Kinetic Energy Transport in Rayleigh--B\'enard Convection

Abstract

The kinetic energy balance in Rayleigh--B\'{e}nard convection is investigated for the Prandtl number range \(0.01\le Pr \le 150\) and for fixed Rayleigh number \(Ra=5\cdot10^{6}\). The kinetic energy balance is divided into a dissipation, a production and a flux term. We discuss profiles of all terms and find that the different contributions to the energy balance can be spatially separated into regions where kinetic energy is produced and where kinetic energy is dissipated. Analysing the Prandtl number dependence of the kinetic energy balance, we show that the height-dependence of the mean viscous dissipation is closely related to the flux of kinetic energy. We show that the flux of kinetic energy can be divided into four additive contributions, each representing a different elementary physical process (advection, buoyancy, normal viscous stresses and viscous shear stresses). The behaviour of these individual flux contributions is found to be surprisingly rich and exhibits a pronounced Prandtl number dependence. Different flux contributions dominate the kinetic energy transport at different depth, such that a comprehensive discussion requires a decomposition of the domain into a considerable number of sub-layers. On a less detailed level, our results reveal that advective kinetic energy fluxes play a key role in balancing the near-wall dissipation at low Prandtl number, whereas normal viscous stresses are particularly important at high Prandtl number. Finally, our work reveals that classical velocity boundary layers are deeply connected to the kinetic energy transport, but fail to correctly represent regions of enhanced viscous dissipation.