3.1. Temporally and spatially averaged turbulence statistics

The canopy boundary layer and inner flow within the surrogate seagrass exhibited significant departure from the rigid counterparts, which were modulated largely by the blade swaying. Basic flow statistics within and above the flexible and rigid canopies for a representative case at $Re = 7.0 \times 10^{4}$ (i.e. F3 and R3 scenarios; table 1) evidence the impact of the surrogate seagrass on the flow. This is illustrated in figure 3 with the dimensionless streamwise velocity, $U/U_{\infty }$, kinematic shear stress, $-\langle u'w'\rangle /U_{\infty }^{2}$, and turbulence intensity of the streamwise, ${\sigma _u} /U_{\infty }$, and vertical, $\sigma _w /U_{\infty }$ components. Here, $()'$ indicates temporal fluctuations and $\sigma$ and $\langle \rangle$ denote the standard deviation and time-averaging operators.

Figure 3.

Time-averaged streamwise velocity $U/U_{\infty }$, kinematic shear stress, $-\langle {u'w'}\rangle /U_{\infty }^{2}$, and turbulence intensity of the streamwise, $I_u = {\sigma _u}/U_{\infty }$, and vertical, $I_w = {\sigma _w}/U_{\infty }$, velocity components within and above the surrogate seagrass (a,c,e,g) and rigid canopy (b,d,f,h) at $Re = 7.0 \times 10^{4}$ (F3 and R3 scenarios; table 1). The horizontal, white dashed lines approximately mark the inner (within canopy) and outer flows.

The deflected blades significantly restricted the mass flux within the surrogate seagrass and enhanced the turbulence surrounding the top as compared with the rigid canopy. This resulted in a sharper mean shear that promoted unsteady exchange with the external flow, resulting in increased kinematic shear stress and turbulence levels. This, in turn, promoted oscillations of the canopy elements, which enhanced unsteady flow exchange and flow fluctuations near the top of the surrogate seagrass. The degree of such two-way interaction between the flow and blade motions is embodied in the Cauchy number. Evaluation of scenarios encompassing a range of $Ca$ provides context for natural flows, and offers a way to uncover several processes, including the control of flow fluctuations and particulate transport.

Double averaging in the streamwise direction and time (figure 4) illustrates distinct features of the bulk flow statistics across the vertical span for a range of ${Re}$ in the seagrass and rigid canopy. The vertical axis is split into two non-dimensional regions to aid understanding of the effect of the deflection of the flexible canopy. The canopy region is normalised by the mean deflected canopy height, i.e. $z/h_d$; whereas the region above the canopy is normalised as $z_{A}=(z-h_d)/h_c+1$. This factor sets the relative canopy top as a secondary origin. Note that the non-dimensional streamwise velocity, $U/U_{\infty }$ profiles collapse very well within and above the rigid canopy and $z_{A}$ is not normalised by the boundary layer depth, $\delta$, to stress the flow variability induced by the flexible canopy. In contrast to the seagrass, the rigid canopy did not alter development of the boundary layer at the $Re$ analysed. It is also worth highlighting the changing profiles within the seagrass canopy with changes in $Re$. This is further illustrated by the velocity change $\Delta U/U_{\infty }=(1-U_c/U_{\infty })$ within and above the canopies (figure 5a), which is approximately constant for the rigid canopy but decreases with $Re$ in the seagrass experiments.

Figure 4.

Space- and time-averaged vertical profiles of the normalised (a) streamwise velocity $\bar {U}/U_{\infty }$, and (b) kinematic shear stress $-\langle \overline {u'w'}\rangle /U_{\infty }^{2}$. Turbulence intensity of the streamwise (c) $\overline {I_u} = \overline {\sigma _u} /U_{\infty }$ and vertical (d) $\overline {I_w} = \overline {\sigma _w}/U_{\infty }$ velocity components. The overbar denotes space-averaging. Vertical axis is normalised by the deflected canopy height $h_d$, and the shaded area represents the data within the canopy.

Figure 5.

