Aspect ratio dependence of heat transfer and large-scale flow in turbulent convection

Abstract

The heat transport and corresponding changes in the large-scale circulation (LSC) in turbulent Rayleigh-Bénard convection are studied by means of three-dimensional direct numerical simulations as a function of the aspect ratio Γ of a closed cylindrical cell and the Rayleigh number Ra. The Prandtl number is P r = 0.7 throughout the study. The aspect ratio Γ is varied between 0.5 and 12 for a Rayleigh number range between 10 7 and 10 9 . The Nusselt number N u is the dimensionless measure of the global turbulent heat transfer. For small and moderate aspect ratios, the global heat transfer law N u = A × Ra β shows a power law dependence of both fit coefficients A and β on the aspect ratio. A minimum of N u(Γ) is found at Γ ≈ 2.5 and Γ ≈ 2.25 for Ra = 10 7 and Ra = 10 8 , respectively. This is the point where the LSC undergoes a transition from a single-roll to a doubleroll pattern. With increasing aspect ratio, we detect complex multi-roll LSC configurations in the convection cell. For larger aspect ratios Γ > ∼ 8, our data indicate that the heat transfer becomes independent of the aspect ratio of the cylindrical cell. The aspect ratio dependence of the turbulent heat transfer for small and moderate Γ is in line with a varying amount of energy contained in the LSC, as quantified by the Karhunen-Loève or Proper Orthogonal Decomposition (POD) analysis of the turbulent convection field. The POD analysis is conducted here by the snapshot method for at least 100 independent realizations of the turbulent fields. The primary POD mode, which replicates the time-averaged LSC patterns, transports about 50% of the global heat for Γ ≥ 1. The snapshot analysis enables a systematic disentanglement of the contributions of POD modes to the global turbulent heat transfer. Although the smallest scale -the Kolmogorov scale ηK -and the largest scale -the cell height H -are widely separated in a turbulent flow field, the LSC patterns in fully turbulent fields exhibit strikingly similar texture to those in the weakly nonlinear regime right above the onset of convection. Pentagonal or hexagonal circulation cells are observed preferentially if the aspect ratio is sufficiently large (Γ > ∼ 8).

New perspectives in turbulent Rayleigh-Bénard convection

“…When convection sets in, plumes of the same type (cold or hot) have the tendency to cluster giving rise to a large-scale flow [84]. It has been found in experiments [85] and DNS [27] that the morphology of the large-scale circulation depends on the form of the cell and the aspect ratio, particularly at intermediate Rayleigh numbers. For example in a cell with aspect ratio of one essentially one LSC roll is expected.…”

“…In an extension of the theory [32] a full functional dependence for the relations Nu(Ra, P r) and Re(Ra, P r) was suggested that incorporates the varying ratio of both BL thicknesses with respect to each other when the Prandtl number is changed, i.e., with (11) and (27) …”

“…The coefficients c 1 to c 4 and the cut-off Reynolds number Re c follow when experimental data are matched by a nonlinear fit. An additional sixth fit parameter is the prefactor a from (27). The resulting Reynolds number dependence was further discussed in [33].…”

“…In refs. [27,77,78] new criteria have been suggested which incorporate the strong vertical variation of gradients and thus of the kinetic energy dissipation rate. Based on Blasius theory the following estimates for 10 6 ≤ Ra ≤ 10…”

“…In case of the Earth mantle and solar convection the circumferences at mid height have been taken for L. Vapor, water and ice; Anelastic limit Solar convection zone [12,13] hot at the bottom for T /ΔT = 0.7 in red, cold at the top for T /ΔT = 0.3 in blue. Parameters are here: Ra = 10 9 , P r = 0.7 and Γ = 3 (see [26,27] for the corresponding simulations).…”