3.1.1. Instantaneous vortical structures and mean flow statistics

Figure 4 illustrates the instantaneous three-dimensional vortical structures for the three different aspect ratios, identified through the $Q$ criterion with $Q=0.1$. Since the flows exhibit symmetry, the evolution processes of various vortex structures on the upper side of the cylinder will be described only. For $L/D=5$ in figure 4(a), the onset of KH instability is observed in the LE shear layer, resulting in the formation of spanwise KH vortices. The subsequent secondary instability of the KH vortex indicates its limited persistence over a short range. This secondary destabilized KH vortex then transforms into the LE vortices, as also noted in the study by Zhang et al. (Reference Zhang, Kareem, Xu and Jiang2023). However, the present DNS results provide clearer identification of hairpin vortices compared with previous studies. These hairpin vortices arise from the LE vortices due to the background mean shear. The intense shear effects cause regions of high momentum to form the head of the hairpin vortices, while regions of low momentum develop into their legs. This generation process of hairpin vortices is akin to that observed in the interaction between the wake of a circular cylinder and a wall boundary at a very small gap ratio (Li et al. Reference Li, Wang, Qiu, Wu, Zhou, Fu and Liu2022a). Although these two flow configurations are different, the hairpin vortices in both flows are evolved from KH vortices. Furthermore, the LE vortices composed of multiple hairpin vortices appear more irregular than those observed at lower Reynolds numbers in the studies of Hourigan et al. (Reference Hourigan, Thompson and Tan2001) and Chiarini et al. (Reference Chiarini, Quadrio and Auteri2022b). It should be noted that the scale of hairpin vortices appears larger than the results reported by Cimarelli et al. (Reference Cimarelli, Leonforte and Angeli2018b), Chiarini & Quadrio (Reference Chiarini and Quadrio2021) and Chiarini et al. (Reference Chiarini, Gatti, Cimarelli and Quadrio2022a), which can be attributed to the higher Reynolds numbers employed in their investigations.

Figure 4. Instantaneous three-dimensional vortical structures visualized by isosurfaces of $Q=0.1$, coloured with the instantaneous streamwise velocity $u/U_0$: (a) $L/D=5$; (b) $L/D=10$; (c) $L/D=15$.

The mean shear, denoted as $\tau = ({1}/{Re}) ({\partial U}/{\partial y})$, is illustrated in figure 5. The figure clearly demonstrates the presence of strong shear in the LE shear layer and the forward wall boundary layer after the reattachment. Furthermore, it can be observed that as the aspect ratio increases, the duration of the mean shear becomes longer, resulting in a greater amount of shear experienced by the flow. Consequently, the intensified mean shear in cases with larger aspect ratios significantly impacts the evolution of hairpin vortices.

Figure 5. Mean shear $\tau /U_0^2$: (a) $L/D=5$; (b) $L/D=10$; (c) $L/D=15$.

The vortical structures associated with larger aspect ratios have received limited attention in previous researches. Figure 4(b,c) demonstrate that the generation and three-dimensional instability processes of the KH vortices at $L/D=10$ and $15$ are similar to those observed at $L/D=5$. Nevertheless, the formation of KH vortices is observed to take place at larger distances from the LE as the aspect ratio increases. The evolution of KH rolls into hairpin vortices along the shear layer is less abrupt as the aspect ratio increases, although the mean shear increases. The increased streamwise dimension of the rectangular cylinder allows the hairpin vortices to travel farther downstream along the wall at larger aspect ratio. As they progress downstream, the hairpin vortices experience stretching due to the enhanced mean shear. The high-momentum heads of the hairpin vortices tend to be lifted up, leading to the formation of hairpin vortex packets for both aspect ratios ($L/D=10$ and $15$). Furthermore, despite a portion of the packet being convected downstream of the TE, the hairpin vortex packet retains its shape due to memory effects.

For all three aspect ratios, the TE vortex is generated and subsequently sheds into the wake. However, it is evident that the wake for $L/D=5$ exhibits larger-scale oscillations in the vertical direction compared with the other two aspect ratios. At the current medium Reynolds number, the oscillation in the $L/D=5$ flow appears to be more apparent than what was observed in the studies conducted by Cimarelli et al. (Reference Cimarelli, Leonforte and Angeli2018b), Chiarini & Quadrio (Reference Chiarini and Quadrio2021) and Chiarini et al. (Reference Chiarini, Gatti, Cimarelli and Quadrio2022a) at a high Reynolds number ($Re=3000$). This can be observed not only in the instantaneous vortical structures, but also quantitatively reflected in $C_{L,rms}$. Specifically, at $L/D=5$, the present work reports a value of $C_{L,rms}=0.71$, as shown in table 1. This is larger than the value of $C_{L,rms}=0.29$ obtained under the same aspect ratio for $Re=3000$ (Chiarini & Quadrio Reference Chiarini and Quadrio2021). Furthermore, the wakes for all three aspect ratios exhibit a Kármán-like vortex street formation, which has rarely been paid close attention to in studies investigating high Reynolds number (Cimarelli et al. Reference Cimarelli, Leonforte and Angeli2018b; Chiarini & Quadrio Reference Chiarini and Quadrio2021; Chiarini et al. Reference Chiarini, Gatti, Cimarelli and Quadrio2022a).

Figure 6(a,c,e) present the time and spanwise-averaged streamwise velocity $U/U_0$ for $L/D=5$, $10$ and $15$, respectively, in pseudocolour with the superimposition of streamlines. Due to the flow symmetry, the discussion of mean flow statistics focuses solely on the upper side of the cylinder as well. The observations reveal a consistent flow behaviour across all three aspect ratios. The flow separates at the LE and subsequently reattaches downstream on the upper side of the rectangular cylinder. This phenomenon gives rise to the formation of separation bubbles. The streamwise scales of these separation bubbles are found to be $4.42D$, $6.28D$, $7.12D$ for aspect ratios of $L/D=5$, 10, 15, respectively. Additionally, the vertical scales are determined to be $0.92D$, $1.04D$, $1.10D$. This indicates that both the streamwise and vertical scales of the separation bubble increase with an augmentation in the aspect ratio. Notably, it is worth mentioning that, unlike the mean flow characteristics observed in the studies for $L/D=5$ conducted by Cimarelli et al. (Reference Cimarelli, Leonforte and Angeli2018b), Chiarini & Quadrio (Reference Chiarini and Quadrio2021) and Chiarini et al. (Reference Chiarini, Gatti, Cimarelli and Quadrio2022a), there is only one primary separation bubble above the upper side of the rectangular cylinder in the present results. In those previous studies, a secondary counter-rotating bubble was observed beneath the primary bubble at higher Reynolds number ($Re=3000$) (Cimarelli et al. Reference Cimarelli, Leonforte and Angeli2018b; Chiarini & Quadrio Reference Chiarini and Quadrio2021; Chiarini et al. Reference Chiarini, Gatti, Cimarelli and Quadrio2022a). Consequently, at the current Reynolds number, the reverse flow induced by the primary bubble generates a boundary layer that moves directly upstream without a secondary detachment of this reverse boundary layer.

