3.1. The deformation of a cantilever plate under uniform loading

First, we verify the codes for pure deformation of a wall-mounted flexible stem with a rectangular cross-section under uniform loading. As modelled in figure 5, the cantilever plate is wall mounted under uniform loading. The geometric parameters for the plate are the length ( $l$ ) of 5 cm, the width ( $b$ ) of 1 cm and the thickness ( $h$ ) of 0.2 cm. The plate density is 0.678 g cm^– $^3$ , the elastic modulus is 5 $\times$ $10^5$ Pa and Poisson’s ratio is 0.4. The uniform loading varies in the range of $q=$ 0.0001–0.0008 N cm^–1. The plate is divided into elements, i.e. 10, 25 and 50. The corresponding grid in the vertical direction (along the length) is ${\rm d}x_s$ . For different situations, the time step is controlled to be smaller than the critical time step ${\rm d}t_s^0$ calculated using the equation of ${\rm d}t_s^0 = {\rm d}x_s/\sqrt{E/\rho_s}$ , where E is Young’s modulus, and $\rho_s$ is the plate density. The structural damping is set as $\zeta=$ 0.

Based on the deformation theory of a plate under uniform loading ( $q$ ), the final deformation curve follows the equation

(3.1) \begin{equation} v(x)=-\frac {q x^2}{24 E I}\left (x^2-4 x l+6 l^2\right ), \end{equation}

where $x$ is the distance of one point to the fixed end (positive in the $x$ direction), $v(x)$ is the final deformation of the plate and $I=bh^3/12$ is the cross-sectional inertial moment of the plate.

Figure 6. Comparison of the transverse shift of the tip with the theoretical value for a plate under uniform loading. Here, Case_n10q1 represents the case of the plate divided into 10 elements and under the loading of $q = 0.0001$ N cm^–1. The naming scheme is applied for other cases. For example, Case_n25q8 denotes the case of the plate divided into 25 elements and under the loading of $q = 0.0008$ N cm⁻¹.

Figure 7. ( $a$ ) The relative difference of the deformation at the free plate end versus the dimensions of the uniform loading, and ( $b$ ) the relative difference of the vibration frequency of the plate with different elements versus the uniform loading.

Figure 6 shows the comparison of the simulated results and theoretical results. It is seen that the model for the plate motion can accurately predict the deformation of the plate. Further, the number of elements in the simulation is also essential. The difference (in percentage) between the simulation and the theoretical value becomes higher as it gets closer to the free end of the plate, which is related to the deformation growing smaller. Thus, it requires a higher number of elements. As shown in figure 7, after the element number exceeds 25, the simulated deformation and vibration frequency match very well the theoretical results. The relative difference generally gets smaller when the loading is higher. Here, the relative differences for the deformation and vibration frequency are defined as $\Delta v_{tip}=(v_{tip}-v_{tip}^0 )/v_{tip}^0$ and $\Delta f_{1}=(f_{1}-f_{1}^0 )/f_{1}^0$ , respectively, where $v_{tip}$ is the simulated deformation, $v_{tip}^0$ is the theoretical solution, $f_{1}$ is the simulated vibration frequency and $f_{1}^0$ is the theoretical value.

3.2. Flexible plate in uniform channel flow

In this section, we validate the codes through a benchmark case of a wall-mounted flexible plate in uniform channel flow. Interactions between the fluid and plate are considered. Luhar & Nepf (Reference Luhar and Nepf2011) carried out a series of model experiments in the channel flow on a flexible plate, widely used to compare numerical or experimental results. To check the fidelity of our present numerical methodology, we compare our numerical results with those of Luhar & Nepf (Reference Luhar and Nepf2011).

In our present simulation, the setting of the computational domain is very similar to the water flume experiment in Luhar & Nepf (Reference Luhar and Nepf2011), see their figure 3. Based on this, the computational domain is 40 cm $\times$ 16 cm $\times$ 16 cm in length, width and height, respectively. The water depth in the channel is 16 cm. In the inlet, the incoming flow is a uniform velocity $U_\infty$ , while the velocity condition is applied in the outlet. For the top and lateral walls, free-slip boundaries are adopted. The bottom is a no-slip boundary and the wall function developed by Werner & Wengle (Reference Werner, Wengle, Durst, Friedrich, Launder, Schumann and Whitelaw1993) was used. The geometric parameters for the flexible plate are the length ( $l$ ) of 5 cm, width ( $b$ ) of 1 cm and thickness ( $h$ ) of 0.2 cm, respectively. The density of the flexible plate is 0.678 g cm^– $^3$ , the elastic modulus is $5 \times 10^5$ Pa and the Poisson ratio is 0.4.

Before the validation, we first carry out a convergence test. A uniform mesh is applied around the flexible plate, and a large region is adopted to cover the possible motion of the plate. Outside this region, the stretched mesh with the constant ratio (not larger than 1.02) is adopted. Therefore, the total meshes is affordable in the present simulations. Four uniform meshes are selected, i.e. $\Delta h/b=$ 0.2, 0.1, 0.0625 and 0.05. Here, $h$ is the minimum mesh. The dimension ( ${\rm d}x_s$ ) of the solid element is the same as the minimum mesh adopted. The fluid time step ( ${\rm d}t_f$ ) for each situation is smaller than the critical time step ( ${\rm d}t_f^0$ ) calculated through the equation below

(3.2) \begin{equation} \mathrm{d} t_f^0=\frac {\min \left ({\rm d} x_f, {\rm d} x_s\right )}{\sqrt {E / \rho _s}+2 U_{\infty }} ,\end{equation}

where ${\rm d}x_f$ is the fluid mesh, ${\rm d}x_s$ is the dimension of the solid element, $E$ is Young’s modulus, $\rho _s$ is the solid (or structure) density and $U_\infty$ is the incoming flow velocity. The Reynolds number is $Re_b=$ 1600 when the width ( $b$ ) of the plate is used as the characteristic length.

Figure 8( $a$ ) compares the simulation results and those ( $D_x/b=$ 2.14, $D_z/b=$ 0.59 and $C_d=$ 1.03) of model experiments in Luhar & Nepf (Reference Luhar and Nepf2011) under different mesh spacings. Here, $D_x$ is the deflection of the plate at the free end in the streamwise direction, $D_z$ is the height reduction of the plate at the free end and $C_d$ is the drag force because of the flow around the vegetation. It is seen that the relative difference between the present simulation and the experiments of Luhar & Nepf (Reference Luhar and Nepf2011) becomes lower as the mesh is finer. When the mesh is no larger than $\Delta h/b=$ 0.0625, the relative errors for $D_x/b$ and $C_d$ are lower than 1 $\, \%$ and smaller than 6 $\, \%$ for $D_z/b$ . Thus, our numerical results agree well with the experiments of Luhar & Nepf (Reference Luhar and Nepf2011), suggesting a high fidelity of the present numerical methodology. Further, the mesh with $h/b$ smaller than 0.0625 is an appropriate choice for our following simulations.

The incoming flow velocities, including $U_\infty=$ 3.6, 11.0, 16.0 and 27.0 cm s^–1, are simulated and compared with the experiments in Luhar & Nepf (Reference Luhar and Nepf2011) for further validation. The mesh with $\Delta h/b=$ 0.0625 is adopted. Figure 8( $b$ ) compares the drag force ( $F_d$ ) acting on the plate of the present simulation with both model predictions and experiment results. It is seen that the present simulation results are well matched with those in the experiments. This again indicates that our present methodology has high fidelity in simulating flexible plates in uniform channel flow.

