Consistent calculation of heat fluxes through solid boundaries applied to natural convection problems

Calculations of heat fluxes through fixed temperature walls from CFD results have become of great importance in many industrial applications. The objective of this work is to present a consistent numerical technique to compute heat fluxes through isothermal boundaries. In the present paper, we consider a stabilized PSPG/SUPG finite element scheme for the steady Navier-Stokes equations for variable density flows. Three variants are considered which differ by the treatment of the convective terms in the momentum and energy equations, i.e. a convective formulation, a corrected convective formulation and a conservative formulation. A pseudo Newton method is employed as non linear solver. A numerical technique to compute the boundary heat fluxes consistent with the finite element formulation is then presented, as well as the expression obtained using the gradient of the finite element approximation T h. To illustrate the effect of the formulation, numerical simulations of natural convection of air in 2D and 3D cubic cavities with large temperature differences between opposite walls are carried out. The effects of the finite element formulation, of the expression for the calculation of the heat flux and of mesh refinement are presented. The results demonstrate the superior accuracy and convergence of the proposed numerical technique for the heat flux computation. Keywords— Finite Elements,Consistent Calculations,Heat Fluxes