3.1. Mean flow

Given the turbulent nature of the flow fields, we decompose the velocity $\boldsymbol{u}$ into a mean component $\boldsymbol{U}$ and a fluctuating component $\boldsymbol {u'}$ . Additionally, the mean flow along the canopy edge is known to develop a cellular structure (Unigarro Villota et al. Reference Unigarro Villota, Ghisalberti, Philip and Branson2023) similar to what is observed along streamwise-elongated roughness elements (Stroh et al. Reference Stroh, Schäfer, Frohnapfel and Forooghi2020). Such a structure emerges clearly from our simulations, and we thus better isolate it, increasing our statistics and enforcing the symmetry exhibited by the system between the left and right halves of the domain. This treatment is adopted only to enhance the intrinsic features of the mean flow, whereas the instantaneous fluctuations do not satisfy any symmetry constraint and their complex three-dimensional structure is fully retained. The mean field $\boldsymbol{U}$ is thus attained averaging $\boldsymbol{u}$ in time, along the only homogeneous direction $x$ , and exploiting symmetry with respect to the middle of the vegetation gap. Such a peculiar average is denoted with angle brackets $\langle \boldsymbol{\cdot }\rangle$ throughout the remaining part of the manuscript. At every time instant and at every point in space,

(3.1) \begin{equation} \boldsymbol {u} (\boldsymbol{x},t) = \langle \boldsymbol{u} \rangle (y,z) = \boldsymbol{U} (y,z) + \boldsymbol {u'} (\boldsymbol{x},t). \end{equation}

Figure 2. Bi-dimensional mean flow established above and within the stems for the two different values of $\textit{Ca}$ considered in our study. The total velocity $\boldsymbol{u}$ is averaged in time, along the $x$ axis, and made symmetric with respect to the middle of the vegetation gap, yielding $\boldsymbol{U}(y,z)=\{U,V,W\}$ . The same averaging procedure is adopted to compute the mean streamwise vorticity $\varOmega _x$ and the envelope of the deflected stem tips, reported as a black line.

We thus commence our analysis observing the fully bi-dimensional flow field $\boldsymbol{U}$ established above and between the stems along with the mean envelope of the deflected stem tips, as reported in figure 2. In fact, averaging in time and along the homogeneous $x$ direction, the vertical coordinate of the tips belonging to stems located in tiles at the same spanwise location (i.e. within bins of size $\Delta S$ along the $z$ axis), we obtain a function $\langle \eta \rangle (z)$ representing the mean envelope of the deflected stem tips. We overlay this function to our plots (i.e. figures 2, 4 and 5) to indicate, on average, the interface between the vegetation and the open flow above. At the lateral edge of the vegetated region, we close the envelope by drawing a straight line from the average position of the last stem tip down to its root.

The streamwise component $U$ increases moving away from the wall, reaching its maximum at the middle of the vegetation gap; a slightly higher value is attained for the case at $\textit{Ca}=10$ due to the higher blockage effect of the more rigid stems. The dense canopy considered here significantly depletes the mean flow velocity within its interior, as typically observed in tightly packed porous media (see e.g. Rosti, Cortelezzi & Quadrio Reference Rosti, Cortelezzi and Quadrio2015), yielding a sharp velocity increase emerging from its tip. The transition appears slightly less sharp in the more compliant case ( $\textit{Ca}=100$ ), where the outer flow penetrates further into the canopy array. The wall-normal component $V$ denotes the generation of an upwelling in correspondence to the interface between the vegetated and non-vegetated regions. A portion of the lifted fluid is deflected over the canopy, and thus impinges on the stems inducing their deflection (as clearly appreciated in the most flexible case, $\textit{Ca}=100$ ). The remainder is deviated towards the centre of the vegetation gap, from where it descends obliquely in the direction of the canopy. This is confirmed observing the spanwise component $W$ , which, together with $V$ , highlights the formation of an intense circulating flow at the canopy edge. The streamwise-oriented vortex formed at the canopy edge draws fluid inside the canopy from the side, and thus ejects it upward in the previously discussed upwelling. In the most flexible case, at $\textit{Ca}=100$ , the flapping motion of the stems is more intense and fluid is thus pumped from the canopy upward, enhancing the upwelling. A closer inspection of figure 2 also reveals a region of weaker circulation moving away from the canopy towards the centre of the gap, counterrotating with respect to the canopy-edge vortex. This is further confirmed by the plots of the streamwise vorticity of the mean flow $\varOmega _x$ , which also corroborate the previous description of the canopy-edge vortex. Note that positive values of $\varOmega _x$ correspond to clockwise circulation in the $y$ – $z$ plane.

Our observations are in qualitative agreement with the mean flow features reported by experiments conducted in partially obstructed channels with a canopy of width comparable to that of the gap (Yan et al. Reference Yan, Duan, Zhang and Wang2023). Furthermore, the formation of an upwelling similar to that reported here is also observed at the interface between a smooth and a rough bed (see case $h=0$ of Stroh et al. Reference Stroh, Schäfer, Frohnapfel and Forooghi2020). Yet, it is interesting to notice that this scenario can be completely upset in the case of a less wide canopy, where the effect of the lateral confinement is felt more strongly at the interface between the vegetated and non-vegetated regions. In this case, experiments (Unigarro Villota et al. Reference Unigarro Villota, Ghisalberti, Philip and Branson2023) report the formation of a downwelling at the interface, with fluid being drawn upward from deeper regions of the canopy. The direction of rotation of the canopy-edge vortex is therefore inverted with respect to what we report.

