Flow pattern and heat transfer rate in Rayleigh–Bénard convection

The three-dimensional Rayleigh–Bénard convection is simulated numerically using the lattice Boltzmann method. Flow patterns are observed and the heat transfer rate is estimated in terms of the Nusselt number. The dependence of the Nusselt number on the Rayleigh number is shown to agree well with that obtained by the two-dimensional calculations of the Navier–Stokes equations. It is shown that several roll patterns with different wave numbers and heat transfer rates are established even though the ratio of the horizontal size to the vertical size is a multiple of 2. Two types of oscillatory roll patterns are shown: one is with oscillatory heat transfer rate and the other is with the constant heat transfer rate. It is found that the square pattern is possible under the same condition for the stable or oscillatory roll pattern. The heat transfer rate decreases with decreasing wave number.