2.1. Two-dimensional velocity evolution: submerged canopy of infinite width

When unidirectional flow encounters a 2-D submerged canopy, velocity decreases within the canopy ($0\leqslant z < h$, with $h$ the canopy height), diverting flow to the region above the canopy, such that the velocity increases above the canopy ($h\leqslant z < H$, with $H$ the water depth; see figure 1). Note that $x$ and $z$ are the streamwise and vertical coordinates, respectively. Previous studies have used a two-layer model to characterize the fully developed flow over a submerged canopy (e.g. Konings, Katul & Thompson Reference Konings, Katul and Thompson2012; Chen et al. Reference Chen, Jiang and Nepf2013). The velocities within ($U_{1}(x$)) and above ($U_{2}(x$)) the canopy are assumed to be vertically uniform.

Figure 1.

(a) Flow adjustment along a 2-D submerged canopy. Vertical grey bars denote the canopy elements. Dashed lines display the development of the vertical shear layer, which contains vortices generated by Kelvin–Helmholtz instability. Here $\delta _{e}$ is the penetration length in the fully developed region, defined as the distance the shear layer vortices penetrate into the canopy; $X_D$ is the adjustment length. (b) Two-layer model with vertically uniform velocity within each layer. Within the adjustment region the vertically averaged velocity in the canopy and in the overflow are $U_{1}(x)$ and $U_{2}(x)$, respectively, and $U_{1f}$ and $U_{2f}$ are the velocities in the fully developed region, respectively. Here $U_{\infty }$ is the depth-averaged velocity upstream of the canopy.

For a submerged canopy of rigid cylinders, the fully developed, in-canopy velocity is described by (21) in Chen et al. (Reference Chen, Jiang and Nepf2013), i.e.

(2.1)\begin{equation} \frac{U_{1f}}{U_{\infty}} = \frac{1}{1-\dfrac{h}{H}\phi+\sqrt{\dfrac{C_{D}ah}{2C_{2Dim}(1-\phi)}\left(\dfrac{H-h}{H}\right)^{3}}}, \end{equation}

with cylinder diameter $d$ and area density $n$ ($\textrm {cyl.}\ \textrm {m}^{-2}$), such that the solid volume fraction $\phi =({{\rm \pi} }/{4})nd^{2}$. Here $C_{D}$ is the drag coefficient on the canopy elements, and $a=nd$ is the frontal area per canopy volume. Equation (2.1) requires that the canopy drag is sufficient to produce a shear layer profile, which occurs for dimensionless meadow density $ah > 0.1$ (e.g. Belcher, Jerram & Hunt Reference Belcher, Jerram and Hunt2003; Nepf et al. Reference Nepf, Ghisalberti, White and Murphy2007). The coefficient $C_{2Dim}$ characterizes the turbulent momentum exchange between the two layers. The subscript ‘2Dim’ was chosen to represent 2-D canopies. Based on the roughness scaling, previous studies (e.g. Gioia & Bombardelli Reference Gioia and Bombardelli2001; Konings et al. Reference Konings, Katul and Thompson2012) have suggested that

(2.2)\begin{equation} C_{2Dim} = K_{c}\left(\frac{\delta_{e}}{H}\right)^{1/3}, \end{equation}

with $\delta _{e}$ the penetration length, defined as the vertical extent of the shear layer within the canopy (figure 1a), and $K_{c}$ is an empirical constant which, for rigid canopies, is $K_{c}=0.07\pm 0.02$ (Chen et al. Reference Chen, Jiang and Nepf2013). The penetration length is inversely proportional to canopy drag, $\delta _{e}\sim (C_{D}a)^{-1}$ (Nepf et al. Reference Nepf, Ghisalberti, White and Murphy2007), with scale constant $0.3\pm 0.1$ determined from data across a wide range of physical scale $10^{-3}<(C_{D}a)^{-1}<10^{2}$, (Ghisalberti & Nepf Reference Ghisalberti and Nepf2009). The penetration length is also constrained by the water depth and the canopy height (Nepf & Vivoni Reference Nepf and Vivoni2000; Konings et al. Reference Konings, Katul and Thompson2012), such that

(2.3)\begin{equation} \delta_{e} = \min\left(\frac{0.3\pm 0.1}{C_{D}a}, H-h, h\right). \end{equation}

Chen et al. (Reference Chen, Jiang and Nepf2013) suggested that $U_{1}(x)$ decays exponentially from the canopy leading edge over the adjustment length $X_{D}$, which is defined as the distance required for the in-canopy velocity to achieve 95 % of the transition from upstream $U_{\infty }$ to fully developed $U_{1f}$, i.e.

