3.1.1. Instantaneous 3-D vortex structures

Figure 4.

Instantaneous 3-D vortical structures visualized by isosurfaces of $Q=0.1$ at $G/D=0.1$ , coloured with the instantaneous streamwise velocity $u/U_0$ : $(a)$ $L/D=5$ ; $(b)$ $L/D=10$ ; $(c)$ $L/D=15$ .

Figure 5.

Instantaneous 3-D vortical structures visualized by isosurfaces of $Q=0.1$ at $G/D=0.3$ , coloured with the instantaneous streamwise velocity $u/U_0$ : $(a)$ $L/D=5$ ; $(b)$ $L/D=10$ ; $(c)$ $L/D=15$ .

The instantaneous 3-D vortex structures for all flows are identified by the criterion with $Q=0.1$ . Figure 4 illustrates the instantaneous vortices for $G/D=0.1$ . The FNRC exhibits distinct asymmetry compared with the FIRC (Zhang et al. Reference Zhang, Kareem, Xu and Jiang2023; Li et al. Reference Li, Wang, Qiu, Zhou, Fu and Liu2024b ) and aligns more closely with the FNCC (Li et al. Reference Li, Wang, Qiu, Wu, Zhou, Fu and Liu2022a ) with the same $G/D$ , featuring a turbulent boundary layer downstream rather than the wake of an isolated blunt body. The KH instability in the upper leading edge (ULE) shear layer results in the formation of a two-dimensional upper KH (UKH) vortex. The LE/TE shear layer is the structure that detaches from the LE/TE corner (Cimarelli et al. Reference Cimarelli, Leonforte and Angeli2018b ; Chiarini et al. Reference Chiarini, Quadrio and Auteri2021; Cimarelli et al. Reference Cimarelli, Corsini and Stalio2024; Li et al. Reference Li, Wang, Qiu, Zhou, Fu and Liu2024b ). Moreover, there is no flow separation at the lower leading edge (LLE), leading to the absence of the vortices in the gap. As illustrated in figure 4 $(a)$ , for $L/D=5$ and $G/D=0.1$ , the near-wall effect causes the UKH vortices to form and merge farther downstream compared with the FIRC (Zhang et al. Reference Zhang, Kareem, Xu and Jiang2023; Li et al. Reference Li, Wang, Qiu, Zhou, Fu and Liu2024b ), resulting in a larger streamwise scale of the mean LLE recirculation (shown in § 3.1.2). The 3-D instability of the ULE vortex manifests downstream of the UTE, rather than on the upper side of the rectangular cylinder, highlighting a significant contrast with the flow characteristics observed in the FIRC (Zhang et al. Reference Zhang, Kareem, Xu and Jiang2023; Li et al. Reference Li, Wang, Qiu, Zhou, Fu and Liu2024b ). As the ULE vortices interact with the downstream reverse wall boundary layer, hairpin vortices eventually form. For $L/D = 10$ and $15$ , as shown in figures 4 $(b)$ and 4 $(c)$ , the UKH vortices interact with the reverse boundary layer on the upper cylinder surface, evolving into hairpin vortices. For all these three flows, upon reattachment to the plate, the flow transitions into a wall boundary layer, with hairpin vortex packets forming farther downstream.

For $G/D=0.3$ , the instantaneous 3-D vortical structures are shown in figure 5. The KH instability in the ULE shear layer is observed across all three flows, leading to the formation of KH vortices. The absence of vortical structures within the gap indicates significant suppression of the LLE shear layer. In contrast to $G/D=0.1$ , the lower trailing edge (LTE) shear layer extends from the LTE for $L/D=5$ and $L/D=10$ (in figure 5 $a$ and 5 $b$ ). However, at $L/D=15$ , the LTE shear layer is negligible, as shown in figure 5 $(c)$ . At $L/D=5$ , as shown in figure 5 $(a)$ , the UKH vortex forms farther downstream on the upper side of the cylinder. Consequently, the merged UKH vortex undergoes 3-D instability and evolves into the ULE vortex downstream. No hairpin vortices are present on the upper side, similar to the flow at $G/D=0.1$ . However, at $L/D=10$ and $15$ (in figure 5 $b$  and  $c$ ), hairpin vortex packets appear on the upper side, which contrasts with the flow at $G/D=0.1$ . Furthermore, the distribution of ULE vortices becomes more intricate. When the gap ratio increases to $G/D = 0.9$ , as shown in figure 6, the influence of the upstream wall boundary layer diminishes, leading to the formation of a distinct LLE shear layer. In the gap, vortices are observed across all three $L/D$ configurations, indicating that KH vortices shed from the LLE shear layer, unlike at $G/D=0.1$ and $0.3$ . Both the LTE and TSVs are identifiable and closely related to the LLE vortex. The complexity of the vortical structures on the upper side of the cylinder increases.

Figure 6.

Instantaneous 3-D vortical structures visualized by isosurfaces of $Q=0.1$ at $G/D=0.9$ , coloured with the instantaneous streamwise velocity $u/U_0$ : $(a)$ $L/D=5$ ; $(b)$ $L/D=10$ ; $(c)$ $L/D=15$ .

3.1.2. Mean flow characteristics

The presence of a strong shear layer induced by the plate wall results in pronounced mean shear characteristics near the rectangular cylinder. Consequently, an investigation of these mean shear characteristics is undertaken in this section. Exploiting the spanwise homogeneity, the mean shear tensor in the $x{-}y$ plane is decomposed, which provides a comprehensive assessment of all non-zero components of the 3-D mean strain rate tensor ( $S_{xy}$ ) (Fang & Tachie Reference Fang and Tachie2020),

(3.1) \begin{equation} S_{xy}=\begin{bmatrix} \displaystyle\frac {\partial U}{\partial x} & \displaystyle\frac {1}{2}\left(\displaystyle\frac {\partial U}{\partial y}+ \displaystyle\frac {\partial V}{\partial x}\right) \\[12pt] \displaystyle\frac {1}{2}\left(\displaystyle\frac {\partial U}{\partial y}+ \displaystyle\frac {\partial V}{\partial x}\right)& \displaystyle\frac {\partial V}{\partial y} \end{bmatrix} =\textbf{M} \begin{bmatrix} \sigma _1 & 0\\ 0 & \sigma _2 \end{bmatrix} \textbf{M}^T, \end{equation}

where $\sigma _1$ and $\sigma _2$ are eigenvalues of $S_{xy}$ , while $\textbf{M}$ denotes the matrix of eigenvectors of $S_{xy}$ . Due to the incompressibility and the spanwise homogeneity of the mean flow, it follows that $\sigma _1 + \sigma _2 = \partial U/\partial x +\partial V/\partial y =0$ . Consequently, it can be inferred that $\sigma _1 = -\sigma _2$ , with $\sigma _1$ defined as positive to represent the magnitude of principal stretching. The time- and spanwise-averaged streamwise velocity ( $U/U_0$ ) fields for different cases are presented in figures 7 $(a,c,e)$ –9 $(a,c,e)$ . Meanwhile, the fields of principal stretching $\sigma _1$ , which reflect the distribution of mean shear strength, are displayed in figures 7 $(b,d,f)$ –9 $(b,d,f)$ .

Figure 7.

Time- and spanwise-averaged streamwise velocity $U/U_0$ in pseudocolour with the superimposition of streamlines for $G/D=0.1$ : $(a)$ $L/D=5$ ; $(c)$ $L/D=10$ ; $(e)$ $L/D=15$ . The principal stretching $\sigma _1$ for $G/D=0.1$ : $(b)$ $L/D=5$ ; $(d)$ $L/D=10$ ; $(f)$ $L/D=15$ . The cures in (b), (d) and (f) are the local streamlines.

Figure 8.

