4.1. Wake structure related to patch density and aspect ratio

Our experimental measurements of hydrodynamic fields around a centre-channel finite patch illustrate how flow velocity adjustment and wake structure respond to the vegetation patch under different cumulative resistance. The patch cumulative resistance is defined as the combined effect of the solid volume fraction ($\varPhi$) and aspect ratio ($L/W$), the latter of which has not yet drawn enough attention in previous studies. However, the aspect ratio of the vegetation patch may vary significantly in nature due to vegetation colonization dynamics and different types of aquatic vegetation morphology (Gurnell Reference Gurnell2014; Gran et al. Reference Gran, Tal and Wartman2015; Biggs et al. Reference Biggs, Nikora, Gibbins, Fraser, Green, Papadopoulos and Hicks2018; Yamasaki et al. Reference Yamasaki, de Lima, Silva, Preza, Janzen and Nepf2019). The increase in patch density and aspect ratio can both decrease the pore velocity and, thus, increase the main flow velocity by meeting the balance between momentum flux, high drag and gravitational force in the patch. This enhances the velocity difference or gradient transversely, thereby promoting flow instability in the wake. However, it is identified that the global response of flow to these two factors are in essence distinct due to the shifting of the interaction between the patch and flow. From the perspective of momentum conservation, patch elongation affects the flow structure by expanding the reach of its source effect instead of intensifying this effect. In contrast, the increasing patch density could locally enhance the drag force and dispersive stress on the local flow.

Beyond the trailing edge, a steady wake region can be raised for short and sparse patches, where momentum transport is dominated by advection effect (Zong & Nepf Reference Zong and Nepf2012; Kim, Kimura & Park Reference Kim, Kimura and Park2018). The steady wake length ($L_1$) would be shortened with an early onset of vortex street. Wake vortex street can be found directly downstream of the patch (i.e. $L_1$ = 0) when $L > L_a$ under large $\varPhi$ (e.g. case D3) or large $L/W$ (e.g. cases A4, A5, B4, B5), or a combination of $\varPhi$ and $L/W$ (e.g. cases C3, D4, D5). According to figure 12(a,b) and table 2, both $U_1/U_\infty$ and $L_1/W$ decrease as $\varPhi$ increases within uncertainty. For the A cases ($\varPhi$ = 0.009), both $U_1/U_\infty$ and $L_1/W$ show a continuous decrease trend as $L/W$ increases, while $U_1/U_\infty$ presents a recovery trend for other cases with higher density ($\varPhi > 0.009$, i.e. cases B, C, D). The unsteady wake length ($L_2$) is dependent on both the velocity recovery rate, $(U_\infty -U_1)/L_2$, and initial velocity deficit, ($1-U_1/U_\infty$), at the wake vortex initiation point. The velocity recovery rate describes the rate of momentum input outside the wake while the initial velocity deficit describes the momentum required for full recovery. However, these two factors are interdependent and further complicate this issue. The unequal contributions of turbulence and velocity difference lead to a non-monotonic trend of $L_2$ to $\varPhi$ and $L/W$. Regarding the influence of the patch width, Bw cases with a width of 0.21 m present larger $L_1$ and $L_2$, a smaller velocity difference as well as recovery rate compared with B cases with a width of 0.09 m, indicating that patch width may have a negative effect on the generation of the vortex street. From the aspect ratio perspective, widening the patch performs like shortening the patch. However, only two patch widths are investigated here (not shown in the figure), a more detailed study with varying patch widths is recommended to confirm this finding.

Figure 12.

The impact of $\varPhi$ (i) and aspect ratio ($L/W$, (ii)) on (a) normalized steady wake length, $L_1/W$, (b) normalized steady wake velocity, $U_1/U_\infty$, (c) wake recovery rate, $(U_\infty -U_1)/L_2$, (d) wake recovery length, $L_2$.