(2.4)\begin{equation} \left\{\begin{array}{@{}ll} U_{1}(x) = U_{1f} + (U_{\infty}-U_{1f})\,\textrm{e}^{-{3x}/{X_{D}}}, \\ \dfrac{X_{D}}{L_{c}} = A_{1}[1+A_{2}]C_{D}ah, \end{array} \right. \end{equation}

in which $L_{c} =({2(1-\phi )})/{C_{D}a}$ is the canopy drag length scale (Belcher et al. Reference Belcher, Jerram and Hunt2003). Using the empirical coefficients determined by Chen et al. (Reference Chen, Jiang and Nepf2013), $A_{1}=1.5\pm 0.2$ and $A_{2}=2.3\pm 0.2$, (2.4) can be written as

(2.5)\begin{equation} \frac{X_{D}}{h} = (6.9\pm 1.1)(1-\phi) + \frac{3.0\pm 0.4}{C_{D}ah}(1-\phi). \end{equation}

The exponential decay model works well for sparse canopies, but it overestimates the velocity for dense canopies ($ah > 1.4$, figure 15 in Chen et al. Reference Chen, Jiang and Nepf2013). In the present study, rather than assuming exponential decay, an analytical solution to $U_{1} (x)$ was derived, appropriate for any density $ah > 0.1$. The continuity and layer-averaged momentum equations in the developing region can be written as follows (see also (18)–(20) in Chen et al. Reference Chen, Jiang and Nepf2013).

Continuity:

(2.6)\begin{equation} U_{1}(x)h(1-\phi) + U_{2}(x)(H-h) = U_{\infty} H. \end{equation}

Conservation of momentum in the canopy layer:

(2.7)\begin{align} \rho\left[U_{1}(x)\frac{\partial U_{1}}{\partial x}+W_{1}(x)\left(\frac{\partial U}{\partial z}\right)_{1}\right]h &= \rho gh\frac{\partial H}{\partial x} - \frac{\rho C_{D}ah}{2(1-\phi)}U_{1}(x)^{2} \nonumber\\ &\quad + \rho C_{2Dim}(U_{2}(x) - U_{1}(x))^{2}. \end{align}

Conservation of momentum in the overflow layer:

(2.8)\begin{align} \rho\left[U_{2}(x)\frac{\partial U_{2}}{\partial x}+W_{2}(x)\left(\frac{\partial U}{\partial z}\right)_{2}\right](H-h) = \rho g(H-h)\frac{\partial H}{\partial x} - \rho C_{2Dim}(U_{2}(x)-U_{1}(x))^{2} . \end{align}

The vertical velocity ($W$) and vertical gradient of $U({\partial U}/{\partial z})$ at the shared interface ($z=h$) must be the same, such that $W_{1}(x)=W_{2}(x)$ and $({\partial U}/{\partial z})_{1}=({\partial U}/{\partial z})_{2}$. The vertical transport and the pressure terms can be cancelled with the manipulation ((2.7) and (2.8) $\times \,h/({H-h})$), resulting in

(2.9)\begin{align} \left[U_{1}(x)\frac{\partial U_{1}(x)}{\partial x} - U_{2}(x)\frac{\partial U_{2}(x)}{\partial x}\right] h = C_{2Dim}(U_{2}(x) - U_{1}(x))^{2}\frac{H}{H-h} - \frac{C_{D}ah}{2(1-\phi)}U_{1}(x)^{2}. \end{align}

Combining (2.6) and (2.9), an expression for $U_{1} (x)$ was determined, i.e.

(2.10)\begin{align} &\frac{\partial U_{1}(x)}{\partial x}h\frac{U_{1}(x)[(H-h)^{2}-(1-\phi)^{2}h^{2}]+U_{\infty}Hh(1-\phi)}{(H-h)^{2}} \nonumber\\ &\quad =C_{2Dim}\left[U_{\infty}-U_{1}(x)\left(1-\frac{h}{H}\phi\right)\right]^{2} \left(\frac{H}{H-h}\right)^{2}\nonumber\\ &\qquad - \frac{C_{D}ah}{2(1-\phi)}U_{1}(x)^{2} . \end{align}

In the fully developed region, ${\partial U_{1}(x)}/{\partial x}=0$ and (2.10) correctly reverts to (2.1). The analytical solution to (2.10) was derived as follows. Let

