2.1. Flow facility

The test section of the water flume has a length, width and height of 5.0 m, 0.68 m and 0.50 m, respectively. Two centrifugal pumps operate in parallel to circulate the flow in the flume. The reservoir of the flume consists of guide vanes, a honeycomb structure, and two layers of mesh and is connected via a small contraction section to the downstream test section. To provide optical access, the bottom and side walls of the test section are made of glass. The free-stream speed was set to ${U_\infty } = 0.396\ \textrm{m}\ {\textrm{s}^{ - 1}}$ during all the measurements.

An Ahmed body was mounted upside down on a flat plate as seen in figure 1(a). The flat plate had a total length of 700 mm and a width of 620 mm. It featured an optimized leading-edge (LE) profile and a tapered trailing edge (Hanson, Buckley & Lavoie Reference Hanson, Buckley and Lavoie2012). The Ahmed body was located along the spanwise centre of the test section, 195 mm above the bottom wall of the flume, and 126 mm below the free surface of the water flume. These distances minimized the interference of the wake flow with the boundary layer on the flume walls and the free-surface waves.

Figure 1. (a) Schematic of the experimental configuration showing the test section of the water flume, the flat plate and the Ahmed body. (b) An upside-down schematic of the flat plate, Ahmed body, the laser sheet and two cameras used for planar PIV. The coordinate system is shown in the enlarged view of the model with the origin, O, at the centre of the rear face. (c) An upside-down schematic of the tomo-PIV configuration showing the four high-speed cameras, the laser light (green) and the measurement volume $V$ (red).

The LE of the Ahmed body was at a distance of l ₁ = 259 mm downstream of the plate's LE. At this location, the Reynolds number, Re, of the boundary layer based on l ₁ and U_∞, is 1.0 × 10⁵, slightly smaller than the critical Reynolds number for natural laminar-to-turbulent transition. Preliminary planar PIV measurements without the Ahmed body showed that the shape factor at this location was 1.86, confirming a transitional state (Schlichting & Gersten Reference Schlichting and Gersten2016). Therefore, a tripping tape with a rectangular cross-section of 1 mm × 4.5 mm (wall-normal height × streamwise length) was installed at l ₂ = 160 mm downstream of the plate LE. The location of the tape (l ₂) and height (h) were selected to ensure the laminar-to-turbulent transition according to the criteria described by Braslow & Knox (Reference Braslow and Knox1958) and Schlichting & Gersten (Reference Schlichting and Gersten2016). In addition, PIV measurements showed a shape factor of 1.47 at l ₁ (without the Ahmed body), indicating that the trip wire resulted in a turbulent boundary layer. When the Ahmed body was not installed, the boundary layer thickness (based on 99 % of U_∞) at l ₂ was δ ₂ = 5.7 mm, and at l ₁ was δ ₁ = 8.6 mm. The distance between the tripping tape and the Ahmed body is 17.5δ ₂, which allows sufficient flow development before the flow reaches the Ahmed body (Elsinga & Westerweel Reference Elsinga and Westerweel2012).

The Ahmed body featured a flat back (no taper at the rear face) and had a 1 : 9.6 scale with respect to the original model of Ahmed et al. (Reference Ahmed, Ramm and Faltin1984). The model was machined from acrylic using a computerized numerical control router. The front and rear faces were painted in black to reduce reflections of the laser light during the experiments. The total length L of the model was 109.0 mm, and the height H and width W of the rear face were 30.0 mm and 40.0 mm, respectively. The uncertainty in these dimensions is approximately ±0.03 mm, accounting for potential thermal expansion or contraction of the model. The Reynolds number, defined as Re_H = U_∞H/ν, is 10 000. This Re_H matches the Re_H of recent numerical simulations by Podvin et al. (Reference Podvin, Pellerin, Fraigneau, Evrard and Cadot2020, Reference Podvin, Pellerin, Fraigneau, Bonnavion and Cadot2021). Podvin et al. (Reference Podvin, Pellerin, Fraigneau, Evrard and Cadot2020) employed POD analysis on the flow field derived from their numerical simulation and noted a good agreement with the experimental findings reported by Evrard et al. (Reference Evrard, Cadot, Herbert, Ricot, Vigneron and Délery2016) at a higher $R{e_H} = 4.0 \times {10^5}$. Consequently, based on the observations of Podvin et al. (Reference Podvin, Pellerin, Fraigneau, Evrard and Cadot2020), the flow structures characterized in the current investigation may hold relevance at higher Re_H. The model was connected to the plate using four cylinders with a 3 mm diameter. The clearance between the model and plate, C, was set to 11.2 mm. The clearance-to-height ratio of the current set-up is C/H = 0.37, which is larger than the C/H = 0.17 used by Ahmed et al. (Reference Ahmed, Ramm and Faltin1984). The larger clearance is chosen to compensate for the large δ ₁/H = 0.29 of the current experiment. In general, Grandemange et al. (Reference Grandemange, Gohlke and Cadot2013) and Cadot, Evrard & Pastur (Reference Cadot, Evrard and Pastur2015) showed that bi-stability is present when C/H > 0.1.

As shown from the enlarged view of the Ahmed body in figure 1(b), the origin of Cartesian coordinate system O is at the middle of the rear face. In the following sections, the directions of the axes are defined as x in the streamwise direction, y in the spanwise direction and z in the direction normal to the flat plate. The components of instantaneous velocity are defined as U, V and W, following the same order as the coordinate system axes. Lastly, the lowercase letters u, v and w represent the fluctuating velocity components. The mean value and the absolute magnitude of any quantity are indicated as $\langle \cdots \rangle $ and $|\cdots |$, respectively.

2.2. Planar particle image velocimetry

A planar PIV experiment was carried out to characterize the mean flow upstream and downstream of the Ahmed body. The PIV system consisted of two Imager ProX (LaVision GmbH) 14-bit CCD cameras with a pixel size of 7.4 × 7.4 μm². Both cameras were equipped with Nikon lenses with a focal length of f = 105 mm and an aperture setting of f/5.6. The upstream and downstream measurements were conducted simultaneously using a common laser sheet aligned with the x–y plane, as shown in figure 1(b). The first camera imaged the field-of-view (FOV) indicated as FOV1 upstream of the Ahmed body, while the second camera imaged FOV2 downstream of the model. Both cameras imaged the flow using cropped sensors with 2047 × 1001 pixel. FOV1 and FOV2 had similar dimensions of 146.4 × 71.5 mm² and 149.8 × 73.2 mm², respectively. The digital resolution of FOV1 was 71.5 μm pixel⁻¹, and the digital resolution of FOV2 was 73.2 μm pixel⁻¹. A dual-cavity Nd:YAG laser (Gemini, NewWave Research) with 110 mJ pulse⁻¹ was used as the illumination source. Combining two cylindrical and two spherical lenses converted the laser beam into a wide sheet with 1 mm thickness covering FOV1 and FOV2. Two sets of 1500 double-frame images were collected at an acquisition rate of 9.4 Hz and with a laser pulse separation of Δt = 1500 μs.

Glass beads with the thin silver coating (SG02S40 Potters Industries) were used as tracer particles for planar PIV. These monodisperse tracers have a mean diameter of d_p = 2 μm and a density of ${\rho _p} = 4\ \textrm{g}\ \textrm{c}{\textrm{m}^{ - 3}}$ (Das & Ghaemi Reference Das and Ghaemi2021). Their response time, ${t_p} = {\rho _p}d_p^2/18\mu$, and settling velocity, ${u_p} = ({\rho _p} - \rho )d_p^2g/18\mu$, are 9.8 × 10⁻⁷ s and 7.2 × 10⁻⁶ m s⁻¹, respectively. Here, ρ and μ are water's density and dynamic viscosity, and g is the gravitational acceleration. Assuming that the flow time scale is estimated based on the vortex-shedding period (~7 × H/U_∞), the Stokes number of these tracers, Stk = t_p/t_f, is 1.8 × 10⁻⁶, which is smaller than the recommended value of 0.1 (Samimy & Lele Reference Samimy and Lele1991). The Froude number Fr is also estimated as u_p/U_∞, which is 1.8 × 10⁻⁵, several orders of magnitude smaller than unity (Bewley, Sreenivasan & Lathrop Reference Bewley, Sreenivasan and Lathrop2008).