Figure 8. ( $a$ ) The relative errors in hydrodynamic forces with different $b/h$ values and ( $b$ ) comparison of the drag force with prediction values and experimental results in Luhar & Nepf (Reference Luhar and Nepf2011).

3.3. Flexible stem under uniform loading

In this section, the deformation, including the tip position in the $x$ and $z$ directions and inclination angle of the chord line (i.e. between the tip and root) and horizontal line of a wall-mounted stem under uniform loading, is investigated and verified against the theoretical value, see figure 9( $a$ ). The aspect ratio is AR $= 10$ and the uniform loading $q$ is given according to the drag coefficient on the rigid stem. Figure 9(b–d) compares our simulation results with the theoretical value. It shows a good agreement for all three deformation quantities, thus suggesting a high fidelity.

As shown in figure 9( $b$ , $c$ ), the tip position in either direction is dominated by stem stiffness while it is independent of $Re$ . With increasing $K$ , $x_{tip}/l$ decreases while $h_{tip}/l$ increases gradually, corresponding to the state changing from upright to lodging. However, when lg( $K$ ) $\gt$ 0, $h_{tip}/l$ remains almost constant while $x_{tip}/l$ decreases approximately linearly when the log–log coordinate is used. When lg( $K$ ) $\lt$ –3, $h_{tip}/l$ follows the relationship $h_{tip}/l$ $\sim$ $K^{1/3}$ . This relationship was reported in Luhar & Nepf (Reference Luhar and Nepf2011).

As shown in figure 9( $d$ ), the inclination angle $\theta$ grows with $K$ , following the theoretical line. When $K$ is large, the stem maintains upright, thus $\theta$ $\approx$ 90 $^\circ$ ; while $K$ is extremely small, the stem maintains nearly parallel to the wall, $\theta$ approaches 0 $^\circ$ , indicating a lodging state.

Figure 9. ( $a$ ) Schematic of a wall-mounted flexible stem and deformation quantities, ( $b$ , $c$ ) the tip position in the $x$ and $z$ directions with $K$ , and ( $d$ ) the inclination angle $\theta$ with $K$ . The theoretical value is superimposed for comparison.

Figure 10. ( $a$ , $c$ ) Comparison of the time history of the inclination angle ( $\theta$ ) of the present study with Zhang et al. (Reference Zhang, He and Zhang2020), and ( $b$ , $d$ ) time history of the inclination angle ( $\theta$ ) at different grid spacings. ( $a$ , $b$ ) Are for the case of $Re = 400$ and $\gamma = 0.008$ , and ( $c$ , $d$ ) are for the case of $Re = 400$ and $\gamma = 0.004$ .

3.4. Dynamics of a flexible filament in uniform flow

In this section, we validate the dynamics of a flexible filament by comparing our simulation results with those in Zhang et al. (Reference Zhang, He and Zhang2020). The computation domain and parameter settings are the same as those in Zhang et al. (Reference Zhang, He and Zhang2020). Two cases are selected: one is at $\gamma = 0.008$ and the other is at $\gamma = 0.004$ , with $\beta$ and $Re$ being constant at 1.0 and 400, respectively. Here, $ Re = {U_{\infty } L}/{v}$ , $\beta = {\rho _s \delta_1 }/{\rho _f L}$ and $ \gamma ={B}/{\rho _f U_{\infty }^2 L^3}$ , where $B$ is the bending rigidity, $L$ is the length of the filament, $\rho _s$ is the filament density and $\delta_1$ is the thickness of the filament.

The time history of the inclination angle ( $\theta$ ) is plotted to check the consistency of the filament dynamics. Figure 10( $a$ , $c$ ) shows the comparison of our simulation results with those in Zhang et al. (Reference Zhang, He and Zhang2020). Note that the results in figure 10( $a$ , $c$ ) are using the mesh resolution of 1/8. It is seen that our present simulation agrees well with those in Zhang et al. (Reference Zhang, He and Zhang2020) for the two cases. However, differences are also observed. For the first case (i.e. $\gamma = 0.008$ ), the dominant frequency of the angular oscillation is slightly smaller than that in Zhang et al. (Reference Zhang, He and Zhang2020). For the second case (i.e. $\gamma = 0.004$ ), the filament is much closer to the wall and the dynamics of the filament is influenced by the mesh resolution near the wall. The slight difference is at the smaller peaks of the angular oscillation. The mesh convergence is also shown in figure 10( $b$ , $d$ ). For the first case, the angular oscillations for the meshes of 1/16 and 1/32 are well matched. This indicates that the mesh of 1/16 is appropriate for the present simulation. However, for the second case, the angular oscillation is convergent when the mesh is not coarser than 1/32, see figure 10( $d$ ). This means that the latter case requires a finer mesh. For a more flexible filament, its tip goes much closer to the wall, and accordingly, it requires a finer mesh to deal well with the collision process. We also check the influence of the non-dimensional time step (not shown here). To achieve a convergent result, in our present simulation significantly small time steps are adopted. The difference observed on halving the adopted time step is insignificant.

3.5. The non-dimensional time step for a wall-mounted stem in uniform flow

In this section, we check the influence of the non-dimensional time step in the simulation of a wall-mounted flexible stem. The aspect ratio is constant at AR $= 10$ and the Reynolds number is $Re = 400$ . As shown in table 1, by reducing the non-dimensional time step by half, the largest difference for $A_x$ is only 3.2 $\, \%$ .

Table 1. Comparison of the oscillation amplitudes in three directions and dominant frequency in the transverse direction at different non-dimensional time steps for a wall-mounted flexible stem at $Re = 400$ and lg( $K) = -0.5$ .

3.6. Other validation

Another validation, such as flow over a rigid circular cylinder in unbounded flow with the $Re$ up to 1000, has been provided in our previous work (Chen et al. Reference Chen, Ji, Alam and Yan2022). The mesh settings in the present study are approximately the same as those adopted in the direct numerical simulation. The convergence analysis of the grid spacing and non-dimensional time step has been conducted in our previous work.

5.1. Statistics of fluid forces and vortex shedding frequency

Figure 11 shows the vertically averaged fluid forces on the rigid stem at $Re = 200-1000$ . Here, the drag and lift coefficients are defined as $C_d=2F_d/\rho U_\infty ^2 Dl$ and $C_l=2F_l/\rho U_\infty ^2 Dl$ , respectively. As shown in figure 11( $a$ ), the mean drag coefficient ( $\bar {C}_d$ ) is 1.12 at $Re = 200$ and gradually decreases as $Re$ increases. The $\bar {C}_d$ values for a circular cylinder in an unbounded flow (represented by triangular symbols) are slightly higher than those for the wall-mounted case. This discrepancy arises because, in the wall-mounted case: (i) the oscillating shear layers near the junction are suppressed by the wall, and (ii) the free end delays flow separation, resulting in a less negative wake pressure (Luo, Gan & Chew Reference Luo, Gan and Chew1996). As discussed later, the downwash flow from the free end pushes the shear layers away from the stem base, contributing to the reduction in $\bar {C}_d$ .

Figure 12. Power spectral density (PSD) of the vertically averaged lift coefficient of a wall-mounted rigid stem within $Re = 200-1000$ . Panels show ( $a$ ) $Re = 200$ , ( $b$ ) $Re = 400$ , ( $c$ ) $Re = 600$ , ( $d$ ) $Re = 800$ and ( $e$ ) $Re = 1000$ . The inset in ( $b$ ) shows the spectrum of the lift coefficient of an infinite circular cylinder at $Re = 400$ .