4.2. Coexistence between KH vortex street and wake vortex street

According to the observations in the current experiment (figure 11) and previous studies (Rominger & Nepf Reference Rominger and Nepf2011), two parallel KH vortex streets tend to be triggered due to flow instability along the lateral interfaces. Therefore, the correlation between the KH vortex streets and the wake vortex streets is of great interest. A straightforward way is to use a dye tracer to track the vortex evolution, for example, releasing dye tracers simultaneously along the patch edge and at the wake. However, this method is not effective due to the strong dispersion of dye tracers within water flows. Alternatively, the frequency of the two kinds of vortices may be another indicator to imply their connection.

Similar to the Kármán vortex description, the presence of a KH vortex can be determined through the velocity power spectra ($S_{vv}$) distribution along the patch lateral edge ($x = 0 - L$, $y = W/2$, figure 13). The dashed lines in the figures indicate their perspective interior adjustment length ($L_a$). As shown in figure 13, a peak in $S_{vv}$ distribution demonstrates the initiation of the KH vortex (White & Nepf Reference White and Nepf2007; Caroppi et al. Reference Caroppi, Gualtieri, Fontana and Giugni2020; Proust & Nikora Reference Proust and Nikora2020; Yan et al. Reference Yan, Duan, Wai, Li and Wang2022). Specifically, when approaching $L_a$, an instability structure starts to form (indicated by yellow in figure 13). Then, over $L_a$, KH vortex streets can be observed (indicated by red in figure 13), which is consistent with the study of Rominger & Nepf (Reference Rominger and Nepf2011). As the flow travels downstream, the enhanced velocity gradient at the patch lateral edge and weakened flow diversion contribute to the generation of a KH vortex street. This evolution will be intensified through the momentum exchange between the parallel KH vortex streets across the patch width (Rominger & Nepf Reference Rominger and Nepf2011), given by the stronger peaks at the B cases ($W = 0.09$ m) relative to the Bw cases ($W = 0.21$ m). Moreover, the presence of these peaks also complies with the occurrence of in-patch peaks (figure 11). Therefore, the Kármán-like structure within the patch is proved to be a result of the interaction of the two parallel KH vortex streets. This in-patch Kármán-like structure will persist after the patch as a Kármán vortex street. Based on the analysis of the results presented in figure 14, when a KH vortex occurs, a peak presents at $f_{KH} = 0.12$–$0.17$ Hz in $S_{vv}$, with an average of 0.14 Hz, consistent with the mean value of $f_{VK}$ = 0.15 Hz for the patch-scale Kármán vortex street, indicating that the KH vortex does have some similarity with the Kármán vortex.

Figure 13.

Power spectra ($S_{vv}$) along the lateral edge of the patch ($x = 0 \sim L, y = W/2$). The dashed lines indicate the interior adjustment region ($L_a$). Note that the width ($W$) for the Bw cases is 0.21 m and 0.09 m for cases A–D.

Figure 14.

Comparison between $f_{VK}$ and $f_{KH}$. Here $f_{VK}$ is the frequency of the von Kármán vortex and $f_{KH}$ is the frequency of the KH instability. Both are measured from velocity power spectra. The solid line indicates $f_{VK} = f_{KH}$ and the dashed lines indicate the 80 % confidence intervals.

Furthermore, the results of the normalized Reynolds shear stress at $x = 0-L$, $y = W/2$ for all cases are shown in figure 15. The normalized shear Reynolds stress at the end of the trailing edge shows an increasing trend as the aspect ratio and solid volume fraction increase, indicating the formation and growth of the KH vortex. This result is consistent with the peak values presented in the velocity power spectra above. However, when the length of case D ($\varPhi = 0.040$) extends from $L$ = 1.0 m to 1.4 m, a sudden drop occurs, which is mainly caused by the strong interaction between two KH coherent structures developed along the parallel edges of the patch. This interaction leads to the formation of an in-patch Kármán-like vortex, which results in vortex intensity dissipation and velocity recovery as indicated in figures 5 and 6. This Kármán-like vortex has also been observed in the study of cylinders in tandem (Hosseini et al. Reference Hosseini, Griffith and Leontini2020). Moreover, this in-patch Kármán-like structure may explain why natural patches may not keep elongating downstream unrestrictedly but present an equilibrium length. Meanwhile, there exists a threshold value for normalized Reynolds shear stress, $-\overline {u'v'}/u_*^2$ (shown by the dashed line in figure 15), indicating the presence of a KH vortex street. More importantly, according to figure 15, a Kármán vortex will certainly be detected if a KH vortex presents, but a Kármán vortex can also form in the absence of a KH vortex. Therefore, here we consider that a KH vortex is a positive but not necessary condition for Kármán vortex initiation.

