4.1. Wall-pressure mean field

The $C_p$ distribution across the 25 taps is shown in figure 4(a). The approximate location of critical points identified from flow visualisation of Williams et al. (Reference Williams, Samuell, Sarwas, Robbins and Ferrante2020) are overlaid in green – stable foci at $(x/H,\, y/H) \in \{(1.65,\, \approx \pm 1.8)\}$ and two saddles $(x/H,\,y/H)\in \{(1.25,\,0),\,(4.2,\,0)\}$ . The approximate location and range of the saddle points along the centreline, as identified by Annamalai (Reference Annamalai2022), are also marked, along with the top-down PIV FOV for the present study. The $C_p$ distribution is approximately symmetric about the centreline $y/H=0$ and consistent with earlier studies (Robbins et al. Reference Robbins, Samuell, Annamalai and Williams2021). Profiles of the centreline $C_p$ evolution ( $y/H=0$ ) and off-centreline planes $y/H=\pm 1.47$ are highlighted in figure 4(b), showcasing this symmetry.

Figure 4.

(a) Mean $C_p$ variation over the bump. Flow from left to right. Tap numbers labelled. Symbols along centreline $y/H=0$ represent mean (+), upstream extent (X) and downstream extent (*) of separation and downstream saddle points as determined by Annamalai (Reference Annamalai2022). Approximate location of critical lines overlaid, based on shear-stress wall topology of Williams et al. (Reference Williams, Samuell, Sarwas, Robbins and Ferrante2020). (b) Profiles of $C_p$ along centreline $y/H=0$ and off-centreline planes $y/H=\pm 1.47$ , demonstrating mean symmetry. Lines joining dots drawn to assist visibility only. Locations of upstream and downstream surface-concavity change marked as vertical dashed lines with bump profile (top).

The interplay between surface concavity and the 3-D evolution of the wall-pressure field is illustrated in figure 4(b). At $x/H=1.65$ , the wall curvature changes from convex to concave. Upstream of this location, the wall-pressure distribution shows strong pressure recovery associated with the APG region imposed by the bump geometry, and the flow separates along the spanwise front. In particular, the centreline wall-pressure gradient, expressed here as $\partial C_p/\partial x$ , rapidly decreases and remains small over $x/H\in [1.65,\approx 2.5]$ . This plateau in the centreline $C_p$ is interpreted as a wall-pressure signature of the separated/recirculating region near the surface-concavity change. It is noteworthy that this region is dominated by the surface vortices, as illustrated by the critical lines sketched in figure 4(a).

In the concave region, for $x/H\gt 1.65$ , the centreline wall-pressure gradient becomes positive again, although it remains weaker than in the upstream convex region. For $x/H\gt 3$ , the wall-pressure gradient appears approximately uniform across the span. Meanwhile, the off-centreline $\partial C_p/\partial x$ remains positive near the concavity-change location, suggesting that the wall-pressure response remains spanwise non-uniform past separation. These trends are consistent with the measurements of Gray et al. (Reference Gray, Gluzman, Thomas, Corke, Lakebrink and Mejia2021, Reference Gray, Lakebrink, Thomas, Corke, Gluzman and Straccia2023b ) at a slightly higher Reynolds number ( $\textit {Re}_H=335\,000$ ). Moreover, both the present pressure measurements and prior experiments on this bump (Williams et al. Reference Williams, Samuell, Sarwas, Robbins and Ferrante2020; Gray et al. Reference Gray, Corke, Thomas, Gluzman and Straccia2023a ) exhibit a dip in the spanwise distribution of $C_p$ at the apex plane ( $x/H=0$ ) when examined along the spanwise plane $x=0$ – a feature absent in 2-D simulations of the same geometry (Iyer & Malik Reference Iyer and Malik2021, Reference Iyer and Malik2023), in which the geometry and mean response are homogeneous in the spanwise direction. This discrepancy highlights the intrinsically 3-D nature of the wall-pressure field and further motivates the exploration of the unsteady motions associated with that three-dimensionality.

4.2. Three-dimensional velocity mean field

Aspects of the mean-flow fields measured by PIV have been extensively explored by Gray et al. (Reference Gray, Gluzman, Thomas and Corke2022, Reference Gray, Corke, Thomas, Gluzman and Straccia2023a ) at a slightly higher Reynolds number at $\textit {Re}_H=335\,000$ . Here, we provide a concise summary of the present dataset at $\textit {Re}_H = 2.26\times 10^5$ , highlighting its broad similarity to prior studies (e.g. Gray et al. (Reference Gray, Gluzman, Thomas and Corke2022, Reference Gray, Corke, Thomas, Gluzman and Straccia2023a )) despite moderate differences in inflow conditions and Reynolds number. Representative reconstructions of the mean PIV fields relative to the bump geometry are shown in figure 5. For completeness, additional statistics – including Reynolds stresses and turbulence production – are reported in Appendix A.1 for comparison with existing experiments.

Figure 5.

Rendering of mean field from measurements collected at several PIV planes: (a) isometric and (b) birds-eye views. Mean fields from three overlapping FOVs along a single $x$ – $z$ plane are stitched using linear interpolation to produce a planar composite mean field. Measurements are taken at (c) three separate side-on $x$ – $z$ planes at $y/H=0,\,0.75,\,1.47$ . Mean-flow field for the (d) top-down, $x$ – $y$ plane at $z/H=0.53$ . Flow symmetry along centreline plane $y/H=0$ . Contours represent streamwise mean velocity $U/U_{\infty }$ . Magenta dots on wall represent pressure sensor locations. Select pressure taps are labelled. The symbols $+$ , $X$ and $*$ denote the mean, upstream extent and downstream extent of the separation/downstream saddle locations, respectively, as defined in figure 4; their colour is adjusted here for contrast against the velocity contours.

