In this thesis, turbulent flows over canopies in the sparse and dense regimes are examined using direct numerical simulation. The term `canopy' is used to refer to tall roughness elements in the flow. Sparse canopies typically have large element spacings and allow turbulent eddies to penetrate between the elements, whereas dense canopies have small spacings and preclude the penetration of turbulent eddies within them.

In sparse canopies, we consider layouts with rigid elements and spacings larger than the characteristic scales of near-wall turbulence, $s+\gtrsim 100$. We focus on the effect of the canopy on the background turbulence, the part of the flow that remains once the element-induced flow is filtered out. In channel flows, the distribution of the total stress is linear with height. Over smooth walls, the total stress is only the `fluid stress' $\tau f$, the sum of the viscous and the Reynolds shear stresses. In canopies, in turn, there is an additional contribution from the canopy drag, which can dominate within. We find that, for sparse canopies, the ratio of the viscous and the Reynolds shear stresses in $\tau f$ at each height is similar to that over smooth-walls, even within the canopy. From this, a height-dependent scaling based on $\tau f$ 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 $\tau f$, 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 $\tau f$, 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. The present results thus 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 dense canopies studied consist of rigid, prismatic filaments with small spacings. The effect of the height and spacing of the canopy elements on the flow is studied. The flow is composed of an element-coherent, dispersive flow and an incoherent flow, which includes contributions from the background turbulence and from the flow arising from the Kelvin--Helmholtz-like, mixing-layer instability typically reported over dense canopies. For the present canopies, with spacings $s+\approx 3$--$50$, the background turbulence is essentially precluded from penetrating within the canopy. As the elements are `tall', with height-to-spacing ratios $h/s\gtrsim 1$, the roughness sublayer of the canopy is determined by their spacing, extending to $y\approx 2$--$3s$ above the canopy tips. The dispersive velocity fluctuations are observed to also depend mainly on the spacing, and are small deep within the canopy, where the footprint of the Kelvin--Helmholtz-like instability dominates. The instability is governed by the canopy drag, which sets the shape of the mean velocity profile, and thus the shear length near the canopy tips. For the tall canopies considered here, this drag is governed by the element spacing and width, that is, the planar layout of the canopy. The mixing length, which determines the lengthscale of the instability, is essentially the sum of its height above and below the canopy tips. The former remains roughly the same in wall-units and the latter is linear with $s$ for all the canopies considered. For very small element spacings, $s+\lesssim 10$, the elements obstruct the fluctuations and the instability is inhibited. Within the range of $s+$ of the present canopies, the obstruction decreases with increasing spacing and the signature of the Kelvin--Helmholtz-like rollers intensifies. For sparser canopies, however, the intensification of the instabilities ceases as the assumption of a spatially homogeneous mean flow breaks down. For the present, dense configurations, the canopy depth also has an influence on the development of the instability. For shallow canopies, $h/s\sim 1$, the lack of depth blocks the Kelvin--Helmholtz-like rollers. For deep canopies, $h/s\gtrsim 6$, the rollers do not perceive the bottom wall and the effect of the canopy height on the flow saturates.

Two approaches based on linear stability analysis are proposed to capture the Kelvin--Helmholtz-like instability over dense canopies. The first approach models the canopy as an anisotropic permeable substrate whose wall-normal permeability, $Ky$, is larger than its streamwise permeability, $Kx$. This model predicts that the instability over canopies is governed by the geometric mean of the two permeabilities, $Kx+Ky+$. We also use this model to study the effect of the mean inclination of the canopy elements on the instability. The second approach models the canopy using a drag force in the momentum equation. This model shows that two competing effects, originating from the canopy drag, govern the growth of the instability. Increasing the canopy drag results in a stronger inflection point in the mean velocity profile, which enhances the instability, while at the same time, it also inhibits fluctuations within the canopy, suppressing the instability. We also analyse the stability of the mean profiles obtained from the DNS of dense canopy flows. Using this analysis, we show that the shear-layer thickness within the canopy, which determines the streamwise wavelength of the instability, also scales with the element spacing.