Turbulent Heat Flux and Temperature Variance Dissipation Rate in Natural Convection in Lead-Bismuth

Results of a direct numerical simulation (DNS) for Rayleigh-Bénard convection for the Rayleigh number Ra = 10⁵ in a fluid with the Prandtl number Pr = 0.025, which corresponds to liquid lead-bismuth, are used to analyze the turbulent heat flux and the temperature variance dissipation rate. The results indicate that application of a thermal or a mixed timescale may considerably improve gradient diffusion and algebraic heat flux models at these Rayleigh and Prandtl numbers. Therefore, a good approximation of the temperature variance dissipation rate is required. The standard temperature variance dissipation rate model is investigated using the DNS results. The analysis of the standard model shows the importance of wall functions and qualitatively good predictions by the model for this type of flow. Quantitatively, the model overpredicts the temperature variance dissipation rate evaluated from the results of DNS by ~25%. The two-point correlation method is used to derive new models for the temperature variance dissipation rate. Comparison with DNS results shows qualitatively and quantitatively good predictions by the new models. These new models lead therefore to an increased accuracy of the turbulent heat flux models for this type of flow.