A model for the turbulent diffusion of turbulent kinetic energy in natural convection

Abstract

The widely used standard Reynolds Averaged Navier-Stokes models, e.g. 1-equation or 2-equation models, use the transport equation for the turbulent kinetic energy. They are known to be problematic in describing thermally stratified flow. In the transport equation for the turbulent kinetic energy the turbulent diffusion term is modeled with the gradient-diffusion approximation which is inadequate in internally heated fluid layers and Rayleigh-Benard convection. These flow types are explained by means of direct numerical simulation (DNS) data. The data also include a new simulation of internally heated fluid layers with Rayleigh number R a =10 9 and Prandtl number Pr = 7.0. This simulation is performed using the TURBIT code. One of the possible deficiencies in the gradient-diffusion model for the turbulent diffusion of the turbulent kinetic energy is discussed using the direct numerical simulation data. Based on this study and the investigations in meteorology, extended forms of the gradient diffusion model for the turbulent diffusion are derived. For this deduction, the different closure terms in the turbulent diffusion, namely the velocity-fluctuation triple correlation and the velocity-pressure fluctuation correlation, are modeled separately. Coupling of these models results in a Reynolds Averaged Navier Stokes model for the turbulent diffusion. In this model a variable Daly and Harlow model for the buoyancy contribution, i.e. the turbulent convection of the heat flux, is used to derive an extended Reynolds Averaged Navier Stokes model 1 for the turbulent diffusion. Based on an analysis of the transport equation for the buoyancy contribution a Daly and Harlow extended model for this term is obtained. Incorporating this extension in the model for the turbulent diffusion gives an extended Reynolds Averaged Navier Stokes model 2 for the turbulent diffusion. The modified or new models for the closure terms in the turbulent diffusion are validated using the direct numerical simulation data of internally heated fluid layers and Rayleigh-Benard convection. The Daly and Harlow extended model for the buoyancy contribution is also tested on both flow types. Also, the extended models 1 and 2 for the turbulent diffusion are analyzed and validated using the direct numerical simulation data of internally heated fluid layers. Their performance is also tested in Rayleigh-Benard convection. The model 1 shows an acceptable improvement in comparison to the gradient-diffusion model for the turbulent diffusion in internally heated fluid layers. In Rayleigh-Benard convection a small improvement is observed. The model 2 gives a slight improvement over model 1 in certain height points in these flow types. The resulting 3-equation model should lead to more accurate calculations for buoyant convection in fluid layers involving both stable and unstable stratification.