Figure 6. Time- and spanwise-averaged streamwise velocity $U/U_0$ in pseudocolour with the superimposition of streamlines: (a) $L/D=5$; (c) $L/D=10$; (e) $L/D=15$. The spanwise-averaged r.m.s. of the streamwise fluctuation $u'_{rms}/U_0$: (b) $L/D=5$; (d) $L/D=10$; (f) $L/D=15$. The black crosses are designed to monitor the local power spectral densities (PSDs) in the LE shear layer, the boundary layer over the upper side of the rectangular cylinder, and the wake. The inverted triangle is designed to monitor feedback signal. The orange and magenta lines are designed to track the fluctuations in the LE shear layer and the boundary layer, respectively.

The flow undergoes a conversion into a boundary layer downstream following the reattachment point, and subsequently separates at the TE. As a result, another bubble forms in the wake region immediately after the TE due to the shedding of the TE vortex. The streamwise scale of these bubbles are $0.73D$, $1.16D$ and $1.31D$ for $L/D=5$, 10, 15, respectively. Therefore, similarly to the bubble originating from the LE, the streamwise scale of this wake bubble also increases as the $L/D$ rises.

The spanwise-averaged r.m.s. of the streamwise velocity fluctuation, $u'_{rms}/U_0$, is depicted in figure 6(b,d,f) for aspect ratios $L/D=5$, $10$ and $15$, respectively. It is evident from the figure that the initial LE shear layer exhibits laminar flow characteristics. However, with the generation and destabilization of the KH vortex, there is a significant increase in the intensity of velocity fluctuation. For aspect ratios $L/D=10$ and $L/D=15$, the intensity of $u'_{rms}/U_0$ gradually diminishes within the downstream boundary layer. In contrast, this trend is not as pronounced for $L/D=5$ since the distance between the reattachment point and the TE is comparatively small at this aspect ratio. As the flow detaches at the TE and forms the TE vortex, there is a subsequent increase in the intensity of $u'_{rms}/U_0$ in the wake region just behind the TE. Figure 7 illustrates the profiles of the spanwise-averaged intensity of vertical velocity fluctuation $v'_{rms}/U_0$ at various streamwise locations in the near wake. A comparison of the three aspect ratios demonstrates that the intensity of wake oscillation is significantly greater for $L/D=5$ in contrast to $L/D=10$ and $15$. This finding confirms that the wake strength is significantly greater for an aspect ratio of $L/D=5$, as illustrated in figure 4. In addition, it is worth noting that while the intensity $v'_{rms}/U_0$ for $L/D=15$ is indeed larger than that for $L/D=10$, this discrepancy can be attributed to the different vortex-shedding patterns investigated in § 3.3.1.

Figure 7. Profiles of the spanwise-averaged vertical velocity fluctuation intensity $v'_{rms}/U_0^2$ at different streamwise locations in the near wake: (a) $x'/D=1$; (b) $x'/D=3$; (c) $x'/D=5$, where $x'=x-L$.

3.1.2. Flow processes with typical frequencies

The inherent instabilities in the complex flow give rise to distinct frequencies that correspond to different physical phenomena. As a result, investigations have been conducted on the frequency characteristics associated with various vortex dynamics in the LE shear layer, boundary layer and wake regions. The signals of the streamwise velocity fluctuations are extracted from the points marked by black crosses in figure 6(b,d,f).

Figure 8 illustrates the fluctuation tracking using PSD. The PSDs have been averaged across the spanwise direction to identify dominant frequencies. A second Strouhal number, based on the vertical dimension of the rectangular cylinder, is defined as $St=f\kern0.7pt D/U_0$. In each flow scenario, a characteristic fundamental frequency peak denoted as $St_0$ is clearly observed in all velocity signals, both in the LE and TE shear layer regions. Therefore, it can be inferred that the LE and TE vortices shed at the same frequency ($St_0$), indicating the presence of frequency locking in all three flows with different aspect ratios at $Re=1000$. This observation aligns with previous studies of Nakamura et al. (Reference Nakamura, Ohya and Tsuruta1991) and Zhang et al. (Reference Zhang, Kareem, Xu and Jiang2023), which reported similar findings. However, the experimental investigation conducted by Liu & Zhang (Reference Liu and Zhang2015) did not observe frequency locking, possibly due to the influence of blockage ratios in the open-water channel. Furthermore, the topological structure of the DMD for the shedding of LE and TE vortices is presented in figure 9. These structures correspond to the frequency $St_0$ in all three flow cases. The LE vortex formation exhibits alternating structures dominantly in this mode, arranged regularly in the streamwise direction. This indicates that the DMD mode captures the coherent structures of the LE vortex. In the wake region behind the TE, the DMD mode is dominated by the Kármán-type vortex street, representing the coherent structures of the TE vortex. Notably, the distance between adjacent topologies in the wake for $L/D=5$ is significantly larger than that in the other two flows, suggesting a lower shedding frequency for $L/D=5$ compared with $L/D=10$ and $15$, consistent with the PSD results presented in figure 8. Consequently, the findings from this DMD mode analysis confirm the existence of frequency locking in all three flow scenarios.

Figure 8. Local PSDs of streamwise velocity fluctuation in the LE shear layer, the boundary layer and the wake: (a,b) $L/D=5$; (c,d) $L/D=10$; (e,f) $L/D=15$. The PSDs are calculated at crosses 1–7 in figure 6 and remain marked by the same numbers as in figure 6. The amplitudes of PSDs for crosses 2–4 and 6–7 are amplified step-by-step by $10^3$ along the serial number. Labels $St_L$, $St_0$, $St_{FB}$ and $St_{KH}$ denote the low frequency, vortex-shedding frequency, feedback frequency and KH fluctuation frequency, respectively. The dashed lines define the frequency bands related to the typical frequencies.

Figure 9. Isosurfaces of the real part of the DMD mode for the streamwise component, corresponding to the LE and TE vortex-shedding frequency. The red parts are positive and the blue parts are negative: (a) $L/D=5$; (b) $L/D=10$; (c) $L/D=15$.

In the LE shear layers (marked as crosses 1–3 in figure 6), multiple types of frequency peaks can be observed in figure 8(a,c,e) for different values of $L/D$ (5, 10 and 15). Of particular interest is the relatively large range of frequencies denoted as $St_{KH}$, which corresponds to the KH instability in the LE shear layer. Similar high-frequency behaviour with a wide bandwidth has also been observed in flows at high Reynolds numbers in previous studies (Cimarelli et al. Reference Cimarelli, Leonforte and Angeli2018a; Moore et al. Reference Moore, Letchford and Amitay2019; Chiarini & Quadrio Reference Chiarini and Quadrio2021). For the case of $L/D=5$, the value of $St_{KH}$ is close to 3$St_0$, indicating that every three KH vortices are destabilized and merge into one LE vortex during one shedding cycle of the LE vortex. This flow process was also reported by Zhang et al. (Reference Zhang, Kareem, Xu and Jiang2023) for the same aspect ratio. However, the bandwidth of the KH instability frequency in their study appears to be narrower compared with the current work. On the other hand, for $L/D=10$ and $L/D=15$ in our study, it is observed that $St_{KH}$ is close to 2$St_0$. This suggests that every two KH vortices merge into one LE vortex during one shedding cycle of the LE vortex for these flow configurations. In the reverse boundary layer, monitored at cross 4 in figure 6, the prominence of the KH instability peak diminishes progressively as $L/D$ increases. A similar trend is also observed in the downstream boundary layer. This indicates that the KH instability is primarily present in the LE shear layer and its significance decreases in the downstream boundary layer.

