3.1.1. Wake classification

By varying flow blockage and wake stability parameters, four wake regimes are observed behind an array of cylinders in the present experimental program: (i) vortex street wake (VS); (ii) ‘steady + shedding’ wake (SS); (iii) Kelvin–Helmholtz instability wake (KH); and (iv) coupled individual wake (CI).

Figure 4 illustrates the key flow characteristics in each wake regime. The first column shows a snapshot of dye streaks to visualise the instantaneous wake structure (see also the supplementary movies 1–5 available at https://doi.org/10.1017/jfm.2024.338), the second column presents C_V to indicate the time-averaged wake structure and the third column plots the power spectral density of the transverse velocity at (x/D, y/D) = (8.5, 0.5).

Figure 4. Wake regimes are well distinguished by the instantaneous wake structure (snapshot of dye streaks), the time-averaged flow structure (field of C_V) and power spectral density (S_vv) of transverse velocity. The incoming flow is directed upwards. Transverse velocities v for calculating the spectra were sampled at (x/D, y/D) = (8.5, 0.5). The four rows present (in order) cases H1, F1, C1 and A1. The steady wake region in panel (e) is roughly indicated by the black arrow. The series of KH vortices formed in the array shear layer in regime KH in panel (g) is marked by grey arrows.

Regime VS is characterised by a classical von Kármán vortex street formed immediately behind the array (figure 4a). The time-averaged wake structure behind the cylinder array (figure 4b) is similar to the classical wake of a single solid cylinder (Zdravkovich Reference Zdravkovich1997). At any given streamwise location, the value of C_V is higher near the wake boundaries than along the wake centreline. Near the wake boundaries, von Kármán vortices continuously entrain the ambient uncoloured fluid into the wake, leading to a sharp change in temporal variation of colour intensity C_V from nearly zero to very high values. The von Kármán vortex street behind the cylinder array creates a sharp peak in the spectrum of transverse velocity (figure 4c). The corresponding array Strouhal number (defined as $S{t_D} = fD/{U_\infty }$) for the case shown is 0.23, which is close to the value $(S{t_D} \approx 0.2)$ of an isolated solid cylinder in a vertically unbounded flow (Sumer & Fredsøe Reference Sumer and Fredsøe1997). Cylinder arrays in the VS regime thus behave similarly to a single solid obstacle in terms of the wake structure.

The ‘steady + shedding’ wake (SS) is characterised by a steady wake region attached behind the array, followed downstream by a von Kármán vortex street. The two array shear layers remain stable in the steady region, before interacting to result in the vortex street downstream. The merging of the dye streaks at the wake centreline (at x/D ≈ 5 in figure 4d) coincides with the end of the steady wake region, as demonstrated by Zong & Nepf (Reference Zong and Nepf2012). The steady wake region is delineated approximately by a region of nearly zero C_V in figure 4(e). The vortex street is associated with a sharp peak in the spectrum at St_D = 0.21 (figure 4f), which is again similar to that of a single solid cylinder. The formation of a steady wake region immediately downstream of the array is the key feature that differentiates regime SS from regime VS.

The regime KH is identified by the occurrence of a Kelvin–Helmholtz (KH) instability in the array shear layers, through flow visualisation and analysis of the velocity spectrum (described in more detail in § 3.1.3). The array-scale KH vortices (grey arrows in figure 4g) are distinctly different from those in regimes SS and VS in four ways:

1. (i) the two dye streaks do not mix together at the location where the vortices are initially formed, suggesting the vortices are not generated through the direct interaction between the two shear layers, i.e. vortex shedding;

2. (ii) the rows of coherent vortices within each array shear layer are not shed from alternating sides of the array (see also the online supplementary movie 3), which indicates that there is minimal temporal correlation between the coherent structures across the two shear layers. This is distinct from that observed in the SS regime where the von Kármán vortices are shed alternately from either side of the array (supplementary movie 2);

3. (iii) at a given streamwise location, the KH vortices are barely perceptible within either array shear layer at some instants, whereas at other instants, discrete KH vortices had developed in the shear layer (see supplementary movies 3 and 4). This indicates the irregular generation of coherent vortices in array shear layers;

4. (iv) the frequency content of the wake is different. In contrast to the narrow-banded peak in regimes SS and VS (figure 4c,f), a broadband peak is detected within the array shear layers in the KH regime (figure 4i). The broadband peak, defined as the range in which the power spectral density is larger than 20 % of its maximum, is in the range of approximately $0.16 < fD/{U_\infty } < 0.41$. The broadband peak is indicative of a KH instability in shear flows with a convective nature (Cardell Reference Cardell1993; Prasad & Williamson Reference Prasad and Williamson1997; White & Nepf Reference White and Nepf2007).

In the coupled individual wake (CI), the wakes from the elements of the array are the dominant flow feature in the array wake. The array shear layers still exist in this regime, but they are stable, such that no KH instability or vortex shedding occurs. This is indicated by the similarity of downstream evolution of dye streaks to that of the case without the array (see figures 3a and 4j). The dye streaks expand in the transverse direction smoothly downstream, without rolling up into any coherent vortex structures (figure 4j). The C_V value is high near the array and decreases along the wake for regime CI (figure 4k), which is indicative of lower lateral mixing rate downstream and hence no instability developing in the array shear layers in regime CI. The stabilisation of the wake flow is also confirmed by the velocity spectrum, with no identifiable dominant frequency across the entire frequency range (figure 4l).