Figure 3. Wall-normal profiles of the mean streamwise velocity, normalised with the local bulk velocity $U_{bl}$ as defined in the text. Colours ranging from blue to red denote different positions along the $z$ axis. For comparison, we also report the profiles of an open channel (dashed blue line) and of full canopies (dashed red line) with the same parameters as those considered here, as measured in our former study (Foggi Rota et al. Reference Foggi Rota, Monti, Olivieri and Rosti2024b ).

Next, we further characterise the streamwise component of the mean velocity field observing its wall-normal profiles at different positions along the $z$ axis, $U(y;z)$ as reported in figure 3. The profiles are normalised with respect to the local bulk velocity,

(3.2) \begin{equation} U_{bl}(z) = \int _0^H U(y,z)\, {\textrm d}y, \end{equation}

and compared with those we measured in an open channel and in full canopies with the same parameters as those considered here (Foggi Rota et al. Reference Foggi Rota, Monti, Olivieri and Rosti2024b ). Starting from the most rigid case, at $\textit{Ca}=10$ , we observe that the profile at the middle of the canopy resembles that found in the case of a full canopy. Analogously, the profile at the middle of the gap has a shape similar to what is found in an open channel. We thus deduce that, in this case, the vegetated and non-vegetated regions are wide enough to attain fully developed conditions at their respective middles, as far as the mean flow is concerned. At the interface between the two regions, located at $L_z/4$ , the profile is non-monotonous, with the lower fluid moving faster than the one on top. This effect is a consequence of the bi-dimensionality of the mean flow, and follows from the presence of the canopy-edge vortex. High-speed fluid is drawn from the gap towards the lower region of the canopy by the vortex, through the vegetation interface. As it ascends in the upwelling, the fluid looses streamwise momentum due to the interaction with the stems and the non-monotonous trend of the profile is attained. Finally, upon exiting the canopy from the top, the fluid accelerates again due to the reduced drag.

The situation in the most flexible case, at $\textit{Ca}=100$ , appears qualitatively different. The profile at the middle of the gap still resembles that found in an open channel, but the one at the middle of the canopy deviates from that of a full canopy. Increasing the compliance of the stems, perturbations from the edge penetrate deeper in the canopy along the transversal direction and the mean flow is consequently depleted within the canopy region, compared with the full canopy case. The velocity far above the canopy is higher than in the full canopy merely due to mass conservation arguments. We also notice that, in this case, the monotonous growth of the profile is preserved at all spanwise locations. The increased updraught induced by the enhanced flapping motion of the stems, pumping the fluid upward, ensures that the high-speed fluid drawn by the canopy-edge vortex from the gap into the canopy emerges quicker, with a diminished loss of streamwise-momentum against the stems. Noticeably, looking back at figure 2, the canopy-edge vortex itself is not significantly affected by the increased flexibility of the stems. Yet, in this case, the stems are deflected forward and sway in the flow (as discussed later in § 3.4), exerting less drag on the fluid.

Figure 4. Turbulent kinetic energy $k$ and normal components of the Reynolds’ stress tensor in the spanwise $y$ – $z$ plane for the two different values of $\textit{Ca}$ considered in our study. The fluctuations are averaged in time, along the $x$ axis, and made symmetric with respect to the middle of the vegetation gap. The same averaging procedure is adopted to compute the mean envelope of the deflected stem tips, reported as a black line.

3.2. Fluctuations and stress balances

We now turn our attention to the fluctuating velocity field $\boldsymbol {u'} (\boldsymbol{x},t)$ , introduced in (3.1) as the deviation from the symmetric mean flow specified in § 3.1. Angled brackets, introduced there, denote the average employed to isolate it. First, we focus on the spatial distribution of the turbulent kinetic energy $k = 0.5 (\langle u' u' \rangle + \langle v' v' \rangle + \langle w' w' \rangle )$ in the spanwise $y$ – $z$ plane, as shown in the top panels of figure 4. In the most rigid case, $\textit{Ca}=10$ , $k$ appears more intense in proximity of the bottom wall within the gap, and along the interface between the vegetated and non-vegetated regions, attaining slightly lower values over the canopy. In the most flexible case, $\textit{Ca}=100$ , instead, the magnitude of $k$ over the canopy is comparable to that attained over the wall within the gap, suggesting that turbulence over the canopy is enhanced by the flapping motion of the stems.