The r.m.s. drag coefficient ( $C_d^{\prime }$ ), shown in figure 11( $b$ ), exhibits a different behaviour with $Re$ . At $Re = 200$ , $C_d^{\prime }$ is approximately zero (0.001). For higher $Re$ values, $C_d^{\prime }$ stabilises around 0.08, which is significantly higher than at $Re = 200$ . This suggests a fundamental difference in the vortical structures between $Re = 200$ and $Re$ $\gt$ 200. The $C_d^{\prime }$ values for an infinite circular cylinder at the corresponding $Re$ are also plotted in figure 11( $b$ ). It is seen that the $C_d^{\prime }$ values in triangular symbols are much higher than those for the wall-mounted case, implying that the vortical structures near the free end and junction supress shear layer fluctuations (Baban, So & Otugen Reference Baban, So and Otugen1989; Wang & Zhou Reference Wang and Zhou2009; Kumar & Tiwari Reference Kumar and Tiwari2019).

Due to the rotational symmetry of circular cross-section, the time-averaged lift coefficient ( $\bar {C}_l$ ) is zero and will not be discussed further. As shown in figure 11( $c$ ), the r.m.s. lift coefficient ( $C_l^{\prime }$ ) for the wall-mounted stem is relatively large at $Re = 200$ , with a value of 0.023. However, at higher $Re$ , $C_l^{\prime }$ stabilises at approximately 0.005. The transitions for $C_d^{\prime }$ and $C_l^{\prime }$ from $Re = 200$ to $Re$ $\gt$ 200 exhibit contrasting behaviours, highlighting the influence of different vortical structures in these two $Re$ clusters. These two $Re$ clusters will be analysed further to explore the roles of different flow structures in the transition. The $C_l^{\prime }$ values for the infinite circular cylinder, represented by triangular symbols in figure 11( $c$ ), are much higher due to the reduced fluctuations in the shear layers of the wall-mounted stem.

The vortex shedding frequency is a key indicator of flow physics and is significantly influenced by the vortex dynamics at both the free end and the junction (Liu, So & Cui Reference Liu, So and Cui2005; Yauwenas et al. Reference Yauwenas, Porteous, Moreau and Doolan2019). Figure 12 shows the spectra of the vertical-averaged lift coefficient at $Re = 200-1000$ , representing the vortex shedding frequency. As $Re$ increases, the spectral characteristics exhibit three distinct patterns. At $Re = 200$ , three frequencies are observed, but two have trivial magnitudes, as shown in figure 12( $a$ ). The dominant frequency is $f^\ast (= \textit{fD} / U_\infty) = 0.285$ . However, at $Re = 400$ , the dominant frequency decreases to $f^\ast = 0.182$ , significantly lower than at $Re = 200$ . Compared with the unbounded case, the dominant frequency is slightly lower, see the inset in figure 12( $b$ ). A secondary frequency, $f^\ast = 0.012$ , is approximately one order of magnitude lower than the dominant one. At $Re = 600$ , only one low frequency is detected at $f^\ast = 0.021$ , as shown in figure 12( $c$ ). For $Re = 800-1000$ , no dominant frequency is observed, as multiple low-frequency components below 0.03 exhibit comparable but weak magnitudes, see figure 12( $d$ , $e$ ).

Based on the features of hydrodynamic forces and vortex shedding frequencies, the investigated $Re$ can be categorised into three regimes. The first regime (R1) corresponds to $Re = 200$ , where vortex shedding is periodic, and correspondingly, a regular lift spectrum appears. The second regime (R2) occurs at $Re = 400$ , where both low and high frequencies appear due to different vortical structures. The third regime (R3) spans at $Re = 600-1000$ , where only low-frequency components are present. A slight distinction exists between $Re = 600$ and $Re = 800-1000$ in that the lift spectrum becomes chaotic at higher $Re$ . These variations are linked to the evolution of different 3-D vortical structures.

Figure 13. Three-dimensional vortical structures behind the wall-mounted rigid stem at ( $a$ ) $Re=$ 200, ( $b$ ) $Re=$ 400, ( $c$ ) $Re=$ 600, ( $d$ ) $Re = 800$ and ( $e$ ) $Re = 1000$ . The $Q$ -criterion is adopted to visualise the vortex structures (Hunt et al. 1988), i.e. $Q = 5$ , 20, 45, 80 and 125 for panels (a–e). The arrow indicates the flow direction (i.e. from left to right).

Figure 14. The vorticity field of a wall-mounted stem at the vertical position of $z/D=$ 0.1. (a–c) For a rigid stem and ( $d$ , $e$ ) for a flexible stem. Panels show ( $a$ ) $Re=$ 200, ( $b$ ) $Re = 400$ , ( $c$ ) $Re = 600$ , ( $d$ ) $Re=$ 200 and lg( $K) = -0.5$ and ( $e$ ) $Re = 600$ and lg( $K) = -2.0$ . The HV in each plot is marked; (i) and (ii) at different instants. The legs of the HV system, visualised by the $x$ -vorticity at $x/D = 0$ , are marked in the subplot.

5.2. Details of the 3-D vortical structures

The vortical structures in the studied $Re$ range are shown in figure 13. Since the AR exceeds the critical value (Bourgeois et al. Reference Bourgeois, Sattari and Martinuzzi2011, Reference Bourgeois, Sattari and Martinuzzi2012), the shear layer from the front side is reattached onto the free-end surface due to flow reversal (Pattenden et al. Reference Pattenden, Turnock and Zhang2005; Krajnovic Reference Krajnovic2011; Essel et al. Reference Essel, Tachie and Balachandar2021). The leading-edge vortex emerges at the free end surface, while near the bottom, horseshoe vortices form upstream of the stem, wrapping around its sides and trailing behind it (Hussein & Martinuzzi Reference Hussein and Martinuzzi1996; Simpson Reference Simpson2001; Wang & Zhou Reference Wang and Zhou2009; Chang et al. Reference Chang, Constantinescu and Tsai2017, Reference Chang, Constantinescu and Tsai2020) due to the adverse pressure gradient. These vortices tend to drive the flow downward inside its vortical trails. Figure 14 shows the horseshoe vortex (HV) system circling the stem base at a vertical position of $z/D = 0.1$ . Within the studied $Re$ , the HV system is characterised by a single necklace vortex, as marked in figure 14. In the current simulation, both $Re$ and $\delta /D$ increase simultaneously as the incoming flow velocity rises. As expected, the HV grows with increasing $Re$ , exhibiting greater strength and a longer extension into the wake. This suggests that the dynamics of the HV system is affected by $Re$ , the characteristics of the upstream boundary layer and the shape of the cylinder (Baker Reference Baker1979; Seal et al. Reference Seal, Smith, Akin and Rockwell1995). As shown in figure 14( $a$ ), the HV at $Re = 200$ remains nearly constant, with weak interactions with the two side shear layers and no noticeable amalgamation. However, at higher $Re\, (=400-600)$ , the HV legs extend further downstream, see figure 14( $b$ , $c$ ). At $Re=$ 400, the HV system encircles the two side shear layers, and their interactions intensify. In the wake, vortices shed from the HV system and the two side shear layers mix together, and amalgamation occurs. At $Re=$ 600, the interactions between the HV system, the shear layers on either side of the stem and the resulting vortices become more pronounced. The conditions at $Re= 800-1000$ are very similar to those at $Re=$ 600 and are therefore not shown. The HV system could be either laminar or turbulent. The laminar type is typically classified into five, transiting from steady to periodic by increasing $Re$ and/or $\delta /D$ (Greco Reference Greco1990; Seal et al. Reference Seal, Smith and Rockwell1997; Wei, Chen & Du Reference Wei, Chen and Du2001; Kirkil & Constantinescu Reference Kirkil and Constantinescu2012). In the present simulation, the HV system remains laminar, with a steady necklace vortex persisting at $Re= 200-1000$ due to the combined effects of lower $Re$ and a thin $\delta /D$ (Wei et al. Reference Wei, Chen and Du2001). In Kirkil & Constantinescu (Reference Kirkil and Constantinescu2012), where the boundary layer thickness is $\delta /D \cong 0.6$ , the laminar HV system transitions from steady to periodic oscillating and then to periodic breakaway as $Re$ increases. In the flexible case, the HV system also remains steady but is significantly influenced by the redistribution of vortical structures. As will be shown later, the flexible stem experiences stronger vortex shedding near the tip due to larger amplitudes and significant deformation. At $Re=$ 200, the HV system remains stable but its vortex legs extend a shorter distance downstream compared with the rigid stem, see figure 14( $d$ ). A similar behaviour is noticed at $Re=$ 600, see figure 14( $e$ ). It should be noted that as the stem becomes more flexible, the HV system is strongly affected because the two side shear layers (or separated vortices) move closer to the wall, influencing the HV system’s development. More details of the interactions between different vortex structures will be discussed later.