The differences in the dye streak pattern, spatial C_V distribution and spectrum between these four regimes demonstrate the generation of stable array shear layers in regime CI and unstable array shear layers creating coherent vortex structures in regimes KH, SS and VS. A summary of principal features for these four regimes is given in table 4, together with a schematic of the wake structure for each regime.

Table 4. Characteristic flow features of the four wake regimes behind an array of cylinders. Dashed circle represents the edge of a cylinder array, and the incident flow points upwards. Shear layers are considered ‘unstable’ when they generate coherent vortex structures.

3.1.2. Controlling influence of ${\varGamma _D}$ and S_C on wake structure

The wake flow regimes observed behind 71 large arrays of cylinders $(D/d \approx 42.5)$ in shallow flow are mapped out in the (${\varGamma _D}$, S_C) parameter space in figure 5 (solid symbols), combined with existing data (hollow symbols). From the bottom-right corner to the top-left corner of the ${\varGamma _D}\unicode{x2013} {S_C}$ plane, the wake structure varies from a vortex street wake (VS) to a ‘steady + shedding’ wake (SS), then to a Kelvin–Helmholtz instability wake (KH), and finally to a ‘coupled individual wake’ (CI). This indicates that increases in S_C and reductions in ${\varGamma _D}$ tend to suppress the generation of large-scale coherent vortex structures, resulting in more stable wakes.

Figure 5. Separation, in the ${\textrm{S}_C}\unicode{x2013} {\varGamma _D}$ parameter space, of wake regimes in flows through an array of cylinders. Filled and unfilled markers represent data from this work and previous studies, respectively. Six filled symbols with black edges represent the pairs of cases tested with arrays of both large and small dimensionless array sizes (D/d). The dashed ellipses denote two cases with close values of ${\varGamma _D}$ or S_C, but with an approximately 50 % difference in D/d. This chart shows that a wake transition can be caused by variation of either ${\varGamma _D}$ or S_C and allows prediction of the wake regime behind an array of cylinders.

Figure 5 shows that data points generated with these large arrays and compiled from the literature, associated with each regime, cluster together within the parameter space. Data points of each regime are separated, forming clear boundary lines between them (as delineated by dashed lines). This clear separation suggests that the wake structure is, to first order, controlled by flow blockage and wake stability parameters, as hypothesised in § 1.

The curved regime boundaries between regimes implies that the wake structure behind a cylinder array depends on both ${\varGamma _D}$ and S_C nonlinearly. With increasing S_C, the regime boundary lines become increasingly horizontal (figure 5). This indicates that wake transition depends increasingly on S_C as it increases. Figure 5 suggests that there is an upper limited for S_C ≈ 0.6 above which only regime CI is found. This is expected as for ${S_C}\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel> \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }0.6$ the wake flow is fully stabilised without any oscillation for an isolated cylinder (Chen & Jirka Reference Chen and Jirka1997). The isolated cylinder, however, has much stronger shear layers than the cylinder array (demonstrated later in § 4.2) and, unlike the porous array, there is no stabilising influence of bleeding flow on the wake. This suggests that S_C ≈ 0.6 is conservative for the cylinder array, above which all the instability associated with array shear layers will be completely suppressed. Increasing ${\varGamma _D}$ can have a similar effect to decreasing S_C in causing transitions between regimes. Because of the nonlinear dependency, it is difficult to collapse ${\varGamma _D}$ and S_C down to one single parameter that defines the wake structure.

As bleeding flow has controlling influence on the wake structure (Wood Reference Wood1967; Zong & Nepf Reference Zong and Nepf2012; Chang & Constantinescu Reference Chang and Constantinescu2015), especially at low wake stability parameter, the shape of the boundary lines between different regimes can be interpreted by scaling terms in the momentum equation. Within the array, the time-mean flow in the streamwise direction is governed by the following equation:

(3.1) \begin{equation}\bar{u}\frac{{\partial \bar{u}}}{{\partial x}} + \bar{v}\frac{{\partial \bar{u}}}{{\partial y}} =- \frac{1}{\rho }\frac{{\partial \bar{p}}}{{\partial x}} - \frac{{{C_d}a}}{{2(1 - \phi )}}\bar{u}{({\bar{u}^2} + {\bar{v}^2})^{1/2}} - \frac{{{C_f}}}{{2h}}{ \bar{u}({\bar{u}^2} + {\bar{v}^2})^{1/2}},\end{equation}

where $\bar{p}$ is the time-mean pressure. Similarly, the diverted flow in the streamwise direction at either side of the array is controlled by

(3.2) \begin{equation}\bar{u}\frac{{\partial \bar{u}}}{{\partial x}} + \bar{v}\frac{{\partial \bar{u}}}{{\partial y}} =- \frac{1}{\rho }\frac{{\partial \bar{p}}}{{\partial x}} - \frac{{{C_f}}}{{2h}}{\bar{u}({\bar{u}^2} + {\bar{v}^2})^{1/2}}.\end{equation}

