Heat-transfer scaling at moderate Prandtl numbers in the fully rough regime

In the fully rough regime, proposed models predict a scaling for roughness heat-transfer coefficient, e.g. Stanton number ${St}_k \sim (k^+)^{-p} {Pr}^{-m}$ where exponent values $p$ and $m$ are model dependent, giving diverse predictions. Here, $k^+$ is Reynolds ${Pr}$ Prandtl number. To clarify this ambiguity, we conduct direct numerical simulations of forced convection over three-dimensional sinusoidal surface spanning $k^+ = 5.5$ – $111$ numbers ${Pr} 0.5$ , 1.0 2.0. These unprecedented parameter ranges reached by employing minimal channels, which resolve sublayer at an affordable cost. We focus on phenomenologies, fall into two groups: $p=1/2$ (Owen & Thomson, J. Fluid Mech. vol. 15, issue 3, 1963, pp. 321–334; Yaglom Kader, 62, 1974, 601–623) $p=1/4$ (Brutsaert, Water Resour. Res. 11, 4, 1975 b 543–550). Although find mean heat transfer favours scaling, Prandtl–Blasius boundary-layer ideas associated with Reynolds–Chilton–Colburn analogy that underpin can remain apt description flow locally in regions exposed to high shear. Sheltered regions, meanwhile, violate behaviour instead dominated reversed flow, no clear correlation between momentum evident. The overall picture then not encapsulated one singular mechanism or phenomenology, but rather ensemble different behaviours locally. implications approach Reynolds-analogy-like bulk measures Nusselt also examined, evidence pointing onset regime transition even-higher numbers.