The evolution of 3-D vortical structures at $Re=$ 200 is shown in figure 13( $a$ ). Due to the high AR, the shear layers are allowed to develop and separate from the stem (Fröhlich and Rodi, 2004; Pattenden et al. Reference Pattenden, Turnock and Zhang2005; Tsutsui Reference Tsutsui2012). The spanwise vortices exhibit a slanted arrangement with an inclination angle of roughly 32.6 $^\circ$ (measured between the vertical axis and the slanted shear layer line) due to the influences of downwash flow from the free end (Lee Reference Lee1997; Fröhlich and Rodi, 2004). In the near wake, the spanwise vortices are significantly bent and stretched. Upwash and downwash flow entrain free-stream flow into the wake, pushing the spanwise vortices away from the centreline. Meanwhile, the streamwise vortices gradually develop. As a result of vortex twisting and interaction, the deformation of the spanwise and streamwise vortices becomes more evident further downstream. However, the interactions between these vortices weaken as they move apart. Compared with $Re = 200$ , the 3-D vortical structures at $Re = 400$ are more complex, featuring finer-scale vortices. Despite these differences, similarities remain, such as the presence of downwash flow, see figure 13( $b$ ). At this $Re$ , the spanwise vortices take over a longer length, but their formation region is larger near the wall. Near the free end, spanwise vortices are absent, while streamwise vortices develop more quickly due to increased flow velocity. As a result, vortex interactions intensify, leading to faster dissipation.

At $Re=$ 600–1000, the vortex dynamics is characterised by finer-scale vortices. As shown in figure 13(c–e), spanwise vortices separate from the stem simultaneously and remain approximately parallel. Compared with the lower $Re$ cases (200–400), the finer-scale spanwise vortices exhibit more irregular behaviour in the near wake, as indicated by significantly low dominant frequencies. Similarly, the streamwise vortices are smaller in scale and mix with the spanwise vortices (Zhang et al. Reference Zhang, Cheng, An and Draper2021), leading to faster dissipation. Although the downwash flow persists, it is considerably weaker, as indicated by the formation of spanwise vortices closer to the free end.

Figure 15 shows the $z$ -vorticity contours at different vertical positions. It is evident that vortex shedding varies significantly along the length within the studied $Re$ range. At $Re=$ 200, vortex shedding is regular near the junction. As shown in figure 15( $a$ -i), the vortex shedding in this region (i.e. $z/D=$ 0.5) is characterised by the formation of horseshoe and hairpin vortices in the wake. The hairpin vortex comes from disturbances in the stretched vortices affecting the boundary layer over the stationary wall, a behaviour similar to the bypass transition observed over a circular cylinder (He et al. Reference He, Wang, Pan, Feng, Gao and Rinoshika2017). The shear layers on both sides extend further downstream, and the separated vortices gradually dissipate in the far wake. At the middle plane (i.e. $z/D = 5.0$ ), vortices shed alternately from both sides of the stem. In the near wake, they appear nearly identical, apart from their opposite rotation directions, see figure 15( $a$ -ii). Further downstream, these vortices become increasingly irregular due to the bending and twisting of spanwise vortices and the emergence of stretched streamwise vortices, leading to a broader wake. Higher up, i.e. $z/D=$ 7.5, the shear layers weaken, and fluctuations of shear layers occur farther downstream, with dissipation occurring more rapidly than at the middle plane, see figure 15( $a$ -iii). Near the free end, i.e. $z/D = 9.5$ , the shear layers on both sides remain stable, with no visible vortices in the wake, forming a symmetric wake, see figure 15( $a$ -iv).

At $Re=$ 400, vortex strength increases compared with $Re=$ 200. As shown in figure 15( $b$ -i), the HV near the junction is more intense and extends farther downstream. Compared with $Re=$ 200, the shear layers on both sides of the stem extend further downstream. The hairpin vortex appears in the near wake but exhibits greater irregularity. Interaction between the hairpin vortex and spanwise vortices occurs where the spanwise vortices separate. At the midspan (i.e. $z/D=$ 5.0), the vortex shedding pattern is similar to that at $Re=$ 200, although the vortices are slightly smaller, see figure 15( $b$ -ii). In the near wake, vortices become finer and more irregular before the wake widens. At $z/D=$ 7.5, weak fluctuations in the shear layers are observed, see figure 15( $b$ -iii). Further downstream, the shear layers thin out due to dissipation. Near the free end, i.e. $z/D = 9.5$ , the shear layers remain stable, with no vortices shed into the wake, see figure 15( $b$ -iv). Compared with $Re = 200$ , the downwash flow at $Re = 400$ is stronger but extends over a shorter length.

As shown in figure 15(c–e), the vortex shedding patterns at the same vertical position remain similar for $Re = 600-1000$ . As $Re$ increases, vortices become stronger but more irregular. As shown in figure 15( $c$ -i, $d$ -i, $e$ -i), the HV intensifies, and its legs extend further downstream. Vortex shedding occurs at a relatively shorter downstream position with increasing $Re$ . The formed vortices subsequently interact with the HV. Further downstream, the vortices become finer and dissipate more quickly, leading to a wider wake. At midspan, vortex formation shifts further downstream compared with the low $Re$ case, see figure 15( $c$ -ii, $d$ -ii, $e$ -ii). At higher vertical position, i.e. $z/D = 7.5$ , the wake narrows significantly. Near the free end, the shear layers are slightly unstable with small vortices formed in the wake for Re = 800–1000, see figure 15( $c$ -iv, $d$ -iv, $e$ -iv).

Figure 15. The vorticity contours ( $\omega _z^*=({D}/{2 U_{\infty })}(({\partial v}/{\partial x})- ({\partial u}/{\partial y})$ ) of a single stem at several vertical positions. Panels show ( $a$ ) $Re = 200$ , ( $b$ ) $Re = 400$ , ( $c$ ) $Re = 600$ , ( $d$ ) $Re = 800$ and ( $e$ ) $Re = 1000$ .

5.3. The downwash and upwash flow

In §§ 5.1 and 5.2 , we discussed the effects of downwash and upwash flow on the vortex dynamics. Figure 16 shows side views of these flows. For each $Re$ , the separated flow from the free end enters the near wake at a nearly constant angle relative to the streamwise direction (Krajnovic Reference Krajnovic2011). As shown in figure 16( $a$ ), at $Re = 200$ , the flow from the free end moves downward directly along the bottom (i.e. $x/D = 1.0$ ), indicating that the downwash flow initiates immediately behind the stem but weakens gradually. Beyond $x/D$ $\geqslant$ 4.0, the downwash flow becomes nearly negligible.