Time- and spanwise-averaged streamwise velocity $U/U_0$ in pseudocolour with the superimposition of streamlines for $G/D=0.3$ : $(a)$ $L/D=5$ ; $(c)$ $L/D=10$ ; $(e)$ $L/D=15$ . The insets in (a), (c) and (e) are the close-up views. The black arrow marks the clockwise recirculation, while the magenta ones mark the anticlockwise recirculation. The principal stretching $\sigma _1$ for $G/D=0.3$ : $(b)$ $L/D=5$ ; $(d)$ $L/D=10$ ; $(f)$ $L/D=15$ . The cures in (b), (d) and (f) are the local streamlines.

For $G/D = 0.1$ , a recirculation zone forms upstream of the rectangular cylinder in all flow cases, as shown in figure 7 $(a,c,e)$ . The streamwise scale of this recirculation, which exceeds $3.7D$ for all three flows, is significantly larger than the scale observed in the FNCC, where it is smaller than $2D$ (Li et al. Reference Li, Wang, Qiu, Wu, Zhou, Fu and Liu2022b). This scale can be determined by the location of the reattachment point of the recirculation. On the upper side of the cylinder, a LE recirculation forms, similar to the FIRC (Zhang et al. Reference Zhang, Kareem, Xu and Jiang2023; Li et al. Reference Li, Wang, Qiu, Zhou, Fu and Liu2024b ), but with no recirculation on the lower side, creating an asymmetry not seen in the FIRC (Cimarelli et al. Reference Cimarelli, Leonforte and Angeli2018b ; Chiarini et al. Reference Chiarini, Gatti, Cimarelli and Quadrio2022a ; Li et al. Reference Li, Wang, Qiu, Wu, Zhou, Fu and Liu2022a ; Zhang et al. Reference Zhang, Kareem, Xu and Jiang2023; Cimarelli et al. Reference Cimarelli, Corsini and Stalio2024). For $L/D = 5$ , the LE recirculation nearly spans the entire upper surface of the cylinder without a distinct reattachment point, differing from the FIRC (Cimarelli et al. Reference Cimarelli, Leonforte, Angelis, Crivellini and Angeli2019a ; Chiarini & Quadrio Reference Chiarini and Quadrio2021; Chiarini et al. Reference Chiarini, Quadrio and Auteri2021; Zhang et al. Reference Zhang, Kareem, Xu and Jiang2023; Li et al. Reference Li, Wang, Qiu, Zhou, Fu and Liu2024b ). Immediately downstream of the cylinder, only a LTE recirculation is observed, with a significant wall recirculation also present, similar to the FNCC (He et al. Reference He, Pan and Wang2013a ,Reference He, Wang and Pan b , Reference He, Wang, Pan, Feng, Gao and Rinoshika2017; Li et al. Reference Li, Wang, Qiu, Wu, Zhou, Fu and Liu2022b), but with a smaller streamwise scale across all $L/D$ configurations. This difference arises because KH instability in the upper shear layer in the FNCC develops farther downstream (Li et al. Reference Li, Wang, Qiu, Wu, Zhou, Fu and Liu2022a ). The distribution of recirculations resembles that in backward-facing step flows (Ma & Schröder Reference Ma and Schröder2017; Li et al. Reference Li, Zhang, Liu, Wu and Gao2020; Ma, Tang & Jiang Reference Ma, Tang and Jiang2020), suggesting that the FNCC shear layer is more stable than the UTE shear layer in the current FNRC at the same $G/D$ , mainly due to the absence of upstream vortex disturbances. At $L/D = 5$ , the LE recirculation connects with the downstream wall recirculation, forming a continuous recirculation zone, allowing fluid to return from the downstream region to the upper side upstream of the cylinder. As shown in figure 7 $(b,d,f)$ , the upstream wall boundary layer, ULE shear layer and UTE shear layer (for $L/D = 10, 15$ ) exhibit strong shear characteristics. The upstream wall boundary layer rises upon encountering the rectangular cylinder, contributing to the formation of the upstream recirculation.

As $G/D$ increases to $0.3$ , the upstream recirculation persists, accompanied by the downstream wall recirculation, as illustrated in figure 8. The gap flow significantly enhances, obstructing the connection between the wall recirculation and the TE, in contrast to $G/D = 0.1$ . For $L/D = 5$ , the LE recirculation covers nearly the entire upper surface, with a distinct reattachment point and two recirculations behind the cylinder, differing from $G/D = 0.1$ . For $L/D = 10$ and $15$ , only one LTE recirculation is observed, as depicted in figures 8(c) and 8(e). Compared with the FNCC (He et al. Reference He, Pan and Wang2013a,b, Reference He, Wang, Pan, Feng, Gao and Rinoshika2017; Li et al. Reference Li, Wang, Qiu, Wu, Zhou, Fu and Liu2022b), a novel feature is the occurrence of multiple flow separations at the downstream wall recirculation, as seen in the insets of figure 8(a,c,e). At $L/D = 5$ , only an anticlockwise recirculation is observed beneath the wall recirculation (marked by the magenta arrow). However, at $L/D = 10$ and $15$ , both anticlockwise (marked by the magenta arrow) and clockwise (marked by the black arrow) recirculations appear. The distributions of $\sigma _1$ in figure 8 $(b,d,f)$ are similar to those at $G/D = 0.1$ , although the shear strength beneath the rectangular cylinder is slightly higher for all three cases at $G/D = 0.3$ .

At $G/D = 0.9$ , the upstream wall recirculation weakens, and the gap flow increases significantly, as shown in figure 9 $(a,c,e)$ . Unlike at $G/D = 0.1$ and $0.3$ , a LLE recirculation forms, with both its streamwise and wall-normal scales smaller than those of the ULE recirculation, indicating the establishment of the LLE shear layer. The recirculation pair behind the cylinder, though not perfectly symmetrical, resembles that in the FIRC, with two distinct recirculations, signalling a diminished near-wall effect. As shown in figure 9 $(b,d,f)$ , the upstream wall boundary layer lifting is reduced and no longer contacts the cylinder. Additionally, figure 9 $(a,c,e)$ show a small clockwise recirculation (marked by the black arrow) beneath the LLE recirculation, suggesting that the LLE vortex induces wall boundary layer separation. The downstream wall recirculation persists, indicating continued separation of the downstream wall boundary layer, but its scale is significantly reduced.

Figure 9.

Time- and spanwise-averaged streamwise velocity $U/U_0$ in pseudocolour with the superimposition of streamlines for $G/D=0.9$ : $(a)$ $L/D=5$ ; $(c)$ $L/D=10$ ; $(e)$ $L/D=15$ . The insets in (a), (c) and (e) are the close-up views. The black arrow marks the clockwise recirculation. The principal stretching $\sigma _1$ for $G/D=0.9$ : $(b)$ $L/D=5$ ; $(d)$ $L/D=10$ ; $(f)$ $L/D=15$ . The cures in (b), (d) and (f) are the local streamlines.

Figure 10.

The quantified size of the ULE recirculation: $(a)$ the streamwise scale $x_r$ ; and $(b)$ wall-normal scale $y_h$ , compared with these for the FIRC at the same $Re$ (Li et al. Reference Li, Wang, Qiu, Zhou, Fu and Liu2024b ), respectively.

Figure 11.

$(a)$ The local angle $ \theta$ between the ULE shear layer and the $x$ coordinate. $(b)$ The local principal stretching $\sigma _1$ along the ULE shear layer for $L/D=10$ and $15$ . The values for $L/D=15$ are increased by 10. The pentagrams represent the locations where the flows reattach.