The fractional harmonic $St_{FB}$ corresponds to the feedback of the LE shear layer. Previous studies have not reported this fractional harmonic, nor have they explored the related physical mechanism involved in the flow process. Therefore, the present work aims to investigate the flow mechanism associated with $St_{FB}$. To analyse this, the FMD is applied to extract the characteristic frequency and reconstruct the flow fluctuations. For brevity, only the flow process of $St_{FB}$ at $L/D=5$ will be described, as the same flow mechanism is present in the other two cases. The signal for $St_{FB}$ is extracted from the point marked by an inverted triangle in figure 6(b). For comparison, the signal for $St_0$ is also extracted from the point marked cross 3 in the plane $z/D={\rm \pi}$ depicted in figure 6(b). Figure 10 illustrates the time signals of streamwise velocity fluctuation for $St_0$ and $St_{FB}$. As depicted in figure 10, the fractional harmonic ($St_{FB}$) is present at $t=t_a$, but it is absent at $t=t_b$.

Figure 10. The time signals of streamwise velocity fluctuation for $St_0$ and $St_{FB}$ at $L/D=5$. The blue and orange curves are the signals of the FMD mode extracted from the cross 3 for $St_0$ and the inverted triangle for $St_{FB}$ in the plane $z/D={\rm \pi}$ presented in figure 6(b), respectively. Here $t_a$ and $t_b$ mark the typical instances for the presence and absence of the fluctuation of $St_{FB}$, respectively.

Figures 11(a,c,e) and 12(a,c,e) illustrate the evolution of the reconstructed instantaneous streamwise velocity fluctuation field for the presence and absence of $St_{FB}$, respectively. When $St_{FB}$ is prominent (around time $t_a$), fluctuations related to $St_{FB}$ appear below the LE shear layer presented in figure 11(a,c,e) and are marked by an ellipse. In contrast, the fluctuations related to $St_{FB}$ are pretty weak at the same location illustrated in figure 12(a,c,e) when $St_{FB}$ is absent (around time $t_a$). For additional insight into the flow process of $St_{FB}$, the instantaneous spanwise vorticity fields are shown in figures 11(b,d,f) and 12(b,d,f) corresponding to the presence and absence of $St_{FB}$, respectively. As shown in figure 11(b,d,f), it can be observed that the LE shear layer splits into two parts after impinging on the wall boundary layer when $St_{FB}$ is present. One branch of this LE shear layer convects downstream towards the free flow, while another branch moves upstream along the reverse boundary layer and is marked by feedback in figure 11(b,d,f). On the other hand, when $St_{FB}$ is absent as shown in figure 12(b,d,f), the LE shear layer convects downstream instead of splitting into two parts after impinging on the wall. This behaviour could be caused by the rapid lift-up of the reverse boundary layer. Therefore, the flow process of $St_{FB}$ involves one part of the split LE shear layer moving upstream along the reverse boundary layer. Moreover, the feedback causes an interaction between the upstream LE shear layer and the reverse flow, generating local fluctuations.

Figure 11. The reconstructed instantaneous streamwise velocity fluctuations related to $St_{FB}$ and the instantaneous spanwise vorticity fields in the plane $z/D={\rm \pi}$ for $L/D=5$. (a,c,e) The reconstructed velocity fields and (b,d,f) the vorticity fields. Here (a,b) $t=t_a$; (c,d) $t=t_a+2\Delta t$; (e,f) $t=t_a+4\Delta t$, where $\Delta t=0.25D/U_0$ and $t_a$ is presented in figure 10.

Figure 12. The reconstructed instantaneous streamwise velocity fluctuations related to $St_{FB}$ and the instantaneous spanwise vorticity fields in the plane $z/D={\rm \pi}$ for $L/D=5$. (a,c,e) The reconstructed velocity fields and (a,d,f) the vorticity fields. Here (a,b) $t=t_b$; (c,d) $t=t_b+2\Delta t$; (e,f) $t=t_b+4\Delta t$, where $\Delta t=0.25D/U_0$ and $t_b$ is presented in figure 10.

3.2.1. Transitional characteristics

As shown in figure 4, the formation of hairpin vortex packets is observed downstream at $L/D=10$ and $15$, indicating a potential transition in the boundary layer on the upper or lower side of the rectangular cylinder. To quantify this transition, the shape factor $H$ is employed as an indicator of the laminar-to-turbulent transition. The calculation of $H$ is performed as follows:

(3.1)\begin{equation} H= \frac{\delta^{*}}{\theta}, \end{equation}

where, $\delta ^{*}$ and $\theta$ are displacement thickness and momentum thickness, respectively. Here $\delta ^{*}$ and $\theta$ are defined as

(3.2a,b)\begin{equation} \delta^{*}=\int_0^{\delta} \left(1-\frac{U}{U_{\delta}}\right){{\rm d}\kern0.05em y},\quad \theta=\int_{0}^{\delta} \frac{U}{U_{\delta}}\left(1-\frac{U}{U_{\delta}}\right){{\rm d}\kern0.05em y}, \end{equation}

where $\delta$ is the boundary layer thickness and $U_\delta$ is the velocity at the boundary layer edge. For the downstream boundary layer following reattachment, the boundary layer edge is determined using the standard assumption, where the velocity equals $0.99U_0$. However, for the upstream reverse boundary layer, the boundary layer edge is defined as the separatrix between positive $\omega _Z$ and negative $\omega _Z$, where $\omega _Z$ represents the time- and spanwsie-averaged spanwise vorticity. This approach has been adopted by Wang et al. (Reference Wang, Feng, Wang and Li2018) and Wang & Wang (Reference Wang and Wang2021b). The evolution of $H$ (shown in figure 13) illustrates that the curves of $H$ for all three aspect ratios progressively cross the line $H=2.59$ (the laminar stage) along the streamwise direction after the reattachment point. For $L/D=10$ and $15$, the shape factor tends to approach $H=1.4$, indicating the attainment of the turbulent stage in the boundary layer. However, the shape factor curve for $L/D=5$ does not approach $H=1.4$ due to the proximity of the reattachment point to the TE. Moreover, the shape factor curve for $L/D=15$ gradually deviates from $H=1.4$ towards $H=2.59$ near the TE, which is a result of the relaminarization process discussed in § 3.2.2.

Figure 13. Shape factor of the wall boundary layer for different aspect ratios. The shape factor of the turbulent boundary layer with zero pressure gradients ($H=1.4$) and laminar Blasius boundary layer ($H=2.59$) are provided as a dashed line and dash–dotted line, respectively. The dotted lines present the locations of reattachment in the streamwise direction for the three aspect ratios. Here RP stands for the reattachment point.

