3.1.1. Infinite plate regime

Figure 5 illustrates the canonical interaction between an infinite planar plate and a vortex, a scenario well documented in existing studies. The plate dimensions of $4 \,D_n$ , $5 \,D_n$ and $10 \,D_n$ are effectively classified as infinite, with their vortex interactions reflecting identical characteristics. Detailed quantitative analysis of this categorisation follows in later sections. Please see supplementary movie 1 available at https://doi.org/10.1017/jfm.2026.11137 for the full flow visualisation recording. The PVR’s effect is observable in the vorticity distribution, where it induces flow along the plate, adhering to the no-slip condition and thereby generating a boundary layer with opposite-sign vorticity (figure 5 a,b). The PVR then stretches over the plate and its core alters as a result of the stretching effect of the interaction. An adverse pressure gradient causes the induced boundary layer to separate and erupts away from the surface (figure 5 c), a phenomenon also reported by Doligalski (Reference Doligalski1980). Subsequently, the vortex rebounds from the surface, deformed by stretching along the wall interacting with the erupted boundary layer, leading to a distinct rolled-up SVR in figure 5(d). As depicted in figure 5(e), both PVR and SVR orbit above the plate, in agreement with Cerra & Smith (Reference Cerra and Smith1983). The shape of the PVR is more relaxed, now farther out of the wall influence. It later returns to the plate (figure 5 f), where the second vortex stretching phase begins, caused by the SVR’s orbit and image-induced motion, resulting in another collision. However, the PVR has diminished in size and strength. Finally, figure 5(g) depicts the subsequent vortex rebound, with a coherent TVR forming from the ejected boundary layer. The viscous rebound mechanism arrests lateral macro-structure development, averting any edge-induced influence. These observations align with the descriptions found in previous studies (Boldes & Ferreri Reference Boldes and Ferreri1973).

Figure 5. Vortex ring collision with the $5\, D_n$ plate. Contours of vorticity non-dimensionalised by the ratio of nominal circulation to the squared nozzle diameter (left column). Time-matched flow visualisation pictures (right column) with important features identified. Results are shown for (a) $t^*\,=\,-1$ , (b) $t^*\,=\,0$ , (c) $t^*\,=\,1$ , (d) $t^*\,=\,2.2$ , (e) $t^*\,=\,4$ , (f) $t^*\,=\,5$ and (g) $t^*\,=\,6.5$ .

3.1.2. Quasi-infinite plate regime

As the dimensions of the plate are reduced, unique flow dynamics manifest, necessitating the identification of a new regime classification. However, further quantitative analysis is needed to evaluate the extent to which these differences influence the evolution of the flow field. Figure 6 illustrates the observations for the $2.5\,D_n$ plate, which serves as a representative for the quasi-infinite regime that also includes the $3\,D_n$ plate. Please see supplementary movie 2 for the full flow visualisation recording. Figure 6(a,b) illustrates the convection of the free PVR towards the plate. The interaction of the PVR with the flat plate is evident in the vorticity contours, showing induced flow along the plate, leading to a boundary layer of vorticity opposing that of the PVR. Notably, there is vorticity also generated at the plate’s edge. The PVR exhibits stretching along the plate surface, accompanied by significant deformation, mirroring the infinite plate interaction (figure 6 c). Due to an adverse pressure gradient, the induced boundary layer separates and erupts from the wall, akin to the infinite plate scenario. However, the intriguing behaviour of edge-generated vorticity is its induction towards the separating boundary layer. The vorticity contours indeed illustrate a cohesive secondary structure that incorporates both the separated boundary layer and some edge-generated vorticity. Figure 6(d) displays the vortex rebounding from the wall and the significant deformation of the core endured by the PVR is evident. A distinct SVR begins to emerge, with edge-generated vorticity blending with the separated boundary layer, though the extent of this integration remains to be quantified. Figure 6(e) depicts the orbiting PVR and SVR pair rebounding, and at this stage, the boundary layer is detaching over the plate edge. The counter-rotating vortex pair is observed to follow a looping rebound path, reaching its apex in figure 6(f). By then, the PVR core, distanced from the near-wall stretching, has reduced in size – an indication of circulation decay that lacks a quantitative measure. Finally, figure 6(g) marks the second phase of vortex stretching. The PVR re-engages with the plate’s influence, initiating the formation of a coherent TVR. Formed from edge-separated vorticity, this TVR is situated lower, extending over the plate’s edge. The macro-structural progression within this plate size domain mirrors the vortex phases observed in the infinite regime, albeit the PVR induces flow over the plate edge initially. However, the separation of the boundary layer noted in the infinite regime is observed before the edge-generated vorticity significantly impacts the initial rebound. While the PVR and SVR both rebound, the boundary layer’s separation reaches the plate’s edge, allowing edge-generated vorticity to impact the second rebound event. It remains to be determined if this edge effect can be quantitatively verified.

