3.1. Identification of different dynamic modes

Before introducing the evolution of large-scale vortices, it is necessary to classify different dynamic modes in the current parameter space, which lends itself to understand the discussions in the following sections. As stated in our previous study (Ni et al. Reference Ni, Ji, Xu, Zhang and Liang2023), distinct Monami modes can be further identified according to the fluctuation features of KH instability onset position $n_o$ , which is determined by the following criterion:

(3.1) \begin{align}& {{n}_{o}}\left ( t \right )=\left \{ {{n}_{o}}|\widetilde {\theta }(n={{n}_{o}},t)\geqslant {{1}^{\circ }}\wedge \widetilde {\theta }(n\lt {{n}_{o}},t)\lt {{1}^{\circ }} \right \}\!,\end{align}

(3.2) \begin{align}&\qquad\,\, \widetilde {\theta }(n,t)=\theta (n,t)-\frac {1}{\Delta t}\int _{{{t}_{0}}}^{{{t}_{0}}+\Delta t}{\theta (n,t)}{\rm d}t, \end{align}

where $\widetilde {\theta }(n,t)$ is the fluctuation amplitude of the vegetation inclination angle of element $n$ at time $t$ . To improve the clarity of results, since the location of the first vegetation is fixed in all simulation cases, $x = (n - 1)\boldsymbol{\cdot} D$ is used to convert $n$ to the distance from the first vegetation $x$ , which is further non-dimensionalised with the canopy length $L_a$ to make $x/L_a$ range from 0 to 1. The contours of instantaneous z-vorticity of different modes are shown in figure 3, while figure 4 gives the spatial and temporal variations in tilt angle fluctuations across the array, which helps to visualise the KH instability onset position. Note that the z-vorticity is nearly zero within the vegetation layer in most cases shown in figure 3, primarily due to the 2-D simulation used in this study. Different from the 3-D canopy that is semi-permeable and allows some through-flow, the 2-D canopy here provides full flow blockage. Therefore, the gap between the adjacent vegetation elements can be seen as a narrow cavity, the flow inside is extremely weak, resulting in almost zero vorticity.

Figure 3.

Contours of instantaneous z-vorticity of several cases representing (a) regular Monami, (b) quasi-regular Monami_A, (c) quasi-regular Monami_B, (d) irregular Monami, (e, f) quasi-regular Monami_C, and (g) cavity oscillation modes.

Figure 4.

Spatial and temporal variations in inclination angle fluctuations across the array in (a) regular Monami, (b) quasi-regular Monami_A, (c) quasi-regular Monami_B, (d) irregular Monami, (e) quasi-regular Monami_C and (f) cavity oscillation modes. The red solid line denotes the KH instability onset position $x_o/L_a$ .

In regular Monami mode, as shown in figure 3(a), due to the fixed KH instability onset position, there is no low-frequency flow and no vortex merging appears as the KH vortex propagates downstream. This is also reflected in figure 4(a), where the KH instability onset position marked by a red solid line does not vary with time and the stripes are continuous without interruption. Note that each stripe in the plot represents a Monami wave propagating downstream with time. In regular Monami mode, the vegetation elements deflect by the same degree every time a large-scale vortex passes. In quasi-regular Monami_A mode, a single low frequency is caused by the regular fluctuation of KH instability onset position shown in figures 3(b) and 4(b), which can induce vortex merging to occur regularly in the downstream. However, the regular vortex merging that should have occurred does not appear in the case $\gamma$ 0.04d0.4, as highlighted by a black dashed box in figure 3(b). This is just because the length of the array is limited and the KH vortices leave the array before they interact with each other. Quasi-regular Monami_B mode is similar to quasi-regular Monami_A mode, and the difference is that the low frequencies in the former are both the single low frequency and its significant harmonics, which can eventually lead to complex interactions between KH vortices shown in figure 3(c). This is also seen in figure 4(c), where the interruptions of stripes indicating the merge between KH vortices occur almost randomly after $x/L_a=0.60$ . The irregularity is more obvious in irregular Monami mode, since the KH instability onset position fluctuates irregularly with time, which is shown in figures 3(d) and 4(d). For the quasi-regular Monami_C mode emerging at relatively large $\gamma$ , the KH instability onset position no longer changes with the vortex merging still exhibiting downstream, as shown in figures 3(e), 3( f) and 4(e). In these quasi and irregular Monami modes, the vegetation elements deform by different degrees each time a large-scale vortex passes, influenced by low-frequency effects. All the above-mentioned modes are Monami modes with $x_o/L_a\gt 0$ , which means that there is a shear layer with KH instability before KH vortices are excited. However, note that if $x_o/L_a=0$ , indicating that the swaying of vegetation begins from the first vegetation, this implies that the dynamic mode under this circumstance is the cavity oscillation (Zhang et al. Reference Zhang, Li and Zhang2022), as shown in figures 3(g) and 4( f).

Figure 5.

A map for the distribution of the different dynamic modes in the parameter space of ( $\gamma$ , $d$ ). The green line is the diagonal line of region IV, where $\gamma$ and $d$ play an equivalent role.

The distribution of the above-mentioned dynamic modes in the current parameter space is shown in figure 5. Note that the bending rigidity $\gamma$ in our previous study ranging from 0.04 to 0.22 is part of that in this work, so the divided regions I–IV are identical to those of Ni et al. (Reference Ni, Ji, Xu, Zhang and Liang2023). In region I, four Monami modes with a scattered distribution are observed due to the competition between $\gamma$ and $d$ . Increasing $d$ leads to quasi-regular Monami_B mode dominating in region II, while increasing $\gamma$ results in a prevalent distribution of regular Monami mode in region III. The irregular Monami mode is near the diagonal line of region IV. Moving away from this line in the direction of increasing $\gamma$ and $d$ , the dynamic modes transform to regular Monami mode and quasi-regular Monami_B mode, respectively. The expanded part in the present study is further divided into regions V and VI, which are mainly filled with quasi-regular Monami_C and cavity oscillation modes, respectively. The cavity oscillation modes only occur when both $\gamma$ and $d$ are large, which is consistent with the work done by Zhang et al. (Reference Zhang, Li and Zhang2022).

