Ultimate regime in Rayleigh-Darcy Convection

Abstract

DNS of Rayleigh-Darcy convection in a 3D porous domain is performed at Ra $\in [10^3, 10^6]$ to investigate heat-transfer scaling, thermal boundary-layer dynamics, and flow-structure evolution in the unexplored ultimate regime. The Nu exhibits an approximately linear dependence on Ra throughout the investigated range. However, a distinct change in slope is observed at $Ra \approx 4\times10^5$, indicating the onset of the ultimate regime. For $Ra \leq 2.5\times10^5$, our scaling is 6.25% lower than that reported by \cite{hewitt2014high}, while for $Ra \geq 4\times10^5$ our results are within 1.24% of the extrapolated ultimate-regime prediction of \cite{pirozzoli2021towards}. Analysis of thermal structure reveals formation of near-wall protoplumes that merge into large-scale columnar megaplumes. With increasing Ra, the size of the protoplumes decreases, whereas the numbers increase, thus enhancing boundary-layer convection and heat transport. The thermal boundary-layer thickness scales as ~ Ra^{-1} and ~ Nu^{-1}, corroborating the persistence of linear heat-transfer scaling in the ultimate regime. The thermal dissipation is found to be increasingly shifting from the boundary layer to the bulk with increasing $Ra$, further indicating that the finer protoplumes efficiently transport heat from walls to bulk. The flow structures are quantified using the dominant length scale using the mean wavenumber ($\overline{k}$). It exhibits linear variation with $Ra$ for near-wall structures, with a higher slope in the ultimate regime, signifying finer protoplumes. At the mid-plane, a weaker scaling suggests that the megaplumes also become finer with increasing $Ra$ in the ultimate regime, thus leading to efficient heat transport in the bulk.