(2.11)\begin{equation} \left\{\begin{array}{@{}l} \alpha = \dfrac{[(H-h)^{2} - (1-\phi)^{2}h^{2}]h}{(H-h)^{2}}, \\ \beta = C_{2Dim}\left(\dfrac{H}{H-h}\right)^{3} \left(1-\dfrac{h}{H}\phi\right)^{2}-\dfrac{C_{D}ah}{2(1-\phi)}, \\ A' = \dfrac{Hh(1-\phi)}{(H-h)^{2} - (1-\phi)^{2}h^{2}}U_{\infty}, \\ B' = \dfrac{-2\left(1-\dfrac{h}{H}\phi\right)}{\left(1-\dfrac{h}{H}\phi\right)^{2} - \dfrac{C_{D}ah}{2C(1-\phi)}\left(\dfrac{H-h}{H}\right)^{3}}U_{\infty}, \\ C' = \dfrac{1}{\left(1-\dfrac{h}{H}\phi\right)^{2} - \dfrac{C_{D}ah}{2C(1-\phi)}\left(\dfrac{H-h}{H}\right)^{3}}U_{\infty}^{2}, \end{array} \right. \end{equation}

and substitute (2.11) into (2.10), we get

(2.12)\begin{equation} \frac{\alpha}{\beta}\frac{U_{1}(x)+A'}{U_{1}(x)^{2}+B'U_{1}(x)+C'}\,\textrm{d} U_{1}(x) = {\textrm{d} x} . \end{equation}

Integrating (2.12) yields

(2.13)\begin{equation} \left\{\begin{array}{@{}l} f(U) = \displaystyle\int\dfrac{U+A'}{U^{2}+B'U+C'}\,\textrm{d}U = \dfrac{2A'-B'}{\sqrt{4C'-B'^{2}}}\arctan\dfrac{B'+2U}{\sqrt{4C'-B'^{2}}} \\ \qquad \quad + \dfrac{1}{2}\ln(U^{2}+B'U+C'),\\ f(U_{1}(x)) - f(U_{\infty}) = \dfrac{\beta}{\alpha}x, \end{array} \right.. \end{equation}

Note that (2.11) can be simplified if $\phi \ll 1$. The simplified expression is shown in Appendix A.1. With ${\partial U_{1}(x)}/{\partial x}$ given in (2.10), forward Euler's method was used to compute $U_{1} (x)$ with the initial condition $U_{1}(x=0)=U_{\infty }$, which is justified in § 4.4.

A scale analysis of (2.7) was used to provide a framework for the dependence of $X_{D}$ on pressure, drag and shear forcing terms. First, $U_{2}$ was scaled with $U_{\infty }$, and $(U_{2}-U_{1}) = ({H}/({H-h}))(U_{\infty }-U_{1})\sim ({H}/({H-h}))U_{\infty }$. Second, the distance from the leading edge, $x$, was scaled with $X_{D}$. Third, based on previous studies (e.g. Rominger & Nepf Reference Rominger and Nepf2011), the pressure terms $\rho gh({\partial H}/{\partial x})$ scale with ${\rho U_{\infty }^{2}C_{D}ah}/{X_{D}}$. Finally, $\phi$ was neglected for simplicity. With these scales, (2.7) yields

(2.14)\begin{equation} \left.\begin{gathered} -\frac{U_{\infty}^{2}}{X_{D}} \sim \frac{C_{D}ahU_{\infty}^{2}}{X_{D}}+ C_{2Dim}\frac{U_{\infty}^{2}}{h}\left(\frac{H}{H-h}\right)^{2}-C_{D}aU_{\infty}^{2}, \\ \qquad\quad pressure\qquad \qquad \qquad shear\qquad\qquad drag\\ \frac{X_{D}}{h}\sim\frac{C_{D}ah+1}{C_{D}ah-C_{2Dim}\left(\dfrac{H}{H-h}\right)^{2}}. \end{gathered}\right\}\end{equation}

This shows that the inclusion of the turbulent shear stress at the top of the canopy, which was excluded by Chen et al. (Reference Chen, Jiang and Nepf2013), extends the adjustment length, which is consistent with the tendency to accelerate the velocity within the canopy. Note that when the drag term is much greater than the shear term, (2.14) reverts to the form proposed by Chen et al. (Reference Chen, Jiang and Nepf2013), shown in (2.5).

2.2. Three-dimensional velocity evolution: submerged canopy of finite width

For 3-D canopies, the flow was separated into the canopy region, with velocity $U_{1,3D} (x)$, and the open-channel (outer) region, with velocity $U_{2,3D} (x)$, with the averaging now defined over the fraction of cross-sectional area occupied by the canopy (grey in figure 2) and outside the canopy (white in figure 2), respectively. Since the model assumes a spatially uniform velocity outside the canopy, $U_{2,3D} (x)$, the accuracy of the model will diminish if the edges of the canopy are too close to the boundaries (${h}/{H}\approx 1$ or ${b}/{W}\approx 1$). In that case, the flow velocity on the top of the canopy might be significantly different from that in regions next to the lateral interfaces. The analytical solution in the following is strictly valid for cases with small ${h}/{H}$ and ${b}/{W}$. In our experiments, different values of ${h}/{H}$ (${h_{d}}/{H}$ for 3-D flexible canopies) and ${b}/{W}$ were considered to explore the range of validity of the 3-D analytical model. For ${h_{d}}/{H}$ and ${b}/{W}$ as large as 0.52 and 0.51, the velocity in the regions above and adjacent to the canopy are comparable, supporting the model assumption of a spatially uniform velocity outside the canopy.

