Scaling laws for Rayleigh–Bénard convection between Navier-slip boundaries

Abstract

We consider the two-dimensional Rayleigh–Bénard convection problem between Navier-slip fixed-temperature boundary conditions, and present a new upper bound for the Nusselt number ( ${\textit {Nu}}$ ). The result, based on a localization principle for the Nusselt number and an interpolation bound, exploits the regularity of the flow. On one hand our method yields a shorter proof of the celebrated result of Whitehead & Doering (Phys. Rev. Lett., vol. 106, 2011, 244501) in the case of free-slip boundary conditions. On the other hand, its combination with a new, refined estimate for the pressure gives a substantial improvement of the interpolation bounds in Drivas et al. (Phil. Trans. R. Soc. A, vol. 380, issue 2225, 2022, 20210025) for slippery boundaries. A rich description of the scaling behaviour arises from our result: depending on the magnitude of the Prandtl number ( ${\textit {Pr}}$ ) and slip length, our upper bounds indicate five possible scaling laws (where ${\textit {Ra}}$ is the Rayleigh number): ${\textit {Nu}}\sim (L_s^{-1}\,{\textit {Ra}})^{{1}/{3}}$ , ${\textit {Nu}}\sim (L_s^{-2/5}\,{\textit {Ra}})^{{5}/{13}}$ , ${\textit {Nu}}\sim {\textit {Ra}}^{{5}/{12}}$ , ${\textit {Nu}}\sim {\textit {Pr}}^{-1/6}(L_s^{-4/3}\,{\textit {Ra}})^{{1}/{2}}$ and ${\textit {Nu}}\sim {\textit {Pr}}^{-1/6}(L_s^{-1/3}\,{\textit {Ra}})^{{1}/{2}}$ .