4.1. Flow properties

Figure 5 shows time-averaged velocity profiles for $-1\le z/D\le 1.4, x/D=1.0$, 2.0 and $y/D=0$ at each $\alpha$. Comparison is made with the time-averaged velocity profile obtained by Yokota et al. (Reference Yokota, Ochiai, Ozawa, Nonomura and Asai2021) only for the case of $\alpha =0^\circ$ in figure 5(a,b). The $z$ component agrees well at $z/D\le 0.5$ regardless of $x/D$, but it does not agree outside this region, and the trends are different. This is caused by the problem of particle introduction in the free-stream region in the velocity field measurements of the previous study, which led to the occurrence of no-particle regions sometimes. In addition, the acrylic wall used for the test section wall was scratched and unnatural changes were observed in some areas. The trends in the $x$ component agree well, but bias errors are observed in the positive velocity direction. Yokota & Nonomura (Reference Yokota and Nonomura2024) considered that a factor for this bias error was related to the size of the particle on the image. The optical set-up for the stereo-PIV measurements in the present study was the same as in the previous study, and the bias error could be attributed to the same factor. However, the error in the present study was about 6.3 % of the free stream for $x/D=1.$0 and $z/D=0$, which is smaller than in the previous study.

Figure 5.

The time-averaged velocity profiles for each component (red: $x$, blue: $y$, yellow: $z$) in the case of (a,b) $\alpha =0^\circ$, (c,d) 5$^\circ$, (e,f) 10$^\circ$ and (g,h) 15$^\circ$ at (a,c,e,g) $x/D=1.0$ and (b,d,f,h) 2.0. The solid lines and the dotted lines represent the results in the present study and the previous study (Yokota et al. Reference Yokota, Ochiai, Ozawa, Nonomura and Asai2021), respectively.

A reverse flow region due to the recirculation region is observed regardless of $\alpha$ near the centre of the wake at $x/D=1.0$. In the following discussion, the positive and negative sides of the $z$ axis are referred to as the top and bottom sides, respectively. Figure 5(a,c,e) shows that the position with minimum $u_x$ moves in the positive direction of the $z$ axis with increasing $\alpha$ for the range of 0$^\circ$–10$^\circ$. A sign switch in $u_z$ is observed near the position, indicating that the flow is entrained towards the centre of the wake. The profile also loses symmetry to $z/D=0$, especially the weaker reverse flow on the bottom side can be seen at $\alpha =10^\circ$. However, the relationship is reversed at $\alpha =15^\circ$, where the reverse flow on the bottom side is stronger. Furthermore, the $u_z$ profile is also changed, and there is a flow from near the centre of the wake towards the free-stream side. The velocity profile at $x/D=2.0$ in figure 5(b,d,f,h), which corresponds to the outside of the recirculation region, shows a similar trend to the case of $x/D=1.0$, but the position with minimum $u_x$ moves in the positive $z$ direction compared with the case of $x/D=1.0$. Tian et al. (Reference Tian, Hu, Lu and Yang2017) and Gao et al. (Reference Gao, Tao, Tian and Yang2018) showed that the vortex structures formed in the wake convect away from the $x$ axis as $x/D$ increases, which is consistent with the results of the present study.

Figure 6 shows the $z$-direction profile of turbulence statistics. Figure 6(a,c,e,g) shows the turbulent kinetic energy $k_{3C}$ and figure 6(b,d,f,h) shows the root-mean-square (RMS) values of velocity fluctuations $u_{j,RMS}$ for each component. The calculation is as follows, which is the same as in Yokota & Nonomura (Reference Yokota and Nonomura2024):

(4.1)$$\begin{gather} k_{{3C}} = \frac{1}{2}\sum_j(u_{j, {{RMS}}}/U)^2, \end{gather}$$

(4.2)$$\begin{gather}u_{j, {{RMS}}} = \sqrt{\frac{1}{N_{PIV}}{\sum_{n=1}^{N_{PIV}}}{u'_{j, n}}^2}. \end{gather}$$

The results of Yokota et al. (Reference Yokota, Ochiai, Ozawa, Nonomura and Asai2021) are plotted in figure 6(b) for comparison, as with the time-averaged velocity.

Figure 6.

The profiles of (a,c,e,g) the turbulent kinetic energy $k_{3C}$ and (b,d,f,h) the RMS of velocity fluctuations $u_{j, RMS}$ for each component (red: $x$, blue: $y$, yellow: $z$) in the case of (a,b) $\alpha =0^\circ$, (c,d) 5$^\circ$, (e,f) 10$^\circ$ and (g,h) 15$^\circ$. The solid lines and the dotted lines represent the results in the present study and the previous study (Yokota et al. Reference Yokota, Ochiai, Ozawa, Nonomura and Asai2021), respectively.