Figure 16. Instantaneous vertical velocity ( $w^{\ast }$ ) at the $y$ ( $=8D$ ) plane for $Re= 200{-}600$ . Panels show ( $a$ ) $Re = 200$ , ( $b$ ) $Re=$ 400, ( $c$ ) $Re=$ 600 and ( $d$ ) $Re=$ 1000. Flow is from left to right. The rectangle in each plot indicates the stem position.

At $Re = 400$ , the downwash flow extends slightly farther downstream. As shown in figure 16( $b$ ), downwash begins just past the free end but primarily affects the lower part of the stem at $x/D = 1.0$ . A slight upward flow is observed at the upper part, indicating the presence of upwash flow. The downwash flow weakens quickly and disappears around $x/D = 8.0.$

At $Re = 600$ , the downwash flow behaves similarly, as shown in figure 16( $c$ ). Behind the stem, an upward flow is observed, pushing the downwash flow further downstream. Between $x/D = 4.0$ and 8.0, the downwash flow weakens but spreads toward the wall. In contrast, at $Re = 800-1000$ , the downwash flow reaches the wall, whereas at $Re = 600$ , it does not, see figure 16( $c$ , $d$ ).

Upwash flow appears at $Re = 400{-}1000$ , where it coexists with downwash flow. At $Re = 600$ , the upwash flow originates from the junction and is associated with the development of the near-wake base vortex (Etzold & Fiedler Reference Etzold and Fiedler1976; Essel et al. Reference Essel, Tachie and Balachandar2021). As indicated in figure 1, the base vortex consists of a pair of streamwise counter-rotating vortices near the base of the stem, influenced by $Re$ and $\delta /D$ (Sumner et al. Reference Sumner, Heseltine and Dansereau2004; Wang et al. Reference Wang, Zhou, Chan and Lam2006; Wang & Zhou Reference Wang and Zhou2009). As shown in figure 16( $c$ , $d$ ), the upwash flow intensifies with increasing $Re$ . However, at $Re = 400$ , the upwash flow is only present in the upper region (see figure 16 $b$ ), exhibiting a different behaviour from higher $Re$ cases.

5.4. Three-dimensional wake instability comparison with the infinite cylinder case

In this section, we compare the 3-D wake instability of a wall-mounted cylinder with that of an infinite cylinder to deepen our understanding of the effects of the impermeable wall and free end. For flow past an infinite circular cylinder, the wake exhibits distinct patterns due to the presence of inherent 3-D wake instabilities (i.e. modes A and B) as $Re$ increases from approximately 190 to 300 (Williamson Reference Williamson1996; Jiang et al. Reference Jiang, Cheng, Draper, An and Tong2016). The transition from mode A to mode B has been well documented. Notably, mode A remains stable only for a short period before transitioning into a more stable pattern, mode A^*, which exhibits significant dislocation. At $Re$ $\leqslant$ 220, mode A^* dominates, while modes A^* and B coexist in the range of 230 $\leqslant$ $Re$ $\leqslant$ 265. When $Re$ $\geqslant$ 270, mode A^* disappears, and mode B becomes the dominant instability. Modes A^* and B exhibit distinct characteristics. Mode A^*, induced by an elliptic instability of the primary vortex cores, features streamwise vortices that are out of phase between neighbouring braids. Due to spanwise waviness, the primary vortex cores become significantly stretched. In contrast, mode B, caused by the hyperbolic instability of the braid shear layer, exhibits an in-phase pattern. Compared with mode A, the primary vortices in mode B are more stable, with no noticeable spanwise waviness.

As discussed in § 5.2, the vortex dynamics of a wall-mounted cylinder is significantly altered by the presence of the impermeable wall and free end. A key feature of this configuration is the suppression of vortex formation near both the cylinder base and the free end. Consequently, the separation of the spanwise shear layers typically occurs in a slanted manner, see figure 13( $a$ , $b$ ). Compared with the infinite cylinder case, the wake transition of the wall-mounted cylinder is influenced by additional factors, including $Re$ , AR and $\delta /D$ . Kumar & Tiwari (Reference Kumar and Tiwari2019) found that, at AR $=5$ , no vortex shedding occurs when $Re$ $\leqslant$ 100, indicating a symmetric wake structure. When $Re$ $\geqslant$ 150, spanwise vortex shedding emerges, with vortices transitioning into hairpin-like structures. The critical $Re$ for the onset of vortex shedding increases as AR decreases (Saha Reference Saha2013). It is expected that if the wall-mounted cylinder has a much larger AR, modes A^* (or A) and B would likely emerge. However, for a short cylinder, the wake structures are primarily governed by downwash and upwash flows (Zhu et al. Reference Zhu, Wang, Wang and Wang2017), resulting in a distinctly different flow behaviour.

In the present simulation, we observe two patterns of 3-D wake instabilities. As shown in figure 13( $a$ ), the streamwise vortices primarily result from the suppression of shear layer separation. In the near wake, the separated vortices undergo significant stretching before transitioning into the hairpin-like structures. At higher $Re\, (=400)$ , the formation of streamwise vortices resembles mode A^*. As shown in figure 13( $b$ ), streamwise vortices appear after the spanwise vortices separate from the shear layers, exhibiting large dislocations. The emergence of mode-A^*-like instability is closely associated with slanted vortex shedding. However, in the near wake, the formed streamwise vortices undergo significant stretching, preventing the mode-A^*-like vortices from persisting farther downstream. Additionally, mode B, which typically appears in an in-phase pattern, is absent in the wall-mounted case. This absence may be attributed to the slanted nature of vortex shedding.

It is important to note that the 3-D wake instabilities observed in this study apply specifically to the simulated conditions. Further investigation in needed to explore these phenomena in great detail.

6.1. Characteristics of shape deformation and stem response

The mean position of the stem tip in different directions serves as a crucial indicator of hydrodynamic loading. Figure 17 shows the mean position, vibration amplitude and frequency of the stem tip in different directions. The mean position shifts are defined as $\Delta x^{\ast }=(\bar {x}_t - \bar {x}_{t,0})/D$ , $\Delta y^{\ast }=(\bar {y}_t- \bar {y}_{t,0})/D$ and $\Delta z^{\ast }=(\bar {l}_t- \bar {l}_{t,0})/D$ , where the subscript $0$ denotes initial values. Due to the circular cross-section’s rotational symmetry, transverse shifts remain zero in most simulations and are not further discussed. As shown in figure 17( $a$ ), for $Re=$ 200–1000, the streamwise shift ( $\Delta x^{\ast }$ ) follows a consistent trend. Excluding divergent cases, the best-fit nonlinear equation is $\Delta x^*=9.4 / (1+e^{1.96(\lg (K)+1.88)} )$ with a correlation coefficient ( $R^2$ $\gt$ 0.99), see figure 17( $a$ ). This suggests that the streamwise deformation is independent of $Re$ . Similarly, the vertical shift ( $\Delta z^{\ast }$ ) follows a similar equation: $\Delta z^*=8.1 / (1+e^{2.42(\lg (K)+2.54)} )$ , see figure 17( $b$ ). Luhar & Nepf (Reference Luhar and Nepf2011) proposed a scaling law for the effective length of the blade under uniform flow. The effective length could be scaled as $l_e/l\sim K^{1/3}$ . Here, the effective length ( $l_e$ ) is defined as the length of a rigid vertical blade that generates the same drag as the flexible blade of total length ( $l$ ). In the present study, the deflected length ( $l_d$ ) is scaled as $l_d/l\sim K^{1.05}$ . The exponent is higher than that in Luhar & Nepf (Reference Luhar and Nepf2011). We assume the difference is likely caused by three factors: (i) the stem in the present study is circular cross-sectional, (ii) the consistently higher deflected height compared with effective height (Luhar & Nepf Reference Luhar and Nepf2011; Luhar Reference Luhar2012) and (iii) for a circular stem, the transverse vibration leads to stronger spanwise vortices, contributing to higher mean drag (Sarpkaya Reference Sarpkaya2004).