To investigate the mechanism of boundary layer transition, we examine the profiles of spanwise-averaged streamwise velocity fluctuation intensity at various streamwise locations after reattachment (see figure 14). The grey solid curve represents the optimal disturbance theory proposed by Luchini (Reference Luchini2000). This theory is further supported by the disturbance distributions of bypass transition induced by free stream turbulence (FST) in the experimental study of Matsubara & Alfredsson (Reference Matsubara and Alfredsson2001). Additionally, the $u^{\prime }_{rms}$ profiles obtained from the wall boundary layer transition induced by a wake of a circular cylinder with large gap ratios are consistent with the theory (Ovchinnikov, Piomelli & Choudhari Reference Ovchinnikov, Piomelli and Choudhari2006; Mandal & Dey Reference Mandal and Dey2011; He et al. Reference He, Wang and Pan2013). However, for small gap ratios, these profiles deviate from the optimal disturbance theory due to the strong direct interaction between the boundary layer and the wake, as reported by Li et al. (Reference Li, Wang, Qiu, Wu, Zhou, Fu and Liu2022a). In present flows, the $u^{\prime }_{rms}$ profiles also deviate from the optimal disturbance theory, indicating that the boundary layer transitional mechanism here differs from the bypass transition induced by FST. This deviation may be attributed to the significant interaction between the boundary layer and the KH vortex during the destabilization of the KH vortex. Furthermore, another noteworthy observation is the gradual appearance of a second peak in the $u^{\prime }_{rms}$ profiles within the boundary layer at streamwise locations corresponding to $L/D=10$ and $15$, as depicted in figure 14(b,c), respectively. This second peak signifies the occurrence of boundary layer transition. However, at $L/D=5$, the $u^{\prime }_{rms}$ profiles exhibit only one peak near the wall. This observation aligns with the findings regarding the shape factor, indicating that the boundary layer fails to transition at this particular aspect ratio.

Figure 14. Profiles of the spanwise-averaged streamwise velocity fluctuation intensity normalized by the maximum value of $u^{\prime }_{rms}$ at different streamwise locations: (a) $L/D=5$; (b) $L/D=10$; (c) $L/D=15$. The height in the vertical direction ( $y^{\prime }=y-0.5D$) is normalized by the displacement thickness $\delta ^*$. The thicknesses of the boundary layer are marked by diamonds. The grey solid curve is the optimal disturbance growth theory from Luchini (Reference Luchini2000).

Figure 15(a,b) depict the streamwise velocity fluctuation structures for $L/D=10$ and $15$, respectively. The negative streaks emerge as low-speed structures, a phenomenon observed in both the plane turbulent boundary layer with zero pressure-gradient (Dennis & Nickels Reference Dennis and Nickels2011a,Reference Dennis and Nickelsb) and the transition of the wall boundary layer induced by a wake of a circular cylinder with small gap ratios (Li et al. Reference Li, Wang, Qiu, Wu, Zhou, Fu and Liu2022a). On the other hand, the high-speed structures include positive streaks as well as local motions exhibiting an arch-like shape. The arch-like shape of the high-speed structures seems to be inherited by the shape of the hairpin vortices. They induce a velocity field, which pushes fast flow towards the wall between their legs, and slow flow farther from the wall in the outer region. These arch-like structures straddle the low-speed streaks, resembling the behaviour of hairpin vortices observed in experimental studies (Dennis & Nickels Reference Dennis and Nickels2011a,Reference Dennis and Nickelsb). Notably, for $L/D=15$, the low-speed streaks are lifted up near the TE, which is consistent with the motion of hairpin vortices depicted in figure 4. This lift-up mechanism can be attributed to the strong mean shear (Zaki & Saha Reference Zaki and Saha2009; Monokrousos, Åkervik & Henningson Reference Monokrousos, Åkervik, Brandt and Henningson2010). Additionally, the second peak of fluctuation observed in figure 14(b,c) for $L/D=10$ and $15$, respectively, may be attributed to the sheltering effect caused by the mean shear (Hunt & Durbin Reference Hunt and Durbin1999; Jacobs & Durbin Reference Jacobs and Durbin2001; Li et al. Reference Li, Wang, Qiu, Wu, Zhou, Fu and Liu2022a). Figure 15(c,d) illustrate the DMD mode corresponding to the low frequency $St_L$, indicating that the low-frequency mode is associated with the streaks present inside the boundary layer. This suggests that low-frequency disturbances penetrate the sheltering edge, giving rise to the generation of streaks (Wang, Mao & Zaki Reference Wang, Mao and Zaki2019).

Figure 15. (a,b) Isosurfaces of the streamwise velocity fluctuations and (c,d) isosurfaces of the real part of DMD mode corresponding to $St_L$: (a,c) $L/D=10$; (b,d) $L/D=15$. The red parts are positive and the blue parts are negative.

The disturbance energy, $E_{rms}$, is used to quantify the growth of disturbance strength in the LE shear layer and the wall boundary layer along the streamwise direction. It is defined as $E_{rms}(St_{*})=[(u^\prime _{rms}(St_{*}))^2+(v^\prime _{rms}(St_{*}))^2+(w^\prime _{rms}(St_{*}))^2]/U^2_0$. Figure 16 presents the results of $E_{rms}$ for different aspect ratios, including several components ($E_{rms}(St_{0})$, $E_{rms}(St_{KH})$, $E_{rms}(St_{L})$, $E_{rms}(St_{FB})$) associated with typical frequencies extracted by FMD within the same bandwidth. In the LE shear layer, the overall behaviour of the total disturbance energy $E_{rms}(St_{all})$ exhibits similarities, and it can be divided into four regimes along the streamwise direction for all three flows. In regime I, the LE shear layer is predominantly laminar, as indicated by figures 4 and 6. Therefore, $E_{rms}(St_{all})$ experiences multiple orders of magnitude growth along the streamwise direction in this regime. As the flow progresses, the LE shear layer enters regime II, where $E_{rms}(St_{all})$ grows exponentially following the scaling law $E_{rms}(St_{all})\sim {\rm e}^{2x/D}$. Recalling the evolutions of vortices shown in figure 4, it can be inferred that the formation of the KH vortex results in this growth. In regime III, a second exponential growth stage occurs with $E_{rms}(St_{all})$ increasing according to $E_{rms}(St_{all})\sim {\rm e}^{0.6x/D}$. The three-dimensional destabilization of the KH vortices leads to the development of hairpin vortices (LE vortices), which is the primary cause of the second growth in this regime. This mechanism is similar to the transition flow induced by a wake of a circular cylinder with gap ratios less than or equal to 1.8 (He et al. Reference He, Wang and Pan2013; Li et al. Reference Li, Wang, Qiu, Wu, Zhou, Fu and Liu2022a). Moreover, the three-dimensional instabilities cause a redistribution of disturbance energy during the second exponential growth, as also observed in the transition induced by a circular cylinder wake (Li et al. Reference Li, Wang, Qiu, Wu, Zhou, Fu and Liu2022b). The three-dimensional instabilities eventually lead to the nonlinear saturation of $E_{rms}(St_{all})$. Before entering regime IV, where the LE shear layer begins to reattach to the wall and evolve into the wall boundary layer, $E_{rms}(St_{all})$ reaches its maximum value and saturates. In regime IV, as the hairpin vortices takes shape, $E_{rms}(St_{all})$ gradually decreases following an exponential decay with $E_{rms}(St_{all})\sim {\rm e}^{-0.75x/D}$. As the flow continues downstream, transitioning into the wall boundary layer, it is noteworthy that $E_{rms}(St_{all})$ continues to decrease for $L/D=10$ and $15$, while remaining stable for $L/D=5$. This difference can be attributed to the different relaminarization processes of the boundary layer, which will be further investigated in subsequent subsections. For $L/D=10$ and $15$, the transition of the boundary layer occurs downstream after reaching the saturation state. This confirms that the transitional mechanism in these flows differs from bypass transition induced by FST, probably due to different receptivity processes (Morkovin Reference Morkovin1969; Westin et al. Reference Westin, Boiko, Klingmann, Kozlov and Alfredsson1994; Westin & Henkes Reference Westin and Henkes1997; Kendall Reference Kendall1998; Jacobs & Durbin Reference Jacobs and Durbin2001; Matsubara & Alfredsson Reference Matsubara and Alfredsson2001; Fransson, Matsubara & Alfredsson Reference Fransson, Matsubara and Alfredsson2005).