Figures 7 to 9 show that the size of the ULE recirculation changes with increasing $G/D$ . The streamwise ( $x_r$ ) and wall-normal ( $y_h$ ) scales of the ULE recirculation are quantified in figures 10 $(a)$ and 10 $(b)$ , respectively, with comparison with the LE recirculation in the FIRC at the same $Re$ . Here $x_r$ is determined by the reattachment point, where the wall friction coefficient changes from a negative value to a positive value. Since the streamwise velocity is negative within the recirculation, $y_h$ is determined by the point with the maximum wall-normal position within the recirculation. For $L/D = 5$ , the ULE recirculation nearly covers the entire upper surface, resulting in a significantly larger $x_r$ compared with the FIRC (Li et al. Reference Li, Wang, Qiu, Zhou, Fu and Liu2024b ). As shown in figure 10 $(a)$ , the streamwise scale $x_r$ for $L/D = 10$ and $15$ is also larger than in the FIRC, indicating that the near-wall effect delays reattachment. In contrast, the wall-normal scale $y_h$ , shown in figure 10 $(b)$ , is smaller than in the FIRC (Li et al. Reference Li, Wang, Qiu, Zhou, Fu and Liu2024b ). Notably, $y_h$ increases with $G/D$ , reflecting a reduced near-wall effect as $G/D$ increases. Additionally, for the same $G/D$ , a larger streamwise span of the cylinder results in a larger $y_h$ for the ULE recirculation.

To understand the factors influencing the scale of the ULE recirculation, the local angle $\theta$ between the ULE shear layer and the $x$ axis is calculated. The position of ULE shear layer is defined by the local mean streamline, as shown in figures 7 to 9. Figure 11 $(a)$ shows that the ULE shear layer initially deflects upward but this deflection decreases downstream. The initial deflection angle is nearly uniform for the same $G/D$ , with minimal dependence on $L/D$ . The mean shear resulting from the upstream wall boundary layer suppresses the wall-normal growth of the ULE recirculation. Therefore, smaller $G/D$ corresponds to a smaller deflection angle, indicating a diminishing influence of the upstream wall boundary layer as $G/D$ increases. This explains the increasing wall-normal scale of the ULE recirculation with higher $G/D$ , as shown in figure 10 $(b)$ . Figure 11 $(b)$ shows that the local principal stretching $\sigma _1$ for $L/D = 10$ and $15$ behaves similarly across all cases. Initially, the mean shear increases to a maximum before decreasing to a minimum near the reattachment point. As the $G/D$ increases, the initial principal stretching becomes more pronounced, while the decay occurs more steeply. A larger $G/D$ also results in a larger $y_h$ , and although the decay is most rapid when $G/D = 0.9$ , the larger $y_h$ causes the decay to reach its minimum at a more downstream location. In contrast, the flow with $G/D = 0.3$ reaches its minimum value and reattaches at a more upstream position, which explains the smaller streamwise scales of the ULE recirculation observed for $G/D = 0.3$ , as shown in figure 10(a). The flow with $G/D = 0.1$ , on the other hand, experiences the weakest decay, yet its wall-normal scale of the ULE recirculation is similar to that of the flow with $G/D = 0.3$ . As a result, the flow with $G/D = 0.1$ ultimately decays to its minimum value, with reattachment occurring at the farthest downstream location.

Figure 12.

$(a)$ The maximum mean streamwise velocity, $U_{max}$ , over the upper surface of the cylinder at a given $x$ position, where $x/D=0\sim x_r/D$ . The values for $L/D=15$ are increased by $0.25$ . $(b)$ The mean pressure coefficient $C_p$ , referring to the free stream pressure, along the upper LE shear layer. The values for $L/D=15$ are increased by $1$ .

To further illustrate these mean flow features, $U_{max}={\rm max}_{\it{y/D}\gt \rm {1}+\it{G/D}}\it{U}(\it{x/D,y/D})$ is extracted and shown in figure 12 $(a)$ . Here $U_{max}$ represents the flow velocity along the ULE shear layer (Chiarini & Auteri Reference Chiarini and Auteri2023). By examining figures 11 $(a)$ and 12 $(a)$ together, it is evident that as the streamline passes over the upper LE corner, the curvature increases, resulting in a higher flow acceleration. This observation aligns with the findings in the study of Chiarini & Auteri (Reference Chiarini and Auteri2023). Meanwhile, in the current study, an increase in $G/D$ results in greater flow acceleration for the same $L/D$ . As a result, $U_{max}$ increases with an increase in $G/D$ . It should be noted that this trend contrasts with the findings of Chiarini & Auteri (Reference Chiarini and Auteri2023), who explores the FNRC with low $Re$ and a moving wall. This difference is likely driven by $Re$ , as the LE flow separation in the FIRC exhibits distinct behaviour at current moderate $Re$ (Zhang et al. Reference Zhang, Kareem, Xu and Jiang2023; Li et al. Reference Li, Wang, Qiu, Zhou, Fu and Liu2024b ) compared with lower $Re$ (Chiarini et al. Reference Chiarini, Quadrio and Auteri2021). Additionally, the effect of the incoming boundary layer is significant, especially for small $G/D$ . However, to a significant extent, this observation helps clarify why an increase in $G/D$ leads to a larger $y_h$ , giving that the flow velocity decreases as one approaches the wall. After undergoing an approximate plateau, $U_{max}$ gradually decreases, signalling that the flow is nearing reattachment. The attenuation of velocity is closely linked to local pressure variations. As shown in figure 12 $(b)$ , the flow experiences a reversed pressure gradient prior to reattachment. Furthermore, for the same $L/D$ , the larger $G/D$ , the greater the reversed pressure gradient. However, similar to the decay of $\sigma _1$ , although the reversed pressure gradient at $G/D = 0.9$ is stronger than at $G/D = 0.3$ , the flow at $G/D = 0.9$ has a larger $U_{{max}}$ , resulting in a longer decay process. Ultimately, the flow at $G/D = 0.3$ decays to its minimum value at the closest point, while the flow at $G/D = 0.1$ decays at the furthest point, due to its significantly smaller reversed pressure gradient and the fact that the reversed pressure gradient appears at a later location.

The spanwise-averaged root-mean-square (r.m.s) of the streamwise velocity fluctuation, $u'_{rms}/U_0$ , is shown in figure 13. The fluctuations near the LE are relatively weak, suggesting laminar flow characteristics. However, as the KH vortices develop, there is a substantial increase in the velocity fluctuations. At $G/D = 0.1$ and $0.3$ (in figures 13 $a$ and 13 $b$ ), the distributions of $u'_{rms}/U_0$ are similar. Strong fluctuations occur primarily on the upper side of the cylinder and downstream, with negligible fluctuations below it, indicating that the LLE and LTE shear layers are very weak at these smaller $G/D$ values. When $G/D$ increases to $0.9$ (in figure 13 $c$ ), fluctuations extend from the LLE, although they are weaker than those on the upper side. Weak fluctuations also appear beneath the cylinder on the plate wall. This behaviour aligns with the mean streamwise velocity distribution (in figure 9), where the LLE shear layer forms, separates and induces the separation of the wall boundary layer. Distinct fluctuations also extend from the LTE, indicating alternative TE vortex shedding and contributing to the separation of the plate wall boundary layer near the LTE. Overall, as the cylinder moves away from the wall, fluctuations gradually increase, suggesting that the near-wall effect on vortex generation and evolution diminishes with increasing $G/D$ .

Figure 13.

The spanwise-averaged the r.m.s. of the streamwise fluctuations $u'_{rms}/U_0$ : $(a)$ $G/D=0.1$ ; $(b)$ $G/D=0.3$ ; $(c)$ $G/D=0.9$ ; (i) $L/D=5$ ; (ii) $L/D=10$ ; (iii) $L/D=15$ . The fluctuations from these azure points are extracted to calculate the local power spectral densities (PSDs).