Figure 17. Statistics of the stem response with different stiffness ratios at $Re=$ 200–1000. ( $a$ , $b$ ) The mean position shift, ( $c$ ) the transverse amplitude and ( $d$ ) dominant frequency of the transverse displacement. The solid lines in ( $a$ ) and ( $b$ ) denote the fitted equations and the correlation coefficients ( $R^2$ ) are both 0.997. The solid lines in ( $d$ ) represent the first natural frequency ( $f_{n,1}$ ) at different $Re$ values. The arrow indicates the increase of $Re$ from 200 to 1000.

Despite significant deformation in streamwise and vertical directions, vibrations in these directions remain negligible. No further discussion on them is presented. The transverse amplitude is defined as $A^* = \sqrt {2} y_{rms} / D$ , where $y_{rms}$ is the r.m.s. transverse displacement. Figure 17( $c$ ) shows that the transverse response does not exhibit a clear dependence on lg( $K$ ), suggesting that vortex shedding is not persistently locked to the natural frequency. The $i^{}$ th natural frequency of the flexible stem is given by

(6.1) \begin{equation} f_{n,i}=\frac {k_i^2}{2 \pi } \sqrt {\frac {E I}{m+m_a}}, \end{equation}

where $i$ is the mode number, $m$ is the structure mass, $m_a$ is the added mass of the fluid defined as $m_a = \pi D^2 \rho _f l /4$ and $k_i$ is the dimensionless frequency parameter of the $i^{}$ th mode. The first and second modal parameters are $k_1=$ 1.875 and $k_2=$ 4.694 (Blevins Reference Blevins2016). For conciseness, only the first natural frequency (i.e. $f_{n,1}$ ) is shown in figure 17( $d$ ). At specific lg( $K$ ) values, the dominant frequency aligns with $f_{n,1}$ , with locking to $f_{n,2}$ observed at $Re=$ 200 (not shown).

The highest amplitude occurs at lg( $K$ ) = −0.5. At the lowest lg( $K$ ), transverse amplitude is tiny as the stem yields more readily to the incoming flow, exhibiting ‘high flexibility-induced rigidity’ (He, Liu & Shen Reference He, Liu and Shen2022). The dominant frequency (i.e. $f^*$ ) of transverse displacement is shown in figure 17( $d$ ). At $Re=$ 200, the dominant frequency grows roughly with lg( $K$ ). However, for $Re=$ 400–1000, dominant frequencies are irregular, mostly clustering around $f^*=$ 0.1. As discussed in § 5, more complex vortical structures at the higher $Re$ influence displacement. Unlike an infinite cylinder experiencing VIV, where $f^*$ $\gt$ 0.2 for $Re$ $\gt$ 200 (Wang, Xiao & Incecik Reference Wang, Xiao and Incecik2017; Gsell, Bourguet & Braza Reference Gsell, Bourguet and Braza2019; Yu et al. Reference Yu, Wang, Bao, Xiao, Li, Incecik and Lin2024), vortex shedding in a wall-mounted flexible cylinder is significantly slower due to junction-induced vortex retardation.

Figure 18. The regimes of the response are defined based on ( $a$ ) the transverse amplitude and ( $b$ ) the spectral features of the displacement. Here, the minor vibration is defined as $A^*$ $\lt$ 0.2, medium vibration is defined as 0.2 $\leqslant$ $A^*$ $\leqslant$ 0.6 and substantial vibration is defined as $A^* \gt$ 0.6. The phase portrait of the transverse displacement is applied to identify the types.

Figure 19. Time histories (left column), power spectral density (middle column) and phase portraits of the transverse displacement (right column) at the stem tip at different $Re$ and lg( $K$ ). Panels show ( $a$ ) $Re=$ 200 and lg( $K$ ) = 0, ( $b$ ) $Re=$ 800 and lg( $K$ ) = –1.5, ( $c$ ) $Re=$ 400 and lg( $K$ ) = –3.0, ( $d$ ) $Re=$ 600 and lg( $K$ ) = –0.5, ( $e$ ) $Re=$ 600 and lg( $K$ ) = –2.5 and ( $f$ ) $Re=$ 600 and lg( $K$ ) = –3.5. Note that the discrete points are used in the phase portraits.

Based on transverse amplitude and displacement characteristics, responses are categorised into three types, see figure 18. Based on the amplitude, the three types are minor vibration (i.e. $A^*$ $\lt$ 0.2), medium vibration (i.e. 0.2 $\leqslant$ $A^*$ $\leqslant$ 0.6) and substantial vibration (i.e. $A^* \gt$ 0.6), see figure 18( $a$ ). Minor vibration primarily occurs in three regions: lg( $K$ ) = 0, lg( $K$ ) = –1.0–1.5, and lg( $K$ ) = –2.5–3.5 (for $Re=$ 200 only). In the first region, the stem is relatively rigid, with small transverse amplitude, and vortex shedding dominates vibration ( $f^*=$ 0.205), see figure 19( $a$ -i). With the same $K$ but at a higher $Re$ , stronger vortex shedding increases the hydrodynamic loads, slightly amplifying the vibration while still remaining in the minor vibration category. In the second region, the vortex shedding frequency deviates from the stem’s first natural frequency. As shown in figure 19( $b$ -i, $b$ -ii), the dominant frequency ( $f^*=$ 0.047) in this region is much lower and significantly differs from the natural frequency. Unlike the first region, multiple comparable low frequencies are observed, even though the transverse amplitude is higher. The third region occurs at lg( $K$ ) $\lt$ –2.0 and $Re=$ 200, where the stem approaches a lodging state, and the vortical structures remain regular, as discussed in § 6.3.

Medium vibration mainly appears at lg( $K$ ) $\leqslant$ –2.0, see figure 18( $a$ ). As shown in figure 19( $c$ -i), although the transverse amplitude is slightly above $A^*=$ 0.2, the displacement remains regular. The dominant frequency is $f^*=$ 0.115, with a magnitude significantly higher than those of other frequency components, see figure 19( $c$ -ii). Due to substantial deformation, the vortex shedding frequency in this case is lower.

Substantial vibration occurs in two regions: lg( $K$ ) = –0.5–1.0 with $Re=$ 200–1000 and g( $K$ ) = –2.0–3.0 with $Re=$ 600–800, see figure 18( $a$ ). In the first region, the transverse amplitude is larger than in the second region, which may indicator better synchronisation between stem vibration and vortex shedding. In the first region, the displacement is periodic, and the stem vibration is dominated by a single frequency ( $f^*=$ 0.137), see figure 19( $d$ -i, $d$ -ii). In contrast, in the second region, the displacement is chaotic, and multiple low-frequency components dominate stem vibration, see figure 19( $e$ -i, $e$ -ii). The dominant frequency is $f^*=$ 0.04, which is significantly lower than the stem’s natural frequency.

