4.1. Flow properties

Figure 4 shows the temporal and azimuthal averaged velocity profiles at $x/D = 1.0$ and 2.0 for the wake of the cylinder with each $L/D$. The results for free-stream and radial velocities are compared with the PIV results of Yokota et al. (Reference Yokota, Ochiai, Ozawa, Nonomura and Asai2021); (two dimensional two component), except for $x/D = 2.0$ for $L/D = 1.5$ and 2.0, respectively. They compared the length of the recirculation region behind the cylinder with that behind the disk obtained in previous studies and found good agreement; therefore, the data obtained by them are reliable. The figures illustrate that the velocity in the free-stream direction in all cases deviates significantly in the positive direction compared with the previous study, with a maximum difference of approximately 30 % of the free-stream velocity, even though the trend is in agreement with that of the previous study. The error is considered to be due to the small size of the particle on the image. The effect of peak locking is more apparent when the particle image is small, resulting in bias errors. A camera arrangement such as in the present study, where the camera captures the base of the cylinder, increases the distance between the measurement plane and the camera, leading to a reduction in the spatial resolution. The measurements were carried out with the highest possible lens f-value for preventing peak locking, but the f-number was set to 8 considering the brightness of the particle image and the effect of reflected light on the model. Particularly, the low spatial resolution in the free-stream direction in the stereo PIV measurements of the present study is considered to be responsible for the large errors. The two components corresponding to the in-plane components are in good agreement with the results of the previous study for the $r$ component, while the $\theta$ component is almost zero regardless of the $r$ position, indicating the axisymmetry of the wake of the cylinder. The velocity profiles of the previous study show an unnatural high-wavenumber oscillation in some places due to the time during the measurement when particles could not be introduced into the free-stream region and the scratched lines on the acrylic wall used in the test section.

Figure 4. The time-averaged velocity profiles for each component (red: $x$, blue: $r$, yellow: $\theta$) in the case of (a,b) $L/D=1.0$, (c,d) 1.5 and (e,f) 2.0 at (a,c,e) $x/D=1.0$ and (b,d,f) 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.

Next, the profiles of the turbulence statistics are discussed. Figure 5 shows the profile of turbulent kinetic energy $k_{{3C}}$ calculated from the three components measured in the present study and the r.m.s.values of fluctuations of each velocity component are shown. The calculations are as follows:

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

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

All statistics for the data obtained in the present study are averaged in the azimuthal direction. The r.m.s.values are compared with the results of the previous study in the same way as with the case of time-averaged velocity profiles.

Figure 5. The profiles of (a,c,e) the turbulent kinetic energy $k_{3C}$ and (b,d,f) the r.m.s. of velocity fluctuations $u_{j, rms}$ for each component (red: $x$, blue: $r$, yellow: $\theta$) in the case of (a,b) $L/D=1.0$, (c,d) 1.5 and (e,f) 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.

Figures 5(a,c,e) show that $k_{{3C}}$ is large at $r/D<0.5$, which is behind the cylinder and decreases rapidly in the free-stream region. This tendency is more noticeable at $x/D=1.0$, which is close to the cylinder. In addition, for a given $L/D$, $k_{{3C}}$ decreases over the entire $r$ position downstream, and the profile also becomes linear from a curve with a large change from the free-stream region to the wake. This is considered to indicate that small vortices or large vortex structures in the turbulence of the wake of the cylinder are convected and weakened by viscous dissipation. Furthermore, a comparison of figures 5(a,c,e) illustrates that $k_{{3C}}$ decreases as $L/D$ increases. The reason for this $L/D$ dependence is considered to be related to the reattachment of flow separated at the leading edge to the curved surface of the cylinder. A non-reattaching flow with no flow reattachment occurs at $L/D=1.0$, while a reattaching flow is formed at $L/D=1.5$ and 2.0. This flow classification was made by Yokota et al. (Reference Yokota, Ochiai, Ozawa, Nonomura and Asai2021) based on the time-averaged velocity field, and they reported that the presence or absence of flow reattachment switches at $L/D=1.5$. Note that there is intermittency in flow reattachment for $L/D=1.5$. Higuchi et al. (Reference Higuchi, Sawada and Kato2008) reported that the vortex structure formed in the separated shear layer in the case of the reattaching flow is smaller than those in the non-reattaching flow and that the instantaneous velocity also decreases as it approaches the trailing edge. Yokota et al. (Reference Yokota, Ochiai, Ozawa, Nonomura and Asai2021) showed the change by $L/D$ in power spectral densities of velocity fluctuations that are consistent with the report of Higuchi et al. (Reference Higuchi, Sawada and Kato2008) and discussed that small vortices formed in the shear layer produce low velocity fluctuations behind the cylinder. The process by which small eddies are formed can be considered as follows. Yokota et al. (Reference Yokota, Asai and Nonomura2022) reported that KH vortices in the separated shear layer grow, merge and form a relatively large vortex structure, and $Q$-criterion distributions and PSD showed that this vortex structure is even formed before reattachment in the case of $L/D=1.5$ and 2.0. Reattachment of the flow could result in this vortex impinging on the curved surface, collapsing the large-scale vortex structure and breaking it up into smaller vortices. The kinetic energy of the flow may also be reduced by viscous drag due to the formation of a turbulent boundary layer on the surface downstream of the reattachment point. Hence, the velocity fluctuations are considered to be smaller in the case of the reattaching flow than in the case of the non-reattaching flow because the smaller vortices with reduced kinetic energy are shed from the trailing edge of the cylinder. In short, $k_{{3C}}$ is considered to be smaller in the order of $L/D=1.0$, where a steady large-scale structure is observed, $L/D=1.5$, where an intermittent large-scale structure is observed, and $L/D = 2.0$, where the vortex structure is small.

The profiles of r.m.s. values of the velocity fluctuations in each direction are shown in figures 5(b,d,f). The results of Yokota et al. (Reference Yokota, Ochiai, Ozawa, Nonomura and Asai2021) are plotted together in these figures for comparison. However, as data for $x/D=2.0$ could not be obtained for the cases $L/D=1.5$ and 2.0 in the previous study, comparisons were not made at that position. As shown in figures 5(d,f), for $L/D=1.5$ and 2.0, the magnitude of the radial velocity fluctuations differs from the previous study for $r/D > 0.6$, while the profiles are consistent for the other positions and cases. Unnatural changes in the $r$ direction are also observed, which could have been caused by problems in PIV measurements due to the particle introduction and the scratched lines on the acrylic wall in the previous study, as described above. The streamwise velocity fluctuations are larger at locations where the change in the time-averaged streamwise velocity in the $r$ direction is greater, i.e. at the shear layer location, and are particularly large for $L/D=1.0$, which is in the case of the non-reattaching flow. On the other hand, the radial and circumferential velocity fluctuations are large at the wake centre and decrease towards the positive $r$ direction. These velocity fluctuations in each direction show a profile similar to the normalised turbulent stresses in the disk wake reported by Nidhan et al. (Reference Nidhan, Chongsiripinyo, Schmidt and Sarkar2020). Here, their paper shows results for $20< x/D<120$. The change in profile in the $r$ direction becomes more gradual downstream of any $L/D$, resulting in a linear turbulent kinetic energy distribution downstream, as shown in figures 5(a,c,e).