Figure 2.

Cross-section of a channel with a submerged canopy of finite width $2b$ (3-D canopy). The channel width is $2W$. The canopy, shown with grey shading, has velocity $U_{1,3D} (x)$), and the unobstructed outer flow region, shown in white, has velocity $U_{2,3D} (x)$. The vertical bars denote canopy elements. The canopy height is $h$. The water depth is $H$.

For the 3-D submerged canopy, three interfaces must be considered, introducing new terms into the momentum equations. The additional subscript ‘3D’ was added to distinguish this as the 3-D solution.

Continuity:

(2.15)\begin{equation} U_{1,3D}(x)bh(1-\phi) + U_{2,3D}(x)(WH-bh) = U_{\infty}WH. \end{equation}

Conservation of momentum in the canopy region (region 1):

(2.16)\begin{align} &\rho\left[U_{1,3D}(x)\frac{\partial U_{1}(x)}{\partial x} + V_{1,3D}(x)\left(\frac{\partial U}{\partial y}\right)_{1} + W_{1,3D}(x)\left(\frac{\partial U}{\partial z}\right)_{1}\right]bh \nonumber\\ &\quad =\rho gbh\frac{\partial H}{\partial x} - \frac{\rho C_{D}ahb}{2(1-\phi)}U_{1,3D}(x)^{2} + \rho C_{3Dim}(b+h)[U_{2,3D}(x) - U_{1,3D}(x)]^{2} . \end{align}

Conservation of momentum outside the canopy (region 2):

(2.17)\begin{align} & \rho\left[U_{2,3D}(x)\frac{\partial U_{2}(x)}{\partial x} + V_{2,3D}(x)\left(\frac{\partial U}{\partial y}\right)_{2} + W_{2,3D}(x)\left(\frac{\partial U}{\partial z}\right)_{2}\right] (WH - bh) \nonumber\\ &\quad =\rho g(WH-bh)\frac{\partial H}{\partial x} - \rho C_{3Dim}(b+h)[U_{2,3D}(x) - U_{1,3D}(x)]^{2}. \end{align}

Similar to 2-D canopies, the advective transport of momentum between the two zones, represented by the second and third terms of (2.16) and (2.17), must be equal at the interface between the two zones. The vertical velocity ($W$) and vertical gradient of $U({\partial U}/{\partial z})$ at the shared interface ($z = h$) must be the same, $W_{1,3D}(x)=W_{2,3D}(x)$ and $({\partial U}/{\partial z})_{1}=({\partial U}/{\partial z})_{2}$. Similarly, lateral velocity and its lateral gradient are continuous across the boundary between the two regions, such that $V_{1,3D}(x)=V_{2,3D}(x)$ and $({\partial U}/{\partial y})_{1}=({\partial U}/{\partial y})_{2}$. Then, the advective transport terms and the pressure terms can be cancelled with the manipulation ((2.16) and (2.17) $\times \,bh/({WH-bh})$).

Here, $C_{3Dim}$ is the coefficient that characterizes the turbulent momentum exchange between the canopy and outer flow. The 3-D canopy has three interfaces with the outer flow, two along the side boundaries and one at the top. The penetration length scale, $\delta _{e}$, and the coefficient, $K_{c}$, were assumed to be the same on every interface. This is supported by data in White & Nepf (Reference White and Nepf2008), who considered momentum exchange at the lateral edge of a cylinder array. They found $\delta _{e}\approx 0.5(C_{D}a)^{-1}$, which is close to (2.3), and $K_{c}=0.06\pm 0.03$ (based on table 1 of that paper), which is similar to the value observed at the top interface ($K_{c}=0.07\pm 0.02$, Chen et al. Reference Chen, Jiang and Nepf2013), so that the application of a single roughness on all three interfaces of a 3-D canopy was reasonable. Here, $K_{c}=0.07\pm 0.02$ was used for all interfaces. To represent the average dimension of the outer flow, the water depth, $H$, was replaced with the hydraulic radius, $R$, defined as the channel cross-section divided by its wetted perimeter, $R=(({H\times 2W})/({2H+2W}))={HW}/({H+W})$, and then (2.2) becomes

(2.18)\begin{equation} C_{3Dim} = K_{c}\left(\frac{\delta_{e}}{R}\right)^{1/3}. \end{equation}

Note that as the channel width ($2W$) approaches infinity, (2.18) appropriately reverts (2.2). The penetration length is also constrained by the water depth, the canopy height, the channel width and the canopy width. Similar to (2.3), for 3-D canopies,

(2.19)\begin{equation} \delta_{e}=\min{\left(\frac{0.3\pm 0.1}{C_{D}a},H-h,h,W-b,b\right)}. \end{equation}

