Horizontal convection in a rectangular enclosure driven by a linear temperature profile

The horizontal convection in a square enclosure driven by a linear temperature profile along the bottom boundary is investigated numerically by using a finite difference method. The Prandtl number is fixed at 4.38, and the Rayleigh number Ra ranges from 107 to 1011. The convective flow is steady at a relatively low Rayleigh number, and no thermal plume is observed, whereas it transits to be unsteady when the Rayleigh number increases beyond the critical value. The scaling law for the Nusselt number Nu changes from Rossby’s scaling Nu ∼ Ra1/5 in a steady regime to Nu ∼ Ra1/4 in an unsteady regime, which agrees well with the theoretically predicted results. Accordingly, the Reynolds number Re scaling varies from Re ∼ Ra3/11 to Re ∼ Ra2/5. The investigation on the mean flows shows that the thermal and kinetic boundary layer thickness and the mean temperature in the bulk zone decrease with the increasing Ra. The intensity of fluctuating velocity increases with the increasing Ra.

The curved kinetic boundary layer of active matter

A body submerged in active matter feels the swim pressure through a kinetic accumulation boundary layer on its surface.

Prandtl and Rayleigh number dependence of heat transport in high Rayleigh number thermal convection

Results from direct numerical simulation for three-dimensional Rayleigh–Bénard convection in samples of aspect ratio $\Gamma = 0. 23$ and $\Gamma = 1/ 2$ up to Rayleigh number $\mathit{Ra}= 2\ensuremath{\times} 1{0}^{12} $ are presented. The broad range of Prandtl numbers $0. 5\lt \mathit{Pr}\lt 10$ is considered. In contrast to some experiments, we do not see any increase in $\mathit{Nu}/ {\mathit{Ra}}^{1/ 3} $ with increasing $\mathit{Ra}$, neither due to an increasing $\mathit{Pr}$, nor due to constant heat flux boundary conditions at the bottom plate instead of constant temperature boundary conditions. Even at these very high $\mathit{Ra}$, both the thermal and kinetic boundary layer thicknesses obey Prandtl–Blasius scaling.

Scaling in thermal convection: a unifying theory

A systematic theory for the scaling of the Nusselt number Nu and of the Reynolds number Re in strong Rayleigh–Bénard convection is suggested and shown to be compatible with recent experiments. It assumes a coherent large-scale convection roll (‘wind of turbulence’) and is based on the dynamical equations both in the bulk and in the boundary layers. Several regimes are identified in the Rayleigh number Ra versus Prandtl number Pr phase space, defined by whether the boundary layer or the bulk dominates the global kinetic and thermal dissipation, respectively, and by whether the thermal or the kinetic boundary layer is thicker. The crossover between the regimes is calculated. In the regime which has most frequently been studied in experiment (Ra [lsim ] 1011) the leading terms are Nu ∼ Ra1/4Pr1/8, Re ∼ Ra1/2Pr−3/4 for Pr [lsim ] 1 and Nu ∼ Ra1/4Pr−1/12, Re ∼ Ra1/2Pr−5/6 for Pr [gsim ] 1. In most measurements these laws are modified by additive corrections from the neighbouring regimes so that the impression of a slightly larger (effective) Nu vs. Ra scaling exponent can arise. The most important of the neighbouring regimes towards large Ra are a regime with scaling Nu ∼ Ra1/2Pr1/2, Re ∼ Ra1/2Pr−1/2 for medium Pr (‘Kraichnan regime’), a regime with scaling Nu ∼ Ra1/5Pr1/5, Re ∼ Ra2/5Pr−3/5 for small Pr, a regime with Nu ∼ Ra1/3, Re ∼ Ra4/9Pr−2/3 for larger Pr, and a regime with scaling Nu ∼ Ra3/7Pr−1/7, Re ∼ Ra4/7Pr−6/7 for even larger Pr. In particular, a linear combination of the ¼ and the 1/3 power laws for Nu with Ra, Nu = 0.27Ra1/4 + 0.038Ra1/3 (the prefactors follow from experiment), mimics a 2/7 power-law exponent in a regime as large as ten decades. For very large Ra the laminar shear boundary layer is speculated to break down through the non-normal-nonlinear transition to turbulence and another regime emerges. The theory presented is best summarized in the phase diagram figure 2 and in table 2.