Figure 6. Vortex ring collision with the $2.5\, D_n$ plate. Contours of vorticity non-dimensionalised by the ratio of nominal circulation to the squared nozzle diameter (left column). Time-matched flow visualisation pictures (right column) with important features identified. Results are shown for (a) $t^*\,=\,-1$ , (b) $t^*\,=\,0$ , (c) $t^*\,=\,1$ , (d) $t^*\,=\,2.2$ , (e) $t^*\,=\,-4$ , (f) $t^*\,=\,5$ and (g) $t^*\,=\,6.5$ .

Figure 7. Vortex ring collision with the $2\, D_n$ plate. Contours of vorticity non-dimensionalised by the ratio of nominal circulation to the squared nozzle diameter (left column). Time-matched flow visualisation pictures (right column) with important features identified. Results are shown for (a) $t^*\,=\,-1$ , (b) $t^*\,=\,0$ , (c) $t^*\,=\,1$ , (d) $t^*\,=\,2.2$ , (e) $t^*\,=\,-4$ , (f) $t^*\,=\,5$ and (g) $t^*\,=\,6.5$ .

3.1.3. Finite plate regime

In the finite plate regime, the PVR is truncated by the separated boundary layer, preventing its extension towards the plate’s periphery. In the quasi-infinite regime, it is evident that while the edge serves as a site for secondary vorticity genesis, its influence on the overall flow field is minimal. The finite plate regime, depicted in figure 7, highlights the full potential of the plate edge and its effect on the classical vortex ring–wall interaction. Please see supplementary movie 3 for the full flow visualisation recording. Figure 7(a,b) illustrates the free PVR convecting towards the plate, with vorticity contours indicating the edge’s role in generating vorticity that rolls up over the plate’s side. By examining the trajectory of the orange dye in the visualisations between the frames, rotational strain is evident. This phenomenon observed in the quasi-infinite plate scenario is more enhanced than in the infinite plate regime due to the edge’s proximity to the PVR core. The next time frame (figure 7 c) marks a significant deviation from the quasi-infinite case, where boundary-layer separation isolated the PVR from edge-generated vorticity, which was partly entrained into the SVR. However, in this regime, the edge is the exclusive source of rolled-up secondary vorticity. The ensuing edge-generated vorticity is robust enough to induce a boundary layer of opposite-sign vorticity along the plate edge. Figure 7(d) portrays the PVR that has convected over the edge of the wall and stretched considerably but is still undergoing rebound. This rebound is attributed not to boundary-layer eruption but to the dynamics of a counter-rotating vortex pair of unequal strengths with the weaker SVR rotating around the stronger PVR. Figure 7(e) illustrates the orbiting PVR–SVR pair rebounding away from the plate and interestingly, the PVR has stirred-up notable secondary vorticity on the plate’s edge, that was not entrained into the SVR. Both the PVR’s core size and its strength appear to have diminished, as shown by the vorticity contours. In figure 7(f) the PVR–SVR pair rebound persists, with some loss in strength and core diffusion evident within the SVR. At the final non-dimensional time mark (figure 7 g), the interaction shows limited progression compared with the preceding regimes, lacking a second vortex stretching phase.