The flow accelerates over the apex at $(x/H,y/H)=(0,0)$ , reaching a peak velocity of approximately $1.5\,U_\infty$ ( ${\approx} 67.5$ m s $^{-1}$ ), and remains symmetric in the mean about the centreline plane ( $y/H=0$ ). Side-on PIV planes at $y/H = 0$ , $0.75$ and $1.47$ reveal a well-defined separated region on the leeward side of the bump. Using conditional averaging based on reverse-flow fraction of the centreline side-on PIV data (see Annamalai (Reference Annamalai2022) for details), the upstream saddle (mean separation point) is located at $x/H=1.25$ and fluctuates over $x/H\in [1.06,\,1.43]$ , while the downstream saddle lies at $x/H=4.2$ with fluctuations over $x/H\in [3.58,\,4.81]$ . The corresponding mean reverse-flow length based on the Euclidean distance of mean centreline upstream and downstream saddle locations is $L_{\textit{sep}}\approx 0.226$ m ( $=2.7H=0.21\,L$ ). Here, the separated-region width is defined kinematically as the transverse extent of negative mean streamwise velocity in the PIV measurement plane. This reverse-flow width is distinct from the full wall-shear separation footprint inferred from oil-flow visualisation or skin-friction-line topology (Williams et al. Reference Williams, Samuell, Sarwas, Robbins and Ferrante2020; Gray et al. Reference Gray, Lakebrink, Thomas, Corke, Gluzman and Straccia2023b ), which also captures lateral near-wall deflection and the surface critical-point structure associated with the separated topology.

These values are of the same order as Gray et al. (Reference Gray, Corke, Thomas, Gluzman and Straccia2023a ), who – at a higher Reynolds number ( $\textit {Re}_H=3.35\times 10^5$ or $\textit {Re}_L=4.0\times 10^6$ at $U_{\infty }=69$ m/s) – report a centreline mean separation at $x=0.08L=0.95H$ and a downstream saddle (although Gray et al. (Reference Gray, Corke, Thomas, Gluzman and Straccia2023a ) refer to this location as a reattachment point, we label it as a downstream saddle since the mean field shows no canonical recirculation bubble consistent with true reattachment) near $x=0.36L=4.3H$ based on mean PIV data, giving $L_{\textit{sep}} \approx 0.30L = 3.6H$ . The slightly larger $L_{\textit{sep}}$ in Gray et al. (Reference Gray, Corke, Thomas, Gluzman and Straccia2023a ) is plausibly attributable to $\textit {Re}$ effects on shear-layer deflection – i.e. the separation zone grows with $\textit {Re}$ (Williams et al. Reference Williams, Samuell, Robbins, Annamalai and Ferrante2021). In what follows, we adopt the present estimate of mean separation at $x = 1.25\,H \approx 0.10L$ with $L_{\textit{sep}}\approx 2.7H=0.21\,L$ at $\textit {Re}_H=2.26\times 10^5$ .

The critical-point topology observed in the side-on and top-down mean fields is consistent with the skin-friction critical points identified in figure 4(a). In particular, the centreline side-on mean field shows that the upstream saddle coincides with the onset of mean-flow separation, whereas the downstream saddle anchors a bifurcation line that extends vertically in the $x$ – $z$ plane. Outboard flow descends toward the wall and converges onto this downstream saddle. Downstream of this second saddle, the flow experiences an upwash at the centreline – see the top side-on mean-flow field in figure 5(c). This bifurcation line terminates at a third saddle visible in the top-down field at $z/H=0.53$ (figure 5(d)). In this sense, the top-down PIV streamlines should be interpreted as planar projections of truly 3-D mean streamlines; care must therefore be exercised when drawing conclusions about dynamical features of the flow from projected streamline patterns alone. The resulting critical-line arrangement matches the well-known owl-face-of-the-first-kind topology (Perry & Hornung Reference Perry and Hornung1984; Williams et al. Reference Williams, Annamalai, Ozoroski, Roy and Lowe2022), consistent with oil-film interferometry measurements by Gray et al. (Reference Gray, Corke, Thomas, Gluzman and Straccia2023a ).

The top-down mean field highlights two dynamically distinct regions separated by a mean separatrix. Within the separated zone, the measurement plane is submerged within the reverse-flow region and mean motion is upward/outward from the plane. Outside the separatrix, the plane primarily samples fluid above the separated zone. In the mean sense, fluid is coming out of the plane within the separated zone, while fluid is going into the plane in the forward-flow regions due to the increasing wall-ward deflection of the shear layer away from the centreline.

The reverse-flow region exhibits strong three-dimensionality: its spanwise extent decreases rapidly with $|y|/H$ and the flow becomes predominantly attached for $|y|/H \gtrsim 1.5$ , even though the bump shoulders lie at $|y|/H \approx 4$ . This yields a separation-zone aspect ratio $B_w/L_{\textit{sep}} \approx 2.83$ , underscoring the inadequacy of quasi-2-D assumptions commonly used in simplified simulations (Williams et al. Reference Williams, Annamalai, Ozoroski, Roy and Lowe2022).