Finally, combining (2.15), (2.16) and (2.17), we get

(2.20)\begin{align} & \frac{\textrm{d} U_{1,3D}(x)}{\textrm{d} x}bh\frac{U_{1,3D}(x)[(WH-bh)^{2}-(1-\phi)^{2}(bh)^{2}]+ U_{\infty}WHbh(1-\phi)}{(WH-bh)^{3}} \nonumber\\ &\quad = C_{3Dim}(b+h)\left[U_{\infty}-U_{1,3D}(x)\left(1-\frac{bh}{WH}\phi\right)\right]^{2} \left(\frac{WH}{WH-bh}\right)^{3} \nonumber\\ &\qquad - \frac{C_{D}abh}{2(1-\phi)}U_{1,3D}(x)^{2}. \end{align}

In the fully developed region, ${\textrm {d} U_{1,3D} (x)}/{\textrm {d} x}=0$, from which (2.20) predicts the fully developed in-canopy velocity, $U_{1f,3D}$,

(2.21)\begin{equation} \frac{U_{1f,3D}}{U_{\infty}} = \frac{1}{1-\dfrac{bh}{WH}\phi+\sqrt{\dfrac{C_{D}abh}{2C_{3Dim}(b+h)(1-\phi)}\left(\dfrac{WH-bh}{WH}\right)^{3}}}. \end{equation}

The complete analytical solution to (2.20) is shown in Appendix A.2.

2.3. Flexible canopies

Some canopies, such as a seagrass meadow, consist of flexible blades which bend in response to flow. The bending of blades, called reconfiguration, can be described by two dimensionless parameters (e.g. Luhar & Nepf Reference Luhar and Nepf2011). The Cauchy number ($Ca$, (2.22)) is the ratio of hydrodynamic drag to the restoring force due to blade stiffness, and the buoyancy parameter ($B$, (2.23)) is the ratio of buoyancy force to the restoring force, i.e.

(2.22)\begin{gather} Ca = \frac{\frac{1}{2}C_{D}\rho w_{b}\,\textrm{d}U^{2}l^{3}}{EI}, \end{gather}

(2.23)\begin{gather}B = \frac{\Delta\rho g\,\textrm{d}tl^{3}}{EI}. \end{gather}

Here, $w_{b}$ and $t$ are the blade width and thickness, respectively, $\rho$ is the density of water, $U$ is the velocity acting on the blade, $\Delta \rho$ is the difference in density between the water and the blade, $E$ is the Young's modulus and $I={w_{b}t^{3}}/{12}$ is the bending moment of inertia. The vertical distance between the blade tip and the blade base is called the deflected blade height, $h_{d,b}$. Combining experiments and simulation, Luhar et al. (Reference Luhar, Infantes, Orfila, Terrados and Nepf2013) described the deflected blade height (4) of individual artificial seagrass blades under unidirectional currents as a function of $Ca$ and $B$, i.e.

(2.24)\begin{equation} \frac{h_{d,b}}{l} = 1 - \frac{1-Ca^{-{1}/{4}}}{1+Ca^{-{3}/{5}}(4+B^{{3}/{5}})+Ca^{{-}2}(8+B^{2})}. \end{equation}

Furthermore, the drag on a flexible blade can be described by an effective blade length, $l_{e,b}$, which is the length of a rigid blade that experiences the same drag as a flexible blade of length $l$. Reconfiguration reduces drag, such that $l_{e,b} < l$. Combining experiments and simulation, Luhar & Nepf (Reference Luhar and Nepf2011, (16)) described

(2.25)\begin{equation} \frac{l_{e,b}}{l} = 1-\frac{1-0.9Ca^{-{1}/{3}}}{1+Ca^{{2}/{3}}(8+B^{{3}/{2}})}. \end{equation}

For the flexible canopies, the deflected canopy height, $h_{d}$, was calculated as the sum of the deflected blade height, $h_{d,b}$ (by (2.24)), and the sheath height, $l_{s}$, i.e.

(2.26)\begin{equation} h_{d} = h_{d,b} + l_{s}. \end{equation}

The flexible canopy drag was defined as $C_{D}al_{e} = C_{D}nA_{f,e}$, with $A_{f,e}$ the effective frontal area per plant with deflected blades,

(2.27)\begin{equation} \left.\begin{gathered} C_{D}al_{e} = C_{D}nA_{f,e} = n(6C_{D,plate}l_{e,b}b + C_{D,cylinder}l_{s}d),\\ \qquad\qquad\qquad\qquad\qquad blades\qquad\qquad sheath \end{gathered}\right\} \end{equation}

with $l_{e,b}$ estimated from (2.25).