3.2. Determination of the size and strength of large-scale vortices

To better quantify the evolution of large-scale vortices across the canopy, reasonable assumptions need to be made, and then appropriate physical quantities are chosen to characterise the position and strength of the vortex.

It is necessary to clarify at the outset that the instantaneous geometry and intensity of different large-scale vortices passing a specific location, such as $x/L_a=0.40$ in figure 3(c), vary from time to time due to low-frequency effects discussed in § 3.1. Since instantaneous values cannot capture the long-term characteristics of the vortex, the values of the large-scale vortices’ size and intensity presented in this study are statistical averages derived from multiple vortices passing specific locations over an extended period. Based on the contours shown in figure 3, the large-scale vortex in this study can be considered to be centrosymmetric about the vortex core, as shown in figure 6(a). Due to its clockwise rotation around the vortex core, the streamwise and vertical velocity fluctuation intensities $\boldsymbol{\sigma }_u$ and $\boldsymbol{\sigma }_v$ (the root mean square of the streamwise and vertical velocity fluctuation) exhibit local minima and maxima at the height of the vortex core, respectively, as indicated by the green dashed lines in figure 6(b). Additionally, at the upper and lower boundaries of the vortex, $\boldsymbol{\sigma } _u$ is expected to attain local maxima. For a large-scale vortex, such as at $x/L_a=0.50$ shown in figure 3(a), whose movement is not influenced by the vegetation, the local maximum values of $\boldsymbol{\sigma } _u$ at the upper and lower boundaries are approximately equal, as seen in figure 6(b). In contrast, for a large-scale vortex propagating downstream at the top of the canopy, like those at $x/L_a=0.30$ and 0.40 in figure 3(a), the vortex’s movement can be influenced by the canopy. This results in $\boldsymbol{\sigma }_u$ deviating from its expected trend and forming a maximum at the canopy’s top, as shown in figure 6(b). In such cases, the lower boundary of the large-scale vortex is difficult to identify. Given the interaction between the large-scale vortex and the vegetation, the lower boundary of the large-scale vortex is excluded from the discussion. The vertical size of the large-scale vortex $\Delta y$ is defined as the height difference between the upper boundary and the core of the vortex, that is, half of its actual vertical size. In addition, the strength and lower boundary of the VO vortex shed by vegetation oscillation are also determined by this means. Its upper boundary is not considered also owing to the interaction with the large-scale vortex. However, due to the small (an order of magnitude smaller than the large-scale vortex based on our results) and irregular size of the VO vortex, this method is somewhat rough. Despite this, the VO vortex is considered when necessary to assist in the analysis of large-scale vortices.

Figure 6.

(a) Schematic diagram of the position, scale and strength of the large-scale vortex. (b) Vertical distributions of the streamwise velocity fluctuation intensity $\boldsymbol{\sigma }_u$ and vertical velocity fluctuation intensity $\boldsymbol{\sigma }_v$ (the root mean square of the streamwise and vertical velocity fluctuation) at some selected positions in the case $\gamma$ 0.20d0.4, where the black dotted line, the green dashed line and the cyan dashed line denote the undeflected vegetation height, the vortex core positions and the boundary positions of the vortex, respectively. Note that the dashed lines above $y/L = 1.0$ represent the positions of the large-scale vortex, while the dashed lines below $y/L = 1.0$ represent the positions of the VO vortex.

Note that the above-mentioned method is only used in the relatively stable flow field, that is, from the initial formation of large-scale vortices to the complex interactions between large-scale vortices. Taking the case $\gamma$ 0.04d1.2 as an example, as shown in figure 4(c), a position slightly downstream of the KH instability onset position is taken as the vortex identification initial position, as marked by the red dotted line in figure 4(c). Although there are stripe interruptions around $x/L_a=0.40$ –0.60, which means the merging between large-scale vortices, the flow field is still relatively stable. After $x/L_a=0.60$ , as marked by the blue dotted line in figure 4(c), the interruptions of stripes become frequent and messy, indicating that complex interactions between large-scale vortices occur, which leads to chaos in the flow field and the geometry of large-scale vortices is no longer identified. Therefore, in this study, the change of the position, size and strength of the large-scale vortex is discussed only in the range between the red dotted line and the blue dotted line. In addition, to minimise the interference of vegetation swaying, the flow field data at position $x/L_a$ are taken from the position half-gap distance d upstream of this position.

Other methods to determine the large-scale vortex such as through the distributions of z-vorticity and Q-criterion are also considered. Assuming that the vertical profile of $\omega _z$ and $Q$ is a Gaussian distribution, it is easy to define the core, boundary and strength of the large-scale vortex. However, in the context of the current study, compared with the above-mentioned approach, using $\omega _z$ , for one thing, can introduce more artificial interference, and for another, cannot reveal the vorticity induced by the rotation of the large-scale vortex alone under the influence of the stable shear layer. The more stringent Q-criterion overcomes these two disadvantages, but shows a less obvious tendency in vortex intensity between different stages. These significant drawbacks are discussed in detail in Appendix A.

3.3. Evolution zones and sequences of large-scale vortices across the canopy

In this section, the evolution of large-scale vortices across the canopy with different bending rigidity and gap distance are systematically investigated. According to the characteristics of the large-scale vortices during their evolution process, four evolution zones and seven evolution sequences are identified. In addition, the underlying mechanisms of the evolution of large-scale vortices in each zone are elucidated.

