Simulation of Rayleigh-Benard convection
by finite elements method
I.A. Ermolaev, A.I. Zhbanov, V.S. Koshelev
Department of Physics, Saratov State University, Saratov, Russia
There are a significant number of research papers on small-dimensional models of Rayleigh-Benard convection in horizontal layer of liquid (gas) heated from below. Especially, many works are devoted to Lorenz system [1] which is limited by the first dimensional harmonics of speed components and zero, first and second space harmonic of temperature, is singled out from more complete system of fluid mechanics equations (Boussinesq's approach).
Curre's model [2] can be considered as a more complete model of Rayleigh's convection. It generalizes Lorenz system, having order 14, instead of 3. However, it remains unknown which of the conclusions for these small- dimensional approximations are valid for the full system of Boussinesq's equations. Some reseach papers (e.g., [3], [4], [5]) point out that even qualitative features of the solitons' behaviour strongly depend on the character of approximation, and they do not confirm (e.g., [6]) the conclusions about the sequence of bifurcations in Lorenz's and Curry's model.
This work presents the numerical solution of non-stationary task of Rayleigh-Benard's convection in the horizontal layer of air. As an initial model we used the fluid mechanics equations in Boussinesq's approach described in variables of "vorticity vector - stream function - temperature" in two-dimensional cartesian system of coordinates. The entry conditions were as follows: the conditions of the absence of current and the beginning field of temperatures. The boundary conditions could be given as Dirichlet's, Neuman's or Newton's conditions (the impedance).
The numerical method used was Galerkin's finite elements method, enabling to take into account such complicated dynamics factors as the curvature and thermal properties of borders and the presence of the internal sources of warmth, and etc.
The results of direct numerical simulation of natural convection in horizontal layer of air with heat-conductives borders heated from below thermal flow of varying density have been obtained.