In summary, both the flow visualisation and vorticity contours indicate nuanced finite-edge effects for plates in the quasi-infinite size category, with more pronounced variations noted in the finite size category. Within the finite plate size domain, the mechanism underlying vortex–wall collisions and subsequent rebounds displays distinctive characteristics. The proximity of the plate edge to the incoming PVR leads to the formation of an edge-generated vortex. as the PVR approaches the edge before the boundary layer separates. The resultant vortex pair follows a rebounding trajectory, although it initially extends beyond the plate edge. The pair of cores exhibit slower orbital movement compared with those observed in the infinite and quasi-infinite size categories. Notably, in this regime, for the same non-dimensional time of observation, there was no second rebound. Further quantitative analysis is required to thoroughly comprehend the mechanics of these interactions.

3.2.1. Infinite plate regime

For the infinite plate regime, as illustrated in figure 8(a), the primary circulation initially increases from zero as the PVR enters the field of view. At $t^* = 0$ , the PVR has a nominal normalised circulation of 1, as defined in § 2.3.1. During the FT phase, the circulation remains nearly constant since its only decay mechanism is due to viscous diffusion, which occurs on a much longer time scale. Upon contacting the wall, the interaction transitions from FT to VSI, and, consistent with Chu et al. (Reference Chu, Wang and Hsieh1993), the PVR’s strength decreases to $\approx 60\,\%$ of its initial strength due to cross-core cancellation between PVR core vorticity and opposite-sign boundary-layer vorticity. The core’s area shrinks to conserve the volume of the growing toroid, leading to heightened vorticity (due to conservation of angular momentum), and thus, leads to greater cross-core cancellation. This VSI phase spans approximately from $1 \lessapprox t^* \lessapprox 4$ . Next, the adverse pressure gradient induced by the PVR causes the boundary layer to separate, initiating the VRI phase ( $4 \lessapprox t^* \lessapprox 8$ ), where the circulation decay relaxes. During VRI, the separated boundary layer rolls up into the SVR and pushes the PVR away from the wall’s strong vorticity gradients. After the SVR forms and begins orbiting the PVR, the PVR experiences reduced dissipation from wall interaction and only feels mild cross-core cancellation. After this relaxation phase, another stretching phase occurs at around $t^* \approx 8$ , though with weaker wall interaction and significantly lower circulation decay. The VSII phase gives way to the VRII phase, marking the time point where tracking becomes difficult due to structural breakdown. Meanwhile, the SVR’s circulation evolves clearly, reaching peak strength transitioning from VSI to VRI, with a maximum of roughly $20\,\%$ of the initial PVR strength for ${\textit{Re}}_\varGamma = 1280$ , aligning with Chu et al. (Reference Chu, Wang and Hsieh1993). The SVR then slowly loses strength through diffusion. Outside of the influence of the wall, the dominant source of circulation decay is viscous diffusion, and weaker vortices are more prone to viscous core diffusion, having less rotational inertia to counteract viscous effects.

Figure 8. (a) Normalised circulation data as a function of non-dimensional time for the PVR and SVR for interactions with infinite-surface plates at ${\textit{Re}}_{\varGamma }=1280$ . The phases of the vortex ring collision are labelled with arrows. (b) Vortex-core trajectories for the PVR (circles) and SVR (triangles) of all plates in the infinite plate regime at ${\textit{Re}}_{\varGamma }=1280$ from $t^*=-1$ to $t^*=11$ for the PVR and from $t^*=2.5$ to $t^*=5$ for the SVR, where $x_i$ is the x value of the PVR at the first instant of tracking.