Figure 15.

Normalized Reynolds stress at the position of $(x, y) = (L, W/2)$. The dashed line indicates the threshold value for the normalized Reynolds stress of the presence of the KH vortex. Here ‘VK’ indicates wake vortex street; ‘KH’ indicates KH vortex street; ‘Y’ indicates yes/presence; ‘N’ indicates no/absence.

4.3. Wake vortex street formation thresholds

The generation of the wake vortex street may significantly impact the flow velocity distribution, scalar convection and dispersion, sediment deposition and morphodynamics in the wake (Follett & Nepf Reference Follett and Nepf2012; Zong & Nepf Reference Zong and Nepf2012; Liu et al. Reference Liu, Hu, Lei and Nepf2018). Therefore, a threshold indicator of the wake vortex street is desirable. Considering the available implementations for different scenarios, three groups of threshold parameters are identified based on the results and findings above.

It has been noted that the initiation and growth of large-scale coherent structures are related to the velocity asymmetry imposed by factors such as the topographic forcing in the transition region of the compound channel (Dupuis et al. Reference Dupuis, Proust, Berni and Paquier2017; Proust et al. Reference Proust, Fernandes, Leal, Rivière and Peltier2017), and multilayered density stratification (Caulfield et al. Reference Caulfield, Peltier, Yoshida and Ohtani1995). Similarly, the cessation of the wake vortex street has been shown to be related to the velocity difference (or gradient) induced by the porous patch because a high-velocity bleed flow can prevent the shear layers developing along two sides to reach each other. Intuitively, the parameter combination of $U_1$ and $U_2$ (see definition in figure 1) at the wake vortex onset location is available for the threshold identification following Zong & Nepf (Reference Zong and Nepf2012). That is, when ($U_2-U_1$) is too small, wake instability is unlikely to be triggered to form wake vortex streets. According to our experiments, a distinct vortex street appears when $U_1/U_2 <$ 0.5, which is different from the result ($U_1/U_2 < 0.35$) from experiments conducted by Zong & Nepf (Reference Zong and Nepf2012) and numerical experiments by Chang & Constantinescu (Reference Chang and Constantinescu2015) regarding a circular patch (i.e. $L/W = 1$).

In addition to vegetation density as stated in previous studies, the wake instability has been found to be related to the patch aspect ratio ($L/W$) based on our experiments. Under the same density investigated in this study, the vortex generation is promoted by patch length ($L$) and suppressed by patch width ($W$). Therefore, an integrated non-dimensional threshold value can be determined as $U_1/U_2 \times (L/W)^{-1}$, which is more appropriate for the threshold identification. Therefore, a threshold map for the wake vortex generation can be formed as shown in figure 16. Using a linear boundary $U_1/U_2 \times (L/W)^{-1} = 0.31$, no vortex street will be detected within the shaded area, while vortex street can be identified within the non-shaded area (figure 16a). This threshold incorporating the effect of the aspect ratio can express well the results of existing works (Zong & Nepf Reference Zong and Nepf2012) that are special cases of this expression. Alternatively, we can also use $(U_2-U_1)/U_\infty$ to represent the normalized velocity difference (figure 16b). In this case, the non-dimensional parameter is defined as $(U_2-U_1)/U_\infty \times L/W$ and it shifts to 0.84. That is, the vortex street occurs when $(U_2-U_1)/U_\infty \times L/W > 0.84$.

Figure 16.

Aspect ratio ($L/W$) versus velocity difference (a) $U_1/U_2$; (b) $(U_2-U_1)/U_\infty$ at $x = L + L_1$. The shaded area indicates the absence of the wake vortex street according to the threshold value.

