Scaling relations for heat and momentum transport in sheared Rayleigh-Bénard convection

We provide scaling relations for the Nusselt number Nu and the friction coefficient CS in sheared Rayleigh-B & eacute;nard convection, i.e. in Rayleigh-B & eacute;nard flow with Couette- or Poiseuille-type shear forcing, by extending the Grossmann & Lohse (J. Fluid Mech., vol. 407, 2000, pp. 27-56, Phys. Rev. Lett., vol. 86, 2001, pp. 3316-3319, Phys. Rev. E, vol. 66, 2002, 016305, Phys. Fluids, vol. 16, 2004, pp. 4462-4472) theory to sheared thermal convection. The control parameters for these systems are the Rayleigh number Ra, the Prandtl number Pr and the Reynolds number Re-S that characterises the strength of the imposed shear. By direct numerical simulations and theoretical considerations, we show that, in turbulent Rayleigh-B & eacute;nard convection, the friction coefficients associated with the applied shear and the shear generated by the large-scale convection rolls are both well described by Prandtl's (Ergeb. Aerodyn. Vers. G & ouml;tt., vol. 4, 1932, pp. 18-29) logarithmic friction law, suggesting some kind of universality between purely shear-driven flows and thermal convection. These scaling relations hold well for 10(6) <= Ra <= 10(8), 0.5 <= Pr <= 5.0, and 0 <= Re-S <= 10(4).