(a) Velocity change within canopy, (b) mixing layer thickness, (c) vortex penetration depth relative to the undeflected canopy height (left axis) and vortex penetration relative to the deflected canopy height (right axis). Data are shown for both the flexible seagrass and rigid canopy.

A distinct effect induced by the surrogate seagrass is the relative height of the maximum kinematic shear stress (figure 4b), which does not occur within the vicinity of the mean canopy top due to the particular modulation of the blade oscillations. Note the lower kinematic shear stress compared with, e.g. Ghisalberti & Nepf (Reference Ghisalberti and Nepf2006) and Chung et al. (Reference Chung, Mandel, Zarama and Koseff2021). This is due to the shorter canopy length $L_c$ and the relative FOV location. Indeed, the canopy length $L_c = 1.435$ m (figure 2a) is shorter than those in Ghisalberti & Nepf (Reference Ghisalberti and Nepf2006) and cases in Chen et al. (Reference Chen, Jiang and Nepf2013) and Chung et al. (Reference Chung, Mandel, Zarama and Koseff2021). As pointed out by Chung et al. (Reference Chung, Mandel, Zarama and Koseff2021), the maximum shear stress increases with $L_c$. Additionally, our FOV measured the developing region of the mixing layer. Chen et al. (Reference Chen, Jiang and Nepf2013) showed that the development length $X_* \approx (8 \pm 2) {U_vL_S}/{u_*}|_{h_c}$, where $U_v, L_S$ and $u_*$ are the vortex convection velocity, shear length scale and friction velocity at the canopy top; Chung et al. (Reference Chung, Mandel, Zarama and Koseff2021) showed that the shear length scale can be estimated as $L_{s}=\bar {U}/{\partial \bar {U}}/{\partial z}|_{h_c}$, resulting in $L_s \approx 0.04$ m and $X_* \approx 1.75 \pm 0.44$ m for the R1 case. This indicates that our FOV is in the developing region of ${\approx }0.72X_*$, which also leads to lower kinematic shear stress, as shown in Chen et al. (Reference Chen, Jiang and Nepf2013) and Chung et al. (Reference Chung, Mandel, Zarama and Koseff2021). It is worth stressing the importance of this region given the heterogeneity or patchiness of natural canopies affected by various topographic factors and morphological features, including width, shape factor, rigidity and orientation of the blades. Remarkably, the associated streamwise component, expressed here as turbulence intensity $\bar {I}_u$, occurred below the mean canopy height. The particular locations of the maximum in $-\langle {u'w'}\rangle$ at $z/h_d>1$ and $\langle {u'u'}\rangle$ at $z/h_d<1$ in the seagrass indicate a transport modulated by the motion of the blades, which is very different from the rigid counterpart and represent a signature of the distinct scale-to-scale dynamics. The higher turbulence momentum flux above the seagrass canopy, indicated by larger magnitude $-\langle \overline {u'w'}\rangle$, suggests greater mass transfer within the canopy. The deflection of flexible blades during the passage of coherent vortices reduces the exposed blade surface area, and thus the potential uptake of e.g. dissolved inorganic nitrogen relative to separated blades during weaker flows (Weitzman et al. Reference Weitzman, Aveni-Deforge, Koseff and Thomas2013). Furthermore, the extent of the $-\langle \overline {u'w'}\rangle$ within the canopies (figure 4b) provides an indication of the vortex penetration depth, ${\delta }_e$, given by the distance below $h_d$ whereby kinematic stress decreases to 10 % of the maximum (Nepf & Vivoni Reference Nepf and Vivoni2000). Evaluation of vortex penetration depth normalised by deflected canopy height ($h_d$) indicates vortex penetration towards the bed was smaller for the flexible canopy than the rigid canopy at lower $Re$, but increased with larger $Re$ (figure 5c).

