2.1. Fluid phase

Fluid motion is governed by the unsteady three-dimensional Navier–Stokes equations for a Newtonian viscous fluid of constant density,

(2.1a) \begin{align} \frac {\partial \boldsymbol{u}}{\partial t} + \nabla \mathbin {\boldsymbol{\cdot }}\left (\boldsymbol{u}\mathbin {\otimes }\boldsymbol{u}\right ) & = \frac {1}{\rho }\nabla \mathbin {\boldsymbol{\cdot }}\boldsymbol{\sigma } + \boldsymbol{f} ,\end{align}

(2.1b) \begin{align} \nabla \mathbin {\boldsymbol{\cdot }}\boldsymbol{u} & = 0 , \\[6pt] \nonumber\end{align}

where $\boldsymbol{u} = (u, v, w)^\top$ is the velocity vector with its components along the Cartesian coordinates $x$ , $y$ , $z$ , while $t$ represents the time, $\rho$ the fluid density and $\boldsymbol{f}$ a mass-specific force. The latter is obtained as

(2.2) \begin{equation} \boldsymbol{f} = \boldsymbol{f_{d}} + \boldsymbol{\boldsymbol{f_{{\!\Gamma }}}} , \end{equation}

with $\boldsymbol{f_{d}} = (f_{d}, 0, 0)^\top$ , a volume force driving the flow. The second term, $\boldsymbol{\boldsymbol{f_{{\!\Gamma }}}}$ , results from the fluid--structure coupling, as described below. The hydrodynamic stress tensor reads

(2.3) \begin{equation} \boldsymbol{\sigma } = \boldsymbol{\sigma }_{h} = -p\, \boldsymbol{\mathsf{I}} + \rho \nu 2 \unicode{x1D64E} ,\qquad \unicode{x1D64E} = \tfrac {1}{2} (\nabla \boldsymbol{u} + \nabla \boldsymbol{u}^\top ), \end{equation}

where $p$ is the pressure, $ \boldsymbol{\mathsf{I}}$ the identity matrix, $\nu$ the kinematic viscosity and $ \unicode{x1D64E}$ the strain rate tensor.

The Navier–Stokes equations were solved with a second-order finite volume approach on a staggered Cartesian grid for the spatial discretization, and a semi-implicit second-order scheme for the time integration (Kempe & Fröhlich Reference Kempe and Fröhlich2012; Tschisgale et al. Reference Tschisgale, Kempe and Fröhlich2017, Reference Tschisgale, Kempe and Fröhlich2018). An LES approach was employed, using the Smagorinsky subgrid-scale model, with the Smagorinsky constant $C_{\textit{S}}$ , to compute the subgrid-scale viscosity $\nu _{{\textit{sgs}}}$ , so that the stress tensor in (2.3) is redefined as (Smagorinsky Reference Smagorinsky1963)

(2.4) \begin{equation} \boldsymbol{\sigma } = \boldsymbol{\sigma }_{h} - \boldsymbol{\tau }_{{\textit{sgs}}} ,\qquad \boldsymbol{\tau }_{{\textit{sgs}}} = -\rho \nu _{{\textit{sgs}}} 2 \unicode{x1D64E} + \tfrac {1}{3}\textrm{tr}({\boldsymbol{\tau }_{{\textit{sgs}}}}) \boldsymbol{\mathsf{I}} . \end{equation}

2.2. Blades

The model canopy is composed of elastic, slender ribbons of length $L$ , width $W$ and thickness $T$ , with $L \gt W \gg T$ . As in Tschisgale et al. (Reference Tschisgale, Löhrer, Meller and Fröhlich2021), the motion of these ribbons is modelled by a geometrically exact Cosserat rod, a one-dimensional model suitable for rods undergoing large deflections that can be considered the geometrically nonlinear generalization of a Timoshenko–Reissner beam Lang, Linn & Arnold (Reference Lang, Linn and Arnold2011). A Cosserat rod is capable of representing bending, torsion, as well as extension and shearing (Lang et al. Reference Lang, Linn and Arnold2011). The motion of such a Cosserat rod can be expressed by the equations for the linear and angular momentum (Simo Reference Simo1985; Antman Reference Antman1995; Lang et al. Reference Lang, Linn and Arnold2011)

(2.5a) \begin{align} \rho _{s}A\boldsymbol{\ddot {c}} & = \frac {\partial \boldsymbol{\overset {\,\scriptscriptstyle \triangle }{f}}}{\partial s} + \boldsymbol{\overset {\,\scriptscriptstyle \triangle }{f}}, \end{align}

(2.5b) \begin{align} \rho _{s} \left (\unicode{x1D644}\mathbin {\boldsymbol{\cdot }}\boldsymbol{\dot {\omega }} + \boldsymbol{{\omega }}\times \unicode{x1D644}\mathbin {\boldsymbol{\cdot }}\boldsymbol{{\omega }} \right ) & = \frac {\partial \boldsymbol{\overset {\,\scriptscriptstyle \triangle }{\,m}}}{\partial s} + \frac {\partial \boldsymbol{c}}{\partial s} \times \boldsymbol{\overset {\,\scriptscriptstyle \triangle }{f}} + \boldsymbol{\overset {\,\scriptscriptstyle \triangledown }{m}} , \end{align}

where $\boldsymbol{c} = \boldsymbol{c} ({s, t} )$ is the position of the skeleton line in the laboratory coordinate system, with $s$ the coordinate along the skeleton line of the structure, such that $\partial \boldsymbol{c}/\partial s$ gives the local orientation of the structure and its second temporal derivative $\boldsymbol{\ddot {c}}$ the acceleration. The translational and rotational inertia of cross-sectional segments are given through the density $\rho _{s}$ of the structure material, the cross-sectional area $A$ of the blade, and the tensor of inertia $ \unicode{x1D644}$ . The angular velocity of the cross-sections of a rod is denoted by $\boldsymbol{\omega } ({s, t} )$ , while internal forces and torques are denoted by $\boldsymbol{\overset {\,\scriptscriptstyle \triangle }{f}}$ and $\boldsymbol{\overset {\,\scriptscriptstyle \triangle }{\,m}}$ , respectively. The arc-length-specific forces and torques imposed from the exterior onto the structure are given by $\boldsymbol{\overset {\,\scriptscriptstyle \triangledown }{f}}$ and $\boldsymbol{\overset {\,\scriptscriptstyle \triangledown }{m}}$ , respectively.

The internal forces $\boldsymbol{\overset {\,\scriptscriptstyle \triangle }{f}}$ and moments $\boldsymbol{\overset {\,\scriptscriptstyle \triangle }{\,m}}$ obey viscoelastic material behaviour (Lang & Linn Reference Lang and Linn2009; Lang et al. Reference Lang, Linn and Arnold2011). Calculated in the material reference frame denoted by the index $0$ , these forces are defined as

(2.6) \begin{equation} \boldsymbol{\overset {\,\scriptscriptstyle \triangle }{f}}_{\!{\textrm{0}}} = \unicode{x1D63E}_{\varepsilon } \mathbin {\boldsymbol{\cdot }} \boldsymbol{\varepsilon _{{\textrm{0}}}} + 2 \unicode{x1D63E}_{\dot {\varepsilon }} \mathbin {\boldsymbol{\cdot }} \boldsymbol{\dot {\varepsilon }_{{\textrm{0}}}} ,\qquad \boldsymbol{\overset {\,\scriptscriptstyle \triangle }{\,m}_{{\textrm{0}}}} = \unicode{x1D63E}_{\kappa } \mathbin {\boldsymbol{\cdot }} \boldsymbol{\kappa _{{\textrm{0}}}} + 2 \unicode{x1D63E}_{\dot {\kappa }} \mathbin {\boldsymbol{\cdot }} \boldsymbol{\dot {\kappa }_{{\textrm{0}}}}, \end{equation}

with the Hookean-like matrices

(2.7) \begin{equation} \unicode{x1D63E}_{\varepsilon } = \textrm{diag} ({k_{{s}_1}GA, k_{{s}_2}GA, EA}) ,\qquad \unicode{x1D63E}_{\kappa } = \textrm{diag}({EI_1, EI_2, k_{t}GI_3}) , \end{equation}

the strain vector $\boldsymbol{\varepsilon }$ and the curvature vector $\boldsymbol{\kappa }$ . The geometric properties $I_1$ and $I_2$ in (2.7) contain the area moments with respect to the spanwise and the blade-normal axes, respectively, while $I_3$ is the polar area moment (Linn, Lang & Tuganov Reference Linn, Lang and Tuganov2013). The matrices $ \unicode{x1D63E}_{\dot {\varepsilon }}$ and $ \unicode{x1D63E}_{\dot {\kappa }}$ , provide contributions of dissipative internal force and torque. This dissipation, i.e. damping, was assumed to be negligible due to the elevated Cauchy number. With fluid drag vastly superior to restoring elastic forces, material damping of the blades was concluded to be negligible compared with the damping executed by the surrounding fluid, as in all such studies (Sundin & Bagheri Reference Sundin and Bagheri2019; Wang et al. Reference Wang, He, Dey and Fang2022; Monti et al. Reference Monti, Olivieri and Rosti2023; Foggi Rota et al. Reference Foggi Rota, Monti, Olivieri and Rosti2024b ).

The factors ${k_{\textit{s}_1}}, {k_{\textit{s}_2}} \in [0,1]$ in (2.7) are Timoshenko shear correction factors that depend on the geometry of the cross-section and serve to account for the non-uniformity of stresses and strain within cross-sections (Cowper Reference Cowper1966). The factor $k_{t} \in [0,1]$ approximates the effect of torsional out-of-plane warping (Linn et al. Reference Linn, Lang and Tuganov2013). In fact, the Cosserat model, given its one-dimensional formulation, is inherently unable to represent this effect, since it would require deformable cross-sections. The shear correction factors are a standard approach to cope with this issue, with numerous expressions for obtaining these values proposed in the literature (Cowper Reference Cowper1966; Kaneko Reference Kaneko1975; Hutchinson Reference Hutchinson2000; Gruttmann & Wagner Reference Gruttmann and Wagner2001). The effect of warping was addressed, e.g. by Dong, Alpdogan & Taciroglu (Reference Dong, Alpdogan and Taciroglu2010); Freund & Karakoç (Reference Freund and Karakoc2016), extending Saint–Venant’s theory of uniform torsion (Simo & Vu-Quoc Reference Simo and Vu-Quoc1991).

The spatial discretization of the Cosserat rods is accomplished by representing the structure in the form of straight, possibly twisted, longitudinal segments of length $L_{e}$ as sketched in figure 1(b). A staggered finite difference scheme is employed, with quaternions ( $\boldsymbol{q}$ in figure 1 b) defined at the centre points of the elements, describing their orientation and angular velocity. Linear velocity and position are defined at the edges of the segments, as illustrated in figure 1(a) (Lang et al. Reference Lang, Linn and Arnold2011; Tschisgale, Thiry & Fröhlich Reference Tschisgale, Thiry and Fröhlich2019). The numerical solution of the structure is advanced in time within the temporal loop for the fluid. In an inner loop the equations are solved by means of RADAU5 (Hairer & Wanner Reference Hairer and Wanner1996), which is based on an implicit Runge–Kutta method of order 5. Further details are available in Lang & Linn (Reference Lang and Linn2009); Lang et al. (Reference Lang, Linn and Arnold2011); Linn et al. (Reference Linn, Lang and Tuganov2013).

2.3. Fluid–structure coupling

Fluid phase and elastic blades are coupled by a no-slip condition at the surface of the blades. This is accomplished by the IBM of Tschisgale & Fröhlich (Reference Tschisgale and Fröhlich2020). In this framework, Lagrangian marker points are distributed across the coupled surfaces, with every point attributed the velocity of its respective location on the ribbon. In each time step the fluid velocity is interpolated to determine the coupling forces at the marker points via the so-called direct forcing approach (Mohd-Yusof Reference Mohd-Yusof1997; Fadlun et al. Reference Fadlun, Verzicco, Orlandi and Mohd-Yusof2000; Tschisgale & Fröhlich Reference Tschisgale and Fröhlich2020). These coupling forces are spread to the proximate Eulerian grid points providing the coupling force term $\boldsymbol{\boldsymbol{f_{{\!\Gamma }}}}$ in (2.2). Both interpolation and spreading are accomplished with the three-dimensional smoothed delta function proposed by Roma, Peskin & Berger (Reference Roma, Peskin and Berger1999). The same delta function is used to dampen the subgrid-scale viscosity near the blades as required by the Smagorinsky model employed.

