h‐adaptive finite element solution of high Rayleigh number thermally driven cavity problem

An h‐adaptive finite element code for solving coupled Navier‐Stokes and energy equations is used to solve the thermally driven cavity problem. The buoyancy forces are represented using the Boussinesq approximation. The problem is characterised by very thin boundary layers at high values of Rayleigh number (>10⁶). However, steady state solutions are achievable with adequate discretisation. This is where the auto‐adaptive finite element method provides a powerful means of achieving optimal solutions without having to pre‐define a mesh, which may be either inadequate or too expensive. Steady state and transient results are given for six different Rayleigh numbers in the range 10³ to 10⁸ for a Prandtl number of 0.71. The use of h‐adaptivity, based on a posteriori error estimation, is found to ensure a very accurate problem solution at a reasonable computational cost.