Next, we observe separately the normal components of the Reynolds stress tensor $\langle \boldsymbol {u'} \boldsymbol {u'} \rangle$ (in figure 4). Predictably, the most significant contribution to $k$ at the bottom wall within the gap comes from the streamwise component $\langle u' u' \rangle$ , capturing the velocity fluctuations associated with the formation of low- and high-speed streaks. Those are the fingerprint of the near-wall cycle (Jiménez & Pinelli Reference Jiménez and Pinelli1999), active there. Here, $\langle u' u' \rangle$ appears depleted over the canopy, but still remains significant, compatibly with the formation of the large streak-like structures held responsible for the coherent swaying of the stems and the propagation of monami waves (Monti et al. Reference Monti, Olivieri and Rosti2023). The dominant components over the canopy are nevertheless $\langle v' v' \rangle$ and $\langle w' w' \rangle$ , symptomatic of the Kelvin–Helmholtz rollers populating the shear-layer (Finnigan Reference Finnigan2000). The magnitude of $\langle v' v' \rangle$ and $\langle w' w' \rangle$ drops moving towards the middle of the gap, with a slightly slower decay in the most rigid case, $\textit{Ca}=10$ , where the more rigid stems induce the formation of a sharper shear-layer. Thus, in this case, the rollers thus extend further from the canopy towards the gap.

Figure 5. Extra-diagonal components of the Reynolds’ stress tensor in the spanwise $y$ – $z$ plane, for the two different values of $\textit{Ca}$ considered in our study. The fluctuations are averaged in time, along the $x$ axis, and made (anti-)symmetric with respect to the middle of the vegetation gap. The same averaging procedure is adopted to compute the mean envelope of the deflected stem tips, reported as a black line.

We now look at the extra-diagonal terms of the Reynolds’ stress tensor, as shown in figure 5. The $\langle u'v' \rangle$ component peaks above the canopy, where the Kelvin–Helmholtz rollers induce correlated fluctuations of the streamwise and wall-normal velocities. The $\langle u'w' \rangle$ component, instead, denotes regions where fluid with high and low streamwise momentum is exchanged between the canopy and the gap. High-speed fluid from the gap, entrained by the canopy-edge vortex, approaches the canopy from the side. Meanwhile, low-speed fluid emerging from the canopy deviates towards the gap above the canopy edge. These two mechanisms give rise to $u'w'$ events of the same sign, generating the regions of intense $\langle u'w' \rangle$ activity visible in the plots. In the bottom panels of figure 5, the joint fluctuations of $\langle v'w' \rangle$ peak close to the core of the canopy-edge vortex, revealing its unsteady nature.

The role of the turbulent shear stresses is better understood within the context of the shear stress balance, which we therefore elucidate. Let us consider the momentum equation along the streamwise direction,

(3.3) \begin{equation} \frac {\partial u}{\partial t} + u \frac {\partial u}{\partial x} + v \frac {\partial u}{\partial y} + w \frac {\partial u}{\partial z} = - \frac {1}{\rho _f}\frac {\partial p}{\partial x} + \frac {\mu }{\rho _f} \left ( \frac {\partial ^2 u}{\partial x^2} + \frac {\partial ^2 u}{\partial y^2} + \frac {\partial ^2 u}{\partial z^2}\right ) + f_{s,x} + f_{b,x}. \end{equation}

For simplicity, we rename the force $- f_{s,x}$ exerted by the stems on the fluid $d$ , for drag, and incorporate the forcing $f_{b,x}$ into a modified pressure gradient $\partial \tilde {p} / \partial x$ . Introducing the decomposition in (3.1), averaging and reordering the terms, (3.3) rewrites as

(3.4) \begin{align} \frac {\mu }{\rho _f} \left ( \frac {\partial ^2 \langle u \rangle }{\partial y^2} + \frac {\partial ^2 \langle u \rangle }{\partial z^2}\right ) &= \frac {1}{\rho _f} \left \langle \frac {\partial \tilde {p}}{\partial x} \right \rangle + \langle u \rangle \frac {\partial \langle u \rangle }{\partial x} + \left \langle u' \frac {\partial u'}{\partial x} \right \rangle + \langle v \rangle \frac {\partial \langle u \rangle }{\partial y} + \nonumber \\ &\quad +\, \left \langle v' \frac {\partial u'}{\partial y} \right \rangle + \langle w \rangle \frac {\partial \langle u \rangle }{\partial z} + \left \langle w' \frac {\partial u'}{\partial z} \right \rangle + \langle d \rangle . \end{align}

Integrating (3.4) between the bottom wall ( $y = 0$ ) and a generic position $y$ in the wall-normal direction, there follows

(3.5) \begin{align} \frac {\tau _w}{\rho _f} &= \frac {\mu }{\rho _f} \frac {\partial \langle u \rangle }{\partial y} \bigg |_y + \frac {\mu }{\rho _f} \int _0^y \frac {\partial ^2 \langle u \rangle }{\partial z^2}\, {\textrm d}y - \frac {1}{\rho _f} \left \langle \frac {\partial \tilde {p}}{\partial x} \right \rangle y - \int _0^y \left \langle u' \frac {\partial u'}{\partial x} \right \rangle\, {\textrm d}y - \int _0^y \langle v \rangle \frac {\partial \langle u \rangle }{\partial y}\, {\textrm d}y \nonumber \\ &\quad -\, \int _0^y \left \langle v' \frac {\partial u'}{\partial y} \right \rangle \,{\textrm d}y - \int _0^y \langle w \rangle \frac {\partial \langle u \rangle }{\partial z}\, {\textrm d}y - \int _0^y \left \langle w' \frac {\partial u'}{\partial z} \right \rangle {\textrm d}y - \int _0^y \langle d \rangle \,{\textrm d}y, \end{align}

where the shear stress at the wall, $\tau _w$ , stems from

(3.6) \begin{equation} \frac {\mu }{\rho _f} \int _0^y \frac {\partial ^2 \langle u \rangle }{\partial y^2} \,{\textrm d}y = \frac {\mu }{\rho _f} \frac {\partial \langle u \rangle }{\partial y} \bigg |_y - \frac {\mu }{\rho _f} \frac {\partial \langle u \rangle }{\partial y} \bigg |_0 = \frac {\mu }{\rho _f} \frac {\partial \langle u \rangle }{\partial y} \bigg |_y - \frac {\tau _w}{\rho _f}. \end{equation}

