Scaling in Rayleigh–Bénard convection

We consider the Nusselt–Rayleigh number problem of Rayleigh–Bénard convection and make hypothesis that velocity thermal boundary layer widths, $\delta _u$ _T$ , in absence a strong mean flow are controlled by dissipation scales turbulence outside layers and, therefore, order Kolmogorov Batchelor scales, respectively. Under this assumption, we derive $Nu \sim Ra^{1/3}$ high $Ra$ limit, independent Prandtl number, _T/L Ra^{-1/3}$ _u/L Ra^{-1/3} Pr^{1/2}$ where $L$ is height cell. The scaling relations valid as long not too far from unity. For $Pr 1$ more general ansatz, _u \nu ^{\alpha }$ $\nu$ kinematic viscosity assume $\sim u^3/L$ $u$ characteristic turbulent velocity. these assumptions show Ra^{\alpha /(3-\alpha )}$ implying Ra^{1/5}$ if were Blasius Ra^{1/2}$ (with some logarithmic correction) it standard shear layer. It argued will retain intermediate $\alpha = 3/4$ limit .