The mixing layer thickness, $t_{ml} = z_2-z_1$, where $z_1$ and $z_2$ denotes the height of the 99 % free-stream and in-canopy velocities, remained nearly constant for the rigid canopy with $Re$ (figure 5b), with an asymmetrical vertical distribution with approximately one third below $h_c$, similar to the observations of Ghisalberti & Nepf (Reference Ghisalberti and Nepf2004). Ghisalberti & Nepf (Reference Ghisalberti and Nepf2002) found monami occurred when $t_{ml}/h_d > 1.5 - 2.1$, which is the case here, whereby $t_{ml}/h_d$ ranges between 1.4 and 2.0, with the lowest $Re$ condition not initiating coherent waving representative of monami. Ghisalberti & Nepf (Reference Ghisalberti and Nepf2002) also noted that waving increases the vortex penetration depth (normalised by $h_d$) and suggested that waving canopies induce a weaker momentum sink compared with rigid cases. However, Nepf & Ghisalberti (Reference Nepf and Ghisalberti2008) reported no difference in dimensional $\delta _{e}$ vs waving and un-waving canopies. Alternatively, when evaluated in dimensional terms based on the fixed dimension of $h_c$, the flexible canopy vortex penetration depth does not increase with $Re$ and remains smaller than the rigid canopy, revealing that the flexible canopy limits the depth of stresses (see figure 5c). It is worth pointing out that Okamoto & Nezu (Reference Okamoto and Nezu2009) inspected penetration of coherent structures within a flexible canopy composed of an array of single elements and suggested that, regardless of differences in vegetation morphology, the role of canopy reconfiguration associated with flexibility plays a dominant role in reducing the vertical penetration of stresses. However, Wilson et al. (Reference Wilson, Stoesser, Bates and Pinzen2003) noted the role of foliage morphology, where a smaller penetration of turbulent stresses occurred for rods with fronds (similar to flexible blades) than rods alone, and suggested that the presence of the fronds limited the momentum exchange between the canopy and overlying flow.

The redistribution of the Reynolds stresses induced by the seagrass can be further assessed by inspecting the velocity fluctuations into four quadrants, namely $Q_1$: $u' >0$, $w'>0$ (outward interactions), $Q_2$: $u'<0$, $w'>0$ (ejections), $Q_3$: $u'<0$, $w'<0$ (inwards interactions) and $Q_4$: $u'>0$, $w'<0$ (sweeps). Specifically, figure 6 illustrates the spatial distribution of the sweep-to-ejection ratio, $Q_4/Q_2$, for selected cases. Consistent with Yue et al. (Reference Yue, Meneveau, Parlange, Zhu, van Hout and Katz2007), Poggi et al. (Reference Poggi, Porporato, Ridolfi, Albertson and Katul2004) and Chen et al. (Reference Chen, Jiang and Nepf2013), the base cases with the rigid canopy exhibited a predominance of sweeps within the canopy for all $Re$ tested. However, ejections dominated within the surrogate seagrass with comparatively strong sweeps in the vicinity of the mean canopy height. Note also the distinct modulation of the blade motions in the distribution of $Q4/Q2$ in the boundary layer. Although sweep-dominated events appear on the rigid canopy at the $Re$ studied, those were only present in the lower half of the flexible canopy ($z/h_d < 0.5$) at the lowest ${Re}$ (figure 6a); the minimal blade deformation at that $Re$ made it comparable to that in the rigid canopy. Figure 6 also shows a phenomenon not previously reported, namely, the transition to ejection-dominated events with $Re$ within the seagrass. The upward motion of the blades associated with the ejection at the rear of vortices may result in ejection events. Likewise, the deflection of blades prevents penetration of sweep events as per the rigid canopy. This is also consistent with the arguments by Okamoto et al. (Reference Okamoto, Nezu and Sanjou2016), who pointed out that coherent structures do not extend into the lower canopy region under monami.

Figure 6.

Ratio of total contribution of sweep ($Q4$) and ejection ($Q2$) events for the seagrass scenarios (a) F1, (b) F3, (c) F5 and rigid canopy cases (d) R1, (e) R3, (f) R5. Dashed white lines indicate the mean canopy height.

