Two-dimensional Darcy–Bénard convection evolving in Fourier space

Abstract

Nonlinear transient convection in a porous rectangle heated from below is studied by an analytically based method. Fourier series for the temperature and streamfunction are applied, where each Fourier coefficient evolves in time according to a coupled set of ordinary differential equations. The mathematical method can be considered as a recursive nonlinear mapping in Fourier space from a given state at a time t to an updated state at time t + dt, with an infinitesimal time increment. This nonlinear evolution in Fourier space requires normal-mode compatible boundary conditions along the entire boundary. Each Fourier coefficient gets three contributions during its updating in time: (i) one decay term due to thermal diffusion, governed by linear theory; (ii) one growth term due to buoyancy, governed by linear theory; and (iii) quadratic nonlinearities from the convection term in the heat equation, involving all pairwise interactions between the Fourier modes. We present numerical computations with a standard Runge–Kutta method, with Rayleigh numbers up to ten times the well-known critical value [Formula: see text]. Our plots are produced with Fourier series truncated to include 15 normal modes in both the horizontal and vertical directions. For validation of our method, we present tables for the Nusselt number of steady convection, with a higher number (up to 20) of normal modes included in the truncated system of equations. Our computations of transient nonlinear convection lead to steady states. A final steady state is not unique for a given geometry, but depends on the initial state and the Rayleigh number. This ambiguity of steady states is depicted by a hysteresis loop. The Malkus hypothesis of maximal heat transfer is put into perspective. This hypothesis does not pick a preferred cell width, but it nevertheless constrains the hysteresis loop.