Integrating again (3.4) along the wall-normal direction, this time between the bottom wall ( $y = 0$ ) and the top surface ( $y = H$ ), where the free-slip condition is enforced, we find

(3.7) \begin{align} \frac {\tau _w}{\rho _f} &= \frac {\mu }{\rho _f} \int _0^H \frac {\partial ^2 \langle u \rangle }{\partial z^2} \,{\textrm d}y - \frac {1}{\rho _f} \left \langle \frac {\partial \tilde {p}}{\partial x} \right \rangle H - \int _0^H \left \langle u' \frac {\partial u'}{\partial x} \right \rangle \,{\textrm d}y - \int _0^H \langle v \rangle \frac {\partial \langle u \rangle }{\partial y} \,{\textrm d}y \nonumber \\[4pt] &\quad-\, \int _0^H \left \langle v' \frac {\partial u'}{\partial y} \right \rangle \,{\textrm d}y - \int _0^H \langle w \rangle \frac {\partial \langle u \rangle }{\partial z}\, {\textrm d}y - \int _0^H \left \langle w' \frac {\partial u'}{\partial z} \right \rangle \,{\textrm d}y - \int _0^H \langle d \rangle \,{\textrm d}y. \end{align}

Equating the right-hand-sides of (3.5) and (3.7), then de facto removing $\tau _w$ , we find the desired balance

(3.8) \begin{align} & - \underbrace {\left \langle \frac {\partial \tilde {p}}{\partial x} \right \rangle (H - y)}_{\tau } = \underbrace {\rho _f \int _y^H \langle D \rangle \,{\textrm d}y}_{D} + \underbrace {\mu {\frac {\partial \langle u \rangle }{\partial y}}\bigg |_y}_{\tau _1} - \underbrace {\mu \int _y^H \frac {\partial ^2 \langle u \rangle }{\partial z^2} \,{\textrm d}y}_{\tau _2} + \nonumber \\ &\quad +\, \underbrace {\rho _f \int _y^H \left [ \langle v \rangle \frac {\partial \langle u \rangle }{\partial y} + \langle w \rangle \frac {\partial \langle u \rangle }{\partial z} \right ] \,{\textrm d}y}_{\tau _3} + \underbrace {\rho _f \int _y^H \left [ \left \langle u' \frac {\partial u'}{\partial x} \right \rangle + \left \langle v' \frac {\partial u'}{\partial y} \right \rangle + \left \langle w' \frac {\partial u'}{\partial z} \right \rangle \right ]\, {\textrm d}y}_{\tau _4}, \end{align}

where the combination of all the different contributions balances the linear profile of the total shear stress $\tau _{\textit{tot}}(y) = - \langle {\partial \tilde {p}} /{\partial x} \rangle (H - y)$ , with a linear slope set by the driving pressure gradient. In particular:

1. (i) $D$ is the mean canopy drag;

2. (ii) $\tau _1$ is the viscous shear stress;

3. (iii) $\tau _2$ is the viscous diffusion of streamwise momentum in the spanwise direction;

4. (iv) $\tau _3$ is the advection of streamwise momentum by the mean flow in the $y$ – $z$ plane;

5. (v) $\tau _4$ is the turbulent shear stress.

The contributions reported here are all functions of the spanwise and wall-normal coordinates. To ease the comparison with the homogeneous cases and with results typically reported in the literature (Kim, Moin & Moser Reference Kim, Moin and Moser1987), we average them along the spanwise direction enforcing the periodicity of the domain. An over-bar denotes the newly introduced average operator. Each average contribution is reported in figure 6, both at $\textit{Ca}=10$ and $\textit{Ca}=100$ , and compared with the cases of a full canopy and of an open channel with no vegetation. Note that, trivially, there is no average momentum diffusion nor advection ( $\overline {\tau _2}=\overline {\tau _3}=0$ ) in the full canopy case, while in the case of an open channel, also the canopy drag is null ( $\overline {\tau _2}=\overline {\tau _3}=\overline {D}=0$ ).

Figure 6. Wall-normal profiles of the shear-balance terms computed from our simulations of a full canopy (canopy 100 %), a partially obstructed channel (canopy 50 %) and an open channel (canopy 0 %), with the first and the last from our former investigation (Foggi Rota et al. Reference Foggi Rota, Monti, Olivieri and Rosti2024b ). All flow parameters are matched between each set of simulations, and the different contributions are averaged along the periodic directions. We also ensure the matching of the structural parameters in the vegetated cases, for which we report two different values of stem flexibility. The combination of all the different contributions balances the linear profile of the total shear stress $\overline {\tau _{\textit{tot}}}$ , in particular: $\overline {\tau _1}$ is the viscous shear stress, $\overline {\tau _2}$ is the viscous diffusion of streamwise momentum in the spanwise direction, $\overline {\tau _3}$ is the advection of streamwise momentum by the mean flow in the $y$ – $z$ plane, $\overline {\tau _4}$ is the turbulent shear stress and $\overline {D}$ is the mean canopy drag.

