3.1. Flow regime transition visualisation

In the range $420 \lesssim Re \lt 650$ , the isolated sphere wake is inherently asymmetric, characterised by ‘irregular’ shedding and spanwise-inclined vortices (Tiwari et al. Reference Tomboulides and Orszag2019, Reference Tsutsui2020; Frantz et al. Reference Frantz, Mimeau, Salihoglu, Loiseau and Robinet2025). However, this spanwise asymmetric wake is suppressed by the wall proximity. As $G/D$ decreases, the near wake recovers mirror symmetry about the mid-plane ( $z/D = 0$ ), eliminating the azimuthal oscillations typical of the unbounded flow.

Figure 4 visualises the structural transitions using the $Q$ -criterion coloured by ${u^{*}} = \overline {u}/U_{\infty }$ (Li et al. Reference Li, Luo, Zhao, Shang and Fan2022; Zhang et al. Reference Zhang, Xia, Qiu, Dong and Liu2023, Reference Zhang, Ma, Xia, Qiu and Liu2024; Kumar et al. Reference Kumar, Das and Tiwari2024). As $G/D$ decreases, the near-wake symmetry states undergo a transition. At $G/D = 0.8$ (in figure 4 a), the flow retains the spanwise asymmetric hairpin shedding characteristic of unbounded wakes (Mittal et al. Reference Mittal, Wilson and Najjar2001; Tiwari et al. Reference Tiwari, Bale, Patwardhan, Nandakumar and Joshi2019; Yin & Ong Reference Yin and Ong2020). Reducing the gap to $G/D = 0.5$ (in figure 4 b) induces a hybrid state where wall proximity begins to suppress near-wall shedding, though upper-wake asymmetry persists. By $G/D = 0.3$ (in figure 4 c), this suppression becomes dominant, leading to the re-emergence of symmetrical formations. Finally, at $G/D = 0.1$ (in figure 4 d), all spanwise asymmetry is eliminated, and the wake resolves into a steady, mirror-symmetric double-threaded structure. It should be emphasised that the present wake regime is specific to a laminar boundary layer.

Figure 4.

Instantaneous 3-D near-wake vortical structures at $Re=500$ for different $G/D$ . Structures are visualised using iso-surfaces of the normalised second invariant $Q^{*} = QD^{2}/{U_{\infty }^{2}} = 2.5{\times }10^{-3}$ , coloured by the normalised streamwise velocity ${u^{*}} = \overline {u}/U_{\infty }$ . (a,b) Asymmetric hairpin vorteices for $G/D =0.8, 0.5$ . (c) Symmetric hairpin vortex for $G/D = 0.3$ . (d) Symmetric double-threaded structure for $G/D = 0.1$ .

Figure 4. Long description

Panel A: A 3D visualization shows asymmetric hairpin vortices and vortex rings. The axes are labeled x/D, y/D, and z/D. The structures are colored by the normalized streamwise velocity. Panel B: Another 3D visualization depicts asymmetric hairpin vortices and vortex rings, similar to Panel A but with different configurations. Panel C: A 3D visualization illustrates symmetric hairpin vortices and vortex rings. Panel D: A 3D visualization displays symmetric double-threaded vortices and base vortices. The top view of each panel provides a different perspective of the vortical structures.

Complementing the instantaneous observations, figure 5 presents the time-averaged results of this transition using the dimensionless streamwise velocity $\overline {{u}^{*}}$ , mean pressure $\overline {{p}^{*}} = {\overline {{p}}/0.5{\rho }U_{\infty }^{2}}$ and streamwise vorticity $\overline {{\omega }_{x}^{*}} = {\overline {\omega }_{x}}D/U_{\infty }$ on cross-stream ( $y{-}z$ ) planes at $x/D = 1.0, 2.0, 5.0$ for $Re = 500$ . Notably, despite the strong instantaneous asymmetry observed earlier, the mean flow preserves an approximately planar-symmetric vortex organisation across all gap ratios (van Hout et al. Reference van Hout, Eisma, Elsinga and Westerweel2018). This confirms that the dominant symmetry breaking is inherently unsteady and phase-dependent, rather than a result of persistent mean flow deflection. Additionally, the mean structure is dominated by a counter-rotating pair of primary wake tip vortices ( $\textrm{WTV}_{P}$ ), comprising upper $\textrm{UTV}_{P}$ and lower $\textrm{LTV}_{P}$ parts, alongside wall-induced vortices ( $\textrm{WSV}$ ) (in figure 5 c). At $G/D = 0.8, 0.5$ (in figures 5 a–c), the wake tip vortices remain spatially separated from the wall-induced structures, maintaining distinct identities. As $G/D$ decreases, the lower wake component ( $\textrm{LTV}_{P}$ ) extends towards the wall. By $G/D = 0.3$ (in figure 5 c), the $\textrm{LTV}_{P}$ and $\textrm{WSV}$ merge into a new structure. This merger spatially coincides with the marked recovery of wake symmetry. At $G/D = 0.1$ (in figure 5 c), a fully symmetric state is obtained, characterised by stable paired $\textrm{WTV}_{P}$ , arch vortices ( $\textrm{AV}$ ), and base vortices ( $\textrm{BV}$ ), consistent with wall-mounted bluff-body wakes (Hajimirzaie et al. Reference Hajimirzaie, Tsakiris, Buchholz and Papanicolaou2012, Reference Gushchin, Kostomarov, Matyushin and Pavlyukova2014; Tu et al. Reference Tu, Wang, Gao, She, Wang, Wang and Wei2023).

Figure 5.

Wake cross-sections at $x/D = 3.5$ for different $G/D$ , shown using the mean streamwise velocity $\overline {u^*} = \overline {u}/U_{\infty }$ and mean pressure $\overline {p^*} = \overline {p} / 0.5 {\rho } U_{\infty }^2$ in (a,b). The white dashed circle represents the projected outline of the sphere on the plotted plane. (c) Time-averaged streamwise vorticity $\overline {{\omega }_{x}^{*}} = {\overline {\omega }_{x}}D/U_{\infty }$ on planes $x/D = 2.0, 5.0$ . Pink dash-dotted lines indicate $z/D = 0$ .

3.2. Quantitative classification of near-wake regimes

To complement the qualitative findings in § 3.1, we establish a quantitative classification of the symmetry transitions for four statistical indicators. In figure 6, the classification relies on the spanwise velocity skewness $Skew_{w}={\langle w^3 \rangle }/{{\langle w^2 \rangle }^{{3}/{2}}}$ on the recirculation region for the $y/D = G/D + 0.5$ plane (in figure 6 a), spanwise lateral-force coefficient skewness $Skew_{C_{Z}}={\langle C_{Z}^3 \rangle }/{{\langle C_{Z}^2 \rangle }^{{3}/{2}}}$ (in figure 6 b) (Mittal et al. Reference Mittal, Wilson and Najjar2001; Tiwari et al. Reference Tiwari, Pal, Bale, Minocha, Patwardhan, Nandakumar and Joshi2020), the mean spanwise force $\overline {C_{Z}}$ (in figure 6 c), and the fluctuation intensity of the lateral force $C_{Z,rms}'$ (in figure 6 d). Here, these metrics effectively characterise the degree of spanwise asymmetry and the resulting lateral-force deflection. The results confirm that increasing wall proximity progressively suppresses asymmetric shedding, and identify a specific critical $G/D$ . Based on these $G/D$ values, we map the transition into four distinct near-wake regimes.

Figure 6.

