Use of the finite-element method for natural convection in a horizontally confined infinite layer of fluid

A finite element method is used to compute the natural convection in a horizontally confined, infinite layer of fluid, heated from below and cooled from above. The behavior of the fluid is modelled in terms of a two-dimensional roll cell with a square cross section. Three different sizes of triangular elements are tested for Pr = 10 and a range of Rayleigh numbers from the critical value up to 3000. The Nusselt numbers obtained by extrapolation to zero element size are found to agree well with the experimental values of Silveston (1958). The critical Rayleigh numbers computed for the three element sizes are found to be in close agreement with the theoretical value of 1708. These results indicate that the truncation error for finite element computations is significant and that computations for two or more element sizes, followed by extrapolation to zero element size, are essential.