Figure 6(a) shows that $k_{3C}$ reaches a maximum around $z/D=\pm 0.5$ for $\alpha =0^\circ$ and $x/D=1.0$. This $z$ position is close to the shear layer position as shown in figure 5(a). Large velocity fluctuations associated with the recirculation bubble pumping and the large-scale vortex shedding have been reported near the shear layer in the time-averaged field (Yokota et al. Reference Yokota, Ochiai, Ozawa, Nonomura and Asai2021). This is because the shear layer position fluctuates in the radial direction of the cylinder due to pressure fluctuations caused by the characteristic phenomena, and it experiences high velocities in the free-stream region and low velocity flows inside the recirculation region at that position. The RMS profile of $u_x'$ at $x/D=1.0$ in figure 6(b) supports this discussion. Moreover, figure 6(a,c,e) shows that the symmetry of the profile at $x/D=1.0$ is broken when $\alpha$ is increased in the range of $0\leq \alpha \leq 10^\circ$, and the position with the maximum $k_{3C}$ is on the top side. Furthermore, the maximum $k_{3C}$ gradually increases as $\alpha$ increases. Within this range of $\alpha$, it can be observed that the maximum $k_{3C}$ at $x/D=2.0$ is near the centre of the wake for a given $\alpha$, as shown in figure 6(a,c,e). Similarly, the RMS of the $u_y'$ and $u_z'$ in figure 6(b,d,f,h) shows an almost constant profile near the wake centre for $x/D=1.0$, while they become larger there for $x/D=2.0$. In the investigation in the wake of a free-stream-aligned circular cylinder with $L/D=1.0$ (Yokota et al. Reference Yokota, Ochiai, Ozawa, Nonomura and Asai2021), the PSD distribution of velocity fluctuations associated with the large-scale vortex shedding showed large $u_z'$ at the wake centre in addition to large $u_x'$ near the shear layer in the downstream of the recirculation region.

Interesting results at $\alpha =15^\circ$ are shown as with the time-averaged velocity profiles. Figure 6(g) shows that $k_{3C}$ on the top side is smaller, and a comparable magnitude of $k_{3C}$ occurs on the bottom side. Yokota & Nonomura (Reference Yokota and Nonomura2024) reported that $k_{3C}$ downstream of the cylinder base is generally lower than in the case of non-reattaching flow when the reattachment of the separated shear layer occurs. This was considered to be due to the collapse of the relatively large vortex structure formed before reattachment by impingement on the curved surface and the reduction of the vortex strength by viscous force. A similar trend was observed in the present study for $0\leq \alpha \leq 10^\circ$, but the results for $\alpha =15^\circ$ show a different trend from this report. On the other hand, the turbulence intensity distribution near the windward curved surface at $\alpha =15^\circ$, shown by Yokota et al. (Reference Yokota, Asai and Nonomura2023), decreased downstream from the reattachment point. This result supports the discussions of Yokota & Nonomura (Reference Yokota and Nonomura2024). The strong velocity fluctuations in the bottom side for $\alpha =15^\circ$ and $x/D=1.0$ are discussed further in § 4.2 with the results of the SPOD analysis.

4.2. Characteristic SPOD modes

The spectra of the field of velocity fluctuation obtained by the SPOD for each $\alpha$ are shown in figure 7. These spectra were obtained by integrating the eigenspectra of modes $n_{POD}=1\unicode{x2013}4$ because the contribution is small for $n_{POD}\ge 5$. The red and blue lines in the figure are spectra calculated from the field of velocity fluctuations at $x/D=1.0$ and 2.0, respectively. At first, a clear peak is identified from the spectra regardless of $\alpha$. The peak frequencies are slightly different for $x/D=1.0$ and 2.0, but the peak frequency at $x/D=2.0$ is highlighted in the figure with a dashed line. A peak at $St=0.129$ is observed in the non-reattaching flow at $\alpha =0^\circ$ and 5$^\circ$. A peak of velocity fluctuations due to the large-scale vortex shedding at $St\approx 0.13$ was observed in the previous study on the wake of a free-stream-aligned circular cylinder ($\alpha =0^\circ$) (Yokota & Nonomura Reference Yokota and Nonomura2024), and the peak at $St=0.129$, which appears in the $\alpha$ range with the non-reattaching flow in the present study, is considered to be the result of this phenomenon. In the case of a reattaching flow at $\alpha =10^\circ$ and 15$^\circ$, in contrast, a peak appears at $St=0.143$. Calvert (Reference Calvert1967) investigated the change in vortex shedding frequency with respect to $\alpha$ in a disc wake and reported that the vortex shedding frequency increases with increasing $\alpha$. The same trend is observed in the present study, and the fluctuations at $St=0.143$ are also considered to correspond to the large-scale vortex shedding. However, differences in the change in the vortex shedding frequency with respect to $\alpha$ are observed between the disc wake and the wake of the cylinder with $L/D=1.0$. Calvert (Reference Calvert1967) reported that the vortex shedding frequency of the disc wake increases monotonically as $\alpha$ increases, whereas the vortex shedding frequency tends to vary stepwise between the non-reattaching and reattaching flows in the present study. The peak frequency of wake position fluctuations differs in the $y$ and $z$ directions, as will be discussed in § 4.3, suggesting that the frequency of large-scale vortex shedding depends on the direction. The peak frequency in the $y$ position varies little with $\alpha$, but the difference of it in the $z$ position is large for the non-reattaching and reattaching flows. The peak frequencies in the spectra in figure 7 are approximately between the peak frequencies in each direction, which results in a step-like change in the peak frequency. The velocity fluctuations by the large-scale vortex shedding become smaller as $\alpha$ increases, which can be observed in figure 7.

Figure 7.

Mode-integrated spectra obtained from velocity fluctuations at $x/D=1.0$ and 2.0 in the case of (a) 0$^\circ$, (b) 5$^\circ$, (c) 10$^\circ$ and (d) 15$^\circ$.