Near-wake regime classification for a wall-proximity sphere over $(Re, G/D) = (420{-}600, 0.1{-}1.5)$ . (a) Spanwise-velocity skewness $Skew_{w}$ evaluated at the recirculation region on planes $y = G+D/2$ orthogonal to the mid-plane $z/D = 0$ . (b) Spanwise lateral-force skewness $Skew_{C_{Z}}$ versus $G/D$ . (c,d) Time-averaged spanwise lateral force $\overline {C_{Z}}$ and its fluctuation intensity $C_{Z,rms}^{\prime }$ as functions of $G/D$ .

Figure 6. Long description

Panel A: A line graph shows the spanwise-velocity skewness evaluated at the recirculation region on planes orthogonal to the mid-plane. The x-axis represents the gap ratio (G/D) ranging from 0 to 1.6, and the y-axis represents the skewness of the spanwise velocity (Skew_w) ranging from -1.0 to 1.0. Data points are plotted for three different Reynolds numbers (Re = 420, Re = 500, Re = 600), with distinct markers for each. The graph is divided into symmetric and asymmetric regimes. Panel B: A line graph shows the spanwise lateral-force skewness versus the gap ratio (G/D). The x-axis represents the gap ratio (G/D) ranging from 0 to 1.6, and the y-axis represents the skewness of the spanwise lateral force (Skew_Cz) ranging from -1.6 to 1.6. Data points are plotted for three different Reynolds numbers (Re = 420, Re = 500, Re = 600), with distinct markers for each. The graph is divided into symmetric and asymmetric regimes. Panel C: A line graph shows the time-averaged spanwise lateral force (Cz) as a function of the gap ratio (G/D). The x-axis represents the gap ratio (G/D) ranging from 0 to 1.6, and the y-axis represents the time-averaged spanwise lateral force (Cz) ranging from -0.02 to 0.02. Data points are plotted for three different Reynolds numbers (Re = 420, Re = 500, Re = 600), with distinct markers for each. Panel D: A line graph shows the fluctuation intensity of the spanwise lateral force (Cz,rms) as a function of the gap ratio (G/D). The x-axis represents the gap ratio (G/D) ranging from 0 to 1.6, and the y-axis represents the fluctuation intensity of the spanwise lateral force (Cz,rms) ranging from 0 to 0.04. Data points are plotted for three different Reynolds numbers (Re = 420, Re = 500, Re = 600), with distinct markers for each. The graph is divided into symmetric steady, symmetric unsteady, asymmetric unsteady I, and asymmetric unsteady II regimes.

The near-wake behaviour ( $420 \lesssim Re \leqslant 600$ ) is delineated into four distinct regimes in figure 6. At $0.1 \leqslant G/D \lesssim 0.28$ , the flow resides in the symmetric steady regime, exhibiting vanishing skewness and fluctuations ( $Skew \approx 0$ , $C_{Z,rms}' \approx 0$ ). This corresponds to the steady double-threaded structure in figure 4(d). As the gap increases to $0.28 \lesssim G/D \lesssim 0.35$ , the wake enters the symmetric unsteady regime. While mean mirror symmetry persists ( $Skew \approx 0$ , $\overline {C_{Z}} \approx 0$ ), consistent with prior experiments (van Hout et al. Reference van Hout, Eisma, Elsinga and Westerweel2018), significant fluctuations emerge, associated with unsteady shear-layer motion (Tiwari et al. Reference Tiwari, Bale, Patwardhan, Nandakumar and Joshi2019; Yin & Ong Reference Yin and Ong2020). For $G/D \gtrsim 0.35$ , the wake becomes asymmetric and unsteady. This domain is further subdivided into regimes I ( $0.35 \lesssim G/D \lt 0.8$ ) and II ( $0.8 \lesssim G/D \leqslant 1.5$ ) based on the direction of spanwise deflection. Crucially, the opposite signs of $Skew_{w}$ and $\overline {C_Z}$ between these regimes indicate that wake deflection is non-monotonic, suggesting a reversal of the wake-deflection direction as the sphere approaches the wall, highlighting a complex interaction before the final symmetry restoration ( $G/D \to 0.1$ ).

These statistics substantiate the structural observations detailed in § 3.1. The symmetric regimes are characterised by coherent, bilaterally symmetric structures, such as aligned vortex loops and tip–vortex pairs (Hajimirzaie et al. Reference Hajimirzaie, Tsakiris, Buchholz and Papanicolaou2014; Tiwari et al. Reference Tiwari, Bale, Patwardhan, Nandakumar and Joshi2019). In contrast, the asymmetric regimes exhibit distorted, spanwise-inclined vortices undergoing azimuthal oscillations. Crucially, the statistical metrics quantitatively capture the shifting dominance of wall proximity. In the stronger interaction zone (asymmetric regime I, $0.35 \lesssim G/D \lt 0.8$ ), the wake displays negative $Skew_{w}$ coupled with positive $Skew_{C_{Z}}$ , reflecting a specific azimuthal bias imposed by intense wall confinement. As wall influence diminishes in asymmetric regime II ( $G/D \gtrsim 0.8$ ), a sign inversion occurs ( $Skew_w \gt 0$ , $Skew_{C_Z} \lt 0$ ). This inversion quantitatively confirms the wake reorientation (deflection reversal) observed in the flow structure. Furthermore, the relaxation of confinement in regime II ( $G/D \geqslant 1.0$ ) results in amplified lateral-force fluctuations ( $C'_{Z,rms}$ ) compared to the wall-constrained regime I, supporting prior findings on wake alignment stability (van Hout et al. Reference van Hout, Eisma, Elsinga and Westerweel2018). Additionally, for $Re=420, 500, 600$ , the four wake regimes and their sequence with decreasing $G/D$ remain qualitatively similar. This suggests that $G/D$ is the primary parameter controlling the asymmetric-to-symmetric transition, whereas $Re$ mainly modulates the levels of skewness, mean lateral force, and lateral-force fluctuations.

Figure 7.

Relative TKE contents $e_{i}^{*}$ (red) and cumulative TKE portions ${\sum }_{1}^{i}e_{j}^{*}$ (blue) of the first $100$ POD modes at $Re=500$ .

4.1. The coherent structures of dominant spatial asymmetry and symmetry

This section aims to extract the coherent structures associated with spatial spanwise asymmetry and symmetry, including the upper hairpin vortex ( $\textrm{UHV}$ ), lower hairpin vortex ( $\textrm{LHV}$ ), longitudinal vortex ( $\textrm{LV}$ ), and wall-secondary vortex ( $\textrm{WSV}$ ). Since figure 6 shows that the overall regime sequence is qualitatively similar over $Re=420, 500, 600$ , $Re=500$ is selected as a representative case for the detailed mechanistic analysis. The POD (Wang et al. Reference Wang, Pan and Wang2022) is applied to extract the 3-D dominant coherent structures responsible for the observed changes in spatial symmetry. The snapshot POD method is applied to $1200$ – $1500$ instantaneous 3-D velocity fields $(u, v, w)$ over the domain $x/D = [-0.5, 15]$ , $y/D = [0, 5]$ and $z/D = [-2.5, 2.5]$ . Modes are ranked by their contribution to turbulent kinetic energy (TKE), $e_{i}$ , and the normalised modal energy is defined as $e_i^{*}=e_i / \sum e_j$ .

