Dissipation Layers in Rayleigh-B\'enard Convection: A Unifying View

Dissipation Layers in Rayleigh-Bénard Convection: A Unifying View

K. Petschel, S. Stellmach, M. Wilczek, J. Lülff, and U. Hansen

Phys. Rev. Lett. 110, 114502 – Published 13 March 2013

Abstract

Boundary layers play an important role in controlling convective heat transfer. Their nature varies considerably between different application areas characterized by different boundary conditions, which hampers a uniform treatment. Here, we argue that, independent of boundary conditions, systematic dissipation measurements in Rayleigh-Bénard convection capture the relevant near-wall structures. By means of direct numerical simulations with varying Prandtl numbers, we demonstrate that such dissipation layers share central characteristics with classical boundary layers, but, in contrast to the latter, can be extended naturally to arbitrary boundary conditions. We validate our approach by explaining differences in scaling behavior observed for no-slip and stress-free boundaries, thus paving the way to an extension of scaling theories developed for laboratory convection to a broad class of natural systems.

Received 20 December 2012

DOI:https://doi.org/10.1103/PhysRevLett.110.114502

Authors & Affiliations

K. Petschel^1,†, S. Stellmach¹, M. Wilczek^2,*, J. Lülff², and U. Hansen¹

¹Institut für Geophysik, Westfälische Wilhelms-Universität Münster, D-48149 Münster, Germany
²Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, D-48149 Münster, Germany

^*Present address: Department of Mechanical Engineering, The Johns Hopkins University, Baltimore MD 21218, USA.
^†klaus.petschel@uni-muenster.de

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Vol. 110, Iss. 11 — 15 March 2013

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Figure 1
Temporally averaged depth profiles of the horizontally averaged viscous dissipation rate (red solid line) and the thermal dissipation rate (blue dashed line) for no-slip (upper graph) and stress-free (lower graph) boundaries and for different Prandtl numbers. The horizontal axis is scaled by the corresponding globally averaged dissipation rates [cf. Eq. (4)]. The thickness of the newly defined dissipation layers is illustrated for each Prandtl number and boundary condition, respectively.Reuse & Permissions
Figure 2
Thickness of the dissipation layers versus the Prandtl number for no-slip (upper graph) and stress-free (lower graph) boundaries. Red circles represent the thermal dissipation layer while blue squares denote the viscous dissipation layer. The gray arrows indicate the crossover of the thicknesses of both dissipation layers ( $λ_{T, DL} = λ_{u, DL}$ ). The classical thickness definitions of the thermal and viscous boundary layer, $λ_{T, BL}$ and $λ_{u, BL}$ , are denoted by green triangles and orange crosses, respectively. Note that for stress-free boundaries, the classical viscous boundary layer definition cannot be applied.Reuse & Permissions
Figure 3
Thickness of different definitions of a viscous layer versus the Reynolds number $Re = \sqrt{⟨ u^{2} ⟩_{V}} / \Pr$ . The orange crosses denote a classical definition of a viscous boundary layer thickness $λ_{u, BL}$ , while the blue squares and the red circles denote the dissipation layer thickness $λ_{u, DL}$ for stress-free and no-slip boundary conditions, respectively.Reuse & Permissions
Figure 4
Ratio of the DL contribution of the globally averaged kinetic dissipation rate $ϵ_{u}$ versus the Prandtl number. The blue squares denote results of simulations with stress-free boundaries, while red circles denote results with no-slip boundary conditions. The gray arrows divide the Prandtl number range into a bulk and a dissipation layer dominated regime. For completeness, the decomposition employing the classical boundary layer definition is represented by orange crosses.Reuse & Permissions
Figure 5
Nusselt number versus the Prandtl number for no-slip (upper graph) and stress-free (lower graph) boundaries. The vertical dashed line indicates the change of hierarchies, i.e., the Prandtl number where the thermal and the viscous dissipation layers are of the same thickness. The color-coded background represents the ratio of the DL contribution of $ϵ_{u}$ in Fig. 4, where light gray indicates a bulk dominated regime and dark gray indicates a dissipation layer dominated regime. Data fits for the no-slip case are in good agreement with the predictions by Grossmann and Lohse [14] ( $\sim \Pr^{1 / 5}$ for low Prandtl numbers).Reuse & Permissions

Physical Review Letters

Dissipation Layers in Rayleigh-Bénard Convection: A Unifying View

K. Petschel, S. Stellmach, M. Wilczek, J. Lülff, and U. Hansen

Phys. Rev. Lett. 110, 114502 – Published 13 March 2013

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