The region with the largest sweep-dominated motions was consistently located at the vicinity of the mean deformed seagrass height, $h_d$, in agreement with other findings (Nezu & Sanjou Reference Nezu and Sanjou2008; Marjoribanks et al. Reference Marjoribanks, Hardy, Lane and Parsons2017). The differences in the magnitude of the sweep-to-ejection ratio between the surrogate seagrass and rigid canopy are likely due to the ability of the blade motion to promote turbulent transport. Bailey & Stoll (Reference Bailey and Stoll2016) suggested that a canopy impedes a vortex to draw fluid from below, thus limiting the presence of ejections near the canopy top. In contrast, sweeps dominate due to their ability to draw flow fluctuations from the unobstructed and higher momentum flow above. The deflection of blades also increases the canopy top blockage area, producing an apparent increase of the canopy density, which likely increases sweep dominance (Poggi et al. Reference Poggi, Porporato, Ridolfi, Albertson and Katul2004). In contrast, the upright blades of the rigid canopy do not provide the same top-down area blockage and constraint. Importantly, this suggests that the streamlining of the flexible canopy blades under sufficient flow results in an effective barrier to larger-scale turbulence.

3.2. On the coherent motions and flow–canopy interaction

First, we explore the combined redistribution, enhancement and damping of turbulence across relevant scales modulated by the blade motions within and above the canopy with the one-dimensional compensated spectra of the streamwise velocity fluctuations, $f \phi (f)$, where $f$ is the frequency. Figure 7 shows this quantity normalised by the maximum value, $f{\phi }^{*}={f\phi }/\max \{f\phi \}$, throughout the vertical span at nearly the centre of the FOV ($x/h_c = 10.25$). A low-pass filter with a 10 Hz cutoff frequency is applied to the signal; over that frequency, the energy contribution is minor. The spectral velocity distributions within the rigid canopy at three $Re$ show a comparatively dominant energy at a frequency $f_0=f_0(Re)$, corresponding to vortex shedding from the rigid structures, where the Strouhal number is $St=f_0d_r/U_1\approx 0.19 - 0.20$ (Norberg Reference Norberg1994). Such motions are missing within the seagrass; there, the turbulent energy plays a minor role compared with the above-canopy turbulence and that within the rigid canopy.

Figure 7.

Compensated one-dimensional spectra of the streamwise velocity fluctuations along the vertical span at the centre of the FOV $x/h_c = 10.25$ for (a) F1, (b) F3 and (c) F5 seagrass scenarios, and (d) R1, (e) R3 and (f) R5 rigid canopy cases. Dashed lines indicate the canopy top ($h_d$), and dotted lines denote the location of $z_1$ and $z_2$.

Generation of KH instability (Ghisalberti & Nepf Reference Ghisalberti and Nepf2002; Okamoto & Nezu Reference Okamoto and Nezu2009; Okamoto et al. Reference Okamoto, Nezu and Sanjou2016; Marjoribanks et al. Reference Marjoribanks, Hardy, Lane and Parsons2017) appears to be associated with enhanced energy above the canopy. Considering Ghisalberti & Nepf (Reference Ghisalberti and Nepf2002), the KH frequency, $f_{KH}$, can be estimated as follows:

(3.1a,b)\begin{equation} f_{KH}= St\frac{\bar{U}}{\theta}, \quad \theta =\int^{\infty }_{-\infty}{\left[\frac{1}{4}-{\left(\frac{U-\bar{U}}{\Delta U}\right)}^{2}\right]{\rm d}z,} \end{equation}

where $\bar {U}=(U_{\infty }+U_c)/2$ (see figure 1), $\theta$ is the momentum thickness and $\Delta U = U_{\infty } - U_c$ is the bulk shear magnitude. In scenarios of unforced mixing layers $St\approx 0.032$, and varies modestly with the velocity ratio $R=\Delta U/({2\bar {U}})$ by up to 5 % between $R = 0$ and 1 (Ho & Huerre Reference Ho and Huerre1984). The canopies, however, induce a distinct forcing such that the $St = 0.032$ for parallel, unforced flows is not quite applicable. Indeed, Mandel et al. (Reference Mandel, Gakhar, Chung, Rosenzweig and Koseff2019) obtained $St = 0.064$ for a rigid canopy. This illustrates the possible differences between predicted and measured frequencies presented in table 2. Another characteristic frequency of interest is the undampened natural blade frequency, estimated by