To improve the image's signal-to-noise ratio, the minimum intensity of all images was calculated and then subtracted from each image. The resulting images were then normalized using the average intensity of the images. The 2-D vector fields were obtained through double-frame cross-correlation with a multi-pass algorithm in Davis 8.3 (LaVision GmbH). The final interrogation window was 32 × 32 pixels (2.2 × 2.2 mm²). A window overlap of 75 % was applied between neighbouring interrogation windows, resulting in 55 and 73 vectors along the height and width of the model, respectively. Approximately 3 % of the velocity vectors were identified as spurious and were removed based on the universal outlier detection method (Westerweel & Scarano Reference Westerweel and Scarano2005).

2.3. Tomographic particle image velocimetry

The tomo-PIV system consisted of four high-speed cameras (Phantom v611, Vision Research) imaging the illuminated volume from the camera arrangement shown in figure 1(c). The cameras feature a complementary metal-oxide-semiconductor sensor with 1280 × 800 pixels; each pixel 20 × 20 μm². The two upstream cameras had a small viewing angle of approximately 5° with respect to the z-axis, while the two downstream cameras had a larger viewing angle of approximately 35°. To minimize image distortion at high imaging angles, the latter two cameras imaged through two water-filled prisms installed on the glass wall of the flume. The cameras had a digital resolution of approximately 70 μm pixel⁻¹ and a magnification of 0.3. All cameras had Nikon lenses ( f = 105 mm) and Scheimpflug adapters. An aperture setting of f/22 was used for all cameras, resulting in a depth-of-focus of approximately 66 mm.

A dual-cavity Nd:YLF laser (DM20-527DH, Photonics Industries) with 20 mJ pulse⁻¹ illuminated the measurement volume. The laser light was directed perpendicular to the flat plate (in the negative y direction) from the bottom glass wall of the water flume. The beam passed through multiple cylindrical lenses to first form an elliptical cross-section. Then four knife edges at the bottom wall of the flume cropped the edges of the expanded beam. This resulted in a rectangular cross-section and a relatively ‘top hat’ intensity profile across its cross-section. The illuminated volume had a cross-section of (Δx, Δz) = (85 mm, 67 mm) at y = −H/2 and (81 mm × 65 mm) at y = +H/2 plane. The measurement volume, indicated by the red colour in figure 1(c), had dimensions of 78 × 45 × ~66 mm³ (2.6H × 1.5H × 2.2H) in x, y and z, respectively.

To address the limited energy of the high-repetition laser, we used relatively large polystyrene spheres (TS250 Dynoseeds) with a mean diameter of d_p = 250 μm and density of 1.05 g cm⁻³. The larger tracers increase the intensity of the scattered light $({\sim} d_p^2)$, while their nearly neutral density in water results in a small response time of ${t_p} = 4 \times {10^{ - 3}}\ \textrm{s}$. Based on the vortex-shedding time scale, the estimated Stk number of these particles is 7.6 × 10⁻³, which is smaller than the maximum recommended value of 0.1 (Samimy & Lele Reference Samimy and Lele1991). The settling velocity of the tracers is also 1.9 × 10⁻³ m s⁻¹, and therefore their Fr is 4.7 × 10⁻³, which is also several orders of magnitude smaller than unity (Bewley et al. Reference Bewley, Sreenivasan and Lathrop2008). Based on visual inspection of the tomo-PIV images, the particle image diameter is between 3 and 5 pixels, and the seeding density is 0.03 particles per pixel. Combining these two parameters results in the maximum recommended source density of N_s = 0.3 (Scarano Reference Scarano2012).

The imaging system was calibrated using a dual-plane 3-D calibration plate (type 10, Lavision GmbH) that was traversed to five z locations with 12 mm spacing. The mapping function of measurement volume was initially constructed using a third-order polynomial fit in Davis 8.4 (LaVision GmbH). To improve the calibration accuracy, this was followed with a volume self-calibration procedure (Wieneke Reference Wieneke2008). Using self-calibration, the root-mean-square of the calibration error reduced from 1.3 to 0.05 pixels, which is below the recommended 0.1 pixels.

The tomo-PIV images were recorded as double-frame and time-resolved single-frame images. The double-frame images allowed for obtaining a longer duration of non-correlated data at a low acquisition frequency for faster statistical convergence. In contrast, the single-frame data at a higher acquisition frequency allowed for resolving the high-frequency dynamics of the flow. For double-frame recording, 3 datasets of 3107 double-frame images were collected at 25 Hz using a laser pulse separation of Δt = 3500 μs. The total duration of each set is 124.3 s, which is equivalent to 1641 × H/U_∞ (larger than the bi-stability time scale of ~1000H/U_∞). For time-resolved measurements, 10 datasets of 6212 single-frame images were collected at a recording rate of 280 Hz. Each time-resolved dataset was 22.2 s, which is 292 × H/U_∞ (shorter than the bi-stability time scale). The time-resolved data can be used for spectral analysis of the flow between St_H of 0.003 and 21.2 (half the acquisition frequency). A similar maximum particle displacement of around 18 pixels was obtained for both double-frame and single-frame images. The 10 time-resolved datasets provide the stochastic ensemble required for SPOD computations.

For processing the tomo-PIV images, first, the minimum of all images was subtracted from each image. This was followed by subtracting the local minimum of image intensity within a kernel size of 5 pixels. The images were also normalized using a local average with a kernel of 100 pixels. The measurement volume was reconstructed using the fast multiplicative algebraic reconstruction technique (fast-MART) in Davis 8.4 (LaVision GmbH) following the algorithm of Atkinson & Soria (Reference Atkinson and Soria2009). The reconstructed volume has 1116 × 645 × 931 voxels in x, y, and z directions. The vector field was obtained using the direct correlation in Davis 8.4 (LaVision GmbH). The final interrogation volume was 48 × 48 × 48 voxels (3.36 × 3.36 × 3.36 mm³) with a 75 % overlap, resulting in 93 × 54 × 78 vectors in the x, y and z directions, respectively. The spurious vectors formed approximately 2 % of the total number of vectors. These vectors were detected using the universal outlier detection (Westerweel & Scarano Reference Westerweel and Scarano2005) and replaced using linear interpolation. Finally, the measurement volume, $V$, was further cropped to 83 × 47 × 56 vectors (2.3H × 1.3H × 1.6H) to remove noisy borders. The fast-MART and direct correlation processes for each single-frame dataset (6213 images) were carried out using a 16-core processor (AMD Ryzen 9 5950X) over a duration of 2 weeks.

The bi-stability motion, discussed earlier, leads to prolonged periods of spanwise asymmetry in the flow field. Consequently, achieving spanwise symmetry of time-averaged velocity fields becomes challenging, particularly for tomo-PIV measurements that generate large datasets and require intensive computations. To address this challenge, we adopt the technique utilized by Podvin et al. (Reference Podvin, Pellerin, Fraigneau, Evrard and Cadot2020) to enforce spanwise symmetry. This involves appending the original vector fields of the 10 time-resolved tomo-PIV datasets with 10 additional copies, each flipped along the spanwise direction. The copied vector fields also undergo a reversal of the spanwise velocity component to ensure a divergence-free flow field. A similar symmetrization process is also applied to the 3 double-frame tomo-PIV datasets. The resulting symmetry-enforced database eliminates any spanwise asymmetry in time-averaged fields and modal decomposition and improves the statistical convergence of the results.

