A note on upper bound for heat transport in two-dimensional Rayleigh-Benard convection

摘要

We revisit the upper bound on heat transport in two-dimensional Rayleigh-Benard convection between stress-free isothermal boundaries by imposing the global balance of heat transfer. In previous studies [Whitehead & Doering, Phys. Rev. Lett., 106, 244,501 2011; Wen et al., Phys. Rev. E, 92, 043012 2015], the upper bounds on heat flux (or equivalently the Nusselt number Nu) are yielded from a background temperature field r. However, those upper bounds violate with the value of definition: Nu = -dT/dz|_(z=top/bottom wall), where T is the temporal-horizontal averaged temperature obtained from the upper bound analysis, due to the unbalanced global heat transfer. By enforcing the balance of global heat transfer, we are able to amend the mean temperature profile such that the inconsistency between the upper bound and the definition is corrected. A rigorous proof followed by a numerical solution of the upper bound problem also demonstrates that the upper bound is improved by the imposed global balance of heat transfer, which gives Nu approx< 0.105Ra~(5/12).