(3.2)\begin{equation} f_n\approx C_n\sqrt{{EI}/{l^{4}\left({\rho }_vbd+{\rho }_fC_M({\rm \pi} b^{2}/4)\right)}}, \end{equation}

with $C_n= 0.56$ and $C_M\approx 1$ (Luhar & Nepf Reference Luhar and Nepf2016), which results in $f_n\approx 0.34$ Hz. Recent numerical simulations by O'Connor & Revell (Reference O'Connor and Revell2019) indicate that canopy motion is a coupled response between the natural structure (vegetation blade) and coherent flow motions. Both $f_{n}$ and $f_{KH}$ may coexist if they differ sufficiently; however, a lock-in phenomenon and canopy waving may occur when $f_{n}$ and $f_{KH}$ are similar. The transition of two modes to lock-in behaviour is captured in figure 8(b), where a weaker shear is seen to lead to a smaller $f_{KH}$ at $t \geqslant 8\ {\rm s}$, resulting in synchronisation.

Table 2.

Characteristic parameters in the surrogate seagrass and rigid canopy. Strouhal number $St$, velocity ratio $R$, estimated blade natural resonance frequency $f_n$, predicted KH frequency $f_{KH}$ and time-averaged peak spectral frequency at $z/h_d=1.05$, $f_{max}$. NA: not applicable.

Figure 8.

Wavelet representation of the streamwise velocity within and immediately above the flexible (F5) and rigid (R5) canopies at $Re=1.1\times 10^{5}$. (a) R5, $z/h_d=1.05$, (b) F5, $z/h_d=1.05$, (c) R5, $z/h_d=0.5$, (d) F5, $z/h_d=0.5$. White dashed lines indicate the cone of influence, and horizontal dotted lines mark $f_{KH}$.

A complementary inspection into the time–frequency domain with a Morlet wavelet analysis, reveals insightful features of the turbulent fluctuations in the canopy mixing layer and compact signatures within the period of the measurements, which can be obscured with velocity spectrum analysis. The cases for the highest $Re$ (seagrass, F5, and rigid canopy, R5) scenarios are shown in figure 8 within the canopy at $z/h_d=0.5$ and immediately above the mean canopy height at $z/h_d=1.05$. The frequency associated with the predicted KH instability, $f_{KH}$, is indicated with horizontal dashed lines.

Note that the rigid canopy exhibits relatively energetic fluctuations at a frequency $f\geqslant f_{KH}$ with higher persistence at $z/h_d=1.05$ (figure 8a). Remarkably, the velocity fluctuations in the surrogate seagrass exhibit two dominant, stronger signatures than those of the rigid canopy. One is a non-persistent, shear-induced $f_{KH}$, whereas the other is a persistent, blade-modulated signal at a lower frequency. These two distinct motions induced secondary motions between these frequencies for a short time (see figure 8b). Such an aperiodic phenomenon evidences additional nonlinearities induced by the large deformations of the blades (Jin et al. Reference Jin, Kim, Hong and Chamorro2018).

Spatio-temporal characteristics of the flow fluctuations within the canopies at $z/h_d=0.5$ reveal distinct energetic processes contributing to the dynamics. Numerical simulations by Marjoribanks et al. (Reference Marjoribanks, Hardy, Lane and Parsons2017) noted that $f_{KH}$ was not persistent within the mixing layer in semi-rigid and flexible canopies. Here, in the rigid canopy, the flow exhibits dominance of the cylinder shedding frequency; it is not continuous across the timespan (see figure 8c). In contrast, flow in the seagrass shows the signature of KH motions at times during coordinated blade motions (see supplementary movie 1), which promoted the generation of KH-like motions into the canopy (see figure 8d). Under uncoordinated blade motions, there is a lack of KH-like motions. The averaged vortex penetration remains reduced by the seagrass under uncoordinated blade motions.

