4.1. Estimation results using synchronized pressure data

In applying (2.16) and (2.17), it was mentioned that identifying the non-vanishing correlations and setting the remaining to zero improves the estimation. Here, this statement is considered more closely. Tables 1 and 2 present the covariance magnitudes between selected velocity and pressure POD/SPOD modes at $z = 1.83$ , respectively. Mode pairs associated with the same dominant frequency(ies) in both fields are separated using sidelines (e.g. { $a_{u}^{1},a_{u}^{2}$ } and { $a_{p}^{1},a_{p}^{2}$ } are associated with $f_{s}$ ).

Table 1.

Covariance magnitudes between selected velocity and pressure POD modes at $z=1.83$ . Mode pairs (mode) identified with the same frequency in both fields are separated with sidelines. The highest covariance magnitudes between each velocity mode and all pressure modes is bolded. Magnitudes less than 0.005 are replaced by 0.

Table 2.

Covariance magnitudes between selected velocity and pressure SPOD modes at $z=1.83$ . Mode pairs identified with the same frequency in both fields are separated with sidelines. The highest covariance magnitudes between each velocity mode and all pressure modes is bolded. Magnitudes less than 0.005 are replaced by 0.

In table 1, the covariance magnitudes are highest between modes in the same mode subspace. The covariance of higher-ranked pressure and velocity POD modes were negligible. Furthermore, it was verified that in performing the estimation, contributions due to correlations between the selected antisymmetric and symmetric modes or between the slow-varying and fast-varying symmetric modes were negligible. This is consistent with the findings of Hosseini et al. (Reference Hosseini, Martinuzzi and Noack2015). Therefore, for estimation of the first two antisymmetric modes ( $\hat {a}_u^k$ ; $k=1$ and 2), only correlations with $a_p^1$ and $a_p^2$ are considered

(4.1) \begin{equation} \hat {a}_{u}^{k}(t_{est})= \sum _{n=1}^{2} a_{p}^{n}(t_{est}-\tau _{a}^{n})\frac {\langle a_{p}^{n}(t-\tau _{a}^{n}) \, a_{u}^{k}(t) \rangle }{\lambda _{a_{p}^{n}}} \:. \end{equation}

Similarly, for estimation of the first two modes of the fast-varying symmetric mode subspace ( $\hat {s}_u^{k}$ ; $k=1$ and 2), associated with the second harmonics of the Kármán vortex shedding

(4.2) \begin{equation} \hat {s}_{u}^{k}(t_{est})= \sum _{n=1}^{2} s_{p}^{n}(t_{est}-\tau _{s}^{n})\frac {\langle s_{p}^{n}(t-\tau _{s}^{n}) \, s_{u}^{k}(t) \rangle }{\lambda _{s_{p}^{n}}} \:. \end{equation}

For estimation of the slow-varying symmetric mode $\hat {s}_{u}^{\Delta }$ , associated with the low-frequency dynamics, (2.17) is used.

Table 2 presents covariance magnitudes for the SPOD modes. Here, covariances between antisymmetric and symmetric modes are negligible. Covariances of higher-ranked pressure modes with the selected velocity modes were found to be similarly negligible.

For the antisymmetric field, covariances between modes associated with different frequencies in the velocity and pressure fields (e.g. $a_u^3$ and $a_p^1$ ) are not negligible. This is more noticeable when considering the covariance magnitudes for $a_u^3$ , $a_u^4$ , $a_u^5$ and $a_u^6$ . Therefore, for estimation of each antisymmetric SPOD modes, six pressure modes are considered. For $\hat {a}_u^k$ ( $k=1, 2, \ldots , 6$ )

(4.3) \begin{equation} \hat {a}_{u}^{k}(t_{est})= \sum _{n=1}^{6} a_{p}^{n}(t_{est})\frac {\langle a_{p}^{n}(t) \, a_{u}^{k}(t) \rangle }{\lambda _{a_{p}^{n}}}. \end{equation}

For the symmetric field, covariance magnitudes between modes corresponding to the low-frequency signature and second harmonics are negligible, which further reflects the separation of these dynamics using SPOD. Therefore, in estimating each symmetric SPOD mode, only the pressure modes yielding covariance coefficients above the threshold are considered. For modes associated with the low-frequency dynamics ( $\hat {s}_u^k$ ; $k=1$ and 2)

(4.4) \begin{equation} \hat {s}_{u}^{k}(t_{est})= \sum _{n=1}^{2} s_{p}^{n}(t_{est}-\tau _{s}^{n})\frac {\langle s_{p}^{n}(t-\tau _{s}^{n}) \, s_{u}^{k}(t) \rangle }{\lambda _{s_{p}^{n}}}. \end{equation}

For modes associated with the second harmonics of the Kármán vortex shedding ( $\hat {s}_u^k$ ; $k=3$ and 4)

(4.5) \begin{equation} \hat {s}_{u}^{k}(t_{est})= \sum _{n=3}^{4} s_{p}^{n}(t_{est})\frac {\langle s_{p}^{n}(t) \, s_{u}^{k}(t) \rangle }{\lambda _{s_{p}^{n}}}. \end{equation}

It is noted that the time delay $\tau _{a}^{n}$ and $\tau _{s}^{n}$ of (4.3) and (4.5) are smaller than the sampling interval and are thus omitted.

Figure 4.

Comparison of actual (dashed blue, $a_u$ ) and corresponding estimated (solid red, $\hat {a}_u$ ) antisymmetric temporal coefficients at $z=1.83$ . Mode numbers correspond to the bases introduced in figure 2. Left side is with POD and right side is with SPOD. The reference pressure signal is naturally synchronized with the velocity data; $\Delta (t_{est})=1/f_{PIV}$ throughout this study.

Figure 4 shows excerpts of time series of selected antisymmetric velocity POD/SPOD modes at $z = 1.83$ . Actual and estimated temporal coefficients are shown by dashed blue and solid red lines, respectively. Table 3 presents the TKE contribution of the actual ( $\lambda _{a_{u}^{i}}$ ) and estimated ( $\lambda _{\hat {a}_{u}^{i}}$ ) modes.

Table 3.

Represented TKE ( $\lambda$ ) with the actual and estimated antisymmetric modes at $z=1.83$ . Magnitudes are rounded to the nearest decimal.

In figure 4, the phase of all temporal coefficients are well preserved. However, comparing figures 4(a) and 4(b), $\hat {a}_u^1$ in SPOD more accurately represents $a_u^1$ . The maximum cross-correlation magnitudes ( $r_{max}$ ), mentioned in the top right corner of each panel, is used as a coarse measure to assess the similarity between actual and estimated temporal coefficients. In figure 4(a), $r_{max}$ is 95.7 %, whereas in figure 4(b), it is near 100 %.