In the cases with vegetation, the shear stress at the wall is given by the sum of the mean canopy drag and the viscous shear stress there, $\overline {\tau _w} = \overline {\tau _{\textit{tot}}}(0) = \overline {D}(0) + \overline {\tau _1}(0)$ , as all the other contributions vanish due to the no-slip and no-penetration boundary conditions. The relative magnitude of $\overline {D}(0)$ and $ \overline {\tau _1}(0)$ is likely set by the ratio between the volume occupied by stems and that occupied by stems and fluid within the canopy (i.e. the volume fraction), which for us is approximately $16\,\%$ . Moving upward, away from the wall, the canopy drag $\overline {D}$ and the viscous shear $\overline {\tau _1}$ decrease, while all the other components grow due to the onset of the mean flow and of the turbulent fluctuations. Notably, the growth of the turbulent shear $\overline {\tau _4}$ is non-monotonous in the most rigid case, $\textit{Ca}=10$ , while it appears more regular at $\textit{Ca}=100$ , compatibly with an increased turbulent activity within the canopy in the latter case due to the flapping motion of the stems and the deeper penetration of external fluctuations. The profiles of the viscous shear $\overline {\tau _1}$ and of the turbulent shear $\overline {\tau _4}$ differ from those observed in a full canopy due to the averaging across the vegetated and non-vegetated regions. Nevertheless, the viscous shear $\overline {\tau _1}$ always exhibits a local maximum in correspondence of the drag discontinuity at the canopy tip, where the canopy drag $ \overline {D}$ goes to zero. The peak of $\overline {\tau _1}$ is more spread in the most flexible case since the enhanced compliance of the stems to the flow yields a weaker shear-layer at their tip. The mean flow advection $\overline {\tau _4}$ grows until saturation is attained above the canopy, where the mean flow is more intense. Viscous diffusion $\overline {\tau _2}$ is negligible everywhere. Finally, all contributions decay to zero at the free-slip surface to satisfy the symmetry constraint.

Figure 7. Wall-normal integrals of the averaged shear-balance terms computed from our simulations of a full canopy (canopy 100 %), a partially obstructed channel (canopy 50 %) and an open channel (canopy 0 %), with the first and the last from our former investigation (Foggi Rota et al. Reference Foggi Rota, Monti, Olivieri and Rosti2024b ). The different contributions sum to the total pressure gradient needed to sustain a fully turbulent flow at ${Re}_b=5000$ in each set-up. All flow parameters are matched between each set of simulations. We also ensure the matching of the structural parameters in the vegetated cases, for which we report two different values of stem flexibility. $\overline {\tau _1}$ is the viscous shear stress, $\overline {\tau _2}$ is the viscous diffusion of streamwise momentum in the spanwise direction, $\overline {\tau _3}$ is the advection of streamwise momentum by the mean flow in the $y$ – $z$ plane, $\overline {\tau _4}$ is the turbulent shear stress and $\overline {D}$ is the mean canopy drag.

A more direct quantification of the resistance opposed by the substrate to the flow is achieved integrating the averaged stress components along the wall-normal direction, attaining different scalar contributions which sum to the driving pressure gradient. Those are shown in the histograms of figure 7 for the different cases considered. We first notice that, expectably, the highest drag is attained in the most rigid case of the full canopy, followed by the more compliant one, the two partially obstructed channels ordered by increasing $\textit{Ca}$ and finally the open channel case. The viscous shear contribution $\overline {\tau _1}$ is small and essentially unchanged across all cases, while the viscous diffusion $\overline {\tau _2}$ is totally negligible in agreement with the previous plots (figure 6). The advection of streamwise momentum $\overline {\tau _3}$ accounts for part of the total drag in the partially obstructed channel cases, slightly decreasing for the most compliant stems, while the most significant contribution comes from the turbulent shear stress $\overline {\tau _4}$ in all cases. Differently from the full canopy, where the turbulent shear is depleted increasing the flexibility of the stems, this does not appear to be the case in the partially obstructed channel. There, the turbulent shear generated by the discontinuity between the vegetated and the non-vegetated regions overshadows the effect of a variation in the stem flexibility. The flexibility effect is thus limited to the canopy drag $\overline {D}$ , which undergoes a decrease of more than $50\,\%$ increasing from $\textit{Ca}=10$ to $\textit{Ca}=100$ mainly due to the deflection of the stems and the consequent reduction of the frontal area of the canopy.

Figure 8. Instantaneous flow velocity fluctuations in a wall-parallel plane at $y=0.01H$ of our partially obstructed channels, with the flow going from left to right. The left column refers to the case with the most rigid stems, the right column to that with the most compliant ones. In the first row, we show the streamwise velocity component, ranging from blue to red within $\pm 0.5 U_b$ ; in the second row, we show the wall-normal velocity component, ranging from blue to red within $\pm 0.2 U_b$ ; in the third row, we show the spanwise velocity component, ranging from blue to red within $\pm 0.5 U_b$ .