Inspection of selected instants offers insight into the underlying effects of the sweeps and ejections. In particular, the seagrass F5 case is considered herein, using two points for interrogation, with one above and one within the canopy at nearly the centre of the FOV, ($x/h_c$, $z/h_d)=(10.25, 1.3)$ and (10.25, 0.4). From the resultant time series of $u'/U_{\infty }$, $w'/U_{\infty }$, and associated $-\langle u'w'\rangle /U_{\infty }$, we can observe instants with sweeps (e.g. $t= 4.25$ s, 5.91 and 7.30 s) and ejections (e.g. $t=5.28$ s and 6.46 s). The time series and selected times are indicated in figure 9(a). The red lines correspond to instants with alternating $u'$ above and within the canopy. Above the canopy, the sweep events correspond to positive $u'$ and negative $w'$, while ejection events express the opposite trends. The $u'w'$ values peaked with each event, yet the weak ejection event at $t = 5.28$ s represents a notably lower Reynolds shear stress. The behaviour within the canopy is less clear, which may suggest a lag in processes between the two regions.

Figure 9.

(a) Selected snapshots of the streamwise velocity fluctuations for the F5 case; sweeps at $t h_d/ U_{\infty } = 0$, $\approx$0.2 and $\approx$0.4 and ejections at intermediate times $t h_d/U_{\infty }\approx 0.1$, and $\approx$0.3. The blades around the top are shown with black lines. (b) Time series of streamwise, $u'/U_{\infty }$, and vertical, $w'/U_{\infty }$, velocity fluctuations and kinematic shear stress, $-\langle u' w'\rangle /U_{\infty }$, at $z/h_d = 1.3$ and $z/h_d= 0.4$ at $x/h_c = 10.25$. The vertical red lines correspond to the instants in (a).

The associated whole flow fields at the instants selected are shown in figure 9(b). The canopy experienced a depression of the blades in correspondence with the sweep events, followed by upward motion during ejection events. This waving motion of the canopy is comparable to previously reported monami processes (Ikeda & Kanazawa Reference Ikeda and Kanazawa1996; Ghisalberti & Nepf Reference Ghisalberti and Nepf2002). Details of the motions are shown in the supplementary movie 1.

Specific, quantitative insight into the key coherent structures and their dynamics can be obtained with SPOD. As a data-driven modal analysis technique, SPOD utilises empirical data and combines the merits of the traditional space-only POD and dynamic mode decomposition (DMD). SPOD extracts structures that are spatially and temporally coherent instead of the spatial-only coherence of POD. The SPOD modes are also optimal-averaged DMD modes (Tu et al. Reference Tu, Rowley, Luchtenburg, Brunton and Kutz2014), which contain the inherent energy ranking and form on an orthogonal basis for the flow field, like the POD-based techniques first proposed by Lumley (Reference Lumley1970). Herein, the SPOD approach follows Sieber, Paschereit & Oberleithner (Reference Sieber, Paschereit and Oberleithner2016).

First, the so-called Welch method is used to construct an ensemble of realisations of the temporal Fourier transform of the data from a single time series consisting of $N_T$ snapshots by breaking it into several segments. Each of these consists of $N_{FFT}$ snapshots, and overlaps with the next segment are preferred to account for the statistical variability of the turbulent flow. Here, $N_{T} = 9000$, is the number of the zero-padding snapshot series, and $N_{FFT} = 1024$, which allowed us to achieve a resolution in the frequency domain of $\approx$1 Hz. A discrete Fourier transform is then applied on the separated time-dependent segment realisations, $q^{k}_{T_j}$, where $k$ stands for the $k{\rm th}$ realisation of the vector of observations, and $T_j$ is the instant. It allows us to obtain the coefficient $\hat {q}^{k}_{f_m}$ in the Fourier domain, where