The blades are very slender, i.e. their thickness $T$ is much smaller than the step size of the Eulerian grid, $T\lt \unicode{x1D6E5} _{g}$ , with $\unicode{x1D6E5} _{g} = \sqrt [3]{\unicode{x1D6E5} _x\,\unicode{x1D6E5} _y\,\unicode{x1D6E5} _z}$ . Hence, each segment of the rod model is attributed to a quadrangular element of length $L_{e}$ , width $W$ and of vanishing thickness (figure 1 a). The precise length of such an element is permitted to vary by the algorithm employed, as the structure elements can be stretched, but this form of deformation is negligible in the present application. Immersed-boundary method coupling is achieved on the basis of $\left \lceil 2\,L_{e}/\unicode{x1D6E5} _{g} \right \rceil \times \left \lceil 2\,W/\unicode{x1D6E5} _{g} \right \rceil$ Lagrangian marker points spread over this element, as illustrated in figure 1(c). Each marker represents the corresponding surface tile of area ${S_m}_l\approx (\unicode{x1D6E5} _{g}/2)^2$ and is attributed with a mass ${m_m}_l = \rho \,{S_m}_l\,\unicode{x1D6E5} _{g}$ . The safety factor of $2$ introduced here positions the markers separated by half the Eulerian step size, ensuring that ${S_m}_l\,\unicode{x1D6E5} _{g} \leqslant \unicode{x1D6E5} _{g}^3$ , which is required by the IBM (Tschisgale, Kempe & Fröhlich Reference Tschisgale, Kempe and Fröhlich2018). Observe that in addition to the situation shown in the illustrative sketches of figure 1, the cross-sections of the rods are free to rotate, resulting in possibly twisted and asymmetrically stretched blade elements. This is accounted for when distributing the marker points by interpolating the orientation of the cross-sections along the blade centreline as needed. More details of the immersed-boundary coupling are given by Tschisgale et al. (Reference Tschisgale, Kempe and Fröhlich2018); Tschisgale & Fröhlich (Reference Tschisgale and Fröhlich2020).

Figure 1.

Representations of the flexible blades in the numerical method with an arbitrary element being highlighted. (a) Geometric representation of the rod elements accounting for their vanishing thickness; (b) one-dimensional representation of the corresponding discretized Cosserat rod with the staggered locations of solution variables indicated: orientation and angular velocity, position and linear velocity. (c) Marker points employed in the IBM (), with the surface patch associated with a single marker point at an arbitrary position ${\boldsymbol{x}_m}_l$ being highlighted. The sketches are not to scale, with the thickness $T$ and the length $L_{e}$ exaggerated.

2.4. Structure–structure interaction

When moving inside the canopy, the flexible blades collide very frequently with each other. This is accounted for by imposing appropriate collision forces on colliding blade segments, determined via the constraint-based collision model of Tschisgale et al. (Reference Tschisgale, Thiry and Fröhlich2019). To this end, pairs of blade elements with the shortest distance smaller than $2\unicode{x1D6E5} _{g}$ are identified. For each pair, the relative velocity between the two contact points, one on each element, and by the end of the current time step is estimated. This relative velocity accounts for the current velocity and any expected acceleration of the two elements due to external and interior loads. If the elements are found to be approaching one another, a collision impulse is computed and subsequently imposed on both elements in the structure solver, and with opposite sign, to fulfil the kinematic and dynamic conditions. Since an element can be involved in more than one collision, the computation of this collision impulse requires an iterative procedure. A coefficient of restitution equal to zero was used that is motivated by the soft material considered and the dominance of lubrication forces. In the model, the tangential components obey the Coulomb friction law. Here, friction coefficients of zero were employed, which corresponds to slippery contact. In addition to the forces between the blades upon direct contact, unresolved lubrication forces are represented by the lubrication model of Tschisgale (Reference Tschisgale2020) if blade segments are closer than $4\unicode{x1D6E5} _{g}$ . Both methods were validated in the respective references, and were successfully employed in a previous study by the present authors (Tschisgale et al. Reference Tschisgale, Löhrer, Meller and Fröhlich2021). Several enhancements of the collision model were devised in the course of the present study that will be presented in a forthcoming publication (Löhrer et al. Reference Löhrer, Krause and Fröhlich2025).

3.1. Physical problem

The set-up investigated is based on a configuration previously studied experimentally by Guiot De La Rochère et al. (Reference De La Rochère, Léo, Delphine, Soundar, Löhrer, Fröhlich and Rivière2022) and simulated numerically by the present authors (Löhrer et al. Reference Löhrer, Guiot de la Rochère, Doppler, Puijalon and Fröhlich2022, Reference Löhrer, Doppler, Puijalon, Rivière, Jerome and Fröhlich2020). In this study, blades made up of low-density polyethylene (LDPE) plastic film were attached to the base plate of a flume in a newly proposed order to avoid channelling and other regularities as much as possible. This situation naturally featured sidewalls. To better address fundamental issues of the interaction between the outer flow and the canopy, the present configuration was devised by considering the same canopy in the corresponding fully developed and infinitely extending situation, i.e. without any sidewalls, such that the flow is statistically homogeneous in both streamwise and spanwise directions. This lends itself very well to be represented efficiently by periodic boundary conditions as detailed below. The properties of the blades and their arrangement were left unchanged with respect to Guiot De La Rochère et al. (Reference De La Rochère, Léo, Delphine, Soundar, Löhrer, Fröhlich and Rivière2022). Hence, the problem of interest is an open channel of height $H$ , with a steady bulk velocity $U$ and a canopy along the bottom no-slip wall. The relevant physical parameters are provided in table 2 and the placement of the structures is illustrated in figure 2.

Table 2.

Dimensional physical parameters defining the set-up constituted by the fluid and blades that form the canopy.

Figure 2.

Placement of the structures on the bed, indicated by their root lines.

Table 3.

Dimensionless parameters of the present canopy. Upper part: independent parameter numbers based on the a priori defined physical parameters in table 2, with $\varDelta\rho = \rho _{s} - \rho$ . Lower part: deduced additional dimensionless numbers and numbers obtained from the simulation results according to definitions provided in the text.

The problem is characterized by the dimensionless numbers in table 3, of which the independent numbers can be determined a priori, whereas the additional dimensionless numbers were determined a posteriori from the simulation data. The variable $h$ represents the mean canopy height, constituting the average height of the fluid--canopy interface, and computed as discussed in § 6.4 below.

The hydraulic density of the canopy is quantified by its roughness density $\lambda$ . Owing to reconfiguration, which is flow dependent and not known in advance, this quantity cannot be determined a priori. To provide a classification at this point, simulation data reported below are used here. Evaluating (1.3) with $h^\ast = 0.25\,H$ , which is the tip height of the averaged blade geometry identified in § 6.1 below, yields $\lambda = 0.43$ . When, instead, determined from the averaged instantaneous frontal area of the blades, a larger value of $\lambda = 0.56$ was obtained, computed as

(3.1) \begin{equation} \lambda = {\left \langle {\frac {1}{A_{s}}\int \limits _{{s=0}}^{{L}}\!\boldsymbol{e}_{x}\mathbin {\boldsymbol{\cdot }}\boldsymbol{n}({s, t})\,W\,\textrm{d} s \,}\right \rangle } ,\end{equation}

where $A_{s}$ is the bed area per blade (total bed area divided by total number of blades), $\boldsymbol{e}_{x}$ is the unit vector in the streamwise direction, $\boldsymbol{n}$ is the instantaneous unit vector normal to the surface of the blade at a given arc-length position $s$ and $\langle {\cdot } \rangle$ denotes the average in time and over all blades. Either value of $\lambda$ is well above the thresholds provided in the literature, which is approximately $0.23$ according to Nepf (Reference Nepf2012b ) and $0.15$ according to Monti et al. (Reference Monti, Omidyeganeh, Eckhardt and Pinelli2020), so that the present canopy is classified as dense.

The fluid--structure interaction is characterized by the (nominal) Cauchy number $\textit{Ca}$ and by the reduced velocity $U_{r}$ . Definitions are provided in table 3. The latter relates the fundamental natural frequency of the blades to the time scale of the turbulent shear. The natural frequency of the blades can be estimated as (Han, Benaroya & Wei Reference Han, Benaroya and Wei1999)

(3.2) \begin{gather} f_{n} = \frac {1}{\alpha }\sqrt {\frac {\textit{EI}/L^3}{m}},\qquad \alpha = \frac {2\pi }{{1.875}^2}, \end{gather}

where $m = m_{s} + m_{a}$ with $m_{s} = \rho _{s} L WT$ the mass of the structure and $m_{{a}} = \rho L \pi (W/2)^2$ the added mass created by the surrounding fluid (Dong Reference Dong1978). Owing to the extreme slenderness of the cross-sections this added inertia accounts for $m_{a}/m = 99.5\,\%$ of the effective mass. With a flexural rigidity of $EI = 7.8\cdot 10^{-8}\,\textrm{N}\,\textrm{m}^{2}$ this gives a period $T_{n} = f_{n}^{-1} = {139}H/U$ . It turns out that this time scale is far beyond the dominating fluid time scale, so that it is not physically relevant. The issue relates to the very high Cauchy number of $\textit{Ca}={25\,000}$ reflecting the very small stiffness of the blades.

Analogous to Sundin & Bagheri (Reference Sundin and Bagheri2019), the forcing time scale of turbulent wall shear $T_{\textit{f}}$ was estimated for the present case based on frequency spectra of wall shear in smooth channel flows determined by Hu, Morfey & Sandham (Reference Hu, Morfey and Sandham2006). There, frequency-weighted spectra of shear stress components were reported to peak at $T_{\textit{f}}^{-1} = {0.075}\,u_{\tau }^2/(2\pi \nu )$ for the streamwise component and $T_{\textit{f}}^{-1} = {0.23}\,u_{\tau }^2/(2\pi \nu )$ for the spanwise component in direct numerical simulations (DNS) of channel flows with $360\leqslant \textit{Re}_\tau \leqslant 1440$ . As in Sundin & Bagheri (Reference Sundin and Bagheri2019), it is assumed that these frequency peaks apply also in channel flows with the friction Reynolds number of the present case. Taking the effective friction velocity of the present case, determined at the virtual origin of the overflow $u_{\tau ,{vo}}$ (cf. § 5.2 below), gives $\textit{Re}_\tau = {Re_{\tau ,{vo}}} = {2638}$ . This yields a value of $T_{\textit{f}}^{-1} = {0.075}\,u_{\tau }^2/(2\pi \nu ) = {0.2}\,H/U$ . The reduced velocity then evaluates to $U_{r} = {\alpha ^{-1}T_{n}}/{T_{\textit{f}}} = {395}$ . Alternatively, a reduced velocity can be obtained by analysing spectra of blade tip motion, as done in § 6.3 below. This yielded $U_{r}\approx 28$ . Either way, the elevated value of $U_{r}$ suggests a decoupling of fluid time scales from the natural elastic time scale of the blades, related to the very high Cauchy number.

The buoyancy number, defined in table 3, compares the buoyancy forces to the elastic forces. Its value is $B = 140$ in the present situation so that, in fact, buoyancy provides a mechanism counteracting reconfiguration, which is much stronger than the restitutive effect of the elastic forces. On the other hand, it is still small compared with the fluid drag, as expressed by $B/\textit{Ca} = \varDelta\rho WTL\,g_{y}/(\rho/2U^2WL) \ll 1$ .

3.2. Numerical parameters

The essential numerical parameters are summarized in table 4. The Cosserat rods were discretized with $N_{e} = 108$ elements each. A number of $N_{L_{e}}\times N_{W} = 6 \times 32$ marker points were used on each element of the blades. The correction factors for shearing and warping in (2.7) were determined according to Freund & Karakoç (Reference Freund and Karakoc2016), resulting in $k_{{s}1} = k_{{s}2} = 0.83$ , and $k_{t} = 7 \cdot 10^{-5}$ . The Smagorinsky constant $C_{S} = 0.15$ was chosen equal to the value already employed in other LES of canopy flows (Okamoto & Nezu 2010b; Li & Xie Reference Li and Xie2011; Gac 2014; Tschisgale et al. Reference Tschisgale, Löhrer, Meller and Fröhlich2021), and similar to $0.17$ in Marjoribanks et al. (Reference Marjoribanks, Hardy, Lane and Parsons2014, Reference Marjoribanks, Hardy, Lane and Parsons2017).

Table 4.

Overview over numerical parameters used for the present simulation.