4.2. Characteristic fluctuations

4.2.1. Eigenspectra

Figure 6 shows the eigenspectra obtained for each $L/D$ and $x/D=1.0$ and 2.0 by applying the modal decomposition described in § 3.3 to the velocity fluctuation field. The range is for the azimuthal mode $m=0\unicode{x2013}10$ and leading POD mode $n=1\unicode{x2013}4$. At first, eigenvalues become smaller as $L/D$ increases. This is a similar $L/D$-dependent trend to that of $k_{{3C}}$ shown in the figure, corresponding to the large velocity fluctuations in the non-reattaching flow and the smaller fluctuations in the reattaching flow. Regardless of $L/D$ and the $x$ position, the dominant mode is the $m=1$ mode, which is larger than the other azimuthal modes, especially for $L/D=1.0$ and 1.5. On the other hand, for $L/D=2.0$, the magnitudes of the eigenvalues of $m=1$ and the second contributor, $m=2$, are comparable. The order of contribution of azimuthal modes other than $m=1$ depends on $L/D$ or $x/D$. Comparing the magnitude of the eigenvalues of the $n=1$ mode for each azimuthal mode, the order of contribution for higher azimuthal wavenumbers than the $m=1$ mode decreases with increasing wavenumbers while the order of contribution of the axisymmetric mode $m=0$ varies with $L/D$ or $x/D$. The contribution order of the axisymmetric mode for $L/D=1.0$ is fifth after $m=4$ at $x/D=1.0$, sixth after $m=5$ at $x/D=1.4$ (not shown) and seventh after $m=6$ at $x/D=2.0$.

Figure 6. The eigenspectra in the case of (a,b) $L/D=1.0$, (c,d) 1.5 and (e,f) 2.0 at (a,c,e) $x/D=1.0$ and (b,d,f) 2.0.

Since the present study focuses on the large-scale wake structure in the non-reattaching flow, the subsequent discussion will be mainly on the wake behind the cylinder with $L/D=1.0$. Nidhan et al. (Reference Nidhan, Chongsiripinyo, Schmidt and Sarkar2020) applied spectral POD (SPOD) to velocity fluctuation fields in the wake of a disk which forms a non-reattaching flow, and showed eigenspectra integrated in the frequency direction. Their results show that the $m=0$ mode contributes third after the $m=2$ mode at $x/D=0.1$ and fourth after the $m=3$ mode at $x/D\geq 1.0$. Here, the results are compared for a disk and a cylinder with $L/D=1.0$, which occur for the non-reattaching flow. Kuwata et al. (Reference Kuwata, Abe, Yokota, Nonomura, Sawada, Yakeno, Asai and Obayashi2021) showed that the distance from the leading edge of the cylinder to the downstream end of the recirculation region varies almost linearly with $L/D$ from the results of Fail, Lawford & Eyre (Reference Fail, Lawford and Eyre1957) and Yokota et al. (Reference Yokota, Ochiai, Ozawa, Nonomura and Asai2021) and them. Although the distance between $L/D=0$ and 1.0 differs by $0.34D$, this difference is not considered to be significant, and the result of Nidhan et al. (Reference Nidhan, Chongsiripinyo, Schmidt and Sarkar2020) at $x/D=2.0$ is compared with that of $L/D=1.0$ at $x/D=1.0$, with the same distance from the leading edge. This position is inside the recirculation region in both cases. As mentioned above, the order of contribution of the azimuthal modes for a disk is $1\geq 2\geq 3\geq 0\geq \cdots$, whereas for a cylinder the order is $1\geq 2\geq 3\geq 4\geq 0\geq \cdots$ as shown in figure 6(a). This suggests that even if the flow structure is similar, the order of contribution of the azimuthal modes changes with $L/D$.

Next, we consider a large-scale wake structure that corresponds to each mode in the case of $L/D=1.0$. Yokota et al. (Reference Yokota, Ochiai, Ozawa, Nonomura and Asai2021) reported that large-scale vortex shedding appears in the wake at $L/D\leq 1.5$, which causes velocity fluctuations that are antisymmetric with respect to the cylinder axis. Since the amplitude of the $m=1$ mode is larger than those of the other azimuthal modes at $L/D=1.0$ and 1.5, as mentioned above, and the $m=1$ mode shows an antisymmetric distribution with respect to the cylinder axis, this mode is considered to include the velocity fluctuations by the large-scale vortex shedding. Furthermore, steady symmetry-broken flow, represented by the $m=1$ mode, has been observed in the wake of a disk and an axisymmetric bluff body, which is also considered to be a factor in the dominance of the $m=1$ mode. The next point to note is that the energy of the $m=0$ mode is large only when $L/D=1.0$. The $m=0$ mode is an axisymmetric velocity fluctuation and the relatively high energy in the recirculation region at $x/D=1.0$ suggests that this mode represents recirculation bubble pumping. Furthermore, the $m=2$ mode with a large contribution is considered to correspond to the double-helix structure reported by Johansson & George (Reference Johansson and George2006b) and Nidhan et al. (Reference Nidhan, Chongsiripinyo, Schmidt and Sarkar2020) for the disk wake or the streak-like fluid structure reported by Nekkanti et al. (Reference Nekkanti, Nidhan, Schmidt and Sarkar2023) and Zhang & Peet (Reference Zhang and Peet2023), but as there are no reports on this structure in previous studies on cylinders, it is discussed together with spatial distributions of modes in § 4.2.2.

4.2.2. Eigenfunctions and mode coefficients

The eigenfunctions of each velocity component at $x/D=1.0$ and 2.0 for mode $(m,n)=(0\unicode{x2013}2,1)$ in the case of $L/D=1.0$, which forms the non-reattaching flow focused on in the previous section, are shown in figure 7. In addition, the eigenfunctions for the mode $(m,n)=(0,2)$, which show characteristic spatial patterns of azimuthal shear mode, are also presented. The previous study by Zhang & Peet (Reference Zhang and Peet2023) suggested that the planar symmetric vorticity is twisted by the azimuthal shear mode in the initial stages of vortex shedding. The contribution of this mode to velocity fluctuations shown in figures 6(a,b) is small, but it is considered to be related to the large-scale vortex shedding. Therefore, the present study also focuses on it. Velocity fluctuations related to the azimuthal shear mode are also observed for mode $(m,n)=(0,4)$ in the range shown in figure 6. Figures 7(b,d,f,h) show that the eigenfunctions of each mode are almost the same regardless of the $x$ position. Moreover, figure 9 shows the PSD of the real part of the time-series mode coefficients for these modes at $x/D=1.0$ and 2.0.

Figure 7. The eigenfunctions for mode (a) $(m,n)=(0,1)$, (c) $(m,n)=(0,2)$, (e) $(m,n)=(1,1)$ and (g) $(m,n)=(2,1)$ at $x/D=1.0$ in the case of $L/D=1.0$ and (b,d,f,h) their amplitude normalised by maximum in $|{\boldsymbol{\mathsf{W}}}^{-1}{\boldsymbol {U}}_{m, n}|$ at $x/D=1.0$, 1.4 and 2.0, which is expressed by solid lines, dotted lines and single-pointed lines, respectively. The red, blue and yellow lines represent the $x$, $r$ and $\theta$ components, respectively.

Figure 7(a) shows that for axisymmetric modes $(m,n)=(0,1)$, the $x$ component is dominant, with large fluctuations in the recirculation region. The $r$ component also fluctuates in phase with the velocity fluctuations in the $x$ direction in the free-stream region. On the other hand, the $\theta$ component shows less fluctuation than the other two components, but downstream, fluctuations indicative of flow rotation can be seen in the centre of the wake, which is shown in figure 7(b). The recirculation bubble pumping is a flow structure that produces axisymmetric velocity fluctuations, and it was reported that large fluctuations appear in the $x$ component for $St<0.05$ (Yokota et al. Reference Yokota, Ochiai, Ozawa, Nonomura and Asai2021). It is also known from investigations of an axisymmetric bluff body that the recirculation bubble pumping is a local mode appearing in the recirculation region behind the body (Zhang & Peet Reference Zhang and Peet2023).