The trajectory data in figure 8(b) display vortex phases aligned with those in the circulation data. During the FT phase the primary core is observed to approach the plate directly. Upon nearing the wall, the PVR initiates vortex stretching (VSI), with the influence of the PVR’s image slowly increasing and causing stretching. The primary core then rebounds from the wall, maintaining lateral motion, achieves a peak height and then reapproaches, rebounding again due to the TVR’s influence. At this Reynolds number, coherent structures deteriorate after the second rebound, making feature tracking challenging. The SVR’s orbital trajectory, an arc around a relatively stationary PVR, reflects the weaker core of the SVR. The observed minor differences are attributed to case variability, not geometry, and are discussed in § 2.4. The $10\,D_n$ plate’s PVR does not stretch or rebound as high as the other plates, influencing the SVR’s trajectory and differentiating orbiting distances among plate sizes. However, a second rebound loop for the $10\,D_n$ plate coincides with the $4\,D_n$ plate. The trajectory of the $5\,D_n$ plate mirrors that of the $4\,D_n$ plate, but with a truncated, stationary second loop. Notably, a phase lag exists between vortex phases in trajectory versus circulation data: e.g. at $t^* \approx 4$ , the PVR peaks in the rebound while circulation indicates the onset of this phase. This potential phase lag, previously unnoticed in the literature, merits future study. Since the circulation tracked is just that of the swirling core, it is plausible that the flow properties of the core do not instantly respond to wall effects. A similar phase lag in a vortex ring–heated-wall collision study Jabbar & Naguib (Reference Jabbar and Naguib2019) suggested the inability of the flow properties to respond instantly to the wall proximity. However, since the Nusselt number at the wall, the property scrutinised in that paper, is not a vortex-core property like circulation, this analogy might not apply.

3.2.2. Quasi-infinite plate regime

The circulation data for the quasi-infinite class of plate sizes, shown in figure 9(a), helps quantify the significance of the edge effect’s impact. Overall, the vortex structures’ circulation dynamics in the quasi-infinite plate regime closely resembles that of infinite plates, with all five vortex phases represented on the graph with the same timing as in figure 8(a). There are three minor distinctions: firstly, after the VSI phase, the PVR decay is slightly greater, especially for the $2.5\,D_n$ finite plate. This higher decay links to the second difference, where the SVR strength is about $10\,\%$ higher, reflecting a trade-off against higher cross-core annihilation due to enhanced secondary structures arising partly from edge-generated vorticity, although this contribution is limited and does not significantly boost the SVR. The third difference lies in the final circulation after the VSII and VRII phases, where the PVR circulation is higher for both plate sizes compared with infinite plates, despite a larger decay during the initial phase. Indeed, the initial decay decreases the ratio of primary circulation to secondary circulation, increasing the orbital period of the PVR-SVR system and slowing the progression of the system relative to the infinite plate case. Furthermore, the PVR’s position at the plate edge during the final stretching phase leads to reduced plate interaction and less cross-core annihilation, thereby preserving more PVR strength in this regime.

Figure 9. (a) Normalised circulation data as a function of non-dimensional time for the PVR and SVR for interactions with quasi-infinite-surface plates at ${\textit{Re}}_{\varGamma }=1280$ . The phases of the vortex ring collision are labelled with arrows. (b) Vortex-core trajectories for the PVR (circles) and SVR (triangles) of all plates in the quasi-infinite plate regime at ${\textit{Re}}_{\varGamma }=1280$ from $t^*=-1$ to $t^*=11$ for the PVR and from $t^*=2.5$ to $t^*=5$ for the SVR, where $x_i$ is the x value of the PVR at the first instant of tracking.

The trajectory data for the quasi-infinite plate regime, as shown in figure 9(b), align closely with those of the infinite plate, except for a notable variation during the PVR’s second rebound event near the plate’s edge. In the circulation data, this position was associated with reduced strength decay. The trajectory plot indicates that the second rebound loops exhibit additional edge effects on the vortex ring–wall collision. The $3\,D_n$ plate shows a trajectory that leads the PVR to a larger height, with the apex appearing in the second rebound rather than the first. For the $2.5\,D_n$ plate, where the PVR interacts with the edge sooner, the second rebound is vertically upward. This illustrates that the edge-generated vorticity influences the PVR differently from the surface boundary-layer-separated vorticity. This distinction is subtle as it emerges late in the collision when the PVR is significantly weakened, explaining the regime’s ‘quasi-infinite’ classification. The primary phases of the vortex ring–wall collision resemble those in the infinite case. Furthermore, when the PVR eventually reaches the edge, it is too weak to respond in a manner that has a significant impact on the flow field.