We propose (and later test) that flow development in a flexible canopy can be described by the governing equations developed above, with two adjustments. First, in the terms related to canopy geometry, the canopy height, $h$, was replaced by the deflected height, $h_{d}$. Second, the canopy drag ($C_{D} ah$) is replaced by $C_{D}al_{e}$, defined in (2.27). Making these changes in (2.10) and (2.20), the canopy velocity in flexible 2-D ($U_{1}$) and 3-D ($U_{1,3D}$) canopies, can be expressed, respectively, as

(2.28)\begin{align} &\frac{\textrm{d} U_{1}(x)}{\textrm{d} x}h_{d}(x)\frac{U_{1}(x)[(H-h_{d}(x))^{2}-(1-\phi)^{2}h_{d}(x)^{2}]+U_{\infty}Hh_{d}(x)(1-\phi)}{(H-h_{d}(x))^{2}} \nonumber\\ &\quad = C\left[U_{\infty}-U_{1}(x)\left(1-\frac{h_{d}(x)}{H}\phi\right)\right]^{2} \left(\frac{H}{H-h_{d}(x)}\right)^{3} - \frac{C_{D}al_{e}(x)}{2(1-\phi)}U_{1}(x)^{2}, \end{align}

(2.29)\begin{align} &\frac{\textrm{d} U_{1,3D}(x)}{\textrm{d} x}bh_{d}(x) \frac{U_{1,3D}(x)[(WH-bh_{d})^{2}-(1-\phi)^{2}(bh_{d}(x))^{2}]+U_{\infty}WHbh_{d}(x)(1-\phi)}{(WH-bh_{d}(x))^{2}} \nonumber\\ &\quad = C_{3Dim}(b+h_{d}(x))\left[U_{\infty}-U_{1,3D}(x)\left(1-\frac{bh_{d}(x)}{WH}\phi\right)\right]^{2} \left(\frac{WH-bh_{d}(x)}{WH}\right)^{3}\nonumber\\ &\qquad - \frac{C_{D}abl_{e}(x)}{2(1-\phi)}. \end{align}

Because $h_{d} (x)$ and $l_{e} (x)$ are functions of the velocity (through $Ca$), forward Euler's method was used to compute $U_{1} (x)$ and $U_{1,3D} (x)$. The computation of the flow development is described in Appendix A.3. In the fully developed region the deflected height and the effective length become constant ($h_{d} (x)=h_{df}$, $l_{e} (x)=l_{ef}$). Also, ${\partial U_{1}(x)}/{\partial x}=0$ and ${U_{1,3D} (x)}/{\partial x}=0$, leading to

(2.30)\begin{gather} {\rm 2-D\; flexible\; canopy:} \frac{U_{1f}}{U_{\infty}} = \frac{1}{1-\dfrac{h_{df}}{H}\phi + \sqrt{\dfrac{C_{D}al_{ef}}{2C_{2Dim}(1-\phi)}\left(\dfrac{H-h_{df}}{H}\right)^{3}}}, \end{gather}

(2.31)\begin{gather} {\rm 3-D\; flexible\; canopy:} \frac{U_{1f,3D}}{U_{\infty}} = \frac{1}{1-\dfrac{bh_{df}}{WH}\phi + \sqrt{\dfrac{C_{D}abl_{ef}}{2C_{3Dim}(b+h_{df})(1-\phi)}\left(\dfrac{WH-bh_{df}}{WH}\right)^{3}}}, \end{gather}

in which $h_{df}$ and $l_{ef}$ denote the deflected height and the effective length in the fully developed region. Because $h_{df}$ and $l_{ef}$ depend on the flow velocity, (2.30) and (2.31) were applied in an iterative fashion to calculate $U_{1f}$ and $U_{1f,3D}$.

2.4. Unbounded domain

Field conditions are better represented by a canopy of finite width within an unbounded domain, corresponding to $W \gg b$, or ${b}/{W}\sim 0$, for which (2.21) reduces to

(2.32)\begin{equation} \frac{U_{1f,3D}}{U_{\infty}} = \frac{1}{1+\sqrt{\dfrac{C_{D}ahb}{2C_{3Dim}(b+h)(1-\phi)}}} . \end{equation}

For $b\gg h$, the geometry of the canopy could be considered to approach that of a 2-D canopy. With these limits, (2.32) reduced to

(2.33)\begin{equation} {\rm 3-D\; canopy:} \frac{U_{1f,3D}}{U_{\infty}} = \frac{1}{1+\sqrt{\dfrac{C_{D}ah}{2C_{3Dim}(1-\phi)}}}, \end{equation}

which differed from the original 2-D solution, repeated here for comparison;

(2.34)\begin{equation}{\rm 2-D\; canopy:} \frac{U_{1f}}{U_{\infty}} = \frac{1}{1-\dfrac{h}{H}\phi+\sqrt{\dfrac{C_{D}ah}{2C_{2Dim}(1-\phi)}\left(\dfrac{H-h}{H}\right)^{3}}}. \end{equation}

The difference arises from the assumed confinement of the flow domain. A channel-spanning 2-D canopy (2.34) is confined by the channel walls, such that continuity required that flow diverted out of the canopy must be incorporated into the overflow, such that the overflow velocity ($U_{2}$) increased as the flow within the canopy ($U_{1}$) decreased. For 3-D canopies, when $W\gg b$ and $b\gg h$, only a small portion of the channel is occupied by the canopy, such that the flow velocity outside the canopy is still approximately equal to $U_{\infty }$.

