Heat transfer in rough-wall turbulent thermal convection in the ultimate regime

Heat and momentum transfer in wall-bounded turbulent flow, coupled with the effects of wall roughness, is one of the outstanding questions in turbulence research. In the standard Rayleigh-Bénard problem for natural thermal convection, it is notoriously difficult to reach the so-called ultimate regime in which the near-wall boundary layers are turbulent. Following the analyses proposed by Kraichnan [Phys. Fluids 5, 1374 (1962)] and Grossmann and Lohse [Phys. Fluids 23, 045108 (2011)], we instead utilize recent direct numerical simulations of forced convection over a rough wall in a minimal channel [MacDonald et al., J. Fluid Mech. 861, 138 (2019)] to directly study these turbulent boundary layers. We focus on the heat transport (in dimensionless form, the Nusselt number Nu) or equivalently the heat transfer coefficient (the Stanton number Ch). Extending the analyses of Kraichnan and Grossmann and Lohse, we assume logarithmic temperature profiles with a roughness-induced shift to predict an effective scaling of Nu∼Ra0.42, where Ra is the dimensionless temperature difference, corresponding to Ch∼Re−0.16, where Re is the centerline Reynolds number. This is pronouncedly different from the skin-friction coefficient Cf, which in the fully rough turbulent regime is independent of Re, due to the dominant pressure drag. In rough-wall turbulence, the absence of the analog to pressure drag in the temperature advection equation is the origin for the very different scaling properties of the heat transfer as compared to the momentum transfer. This analysis suggests that, unlike momentum transfer, the asymptotic ultimate regime, where Nu∼Ra1/2, will never be reached for heat transfer at finite Rayleigh number.