Secondly, the low-frequency region of figure 7 shows larger fluctuations at $x/D=1.0$ than that at $x/D=2.0$ regardless of $\alpha$. Here, $x/D=1.0$ corresponds to the inside of the recirculation region. Although no distinct peaks appear as with one at the vortex shedding frequency, there is a clear difference in the magnitude of the fluctuations inside and outside the recirculation region, and this characteristic change should be focused upon. Yokota & Nonomura (Reference Yokota and Nonomura2024) identified the $m=0$ ($m$: azimuthal wavenumber) mode of the recirculation bubble pumping, the $m=1$ mode similar to the very-low-frequency (VLF) mode and the $m=2$ mode of the streak structure in the low-frequency region of $St\le 0.05$. The recirculation bubble pumping is a phenomenon in which the recirculation region expands or contracts in the free-stream direction, and the fluctuations are more noticeable for $\alpha =0^\circ$ in the case of a non-reattaching flow. The VLF mode, which has an $m=1$ structure inside the recirculation region and transitions to an $m=2$ structure near the downstream end of the recirculation region, has been identified in the wake of an axisymmetric bluff body (Zhang & Peet Reference Zhang and Peet2023). The recirculation bubble pumping or $m=1$ mode, similar to the VLF mode, shows significant fluctuations inside the recirculation region. The fluctuations in the low-frequency region decrease with increasing $\alpha$ in the range of $\alpha \leq 10^\circ$, while the fluctuations increase at $\alpha =15^\circ$.

The discussions of the fluctuations in the low-frequency region and vortex shedding frequency focused on above are continued with the eigenfunctions at $x/D=2.0$. The modes in the low-frequency region are discussed first.

Figures 8 and 9 show the real part, amplitude and phase of the spatial mode for $St=0.019, n_{POD}=1$ at each $\alpha$, and the odd and even rows show the $u_x'$ and $u_z'$, respectively. The figure shows that the modes are symmetric with respect to $y=0$ at any $\alpha$. Large $u_x'$ is observed inside the recirculation region for $\alpha =0^\circ$ and 5$^\circ$ in figure 8(a,g), with almost axisymmetric fluctuations with respect to the wake centre. Yokota & Nonomura (Reference Yokota and Nonomura2024) showed a similar distribution in the first POD mode with $m=0$ in the wake of a cylinder with $L/D=1.0$ and $\alpha =0^\circ$. They also identified large fluctuations at $St\approx 0.024$ in the PSD of mode coefficients and considered that these were caused by the recirculation bubble pumping. Therefore, the large fluctuations in the low-frequency region observed in figure 7 are considered to be related to the recirculation bubble pumping. The recirculation region is shorter/longer in the free-stream direction compared with the mean field when $u_x'$ is positive/negative (Yokota et al. Reference Yokota, Ochiai, Ozawa, Nonomura and Asai2021; Yokota & Nonomura Reference Yokota and Nonomura2024).

Figure 8.

(a,d,g,j) The real part, (b,e,h,k) amplitude and (c,f,i,l) phase of eigenfunctions of (a–c,g–i) $u_x$ and (b–f,j–l) $u_z$ fluctuations at $St=0.019$ for $x/D=2.0$ in the case of (a–f) 0$^\circ$ and (g–l) 5$^\circ$. The white masking is applied at the position where the mode amplitude is less than 0.005 and satisfies $\sqrt {(y/D)^2+(z/D)^2}>1$.

Figure 9.

(a,d,g,j) The real part, (b,e,h,k) amplitude and (c,f,i,l) phase of eigenfunctions of (a–c,g–i) $u_x$ and (b–f,j–l) $u_z$ fluctuations at $St=0.019$ for $x/D=2.0$ in the case of (a–f) 10$^\circ$ and (g–l) 15$^\circ$. The white masking is applied at the position where the mode amplitude is less than 0.005 and satisfies $\sqrt {(y/D)^2+(z/D)^2}>1$.

A similar structure can be observed for $\alpha =10^\circ$ as shown in figure 9(a), but the phase difference between the fluctuations at the top and bottom sides is larger, which suggests that the timing of expansion and contraction of the recirculation region at the top and bottom sides is different. In addition, the spatial pattern of $u_x'$ on the top side does not take a maximum at $y=0$ for $\alpha =15^\circ$. Moreover, $u_z'$ comparable to $u_x'$ is found between the top and bottom sides with the recirculation region scaling, which is apparent in figure 9(d,j). The strong fluctuations appearing at $y/D=\pm 1.0$ are ignored in the discussion because they are caused by the error vectors due to the reflected laser light on the model during the stereo-PIV measurements. Although similar $u_z'$ is also found for $\alpha =5^\circ$, the magnitude of the fluctuations is smaller relative to $u_x'$ for $\alpha =10^\circ$ and 15$^\circ$. This $u_z'$ is not present for $\alpha =0^\circ$. Therefore, this $u_z'$ is present at $\alpha \neq 0^\circ$ and is considered to be more apparent when a reattaching flow is formed. The cause of the $u_z'$ can be explained by the phase difference between the expansion and contraction of the recirculation flow on the top and bottom sides. Figure 10 shows the schematics of the change in the recirculation bubble pumping by $\alpha$. Figure 10(a) illustrates the recirculation bubble pumping at $\alpha =0^\circ$. Since the recirculation flow on the top and bottom side contracts simultaneously, the equivalent entrainment occurs symmetrically towards the cylinder axis behind the recirculation region. As a result, $u_z'$ becomes zero at the downstream end of the recirculation region due to the balance of the flow. On the other hand, in the case of $\alpha \neq 0^\circ$, the recirculation flow is asymmetric to the $xy$ plane, resulting in a difference in the entrainment strength in the $z$ direction, as shown in figure 10(b). From the above reasons, $u_z'$ shown in the figures 8(j) and 9(d,j) is considered to appear.

Figure 10.

The schematic of the recirculation bubble pumping at (a) $\alpha =0^\circ$ and (b) $\alpha \neq 0^\circ$.

