The Attractors of the Rayleigh-Bénard Flow of a Rarefied Gas S.Stefanov ∗ , V. Roussinov ∗ and C. Cercignani † ∗ Inst. of Mech., Bulg. Acad. of Sciences, Sofia, Bulgaria † Dipart. di Matematica, Polit. di Milano, Italy Abstract. We have studied the Rayleigh-Bénard flow of a rarefied gas for Kn ∈ [1.0 × 10 −3 , 4 × 10 −2 ] , Fr ∈ [1.0 × 10 −1 , 1.5 × 10 3 ] and a fixed temperature ratio T c /T h = 0.1. The calculations are performed by both the DSMC and the numerical solution of Navier-Stokes equations, with a remarkable agreement between the methods. We exhibit chaotic behavior and also a hysteresis cycle. INTRODUCTION The formation and development of convection flows in a fluid confined between two horizontal parallel plates with the bottom plate heated from below is a classical problem known in hydrodynamics as the Rayleigh-Bénard problem. The first numerical calculations of the RB convection of a rarefied gas performed either by the DSMC method ([1, 2]) or by solving the BGK equation [3] show that the transition from pure conduction to convection occurs for temperature gradients larger than a certain critical value and for sufficiently low Knudsen numbers . A variety of studies [2, 3, 4, 5, 6, 7] report about a stable vortex formation for Knudsen numbers Kn (= ℓ 0 /L)= 0.01 − 0.05 and various magnitudes of temperature ratio and Froude number Fr (= V 2 th /gL). Here g is the acceleration of gravity, V th the most probable molecular speed, L the distance between the hot (with temperature T h ) and cold (with temperature T c ) plates. These investigations can be separated into two groups according on the aim pursued by the authors. In the first group of articles [1, 4, 5] the authors try to eliminate the density gradient by choosing the acceleration of gravity g consistent with constant density in the convection-free (pure conduction) solution. Thus, the flow is assumed to be very close to the Boussinesq approximation conditions and, consequently, the Rayleigh number Ra (in this case, the unique non-dimensional parameter determining the transition from pure conduction to a convection flow) can be used and compared with the critical value Ra c (for no-slip boundary conditions Ra c = 1708) obtained by linear stability analysis of the Oberbek-Boussinesq equation. In this approach, however, the choice of the governing parameters is restricted. The authors of the second group of articles[2, 3, 6, 8] consider the problem for a set of freely varying independent parameters and investigate the effect of gas stratification (a natural formulation for a rarefied gas). Thus, the density of the purely conduction state might increase when moving toward the cold plate in the case of weak gravity, increase when moving toward the hot plate for strong gravity or, as shown further in the paper, be non-monotonic for some intermediate values. All these papers show that for such conditions the onset of instability cannot be determined by a single non-dimensional parameter Ra. In the present research we investigated numerically the RB flow for a set of the non-dimensional parameters in the intervals Kn ∈ [1.0 × 10 −3 , 4 × 10 −2 ] , Fr ∈ [1.0 × 10 −1 , 1.5 × 10 3 ]. For most of the computations the third non- dimensional parameter, the temperature ratio was fixed to T c /T h = 0.1, corresponding to a large temperature difference ( T h serves as reference temperature), for which the RB system is believed to reach most of the possible final states (attractors). © 2003 American Institute of Physics 0-7354-0124-1/03/$20.00 CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz 194