Figure 16. The streamwise growth of the fluctuations within different frequency bands in the LE shear layer and the boundary layer: (a) $L/D=5$; (b) $L/D=10$; (c) $L/D=15$. The fluctuations are extracted from the orange line (for the LE shear layer) and magenta line (for the boundary layer) in figure 6(b,d,f). The LE shear layer is divided into four regimes labelled I–IV, and the last regime corresponds to the wall boundary layer. Here $St_L$, $St_0$, $St_{FB}$ and $St_{KH}$ denote the low frequency, vortex shedding frequency, feedback frequency and KH fluctuation frequency as presented in figure 8.

The disturbance energy of several typical frequencies also increases rapidly and then decreases gradually along the LE shear layer, shown in figure 16, except for $E_{rms}(St_{0})$ at $L/D=5$. The large-scale oscillation in the vertical direction at $L/D=5$ shown in figure 4, as well as the small distance between the reattachment point and the TE shown in figure 6, could be reason for the increase of disturbance energy $E_{rms}(St_{0})$ inside the boundary layer. The LE shear layer plays a crucial role in the rapid amplification of disturbance energy at different frequencies from regime I to III. At $L/D=10$ and $15$, the disturbance energy $E_{rms}(St_{L})$ for the low frequency $St_L$ is dominant near the LE. As the flow develops in the streamwise direction, it is exceeded by the disturbance energy $E_{rms}(St_{0})$, while the disturbance energy $E_{rms}(St_{0})$ is always dominant at $L/D=5$, rather than $E_{rms}(St_{L})$. As a matter of fact, the instantaneous reattachment point exhibits periodic oscillations around the mean reattachment point. Therefore, the separated shear layer cannot always reattach to the upper/lower surfaces of the cylinder, meaning that the instantaneous reattachment point falls outside the side surface temporarily (Zhang et al. Reference Zhang, Kareem, Xu and Jiang2023). Hence, the intermittent flapping behaviour of the shear layer destroys the integrity of the LE separation bubble, which directly hinders the development of low frequency disturbance. Therefore, the LE shear layer amplifies the $St_0$ disturbance preferentially rather than the $St_L$ disturbance, which can be an important connection with the failure of the flow transition at this aspect ratio. The disturbance energies of the high frequency $St_{KH}$ and $St_{FB}$ are weaker than those of $St_L$ and $St_0$ at $L/D=10$ and $15$, while the development of the $St_{FB}$ disturbance is almost of the same magnitude as that of $St_L$ at $L/D=5$. It could be due to the sheltering effect at higher aspect ratios, closely related to boundary layer transition.

3.2.2. Relaminarization of the boundary layer

In order to further explore the characteristics of the boundary layer, the mean streamwise velocity profiles in wall units at different streamwise locations are presented in figure 17. Both flows ($L/D=10$ and $15$) exhibit turbulent boundary layer characteristics. The profiles very close to the wall agree well with $U^{+}=y^{\prime +}$, while those in the outer region follow the log-law. In the case of $L/D=10$ shown in figure 17(a), the velocity profiles still satisfy the classical slope of $1/\kappa =1/0.4$ at $y^{\prime +}>{\rm 30}$. However, for $10< y^{\prime +}<30$, the profiles exhibit a slope of $1/\kappa =1/0.7$, and the velocity values are significantly smaller compared with that in the buffer layer for a normal turbulent boundary layer. For $L/D=15$ shown in figure 17(b), the slopes of the velocity profiles change from $1/\kappa =1/0.4$ to $1/\kappa =1/0.7$ due to the larger streamwise space provided for boundary layer development. These low-slope velocity profiles are reminiscent of boundary layer relaminarization or reverse transition, related to FPG (Spalart & Watmuff Reference Spalart and Watmuff1993; Na & Moin Reference Na and Moin1998; Manhart & Friedrich Reference Manhart and Friedrich2002; Dixit & Ramesh Reference Dixit and Ramesh2008; Coleman, Rumsey & Spalart Reference Coleman, Rumsey and Spalart2018). Previous studies mainly focused on the effects of the adverse pressure gradient (APG) and FPG on the boundary layer with flows over bumps and hills (Webster, DeGraaff & Eaton Reference Webster, DeGraaff and Eaton1996; Wu & Squires Reference Wu and Squires1998; Cavar & Meyer Reference Cavar and Meyer2011; Matai & Durbin Reference Matai and Durbin2019; Balin & Jansen Reference Balin and Jansen2021). In the present study, the effect of FPG on the development of the wall boundary layer on the upper/lower side of the rectangular cylinder is investigated, which has not been reported in previous studies.

Figure 17. Mean streamwise velocity profiles in wall units at different streamwise locations: (a) $L/D=10$; (b) $L/D=15$. The height in the vertical direction is $y^{\prime }=y-0.5D$. The log-law expression used here is $U^{+}=1/\kappa \ln (\kern0.7pt {y}^{\prime +})+{\rm B}_0$, where ${\rm B}_0$ is the logarithmic layer intercept constant.

To observe the relaminarization phenomenon, it is essential to closely examine the region near the wall. In this regard, we investigate the profiles of the time- and spanwise-averaged pressure coefficient $C_p$ and skin friction coefficient $C_f$ on the surface of a rectangular cylinder (see figure 18a,b), given by

(3.3a,b)\begin{equation} C_p=\frac{\bar{P} -\bar{P}_{ref}}{0.5\rho U^{2}_0},\quad C_f=\frac{\tau_w}{0.5\rho U^{2}_0}, \end{equation}

where $\bar {P}_{ref}$ is the reference free stream pressure, $\bar {P}$ the mean surface pressure and $\tau _w$ the mean wall shear stress. Prior to reattachment, an apparent FPG is observed, as depicted in figure 18(a). Following reattachment, a small section of APG still exists, but it is overcome by a FPG, which persists until the TE. Notably, the FPG is significantly intensified near the TE. Figure 18(b) illustrates that initially, the inverse boundary layer leads to a negative $C_f$. However, after reattachment, $C_f$ develops positively due to the forward boundary layer. The enhanced FPG near the TE contributes to a rapid increase in $C_f$, indicating further acceleration of the flow.

