Free Convection of a Heat Generating Fluid in a Hemispherical Closed Volume

From a theoretical analysis and results of previous studies, a model of the free convection of a heat- generating fluid in a hemispherical closed volume with a completely isothermal boundary is developed. By applying analytical estimates, we establish that at values of modified Rayleigh numbers $\mathrm{Ra}_{\mathrm{I}} > 10^{13}$, the flow in the entire volume is turbulent, which substantially simplifies the problem. Integrated relations for heat emission through the upper and lower parts of the boundary of the hemispherical capacity are obtained as a result. Results of numerical calculations have shown that the majority of heat is removed through the spherical surface downwards. The ratio of the heat flux through the upper horizontal boundary to heat flux through the lower boundary decreases from 0.5 to 0.3 in the range of modified Rayleigh numbers $\mathrm{Ra}_{\mathrm{I}}$ from $10^9$ to $10^{17}$.