Large departures from Boussinesq approximation in the Rayleigh–Bénard problem

Abstract

Flows involving natural convection are generally studied employing the Boussinesq approximation of the Navier–Stokes equations. This paper deals with large departures from this limit by considering the two-dimensional, periodic Rayleigh–Bénard problem as an example. The main effect investigated here is a strong variation in the density considered primarily as a function of temperature as it is accounted for by the low Mach number equations. Additionally, the effect of temperature-dependent viscosity and heat conductivity is considered. Both types of departure are measured by the nondimensional temperature gradient. They destroy the midplane symmetry of the problem and lead to quantitative and qualitative changes of the flow, in particular in the bifurcation from the conduction state. The present study illustrates how three different approaches—by linear stability analysis, weakly nonlinear analysis, and direct simulation—permit the investigation of complementary aspects of the problem. The first is used to determine the critical Rayleigh number for the onset of convection. The second is well suited for investigating the nature of the bifurcation and shows that the flow in the form of rolls becomes subcritical if the non-Boussinesq character is strong enough. Direct simulations of the flow field confirm this behavior and permit the study of situations far from criticality. It turns out that non-Boussinesq effects are more and more confined to small parts of the domain as the Rayleigh number increases, so that their influence on mean properties of the flow diminishes.