The computational domain used for the present simulations is a cuboid of extensions $L_x\times L_y \times L_z = {9.22}H \times H \times {4.03}H$ in the streamwise, the wall-normal and the spanwise direction, respectively, containing $N_{s}=672$ blades. Periodic boundary conditions were imposed in the $x$ and the $z$ direction. The flow was driven by the spatially constant volume force $\boldsymbol{f_{d}} = (f_{d}, 0, 0)^\top$ adjusted in each time step so as to obtain the desired bulk flow rate. The domain extensions in the periodic directions comply with the recommendations of Moser, Kim & Mansour (Reference Moser, Kim and Mansour1999); MacDonald et al. (Reference MacDonald, Chung, Hutchins, Chan, Ooi and García-Mayoral2017). In § 7.2 below it is demonstrated that decorrelation of velocity fluctuations was obtained in the computed flow, providing an a posteriori justification of the period length.

The mesh is almost isotropic and uniformly spaced in all three directions, with $N_x \times N_y \times N_z= 2048 \times 250 \times 1024$ cells, corresponding to $W/\unicode{x1D6E5} _z = {24.4}$ cells per width of a blade. With the inner boundary layer inside the canopy characterized by a friction velocity of $u_{\tau,{i}} = 0.026\,U$ (cf. (5.6) below) the extensions of the grid cells in inner units are $\unicode{x1D6E5} _x^{\textrm{+}} \times \unicode{x1D6E5} _y^{\textrm{+}} \times \unicode{x1D6E5} _z^{\textrm{+}} = 2.4 \times 2.1 \times 2.1$ . Owing to the staggered grid, the first streamwise velocity information is located at $\unicode{x1D6E5} _y^{{\textrm{+}}}/2 \approx 1$ . The time step size was set to $\varDelta{}t= 3.125 \cdot 10^{-4}\,\textrm{s}$ , which resulted in a maximum Courant–Friedrichs–Lewy (CFL) number fluctuating around $C_{\textit{CFL}}=0.28$ .

Starting at $t=0$ from some artificial turbulence, and the blades moderately inclined such that they just fit into the computational domain, the simulation was conducted over a startup duration of $T_{\textit{st}} = 80.0\,\textrm{s} = 66.1\,HU^{-1}$ to establish the fully developed state, with the first $60.5\,\textrm{s}=50.0\,HU^{-1}$ using a grid coarser by a factor of 2 in each direction. After reaching the statistically steady state, statistics were collected over a duration of $T_{av} = 88.3\,\textrm{s} = 73.0\,HU^{-1}$ . With the highly optimized code employed, the production run took a total of 1.3 Mcore-h on the CPU-based HPC clusters Taurus (ZIH, Dresden) and CLAIX-2018 (ITC RWTH, Aachen) used for the first and second part of the run, respectively.

Time-averaged fluid velocity, pressure and Reynolds stress components were collected at a sampling rate of $1/(200{\unicode{x1D6E5} }t) = 19.4\,UH^{-1}$ . The geometry of the blades, as well as coarsened velocity and pressure fields were stored at the same sampling rate to enable complementary, time-resolved analysis in post-processing. The coarsening strategy involves trilinear interpolation on a homogeneous grid, coarsened by a factor of 2 in each direction, and with an additional layer of points close to the bottom no-slip wall at the same distance as the wall-closest cell face centres in the computational grid. As a result, $1407$ sets of field data and canopy blade data were produced at constant intervals during the averaging window for the analysis presented below.

3.3. Validation

The IBM employed for the present study was validated by Tschisgale & Fröhlich (Reference Tschisgale and Fröhlich2020) on the basis of three test cases. (i) The first case is the fluid--structure interaction benchmark case of Turek & Hron (Reference Turek, Hron, Bungartz and Schäfer2006) that exhibits the vortex-induced oscillation of a rod in the wake of an immobile obstacle placed in a laminar flow. The time-dependent displacement of the tip was shown to agree with a number of other studies from the literature. (ii) The second case is the configuration experimentally studied by Luhar & Nepf (Reference Luhar and Nepf2011), where a rod is fixed at one end and placed perpendicular to a constant flow. For several values of the free-stream velocity considered, the drag force on the rod agrees well with that measured in the reference experiment. (iii) The third case features an entire canopy at a medium Cauchy number later studied in detail by Tschisgale et al. (Reference Tschisgale, Löhrer, Meller and Fröhlich2021). The simulation was able to reproduce the monami also observed by Okamoto & Nezu (2010a) in their experiments, and mean profiles of velocity and turbulent shear closely match those measured.

The spatial discretization of the present study was chosen on the basis of the grid resolution study conducted for this third validation case. The bulk Reynolds number and friction Reynolds number characterizing this case are included in table 1. The case was well resolved with $336$ equi-sized grid cells normal to the wall translating to $W/\unicode{x1D6E5} _{g} = 12.8$ fluid cells per width of a blade in that study versus a ratio of $W/\unicode{x1D6E5} _{g} = 24$ in the present set-up. The CFL number employed is significantly smaller than the value of $0.5$ found sufficient by Tschisgale et al. (Reference Tschisgale, Löhrer, Meller and Fröhlich2021). The subgrid-scale model and the model constant employed are identical to those of Tschisgale et al. (Reference Tschisgale, Löhrer, Meller and Fröhlich2021), where halving and doubling the value of the constant had practically no effect on the results. Additional validations of the collision model can be found in Tschisgale et al. (Reference Tschisgale, Thiry and Fröhlich2019).

Lacking suitable reference data for the present configuration, it is delicate to assess the numerical accuracy. For instance, DNS of the present problem are not feasible due to excessive computational resources required. This results from the fine spatial and temporal scales present in the interstitial flows between blades and the collisions between them. Appendix B provides a detailed discussion of this issue.

Figure 3.

Instantaneous velocity components at $t=139.1\,UH^{-1}$ : (a) streamwise, (b) vertical, (b) spanwise; all in perpendicular slices, of which the horizontal plane is located at the height of the mean canopy hull $y = h$ . () Positions of the slices; ( ) intersections of the canopy blades with the plane displayed; () intersections with the canopy hull height $y = \bar {y}({x,z;t})$ , introduced in § 6.4. Animations of these figures are provided in movies 1, 2 and 3 of the supplementary material available at https://doi.org/10.1017/jfm.2025.407.

4.1. Two-dimensional views

This section provides an impression of the instantaneous flow before it is analysed statistically. Visualizations juxtaposing the three velocity components are reported in figure 3. Large-scale velocity structures of high- and low-speed streaks can be observed, elongated in the streamwise direction. Their spanwise extent appears to be about $H$ , with a preference of staggered ordering in $x$ . The vertical plane $z=\textrm{const.}$ in the upper part of the figure highlights the substantial level of fine-scale turbulence. It also reveals large-scale velocity patterns hinting towards coherent velocity structures with slow fluid transported to the upper boundary. In some regions the blades are pushed towards the bottom wall (e.g. $1.5\,H \le x \le 3.5\,H$ in the centre plane) while elsewhere they move upwards, far into the outer flow (as seen around $x=5H$ ). This is supported by the spanwise plane ( $x=\textrm{const.}$ ), where intrusions of blades and slow fluid into the outer flow are well discernible.

Together with the corresponding plot of $u$ , the spanwise plane of the $v$ velocity in the left part of figure 3(b) suggests ejections generating streamwise vorticity. Sweep events are also readily visible. The same holds for the centre plane in the upper part. The instantaneous spanwise velocity component in figure 3(c) reveals alternating positive and negative values stemming from large-scale vortices. Close to $x=5.5H$ and further downstream, for example, inclined patterns of such vortices are seen, reminiscent of what is known from similar configurations of flows over rough beds (Detert Reference Detert2008). Also near the upper boundary, strong spanwise velocity can be observed, partly in conjunction with high streamwise velocity, e.g. at around $x=3H$ , hinting towards splat events. The horizontal planes of figures 3(b) and 3(c) illustrate again the high level of turbulence and a fairly large irregularity of the flow, together with a certain number of coherent events.

In the centre planes, $z=0$ , of figure 3, the blades take on a relatively streamlined shape in several places such that their motion is rather limited in the streamwise direction, since they are already oriented streamwise to a large extent. Vertical movement is restricted by the bottom wall, and often occurs when blades undulate under passing flow structures. In contrast, the blades are relatively free to move in the spanwise direction, which they do when strong high-speed sweeps reach down to the channel floor, displacing canopy blades to the sides. This is reflected by large patches of elevated spanwise velocity in figure 3(c) and the accumulation of blades in low-momentum volumes, for example, in figure 3(a) around $x=4.5H$ , $z=2H$ . Since blades and fluid are coupled through no-slip conditions, and since the velocity is dominated by the streamwise component, the interstitial space underneath the blades corresponds to a volume of low fluid momentum, readily discernible in the upper part of figure 3(a). While the blades are massively pushed towards the wall in some places, and straightened by this impact of high-velocity fluid, they also tend to undulate significantly in other places, as seen in the cuts of figure 3, particularly in the horizontal cuts of figures 3(b) and 3(c).

4.2. Three-dimensional views

Figure 4 provides a perspective view at another instant in time. It comprises several contributions: two-dimensional streamwise velocity along vertical planes at the periodic boundaries, three-dimensional positions of blades, fluctuations of streamwise velocity and vortex structures highlighted by isosurfaces of $p'$ (Robinson Reference Robinson1991; Jeong & Hussain Reference Jeong and Hussain1995). Some of the features were annotated by hand to facilitate the discussion. Figure 30 in Appendix D displays separate pictures of the components with better visibility. In figure 4 and even better in figure 30(a), it can be seen that the blades are bent slightly upwards in some places, but also bent downwards in a few other locations, not so readily visible here. Overall, upward bending appears to be more frequent and certainly not uninfluenced by the slight buoyancy the structures have, with $\varDelta\rho /\rho = {-0.078}$ . Figures 4 and 30(a) show more clearly than in figure 3 that in some locations, and varying in time, the blades undergo massive twisting, distortion and reorientation of their centrelines in the spanwise direction. Figures 4 and 30(a) also show that the motion of canopy elements can be related to the passing of high- and low-speed fluid, as well as vortex structures. Some of these prototypical, characteristic vortex structures are annotated in figure 4. These include spanwise rollers due to the KH instability in the canopy-induced shear layer (Raupach et al. Reference Raupach, Finnigan, Brunet, Garratt and Taylor1996; Bailey & Stoll Reference Bailey and Stoll2016; Singh et al. Reference Singh, Bandi, Mahadevan and Mandre2016). They are ideally oriented spanwise. At later stages they are observed to rotate around the vertical axis and the streamwise axis before disintegrating. Sometimes these rollers then become nearly aligned with the streamwise axis, labelled ‘streamwise-oriented’ in figure 4. Head-up (HU) hairpins tend to populate low-speed streaks, but are rather delicate and short lived. This relation between low-speed streaks and hairpins is well known from turbulence over smooth walls (Theodorsen Reference Theodorsen1955; Adrian Reference Adrian2007). The combined KH--HU structures identified by Tschisgale et al. (Reference Tschisgale, Löhrer, Meller and Fröhlich2021) to account for particularly pronounced sweep events and associated with the monami are not discernible here.

Figure 4.

Annotated flow visualization, showing the geometry of the canopy blades coloured by their surface height $\gamma _{y}$ , volumes of $\lvert u'\rvert \gt {0.5}U$ (red and blue clouds), the streamwise velocity $u$ in vertical slices, and iso-pressure surfaces with $p' = -0.1\rho U^2 = \textrm{const.}$ , coloured by $y$ . The contours of some of the more distinct instantaneous vortex structures were drawn by hand on top of the rendering with additional arrows indicating their direction of rotation. They are classified as KH rollers, head-up (HU) hairpins and predominantly streamwise-oriented (SO) rollers. An animation of this figure without annotations is provided in movie 4 of the supplementary material.