Modes with vortex shedding frequencies are discussed next. Figure 11 shows the spatial modes of each velocity component at $St=0.129$ and $n_{POD}=1\unicode{x2013}4$ for $\alpha =0^\circ$. Figure 11(a,b) shows large $u_x'$ distributions that are symmetric and antisymmetric with respect to $y=0$, respectively. These modes correspond to vortex shedding in the $z$ and $y$ directions, respectively, and the negative $u_x'$ region is considered to correspond to the vortex shedding position. A similar $u_x'$ distribution appears in the first POD mode with $m=1$ in the investigation of Yokota & Nonomura (Reference Yokota and Nonomura2024). They also reported that the flow appears near the wake centre from the positive $u_x'$ region to the negative $u_x'$ region and vice versa outside, from the negative $u_x'$ region to the positive $u_x'$ region. In other words, vortices with positive and negative streamwise vorticity are observed for $y>0$ and $y<0$, respectively, in the case of the $u_x'$ distribution shown in figure 11(a). These vortices are represented by the $u_y'$ and $u_z'$ distributions in figure 11(h,i) for the symmetric $u_x'$ distribution and in figure 11(g,j) for the antisymmetric $u_x'$ distribution. The reason why the distribution showing these vortices cannot be represented by a single mode is that the decomposition into symmetric and antisymmetric modes results in a symmetric or antisymmetric distribution of fluctuations in all velocity components for a single mode.

Figure 11.

Eigenfunctions of (a–d) $u_x$, (e–h) $u_y$ and (i–l) $u_z$ fluctuations of the (a,e,i) first, (b,f,j) second, (c,g,k) third and (d,h,l) fourth modes at $St=0.129$ for $\alpha =0^\circ$ and $x/D=2.0$.

The shape of the mode pattern changes with $\alpha$ when it is not 0$^\circ$, but a structure is basically similar in the $\alpha$ range of the present study. Figures 12 and 13 show the symmetric and antisymmetric $u_x'$ modes at the vortex shedding frequencies, respectively. The width in the $y$ direction of the region with large fluctuations varies with $\alpha$. The width is wide on the top side when $\alpha >0^\circ$, while the width is narrow on the bottom side. The phase distributions in figure 12(f,i) indicate that spatial pattern shifts in the positive $z$ direction in the symmetric mode for $\alpha =5^\circ$ and 10$^\circ$. Here, a pattern shift occurs in the direction of phase decrease. These are more noticeable for $\alpha =10^\circ$. However, the shift is observed only on the top side for $\alpha =15^\circ$, i.e. not on the bottom side. Instead of this, a shift to the positive $z$ direction is observed on the bottom side of the antisymmetric mode at this $\alpha$. Although not shown in the paper, the trend in the spatial patterns of $u_y'$ and $u_z'$ in figure 11(e–l) is similar to that of $u_x'$, where the width in the $y$ direction of the region with large fluctuations expands/shrinks at the top/bottom sides as $\alpha$ increases.

Figure 12.

(a,d,g,j) The real part, (b,e,h,k) amplitude and (c,f,i,l) phase of eigenfunctions of $u_x$ fluctuations symmetric to $y=0$ at the peak frequency for $x/D=2.0$ in the case of (a–c) 0$^\circ$, (d–f) 5$^\circ$, (g–i) 10$^\circ$ and (j–l) 15$^\circ$. The white masking is applied at the position where the mode amplitude is less than 0.005 and satisfies $\sqrt {(y/D)^2+(z/D)^2}>1$.

Figure 13.

(a,d,g,j) The real part, (b,e,h,k) amplitude and (c,f,i,l) phase of eigenfunctions of $u_x$ fluctuations antisymmetric to $y=0$ at the peak frequency for $x/D=2.0$ in the case of (a–c) 0$^\circ$, (d–f) 5$^\circ$, (g–i) 10$^\circ$ and (j–l) 15$^\circ$. The white masking is applied at the position where the mode amplitude is less than 0.005 and satisfies $\sqrt {(y/D)^2+(z/D)^2}>1$.

The large $k_{3C}$ on the bottom side appearing for $\alpha =15^\circ$ shown in figure 6(g) is discussed here. This large fluctuation on the bottom side is highly contributed by $u_x'$, which is evident from figure 6(h). The distributions of $u_x'$ due to the recirculation bubble pumping and the large-scale vortex shedding have been presented in this section. The mode of $u_x'$ at the vortex shedding frequency shows comparable magnitudes on the top and bottom sides, which is shown in figure 12(k). Furthermore, the fluctuations at the vortex shedding frequencies decrease monotonically with increasing $\alpha$, as shown in figure 7. In contrast, the $u_x'$ on $y=0$ by the bubble pumping in figure 9(h) is larger on the bottom side and smaller on the top side. The low-frequency fluctuations are larger for 15$^\circ$ when comparing $\alpha =10^\circ$ and 15$^\circ$, which is discussed above. From the above, it is considered that the reason for the large fluctuations on the bottom side for $\alpha =15^\circ$ is the low-frequency expansion and contraction of the recirculation region on the bottom side.