Figure 7 compares the POD energy spectra for $G/D = 0.8, 0.5, 0.3$ . The rapid energy decay is consistent with the multi-scale 3-D nature of the wake (Tu et al. Reference van Hout, Eisma, Elsinga and Westerweel2023, Reference van Oudheusden, Scarano, Hinsberg and Watt2024). Nevertheless, the first 100 modes consistently capture approximately $80{-}90\,\%$ of the total TKE, providing a compact representation of the energetically dominant motions. Notably, as $G/D$ decreases, energy condenses markedly into the first two modes, indicating a transition towards a more organised state dominated by large-scale coherent motions. Accordingly, we reconstruct the flow using the first $100$ modes,

(4.1) \begin{equation} \boldsymbol{u}_{\textit{reconstruct}}(x,y, z,t)=\overline {\boldsymbol{u}} + \sum _{i=1}^{100} \alpha _{i}(t)\, \boldsymbol{\varPsi }_{i}(x,y,z), \end{equation}

and later further filter the expansion to isolate the structures associated with the primary shedding frequency.

Figure 8.

Spectra of the temporal coefficients of the first six POD modes at $Re=500$ , based on streamwise (solid lines), wall-normal (dash-dotted lines) and spanwise (dotted lines) velocities: (a,d) $G/D=0.8$ , (b,e) $G/D=0.5$ , (c,f) $G/D=0.3$ . Plots (d–f) compare the original spanwise motion with the POD-reconstructed spanwise motion. Modes 1–2 and modes 1–4 correspond primarily to $St_1$ and $St_1+St_2$ at $G/D =0.8, 0.5$ , while modes 1–2, modes 1–4 and modes 1–6 correspond primarily to $St_1$ , $St_1+2St_1$ and $St_1+2St_1+St_2$ at $G/D =0.3$ . (g) The corresponding near-wake symmetry metric, highlighting the roles of $St_1$ and $St_2$ in the temporal modulation and spatial asymmetry/symmetry characteristics of the near wake.

Figure 8. Long description

Panel A: A line graph showing the spectra of the temporal coefficients of the first six POD modes at G/D = 0.8, based on streamwise (solid lines), wall-normal (dash-dotted lines), and spanwise (dotted lines) velocities. The x-axis represents the Strouhal number (St) on a logarithmic scale, and the y-axis represents the normalized magnitude of the temporal coefficients. Panel B: A similar line graph for G/D = 0.5. Panel C: Another line graph for G/D = 0.3. Panel D: A line graph comparing the original spanwise motion with the POD-reconstructed spanwise motion for G/D = 0.8. The x-axis represents the non-dimensional time (tD/U∞), and the y-axis represents the integrated spanwise velocity. Panel E: A similar comparison for G/D = 0.5. Panel F: Another comparison for G/D = 0.3. Panel G: A scatter plot showing the near-wake symmetry metric as a function of G/D, highlighting the roles of different modes in the temporal modulation and spatial asymmetry/symmetry characteristics of the near wake. The x-axis represents the ratio G/D, and the y-axis represents the skewness of the spanwise velocity.

To characterise the temporal content of the dominant modes, we compute Fourier spectra of the temporal coefficients of the first six modes (each contributing ${\gt} 2\,\%$ of the TKE). Figures 8(a–c) show that modes 1 and 2 are associated with $St_1$ , indicating that the first two POD modes form a pair representing the dominant periodic component of primary vortex shedding. In the asymmetric regimes II and I, a low-frequency $St_2$ appears in higher-order modes, suggesting the presence of slow temporal variations in the reconstructed signal. The coexistence of $St_1$ and $St_2$ is consistent with previously reported multi-frequency or quasi-periodic shedding (Tiwari et al. Reference Tiwari, Bale, Patwardhan, Nandakumar and Joshi2019; Frantz et al. Reference Frantz, Mimeau, Salihoglu, Loiseau and Robinet2025). In the symmetric unsteady regime, modes 3 and 4 show a clear $2St_1$ contribution, suggesting a coherent harmonic response associated with the primary shedding, while the $St_2$ component becomes weaker and shifts to higher-order modes. Figures 8(d–f) compare the semi-integrated spanwise velocity from the original field and POD-reconstructed fields. The signal is defined as

(4.2) \begin{equation} I(t) = \int _{V_{z\lt 0}} w\, \text{d}V + \int _{V_{z\gt 0}} w\, \text{d}V \end{equation}

to characterise the spanwise motion, where $V_{z\lt 0}$ and $V_{z\gt 0}$ denote the two spanwise halves of the selected near-wake region. The POD reconstructions are obtained using different combinations of modes associated with the dominant frequencies. For all $G/D$ , modes 1–2, which are dominated by $St_1$ , are used for the $St_1$ reconstruction. At $G/D = 0.8, 0.5$ , modes 1–4 are used for the $St_1+St_2$ reconstruction. At $G/D = 0.3$ , modes 1–4 and 1–6 are used for the $St_1+2St_1$ and $St_1+2St_1+St_2$ reconstructions, respectively. With this mode-based definition, the first two modes reproduce the main periodic variation associated with the primary shedding. The inclusion of higher-order modes improves the representation of the slower phase and amplitude variations, or of the $2St_1$ harmonic in the symmetric unsteady case. The results suggest that $St_1$ captures the main periodic variation, and $St_2$ improves the representation of the slower phase and amplitude variations. The skewness metric in figure 8(g) further shows that the reconstructions follow similar overall spatial asymmetry/symmetry trends (in figure 6 a). This suggests that $St_2$ mainly contributes to temporal modulation in the present reconstruction, rather than independently determining the dominant spatial asymmetry/symmetry of the near wake.

To identify the coherent structures that characterise the spanwise asymmetry or symmetry of the near wake, we reconstruct the flow using the mean field and the dominant energetic modes associated with $St_1$ . This reconstruction is used to extract the dominant $St_1$ structural skeleton of the near wake, rather than to represent the full instantaneous dynamics. Figure 9 shows that the reconstructed structures retain the main three-stage evolution observed as $G/D$ decreases. In the asymmetric regime II ( $G/D=0.8$ , in figures 9 a,d), the reconstruction captures alternating shedding of $\textrm{UHV}$ and $\textrm{LHV}$ , consistent with figure 4(a). This alternation helps to maintain a spanwise phase offset, and contributes to the observed asymmetric deflection. In the asymmetric regime I ( $G/D=0.5$ , in figures 9 b,e), suppression of the lower hairpin head converts the lower structure into the $\textrm{LV}$ (in figure 4 b). Its interaction with the $\textrm{WSV}$ reshapes the induced velocity field, and contributes to the reversal of the spanwise deflection. In the symmetric unsteady regime ( $G/D=0.3$ , in figures 9 c,f), the reconstruction captures the merger of the $\textrm{LV}$ s with the $\textrm{WSV}$ , weakening the spanwise phase offset, and yielding a nearly mirror-symmetric near wake. Meanwhile, the initial convection velocity remains nearly constant at $u_0 \approx 0.8U_\infty$ , while the near-wake wavelength contracts from $\lambda _0/D \approx 4.45$ ( $G/D=0.8$ ) to $\lambda _0/D \approx3.95$ ( $G/D=0.3$ ). This shortening of $\lambda _0$ accounts for the observed increase in the primary frequency through $f_1 \approx u_{0}/\lambda _0$ . Overall, the above analysis indicates that the observed spatial asymmetry/symmetry is reflected not only in the mean flow, but also in the dominant coherent structures associated with the primary shedding.

Figure 9.