3.3. Turbulent structures and events

Different regions of the flow are populated by different kinds of coherent turbulent motions, interacting one with the other. We start observing those close to the bottom wall, slicing the domain with a wall-parallel plane at $y=0.01H$ ; the different components of the fluctuating velocity $\boldsymbol {u'}$ there are shown in figure 8. The streamwise component $u'$ within the vegetation gap (the central ‘band’ of the domain) highlights the formation of the typical low- and high-speed streaks found in wall turbulence (Smith & Metzler Reference Smith and Metzler1983), while within the canopy, we observe large regions of coherent motion extended in the spanwise direction. Their coherence is reduced increasing the flexibility of the stems, due to the disruptive effect of their flapping. A similar effect is observed for the spanwise component $w'$ , where, nevertheless, the coherent regions within the canopy exhibit a more oblique arrangement. Comparing $u'$ and $v'$ within the canopy, we better understand their spatial coherence with the formation of local zones of flow divergence (Monti et al. Reference Monti, Omidyeganeh, Eckhardt and Pinelli2020) where the vertical momentum penetrating from the canopy tip gets redistributed along the wall-parallel directions. Accordingly, the wall-normal component $v'$ within the canopy close to the wall is small and incoherent, since it gets disrupted by the stems. Only in the most flexible case do we witness few intense jets of wall-normal velocity reaching the wall. This is not the case for the flow within the vegetation gap: there, all the velocity components exhibit the most intense fluctuations approaching the interface between the vegetated and non-vegetated regions, due to the enhanced turbulent activity.

Figure 9. Visualisations of the instantaneous flow field in our partially obstructed channels, with the flow going from left to right aligned with the vegetation gap. The left column refers to the case with the most rigid stems, the right column to that with the most compliant ones. In all plots, the elevation of the stems between $y=0$ and $y=0.25H$ is denoted varying their colour continuously from green to white. In the first row, we have filtered the fluctuating velocity field $\boldsymbol {u'}$ as discussed in the main text, and, thus, we show the iso-surfaces of its streamwise (red) and spanwise (blue) vorticity $\tilde {\boldsymbol{\omega '}}$ at a fixed value of $1 U_b/H$ . In the second row, instead, we directly report the iso-surfaces of the streamwise velocity fluctuations $u'$ at $-0.3 U_b$ (blue) and $0.3 U_b$ (red).

We now investigate the turbulent structures generated above the canopy tip and in the vegetation gap. The presence of intense vortices above the vegetation (and rough/porous substrates in general) is well established and understood (Finnigan Reference Finnigan2000; Jiménez et al. Reference Jiménez, Uhlmann, Pinelli and Kawahara2001), yet their extension towards the vegetation gap and their modulation with the stem flexibility is yet to be clarified in the present case. To extract them, we thus deploy two different box filters: the first one, with a preferential streamwise orientation, has a cuboidal kernel made by $61\times 15\times 15$ grid cells along the streamwise, wall-normal and spanwise directions, respectively. The second one, with a preferential spanwise orientation, has a cuboidal kernel made by $15\times 61\times 15$ grid cells. After applying the first filter to an instantaneous field of the fluctuating velocity $\boldsymbol {u'}$ , we thus compute its streamwise vorticity $\tilde {\omega _x}$ and report its iso-surfaces in red in the top panels of figure 9. Similarly, after applying the second filter to the same instantaneous field of the fluctuating velocity  $\boldsymbol {u'}$ , we compute its spanwise vorticity $\tilde {\omega _z}$ and report its iso-surfaces in blue in the top panels of figure 9. Qualitatively similar results are attained varying the kernel size of the filter within a reasonable range. We are thus able to observe intense spanwise-oriented rollers spanning the flow region immediately above the canopy tip, inducing secondary streamwise-oriented vortices (much like in figure 12 of Finnigan Reference Finnigan2000). In the most rigid case, the strong drag discontinuity at the canopy tip induces the formation of intense rollers, eventually able to merge across the vegetation gap and to span almost the whole domain transversally. In the most flexible case, instead, the rollers appear limited to the vegetated region and its immediate neighbourhood.

Also for the streamwise-oriented vortices, we observe a more homogeneous distribution throughout the domain in the case at $\textit{Ca}=100$ . One of the processes generating them is in fact related to the secondary instability of the spanwise rollers (Finnigan Reference Finnigan2000), and they thus occupy similar regions. In between pairs of streamwise-oriented vortices, streak-like regions of high and low streamwise velocity $u'$ are generated. We thus visualise them in the bottom panels of figure 9 for the different values of stem rigidity we considered, denoting in blue a negative velocity fluctuation and in red a positive one. Indeed, we observe large elongated structures over the canopy, visibly responsible for the deflection of the stems below them in the most flexible case. We also appreciate an alternative visualisation of the ‘typical’ streaks close to the wall within the gap, with respect to that provided in figure 8. At the interface between the vegetated and non-vegetated regions, in the most rigid case, we notice the generation of larger structures likely associated to edge effects, yet we cannot confirm their presence in the most flexible case.