3.2.3. Finite plate regime

In the finite plate regime, the vortex ring’s interaction with a wall significantly differs from that in the infinite plate regime (figure 10 a). Initially, all primary circulations stabilise near their nominal values as the PVR becomes visible during the FT phase. However, upon entering the VSI phase, primary circulation decay rates vary; smaller plates show more rapid decay. The SVR begins forming early in the VSI phase compared with the infinite plate case where the SVR originates from the wall boundary layer once the vortex rebounds. In quasi-infinite plates, some vorticity rolls up over the plate edge but it is not strong enough to be tracked, delaying SVR formation until the boundary layer entrains the edge-generated vorticity during the rebound phase. As such, unique to the finite plate regime, a strong edge-generated SVR appears during the VSI phase. Over time, the finite plate regime SVR outgrows the infinite regime’s SVR, doubling in strength. This robust secondary vorticity disrupts the PVR through cross-core cancellation, which weakens significantly and stretches along the plate edge. For the $1.5\,D_n$ plate, the PVR strength diminishes by nearly half by the rebound phase compared with the infinite plate case. The SVR strength also decays quickly due to proximity to opposite-sign vorticity cores. The PVR and edge-generated SVR interaction weakens both as they collide at the plate edge. For the $1.5\,D_n$ plate, the rapid circulation decay is further explained by trajectory dynamics. Upon reaching the VRI phase, the interaction diverges further from the infinite plate regime behaviour. In the $1.5\,D_n$ case, both the PVR and SVR gradually weaken over time due to viscous core diffusion, without secondary wall collisions, as revealed by trajectory analysis. For the $2\,D_n$ plate, the PVR circulation appears to decay faster towards the very end of the tracking time, suggesting another plate collision and the onset of a VSII phase, confirmed by vorticity contours indicating impending PVR re-impact with the plate edge and subsequent core annihilation. Trajectory data will elaborate on this return. The absence of a strong VSII phase and a VRII phase altogether has a clear footprint in the circulation data. While the PVR circulation of the infinite reference decays again around $t^*\,\approx \,9$ , the $2\,D_n$ plate’s PVR circulation strength maintains a near-steady evolution, surpassing that of the infinite reference PVR. While the $1.5\,D_n$ interaction PVR maintains a decay rate similar to the $2\,D_n$ plate’s PVR, the extreme initial strength decay of the $1.5\,D_n$ interaction prevents that PVR circulation from overtaking that of the infinite reference in the same amount of time.

Figure 10. (a) Normalised circulation data as a function of non-dimensional time for the PVR and SVR for interactions with finite-surface plates at ${\textit{Re}}_{\varGamma }=1280$ . The phases of the vortex ring collision are labelled with arrows. (b) Vortex-core trajectories for the PVR (circles) and SVR (triangles) of all plates in the finite plate regime at ${\textit{Re}}_{\varGamma }=1280$ from $t^*=-1$ to $t^*=11$ for the PVR and from $t^*=0$ to $t^*=7$ for the SVR, where $x_i$ is the x value of the PVR at the first instant of tracking.

Trajectory maps of finite plate regime vortex cores (figure 10 a) highlight the contrasting vortex–wall interactions. For the $2\,D_n$ plate, the first deviation from the infinite plate interaction is observed as the PVR core approaches the plate more closely during the vortex stretching phase, intensifying secondary vorticity formation, thus augmenting the boundary-layer vorticity. This results in a pronounced circulation decay of the PVR (figure 10 a). In the finite plate regime, the PVR’s rebound loop exhibits greater amplitude, occurs further from the plate’s centre and has a different inclination from that of infinite and quasi-infinite plates. While the infinite plate’s boundary-layer separation arrests the horizontal momentum of the PVR, finite plates do not impede the horizontal motion of the PVR resulting in it spilling over the edge. At this point the PVR experiences a rebounding-like event along a curvilinear path driven by the dynamics of an unequal counter-rotating vortex pair, as discussed in Leweke, Le Dizès & Williamson (Reference Leweke, Le Dizès and Williamson2016). The SVR orbits similarly to the infinite and quasi-infinite plate regimes but appears below the plate’s surface at its edge. On the $1.5\,D_n$ plate, because the PVR has a larger toroidal diameter than the plate, the interaction results in a clipped trajectory that despite some similarity to the conventional rebound loop results in extreme deformation. Tracking data reveal more variation across trials, complicating continuous trajectory assessment. However, the overall trajectory dynamics for small plate sizes remains clear: the PVR impinges on the plate closer and stretches along and over the edge of the plate, while the SVR forms below and around the plate before encircling its PVR counterpart in a notably different rebound behaviour, with the vortex system of the smallest plate interaction failing to re-engage with the surface of the plate.