4.2.1. Top-down flow-field turbulence characteristics

We examine the turbulence characteristics in the top-down plane at $z/H=0.53$ , which intersects the separated shear layer over much of its streamwise extent and provides direct access to spanwise variations within the recirculating region.

Figure 6.

Reynolds stresses overlaid with mean streamlines: $(a)\,\,\overline {u^{\prime 2}}$ with tap numbers labelled*, $(b)\,\,\overline {v^{\prime 2}}$ , $(c)\,\,\overline {u'v'}$ and $(d)$ (planar) turbulence production. *Magenta dots show location of wall-pressure taps projected onto top-down plane at $z/H=0.53$ .

The streamwise Reynolds-stress $\overline {u^{\prime 2}}$ peaks along the separated shear layer, consistent with enhanced velocity fluctuations across the high-shear interface (figure 6 a). The spanwise Reynolds stress $\overline {v^{\prime 2}}$ attains its maximum magnitude near the centreline but remains approximately 40 % lower than $\overline {u^{\prime 2}}$ (figure 6 b), indicating weaker spanwise fluctuations relative to the streamwise component. The Reynolds shear stress $\overline {u'v'}$ exhibits its largest magnitudes at the streamwise location where the $z/H=0.53$ plane intersects the shear layer, and the corresponding production term (figure 6 d) forms an off-centre, streamwise-elongated band aligned with regions of strong mean shear. This pattern reflects the 3-D deflection of the shear layer toward the wall with increasing spanwise distance from the centreline.

The regions of strongest turbulence production and Reynolds-stress anisotropy occur away from the centreline, between the pressure-tap planes, rather than along the centreline-symmetry plane itself. This spatial localisation suggests that the most energetic unsteady motions are expected to originate in these off-centreline regions, a point that is borne out by the wall-pressure spectra and dominant POD modes examined in § 5.

Taken together, these observations establish that the present mean flow shares similar topological features reported in previous studies on slender 3-D bumps and hills. The separation topology, saddle locations, shear-layer deflection and spanwise contraction of the separated region are strongly consistent across Reynolds numbers and facilities. This correspondence provides confidence that the unsteady dynamics characterised later in this paper are representative of the broader class of 3-D separated flows over smooth bump-type geometries.

5.1. Separation dynamics from wall pressure

Figure 7.

Variance of $C_p$ ; refer to figure 4(a) for axes’ orientation. Location of downstream surface-concavity change marked as vertical dashed line. (a–g) Premultiplied wall-pressure spectra from selected tap groups highlighting the dominant frequency bands within the separated region. Vertical dotted lines mark the characteristic frequencies discussed in the text. Spectra were computed from a $4.85\times 10^6$ -sample stationary record at 31.25 kHz using Welch’s method with Hamming windows and 50 % overlap. The VLF/LF panels use 8.39 s windows $(\Delta f=0.119\,\mathrm{Hz},\,\Delta St_H=2.0\times 10^{-4})$ , while the high-frequency panels use 1.05 s windows $(\Delta f=0.954\,\mathrm{Hz},\,\Delta St_H=1.6\times 10^{-3})$ .

A spatial overview of the variance in $C_p$ , used here as a measure of wall-pressure fluctuation intensity, is provided in figure 7, together with premultiplied pressure spectra from selected tap groups. The $C_p$ fluctuation intensity is low upstream of the bump and increases along the upstream slope, with the largest levels observed in the recirculating region, particularly near the surface-concavity change (see tap 10). The distribution is approximately symmetric about the centreline, consistent with the mean-pressure field. Wall-pressure spectra reveal several dynamically important frequency bands spanning multiple orders of magnitude. In the separated region, five dominant broadband features are identified, centred approximately at 1, 13.5, 20, 135 and 200 Hz. These correspond to Strouhal numbers $St_H=fH/U_{\infty }=1.7\times 10^{-3}$ , 0.023, 0.034, 0.23 and 0.34, respectively.

The 1 Hz signal is strongly expressed off-centreline (see figure 7(a,d,f)), with the exception of locations near the surface-concavity change (taps 23/9/24). No other VLF peaks ( $f\lt 20$ Hz) appear in these regions. In contrast, the 13.5 Hz peak occurs only along the centreline and only upstream of mean separation and the concavity change – see figure 7(b). In particular, between taps 7 and 8 ( $x/H\in [0, 0.73]$ ) – upstream of mean centreline separation where the APG is strongest (figure 4 b) – the spectral energy at 13.5 Hz increases by roughly an order of magnitude.

A localised LF signature near 20 Hz is detected primarily in the tap-23/9/24 plane ( $x/H=1.47$ ), near the surface-concavity change – figure 7(c). Its weaker spectral signature, coupled with the known presence of surface vortices in this region (Williams et al. Reference Williams, Samuell, Sarwas, Robbins and Ferrante2020), suggests that the 20 Hz feature could be associated with their motion. The stronger high-frequency content at centreline tap 9, compared with off-centreline taps 23/24, may indicate the onset of smaller-scale shear-layer structures near the centreline.

The highest-frequency spectral peaks (135 Hz, 200 Hz) occur downstream of separation along the centreline, where vortex shedding is expected to occur – figure 7(e,g). The 200 Hz peak only occurs immediately after separation (tap 10), while the 135 Hz peak emerges thereafter and persists across taps 11–14.