2.4. Modal decomposition

The POD computations are carried out based on the snapshot method of Sirovich (Reference Sirovich1987) and using the 6 symmetry-enforced double-frame tomo-PIV datasets. The vector fields obtained from the double-frame images are temporally uncorrelated, resulting in faster statistical convergence of the POD modes. To begin the POD computations, three instantaneous velocity fluctuations (u, v and w) from all the grid points of each tomo-PIV snapshot were stacked into a column vector ${\boldsymbol{x}_i} \in {R^N}$. Here, N equals the number of flow variables times the number of grid points (3 × 218 456). The x_i vectors were then grouped together in a 2-D matrix ${\boldsymbol{\mathsf{X}}} = [{\boldsymbol{x}_1},{\boldsymbol{x}_2}, \ldots ,{\boldsymbol{x}_M}] \in {R^{N \times M}}$ where M = 18 642 is the total number of snapshots. The singular value decomposition of the correlation matrix, X^T X, was computed to obtain UσV^T. The parameters U and V are the left and right matrices of eigenvectors, and σ is the eigenvalues matrix. The matrix of spatial modes is $\boldsymbol{\varPhi}$ = XUσ^−1/2, where each spatial POD mode, $\varPhi$_j, is a column of $\boldsymbol{\varPhi}$. The energy of each POD mode, σ_j, is obtained from the diagonal components of σ. This method is equivalent to carrying out an eigendecomposition of X^T X according to X^T XU = Uσ^−1/2 and calculating the spatial modes following $\boldsymbol{\varPhi}$ = XUσ^−1/2.

An SPOD code was developed in MATLAB (MathWorks) based on the SPOD algorithm detailed by Towne et al. (Reference Towne, Schmidt and Colonius2018). The computations used the 20 symmetry-enforced time-resolved tomo-PIV datasets, each consisting of 6212 snapshots recorded at 280 Hz. Each dataset is divided into 3 equal segments with 50 % overlap, which results in a total of S = 60 segments. The data within each data segment are organized in a 2-D matrix, ${{\boldsymbol{\mathsf{X}}}_k} = [{\boldsymbol{x}_1},{\boldsymbol{x}_2}, \ldots ,{\boldsymbol{x}_M}] \in {R^{N \times M}}$, where k is the segment's index varying from 1 to 60. Similar to POD, x_i is a column-wise vector of the tomo-PIV snapshot with a length of N = 3 × 218 456. The total number of snapshots within each segment, M, equals 3106 since the segment length is half the size of the time-resolved dataset. The current segmentation procedure is slightly different from the algorithm described by Towne et al. (Reference Towne, Schmidt and Colonius2018), in which one long dataset was divided into multiple segments. This difference is imposed by the finite onboard memory of high-speed cameras, which prevents acquiring longer datasets.

In addition, due to the limited memory of the processing computers, each data segment, X_k, is processed separately. The rows of X_k matrices are multiplied by a Hamming window and then converted to the frequency domain by performing the fast Fourier transform operation. The Fourier representation of each X_k is ${{\boldsymbol{\mathsf{Q}}}_k} = [{\boldsymbol{q}_1},{\boldsymbol{q}_2}, \ldots ,{\boldsymbol{q}_K}] \in {C^{N \times K}}$, where q_l is a column-wise vector of Fourier coefficients at frequency index l, varying from 1 to M/2 = 1553. The index l = 1 corresponds to a frequency of 0 Hz, while the last index of l corresponds to 140 Hz (St_H = 10.6). The resolution of the spectral domain is also 0.09 Hz (equivalent to ΔSt_H of 0.007). Next, the column vectors q_l corresponding to frequency index l from all Q_k matrices are grouped into a single matrix, ${{\boldsymbol{\mathsf{Q}}}_l^{\prime}} = [\boldsymbol{q}_l^1,\boldsymbol{q}_l^2, \ldots ,\boldsymbol{q}_l^S] \in {C^{N \times S}}$. Once ${{\boldsymbol{\mathsf{Q}}}_l^{\prime}}$ is computed, matrix C_l, for each frequency index l, is constructed as ${{\boldsymbol{\mathsf{C}}}_l} = (1/S){\boldsymbol{\mathsf{Q}}}^{\prime T}_l{{\boldsymbol{\mathsf{Q}}}_l^{\prime}}$ using the method of snapshots (Sirovich Reference Sirovich1987). Finally, the SPOD modes and eigenvalues are computed following ${{\boldsymbol{\mathsf{C}}}_l}{\boldsymbol{\phi }_l} = {\boldsymbol{\phi }_{\!l}}{\boldsymbol{\lambda }_l}$ and ${\boldsymbol{\varPsi }_{\!l}} = {{\boldsymbol{\mathsf{Q}}}_l}{\boldsymbol{\phi }_l}$. Here, ϕ_l is the eigenvector of C_l, and $\boldsymbol{\varPsi}\!$_l includes the SPOD modes at frequency index l. The matrix $\boldsymbol{\varPsi}\!$_l has the form of $[\varPsi _l^1,\varPsi _l^2, \ldots ,\varPsi _l^S] \in {C^{N \times S}}$. The eigenvalue $\lambda _l^n$ corresponding to SPOD mode n at wavenumber l is embedded in the diagonal elements of λ_l.

In the analysis of the subsequent sections, the SPOD modes are used to reconstruct a periodic reduced-order model (ROM) at selected frequencies (Nekkanti & Schmidt Reference Nekkanti and Schmidt2021). The ROM model of the U component at frequency index, l, is computed according to

(2.1)\begin{equation}{U_{ROM}} = \langle U\rangle /{U_\infty } + \mathcal{R}e({\varPsi _{u,l}}/{U_\infty } \times {\textrm{e}^{\textrm{i}\varPhi }}).\end{equation}

Here, $\mathcal{R}e$ is the real part of a complex number. For any grid point of the measurement domain, $\langle U\rangle$ is the average velocity, $\varPsi_{u}$_,l denotes the streamwise component of SPOD mode at frequency index l. The phase $\varPhi$ is also defined as 2${\rm \pi}$f_lt in which f_l is the frequency (in Hz) corresponding to index l. The duration of each ROM cycle is different as it is equal to 1/f_l. Similar equations based on $\varPsi_{v}$_,l and $\varPsi_{w}$_,l are also used to reconstruct ROMs of the V and W components indicated as V_ROM and W_ROM, respectively. In addition, to reconstruct the ROM of the velocity fluctuations, (2.1) is reduced to

(2.2)\begin{equation}{u_{ROM}} = \mathcal{R}e({\varPsi _{u,l }}/{U_\infty } \times {\textrm{e}^{\textrm{i}\varPhi }}).\end{equation}

Similar equations are also used to reconstruct v_ROM and w_ROM.

6.1. Bi-stable motions

Figure 8 depicts the spatial organization of the first SPOD mode at St_H = 0.007 in zone A. This SPOD mode features two antisymmetric structures that extend the entire streamwise length of the domain, with their axes positioned slightly above the mid-height plane at z/H ~ 0.2. In figures 8(a) and 8(c), smaller and weaker structures are also visible closer to the wall at z/H ~ −0.4. The observed symmetry-breaking pattern provides strong evidence of a spanwise asymmetry in the wake that undergoes switching. The x-component of the reduced-order model, u_ROM, generated using this SPOD at St_H = 0.007, can be viewed in supplementary movie 1 available at https://doi.org/10.1017/jfm.2024.288. The movie reveals that the two structures switch signs at phase $\varPhi$ = ${\rm \pi}$ and then revert back to their initial sign at $\varPhi$ = 2${\rm \pi}$.

Figure 8. Spatial organization of the first SPOD mode at St_H = 0.007 in (a) isometric view, (b) top view and (c) back view. The red and blue isosurfaces show $\varPsi_{u}$_,l/U_∞ = ±0.25 for frequency index l = 2.

