the bottom is insulating, and they range from 0. 78 to 0. 82 when the top and bottom are fixed at equal temperatures. In the latter configuration, the fraction of internally produced heat flowing outward across the bottom boundary falls quite slowly as heating is strengthened. When this fraction is fit to a power of the heating rate, the fit exponents lie between − 0. 049 and − 0. 099.