Analysis and modelling of the turbulent diffusion of turbulent heat fluxes in natural convection

In stably and unstably stratified fluid layers there are often highly anisotropic and counter gradient heat fluxes occurring. Standard heat flux models as the isotropic k–ε–σt model need to be improved for representing such behaviour. More complex algebraic models or even in some cases the full transport equations for the turbulent heat fluxes are therefore required. There, a triple correlation appears as an important closure term in the turbulent diffusion. Usually, this is modelled following Daly and Harlow, which has already been found to be not sufficiently accurate in buoyant flows. In this paper, some of the salient features of an internally heated fluid layer (IHL) and of Rayleigh–Bénard convection (RBC) are discussed basing on direct numerical simulation (DNS) data. In IHL a counter gradient heat flux occurs over a wide region. The transport equation for the triple correlation is analyzed using the DNS data. Based on this study a Reynolds-Averaged Navier Stokes (RANS) model for this closure term is derived which covers the influence of the fluid Prandtl number (Pr) and of buoyancy. The model is validated using the DNS data of both RBC and IHL for different Rayleigh and Prandtl numbers.