Modeling of heat transfer at local convection over a vertical through flow in a two-layered air-heat generating porous system

Convective heat transfer in an air layer partially filled with a heat-generating granular porous medium is studied. There is a slow seepage of air through the layer in the vertical direction with a constant velocity. Equal temperatures are maintained at the outer solid permeable boundaries, and the heat source strength is constant within the porous sublayer and is proportional to the solid volume fraction. The permanent heat generation within the porous sublayer, combined with the vertical throughflow, causes a nonlinear thermal profile which is conducive for convection to occur. The Boussinesq approximation and Darcy's law are used to describe this convection. Numerical solution of the nonlinear convective problem is obtained on the basis of Newton's method. In the limiting case, the numerical data for the onset of convection are compared with the results of the earlier paper of the authors, where a linear stability theory and a method for constructing of the fundamental system of partial solution vectors have been applied, and with the data by other authors. The stationary regimes of local convection, which occurs in an “air - heat-generating porous medium-air” system over the basic vertical throughflow, and its effect on the heat transfer from the porous air sublayer with the growth of supercriticality are studied. It is shown that, depending on the velocity of the basic throughflow (the Peclet number), convection excitation can be both soft (due to supercritical pitchfork bifurcation) and hard (when the loss of stability of the basic throughflow is accompanied by subcritical pitchfork bifurcation that gives rise to an unstable secondary convective regime). This secondary regime is replaced by a stable tertiary convective regime with increasing supercriticality. It has been found that the total heat transfer rate for the upward basic throughflow exceeds that for the downward basic throughflow significantly, and that local convection at any direction of this throughflow increases the heat transfer rate in the system. An increase in the Nusselt number with the growth of supercriticality is recorded. However, a noticeable contribution of local convection to the total heat transfer is observed only when all values of the Pecklet number are negative and its positive values are lower than 2.