Analysis of thermal dissipation rates in turbulent Rayleigh–Bénard convection

Abstract

Direct numerical simulations (DNS) of turbulent Rayleigh–Bénard convection in a wide cylindrical container (aspect ratio $\Gamma\,{=}\,10$) with a lateral wall have been performed for the first time for the Rayleigh numbers $10^5$, $10^6$ and $10^7$ and Prandtl number $Pr\,{=}\,0.7$. Evaluating the thermal dissipation rates from the generated DNS data, the formation and development of the thermal plumes and their interaction are highlighted. Two new functions $\sigma$ and $\tau$ are defined to determine quantitatively the role of the turbulent background. Evaluating these functions from the DNS data, it is shown that the turbulent background pushes the thermal plumes back and that its contribution to the volume-averaged thermal dissipation rate increases with the Rayleigh number. Further, it is proven analytically that the ratio of the area-averaged (over the top or the bottom plates) to the volume-averaged thermal dissipation rate is greater than or equal to the Nusselt number for all aspect ratios, and Prandtl and Rayleigh numbers.