Falkner–Skan boundary layer approximation in Rayleigh–Bénard convection

• Published in: Journal of fluid mechanics 2013-09, Vol.730, p.442-463
• Main Authors: Shishkina, Olga, Horn, Susanne, Wagner, Sebastian
• Format: Article
• Language: English
• Subjects: Approximation; Boundary layer; Boundary layers; Buoyancy-driven instability; Circulation; Computational fluid dynamics; Convection; Exact sciences and technology; Fluid dynamics; Fluid flow; Fundamental areas of phenomenology (including applications); Hydrodynamic stability; Mathematical analysis; Physics; Shear stress; Turbulence; Turbulent flow
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2013.347

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} $ .