Antisymmetric modes associated with the accompanying frequencies $f_{ac1}$ and $f_{ac2}$ can only be estimated with SPOD. With POD, the energetic contents around these frequencies are mixed with that of the primary shedding at $f_{s}$ and therefore, the related dynamics cannot be estimated separately. With SPOD, however, a separate estimation of higher-ranked modes ( $\hat {a}_u^3$ to $\hat {a}_u^6$ ), associated with $f_{ac1}$ and $f_{ac2}$ , is feasible. The correlations between corresponding actual and estimated temporal coefficients of these SPOD modes are lower than that of the first harmonic ( $r_{max}$ of 90.3 % and 75.8 % for $a_{u}^{3}$ and $a_{u}^{5}$ , respectively). Moreover, in figures 4(c) and 4(d), there are instances where the amplitude of $a_{u}^{3}$ and $a_{u}^{5}$ is not well captured by the corresponding estimated modes (e.g. $350\lt t_{est}\lt 400$ ). However, further investigation (presented later in this section) shows that the critical flow features (i.e. critical points and velocity gradients) are well preserved between the actual and estimated fields.

In table 3, the first two antisymmetric POD modes represent more TKE than the six selected SPOD modes (45.2 % vs. 42.6 %). However, for the estimated fields, more TKE is recovered with the selected SPOD basis (40.2 % vs. 39.0 % in POD), which is a result of high correlations between the actual and estimated temporal coefficients. The first two antisymmetric SPOD modes retain about 37.8 % TKE (out of 38.8 %) while the first two POD modes only retain 39.0 % TKE (out of 45.2 %). Including the TKE contributions retained by the other four estimated antisymmetric SPOD modes, associated with the accompanying frequencies, further favours the selection of SPOD over POD.

Figure 5.

Comparison of actual (dashed blue, $s_u$ ) and corresponding estimated (solid red, $\hat {s}_u$ ) symmetric temporal coefficients at $z=1.83$ . Mode numbers correspond to the bases introduced in figure 2. Left side is with POD and right side is with SPOD. The reference pressure signal is naturally synchronized with the velocity data.

Table 4.

Represented TKE ( $\lambda$ ) with the actual and estimated symmetric modes at $z=1.83$ . Magnitudes are rounded to the nearest decimal.

Figure 5 and table 4 present similar information to figure 4 and table 3, but for the symmetric field, at $z = 1.83$ . In comparing the cumulative TKE contribution of symmetric modes associated with the slow-varying signature, the contribution of $s_{u}^{1}$ and $s_{u}^{2}$ of SPOD is higher than $\hat {s}_u^{\Delta }$ of POD (3.9 % vs. 2.6 %). This is mainly attributed to a more successful separation of the low-frequency dynamics. For estimation of these modes, both POD- and SPOD-based techniques provide similar results, with the magnitude of $r_{max}$ in figure 5(c) being 4 % higher than in figure 5(a). However, as a result of better separation of the dynamics with SPOD, the estimation of the two SPOD modes shows much less high-frequency jitter (compare the smoothness of $\hat {s}_u^{\Delta }$ in figure 5(a) and $\hat {s}_u^1$ in figure 5(c) in the range $1100\lt t_{est}\lt 1200$ ).

Estimation of the modes representing the second harmonics of the Kármán vortex shedding ( $s_{u}^{1}$ and $s_{u}^{2}$ with POD and $s_{u}^{3}$ and $s_{u}^{4}$ with SPOD) is significantly improved with SPOD. In figure 5(b), neither the pattern nor amplitude of the POD mode $s_u^1$ is correctly captured by $\hat {s}_{u}^{1}$ , which results in a relatively poor $r_{max}$ of 69.1 %. In comparison, in figure 5(d), the $r_{max}$ between the SPOD mode $s_{u}^{3}$ and its estimation $\hat {s}_{u}^{3}$ is 96.2 %, which is significantly higher. In terms of retaining TKE with the estimated modes (table 4), the POD modes $\hat {s}_{u}^{1}$ and $\hat {s}_{u}^{2}$ only retain 3.0 % out of 5.7 % TKE, whereas the SPOD modes $\hat {s}_{u}^{3}$ and $\hat {s}_{u}^{4}$ retain the same amount (3.0 %), but out of 3.2 % TKE. It should be re-iterated that the higher TKE contribution of the POD modes $s_u^1$ and $s_u^2$ compared with that of the SPOD modes $s_u^3$ and $s_u^4$ (5.7 % vs. 3.2 %) is mainly due to the low-frequency energy contents that are not separated from these POD modes (compare PSDFs of the POD mode $s_u^1$ and SPOD mode $s_u^3$ in figure 2).

Altogether, considering all five selected POD modes and ten selected SPOD modes, 45.0 % TKE is recovered with the estimated POD modes (out of 53.6 %) and 47.7 % TKE is recovered with the estimated SPOD modes (out of 49.8 %). Effectively, SPOD recovers more of the signal energy. Furthermore: (i) flow dynamics of the fundamental mode pair at $f_{s}$ can be presented with less noise; (ii) weaker dynamics, such as the ones associated with the accompanying frequencies at $f_{ac1}$ and $f_{ac2}$ , can be estimated separately; and (iii) the modal dynamics associated with both low- and high-frequency energetic contents in the symmetric field is presented more accurately, especially in the case of the second harmonics of the Kármán vortex shedding.

Next, the quality of the estimated velocity modes, presented in figures 4 and 5, is further evaluated based on the residuals. Figure 6 presents the error of the instantaneous estimated velocity fields from partial reconstructions using POD (top) and SPOD (bottom) modes at $z=1.83$ . Similarly to Hosseini et al. (Reference Hosseini, Martinuzzi and Noack2015), the error is defined as

(4.6) \begin{equation} E_{c}=\frac {\int _{y}(\textbf {u}_{c}-\hat {\textbf {u}}_{c}).(\textbf {u}_{c}-\hat {\textbf {u}}_{c}) \: {\rm d}\textbf {A}}{\overline {\int _{y} \textbf {u}_{c} \: . \: \textbf {u}_{c} \: {\rm d}\textbf {A}}} \:, \end{equation}

where $\textbf {u}_{c}$ and $\hat {\textbf {u}}_{c}$ are the partial reconstruction using the actual and estimated modes, respectively, and the over-line denotes the time-averaging operator. In figure 6(a), $E_{c}$ is the error between $\textbf {u}_{c}$ and $\hat {\textbf {u}}_{c}$ , reconstructed from the two antisymmetric modes and the mean field velocity. A lower $E_{c}$ indicates a more accurate sensor-based estimation. The horizontal lines in figure 6 show the mean $E_{c}$ value of the signal ( $\overline {E}_{c}$ ) in the same colour as the related $E_{c}$ signal.

Figure 6.

Error of the estimated velocity fields reconstructed using different selected POD ((a) and (b)) and SPOD ((c) and (d)) modes at $z=1.83$ . The horizontal lines show the average values in the same colour as the signal data points. The reference pressure data used for estimation are naturally synchronized (acquired simultaneously) with the velocity measurements. Note the difference between the vertical axes range.