Reconstructed near-wake structures at $Re = 500$ using the mean flow and the first two POD modes. Iso-surfaces of $Q^{*} = 0.0025$ are visualised alongside contours of the normalised streamwise velocity $u^{*}$ . Streamwise vorticity and in-plane velocity vectors $(v^*, w^*)$ are plotted on the $y{-}z$ plane. Sampling locations are $x/D = 2.7, 3.05, 3.5$ for (a–c), and $x/D = 2.5, 2.85, 3.25$ for (d–f). Here, (a,d) $G/D = 0.8$ , (b,e) $G/D = 0.5$ , (c,f) $G/D = 0.3$ .

4.2. Vorticity transport analysis of the coherent structures governing the symmetry transition

While § 4.1 delineated the structural evolution from asymmetry to symmetry, the physical mechanism underlying this transition remains to be further quantified. Specifically, the role of the $\textrm{WSV}$ in weakening the spanwise asymmetry as $G/D$ decreases remains to be identified. To address this, we examine the vorticity transport equation (Wu et al. Reference Wu, Ma and Zhou2006), as vorticity provides a physically grounded description of the dynamical evolution of fluid motion (Kumar et al. Reference Kumar, Meneveau and Eyink2023). This transport equation governs the evolution of coherent structures through convection, stretching and tilting (Chen et al. Reference Chen, Zhou, Werner, Cheng and Wu2023; Tu et al. Reference Tu, Wang, Gao, She, Wang, Wang and Wei2023; Chen & Wu Reference Chen and Wu2024; da Silva et al. Reference da Silva, Sumner and Bergstrom2024). In dimensionless Cartesian coordinates, it is expressed as

(4.3) \begin{equation} \frac {{\partial {\boldsymbol{\omega }}}}{\partial {t}} +({\boldsymbol{u}}\boldsymbol{\cdot }{\boldsymbol{\nabla }}){\boldsymbol{\omega }} = ({\boldsymbol{\omega }}\boldsymbol{\cdot }{\boldsymbol{\nabla }}){\boldsymbol{u}} +\frac {{{\nabla} }^{2}{\boldsymbol{\omega }}}{Re}. \end{equation}

We focus on the streamwise ( $\omega _x$ ) and wall-normal ( $\omega _y$ ) components. The $\omega _x$ component characterises the hairpin legs and longitudinal vortices ( $\textrm{LV}$ ), whereas $\omega _y$ captures the vertical wake–wall interactions (Niu et al. Reference Niu, Dou, Zhou and Xu2024). In contrast, the spanwise component ( $\omega _z$ ) is largely dominated by the background mean shear. The component-wise budgets are given by

(4.4) \begin{equation} \frac {\partial {{\omega }_{x}^*}}{\partial {t}} + \underbrace {{u^*}\frac {\partial {{\omega }_{x}^*}}{\partial {x^*}}}_{C_{xx}} + \underbrace {{v^*}\frac {\partial {{\omega }_{x}^*}}{\partial {y^*}}}_{C_{xy}} + \underbrace {{w^*}\frac {\partial {{\omega }_{x}^*}}{\partial {z^*}}}_{C_{xz}} = \underbrace {{{\omega }_{x}^*}\frac {\partial {u^*}}{\partial {x^*}}}_{S_{x}} + \underbrace {{{\omega }_{y}^*}\frac {\partial {u^*}}{\partial {y^*}}}_{T_{{y}\Rightarrow {x}}} + \underbrace {{{\omega }_{z}^*}\frac {\partial {u^*}}{\partial {z^*}}}_{T_{{z}\Rightarrow {x}}} + \frac {{{\nabla} }^{2}{{\omega }_{x}^*}}{Re}, \end{equation}

(4.5) \begin{equation} \frac {\partial {{\omega }_{y}^*}}{\partial {t}} + \underbrace {{u^*}\frac {\partial {{\omega }_{y}^*}}{\partial {x^*}}}_{C_{yx}} + \underbrace {{v^*}\frac {\partial {{\omega }_{y}^*}}{\partial {y^*}}}_{C_{yy}} + \underbrace {{w^*}\frac {\partial {{\omega }_{y}^*}}{\partial {z^*}}}_{C_{yz}} = \underbrace {{{\omega }_{x}^*}\frac {\partial {v^*}}{\partial {x^*}}}_{T_{{x}\Rightarrow {y}}} + \underbrace {{{\omega }_{y}^*}\frac {\partial {v^*}}{\partial {y^*}}}_{S_{y}} + \underbrace {{{\omega }_{z}^*}\frac {\partial {v^*}}{\partial {z^*}}}_{T_{{z}\Rightarrow {y}}} + \frac {{{\nabla} }^{2}{{\omega }_{y}^*}}{Re}, \end{equation}

where $C_{ij}$ (with ${i, j} \in \{{x, y, z}\}$ ) denotes the convection of $i$ -direction vorticity by the $j$ -direction velocity component. The source terms are categorised as stretching ( $S_i$ ), representing the extension or compression of vortex lines, and tilting ( $T_{{i} \Rightarrow {j}}$ ), representing the reorientation of vorticity from direction $i$ to $j$ . To quantify the key contribution to the recovery of spatial symmetry, we define the spanwise net budget as $\varSigma_{M} = M_{+z} + M_{-z}$ . Since a mirror-symmetric wake requires an antisymmetric distribution of $\omega _x$ and $\omega _y$ , the positive- and negative-spanwise contributions of the corresponding budget terms are expected to largely cancel, giving $\varSigma_{M} \approx 0$ . Thus a non-zero residual serves as a quantitative metric of dynamical asymmetry. Additionally, we define $I_{M}(y, \theta ) = \varSigma_{M}/\max_{y \in Y(\theta )}(|M_{+z}|+|M_{-z}|)$ , where $Y(\theta )$ denotes the selected wall-normal region at phase $\theta$ . Based on the symmetric state and the sensitivity tests, $|I_M| \lesssim 5\times 10^{-3}$ is taken as the operational near-zero range (see figure 22 in Appendix C). In this range, $\varSigma_M$ is treated as approximately zero in the present study.

To clarify the main contributors to the above transition, we examine the specific terms governing spanwise evolution for wall-normal and spanwise convection ( $C_{xy}$ , $C_{yy}$ , $C_{xz}$ , $C_{yz}$ ), stretching ( $S_x$ , $S_y$ ) and tilting of $\omega _z$ into $\omega _x$ and $\omega _y$ ( $T_{z \Rightarrow x} = \omega _{z}\, \partial u/ \partial z$ , $T_{z \Rightarrow y} = \omega _{z}\, \partial v/ \partial z$ ). These budget distributions (in figures 10 and 11) are derived by first computing each vorticity transport term from the original instantaneous velocity fields (30 primary vortex-shedding cycles), and the resulting budget terms are then phase-averaged with respect to $St_1$ . This helps to retain the spatial asymmetry/symmetry of the primary shedding while reducing the temporal modulations associated with $St_2$ (see figure 21 in Appendix C). The phase-averaged analysis is performed over six phase angles, defined by

(4.6) \begin{equation} \theta = \arctan (a_1(t)/ \sqrt {2\lambda _1},\ a_2(t)/ \sqrt {2\lambda _2}), \end{equation}

with phase interval $\Delta {\theta }=\pi /3$ . For a selected physical quantity $M$ , the integrated contribution at a given wall-normal location is evaluated as

