Exploring the nexus among roughness function, apparent slip velocity and upscaling coefficients for turbulence over porous/textured walls

The interaction between a turbulent flow and a porous boundary is analysed with focus on the sensitivity of the roughness function, $\Delta U^+$ , to the upscaled coefficients characterizing the wall. The study is aimed at (i) demonstrating that imposing effective velocity boundary conditions at a virtual plane boundary, next to the physical one, can efficiently simplify the direct numerical simulations (DNS); and (ii) pursuing correlations to estimate $\Delta U^+$ a priori , once the upscaled coefficients are calculated. The homogenization approach employed incorporates near-interface advection via an Oseen-like linearization, and the macroscopic coefficients thus depend on both the microstructural details of the wall and a slip-velocity-based Reynolds number, $Re_{slip}$ . A set of homogenization-simplified DNS is run to study the channel flow over transversely isotropic porous beds, testing values of the grains’ pitch within $0\lt \ell ^+\lt 40$ . Reduction of the skin-friction drag is attainable exclusively over streamwise-aligned inclusions for $\ell ^+$ values up to $20{-}30$ . The drag increase over spanwise-aligned inclusions (or streamwise-aligned ones at large $\ell ^+$ ) is accompanied by enhanced turbulence levels, including intensified sweep and ejection events. The root-mean-square of the transpiration velocity fluctuations at the virtual plane, $\tilde V_{rms}$ , is the key control parameter of $\Delta U^+$ ; our analysis shows that, provided $\tilde V_{rms} \lesssim 0.25$ , then $\tilde V_{rms}$ is strongly correlated to a single macroscopic quantity, $\Psi$ , which comprises the Navier-slip and interface/intrinsic permeability coefficients. Fitting relationships for $\Delta U^+$ are proposed, and their applicability is confirmed against reference results for the turbulent flow over impermeable walls roughened with three-dimensional protrusions or different geometries of riblets.