The barycentre position during the full cycle of the ROM generated using the SPOD mode at St_H = 0.007 is shown in figure 9. The variation of the barycentre in the y–z plane is presented in figure 9(a), and the variation in the x–z plane is shown in figure 9(b). The phases of the ROM, when y_b is at the most negative location, zero and the most positive location, are indicated with $\varPhi$₁ = 0.1${\rm \pi}$, $\varPhi$₂ = 0.6${\rm \pi}$ and $\varPhi$₃ = 1.1${\rm \pi}$, as seen in figure 9(a). Note that reference phase of $\varPhi$ = 0 does not have any physical significance and merely acts as a reference point. In other words, a constant value can be subtracted from $\varPhi$₁, $\varPhi$₂, $\varPhi$₃. Although the trace of the barycentre forms a closed loop, it appears as a line because it retraces its own path. Therefore, the trajectory follows a line that slightly moves between z_b/H ~ −0.04 and −0.02, consistent with the single most probable z_b/H position observed in figure 4. In contrast, y_b/H varies over a large range from y_b/H = −0.18 to 0.18. This variation is larger than the range observed in JPDFs of y_b/H in figure 4. The discrepancy suggests that SPOD modes at other frequencies reduce the range of spanwise motion and the asymmetry caused by the first SPOD mode at St_H = 0.007. Moreover, figure 9(b) demonstrates that the motion of the barycentre in the x–z plane is negligible. Therefore, the SPOD mode at St_H = 0.007 mainly causes the wake to fluctuate in the y-direction, which is consistent with the definition of the bi-stable motion.

Figure 9. The variations of the wake barycentre based on ROM obtained using the SPOD mode at St_H = 0.007. (a) The trace of z_b/H versus y_b/H, and (b) z_b/H versus x_b/H.

Figure 10 provides a visualization of the wake topology of the SPOD mode at St_H = 0.007 for $\varPhi$₁, $\varPhi$₂ and $\varPhi$₃ phases. The vortical structures and separation bubble are shown in figure 10(a–c), and y–z cross-sections that display the streamlines and mode values are seen in figure 10(d–f). To visualize the direction of rotation for streamwise vortices, a modified parameter called Q′ is used in conjunction with the Q-criterion. Specifically, Q′ was defined as ${\omega _x}/|{\omega _x}|\times [Q{H^2}/U_\infty ^2]$, where ω_x is streamwise vorticity. Given that Q’ has the same absolute value as Q, it identifies vortical structures similar to those identified by Q, but its sign denotes the direction of streamwise vorticity. Specifically, a positive Q’ corresponds to a positive ω_x, whereas a negative Q’ corresponds to a negative ω_x.

Figure 10. Visualizations of the ROM reconstructed from the SPOD mode at St_H = 0.007 at (a,d) $\varPhi$₁ = 0.1${\rm \pi}$, (b,e) $\varPhi$₂ = 0.6${\rm \pi}$ and (c, f) $\varPhi$₃ = 1.1${\rm \pi}$. In (a–c), the green isosurface shows U_ROM = 0, red shows Q′ = 0.48 and blue shows Q′ = −0.48. In (d–f), 2-D streamlines correspond to V_ROM and W_ROM components of the ROM, and the contours show u_ROM at the cross-flow plane of x/H = 1.33.

At $\varPhi$₁ = 0.1${\rm \pi}$ in figure 10(a), vortical structures form a toroidal vortex and a streamwise vortex that extend downstream. The streamwise vortex (blue) rotates in a clockwise direction due to its negative ω_x, which is also evident at (y/H, z/H) = (0, 0.3) of the y–z cross-section shown in figure 10(d). This vortex results in a strong spanwise motion in the negative y direction. Inspection of the streamlines in figure 10(d) shows that, in addition to this vortex, there is evidence of a weak counterclockwise vortex at (y/H, z/H) = (0.3, −0.1) that did not appear in the 3-D visualization of figure 10(a). The toroid in figure 10(a) is skewed, with the side at negative y/H shifted in the downstream direction, and the separation bubble tilted towards negative y/H. This asymmetry is consistent with the barycentre location being at the most negative location at $\varPhi$₁ in figure 9(a). At $\varPhi$₂ = 0.6${\rm \pi}$ in figure 10(b), the wake symmetry is recovered, and the Q′ isosurface shows a symmetrical toroid around the separation bubble, with no evidence of strong streamwise vortices. Figure 10(e) shows that the magnitude of the SPOD mode ($\varPsi$_u,_l) is negligible, with the mean velocity dominating the ROM. A stagnation line also appears in the streamlines of figure 10(e). At phase $\varPhi$₃ = 1.1${\rm \pi}$, the separation bubble is tilted toward positive y, and a strong streamwise vortex with counterclockwise rotation appears in figure 10(c). Generally, the flow topology at phase $\varPhi$₃ is opposite to that of $\varPhi$₁, with the separation bubble tilted towards the positive y direction. Supplementary movie 2 provides additional perspectives of the vortical structures and separation bubble at different viewing angles and phases. The top view of the movie clearly shows the streamwise vortex remains at the spanwise centre while changing its rotation direction as the separation bubble shifts from one spanwise side to the other.

The dynamics of the bi-stable motion shown in figure 10 is relatively consistent with the vortical structures of asymmetric states identified by Pavia et al. (Reference Pavia, Passmore and Sardu2018) using stereoscopic PIV in cross-flow planes, and by Pavia et al. (Reference Pavia, Passmore, Varney and Hodgson2020) through flow reconstruction using 3-D POD modes. In both investigations, an upper streamwise vortex dominates over a weaker bottom streamwise vortex. Furthermore, Pavia et al. (Reference Pavia, Passmore, Varney and Hodgson2020) demonstrated that the streamwise vortices originate from the skewed toroid, consistent with visualizations of figure 10. However, according to the numerical studies conducted by Dalla Longa et al. (Reference Dalla Longa, Evstafyeva and Morgans2019) and Fan et al. (Reference Fan, Chao, Chu, Yang and Cadot2020), large hairpin vortices separate from the toroid during the switching of asymmetric states. The SPOD mode at St_H = 0.007 does not capture these hairpin vortices. It is plausible that the hairpin shedding is captured by the SPOD mode at higher St_H.

The backflow probabilities, γ, have been calculated using the U_ROM of the first SPOD mode at St_H = 0.007, and their contours are depicted in figures 11(a) and 11(b) at the z/H = 0 and x/H = 0.5 planes, respectively. In figure 11(a), the darkest red contour (γ = 100 %) represents the area that always contains a backward flow, whereas the lightest red contour indicates the maximum extent of the backflow region caused by the oscillations of the bi-stable motion. Notably, the contours demonstrate that the backflow region extends and reaches x/H = 1.7 when the wake is skewed in the y direction. Figure 11(b) shows a relatively symmetrical distribution of γ in the cross-flow plane at x/H = 0.5 with no bi-stable motion in the vertical direction.

Figure 11. The backflow probability, γ, obtained from U_ROM of the first SPOD mode at St_H = 0.007 in the (a) z/H = 0 and (b) x/H = 0.5 planes.

Figure 11(a) shows that the separation bubble length, L_s, is longer for asymmetric states than for the symmetric state. Based on the ROM, the average L_s for the asymmetric states with y_b ≠ 0 is 1.48H, longer than the L_s of 1.24H for y_b = 0 state. This is in agreement with Grandemange et al. (Reference Grandemange, Gohlke and Cadot2013), who reported the length of the conditionally averaged asymmetric wake is slightly longer than the mean separation bubble length. Grandemange et al. (Reference Grandemange, Gohlke and Cadot2014) also found that the L_s of a stable asymmetric wake forced by a small rod placed in the wake flow becomes 4 % longer than the L_s of the undisturbed wake. However, the shorter separation bubble length at y_b = 0 in figure 11(a) may contradict some findings of Grandemange et al. (Reference Grandemange, Gohlke and Cadot2014) and Haffner et al. (Reference Haffner, Borée, Spohn and Castelain2020). Grandemange et al. (Reference Grandemange, Gohlke and Cadot2014) generated a stable symmetric wake by placing a small rod along the spanwise centreline of the wake, resulting in an even longer L_s than that of the asymmetric states. Haffner et al. (Reference Haffner, Borée, Spohn and Castelain2020) also reported a 2 % longer separation bubble length during wake switching between the two asymmetric states. The apparent discrepancy could be due to the presence of other motions in the symmetric states obtained by Grandemange et al. (Reference Grandemange, Gohlke and Cadot2014) and Haffner et al. (Reference Haffner, Borée, Spohn and Castelain2020) or the method used to obtain the symmetric states in these studies.