Figure 18. Time- and spanwise-averaged pressure coefficient profiles $C_p$ (a) and skin friction coefficient profiles $C_f$ (b) on the surface of the rectangular cylinder.

The time- and spanwise-averaged pressure coefficient field is depicted in figure 19, illustrating the formation of pressure gradients, including the APG and FPG. Notably, regions of low pressure emerge in areas where the LE vortex or TE vortex is generated. Consequently, the flows first undergo an APG, developing along the streamwise direction, which directly leads to transition ($L/D=10, 15$) or transition tendency ($L/D=5$) in the boundary layer. Subsequently, the flow encounters a FPG, leading to its acceleration and ultimately facilitating the relaminarization of the boundary layer.

Figure 19. Time and spanwise-averaged pressure coefficient field $C_p$: (a) $L/D=5$; (b) $L/D=10$; (c) $L/D=15$.

Two parameters are applied here to examine the accelerated flow quantitatively. The first parameter is the acceleration parameter $\Delta P$ (Matai & Durbin Reference Matai and Durbin2019; Balin & Jansen Reference Balin and Jansen2021; Uzun & Malik Reference Uzun and Malik2021), and the second one is the relaminarization parameter $K$ (Balin & Jansen Reference Balin and Jansen2021; Uzun & Malik Reference Uzun and Malik2021), defined as

(3.4a,b)\begin{equation} \Delta P=\frac{\nu}{\rho u_\tau ^3}\frac{\partial \bar{P}}{\partial x},\quad K=\frac{\nu}{U_e^2}\frac{\partial U_e}{\partial x}, \end{equation}

where $u_\tau$ is the wall friction velocity, $\bar {P}$ the mean surface pressure and $U_e$ the boundary layer edge velocity. It should be noted that $\Delta P$ accounts for the wall region of boundary layer because of the friction velocity incorporated, but $K$ accounts for the outer region of the boundary layer, based on the boundary layer edge velocity (Uzun & Malik Reference Uzun and Malik2021). Furthermore, when $\Delta P$ is less than $-0.018$, a departure from the logarithmic layer takes place, which is believed to indicate the onset of relaminarization (Patel Reference Patel1965; Patel & Head Reference Patel and Head1968; Uzun & Malik Reference Uzun and Malik2021). The threshold value of $K=3\times 10^{-6}$ has been widely accepted as a point above which relaminarization can occur (Narayanan & Ramjee Reference Narayanan and Ramjee1969; Narasimha & Sreenivasan Reference Narasimha and Sreenivasan1973; So & Mellor Reference So and Mellor1975; Balin & Jansen Reference Balin and Jansen2021; Uzun & Malik Reference Uzun and Malik2021).

The variations of these two parameters along the streamwise direction are shown in figure 20. At the boundary layer wall region, the parameter $\Delta P$ reaches its critical value for all three flows and remains below the threshold until the TE. This indicates strong accelerations in the near-wall region for these three situations, which have a strong relationship with the FPG. These findings are consistent with the results of $C_p$ and $C_f$ shown in figures 18 and 19. For $L/D=10$ and $15$, the parameter $K$ reaches the threshold as the flow progresses in the streamwise direction. Although $K$ initially increases and then decreases after exceeding its corresponding threshold, it always remains greater than this value until the TE. Therefore, relaminarization occurs in the outer region of the boundary layer for these two flows. However, the parameter $K$ for $L/D=5$ cannot reach the threshold because the boundary layer is in a transitional state, as illustrated in figures 13 and 14. Furthermore, $\Delta P$ reaches the corresponding threshold earlier than $K$, indicating that the FPG in the near-wall region is stronger than in the outer region for $L/D=10$ and $15$. The flow of $L/D=15$ experiences acceleration by the FPG over a longer distance compared with the flow of $L/D=10$, resulting in a more prominent boundary layer relaminarization phenomenon for $L/D=15$. As presented in the result of $H$ (see figure 13), the shape factor curve for $L/D=15$ deviates from that of a turbulent boundary layer and approaches that of laminar flow. However, this deviation is not observed in the flow of $L/D=10$. Similarly, the mean streamwise velocity profiles at $L/D=10$ still exhibit a slope of $1/\kappa =1/0.4$ in the outer region, while they have transitioned to $1/\kappa =1/0.7$ for $L/D=15$.

Figure 20. Acceleration parameter $\Delta P$ and relaminarization parameter $K$: (a) $L/D=5$; (b) $L/D=10$; (c) $L/D=15$.

The influence of the FPG on the turbulent structures in the boundary layer is being further investigated, with a particular focus on its impact on the geometrical characteristics of the hairpin vortex packet. To analyse this influence, the two-point correlation $\rho _{uu}$ is employed (Dennis & Nickels Reference Dennis and Nickels2011a), and defined in the $x$–$y$ plane as

(3.5)\begin{equation} \rho_{uu}(x_{ref}, y_{ref}, x, y^\prime)=\frac{\langle u^\prime(x, y^\prime) u^\prime(x_{ref}, y_{ref})\rangle }{\sqrt{\langle u^{\prime 2}(x,y^\prime)\rangle \langle u^{\prime 2}(x_{ref},y_{ref})\rangle }}, \end{equation}

where $(x_{ref}, y_{ref})$ is the reference point. In this study, specific reference points are chosen to analyse the influence of the FPG on the hairpin vortex packet. For the case with $L/D=10$, the reference points are $(8,0.1\delta )$ and $(9,0.1\delta )$, while for $L/D=15$, the reference points are $(12,0.1\delta )$ and $(14,0.1\delta )$, where $\delta$ represents the local boundary layer thickness. The results are presented in figure 21. The streamwise scales of the hairpin vortex packet gradually increase as it develops downstream in both flows. This growth is attributed to the generation of new hairpin vortices by the background mean shear. Interestingly, the inclination angle $\alpha$ of the hairpin vortex packet decreases during the downstream development for both $L/D=10$ (from $16.77^{\circ }$ to $15.84^{\circ }$) and $L/D=15$ (from $13.94^{\circ }$ to $10.03^{\circ }$). This decrease in inclination angle can be attributed to the influence of the FPG. Furthermore, it is observed that the inclination angle of the hairpin vortex packet at locations experiencing less FPG (figure 21a–c) is significantly larger than that in the zero pressure-gradient boundary layer (between $8^{\circ }$ and $12^{\circ }$) (Adrian, Meinhart & Tomkins Reference Adrian, Meinhart and Tomkins2000; Dennis & Nickels Reference Dennis and Nickels2011a). This difference is primarily due to the rapid lift-up of the hairpin vortex caused by the strong shear over a short streamwise distance after reattachment, as discussed in § 3.1.1.