It can be seen from figure 3 that the evolution processes of large-scale vortices are different. In fact, they represent seven distinctive evolution sequences across the canopy and each sequence contains different evolution zones.

In the following, the characteristics of vortex evolution across the canopy are first introduced by taking the typical case $\gamma$ 0.20d0.4 as an example which is classified as regular Monami mode in § 3.1 without any low-frequency in the flow field, and three important zones are identified. Figure 7 gives the variations of the upper boundary, core position, vertical size and intensity of the KH vortex under this case, where the left y-axis is used for the vortex position and size, and the right y-axis is used for the vortex intensity. It is shown in figure 7(a) that as the KH vortex moves downstream from $x/L_a = 0.25$ to 0.38, the position of the vortex slowly rises, and the vortex size and intensity gradually increase, and finally develop to the maximum. Based on the above-mentioned vortex features, this zone is referred to as the ‘developing’ zone, or zone D for short. While in the range of $x/L_a = 0.45$ –1.00, the vortex slowly rises to a fixed height, and its intensity gradually decreases while its size remains almost unchanged. In accordance with the characteristics of the vortex, this zone is referred to as the ‘dissipation’ zone, or zone ${D}^*$ for short. Between these two zones, there is a short region around $x/L_a = 0.40$ –0.45, where the vortex position is significantly uplifted. In the process of this rapid rise, the vortex intensity shrinks rapidly and the vortex size also decreases. This narrow region can be seen as a transition from zone D to ${D}^*$ , which is classified as zone T.

Figure 7.

Variations of the upper boundary, core position, size and intensity of the large-scale vortex across the canopy under various cases. The solid, semi-hollow and hollow symbols denote zones D, T and ${D}^*$ , respectively. Zone I is not labelled, since the geometry of large-scale vortices is unable to be identified due to the complex interactions.

Figure 7(b–f) gives the variations of the characteristics of the large-scale vortex in the other cases shown in figure 3. As for the case $\gamma$ 0.04d0.4, it is seen from figure 7(b) that the size and intensity of the KH vortex continue to increase since its initial formation, i.e. the KH vortex is always in zone D. The same is true for the case $\gamma$ 0.04d1.2 except that zone D can only continue up to the vicinity of $x/L_a = 0.60$ , after which the complex interactions between the KH vortices take place, as shown in figure 3(c). The zone after $x/L_a = 0.60$ is called the ‘interaction’ zone, denoted as zone I for short. Similarly, the KH vortex goes through the zones D and ${D}^*$ sequentially and enters zone I after $x/L_a = 0.40$ in the case $\gamma$ 0.20d1.2. As shown in figure 7(e), it is pertinent to emphasise that the changes in KH vortex characteristics are obviously different from those in the cases discussed previously. The differences are mainly manifested in three aspects. First, the intensity of the KH vortex gradually decreases, which means that it directly enters zone ${D}^*$ after its initial formation. Second, the size of the KH vortex does not change remarkably, but remains almost the same. Third, the KH vortex position does not undergo any significant uplifting when moving downstream. The case shown in figure 3(f) is similar to the case $\gamma$ 0.40d0.4, except that the dissipation process of the KH vortex is quickly interrupted by the emergence of zone I and, for conciseness, the variations of the KH vortex in the case $\gamma$ 0.40d1.2 are not shown separately in figure 7. As shown in figure 7(f), the intensity and size of the CO vortex in the cavity oscillation mode gradually decrease, which is similar to that in the case $\gamma$ 0.40d0.4.

A brief summary of the definitions of the four zones is provided here. The definitions of D, T, ${D}^*$ and I are taken from the words ‘developing’, ‘transition’, ‘dissipation’ and ‘interaction’, respectively. To avoid conflict, the ‘dissipation’ zone is referred to as ${D}^*$ . These zones are defined based on the size and intensity characteristics of the large-scale vortex during its evolution. In zone D, where the large-scale vortex is developing, the position of the vortex rises, and both its size and intensity increase until reaching a maximum. In zone ${D}^*$ , where the large-scale vortex is dissipating, its size gradually shrinks to a fixed value and its strength diminishes. In zone T, the transition from zone D to ${D}^*$ occurs rapidly, and the large-scale vortex is characterised by a rapid decrease in its strength and size, along with a swift upward lift in position. In zone I, where large-scale vortices interact, the geometry of large-scale vortices becomes difficult to identify. According to the types of large-scale vortices and the zones they go through successively, the evolution sequences of large-scale vortices in figure 3(a–g) can be named as $\mathrm{KH\_DTD^*}$ , KH_D, KH_DI, $\mathrm{KH\_DTD^*I}$ , $\mathrm{KH\_D^*}$ , $\mathrm{KH\_D^*I}$ and $\mathrm{CO\_D^*}$ , respectively, and the typical case $\gamma$ 0.20d0.4 discussed thoroughly in this section is a $\mathrm{KH\_DTD^*}$ sequence.

Figure 8.

Contours of instantaneous z-vorticity in the case $\gamma$ 0.20d0.4 indicating the detailed interactions between the KH vortices and VO vortices. The VO vortex shed by the vegetation element at $x/L_a = 0.32$ is labelled by number ‘1’.