3.2.1. The $G/D=0.1$ cases

The vortex distribution on the upper side of the cylinder at $L/D=5$ and $G/D=0.1$ differs significantly from the FIRC at the same $Re$ (Zhang et al. Reference Zhang, Kareem, Xu and Jiang2023; Li et al. Reference Li, Wang, Qiu, Zhou, Fu and Liu2024b ) (in figure 4), necessitating a detailed examination of the vortex dynamics, particularly the formation of the ULE vortex. The evolution of the vortex dynamics is shown in figure 15 $(a,b)$ , with the instantaneous pressure distribution at $z/D=\pi$ in figure 15 $(c,d)$ . At $t=t_0$ , a merged UKH vortex forms near the UTE (in figure 15 $a$ ), creating a low-pressure region and inducing a reverse flow beneath it, as seen in figure 15 $(c)$ . Meanwhile, the UKH vortex undergoes 3-D instability and evolving into the ULE vortex. This reverse flow, encountering an adverse pressure gradient (in the inset of figure 15 $c$ ), causes the separation of the upper boundary layer, forming an anticlockwise-rotating SV on the upper side of the cylinder (USV), marked by the magenta arrow in figure 15 $(a)$ . This USV is absent in the FIRC (Zhang et al. Reference Zhang, Kareem, Xu and Jiang2023; Li et al. Reference Li, Wang, Qiu, Zhou, Fu and Liu2024b ). The upstream motion generated by the downstream wall recirculation enhances the local reverse flow, which is a key factor contributing to the generation of the USV. At $t=t_0 + 12 \Delta t$ (in figures 15 $b$ and 15 $d$ ), the ULE vortex detaches from the cylinder, displaying strong 3-D features and interacting with the wall recirculation. While the initial USV disappears, a new one forms due to the induction of the newly generated UKH vortex. The wall recirculation induced by the new UKH vortex passes over the UTE (in figure 15 $b$ ). This interaction directly contributes to the integration of the ULE recirculation and the downstream wall recirculation in the mean field (in figure 7 $a$ ) and prevents the formation of UTE vortices in the current flow. To confirm this phenomenon, 3-D FTLEs are employed.

Figure 15.

(a,b) The process of the LLE vortex developing from the UKH vortex at $L/D=5$ and $G/D=0.1$ . (c,d) The corresponding instantaneous distribution of the pressure with the superimposition of instantaneous streamlines at $z/D=\pi$ . Here (a,c) $t=t_0$ ; (b,d) $t=t_0+12\Delta t$ , where $\Delta t=0.25D/U_0$ . The vortical structures are visualized by isosurfaces of $Q=0.1$ , coloured with the instantaneous spanwise vorticity $\omega _z$ . The magenta arrow in (a) and (b) marks the SV with positive $\omega _z$ on the upper side of the cylinder. The insets in $(c)$ are the close-up views.

Figure 16.

The vortex evolution process revealed by LCSs presented in the slices at $G/D=0.1$ : (a,b) the ULE vortex for $L/D=5$ , (c,d) the UTE vortex for $L/D=10$ . Here $(b)$ and $(d)$ are delayed by $8\Delta t$ and $4\Delta t$ compared with $(a)$ and $(c)$ , respectively, where $\Delta t=0.25D/U_0$ .

As shown in figure 16 $(a,b)$ , the UKH vortex evolves into the ULE vortices as it passes over the UTE, but no UTE vortex is formed during this process. This confirms that UTE vortices do not form for $L/D=5$ and $G/D=0.1$ . While the reverse uplift of the downstream wall recirculation is observed. The formation of UTE vortices at larger aspect ratios is also examined. For $L/D = 15$ , the flow characteristics are similar to those for $L/D = 10$ . Therefore, only the vortex dynamics near the UTE for $L/D = 10$ are shown in figure 16 $(c,d)$ . At time $t=t_0$ (in figure 16 $c$ ), the ULE vortex forms a hairpin vortex near the UTE, while the UTE shear layer rolls up, initiating the formation of a UTE vortex. This formation is driven by the KH instability of the UTE shear layer, similar to backward-facing step flows (Ma & Schröder Reference Ma and Schröder2017; Li et al. Reference Li, Zhang, Liu, Wu and Gao2020; Ma et al. Reference Ma, Tang and Jiang2020). By $t=t_0+4\Delta t$ (in figure 16 $d$ ), the UTE vortex fragments due to interactions with the wall recirculation, and is entrained into the wake and dissipates. In contrast, the ULE vortex retains its hairpin vortex shape and continues to evolve downstream, becoming the dominant feature in the plate boundary layer.

3.2.2. The $G/D=0.3$ cases

To examine the development of the UTE and LTE shear layers at $G/D=0.3$ , FTLEs are also used for visualization. The evolution for $L/D=5$ is illustrated in figure 17 $(a,b)$ . At $t=t_0$ (in figure 17 $a$ ), the UTE shear layer rolls up, forming the UTE vortex. This process contrasts with the flow at $G/D=0.1$ for the same $L/D$ . Moreover, the UTE shear layer deflects downwards, contributing to the formation of an upper recirculation (in figure 8 $a$ ). As the flow progresses, the UTE vortex is entrained into the wake and disappears (in figure 17 $b$ ). Meanwhile, the LTE shear layer lifts but does not roll up, preventing the formation of the LTE vortex in this flow. The formation of the UTE vortex at $L/D=10$ is shown in figure 17 $(c,d)$ . The flow at $L/D=15$ exhibits similar behaviour, so it is not presented here. At $t=t_0$ (in figure 17 $c$ ), the UTE shear layer rolls up, generating two UTE vortices simultaneously. This double-TE vortex formation has not been reported in previous studies on the flows around rectangular cylinders. The UTE shear layer forms via KH instability, similar to the flow at $L/D=10$ with $G/D=0.1$ , and does not deflect downstream, resulting in no upper recirculation behind the cylinder (in figures 8 $c$ and 8 $e$ ). By $t=t_0+8\Delta t$ (in figure 17 $d$ ), the upstream UTE vortex interacts with the ULE vortex, and breaks up. While the downstream one successfully develops into hairpin vortices. Despite the lifting of the LTE shear layer, no LTE vortex forms. This contrasts with the FNCC at $G/D=0.3$ (Li et al. Reference Li, Wang, Qiu, Wu, Zhou, Fu and Liu2022a) and even at lower $G/D=0.25$ (He et al. Reference He, Wang, Pan, Feng, Gao and Rinoshika2017), where the lower wake vortex forms and sheds, producing alternating vortex shedding dynamics.

Figure 17.

The evolution process of the UTE vortex revealed by LCSs presented in the slices at $G/D=0.3$ : (a,b) $L/D=5$ ; $(c,d)$ $L/D=10$ . Here $(b)$ and $(d)$ are delayed by $6\Delta t$ and $8\Delta t$ compared with $(a)$ and $(c)$ , respectively, where $\Delta t=0.25D/U_0$ .

Figure 18.

Close-up view of the instantaneous SV near the TE (TSV) and the vortex with positive spanwise vorticity on the plate wall (WPV) at $G/D=0.3$ , visualized by isosurfaces of $Q=0.05$ and coloured with the instantaneous spanwise vorticity $\omega _z$ , along with the slices showing the contours of instantaneous pressure with the superimposition of streamlines at $z/D=0$ : $(a)$ $L/D=5$ ; $(b)$ $L/D=10$ . The azure rotation arrow represents local clockwise motion; the red rotation arrow represents local anticlockwise motion.