Fluctuations due to the recirculation bubble pumping are observed at $St\approx 0.024$ in the PSD of this mode at $x/D=1.0$ (figure 9a). The figure shows that this mode has a difference in the amplitude of fluctuations in the low-frequency region of $St\approx 0.024$ inside and outside the recirculation region. In contrast, the fluctuation levels do not change significantly in the other frequency regions. Therefore, the change in fluctuation level in the low-frequency region corresponds to the energy change in the $x$ direction of this mode shown in the eigenspectra (figures 6a,b). The fact that the variation at $St\approx 0.024$ is large at $x$ positions corresponding to the inside of the recirculation region suggests that this variation is due to the recirculation bubble pumping (Berger et al. Reference Berger, Scholz and Schumm1990; Yang et al. Reference Yang, Liu, Wu, Liu and Zhang2015; Yokota et al. Reference Yokota, Ochiai, Ozawa, Nonomura and Asai2021), which is a local mode near the recirculation region. The length of the recirculation region is longer than that in the time-averaged field when a negative fluctuation of the $x$ component in the recirculation region occurs. At the same time, the width of the recirculation region also increases, and a negative fluctuation of the $r$ component is considered to be observed in the free-stream region, corresponding to flow toward the centre. The PSD of this mode also shows a plateau at $St \approx 0.23$. A plateau at $St\approx 0.2$ has been identified in axisymmetric modes in the reports by Nidhan et al. (Reference Nidhan, Chongsiripinyo, Schmidt and Sarkar2020) and Nekkanti et al. (Reference Nekkanti, Nidhan, Schmidt and Sarkar2023), but it is not clear what kind of fluid phenomena they correspond to and its spatial structure. However, as shown by Nidhan et al. (Reference Nidhan, Chongsiripinyo, Schmidt and Sarkar2020), Nekkanti et al. (Reference Nekkanti, Nidhan, Schmidt and Sarkar2023) and the results of the present study, there is a large fluctuation in the frequency around it from near wake to far wake, which is important for the analysis of the wake. The relationship between the fluctuations and them in other modes is discussed in § 4.4.

Figure 7(c) shows that the $\theta$ component is dominant for the axisymmetric mode $(m,n)=(0,2)$, while the fluctuations of the other two components are very small. The sign of the $\theta$ component is opposite in the centre and outer region, representing a nesting structure of swirl flow in opposite directions. The moments at the cross-section are calculated by the following equation (Zhang & Peet Reference Zhang and Peet2023):

(4.3)\begin{equation} M(x, t) = \int_{r_1}^{r_2} \int_{0}^{2{\rm \pi}} \frac{u_\theta(x, r, \theta, t)}{U}\left(\frac{r}{D}\right)^2 \,\text{d}\theta \,\text{d}r. \end{equation}

When $Z_{0,2}=0.01>0$, the moment calculated by the equation is $-0.0070$ at $x/D=1.4$, which corresponds to the clockwise direction moment. Figure 8 shows the profiles of $u_\theta '$ and moment in the $r$ direction for this case. The absolute values of the maximum and minimum velocity seem different, and the positive velocity fluctuation appears to be much larger than the negative velocity fluctuation. However, the angular momentum of positive and negative flows, which is obtained by multiplying the moment arm and integrating it in the azimuthal direction, is almost the same as each other. In this case, the angular moment of positive $u_\theta$ fluctuation at $r/D<0.5$ is 0.0136, and the angular moment of negative $u_\theta$ fluctuation at $r/D>0.5$ is $-0.0206$. This mode was called an azimuthal shear in the previous study as noted below, and the present study illustrates quantitative discussion for the first time. Yang et al. (Reference Yang, Liu, Wu, Zhong and Zhang2014) observed a non-planar-symmetric vorticity distribution called the Yin-Yang pattern in a disk wake, and Zhang & Peet (Reference Zhang and Peet2023) suggested that the plane-symmetric vortex may be twisted by azimuthal shear during vortex formation or in the early stages of vortex shedding, forming a Yin-Yang pattern. The rotational processes were reported to only appear in the recirculation region in the wake of an axisymmetric bluff body, although this cannot be elucidated in the present study as it was only conducted in the region near the recirculation region. The discussion above implicates that this mode of azimuthal shear is considered to be strongly related to vortex shedding and will be further investigated in § 4.3. The characteristic frequencies have not been reported for velocity fluctuations due to the azimuthal shear mode in previous studies. The PSD of this mode, shown in figure 9(b), decreases in the entire frequency domain as $x/D$ increases, but no peaks are observed, and no characteristic trend is observed, unlike the PSDs of other modes.

Figure 8. The $r$ direction profiles of (a) $u_\theta '$ caused by mode $(m,n)=(0,2)$ when $Z_{0,2}=0.01$ at $x/D=1.4$, and (b) its moment $M$ before integration in the $r$ direction.

Figure 9. The power spectral densities of the real part of the mode coefficients for mode (a) $(m,n)=(0,1)$, (b) $(m,n)=(0,2)$, (c) $(m,n)=(1,1)$ and (d) $(m,n)=(2,1)$ at $x/D=1.0$ and 2.0 in the case of $L/D=1.0$.

Figure 7(e) shows that for the antisymmetric mode $(m,n)=(1,1)$, the $u_x$ fluctuation is maximum at the outer edge of the recirculation region. Negative fluctuation regions in the $x$ component correspond to the wake position, with larger fluctuations indicating that the wake position is further away from the central axis of the cylinder. The $u_r$ fluctuations occur at the same position as the $u_x$ fluctuations, but in opposite phase, and this causes the Reynolds stress $-\overline {u_x'u_r'}$. The $\theta$ component has a large fluctuation at a $90^\circ$ deviation in the azimuthal direction from the $x$ and $r$ components. It can be seen that, when considered together with the $r$ and $\theta$ components, the distributions shown in these figures create vortices with opposite sign streamwise vorticity in the $z>0$ and $z<0$ regions. In the region between the two vortices, the flow is from the $u_x$ acceleration region to the deceleration region, and the flow outside each vortex is from the $u_x$ deceleration region to the acceleration region.

The PSD of this mode shown in figure 9(c) shows that there is a clear peak at $St=0.129$. This fluctuation frequency is in good agreement with the large-scale vortex shedding in the wake of the free-stream-aligned circular cylinder reported by previous studies (Berger et al. Reference Berger, Scholz and Schumm1990; Yang et al. Reference Yang, Liu, Wu, Liu and Zhang2015; Nidhan et al. Reference Nidhan, Chongsiripinyo, Schmidt and Sarkar2020; Yokota et al. Reference Yokota, Ochiai, Ozawa, Nonomura and Asai2021). Peaks at this frequency appear from near wake to far wake, indicating that large-scale vortex shedding is a global fluctuation (Johansson & George Reference Johansson and George2006b; Nidhan et al. Reference Nidhan, Chongsiripinyo, Schmidt and Sarkar2020; Nekkanti et al. Reference Nekkanti, Nidhan, Schmidt and Sarkar2023). However, the intensity of the fluctuations weakens as $x/D$ increases, as can be seen from the present study and their results. Similarly, the fluctuations weaken in the low-frequency region of $St\leq 0.05$ as $x/D$ increases. Low-frequency fluctuations of $m=1$ modes have not been discussed in flow around a free-stream-aligned circular cylinder, but the study of the wake of an axisymmetric bluff body (Zhang & Peet Reference Zhang and Peet2023) has identified fluctuations with the same spatial structure as a very-low-frequency (VLF) mode, with $St$ of the order of $10^{-2}$. It is known that the VLF mode exists as an $m=1$ mode near the object, but switches to an $m=2$ mode around the downstream end of the recirculation region. The relationship between $m=1$ and $m=2$ modes is also discussed in § 4.4.