Although the impact of patch length on velocity adjustment has been considered in $U_1$, $L$ incorporated here is a consideration for the impact of the KH instability rather than a normalization purpose. Under a low-density patch with sufficient length, the reduced velocity within the patch enhances the lateral velocity gradient as travelling downstream. This strengthened velocity gradient may trigger the presence of KH instability. Before the generation of an in-patch Kármán-like vortex due to the interaction of the KH vortex, the formed KH instability has positive effects on the wake vortex street formation as well. For example, case C1 presented a close but slightly higher velocity gradient ($(U_2-U_1)/U_\infty /W = 5.6\,{\rm m}^{-1}$) than that in case Bw4 $((U_2-U_1)/U_\infty /W = 4.7\,{\rm m}^{-1})$, but the wake vortex presents after case Bw4 rather than case C1. It is possibly caused by the energy contributed by the KH instability formed along the patch lateral edges in case Bw4, indicated by yellow in figure 13, even though it is weak. In addition, in the last case in the experiment of Zong & Nepf (Reference Zong and Nepf2012), with $(U_2-U_1)/U_\infty /W = 4.3\,{\rm m}^{-1}$ (where $D = 0.12$ m is the $W$ in this case), no wake vortex occurs, though its velocity gradient is higher than that of the vortex-generated case Bw3 $((U_2-U_1)/U_\infty /W = 3.6\,{\rm m}^{-1})$ in our experiment. Thus, the length of the patch should be incorporated into the threshold indicator to consider the positive contribution from the KH instability.

From a viewpoint of practical implementation, the sampling of $U_{e1}$ and $U_{e2}$ at the cross-section of the trailing edge ($x = L$) is much easier compared with $U_1$ and $U_2$ by locating the onset of vortex initiation ($x = L + L_1$). Chen et al. (Reference Chen, Ortiz, Zong and Nepf2012) demonstrated that there existed some relationship between $U_1$ and $U_{e1}$. In this sense, we also seek another threshold in the form of $U_{e1}$ and $U_{e2}$. Similar to the above manipulation, a threshold map can be established for wake vortex street prediction as shown in figure 17. The map implies that the vortex street forms at $U_{e1}/U_{e2} \times (L/W)^{-1} <$ 0.35 (except for case D1) or $(U_{e2}-U_{e1})/U_\infty \times L/W > 0.55$.

Figure 17.

Aspect ratio ($L/W$) versus velocity difference (a) $U_{e1}/U_{e2}$; (b) $(U_{e2}-U_{e1})/U_\infty$ at $x = L$. The shaded area indicates the absence of the wake vortex street according to the threshold value.

Previous studies have reported that the solid volume fraction ($\varPhi$) of the porous obstacle can also be a simple indicator for vortex street generation. According to the study of Castro (Reference Castro1971), the vortex street ceases to form after a slim perforated plate ($L/W \ll 1$) when $\varPhi <$ 0.8. For a circular porous array ($L/W = 1$), Zong & Nepf (Reference Zong and Nepf2012) reported that the cessation occurs at $\varPhi < 0.04$, which is very close to the limit ($\varPhi < 0.05$) found in the numerical modelling by Nicolle & Eames (Reference Nicolle and Eames2011). The inconsistent observations are due to the overlook of the impact of aspect ratios. A smaller aspect ratio leads to a higher critical value. This is because $\varPhi$ only considers the local hydraulic resistance while the patch dimensions consider the reach of the hydraulic resistance. Considering the cumulative effect of patch dimensions, the patch aspect ratio, $L/W$, is incorporated to be $\varPhi \times L/W$. In the present experiments, increasing the cumulative hydraulic resistance (by $\varPhi$ or $L/W$), through altering the velocity shear in the wake, could play a critical role in promoting the formation of wake vortex streets.

