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We perform a direct numerical simulation (DNS) of a turbulent channel flow over porous walls. In the fluid region the flow is governed by the Navier-Stokes equations, while in the porous layers is governed by the Volume-Averaged Navier-Stokes equations derived by Whitaker [1, 2]. The latter equations are obtained by volume-averaging the microscopic flow field over a small volume,in order to model a macroscopic, or volume-averaged, flow field. The volume-averaging technique allows to treat the porous medium as a continuum.

To formulate numerically this problem, we derive and implement the v-η formulation
of the Navier-Stokes and Volume-Averaged Navier-Stokes equations. At the interface between the porous layers and the fluid region, we impose the momentum
transfer conditions proposed by Ochoa-Tapia and Whitaker [3, 4]. The DNS solver used to integrate the coupled evolution equations is a substantial extension
of an existing solver [5]. Our solver uses a Fourier discretization in the streamwise and spanwise directions, and a compact, explicit high-order, finite difference
discretization in the wall-normal direction. Time integration is performed using a semi-implicit method, a combination of the third-order Runge-Kutta scheme and
the second-order Crank-Nicholson scheme. We perform extensive DNSs at two Reynolds numbers: a very low Reynolds number, Re_{τ} = 65 and an intermediate one, Re_{τ} = 200. For both turbulent flows we analyze the
turbulence statistics and the flow fields. For the intermediate case, at Re_{τ} = 200, we perform a parametric study, where we vary the height of the porous layers and
the coefficient of the momentum transfer conditions. The results are compared with the DNS of a turbulent channel flow over impermeable walls. For the very low Reynolds
number, Re_{τ} = 200, we investigate if the porous wall can sustain turbulent flows.

We consider a turbulent channel flow, at a bulk Reynolds number of 3100 (Re_{τ} = 200), over two identical porous walls, whose half-heights are h_{p} = 0.5. The permeability of
the porous layers is σ = 0.004, the porosity is ε = 0.6, and the coefficient of the momentum transfer conditions is set equal to zero. The computation is carried
out on a grid of 128×(128 + 128 + 128)×128 points, in the x, y and z directions respectively, on a computational domain of 2π × (1 + 2 + 1) × π. The resolution
of the numerical simulation is x^{+} ≈ 10 in wall units in the streamwise direction, z^{+} ≈ 5 in the spanwise direction, and with a minimum y^{+} in the wall-normal
direction which is less than 1.

The figure on the left shows the mean velocity profiles as Δ u = u - u_{i} versus the logarithm of the distance from the interface y-1 expressed in wall-units.
The main differences between the two turbulent flows are present in the logarithmic region, where the curve of the porous case is lower than the case with solid walls. The figure
on the right shows the Reynolds stresses of the turbulent channel flow over porous walls. The presence of the porous walls affects the Reynolds stresses, that are enhanced in the
viscous sub-layer. Moreover, in the buffer/log layer anisotropy is reduced.

The two figures show the isosurfaces of velocity fluctuation u'^{+} = ± 4, with impermeable (left) and porous (right) walls. We observe that, when the wall is porous, the development of elongated streaks is prevented.

The figure above shows the two-point correlation of u in x, at different distances from the wall. The thin and thick lines are used for the case with impermeable and porous walls, respectively. When the wall is porous, the length of correlation of the streamwise velocity component is reduced in the near wall region.

**Bibliography**

[1] S. Whitaker. __Flow in porous media i: A theoretical derivation of darcy’s law__. *Transport in porous media*, 1(1):3–25, 1986.

[2] S. Whitaker. __The forchheimer equation: a theoretical development__. *Transport in Porous media*, 25(1):27–61, 1996.

[3] J.A. Ochoa-Tapia and S. Whitaker. __Momentum transfer at the boundary between a porous medium and a homogeneous fluid-i: theoretical development__. *International Journal of Heat and Mass Transfer*, 38(14):2635–2646, 1995.

[4] J.A. Ochoa-Tapia and S. Whitaker. __Momentum transfer at the boundary between a porous medium and a homogeneous fluid-ii: comparison with experiment__. *International Journal of Heat and Mass Transfer*, 38(14):2647–2655, 1995.

[5] P. Luchini and M. Quadrio. __A low-cost parallel implementation of direct numerical simulation of wall turbulence__. *Journal of Computational Physics*, 211(2):551–571, 2006.