Rayleigh number scaling in numerical convection

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Abstract
Using direct simulations of the incompressible Navier-Stokes equations with rigid upper and lower boundaries at fixed temperature and periodic sidewalls, scaling with respect to Rayleigh number is determined. At large aspect ratio (6:6:1) on meshes up to 288 × 288 × 96, a single scaling regime consistent with the properties of ‘hard’ convective turbulence is found for Pr = 0.7 between Ra = 5 × 104 and Ra = 2 × 107. The properties of this regime include Nu ∼ RaβT with βT = 0.28 ≈ 2/7, exponential temperature distributions in the centre of the cell, and velocity and temperature scales consistent with experimental measurements. Two velocity boundary-layer thicknesses are identified, one outside the thermal boundary layer that scales as Ra−1/7 and the other within it that scales as Ra−3/7. Large-scale shears are not observed; instead, strong local boundary-layer shears are observed in regions between incoming plumes and an outgoing network of buoyant sheets. At the highest Rayleigh number, there is a decade where the energy spectra are close to k−5/3 and temperature variance spectra are noticeably less steep. It is argued that taken together this is good evidence for ‘hard’ turbulence, even if individually each of these properties might have alternative explanations.