5.1. Averaging

Based on our previous experience (Vowinckel et al. Reference Vowinckel, Nikora, Kempe and Fröhlich2017a ,Reference Vowinckel, Nikora, Kempe and Fröhlich b ), the turbulent velocity field $\boldsymbol{u} ({\boldsymbol{x}, t} )$ is now decomposed according to the double-averaging scheme (Wilson & Shaw Reference Wilson and Shaw1977; Raupach & Shaw Reference Raupach and Shaw1982). Following Brunet (Reference Brunet2020), this reads

(5.1a) \begin{gather} \boldsymbol{u} = \langle \boldsymbol{u}\rangle + {\langle \boldsymbol{u}\rangle }^{\prime \prime }_{t} + \boldsymbol{u}^{\prime }, \end{gather}

with

(5.1b) \begin{gather} {\langle\boldsymbol{u}\rangle} := \langle \boldsymbol{u} \rangle _{txz} = \langle \langle \boldsymbol{u} \rangle _t\rangle _{xz} \end{gather}

the double-averaged velocity field and

(5.1c) \begin{align}& {\boldsymbol{u}}^{\prime \prime } :=\boldsymbol{u} - {\langle \boldsymbol{u}\rangle }_{xz} , \\[-12pt] \nonumber \end{align}

(5.1d) \begin{align}&\,\, {\boldsymbol{u}}^{\prime } := \boldsymbol{u} - {\langle \boldsymbol{u}\rangle }_{t} \\[12pt] \nonumber \end{align}

the deviation from the spatial average and from the temporal average, respectively. The notation $\langle \ldots \rangle$ designates the average in space and time, while ${\langle \ldots \rangle }_{t}$ , ${\langle \ldots \rangle }_{xz}$ indicate partial averages in time and over horizontal planes, respectively. With statistical stationarity as well as spatial homogeneity in horizontal planes, double averaging the momentum equation (2.1a ) yields

(5.2a) \begin{align} 0 = {\left \langle {f_{d}}\right \rangle } - \frac {\partial \left \langle u^{\prime \prime }v^{\prime \prime }\right \rangle }{\partial y} + \nu \frac {\partial ^{2} {\left \langle {u}\right \rangle }}{\partial y^2} + f_{vx}, \\[-24pt] \nonumber \end{align}

(5.2b) \begin{align} 0 = -\frac {1}{\rho }\frac {\partial \left \langle p\right \rangle }{\partial y} - \frac {\partial \left \langle v^{\prime \prime }v^{\prime \prime }\right \rangle }{\partial y} + f_{vy} , \\[-24pt] \nonumber \end{align}

(5.2c) \begin{align} 0 = f_{vz}, \\[6pt] \nonumber \end{align}

for the three components. The LES contributions due to $\boldsymbol{\tau }_{{\textit{sgs}}}$ have been disregarded in this equation because they are very small compared with the gradients of resolved turbulent stresses as discussed in Appendix C. The double-averaged pressure gradient is zero in the present set-up, ${\partial { \langle {p} \rangle }}/{\partial x}=0$ , since the flow is driven by the mass-specific force $f_{d}$ in the momentum balance equation (2.1a ). The specific force vector $\boldsymbol{f_v} = (f_{vx}, f_{vy}, f_{vz})^\top$ is an averaged quantity. It compensates the force due to vegetation drag and is defined as

(5.3) \begin{align} \boldsymbol{f_v} = -\frac {1}{\rho } {\langle \nabla {\langle p\rangle }^{\prime \prime }_{t}\rangle }_{xz} + \nu\langle\nabla^2\langle\boldsymbol{u}\rangle_t''\rangle_{xz} + {\left \langle {\boldsymbol{\boldsymbol{f_{{\!\Gamma }}}}}\right \rangle } , \end{align}

where the first two terms on the right-hand side are a consequence of differentiation and spatial averaging not being commutable (Finnigan Reference Finnigan2000). Integration of (5.2a ) in $y$ by accounting for the boundary conditions of the present problem yields

(5.4) \begin{equation} H{\left \langle {f_{d}}\right \rangle } =- \left .\nu \frac {\partial {\left \langle {u}\right \rangle }}{\partial y}\right |_{y=0} - \int _{{0}}^{{H}}\!f_{vx}\,\textrm{d} y , \end{equation}

demonstrating that the average driving force $ \langle {f_{d}} \rangle$ compensates both viscous wall shear drag and the drag exerted by the vegetation elements.

In their original definition, double-averaging schemes employ spatial averages that spare the canopy elements by recognizing the fluid domain as a connected region that is interrupted by the embedded canopy elements (Raupach & Shaw Reference Raupach and Shaw1982). In such a context the first two terms in (5.3) would represent the form drag and the viscous drag force, respectively. Here, the situation is different due to the numerical method employed. The blades are dynamically coupled via the IBM coupling force term $\boldsymbol{\boldsymbol{f_{{\!\Gamma }}}}$ and blade surfaces are not represented as fluid domain boundaries. Hence, spatial averaging is done over the entire continuous $x$ -- $z$ plane, and the averaged IBM force $ \langle {\boldsymbol{\boldsymbol{f_{{\!\Gamma }}}}} \rangle$ carries the mean canopy drag. As a result, the first two right-hand side terms in (5.3) drop out and

(5.5) \begin{equation} \boldsymbol{f_v} = {\left \langle {\boldsymbol{\boldsymbol{f_{{\!\Gamma }}}}}\right \rangle } . \end{equation}

This is, indeed, seen with good accuracy when evaluating all terms in (5.3) numerically from the simulation data.

5.2. Profiles of mean flow and Reynolds stresses

Figure 5.

Average velocity and Reynolds stress components, with characteristic heights indicated as defined in the text.

Figure 6.

Profiles of mean streamwise velocity. (a) Mean velocity in inner units according to (5.6), ${\langle {u}\rangle }^{{\textrm{+}}} = {\langle {u}\rangle } / u_{\tau ,i}$ , over $y^{{\textrm{+}}} = y / (\nu /u_{\tau ,i})$ . (b) Mean velocity according to (5.8), ${\langle {u}\rangle }^{+,vo} = {\langle {u}\rangle } / u_{\tau ,{vo}}$ , over $y^{+,{vo}} = (y - y_{{{vo}}}) / (\nu /u_{\tau ,{vo}})$ . The logarithmic profile according to (5.7) is drawn over the same $y$ range as in figure 5. Also included are profile data from a smooth channel flow at the high Reynolds number $\textit{Re}_\tau = 5186$ Lee & Moser (Reference Lee and Moser2015), and data from a canopy flow of (Monti et al. Reference Monti, Omidyeganeh, Eckhardt and Pinelli2020), case DE, which has the same roughness density $\lambda = 0.56$ . Black dots mark intersections with the vertical broken lines.

Figure 5 presents one-point statistics of the three fluid velocity components. The profile of the mean streamwise velocity is similar to that of a dense canopy with medium flexible blades; see, e.g. Tschisgale et al. (Reference Tschisgale, Löhrer, Meller and Fröhlich2021). Starting at the solid wall, $y=0$ , the velocity profile is characterized by the internal friction velocity and length scale

(5.6) \begin{equation} u_{\tau,{i}} = \sqrt {\frac{\tau_{w}}{\rho} }, \quad \tau_{w} = \left .\rho \nu \frac {\partial {\left \langle {u}\right \rangle }}{\partial y}\right |_{y=0} ,\quad \ell _\tau = \frac{\nu}{u_{\tau ,i}}, \end{equation}

which can be expected to match the canonical near-wall profile ${ \langle {u} \rangle }^{{\textrm{+}}} := {\langle {u}\rangle }/u_{\tau ,{i}} = y^{{\textrm{+}}} := y/\ell _\tau$ (Kundu Reference Kundu1990). Figure 6(a) provides a corresponding plot in these inner units. The canonical relation ${ \langle {u} \rangle }^{{\textrm{+}}} = y^{{\textrm{+}}}$ only holds in a very limited range of the viscous sublayer where the skin frictional drag dominates the drag of the blades (Monti, Omidyeganeh & Pinelli Reference Monti, Omidyeganeh and Pinelli2019). Observing that the classical velocity profile reproduced by the DNS data features this behaviour, the linear relation holds for the present case up to $y^+\approx 2$ where the present data match the DNS. The canopy layer is further characterized by a lower inflection point at $y=y_{\textit{lip}}$ and an upper inflection point at $y=y_{\textit{uip}}$ . Above the mean canopy edge the flow starts to resemble that over a rough wall (Ghisalberti & Nepf Reference Ghisalberti and Nepf2006), and the velocity obeys the displaced logarithmic law in a zone above $y = 2h$ (Nepf Reference Nepf2012a ),

(5.7) \begin{align} \frac {{\left \langle {u}\right \rangle }}{u_{\tau ,{vo}}} = \frac {1}{\kappa }\log\! \left ({\frac {y-y_{{{vo}}}}{\ell_{\tau,vo}}}\right ) + A - \varDelta{}U_{vo}^+ \ \text{,} \end{align}

with the von Kármán constant $\kappa$ , the displacement height $y_{{{vo}}}$ , the smooth channel offset $A=5.5$ (Nezu & Nakagawa Reference Nezu and Nakagawa1993; Monti et al. Reference Monti, Nicholas, Omidyeganeh, Pinelli and Rosti2022), the roughness function $\varDelta{}U_{vo}^+$ (Clauser Reference Clauser1954; Hama Reference Hama1954) and the relations (Nepf & Vivoni Reference Nepf and Vivoni2000; Nepf Reference Nepf2012a ; Monti et al. Reference Monti, Omidyeganeh and Pinelli2019)

(5.8) \begin{gather} u_{\tau ,{vo}} = \sqrt {\frac{\widetilde {\tau }|_{y=y_{{{vo}}}}}{\rho} } ,\qquad \ell_{\tau,vo} = \frac{\nu}{ u_{\tau ,{vo}}} . \end{gather}

This is illustrated in figure 6(b). The total shear stress $\widetilde {\tau }$ is obtained by integrating (5.2a ) from $y$ to $H$ , giving

(5.9) \begin{gather} \widetilde {\tau } := \tau -\ \rho \int \limits _{{\tilde {y} = y}}^{{H}}\!f_{vx}\,\textrm{d} \tilde {y} \, = -H{{\left \langle {f_{d}}\right \rangle }}_t\left (1-\frac {y}{H}\right ) , \end{gather}

where

(5.10) \begin{gather} \tau := \rho \nu \frac {\partial {\left \langle {u}\right \rangle }}{\partial y} - \rho {\left \langle {u^{\prime \prime }v^{\prime \prime }}\right \rangle } \end{gather}

is the sum of the laminar and the turbulent shear stress. The length $\ell_{\tau,vo}$ plays the same role as the roughness length in rough-wall boundary layers (Chung et al. Reference Chung, Hutchins, Schultz and Flack2021). By fitting (5.7) and (5.8) with $\kappa =0.41$ (Nezu & Nakagawa Reference Nezu and Nakagawa1993) to the simulated mean flow profile in the range $2.5h\leqslant y \leqslant 0.8H$ , the values $y_{{{vo}}} = {0.18}H$ , $\varDelta{}U_{vo}^+ = {15.4}$ , $u_{\tau ,{vo}}= {0.16}U$ were found for the present case. The given range in $y$ was chosen since the terms in the streamwise momentum balance (5.2a ) are approximately constant in this region (see figure 8). When, instead, also fitting $\kappa$ the values $\kappa = {0.38}$ , $y_{{{vo}}} = {0.13}H$ , $\varDelta{}U_{vo}^+ = {17.4}$ , $u_{\tau ,{vo}} = {0.17}U$ were obtained. The fitted value of $\kappa$ differs from the canonical value only slightly, to an extent comparable to the range also observed in conventional wall-bounded flows, e.g. the range $0.38 \le \kappa \le 0.42$ for pipe flows (Bailey et al. Reference Bailey, Vallikivi, Hultmark and Smits2014). Hence, the canonical value of $\kappa = 0.41$ is used below, while observing that varying the value of $\kappa$ has a relatively large impact on the displacement height $y_{{{vo}}}$ , as noticed by Monti et al. (Reference Monti, Omidyeganeh and Pinelli2019) as well. The value of $y_{{{vo}}}={0.18}H$ determined for the present case is slightly inferior to the canopy height $h= {0.20}H$ obtained in § 6.4. According to the heuristic model of Nicholas et al. (Reference Nicholas, Monti, Omidyeganeh and Pinelli2023), $h$ provides the upper limit of $y_{{{vo}}}$ . While their model was based on cases with rigid canopy elements of maximum height $h$ , it was not clear at the start if and how this rule extends to the present case where the instantaneous canopy height is not constant. The present data, however, suggest that this model may also hold for the situation of very flexible, slightly buoyant canopies, at least it does so in the case investigated here. According to Monti et al. (Reference Monti, Omidyeganeh, Eckhardt and Pinelli2020) the relationship $y_{{{vo}}} \gt y_{\textit{lip}}$ complies with the present configuration being located in the dense regime, which is corroborated by ${\partial { \langle {u} \rangle }}/{\partial y}\approx 0$ at the height of the lower inflection point, a typical feature associated with the wake zone (Nepf & Vivoni Reference Nepf and Vivoni2000; Poggi et al. Reference Poggi, Porporato, Ridolfi, Albertson and Katul2004; Ghisalberti & Nepf Reference Ghisalberti and Nepf2006; Sanjou Reference Sanjou and Bucur2016).

