Boundary and interior layers in turbulent thermal convection in cylindrical containers

Turbulent convection of fluids heated from below and cooled from above, which is known in literature as Rayleigh–Bénard convection (RBC) [1]–[2], is one of the classical problems in fluid dynamics. The most interesting examples of this process are convection in atmospheres, in oceans, on surfaces of stars. There are three main parameters characterizing fluid motion in RBC: the Rayleigh number Ra = αgH3∆T/(κν), the Prandtl number Pr = ν/κ and the aspect ratio of the container Γ = D/H. Here α denotes the thermal expansion coefficient, g the gravitational acceleration, ∆T the temperature difference between the bottom and the top plates, κ the thermal diffusivity, ν the kinematic viscosity and H the height and D the diameter of the Rayleigh cell. Numerous scientifical and industrial problems require a better understanding of the physics of Rayleigh– Bénard convection for the Rayleigh numbers up to 1020 and large aspect ratios. The diffusion coefficients in the governing Navier–Stokes and the heat equations, which are inversely proportional to the square root of Ra, are very small. (For example, for 105 ≤ Ra ≤ 109, Pr = 0.7 and Γ = 10 the diffusion coefficient in the Navier-Stokes equation varies from 8.36 · 10−5 to 8.36 · 10−7.) Therefore the solutions – both the temperature and the velocity fields – have very thin boundary layers near the horizontal walls and interior layers which are recognized as large coherent structures in vizualizations of the flows. Experimental studies of RBC show that above but close to the onset of convection (Ra ≈ 1.7×103) visible flow patterns reflect straight rolls with certain defects induced by the sidewalls [1]. Further above the onset for Pr ≤ 1 the spiral–defect chaos evolves [3] and for Ra ≈ 6.8×103 hexagon patterns occur [4]. Both types of polygon convection cells – those with rising (l-cells) and those with descending (g-cells) motion in the center – can coexist with the spirals [5]. When the Rayleigh number exceeds a value of order 104 the spoke patterns [6] evolve, which tend to be nearly stationary for lower Ra close to the onset of this type of convection and appear chaotically when the Rayleigh number exceeds 105. A further increase of Ra tears off unstable spokes to form more independent large scale flow structures, which are called thermal plumes and are generated from the horizontal thermal boundary layers and driven by buoyancy. The thermal plumes play an important role in the moderate-Rayleigh-number regime that begins at Ra = 105. Close to Ra = 108 the large-scale circulation initiated by thermal plumes develops [7]. The thermal boundary layers, the borders of coherent interior flow structures and the turbulent background are indicated, respectively, by high, moderate and small values of the temperature gradient norm and, hence, by large, moderate and small values of the thermal dissipation rate. In contrast to experimental studies of thermal convection, in which only restricted information is available, numerical simulations enable to investigate quantitatively the boundary and interior layers and turbulent background, since they provide all details of the flow fields needed to calculate the dissipation rates.