3.1. Model canopies

Canopies were constructed from model plants held in a staggered array. The rigid canopies consisted of rigid cylinders 6.4 mm in diameter ($d$), 7 cm tall ($h$), with area density $n$ ($\textrm {cyl}.\ \textrm {m}^{-2}$). The dimensionless canopy density $ah = ndh$. The flexible canopy consisted of model plants that were geometrically and dynamically similar to the seagrass Zostera marina (figure 3). Each plant consisted of six LDPE (low-density polyethylene) blades ($\textrm {density}= 920\ \textrm {kg}\ \textrm {m}^{-3}$; $E = 3\times 10^{8}\ \textrm {Pa}$) with length $l_{b}=13$ cm, width $w_{b}=3$ mm and thickness $t=0.1$ mm, attached to a rigid, circular cylinder, representing the plant sheath, which is rigid. The sheath protruded $l_{s}=1$ cm above the baseboard. The frontal area of each model plant was $A_{f}=l_{s}d + 6w_{b}l_{b}$ with $d=6.4$ mm. With $n$ plants per $\textrm {m}^{2}$, $ah=nA_{f}$. The cylinder (or plant) density ($n$), dimensionless canopy density ($ah$) and channel-average flow velocity ($U_{\infty }$) are summarized in table 1 (2-D canopies) and table 2 (3-D canopies).

Figure 3.

Side view of recirculating water channel with 8 m long submerged canopy that covered the channel width (2-D canopy). Figure is not to scale. Velocity measured with Nortek Vectrino along the centreline of the channel.

Table 1.

Experimental conditions for 2-D canopies. Here $n =$ shoots per bed area; full canopy height, $h$; and deflected height, $h_{d}$, for flexible canopies only. For the rigid plants, $ah=ndh$ and $\phi ={\rm \pi} /4 nd^{2}$. For the flexible plants, the frontal area of each flexible plant was $A_{f}=l_{s}d+6w_{b}l_{b}$, $ah=nA_{f}$ and $\phi =({l_{s}({{\rm \pi} }/{4})nd^{2}+6l_{b}nbt})/({l_{s}+l_{b}})$. Each flexible plant had six blades, so that $n=520\ \textrm {plants}\ \textrm {m}^{-2}$ corresponded to $3120\ \textrm {blades}\ \textrm {m}^{-2}$.

Table 2.

Experimental conditions for 3-D submerged canopies. Here $n=\text {shoots}$ per bed area; full canopy height, $h$, canopy width, $2b$; channel width $2W=120$ cm; and deflected height, $h_{d}$, for flexible canopies only. For rigid plants, $ah = ndh$ and $\phi =({{\rm \pi} }{4})nd^{2}$. For the flexible plants, the frontal area per plant was $A_{f}=l_{s}d+6w_{b}l_{b}$, $ah=nA_{f}$ and $\phi =({l_{s}({{\rm \pi} }/{4})nd^{2}+6l_{b}nbt})/({l_{s}+l_{b}})$. Each flexible plant had six blades, so that $n = 775\ \textrm {plants}\ \textrm {m}^{-2}$ corresponded to $4650\ \textrm {blades}\ \textrm {m}^{-2}$.

3.2. Two-dimensional canopies

Two-dimensional canopy experiments were conducted in a 24 m long, 38 cm wide recirculating water channel (figure 4). The water depth was $H = 36$ cm. The model canopy was 8 m long. Vertical profiles of velocity were measured using a Nortek Vectrino, which was placed between two adjacent model plants at mid-width of the channel. Previous studies using staggered arrays have shown that the mean velocity at this point was within 5 % of the lateral average across a 2-D canopy (see figure 2 in Chen et al. Reference Chen, Jiang and Nepf2013). Vertical profiles were measured at multiple streamwise locations starting upstream of the canopy, $x=-3h$ to $x=84h$, and with 1 cm intervals from the bed ($z = 0$). For each measurement, velocity was collected for $T=6$ min at a sampling rate of 200 Hz. The depth-averaged velocity upstream of the canopy, $U_{\infty }$, was determined from a vertical profile measured at 1 m upstream of the leading edge.

Figure 4.

Canopy of flexible plants, dynamically and geometrically similar to Z. marina. Each plant consists of six LDPE blades attached to a rigid sheath (grey in image). One plant is painted in black to enhance visualization.

