Turbulent flows over porous and rough substrates

Turbulent flows over porous substrates are studied via a systematic exploration of the dependence of the flow properties on the substrate parameters, including permeability $K$, grain size $L$, and depth $h$. The study uses direct numerical simulations for staggered-cube substrates with $L^+\approx10$ - $50$, $\sqrt{K}/L\approx0.01$ - $0.25$, and depths from $h=O(L)$ to $h\gg L$, ranging from typical impermeable rough surfaces to deep porous substrates. The results indicate that the permeability has significantly greater relevance than the grain size for the properties of the overlying flow, including the mean-flow slip and the shear across the interface, the drag increase relative to smooth-wall flow, and the statistics and spectra of the overlying turbulence, whereas the direct effect of grain size is only noticeable near the interface as grain-coherent flow fluctuations. The substrate depth also has a significant effect, with shallower substrates suppressing the effective transpiration at the interface excited by pressure fluctuations. We propose an empirical `equivalent permeability' $K_{eq}^t$, that incorporates this effect and scales well the overlying turbulence for substrates with different depths. Based on this, we propose a conceptual $h^+$-$\sqrt{K^+}$ regime diagram where turbulence transitions smoothly from that over impermeable rough surfaces with $h=O(L)$ to that over deep porous substrates with $h^+\gtrsim50$, with the latter limit determined by the typical lengthscale of the overlying pressure fluctuations.