The Reynolds shear stress in figure 5(b) has a linear behaviour in the free flow for $y/H\gtrapprox {0.45}$ and a nonlinear behaviour inside the canopy for $y\lt y_{\textit{uip}}$ . In lower Cauchy flows reported in the literature, the profiles of $-{ \langle {u^{\prime \prime }v^{\prime \prime }} \rangle }$ peak at a height of $y\approx h^\ast$ , with

(5.11) \begin{equation} h^\ast := \left. \langle c_y \rangle \right|_{s=L} \end{equation}

the average vertical coordinate of the blade tips (Nepf Reference Nepf2012a ; Tschisgale et al. Reference Tschisgale, Löhrer, Meller and Fröhlich2021). Here, $-{ \langle {u^{\prime \prime }v^{\prime \prime }} \rangle }$ is maximum at a more elevated position, $y = 0.34H \gt h^\ast = 0.25H$ , with the associated maximum magnitude consistent with the values provided by Nepf (Reference Nepf2012a ). Normalized by the velocity difference $\varDelta{}U := { \langle {u} \rangle }|_{y=H} - { \langle {u} \rangle }|_{y=y_{\textit{lip}}} = {1.34}\,U$ this gives $-{\langle {u^{\prime \prime }v^{\prime \prime }} \rangle }|_{\textit{max}} / \varDelta{}U^2 = {0.010}$ , which is in the range reported, from $0.017$ for a canopy with rigid blades to $0.008$ for a canopy with flexible blades and monami occurring. The quadrant analysis in § 7.1 below indicates that sweeps alone would produce a peak at $y\approx h^\ast$ , while the peak of the ejection-type contribution is more elevated. The profiles of $\langle {v^{\prime \prime }v^{\prime \prime }} \rangle$ and $\langle {w^{\prime \prime }w^{\prime \prime }} \rangle$ in figures 5(d) and 5(e) have maxima at a similar height, $y = 0.34H = h^\ast + 0.09\,H$ , or $y/h^\ast = 1.36$ , which compares well with the range $-0.02 \le (y - h^\ast) / H \le 0.18$ , or $0.98 \le y/h^\ast \le 1.83$ reported by Ghisalberti & Nepf (Reference Ghisalberti and Nepf2002) for $\langle {v^{\prime \prime }v^{\prime \prime }} \rangle$ based on experiments with flexible blades. The maximum of $ \langle {u^{\prime \prime }u^{\prime \prime }} \rangle$ in figure 5(c) is lower, near $y=h$ . These profiles of $ \langle {u} \rangle$ , $ \langle {u^{\prime \prime }u^{\prime \prime }} \rangle$ , $ \langle {v^{\prime \prime }v^{\prime \prime }} \rangle$ , $ \langle {w^{\prime \prime }w^{\prime \prime }} \rangle$ and $ \langle {w^{\prime \prime }w^{\prime \prime }} \rangle$ are all qualitatively comparable to those found in lower-Cauchy-number flows, such as the situation with $\textit{Ca}=17$ in Tschisgale et al. (Reference Tschisgale, Löhrer, Meller and Fröhlich2021), with the exception of an additional particularly pronounced peak at $y\approx h$ being present in the low-Cauchy-case profiles of $ \langle {u^{\prime \prime }u^{\prime \prime }} \rangle$ , $ \langle {v^{\prime \prime }v^{\prime \prime }} \rangle$ and $ \langle {w^{\prime \prime }w^{\prime \prime }} \rangle$ .

5.3. Canopy drag

The parameter $\unicode{x1D6E5} {}U^+_{{vo}}$ in (5.7) is a first indicator of the friction exerted by the vegetation layer. In general, it accounts for friction due to roughness and increases monotonically with the surface roughness (Chung et al. Reference Chung, Hutchins, Schultz and Flack2021). For canopies, where $\lambda$ plays the role of surface roughness $\varDelta{}U_{vo}^+$ increases monotonically with $\lambda$ (Monti et al. Reference Monti, Omidyeganeh, Eckhardt and Pinelli2020). The outer velocity profile of case DE of Monti et al. (Reference Monti, Omidyeganeh, Eckhardt and Pinelli2020), included in figure 6, underlines this relationship as it results in the same value of $\unicode{x1D6E5} {}U^+_{{vo}}$ for the same solidity $\lambda = 0.56$ as in the present case. Apart from sharing the same value of $\lambda$ , this case is very different, because it features upright, rigid canopy elements. Monti et al. (Reference Monti, Nicholas, Omidyeganeh, Pinelli and Rosti2022) found drag to also depend on the inclination of the rigid blades, but with the dependency vanishing for highly inclined rods, which is the case here due to the large Cauchy number.

Figure 7.

Shear stress contributions according to (5.9).

Drag is further investigated by evaluating (5.9). This equation demonstrates that, due to the vegetation drag, $\tau ({y} )$ alone does not obey the linear profile observed in non-vegetated turbulent channel flows. The individual contributions to the total shear stress are analysed in figure 7. Laminar stress is negligible throughout, except very close to the bottom wall. The contribution due to canopy drag dominates turbulent shear stress in the canopy region for $y \lessapprox y_{v\textit{o}}$ . Actually, canopy drag and turbulent shear stress are exactly equal at this position. The former is quantified by evaluating the friction coefficient based on the mean driving force $ \langle {f_{d}} \rangle$ in (5.4) that compensates the drag of the bottom wall as well as the drag of the blades:

(5.12) \begin{equation} C_{\textit{f}} = \frac {\rho H{\left \langle {f_{d}}\right \rangle }}{\tfrac {1}{2}\rho U^2} . \end{equation}

Evaluating $ \langle {f_{d}} \rangle$ from the simulation yields $C_{\textit{f}} \approx 0.063$ . It can be used to define a shear stress velocity $u_{\tau }$ via

(5.13) \begin{equation} C_{\textit{f}} = 2\left (\frac {u_{\tau }}{U}\right )^2 = 2 \left (\frac {\textit{Re}_\tau }{\textit{Re}_H}\right )^2, \end{equation}

with $\textit{Re}_\tau = u_{\tau }H/\nu$ and $\textit{Re}_H = UH/\nu$ . This gives $\textit{Re}_\tau = (C_{\textit{f}} / 2)^{{1}/{2}} \textit{Re}_H \approx 3550$ , a friction Reynolds number that would be characteristic only in the absence of the canopy elements with the entire driving force acting exclusively against wall shear at the bottom wall. For reference, the friction coefficient of a smooth turbulent channel flow of equal bulk Reynolds number is (Pope Reference Pope2000)

(5.14) \begin{equation} C_{\textit{f}}^0 \approx 0.0549 (\textit{Re}_H^0)^{-0.24} . \end{equation}

For the present case, this evaluates to $C_{\textit{f}}^0\approx 0.0051$ and indicates a significant drag increase due to the presence of the canopy by a factor of $C_{\textit{f}}/C_{\textit{f}}^0 = 12.4$ .

Figure 8 compares the different terms of the streamwise momentum balance (5.2a ). Vegetation drag ( $f_{vx} \leqslant 0$ in this setting) is the predominant force up to $y\approx 2h\approx 0.4H$ , with a local minimum of the absolute drag at the lower inflection point of the velocity profile and a local maximum at the upper inflection point. The turbulent shear stress-induced drag $-\partial {\langle {u^{\prime \prime }v^{\prime \prime }} \rangle }/\partial y$ balances the canopy drag and the driving force $ \langle {f_{d}} \rangle$ . Since $ \langle {f_{d}} \rangle$ is constant in space, and due to the small contribution of viscous drag, the turbulent drag exhibits the same behaviour as $f_{vx}$ , just in the opposite direction. Viscous drag is sizeable only inside the canopy with the expected negative contribution near the bottom wall. Between the two inflection points, however, the curvature of ${ \langle {u} \rangle } ({y} )$ changes from convex to concave, so that the viscous drag changes its sign now being positive in this range, which is unusual.

Figure 8.

Individual terms in the double-averaged streamwise Navier–Stokes equation (5.2a ), balancing the spatially constant volume force $\langle {f_{d}}\rangle$ driving the flow. The term $-\partial {\langle {u'v'}\rangle }/\partial y$ was added for illustration.

In figure 9 the canopy drag profile is compared with profiles related to the canopy density, namely the canopy mean solid volume fraction $\alpha$ , and the height-specific blade frontal area per base area $a$ that is proportional to the pressure drag. All attain their maximum around the second inflection point of the velocity profile, $y=y_{\textit{uip}}$ . The relative frontal area $a$ has a narrower distribution than the drag profile, more concentrated over lower elevations. As a result, this curve cannot be rescaled such that it matches the curve of the total drag. With the present numerical method it is not possible to distinguish between pressure forces and viscous forces, since only the IBM force is provided. But the difference in shape of the two curves of $a$ and $f_{vx}$ suggests that particularly in the upper part of the canopy a larger portion of the drag is due to viscous forces.

Figure 9.

Average canopy drag profile $f_{vx}$ compared with the solid volume fraction $\alpha$ and the height-specific blade frontal area per base area $a$ . All quantities are scaled by their respective maximum absolute value that, for all of them, occurs near the second inflection points of the velocity profile, $y_{\textit{uip}}$ . The respective values are provided in the legend.

6.1. Average blade geometry and shape fluctuations

Figure 10.

Statistics of the blade position. (a) Mean vertical position of the blade (black line) and PDF of the blade position in the $x$ -- $y$ plane (colour plot); (b) PDF of the vertical position of the blade tip, $c_y({s=L})$ , with $h^\ast := {\langle {c_y}\rangle }|_{s=L}$ . (c) Mean spanwise position of the blade (black line) and PDF of the blade position in the $x$ -- $z$ plane (colour plot); (d) PDF of the spanwise position of the blade tip, $c_z({s=L})$ .

The instantaneous shape of each blade is described by the position along its centreline relative to the point of fixation, $\boldsymbol{c}_{n} = \boldsymbol{c}_{n} ({s, t} )$ , where $n$ is the number identifying the blade and $s \in [0, L]$ is the arc-length coordinate, oriented from root to tip, such that $\boldsymbol{c}_{n} ({0, t} ) = (0, 0, 0)^\top$ . The average blade geometry is

(6.1) \begin{gather} {\langle {\boldsymbol{c}}\rangle }({s}) = \frac {1}{N_{s}}\sum _{n=1}^{N_{s}}{{\left \langle {\boldsymbol{c}_{n}}\right \rangle }}_t({s}), \end{gather}

and the fluctuations of the shape with respect to the time average are

(6.2) \begin{gather} \boldsymbol{c}_{n}'({s, t}) = \boldsymbol{c}_{n}({s, t}) - {{\langle {\boldsymbol{c}_{n}}\rangle }}_t({s}). \end{gather}

Both quantities are reported in figure 10 in terms of shape and probability density function (PDF). The average centreline is a highly reconfigured, streamlined curve with a weak upward turn. The latter may be related to the slight buoyancy of the material, but collisions between blades may also play a role. A conclusive assessment requires more analysis and is left for future work. The average spanwise position of the tip relative to the point of fixation is slightly larger than zero, ${ \langle {c^{\ast }_z} \rangle } \approx 0.02\,H$ , which is $1\,\%$ of $L$ . The Roman and Arabic numbers in figure 2 indicate different spanwise positions of the blade rows. It is seen that rows 1, 2, 3 and I, II, III, IV are gradually shifted in the positive $z$ direction bit by bit. Hence, the pattern is not entirely symmetric with respect to reflection in $z$ , potentially producing a slight shift of the velocity in the spanwise direction. The computed mean velocity does indeed have a spanwise component, $ \langle {w} \rangle$ , which is not exactly $0$ , but a factor of $100$ smaller than $\langle {u} \rangle$ (figure 5 a). Since the blades are extremely flexible, this may have induced the slight shift in the positive $z$ direction observed in figures 10(c) and 10(d). It is, however, very small, so that on average the flow can still be considered symmetric in $z$ and the evaluation be done on this basis.