4.3. Behaviour of wake position

Investigations on the wake and vortex shedding position in the flow around a bluff body have been carried out for various geometries (Grandemange, Gohlke & Cadot Reference Grandemange, Gohlke and Cadot2013; Rigas et al. Reference Rigas, Oxlade, Morgans and Morrison2014, Reference Rigas, Morgans, Brackston and Morrison2015; Yang et al. Reference Yang, Liu, Wu, Zhong and Zhang2014; Zhang & Peet Reference Zhang and Peet2023; Yokota & Nonomura Reference Yokota and Nonomura2024). Yokota & Nonomura (Reference Yokota and Nonomura2024) investigated the large-scale vortex shedding in the wake of a free-stream-aligned circular cylinder and clarified that vortex shedding patterns can be roughly classified into the following three types: anticlockwise/clockwise circular and flapping patterns. A trend in the distance from the wake centre to the vortex shedding position depending on the vortex shedding pattern was also suggested, and it was found that the distance is stabilised at a certain distance when the vortex shedding exhibits a circular pattern. Furthermore, low-frequency fluctuations in the distance were related to the recirculation bubble pumping. In this section, the change in the vortex shedding position with respect to $\alpha$ is investigated.

The previous studies (Grandemange et al. Reference Grandemange, Gohlke and Cadot2013; Yang et al. Reference Yang, Liu, Wu, Zhong and Zhang2014; Yokota & Nonomura Reference Yokota and Nonomura2024) assumed the barycentre of momentum deficit calculated from the velocity distribution in the free-stream direction (before Reynolds decomposition) as the wake position. However, as mentioned above, error vectors caused by laser reflections on the model appeared in the present study. Not shown in the paper, the time-averaged velocity field is particularly affected by reflected light. The wake positions calculated by the equations used in previous studies are determined by fluctuations from the time-averaged velocity field, with greater weights for positions with negative velocity fluctuations. Therefore, the positions calculated from the region with negative velocity fluctuations are considered to be available for qualitative discussions. The present study calculates the barycentre of the region with negative velocity fluctuations from the instantaneous velocity fluctuation field reconstructed from the SPOD modes using the following equation and treats this as the wake position:

(4.3)$$\begin{gather} y_m(t) = \frac{\displaystyle \int yu'(y, z, t)/U \,{\rm d}S}{\displaystyle \int u'(y, z, t)/U \,{\rm d}S}, \end{gather}$$

(4.4)$$\begin{gather}z_m(t) = \frac{\displaystyle \int zu'(y, z, t)/U \,{\rm d}S}{\displaystyle \int u'(y, z, t)/U\, {\rm d}S}, \end{gather}$$

(4.5) $$\begin{gather}u'(y, z, t) = \begin{cases} 0 & \text{if }\widetilde{u'_x}(y, z, t)\ge0, \\ \widetilde{u'_x}(y, z, t) & \text{if }\widetilde{u'_x}(y, z, t)<0. \\ \end{cases} \end{gather}$$

Here the examined area was set as $-1.2\leq y/D\leq 1.2$ and $-1.2\leq z/D\leq 1.4$.

Figure 14 illustrates the wake position for 24.7 s at $x/D=2.0$ for each $\alpha$. Figure 14(c,d) shows that the range of it in the $y$ direction is narrower for $\alpha =10^\circ$ and 15$^\circ$ than for 0$^\circ$ and 5$^\circ$. Gao et al. (Reference Gao, Tao, Tian and Yang2018) reported that the vortex shedding position appears helical for small $\alpha$ when the disc wake is CS, but a flapping pattern appears when $\alpha$ is increased. The disc is rotated around the $z$ axis in the previous study, and a plot of the aerodynamic force coefficients $C_y$ and $C_z$ acting on the disc is observed such that the fluctuations of $C_z$ become smaller as $\alpha$ increases. Since the cylinder was angled around the $y$ axis in the present study, the trend of the range of the wake position with $\alpha$ is in agreement with the study by Gao et al. (Reference Gao, Tao, Tian and Yang2018).

Figure 14.

The wake position at $x/D=2.0$ in the case of (a) 0$^\circ$, (b) 5$^\circ$, (c) 10$^\circ$ and (d) 15$^\circ$.

The ellipses centred on $\overline {y_m}$ and $\overline {z_m}$ are shown in figure 15(a) with the standard deviations of $y_m$ and $z_m$ as the major and minor axes, respectively, for ease of understanding the change in the wake position with the angle of attack. Figure 15(b) shows those standard deviations and the variation of $\overline {z_m}$ with $\alpha$. The figure shows that $\overline {z_m}$ moves in the positive $z$ direction with increasing $\alpha$ in the range of $\alpha \leq 10^\circ$, which is consistent with the trend of wake position by changing $\alpha$ in the previous studies (Tian et al. Reference Tian, Hu, Lu and Yang2017; Gao et al. Reference Gao, Tao, Tian and Yang2018). The $\overline {z_m}$ is closer to the origin at $\alpha =15^\circ$ than at 10$^\circ$. This is considered to be caused by the greater deceleration of the velocity in the $x$ direction at the bottom side (figure 5g,h). Nevertheless, the ellipse at $\alpha =15^\circ$ is almost encompassed by the ellipse at 10$^\circ$, as shown in figure 15(a), which can be explained by the small fluctuations of the wake position due to the large-scale vortex shedding at 15$^\circ$. Yokota et al. (Reference Yokota, Ochiai, Ozawa, Nonomura and Asai2021) investigated the wake of a free-stream-aligned circular cylinder and reported that the velocity fluctuations due to the large-scale vortex shedding were smaller for $L/D=2.0$ than for $L/D=1.5$. Here, the intermittent reattachment and steady reattachment are considered to occur for $L/D=1.5$ and 2.0, respectively. An analogous difference between $\alpha =10^\circ$ with intermittent reattachment and 15$^\circ$ with steady reattachment is considered to appear. The trend of $\sigma _{y_m}$ with respect to $\alpha$ is also consistent with the above observation that the range of $y_m$ is narrower.

Figure 15.