The $\overline {E}_{c}$ of the SPOD reconstruction of the antisymmetric field (i.e. $\textbf {U}+\sum _{n=1}^{6}a_{u}^{n}\boldsymbol {\mathit {\phi }}_{a}^{n}$ ) is significantly lower than that of the corresponding POD reconstruction (0.03 vs. 0.11, as shown with horizontal black and red lines in figure 6(c) and figure 6(a), respectively). Considering contributions of individual antisymmetric SPOD mode pairs in figure 6(c), the $E_{c}$ with higher-ranked mode pairs is higher (i.e. with $\textbf {U}+\sum _{n=1}^{2}a_{u}^{n}\boldsymbol {\mathit {\phi }}_{a}^{n}$ , $\textbf {U}+\sum _{n=3}^{4}a_{u}^{n}\boldsymbol {\mathit {\phi }}_{a}^{n}$ and $\textbf {U}+\sum _{n=5}^{6}a_{u}^{n}\boldsymbol {\mathit {\phi }}_{a}^{n}$ , $\overline {E}_{c} \approx$ 0.01, 0.07 and 0.12, respectively), which will be shown in § 5 to reflect momentary intervals of disrupted vortex shedding. However, since the energetic contribution of the higher-ranked mode pairs are significantly lower than that of the first harmonic, the higher $R_c$ of the modes associated with the accompanying frequency modes do not significantly affect the overall residual of $\textbf {U}+\sum _{n=1}^{6}a_{u}^{n}\boldsymbol {\mathit {\phi }}_{a}^{n}$ .

For the symmetric field, reconstructions with slow-varying mode(s) (i.e. $\textbf {U}+s_{u}^{\Delta }\boldsymbol {\mathit {\phi }}_{s}^{\Delta }$ with POD and $\textbf {U}+\sum _{n=1}^{2}s_{u}^{n}\boldsymbol {\mathit {\phi }}_{s}^{n}$ with SPOD), $\overline {E}_{c}$ is low with both approaches (0.04 vs. 0.02, respectively). However, for the fast-varying modes associated with the second harmonics (i.e. $\textbf {U}+\sum _{n=1}^{2}s_{u}^{n}\boldsymbol {\mathit {\phi }}_{s}^{n}$ with POD and $\textbf {U}+\sum _{n=3}^{4}s_{u}^{n}\boldsymbol {\mathit {\phi }}_{s}^{n}$ with SPOD), the performance of the SPOD-based estimation is significantly better ( $\overline {E}_{c} \approx$ 0.25 compared with 0.03). In figure 6(b), $E_{c}$ magnitudes of $\textbf {U}+\sum _{n=1}^{2}s_{u}^{n}\boldsymbol {\mathit {\phi }}_{s}^{n}$ POD reconstruction are typically close or higher than 0.75, whereas they remain an order of magnitude smaller with the SPOD reconstruction $\textbf {U}+\sum _{n=3}^{4}s_{u}^{n}\boldsymbol {\mathit {\phi }}_{s}^{n}$ in figure 6(d) (note the difference in the vertical axes range between figures 6(b) and 6(d)). Finally, for reconstructions composed of all selected modes, $\overline {E}_{c}$ for the POD modes is approximately 0.12, and nearly 4 times lower for SPOD.

Figure 7 shows iso-contours of $v$ -component velocity, overlaid with sectional streamlines, from the measured (columns 1 and 3) and reconstructed (columns 2 and 4) flow fields at $z=1.83$ for a randomly selected PIV snapshot (POD reconstructions are not shown for brevity). The panels correspond to $t_{est}=500$ in figure 6. Supplementary video files no. 1 and no. 2 provide reconstructions corresponding to the data shown in figure 7.

Figure 7.

Iso-contours of the instantaneous lateral velocity component, $v$ , overlaid with sectional streamlines at $z=1.83$ : columns 1 and 3 are reconstructions using selected SPOD modes; columns 2 and 4 show similar information with the estimated SPOD modes. All figures are plotted at the same time instant, corresponding to $t_{est}=500$ in figure 6. The same reference pressure data are used for estimation.

In figure 7(k,l), the low-order reconstructions with all considered actual and estimated SPOD modes show very good similarity, both in terms of recovering the $v$ -velocity contour patterns as well as type and location of streamline critical points. Similarly, a good agreement exists between the partial reconstructions with individual mode pairs. It is only in the case of the second pair of accompanying frequency mode pairs, figure 7(i,j), that discrepancies are easily observable between the reconstructions with actual and estimated modes, which is consistent with the results in figure 6(b). However, this has little effect when considering the reconstructions with all selected modes.

Figure 8 shows an example of the instantaneous relative reconstruction error map, corresponding to the instantaneous $v$ -component velocity in figure 7, where the error parameter, $E_v$ , is defined as

(4.7) \begin{equation} E_{v}=\sqrt {\frac {(v_{c}-\hat {v}_{c})^2}{\textbf {u}_{c} \cdot \textbf {u}_{c}}}. \end{equation}

For each panel of figure 8(a) to (e), the difference between partial reconstructions of the $v$ -component velocity field using actual and estimated modes is shown. For example, figure 8(a) is obtained as the difference of $v_c$ in figure 7(a) and $\hat {v}_c$ in figure 7(b). Figure 8(f) shows the difference for the fields using the 10-mode reconstruction. For all panels, $\textbf {u}_c$ is the velocity field reconstructed from 10 actual modes.

Figure 8.

Error, $E_v$ , between actual and estimated instantaneous $v$ -component velocity, corresponding to the time instant shown in figure 7 (i.e. $t_{est}=500$ in figure 6). Panels (a), (b), (c), (d), (e) and (f) correspond to the $E_v$ computed from figure 7(a)-(b), (c)-(d), (e)-(f), (g)-(h), (i)-(j) and (k)-(l), respectively.

In figure 8, high reconstruction errors occur in one of two scenarios: (i) when there is even a small difference between actual and estimated partial reconstructions of the most energetic coherent motion, such as the Kármán vortex shedding in figure 7(a) and (b) resulting in figure 8(a); and (ii) when there is a relatively large difference between the actual and estimated coefficients, even if the identified coherent motion is weak, such as ( $a_u^5,a_u^6$ ) in figure 7(i) and (j) resulting in figure 8(e). It should be re-iterated that these panels are spatial distributions of the reconstruction error at a particular instant ( $t_{est}=500$ in figure 6), which was chosen to represent a high error level case of the full reconstruction. As can be seen from the sample time interval in figure 6, the error in the full reconstruction is generally significantly smaller than at these extreme instants. Notwithstanding, the error levels are generally less compared with the ones reported in other studies, for example by Deng et al. (Reference Deng, Chen, Liu and Kim2019).

In this section, the performance of the SPOD-based sensor estimation technique was compared with the POD-based counterpart. The reference pressure data were naturally synchronized (acquired concurrently) with the PIV measurements. However, in cases where a 3-D velocity field is to be reconstructed from individual uncorrelated (acquired at different times) planar velocity measurements, a common reference pressure signal needs to be used to synchronize the 3-D estimation. Therefore, the next section will focus on evaluating the performance of the proposed methodology when reconstructing the flow using data from uncorrelated planes.