When the array flow blockage (${\varGamma _D}$) is low, it is normally assumed that the pressure gradient $(\partial \bar{p}/\partial x)$ approaches zero and hence the pressure terms in both (3.1) and (3.2) drop out; if the array flow blockage is high, the flow will become fully developed with the streamwise pressure gradient in the array equal to that in the diverted flow, and hence the pressure gradients in (3.1) and (3.2) are cancelled out (Rominger & Nepf Reference Rominger and Nepf2011). To first approximation, the fraction of flow passing through the array (bleeding flow velocity, ${\bar{u}_b}$) therefore depends on the ratio between the flow resistances in the array and in the diverted flow as

(3.3) \begin{equation}\frac{{{{\bar{u}}_b}}}{{{U_\infty }}} = f\left( {\frac{{ - \dfrac{{{C_f}}}{{2h}}{{\bar{u}({{\bar{u}}^2} + {{\bar{v}}^2})}^{\frac{1}{2}}}}}{{ - \dfrac{{{C_d}a}}{{2(1 - \phi )}}{{\bar{u}({{\bar{u}}^2} + {{\bar{v}}^2})}^{\frac{1}{2}}} - \dfrac{{{C_f}}}{{2h}}{{\bar{u}({{\bar{u}}^2} + {{\bar{v}}^2})}^{\frac{1}{2}}}}}} \right)\!.\end{equation}

Simplifying (3.3) gives

(3.4)\begin{equation}{f^{ - 1}}\left( {\frac{{{{\bar{u}}_b}}}{{{U_\infty }}}} \right) = \frac{{{S_C}}}{{{S_C} + {\varGamma _D}}}.\end{equation}

Rearranging (3.4) yields

(3.5)\begin{equation}{S_C} = \frac{{{f^{ - 1}}\left( {\dfrac{{{{\bar{u}}_b}}}{{{U_\infty }}}} \right)}}{{1 - {f^{ - 1}}\left( {\dfrac{{{{\bar{u}}_b}}}{{{U_\infty }}}} \right)}}{\varGamma _D}.\end{equation}

With a critical value of ${f^{ - 1}}({\bar{u}_b}/{U_\infty })$, the relationship in (3.5) for the bleeding flow can be mapped out into the S_C–${\varGamma _D}$ space, which sets a boundary line between the two adjacent regimes in the regime chart in figure 5. Clearly, for a constant value of ${f^{ - 1}}({\bar{u}_b}/{U_\infty })$, S_C is proportional to ${\varGamma _D}$ and hence the boundary line between each of the two adjacent regimes (e.g. CI, KH) should be a straight line in the log-scale regime chart. This is valid only when the wake stability parameter is low and the influence of bleeding flow dominates the wake transition. For a high value of S_C, the bed friction not only affects the bleeding flow velocity (as described in (3.5)) but also plays an important role in suppressing the instabilities in the far wake. This suppression effect of bed friction on the instabilities has been demonstrated for the single mixing layer (Chu et al. Reference Chu, Wu and Khayat1991) and the wake behind an isolated cylinder (Chen & Jirka Reference Chen and Jirka1995, Reference Chen and Jirka1997) in shallow water. This suppression effect will be demonstrated for an array of cylinders later in § 4.1.2. With this bed friction effect on wake instabilities, the boundary lines between adjacent regimes should deviate downwards from linear lines for a high value of S_C.

These scaling arguments are consistent with the experimental observations. Figure 5 shows that the boundary lines between different regimes are close to linear at ${S_C} \mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel< \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }0.1$, whereas boundary lines increasingly deviate downwards from linear lines to be horizontal with the increase of S_C for ${S_C} \mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel> \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }0.1$. This indicates that the wake transitions for ${S_C} \mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel< \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }0.1$ and ${S_C} \mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel> \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }0.1$ are primarily governed by the array-induced flow blockage and shallowness, respectively.

The regime chart in figure 5 is initially established by varying the water depth while keeping the array diameter virtually constant $(D \approx 42.5\ \textrm{cm)}$ since it is more effective to change S_C through varying h rather than D. This means that the observed wake transition along the S_C-axis of the regime chart reflects the influence of limited water depth rather than the complete range of S_C. To further demonstrate the controlling influence of S_C and ${\varGamma _D}$, here, the dependence of this regime chart on the array size $D/d$ is explored. In particular, the model array diameter was halved in the particular cases that fell near regime boundaries (symbols with black edge in figure 5), while maintaining the same values of S_C, ${\varGamma _D}$ and d (cases ‘72–77’ listed in table 3).

Flow visualisation shows that halving the array diameter has minimal impact on the wake structure. In total, six pairs of scenarios in the neighbourhood of regime boundaries were examined (see table 5). It is seen that flows with the same governing parameter set (${\varGamma _D}$, S_C) but with an approximately 50 % difference in D/d display the same wake form. This independence of wake structure on the dimensionless array size is also confirmed through measurement of the transverse profile of mean velocity in the wake. There is good agreement of transverse profiles of streamwise velocity $(\bar{u}/{U_\infty })$ at x/D = 1 between the large and small arrays with the same values of ${\varGamma _D}$ and S_C (see figure 6).

Table 5. Comparison of visualised wake regimes for pairs of cases with the same values of ${\varGamma _D}$ and S_C, but different size (D/d). Note that the cylinder diameters for all cases are d = 1 cm. In all pairs, there is no change of wake form when the array diameter is halved, confirming that the wake structure is primarily controlled by ${\varGamma _D}$ and S_C.

