Boundary layer structure in turbulent Rayleigh–Bénard convection

Abstract

The structure of the boundary layers in turbulent Rayleigh-Bénard convection is studied by means of three-dimensional direct numerical simulations. We consider convection in a cylindrical cell at an aspect ratio one for Rayleigh numbers of Ra = 3×10 9 and 3×10 10 at fixed Prandtl number P r = 0.7. Similar to the experimental results in the same setup and for the same Prandtl number, the structure of the laminar boundary layers of the velocity and temperature fields is found to deviate from the prediction of the Prandtl-Blasius-Pohlhausen theory. Deviations decrease when a dynamical rescaling of the data with an instantaneously defined boundary layer thickness is performed and the analysis plane is aligned with the instantaneous direction of the large-scale circulation in the closed cell. Our numerical results demonstrate that important assumptions which enter existing classical laminar boundary layer theories for forced and natural convection are violated, such as the strict two-dimensionality of the dynamics or the steadiness of the fluid motion. The boundary layer dynamics consists of two essential local dynamical building blocks, a plume detachment and a post-plume phase. The former is associated with larger variations of the instantaneous thickness of velocity and temperature boundary layer and a fully three-dimensional local flow. The post-plume dynamics is connected with the large-scale circulation in the cell that penetrates the boundary region from above. The mean turbulence profiles taken in localized sections of the boundary layer for both dynamical phases are also compared with solutions of perturbation expansions of the boundary layer equations of forced or natural convection towards mixed convection. Our analysis of both boundary layers shows that the near-wall dynamics combines elements of forced Blasius-type and natural convection.

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

“…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. However, the method has not been tested experimentally so far in a 3D system.…”

New perspectives in turbulent Rayleigh-Bénard convection

“…Currently, DNS for cylindrical cells with Γ = 1/2 can reach Ra = 2× 10 12 [79]. Since the numerical effort grows with Γ 2 when larger aspect ratio cells are considered, recent DNS at Γ = 1 achieved Ra = 3 × 10 10 only, but requiring almost the same number of grid points [80] as the former case for Γ = 1/2.…”

Boundary layer structure in confined turbulent thermal convection

“…addition, the horizontal flow is generated by the vertical hot/cold current that bends at the plates and bends again after having swept them; this phenomenon is accompanied by a streamwise pressure gradient that, in fact, is found in the numerical simulations by Shi et al (2012). In view of how different the conditions of the Rayleigh-Bénard flow are with respect to the Blasius boundary layer it is surprising how much the boundary layer profiles resemble each other.…”

“…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.…”

“…In the paper by Shi et al (2012) the flow in a cylindrical cell of aspect ratio Γ = 1 at Pr = 0.7 is simulated for Ra = 3 × 10 9 and Ra = 3 × 10 10 ; the structure of the boundary layers has been analysed by collecting time series of temperature and velocity components on various vertical arrays for different positions over the plates. The main findings of this study are that the boundary layer vertical structure has some analogies with the Blasius profile although there are also evident differences.…”