Convection Flip in the Rayleigh-Bénard Problem in a Small System

Abstract

Molecular dynamics study of the Rayleigh-Benard system was explored about ten years ago by Mareschal et al. 1), 3) and by Rapaport. 2) The former researchers demonstrated that fully developed convective rolls could be observed even in a small system made up of a few thousands of molecules. In the present study, we focus on an unsteady behavior of convective roll observed near the conventional critical point. Our system consists of 1600 hard disks confined to a square box at a number density of 0.2. The temperatures of the upper and the lower walls are T0(1 − ∆T/2) [K] and T0(1 +∆T/2) [K], respectively, and a gravitational field is introduced so that a disk moving between these two walls experiences a zero net energy change. Initially, the disks are uniformly spaced and their velocities are sampled from the Maxwellian distributions with linearly interpolated temperatures between the two thermal walls. Disks are assumed to be reflected specularly on the walls; however, thermal effect is taken into account to the normal component of the velocity departing from the upper and the lower walls. A series of experiments was performed by changing the value of ∆T from 0.2 to 1.8. The square box is devided into small cells and the variations of local velocity in each of these cells are successively measured by taking an average in an appropriate time scale. General observations are summarized as follows. Spontaneous single-roll structure clearly appears when ∆T > 1; at first, both clockwise and counterclockwise convections are observed irregularly. As ∆T increases, the average lifetime of a convective roll pattern, τroll, is prolonged and finally no flip motion occurs on an observational time scale τobs. We realize that there is a short range of ∆T , instead of a distinct critical point, between no-roll and fully developed single-roll states. To characterize the state in this transitional phase, we introduce two parameters. Let M∗ be the sum of nondimensional angular momentum of all disks around the center of the box. This is, so to speak, an order parameter which describes the transformation from structureless no-roll (disordered) state to fully developed singleroll (ordered) state. Using this, we introduce new parameters M1 and M2 such that