Root-mean-square (r.m.s.) fluctuations from the averaged shape of the blades are shown in figure 11(a) for all three coordinates. Fluctuations in the streamwise direction ${\langle {c_x'c_x'}\rangle }$ are overall lowest, followed by the vertical component ${\langle {c_y'c_y'} \rangle }$ and fluctuations in the spanwise position ${\langle {c_z'c_z'} \rangle }$ . The latter increase almost linearly with the arc-length distance from the root. They are smaller than the vertical fluctuations over about a third of the blade length until $s={0.44}\,H={0.28}\,L$ and then surpasses it with a ratio of about $1.56$ reached at the tip. Here, the geometric confinement by the presence of the bottom wall and by other blades between a given blade and the wall is an important contribution, as backed by figures 10(a) and 10(c). While fluctuations of the blades in the vertical direction appear to be drastically inhibited, spanwise excursions of the blades, in particular if collective, do not experience the same restriction. Whether and how the cross-sectional shape of the blades plays a role here must be left for future studies. Streamwise and spanwise deflections are overall well correlated, with the correlation coefficient of the shape fluctuations in both directions $-\rho _{c_x'c_y'}\gtrapprox 0.5$ in figure 11(b). Owing to the reflectional symmetry of the problem in $z$ , the correlations $\rho _{c_xc_z}$ and $\rho _{c_yc_z}$ vanish, which has been verified with the present data.

Figure 11.

Fluctuations of the blade centreline geometry. (a) Variability of the blade centrelines, measured as the r.m.s. value of the respective fluctuations in shape and in the streamwise, vertical and spanwise position. (b) Correlation coefficient of the streamwise and vertical shape fluctuations.

6.2. Modes of blade shape motion

Given its pronounced flexibility, the instantaneous centreline of a blade is expected to undulate as it interacts with the turbulent overflow. To quantify the relative importance of contributions at different length scales, the instantaneous shape of each blade, expressed as the deviation from the mean shape $\boldsymbol{c}' = (c_x', c_y', c_z')^\top$ , was decomposed into fundamental modes $\varphi _n ({s} )$ , $n=1,2,\dots$ ,

(6.3) \begin{equation} \boldsymbol{c}'({s, t}) = \sum _{n=1}^{N_{n}} \widetilde{\boldsymbol{c}}'_n({t}) \, \varphi _n({s}) + \boldsymbol{\varepsilon }({s}) , \end{equation}

where the vector $\widetilde {\boldsymbol{c}'}_n = (\widetilde {c_x'}_n, \widetilde {c_y'}_n, \widetilde {c_z'}_n)^\top$ assembles the respective magnitudes. With $\boldsymbol{c}'$ being smooth in space, the error truncation $\boldsymbol{\varepsilon }$ vanishes for $N_{n}\to \infty$ . The mode shapes $\varphi _n ({s} )$ were chosen to represent the modes of vibration of an unloaded cantilevered Euler--Bernoulli beam (Timoshenko Reference Timoshenko1937; Hadley Reference Hadley2024); details are provided in Appendix E. The relative importance of the different modes is evaluated in figure 12, comparing the r.m.s. values of the modal amplitudes. Both vertical and spanwise components are dominated by the first mode. The amplitudes of the higher modes decrease monotonically with increasing mode number. Comparing the two directions, first mode deviations from the mean shape are more pronounced in the spanwise direction than vertically. This is likely to be just a consequence of the presence of the wall, as discussed above.

Figure 12.

The r.m.s. amplitudes associated with the first ten modes of the blade centreline motion.

6.3. Linking blade tip motion to fluid velocity, effective stiffness

To condense information, the motion of the blades is now investigated by studying the movement of their tips. This approach is commonly applied to lower-Cauchy-number scenarios where the position of a tip can be considered representative of the overall deformation of the blade (Tschisgale et al. Reference Tschisgale, Löhrer, Meller and Fröhlich2021; Foggi Rota et al. 2024b). There, an oscillator model can be used to investigate the coupling between fluid and canopy, as detailed in Appendix F. The novel issue addressed here is the fact that the blades interact directly with each other in the present situation as they are closely stacked most of the time due to their large deformation. The strong inter-blade forces result in an effective stiffness of the canopy as a whole, much larger than the stiffness of an individual blade. This effective stiffness is one which would be obtained by replacing the length of the blade, $L$ , with a shorter length $l$ in the range $h/2\lessapprox l \lessapprox h$ , while maintaining the other parameters of the blades. This amounts to an increase of stiffness between the single blade and the reconfigured canopy by a factor of about 8, which is large. Hence, for this type of canopy, the effect is sizeable and has to be considered when modelling.

6.4. The canopy hull

In low- to medium-Cauchy-number scenarios, the local canopy height is commonly identified with the height of the blade tips (Tschisgale et al. Reference Tschisgale, Löhrer, Meller and Fröhlich2021; Monti et al. Reference Monti, Olivieri and Rosti2023). The situation is more complex with the present configuration, due to the high flexibility of the canopy blades requiring a more sophisticated approach. To this end, the canopy hull was defined as the vertical position of the most elevated blade segment for a given position $(x, z)$ with an appropriate smoothing procedure to fill gaps smaller than the length scale targeted; details are provided in Appendix G. The canopy hull is denoted $\bar {y} = \bar {y} ({x,z;t} )$ . It serves to provide the mean canopy height by averaging in time and space, i.e.

(6.4) \begin{equation} h := {\left \langle {\bar {y}}\right \rangle } . \end{equation}

In the present simulation this yields $h = 0.20\,H$ , which is the numerical value reported in the figures above. An instantaneous snapshot of this quantity is shown in figure 13 for illustration.

Figure 13.

Instantaneous snapshot of the smoothed canopy hull height at $t=139.1\,UH^{-1}$ , the same instant as in figure 3. An animation of this figure is provided in movie 5 of the supplementary material.

Figure 14.

Profiles characterizing the vertical distributions of canopy elements. The broken line () denotes the horizontally averaged solid volume fraction $\alpha$ , and the continuous line () the PDF of the smoothed hull height $\phi _{\bar {y}}$ . Both are scaled by their maximum absolute values given in the legend. These occur near the second inflection point of the velocity profile, $y_{\textit{uip}}$ . The symbol $P$ designates a percentile.

The PDF of the hull height, $\phi _{\bar {y}}$ , is presented in figure 14. The annotated $5\,\%$ and $95\,\%$ percentiles of $\bar {y}$ are identical with the equally valued percentiles of $\bar {y}'$ and $\bar {y}^{\prime \prime }$ , when shifted by $h$ , since ${{\langle {\bar {y}} \rangle }}_t\approx {\langle {\bar {y}}\rangle }_{xz}\approx { \langle {\bar {y}} \rangle }$ upon statistical convergence. They serve as threshold heights for the definition of trigger events in the conditional averaging below. The normalized solid volume fraction $\alpha$ , obtained by horizontally averaging the blade positions, is shown for comparison. It is strongly concentrated around its maximum value slightly below the upper inflection point $y_{\textit{uip}}$ . The profile of $\phi _{\bar {y}}$ , in comparison, is broader and has its maximum below $h=0.2 H$ . This results from the rare, far excursions of $\bar {y}$ to high elevations, up to $0.6 H$ . By definition, $\bar {y}$ is located at a higher elevation than the mean volume fraction $\alpha$ . From a physical perspective, the uppermost blade determines the position of the canopy–flow interface to a large extent as it provides a kind of shadow to the blades and the interstitial flow underneath. This is easily visible, e.g. in the centre plane plot of figure 3(a) around $5 \lt x/H \lt 5.5$ . Such an approach is deemed more appropriate than a definition of the interface based on a threshold value of some instantaneous, smoothed local volume fraction.

7.1. Quadrant analysis

The Reynolds stresses were reported in figure 5 and discussed in § 5.3. Of these, the turbulent shear stress $\langle {u'v'} \rangle$ is particularly relevant because it represents the turbulent momentum exchange between the canopy region and the outer flow. Above, it was demonstrated that ${\langle {u'v'} \rangle }\approx {\langle {u^{\prime \prime }v^{\prime \prime }} \rangle }$ , so that in the following only the latter is discussed for convenience. Mathematically, the magnitude of $ \langle {u^{\prime \prime }v^{\prime \prime }} \rangle$ is the result of the two fluctuating components being statistically anti-correlated. The degree of (anti-)correlation is reported in figure 15(a) in terms of the correlation coefficient

(7.1) \begin{equation} \rho _{u^{\prime \prime }v^{\prime \prime }} = \frac {{\left \langle {u^{\prime \prime }v^{\prime \prime }}\right \rangle }}{\sqrt {{\left \langle {u^{\prime \prime }u^{\prime \prime }}\right \rangle }{\left \langle {v^{\prime \prime }v^{\prime \prime }}\right \rangle }}} . \end{equation}

Its shape is different from $\langle {u^{\prime \prime }v^{\prime \prime }} \rangle$ due to the division by the magnitude of the $u$ and $v$ fluctuations, but very instructive. While $\left \lvert {\langle {u^{\prime \prime }v^{\prime \prime }} \rangle }\right \rvert$ is maximum around $y\approx 0.34\,H$ , the correlation coefficient $\rho _{u^{\prime \prime }v^{\prime \prime }}$ has its absolute maximum at a much higher position of $y\approx 0.5\,H$ . Above the bottom wall, $-\rho _{u^{\prime \prime }v^{\prime \prime }}$ increases approximately linearly, then flattens between the lower and upper inflection points, remaining about constant in the surrounding of the upper inflection point before increasing again with height.

Figure 15.

Profiles related to the Reynolds shear stress. (a) Correlation coefficient of the streamwise and vertical velocity fluctuations over height. (b) Reynolds shear stress component by quadrants, computed according to (7.2).

To analyse this interplay in more detail, a quadrant analysis (Wallace, Eckelmann & Brodkey Reference Wallace, Eckelmann and Brodkey1972; Willmarth & Lu Reference Willmarth and Lu1972) was performed. Defining a Cartesian plane spanned by $u^{\prime \prime }$ and $v^{\prime \prime }$ , instantaneous fluctuations can be associated with the respective quadrants $i\in \{1,2,3,4\}$ by means of the function

(7.2) \begin{equation} \delta _i({\boldsymbol{x}, t}) = \begin{cases} 1 & \textrm{if}\ (i = 1) \wedge (u^{\prime \prime } \gt 0) \wedge (v^{\prime \prime } \gt 0)\quad {\text{(outward interaction)}},\\ 1 & \textrm{if}\ (i = 2) \wedge (u^{\prime \prime } \lt 0) \wedge (v^{\prime \prime } \gt 0)\quad {\text{(ejection)}}, \\ 1 & \textrm{if}\ (i = 3) \wedge (u^{\prime \prime } \lt 0) \wedge (v^{\prime \prime } \lt 0)\quad {\text{(wallward interaction)}},\\ 1 & \textrm{if}\ (i = 4) \wedge (u^{\prime \prime } \gt 0) \wedge (v^{\prime \prime } \lt 0)\quad {\text{(sweep)}} ,\\ 0 & \textrm{otherwise}, \end{cases} \end{equation}

so that the total Reynolds shear stress can be decomposed as

(7.3) \begin{equation} {\left \langle {u^{\prime \prime }v^{\prime \prime }}\right \rangle } = \sum _{i=1}^4 {\left \langle {u^{\prime \prime }v^{\prime \prime }\delta _i}\right \rangle } . \end{equation}

Figure 15(b) shows the magnitude of each contribution over height. Both ejections and sweeps behave similarly, but differ in the details. The magnitude of sweeps ( $i=4$ ), bringing fast outer fluid towards and into the canopy, increases almost linearly from the upper surface towards the average canopy height, where it attains a maximum slightly above $h$ . Below this elevation, a fairly linear decrease towards the bottom wall is observed. The ejections ( $i=2$ ) are less vigorous within the canopy, exhibiting a slow linear increase between $y_{\textit{lip}}$ and somewhat below $y_{\textit{uip}}$ , i.e. where $ \langle {u} \rangle$ is roughly constant (cf. figure 5 a). Then, a stronger increase towards a maximum is seen, at around $0.36H \gt h$ , i.e. above the canopy. Its magnitude is roughly the same as for the sweep contribution. The contributions from the other two quadrants behave similarly over $y$ with a maximum around $y=h$ of magnitude $2.7$ times smaller than that of the sweeps. As known from canopy flows in general, ejections contribute more to exchange of momentum than sweeps above the canopy, here $y\gtrapprox 0.3H\approx 1.5h$ , whereas sweeps provide a larger contribution than ejections in the layer below (Poggi et al. Reference Poggi, Porporato, Ridolfi, Albertson and Katul2004; Patton & Finnigan Reference Patton, Finnigan and Fernando2012). This dominance of sweeps is a characteristic of dense canopies, which is not observed in sparse canopies (Poggi et al. Reference Poggi, Porporato, Ridolfi, Albertson and Katul2004; Patton & Finnigan Reference Patton, Finnigan and Fernando2012). While smooth boundary layers also exhibit a sweep-dominated layer for $y^{{\textrm{+}}}\lt 15$ (Ong & Wallace Reference Ong and Wallace1998), in the present situation this layer is much thicker, as concluded from the value of $y^{+,{vo}} = (y - y_{{{vo}}}) / (\nu /u_{\tau ,{vo}}) \approx 386$ .

