Stability analysis for three-dimensional Rayleigh–Bénard convection with radiatively participating medium using spectral methods

In this study, a fluid subject to combined natural convection and radiation is studied by employing the Boussinesq approximation of the Navier–Stokes equations. The solution for the flow field within a three-dimensional rectangular enclosure is found numerically using a spectral method. The equation of radiation transfer for the participating medium is analyzed by the exact integral formulation. Black boundaries and a gray medium are prescribed. Linear stability analysis and weakly nonlinear analysis are used to determine the critical Rayleigh number for the onset of convection in the combined mode. For the system with flow, a modified second-order time splitting method and a spectral collocation method are introduced to minimize the errors in the computation. From numerical simulation and stability analysis, insight into the effect of radiation on this classical problem can be accomplished. The results show that the presence of a radiative source changes the static temperature gradient of the fluid, and generally results in increasing the flow critical values. The influences of the conduction–radiation parameter, Rayleigh number, and optical thickness on flow instabilities and bifurcations are discussed.