(4.7) \begin{equation} M(j)=\sum M(x,j,z)\,\Delta S(x,j,z), \end{equation}

where $M$ denotes the selected term in the vorticity transport equation, and $\Delta S(x,j,z)$ is the local area element of the vortex-associated slice at the grid point $(x,z)$ for the wall-normal index $j$ . The instantaneous 3-D vortex-core region is identified as the connected component in the domain $x/D \times y/D \times z/D=[0,5]\times [0,5]\times [-2.5,2.5]$ . In each $+z$ and $-z$ half-domain, this region is defined as the 3-D connected component that contains the maximum $Q_{max }$ , satisfies $Q(x,y,z,t) \gt \alpha\, Q_{rms}(x,y,z)$ (Zhang et al. Reference Zhang, Xia, Qiu, Dong and Liu2023) with $\alpha = 0.25$ (see figure 20 in Appendix C), and meets the prescribed vorticity-sign conditions ( $\omega _x$ and $\omega _y$ ). For the $\textrm{UHV}$ , $(\omega _x,\omega _y)=(+,+)$ on the $+z$ side, and $(\omega _x,\omega _y)=(-,-)$ on the $-z$ side. For the near-wall structures ( $\textrm{LHV}$ , $LV-WSV$ and $LV+WSV$ ), $(\omega _x,\omega _y)=(-,+)$ on the $+z$ side, and $(+,-)$ on the $-z$ side. The summation is performed over the cross-sectional area obtained by intersecting the identified 3-D connected vortex region containing the vortex core with the corresponding $x$ – $z$ plane.

Figure 10.

Phase-averaged vorticity transport budget terms integrated over the selected vortex-core regions of the $\textrm{UHV}$ at $Re=500$ . (a,d,g) The streamwise-vorticity budget ( $\omega _x^*$ ). (b,e,h) The wall-normal-vorticity budget ( $\omega _y^*$ ) at $\theta =\pi /3$ . (c,f,i) The spanwise-vorticity tilting terms $\sum T_{z\Rightarrow x}^*$ and $\sum T_{z\Rightarrow y}^*$ at $\theta =0,\pi /3,2\pi /3$ . (j,k,l) The corresponding wall-normal profiles of wall-normal velocity $v^*$ and net spanwise velocity $\sum w^*$ around the $\textrm{UHV}$ vortex core at $\theta =0, \pi /3, 2\pi /3$ for three $G/D$ . At each wall-normal position ( $y/D$ ), the identified instantaneous 3-D connected vortex-core region is intersected with the corresponding $x$ – $z$ plane, and the budget terms are integrated over the resulting cross-section. Solid and dashed curves denote the $+z$ and $-z$ side contributions, respectively, while the symbols denote their sum ( $\sum$ ). Black curves represent the integrated vorticity components, namely $\omega _x^*$ in (a,d,g), and $\omega _y^*$ in (b,e,h). Green curves denote vortex stretching ( $S_x^*,S_y^*$ ), and pink curves denote wall-normal convection ( $C_{xy}^*,C_{yy}^*$ ). In (a–i), red and blue curves represent the spanwise-vorticity tilting terms $T_{z\Rightarrow x}^*$ and $T_{z\Rightarrow y}^*$ , respectively. Here, (a–c,j) $G/D=0.8$ , (d–f,k) $G/D=0.5$ , (g–i,l) $G/D=0.3$ .

Figure 10. Long description

Panel A: A line graph shows the streamwise-vorticity budget with the y-axis labeled as ‘Tilting/stretching’ and the x-axis labeled as ‘y/D’. The graph includes multiple data series represented by different colors and symbols. Panel B: A line graph displays the wall-normal-vorticity budget with the y-axis labeled as ‘Vorticity’ and the x-axis labeled as ‘y/D’. Panel C: A scatter plot shows the spanwise-vorticity tilting terms with the y-axis labeled as ‘Sum of Tilting Terms’ and the x-axis labeled as ‘y/D’. Panel D: A line graph illustrates the streamwise-vorticity budget with the y-axis labeled as ‘Tilting/stretching’ and the x-axis labeled as ‘y/D’. Panel E: A line graph presents the wall-normal-vorticity budget with the y-axis labeled as ‘Vorticity’ and the x-axis labeled as ‘y/D’. Panel F: A scatter plot depicts the spanwise-vorticity tilting terms with the y-axis labeled as ‘Sum of Tilting Terms’ and the x-axis labeled as ‘y/D’. Panel G: A line graph shows the streamwise-vorticity budget with the y-axis labeled as ‘Tilting/stretching’ and the x-axis labeled as ‘y/D’. Panel H: A line graph displays the wall-normal-vorticity budget with the y-axis labeled as ‘Vorticity’ and the x-axis labeled as ‘y/D’. Panel I: A scatter plot illustrates the spanwise-vorticity tilting terms with the y-axis labeled as ‘Sum of Tilting Terms’ and the x-axis labeled as ‘y/D’. Panel J: A line graph shows the wall-normal profiles of wall-normal velocity with the y-axis labeled as ‘Velocity’ and the x-axis labeled as ‘y/D’. Panel K: A line graph presents the wall-normal profiles of wall-normal velocity with the y-axis labeled as ‘Velocity’ and the x-axis labeled as Șy/D’. Panel L: A line graph depicts the wall-normal profiles of wall-normal velocity with the y-axis labeled as ‘Velocity’ and the x-axis labeled as ‘y/D’.

Figure 11.

Phase-averaged vorticity transport budget terms integrated over the selected vortex-core regions of the near-wall wake structures at $Re = 500$ . These structures include the $\textrm{LHV}$ at $G/D=0.8$ , the longitudinal vortex and wall-secondary vortex ( $LV-WSV$ ) at $G/D=0.5$ , and the merged structure ( $LV+WSV$ ) at $G/D=0.3$ . Plots (a,d,g) and (b,e,h) show the streamwise ( $\omega _x^*$ ) and wall-normal ( $\omega _y^*$ ) vorticity budgets at $\theta =4\pi /3$ , respectively. Plots (c,f,i) show the integrated spanwise-vorticity tilting terms $\sum T_{z\Rightarrow x}^*$ and $\sum T_{z\Rightarrow y}^*$ at $\theta =\pi , 4\pi /3, 5\pi /3$ . Plots (j,k,l) show the corresponding wall-normal profiles of $v^*$ and net spanwise velocity $\sum w^*$ around the near-wall vortex core at $\theta =\pi , 4\pi /3, 5\pi /3$ for three $G/D$ values. At each $y/D$ , the identified instantaneous 3-D connected vortex-core region is intersected with the corresponding $x$ – $z$ plane, and the budget terms are integrated over the resulting cross-section to obtain the profiles plotted as functions of $y/D$ . Solid and dashed curves denote the $+z$ and $-z$ side contributions, respectively, while symbols denote their sum ( $\sum$ ). Black curves represent the integrated vorticity components, namely $\omega _x^*$ in (a,d,g), and $\omega _y^*$ in (b,e,h). Green curves denote vortex stretching ( $S_x^*,S_y^*$ ), and pink curves denote wall-normal convection ( $C_{xy}^*,C_{yy}^*$ ). In (a–i), red and blue curves represent the spanwise-vorticity tilting terms $T_{z\Rightarrow x}^*$ and $T_{z\Rightarrow y}^*$ , respectively. Here, (a–c,j) $G/D=0.8$ , (d–f,k) $G/D=0.5$ , (g–i,l) $G/D=0.3$ .