To further explore the mechanisms related to the variations of a large-scale vortex in each zone, the contours of instantaneous z-vorticity in the range of ( $x/L_a$ , $tU_\infty /L$ ) = (0.24–0.40, 1128.3–1131.7) in the case $\gamma$ 0.20d0.4 are given in figure 8. Taking the vegetation element at $x/L_a = 0.32$ as an example, as shown in figure 8(a), before the KH vortex interacts with it, the vegetation element sways to form VO vortex 1. When the KH vortex moves to the location $x/L_a = 0.32$ , as shown in figure 8(b), due to the rotation direction of the upper boundary of vortex 1 (x-positive direction) being exactly opposite to that of the lower boundary of the KH vortex (x-negative direction), strong horizontal shearing occurs at the vortex–vortex junction. As the KH vortex continues to move downstream, the VO vortex gradually merges with the KH vortex and finally becomes part of the KH vortex, as shown in figure 8(c–f). In fact, within the illustrated range, the KH vortex continuously interacts and merges with several VO vortices as it moves downstream, which can be regarded as the main vorticity source to facilitate the increase in both size and intensity of the KH vortex in zone D. Conversely, the KH vortex conveys its energy to the surrounding fluid and vegetation array, thus helping the vegetation element to maintain its large-amplitude oscillation, which in turn can supply more vorticity to the KH vortex. It is also seen that the positive vorticity formed by the vegetation oscillation merges with each other and converges at the position below the upstream of each KH vortex. Since this portion of the vortex rotates counterclockwise, it imparts an upward velocity to the KH vortices. At approximately $x/L_a = 0.40$ , shown in figure 8, the upward velocity is sufficiently large to uplift the KH vortices significantly and make them detach from the vegetation canopy. In this rapid rising process, i.e. in zone T, due to the larger convective velocity at higher positions, the KH vortex is horizontally stretched by the high-velocity flow, with corresponding compression in the vertical direction. This leads to a reduction in the vertical size of the KH vortex in zone T. The intensity of the KH vortex also decreases rapidly during uplifting and horizontal stretching, as shown in figure 7(a) from $x/L_a = 0.40$ to 0.45. After that, since the KH vortex is at a distance from the vegetation, the KH vortex and the VO vortex cannot interact anymore in zone ${D}^*$ . During the downstream convection, the KH vortex has no vorticity supplement, but only gradually dissipates owing to the fluid viscosity.

However, in the cases shown in figures 3(e) and 3(f), after the formation of large-scale vortices, even though they also interact with the VO vortices, because of the large bending rigidity, the dissipation of the large-scale vortices caused by the vegetation swaying is greater than the vorticity supplement from the VO vortices to the large-scale vortices. As a result, the large-scale vortices enter zone ${D}^*$ since they are generated.

Finally, as mentioned in § 3.1, the appearance of zone I is caused by the low frequency in the flow, which eventually results in complex interactions between KH vortices in the rear portion of the array.

3.4. Influences of bending rigidity and gap distance on the evolution of large-scale vortices

In the previous section, different evolution zones and sequences of large-scale vortices are classified. In this section, the effects of bending rigidity $\gamma$ and gap distance d on the evolution of large-scale vortices are discussed, including the effects on the evolution sequences, growth rate, dissipation rate, maximum strength and maximum size.

As mentioned in the last section, based on the types of large-scale vortices and the zones they enter successively, different evolution sequences of large-scale vortices can be identified. The distribution of the large-scale vortex evolution sequences in the parameter space is shown in figure 9, where the previously identified six regions are divided. On the whole, roughly bounded by $\gamma$ = 0.28, in the region where $\gamma \lt 0.28$ , the large-scale vortex first experiences zone D after its formation. When $\gamma \geqslant 0.28$ , most of the large-scale vortices will directly enter zone ${D}^*$ . This reflects that $\gamma$ has a decisive influence on the evolution of large-scale vortices. In particular, when both $\gamma$ and d are large, i.e. in region V, the large-scale vortex is the CO vortex under the cavity oscillation mode. The distributions show obvious regularity in the region where $\gamma \lt 0.28$ , which can be explained in combination with figure 5. In region II, because there are low frequencies in quasi-regular Monami_B mode which lead to the complex interactions between the KH vortices, almost all cases exhibit the KH_DI sequence. In region III, the $\mathrm{KH\_DTD^*}$ sequence prevails because of the absence of low frequency in regular Monami mode. Similar to irregular Monami modes, the $\mathrm{KH\_DTD^*I}$ pattern is mainly distributed on the diagonal of region IV. Moving away from the diagonal in the direction of increasing $\gamma$ , the evolution sequence changes from $\mathrm{KH\_DTD^*I}$ to $\mathrm{KH\_DTD^*}$ in region III, while moving away from the diagonal in the direction of increasing d, the evolution sequence shifts from $\mathrm{KH\_DTD^*I}$ to KH_DI in region II. In region I, when both $\gamma$ and d are small, almost all cases contain low-frequency flow, which eventually should have resulted in the appearance of zone I. However, the KH instability onset position is very close to the end of the array in region I (Ni et al. Reference Ni, Ji, Xu, Zhang and Liang2023), which shortens the effective length for the evolution of KH vortices. As a result, the KH vortex has already left the array before zone I appears such as the case $\gamma$ 0.04d0.4, which has been mentioned in § 3.1. It can be inferred that if the length of the array is long enough, the sequences KH_D and $\mathrm{KH\_DTD^*}$ in region I will turn into sequences KH_DI and $\mathrm{KH\_DTD^*I}$ , respectively. The potential doubt about why evolution sequences in region II are KH_DI instead of $\mathrm{KH\_DTD^*I}$ is also explained here. Because of the relatively small $\gamma$ and large d in this region, the KH vortex gets less supplement from the VO vortex at the same convection distance, resulting in a low growth rate of the KH vortex, which will be further discussed. The influence mechanism of parameters is briefly summarised. Increasing $\gamma$ can enhance the dissipation caused by vegetation swaying, thus zone D turns into ${D}^*$ at large $\gamma$ . Increasing d can accelerate the emergence of low-frequency flow, resulting in the interactions between KH vortices and the appearance of zone I. In essence, $\gamma$ and d play a role in modulating the features of low frequency, the KH instability onset position, and the growth rate of KH vortices, which lead to the dynamic modes and evolution sequences showing similar distributions in the parameter space.

