Abstract
Rayleigh-Bénard convection (RBC) is a classic model for thermal convection which plays an important role in nature and industry. We conducted a comprehensive study of the mean velocity, mean temperature, and temperature variance profiles in turbulent RBC in a vertical thin disk by direct numerical simulation. The study considers the Prandtl numbers (Pr) ranging from 0.17 to 4.4 and the Rayleigh numbers (Ra) varying between 5×108 and 1×1010 with aspect ratios (thickness over height) of 0.1 and 0.2. First, we introduce the open-source high-accuracy program Nek5000 and perform a systematic test of the simulation for turbulent RBC. The flow properties calculated numerically are in good agreement with the experimental measurements. As we fix Ra at 5×109 with an aspect ratio of 0.1 and lower Pr from 4.4 to 0.17, the instantaneous flow field becomes progressively more coherent with the mean flow. The large-sale circulation (LSC) changes from a rigid-body rotation to a near-wall turbulent jet. We fix Pr at 0.17 and vary Ra from 5×108 to 1×1010 with an aspect ratio of 0.2. Two regions are identified, a boundary layer (BL) region where turbulent diffusion is balanced by molecular diffusion, and a bulk region where turbulent diffusion is balanced by mean convection. In low-Pr RBC, thermal BL is dictated by the viscous BL. Within the viscous BL which lies beneath the thermal BL, turbulent diffusivity grows according to an exact cubic power-law. Outside the viscous BL, it shows systematic deviation from the cubic power-law. In the bulk region, due to the faster LSC and shorter plume lifetime, there is less chance for plumes to survive in the bulk and the LSC is only accelerated on the edge. This gives rise to form a strong near-wall turbulent jet flow. Last, we constructed our modified model for turbulent diffusivity by considering the buoyancy effect on turbulent fluctuations, and explained the behavior of the turbulent diffusivity outside the viscous BL. In addition, we proposed a model for the thermal dissipation rate by charactering dissipation time scale. Based the modeling, the whole temperature variance profile is solved.
Keywords: Rayleigh-Bénard convection, boundary layer, turbulent fluctuations, near-wall jet