The phase portrait provides a geometric representation of a dynamical system’s trajectories in the phase plane and is used to confirm the system dynamics (Jordan & Smith Reference Jordan and Smith2007). In this study, phase portraits of displacement are employed to illustrate the system dynamics (Chang & Modarres-Sadeghi Reference Chang and Modarres-Sadeghi2014). Based on the characteristics of transverse displacement portraits, the response is classified into three categories, i.e. periodic, regular (quasiperiodic) and chaotic. The distribution of these categories is shown in figure 18( $b$ ). The spectral content of the displacement serves as a supplementary indicator. Periodic vibration is characterised by a dominant frequency with a significantly higher magnitude than other frequencies, see figure 19( $d$ -ii, $d$ -iii). Synchronisation between vortex shedding and stem vibration is observed. Comparing figures 18( $a$ ) and 18( $b$ ), the first periodic region coincides with the first substantial vibration region, suggesting that higher amplitudes facilitate periodic vortex shedding. The second periodic region appears at lg( $K$ ) = –2.0–3.0 but only for $Re=$ 200. In this region, the transverse amplitude is moderate, and vortex shedding remains periodic. Beyond $Re=$ 200 finer vortices emerge, and low-frequency components appear. A more flexible stem is more susceptible to turbulent fluctuation, making periodic vibration nearly impossible when lg( $K$ ) $\lt$ –1.0.

Regular vibration is observed in three regions, see figure 18( $b$ ). The first region occurs at lg( $K$ ) = 0 with $Re=$ 200–1000, where the stem is relatively rigid, and its vibration is regulated by vortex shedding, maintaining a steady rhythm, see figure 19( $a$ -ii). The phase portrait in figure 19( $a$ -iii) indicates that stem vibration follows a nearly constant trajectory. The second region, at lg( $K$ ) = –1.0–2.0 with $Re=$ 600–1000, exhibits transverse amplitude ranging from minor to substantial. As shown in figure 19( $c$ -ii), the dominant frequency is $f^*=$ 0.115, while other frequency components are insignificant in magnitude. The phase portrait in figure 19( $c$ -iii) is highly regular. The third region, at lg( $K$ ) = –3.0–3.5 with $Re=$ 400–800, shows an intermediate stem amplitude with a single frequency, $f^*=$ 0.129, see figure 19( $f$ -ii).

The chaotic regime spans lg( $K) = -1.0-3.5$ and $Re = 200-1000$ , see figure 18( $b$ ). A comparison of figures 18( $a$ ) and 18( $b$ ) reveals that chaotic vibration can occur regardless of whether the transverse amplitude is small, medium or substantial. At moderate lg( $K)\, (= -1.0-1.5)$ , the transverse amplitude is small, but multiple low frequencies appear, see figure 19( $b$ -i, $b$ -ii). The phase portrait in figure 19( $b$ -iii) further confirms this chaotic behaviour. At lower lg( $K)\, (= -2.0-3.5)$ , the transverse amplitude may be medium or substantial, but stem vibration is strongly influenced by turbulent structures, leading to chaotic behaviour, see figure 19( $e$ -i, $e$ -ii). The dominant frequency ( $f^* = 0.04$ ) is significantly low, with several comparable low-frequency component. The phase portrait in figure 19( $e$ -iii) indicates no fixed trajectory for the stem to follow.

6.2. Hydrodynamic forces and comparison with the rigid stem

The hydrodynamic forces on the flexible stem are key factors influencing stem oscillation and shape deformation. Figure 20 shows the variation of hydrodynamic forces on the flexible stem as a function of $Re$ and lg( $K$ ). For comparison, results of a rigid stem are superimposed. As shown in figure 20( $a$ ), the mean drag coefficient ( $\bar {C}_d$ ) generally decreases as lg( $K$ ) decreases, indicating that $\bar {C}_d$ is primarily affected by stem deformation. As mentioned in § 6.1, when stiffness is significantly low, the deflected height of the stem decreases, resulting in a more streamlined shape and a corresponding reduction in $\bar {C}_d$ drops. For the two cases with the highest stiffnesses, i.e. lg( $K$ ) = 0 and −0.5, the $\bar {C}_d$ of the flexible stem is higher than that of the rigid stem. In these cases, shape deformation is minimal, but stem oscillation develops, particularly for lg( $K$ ) = –0.5. This suggests that transverse oscillation has a significant impact on $\bar {C}_d$ when the stem is relatively rigid.

The r.m.s. drag and lift coefficients vary significantly with lg( $K$ ). As shown in figure 20( $b$ ), the r.m.s. drag coefficient ( $C_d^{\prime }$ ) increases gradually with increasing lg( $K$ ). However, it is slightly lower for the cases with lg( $K$ ) = 0 and −1 due to weak transverse oscillation. For a flexible stem with identical deformation, stronger transverse oscillation enhances vortex shedding, leading to higher $C_d^{\prime }$ . As shown in figure 20( $c$ ), in most cases, the r.m.s. lift coefficient ( $C_l^{\prime }$ ) on the flexible stem is comparable to that of the rigid stem. However, it is significantly higher for lg( $K$ ) = 0, −0.5 and −2.0, where transverse amplitude is larger. This suggests that $C_l^{\prime }$ is strongly influenced by transverse oscillation.

Figure 20. Hydrodynamic forces on a wall-mounted flexible stem with lg( $K$ ) at $Re=$ 200–1000. ( $a$ ) Mean drag ( $\bar {C}_d$ ), ( $b$ ) r.m.s. drag ( $C_d^{\prime }$ ) and ( $c$ ) r.m.s. lift ( $C_l^{\prime }$ ). In each plot, the horizontal lines (with the same colours as the flexible case) denote the corresponding forces on the rigid stem.

6.3. Characteristics of the 3-D vortical structures at different regimes

The 3-D vortical structures provide valuable insights into the physics of different responses and hydrodynamic loads acting on the flexible stem. Figure 21 shows the detailed 3-D vortical structures at $Re=$ 200 with lg( $K$ ) = 0–3.5. The vortex dynamics of the flexible stem, including vortex arrangement and shedding behaviour, differs significantly from the rigid case. At the highest lg( $K$ ), the stem experiences minimal deformation and behaves similarly to a rigid stem (Luhar & Nepf Reference Luhar and Nepf2011, Reference Luhar and Nepf2016; Jacobsen et al. Reference Jacobsen, Bakker, Uijttewaal and Uittenbogaard2019). As shown in figure 21( $a$ ), at lg( $K$ ) = 0, the wake resembles that of a rigid stem, although some alterations occur due to stem vibration. The spanwise vortex shedding occurs at an angle of approximately 6 $^\circ$ relative to the wall surface, which is considerably lower than the $\approx$ 21 $^\circ$ observed in the rigid case. The downwash flow in the flexible case has a weaker effect on vortex formation, and similar to the rigid case, the spanwise vortices mainly appear around the midspan. At lg( $K$ ) = –0.5, the stem oscillates with a large transverse amplitude ( $A^*=$ 0.79), leading to significant changes in vortical structures. The shift in vortex shedding is associated with the increased transverse amplitude, which enhances vortex formation near the tip despite minor deformation. Consequently, the downwash flow effect becomes negligible. As shown in figure 21( $b$ ), vortex shedding initiates at the free end and weakens closer to the wall, ceasing at approximately 2.8 $D$ from the wall.

As lg( $K$ ) decreases further, stem deformation becomes pronounced, altering the 3-D vortex distribution. As shown in figure 21( $c$ ), deformation leads to a more evenly distributed flow along the stem length. Near the free end, vortex shedding remains approximately parallel, but further downstream, the separated vortices stretch into hairpin-vortex like structures (Robinson Reference Robinson1991). Additionally, vortices from both sides of the stem initially connect during the formation, as marked in figure 21( $c$ ), although these connections disappear downstream due to increased vortex head velocity. In the far wake, the vortices are fully hairpin-vortex like.