This progression suggests a redistribution of spectral energy from higher to mid-frequencies along the separated shear layer. As noted by Manohar et al. (Reference Manohar, Williams, Martinuzzi and Morton2023), the decrease from 200 to 135 Hz may reflect vortex amalgamation during downstream advection, a phenomenon often observed in separated flows (Cherry et al. Reference Cherry, Hillier and Latour1984). The gradual weakening of the 135 Hz peak farther downstream may indicate that vortices are moving away from the wall as they are advected downstream. This is supported by the centreline upwash observed in the side-on mean PIV fields (figure 5(c), top).

Figure 8.

(a) Cross-correlation of fluctuating surface-pressure signals across centreline-symmetry plane $y/H=0$ . Point A marks the maximum (positive) correlation between taps 23 and 24. Pressure signals between taps are correlated in the order presented in parenthesis shown in the legend. Signals low-pass filtered with 400 Hz cutoff, and resampled at 800 Hz. (b) Normalised PDFs of anti-symmetric field across symmetric tap pairs, compared with the standard normal distribution.

Figure 8. Long description

Panel A: A line graph shows the cross-correlation of fluctuating surface-pressure signals across the centerline-symmetry plane. The x-axis is labeled with the dimensionless time variable τU∞/H, ranging from -5000 to 5000. The y-axis is labeled Rij, ranging from -0.6 to 0.3. Multiple lines represent different pairs of pressure taps, with labels (21, 22), (23, 24), (25, 26), and (27, 30). Point A marks the maximum positive correlation between taps 23 and 24. The signals are low-pass filtered with a 400 Hz cutoff and resampled at 800 Hz. Panel B: A line graph displays normalized probability density functions (PDFs) of the anti-symmetric field across symmetric tap pairs. The x-axis is labeled (a - μa)/σa, ranging from -4 to 4. The y-axis is labeled PDF, ranging from 0 to 0.5. Multiple lines represent different tap pairs (21/22, 23/24, 25/26) and are compared with the standard normal distribution.

Cross-correlations between wall-pressure signals measured at various taps provide insight into the nature of the coherent motions across the bump. The cross-correlation between discrete fluctuating pressure signals pertaining to two taps, $p'_i(t)$ and $p'_j(t)$ , is defined as

(5.1) \begin{align} R_{ij}(\tau )= \overline {p'_i(t)\,p'_j(t + \tau )}\,\,/\,\,\sqrt {\overline {p^{\prime 2}_i(t)} \boldsymbol{\cdot }\overline {p^{\prime 2}_j(t)}}, \end{align}

as shown in figure 8(a) for pressure-tap pairs either side of the centreline.

Interestingly, taps 23 and 24 are the only pair exhibiting a positive correlation at zero time lag (marked ‘A’), and they lie closest to the estimated locations of the surface-vortex cores. Their proximity to these core regions supports the earlier inference that the ${\sim} 20$ Hz signature is linked to the local surface-vortex/lateral-shear-layer region, because this frequency band is expressed most strongly at these taps. Meanwhile, the positive cross-correlation across the centreline indicates that the associated wall-pressure fluctuations contain a symmetric component about the centreline, consistent with the approximately symmetric configuration of the surface vortices illustrated in figure 4(a).

Negative correlations at zero time-lag are observed across the centreline along the other streamwise locations – taps 21/22, 25/26 and 27/30 – where the 1 Hz motion ( $St_H \approx 10^{-3}$ ) is observed. Interestingly, the corresponding correlations also show a weak long-time modulation with characteristic spacing $O(10^3 H / U_{\infty })$ , most clearly for taps 21/22 near the off-centre bump apex. This time scale is consistent with the VLF band identified in the pressure spectra. These features suggest that the lowest-frequency motion at 1 Hz is a side-to-side oscillation of the separation region that is anti-symmetric relative to the centreline.

The nature of this oscillation remains to be determined – i.e. whether it is meandering/harmonic or more bi-modal, like the wakes of the Ahmed body (Grandemange et al. Reference Grandemange, Gohlke and Cadot2013), 3-D axisymmetric hill (Byun & Simpson Reference Byun and Simpson2006; Garcia-Villalba et al. Reference Garcia-Villalba, Li, Rodi and Leschziner2009) and BeVERLI hill (Gargiulo et al. Reference Gargiulo, Duetsch-Patel, Ozoroski, Beardsley, Vishwanathan, Fritsch, Borgoltz, Devenport, Roy and Lowe2021; Duetsch-Patel et al. Reference Duetsch-Patel2022; MacGregor et al. Reference MacGregor, Gargiulo, Duetsch-Patel, Lavoie and Lowe2023). To assess potential bi-modality, the anti-symmetric component of the wall-pressure field is taken across the centreline-symmetry plane: $a(t)= ({1}/{2})(p_2(t) - p_1(t))$ , where $p_1$ and $p_2$ are pressure signals corresponding to sensors pairs on opposite sides of the centreline. It is computed for the tap pairs 21/22, 23/24 and 26/27, and the z-score normalised probability density functions (PDFs) are plotted in figure 8(b). The PDFs are found to be approximately normally distributed across the entire separated wake. Therefore, unlike the bi-stable wakes, the VLF spanwise motion ( $St_H = 10^{-3}$ ) appears to be a continuous oscillation, or a meandering about a mean state (It is also noted that the $r$ -parameter proposed by Plumejeau et al. (Reference Plumejeau, Delprat, Keirsbulck, Lippert and Abassi2019) – $ r(t) = ({p_2(t) - p_1(t)}/{p_2(t) + p_1(t)})$ – was not used for the bi-modality analysis as taps 25/26 contain several events where $p_2\approx -p_1$ , causing $r$ to become unbounded).