3.3. Three-dimensional canopies

Submerged canopies of finite width were constructed in a 16 m long and 1.2 m wide recirculating channel. The water depth was $H = 20$ cm. The model canopies were 2.4 m long. Four different canopy widths ($2b$) were constructed (table 2, figure 5). Velocity was measured at multiple distances from the leading edge, $x= 1.5h$ to $x=7.4h$. At each streamwise position, the velocity was measured at 18 positions in a grid of six lateral positions and three vertical positions. The lateral distribution of measurement points in the canopy are shown in figure 6 for different arrays. The vertical positions were 2.5, 4 and 6 cm for rigid canopies and 3, 6 and 8 cm for flexible canopies. The measurement positions were mid-way between adjacent rows and columns. The average of the 18 positions defined the in-canopy velocity, $U_{1,3D}$.

Figure 5.

Recirculating channel with a submerged canopy of finite width ($2b$) less than the channel width ($2W = 1.2$ m). Shaded region denotes the canopy, located at the channel centre. (a) Side view; (b) top view; (c) front view. Not to scale.

Figure 6.

Top view of canopy, with plant positions shown with grey circles. Blue circles denote velocity measurement points located mid-way between adjacent rows and columns of plants. At each blue dot, velocity was measured at three vertical positions, $z=2.5$, $4$ and $6$ cm for rigid canopies and $z=3$, $6$ and $8$ cm for flexible canopies. The canopy width is $2b$. (a) Case 5 and 9, velocity measured at $y=\pm \frac {5}{6}b$, $\pm \frac {1}{2}b$, $\pm \frac {1}{6}b$. (b) Case 6 and 10, $y=\pm \frac {13}{14} b$, $\pm \frac {1}{2}b$, $\pm \frac {1}{14}b$. (c) Case 7 and 11, $y=\pm \frac {25}{26} b$, $\pm \frac {15}{26} b$, $\pm \frac {5}{26} b$. The black dashed lines denote the channel and canopy centreline ($y = 0$).

3.4. Estimation of deflected height

The deflected height was measured using images from a digital camera looking through the side of the channel. A black background was taped to the opposite channel wall. Ten images were taken at one-second intervals and used to estimate deflected height at nine positions along the meadow. The measured deflected height was estimated from the average across all the ten photos, and uncertainty was calculated as the standard deviation.

3.5. Validation of the analytical model

The initial adjustment length, $X_{D}$, was defined as the distance from the leading edge to the point at which the canopy velocity reached the 95 % transition from the upstream velocity $U_{\infty }$ to fully developed velocity $U_{1f}$, i.e. $U_{1} (X_{D})=0.05(U_{\infty }-U_{1f} )+U_{1f}$. It was estimated from both the measured and predicted velocity. The measured $X_{D}$ was interpolated between the measured velocities closest to $U_{1}(X_{D})$. For the prediction, the rigid element drag coefficient, $C_{D}$, was estimated for cylinders from the numerical study of staggered arrays by Etminan, Lowe & Ghisalberti (Reference Etminan, Lowe and Ghisalberti2017), and for flat plates, representing the blades, from Ellington (Reference Ellington1991),

(3.1)\begin{equation} \left\{\begin{array}{@{}l} C_{D,cylinder} = 1 + Re_{c}^{2/3}, \\ C_{D,plate} = 1.95 + \dfrac{50}{Re_{w}}, \end{array} \right. \end{equation}

in which the cylinder Reynolds number $Re_{c}=({U_{1}d}/{\nu })(({1-\phi })/({1-\sqrt {{2\phi }/{{\rm \pi} }}}))$ and the plate Reynolds number $Re_{w}={U_{1}w_{b}}/{\nu }$. Since velocity and, thus, Reynolds number, changed along the canopy, $C_{D}$ and its uncertainty were evaluated as follows. First, $C_{D}$ was calculated using the velocity $U_{\infty }$, resulting in $C_{D} (U_{\infty })$, which was then used to make a preliminary prediction of $U_{1f}$. Second, we used $U_{1f}$ to calculate the drag coefficient in the fully developed region, $C_{D} (U_{1f})$. Finally, a mean drag coefficient was calculated using $({U_{\infty }+U_{1f}})/{2}$ as the representative velocity. The uncertainty in $C_{D}$ was evaluated from the maximum, $C_{D} (U_{1f})$, and minimum, $C_{D} (U_{\infty })$. Finally, $U_{1f}$ was recalculated using $C_{D}(({U_{\infty }+U_{1f}})/{2})$, and the resulting $U_{1f}$ was within 5 % of that calculated using $C_{D} (U_{\infty })$. This indicated that the change in $C_{D}$ along the canopy did not significantly impact the prediction, and that the simplification of using $C_{D} (U_{\infty })$ was reasonable.