4.2. Estimation results using asynchronous pressure data

This section starts with a comparison between estimated temporal coefficients at $z=1.83$ , obtained using synchronized and un-synchronized reference pressure data. Then, the estimated signals with asynchronous data are analysed at other heights. For all estimations with the asynchronous pressure data, the reference pressure acquired concurrently with the PIV measurements at $z=0.92$ is used to estimate the velocity at all other planes.

Table 5 presents relative TKE contribution (%) of selected POD/SPOD modes at $z=1.83$ and their estimated counterparts obtained using (i) synchronized pressure data (similar to the data presented in table 3) and (ii) asynchronous pressure data (collected concurrently with the PIV measurements at $z=0.92$ ). The difference between the TKE of the estimated and actual modes are presented by $\Delta \lambda _{a/s(1)}$ and $\Delta \lambda _{a/s(2)}$ . The focus is on evaluating the estimator performance in retaining TKE when asynchronous pressure data are used for estimation.

Table 5.

Relative TKE contribution (%) of temporal modes and their estimated counterparts at $z=1.83$ . Subscript (1) corresponds to the estimated coefficients obtained with synchronized pressure data (collected concurrently with the PIV measurements at $z=1.83$ ) and subscript (2) corresponds to the estimated coefficients obtained with asynchronous pressure data (collected concurrently with the PIV measurements at $z=0.92$ ).

From table 5, in general, the SPOD-based technique retains more TKE of the original signal. For the fundamental harmonic coefficient $\hat {a}_u^1$ , $\Delta \lambda _{a(1)}$ and $\Delta \lambda _{a(2)}$ are approximately 3.3 % and 2.3 % with POD, whereas they are 0.4 % and (-0.8) % with SPOD, respectively. The higher magnitudes of $\lambda _{\hat {a}_{u}^{1}(2)}$ compared with that of $\lambda _{\hat {a}_{u}^{1}(1)}$ with both POD and SPOD reflect a more energetic reference pressure signal in group (2), which is obtained with PIV measurements at $z=0.92$ , compared with that of group (1). Differences between different realizations (trials) of the pressure sequences are within the statistical uncertainty as discussed in Appendix B. A similar pattern is seen for $\hat {a}_u^2$ . For the accompanying frequency modes, the quality of the SPOD estimator does not change significantly using asynchronous reference pressure data (i.e. for $\hat {a}_u^3$ to $\hat {a}_u^6$ , $\Delta \lambda _{a(1)} \approx \Delta \lambda _{a(2)}$ ).

For symmetric modes, the performance of the POD and SPOD-based estimators do not change significantly using the asynchronous reference pressure signal. For the low-frequency modes, the performance is similar between both estimators. For the estimation of the second harmonics of the Kármán vortex shedding, however, the performance of the SPOD-based estimator remains significantly better than the POD-based estimator, using the asynchronous reference pressure signal. Appendix B presents a complementary discussion on the effect of using different sensor data on retaining modal TKE.

To further explore the quality of estimation with POD and SPOD-based techniques, the dynamical (phase) relationship is considered. Figure 9 depicts phase portraits of selected actual (rows 1 and 3) and estimated (rows 2 and 4) coefficients. The phase portraits in the first three columns show data over approximately 250 shedding cycles. The $x$ -axis corresponds to $a_u^1$ (rows 1 and 3) and $\hat {a}_u^1$ (rows 2 and 4). The temporal coefficients for the $y$ -axis were selected illustrating relationships with $a_{u}^{1}$ . Similar patterns are observed with both POD and SPOD temporal coefficients: in (a) and (i), the cyclical relationship between the two modes of a harmonic pair is resolved; in (b) and (j), the parabolic relationship between the shedding strength and the slow-varying component is rendered (Noack et al. Reference Noack, Afanasiev, Morzynski, Tadmor and Thiele2003; Bourgeois et al. Reference Bourgeois, Sattari and Martinuzzi2011); and in (c) and (k), Lissajou portraits are seen, which indicate that the phase dynamics of the $2f_{s}$ harmonic is enslaved to the fundamental harmonic (Bourgeois et al. Reference Bourgeois, Noack and Martinuzzi2013). The trajectories are clearer in the SPOD case, especially with retaining the Lissajou patterns, figure 9(o), which is a result of improved separation of the flow dynamics.

Figure 9.

Phase portraits of selected actual (rows 1 and 3) and estimated (rows 2 and 4) temporal coefficients: rows 1 and 2 are with POD and rows 3 and 4 are with SPOD. Mode numbers correspond to the bases introduced in figure 2. For the last column, only a short excerpt of the time series (covering about four shedding cycles) are shown for better visualization. Note that with POD, { $s_u^1$ , $s_u^2$ } and { $\hat {s}_u^1$ , $\hat {s}_u^2$ }, and with SPOD, { $s_u^3$ , $s_u^4$ } and { $\hat {s}_u^3$ , $\hat {s}_u^4$ } are related to the second harmonics of Kármán vortex shedding.

Figure 10.

(a) Actual (measured) energy content of SPOD modes relative to TKE (%); (b) TKE of actual SPOD modes; and (c) TKE recovered of SPOD modes. Here, $\lambda _{a}^1+\lambda _{a}^{2}$ corresponds to $f_{s}$ , $\lambda _{a}^3+\lambda _{a}^{4}$ to $f_{ac1}$ , $\lambda _{a}^5+\lambda _{a}^{6}$ to $f_{ac2}$ , $\lambda _{s}^1+\lambda _{s}^{2}$ to $f_{L}$ and $\lambda _{s}^3+\lambda _{s}^{4}$ to 2 $f_{s}$ .

The last column of figure 9 presents the phase relationship for a short excerpt of the time series (covering about four shedding cycles) of temporal coefficients associated with the second harmonics of Kármán vortex shedding ({ $s_u^1$ , $s_u^2$ } with POD and { $s_u^3$ , $s_u^4$ } with SPOD). Ideally, the modes corresponding to the second harmonics should exhibit a cyclical relationship, similar to that of the first (fundamental) mode pair. However, the expected cyclical pattern is not resolved in the POD case. In contrast, the cyclical pattern is very well preserved with SPOD (figure 9(l) and figure 9(p)) by using only the linear terms, as in (4.5). Furthermore, with SPOD, the estimated coefficients of the second harmonics are already in phase with those of the first, such that the delay $\tau _{s}^{n}=0$ , unlike with POD where the delay is needed. Therefore, the SPOD-based estimation is more successful in preserving the phase-relations between modes.

Figure 11.

Estimated SPOD temporal coefficients at $z=0.46$ (left side) and 2.75 (right side), calculated using a reference pressure, synchronized with the PIV measurements at $z=0.92$ . Mode numbers correspond to the bases introduced in figure 2.