Figure 6. Comparison of wake profiles of $\bar{u}/{U_\infty }$ at x/D = 1 between the cases of large and small D/d values. The pair of cases in each panel have the same values of ${\varGamma _D}$ and S_C, but an approximately 50 % difference in D/d. There is negligible difference between the pair of velocity profiles in panels (a)–(d), demonstrating the independence of wake structure on the array size.

The independence of this regime chart on dimensionless array size is also confirmed by the consistency of wake regimes between different studies for the same values of $\varGamma_{D}$ and $S_{C}$. Table 6 summarises the broad range of dimensionless array size (D/d), Reynolds numbers defined based on the element diameter (Re_d), array diameter (Re_D) and water depth (Re_h) for independent studies. Despite large difference in values of D/d, Re_d, Re_D and Re_h, near the transition boundaries, the data points with close values of ${\varGamma _D}$, S_C lie in the same wake regimes (CI, KH, SS). For instance, the data points from the present study and those from existing work (Zong & Nepf Reference Zong and Nepf2012; Chang et al. Reference Chang, Constantinescu and Tsai2017; Wang et al. Reference Wang, Zhang, Liang and Gan2019) (marked with dashed ellipses in figure 5) with close values of ${\varGamma _D}$ and S_C all share the same wake regimes despite their D/d values varying by a factor of two. In summary, the similarity of wake structures between arrays with the same values of ${\varGamma _D}$ and S_C but different values of D/d (in the range $3.5 < D/d < 81.9$) demonstrates the controlling influence of flow blockage and wake stability parameters on wake structure, and thus the insensitivity of this regime chart to array size.

Table 6. Summary of array size (D/d), Reynolds number defined based on element diameter (Re_d), array diameter (Re_D) and water depth (Re_h) for different studies shown in figure 5. Note that the velocity upstream of the obstacle is used in the definitions of different Reynolds numbers. Despite the broad range of Re_d, Re_D and Re_h, the data points with the same value of ${\varGamma _D}$ and S_C but different D/d from these different studies lie in the same wake regimes in figure 5.

Provided that one knows the array geometry, and the depth and velocity of the incident flow (allows for calculation of ${\varGamma _D}$ and S_C), this regime chart can be used to predict wake structure behind a cylinder array across entire relevant practical parameter space.

3.1.3. Regime KH

Regimes VS, SS and CI in vertically unbounded flow are well established in the literature and demonstrated for the shallow flow in § 3.1.1. In comparison to this, the identification of the new regime KH is more novel and needs to be further established.

To start it is useful to demonstrate the consistency between the observed peak frequencies in the spectrum of transverse velocity and those predicted from linear stability analysis of a classical mixing layer (e.g. Ho & Huerre Reference Ho and Huerre1984). The analogy between an array shear layer and a mixing layer serves as the basis for this demonstration. The velocity profile in a mixing layer can be described by a hyperbolic tangent function:

(3.6)\begin{equation}\frac{{\bar{u} - \bar{U}}}{{\Delta U}} = \frac{1}{2}\tanh \frac{{y - \bar{y}}}{{2\theta }},\end{equation}

where $\bar{U}$ is the arithmetic mean of ${\bar{u}_1}$ and ${\bar{u}_2}$ (low- and high-stream velocities), $\Delta U = {\bar{u}_2}\unicode{x2013} {\bar{u}_1}$, $\bar{y}$ is the transverse location where $\bar{u} = \bar{U}$, and θ is the momentum thickness of the mixing layer, defined as

(3.7)\begin{equation}\theta = \int_{ - \infty }^\infty {\left[ {\frac{1}{4} - {{\left( {\frac{{\bar{u} - \bar{U}}}{{\Delta U}}} \right)}^2}} \right]\textrm{d}y} .\end{equation}

Velocity profiles in array shear layers within regime KH collapse to this hyperbolic tangent function (figure 7). As the shear layer profile remains self-similar, this collapse is independent of downstream location. The consistency between the experimental and analytical results indicates that the array shear layers are analogous to classical mixing layers.

Figure 7. Comparison of measured mean velocity profiles at x/D = 3 (all cases in regime KH) to the function presented in (3.6). Here, $\bar{y}$ occurs at $y/D \approx \mathrm{\ \pm 0}\textrm{.5}$ in all cases. The agreement between the measured velocity profiles and the hyperbolic tangent profile of a mixing layer is strong, highlighting the validity of the mixing layer analogy for the array shear layers.

In the linear local stability analysis, infinitesimal perturbations of the stream function can be defined in the form:

(3.8)\begin{equation}\psi (x,y,t) = \eta (\kern0.7pt y)\,{\textrm{e}^{ - \textrm{i}(\alpha x - \omega t)}} + \textrm{c}\textrm{.c}\textrm{.,}\end{equation}

where $\omega = 2{\rm \pi} f$ is the angular frequency, α is the wave number, c.c. represents the complex conjugate, and the eigenfunction of $\eta (\kern0.7pt y)$ can be obtained by solving the linear Rayleigh equation (Ho & Huerre Reference Ho and Huerre1984). It has been shown that the dimensionless spatial growth rate (−α_iθ, i subscript denotes imaginary part) of the perturbation peaks at $f_{KH}^\ast= {f_{KH}}\theta /\bar{U} = 0.032$ for the hyperbolic-tangent profile (3.6) (e.g. Monkewitz & Huerre Reference Monkewitz and Huerre1982). This ${f_{KH}}$ is often termed the natural frequency of the mixing layer.

