Turbulent flows over dense canopies consisting of rigid filaments of small size are investigated using direct numerical simulation. The effect of the height and spacing of the canopy elements on the overlying turbulence is studied. In these dense canopies, the element spacing is the key parameter determining the penetration of turbulent fluctuations within the canopy and the height of the roughness sublayer. The flow within and above the canopy is found to become independent of the filament height at large element heightto-spacing ratios, h/s 6. We also study the effect of the canopy parameters on the Kelvin-Helmholtz-like instability typically associated with dense-canopy flows. We find that the instability is inhibited in canopies with small element height-to-spacing ratios, h/s ∼ 1, due to the blocking effect of the wall at the base of the canopy, while a stronger signature of the instability can be observed as the height is increased. Canopies with very small element spacings, s + 10, also inhibit the instability, owing to the large drag they exert on the fluctuations. The streamwise wavelength of the instability depends on the canopy shear-layer thickness, and we show that, for the present dense canopies, this thickness also scales with the element spacing. Some of the effects of the canopy parameters on the instability can be captured by a linear stability analysis.

Turbulent flows within and over sparse canopies are investigated using direct numerical simulations at moderate friction Reynolds numbers Re τ ≈ 520 and 1000. The height of the canopies studied is h + ≈ 110-200, which is typical of some engineering canopies but much lower than for most vegetation canopies. The analysis of the effect of Reynolds number in our simulations, however, suggests that the dynamics observed would be relevant for larger Reynolds numbers as well. In channel flows, the distribution of the total stress is linear with height. Over smooth walls, the total stress is the sum of the viscous and the Reynolds shear stresses, the 'fluid stress' τ f . In canopies, in turn, there is an additional contribution from the canopy drag, which can dominate within. Furthermore, the full Reynolds shear stress has contributions from the dispersive, element-induced flow and from the background turbulence, the part of the flow that remains once the elementinduced flow is filtered out. For the present sparse canopies, we find that the ratio of the viscous stress and the background Reynolds shear stress to their sum, τ f , is similar to that over smooth-walls at each height, even within the canopy. From this, a height-dependent scaling based on τ f is proposed. Using this scaling, the background turbulence within the canopy shows similarities with turbulence over smooth walls. This suggests that the background turbulence scales with τ f , rather than with the conventional scaling based on the total stress. This effect is essentially captured when the canopy is substituted by a drag force that acts on the mean velocity profile alone, aiming to produce the correct τ f , without the discrete presence of the canopy elements acting directly on the fluctuations. The proposed mean-only forcing is shown to produce better estimates for the turbulent fluctuations compared to a conventional, homogeneous-drag model. These results suggest that a sparse canopy acts on the background turbulence primarily through the change it induces on the mean velocity profile, which in turn sets the scale for turbulence, rather than through a direct interaction of the canopy elements with the fluctuations. The effect of the element-induced flow, however, requires the representation of the individual canopy elements.

The behaviour of turbulent flow over anisotropic permeable substrates is studied using linear stability analysis and direct numerical simulations (DNS). The flow within the permeable substrate is modelled using the Brinkman equation, which is solved analytically to obtain the boundary conditions at the substrate-channel interface for both the DNS and the stability analysis. The DNS results show that the drag-reducing effect of the permeable substrate, caused by preferential streamwise slip, can be offset by the wall-normal permeability of the substrate. The latter is associated with the presence of large spanwise structures, typically associated to a Kelvin-Helmholtz-like instability. Linear stability analysis is used as a predictive tool to capture the onset of these drag-increasing Kelvin-Helmholtz rollers. It is shown that the appearance of these rollers is essentially driven by the wall-normal permeability . When realistic permeable substrates are considered, the transpiration at the substrate-channel interface is wavelength-dependent. For substrates with low , the wavelength-dependent transpiration inhibits the formation of large spanwise structures at the characteristic scales of the Kelvin-Helmholtz-like instability, thereby reducing the negative impact of wall-normal permeability.

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