(a) The ellipse drawn by $\overline {y_m}, \overline {z_m}, \sigma _{y_m}$ and $\sigma _{z_m}$ at $x/D=2.0$ for each $\alpha$ and (b) the change of them except with $\overline {y_m}$ by $\alpha$.

The wake position in the polar coordinate system is then calculated from it in the Cartesian coordinate system by the following equation:

(4.6)$$\begin{gather} r_m(t) =\sqrt{(y_m(t)-\overline{y_m})^2+ (z_m(t)-\overline{z_m})^2}, \end{gather}$$

(4.7)$$\begin{gather}\theta_m(t) = \arctan \frac{z_m(t)-\overline{z_m}}{y_m(t)- \overline{y_m}}(-{\rm \pi}\le\theta_m\le{\rm \pi}). \end{gather}$$

The polar coordinate system introduced by Gao et al. (Reference Gao, Tao, Tian and Yang2018) was based on the origin in a fixed Cartesian coordinate system, which makes investigation difficult with regard to fluctuations in $r_m$ and $\theta _m$ when the wake centre deviates from the origin for $\alpha \neq 0^\circ$. Therefore, the present study adopts a polar coordinate system based on the mean wake position, shown in (4.6) and (4.7). Figure 16 shows the PSD calculated from the positional fluctuations of the wake in each direction $y_m, z_m, r_m$ and $\theta _m$, respectively. Here, five runs of velocity data for 9.7 s each were used for the calculation. First, clear peaks appear in the $y_m$ and $z_m$ fluctuations regardless of $\alpha$. The peak frequencies of $y_m$ and $z_m$ coincide at $St=0.129$ for $\alpha =0^\circ$, and the frequencies also coincide with the mode-integrated spectrum in figure 7. On the other hand, the peak frequencies do not coincide for $y_m$ and $z_m$ when $\alpha \neq 0^\circ$. The fluctuation frequency of $z_m$ is higher in all $\alpha \neq 0^\circ$ cases, and the difference is more apparent for $\alpha =10^\circ$ and 15$^\circ$ shown in figure 16(a,b). The $y_m$ fluctuation frequency shows almost no change with $\alpha$. The fluctuation frequency of $z_m$ gradually increases in the range of $0\leq \alpha \leq 10$, and its frequency does not change after the reattachment. Gao et al. (Reference Gao, Tao, Tian and Yang2018) reported that the dominant frequency of $C_y$ fluctuations (corresponding to aerodynamic force fluctuations in the $z$ direction in the present study) increases with increasing $\alpha$ in the PS state, which is consistent with the results of the present study. The isosurface of streamwise vorticity showed that the streamwise wavelength of the vortex structure becomes short as $\alpha$ increases (Gao et al. Reference Gao, Tao, Tian and Yang2018), which may contribute significantly to the increase of the dominant frequency. The peak frequencies in the mode-integrated spectra in figure 7 are approximately in the middle of the peak frequencies of $y_m$ and $z_m$ fluctuations. The magnitude of the fluctuations at the peak frequency of $y_m$ and $z_m$ decreases with increasing $\alpha$, consistent with the trend observed in the mode-integrated spectra. In particular, the fluctuations at the peak frequency of $z_m$ and $St=0.01$ are equivalent at $\alpha =15^\circ$.

Figure 16.

Power spectral densities of fluctuations of (a) $y_m/D$, (b) $z_m/D$, (c) $r_m/D$ and (d) $\theta _m/2{\rm \pi}$ at $x/D=2.0$.

Fluctuations of $r_m$ and $\theta _m$ are discussed next. Figure 16(d) illustrates that the peak frequency of the $\theta _m$ fluctuations is the same as the peak frequency of $z_m$ regardless of $\alpha$. A peak frequency of $r_m$ fluctuations is double that of $z_m$ at $0\leq \alpha \leq 10^{\circ}$, as shown in figure 16(c). Fluctuations are also large at that frequency for $\alpha =15^\circ$, but no peaks are observed. There are three vortex shedding patterns in the wake of a free-stream-aligned circular cylinder, as mentioned earlier. Yokota & Nonomura (Reference Yokota and Nonomura2024) discussed that the positional fluctuations of vortex shedding in the azimuthal direction have a peak at the same frequency as the velocity fluctuation peak regardless of the pattern, while the peak frequency of fluctuation in the radial direction is double the peak frequency of the velocity fluctuation when the pattern exhibits a flapping. Therefore, a flapping pattern of wake position is also observed in the case of a cylinder with $\alpha$, but the magnitude of the fluctuations decreases with increasing $\alpha$. They become comparable to low-frequency fluctuations at $\alpha =15^\circ$, which is the reason why the peak is not observed. The $r_m$ fluctuations are dominated by low-frequency fluctuations regardless of $\alpha$, which was also observed in the study by Yokota & Nonomura (Reference Yokota and Nonomura2024). However, the $r_m$ fluctuations at $St\approx 0.02$ are smaller for the case of $\alpha =10^\circ$ and 15$^\circ$ with the reattaching flow than for the case of $\alpha =0^\circ$ and 5$^\circ$ with the non-reattaching flow. They found that these low-frequency fluctuations at $St\approx 0.02$ were related to the recirculation bubble pumping. Low-frequency fluctuations for $\alpha \neq 0^\circ$ are also considered to be related to the recirculation bubble pumping, but it has not been clear whether this relationship appears for $\alpha =10^\circ$ and 15$^\circ$. This relationship is discussed in § 4.4.