For a patch of rigid emergent cylinders in aligned rectangular configurations with comparable streamwise and spanwise length scales (cases A1–D1, with $L = 0.10$ m and $W = 0.09$ m, $L/W = 1.11$), the transition occurs at $\varPhi = 0.020$–$0.040$, which is close but slightly smaller than those observed in Nicolle & Eames (Reference Nicolle and Eames2011) and Zong & Nepf (Reference Zong and Nepf2012). When the patch elongates downstream, the transition occurs at $\varPhi = 0.013$–$0.020$ for cases with $L/W = 2.22$, which complies with our inference herein. Moreover, for $L/W > 2.22$, threshold values for $\varPhi$ are less than 0.009. Therefore, another threshold map ($\varPhi \times L/W > 0.045$) considering the cumulative resistance caused by $\varPhi$ and $L/W$ is established for wake vortex street prediction as shown in figure 18. However, this threshold will cease to hold true when $\varPhi$ is so low that velocity recovers before the end of the patch, which is not considered in this study. Therefore, the validity of this threshold map may be limited to a certain range of patch density and dimensions, which needs a broader-range investigation. It is better to understand the wake structure together with the velocity gradient. Moreover, this threshold map was developed for a single rectangular vegetation patch of rigid emergent cylinders in an aligned configuration. Hydraulic resistance is expected to be higher in staggered configurations with less flow through patches, resulting in the onset of the vortex street for lower solid volume fractions. Besides, the flexibility, submergence ratio and the presence of leaves in real plants are not considered either, further investigations are expected.

Figure 18.

Aspect ratio ($L/W$) versus solid volume fraction ($\varPhi$). The shaded area indicates the absence of the vortex street according to the threshold value.

In consequence, the following three kinds of thresholds have been established for identifying the formation of wake vortex streets in a channel with vegetation patches of rigid emergent cylinders in an aligned configuration under different densities and aspect ratios, which can be used alternatively based on the availability of corresponding parameters in different scenarios.

1. (i) Quantified by velocity scales measured at the end of the steady wake region ($x = L + L_1$), a Kármán vortex street will appear when

   (4.1)\begin{equation} U_1/U_2 \times (L/W)^{{-}1} < 0.31 \end{equation}
   or
   (4.2)\begin{equation} (U_2 - U_1)/U_\infty \times L/W > 0.84. \end{equation}

2. (ii) Quantified by velocity scales measured at the patch trailing edge ($x = L$), a Kármán vortex street will appear when

   (4.3)\begin{equation} U_{e1}/U_{e2} \times (L/W)^{{-}1} < 0.35 \end{equation}
   or
   (4.4)\begin{equation} (U_{e2}-U_{e1})/U_\infty \times L/W > 0.55. \end{equation}

3. (iii) Quantified by patch characteristics, a Kármán vortex street will appear when

   (4.5)\begin{equation} \varPhi \times L/W > 0.045. \end{equation}

In this connection, this study forms an essential extension with substantial contribution to this scientific research. However, there still exists some limitations in this study. Firstly, only two widths were investigated in this case, the variations caused by varying patch widths and patch blockage ratios ($W/B$) have not been systematically studied. Secondly, the influence of the KH instability on wake structure has not been fully understood. An interesting future test would be expected to split the patch in half with a solid dividing barrier/plate along the patch centreline to prevent any lateral ’communication’ of KH vortices across the patch (essentially forming two independent shear layers and weaker intensity) and see how it would affect KH vortex formation, especially for narrow cases. Thirdly, the combined impact of vegetation density ($\varPhi$) and patch aspect ratio ($L/W$) was investigated under the aligned configuration of rigid and emergent stems with uniform cylinder rods in this experiment. Hydraulic resistance is expected to vary under different configurations. Lastly, in a natural system the configuration will be even more complicated, features may evolve over time and space, with a random configuration, irregular patch shape and a wider range of stem sizes and densities. For instance, the threshold developed in our study failed to predict the formation of the wake vortex after the natural, submerged patch with flexible leaves investigated in the field study of Biggs et al. (Reference Biggs2019). This may attribute to the restriction of this threshold to rigid, emergent vegetation patches with no leaves or the limited measured reach in their study (Biggs et al. Reference Biggs2019). Therefore, the flexibility, submergence ratio, the presence of leaves in real plants and the varying flow conditions, such as incoming velocity and water depth, in natural aquatic systems need to be further systematically investigated.