  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 pitch $L$, and depth $h$. The
study uses direct numerical simulations mainly 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. Based on the
direct-simulation results, we propose an empirical `equivalent permeability'
$K_{eq}^t$, that incorporates this effect and scales well the overlying
turbulence for substrates with different depths, permeabilities, etc. Based on
this, we propose a conceptual $h^+$-$\sqrt{K^+}$ regime diagram where, for any
given substrate topology, 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.
