An improved method of Nusselt number calculation

Summary. Accurate computation of heat flux is a primary motivation for convection calculations, yet heat fluxes have generally been calculated to substantially less than optimal accuracy, given the accompanying temperature and velocity fields. We analyse this problem in the context of the finite element method, using bilinear shape functions to interpolate the nodal values of velocity and temperature throughout the medium. Our method, however, is general and also applies to finite difference and higher order finite elements. Consider (dimensionless) vertical heat flux qz = wT- dT/dz, with w vertical velocity and T temperature. The first term is quadratic within (bilinear) elements and continuous at nodes, while the second term is constant within elements and discontinuous at nodes. Usual practice is to evaluate qz at element centres. We instead assign to each element the volume average of qz over that element. Element values are then projected to nodes using a modification of a technique used for pressure smoothing (Hughes et al.) Our approach reduces the calculated depth variation in horizontally-averaged flux by more than a factor of 10 and shows more rapid convergence of Nusselt number Nu as a function of grid size. It also eliminates the artificial diffusion of heat out of supposed insulating side walls. Key words: Nusselt number, convection, heat flux, finite element method