As shown in figure 8, multiple recirculations are observed downstream of the cylinder. The corresponding instantaneous vortex dynamics is investigated using the clustering method (Jiménez et al. Reference Jiménez, Del, Juan and Flores2004; Dong et al. Reference Dong, Huang, Yuan and Lozano-Durán2020; Wang et al. Reference Wang, Pan, Wang and Gao2022a ,Reference Wang, Pan and Wang b ), previously effective in identifying secondary and tertiary vortices in the FNCC (Li et al. Reference Li, Wang, Qiu, Wu, Zhou, Fu and Liu2022a ). Here, the criteria for identifying significant vortex structures through clustering methods are defined by $Q \gt 0.05$ , with distinct characteristic structures differentiated by the positive or negative value of the spanwise vorticity $\omega _z$ . The extracted structures are shown in figure 18. Since the flow at $L/D=15$ is similar to that at $L/D=10$ (in figure 8), only the instantaneous structures for $L/D=10$ are discussed. As illustrated in figure 18, the vortex structure distributions for $L/D = 5$ and $L/D = 10$ display notable similarities. Secondary vortices with negative $\omega _z$ emerge downstream of the LTE shear layer. These vortices, positioned near the TE, are referred to as TSV. Immediately downstream, on the wall, vortex structures characterized by positive $\omega _z$ are observed, and are designated as WPV. This vortex distribution pattern closely resembles that observed in the FNCC at the same $G/D$ (Li et al. Reference Li, Wang, Qiu, Wu, Zhou, Fu and Liu2022a ), although the in the current flows, especially the secondary vortices, are less pronounced compared with those in the FNCC.

As shown in the inset of figure 18, in both two cases, the TSV originates from the plate boundary layer, driven by an adverse pressure gradient, as depicted by the smaller clockwise motion arrow near the plate wall. When the gap flow interacts with the downstream wall recirculation (the larger clockwise motion arrow), the WPVs forms. However, the dominant sources for WPV formation differ significantly with varying $L/D$ values. For $L/D=5$ (in the inset of figure 18 $a$ ), the wall recirculation is more pronounced, competing effectively with the gap flow, which causes a significant upward deflection of the gap flow. Near the wall, the WPV (the anticlockwise motion arrow) forms as the wall recirculation encounters the TSV. The intense wall recirculation brings two clockwise recirculations into close proximity – the wall recirculation itself and the one induced by the TSV. This proximity causes the wall recirculation to absorb the TSV-induced recirculation within the mean field, as illustrated in figure 8 $(a)$ . Consequently, aside from the wall recirculation, only anticlockwise recirculation corresponding to the WPV is observed in the mean field. When $L/D=10$ (in the inset of figure 18 $b$ ), the wall recirculation diminishes, and the gap flow extends farther downstream following the formation of the TSV. Upon encountering the wall recirculation, the gap flow lifts upwards, creating a local adverse flow that leads to WPV formation. Thus, the WPV primarily originates from the gap flow, differing from the scenario at $L/D=5$ . Meanwhile, the wall recirculation and TSV are effectively blocked by the WPV, resulting in their independent existence in the mean field, as shown in figure 8 $(b)$ . A comparison between the FNCC and FNRC reveals that: in the FNCC, vortices with positive $\omega _z$ are induced by secondary vortices; while in the FNRC, they result from the competition between the gap flow and wall recirculation. The dominant source is influenced by $L/D$ , which explains why these vortices (WPVs) are not named as TV in this study.

3.2.3. The $G/D=0.9$ cases

For $L/D=5$ and $G/D=0.9$ , the formation of the TE vortex is examined in figure 19. At $t=t_0$ (in figure 19 $a$ ), the LTE sheds from the LTE shear layer, while the UTE shear layer rolls up. By $t=t_0+9\Delta t$ (in figure 19 $b$ ), the LTE vortex has shed off and evolved into wake. The UTE shear layer deflects downward, strengthening the UTE vortex. Then, the alternating shedding of upper and lower trailing vortices occurs, differing from the dynamics at $G/D=0.1$ and $0.3$ . Moreover, figure 19 $(b)$ shows the LLE vortex and TSV, with their formation depicted in figure 20.

Figure 19.

The evolution process of the UTE vortex revealed by LCSs presented in the slices for $L/D=5$ and $G/D=0.9$ : $(a)$ $t=t_0$ ; $(b)$ $t=t_0+9\Delta t$ , where $\Delta t=0.25D/U_0$ .

Figure 20.

The vortex dynamics in the gap at $L/D=5$ and $G/D=0.9$ : $(a)$ $t=t_0$ ; $(b)$ $t=t_0+5\Delta t$ ; $(c)$ $t=t_0+10\Delta t$ ; $(d)$ $t=t_0+15\Delta t$ ; $(e)$ $t=t_0+20\Delta t$ ; $(f)$ $t=t_0+25\Delta t$ , where $\Delta t=0.25D/U_0$ . Here $(g)$ , $(h)$ and $(i)$ are partial structures extracted from $(a)$ , $(b)$ and $(c)$ , respectively. The vortical structures are visualized by isosurfaces of $Q=0.1$ , coloured with the instantaneous spanwise vorticity $\omega _z$ . The magenta arrow marks the SV with negative $\omega _z$ on the lower side of the cylinder. The azure arrow marks TV $_1$ .

As shown in figure 20 $(a)$ , at time $t=t_0$ , the LLE shear layer rolls up, forming the leading secondary vortex (LSV $_3$ ) from the plate wall. Notably, the LSV forms before the LLE vortex. The LSV $_3$ undergoes instability, causing plate boundary layer separation and forming the weak TV $_2$ . The LLE vortex advances to $x/D \approx 4$ , where LSV $_2$ forms, while LSV $_1$ is located near the TE, with LSV $_2$ being smaller than LSV $_1$ . At $t=t_0 + 5\Delta t$ (in figure 20 $b$ ), the LE shear layer rolls farther, creating a SV with negative $\omega _z$ , which quickly dissipates downstream. Meanwhile, the enhanced TV $_2$ stabilizes LSV $_3$ , and the spanwise instability of the LLE vortex suppresses the growth of LSV $_2$ . By $t=t_0+10\Delta t$ (in figure 20 $c$ ), a new LLE vortex sheds. While the established one is transitioning into the LTE vortex. This transition does not occur in the FIRC (Cimarelli et al. Reference Cimarelli, Leonforte and Angeli2018b , Reference Cimarelli, Leonforte, Angelis, Crivellini and Angeli2019a ; Chiarini & Quadrio Reference Chiarini and Quadrio2021; Chiarini et al. Reference Chiarini, Quadrio and Auteri2021, Reference Chiarini, Gatti, Cimarelli and Quadrio2022a ; Zhang et al. Reference Zhang, Kareem, Xu and Jiang2023; Cimarelli et al. Reference Cimarelli, Corsini and Stalio2024; Li et al. Reference Li, Wang, Qiu, Zhou, Fu and Liu2024b ). The LSV $_1$ convects downstream while LSV $_2$ fragments. At $t=t_0+15\Delta t$ (in figure 20 $d$ ), a weaker SV (LSV $_4$ ) forms, indicating that each LLE vortex can induce two secondary vortices in one shedding cycle. The LTE vortex interacts with LSV $_1$ , and TV $_2$ flattens and merges into the LTE vortex. At $t=t_0+20\Delta t$ (in figure 20 $e$ ), the LTE vortex lifts, causing LSV $_1$ to rise. The LSV $_2$ , previously fragmented, is enhanced by the LTE vortex. At $t=t_0+25\Delta t$ (in figure 20 $f$ ), the LTE vortex sheds off, and LSV $_1$ deforms and rises. The LSV $_2$ is enhanced but no new SV forms near the LTE. The separation of the wall boundary layer near the LTE results from the established weak LSV. The TV $_2$ does not shed with the LTE vortex; instead, it forms a pseudo-TE vortex, with its shedding corresponding to the evolution of TV $_1$ (in figure 20 $a-c$ ), which is extracted through the clustering method and shown in figure 20 $(g-i)$ . After shedding, TV $_2$ does not rise like the LTE vortex but convects downstream. Double-secondary vortices, similar to those observed in multielement airfoils (Wang, Wang & Kim Reference Wang, Wang and Kim2019; Wang & Wang Reference Wang and Wang2021b ) and simplified circular-cylinder/airfoil configurations (Wang, Wu & Wang Reference Wang, Wu and Wang2023), also appear in the FNCC boundary layer at $G/D=1.0$ and $1.333$ (Wang et al. Reference Wang, He and Wang2024). However, the secondary vortices in this study are smaller and weaker, distinguishing them from previous double-SV configurations.

