Numerical simulation of cellular natural convection in a wide porous box to evaluate thermodynamic wavenumber prediction theories for an infinite horizontal porous layer

When a layer of fluid or fluid-saturated porous media is heated from below so that the temperature gradient exceeds a critical value, the fluid begins to move in a cellular convective pattern, known as Rayleigh-Benard convection, which significantly increases the rate of heat transferred across the layer. Theories to predict the size of the convective cells, which are characterized by the wavenumber of convection because of their periodic nature, are still unproven; and experimental measurements of the wavenumber have been few. This work seeks to verify two thermodynamic theories to predict the wavenumber of convection: one theory based on extremum of the rate of entropy generation caused by both heat transfer and viscous fluid flow through the medium, and another theory employing a stability functional expressing the "generalized excess entropy production," based on the nonequilibrium thermodynamic theory of stability and fluctuations.; For this work numerical simulations are performed for a wide, horizontal, rectangular enclosure filled with fluid and porous media. The modified Darcy-Brinkman-Forchheimer momentum and continuum-model thermal energy equations are solved with a mixed Fourier-Galerkin and finite-difference method. The nonlinearities in the momentum equation are transformed into the Fourier space; and the equations are quasi-linearized about each diagonal mode. The discretized equations are then solved by iteration, alternating the velocity and temperature solutions. The stream function, Nusselt number, entropy generation rate, and generalized excess entropy production functional are calculated from the converged solutions in the discretized Fourier space. The wavenumber of convection is then measured from the variations in the flow velocity field.; Numerical solutions with successively wider boxes demonstrate Nusselt number and wavenumber dependence on the porous media Rayleigh number that approaches the infinite layer behavior in the central region of boxes with aspect ratios of about 15 for Rayleigh numbers between 40 and 200. The Brinkman and Forchheimer momentum terms are seen to only affect the Nusselt number and wavenumber for extreme values. Similar numerical simulations of an infinite horizontal layer were performed that employed the entropy generation rate and generalized excess entropy production stability functional theories to determine the wavenumber for the system. Comparisons with the Nusselt number and wavenumber dependence on the porous media Rayleigh number from the porous box simulations of this work indicate the best agreement with the entropy generation rate theory.