4.2.2. Symmetric vortex shedding and symmetry restoration

Symmetry recovery is completed as the wake enters the symmetric regime ( $G/D = 0.3$ ). In this state, the spanwise gradients ( $\partial u^*/\partial z^*, \partial v^*/\partial z^*$ ) that promoted the previous asymmetries largely vanish (in figures 12 c,f). To quantify this pathway, vorticity transport budgets (in (4.4)–(4.5)) are evaluated over the vortex-core regions associated with the $\textrm{UHV}$ and the coupled $LV+WSV$ structure at the selected phase-averaged angles $\theta =0, \pi /3, 2\pi /3, \pi , 4\pi /3, 5\pi /3$ .

Figure 12.

Phase-averaged spanwise gradients of the streamwise and wall-normal velocities, $\partial u^*/\partial z^*$ and $\partial v^*/\partial z^*$ , integrated over the selected vortex-core regions at $Re = 500$ . (a–c) The $\textrm{UHV}$ during its formation at $\theta =0$ , $\pi /3$ and $2\pi /3$ . (d–f) The near-wall wake structures, including the $\textrm{LHV}$ , $\textrm{LV}$ and $\textrm{WSV}$ , at $\theta =\pi ,4\pi /3,5\pi /3$ . The curve colour changes from dark to light as the phase angle $\theta$ increases. At each $y/D$ , the gradients are integrated over the cross-section of the identified instantaneous 3-D connected vortex-core region on the corresponding $x$ – $z$ plane before phase averaging. At the representative phases $\theta =\pi /3$ and $4\pi /3$ , solid and dashed curves denote the $+z$ and $-z$ side contributions, respectively, while symbols denote their sum ( $\sum$ ). Here, (a,d) $G/D=0.8$ , (b,e) $G/D=0.5$ , (c,f) $G/D=0.3$ .

Figure 12. Long description

Panel A: A line graph shows velocity gradients for G/D = 0.8, UHV. The x-axis is labeled y/D and ranges from 1.6 to 2.2. The y-axis is labeled Velocity - gradients and ranges from -0.8 to 0.8. Different line styles and symbols represent various velocity gradient components. Panel B: A line graph shows velocity gradients for G/D = 0.5, UHV. The x-axis is labeled y/D and ranges from 1.3 to 1.9. The y-axis is labeled Velocity - gradients and ranges from -0.8 to 0.8. Different line styles and symbols represent various velocity gradient components. Panel C: A line graph shows velocity gradients for G/D = 0.3, UHV. The x-axis is labeled y/D and ranges from 1.2 to 1.8. The y-axis is labeled Velocity - gradients and ranges from -0.8 to 0.8. Different line styles and symbols represent various velocity gradient components. Panel D: A line graph shows velocity gradients for G/D = 0.8, LHV. The x-axis is labeled y/D and ranges from 0.1 to 1.1. The y-axis is labeled Velocity - gradients and ranges from -0.4 to 0.4. Different line styles and symbols represent various velocity gradient components. Panel E: A line graph shows velocity gradients for G/D = 0.5, LV - WSV. The x-axis is labeled y/D and ranges from 0.2 to 1.0. The y-axis is labeled Velocity - gradients and ranges from -0.4 to 0.4. Different line styles and symbols represent various velocity gradient components. Panel F: A line graph shows velocity gradients for G/D = 0.3, LV + WSV. The x-axis is labeled y/D and ranges from 0.2 to 0.8. The y-axis is labeled Velocity - gradients and ranges from -0.4 to 0.4. Different line styles and symbols represent various velocity gradient components.

When $\theta = 0, \pi /3, 2\pi /3$ , the spanwise deflection of the $\textrm{UHV}$ differs from the earlier asymmetric regimes. In regimes II and I, the wake is characterised by a distinct net contribution from asymmetric tilting ( $\sum T_{z \Rightarrow y}^* \neq 0$ ), which promotes wake deflection towards $+z$ and $-z$ , respectively. In contrast, the symmetric regime is characterised by the near removal of this spanwise skewness. The budget analysis (in figures 10 g,h,i) reveals a state of an approximately antisymmetric distribution, where the tilting terms on the $+z$ side are equal and opposite to those on the $-z$ side ( $|T_{z {\Rightarrow } x}^{*}(+z)| \approx |T_{z {\Rightarrow } x}^{*}(-z)|$ and $|T_{z {\Rightarrow } y}^{*}(+z)| \approx |T_{z {\Rightarrow } y}^{*}(-z)|$ ). Consequently, the net spanwise tilting residual nearly vanishes ( $\sum T_{z \Rightarrow y}^* \approx 0$ and $\sum T_{z \Rightarrow x}^* \approx 0$ ). This removes the main tilting net residual that sustained the spanwise deflection in the earlier regimes. The result is a $\textrm{UHV}$ with symmetric leg convergence, represented by the antisymmetric distribution of $w^*$ , together with a nearly uniform downward velocity component ( $v^* \lt 0$ ) (in figure 10 l). This marks a clear departure from the tilting mechanisms observed at larger gaps.

When $\theta = \pi , 4\pi /3, 5\pi /3$ , the dynamical characteristics are characterised by a reorganisation of the near-wall structures. Unlike the asymmetric regimes I (where $\textrm{LV}$ and $\textrm{WSV}$ were isolated), this stage features the formation of a unified hybrid structure ( $LV + WSV$ ). This structural fusion is facilitated by a near balance of wall-normal convection ( $C_{xy}^*$ and $C_{yy}^*$ ). The profiles (in figures 11 g,h,i) reveal a bidirectional convection mechanism. Upward convection lifts the near-wall $\textrm{WSV}$ and downward convection transports the $\textrm{LV}$ . These opposing fluxes converge into a central focal band ( $0.5 \lesssim y/D \lesssim 0.6$ ), effectively ‘bridging’ the gap between the wall and the free stream. This convective bridging promotes vorticity exchange, and facilitates the merger of the previously isolated $\textrm{LV}$ and $\textrm{WSV}$ into a single structure. The near balance ( $\sum T_{z \Rightarrow y}^* \approx 0$ and $\sum T_{z \Rightarrow x}^* \approx 0$ ) supports the interpretation that the merger occurs near the centreline, without the lateral drift observed in previous regimes.

The analyses in §§ 4.2.1 and 4.2.2 indicate that wake transitions are strongly influenced by wall-induced modifications to the velocity-gradient field, specifically $\partial u^*/\partial z^*$ and $\partial v^*/\partial z^*$ . As $G/D$ decreases, the critical factor is the spanwise homogenisation of these gradients. This kinematic smoothing substantially weakens the source terms for asymmetric tilting, thereby reducing the directional deflection. Simultaneously, this gradient evolution reconfigures the flow field, allowing wall-normal convection to shift into a ‘bridging’ mode that amalgamates the $\textrm{LV}$ and $\textrm{WSV}$ . Thus while larger gaps allow non-uniform gradients to promote deflection, strong wall confinement promotes kinematic uniformity, thereby favouring a symmetric near-wake state. The complete sequence of structural transitions and their corresponding vorticity-budget signatures is summarised in table 3.

Table 3.

With decreasing $G/D$ , the wall homogenises the background velocity-gradients, which weakens the asymmetric tilting of spanwise vorticity, and enables the wake to recover symmetry.