Figure 21.

The vortex dynamics in the gap at $L/D=15$ and $G/D=0.9$ : $(a)$ $t=t_0$ ; $(b)$ $t=t_0+7\Delta t$ ; where $\Delta t=0.25D/U_0$ . The vortical structures are visualized by isosurfaces of $Q=0.1$ , coloured with the instantaneous spanwise vorticity $\omega _z$ .

The vortex dynamics at $L/D=10$ will be discussed in $\S$ 3.3.1, influenced by two distinct frequencies. For $L/D=15$ , as shown in figure 21, vortex formation is similar to $L/D=5$ , but no double-secondary vortices form from the plate boundary layer. The LKH vortex rapidly evolves into the LLE vortex in the form of a streamwise vortex pair, as shown in figure 21 $(a)$ . Figure 21 $(b)$ indicates the LSV exhibits spanwise instability, reversely inducing lower boundary layer separation and forming a TV, as at $L/D=5$ . Unlike at $L/D=5$ , the LSV does not return to a two-dimensional structure, instead stretching into hairpin-like vortices. The TV strengthens under the influence of TSV but breaks up due to mean shear. Two adjacent TV groups merge near the LTE, forming an LTE vortex, which induces a TSV from the plate boundary layer.

3.3.1. Flow processes with typical frequencies

The PSDs are calculated at the azure points in figure 13, and remain marked by the same numbers. The velocity fluctuations at this location are typically the highest locally, particularly at points 1–3, which are distributed along the shear layer. The characteristic peaks detected at nearly all monitoring points within a flow can be used to identify the corresponding characteristic frequency of the flow. Figure 22 $(a,c,e)$ , show the PSDs at various spatial positions for different $L/D$ ratios at $G/D=0.1$ . For $L/D = 5$ , as illustrated in figure 22 $(a)$ , a prominent fluctuation signal is observed from the LE shear layer extending to the downstream wall recirculation, which corresponds to the fundamental frequency $St_0$ and is characteristic of the LE vortex dynamics. Probes 1–3 show $0.5St_0$ (marked by the red circle), $1.5St_0$ (marked by the blue circle) and $St_{\textit {KH}}$ ( $St_{\textit {KH}}=2St_0$ ), with the subharmonic $0.5St_0$ and fractional harmonic $1.5St_0$ resulting from nonlinear interactions. The $St_{\textit {KH}}$ weakens downstream and represents the fluctuation frequency of KH instability within the ULE shear layer. The relation $St_{\textit {KH}}=2St_0$ indicates that every two UKH vortices merge to form a single ULE vortex during one shedding cycle. In contrast, in the FIRC at $L/D=5$ , every three KH vortices merge into one LE vortex (Zhang et al. Reference Zhang, Kareem, Xu and Jiang2023; Li et al. Reference Li, Wang, Qiu, Zhou, Fu and Liu2024b ). The merging process of UKH vortices for $G/D=0.1$ and $L/D=5$ is shown in figure 23. Instantaneous vortices visualized with isosurfaces of $Q=0.5$ reveal that adjacent UKH vortices shed from the ULE shear layer, merging downstream, as shown in figure 23 $(a-d)$ . After merging, the vortex becomes 3-D and evolves into the ULE vortex. Notably, the UKH vortices merge before undergoing 3-D instability here, whereas in the FIRC, instability occurs first (Zhang et al. Reference Zhang, Kareem, Xu and Jiang2023). The near-wall effect influences both the number and sequence of merging and instability. Furthermore, figure 22 $(a)$ shows consistent characteristic frequencies across all probes, including in the downstream recirculation, indicating that the frequency-locking feature persists in the FNRC with $L/D=5$ and $G/D=0.1$ .

For $L/D=10$ and $G/D=0.1$ , figure 22 $(c)$ shows the fundamental frequency $St_0$ in the PSDs for all probes. Additionally, a new frequency $St_{0*}$ , similar to $St_0$ , is detected in probes 1 and 2 within the ULE shear layer but weakens and disappears downstream. This is interpreted as a pseudoshedding frequency, to be validated by DMD analysis in § 3.3.2. In probe 1, the dominant frequency with a wide bandwidth is $St_0$ , indicating that the ULE vortex primarily originates from a single UKH vortex. The evolution of the ULE vortex from the UKH vortex is shown in figure 24. Notably, adjacent UKH vortices do not merge but directly evolve into ULE vortices in the form of hairpin vortices. In contrast, in the FIRC with $L/D=10$ , every two KH vortices merge to form an LE vortex (Li et al. Reference Li, Wang, Qiu, Zhou, Fu and Liu2024b ). As $L/D$ increases to $15$ , figure 22 $(d)$ shows two distinct shedding frequencies, $St_{01}$ and $St_{02}$ , due to mode competition, which will be discussed further in the DMD analysis in § 3.3.2. The evolution of the UKH vortex into the ULE vortex at $L/D=15$ is similar to $L/D=10$ . For $G/D=0.3$ , the PSDs for different $L/D$ ratios (in figure 22 $b,d,f$ ) are similar to those at $G/D=0.1$ , except at $L/D=15$ , where three-mode competition arises.

Figure 23.

The process of the UKH vortex merging at $G/D=0.1$ and $L/D=5$ : $(a)$ $t=t_0$ ; $(b)$ $t=t_0+2\Delta t$ ; $(c)$ $t=t_0+4\Delta t$ ; $(d)$ $t=t_0+6\Delta t$ , where $\Delta t=0.25D/U_0$ . The vortical structures are visualized by isosurfaces of $Q=0.5$ , coloured with the instantaneous streamwise velocity $u/U_0$ . The black arrow marks the UKH vortex.

Figure 24.

The process of the UKH vortex evolving into the ULE vortex at $G/D=0.1$ and $L/D=10$ : $(a)$ $t=t_0$ ; $(b)$ $t=t_0+10\Delta t$ ; $(c)$ $t=t_0+20\Delta t$ , where $\Delta t=0.25D/U_0$ . The vortical structures are visualized by isosurfaces of $Q=0.5$ , coloured with the instantaneous streamwise velocity $u/U_0$ . The arrow marks the UKH vortex.

Figure 25.

Local PSDs of streamwise velocity fluctuation at different spatial locations for $G/D=0.9$ : (a,b) $L/D=5$ ; (c,d) $L/D=10$ ; (e,f) $L/D=15$ . The PSDs are calculated at azure points in figure 13 $(c)$ and remain marked by the same numbers. The amplitudes of PSDs for azure point 2 $-$ 5 and 7 $-$ 9 are amplified step-by-step by $10^{3}$ along the serial number. Here $St_0$ , $St_{01}$ and $St_{02}$ denote the shedding frequencies of the LE or TE vortex; $St_{0*}$ is the pseudoshedding frequency. The dashed lines define the frequency bands with a bandwidth of $0.02$ , related to the typical frequencies.