Figure 21. The 3-D vortical structures of a wall-mounted flexible stem at $Re=$ 200 and different $K$ values. Panels show ( $a$ ) lg( $K$ ) = 0, ( $b$ ) lg( $K$ ) = –0.5, ( $c$ ) lg( $K$ ) = –1.0, ( $d$ ) lg( $K$ ) = –1.5, ( $e$ ) lg( $K$ ) = –2.0 and ( $f$ ) lg( $K$ ) = –3.0. Flow is from left to right. Here, $Q=$ 5. The plot in ( $b$ ) is the corresponding vertical velocity field at the midplane of the $y$ -direction.

Figure 22. The 3-D vortical structures of a wall-mounted flexible stem at $Re=$ 400 and different $K$ values. Panels show ( $a$ ) lg( $K$ ) = 0, ( $b$ ) lg( $K$ ) = –0.5, ( $c$ ) lg( $K$ ) = –1.0, ( $d$ ) lg( $K$ ) = –1.5, ( $e$ ) lg( $K$ ) = –2.0 and ( $f$ ) lg( $K$ ) = –3.0. Flow is from left to right. Here, $Q=$ 20. The inset in ( $a$ ) is the corresponding vertical velocity field at the midplane of the $y$ -direction.

Figure 23. The 3-D vortical structures of a wall-mounted flexible stem at $Re = 600-1000$ and $K$ values. Panels show ( $a$ ) $Re=$ 600 and lg( $K$ ) = –0.5, ( $b$ ) $Re=$ 600 and lg( $K$ ) = –2.0, ( $c$ ) $Re=$ 800 and lg( $K$ ) = –1.5, ( $d$ ) $Re=$ 800 and lg( $K$ ) = –2.0, ( $e$ ) $Re=$ 1000 and lg( $K$ ) = –0.5, and ( $f$ ) $Re=$ 1000 and lg( $K$ ) = –2.0. Flow is from left to right. Here, $Q=$ 45, 80 and 125 for the cases at $Re=$ 600, 800 and 1000, respectively.

At lg( $K$ ) $\leqslant$ –1.5, significant stem deformation results in spanwise vortex shedding along the entire stem length, although this is weaker than in less deformed cases, see figure 21(d–f). As lg( $K$ ) drops, vortices form progressively closer to the free end. At lg( $K$ ) = –1.5 and –3.0, transverse amplitudes are smaller, and vortical structures are weaker compared with lg( $K$ ) = –2.0, where the transverse amplitude is higher ( $A^*=$ 0.55). Comparing figures 21( $d$ ) and 21( $e$ ), despite a slightly stronger stem oscillation at lg( $K$ ) = –3.0 than at lg( $K$ ) = –1.5, the vortical structures at lg( $K$ ) = –3.0 are weaker due to greater deformation. A common feature across these cases is the slow upward movement of vortices, see the inset in figure 21( $d$ ). For significantly deformed stems, separation of the attached shear layer at the free-end surface occurs, forming a leading-edge vortex and/or free-end arch vortex (observed in high $Re$ flow, e.g. $Re \gtrsim 4\times 10^3$ , see figure 1). As shown in the inset of figure 21( $e$ ), the reattached shear layer at the free-end surface separates, alternately aligned with the vortex shedding. This behaviour is absent in the rigid case. Large deformation makes the attached shear layer more prone to detachment.

Figure 22 shows the organisation of 3-D vortical structures at $Re = 400$ . As in the rigid case, vortices behind the flexible stem at $Re=$ 400 are stronger than at $Re = 200$ . As lg( $K$ ) decreases, stem deformation intensifies, altering the vortical structures. At lg( $K$ ) = 0, deformation is negligible, and vortex shedding remains similar to the rigid case, see figures 13( $b$ ) and 22( $a$ ). The downwash flow near the free end impacts a shorter span in the flexible case. As shown in figure 22( $a$ ), vortex shedding remains approximately parallel but occurs in two phases, potentially influenced by free-end downwash flow and junction upwash flow. At lg( $K$ ) = –0.5, vortex shedding changes drastically, as in the case of $Re=$ 200. Shedding occurs near the free end, with spanwise vortices forming earlier than those closer to the wall, see figure 22( $b$ ). In the near wake, spanwise vortices undergo considerable stretching and slowly move toward the wall. The hairpin vortex on the wall develops and directly interacts with the lower-side vortices from the stem. At lg( $K$ ) = –0.5, stem oscillation is stronger, and vortex strength is higher than at lg( $K$ ) = 0.

For lower lg( $K$ ), deformation increase significantly. As shown in figure 22(c–f), the vortex dynamics is considerably different as the stem tip nears the wall. Similar to $Re=$ 200, vortex shedding shifts toward the free end as deformation increases, with vortex intensity generally weakening. However, transverse oscillation impacts both formation position and vortex strength. At lg( $K$ ) = –2.0, transverse amplitude is larger than at lg( $K$ ) = –1.0 and –1.5, leading to stronger vortices, see figure 22( $c$ , $d$ , $e$ ). At lg( $K$ ) = –3.0, the stem is nearly lodged against the wall, producing a distinct vortex dynamics, see figure 22( $f$ ). Here, spanwise vortices shed only from the stem base and free end. In the near wake, vortices from these positions interact.

Figure 23 shows the arrangement of 3-D vortical structures at $Re = 600{-}1000$ . At these higher $Re$ , vortex shedding occurs along nearly the entire stem length, forming finer vortices. Due to increased instability from lower-part vortices, the wall-developing hairpin vortex strengthens compared with lower $Re\, (=200{-}400)$ cases. As shown in figure 23( $a$ , $b$ ), at $Re=$ 600, vortex shedding extends from top to bottom but is suppressed at the junction due to the presence of a HV. At $Re=$ 800, vortex strength increases further, see figure 23( $c$ , $d$ ). At lg( $K$ ) = –1.5, upper-part vortex shedding aligns closely with the deformed stem, while lower-part vortex shedding occurs but dissipates quickly, see figure 23( $c$ ). Compared with lg( $K$ ) = –1.5, transverse oscillation at lg( $K$ ) = –2.0 is stronger, significantly influencing vortex shedding. As shown in figure 23( $d$ ), upper-part vortex shedding remains nearly parallel, but lower-part vortex shedding is not synchronised, leading to vortices gradually shifting upward downstream. Consequently, interactions between hairpin and spanwise vortices weaken. The stem exhibits a low-frequency back-and-forth motion, resembling a slow wave, as indicated in figure 23( $d$ ). At $Re=$ 1000, vortex strength intensifies further. At high lg( $K$ ), vortex shedding near the tip experiences minor influence from downwash flow, see figure 23( $e$ ). The upper and lower parts of the stem shed vortices at different frequencies. In the near wake, separated vortices are significantly stretched and finer than those at the lower $Re$ . As lg( $K$ ) = –2.0, stem deformation is pronounced, with upper-part vortex shedding nearly parallel, see figure 23( $f$ ). Lower-part vortices interact with the hairpin vortex but dissipate quickly downstream.

Figure 24. The non-dimensional vertical velocity ( $w^*$ ) of a wall-mounted flexible stem at $Re=$ 400 and lg( $K$ ) = –0.5. ( $a$ , $b$ ) instantaneous flow field and ( $c$ ) time-averaged flow field. The stem position in ( $a$ , $b$ ) is roughly denoted by the solid lines.