Figure 12 demonstrates the impact of the SPOD mode at St_H = 0.007 on the separation bubble volume by displaying the fluctuations of the separation volume, ${\varPi ^{\prime}_{ROM}}$, over a 2${\rm \pi}$ phase duration. Here ${\varPi ^{\prime}_{ROM}}$ is calculated as $\varPi$_ROM − $\langle \varPi\rangle$, where $\langle \varPi\rangle$ is the mean separation bubble volume. The figure shows that the volume fluctuations are positive, varying from zero to 0.45$\langle \varPi\rangle$. This figure, together with the 3-D visualizations in figure 10, suggests that as the wake tilts in the spanwise direction towards $\varPhi$₁ = 0.1${\rm \pi}$ and $\varPhi$₃ = 1.1${\rm \pi}$, the separation bubble expands. This is also in agreement with the longer L_s observed in figure 11 during the asymmetric states. The separation bubble returns to a similar size as $\langle \varPi\rangle$ when the wake symmetry is restored at $\varPhi$₂ = 0.6${\rm \pi}$. Additionally, figure 12 reveals that the bi-stable motion results in relatively large variations in volume that can reach up to 45 % of $\langle \varPi\rangle$.

Figure 12. The normalized fluctuations of separation volume $\varPi'\!\big/$$\langle \varPi\rangle$ based on U_ROM obtained using the SPOD mode at St_H = 0.007.

6.2. Swinging/flapping motions

This section characterizes the SPOD modes within zone B covering the range of 0.014 ≤ St_H < 0.123. Figures 13(a)–13(c) illustrate a sample of the first SPOD modes at St_H = 0.034 from different viewing angles. These modes consist of three streamwise-elongated structures, where the two outer structures have the same sign, while the middle one has an opposite sign. This configuration forms a symmetry-preserving pattern with respect to the y = 0 plane. However, there is no symmetry in the vertical direction with respect to the z = 0 plane, as evident in figure 13(c). An animation of the first SPOD mode at St_H = 0.034 is presented in supplementary movie 3. The animation demonstrates that with increasing $\varPhi$, the structures move in the vertical direction while always maintaining spanwise symmetry. As the structures move vertically, they are gradually replaced with streamwise structures of opposite sign. The dynamics of this ROM indicates that this mode can only result in vertical and streamwise motions of the separation bubble.

Figure 13. Spatial organization of the first SPOD mode at St_H = 0.034 (l = 6) in (a) isometric view, (b) top view and (c) back view, and at St_H = 0.041 (l = 7) in (d) isometric view, (e) top view and ( f) back view. The mode at St_H = 0.041 is shown at $\varPhi$ = 0.5${\rm \pi}$. The red and blue isosurfaces show $\varPsi_{u}$_,l/U_∞ = ±0.08.

The first SPOD mode at a slightly higher St_H = 0.041 is shown in figures 13(d)–13( f). This mode is visualized here at $\varPhi$ = 0.5${\rm \pi}$ when the structures are fully extended. In contrast to the previous case, this St_H features only two streamwise-elongated structures that are antisymmetric with respect to the y = 0 plane. The structures are similar to the bi-stability mode presented in figure 8. However, considering that the isosurfaces in figure 13 are displayed at $\varPsi_{u}$_,l/U_∞ = ±0.08, a value smaller than the $\varPsi_{u}$_,l/U_∞ = ±0.25 limit in figure 8, it can be concluded that the structures in figure 13 are notably weaker. The ROM of this mode at St_H = 0.041 in supplementary movie 4 shows the structures become longer with increasing phase $\varPhi$, then gradually fade, and transform to structures with an opposing sign. Smaller and weaker streamwise structures also emerge closer to the wall at z/H ~ −0.4 when $\varPhi$ is approximately 0.5${\rm \pi}$ and 1.5${\rm \pi}$. Therefore, this mode is anticipated to contribute to spanwise oscillations of the wake, although a smaller displacement is expected compared with those observed for the bi-stability mode at St_H = 0.007.

Examining other first and second SPOD modes within the range of 0.014 ≤ St_H < 0.123 shows either spanwise symmetric structures similar to those in figure 13(a–c) or anti-symmetric structures resembling figure 13(d–f). Notably, no specific difference is evident between the first and second SPOD modes, as both exhibit either spanwise symmetric or anti-symmetric structures. Moreover, there is no apparent trend with increasing St_H within the 0.014 ≤ St_H < 0.123 range, except for a decrease in strength and therefore irregularity in the visualization of the SPOD modes. In summary, the first and second SPOD modes with 0.014 ≤ St_H < 0.123 all have symmetric or anti-symmetric structures similar to those shown in figure 13.

Figure 14(a) shows the trajectory of the barycentre in the y–z plane derived from the ROM of the first SPOD mode for all St_H values in the range 0.014 ≤ St_H < 0.123. The trajectory at St_H = 0.034 (shown in red) indicates displacements of the barycentre from z_b/H = −0.06 to −0.02, with no displacement in the y direction. This aligns with the spanwise symmetry observed in the spatial pattern of figure 13(a–c) and supplementary movie 3. For future reference, the phases of the ROM, when the trajectory is at the most negative, zero and most positive z_b/H, are labelled in figure 14 as $\varPhi$₃ = 1.5${\rm \pi}$, $\varPhi$₂ = ${\rm \pi}$ and $\varPhi$₁ = 0.5${\rm \pi}$ (in red font). In contrast, the barycentre trajectory for St_H = 0.041 (shown in black) results in a spanwise motion with no displacement in the vertical direction. Although this barycentre motion is similar to the bi-stability mode in figure 9, the range of motion spans from y/H = −0.05 to 0.05, which is 3.5 times smaller than the spanwise displacement of the bi-stability. The phases of the ROM for the St_H = 0.041 case, when the trajectory is at the most negative, zero and most positive y_b/H, are labelled in figure 14(a) as $\varPhi$₁ = 0.3${\rm \pi}$, $\varPhi$₂ = 0.8${\rm \pi}$ and $\varPhi$₃ = 1.3${\rm \pi}$.

Figure 14. The trajectory of the wake barycentre projected in the (a) y–z and (b) x–z planes obtained from U_ROM for each St_H within 0.014 ≤ St_H < 0.123 range. The red and black lines correspond to the first SPOD modes at St_H = 0.034 and 0.041, respectively. The trajectories for the remaining St_H values of the first SPOD mode are shown in blue.

The trajectories of the remaining St_H values for the first SPOD mode also exhibit similar vertical or horizontal motions of the barycentre in the y–z plane. However, these trajectories (blue lines) are not distinctly visible in figure 14(a) as they are obscured by the red and black lines representing the trajectory of the first SPOD mode at St_H = 0.034 and 0.041. Inspection of the trajectories resulting from the second SPOD mode also shows vertical or horizontal motions and are consequently not shown here for brevity.

The trajectories shown in figure 14(b) within the x–z plane show two types of trajectories: elliptical paths involving both vertical and streamwise motions, and a stationary point with no motion in the streamwise or vertical direction. Due to the presence of vertical displacement, the elliptical paths correspond to the symmetry-preserving modes such as the one at St_H = 0.034. In contrast, the stationary points correspond to the symmetry-breaking modes such as the one at St_H = 0.041 with only spanwise motions. For the elliptical paths the streamwise translations have a similar range as the translations observed in the vertical direction. Examining the trajectories of the second mode (not shown) demonstrates no difference from those of the first SPOD mode in the x–z plane.