The local PSDs for $G/D=0.9$ are presented in figure 25. For $L/D=5$ , figures 25 $(a)$ and 25 $(b)$ show the fundamental frequency $St_0$ is present in all probes, confirming frequency locking and that the characteristic frequency of SV coincides with $St_0$ . A high-frequency band at four times $St_0$ suggests four UKH vortices shed from the ULE shear layer in one LE/TE shedding cycle. The evolution of this process is illustrated in figure 26. At $t=t_0$ (figure 26 $a$ ), UKH $_{1}$ and UKH $_{2}$ form, while the LTE shear layer rolls up to create an LTE vortex. The UKH $_{1}$ evolves into hairpin vortices, while UKH $_{2}$ strengthens the LTE vortex (in figure 26 $b$ and 26 $c$ ). By $t=t_0+15\Delta t$ (in figure 26 $d$ ), UKH $_{3}$ forms and, along with UKH $_{2}$ , merges to form the ULE vortex (in figure 26 $e$ ), with the LTE vortex shedding off. As shown in figure 26 $(f)$ , the merged vortex separates, while a new UKH vortex (UKH $_{4}$ ) begins to form. Notably, although four UKH vortices are generated, not all participate in merging. The PSDs from probes 7 and 8 show harmonics of $St_0$ , attributed to downstream convection of the TV and the enhancement of the weak LSV, resembling the harmonics seen in double-secondary vortices in the FNCC (Wang et al. Reference Wang, He and Wang2024).

Figure 26.

The process of the UKH vortex evolving into the ULE vortex at $G/D=0.9$ and $L/D=5$ : $(a)$ $t=t_0$ ; $(b)$ $t=t_0+5\Delta t$ ; $(c)$ $t=t_0+10\Delta t$ ; $(d)$ $t=t_0+15\Delta t$ ; $(e)$ $t=t_0+20\Delta t$ ; $(f)$ $t=t_0+25\Delta t$ , where $\Delta t=0.25D/U_0$ . The vortical structures are visualized by isosurfaces of $Q=0.5$ , coloured with the instantaneous streamwise velocity $u/U_0$ . The magenta arrow marks the LTE vortex.

The PSDs for $L/D=10$ and $G/D=0.9$ are illustrated in figures 25 $(c)$ and 25 $(d)$ . Two dominant frequencies, $St_{01}$ and $St_{02}$ , are observed. At probe 1, both frequencies are distinct, but at probes 2 and 3, $St_{02}$ becomes dominant, while $St_{01}$ weakens. The wide bandwidth at probes 1 and 2 indicates that a single ULE vortex evolves from a UKH vortex, with shedding frequency $St_{02}$ . At probe 4, downstream, $St_{01}$ reappears, signalling double-mode competition as the flow becomes more complex and wall boundary layer characteristics emerge (in figure 6 $b$ ). At probe 5, near the UTE vortex shedding location, a reduction in the peaks of both frequencies occurs, with $St_{01}$ still detectable and $St_{02}$ disappearing, indicating the UTE vortex shedding frequency is $St_{01}$ . This marks a shift in the characteristic frequency from $St_{02}$ to $St_{01}$ on the upper side of the cylinder. At probe 6, within the LLE shear layer, both $St_{01}$ and $St_{02}$ remain, but no wide bandwidth is observed, suggesting bimodal competition in LLE vortex formation. At probes 8 and 9, associated with the LTE vortex shedding and TSV formation, the frequency is $St_{01}$ , with $St_{02}$ diminishing. This shows that frequency locking is suppressed in the flow at $G/D=0.9$ and $L/D=10$ .

The presence of two dominant frequencies warrants investigation into their effects on vortex dynamics. The signals for $St_{01}$ and $St_{02}$ are extracted via Fourier mode decomposition (Basley et al. Reference Basley, Pastur, Lusseyran, Faure and Delprat2011; Ma et al. Reference Ma, Feng, Pan, Gao and Wang2015; Wang et al. Reference Wang, Feng, Wang and Li2018; Wang & Wang Reference Wang and Wang2021a ,Reference Wang and Wang b ; Li et al. Reference Li, Wang, Qiu, Zhou, Fu and Liu2024b ) from probe 7 at $z/D=\pi$ , as shown in figure 13( $c$ i). The results, presented in figure 27, show strong competition between the two frequencies: when $St_{01}$ fluctuates strongly, $St_{02}$ weakens and vice versa. Specifically, around time $t=t_a$ , the fluctuation is dominated by $St_{01}$ , while $St_{02}$ is weak; the reverse occurs at $t=t_b$ . The vortex dynamics for $L/D=10$ and $G/D=0.9$ in figure 28 resemble those at $L/D=15$ and $G/D=0.9$ (in figure 21), but the SV evolution differs between the two frequencies. When $St_{01}$ is dominant, hairpin-like vortices form from the LSV (in figures 28 $a$ and 28 $b$ ), as seen at $L/D=15$ . In contrast, when $St_{02}$ dominates, no hairpin vortices form, and only spanwise vortices are present (in figures 28 $c$ and 28 $d$ ). Additionally, with $St_{01}$ , the LTE vortex sheds before the TV reaches the LTE, whereas for $St_{02}$ , the LTE vortex has not yet shed at that point. This phase difference results from the competition between the two frequencies but does not affect the characteristic frequencies of the LTE vortex and TSV, as shown in figure 25 $(d)$ at probes 8 and 9.

Figure 27.

The time signals of streamwise velocity fluctuation for $St_{01}$ and $St_{02}$ at $L/D=10$ and $G/D=0.9$ . The azure and orange curves are the signals of the Fourier mode decomposition mode for $St_{01}$ and $St_{02}$ , respectively, which are extracted from the probe 7 (presented in figure 13 c i) in the plane $z/D=\pi$ . Here $t_a$ and $t_b$ mark the typical instances for the strong fluctuations of $St_{01}$ and $St_{02}$ , respectively.

Figure 28.

The instantaneous vortical structures in the gap for $L/D=10$ and $G/D=0.9$ : $(a)$ $t=t_a-8\Delta t$ ; $(b)$ $t=t_a$ ; $(c)$ $t=t_b-8\Delta t$ ; $(d)$ $t=t_b$ , where $\Delta t=0.25D/U_0$ . The vortical structures are visualized by isosurfaces of $Q=0.1$ , coloured with the instantaneous spanwise vorticity $\omega _z$ .

Figure 29.

The process of the UKH vortex evolving into the ULE vortex at $L/D=15$ and $G/D=0.9$ : $(a)$ $t=t_0$ ; $(b)$ $t=t_0+5\Delta t$ ; $(c)$ $t=t_0+10\Delta t$ ; $(d)$ $t=t_0+15\Delta t$ , where $\Delta t=0.25D/U_0$ . The vortical structures are visualized by isosurfaces of $Q=0.5$ , coloured with the instantaneous streamwise velocity $u/U_0$ . The magenta arrow marks the UKH vortex, and the ellipse marks the ULE vortex in the form of the hairpin vortex cluster.

Figure 30.

The process of LLE vortex shedding being suppressed at $L/D=15$ and $G/D=0.9$ : $(a)$ $t=t_0$ ; $(b)$ $t=t_0+2\Delta t$ ; $(c)$ $t=t_0+4\Delta t$ ; $(d)$ $t=t_0+6\Delta t$ ; $(e)$ $t=t_0+8\Delta t$ ; $(f)$ $t=t_0+10\Delta t$ , where $\Delta t=0.25D/U_0$ . The vortical structures are visualized by isosurfaces of $Q=0.1$ , coloured with the instantaneous spanwise vorticity $\omega _z$ . The black arrow marks the LSV, the magenta arrow marks the LLE shear layer.