The performance of the SPOD-based estimator in capturing the modal energy content when using asynchronous reference pressure signals is discussed next. Figure 10(a) shows variation of the TKE contribution of the five most energetic actual SPOD mode pairs relative to the TKE of all modes in each plane. Briefly, $\lambda _{a}^{1}+\lambda _{a}^{2}$ relates to $f_{s}$ , $\lambda _{a}^{3}+\lambda _{a}^{4}$ to $f_{ac1}$ , $\lambda _{a}^{5}+\lambda _{a}^{6}$ to $f_{ac2}$ , $\lambda _{s}^{1}+\lambda _{s}^{2}$ to $f_{L}$ and $\lambda _{s}^{3}+\lambda _{s}^{4}$ to $2f_{s}$ . The contribution of $\lambda _{a}^{1}+\lambda _{a}^{2}$ is dominant for $z\leqslant 2.05$ . Above $z\gt 2.05$ , $\lambda _{s}^{1}+\lambda _{s}^{2}$ notably increases and $\lambda _{a}^{1}+\lambda _{a}^{2}$ notably decreases, such that for $z\gt 3$ , the low-frequency $f_{L}$ motion, originating at the obstacle’s free end, dominates the vortex shedding signature. The variations of $\lambda _{a}^{3}+\lambda _{a}^{4}$ and $\lambda _{a}^{5}+\lambda _{a}^{6}$ closely follow each other. For $z\leqslant 2.05$ , the relative TKE contributions of both mode pairs stay nearly constant; for $2.05\lt z\leqslant 2.75$ , they increase with strengthening of the low-frequency instability; and for $z\gt 2.75$ , they decrease following the weakening of the Kármán vortex shedding. Lastly, $\lambda _{s}^{3}+\lambda _{s}^{4}$ is strongest in the $1.25\lt z\lt 3$ region, which coincides with the loci of streamwise vortical connector strands (Mohammadi et al. Reference Mohammadi, Morton and Martinuzzi2022). These flow patterns will be explained in greater details in § 5.

Figure 10(b) shows the absolute magnitudes of TKE contribution of the converged SPOD modes (from six trials). Figure 10(c) shows similar information for the estimated coefficients using the asynchronous pressure data, measured concurrently with the PIV measurements at $z=0.92$ based on a single trial. The two plots are not overlaid for clarity. It can be seen that the TKE distributions are similar for the estimated and converged set of actual SPOD modes along the cylinder height. However, due to the stochastic nature of the flow during separate measurements, differences exist between the two sets. In Appendix B, TKE contributions from a converged dataset and from single trials are further compared.

Figure 11 presents estimated temporal coefficients at $z=0.46$ (left side) and 2.75 (right side). The root mean square (r.m.s.) of each signal is shown at the top right corner of each panel. It can be seen that the amplitude (strength) and r.m.s. of the $\hat {a}_u^1$ coefficient are significantly smaller at $z=2.75$ compared with $z=0.46$ . In contrast, the amplitude and r.m.s. of $\hat {s}_u^1$ are significantly larger at $z=2.75$ compared with $z=0.46$ . In comparison, those of $\hat {a}_u^3$ , $\hat {a}_u^5$ and $\hat {s}_u^3$ do not show significant changes between the two planes. These observations are consistent with the TKE magnitudes presented in figure 10(c).

It should be clarified that for estimations using asynchronous pressure data, any other reference pressure data (acquired concurrently with PIV measurements at any other heights) could be used. Several tests with other reference pressure data verified that the estimated signals capture the patterns of actual temporal coefficients and correctly retain the TKE of SPOD modes at every other plane. The choice of using the pressure data acquired concurrently with PIV measurements at $z=0.92$ (among some others), however, made it possible to capture unique 3-D physical phenomena which will be further discussed in § 5.

5.1. Mean and phase-averaged 3-D velocity fields

Figure 12 shows educed mean vortex structures as rendered by $Q=0$ criterion ( $Q \geqslant 0$ identifies vortex core following Hunt et al. (Reference Hunt, Abell, Peterka and Woo1978)), coloured by streamwise vorticity, $\Omega _{x}$ . Two pairs of streamwise vortices, dipole ( $D^{+/-}$ ) (Wang et al. Reference Wang, Zhou, Chan and Lam2006; Bourgeois et al. Reference Bourgeois, Sattari and Martinuzzi2011; Hosseini et al. Reference Hosseini, Bourgeois and Martinuzzi2013) and descending vortices ( $DV^{+/-}$ ) (Kindree et al. Reference Kindree, Shahroodi and Martinuzzi2018), extend from the mean lee-ward recirculation region. These vortex pairs have the same sense of rotation on each side of the symmetry plane ( $y=0$ ), and were shown to be separate structures (Kindree et al. Reference Kindree, Shahroodi and Martinuzzi2018; Mohammadi et al. Reference Mohammadi, Morton and Martinuzzi2022).

Figure 12.

Educed mean vortex structures identified by the $Q = 0$ criterion and coloured by streamwise vorticity, $\Omega _{x}$ : black and purple dashed lines correspond to the dipole ( $D^{+/-}$ ) and descending vortices ( $DV^{+/-}$ ), respectively. Dashed arrows indicate an additional pair of post attachment vortices ( $PAV$ ), which were related to interactions of the horse-shoe vortex system and Kármán vortices (Mohammadi et al. Reference Mohammadi, Morton and Martinuzzi2023). Figure is reprinted with permission from Mohammadi et al. (Reference Mohammadi, Morton and Martinuzzi2022).

The dipole and descending vortices are the mean signature of half-loop vortex shedding pattern downstream of the vortex formation region (Bourgeois et al. Reference Bourgeois, Sattari and Martinuzzi2011; Mohammadi et al. Reference Mohammadi, Morton and Martinuzzi2022). Mohammadi et al. (Reference Mohammadi, Morton and Martinuzzi2022) elucidated the vortex topology from the 3-D phase-averaged reconstruction of the flow field. Figure 13(left side) shows phased-averaged iso-surfaces of $Q=0$ at three shedding phases, $\mathit {\phi }_{1}$ , $\mathit {\phi }_{3}$ and $\mathit {\phi }_{10}$ , covering half of the shedding cycle. Figure 13(right side) shows the vortex skeleton schematics inspired by these educed vortex structures. The phase-averaged shedding cycle is rendered in the supplementary video file no. 3.

Figure 13.

Left side: educed phased-averaged vortex structures identified by the $Q = 0$ criterion and coloured by streamwise vorticity, $\langle \Omega _{x} \rangle$ , at phases (a) $\mathit {\phi }_1$ , (c) $\mathit {\phi }_{3}$ and (e) $\mathit {\phi }_{10}$ , where $\mathit {\phi }_n = \mathit {\phi }_0 + n \pi /10$ (with an arbitrary $\mathit {\phi }_0$ ). The dashed lines are the bounds of the dipole and descending vortices as shown in figure 12. Right side: vortex skeleton schematics, inspired by the educed vortex structures on the left.

