Plane-layer Rayleigh-B\'enard convection up to $Ra=10^{11}$: Near-wall fluctuations and role of initial conditions

Abstract

We study turbulent Rayleigh-B\'enard convection through direct numerical simulations in a three-dimensional plane layer of aspect ratio 4 for Rayleigh numbers $10^5 \leq Ra \leq 10^{11}$ and Prandtl number $Pr=0.7$. We summarize the height-dependent statistics of velocity and temperature fluctuations and corresponding scalings with the Rayleigh number. We include an analysis on the role of coherent and incoherent flow regions near the wall for global heat transfer. Furthermore, we investigate the dependence of turbulent transport on a finite-amplitude sinusoidal shear flow added at time $t=0$, which either freely decays in a long transient or remains existent when a steady sinusoidal volume forcing is added. In the latter case, weak logarithmic near-wall layers are formed, however, with von K\'arm\'an and offset constants that differ from standard values. The typical magnitude of both coefficients, and thus a full turbulent boundary layer of velocity and temperature, is re-established only for a switch from sinusoidal to constant pressure gradient driving of the flow. In all cases, except for the constant pressure gradient-driven flow, no enhancement of global turbulent heat and momentum transfer within error bars is detected, even though the sinusoidal amplitude is of the order of the characteristic free-fall velocity.