The broadband peaks in the velocity spectra of regime KH all span $f_{KH}^\ast $ (figure 8). The arithmetic mean value of the lower and upper bounds of the observed KH frequency ($\kern0.7pt f_1^\ast $ and $f_2^\ast $ defined by chosen critical value of ${S_{vv}}/{({S_{vv}})_{max}}( = 0.2)$, dot-dashed line in figure 8) for these three cases is 0.036 ± 0.004, fairly consistent with $f_{KH}^\ast $. Similar agreement between the experimentally observed broadband peak and the natural KH frequency has been observed in the wake of a single solid cylinder with a permeable wake splitter plate (Cardell Reference Cardell1993) and the transverse shear flow adjacent to an infinitely long array of cylinders (White & Nepf Reference White and Nepf2007). This agreement supports the conclusion that the broadband peak observed in regime KH (figure 4i) is indicative of Kelvin–Helmholtz instability, which generates the coherent vortices seen in figure 4(g).

Figure 8. Collapse of spectrum of transverse velocity for three regime KH flows. The power spectral density is normalised by its maximum, and the frequency is normalised as ${f^\ast } = f\theta /\bar{U}$. All the spectra were sampled at y/D = 0.5 upstream of where two array shear layers merge. The horizontal dashed line represents the chosen critical value of S_vv/(S_vv)_max (= 0.2) to determine the upper and lower bounds $(f_1^\ast ,f_2^\ast )$ of the observed KH frequency. The observed broadband peaks always span the normalised natural frequency $(f_{KH}^\ast= 0.032)$, indicating the occurrence of KH instability in these three cases.

In regime KH, Kelvin–Helmholtz vortices appear to be the first vortex structures observed along the wake. Further downstream, the two array shear layers grow and eventually merge at the centreline (figure 4h), raising the question of whether a von Kármán vortex street forms downstream of the merging point. Dye streaks in regime KH did not manifest distinct regular oscillations as those in regime SS, indicative of the absence of von Kármán vortex street (see supplementary movies 3 and 4). To further confirm this, the spectrogram is evaluated at two locations: upstream (figure 9a,c) and downstream (figure 9b,d) of where the two array shear layers merge. The merging location is identified through dye flow visualisation and the increase in streamwise velocity along the wake centreline according to Zong & Nepf (Reference Zong and Nepf2012). Upstream of the merging location, energetic high frequencies (fD/U_∞ > 1), associated with small vortices shed from the array elements, are detected in regime SS (figure 9a). These are randomly distributed in time and at frequencies much higher than that predicted for the KH instability, i.e. f_KH (solid line in figure 9a). Hence, it appears that no KH instability develops within the array shear layers in regime SS. In contrast, lower dominant frequencies (fD/U_∞ < 0.6) are observed in regime KH (figure 9c), which are more concentrated in time and closer to the predicted value of KH instability (f_KH). If there was von Kármán vortex shedding, the instantaneous oscillation frequency and oscillation amplitude would be expected to remain constant with time (at the vortex shedding frequency) as observed in regime SS flow (see figure 9b). However, the primary frequency around fD/U_∞ = 0.2 (dashed line) for regime KH was intermittent over time, with much lower oscillation magnitude relative to regime SS (figure 9d). The intermittency of transverse velocity downstream of the merging point has been confirmed for other regime KH flows (e.g. cases B1, D1, D2, H2) with a quadrupled sample length (1200 s) (spectrogram omitted for simplicity). This is consistent with the observation of an intermittent wake oscillation for a cylinder array with $\varGamma_{D} \approx 2$ by Ball et al. (Reference Ball, Stansby, Allison and Strouhal1996). Therefore, both flow visualisation and wavelet analysis of velocity fluctuations show that in regime KH, the von Kármán vortex street is absent downstream of where the array shear layers merge and start to interact.

Figure 9. Variation of the velocity spectrum with time. The colour bar represents the amplitude of $v^{\prime}(t)$ with units of cm s$^{-1}$. The spectrograms are separated into two groups, i.e. (a,c) upstream and (b,d) downstream of the merging of array shear layers at the centreline. The velocities upstream and downstream of the merging point were sampled at y/D = 0.5, 0, respectively. x/D = (a) 5, (b) 8.5, (c) 1 and (d) 5. The solid lines in panels (a,c) denote the predicted values from linear stability theory $(\,{f_{KH}}D/{U_\infty })$ and the dashed lines in panels (a–d) represent the typical Strouhal number 0.2 of an isolated solid cylinder. Whilst the von Kármán vortex street is characterised by a constant frequency and oscillation amplitude downstream of the merging point in panel (b), the dominant frequencies associated with KH vortices are intermittent over time regardless of location in panels (c,d).

In the two-dimensional numerical simulations of flow through a circular array (He et al. Reference He, Draper, Ghisalberti, An, Branson, Cheng and Ren2022), it was found that the two array shear layers, containing these large-scale KH vortices, interact with each other to form a secondary vortex street further downstream of the merging point. This secondary vortex is not visible in figure 4(g) due to the limited wake length visualised in the shallow flume for the large array. To confirm the presence of secondary vortex street in shallow flow for regime KH, a small array with D/d = 24 is created, allowing to examine the wake structure with length up to 16D downstream of the array.