The Reynolds shear stress is also given by (Wallace Reference Wallace2016)

(7.4) \begin{equation} {\left \langle {u^{\prime \prime }v^{\prime \prime }}\right \rangle } = \int \limits _{-\infty }^{\infty }\int \limits _{-\infty }^{\infty }\!u^{\prime \prime }v^{\prime \prime }\phi _{u^{\prime \prime },v^{\prime \prime }}\,\textrm{d} u^{\prime \prime } \,\textrm{d} v^{\prime \prime } , \end{equation}

where $\phi _{u^{\prime \prime },v^{\prime \prime }}$ is the joint probability density function (JPDF) of $u^{\prime \prime }$ and $v^{\prime \prime }$ . This quantity is reported in figures 16 and 17 for three characteristic elevations in and above the canopy to demonstrate how contributions evolve with height. Representative of the inner canopy region, at the elevation of the upper inflection point $y = y_{\textit{uip}}$ sweeps dominate the contributions of all other quadrants (cf. figure 15 b). The corresponding JPDF in figures 16(a) and 17(a) peak at $u^{\prime \prime } \approx -{\langle {u} \rangle }$ , $v^{\prime \prime } = 0$ , which is concluded to be a signature of the canopy blades. Since the blades are fixed at the bottom, the velocity at their surface close to the floor is small. With $u^{\prime \prime } = u - {\langle {u}\rangle }_{xz} \approx u - {\langle {u} \rangle }$ this can be expected to produce a spike at $u^{\prime \prime } = -{\langle {u} \rangle }$ . With increasing height, ejections start to gain relevance, but sweeps remain the dominant contribution to $-{ \langle {u^{\prime \prime }v^{\prime \prime }} \rangle }$ (cf. figure 15 b). At the average height of the canopy, $y = h$ , $|\langle \delta_4 u^{\prime\prime} v^{\prime\prime} \rangle|$ is approximately maximum. The peak associated with $-{ \langle {u} \rangle }$ is still noticeable, but less pronounced (figures 16 band 17 b). Further into the outer flow, at $y = H/2$ , the correlation $-\rho _{u^{\prime \prime }v^{\prime \prime }}$ is approximately maximum (cf. figure 15 a) and ejections are the predominant mechanism in turbulent momentum exchange (cf. figure 15 b).

Figure 16.

The JPDF of the streamwise and vertical velocity fluctuations, $\phi _{u^{\prime \prime },v^{\prime \prime }}$ , at constant height. (a) At the height of the upper inflection point $y=y_{\textit{uip}}$ ; (b) at the average canopy edge $y=h$ ; (c) at channel half-height $y=H/2$ . The broken line () in (a–b) indicates $u^{\prime \prime } = -{\langle {u}\rangle }$ .

Figure 17.

Magnitude of covariance integrand in (7.4), $\lvert u^{\prime \prime }v^{\prime \prime }\phi _{u^{\prime \prime },v^{\prime \prime }}\,/\,{\langle {u^{\prime \prime }v^{\prime \prime }}\rangle }\rvert$ , at constant height. (a) At the height of the upper inflection point $y=y_{\textit{uip}}$ ; (b) at the average canopy edge $y=h$ ; (c) at channel half-height $y=H/2$ . The broken line () in (a–b) indicates $u^{\prime \prime } = -{\langle {u}\rangle }$ .

7.2. Spatial extent of flow structures

Two-point correlations in space were computed to characterize coherent structures in the flow. The normalized two-point space--time correlation coefficients of two field quantities $p$ and $q$ read (He, Wang & Lele Reference He, Wang and Lele2004)

(7.5) \begin{equation} \rho _{pq}({y; \boldsymbol{r}; \tau }) = \frac {{\langle { p({\boldsymbol{x},t}) \, q({\boldsymbol{x}+\boldsymbol{r},t+\tau })}\rangle }} {p_{{{rms}}}({y})\ q_{{{rms}}}({y})} . \end{equation}

Starting with the fluid velocity, zero-lag auto correlations of streamwise velocity fluctuations are provided in figure 18, showing perpendicular slices of the three-dimensional fields. The streamwise velocity fluctuations are highly correlated in a rather small volume surrounding the origin, with correlations decaying much faster in the spanwise direction than in the streamwise direction, resulting in elongated, streamwise-oriented isocontours. This reflects the convective transport with $\langle {u}\rangle$ in the streamwise direction. In the spanwise direction the volume of positive correlation surrounding the origin is accompanied by a pair of negatively correlated volumes further outward, indicating a spanwise neighbourhood of high- and low-speed streaks at a relatively stable distance. In figure 18(a) the horizontal extent of (anti-)correlated volumes can be seen to increase with the distance from the no-slip channel floor, and towards the free-slip top boundary, while low-intensity, long distance correlations in the streamwise direction tend to be maximum at $y\approx H/2$ . Figure 18(b) shows the correlation of the streamwise velocity fluctuations with the reference value taken at $y=h$ , resulting in an inclined pattern such that with the increasing $r_x$ correlation is maximum at increasingly large values of $r_y\gt h$ . This reflects the presence of velocity structures inclined by a ratio of about 6 : 1, known from smooth- and rough-wall open-channel flows (Nezu & Nakagawa Reference Nezu and Nakagawa1993). The graph also illustrates that substantial coherence with the data at $y=h$ is obtained over the vertical direction, signifying that the coherent structures located at the canopy edge fill a sizeable portion of the submergence to some extent reaching even to the upper boundary.

Figure 18.

Two-point auto-correlation of $u^{\prime \prime }$ . Results are shown for (a) $\rho _{u^{\prime \prime }u^{\prime \prime }}({y; r_x, r_y=0, r_z; \tau =0})$ , i.e. correlations in horizontal slices; (b) $\rho _{u^{\prime \prime }u^{\prime \prime }}({y=h; r_x, r_y, r_z; \tau =0})$ , i.e. correlations with $u^{\prime \prime }$ at the mean canopy height $h$ .

Extracts of $\rho _{u^{\prime \prime }u^{\prime \prime }}$ at the mean canopy height $y = h$ and in the free stream at $y=H/2$ corresponding to the usual two-point correlations are shown in figure 19. Based on the global minima in figure 19(b), the average spanwise spacing between high- and low-speed streaks can be determined to be around $0.75H$ . Concerning the streamwise direction, no such statistical neighbourhood between high- and low-speed streaks is discernible. While decorrelated below half the domain length, streamwise correlations do not exhibit the negative peak that is present in the spanwise direction. Indeed, viewing visualizations of $u^{\prime \prime }$ in figure 3(a) and figure 30(b), instantaneous high-speed streaks appear to vary much more in length than in $z$ . These streaks can become quite long, sometimes reaching the length of the simulation domain, as appears to be the case in Monti et al. (Reference Monti, Omidyeganeh, Eckhardt and Pinelli2020) where $L_x = 6.24H$ .

Figure 19.

Two-point auto-correlations of the streamwise velocity $u^{\prime \prime }$ at $y=h$ and $y=H/2$ , and of the canopy hull height $\bar {y}^{\prime \prime }$ . (a) Correlations in the streamwise direction; (b) correlations in the spanwise direction. The curve of $\rho _{\bar {y}^{\prime \prime }\bar {y}^{\prime \prime }}$ is very close to that of $\rho _{u^{\prime \prime }u^{\prime \prime }}$ .

Evaluating the auto-correlations of the canopy hull height, also shown in figure 19, results in a similar picture as that of $\rho _{u^{\prime \prime }u^{\prime \prime }}$ in figure 18 (b). The present results show that the canopy height is well correlated with streamwise velocity fluctuations. This is seen in figure 20(b), where the cross-correlation coefficient of the two yields a value of $\rho _{\bar {y}^{\prime \prime }u^{\prime \prime }}\approx -0.6$ at the mean canopy height, only slightly inferior to the maximum absolute correlation. As a result, slices of $\rho _{\bar {y}^{\prime \prime }u^{\prime \prime }}$ in figure 20(a) yield essentially the same picture as $\rho _{u^{\prime \prime }u^{\prime \prime }}$ in figure 18(b), just with an opposite sign. Observing the negative sign, this demonstrates that the canopy hull is governed by high-speed streaks pushing it towards the ground, whereas low-speed streaks correlate with higher elevations of the hull.

Figure 20.

Two-point cross-correlation $\rho _{\bar {y}^{\prime \prime }u^{\prime \prime }}({y; r_x, r_z; \tau =0})$ between the vertical position of the canopy hull $\bar {y}^{\prime \prime }({x,z;t}) = \bar {y}({x,z;t}) - {\langle {\bar {y}}\rangle }_{xz}({t})$ and the streamwise velocity fluctuations $u^{\prime \prime } = u^{\prime \prime }({x,y,z;t})$ . (a) Evaluated in perpendicular slices through $r_x = 0$ , $y = h$ , $r_z=0$ ; (b) profile extracted at $r_x = r_z = 0$ .

A different view on the spatial extent of flow structures is provided through energy spectra of streamwise velocity fluctuations. The two-dimensional energy spectra of these quantities, premultiplied by the wavenumber, are shown in figure 21. Doing so produces plots in which equal areas under curves of premultiplied spectra represent equal energies, when plotted against the logarithmically scaled wavenumber (Kim & Adrian Reference Kim and Adrian1999), or wavelength. Maxima in these plots are, therefore, referred to as energy sites in the following (Hutchins & Marusic Reference Hutchins and Marusic2007). They can be understood as wavelengths where turbulence kinetic energy (TKE) is maximally concentrated, while not necessarily resembling maxima in the plain spectra. Energy sites of $\Phi _{u'u'}$ are located at the canopy edge, comparable to the maxima in the profiles of $\langle {u'u'} \rangle$ (figure 5 c). The corresponding dominant wavelengths are $\lambda _x \approx 2.5 H$ in the streamwise direction and $\lambda _z \approx 1 H$ in the spanwise direction, respectively, both at $y\approx h$ . With increasing $y$ the maximum of $\kappa _z\Phi _{u'u'}$ in figure 21(b) shifts to larger wavelengths, approaching a value of $\lambda _z = 2 H$ at $y=H$ . The dominant spanwise wavelengths are consistent with the average spanwise spacing between high- and low-speed streaks identified above, expecting twice that distance to be related to the dominant wavenumber. Indeed, the values of $2\,\textrm{arg min}_{r_z} ({\rho _{u^{\prime \prime }u^{\prime \prime }}|_{y=H/2}} ) \approx 1.5 H$ and $2\,\textrm{arg min}_{r_z} ({\rho _{u^{\prime \prime }u^{\prime \prime }}|_{y=h}} ) \approx 1 H$ match the wavelengths of maximum $\Phi _{u'u'}$ .

The premultiplied spectrum of the canopy hull height $\bar {y}$ is shown in figure 22. It is characterized by a maximum near $\lambda _x \approx 2.5 H$ , $\lambda _z \approx 1 H$ , identical to the maximum locations of fluid velocity spectra at $y=h$ . This, once again highlights the close relationship between streamwise velocity fluctuations and the instantaneous canopy hull height also observed through their correlation in figure 20(b). Additional spectra and further details on their computation are provided in Appendices I and J.

Figure 21.

Premultiplied wavelength power spectral densities of the streamwise fluid velocity component, evaluated for different heights. (a) Streamwise direction; (b) spanwise direction. Heights are marked with () $h$ , () $y_{{{vo}}}$ , () $y_{\textit{lip}}$ , $y_{\textit{uip}}$ .

Figure 22.

Premultiplied power spectral densities of the canopy hull with regard to the streamwise and spanwise wavelength.

