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Falkner–Skan boundary layer approximation in Rayleigh–Bénard convection

Published online by Cambridge University Press:  01 August 2013

Olga Shishkina ,
Susanne Horn  and
Sebastian Wagner
Olga Shishkina*
Affiliation:
DLR - Institute for Aerodynamics and Flow Technology, Bunsenstraße 10, 37073 Göttingen, Germany
Susanne Horn
Affiliation:
DLR - Institute for Aerodynamics and Flow Technology, Bunsenstraße 10, 37073 Göttingen, Germany
Sebastian Wagner
Affiliation:
DLR - Institute for Aerodynamics and Flow Technology, Bunsenstraße 10, 37073 Göttingen, Germany
*
†Email address for correspondence: Olga.Shishkina@dlr.de
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Abstract

To approximate the velocity and temperature within the boundary layers in turbulent thermal convection at moderate Rayleigh numbers, we consider the Falkner–Skan ansatz, which is a generalization of the Prandtl–Blasius one to a non-zero-pressure-gradient case. This ansatz takes into account the influence of the angle of attack $\beta $ of the large-scale circulation of a fluid inside a convection cell against the heated/cooled horizontal plate. With respect to turbulent Rayleigh–Bénard convection, we derive several theoretical estimates, among them the limiting cases of the temperature profiles for all angles $\beta $, for infinite and for infinitesimal Prandtl numbers $\mathit{Pr}$. Dependences on $\mathit{Pr}$ and $\beta $ of the ratio of the thermal to viscous boundary layers are obtained from the numerical solutions of the boundary layers equations. For particular cases of $\beta $, accurate approximations are developed as functions on $\mathit{Pr}$. The theoretical results are corroborated by our direct numerical simulations for $\mathit{Pr}= 0. 786$ (air) and $\mathit{Pr}= 4. 38$ (water). The angle of attack $\beta $ is estimated based on the information on the locations within the plane of the large-scale circulation where the time-averaged wall shear stress vanishes. For the fluids considered it is found that $\beta \approx 0. 7\mathrm{\pi} $ and the theoretical predictions based on the Falkner–Skan approximation for this $\beta $ leads to better agreement with the DNS results, compared with the Prandtl–Blasius approximation for $\beta = \mathrm{\pi} $.

JFM classification

Boundary Layers: Boundary layer structure Turbulent Flows: Turbulent convection
Type
Papers
Information
Journal of Fluid Mechanics , Volume 730 , 10 September 2013 , pp. 442 - 463
Copyright
©2013 Cambridge University Press 

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