To investigate the spanwise motions, the flow filed reconstructed using the SPOD mode at St_H = 0.041 at $\varPhi$₁, $\varPhi$₂ and $\varPhi$₃ is presented in figure 15. The top visualizations of the figure show 3-D vortical structures and the separation bubble, while the lower figures present 2-D contours of streamwise mode magnitude ($\varPsi_{u}$_,l) and streamlines in x/H = 1.33 plane. The vortical structures are visualized using normalized Q-criterion defined as ${Q^\ast } = Q{H^2}/{U_\infty }2$. At $\varPhi$₁, as shown in figure 15(a), the separation bubble is slightly skewed towards the negative y direction, and the toroidal vortex is tilted away from the negative y side of the rear-face. The 2-D streamlines of figure 15(d) show a node that is skewed towards the region of blue isocontour, indicating a greater momentum deficits. In contrast, the reconstructed flow at $\varPhi$₂ in figure 15(b) shows a symmetric separation bubble and toroidal vortex. Inspection of the 2-D streamlines in figure 15(e) reveals that the node is close to the centre of the y–z plane. At $\varPhi$₃ in figure 15(c), the separation bubble and the toroidal vortex are skewed towards positive y, showing a pattern opposite to figure 15(a). The node is also displaced towards positive y as seen in figure 15( f). Supplementary movie 5 provides an animation that corresponds to the flow reconstruction shown in figure 15. The animation demonstrates that as $\varPhi$ increases, the separation bubble and the toroid oscillate between two skewed states. While the toroidal vortex and the separation bubble appears similar to that of the bi-stability in figure 10, a streamwise vortex is not present in the ROM of figure 15 and supplementary movie 5. Therefore, it is concluded that the absence of the streamwise vortex leads to a smaller spanwise motion of the barycentre compared with the spanwise displacement observed in the bi-stability motion.

Figure 15. Visualizations of the ROM reconstructed from the SPOD mode at St_H = 0.041 at (a,d) $\varPhi$₁ = 0.3${\rm \pi}$, (b,e) $\varPhi$₂ = 0.8${\rm \pi}$ and (c, f) $\varPhi$₃ = 1.3${\rm \pi}$. In (a–c), the green isosurface shows U_ROM = 0, red shows Q* = 0.24. In (d–f), 2-D streamlines correspond to V and W components of the ROM, and the contours show u_ROM for l = 7 at the cross-flow plane of x/H = 1.33.

For brevity, the flow field reconstructed using the first SPOD mode at St_H = 0.034 is only depicted in supplementary movie 6. This movie shows vortical structures using Q′ (in red and blue), while the separation bubble is visualized in green. The movie shows a toroidal vortex that undergoes deformation and slightly protrudes in the downstream direction with increasing $\varPhi$. Additionally, the movie illustrates two weak counter-rotating streamwise vortices emerging at $\varPhi$ = ${\rm \pi}$. At this phase, the top view reveals the separation bubble extending downstream, reaching its maximum length. Subsequently, as $\varPhi$ increases, the separation bubble contracts, attaining its minimum length at $\varPhi$ = 2${\rm \pi}$. This motion resembles the expansion and contraction characteristic of the bubble-pumping motion, although the streamwise displacement is small, as seen in figure 14(b). According to the observed barycentre trajectory in figure 14(b) for St_H = 0.034 (the red line), the outward motion of the barycentre in the streamwise direction (+x) occurs along with its upward movement (+z). Conversely, the inward streamwise motion (−x) is coupled with a downward motion (−z) of the barycentre.

The probability of backflow based on the U_ROM of SPOD mode at St_H = 0.041 is depicted in figure 16 for x–y and y–z cross-sections. Overall, the figure shows that the displacements of the backflow region are relatively small, and the edge of the backflow region does not oscillate considerably. The transition from γ = 100 % (red) to γ = 0 (white) occurs over a short length of approximately 0.1H across the spanwise edge of the separation bubble. Therefore, the effect of the first SPOD mode at St_H = 0.041 on the spanwise motion of the separation bubble is negligible compared with the bi-stable mode shown in figure 11.

Figure 16. Backflow probabilities from U_ROM of the SPOD mode at St_H = 0.041 in (a) z/H = 0 and (b) x/H = 0.5 planes.

Figure 17 presents the variations in recirculation volume obtained from ROM for each St_H of the first SPOD mode within the range of 0.014 ≤ St_H < 0.123. Thin blue lines represent all St_H cases, except for St_H = 0.034 and 0.041, which are illustrated with red and black lines, respectively. The figure shows that the separation volume approximately fluctuates in the range ±0.1$\langle \varPi\rangle$, which is significantly smaller than the variations observed due to the bi-stability motion in figure 12. The variation of $\varPi$′ can also become negative for St_H = 0.034, which involves vertical barycentre translation, in contrast to the positive $\varPi$′ observed in figure 12. The figure also shows that the $\varPi$′ variation for St_H = 0.034 is slightly larger than $\varPi$′ variation for St_H = 0.041, characterized by spanwise translation of the barycentre. In general, the results indicate that the motions within 0.123 ≤ St_H < 0.212 result in small the expansion and contraction of the separation bubble.

Figure 17. The normalized fluctuations of separation volume $\varPi'\!\big/$$\langle \varPi\rangle$ obtained from U_ROM of the first SPOD mode for each St_H within 0.014 ≤ St_H < 0.123 range. The red and black lines indicate the U_ROM for St_H = 0.034 and 0.041, respectively, while the blue lines correspond to the other St_H values.

The variations for St_H = 0.041 demonstrate that $\varPi$′ reaches maximum and minimum values at approximately $\varPhi$ = 0.6${\rm \pi}$ and 1.05${\rm \pi}$, respectively, closely aligning with the asymmetric states observed in figures 15(a) and 15(c) at $\varPhi$₁ = 0.5${\rm \pi}$ and $\varPhi$₃ = 1.3${\rm \pi}$. The midpoint between these extrema in figure 17 also approximately coincides with the symmetric state at $\varPhi$₂ = 0.8${\rm \pi}$ in figure 15(b). For St_H = 0.034, the maximum and minimum $\varPi$′ occur at $\varPhi$ = ${\rm \pi}$ and 0, respectively. Upon inspecting the top view in supplementary movie 6, it becomes clear that these phase values correspond to the longest and shortest separation bubbles, indicative of the maximum and minimum separation volume. This observation leads to two conclusions when revisiting figures 14 and 5. First, examination of phase values in figure 14(b) reveals that the maximum and minimum volumes at $\varPhi$ = ${\rm \pi}$ and 0 do not align with the barycentre's maximum and minimum z locations (top and bottom points of the trajectory). In fact, the maximum and minimum volumes correspond to the barycentre being at maximum and minimum x locations when the barycentre crosses z = 0. Second, the correlation between volume fluctuations ($\varPi$′) and the coordinates of the barycentre in figure 5(c) indicates that $\varPi$′ fluctuations within 0.014 ≤ St_H < 0.123 range mainly corresponds to the streamwise oscillations of the wake barycentre. This indicates that the primary cause of volume oscillations is the streamwise motions of the barycentre due to the vertical swinging motion.

The SPOD analysis showed flow motions within the range of 0.014 ≤ St_H < 0.123 result in a small vertical or spanwise movement of the barycentre. The vertical motions were caused by a symmetry-preserving SPOD mode while the spanwise motions were generated by a symmetry-breaking SPOD mode. Both vertical and spanwise motions are referred to as swinging (or flapping) motions here, using the terminology introduced by Pavia (Reference Pavia2019) and Pavia et al. (Reference Pavia, Passmore, Varney and Hodgson2020) to describe movements of the separation bubble that are unrelated to bi-stability motion. The spanwise swinging differs from spanwise bi-stability, as the range of the spanwise motion of the wake barycentre during the swings is several times smaller than the range of spanwise motions of the barycentre induced by the bi-stability. Additionally, visualizations of the flow structures using ROM revealed that the asymmetric state of the swinging motions comprises solely a skewed toroidal vortex, lacking any strong streamwise vortex that was observed in the asymmetric state of the bi-stability motion.