5.2. Characteristics of the breathing and vortex-shedding modes observed from side-on flow fields

Figure 9.

(Adapted from Manohar et al. (Reference Manohar, Williams, Martinuzzi and Morton2023).) Columns 1–3 (from left to right) correspond to the first three spatial POD modes with $u$ -(top) and $w$ -components (bottom). Column 4 shows pre-multiplied power spectral density (PSD) of corresponding original $a^{(1-3)}$ (15 Hz) and super-resolved $\widetilde {a}^{(1-3)}$ (2 kHz) coefficients estimated from unseen pressure inputs spanning the test dataset. The PSDs of each mode are offset by $10^{-1}$ for clarity. Modes 1–3 represent 33.8 %, 7.5 % and 5.2 % of the planar turbulent kinetic energy (TKE), respectively.

Figure 9. Long description

The image contains multiple heatmaps and line graphs. Panel 1 to 3 (top row) show heatmaps of the first three spatial POD modes with u-component. Panel 4 (top row) shows a line graph of pre-multiplied PSDs of corresponding original (15 Hz) and super-resolved (2 kHz) coefficients estimated from unseen pressure inputs spanning the test dataset. Panel 1 to 3 (bottom row) show heatmaps of the first three spatial POD modes with w-component. Panel 4 (bottom row) shows a line graph of pre-multiplied PSDs of corresponding original (15 Hz) and super-resolved (2 kHz) coefficients estimated from unseen pressure inputs spanning the test dataset. The x-axis for the heatmaps is labeled x/H and the y-axis is labeled z/H. The x-axis for the line graphs is labeled St_H = fH/U_infinity and the y-axis is labeled St_H * Ψ_a. The heatmaps use a color scale ranging from -2 to 2 times 10^-4, indicating the intensity of the POD modes. The line graphs show the spectral power density for different modes, with each mode offset for clarity. Modes 1 to 3 represent 33.8 percent, 7.5 percent, and 5.2 percent of the planar TKE, respectively.

It is well known that the dynamics of the shear layer at high Reynolds number are dominated by advecting large-scale vortical structures emerging from Kelvin–Helmoltz instabilities at separation (Cherry et al. Reference Cherry, Hillier and Latour1984; Simpson Reference Simpson1989). To explore the physical sources of the 13.5 and 135–200 Hz frequencies, figure 9 surveys the frequencies of the dominant PIV modes in side-on PIV fields at the bump centreline. The method of Manohar et al. (Reference Manohar, Williams, Martinuzzi and Morton2023) has been used to upsample the low-rate PIV fields using unsteady wall-pressure signals such that otherwise-aliased frequencies can be investigated. The first three spatial POD modes $\varPhi ^{(1-3)}$ with components $\varPhi ^{(1-3)}_u,\,\varPhi ^{(1-3)}_w$ are shown, along with their super-resolved temporal coefficients, $\widetilde {a}^{(1-3)}$ . Following Manohar et al. (Reference Manohar, Williams, Martinuzzi and Morton2023), the dynamics of the first POD mode are dominated by a 13.5 Hz spectral signature that was observed strictly upstream of mean centreline separation in the wall-pressure signals.

Single-mode reconstructions of the super-resolved mode-1 overlaid with the mean field (figure 10) reveal its association with the motion of the separation point. This motion modulates the instantaneous extent of the reverse-flow region, resulting in a vertical motion of the shear layer, or ‘breathing’ motion of the separation zone. The local minimum $a^{(1)}_{\textit{min}}\lt 0$ appears to correspond to a flow state with a smaller reverse-flow region, while the local maximum $a^{(1)}_{\textit{max}}\gt 0$ corresponds to a state with a larger separation zone. The separation point appears to move correspondingly (see supplementary file 2 of Manohar et al. (Reference Manohar, Williams, Martinuzzi and Morton2023)). The intermediate states corresponding to $a^{(1)}_{\textit{min}}\lt a^{(1)}\lt a^{(1)}_{\textit{max}}$ describe a range of flow states associated with the breathing mode. The above results are consistent with the breathing behaviour of the pressure-induced TSB flows studied by Mohammed-Taifour & Weiss (Reference Mohammed-Taifour and Weiss2016), Wu et al. (Reference Wu, Meneveau and Mittal2020) and Fang & Wang (Reference Fang and Wang2024).

Figure 10.

Time series of breathing mode coefficient $a^{(1)}$ with labelled local minimum and maximum, with corresponding mode-1 reconstructions. Dots connected by lines to assist visibility only.

The spatial signatures of modes 2 and 3, along with the super-resolved spectra of their temporal coefficients, show that these modes represent the formation, amalgamation, and downstream advection of large-scale coherent structures tied to the vortex-shedding dynamics in the $135$ – $200$ Hz band. These observations connect the side-on PIV modes 2–3 (figure 9) with the high-frequency wall-pressure signatures identified in figure 7.

The side-on POD analysis is restricted to FOV F1 as labelled in figure 3, where the LF separation dynamics are most clearly resolved. Although POD can be performed separately on the other FOVs F2 and F3, the resulting modes are sensitive to the spatial domain and are therefore not directly comparable across non-identical fields of view. Similarly, POD of a stitched side-on field was avoided because the side-on FOVs were not acquired simultaneously and the stitching procedure can introduce interpolation and overlap artefacts that contaminate the leading energy-ranked modes. The downstream FOVs are instead dominated by shedding-type structures, consistent with the higher-frequency shear-layer dynamics.

