Thermal boundary layer profiles in turbulent Rayleigh-Bénard convection in a cylindrical sample

Abstract

We numerically investigate the structures of the near-plate temperature profiles close to the bottom and top plates of turbulent Rayleigh-Bénard flow in a cylindrical sample at Rayleigh numbers Ra = 10(8) to Ra = 2 × 10(12) and Prandtl numbers Pr = 6.4 and Pr = 0.7 with the dynamical frame method [Zhou and Xia, Phys. Rev. Lett. 104, 104301 (2010)], thus extending previous results for quasi-two-dimensional systems to three-dimensional systems. The dynamical frame method shows that the measured temperature profiles in the spatially and temporally local frame are much closer to the temperature profile of a laminar, zero-pressure gradient boundary layer (BL) according to Pohlhausen than in the fixed reference frame. The deviation between the measured profiles in the dynamical reference frame and the laminar profiles increases with decreasing Pr, where the thermal BL is more exposed to the bulk fluctuations due to the thinner kinetic BL, and increasing Ra, where more plumes are passing the measurement location.

Viscous boundary layer properties in turbulent thermal convection in a cylindrical cell: the effect of cell tilting

“…This is no surprise, since, as the convective flow above the BL becomes more turbulent with increasing Ra, the BL itself will experience stronger fluctuations and hence larger deviations from the laminar case. This finding that the dynamical rescaling method works better for smaller Ra than larger ones is consistent with those found in DNS studies in the same geometry by Stevens et al (2012) for the temperature profile and by Shi et al (2012) and by Scheel et al (2012) for the velocity profile. The second feature is that for all θ and Ra the profiles obtained in the dynamical frame in general show some degree of improvement towards that of Prandtl-Blasius value as compared to those obtained in the laboratory frame.…”

“…In a follow-up study using two-dimensional DNS data, found that the method is also valid for thermal boundary layers and for the case of P r = 0.7 as well. More recently, these authors further shown, again using numerical data, that the method works also in other positions in the horizontal plate other than the central axis (Zhou et al 2011) and in three-dimension (3D) cylindrical cell for moderate values of Ra (Stevens et al 2012). However, Scheel, Kim & White (2012) and Shi, Emran & Schumacher (2012), both using numerical approaches, have found that dynamic scaling works less well in the 3D cylindrical geometry than in the quasi-2D case.…”

Boundary layer structure in confined turbulent thermal convection

“…Xia 2010 andStevens et al 2012) which for the same problem report boundary layer vertical profiles with the same level of agreement with the Prandtl-Blasius solution as the present study, even if the previous work tends to focus more on the similarities. Two examples are shown in figure 1(c,d) and indeed it can be appreciated that the results are very similar and that in the region closest to the wall the profiles are always indistinguishable.…”

“…The thickness of the viscous boundary layer δ v can be smaller than δ T for Pr < 1 and vice versa for Pr > 1. Assuming the correlation Nu 0.08Ra 0.32 it is trivial to estimate that for Pr = O(1) at Ra = 10 8 one obtains δ T ≈ δ v h/60 and at Ra = 10 12 the result is δ T ≈ δ v h/1100; the analysis of the boundary layer structure, therefore, requires large experimental setups (du Puits et al 2007) or high-resolution particle image velocimetry (Sun et al 2008) or high-fidelity direct numerical simulations (Shi, Emran & Schumacher 2012;Stevens et al 2012) for the highest Rayleigh numbers.…”

“…Blue triangles: 'raw' data (laboratory frame); red circles: coordinate rescaled with the instantaneous boundary layer thickness (dynamical frame) (adapted from Zhou & Xia 2010, copyright (2010 by the American Physical Society). (d) Array near the centre of the bottom plate at Pr = 0.7, Ra = 2 × 10 12 and a cylindrical cell of Γ = 1/2 (PB: Prandtl-Blasius) (adapted from Stevens et al (2012Stevens et al ( ), copyright (2012 by the American Physical Society).…”