(3.3a,b)\begin{equation} \hat{q}^{(k)}_{f_{m}}=\sum_{j=0}^{N_{{FF}}-1} q^{(k)}\left(t_{j+1}\right) \exp({-{\rm i} 2 {\rm \pi}j m / N_{{FFT}}}), \quad k={-}N_{{FFT}} / 2+1, \ldots, N_{{FFT}} / 2. \end{equation}

Then, for each frequency $f_{m}$, $\hat {Q}_{f_{m}}$ may be formed as follows:

(3.4)\begin{equation} \hat{Q}_{f_{m}}=\left[\begin{array}{cccc} \mid & \mid & & \mid \\ \hat{q}_{f_{m}}^{(1)} & \hat{q}_{f_{m}}^{(2)} & \cdots & \hat{q}_{f_{m}}^{(N)} \\ \mid & \mid & & \mid \end{array}\right], \quad \hat{Q} \in \mathbb{C}^{M \times N} \end{equation}

where $M$ is the degree of freedom given by the number of spatial points in the $x$- and $z$-directions in each case, and $N$ is the number of realisations. Singular value decomposition is then performed in each $\hat {Q}$ matrix.

We use the variance norm of a two-dimensional, two-component stochastic velocity fluctuations matrix $\underline{u'}(x,z,t) = [u' w']^{\textrm {T}}$ for the SPOD input. Modes are optimised in the mean square value of the velocity fluctuations by utilising singular value decomposition (Taira et al. Reference Taira, Brunton, Dawson, Rowley, Colonius, McKeon, Schmidt, Gordeyev, Theofilis and Ukeiley2017). Further details regarding SPOD implementation are given in Schmidt & Colonius (Reference Schmidt and Colonius2020) and SPOD applications in Towne, Schmidt & Colonius (Reference Towne, Schmidt and Colonius2018) and Schmidt et al. (Reference Schmidt, Towne, Rigas, Colonius and Brès2018).

Application of the SPOD, particularly for the rigid R5 and seagrass F5 cases, reveals particular features of the coordinated dynamics of the flow and seagrass, and the role of energetic coherent motions. Figure 10 shows the SPOD spectra above and within the canopy of the rigid case R5, and demonstrates the mode ranking characteristic of the SPOD method. The separation between the first and lower energy modes is prominent at a lower frequency ($\lesssim$3 Hz) and suggests that the dynamics of large coherent structures can be approximated by the first few modes using a lower rank approximation. A broader band frequency of the dominant KH-type instability occurs above and within the rigid canopy. In contrast, a higher frequency peak is observed within the canopy (figure 10b) in agreement with the two-dimensional spectra in figure 7. The spatial mode shapes associated with the dominant frequencies of the streamwise SPOD modes, $\phi _{u'}$, are illustrated in figure 11, with the red–blue colour representing the spatially correlated patterns. The mode shapes above the canopy tip demonstrate structural features consistent with the shear layer KH instability (table 2); local changes of the convective velocity cause the observed broad frequency distribution. The von Kármán vortex street past the rods exhibited the well-known Strouhal relationship, $St \approx 0.2$.

Figure 10.

Spectral proper orthogonal decomposition (SPOD) spectra of the rigid R5 case of the flow field (a) above and (b) within the canopy, with the darkest (black) line being the first SPOD mode and the lighter lines the subsequent SPOD modes.

Figure 11.

Spatial organisation of the first SPOD modes above the rigid R5 canopy at $f =$ (a1) 0.88 Hz and (a2) 1.07 Hz; and within the canopy at $f=$ (b1) 0.88 Hz and (b2) 3.52 Hz.