Figure 10(a) shows that the two array shear layers grow, driven by KH instability, to merge and interact at x/D ≈ 8, forming a secondary vortex street downstream. Although the secondary vortex street is similar to the von Kármán vortex street in regimes SS and VS in terms of the meandering form (comparing figure 10a with figures 4a,d), the generation of secondary vortex street is very irregular with time. This is evidenced by figure 10(b) where the large-scale oscillations of transverse velocity (such as an example at (x/D, y/D) = (8, 0)), indicating generation of the secondary vortex street, occur in the time range of $t{U_\infty }/D \approx 15\unicode{x2013} 75$ but cease in the range of 75–100.

Figure 10. Characteristics of secondary vortex street for case D3* with (${\varGamma _D}$, S_C) = (8.12, 0.06) in regime KH. (a) Instantaneous wake structure; (b) the time history of $v^{\prime}$ sampled at (x/D, y/D) = (8, 0) (located at the blue dot in panel a). Although the secondary vortex street in panel (a) is similar to the von Kármán vortex street in figure 4(a) in terms of meandering form, the secondary vortex street is generated irregularly over time as indicated by discontinuous oscillations in panel (b).

Physically, the secondary vortex street is a result of interaction between two rows of KH vortices at either side of the array. These KH vortices in each array shear layer are, however, generated irregularly over time, as indicated by irregularity of oscillations in time histories of $v^{\prime}/{U_\infty }$ upstream of the merging point (x/D = 2, 6, not shown). In the two-dimensional flow (He et al. Reference He, Draper, Ghisalberti, An, Branson, Cheng and Ren2022), the irregularly generated KH vortices will merge randomly within either array shear layer, before interacting with merged KH vortices in the other array shear layer. The irregularity of generation of KH vortices and the random amalgamation of KH vortices are expected to be the reason for the irregularity of the secondary vortex street in shallow flow.

The results in figures 9 and 10 combined with numerical results from He et al. (Reference He, Draper, Ghisalberti, An, Branson, Cheng and Ren2022) illustrate the fundamental distinction between regimes KH and SS, although the normalised dominant frequency in the two regimes both approach 0.2 in the far wake (see figure 9b,d). This finding may answer the question raised in previous studies (Zong & Nepf Reference Zong and Nepf2012; Chang & Constantinescu Reference Chang and Constantinescu2015) as to why a dominant frequency close to the typical Strouhal number of 0.2 can be detected despite the absence of a clear vortex shedding pattern when the two array shear layers merge and interact. The Strouhal number of ∼0.2 in regime KH corresponds to the secondary vortex street.

With these distinctive features for regime KH, the evidence for classifying the data points from existing work into regime KH in figure 5 is explained. First, despite the absence of a von Kármán vortex street, the array shear layers have been found to display large-scale undulatory motions (Chang & Constantinescu Reference Chang and Constantinescu2015) (denoted ‘∇’ in figure 5); second, the two array shear layers eventually merge with the formation of a meandering wake flow downstream instead of vortex shedding (Zong & Nepf Reference Zong and Nepf2012) (denoted ‘□’); third, a frequency $(fD/{U_\infty })$, detected in the array wake, is close to that expected for von Kármán vortex shedding $({\varGamma _D} \approx 0.2)$ (Zong & Nepf Reference Zong and Nepf2012; Chang & Constantinescu Reference Chang and Constantinescu2015). It appears that the large-scale undulatory motions and the characteristic frequency $(fD/{U_\infty } \approx 0.2)$are consistent with the KH instability, and a secondary vortex street may be responsible for the wake meandering downstream of merging point. In addition to these cases, there is one case (‘Δ’ near ${\varGamma _D} = 2.2$) from figure 10(c) of Nicolle & Eames (Reference Nicolle and Eames2011), for which two rows of coherent structures are formed without von Kármán vortex shedding and the two rows then interact downstream to develop a vortex street. It appears that this case, originally classified as regime CI by Nicolle & Eames (Reference Nicolle and Eames2011), is also consistent with regime KH. These cases from previous studies are therefore tentatively classified into regime KH (red hollow squares in figure 5).

Regime KH was also observed in the wake of other porous obstacle systems, for example, the wake of a porous plate (Huang & Keffer Reference Huang and Keffer1996) and a screen cylinder (Sun et al. Reference Sun, Azmi, Leontini, Zhu and Zhou2021). Fundamentally, the bleeding flow for the system of porous obstacles acts to suppress the interaction of array shear layers. Accordingly, the absolute instability (Huerre & Monkewitz Reference Huerre and Monkewitz1985), manifested by von Kármán vortex shedding, is suppressed. Following this, flow features associated with the KH instability (convective instability) become dominant in the wake.

3.2.1. Shear layer growth

As the wake structure is a result of the development of array shear layers (Zong & Nepf Reference Zong and Nepf2012), in this section, a comparison of the downstream growth of array shear layers in regimes SS, KH and CI is presented, which helps to reveal the difference in formation mechanism between the three regimes. Note that in regime VS, array shear layers have limited growth behind the array and are not considered here. The shear layer growth is quantified through transverse profiles of the normalised Reynolds shear stress $- \overline {u^{\prime}v^{\prime}} /U_\infty ^2$ (figure 11). Edges of the array shear layers where Reynolds shear stress is less than 20 % of its maximum in a profile at any streamwise location are marked with red dashed lines.

