The Bénard problem for a rarefied gas: Formation of steady flow patterns and stability of array of rolls

Abstract

The two-dimensional Bénard problem for a rarefied gas in a rectangular domain is studied numerically on the basis of kinetic theory. To be more specific, the instability of a stationary stratified gas is investigated by a finite-difference analysis of the Boltzmann–Krook–Welander equation (the so-called BGK model) with the diffuse reflection condition on the top (cooled) and bottom (heated) boundaries and the specular reflection condition on the side boundaries. The study is a continuation of the previous paper by the present authors with H. Motohashi [in Rarefied Gas Dynamics, edited by J. Harvey and G. Lord (Oxford U.P., Oxford, 1995), Vol. 1, p. 135], where the parameter range for which a steady convection exists was clarified, and some examples of steady flow patterns as well as those of flow bifurcation were presented. In the present paper, the attention is focused on the formation of steady flow patterns and the stability of an array of rolls. In particular, the effects of the initial condition and the aspect ratio of the domain on the types of steady convection as well as on the process of its formation and the stability of the double-roll solutions are investigated closely. The existence range of a steady convection is also discussed with some new examples of flow patterns supplementing the previous paper.

On the Rayleigh–Bénard problem in the continuum limit

“…Periodicity 14 or specular reflection [10][11][12][13]15,16 conditions are imposed at the vertical planar boundaries ͑perpendicular to the walls at x =0,L͒. The finite domain together with the boundary conditions prescribed result in a discrete spectrum of perturbation wavelengths.…”

“…Consequently, they noted that the resulting problem could not be completely characterized in terms of the Rayleigh number. Sone and co-workers 15,16 considered the corresponding problem for a Bhatnagar-Gross-Krook model equation. Making use of a finite-difference scheme, they studied the effects of the Knudsen ͑Kn͒ and Froude ͑Fr͒ numbers, the temperature ratio and the geometry of the rectangular domain occupied by the gas.…”

A new iterative wall heat flux specifying technique in DSMC for heating/cooling simulations of MEMS/NEMS

“…(2) in the resulting equation. It yields: U new j;out ¼ U old j;out þ À P j;out À P out Á r j;out a j;out (5) The inlet velocity is also calculated in the same way, as follows:…”

Stability of thermal transpiration flows in rectangular enclosures – Axisymmetric disturbances

“…Instabilities of the former flow is primarily induced by buoyancy forces resulting from a temperature gradient between bottom and top boundaries of a closed rectangular geometry. This phenomenon has been extensively studied by many researchers under the heading of Rayleigh-Bénard (RB) instability [2][3][4]. Stefanov et al [5,6] have investigated long-time behaviour of the RB convection of rarefied gases for varying Knudsen (1 × 10 -3 b Kn b 4 × 10 − 2 ) and Froude (1 × 10 − 3 b Fr b 1.5 × 10 3 ) numbers.…”