Insight from the SPOD spectra above and below the seagrass at the same $Re$ for the F5 case, can be obtained from figure 12. Coherent motions related to the KH frequency dominate above and below the canopy, and the corresponding mode shapes are shown in figure 13. While KH vortices have previously been discussed above seagrass canopies (Ghisalberti & Nepf Reference Ghisalberti and Nepf2002), and coherent vortices have been observed (Nezu & Sanjou Reference Nezu and Sanjou2008), these results provide compelling evidence of KH-type vortices and provide spatio-temporal information. A weaker local peak at 2.25 Hz present above the canopy suggests the existence of a weaker coherent structure (figure 13a2). However, this coherent structure with lower energy content is not able to penetrate the barrier created by the blade motion and thus is not observed for the in-canopy flow. Also, a mismatch exists between the primary mode frequency of the flow above and within the canopy, with the upper flow having the strongest mode at $\approx$0.4 Hz and the lower part at $\approx$0.6 Hz.

Figure 12.

SPOD spectra of the seagrass, F5 case, of the flow field (a) above and (b) within the canopy, with the darkest (black) line being the first SPOD mode and the lighter lines the subsequent SPOD modes.

Figure 13.

Spatial organisation of the first SPOD modes above the canopy at $f =$ (a1) 0.4 Hz and (a2) 2.25 Hz; and within the canopy at $f=$ (b1) 0.6 Hz and (b2) 0.68 Hz.

To uncover particular effects of the blade motion on flow interaction above and within the canopy, SPOD is performed on the full F5 case (figure 14). Close inspection shows that the most energetic modes at $\approx$0.4 Hz captured only the coherent structures above the canopy, whereas monami effects and entrainment of the coherent structures are better captured by the second strongest mode (see also supplementary movie 3). The inset shows the projected second mode time coefficient for the natural blade frequency and the KH frequency. Figure 15 illustrates a sequence every $\Delta th_d/U_{\infty }\approx 0.05$ within one period of the 0.6 Hz projected time coefficient. It reveals the streamwise propagation features and a monami effect with entrainment into the region within the canopy, which is not observed at 0.4 Hz due to a much weaker second mode despite its stronger first mode energy. The lack of sinusoidal-like behaviour exhibited in the 0.4 Hz projected-coefficient second mode, as compared with the 0.6 Hz counterpart, indicates that blade motion is dominated by the KH frequency rather than the natural frequency, consistent with figure 9.

Figure 14.

(a) Full flow field SPOD spectrum on the seagrass F5 case. The darkest to lighter lines indicate the progression from the first to the fourth SPOD modes; (b) projected SPOD mode coefficients, $b_{i,j}(t)$ for the $i{\rm th}$ mode of $j$ frequency.

Figure 15.

Sequence uniformly distributed in a period of the second SPOD mode of $f = 0.6$ Hz for the F5 seagrass scenario.

This description motivates inspection of the instantaneous quadrant analysis data given in figure 16(a). The 0.6 Hz peak supports our previous assumption that the switch from sweep-dominated to ejection-dominated behaviour within the flexible canopy is related to monami and the blade motion. The projected-coefficient first mode at 0.6 Hz is shown in figure 16(b). Combining this with the base analysis of § 3.1 and figure 9, it is inferred that the projected-coefficient period is associated with the canopy blade motion. The conditional sampling method provides a representation of the sweep- or ejection-dominated flow within the canopy during the phase of downward and upward blade motions (figure 17). This supports our conjecture, whereby the presence of strong ejections within the flexible canopies correspond to the upward blade motions, which coincides with the rear of the KH vortex, and previously identified above-canopy ejection events shown in figure 9 and in past research (Ghisalberti & Nepf Reference Ghisalberti and Nepf2002; Okamoto & Nezu Reference Okamoto and Nezu2009). These results reveal the complexity and diversity of within canopy processes, which are driven by the canopy motion.

Figure 16.

(a) SPOD spectra form the quadrant analysis of the seagrass F5 case. The darkest to lighter lines indicate the progression from the first to the fourth SPOD modes; (b) projected SPOD mode coefficients, $b_{1,4}(t)$ for the first mode of 0.6 Hz.

Figure 17.

Conditional average of sweep to ejection ratio during the F5 flexible canopy (a) downward motion phase and (b) upward motion phase.