Direct simulation of two-dimensional Bénard flow with free-slip boundary conditions

Abstract

A fourth-order finite difference algorithm is developed for the direct simulation of two-dimensional Rayleigh-Bénard convection in a horizontally periodic domain of aspect ratio Γ. The free-slip condition prevents the formation of kinetic boundary layers and allow the implementation of an efficient long-stencil scheme for the vorticity equation without additional risk of instability. The required grid size to properly resolve Batchelor’s microscale and therefore avoid aliasing is expressed in terms of the Rayleigh (Ra) and Prandtl (Pr) numbers. It was verified that the method was able to reproduce the Nusselt and Reynolds number scalings as well as the different flow regimes documented in the literature for Γ=5, Pr=10, and Ra≤107. Furthermore, the analytical solution of the Poisson equation in Fourier series is derived and compared with the standard fast Poisson solver. The horizontal wavenumbers decay much slower than the vertical ones, which might be explained by the adopted domain’s aspect ratio. The results also suggest that the largest energy-containing wavenumbers scale with Ra3/8 for large enough Rayleigh numbers.