Langevin equation elucidates the mechanism of the Rayleigh-Bénard instability by coupling molecular motions and macroscopic fluctuations

It is well known that Brownian motion can be described using Langevin equation. In this paper we extend the application of the Langevin equation to the Rayleigh-Bénard (RB) flow, assuming that each molecule in the system is a Brownian particle colliding with its surrounding molecules. The phenomenon of thermal instability, changing from a conductive to a convective state, is well reproduced by Langevin dynamics simulations. The roles of the drag force and the random force terms in the Langevin equation in triggering thermal instability are elucidated via numerical tests. Furthermore, we demonstrate that the strength of the fluctuation correlations increases as the Rayleigh number approaches the critical value, and the characteristics of the fluctuation correlations below the onset of instability foreshadow the form of the convective patterns emerging above the critical point. The Langevin equation, together with the form of the fluctuation correlations, sheds new light on the mechanism of the RB instability.

Figure 1

The temporal evolution of the heat flux through the entire lower plate at various Ra.

Figure 2

Temperature and density fields for $\text{Ra}=1600$ and $\text{Ra}=4000$: (a) temperature field for $\text{Ra}=1600$, (b) density field for $\text{Ra}=1600$, (c) temperature field for $\text{Ra}=4000$, and (d) density field for $\text{Ra}=4000$. Velocity vectors are also shown when a convection pattern is formed at $\text{Ra}=4000$.

Figure 3

Relation between $\mathrm{Nu}\times \mathrm{Ra}$ and Ra. The current results obtained by Langevin dynamics simulations are compared with theoretical and experimental results.

Figure 4

Langevin simulation results of (a) the power spectrum and (b) the correlation function of the density fluctuations in the horizontal direction below the onset of the RB instability. For comparison, the DSMC result of $\text{Ra}=1400$ under the same condition as the Langevin simulation is presented.