Figure 7(g) shows that mode $(m,n)=(2,1)$, as with the mode $(m,n)=(1,1)$, has large $u_x$ fluctuations at the outer edge of the recirculation region and that the region with large fluctuations also moves in the positive direction of the $r$ axis as $x/D$ increases. The $u_r$ fluctuations are also large in the same position as the $u_x$ fluctuations as well as mode $(m,n)=(1,1)$ and are in opposite phases. Hence, mode $(m,n)=(2,1)$ is also a factor producing Reynolds stress $-\overline {u_x'u_r'}$. The $\theta$ component has a large fluctuation at a position $90^\circ$ shifted azimuthally from the other two components, and when considered together with the $r$ component, shows that a total of four vortices appear, with the sign of the streamwise vorticity switching alternately in the azimuthal direction. This structure is also seen in the disk wake (Nekkanti et al. Reference Nekkanti, Nidhan, Schmidt and Sarkar2023) and is very similar to that of the present study. There is an acceleration region of $u_x$ at the boundary of vortices with switching positive to negative streamwise vorticity in the azimuthal direction, and a deceleration region of $u_x$ at the boundary of vortices with negative to positive streamwise vorticity. A very similar structure was found by Nekkanti et al. (Reference Nekkanti, Nidhan, Schmidt and Sarkar2023) in their SPOD analysis of the disk wake, which is a streak structure represented by the first SPOD mode with $m=2$, $St\rightarrow 0$. Another similar spatial structure is the double-helix structure reported by Zhang & Peet (Reference Zhang and Peet2023). In their report, the cross-sectional pattern of the streamwise component obtained by dynamic mode decomposition, which corresponds to the double-helix structure, shows a ‘tail’ that looks like the $x$ component pattern in figure 7(g) twisted in the azimuthal direction. The frequency of velocity fluctuations due to the double-helix structure occurring in the disk wake is $St\approx 0.2$ at $x/D=2.0$ (Nidhan et al. Reference Nidhan, Chongsiripinyo, Schmidt and Sarkar2020) and $St=0.27$ at $x/D=10$ (Nekkanti et al. Reference Nekkanti, Nidhan, Schmidt and Sarkar2023). Hence, a large fluctuation in the PSD of this mode is expected at $St\rightarrow 0$ and $St=0.2\unicode{x2013}0.3$.

The PSDs for this mode shown in figure 9(d) reveal a small broad peak at $St \approx 0.23$. The fluctuations at this frequency become smaller as $x/D$ increases. Zhang & Peet (Reference Zhang and Peet2023) present that the double-helix structure is apparent in the wake of an axisymmetric bluff body at almost twice the frequency of large-scale vortex shedding and that the fluctuations decrease downstream, from the spatial distribution of the POD modes. Similarly, in the disk wake, peaks of fluctuation due to the double-helix structure were observed for $x/D=2.0$ (Nidhan et al. Reference Nidhan, Chongsiripinyo, Schmidt and Sarkar2020) and 10 (Nekkanti et al. Reference Nekkanti, Nidhan, Schmidt and Sarkar2023), but not for $x/D\geq 20$ (Johansson & George Reference Johansson and George2006b; Nidhan et al. Reference Nidhan, Chongsiripinyo, Schmidt and Sarkar2020), which suggests that the fluctuations are smaller as $x/D$ increases in the intermediate wake. Therefore, the fluctuations at $St \approx 0.23$ in the cylinder wake are considered to be caused by the double-helix vortex. On the other hand, the fluctuations increase downstream in the low-frequency region of $St\leq 0.05$, which is opposite to the trend in the fluctuations in the low-frequency region for mode $(m,n)=(1,1)$. Since fluctuations with the same spatial structure as the VLF mode mentioned above make a transition from $m=1$ to $m=2$ around the downstream end of the recirculation region, the reversal of the tendency of the PSDs for $m=1$ and $m=2$ in the low-frequency range to change in the downstream direction is also considered to be indicative of this transition. Another related fluctuation in the low-frequency region of the azimuthal mode $m=2$ is the recirculation bubble pumping. Ohmichi et al. (Reference Ohmichi, Kobayashi and Kanazaki2019) numerically investigated the wake of the re-entry capsule and presented that recirculation bubble pumping is associated with four streaks that appear downstream of the recirculation region by showing corresponding DMD results. In any case, the low-frequency fluctuations of mode $(m,n)=(2,1)$ are considered to indicate four streaks appearing in the wake.

Figure 10 shows the eigenfunctions and the PSDs of the mode coefficients for $L/D=2.0$. Note that both $x/D=1.0$ and 2.0 are the downstream side of the recirculation region because the reattaching flow is formed for $L/D=2.0$ and the recirculation region is short. The eigenfunction for mode $(m,n)=(0,1)$ is similar to those in figure 7(c) representing the azimuthal shear mode, and the eigenfunction for mode $(m,n)=(0,2)$ is also similar to those in figure 7(a) representing the bubble pumping, which indicates that the azimuthal shear mode is dominant in axisymmetric fluctuations for the reattaching flow. The mode order is switched compared with that of the non-reattaching flow because the fluctuations due to recirculation bubble pumping are smaller in the case of the reattaching flow, and the energy is relatively low compared with the azimuthal shear mode. The low-frequency fluctuations for mode $(m,n)=(0,2)$, considered to be related to the recirculation bubble pumping, are smaller than the case with $L/D=1.0$ and $x/D=2.0$. The eigenfunctions for mode $(m,n)=(1\unicode{x2013}2,1)$ are almost the same pattern as those shown in figures 7(e,g). Fluctuations due to the large-scale vortex shedding do not appear in a narrow band like in the case of $L/D=1.0$, but in a broadband fluctuation. Fluctuations due to the double-helix structure also appear at $St=0.2\unicode{x2013}0.3$. Although not shown in the paper, the results for $L/D=1.5$ are similar to those for $L/D=2.0$.

Figure 10. The eigenfunctions for mode (a) $(m,n)=(0,1)$, (c) $(m,n)=(0,2)$, (e) $(m,n)=(1,1)$ and (g) $(m,n)=(2,1)$ at $x/D=1.0$ in the case of $L/D=2.0$ and (b,d,f,h) PSDs of the real part of the mode coefficients at $x/D=1.0$ and 2.0, which is represented by the blue and red lines, respectively. The red, blue and yellow lines represent the $x$, $r$ and $\theta$ components, respectively.

4.3. Change in vortex shedding position

The vortex shedding position has been reported to be related to aerodynamic fluctuations in the lateral direction of a free-stream-aligned circular cylinder by Yokota et al. (Reference Yokota, Ochiai, Ozawa, Nonomura and Asai2021) and Shinji et al. (Reference Shinji, Nagaike, Nonomura, Asai, Okuizumi, Konishi and Sawada2020). Therefore, the comprehension of the vortex shedding position is important for suppressing vibrations acting on the cylinder. In addition, Yang et al. (Reference Yang, Liu, Wu, Zhong and Zhang2014) suggested that fluctuations in the azimuthal position of vortex shedding are associated with the recirculation bubble pumping. The discussion proceeds for the vortex shedding position before investigating this relationship. In the present study, the representative position of the vortex shedding position is expressed by the barycentre of the momentum deficit in the following equation, as used in the previous studies (Grandemange et al. Reference Grandemange, Gohlke and Cadot2013; Yang et al. Reference Yang, Liu, Wu, Zhong and Zhang2014; Gentile et al. Reference Gentile, Schrijer, Van Oudheusden and Scarano2016; Zhang & Peet Reference Zhang and Peet2023):

(4.4)$$\begin{gather} y_m = \frac{\displaystyle\int y(1-u_x/U)\,{\rm d}S}{\displaystyle\int (1-u_x/U)\,{\rm d}S}, \end{gather}$$

(4.5)$$\begin{gather}z_m = \frac{\displaystyle\int z(1-u_x/U)\,{\rm d}S}{\displaystyle\int (1-u_x/U)\,{\rm d}S}. \end{gather}$$

Here $S$ is the examined area, which in the present study is [$-1.2D 1.2D$] in both $y$ and $z$ directions. Figure 4 shows that the time-averaged velocity in the free-stream direction is positively biased, but that the fluctuation components in the present study are in good agreement with the previous study. Therefore, the qualitative discussion should be reasonable.