Table 3. Long description

The table has four rows and four columns. The columns are labeled Near-wake, Asymmetric regime II, Asymmetric regime I, and Symmetric regime. The rows are labeled Flow structures, Vorticity tilting, and Wall-normal convection. Row 1: Flow structures, Persistent asymmetry (UHV deflection, LHV deflection), Asymmetry patterns (UHV deflection, LV deflection, WSV deflection), Symmetric patterns (UHV symmetric, LV merged). Row 2: Vorticity tilting, Primary driver of asymmetry (alternating), Reversed asymmetry (inverted), Restored symmetry (removed). Row 3: Wall-normal convection, Reinforces inclined transport in UHV and LHV, Spanwise-biased transport sustains LV and WSV, Symmetric bridging transport facilitates LV and WSV merger.

Figure 13.

(a) Contours of the pressure Hessian term ( $-{\partial ^2 \overline {p^*}}/{\partial x^* \partial z^*}$ ) on the $y{-}z$ plane, evaluated at $x/D = 2.5, 2.85, 3.35$ (representing the three regimes). Black circles denote the spanwise summation of the pressure-Hessian term. The white dashed circle represents the projected outline of the sphere on the plotted plane. (b) The wall-normal turbulent transport term ( $-{\partial \overline {u'v'}^*}/{\partial y^*}$ ) evaluated in the symmetry plane ( $z/D = 0$ ). (c) The distribution of spanwise Reynolds shear stress $\overline {\tau _{uw}^*} = -\overline {u'w'}^* = -\overline {u'w'}/U_{\infty }^{2}$ on the mid-plane $z/D = 0$ at $Re = 500$ for $G/D = 0.8, 0.5, 0.3$ .

Crucially, the re-equilibration of these gradients involves a corresponding adjustment of the pressure field. To clarify the momentum-balance basis of the observed spanwise-shear homogenisation (in figure 12), we examine the streamwise Reynolds-averaged Navier–Stokes momentum budget (4.8), where the balance of forces constrains the evolution of the shear strain. In (4.8), the mean convection ( $I$ ) is balanced by the streamwise pressure gradient ( $ \textit{II} $ ), turbulent transport ( $ \textit{III}-V$ ) and viscous diffusion. Consistent with the Reynolds stress distribution ( $-\overline {u'w'}^{*}$ , in figure 13 c) and the wall-normal turbulent transport results in figure 13(b), decreasing $G/D$ redistributes momentum transport. Specifically, the intense wall-jet effect amplifies wall-normal turbulent transport ( $ \textit{IV} $ ), while simultaneously suppressing spanwise turbulent transport ( $V$ ) to negligible levels. As the term $(V)$ becomes negligible, the momentum balance is established between the streamwise pressure gradient $-\partial \overline {p^*}/\partial x^*$ $(II)$ , inertial convection $(I)$ and wall-normal turbulent transport $(\textit{IV})$ . Consequently, wall confinement promotes a more spanwise-uniform distribution of the streamwise pressure gradient. We further assess the contribution of pressure reorganisation to symmetry recovery by analysing the pressure Hessian (4.9). This term represents a key pressure-related contribution to the balance governing the evolution of the shear strain ( $\partial \overline {u^*}/\partial z^*$ ):

(4.8) \begin{equation} \underbrace { {\overline {u^*}}\frac {\partial \overline {u^*}}{\partial {x^*}} + {\overline {v^*}}\frac {\partial \overline {u^*}}{\partial {y^*}} + {\overline {w^*}}\frac {\partial \overline {u^*}}{\partial {z^*}} }_{I} = \underbrace { -\frac {\partial \overline {p^*}}{\partial x^*} }_{II} -\left( \underbrace { \frac {\partial \overline {u'u'}^*}{\partial x^*} }_{III} + \underbrace { \frac {\partial \overline {u'v'}^*}{\partial y^*} }_{IV} + \underbrace { \frac {\partial \overline {u'w'}^*}{\partial z^*} }_{V} \right) + \frac {1}{Re} {{\nabla} }^{2}{{\overline {u^*}}}, \end{equation}

(4.9) \begin{align} {-}\frac {\partial ^2 \overline {p^*}}{\partial x^* \partial z^*} \sim \frac {\rm D}{{\rm D}t}\left (\frac {\partial \overline {u^*}}{\partial z^*}\right ) + \left(\frac {\partial \overline {u^*}}{\partial z^*}\frac {\partial {\overline {u^*}}}{\partial {x^*}} + \frac {\partial \overline {v^*}}{\partial z^*}\frac {\partial {\overline {u^*}}}{\partial {y^*}} + \frac {\partial \overline {w^*}}{\partial z^*}\frac {\partial {\overline {u^*}}}{\partial {z^*}}\right)\!. \end{align}

Figure 13(a) presents the transition of the pressure-Hessian distribution in the $y{-}z$ plane for three $G/D$ values. The results reveal that the spanwise-shear homogenisation develops together with the establishment of a spanwise-uniform pressure gradient. This change is reflected in the attenuation of the pressure-Hessian term in (4.9). At $G/D = 0.3$ , the attenuation of this term substantially weakens the pressure-related contribution to the balance governing this spanwise shear strain, $\partial u^*/\partial z^*$ . Therefore, the recovery of wake symmetry involves the reduction of $\partial \overline {u^*}/\partial z^*$ and the weakening of the corresponding pressure-Hessian contribution, thereby reducing the difference in asymmetric vorticity tilting between the two spanwise sides. This finding is consistent with isolated bluff-body studies, in which pressure-gradient reorganisation has been linked to symmetry transitions (Johnson & Patel Reference Johnson and Patel1999). Additionally, in the present Reynolds number range ( $420 \lesssim Re \leqslant 600$ ), the four wake regimes occur in the same sequence as $G/D$ decreases, while variations in $Re$ mainly modify the regime-classification quantities and the tested vorticity transport diagnostics. The detailed momentum-budget and pressure-Hessian analyses presented here are performed at $Re=500$ .

4.3. The weakening of spanwise shear-layer deflection

In unbounded sphere wakes, the onset of symmetry breaking is linked to the structure of the base-flow azimuthal vorticity and its near-wake gradients (Magnaudet & Mougin Reference Magnaudet and Mougin2007; Chiarini et al. Reference Chiarini, Gauthier and Boujo2025). Following the results in § 4.2.2 that wall proximity homogenises the streamwise pressure and background velocity-gradients, we further examine how spanwise-vorticity compression and reorientation contribute to the weakening of spanwise shear-layer deflection as $G/D$ decreases. To elucidate this process, we employ the mean spanwise stretching/compression indicator $\overline {S_z^*}=\overline {\omega _z^*}\, \overline {\partial w^*/\partial z^*}$ to quantify the interaction between mean spanwise vorticity $\omega _{z}^{*}$ and spanwise strain. It provides a local diagnostic of whether the mean strain field tends to amplify or compress spanwise vorticity. This approach is motivated by vortex-line stretching concepts linking mean strain to vorticity amplification (Wu et al. Reference Wu, Ma and Zhou2006), and by recent analyses of wake transition where mean stretching diagnostics successfully characterised the tearing of coherent structures (Verma & Hemmati Reference Verma and Hemmati2024). Here, $\overline {S_{z}^{*}}$ is the mean compression or stretching of spanwise vorticity. The tilting term $\overline {T_{x \Rightarrow z}^*} + \overline {T_{y \Rightarrow z}^*}$ is the mean reorientation of streamwise and wall-normal vorticity into the spanwise direction. Neither quantity is a perturbation growth rate or an eigenvalue-based stability measure.

Figure 14.