7.3. Conditionally averaged sweep and ejection events

The data are now analysed to better understand the dynamic relationship between the motion of the canopy hull and the flow structures. This is an actively discussed topic in the literature with multiple conceptual models developed. The most common model involves a spanwise-oriented KH vortex located near the canopy edge (Nezu & Sanjou Reference Nezu and Sanjou2008; Okamoto & Nezu Reference Okamoto and Nezu2010a ; Nepf Reference Nepf2012a ), linking the canopy-related velocity gradient to the velocity profile of classical mixing layers. The theoretical foundation was extended by Singh et al. (Reference Singh, Bandi, Mahadevan and Mandre2016) who identified a secondary instability mechanism that usually occurs simultaneously when considering the properties of common aquatic vegetation canopies. Instantaneous coherent vortex structures must be expected to exhibit three-dimensional patterns, more general than the two-dimensional flow patterns resulting from a KH instability perpendicular to the mean flow, as such a pattern moves with the mean flow and may be turned around, stretched or deformed otherwise. Indeed, rather complicated three-dimensional large-scale flow patterns associated with sweeps and ejections were identified by Finnigan (Reference Finnigan2000); Nezu & Sanjou (Reference Nezu and Sanjou2008); Finnigan, Shaw & Patton (Reference Finnigan, Shaw and Patton2009); Tschisgale et al. (Reference Tschisgale, Löhrer, Meller and Fröhlich2021).

Inspired by the strategy of Finnigan et al. (Reference Finnigan, Shaw and Patton2009), Tschisgale et al. (Reference Tschisgale, Löhrer, Meller and Fröhlich2021) employed a conditional averaging technique to isolate coherent flow features surrounding patches of locally strongly deflected canopy blades. This choice of the trigger event served to follow the patterns created by the monami phenomenon. The same analysis is now applied to the present data.

Conditional averages were collected following troughs of the instantaneous canopy hull and ridges thereof. To this end, threshold values $y^{\prime \prime }_{\textit{c,t}}$ and $y^{\prime \prime }_{c,r}$ were introduced to determine particularly low or elevated patches of the hull, termed ‘troughs’ and ‘ridges’, respectively. Each of the patches related to particularly low or high elevations of the hull are defined as a region $\Omega _{c}\subseteq \Omega _{xz}$ in the $x$ -- $z$ space. The coordinates of the trigger events are then identified as the centroids of these patch regions, and the resulting tuples of trigger location, time and associated patch area are collected in corresponding sets:

(7.6a) \begin{align} \mathcal{C}_t &= \left \lbrace \left .(\boldsymbol{x}_{c}, t_{c}, A_{c})\right . \mid \left .\boldsymbol{x}_{c} = \textrm{centroid}({\Omega _{c}})\right . \wedge \left .A_{c} = \textrm{area}({\Omega _{c}})\right . \wedge \left .\forall \boldsymbol{x}\in \Omega _{c}: \bar {y}^{\prime \prime }({\boldsymbol{x}, t_{c}}) \leqslant y^{\prime \prime }_{\textit{c,t}}\right . \right \rbrace ,\end{align}

(7.6b) \begin{align} \mathcal{C}_r &= \left \lbrace \left .(\boldsymbol{x}_{c}, t_{c}, A_{c})\right . \mid \left .\boldsymbol{x}_{c} = \textrm{centroid}({\Omega _{c}})\right . \wedge \left .A_{c} = \textrm{area}({\Omega _{c}})\right . \wedge \left .\forall \boldsymbol{x}\in \Omega _{c}: \bar {y}^{\prime \prime }({\boldsymbol{x}, t_{c}}) \geqslant y^{\prime \prime }_{{{c,r}}}\right . \right \rbrace . \end{align}

As shown above, the local canopy hull height is determined by the interaction with the fluid flow field, so that this method can be expected to isolate sweep and ejection events for example.

The threshold values were chosen as the $5\,\%$ and $95\,\%$ percentiles of the fluctuating hull height $\bar {y}^{\prime \prime }$ . The respective heights are indicated in figure 14. Different from medium-Cauchy-number cases, where the tips of the blades define the canopy envelope, even the smoothed canopy hull remains relatively rough. As a result, the ensemble of events in the present situation also contains a relatively large number of events with tiny associated areas $A_{c}$ , typically occurring in the neighbourhood of larger ones. These small events were considered as noise rather than indicators of own flow structures. Weighting events by their areas ensures that contributions of these side detections are also negligible in the resulting average. Still, a lower threshold for the size was introduced, demanding

(7.7) \begin{equation} A_{c} \overset {!}{\geqslant } A_{\textit{crit}}, \end{equation}

with $A_{\textit{crit}} = 4W^2$ . This is physically reasonable and reduces the computational effort substantially without affecting the result, as detailed inAppendix H.

The conditional averages for $\mathcal{C}_t$ and $\mathcal{C}_r$ were computed analogously to Tschisgale et al. (Reference Tschisgale, Löhrer, Meller and Fröhlich2021). The procedure reads

(7.8) \begin{gather} \langle \varphi \rangle _{c}({\boldsymbol{r}}) = \frac {\sum _{{(\boldsymbol{x}_{c}, t_{c}, A_{c})\in \mathcal{C}}} \left . {A_{c}}\, {\varphi ({\boldsymbol{x}_{c} + \boldsymbol{r}, t_{c}})} \right .} {\sum _{{(\boldsymbol{x}_{c}, t_{c}, A_{c})\in \mathcal{C}}}\left .A_{c}\right .} ,\qquad \left \lbrace \begin{aligned}[c] \boldsymbol{x}_{c} &:= (x_{c}, 0, z_{c})^\top, \\ \boldsymbol{r} &:= (r_x, y, r_z)^\top, \\ r_x &\in [-L_x/2, L_x/2], \\ r_z &\in [-L_z/2, L_z/2], \end{aligned}\right . \end{gather}

where $\varphi$ represents the scalar quantity to be averaged, such as pressure $p$ or components of the instantaneous velocity vector $\boldsymbol{u}$ . Equation (7.8) can be applied to all three-dimensional fields. The adaption to two-dimensional data defined in $\Omega _{xz}$ , such as canopy hull information, is straightforward.

The conditionally averaged velocity fields are shown in figure 23 along with the conditionally averaged hull height. Corresponding profiles above the trigger location are assembled in figure 24 and three-dimensional views are shown in figure 25. The conditional averages are symmetric in the spanwise direction. This symmetry results from the reflectional invariance of the set-up in $z$ , while instantaneous realizations may be asymmetrical with only a single pronounced streamwise vortex present, as seen with the instantaneous vortical structures in figure 4. The averaged hull height at the event centre is $\left .\langle \bar {y} \rangle _{c}\right |_{\boldsymbol{r}=0} = 0.06\,H$ for $\mathcal{C}_t$ and $\left . \langle \bar {y} \rangle _{c}\right |_{\boldsymbol{r}=0} = 0.44\,H$ for $\mathcal{C}_r$ .

Figure 23.

Conditionally averaged flow in perpendicular slices at $r_x=0$ , $y = \max ({\langle \bar {y} \rangle _{c}})$ and $r_z=0$ as indicated by the dotted lines. Each slice features streamlines of the in-plane velocity and is coloured by the magnitude of the respective in-plane velocity, denoted ${\langle \boldsymbol{u} \rangle _{c}}_\parallel$ . The solid line () in vertical slices represents the average hull height. (a) Average over trough-centred events, $\mathcal{C}_t$ ; (b) average over ridge-centred events, $\mathcal{C}_r$ .

Figure 24.

Profiles of the conditional average for the ridge-centred condition () $\mathcal{C}_r$ and the trough-centred condition () $\mathcal{C}_t$ . (a) Streamwise velocity, (b) streamwise velocity fluctuation, (c) vertical velocity fluctuation, (d) turbulent shear stress. Profiles in (a–c) were extracted exactly at the centre of the average $r_x = r_z = 0$ , in (d) the position is slightly different, with the profile for $\mathcal{C}_r$ extracted at $x=0.1H$ and that for $\mathcal{C}_t$ extracted at $x=-0.03H$ to capture global maxima. For reference, the ordinary average profiles () of $\langle {u}\rangle$ from figure 5(a) and $-{\langle {u^{\prime \prime }v^{\prime \prime }}\rangle }$ from figure 5(b) are included in (a, d), respectively, as well. The horizontal straight lines refer to the canopy hull at the same position as the respective curves. the height of the hull for $\mathcal{C}_r$ , height of the hull for $\mathcal{C}_t$ , the average hull height $y=h$ .

Figure 25.

Conditionally averaged flow structures. The height of the floor corresponds to the averaged canopy hull height $\langle \bar {y} \rangle _{c}$ , contour surfaces represent velocity fluctuations, with blue denoting $\langle u' \rangle _{c} = -0.1\,U$ and red denoting $\langle u' \rangle _{c} = +0.1\,U$ , and vortex structures of the conditionally averaged velocity field $\lambda _2({\langle \boldsymbol{u} \rangle _{c}}) = -0.1\,U^2/H^2$ coloured by height. (a) Average over trough-centred events, $\mathcal{C}_t$ ; (b) average over ridge-centred events, $\mathcal{C}_r$ . To obtain smooth contours, $\langle u' \rangle _{c}$ and $\langle \boldsymbol{u} \rangle _{c}$ were filtered in $r_x$ and $r_z$ with a Gaussian kernel of standard deviation $\sigma = 0.17H$ .

The flow situation associated with troughs $\mathcal{C}_t$ , in figures 23(a), 24 and 25(a), constitutes a sweep event characterized by a locally increased streamwise velocity ( $ \langle u' \rangle _{c} \gt 0$ in figure 24 b), a downwash ( $ \langle v' \rangle _{c} \lt 0$ in figure 24 c), and, hence, elevated turbulent shear that is maximum between the conditionally averaged and the mean hull height (figure 24 d). The trough in the canopy hull has a spanwise width of $1.04\,H$ measured as the distance between the neighbouring crests. This trough complies with the velocity induced by the streamwise-oriented, counter-rotating vortices in the left panel of figure 23(a) that push the canopy blades apart. These streamwise-aligned vortices in $\langle \boldsymbol{u} \rangle _{c}$ related to $\mathcal{C}_t$ are also represented by isocontours of negative $\lambda _2 ({\langle \boldsymbol{u} \rangle _{c}} )$ in figure 25(a). Their overall appearance is similar to a vortex structure identified by Tschisgale et al. (Reference Tschisgale, Löhrer, Meller and Fröhlich2021), being constricted near the trough, then growing to the sides and upwards further downstream. Note that in Tschisgale et al. (Reference Tschisgale, Löhrer, Meller and Fröhlich2021) the trigger event was located at the root of the locally most deflected blade. The actual trough in the resulting conditional mean then occurred at a value $r_x\gt 0$ instead of $r_x = 0$ .

The conditionally averaged flow situation associated with ridges ( $\mathcal{C}_r$ ) is shown in figures 23(b), 24 and 25(b). Here, the flow features an ejection event characterized by locally reduced streamwise velocity ( $\langle u' \rangle _{c} \lt 0$ in figure 24 b), a substantial uplift surrounding $\boldsymbol{r} = 0$ (figures 23 band 24 c) and substantially increased turbulent shear with a maximum of yet greater magnitude than that of $\mathcal{C}_t$ . This maximum is located just below the conditionally averaged hull height at $y\approx 2h$ . Streamwise-oriented vortices accompany the ridge on either side, rotating in opposite directions compared with the sweep event (left panel of figure 23 b). The width of the ridge can be assessed by the distance between the minima of the conditional hull height on either side (left picture of figure 23 b), which amounts to $0.92\,H$ . This is smaller than the width of the mean trough in figure 23(a).

The mean flows obtained by conditioning the two dominating types of events exhibit marked differences (figure 24 a). Very small velocities occur inside the canopy for the ejections ( $\mathcal{C}_r$ ), reaching $U/2$ only at the boundary with the outer fluid. In contrast, a full profile is observed for the sweeps ( $\mathcal{C}_t$ ) with a much stronger velocity gradient at $ \langle \bar {y} \rangle _{c}$ in this case.

Neighbouring velocity streaks associated with both events in figure 25 are positioned with a spanwise distance of $\approx 0.55\,H$ , matching the distance $r_z$ at which two-point correlations are most anti-correlated in § 7.2 above, although the latter was obtained by averaging over the entire domain, while here averages are conditioned by trigger events. Furthermore, the low-speed streak associated with $\mathcal{C}_r$ and the accompanying high-speed volumes are more pronounced than the high-speed streak with accompanying low-speed volumes in the pattern for $\mathcal{C}_t$ . This matches the observations made when inspecting instantaneous flow structures in animated visualizations corresponding to figure 4: high-speed streaks stand out more than low-speed counterparts, meaning that they are more concentrated and more intense when they appear, while low-speed volumes are larger and more frequent, but also of lower intensity.