Figure 11 shows that the vortex shedding position fluctuates irregularly around the origin. Here, the trajectory is based on the data for 24.7 s. They also show that the circular area drawn by the trajectory becomes larger as $x/D$ increases, indicating that the wake width increases. The centre of the trajectory is shifted from the origin to the negative direction of the $y$ axis for $x/D=2.0$ (figure 11b), which is due to error vectors at the left end of the examined area. Figure 12 shows the probability distributions obtained from the amplitude and argument of mode $(m,n)=(1,1)$ for 24.7 s at $L/D=1.0$ and $x/D=1.4$, which are considered to correspond to the radial and azimuthal positions of the wake position, respectively. Here, the correlation coefficient between the wake radial position $r_m$ and the amplitude of mode $(m,n)=(1,1)$ is 0.72. The distribution of $|Z_{1,1}|$ has a maximum in the bin of 0.022–0.024, as shown in figure 12(a), and the maximum not being at zero indicates that the flow field is quasisteadily asymmetric. This steady symmetry breaking 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). On the other hand, the probability of $\mathrm {arg}\,Z_{1,1}$ shown in figure 12(b) represents an almost uniform distribution. Here, the red broken line in the figure shows the position where the probability is 0.025. Since the number of bins in the histogram is 40, it should match the red dashed line if the probabilities were uniform, but a slight dispersion is observed. The average and standard deviation of the probability are 0.025 and 0.0026, respectively. The uniformity of $\mathrm {arg}\,Z_{1,1}$ means that there is no bias in the azimuthal position of the spatial pattern shown in figure 7(e) and that the mean velocity field represents an axisymmetric distribution about the cylinder axis. Although results are not shown, the probability distribution of $|Z_{1,1}|$ does not show a maximum at zero in the case of $L/D=1.5$ and 2.0, indicating steady symmetry breaking, and $\mathrm {arg}\,Z_{1,1}$ also shows a uniform distribution. Therefore, stationary symmetry breaking appears regardless of shear layer reattachment.

Figure 11. The trajectories of the wake position at (a) $x/D=1.0$ and (b) 2.0 in the case of $L/D=1.0$.

Figure 12. The probability distribution of (a) the amplitude and (b) argument of the mode coefficients for mode $(m,n)=(1,1)$.

The trajectory of the vortex shedding position represents a loop or a flapping-like reciprocating motion, depending on the time. Yang et al. (Reference Yang, Liu, Wu, Zhong and Zhang2014) also showed trajectories of vortex shedding positions under $Re=10^4$, in which closed loops were identified. They suggested that this loop was a helical vortex structure appearing in the wake of the cylinder. However, they also recognised that the shape of the loops varied irregularly with time. If only helical vortices appeared, they would show a circular trajectory as shown by Zhang & Peet (Reference Zhang and Peet2023). Therefore, helical vortices, flapping and a structure mixing them in the wake can be assumed to appear depending on the time at this Reynolds number. Consequently, the velocity fluctuations due to mode $(m,n)=(1,1)$ at $x/D=1.4$, corresponding to large-scale vortex shedding, are visualised three dimensionally using Taylor's hypothesis. Band-pass filtering of the mode coefficients with $0.1\leq St \leq 0.2$ was applied, and large-scale vortex shedding was focused. The parameter $k_{3C}^{1/2}$, calculated from the turbulent kinetic energy, is at the maximum of 28.4 % for $L/D=1.0$ and $x/D=1.4$, and although it is not appropriate to apply Taylor's hypothesis as in the paper of Nekkanti et al. (Reference Nekkanti, Nidhan, Schmidt and Sarkar2023), it was applied because it is useful for understanding changes in the vortex shedding position. Figure 13 shows selected visualisations when helical vortices, flapping or a mixture of both are considered to occur. The left contour plot shows the velocity fluctuations of the $(m,n)=(1,1)$ mode at time $t$, and the isosurface is formed by connecting the points where the fluctuations are $\pm 0.05$. Figures 13(a,b) show anticlockwise and clockwise helices in the view from downstream, respectively, where twisting occurs without significant change in the magnitude of the fluctuations. In other words, the vortex shedding position in the $r$ direction remains the same, but its position changes in the azimuthal direction in a circular pattern. On the other hand, figure 13(c) shows that the positive and negative values switch without changing the azimuthal position, indicating that vortex shedding is performed in a certain plane like flapping motion. Furthermore, figure 13(d) shows a mixture of helical and flapping features and is considered to be an elliptical motion. These four states appear irregularly, as shown in the supplementary material of movie 1 available at https://doi.org/10.1017/jfm.2024.93. These states are also observed for $L/D=1.5$ and $2.0$ from movies 2 and 3 of the supplementary material, respectively. The isosurfaces in these videos are displayed by connecting the points where the fluctuations are $\pm 0.01$.

Figure 13. The snapshots of the pseudo-three-dimensional $u_x'$ map for the mode of $(m,n)=(1,1)$ with the state of (a) anticlockwise circular, (b) clockwise circular, (c) flapping and (d) mixture of circular and flapping at $x/D=1.4$ in the case of $L/D=1.0$.

Figure 14 shows the PSD calculated from the amplitude and the angular change of the spatial pattern of mode $(m,n)=(1,1)$, which are considered to correspond to the radial and azimuthal fluctuations of the vortex shedding position, respectively. The angular variation of the spatial pattern was calculated from the azimuthal position of the deceleration region of the streamwise velocity corresponding to the vortex shedding position. The data used for the calculation of the PSDs were the same as those used for the plot of the trajectories, while each PSD was normalised by its maximum value. Only azimuthal position fluctuations appear when the trajectory of the vortex shedding position is circular, as shown in figures 13(a,b), and a peak is considered to appear at $St=0.129$ in the PSD of the angular variation. However, the angular variation appears as a fluctuation at $St=0.129$ even in the case of flapping, as shown in figure 13(c). The gradient of the angular change, i.e. the angular velocity, is necessary to classify them. Flapping or elliptical loops across the vicinity of the cylinder axis, as shown in figures 13(c,d), are expected to appear as the twice higher fluctuation frequency due to large-scale vortex shedding. This is because the sign of the velocity switches when the velocity fluctuations due to vortex shedding show a flapping pattern, and $|Z_{1,1}|$ goes from near its maximum value to zero and back to a value near its maximum value again. Thus, when the velocity fluctuations advance by half a period, the fluctuations in $|Z_{1,1}|$ advance by one period, and a peak appears at twice the frequency of the fluctuations due to the large-scale vortex shedding. Accordingly, the fluctuations at $St\approx 0.26$ are considered to be due to flapping patterns. The peak at $St\approx 0.26$ is observed regardless of the $x$ position. A similar spectrum of positional fluctuations has been observed immediately behind an axisymmetric bluff body (Zhang & Peet Reference Zhang and Peet2023). Other large fluctuations were observed, appearing at $St\approx 0.02$ and 0.23 of the PSD of the amplitude. The low-frequency fluctuations at $St\approx 0.02$ are considered to be related to the recirculation bubble pumping. The vortex shedding position becomes larger or smaller in the $r$ direction as the size of the recirculation region changes in the streamwise direction. At last, the fluctuations at $St=0.23$, which are comparable to the fluctuations at $St\approx 0.26$ in the case of $x/D=1.0$, are relatively weak compared with the fluctuations at $St=0.26$ as $x/D$ increases, suggesting that these fluctuations are represented inside the recirculation region. The broadband large fluctuations are observed at $St\approx 0.23$ for modes $(m,n)=(0,1)$ and (2,1), as shown in figure 9, and further discussion of the relationship between the modes at $St=0.23$ will be carried out in the next section.

Figure 14. The normalised power spectral densities of the amplitude (red) and the angular change (blue) of spatial pattern of mode $(m,n)=(1,1)$ at (a) $x/D=1.0$, (b) 1.4 and (c) 2.0 in the case of $L/D=1.0$.