Figure 21. Two-point correlations of the streamwise fluctuation velocity in the $x$–$y$ plane: (a,b) $L/D=10$; (c,d) $L/D=15$. The reference points are (a) $(8D,0.1\delta D)$; (b) $(9D,0.1\delta D)$; (c) $(12D,0.1\delta D)$; (d) $(14D,0.1\delta D)$, where $\delta$ is the local boundary layer thickness. The straight dash–dotted line (black) is inclined to show the inclination angle of the correlational structure. The inclination angle $\alpha$ is (a) 16.77$^{\circ }$; (b) 15.84$^{\circ }$; (c) 13.94$^{\circ }$; (d) 10.03$^{\circ }$.

3.3.1. Interaction between the LE vortex and TE vortex

The generation of the LE vortex has been described in detail in § 3.1.1. Here, the shedding process of the TE vortex from the TE shear layer is explored, including the interaction between the LE vortex and TE vortex. To reveal the dynamic behaviours of the LE vortex and TE vortex near the TE, as well as their interaction, the three-dimensional FTLEs is applied. The FTLEs help to visualize the flow patterns and identify the dominant vortices. The results are presented in figures 22–24, which depict the flows in a combination of the $x$–$y$ plane and $z$–$y$ plane slices. The abundant turbulent structures can be observed clearly. It is important to note that the focus of the analysis is on the evolution of the LE vortex (also known as the hairpin vortex) and the TE vortex as observed on the $x$–$y$ plane.

As depicted in figure 22(a) ($L/D=5$), the LE vortex located on the upper side of the cylinder, takes the form of a hairpin vortex and has moved near the TE at time $t=t_0$. Simultaneously, the TE shear layer on the same side of the cylinder rolls up, giving rise to a new upper TE vortex. Indeed, both the LE vortex and the TE vortex on the opposite side have detached from the cylinder together and evolve into the wake. By the time the flow reaches $t=t_0+4\Delta t$ (figure 22b), the upper TE vortex has significantly strengthened and increased in size, while the upper LE vortex has moved above the TE vortex. At $t=t_0+8\Delta t$ (figure 22c), the LE vortex has overtaken the TE vortex and almost completely separated from the cylinder. However, the TE vortex has shed from the TE shear layer induced by the LE vortex. Hence, at this aspect ratio, the TE vortex is generated and shed in phase with the LE vortex arriving at the TE. This shedding pattern is referred to as ipsilateral vortex shedding in this paper. Although it has been observed in two-dimensional flows with the same aspect ratio of $L/D=5$ and low Reynolds number ($Re=400$) (Chiarini et al. Reference Chiarini, Quadrio and Auteri2022c), it is reported for the first time in the three-dimensional flows in this paper.

Figure 22. Interaction between the LE vortex and TE vortex revealed by FTLEs presented in the slices for $L/D=5$: (a) $t=t_0$, $z/D={\rm \pi}$, $x/D=4.5$; (b) $t=t_0+4\Delta t$, $z/D={\rm \pi}$, $x/D=4.0$; (c) $t=t_0+8\Delta t$, $z/D={\rm \pi}$, $x/D=4.0$, where $\Delta t=0.25D/U_0$.

When it comes to the flow with an aspect ratio of $L/D=10$, as depicted in figure 23, the interaction between the LE and TE vortices follows a different process compared with that of $L/D=5$. At time $t=t_0$, as shown in figure 23(a), the lower LE vortex has reached the TE, while the upper LE vortex is positioned farther upstream. The phase difference in the generation of LE vortices on the upper and lower sides contributes to their disparate locations. Simultaneously, the upper TE shear layer has rolled up, forming a new upper TE vortex. Moving forward to $t=t_0+3\Delta t$, illustrated in figure 23(b), the lower LE vortex begins to overtake the TE, and the upper TE vortex strengthens and prepares to shed from the TE shear layer. However, the upper LE vortex approaches the upper TE, accompanied by the rolling up of the lower TE shear layer. Finally, at $t=t_0+8\Delta t$, presented in figure 23(c), the lower LE vortex completely overtakes the TE, and the upper TE vortex sheds off. Meanwhile, the upper LE vortex reaches the TE, and the lower TE vortex grows, albeit being squeezed by the lower LE vortex. Consequently, in this flow, the shedding pattern of the TE vortex occurs in phase with the opposite-side LE vortex reaching the TE. This shedding pattern is referred to as contralateral vortex shedding. It was also observed in a two-dimensional study by Chiarini et al. (Reference Chiarini, Quadrio and Auteri2022c) for the flow with $L/D=7$ at $Re=400$. This paper reports the first observation of contralateral vortex shedding in three-dimensional flows as well. The interaction between the LE vortex and TE vortex at $L/D=15$, as depicted in figure 24, is similar to the ipsilateral vortex shedding observed for $L/D=5$ and will not be discussed further here.

Figure 23. Interaction between the LE vortex and TE vortex revealed by FTLEs presented in the slices for $L/D=10$: (a) $t=t_0$, $z/D={\rm \pi}$, $x/D=8.0$ and $9.3$; (b) $t=t_0+3\Delta t$, $z/D={\rm \pi}$, $x/D=8.3$ and $9.8$; (c) $t=t_0+8\Delta t$, where $\Delta t=0.25D/U_0$, $z/D={\rm \pi}$, $x/D=9.7$ and $7.0$.

Figure 24. Interaction between the LE vortex and TE vortex revealed by FTLEs presented in the slices for $L/D=15$: (a) $t=t_0$, $z/D={\rm \pi}$, $x/D=14.2$; (b) $t=t_0+5\Delta t$, $z/D={\rm \pi}$, $x/D=14.4$; (c) $t=t_0+10\Delta t$, $z/D={\rm \pi}$, $x/D=14.8$, where $\Delta t=0.25D/U_0$.

To further validate the shedding pattern for various aspect ratios, the cross-correlation of filtered streamwise fluctuation velocity between the LE vortex and TE vortex is examined. The cross-correlation is calculated as follows:

(3.6a,b)\begin{equation} R_{AB}(\Delta \tau )=\frac{\langle u^\prime_A(t) u^\prime_B(t+\Delta \tau ) \rangle }{\sqrt{\langle u^{\prime 2}_A(t)\rangle \langle u^{\prime 2}_B(t)\rangle }},\quad R_{AC}(\Delta \tau )=\frac{\langle u^\prime_A(t) u^\prime_C(t+\Delta \tau )\rangle }{\sqrt{\langle u^{\prime 2}_A(t) \rangle \langle u^{\prime 2}_C(t)\rangle }}, \end{equation}

where $u^\prime _A$, $u^\prime _B$ and $u^\prime _C$ are streamwise fluctuation velocities filtered with $St_0$ from the points A, B and C, respectively. The point A is inside the wake where the upper TE vortex sheds, while point B and C are situated in the upper and lower boundary layer of the rectangular cylinder near the TE. At these two points, the upper and lower LE vortices pass through these two points periodically, respectively. The spatial coordinates of the points A, B and C are $(5.5,0.35)$, $(4.8,0.8)$, $(4.8,-0.8)$ for $L/D=5$; $(10.8,0.35)$, $(9.8,0.7)$, $(9.8,-0.7)$ for $L/D=10$; $(16,0.35)$, $(14.8,0.7)$, $(14.8,-0.7)$ for $L/D=15$. The results of cross-correlation are illustrated in figure 25.