To better understand how momentum is exchanged between the canopy and the outer flow, we now turn our attention to the events taking place at the interface between the two (on top and on the side of the canopy). Sweep events at the average position of the canopy tip are defined as flow regions where $u'\gt 0$ and $v'\lt 0$ , while ejections have $u'\lt 0$ and $v'\gt 0$ (Gao et al. Reference Gao, Shaw and Paw1989; Poggi et al. Reference Poggi, Katul and Albertson2004a ). At the left interface between the vegetated and non-vegetated regions, sweeps have $u'\gt 0$ and $w'\lt 0$ while ejections have $u'\lt 0$ and $w'\gt 0$ , yet at the right interface, sweeps have $u'\gt 0$ and $w'\gt 0$ while ejections have $u'\lt 0$ and $w'\lt 0$ . To resolve this ambiguity, we denote by $w^{\prime}_{out}$ the fluctuations of the spanwise velocity going out from the canopy and use for both interfaces the definition natural to the left one. Instantaneous visualisations of the events are provided in figure 10. At the canopy tip, as already noticed in our previous work (Foggi Rota et al. Reference Foggi Rota, Monti, Olivieri and Rosti2024b ), sweep events appear less frequent than ejections, but often have higher intensity. Increasing the flexibility of the stems, the events grow larger and more spatial coherence is attained. An analogous behaviour is found at the lateral interface between the canopy and the gap, where we notice that the most intense events take place far away from the wall, compatibly with the enhanced turbulent activity observed there (see figure 4).

Figure 10. Instantaneous sweep and ejection events at the average position of the canopy tip (larger rectangles) and at the interface between the vegetated and non-vegetated regions (smaller rectangles) for the the two different values of $\textit{Ca}$ considered in our study. We sample the flow on wall-parallel planes (larger rectangles) and streamwise-oriented, wall-normal planes (smaller rectangles) with the mean flow going from bottom to top. Regions where the events are occurring are delimited with black lines, while their magnitude is quantified as $|u'v'|/U_b^2$ or $|u'w^{\prime}_{out}|/U_b^2$ and visualised with a linear colour map ranging from white to orange (ejections, in $[ 0, 0.2 ]$ ) or blue (sweeps, in $[ 0, 0.3 ]$ ).

Figure 11. Isolines of the joint probability density functions (j.p.d.f.s; normalised to a unitary integral over the domain) associated with the fluctuations of the streamwise velocity component and with the velocity component going out from the canopy at the planes selected in figure 10. We consider the mean position of the canopy tip ( $u'v'$ panels) and at the interface between the vegetated and non-vegetated regions ( $u'w^{\prime}_{out}$ panels), for the the two different values of $\textit{Ca}$ employed in our study. Levels are evenly distributed between 0.4 and 6 in 0.4 increments, while the locations of the peaks are denoted by red dots.

To provide a more quantitative description of the events, we now observe the joint probability density functions (j.p.d.f.s) of the fluctuating streamwise velocity component and of the fluctuating velocity component exiting the canopy, within the planes selected here. Those are reported in figure 11. Ejections are the most frequent events at the tip of the canopy, while sweeps are rarer but stronger: the canopy opposes little hindrance to the fluid being lifted from behind the rollers spanning its tip, while only the most intense downwellings in front of the rollers are able to penetrate between the stems. This holds true for both the values of $\textit{Ca}$ we considered, yet in the most flexible case, the j.p.d.f. exhibits a more oval shape as less streamwise momentum is lost by the fluid upon impinging on the stems. Additionally, at the interface between the vegetated and non-vegetated regions, ejections appear slightly more frequently than sweep events, with the latter reaching higher intensities. Yet the difference is significantly less sharp, suggesting that wall-normal Kelvin–Helmholtz-like rollers are unlikely to drive the momentum exchange there.

3.4. Stem reconfiguration and dynamics

Investigating the dynamics of the flow, we have observed how that is affected by a variation in the stem flexibility, understanding the important role played by the reconfiguration and dynamics of the stems. Here, we aim at further characterising these two aspects from the perspective of the structure.

Figure 12. (a,c) Streamwise and (b,d) spanwise deflection of the stems for different spanwise locations; $z=0$ lays at the middle of the vegetated region, while $z=1.15H$ lays at its margin. Results for the most rigid stems are reported in panels (a,b), while panels (c,d) refer to the most flexible ones. The stems oscillate about their time-averaged configuration, with the shaded regions denoting the root mean square of their displacement.

The reconfiguration of flexible structures exposed to an incoming flow is a widely investigated topic due to its capability of reducing the drag experienced by the structure (Alben, Shelley & Zhang Reference Alben, Shelley and Zhang2002; Luhar & Nepf Reference Luhar and Nepf2011; Gosselin Reference Gosselin2019) and to modify the mean flow properties setting an ‘implicit’ boundary condition dependent on the flow itself (Akita, Okabayashi & Takeuchi Reference Akita, Okabayashi and Takeuchi2024). Reconfiguration is typically studied in relatively simple set-ups where the incoming flow, either laminar or turbulent, has no significant transversal components. Instead, here, we investigate how the structure deflects in response to the fully bi-dimensional mean flow established in our set-up. The deflection of the stems, reported in figure 12, is predictably more pronounced in the case at $\textit{Ca}=100$ , where they undergo a finite reconfiguration in the streamwise direction. Moderate reconfiguration along the streamwise direction can also be appreciated for the most rigid stems, while in no case, we observe a significant deflection in the spanwise direction. The spatial pattern with which the stems bend forward, on average, is a consequence of the mean flow to which they are exposed and, in particular, of the canopy-edge vortex. Consequently, the most upright stems (located approximatively at $z=1.07H$ ) are found in the middle of the upwelling described in § 3.1, while those located deeper in the canopy bend more, under the influence of the descending fluid. Furthermore, the stems located closer to the interface between the vegetated and non-vegetated regions achieve a higher deflection, bending under the high-momentum fluid coming from the gap. Once deflected, the stems oscillate about their time-averaged configuration; we thus report the root mean square of their displacement with the shaded regions in figure 12. Noticeably, in the case at $\textit{Ca}=10$ , oscillations of wider amplitude are achieved next to the gap, marking the intense turbulent activity associated with the canopy-edge vortex. This is not the case for the most compliant stems, which all exhibit wider fluctuations of comparable amplitude.