Figure 9.

A map for the distribution of the large-scale vortex evolution sequences in the parameter space of ( $\gamma$ , d).

Figure 10.

Development of large-scale vortex intensity under different cases, where panels (a)–(d) are mainly cases with $\gamma \lt 0.28$ where the first zone of the evolution sequence is zone D and panels (e)–(h) are mainly cases with $\gamma \geqslant 0.28$ where the first zone of the evolution sequence is zone ${D^*}$ . The solid, semi-hollow and hollow symbols denote zones D, T and ${D}^*$ , respectively.

Several cases marked by black dotted boxes in figure 9 are selected to analyse the influence of $\gamma$ and d on the variation of large-scale vortex intensity. Note that since the KH case dominates the parameter space in this study, only KH cases are selected, excluding the CO case. Here, $x/L = (n - 1)\boldsymbol{\cdot} d$ is used to facilitate comparison under different gap distances. To make it easier to distinguish, figure 10(a–d) is mainly the cases with $\gamma \lt 0.28$ where the first zone of the evolution sequence is zone D, while figure 10(e–h) is mainly the cases with $\gamma \geqslant 0.28$ where the first zone of the evolution sequence is zone ${D^*}$ . The solid, semi-hollow and hollow symbols represent the large-scale vortex in zones D, T and ${D}^*$ , respectively. The variation of the large-scale vortex intensity shown in figure 10(a–d) is approximated in the simplified form of a three-segment line, and the slopes of each segment are calculated by the least-square method as $k_{{D}}$ , $k_{{T}}$ and $k_{{D^*}}$ . Since the slope represents the speed of the variation of large-scale vortex intensity along the array, it can be used as an indicator of the development rate. More specific, $k_{{D}}$ denotes the growth rate in zone D, while $|k_{{T}}|$ and $|k_{{D^*}}|$ correspond to the dissipation rates during the rapid uplift in zone T and free convection in zone ${D}^*$ , respectively. Similarly, in the cases shown in figure 10(e–h), the dissipation begins simultaneously with the formation of large-scale vortices, and the dissipation rates can be approximated by $|k_{{D^*}}^1|$ and $|k_{{D^*}}^2|$ . In other words, $|k_{{D^*}}^1|$ and $|k_{{D^*}}^2|$ are the dissipation rates for zone ${D}^*$ in sequences $\mathrm{KH\_D^*}$ and $\mathrm{KH\_D^*I}$ , as distinguished from $|k_{{D^*}}|$ which is the dissipation rate for zone ${D}^*$ in sequences $\mathrm{KH\_DTD^*}$ and $\mathrm{KH\_DTD^*I}$ .

Figure 11.

(a) Growth rate $k_{{D}}$ of large-scale vortex intensity under different cases. Note that $|k_{{ss}}|$ is the growth rate predicted by shear strength of the shear layer. (b) Variation of shear strength of the shear layer in zone D in the same cases with $d$ = 0.4 as those in panel (a).

Figure 11(a) shows the growth rate of large-scale vortex intensity $k_{{D}}$ under different cases. It is seen that with the increase of $\gamma$ , $k_{{D}}$ also increases, and the maximum value is obtained between 0.20 and 0.24. It is useful to further elucidate whether the variation trends in figure 11(a) are fundamentally a result of the changes of shear layer characteristics caused by varying $\gamma$ and $d$ . For a KH vortex in a shear layer, its growth rate can also be estimated based on the shear layer properties, such as the shear strength, which is defined as the ratio of the velocity difference $\Delta U$ to the shear layer thickness $S$ , i.e. $(\Delta U/U_\infty )/(S/L)$ in a dimensionless form (Michalke Reference Michalke1964; Winant & Browand Reference Winant and Browand1974; Ho & Huerre Reference Ho and Huerre1984; Rogers & Moser Reference Rogers and Moser1992). It should be noted that in the current study, the VO vortex significantly influences the streamwise velocity distribution, i.e. the velocity difference inside the canopy induced by VO vortex, which is discussed in detail in § 3.6. Considering the discussion here focuses on the KH vortex, the shear layer thickness $S$ is defined as the distance from the height where the maximum mean streamwise velocity occurs to the vegetation height, with $\Delta U$ representing the corresponding velocity difference between these two heights. Also note that only zone D is included to analyse the growth rate. The variations of the shear strength in the same cases with $d$ = 0.4 as those in figure 11(a) are shown in figure 11(b). It is seen that the vortex intensity predicted by shear strength decreases during evolution in zone D, which is exactly opposite to the observed tendency that the vortex intensity increases in zone D. Given the potentially negative correlation between KH vortex strength and shear strength of the shear layer, the growth rates $|k_{{ss}}|$ between different cases in figure 11(b) are further compared. However, as shown in figure 11(a), the variation tendency of $|k_{{ss}}|$ predicted by shear layer properties is different from that quantified directly by vortex intensity. This implies that it is not the changes to the shear layer caused by different $\gamma$ or $d$ that induce growth rate changes between different cases, and tends to favour other explanations. A reasonable interpretation is that the large-scale vortex can gain more vorticity supplement from the VO vortex with increasing $\gamma$ , but further increasing $\gamma$ leads to a significant rise in dissipation caused by vegetation swaying. Consequently, $k_{{D}}$ first increases at smaller values of $\gamma$ and then decreases as $\gamma$ becomes larger. This interpretation also explains why $k_{{D}}$ decreases as $d$ increases in the range $0.04 \leqslant \gamma \leqslant 0.10$ , which corresponds to the lower $\gamma$ region. This occurs because, as $d$ increases, the number of vegetation elements within the same length decreases, reducing the vorticity supplement of the VO vortex to the large-scale vortex and thus slowing the growth rate of the large-scale vortex intensity. From the perspective of the whole parameter space, the maximum growth rate of large-scale vortex strength is generally taken at medium $\gamma$ and medium $d$ , such as the case $\gamma$ 0.20d0.8.