Figure 17 shows the probability distribution of $r_m$ and $\theta _m$ at $x/D=2.0$ for 24.7 s at each $\alpha$. The $r_m$ and $\theta _m$ used in the calculation of these probability distributions were obtained using (4.6) and (4.7) after bandpass filtering of $y_m$ and $z_m$ with $0.1\le St\le 0.2$ to focus on the large-scale vortex shedding. Figure 17(a,c,e,g) shows that $r_m$ does not have a peak at zero regardless of $\alpha$. This indicates that the flow field is quasisteady asymmetric, which is in agreement with the results obtained by Yokota & Nonomura (Reference Yokota and Nonomura2024). This has also been observed in the wake of an axisymmetric bluff body (Rigas et al. Reference Rigas, Oxlade, Morgans and Morrison2014, Reference Rigas, Morgans, Brackston and Morrison2015). The $r_m$ where the probability distribution is maximum becomes small with increasing $\alpha$, which is considered to correspond to a narrower wake width. In contrast, figure 17(b,d,h) shows that the probability distribution for $\theta _m$ is almost uniform for $\alpha =0^\circ$, 5$^\circ$ and 15$^\circ$, which implies that there is no bias in the wake position. The number of bins is 40 and the probability in each bin for a uniform distribution is 0.025. The red broken line at this value is overlaid on the probability distribution. On the other hand, the results for $\alpha =10^\circ$ shown in figure 17(f) represent that the probability increases around $\pm {\rm \pi}/2$. It was expected that vortex shedding would show a flapping pattern for large $\alpha$ as reported by Gao et al. (Reference Gao, Tao, Tian and Yang2018) and that the probability would increase around $\pm {\rm \pi}/2$. However, such a trend is observed only in the case of $\alpha =10^\circ$ in the present study. Although a reattaching flow is formed for $\alpha \ge 10^\circ$, no increase in probability is observed around $\pm {\rm \pi}/2$ at $\alpha =15^\circ$, suggesting that intermittent flow reattachment causes a bias in the azimuthal position of the wake.

Figure 17.

The probability distribution of (a,c,e,g) $r_m/D$ and (b,d,f,h) $\theta _m$ at $x/D=2.0$ in the case of (a,b) 0$^\circ$, (c,d) 5$^\circ$, (e,f) 10$^\circ$ and (g,h) 15$^\circ$. The mean and standard deviation of $r_m/D$ are shown in (a,c,e,g).

Yokota & Nonomura (Reference Yokota and Nonomura2024) further reported that the vortex shedding pattern causes a trend in $r_m$. Figure 18(a) shows the time history of the wake position at $\alpha =15^\circ$ for 24.7 s. The irregular switching of the rotational direction can be observed. Here, the bandpass filtering of $0.1\le St\le 0.2$ is applied before the calculation of $\theta _m$. Figure 18(b–d) focuses on a short-time variation of $\theta _m$. The red highlighted points in figure 18(b,c) show that the azimuthal position varies smoothly with a positive or negative gradient, respectively. The wake positions for this time period are plotted in figure 18(e,f), which show that they are circular with different rotational directions. The angular velocity is close to the vortex shedding frequency while $r_m$ remains almost constant in this case. On the other hand, the red points in figure 18(d) show step-like changes with near-zero or very large gradients. The change in wake position in figure 18(g) shows a linear trajectory passing near the $\overline {y_m}$ and $\overline {z_m}$. These circular and linear trajectories are considered to correspond to the circular and flapping patterns of vortex shedding reported in the previous study (Yokota & Nonomura Reference Yokota and Nonomura2024). The trend of $\theta _m$ change for each pattern is also similar. Subsequently, using the above features, an analysis of conditional sampling based on the gradient of $\theta _m$, i.e. angular velocity, is carried out as in the previous study. The following conditional equations identify the pattern of vortex shedding:

(4.8) $$\begin{gather} \text{state is} \begin{cases} \text{anticlockwise} & \text{if }\dfrac{\omega_m}{2{\rm \pi}}\ge0, \\ \text{clockwise} & \text{if }\dfrac{\omega_m}{2{\rm \pi}}<0, \\ \text{anticlockwise circular} & \text{if }0.1\le\dfrac{\omega_m}{2{\rm \pi}}\le0.2, \\ \text{clockwise circular} & \text{if }-0.2\le\dfrac{\omega_m}{2{\rm \pi}}\le-0.1, \\ \text{flapping} & \text{if }\left|\dfrac{\omega_m}{2{\rm \pi}}\right|<0.1\text{ or } \left|\dfrac{\omega_m}{2{\rm \pi}}\right|>0.2,\\ \end{cases} \end{gather}$$

(4.9)$$\begin{gather}\omega_m = \frac{{\rm d}\theta_m}{{\rm d}(tU/D)}. \end{gather}$$

No adjustment of the conditional equation was made because the range of the peak frequency variation is small, which is in the range of $0.1\le St\le 0.2$ for all cases, while the peak frequency varies with $\alpha$ and a direction as described earlier. This cutoff frequency is the same as that used in the previous study, and a comparison of the wake position change shown in figure 18(b–d) with the wake azimuthal position change shown in figure 18(e–g) was also made in the present study. The sample points were correctly selected to satisfy the conditions.

Figure 18.

The temporal variation of the wake position at $x/D=2.0$ for (a) the whole measurement time and (b–d) the time of trajectories shown in (e–g), respectively. The red dots correspond to the moment of the trajectories.