In figure 13(a) and figure 13(b), phase $\mathit {\phi }_1$ , the top of recently shed vortex $A$ bends upstream, forming a strand connecting to the incipient forming vortex $A^{+}$ at the obstacle base. This strand induces a downwash along $y=0$ . On the opposing side, vortex $B$ is developing in the base region. Vortex $B^{-}$ has shed from the previous half-cycle and its associated strand has already rotated inward and connected with vortex $A$ , in the region indicated with a blue dashed circle. By phase $\mathit {\phi }_3$ , figure 13(c) and figure 13(d), the half-loop pattern connecting vortices $A$ and $B^{-}$ is more clearly defined. The connector strand of vortex $A$ is still connected to vortex $A^{+}$ , but at a lower elevation. Its axis has rotated towards vortex $B$ , which has grown considerably. As vortex $B$ begins to shed ( $\approx \mathit {\phi }_{5}$ ), vortex $A$ completely detaches from vortex $A^{+}$ and forms a half-loop with vortex $B$ . Note that during this process ( $\mathit {\phi }_{1}$ to $\mathit {\phi }_{10}$ ), the strand remains located within the bounds of the mean dipole (indicated by the black dashed lines) and the connection site remains within the bounds of the descending vortices (indicated by the purple dashed lines). This supports the interpretation that the dipole and descending vortices of figure 12 are the mean signature of the instantaneous half-loop shedding pattern in figure 13.

The phase-averaged reconstruction provides cycle-resolved representation of the coherent motions occurring at a certain frequency, in this case the average shedding frequency, $f_{s}$ , and its harmonics. Consequently, contributions of coherent motions at other frequencies are smeared in the averaging process and their influence is not captured. Hence, the phase average represents the vortex topology during a typical shedding cycle, as can be seen in the reconstructions using the present estimation (see supplementary video file no. 3). In the present flow, however, the vortex shedding process is highly modulated and contributions of coherent motions occurring at other frequencies are expected to be important. In the following section, it will be shown that estimator-based 3-D reconstruction of the flow field makes it possible to study the cycle-to-cycle variations.

5.2. Instantaneous 3-D reconstruction of the flow field

Figure 14 shows the temporal coefficients of velocity modes in the $z=1.83$ plane estimated using reference pressure data collected concurrently with the PIV measurements at $z=0.92$ . The temporal coefficients are ordered based on increasing TKE contribution from bottom to top. For clarity, temporal coefficients of the antisymmetric and symmetric SPOD modes are coloured in blue and red, respectively.

Figure 14.

Time series of estimated temporal coefficients at $z=1.83$ using asynchronous pressure data (collected concurrently with PIV measurements at $z=0.92$ ). Estimated temporal coefficients are ordered from bottom to top based on their TKE contribution. Antisymmetric and symmetric modes are coloured in blue and red, respectively. Four intervals are coloured in grey, reflecting regions where the Kármán vortex shedding is modulated: for $70\lt t_{est}\lt 130$ , $\hat {s}_{u}^{1}$ and { $\hat {a}_{u}^{3}$ , $\hat {a}_{u}^{4}$ } are strong; for $320\lt t_{est}\lt 400$ , { $\hat {s}_{u}^{1}$ , $\hat {s}_{u}^{2}$ } is strong, but { $\hat {a}_{u}^{3}$ , $\hat {a}_{u}^{4}$ } is weak; for $780\lt t_{est}\lt 840$ , { $\hat {a}_{u}^{3}$ , $\hat {a}_{u}^{4}$ } is strong, but { $\hat {s}_{u}^{1}$ , $\hat {s}_{u}^{2}$ } is weak; and for $1550\lt t_{est}\lt 1650$ , both { ${\hat{s}_u^1,\hat{s}_u^2}$ } and { $\hat {a}_{u}^{3}$ , $\hat {a}_{u}^{4}$ } are strong. Intervals between the two sets of dashed lines ( $437\lt t_{est}\lt 446$ and $1594\lt t_{est}\lt 1605$ ) indicate a typical and an atypical vortex shedding period, respectively.

In figure 14, the amplitude of the temporal coefficients show significant cycle-to-cycle variation. Closer inspection suggests amplitude changes for different coefficients coincide. For instance, the variations of the signal amplitude envelope of the second harmonics of the Kármán vortex shedding, { $\hat {s}_{u}^{3}$ , $\hat {s}_{u}^{4}$ }, generally follow those of the fundamental harmonics { $\hat {a}_{u}^{1}$ , $\hat {a}_{u}^{2}$ } (e.g. amplitudes in regions coloured in grey). Also, generally, the first pair of the accompanying frequency modes, { $\hat {a}_{u}^{3}$ , $\hat {a}_{u}^{4}$ }, seems to be more active (amplitudes increase) when the low-frequency modes { $\hat {s}_u^1$ , $\hat {s}_u^2$ } are stronger. Note that $\hat {s}_u^2$ lags $\hat {s}_u^1$ for approximately $2/f_{L}$ or $1/5f_{s}$ (see Appendix C for a discussion on the difference between $s_u^1$ and $s_u^2$ ).

The variations of the strength (amplitude) of the fundamental harmonic of Kármán vortex shedding, { $\hat {a}_{u}^{1}$ , $\hat {a}_{u}^{2}$ }, seem related to the strength of low-frequency modes, { $\hat {s}_u^1$ , $\hat {s}_u^2$ }, and the first pair of the accompanying frequency modes, { $\hat {a}_{u}^{3}$ , $\hat {a}_{u}^{4}$ }. Generally, three main patterns can be identified in figure 14:

1. (i) The Kármán vortex shedding is similar to a typical cycle when neither { $\hat {a}_{u}^{3}$ , $\hat {a}_{u}^{4}$ } nor { $\hat {s}_{u}^{1}$ , $\hat {s}_{u}^{2}$ } is strong, such as the shedding period between the dashed lines about $t_{est} \approx 440$ .

2. (ii) The Kármán vortex shedding is weakly modulated under three circumstances: (a) { $\hat {a}_{u}^{3}$ , $\hat {a}_{u}^{4}$ } and one of the $\hat {s}_{u}^{1}$ or $\hat {s}_{u}^{2}$ modes are strong (e.g. the grey interval about $t_{est}\approx 100$ ); (b) { $\hat {s}_{u}^{1}$ , $\hat {s}_{u}^{2}$ } is strong, but { $\hat {a}_{u}^{3}$ , $\hat {a}_{u}^{4}$ } is weak (e.g. the grey interval about $t_{est}\approx 360$ ); and (c) { $\hat {s}_{u}^{1}$ , $\hat {s}_{u}^{2}$ } is weak, but { $\hat {a}_{u}^{3}$ , $\hat {a}_{u}^{4}$ } is strong (e.g. the grey interval about $t_{est}\approx 810$ ).

3. (iii) The Kármán vortex shedding is strongly modulated (nearly suppressed) when both $\{\hat {s}_{u}^{1}$ , $\hat {s}_{u}^{2}\}$ and { $\hat {a}_{u}^{3}$ , $\hat {a}_{u}^{4}$ } are strong, such as the interval within the dashed lines about $t_{est} \approx 1600$ .