The variations of dissipation rates of the intensity of the large-scale vortices $|k_{{T}}|$ , $|k_{{D^*}}^1|$ , $|k_{{D^*}}|$ and $|k_{{D^*}}^2|$ are shown in figures 12(a)–12(d), respectively. Note that the abscissa in figure 12(a) is the gap distance $d$ and that in figure 12(b–d) is the bending rigidity $\gamma$ . It is seen from figure 12(a) that the dissipation rate during the rapid uplift process in zone T, $|k_{{T}}|$ , is mainly affected by d, and the maximum value is obtained at d = 0.8. In zone ${D}^*$ subsequent to zone T, as shown in figure 12(c), $|k_{{D^*}}|$ is obviously impacted by both $\gamma$ and d, and decreases with the increase of $\gamma$ and d. In addition, on the whole, $|k_{{T}}|$ is an order of magnitude larger than $|k_{{D^*}}|$ . In contrast, the variations of $|k_{{D^*}}^1|$ and $|k_{{D^*}}^2|$ with $\gamma$ and d are irregular. This is mainly because, on one hand, there are few data points in cases like d = 0.8, which may lead to a large fitting error. On the other hand, these parameters, from $\gamma$ = 0.28 to 0.60, have a large jump and are the critical values of the transition to large rigidity. Nevertheless, by comparison, the relationship of dissipation rates can be summarised as $|k_{{T}}|\gt |k_{{D^*}}^1|\gg |k_{{D^*}}|\approx |k_{{D^*}}^2|$ . That is to say, the intensity of the large-scale vortex dissipates fastest in the process of significant uplift and dissipates slowest during the free convection.

Figure 12.

Variations of dissipation rates (a) $|k_{{T}}|$ , (b) $|k_{{D^*}}^1|$ , (c) $|k_{{D^*}}|$ and (d) $|k_{{D^*}}^2|$ in different cases.

Figure 13.

Variations of the (a) maximum intensity $\boldsymbol{\sigma }_{v\_{{max}}}/U_{\infty }$ and (b) maximum size $\Delta y_{\_{{max}}}/L$ of the large-scale vortex in different cases.

Figure 13 displays the maximum intensity $\boldsymbol{\sigma }_{v\_{{max}}}/U_\infty$ and maximum size $\Delta y_{\_{{max}}}/L$ of the large-scale vortex in the evolution process under different cases. For the KH_DI sequence mainly distributed in the region of $\gamma \leqslant 0.10$ , because the KH vortex does not fully develop to the maximum strength and size before entering zone I, it is not involved in the discussion here. It is seen from figure 13 that $\boldsymbol{\sigma }_{v\_{{max}}}/U_\infty$ and $\Delta y_{\_{{max}}}/L$ of the large-scale vortex are primarily affected by $\gamma$ , and decrease with the increase of $\gamma$ as a whole. However, a local peak exhibits near $\gamma$ = 0.34, which is particularly evident in figure 13(a). It is not surprising since the $\mathrm{KH\_DD^*I}$ sequence appears near $\gamma$ = 0.34 as shown in figure 9. Compared with the $\mathrm{KH\_D^*}$ and $\mathrm{KH\_D^*I}$ sequences, the KH vortex in the $\mathrm{KH\_DD^*I}$ sequence increases in its intensity and size after formation, thus leading to a local peak at $\gamma$ = 0.34.

3.5. Influences of large-scale vortices evolution on vegetation swaying

As the large-scale vortex develops along the array, the swaying characteristics of vegetation in different zones are different. In this section, the influences of large-scale vortices evolution on vegetation swaying are simply discussed, including the influences on the mean and standard deviation of the vegetation tilt angle, which reflect the mean angular position and oscillation amplitude, respectively.

Figure 14.

Variations of mean and standard deviation of tilt angle along the array in the same cases as those in figure 7. The mean and standard deviation of tilt angle are marked by black hollow circles and black solid lines, respectively.

Selecting the same cases as those in figure 7, the mean and standard deviation of the vegetation inclination angle in each case are shown in figure 14. It can be seen from figure 14(a–d) that the mean angular position of the vegetation in zone D is greater than zero and gradually increases with the enhancement of large-scale vortices. The swaying amplitude of the vegetation is also gradually increasing as the main trend. In zone T, as shown in figure 14(a), in the process of the rapid uplifting of KH vortices, the mean position of the vegetation swaying quickly returns to zero and the swaying amplitude gradually declines. Then in zone ${D}^*$ , the mean position remains unchanged at zero and the swaying amplitude continues to decline with the attenuation of the KH vortex. A similar process is also shown in figure 14(d), but the difference is that zone ${D}^*$ is interrupted by the appearance of zone I. It is seen from figures 14(c) and 14(d) that the variations of mean position and swaying amplitude in zone I in different cases show different trends. To be specific, in figure 14(c), the mean angular position and swaying amplitude increase, which is the same as those in zone D. While in figure 14(d), the mean angular position remains almost zero and the oscillation amplitude gradually decreases, which is identical to those in zone ${D}^*$ . This means that the characteristics of vegetation swaying in zone I duplicate those in its upstream zone. This is not surprising. If the upstream is zone D, the interactions occur between KH vortices with relatively great size and high intensity, so the chaotic flow field still maintains high intensity causing the vegetation in zone I to sway like that in zone D. As for the case where the upstream is zone ${D}^*$ , on one hand, the KH vortex has detached from the array before entering zone I. On the other hand, the KH vortex is in low intensity. As a result, after the interactions between KH vortices with low strength at a position far from the array, the flow field is very weak, which induces the vegetation to sway in the way like that in zone ${D}^*$ . For the two cases that are always in zone ${D}^*$ under large $\gamma$ , as shown in figures 14(e) and 14(f), on the whole, the mean angular position of the vegetation is almost zero except that in the first and second vegetation in the cavity oscillation mode shown in figure 14(f), and the swaying amplitude is much smaller than that in the cases shown in figure 14(a–d). With the decrease of the strength of the large-scale vortex, the swaying amplitude is also reduced along the way.