Figure 15 shows the temporal variation of the vortex shedding position at $x/D=1.4$, close to the downstream end of the recirculation region. Figure 15(a) plots the azimuthal position over 24.7 s, while figures 15(b)–15(e) show the change in the position over time of the pseudo-three-dimensional maps shown in figures 13(a)–13d), respectively. As in the previous paragraph, band-pass filtered data with $0.1\leq St \leq 0.2$ was used here, and the azimuthal position was calculated. Figure 15(a) shows that the rotational direction of the vortex shedding switches irregularly with a major trend to rotate in the positive direction of the $\theta$ axis at the times shown. Although not shown here for brevity, different runs at the same location had a major trend to rotate in the negative direction of the $\theta$ axis. Yang et al. (Reference Yang, Liu, Wu, Liu and Zhang2015) suggested that switching the direction of rotation of the vortex shedding position is associated with the recirculation bubble pumping. Therefore, conditional sampling was applied to mode $(m,n)=(0,1)$ and its association was investigated in the present study. Figure 16(a,b) (red histograms) shows the results of sampling $Z_{0,1}$ at the time for positive and negative slopes of the position fluctuations of the vortex shedding. The probability distributions of them over the whole measurement time are shown together in the figure for comparison (blue histogram). The $Z_{0,1}$ probability distributions should be biased if the rotational direction of the vortex shedding position is associated with recirculation bubble pumping, but no such bias is observed in the results. In other words, there is no relationship between the rotational direction of the vortex shedding position and the recirculation bubble pumping.

Figure 15. The temporal variation of the vortex shedding position at $x/D=1.4$ for (a) the whole measurement time and (b–e) the time of pseudo-three-dimensional maps shown in figures 13(a)–13(d), respectively. The red dots correspond to the moment of snapshots shown in figure 13.

Figure 16. The probability distribution of $Z_{0,1}$ obtained by sampling under the condition that the state of vortex shedding is (a) anticlockwise, (b) clockwise, (c) anticlockwise circular, (d) clockwise circular and (e) flapping at $x/D=1.4$ in the case of $L/D=1.0$. The red and blue histograms show the results of the conditional sampling and the whole measurement time, respectively.

The present study investigates not only the rotational direction of the vortex shedding position but also whether the state of the recirculation region differs depending on whether the vortex shedding pattern shows a circular or reciprocating pattern, such as flapping. The gradient of the azimuthal position was used as a condition for classifying these patterns. The conditions are as follows:

(4.6)$$\begin{gather} \text{state is} \begin{cases} \text{anticlockwise} & \text{if}\ \dfrac{\omega_m}{2{\rm \pi}}\geq 0, \\[10pt] \text{clockwise} & \text{if}\ \dfrac{\omega_m}{2{\rm \pi}}<0, \\[10pt] \text{anticlockwise circular} & \text{if}\ 0.1\leq \dfrac{\omega_m}{2{\rm \pi}}\le0.2, \\[10pt] \text{clockwise circular} & \text{if}\ -0.2\leq \dfrac{\omega_m}{2{\rm \pi}}\leq -0.1, \\[10pt] \text{flapping} & \text{if}~\left|\dfrac{\omega_m}{2{\rm \pi}}\right|<0.1\ \text{or}\ \left|\dfrac{\omega_m}{2{\rm \pi}}\right|>0.2, \end{cases} \end{gather}$$

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

Here $\theta _m$ is the azimuthal position with the minimum $u_x'$ of mode $(m,n)=(1,1)$. Figures 15(b,c) show that the vortex shedding position varies smoothly if the pattern is circular. On the other hand, figure 15(d) illustrates that discontinuous changes and near-zero gradient changes appear when a flapping pattern appears. The present authors compared the pseudo-three-dimensional field shown in figure 13 with the positional changes in figure 15 and confirmed that the sample points satisfying the conditions were selected appropriately. The points of the circle pattern and the points of the flapping pattern were confirmed to appear when the two patterns are mixed, as shown in figure 15(e). Figures 16(c–e) show the results of sampling $Z_{0,1}$ when the anticlockwise/clockwise circular pattern and the flapping pattern appear, respectively. In the case of the circular pattern shown in figures 16(c,d), the distribution is slightly different from that of $Z_{0,1}$ for the whole measurement time, but there is no clear difference. No significant changes in the distribution of $Z_{0,1}$ compared with the probability distribution of it over the whole measurement time are also observed in figure 16, suggesting that there is no link between the shedding pattern and the recirculation bubble pumping.

The distribution of the azimuthal shear mode is shown at mode $(m,n)=(0,2)$ in the present study, and the previous study (Zhang & Peet Reference Zhang and Peet2023) suggests that the mode is highly associated with vortex shedding. Zhang & Peet (Reference Zhang and Peet2023) reported that the azimuthal shear mode is found within the recirculation region, twisting planar symmetric vortex loops, and exhibiting a vorticity distribution called the Yin-Yang pattern. Here, the pattern is the RSB distribution, which was also found in the disk wake (Yang et al. Reference Yang, Liu, Wu, Zhong and Zhang2014). Therefore, conditional sampling was applied to this mode as well as mode $(m,n)=(0,1)$. The conditions are described in (4.6). Figures 17(a–e) show the results of sampling $Z_{0,2}$ for anticlockwise and clockwise changes, anticlockwise and clockwise circular patterns and flapping, respectively, as in figure 16. At first, figures 17(a,b) present that $Z_{0,2}$ are biased depending on the direction of rotation of the vortex shedding position. It is negatively biased when the vortex shedding is anticlockwise and positively biased when the shedding is clockwise. The bias becomes more apparent when the conditions are divided into circular patterns and flapping, with $Z_{0,2}$ being negatively biased when the vortex shedding shows an anticlockwise circular pattern and positively biased when it shows a clockwise circular pattern, while the probability distribution of $Z_{0,2}$ in the case of flapping is the same as that of $Z_{0,2}$ at all times, centred around zero. The results above show that the azimuthal shear mode tends to appear not when the vortex shedding shows flapping, but when the vortex shedding shows a circular pattern. Additionally, the rotational direction of the vortex shedding is found to determine the direction of the azimuthal shear. The rotational direction of the vortex shedding and the moments generated by the azimuthal velocity fluctuation field of mode $(m,n)=(0,2)$ are also discussed. The bias distributions in figures 17(a–d) show that anticlockwise moments act when the vortex shedding position changes in the anticlockwise direction and clockwise moments act when the vortex shedding position changes in the clockwise direction, which indicates that the direction of the moments matches the change in the vortex shedding position. These relationships are further discussed in § 4.4.

Figure 17. The probability distribution of $Z_{0,2}$ obtained by sampling under the condition that the state of vortex shedding is (a) anticlockwise, (b) clockwise, (c) anticlockwise circular, (d) clockwise circular and (e) flapping at $x/D=1.4$ in the case of $L/D=1.0$. The red and blue histograms show the results of the conditional sampling and the whole measurement time, respectively.

