Rayleigh–Bénard flow of a rarefied gas and its attractors. I. Convection regime

Abstract

In this paper we investigate the long time behavior (final state) of the Rayleigh–Bénard (RB) flow of a rarefied monatomic gas for a set of the nondimensional Knudsen and Froude numbers in the intervals Kn∈[1.0×10−3,4×10−2], Fr∈[1.0×10−1,1.5×103]. For the most part of the computations the third nondimensional parameter, the ratio of the cold and hot wall temperatures is fixed to Tc/Th=0.1, corresponding to a large temperature difference (Th serves as reference temperature), for which the RB system is believed to reach most of the possible final states (attractors). The low Knudsen numbers allow the problem to be investigated numerically by using two completely different methods: direct simulation Monte Carlo (DSMC) method (molecular approach) and finite difference (FD) method (continuum approach based on the model of compressible viscous heat conducting gas with state-dependent transport coefficients). As a result the effect of rarefaction on the onset of convection in the two-dimensional case is studied and the zone of convection is delineated. The gravitational instability of the RB system is analyzed by using the exact solution of pure-conduction state. As a result two analytical conditions for large and small Froude numbers, respectively, are suggested as exterior bounds of the zone of convection. Both DSMC and FD calculations, which are in good agreement in the considered set of the governing parameters, confirm the validity of these estimations. The observed hysteresis loops between the found out co-existing attractors within the zone of convection is another finding of the presented study. The results for the lowest Knudsen number considered (Kn=0.001), where a set of qualitatively new regimes and final states have been found, are presented separately in paper II.