We report a comprehensive study of the mean velocity, mean temperature, and temperature variance profiles in turbulent Rayleigh-Bénard convection (RBC) using direct numerical simulation (DNS). This study is conducted with the Prandtl numbers (Pr) varied from 0.17 to 4.4 and the Rayleigh numbers (Ra) varied between 5 x 10⁸ and 1 x 10¹⁰ in a vertical thin disk with the aspect ratios Γ (thickness over height) of 0.1 and 0.2.

First, we introduced the open-source spectral element method solver Nek5000 and performed a detailed test of mesh resolution for turbulent RBC at Ra = 5 x 10⁹ and Pr = 4.4 in a vertical thin disk with Γ = 0.1. The numerically calculated mean temperature profile agrees well with the experimental data and theoretical prediction. Based on the theory for the mean tempera...[ Read more ]

We report a comprehensive study of the mean velocity, mean temperature, and temperature variance profiles in turbulent Rayleigh-Bénard convection (RBC) using direct numerical simulation (DNS). This study is conducted with the Prandtl numbers (Pr) varied from 0.17 to 4.4 and the Rayleigh numbers (Ra) varied between 5 x 10⁸ and 1 x 10¹⁰ in a vertical thin disk with the aspect ratios Γ (thickness over height) of 0.1 and 0.2.

First, we introduced the open-source spectral element method solver Nek5000 and performed a detailed test of mesh resolution for turbulent RBC at Ra = 5 x 10⁹ and Pr = 4.4 in a vertical thin disk with Γ = 0.1. The numerically calculated mean temperature profile agrees well with the experimental data and theoretical prediction. Based on the theory for the mean temperature profile, we studied the temperature variance profile in the boundary layer (BL) region and mixing zone. We numerically examined the budget terms in the temperature variance equation, which are difficult to measure in experiment. Excellent agreement between the numerical data and theoretical prediction was obtained, which provides a strong support for the modeling in the BL region. In the mixing zone, our DNS data shows a simple two-term between turbulent convection and thermal dissipation. A power law solution for the temperature variance profile is obtained.

When the value of Pr is reduced from 4.4 to 0.17 at Ra = 5 x 10⁹ and Γ = 0.1, the rotation speed of the large-scale circulation (LSC) becomes faster, and the flow region becomes narrower, moving toward the near-wall region. The instantaneous flow field becomes progressively more coherent with a dominant mean flow. The LSC changes from a rigid-body rotation to a near-wall turbulent jet. With fixed Pr = 0.17 and varying Ra from 5 x 10⁸ to 1 x 10¹⁰ in a thin disk with Γ = 0.2, we identified two flow regions with increasing distance from the bottom to the center of the cell, a BL region where molecular diffusion is balanced by turbulent diffusion, and a bulk region where mean convection and turbulent diffusion balance with each other. Within the viscous BL, the turbulent viscosity follows a cubic power law with distance. The velocity profile within the BL can be determined analytically. Outside the viscous BL, the turbulent viscosity deviates from the cubic power law. The velocity profile can be solved by numerical integration of a formal solution with a piecewise fit to turbulent viscosity. For the thermal BL, the turbulent thermal diffusivity is dictated by the viscous BL. Within the viscous BL, which lies beneath the thermal BL, the turbulent diffusivity increases with distance following a cubic power law. Outside the viscous BL, deviations from the cubic power law are observed. The mean temperature profile can be solved in the same manner as the mean velocity profile. Mathematically, the strong shearing by the underneath viscous BL provides a slip boundary condition for the turbulent heat flux, which yields a reduced power law exponent between 2 and 3 for the turbulent diffusivity. Furthermore, we considered the buoyancy effect on the turbulent heat flux far away from the viscous BL and proposed a modified turbulent diffusivity model. This model included the effect of thermal plumes and explained quantitatively the deviations of turbulent diffusivity from the cubic power law outside the viscous BL. Because of the relatively faster LSC and shorter plume lifetime, the thermal plumes have a smaller chance to survive in the bulk region and thus the LSC is confined in the near-wall region. This gives rise to a strong near-wall flow. Based on the two-term balance behavior and a constant turbulent viscosity, we recognized the LSC as a turbulent jet flow and obtained an analytical self-similarity solution for the mean velocity profile.

Finally, we introduced the concept of dissipation time for the temperature variance profile and observed a transition of the scaling behavior of the dissipation time at the edge of the viscous BL. We predicted the scaling exponents based on two physical processes: random diffusion within the viscous BL and buoyancy-enhanced diffusion outside the viscous BL. We solved the temperature variance profile analytically within the viscous BL and in the mixing zone. In the buffer layer between the viscous BL and the mixing zone, the temperature variance profile is solved by numerical integration. Our work thus provided a complete and self-consistent framework for understanding the effect of the viscous BL on the mean and variance temperature profiles, and the large-scale flow dynamics in turbulent RBC in the low-Pr regime.