5.3. Characteristics of spanwise motions

The top-down flow field in the separated region at $z/H = 0.53$ is analysed by splitting the POD basis into antisymmetric (A) and symmetric (S) families with respect to the centreline, allowing the dynamics of spanwise structures in the separated region to be explored. Coefficients of each of these families are written $a^{(i)}_{\textit{A}}$ and $a^{(k)}_{\textit{S}}$ , where $i,k$ are mode numbers. Unless otherwise stated, all POD coefficients are presented in their normalised form, $a/\sqrt {2\,\lambda }$ , with $\lambda$ denoting the corresponding modal energy.

The first three antisymmetric and symmetric spatial modes are shown in figure 11. The most energetic mode is the leading antisymmetric mode, $a_{\textit{A}}^{(1)}$ , contributing around 20 % of the total planar fluctuating kinetic energy (see figure 11(g)). In contrast, the leading symmetric mode, $a_{\textit{S}}^{(1)}$ , contains 3 % of the planar fluctuating energy. The higher-order antisymmetric modes contain less energy than $a_{\textit{S}}^{(1-6)}$ , and are thus disregarded in the following analysis.

Figure 11.

The POD mode shapes and energies. (a–c) First three antisymmetric modes $\boldsymbol{\varPhi }_{\textit{A}}^{(1-3)}$ and, (d–f) first three symmetric modes $\boldsymbol{\varPhi }_{\textit{S}}^{(1-3)}$ . Contours represent vertical vorticity component $\omega _z$ ; streamlines drawn from corresponding modal velocity field. (g) Modal energy distribution – per-mode planar TKE (left, log scale) and cumulative fraction (right, linear) – for antisymmetric (solid) and symmetric (dashed) families. The labels S and A denote the reflectional parity of the underlying in-plane velocity modes, not of the plotted scalar vorticity. Consequently, antisymmetric velocity modes appear symmetric in vorticity, while symmetric velocity modes appear antisymmetric in vorticity.

Figure 12.

(a) Premultiplied spectrum of the antisymmetric mode-1 coefficient; vertical dashed line marks the 1 Hz peak. (b) Probability density of the standardised coefficient with a standard normal reference (solid line); annotations report skewness $0.17 \pm 0.04$ and kurtosis 2.43 (excess $-0.57 \pm 0.08$ ).

Figure 12. Long description

Panel A: A line graph shows the premultiplied spectrum of the antisymmetric mode-1 coefficient. The x-axis represents the Strouhal number (St_H = fH/U_infinity) on a logarithmic scale ranging from 10^-3 to 10^-2, and the y-axis represents the product of the Strouhal number and the spectrum (St_H * Ψ_dλ^(1)) on a logarithmic scale ranging from 10^2 to 10^3. A vertical dashed line marks the 1 Hz peak. Panel B: A histogram displays the probability density function (PDF) of the standardized coefficient. The x-axis represents the standardized coefficient ((a_A^(1)) / sqrt(2λ_1,A - μ) / σ) ranging from -4 to 4, and the y-axis represents the PDF ranging from 0 to 0.5. A solid line represents a standard normal reference. Annotations report skewness (0.17 ± 0.04) and kurtosis (2.43, excess = -0.57, ±0.08).

The dynamics of $a_{\textit{A}}^{(1)}$ are summarised in figure 12. The pre-multiplied PSD of $a^{(1)}_{\textit{A}}$ shows a broadband hump centred at 1 Hz, or $St_H=fH/U_{\infty }\approx 10^{-3}$ , consistent with the off-centreline wall-pressure spectra reported in § 5.1. Note that the 1 Hz-centred spectral hump was absent from the coefficients of all modes (antisymmetric and symmetric) other than $a_{\textit{A}}^{(1)}$ . The standardised PDF of $a^{(1)}_{\textit{A}}$ is near-Gaussian (skew = 0.17), with significantly lighter-than-normal tails (excess kurtosis = −0.57), indicating weak asymmetry and low intermittency – consistent with a LF meandering about the mean, i.e. a continuous lateral wake oscillation without long residence in a preferred state. This matches the conclusion from the bi-modality analysis conducted on the wall-pressure signals in § 5.1, that the $St_H\approx 10^{-3}$ spanwise motion is a meandering mode, which is in stark contrast to the bi-stable wakes observed in other geometries, where the wake prefers more extreme conditions away from the mean, with reasonably long residence times in theses states.

Note that as a check, POD was also performed on the complete top-down fluctuation field without first decomposing the data into symmetric and antisymmetric components. The leading mode of this full-field decomposition is again associated with the VLF spanwise-meandering motion. The symmetric/antisymmetric decomposition is therefore not required to reveal the VLF mode; rather, it is used here to separate the two reflection-symmetry classes and to isolate the lateral meandering dynamics from symmetric stretching/contraction of the separated footprint.

To probe the coupling of dominant spanwise antisymmetric and symmetric motions, the joint distributions of normalised coefficients $(a^{(1)}_{\textit{A}}, a^{(i)}_{\textit{S}}),\,i=1:3$ are examined in figure 13. It is observed that probability distributions are centred near $a_{\textit{A}}^{(1)} = 0$ , showing that the flow spends most of the time close to the symmetric mean position. Interestingly, the structure of the distributions suggests an underlying coherence in the inter-modal dynamics. Two trends emerge:

1. (i) In figure 13(a), $\{a_{\textit{A}}^{(1)},\,a_{\textit{S}}^{(1)}\}$ displays a parabolic structure in the low-probability rims. The distribution is nearly symmetric about $a^{(1)}_{\textit{A}}=0$ : $a_{\textit{S}}^{(1)}$ is largest when $a_{\textit{A}}^{(1)}\approx 0$ and becomes negative for large $|a_{\textit{A}}^{(1)}|$ .