Figure 11. Comparison of the growth of array shear layers in (a) the SS regime (case F1), (b) the KH regime (case C1) and (c) the CI regime (case A1). The red dashed lines mark the edges of shear layers, which are defined by the transverse location where the Reynolds shear stress is less than 20 % of the maximum in the profile. Hashed region represents the steady wake region in the field of view. The shear layer growth rate decreases as the wake flow varies from regime SS to KH and to CI.

Focusing on regime SS first, it is evident that the array shear layer grows slowly close to the array, with width increasing from 0.30D at x/D = 1 to 0.37D at x/D = 3 (figure 11a). It then experiences rapid growth between 3 < x/D < 5, with the width almost doubling. As a result, the two shear layers on either side of the array merge at the centreline near x/D = 5, which is consistent with the flow visualisation (C_V contour) in figure 4(e). From x/D = 3 to 5, the shear layer growth coincides with an increase in the rate of lateral momentum transport across the shear layer, as indicated by an increase in the magnitude of $- \overline {u^{\prime}v^{\prime}} /U_\infty ^2$. This rapid growth is associated with the formation of the von Kármán vortex street, as evidenced by the non-zero $- \overline {u^{\prime}v^{\prime}} /U_\infty ^2$ next to the centreline, the rapid increase in the magnitude of $- \overline {u^{\prime}v^{\prime}} /U_\infty ^2$ along the centreline for 5 < x/D < 8.5 and vortex shedding starting at x/D ≈ 5 (see dye streaks in figure 4d).

In contrast to regime SS, the array shear layers in regime KH grow slowly along the wake (figure 11b), indicating that the KH vortices are less efficient at driving shear layer growth. This results in a delayed merging of the two array shear layers at x/D ≈ 8.5 in regime KH (compare figures 11a and 11b). Again, the linear growth of the shear layers, indicated by velocity measurements, is consistent with that shown by the C_V contours (figure 4h), which indicates merging at the same location (x/D ≈ 8.5).

Finally, the shear layer evolution for regime CI is different from both regimes SS and KH. The shear layer grows initially, with its width tripled from x/D = 1 to 7 (figure 11c). However, the shear layer stops growing further downstream, as shown by the virtually constant shear layer width at x/D > 7 (figure 11c). The maximum value of $- \overline {u^{\prime}v^{\prime}} /U_\infty ^2$ decreases steadily with x/D, suggesting that no instability develops within the array shear layers in regime CI. The magnitude of $- \overline {u^{\prime}v^{\prime}} /U_\infty ^2$ reduces in regime KH and CI compared with SS (figure 11a–c), indicating a decreasing trend in the rate of turbulent lateral momentum transport as flow varies from regime SS to CI. This is further indicative of increasingly stabilised wake flow.

Despite the differences, the shear layers preferentially grow towards the centreline rather than into the diverted flow in regimes SS and KH, as seen from the expansion of two edges in each flow (figure 11a,b). In other words, the shear layer tends to grow into the low-velocity stream rather than the high-velocity stream, which is consistent with plane mixing layers in both vertically unbounded (Champagne, Pao & Wygnanski Reference Champagne, Pao and Wygnanski1976) and shallow (van Prooijen & Uijttewaal Reference van Prooijen and Uijttewaal2002) flows.

The development of array shear layers as predicted by an existing analytical model is in good agreement with the measurements (see figure 12). The model used to predict the shear layer width $\delta (x)$ was developed based on integration of the shallow water equations both for high- and low-velocity sides of a mixing layer (van Prooijen & Uijttewaal Reference van Prooijen and Uijttewaal2002; Van Prooijen Reference van Prooijen2004) and can be expressed by

(3.9)\begin{equation}\frac{{\delta (x)}}{D} = \gamma \frac{{\Delta U}}{{\bar{U}}}\frac{1}{{{S_C}}}\left( {1 - \exp \left( {{S_C}\frac{x}{D}} \right)} \right) + \frac{{{\delta _0}}}{D},\end{equation}

where $\gamma $ is the entrainment coefficient for vertically unbounded flow with an empirical value of $\gamma = 0.085$, and ${\delta _0}$ is initial width of the mixing layer and is approximately ${\delta _0} = h$.

Figure 12. Agreement between the measured and modelled (by (3.9)) development of array shear layer for the cases A1 (regime CI) and C1 (regime KH) both at S_C = 0.04. Superscripts ‘m’ and ‘p’ represents measurements and predictions, respectively.

Figure 12 shows that with increasing flow blockage parameter, the measured shear layer grows faster. The measured widths of array shear layer agree with those predicted by (3.9), with the root mean squared error less than 3 % along the array shear layer for the two cases. This demonstrates the applicability of the analytical model developed based on one single mixing layer to the shear layers generated at either side of the array upstream of where the two shear layers interact.

3.2.2. Steady wake region

As discussed in § 1, the existence and extent of the steady wake region has practical implications for aquatic vegetated systems (e.g. Chen et al. Reference Chen, Ortiz, Zong and Nepf2012; Follett & Nepf Reference Follett and Nepf2012). An understanding of its occurrence and streamwise extent L is therefore practically critical and contributes to fully defining shallow wakes. Given that L = 0 in regime VS and L = ∞ in regime CI, the steady wake region length is only defined for regimes SS and KH.