Figures 13(a,b) show that the magnitude of the fluctuations, i.e. $|Z_{1,1}|$, does not change significantly while the vortex shedding exhibits a circular pattern. Therefore, the same conditional sampling analyses for a circular pattern were conducted and a trend in $|Z_{1,1}|$ was investigated. Here, band-pass filtering of $0.1\leq St\leq 0.2$ was applied to $Z_{1,1}$ in advance. Figures 18(a–e) show the results of sampling of $|Z_{1,1}|$ in the anticlockwise/clockwise, anticlockwise/clockwise circular and flapping patterns, respectively. The probability distribution for the whole measurement time, shown by the blue histogram, is similar to the probability distribution in figure 12(a), and the maximum value is also found in the bin of $0.022\unicode{x2013}0.024$. Figures 18(c,d) show that $|Z_{1,1}|$ concentrates near the maximum value of the probability distribution for the whole measurement time in the case of the circular pattern. On the other hand, the probability is slightly higher outside that area in the case of flapping patterns. Rigas et al. (Reference Rigas, Morgans, Brackston and Morrison2015) reported that the positional fluctuations of the centre of pressure can be explained by the Mexican-hat-shaped potential field of the nonlinear Langevin model at the base of an axisymmetric bluff body with $Re_D=1.88\times 10^5$, and the position in the $r$ direction where the centre of pressure probability distribution is maximal corresponds to the position of the potential well. The probability distribution of $|Z_{1,1}|$ corresponding to the $r$ position of the vortex shedding shown in the blue histogram is similar to theirs, which implies that the vortex shedding position of the wake of the free-stream-aligned circular cylinder can also be explained by a Mexican-hat-shaped potential field. In short, the $r$ position of the vortex shedding has small fluctuations and is stable near the potential well with an almost constant velocity in the azimuthal direction when the vortex shedding draws a circular pattern. In other words, since the vortex shedding position is trapped in the potential well and becomes stable, the $r$ position and angular velocity of the vortex shedding are considered to be almost constant. On the other hand, the vortex shedding position is considered to move through the centre of the potential field on a plane passing the cylinder axis (with a ${\rm \pi}$ shift in azimuthal position) when the vortex shedding draws a flapping pattern. A transition to a flapping pattern is considered to occur when the $r$ position is more distant from the cylinder axis during the fluctuation of the $r$ position of the vortex shedding because it can cross through the potential well and over the potential mountain at the cylinder axis. The reason why the probability is lower than the whole measurement time near the potential well in the case of the flapping pattern shown in figure 18(e) is considered to be due to the uniform sampling of the measurements and the large velocity in the $r$ direction of the vortex shedding position near the potential well.

Figure 18. The probability distribution of $|Z_{1,1}|$ obtained by sampling under the condition that the state of vortex shedding is (a) anticlockwise, (b) clockwise, (c) anticlockwise circular, (d) clockwise circular and (e) flapping at $x/D=1.4$ in the case of $L/D=1.0$. The red and blue histograms show the results of the conditional sampling and the whole measurement time, respectively.

Although not shown in the paper, the change in the azimuthal shear mode or $|Z_{1,1}|$, associated with the vortex shedding pattern shown in figures 17 and 18, is also observed for $L/D=1.5$ and 2.0. Here, the analysis method is the same as described above. Therefore, it is expected that the shear layer reattachment does not change the flow characteristics due to the vortex shedding patterns.

4.4. Relationship between characteristic fluctuations

Previous studies on the wake flow of a disk have discussed which characteristic fluid phenomena each mode corresponds to based on the results of modal decomposition. Similar discussions have been developed in §§ 4.2.2 and 4.3 of this paper. The fluctuations of $Z_{0,1}$ and $|Z_{1,1}|$ in the case of the non-reattaching flow were found to be larger in the low-frequency region of $St\approx 0.024$, while those of $Z_{0,1}$, $Z_{2,1}$ and $|Z_{1,1}|$ were larger at $St\approx 0.23$. Coherence and phase differences were calculated and the relationships between the modes were investigated for $L/D=1.0$. The results shown in figure 19 illustrate the coherence between $Z_{0,1}$–$|Z_{1,1}|$, $Z_{0,1}$–$|Z_{2,1}|$ and $|Z_{1,1}|$–$|Z_{2,1}|$. Here, absolute values are taken for the mode $m\neq 0$ because the mode coefficients are complex values. Another reason for the analysis was that the cylinder wake is statistically homogeneous in the azimuthal direction. Therefore, it was considered that correlations could be found in the absolute values of the mode coefficients corresponding to the amplitude of fluctuations and in quantities such as $|\mathrm {arg}\,Z_{1,1}/{\rm d}t|$ that are independent of the direction of rotation of the spatial pattern.

Figure 19. The coherence between (a,b) $Z_{0,1}$–$|Z_{1,1}|$, (c,d) $Z_{0,1}$–$|Z_{2,1}|$ and (e,f) $|Z_{1,1}|$–$|Z_{2,1}|$ at (a,c,e) $x/D=1.0$ and (b,d,f) 2.0 in the case of $L/D=1.0$. The phase difference at the focused frequency is shown next to the dotted line.

At first, high coherence is observed in the low-frequency region of $St\approx 0.024$. The values are approximately 0.75 between $Z_{0,1}$–$|Z_{1,1}|$ as shown in figures 19(a,b). The coherence between $Z_{0,1}$–$|Z_{2,1}|$ is relatively high downstream from $x/D=1.4$ near the downstream end of the recirculation region with a value of about 0.5 as shown in figures 19(c,d), and the phase difference is similar to that between $Z_{0,1}$–$|Z_{1,1}|$. Here, the result of the coherence at $x/D=1.4$ is not shown. Moreover, relatively high coherence is observed downstream of the recirculation region. The low-frequency fluctuations for $St\leq 0.05$ in $Z_{2,1}$ are larger downstream of the recirculation region, as shown in figure 9(d). This low-frequency fluctuation is considered to be due to streaks, discussed in § 4.2.2. In contrast, fluctuations at $St\approx 0.23$ due to the double-helix structure were also observed at $x/D=1.0$ and 2.0, which corresponds to inside and outside the recirculation region, respectively. For these reasons, the low-frequency fluctuations in $|Z_{2,1}|$, which are correlated with $Z_{0,1}$, are considered to be mainly related to streaks. A similar trend is also observed between $|Z_{1,1}|$–$|Z_{2,1}|$ as shown in figures 19(e,f). Figure 20 summarises the relationship between the length of the recirculation region and the strength of fluctuations in mode $(m,n)=(1,1),(2,1)$ based on the phase difference between $Z_{0,1}$–$|Z_{1,1}|$ and $Z_{0,1}$–$|Z_{2,1}|$. The fluctuations in mode $(m,n)=(1,1),(2,1)$ are strong when the recirculation region is shorter than the mean field due to bubble pumping (red region in the figure). However, the causality between these modes is not clear, and therefore, the transfer entropy between $Z_{0,1}$, $|Z_{1,1}|$ and $|Z_{2,1}|$ was calculated.

Figure 20. The relationship among the length of the recirculation region, $|Z_{1,1}|$ and $|Z_{2,1}|$.

Entropy $H$ in information theory is calculated by the equation

(4.8)\begin{equation} H(X) ={-}\sum_{x \in X} p(x)\log_2p(x), \end{equation}

where $p$ is the probability density function with $x$ as the random variable. If two or more random variables are involved, it is called joint entropy and is calculated as

(4.9)\begin{equation} H(X,Y) ={-}\sum_{x \in X}\sum_{y \in Y} p(x,y)\log_2p(x,y). \end{equation}

The probability distribution in the present study was created by discretizing the data in six bins between the maximum and minimum values of the data and dividing the number of samples in each bin by the total number of sample points. The analysis for setting the number of bins is described in Appendix B. The transfer entropy ${\rm TE}$ is calculated as

(4.10)\begin{align} {\rm TE}_{X \rightarrow Y} &=H(Y(t)|Y(t-\Delta t))-H(Y(t)|Y(t-\Delta t),X(t-\Delta t))\nonumber\\ &=H(Y(t),Y(t-\Delta t))-H(Y(t-\Delta t))\nonumber\\ &\quad-H(Y(t),Y(t-\Delta t),X(t-\Delta t))+H(Y(t-\Delta t),X(t-\Delta t)), \end{align}

where $\Delta t$ is the time lag and was set to 0.03 s ($\Delta tU/D=6.3$) in the calculation of transfer entropy between $Z_{0,1}$, $|Z_{1,1}|$ and $|Z_{2,1}|$. The spurious transfer entropy caused by statistical errors is removed by the equation (Lozano-Durán, Bae & Encinar Reference Lozano-Durá, Bae and Encinar2020; Milani et al. Reference Milani, Ching, Banko and Eaton2020)