Figure 15 shows phase plots of temporal coefficients corresponding to the low-frequency signature { $\hat {s}_u^1$ , $\hat {s}_u^2$ }, first pair of accompanying frequency modes { $\hat {a}_u^3$ , $\hat {a}_u^4$ }, and the corresponding amplitude envelope for each pair (i.e. $\sqrt {(\hat {s}_u^1)^2+(\hat {s}_u^2)^2}$ and $\sqrt {(\hat {a}_u^3)^2+(\hat {a}_u^4)^2}$ , respectively), all against the envelope for the first harmonics of Kármán vortex shedding (i.e. $\sqrt {(\hat {a}_u^1)^2+(\hat {a}_u^2)^2}$ ). Most data points fall in a dense cluster.

Figure 15.

Phase plots of selected estimated temporal coefficients and envelopes of the amplitudes of selected mode pairs, all against the envelope of the first harmonic temporal coefficients, at $z=1.83$ . Estimated coefficients are obtained using asynchronous reference pressure signal (i.e. collected concurrently with PIV measurements at $z=0.92$ ). Green and red data points correspond to the first ( $70\lt t_{est}\lt 130$ ) and last ( $1550\lt t_{est}\lt 1650$ ) grey regions in figure 14, respectively. Green dashed boxes and green arrows are added to help visualizing the bounds of the data points in the first interval. Every second point is plotted for clarity.

Figure 16.

Educed vortex structures, using $Q=0$ criterion, from reconstruction of the flow field with all 10 SPOD velocity modes. Iso-surfaces are coloured with streamwise vorticity, $\Omega _{x}$ . Both time intervals shown on the left and right sides cover approximately half of the shedding cycles, indicated with dashed lines in figure 14. The beginning of these cycles are selected such that they correspond to the phase-averaged reconstruction of phase $\mathit {\phi }_{1}$ in figure 13. For (a)–(c), added annotations are similar to those of figure 13(left side). For (d)–(e), the enclosed regions with yellow dashed-lines indicate loci of un-conventional re-attachments of connector strands. Regions indicated with green and red dashed lines indicate patched of isolated vortices, originating from near the free end.

Data points lying on trajectories outside the cluster can typically be related to changes in modulation or the dynamics of the vortex shedding. In figure 15, two groups of data points are distinguished: (i) the green-coloured data points are related to the weakly modulated motions in the first grey interval ( $70\lt t_{est}\lt 130$ ) in figure 14, and (ii) the red-coloured data points are related to the nearly suppressed cycles in the last grey interval ( $1550\lt t_{est}\lt 1650$ ) in figure 14. To help identifying the bounds of the first group data points, green dashed boxes and green arrows are added. The second group data points are generally on the left side of the green dashed boxes. For the first group, although the magnitude of $\sqrt {(\hat {s}_u^1)^2+(\hat {s}_u^2)^2}$ can go as high as approximately 0.70 in figure 15(a), the magnitude of $\sqrt {(\hat {a}_u^1)^2+(\hat {a}_u^2)^2}$ remains above 1, indicating only a weak modulation to the Kármán vortex shedding. In this interval, most of the contributions to $\sqrt {(\hat {s}_u^1)^2+(\hat {s}_u^2)^2}$ come from $\hat {s}_u^1$ while $\hat {s}_u^2$ remains about zero (see figure 15(b) and figure 15(c) and the time series of figure 14). For the second group, on the other hand, while $\sqrt {(\hat {s}_u^1)^2+(\hat {s}_u^2)^2}$ hovers around 0.5 in figure 15(a), $\sqrt {(\hat {a}_u^1)^2+(\hat {a}_u^2)^2}$ reach very low magnitudes, indicating a significant modulation of the the Kármán vortex shedding. During this interval, both $\hat {s}_u^1$ and $\hat {s}_u^2$ contribute similarly to $\sqrt {(\hat {s}_u^1)^2+(\hat {s}_u^2)^2}$ , which indicates that for the Kármán vortex shedding to be suppressed, both low-frequency modes must be strong. Additionally, during the suppressed cycles, the amplitude envelope of $\hat {a}_u^3$ and $\hat {a}_u^4$ (i.e. $\sqrt {(\hat {a}_u^3)^2+(\hat {a}_u^4)^2}$ in figure 15(d)) is larger when compared with the weakly modulated cycles.

Typical and very strongly modulated (atypical) vortex shedding cycles are compared next. Figure 16 shows educed vortex structures, using the $Q=0$ criterion, from a reconstruction of the flow field using the 10 estimated SPOD velocity modes. Three instances of two time intervals indicated with dashed lines in figure 14 are shown: $437\leqslant t_{est} \leqslant 442$ (typical) and $1594 \leqslant t_{est} \leqslant 1599$ (atypical). In general, the typical shedding cycles last between 9 and 10 time units to complete, whereas the atypical shedding cycles during the $1550\lt t_{est}\lt 1650$ interval take between 10 and 12 time units. The cycles shown in figure 16 take 10 and 11 time units, respectively. Thus, the shown time intervals in both cases cover approximately half of a shedding cycle. For the first interval (typical cycle), the low frequency and first accompanying frequency mode pairs are weak, whereas for the second interval (atypical cycle), both these mode pairs are strong. Supplementary video files no. 4 to no. 7 show the flow field reconstruction for $0 \lt t_{est}\lt 2000$ (over 200 shedding cycles); each file covers 500 time units. Additionally, figure 17 shows the educed vortex structures of figure 16(a) and figure 16(d) from two other angles; figure 18 shows iso-contours of reconstructed $\hat {v}$ - and $\hat {w}$ -velocity components from reconstructions of figure 16, overlaid by sectional pseudo-streamlines, at the symmetry plane ( $y=0$ ); and figure 19 shows similar information to figure 18 at two horizontal planes, $z=1$ and $z=2$ .

Figure 17.

Educed vortex structures, using $Q=0$ criterion, from reconstruction of the flow field with all 10 SPOD velocity modes. Iso-surfaces are coloured with streamwise vorticity, $\Omega _{x}$ . For the plots on left, the time instance is the same as in the figure 16(a) and for the plots on right, the time instance is the same as in the figure 16(d). Annotations in black colour indicate the connection sites between Kármán vortices, and the green arrow indicate the structure associated with post attachment vortices ( $PAV$ ) in figure 12.

Figure 18.

Iso-contours of $\hat {v}$ - and $\hat {w}$ -velocity components from reconstructions shown in figure 16, overlaid with sectional pseudo-streamlines at the symmetry plane ( $y=0$ ). Green arrows indicate pseudo-saddle points. For better correspondence between two figures, letter (c) is not used in addressing the panels.

Figure 19.