Distribution of the mean spanwise vorticity compression $\overline {S_{z}^{*}} = \overline {\omega _{z}^{*}}\, \overline {{\partial w^{*}} /{\partial z^{*}}}$ at $Re = 500$ for different $G/D$ values. Blue and red contours represent negative ( $\omega _{z}^{*}=-0.75,-0.15$ ) and positive ( $\omega _{z}^{*}=+0.75,+0.15$ ) spanwise vorticity, respectively. Here, (a) $x/D =3.5$ , $G/D = 0.8$ ; (b) $x/D =3.5$ , $G/D = 0.5$ ; (c) $x/D =3.5$ , $G/D = 0.3$ ; (d) $x/D =6.5$ , $G/D = 0.1$ .

Figure 14 shows the distribution of $\overline {S_z^*}$ on the $y{-}z$ plane at different streamwise locations. In the upper shear-layer regions, where $\overline {\omega _z^*} \lt 0$ (blue contours), a positive $\overline {S_z^*}$ corresponds to spanwise compression of negative spanwise vorticity. In asymmetric unsteady regime II ( $G/D = 0.8$ , in figure 14 a), $\overline {S_z^*} \gt 0$ exhibits strong intensity biased towards the $+z$ side. This compression supports the upper shear-layer deflection towards $+z$ , and helps to maintain the corresponding near-wake deflection (in figures 16 a,b). In asymmetric unsteady regime I ( $G/D = 0.5$ , in figure 14 b), the compression weakens, and its strongest region shifts to the $-z$ side. This promotes upper shear-layer deflection towards $-z$ , corresponding to the deflection reversal of the near wake (in figures 16 a,b). For the symmetric unsteady regime ( $G/D = 0.3$ , in figure 14 c), the compression becomes nearly symmetric about $z=0$ , and no persistent shear-layer deflection towards either side remains. For the symmetric steady regime ( $G/D = 0.1$ , in figure 14 d), $\overline {S_z^*}$ drops to negligible or negative levels ( $\overline {S_z^*} \lesssim 0$ ). This collapse indicates that spanwise vorticity compression is largely reduced, consistent with the emergence of a steady, symmetric wake. This result is consistent with the attenuation of spanwise force fluctuations observed in figure 6(d).

The lower shear layer ( $\overline {\omega _z^*} \gt 0$ , red contours in figure 14) follows a similar trend, while spanwise vorticity compression corresponds to $\overline {S_z^*} \lt 0$ . At $G/D = 0.8$ , the negative- $\overline {S_z^*}$ region is stronger on one spanwise side, consistent with the deflected lower shear layer. At $G/D = 0.5$ , due to localised negative strain induced by interactions between the upper and near-wall shear layers, compression weakens. When the gap is further reduced to $G/D = 0.3$ , the compression decreases markedly, thereby reducing the spanwise deflection of the lower shear-layer response, consistent with the reduced spanwise fluctuation intensity. At $G/D = 0.1$ , $\overline {S_{z}^{*}} \approx 0$ over most of the selected region, and the remaining compression is too weak to sustain a pronounced one-sided deflection. Complementing this process, figure 15 reveals that the tilting contribution $\overline {T_{x \Rightarrow z}^*}+\overline {T_{y\Rightarrow z}^*}$ tends to locally attenuate $\overline {\omega _z^*}$ , thereby opposing the persistence of spanwise vorticity (in figures 16 c,d). This implies that as $G/D$ decreases, the conversion of streamwise and wall-normal vorticity into the spanwise component is weakened. This provides an additional mechanism through which spanwise fluctuations are reduced and the asymmetric deflection of the shear layer is weakened. Additionally, for $Re=420, 500, 600$ , the four wake regimes and their sequence with decreasing $G/D$ remain qualitatively similar, as shown in figure 6. This suggests that $G/D$ is the primary parameter controlling the asymmetric-to-symmetric transition, whereas $Re$ mainly modulates the levels of skewness, mean lateral force, and lateral-force fluctuations.

Figure 15.

Distribution of the mean tilting contribution $\overline {T_{x \Rightarrow z}^{*}} + \overline {T_{y \Rightarrow z}^{*}}$ at $Re = 500$ for different $G/D$ values. Blue and red represent negative ( $\omega _{z}^{*}=-0.75,-0.15$ ) and positive ( $\omega _{z}^{*}=+0.75,+0.15$ ) spanwise vorticity, respectively. Here, (a) $x/D =3.5$ , $G/D = 0.8$ ; (b) $x/D =3.5$ , $G/D = 0.5$ ; (c) $x/D =3.5$ , $G/D = 0.3$ ; (d) $x/D =6.5$ , $G/D = 0.1$ .

Figure 15. Long description

Panel A: A 3D plot shows the distribution of mean tilting contribution with axes labeled x/D, y/D, and z/D. The color scale ranges from negative to positive spanwise vorticity. The 2D projection below shows a side view, and the bottom plot shows a top view with blue and red representing negative and positive vorticity, respectively. Panel B: Similar to Panel A, this 3D plot and its projections show the distribution for different values, with corresponding side and top views. Panel C: This 3D plot and its projections depict the distribution for another set of values, with side and top views. Panel D: The final 3D plot and its projections show the distribution for the highest set of values, with side and top views.

Figure 16.

Spanwise distributions of the mean spanwise-vorticity compression $\overline {S_z^{*}}$ and the mean vorticity- reorientation contribution $\overline {T_{x\Rightarrow z}^{*}} +\overline {T_{y\Rightarrow z}^{*}}$ in the selected upper and lower shear-layer regions for $G/D = 0.8, 0.5, 0.3, 0.1$ . (a,b) Plots of $\overline {S_z^{*}}$ in the upper and lower shear layers, respectively. (c,d) Plots of $\overline {T_{x\Rightarrow z}^{*}} +\overline {T_{y\Rightarrow z}^{*}}$ in the upper and lower shear layers, respectively. Solid red, dashed black, and dotted purple curves denote $Re=420, 500, 600$ , respectively. Yellow markers indicate the dominant extremum in each selected shear-layer region.

In summary, the distributions of $\overline {S_z^*}$ and the tilting terms indicate that the spanwise deflection of the shear layers is modulated by vorticity compression and reorientation. As $G/D$ decreases, the difference in compression between the $+z$ and $-z$ sides diminishes, and the reorientation towards either side becomes weaker, thereby reducing the ability of the shear layers to sustain a preferred spanwise deflection. These changes provide a physical interpretation of the reduced capacity of the shear layers to sustain spanwise asymmetry during the observed wake transition, including the monotonic decay of the lateral-force fluctuations $C'_{Z,{rms}}$ (in figure 6 d). Specifically, the reduced compression in the lower shear layer weakens its tendency to form asymmetric shedding structures. The arch- and base-vortex structures in figure 4(d) emerge after the persistent spanwise deflection has been substantially reduced. Notably, the persistence of double-threaded structures at $G/D = 0.1$ suggests that streamwise-oriented vortex dynamics may remain active after spanwise asymmetry has been strongly reduced, consistent with related observations for isolated sphere wakes (Ghidersa & Dusek Reference Ghidersa and Dusek2000; Tiwari et al. Reference Tiwari, Bale, Patwardhan, Nandakumar and Joshi2019). Therefore, the transition from asymmetry to symmetry is characterised primarily by the gap-dependent weakening of the spanwise shear-layer deflection, as evidenced by the decay and symmetrisation of $\overline {S_z^*}$ , and the collapse of spanwise vorticity conversion in the lower shear layer. The wall does not merely alter the vortex structure, but also weakens the asymmetric compression and reorientation of spanwise vorticity that supports spanwise wake asymmetry.