The investigated swinging motions in this study includes the St_H range of 0.04 to 0.08, a range previously associated with bubble-pumping motion in studies by Duell & George (Reference Duell and George1999), Khalighi et al. (Reference Khalighi, Zhang, Koromilas, Balkanyi, Bernal, Iaccarino and Moin2001), Volpe et al. (Reference Volpe, Devinant and Kourta2015) and Pavia et al. (Reference Pavia, Passmore and Sardu2018). The current investigation shows that vertical swinging motions lead to streamwise displacements of the barycentre and variations in the separation bubble volume, aligning with the definition of bubble-pumping motion. However, it is important to note that the change in the separation bubble volume is relatively small, constituting only approximately 10 % of the mean separation bubble volume. Similarly, spanwise swinging motions result in a comparable variation in separation volume, even in the absence of any streamwise motion of the barycentre. Furthermore, the expected in-phase fluctuations around the rear face of the Ahmed body, characteristic of the original bubble-pumping motions, are not apparent in the ROM presented in movie 6. Instead, the in-phase pressure fluctuations noted by Duell & George (Reference Duell and George1999) at the two spanwise sides of the rear face align with the dynamics of the spanwise-symmetric SPOD mode at St_H = 0.034. Consequently, a limited version of the bubble-pumping motion, defined as a small contraction and expansion of the separation bubble and in-phase fluctuations only at the spanwise sides of the rear face, is consistent with the vertical swinging motions observed in the St_H range of 0.014 to 0.123. However, the term swinging/flapping motion describes the key feature of the SPOD motions within this St_H range, specifically referring to the vertical and spanwise motions of the barycentre.

6.3. Vortex shedding

This section investigates the SPOD mode for a range of 0.123 ≤ St_H < 0.234, corresponding to zone C. Within this zone, the distinguishing characteristic of the SPOD mode is the manifestation of multiple alternating structures in the streamwise direction. These alternating structures, which exhibit opposite signs in the SPOD modes, result in vortex-shedding behaviour. The ROMs of the flow provide further confirmation of this, revealing the presence of large vortical structures that are shed downstream. Such unique features set these structures apart from those observed in other zones.

In figure 18(a–c), the SPOD mode for a sample St_H = 0.164 is displayed from different viewing angles. This St_H corresponds to the high energy peak seen in the eigenvalue spectrum of figure 7. The SPOD mode displays a pair of structures with opposite signs arranged vertically at the downstream region of the measurement domain. Another pair of smaller structures with opposite signs is also observed closer to the rear face of the Ahmed body. This spatial arrangement bears a resemblance to the third POD mode in figure 6(d), which was attributed to the vertical vortex shedding. The structures of figure 18(a–c) exhibit spanwise symmetry, consistent with utilizing symmetry-enforced data. They also demonstrate an approximate anti-symmetry in the vertical direction with respect to z = 0 plane as seen in figure 18(c). The animation of this SPOD mode is available in supplementary movie 7, showing streamwise advection of pairs of structures with alternating sign. The movie also shows that the structures within each vertical pair are not precisely identical, exhibiting a slight deviation from vertical anti-symmetry.

Figure 18. Visualization of the first SPOD mode at St_H = 0.164 (l = 25) in (a) isometric view, (b) top view and (c) back view, and at St_H = 0.218 (l = 33) in (d) isometric view, (e) top view and ( f) back view. The red and blue isosurfaces show $\varPsi_{u}$_,l/U_∞ = ±0.08 for l = 25 and $\varPsi_{u}$_,l/U_∞ = ±0.05 for l = 33.

A visualization of the first SPOD mode corresponding to the smaller peak of the pre-multiplied spectra of eigenvalues in figure 7 at St_H = 0.218 is presented in figure 18(d–f). This visualization also shows vertical pairs of structures with alternating signs that possess spanwise symmetry. However, the structures of each pair are significantly dissimilar; the downstream pair consists of an ellipsoidal structure at the spanwise centre (in blue), while its lower counterpart consists of an arch-shaped structure (in red). Consequently, there is no antisymmetry in the vertical direction. Nevertheless, the visualization in supplementary movie 8 shows that these structures also advect in the streamwise direction, thus they correspond to the vortex-shedding category.

The barycentre trajectory from the ROM of the modes within the 0.123 ≤ St_H < 0.234 range is depicted in figure 19. All trajectories are represented by thin blue lines, except for St_H = 0.164 and 0.218, which are shown with a thick black and red lines, respectively. The barycentre motions for St_H = 0.164 and 0.218 follow a vertical trajectory in the y–z plane and an oblique trajectory in the z–x plane. Note that many vertical trajectories in the y–z plane are not visible as they are obscured by the thicker black and red lines. This applies to the trajectories for most St_H values, except for a few cases that demonstrate solely spanwise motion in the y–z plane. In these instances, the trajectory in the x–z plane forms a point. Generally, the range of motions is greater in the z-direction compared with the x-direction, with the smallest range observed in the y-direction. This order is in line with the magnitudes of spectral peaks observed at St_H = 0.16 in figure 5(a), showing a larger peak for ${z^{\prime}_b}$, followed by ${x^{\prime}_b}$, and finally ${y^{\prime}_b}$. Overall, the unique feature of trajectories for zone C is the oblique motion of the barycentre in the x–z plane that moves the barycentre upwards and away from the wall, then downwards and closer to the rear face. Therefore, the barycentre motions generated by the vortex-shedding motions are distinct from those caused by bi-stability and swirling motions.

Figure 19. The trajectory of the wake barycentre in the (a) y–z and (b) x–y planes. The coordinates of the barycentre (x_b, y_b, z_b) are obtained from U_ROM of the first SPOD mode for each St_H within 0.123 ≤ St_H < 0.234. The black and red lines indicate the U_ROM for St_H = 0.164 and 0.218. Blue lines show the trace of the remaining St_H values.

Figure 20 shows the spatial organization of vortical structures at three selected phases of the ROM reconstructed using the SPOD mode at St_H = 0.164. These three phases, $\varPhi$₁, $\varPhi$₂ and $\varPhi$₃, are indicated on the barycentre trajectory in figure 19. At $\varPhi$₁ = 0.8${\rm \pi}$, figure 20(a) displays two counter-rotating quasi-streamwise vortices that are connected to the toroidal vortex and extended outwards in the downstream direction. Examination of supplementary movie 9 reveals that these two vortices initiate from the inner-bound of the toroid and divert outwards in the spanwise direction. Consequently, they have vorticity components in both the x and y directions. These quasi-streamwise vortices move downstream and out of the measurement domain with increasing $\varPhi$. At $\varPhi$₂ = 1.3${\rm \pi}$, only the tail of these vortices is visible in the downstream region of figure 20(b) and new streamwise vortices start forming at the toroidal vortex. As the phase increases to $\varPhi$₃ = 1.8${\rm \pi}$, the new pair of quasi-streamwise vortices with an opposite direction of rotation have extended from the toroid as seen in figure 20(c). The new vortex pair extends in the downstream direction, and their rotation is opposite to those observed at $\varPhi$₁. Additionally, supplementary movie 9 shows that the streamwise vortices arise from the inner part of the toroidal vortex.

Figure 20. Visualizations of ROM reconstructed from the first SPOD mode at St_H = 0.164 at (a) $\varPhi$₁ = 0.8${\rm \pi}$, (b) $\varPhi$₂ = 1.3${\rm \pi}$ and (c) $\varPhi$₃ = 1.8${\rm \pi}$. The red isosurface shows Q′ = +0.52, and blue shows Q′ = −0.52.

For brevity, the spatial organization of vortical structures for the SPOD mode at St_H = 0.218 is only shown in supplementary movie 10. The visualization resembles the spatial organization observed at St_H = 0.164, demonstrating quasi-streamwise vortices moving in the downstream direction. Upon analysing other St_H values within zone C, the results indicate that the shedding process primarily involves quasi-streamwise vortices. Therefore, an important observation is that the shedding process mainly consists of quasi-streamwise vortices.

To demonstrate how vortex-shedding results in velocity fluctuations, figure 21 illustrates v_ROM and w_ROM of the SPOD mode at St_H = 0.164. The reconstructions correspond to the three selected phases, $\varPhi$₁, $\varPhi$₂ and $\varPhi$₃. At $\varPhi$₁ = 0.8${\rm \pi}$ in figure 21(a), the velocity field displays two large zones of upward and downward velocity fluctuations along the centreline. The two quasi-streamwise vortices seen in figure 20(a) generate the Y-shaped upward velocity region (in green) in their outer boundary, while they induce a combined zone of downward velocity between them (in blue). These vortices also generate strong spanwise motions in both the positive and negative y directions, as seen in figure 21(d). The vertical and spanwise motions persist and advect downstream, as seen at $\varPhi$₂ = 1.3${\rm \pi}$ in figures 21(b) and 21(e). Similarly, figures 21(c) and 21( f), corresponding to $\varPhi$₃ = 1.8${\rm \pi}$, show similar structures as $\varPhi$₁ but with opposite magnitudes, consistent with the reversal of the direction of rotation of the vortices in figure 20(c). Vertical and spanwise velocity fluctuations for the SPOD mode at St_H = 0.218 exhibit patterns similar to those shown in figure 21. The corresponding figures are not shown for brevity.