2. (ii) In figure 13(c), $\{a_{\textit{A}}^{(1)},\,a_{\textit{S}}^{(3)}\}$ also displays a parabolic structure in the low-probability rims, where $a_{\textit{S}}^{(3)}$ increases with $|a_{\textit{A}}^{(1)}|$ , with the largest $a_{\textit{S}}^{(3)}$ events occurring at the farthest off-centreline positions.

Practically, this suggests that large $|a_{\textit{A}}^{(1)}|$ tends to co-occur with negative $a_{\textit{S}}^{(1)}$ and positive $a_{\textit{S}}^{(3)}$ , noting that the signs of the POD modes are arbitrary and may flip depending on the specific realisation of the decomposition. Moreover, the extreme (min/max) events lie on low-probability rims of the distributions, confirming that large spanwise excursions are rare compared with the near-symmetric state, thereby indicating the stability of the mean field.

The distribution involving $a_{\textit{S}}^{(2)}$ (figure 13 b) does not exhibit a clear geometric trend in its low-probability structure. Unlike those involving $a_{\textit{S}}^{(1)}$ and $a_{\textit{S}}^{(3)}$ , which show a relatively well-defined parabolic structure, the $\{a_{\textit{A}}^{(1)}, a_{\textit{S}}^{(2)}\}$ distribution remains comparatively isotropic and lacks an unambiguous modal relation. We therefore do not extract further dynamical interpretation from this pairing.

Parabolic low-probability loci in the joint PDFs have also been reported for axisymmetric wakes (e.g. cylinder with an elliptic nose cone; figure 18(e) of Pavia et al. Reference Pavia, Varney, Passmore and Almond2019), where they become especially pronounced when the long-time dynamics are isolated. In that case, the parabolic structure arises from a nonlinear coupling between the antisymmetric modes (azimuthally asymmetric, $m=\pm 1$ ) and the symmetric deformation mode (axisymmetric bubble pumping, $m=0$ ) during azimuthal meandering. This behaviour is directly analogous to the meandering dynamics observed here in a reflectionally symmetric bump.

Figure 13.

Joint distributions of POD coefficients comparing $a_{\textit{A}}^{(1)}$ with $a_{\textit{S}}^{(1-3)}$ . Each panel shows the 2-D histogram (120 × 120 bins) of $(a_{\textit{A}}^{(1)},\,a_{\textit{S}}^{(i)})$ after normalisation. Colour indicates fraction of samples per bin. Black contours denote highest-density regions enclosing 50 %, 90 % and 99 % of the probability. Select extreme (minimum/maximum) points marked as coloured symbols; the position of a given extreme point in any of the three distributions corresponds to the same time instant. Axes are equal with common limits.

Instantaneous flow fields at five key modal events are selected from the joint distributions – ‘min A1’, ‘max A1’, ‘max S1’, ‘min S2’ and ‘min S3’. These events correspond to time instants where $a_{\textit{A}}^{(1)}$ is a global minimum/maximum within the time-series of the dataset (‘min/max A1’, respectively), and similarly ‘min S1’ is the time instant corresponding to when $a_{\textit{S}}^{(1)}$ is a global minimum, and so on. Figure 14 shows flow-field reconstructions at these modal events. The single-mode panels (top row) illustrate the effect of each mode added to the mean, whereas the second row displays four-mode reconstructions based on modes A1 and S1–S3.

Before examining the individual S1–S3 events in detail, we note that the single–mode reconstructions in figure 14 (top row) capture the essential spatial features observed in the four–mode reconstructions of figure 14 (bottom row). At the instants corresponding to the extreme events, the dominant symmetric or antisymmetric mode carries most of the energy, such that adding the remaining modes modifies the field only incrementally. This indicates that the extreme events are largely governed by the leading mode in each symmetry class, and therefore the following discussion focuses on the physical interpretation of these dominant deformations rather than on differences between the single–mode and multi–mode reconstructions.

Instants of ‘min/max A1’ appear to correspond to asymmetric wake states with the downstream saddle close to one bump height, $H$ , from the centreline (figure 14; columns min/max A1). Thus, the effect of adding the spanwise-meandering mode $a_{\textit{A}}^{(1)}$ to the mean field is to shift the separated region off-centreline, creating a side-to-side spanwise meandering-type oscillation at $St_H\approx 10^{-3}$ (see Supplementary Movie 1: A1).

Figure 14.

Instantaneous flow-field reconstructions at key modal events identified in figure 13. Columns show (from left): min A1, max A1, max S1, min S2, min S3 added to the mean field. Top row: single-mode- $k$ snapshots $\boldsymbol{U}+\boldsymbol{\varPhi }_{\textit{A/S}}^{(k)}\,a_{\textit{A/S}}^{(k)}$ at the indicated time; titles report the normalised coefficient. Bottom row: multi-mode reconstructions at the same time steps, $\boldsymbol{U}+\boldsymbol{\varPhi }_{\textit{A}}^{(1)}a_{\textit{A}}^{(1)}+\sum _{i=1}^3\boldsymbol{\varPhi }_{\textit{S}}^{(i)}a_{\textit{S}}^{(i)}$ . Colour maps vertical vorticity component $\omega _z$ ; black curves are instantaneous streamlines. Markers in upper-left of top panels identify event type and use the same symbols as figure 13.