3.6. Influences of large-scale vortices evolution on flow characteristics

In this section, the influences of large-scale vortices evolution on the flow characteristics, including the influences on the vertical distributions of both the time-averaged velocity (streamwise and vertical) and the Reynolds stress, are explored.

Figure 15.

Vertical distribution of time-average streamwise velocity $\overline {\boldsymbol{u}}$ and vertical velocity $\overline {\boldsymbol{v}}$ normalised by the incoming flow velocity $U_\infty$ marked in red and blue solid lines at different positions in the same cases as those in figure 7. The meanings of the horizontal lines are the same as those in figure 6(b).

Figure 15 shows the vertical distributions of the time-averaged streamwise velocity $\overline {\boldsymbol{u}}$ and vertical velocity $\overline {\boldsymbol{v}}$ , and the selected cases are the same as those in figure 7. The green and cyan dashed lines above $y/L = 1.0$ are the positions of the large-scale vortex core and upper boundary, respectively, while the lines below $y/L = 1.0$ represent the corresponding positions of the VO vortex. Overall, since the vegetated flow is dominated by downstream convection, $\overline {\boldsymbol{v}}$ is significantly smaller than $\overline {\boldsymbol{u}}$ and can even be an order of magnitude smaller in some positions. It has been mentioned in § 1 that the vertical distribution of $\overline {\boldsymbol{u}}$ in the flow with submerged flexible vegetation approximately satisfies the hyperbolic tangent law, similar to that in figure 1. However, it can be seen from figure 15 that $\overline {\boldsymbol{u}}$ does not obey this distribution law in the current study and the main difference is reflected by the fact that negative values of $\overline {\boldsymbol{u}}$ exist in the $y/L \lt 1.0$ part, which is rarely mentioned in the previous studies. This is mainly because the 2-D vegetation here is fully impermeable, while a 3-D canopy is semi-permeable, allowing some through-flow, rather than providing full flow blockage. Specifically, the position where the maximum negative value $\overline {\boldsymbol{u}}^{-}$ is obtained is exactly the lower boundary of the VO vortex. This is not surprising since the streamwise velocity of the lower boundary of the VO vortex is negative due to its clockwise rotation. Therefore, it is reasonable to consider that the distribution of $\overline {\boldsymbol{u}}$ inside the canopy is dominated by the VO vortex shed by the vegetation oscillation. In different cases, the position of the lower boundary of the VO vortex is different, which mainly depends on $d$ and the swaying amplitude of the vegetation. When $d$ is small, no matter how large is the swaying amplitude of the vegetation, under the obstruction of adjacent vegetation, the VO vortex can only be shed at a position slightly lower than the top of the vegetation, so that the lower boundary of the VO vortex is higher, as shown in figures 15(a), 15(b) and 15(e). The position of the VO vortex in the case $\gamma$ 0.20d0.4 can be seen in figure 8 as a reference. When d is large, if the swaying amplitude of the vegetation is also large, the VO vortex can be shed at a lower position, as shown in figures 15(c) and 15(d). A negative maximum $\overline {\boldsymbol{v}}^{-}$ is also seen in the vertical distribution of $\overline {\boldsymbol{v}}$ and it is reasonable to believe that $\overline {\boldsymbol{v}}^{-}$ also results from the VO vortex since it is always taken near the top of the vegetation canopy, as shown in figure 15(a–e). In the external flow with $y/L \gt 1.0$ , in zones D and ${D}^*$ , the upper boundary of the mixing layer (approximately the position of the maximum streamwise velocity $\overline {\boldsymbol{u}}^{+}$ ) almost coincides with the upper boundary of the KH vortex. The position of the inflection point in the mixing layer is very close to the position of the KH vortex core. That is to say, the vertical distribution of $\overline {\boldsymbol{u}}$ in the free flow layer is controlled by the large-scale vortex, and the thickness and position of the mixing layer are similar to the vertical size and position of the large-scale vortex, respectively, which is consistent with previous conclusions (Ghisalberti & Nepf Reference Ghisalberti and Nepf2002; Poggi et al. Reference Poggi, Porporato, Ridolfi, Albertson and Katul2004; Nepf et al. Reference Nepf, Ghisalberti, White and Murphy2007). Furthermore, the positive maximum $\overline {\boldsymbol{v}}^{+}$ is taken at the upper boundary of the KH vortex due to the asymmetry of strength between the front and rear parts of the KH vortex. In zone D, $\overline {\boldsymbol{v}}^{+}$ increases with the increase of the KH vortex intensity, while in zone ${D}^*$ , $\overline {\boldsymbol{v}}^{+}$ decreases with the attenuation of the KH vortex. This provides another evidence proving that the large-scale vortices dominate the external flow of the canopy. In zone I, the interactions between large-scale vortices make the vertical distributions of both $\overline {\boldsymbol{u}}$ and $\overline {\boldsymbol{v}}$ more uniform.