Figure 21. Visualizations of the w_ROM and v_ROM of the SPOD mode at St_H = 0.164 at (a,d) $\varPhi$₁ = 0.8${\rm \pi}$, (b,e) $\varPhi$₂ = 1.3${\rm \pi}$ and (c, f) $\varPhi$₃ = 1.8${\rm \pi}$. In (a–c), the green isosurface shows w_ROM = +0.1, and blue shows w_ROM = −0.1. In (d–f), yellow isosurface shows v_ROM = +0.1, and pink shows v_ROM = −0.1.

Figure 22 illustrates the variations of the separation bubble volume for all St_H values of zone C. All trajectories are shown with thin blue lines except for St_H = 0.164 and 0.218, which are shown in black and red, respectively. The ROM shows that $\varPi$′ for St_H = 0.164 varies between minimum and maximum values of −0.07$\langle \varPi\rangle$ and +0.13$\langle \varPi\rangle$ occurring at $\varPhi$ = 0.8${\rm \pi}$ and 1.6${\rm \pi}$, respectively. The minimum $\varPi$′ corresponds to the moment shown in figure 20(a), wherein the streamwise vortices have reached full extension and are still connected to the toroidal vortex. As illustrated in figure 21(a), these vortices generate a strong upward motion in the positive z-direction. At $\varPhi$ = 1.6${\rm \pi}$ of the maximum $\varPi$′, as demonstrated in supplementary movie 9, new quasi-streamwise vortices are formed, although not fully extended. However, due to their opposite rotation direction, these vortices induce a downward motion in the z-direction. This observation aligns with the oblique barycentre trajectory illustrated in figure 19(b), where the separation bubble experiences simultaneous streamwise stretching and upward motion, followed by simultaneous streamwise contraction and downward motion. The $\varPi$′ variations for the mode at St_H = 0.218 are smaller than the variations for St_H = 0.164, which is consistent with its weaker structures observed in figure 18. The magnitude of volume fluctuations is comparable to those of the swinging motion seen in figure 17 but smaller than the bi-stable motion illustrated in figure 12.

Figure 22. The normalized fluctuations of separation volume $\varPi'\!\big/$$\langle \varPi\rangle$ obtained from U_ROM of the SPOD modes within 0.123 ≤ St_H < 0.218 range. The black and red lines indicate St_H = 0.164 and 0.218, respectively. The blue lines show other St_H values in zone C.

The present investigation highlights that the shedding of a pair of counter-rotating quasi-streamwise vortices is the primary mechanism behind vortex shedding, leading to regions of spanwise and vertical velocity fluctuations. Notably, flow visualization experiments conducted by Grandemange et al. (Reference Grandemange, Gohlke and Cadot2012) at Re_H = 365 also indicate the presence of alternating quasi-streamwise vortices shed from the centre of the toroid, although they referred to them as vortex loops. Similarly, phase-averaged visualizations by Evstafyeva et al. (Reference Evstafyeva, Morgans and Dalla Longa2017) based on simulations at Re_H = 435 showed that vortex shedding primarily involved quasi-streamwise structures extending from the toroid's centre and skewed towards one spanwise side. Based on analysing the coherence function and spectra of hot-wire measurements at Re_H of 9.2 × 10⁴, Grandemange et al. (Reference Grandemange, Gohlke and Cadot2013) suggested that separate vortex loops shed from the vertical and horizontal edges of the rear face at different frequencies. They explained that the rotational direction of successive vortex loops shed from the horizontal edges alternate, while the rotation direction of successive vortices shed from the vertical edge remained consistent due to the spanwise bi-stability. Grandemange et al. (Reference Grandemange, Gohlke and Cadot2013) noted the difficulty of presenting a combined sketch of the two shedding loops due to their distinct frequencies. The simulation conducted by Lucas et al. (Reference Lucas, Cadot, Herbert, Parpais and Délery2017) also provides some evidence of one-sided vortex loops shed in instantaneous visualizations. However, at the high Re_H of 4.0 × 10⁵, the cluttered nature of the vortical structures made it challenging to accurately characterize the vortex topology. These vortex loops bear resemblance to the shedding of instantaneous hairpin-like structures observed just before the wake switching in numerical simulations by Dalla Longa et al. (Reference Dalla Longa, Evstafyeva and Morgans2019) and Fan et al. (Reference Fan, Chao, Chu, Yang and Cadot2020).

6.4. Shear layer instability

To examine the spatio-temporal dynamics of fluctuations within the 1.024 ≤ St_H < 1.228 range, the SPOD mode at St_H = 1.147 is presented in figure 23. This St_H corresponds to the peak in the eigenvalue spectrum shown in figure 7. Figure 23 shows small structures with alternating signs along the shear layers. Stronger structures are observed in the top and vertical shear layers while the structures in the lower shear layer are weak and barely visible in figure 23. Supplementary movie 11 demonstrates that these structures evolve along the shear layers and gradually diminish as they move downstream.

Figure 23. Spatial organization of the first SPOD mode at St_H = 1.147 in (a) isometric view, (b) top view and (c) back view. The red and blue isosurfaces show $\varPsi_{u}$_,l/U_∞ = ±0.03 for l = 169.

To understand the source of the structures observed in figure 23, their Strouhal number is calculated using the local thickness of the shear layer (ξ), which is obtained based on the thickness of the layer with intense vorticity measured using planar PIV. At x/H = 0.3, ξ is found to be 0.16H for both upper and lower shear layers. The Strouhal number of the structures, calculated as St_ξ = f × ξ/U_∞ with f = 15.14 Hz and ξ = 0.16H, is determined to be 0.18. This value is similar to the Strouhal number of 0.2 obtained from the linear stability theory for Kelvin–Helmholtz instabilities (Monkewitz & Huerre Reference Monkewitz and Huerre1982). Therefore, the analysis suggests that the structures observed in figure 23 are generated by the instabilities of the shear layers. The structures observed in figure 23 are consistent with those identified in the SPOD mode at St_H ≈ 1 by Haffner et al. (Reference Haffner, Borée, Spohn and Castelain2020). They applied SPOD analysis to velocity fields in the streamwise–spanwise plane and light intensity images in the wall-normal–streamwise plane. Their analysis revealed that the spatial pattern of the SPOD mode at St_H ≈ 1 is linked to the interaction between the recirculating flow and the shear layer on the opposite side, leading to the amplification of shear layer instabilities and the roll-up of vortices.

Figure 24(a) displays the isosurface of Q* for St_H = 1.147, computed using the velocity fields obtained from the ROM at $\varPhi$ = 0. The visualization shows two successive toroidal vortical structures that resemble the toroidal vortex of the mean flow. Supplementary movie 12 corresponds to the animation of figure 24, which shows that these toroidal vortices gradually weaken and eventually vanish as they move downstream. In addition, figure 24(b) presents the vertical velocity fluctuations, which consist of small structures along the span of the top and side shear layers. In contrast, the spanwise velocity fluctuations in figure 24(c) are predominantly observed in the side shear layers. After examining the spatial variations of the barycentre and the fluctuations of separation volume, it was found that they are negligible compared with those of previously discussed flow motions. As a result, the corresponding plots are not included here to keep the discussion short.

Figure 24. Visualizations of ROM obtained for the SPOD mode at St_H = 1.147 at $\varPhi$ = 0. (a) The red isosurface shows Q* = 0.32, (b) green and blue isosurface shows w_ROM = +0.035 and −0.035, respectively, (c) yellow and pink isosurfaces show v_ROM = +0.035 and −0.035, respectively.