Figures 19(a,c,e,g) and 19(b,d,f,h) show the probability distributions obtained by sampling $r_m$ when the vortex shedding pattern exhibits an anticlockwise circular pattern or a flapping pattern, respectively. Here, the blue histogram is the probability distribution calculated from the data for the whole measurement time (24.7 s) and is presented for comparison with the results of the conditional sampling. Figure 19(a,c,e,g) shows that the probability is high around the $r_m$ where the blue histogram has a maximum when it shows an anticlockwise circular pattern. In contrast, the probability is low in that region for the flapping patterns in figure 19(b,d,f,h) and is higher in the other regions. Yokota & Nonomura (Reference Yokota and Nonomura2024) suggested that the vortex shedding position follows a Mexican-hat shaped potential field in the wake of a free-stream-aligned circular cylinder, by combining the results of conditional sampling with the discussions in the study of the wake of an axisymmetric bluff body (Rigas et al. Reference Rigas, Morgans, Brackston and Morrison2015). The fluctuations in the $r$ position of vortex shedding are small when the vortex shedding follows a circular pattern and is considered to be stable around the potential well. On the other hand, a flapping pattern is considered to appear when the vortex shedding position moves away from the cylinder axis, allowing it to pass through the potential well and also to cross the potential peaks near the cylinder axis. From the above, the same trends as with the 0$^\circ$ case are observed for $\alpha \neq 0^\circ$ in the range of the present study, and it can be concluded that flow characteristics specific to vortex shedding patterns appear. The results for the clockwise circular pattern are not shown here but are similar to those for the anticlockwise circular pattern.

Figure 19.

The probability distribution of $r_m/D$ obtained by sampling under the condition that the state of vortex shedding is (a,c,e,g) anticlockwise circular and (b,d,f,h) flapping at $x/D=2.0$ in the case of (a,b) 0$^\circ$, (c,d) 5$^\circ$, (e,f) 10$^\circ$ and (g,h) 15$^\circ$. The red and blue histograms show the results of the conditional sampling and the whole measurement time, respectively. The mean and standard deviation of the conditional sampling results are shown in each subplot.

4.4. Relationships between recirculation bubble pumping and large-scale vortex shedding

The position in $r$ of the vortex shedding and the streak strength affect the length of the recirculation region in the wake of a free-stream-aligned circular cylinder (Yokota & Nonomura Reference Yokota and Nonomura2024). The correlation between the recirculation bubble pumping and vortex shedding position was particularly high, showing a coherence of approximately 0.75. Although low-frequency fluctuations of $r_m$ are also observed in the present study, it is unclear whether this relationship exists for $\alpha =10^\circ$ and 15$^\circ$, since the fluctuation energy at $St\approx 0.02$ is lower in figure 16(c). Therefore, the coherence and phase difference between the reconstructed velocity fluctuation field and $r_m$ were calculated by the following equations, and the relationship of those was investigated:

(4.10)$$\begin{gather} {\rm Coh}_{XY}(f) = \frac{|\langle X^*(f)Y(f)\rangle|^2}{\langle X^*(f)X(f)\rangle\langle Y^*(f)Y(f)\rangle}, \end{gather}$$

(4.11)$$\begin{gather}\theta_{XY}(f) = \arctan\frac{{\rm Im}({\langle X^*(f)Y(f)\rangle})}{{\rm Re}({\langle X^*(f)Y(f)\rangle})} \times \frac{180}{\rm \pi}. \end{gather}$$

Here $X$ and $Y$ are the frequency spectra of the velocity and $r_m$ fluctuations, respectively, and the operator $\langle \rangle$ denotes the ensemble average. The FFT analysis length and overlap in this calculation were set to 100 and 50 %, respectively.

Figure 20 shows the coherence and phase difference between the $\widetilde {u'_x}$ and $r_m$ fluctuations in the case with $St=0.019$ and $x/D=2.0$ for each $\alpha$. Figure 20(a) illustrates that the coherence is high near the cylinder axis for $\alpha =0^\circ$, with a value of approximately 0.55. Moreover, the phase difference is approximately 50$^\circ$ in the high coherence region, shown in figure 20(b). The distributions of coherence and phase difference are axisymmetric and similar to the spatial mode pattern related to the recirculation bubble pumping shown in figure 8(a). This indicates that the recirculation bubble pumping is associated with low-frequency fluctuations in $r_m$, which is in agreement with the report by Yokota & Nonomura (Reference Yokota and Nonomura2024). The case of $\alpha =5^\circ$ in figure 20(c,d) shows the very similar coherence and phase difference distribution as the 0$^\circ$ case, but the region with high coherence shifts a bit towards the positive $z$ direction. This is consistent with the change in spatial pattern with $\alpha$ shown in figure 8. The coherence decreases for $\alpha =10^\circ$ in figure 20(e) and is approximately 0.4. In the case of $\alpha =15^\circ$ in figure 20(g), the coherence is much lower, approximately 0.2. Yokota & Nonomura (Reference Yokota and Nonomura2024) discussed that the recirculation region is shorter when the mixing caused by the large-scale vortex shedding is strong, while the recirculation region is longer when the mixing is weak. The $r_m$ increases when the fluctuations of the large-scale vortex shedding are large. Figures 7 and 16 show that the fluctuations at the peak frequency become small with increasing $\alpha$, especially in the case of the reattaching flow. In other words, mixing becomes weak as $\alpha$ increases, and the influence of the $r_m$ on the state of the recirculation region is considered to be weaker.

Figure 20.

(a,c,e,g) Coherence and (b,d,f,h) phase difference between the reconstructed velocity and $r_m$ fluctuations at $St=0.019$ for $x/D=2.0$ in the case of (a,b) 0$^\circ$, (c,d) 5$^\circ$, (e,f) 10$^\circ$ and (g,h) 15$^\circ$.