Boundary layers and wind in cylindrical Rayleigh–Bénard cells

Abstract

We analyse the wind and boundary layer properties of turbulent Rayleigh–Bénard convection in a cylindrical container with aspect ratio one for Prandtl number $\mathit{Pr}= 0. 786$ and Rayleigh numbers ($\mathit{Ra}$) up to $1{0}^{9} $ by means of highly resolved direct numerical simulations. We identify time periods in which the orientation of the large-scale circulation (LSC) is nearly constant in order to perform a statistical analysis of the LSC. The analysis is then reduced to two dimensions by considering only the plane of the LSC. Within this plane the LSC is treated as a wind with thermal and viscous boundary layers developing close to the horizontal plates. Special focus is on the spatial development of the wind magnitude and the boundary layer thicknesses along the bottom plate. A method for the local analysis of the instantaneous boundary layer thicknesses is introduced which shows a dramatically changing wind magnitude along the wind path. Furthermore a linear increase of the viscous and thermal boundary layer thickness along the wind direction is observed for all $\mathit{Ra}$ considered while their ratio is spatially constant but depends weakly on $\mathit{Ra}$. A possible explanation is a strong spatial variation of the wind magnitude and fluctuations in the boundary layer region.

New perspectives in turbulent Rayleigh-Bénard convection

“…The phase shift φ LSC characterizes the mean orientation of the LSC over a finite time interval. As stated above, all DNS [77,112,80] showed that this orientation can fluctuate strongly. Wagner et al [112] provided therefore a comparison of five different functions to determine finite-time interval average of φ LSC which were for example based on the local heat flux J c + J d or the azimuthal vorticity component and found that the heat flux criterion works best.…”

“…This permanent change of the mean direction of the LSC is accompanied by a irregular temporal variation of its mean velocity magnitude. This is also the outcome of very recent high-resolution DNS by Wagner et al [112] and Shi et al [80] in which access to the full three-dimensional velocity field is given. These oscillations have been also measured earlier in the BOI [113].…”

“…The probability density functions of ∂p/∂r and r −1 ∂p/∂φ yield fatter tails when determined in the boundary layers in comparison to the bulk [80]. The boundary layer thicknesses taken in a plane which is aligned with the LSC grows in the mean downstream direction as shown for the data taken from [112] in fig. 6.…”

“…Normalized boundary layer thicknesses of temperature, δ T (r), and velocity, δv(r), across the whole convection cell. Data are from a DNS by Wagner et al [112] for Ra = 10 9 , Γ = 1, and P r = 0.786. They are normalized by δ T = H/(2Nu) and the most probable velocity BL thickness, respectively.They are obtained by time averaging for each r in the plane aligned with the LSC.…”

“…Recent high-resolution DNS of the boundary layer dynamics [80,112] brought further insights on the 3D structure close to the walls. As already said above, both works showed that the LSC is a three-dimensional flow in the cylindrical cell.…”