Conventionally, L represents the distance along the centreline from the array to the point where the two array shear layers merge (Zong & Nepf Reference Zong and Nepf2012). There are two existing methods to quantify L (Zong & Nepf Reference Zong and Nepf2012). One is based on velocity measurements as the distance along the centreline from the array to the point where the streamwise velocity starts to increase. The second is based on dye flow visualisation as the distance from the array to the point where dye streaks emerging from either side of the array merge together at the centreline. Here, it is proposed that the first point of non-zero lateral turbulent momentum transport along the centreline also provides a robust description of steady wake region length.

The three techniques above agree reasonably with each other in the estimation of L. First, it is seen that the normalised Reynolds shear stress remains zero $( - \overline {u^{\prime}v^{\prime}} /U_\infty ^2 < 0.001)$ until x/D = 5.1 and then starts to increase rapidly (figure 13a). Note that the Reynolds shear stress profile was sampled along the line of y/D = 0.05, which was slightly shifted outwards from the centreline where the Reynolds shear stress is always zero due to its anti-symmetry about the centreline. Since the steady wake region is characterised by nearly zero Reynolds shear stress (hashed region in figure 11), the profile of $- \overline {u^{\prime}v^{\prime}} /U_\infty ^2$ gives L equal to 4.6D. Figure 13(b) shows that the streamwise velocity $\bar{u}/{U_\infty }$ first decreases slightly along the line of y/D = 0.05 and then remains at ${\sim} 0.05{U_\infty }$ until the formation of a recirculation bubble between 4.0 < x/D < 4.9; the extent of the recirculation bubble is defined as the region where $\bar{u}/{U_\infty } < 0$ (see the inset of figure 13b), following Zong & Nepf (Reference Zong and Nepf2012). Further downstream, the streamwise velocity continuously increases presumably due to the momentum transport to the centreline induced by the vortex street. The identification based on $\bar{u}/{U_\infty }$ gives L equal to 4.4D. Finally, the coefficient of colour variation (C_V) shows a similar variation with x/D to $- \overline {u^{\prime}v^{\prime}} /U_\infty ^2$ in figure 13(c). The C_V remains almost zero in the near wake and then increases sharply from x/D = 4.6, which gives L = 4.1D. Therefore, the L values estimated by the profile $- \overline {u^{\prime}v^{\prime}} /U_\infty ^2$, $\bar{u}/{U_\infty }$ and C_V along the wake centreline are reasonably consistent. Given that velocity measurements were conducted for only some cases in this study, estimation of L based on flow visualisation (C_V) is employed across the whole testing parameter space.

Figure 13. Comparison between the streamwise variations of (a) normalised Reynolds shear stress $- \overline {u^{\prime}v^{\prime}} /U_\infty ^2$, (b) normalised mean streamwise velocity $\bar{u}/{U_\infty }$ and (c) coefficient of variation C_V for a regime SS flow (case F1). All the data points in panels (a–c) were sampled along the line of y/D = 0.05. The end of the steady wake region is identified as the points where values of $- \overline {u^{\prime}v^{\prime}} /U_\infty ^2$, $\bar{u}/{U_\infty }$ and C_V start to increase in the streamwise direction.

The measured length L combined with observations from existing work indicate that the length generally increases with a decrease of ${\varGamma _D}$ and an increase of S_C (figure 14a). However, the L values of the present study (hexagons) are larger than those of previous studies for a given ${\varGamma _D}$ as the flows in this study are shallower (larger S_C) than others, highlighting the role of shallowness in modifying the wake structure. The measured L combined with that from previous work is generally decreased as the wake flow transits from regime KH (hashed region) to SS, which means that regimes KH and SS can be reasonably differentiated by L.

Figure 14. (a) Variation of the steady region length L as a function of ${\varGamma _D}$ and S_C. Hexagons, the present study; symbols denoting other studies refer to figure 5. Cases in regime KH are in the hashed region; (b) the comparison of measured values of L with the predictions by (3.10). Some cases in panel (a) are not shown in panel (b) due to lack of velocity measurements for these cases. The steady region length generally increases with increasing S_C in panel (a) and is largely underpredicted by (3.10) in panel (b), highlighting the role of shallowness in controlling shear layer growth.

The entrainment theory model developed by Zong & Nepf (Reference Zong and Nepf2012) predicts that the steady wake region has a length

(3.10)\begin{equation}L = \frac{{D/4}}{{{S_\delta }}}\left[ {\frac{{1 + \overline {{u_1}} /\overline {{u_2}} }}{{1 - \overline {{u_1}} /\overline {{u_2}} }}} \right]\!,\end{equation}

where S_δ is a constant empirical entrainment coefficient (0.098) in the absence of shallowness over a wide range of ${\bar{u}_1}/{\bar{u}_2}$ (Champagne et al. Reference Champagne, Pao and Wygnanski1976; Zong & Nepf Reference Zong and Nepf2012). It is seen here that (3.10) may underpredict L (figure 14b) due most likely to the influence of bed friction. In shallow flow, bed friction has been found to be capable of increasingly inhibiting the shear layer growth downstream (van Prooijen & Uijttewaal Reference van Prooijen and Uijttewaal2002), leading to delayed merging of the array shear layers and hence longer steady wake regions. This explains the discrepancy between the measurements and model predictions in figure 14(b).