Figures and the mpeg animation below show how the Rayleigh-Benard instability develops in a rectangular domain (0.06 x 0.01 m^2) where the bottom boundary is kept at 0.5 degrees higher temperature than the upper boundary. The material parameters used in this simulation are those of water. Incompressible flow equations and heat equation are coupled by the density variations due to temperature and transport of heat by convection.

Initially the fluid was at rest and at constant temperature.

The ElmerSolver uses a stabilized finite element formulation to solve the incompressible Navier-Stokes equations and the heat equation with the convection term. In this simulation bilinear quadrilateral finite elements were used. The time integration method used was second order BDF (Backward Differentiation Fourmulaes) with time step size of two seconds. The coupled system within timesteps was solved by the method of sequential iteration.

Here you may load an animation of the development of the velocity field mpeg: [~600 Kb]

If you want to reproduce the results, download the file RayleighBenard.tar.gz and follow the instructions in the file README. You should have the Elmer package installed in your computer, however.

Temperature and velocity fields at 0 s.

Temperature and velocity fields at 100 s.

Temperature and velocity fields at 200 s.

Temperature and velocity fields at 300 s.

Temperature and velocity fields at 400 s.