Iso-contours of reconstructed velocity components using all 10 selected SPOD modes. Quadrant one (top left) shows $\hat {v}$ -component at $z=1$ , quadrant two (top right) $\hat {v}$ -component at $z=2$ , quadrant three (bottom left) $\hat {w}$ -component at $z=1$ , and quadrant four (bottom right) $\hat {w}$ -component at $z=2$ . The iso-contours are overlaid with sectional pseudo-streamlines and iso-contours of $Q=0.01$ (solid green lines). Additionally, at $z=2$ , the bounds of connector strands intersections with the plane are indicated using dashed and dotted lines, where the line colours indicate the sense of rotation of streamwise vorticity, $\Omega _{x}$ , of the strand. Time stamps are the same as in figure 16.

The iso-surfaces in figure 16(a–c) capture the educed vortex structures during $437 \leqslant t_{est} \leqslant 442$ . During this typical cycle, the educed Kármán vortex shedding patterns are very similar to the phase-averaged reconstructions of figure 13. In the half-loop shedding pattern shown in figure 16(a–c), the top of a newly shed Kármán vortex $A$ detaches from $A^{+}$ , re-orients towards the symmetry plane and connects with the back side of a newly formed Kármán vortex $B$ on the opposite side. This connection to the opposite side vortex can be better seen in figure 17(left side), which shows the half-loop pattern formed between vortices $B^{-}$ and $A$ at $t_{est}=437$ (same as in figure 16(a)).

In figure 18, the pseudo-saddle points, indicated by green arrows, show the loci of intersection between two positive bifurcation surfaces: one originating from the top of the cylinder and the other originating from the base plate. For the first interval, figure 18(left side), the position of this saddle point moves between $z=0.75$ and 1.75, which is consistent with its mean location during a phase-averaged cycle, at approximately $z\approx 1.25$ . Furthermore, the iso-contour patterns of $\hat {v}$ -velocity component in figure 18( $\text {a}_{1}$ ) and ( $\text {b}_{1}$ ) are similar to the ones obtained from the phase-averaged field (Mohammadi et al. Reference Mohammadi, Morton and Martinuzzi2022).

The educed vortical structures for the atypical shedding cycle during $1594 \leqslant t_{est} \leqslant 1605$ , figure 16(d–f), are significantly different from either the phase-averaged or typical cycle reconstructions. This is associated with (i) an increase in the amplitudes of { $\hat {s}_u^1$ , $\hat {s}_u^2$ }, which are related to the low-frequency ( $f_{L}$ ) motion from the obstacle’s free end, and (ii) an increase in the strength of interactions between the $f_L$ motion and Kármán vortex shedding ( $f_{s}$ ), which is reflected in the amplitude of coefficients of the first pair of accompanying frequencies { $\hat {a}_u^3$ , $\hat {a}_u^4$ }. As a result, the first and second harmonics of the Kármán vortex shedding become weak (see associated times series in figure 14), the connector strands form at a lower spanwise location (i.e. top of vortex $A$ in figure 16(d) is at $z=2.7$ compared with that in figure 16(a) at $z=3.2$ ), and the associated shedding process (i.e. half-loop shedding pattern) is strongly disrupted, giving rise to regions of vortex dislocation. Here, the backward-tilted top of vortex $A$ remains connected to the newly forming Kármán vortex on the same side, vortex $A^{+}$ (see regions enclosed by yellow dashed lines in figure 16(e) and 16(f)). While this connection moves to lower heights as the shedding cycle continues, it does not re-orient towards the symmetry plane to connect with vortex $B$ . As a result, the so-called half-loop shedding pattern does not form. This vortex connection pattern is further visualized in figure 17(right side), which shows the connection formed between vortices $B^{-}$ and $B$ at $t_{est}=1594$ (same as in figure 16(d)).

This disrupted vortex formation pattern is accompanied by another pair of streamwise vortices, originating from close to the free end of the obstacle. In figure 16(e), the vortex core, enclosed with a green dashed line, has an opposite sense of rotation to the primary vortex and associated connector strand below it. As the shedding process goes on, this vortex core grows (see figure 16(f)), without connecting with the vortical strands below it. Similarly, on the opposite side of figures 16(d) and 16(e), growing vortex cores with an opposite sign of streamwise vortices to the connector strands below them, are enclosed with red dashed lines.

The difference in the orientation of the connector strands between the typical and atypical cycles becomes more clear in comparing figure 19( $\text {a}_{2}$ )-( $\text {c}_{2}$ ) with figure 19( $\text {d}_{2}$ )-( $\text {e}_{2}$ ), where the connectors strands intersections with the horizontal plane at $z=2$ are indicated with red (dashed) and blue (dotted) lines. The line colours reflect the sign of streamwise vorticity of the vortical strand (red, positive and blue, negative). It can be seen that during $437\leqslant t_{est} \leqslant 442$ , only one vortical strand intersects with the $z=2$ plane. For instance, at $t_{est}=437$ , only the red-coloured connector strand (originating from positive $y$ values) crosses the mid-span height, which connects with the opposite Kármán vortex at lower heights. At this time instance, another blue-coloured connector strand is forming, but is still at higher elevations (see figure 16(a)). During $1594 \leqslant t_{est} \leqslant 1599$ , on the other hand, although the newer connector strand (the one closer to the obstacle in each time instance) is located at slightly higher $z$ position compared with the older one, the difference in their spanwise location is small, such that both intersect with the $z=2$ plane at all time instances. The simultaneous presence of these vortical strands around the mid-height of the obstacle induces a strong downwash about the symmetry plane, as evident in figure 19( $\text {d}_{4}$ )-( $\text {f}_{4}$ ), which further dislocates the Kármán vortex structures in the lower half of the wake. These observations are consistent with the differences in iso-contours of $\hat {v}$ - and $\hat {w}$ -velocity components in figure 18.

Finally, the stronger negative $\hat {w}$ -component velocity in the lower half of the obstacle during the atypical cycle alters the topology of streamlines in the symmetry plane. The pseudo-saddle point mentioned earlier in the discussion of figures 18( $\text {a}_{1}$ ) and 18( $\text {a}_{2}$ ), is not detected in figures 18( $\text {d}_{1}$ ) and 18( $\text {d}_{2}$ ). Instead, the positive bifurcation line from the free end extends all the way to the base plate. Additionally, in comparison with figures 18( $\text {b}_{1}$ ) and 18( $\text {b}_{2}$ ), the saddle point in figures 18( $\text {d}_{1}$ ) and 18( $\text {d}_{2}$ ) is at a significantly lower location ( $z\approx 0.35$ compared with 0.75) and a more advanced streamwise position ( $x \approx 3.4$ compared with 2.6), which indicates a longer recirculation region in the atypical case. The location of this saddle point downstream of the mean recirculation length coincides with the beginning of a $\Lambda$ -shaped mean structure observed over the base plate ( $x\gt 4, -1\lt y\lt 1$ ) in figure 12. This structure was associated with intermittent interactions between Kármán vortices and the horse-shoe vortex system (Mohammadi et al. Reference Mohammadi, Morton and Martinuzzi2023), which is partly captured in figure 17(right side).