(4.11)\begin{equation} \widetilde{{\rm TE}}_{X \rightarrow Y} = {\rm TE}_{X \rightarrow Y}-{\rm TE}_{X^{shuffled} \rightarrow Y}, \end{equation}

where $X^{shuffled}$ is the data with $X$ randomly permuted, and the dominant direction of information flow is determined by the net transfer entropy ${\rm TE}_{net}$ calculated by (4.12)

(4.12)\begin{equation} {\rm TE}_{{net},X \rightarrow Y} = \tilde{{\rm TE}}_{X \rightarrow Y}-\tilde{{\rm TE}}_{Y \rightarrow X}. \end{equation}

Positive ${\rm TE}_{{net},X \rightarrow Y}$ indicates that the information flow is dominant in the $X$ to $Y$ direction. In the present study, the data for 24.7 s at $x/D=1.4$ was used to create a discrete probability distribution, and entropy and joint entropy were calculated to compute ${\rm TE}_{net}$. Figure 21(a) shows the results of the net transfer entropy for three different paths between $Z_{0,1}$, $|Z_{1,1}|$ and $|Z_{2,1}|$. Here, $Z_{0,1}$ and $Z_{2,1}$ were low-pass filtered with $St\leq 0.05$ and $Z_{1,1}$ was band-pass filtered with $0.1\leq St\leq 0.2$, respectively, so as to focus on the recirculation bubble pumping, the large-scale vortex shedding and the streaks. Furthermore, calculated $|Z_{1,1}|$ and $|Z_{2,1}|$ were processed with a low-pass filtering of $St\leq 0.05$. The ${\rm TE}_{net}$ between $Z_{0,1}$–$|Z_{1,1}|$ is negative, which indicates that the information flow is dominant in the direction from the amplitude of mode $(m,n)=(1,1)$, which is considered to correspond to the radial position of the vortex shedding position, to bubble pumping represented by mode $(m,n)=(0,1)$. Similarly, the direction from the amplitude of mode $(m,n)=(2,1)$ to mode $(m,n)=(0,1)$ is dominant between $Z_{0,1}$–$|Z_{2,1}|$. The dominant flow of information between $|Z_{1,1}|$–$|Z_{2,1}|$ is from the amplitude of mode $(m,n)=(1,1)$ to the amplitude of mode $(m,n)=(2,1)$, and ${\rm TE}_{net}$ is comparable to it between $Z_{0,1}$–$|Z_{2,1}|$. Information is mainly transferred from the one with a low wavenumber to the one with a high wavenumber only in the $|Z_{1,1}|$–$|Z_{2,1}|$ case. In short, the length of the recirculation region is considered to be affected by changes in the radial position of the vortex shedding and the strength of the streak. The results in figures 19 and 21(a) indicate that the large-scale vortex shedding especially has a significant effect on the bubble pumping. Stronger mixing by flapping or helical structures consequently leads to a shorter recirculation region when flapping or helical structures exhibiting circular patterns of large-scale vortex shedding are stronger, while conversely, weaker mixing by flapping or helical structures results in a longer recirculation region. Hence, the results imply that the low-frequency fluctuations in large-scale vortex shedding strength are mainly producing the bubble pumping.

Figure 21. The net transfer entropy for (a) three different paths between $Z_{0,1}$, $|Z_{1,1}|$ and $|Z_{2,1}|$ and (b) a path between the $Z_{0,2}$ and the vortex shedding states.

Secondly, relatively high coherence between $Z_{0,1}$–$|Z_{1,1}|$ at $St=0.23$, which is apparent in the $x$ position where the recirculation region is located, as shown in figure 19. The relationship between $Z_{0,1}$–$|Z_{1,1}|$ at this frequency implies that the vortex shedding position moves in the $r$-positive direction as the length of the recirculation region increases, which is the reversal of the relationship in the low-frequency region. High-frequency fluctuations in the azimuthal mode $m=0$ have been observed in previous studies (Fuchs, Mercker & Michel Reference Fuchs, Mercker and Michel1979; Berger et al. Reference Berger, Scholz and Schumm1990; Nidhan et al. Reference Nidhan, Chongsiripinyo, Schmidt and Sarkar2020; Nekkanti et al. Reference Nekkanti, Nidhan, Schmidt and Sarkar2023) but neither have been discussed in detail, nor does the present study provide a three-dimensional velocity field. Thus, the discussion of physical relationships is difficult. However, the fact that the relationship is found only in the recirculation region is consistent with the finding by Meliga, Chomaz & Sipp (Reference Meliga, Chomaz and Sipp2009) that the nonlinear interaction between unstable modes occurs only in the recirculation region.

Another feature found only within the recirculation region is the rotational flow reported by Zhang & Peet (Reference Zhang and Peet2023). They suggested that the planar symmetric vortex loops formed behind an axisymmetric body are collapsed by the azimuthal shear mode that appears in the recirculation region, which results in a twisted form of vortex loops known as the Yin-Yang pattern. Mode $(m,n)=(0,2)$, which represents the azimuthal shear mode, is also found in the present study, as shown in the figure, and the results in the section reveal that mode $(m,n)=(0,2)$ is associated with the state of vortex shedding. However, the causality between them is not clear. For this reason, a causality investigation was carried out using transfer entropy in the same way as described above. Since there are three vortex shedding patterns in the calculation, the number of bins for the vortex shedding state is three, and the number of bins for $Z_{0,2}$ is six. Figure 21(b) shows the ${\rm TE}_{net}$ between the $Z_{0,2}$ and vortex shedding states and also represents the interesting profile in the range with small $\Delta t$. The absolute value of ${\rm TE}_{net}$ reaches a maximum at $\Delta tU/D=3.5$, and the dominant direction is from the vortex shedding state to $Z_{0,2}$. However, the information flow in the opposite direction becomes dominant with increasing $\Delta t$. Zhang & Peet (Reference Zhang and Peet2023) suggest that a plane-symmetric vortex is twisted by the azimuthal shear mode and an RSB distribution of vorticity appears, and since the present study also shows that the change in vortex shedding position coincides with the direction of the moment in the cross-section, it is reasonable to assume that the azimuthal shear mode determines the vortex shedding state. In contrast, changes in the vortex shedding position are not expected to generate azimuthal shear or moments. Two vortices with opposite signs of streamwise vorticity in the $yz$ plane do not form an RSB distribution at each other's induced velocities and have zero moments in the cross-section. The relationship between this vortex shedding pattern and the azimuthal shear mode remains to be further investigated.

The above discussion shows that axisymmetric fluctuations, such as bubble pumping, are related to the radial position of vortex shedding and the strength of the streak, especially in the low-frequency range. Specifically, changes in vortex shedding have been suggested to cause the bubble pumping. The large-scale vortex shedding is linked to aerodynamic forces acting in the lateral, pitch and yaw directions of the cylinder, while fluctuations due to bubble pumping are closely related to base pressure (Nonomura et al. Reference Nonomura, Sato, Fukata, Nagaike, Okuizumi, Konishi, Asai and Sawada2018; Shinji et al. Reference Shinji, Nagaike, Nonomura, Asai, Okuizumi, Konishi and Sawada2020; Kuwata et al. Reference Kuwata, Abe, Yokota, Nonomura, Sawada, Yakeno, Asai and Obayashi2021; Yokota et al. Reference Yokota, Ochiai, Ozawa, Nonomura and Asai2021). This suggests the possibility of predicting the aerodynamic forces acting in the lateral and rotational directions by measuring base pressure in axisymmetric bodies such as those focused on in the present study. However, since the causes of the azimuthal shear mode are not known and could not be clarified in the present study, further investigation is required for this point.