Figure 25. Cross-correlation of streamwise fluctuation velocity between the LE vortex and TE vortex: (a) $L/D=5$, points A$(5.5,0.35)$, B$(4.8,0.8)$, C$(4.8,-0.8)$; (b) $L/D=10$, points A$(10.8,0.35)$, B$(9.8,0.7)$, C$(9.8,-0.7)$; (c) $L/D=15$, points A$(16,0.35)$, B$(14.8,0.7)$, C$(14.8,-0.7)$. Circles and diamonds mark the maximum value of $R_{AB}$ and $R_{AC}$, respectively.

It can be observed that the results for $L/D=5$ and $L/D=15$ (in figure 25a,c, respectively) exhibit similar characteristics. In both cases, $R_{AB}$ reaches its maximum first ($\Delta \tau <0$), followed by $R_{AC}$ reaching its maximum ($\Delta \tau >0$) after half of the vortex shedding period. This indicates that the upper LE vortex passes through point B (the upper TE) first, followed by the shedding of the TE vortex on the same side, and then the lower LE vortex passes through point C (the lower TE). Consequently, the phase of the LE vortex passing through the TE coincides with that of the TE vortex shedding on the same side in these two flows. However, at $L/D=10$, $R_{AC}$ achieves its maximum first, and $R_{AB}$ reaches its maximum after half of the vortex-shedding period. Hence, the phase of the LE vortex passing the TE coincides with that of the TE vortex shedding on the opposite side in this flow. These cross-correlation results are consistent with the evolution processes of these vortices presented by FTLEs in figures 22–24. Although these two vortex-shedding patterns have been identified in previous two-dimensional studies at low $Re$ (Chiarini et al. Reference Chiarini, Quadrio and Auteri2022c), it has been verified that they also occur in the present complex three-dimensional flows at medium $Re$. Moreover, contralateral vortex shedding results in stronger interactions between different vortex structures in the wake compared with ipsilateral vortex shedding. This is because the Kármán-type vortex at $L/D=5$ and $L/D=15$ can maintain coherence over a larger streamwise distance in the wake than at $L/D=10$, as shown in figure 9.

3.3.2. Decay of the wake

To investigate the features of the plane turbulent wakes, the streamwise behaviour of the centreline velocity defect $U_s$ is conducted, shown in figure 26. Here $U_s$ is following the equation (Pope Reference Pope2000)

(3.7)\begin{equation} U_{s}(x^{\prime})=\frac{U_0- U(x^{\prime},0)}{U_0}. \end{equation}

Figure 26. Streamwise behaviour of the centreline velocity defect $U_s$ in the wake. Inset: the velocity defect $U_s$ compensated with the power-law scaling $x^{\prime 1/2}$.

The velocity defect curves for $L/D=10$ and $15$ exhibit a significant overlap, indicating similar behaviour. These curves also show higher defects compared with the curve for $L/D=5$ until $x^{\prime }/D=28$ ($x^{\prime }=x-L$). This streamwise position is defined as the point where the local slope of $U_s x^{\prime 1/2}$ is less than 0.01. However, beyond this point, the velocity defects for all three flows converge and become nearly identical. Moreover, in this regime ($x^{\prime }/D>28$), all three flows demonstrate an identical self-similarity characterized by a power law relationship $U_s\sim x^{\prime -1/2}$. This observed self-similarity accords with the theoretical derivation by Pope (Reference Pope2000), and it has also been documented in studies by Chiarini & Quadrio (Reference Chiarini and Quadrio2021) and Cimarelli et al. (Reference Cimarelli, Leonforte and Angeli2018a) in the wake of $L/D=5$ at high $Re$. However, it is important to note that the self-similarity of the plane turbulent wake emerges at a much earlier stage, specifically for $x^{\prime }/D>10$, at the high Reynolds number in the investigations conducted by Chiarini & Quadrio (Reference Chiarini and Quadrio2021) and Cimarelli et al. (Reference Cimarelli, Leonforte and Angeli2018a). This finding suggests that as the Reynolds number increases, the plane turbulent wake reaches the state of self-similarity decay more rapidly.

In order to further investigate the decay characteristics of turbulent wakes, the disturbance energy along the centreline of the wake is examined. Figure 27(a) illustrates that the disturbance energy at all frequencies gradually decays after the separation bubble behind the TE. In the streamwise domain under consideration, the disturbance energy for $L/D=5$ is greater than that for the other two flows. As the wakes decay, the disturbance energies for $L/D=10$ and $15$ become almost equal to each other. During the velocity self-similarity decay regime, the disturbance energy also exhibits self-similar decay, following a trend of $E_{rms}(St_{all})\sim {\rm e}^{-0.028x^{\prime }/D}$. Figure 4 illustrates the presence of significant large-scale oscillations in the vertical direction at $L/D=5$. These oscillations provide a plausible explanation for the observed higher wake disturbance energy compared with the other two flows. To further investigate this phenomenon, the vertical disturbance energy at all frequencies is examined and depicted in figure 27(b). As anticipated, the vertical disturbance energy is indeed greater for the flow of $L/D=5$ when compared with the other flows.

Figure 27. Streamwise development of the centreline fluctuations in the wake. (a) The disturbance energy at all frequencies; (b) the vertical disturbance energy at all frequencies; (c) the disturbance energy at $St_0$; (d) the vertical disturbance energy at $St_0$.

The different vortex-shedding patterns could result in various Kármán-type vortex decay behaviours. Consequently, the disturbance energy at the vortex-shedding frequency $St_0$ is calculated and depicted in figure 27(c). In the near wake region, the disturbance energy at $St_0$ is found to be lowest for $L/D=10$. This can be attributed to the contralateral vortex-shedding pattern, which leads to stronger interactions between turbulent structures in the wake. These interactions directly suppress the development of a Kármán-type vortex, as shown in figure 9. During the velocity self-similarity decay regime, the disturbance energy at $St_0$ exhibits a decay behaviour described by $E_{rms}(St_{0})\sim {\rm e}^{-0.038x^{\prime }/D}$. The absolute value of this exponent is greater than that for the total disturbance energy $E_{rms}(St_{all})$, indicating that the decay of disturbance energy at $St_0$ contributes more significantly to the overall decay of disturbance energy in the wake. In other words, the decay of the wake is primarily driven by the decay of the Kármán-type vortex street. Similarly, the vertical disturbance energy at $St_0$ is calculated and presented in figure 27(d). Its decay process closely resembles that of $E_{rms}(St_0)$, with the lowest values still observed for $L/D=10$. This suggests that the contralateral vortex-shedding pattern preferentially suppresses vertical fluctuations in the wake.