  We perform direct numerical simulations (DNS) of a turbulent channel flow
over porous walls. In the fluid region the flow is governed by the
incompressible Navier--Stokes (NS) equations, while in the porous layers the
Volume-Averaged Navier--Stokes (VANS) equations are used, which are obtained by
volume-averaging the microscopic flow field over a small volume that is larger
than the typical dimensions of the pores. In this way the porous medium has a
continuum description, and can be specified without the need of a detailed
knowledge of the pore microstructure by indipendently assigning permeability
and porosity. At the interface between the porous material and the fluid
region, momentum-transfer conditions are applied, in which an available
coefficient related to the unknown structure of the interface can be used as an
error estimate. To set up the numerical problem, the velocity-vorticity
formulation of the coupled NS and VANS equations is derived and implemented in
a pseudo-spectral DNS solver. Most of the simulations are carried out at
$Re_\tau=180$ and consider low-permeability materials; a parameter study is
used to describe the role played by permeability, porosity, thickness of the
porous material, and the coefficient of the momentum-transfer interface
conditions. Among them permeability, even when very small, is shown to play a
major role in determining the response of the channel flow to the permeable
wall. Turbulence statistics and instantaneous flow fields, in comparative form
to the flow over a smooth impermeable wall, are used to understand the main
changes introduced by the porous material. A simulations at higher Reynolds
number is used to illustrate the main scaling quantities.