It is noteworthy that the shear layer in the present study is significantly offset from the undeflected vegetation. In many previous studies (Ghisalberti & Nepf Reference Ghisalberti and Nepf2002, Reference Ghisalberti and Nepf2006; Nepf Reference Nepf2012), however, the shear layer is near the top of the canopy. This difference is mainly due to the current 2-D simulations. Since the 2-D vegetation in this study is fully impermeable, the KH vortex is deflected upwards, leading to a significant offset from the undeflected vegetation height; while in a 3-D condition, the KH vortices are able to invade into the interior of the vegetation canopy, resulting in shear layer lying near the top of the canopy. In addition, this offset occurring in 2-D studies is seen not only at the location where the KH vortex is excited, such as that in figure 15, but also at the location prior to the onset of the mixing layer instability (Fang et al. Reference Fang, Gong, Revell and O’Connor2022).

Figure 16.

Contour of instantaneous z-vorticity of the front portion of the array in the case $\gamma$ 0.60d1.2.

Figure 17.

Vertical distribution of Reynolds stress $\boldsymbol{\tau }_{uv}$ normalised by the square of incoming flow velocity $U_\infty ^{2}$ marked in red solid lines at different positions in the same cases as those in figure 7. The meanings of the horizontal lines are the same as those in figure 6(b).

It should be noted that in the cavity oscillation mode shown in figure 15(f), because the width of the cavity is larger than the horizontal size of the CO vortex, the CO vortex at its initial evolution stage ( $x/L_a$ = 0.02–0.03) can partially enter the interior of the array. Under this circumstance, the maximum negative velocity $\overline {\boldsymbol{u}}^{-}$ is not caused by the VO vortex, but by the recirculation of the CO vortex within the cavity, as shown in figure 16. The maximum negative velocity $\overline {\boldsymbol{v}}^{-}$ is also caused by the rotation of the CO vortex. This leads to, for one thing, $\overline {\boldsymbol{v}}^{-}$ much larger than that shown in figure 15(a–e) which is caused by the VO vortex with lower intensity. For another, $\overline {\boldsymbol{v}}^{-}$ is no longer obtained near the top of the canopy, but is taken at the core of the CO vortex. With the uplift and rapid dissipation of the CO vortex when moving downstream, the negative $\overline {\boldsymbol{u}}$ inside the vegetation canopy becomes very weak and almost zero, and $\overline {\boldsymbol{v}}^{-}$ also dissipates quickly.

In addition to the time-averaged velocity profiles, the Reynolds stress $\boldsymbol{\tau }_{uv} = -\overline {\boldsymbol{u}^{\prime }\boldsymbol{v}^{\prime }}$ , where $\boldsymbol{u'}$ and $\boldsymbol{v'}$ represent the fluctuations of the streamwise and vertical velocity, is also a critical quantity measuring the vertical momentum exchange and is mainly impacted by the large-scale vortices. Figure 17 gives the vertical distributions of $\boldsymbol{\tau }_{uv}$ in different cases and different zones. It is seen from figure 17(a–e) that in zones D and ${D}^*$ , $\boldsymbol{\tau }_{uv}$ has a positive maximum value ${\boldsymbol{\tau }_{uv}}^{+}$ near the top of the vegetation canopy due to the strong horizontal shear and vertical momentum exchange. As $y/L$ increases, $\boldsymbol{\tau }_{uv}$ gradually decreases to zero near the upper boundary of the KH vortex. In the part of $y/L \lt 1.0$ , owing to the influence of the VO vortex, $\boldsymbol{\tau }_{uv}$ does not gradually decrease to zero like that in figure 1, but first forms a negative maximum ${\boldsymbol{\tau }_{uv}}^{-}$ and then increases to zero. Generally, combined with figure 14, a greater swaying amplitude will result in a larger ${\boldsymbol{\tau }_{uv}}^{-}$ , such as the case $\gamma$ 0.04d1.2 shown in figure 17(c). This indicates that the origin of ${\boldsymbol{\tau }_{uv}}^{-}$ is closely related to the outward and inward interactions led by the VO vortex, where the outward interaction is featured by $\boldsymbol{u'}\gt 0$ and $\boldsymbol{v'}\gt 0$ , while the inward interaction is featured by $\boldsymbol{u'}\lt 0$ and $\boldsymbol{v'}\lt 0$ . In zone I, the position where ${\boldsymbol{\tau }_{uv}}^{+}$ is obtained moves away from the top of the vegetation canopy, and $\boldsymbol{\tau }_{uv}$ reduces to zero until near the top surface because of the chaos in the entire water depth caused by the complex interaction of KH vortices. For the cavity oscillation mode in figure 17(f), there is no apparent ${\boldsymbol{\tau }_{uv}}^{-}$ because of the weak vortex shed by small oscillations of the vegetation.

Affected by the enhancement of large-scale vortices in zone D, ${\boldsymbol{\tau }_{uv}}^{+}$ expands with the increase of KH vortex intensity. It should be noted that in figure 17(a), ${\boldsymbol{\tau }_{uv}}^{+}$ at $x/L_a=0.40$ decreases compared with that at $x/L_a=0.30$ . This is mainly because $x/L_a=0.40$ happens to be the transition zone T. Although the KH vortex intensity reaches the maximum, it has begun to leave the array and the pulsation near the top of the array has started to weaken, thus reducing the Reynolds stress. In zone ${D}^*$ , ${\boldsymbol{\tau }_{uv}}^{+}$ falls with the dissipation of the large-scale vortex and, especially in the cavity oscillation mode, ${\boldsymbol{\tau }_{uv}}^{+}$ drops rapidly to zero in a short range with the attenuation of the CO vortex. The above-mentioned analysis indicates that the vertical momentum exchange in zone D is evidently stronger than that in zone ${D}^*$ .