As $L/D$ increases to 15, the PSDs in figures 25 $(e)$ and 25 $(f)$ show a single fundamental frequency, $St_0$ , across all probes, indicating frequency locking despite the presence of a pseudofrequency related to vortex shedding. In the PSD at probe 1 within the ULE shear layer, the dominant high-frequency bandwidth is $2St_0$ , suggesting that two UKH vortices merge to form one ULE vortex. This is illustrated in figure 29, where two adjacent UKH vortices shed from the ULE shear layer (in figure 29 $a$ ), interact with the ULE recirculation (in figure 29 $b$ ), and evolve into 3-D vortices (in figure 29 $c$ ). They then merge to form a ULE vortex, a hairpin vortex cluster (in figure 29 $d$ ). This process, where UKH vortices first undergo 3-D instability before merging, contrasts with the behaviour at $L/D=5$ and $G/D=0.1$ (in figure 23). Additionally, figure 25 $(f)$ shows a wide bandwidth around $2St_0$ at probe 6, but the merging of LKH vortices into an LLE vortex does not occur here. This suppression occurs because the enhanced LSV inhibits the formation of KH vortices when the LLE shear layer undergoes the second KH instability. As a result, only the first KH instability contributes to the formation of KH vortices within a single shedding cycle. As shown in figure 30, the LKH vortex quickly evolves into the LLE vortex and experiences local instability figure 30( $b{-}d$ ). However, as the LSV strengthens, it suppresses the instability of the LLE shear layer figure 30( $e{-}f$ ), preventing the formation of additional LKH vortices during the same shedding cycle.

3.3.2. Dynamic mode decomposition analysis

Figure 31.

Isosurfaces of the real part of the DMD mode for the streamwise component at $G/D=0.1$ : $(a)$ $St_0$ mode for $L/D=5$ ; $(b)$ $St_0$ mode for $L/D=10$ ; $(c)$ $St_{01}$ mode for $L/D=15$ ; $(d)$ $St_{02}$ mode for $L/D=15$ . (i) The values are $\pm 0.0001$ ; (ii) the values are $\pm 0.0005$ . The red parts are positive and the blue parts are negative. The arrows mark the periodic topological structures.

Figure 32.

Isosurfaces of the real part of the DMD mode for the streamwise component at $G/D=0.3$ : $(a)$ $St_0$ mode for $L/D=5$ ; $(b)$ $St_0$ mode for $L/D=10$ ; $(c)$ $St_{01}$ mode for $L/D=15$ ; $(d)$ $St_{02}$ mode for $L/D=15$ ; $(e)$ $St_{03}$ mode for $L/D=15$ . (i) The values are $\pm 0.0001$ ; (ii) the values are $\pm 0.0005$ . The red parts are positive and the blue parts are negative. The arrows mark the periodic topological structures.

To further investigate the dynamic characteristics of vortex shedding in the current FNRC, the DMD method is employed, akin to the analyses conducted on the FNCC (Li et al. Reference Li, Wang, Qiu, Wu, Zhou, Fu and Liu2022a ) and FIRC (Zhang et al. Reference Zhang, Kareem, Xu and Jiang2023; Li et al. Reference Li, Wang, Qiu, Zhou, Fu and Liu2024 b). This approach aims to explore the spatial characteristics of vortex shedding frequencies across all flows examined. Particular attention will be directed towards flows exhibiting multiple characteristic shedding frequencies, with the DMD method utilized to identify the dominant frequency for each flow.

Figure 31 illustrates the DMD analysis of ULE/UTE vortex shedding for flows with $G/D=0.1$ . The displayed topology represents the streamwise component of the DMD mode, with the structures in figures 31( $a$ i) to 31( $d$ i) showing similarities across different flows. These topological structures exhibit a regular alternating pattern in the streamwise direction, confirming the periodic nature of the flow and the presence of frequency locking. At $L/D=15$ , both $St_{01}$ and $St_{02}$ modes show periodicity, indicating competition between the two modes. However, the $St_{02}$ mode is more pronounced and persists farther downstream (in figure 31 $d$ i). To identify the dominant shedding mode, the isosurface values of the DMD mode are increased, showing stronger fluctuations in figures 31( $a$ ii) to 31( $d$ ii). The DMD topological structures for $St_0$ at $L/D=5$ and $L/D=10$ and $St_{02}$ at $L/D=15$ maintain great coherence, reflecting flow periodicity. However, as the flow enters the plate boundary layer state, the stronger fluctuations dissipate, and the coherence of the $St_{01}$ mode at $L/D=15$ nearly disappears. Thus, the dominant frequency for the flow at $L/D=15$ and $G/D=0.1$ is $St_{02}$ .

For $G/D=0.3$ , the DMD vortex shedding topologies are presented in figure 32. The topological structures in figures 32( $a$ i) to 32( $e$ i) exhibit strong periodicity, similar to those at $G/D=0.1$ , until the flow transitions into the boundary layer state. The $St_{0*}$ mode at $L/D=10$ , representing the pseudoshedding frequency, is shown in figure 34 $(a)$ with identical isosurface values. This mode lacks spatial coherence, explaining its inability to capture flow periodicity and justifying its designation as a pseudoshedding frequency. The topologies for stronger fluctuations, shown in figures 32( $a$ ii) to 32( $e$ ii), reveal that the $St_0$ modes at $L/D=5$ and $L/D=10$ maintain great periodicity. However, at $L/D=15$ , only the $St_{02}$ mode exhibits coherence, indicating it as the dominant mode despite the presence of three competing shedding frequencies. The $St_{0*}$ mode at $L/D=10$ continues to lack spatial coherence, as shown in figure 34 $(b)$ .

Figure 33.

Isosurfaces of the real part of the DMD mode for the streamwise component at $G/D=0.9$ : $(a)$ $St_0$ mode for $L/D=5$ ; $(b)$ $St_{01}$ mode for $L/D=10$ ; $(c)$ $St_{02}$ mode for $L/D=10$ ; $(d)$ $St_0$ mode for $L/D=15$ . (i) The values are $\pm 0.0001$ ; (ii) the values are $\pm 0.0005$ . The red parts are positive and the blue parts are negative. The arrows mark the periodic topological structures.

Figure 34.

Isosurfaces of the real part of the DMD mode for the streamwise component: (a,b) $St_{0*}$ mode for $L/D=10$ and $G/D=0.3$ ; $(c)$ $St_{01}$ mode for $L/D=10$ and $G/D=0.9$ ; $(d)$ $St_{02}$ mode for $L/D=10$ and $G/D=0.9$ . $(a)$ The values are $\pm 0.0001$ ; (b,c,d) the values are $\pm 0.0005$ . The red parts are positive and the blue parts are negative. The arrows mark the periodic topological structures corresponding to streamwise vortices in the gap.

As $G/D$ increases to $0.9$ , all topological structures capture the periodicity of vortex shedding, similar to the smaller gap ratios, as shown in figures 33( $a$ i) to 33( $d$ i). These structures confirm the instability of the LLE shear layer, leading to vortex formation. At $L/D=5$ , the mode exhibits a Kármán vortex street pattern, indicating a reduced near-wall effect. At $L/D=10$ , the $St_{02}$ mode shows greater coherence than $St_{01}$ , with topologies capturing the periodicity of the LE and TE vortices. However, the $St_{01}$ mode captures the periodicity of UTE and LTE vortices, while $St_{02}$ reflects the ULE vortex but loses coherence near the TE. Thus, the $St_{01}$ mode governs UTE and LTE vortices shedding, while $St_{02}$ regulates ULE vortices shedding. This agrees with the PSDs in figures 25 $(c)$ and 25 $(d)$ , confirming the suppression of frequency locking at $L/D=10$ and $G/D=0.9$ . Figures 34 $(c)$ and 34 $(d)$ show the DMD topologies of the stronger fluctuations for $St_{01}$ and $St_{02}$ modes. Near the LTE, the $St_{01}$ mode shows elongated streamwise features from the LSV, while $St_{02}$ lacks these structures. These topologies correlate with the vortices in figures 28 $(a)$ and 28 $(b)$ , indicating competitive coexistence within the LLE shear layer, where $St_{01}$ promotes streamwise vortices and $St_{02}$ suppresses them.