Figure 14. Long description

Panel A: Five contour plots show instantaneous flow-field reconstructions at key modal events. Each plot represents a different event type: min A1, max A1, max S1, min S2, and min S3 added to the mean field. The x-axis and y-axis are labeled as x/H and y/H, respectively. The color map represents the vertical vorticity component, with black curves indicating instantaneous streamlines. Markers in the upper-left corner of each plot identify the event type. Panel B: Five contour plots show multi-mode reconstructions at the same time steps as Panel A. The x-axis and y-axis are labeled as x/H and y/H, respectively. The color map represents the vertical vorticity component, with black curves indicating instantaneous streamlines.

The ‘max S1’ event coincides with an $a^{(1)}_{\textit{A}}=0$ event (see figure 13(a)). In particular, ‘max S1’ corresponds to the event where the saddle point is positioned farthest downstream within the FOV (roughly one $H$ downstream from the mean saddle location), consistent with a symmetric streamwise elongation of the wake (figure 14; col. max S1). Moreover, the ‘min/max A1’ events align with the minima of $a_{\textit{S}}^{(1)}$ , indicating that asymmetric wake states preferentially coincide with upstream-shifted configurations of the separation region (Supplementary Movie 2: S1). Thus, $a_{\textit{S}}^{(1)}$ may be interpreted as a proxy for the streamwise extent of the separation zone. In particular, for the current decomposition, $a_{\textit{S}}^{(1)}\gt 0$ corresponds to a downstream-shifted wake (relative to the mean), whereas $a_{\textit{S}}^{(1)}\lt 0$ indicates an upstream contraction.

The ‘min S2’ and ‘min S3’ events also both occur when the wake is close to the symmetric position ( $a^{(1)}_{\textit{A}}\approx 0$ ) – see figure 13. At large spanwise excursions ( $|a^{(1)}_{\textit{A}}|$ large), the symmetric response is not limited to $a_{\textit{S}}^{(1)}$ : while $a_{\textit{S}}^{(1)}$ carries the largest energy among symmetric modes, the joint distributions show that $|a_{\textit{S}}^{(2)}|$ and $|a_{\textit{S}}^{(3)}|$ likewise become appreciable. In particular, for this decomposition, $a_{\textit{S}}^{(1)}$ follows becomes increasingly negative with $|a_{\textit{A}}^{(1)}|$ implying a contracted wake, $a_{\textit{S}}^{(3)}$ increases with $|a_{\textit{A}}^{(1)}|$ , and $|a_{\textit{S}}^{(2)}|$ departs from zero as the antisymmetric deformation grows. Thus, asymmetric states are generally accompanied by a contracted wake ( $a_{\textit{S}}^{(1)}\lt 0$ ) – at most roughly $0.5\,H$ upstream from the mean saddle location – as well as appreciable higher-order symmetric content.

The ‘min S2’ and ‘min S3’ events also produce qualitatively similar symmetric deformations in the reconstructed fields (figure 14, bottom row). In particular, within the top-down FOV, ‘min S3’ corresponds to the most upstream configuration of the symmetric wake (green circle in figure 13(a)), whereas the largest $a_{\textit{S}}^{(3)}$ amplitudes co-occur with the extreme $|a_{\textit{A}}^{(1)}|$ states (red triangles in figure 13(c)). When modes A1 + S1 + S2 + S3 are superposed (i.e. $\boldsymbol{U}+\boldsymbol{\varPhi }_{\textit{A}}^{(1)}a_{\textit{A}}^{(1)}+\sum _{i=1}^3\boldsymbol{\varPhi }_{\textit{S}}^{(i)}a_{\textit{S}}^{(i)}$ ) at the ‘min S2’ or ‘min S3’ instants, the reconstructed flow exhibits a split of the central saddle point (figure 14, bottom row, cols. min S2/S3). A similar splitting tendency is already visible in the S3 mode alone (figure 11 and Supplementary Movie 3: S3). These observations suggest that the most upstream symmetric state is associated with a saddle-split topology – potentially an unstable intermediate state preceding spanwise wake shifting. Time-resolved measurements would be required to verify this by resolving trajectories in $(a^{(1)}_{\textit{A}}, a^{(1)}_{\textit{S}}, a^{(3)}_{\textit{S}})$ space.

A closer inspection of the multi–mode reconstructions at the ‘min S2’ and ‘min S3’ instants (bottom row of columns 4–5 in figure 14) reveals a topology that is distinct from the owl–face-of-the-first-kind found in the mean skin-friction field (recall figure 4(a)). In both cases, the instantaneous streamline pattern in the $z/H=0.53$ plane is characterised by a centreline focal point flanked by two off-centreline saddles. This topology is locally similar to the downstream portion of the surface-shear-stress pattern of an owl–face-of-the-second-kind in the sense of Perry & Hornung (Reference Perry and Hornung1984), which is associated with the presence of loop vortices emanating from the wall. This instantaneously observed saddle-split topology may therefore be associated with the shedding of large loop vortices. These off-centre saddles occur in the same general region as the elevated Reynolds stresses and turbulence production in figure 6. This spatial correspondence is consistent with the possibility that the saddle-split topology is connected to the lateral limits or reorganisation of the central shear-layer structure, although time-resolved volumetric measurements would be required to establish this interpretation definitively.