Our code tracks the position of each stem tip over time, allowing us to reconstruct the full envelope of the deflected canopy tips at each time instant through a binning procedure. We thus characterise the canopy envelopes through the p.d.f.s of their elevation $\eta (x,y)$ , averaging the p.d.f.s across all time instants. As shown in figure 13(a), for $\textit{Ca}=10$ , the p.d.f. exhibits a monotonous trend, quickly growing to its maximum at the value corresponding to undeflected stem ( $\eta =0.25$ ). For $\textit{Ca}=100$ , instead, the p.d.f. appears significantly broader and centred around $\eta \approx 0.2$ , confirming not only the increased deflection of the stems due to their higher compliance, but also their enhanced swaying motion. Next, in figure 13(b,c), we report the j.p.d.f.s of the elevation gradients. The spatial derivatives of $\eta (x,y)$ have significantly smaller magnitudes at $\textit{Ca}=10$ than at $\textit{Ca}=100$ , consistent with the reduced motion of the stems in the former case. Yet, both plots highlight how the spanwise gradient $\partial _z \eta$ attains larger values than the streamwise one $\partial _x \eta$ , suggesting intense variations in the stem tips configuration along the transverse direction. While, in fact, only the turbulent fluctuations are responsible for rippling the canopy envelope along $x$ , both the turbulent fluctuations and the mean flow inhomogeneity are active along $y$ .

Figure 13. (a) Probability density functions of the canopy envelope elevation $\eta (x,y)$ for the two values of $\textit{Ca}$ considered in our study. (b,c) J.p.d.f.s for the same values of $\textit{Ca}$ of the envelope gradients along the streamwise and spanwise directions.

Figure 14. Magnitude of the temporal response of the stems at different spanwise locations, for (a) the most rigid and (b) the most compliant case. The response is computed as the squared Fourier transform of the spanwise stem-tip velocity, averaged across stems at the same spanwise location. Black lines aid the identification of the dynamical regime at which the response is occurring: $f_{\textit{nat}}$ is defined as in § 2, while $f_{\textit{turb}}\approx 0.5U_b/H$ following previous works (Foggi Rota et al. Reference Foggi Rota, Koseki, Agrawal, Olivieri and Rosti2024a ).

To link the topological features of the canopy envelope with the motion of the stems, we further probe their dynamic response in the frequency domain. We focus on the spanwise velocity of the stem tips as it is the Lagrangian velocity component less affected by the inextensibility of the stems and by the presence of the bottom wall. Sampling its signal for all stems over approximately $50 H/U_b$ , we thus compute its temporal Fourier transform and average along the streamwise direction over all stems within tiles (see § 2.1) occupying the same spanwise location, symmetrically across the gap. The outcome, $\mathcal{F}(w/U_b)^2$ as reported in figure 14, is a map of the response intensity at each frequency as a function of the spanwise location of the stems. To understand the regime of oscillations of the stems, that being either their natural dynamics or a compliant response to the turbulent flow (Foggi Rota et al. Reference Foggi Rota, Koseki, Agrawal, Olivieri and Rosti2024a ), we overlay on the plots a dashed line corresponding to the structural natural frequency $f_{\textit{nat}}$ (as defined in § 2) and a continuous line corresponding to the turbulent frequency of the largest structures in the flow, $f_{\textit{turb}}\approx 0.5U_b/H$ (Foggi Rota et al. Reference Foggi Rota, Koseki, Agrawal, Olivieri and Rosti2024a ). From the plots, it thus appears clear that at $\textit{Ca}=10$ , the stem dynamics is dominated by their natural response, while the case at $\textit{Ca}=100$ is close to the cross-over where $f_{\textit{nat}}\approx f_{\textit{turb}}$ . The more rigid stems oscillate with their natural dynamics and exhibit a wider range of excitation closer to the canopy-edge vortex ( $z/H\gtrsim 1$ ), where the turbulent forcing is more broad-banded. This latter observation appears consistent with the enhanced amplitude of the stem oscillations observed there, as in figure 12. The more flexible stems, instead, are significantly more deflected and thus hindered in their natural dynamics. They therefore approach a regime where their motion is fully dominated by the turbulent forcing and oscillate at a broader range of frequencies centred about $f_{\textit{turb}}$ . The more flexible stems consequently explore a broader range of elevations, as denoted by their p.d.f. in figure 13, and yield a more rippled envelope.