Overall, the circulation tracking data revealed quite a divergent behaviour for the finite plate regime. First, the smaller plate interactions yielded higher strength SVRs that appeared earlier in time because the nature of their formation is edge generation rather than flat plate boundary-layer separation and roll-up. Second, the PVR was seen to decay significantly more upon stretching along the plate. However, the subsequent vortex–wall interaction phases played out more slowly for the $2\,D_n$ plate, not stretching along the plate again until at least $t^*\,\approx \,10$ . There is no evidence that the $1.5\,D_n$ plate re-interacts with the plate. These findings are corroborated by the trajectory tracking. The finite regime PVRs can be seen to impinge more closely upon the surface of the plate, leading to increased vorticity gradients and enhanced cross-core cancellation of the PVR. The SVRs are tracked as coherent structures first appearing underneath the surface of the plate at the edge. The orbits play out more slowly in the finite plate interactions, as dictated by the decreased disparity between the circulation strengths of the primary and secondary cores.

3.3.1. Vortex-core velocity

The motion of the vortex core is expressed in a plate-based reference frame (figure 3), where the $x$ direction is wall tangential and the $y$ direction wall normal. This convention, while equivalent to the global coordinate system, facilitates a more direct physical interpretation of the vortex–wall interaction.

For the infinite plate regime, figure 11(a) shows that the total velocity of the PVR decreases sharply during the VSI phase, demonstrating the wall’s role as a momentum sink and flow obstruction (New et al. Reference New, Xu and Shi2024). By the onset of boundary-layer separation, the PVR’s velocity has substantially reduced, indicating near arrest along its image-induced trajectory. The wall-tangential component $U_T$ (figure 11 d) reveals two clear rebound events. The first peak at $t^* \approx 1.2$ corresponds to vortex stretching, while the return of $U_T$ to zero at $t^* \approx 4$ marks the onset of the first rebound (VRI). A weaker secondary stretching event (VSII) occurs near $t^* \approx 6$ , consistent with circulation decay. The wall-normal velocity $U_N$ (figure 11 g) isolates the rebound behaviour: the vertical velocity decays to zero at $t^* \approx 1$ , reverses sign as the vortex rebounds and then decays again after the second rebound ( $t^* \gt 6$ ), reflecting the progressive loss of coherent momentum. The $10\,D_n$ plate completes this sequence slightly faster due to a weaker induced image system.

Figure 11. Non-dimensionalised total (a–c), wall-tangential (d–f) and wall-normal (g–i) core velocities, denoted by $|\hat {U}|$ , $U_T$ and $U_N$ , respectively, plotted as functions of non-dimensional time for the PVR and SVR at ${\textit{Re}}_{\varGamma }=1280$ . Results are shown for infinite plates (a,d,g), quasi-infinite plates (b,e,h) and finite plates (c,f,i).

In the quasi-infinite regime, trends are broadly similar but with distinct modifications near the plate edge. As shown in figure 11(d,e,h), the total velocity of the $2.5\,D_n$ plate decays more slowly during the VSI phase ( $t^* \gt 2$ ), as the PVR transitions from wall to edge separation. The reduced confinement allows prolonged stretching and sustained higher velocities. This delayed decay introduces a phase lag in both $U_T$ and $U_N$ , postponing the VSII and VRII phases. The enhanced stretching explains the longer wall-tangential excursion of the $2.5\,D_n$ trajectory (figure 9 b). The $3\,D_n$ plate reaches its edge later, experiencing weaker interaction and a correspondingly smaller phase delay. Notably, the wall-normal rebound height in this regime can exceed that of the first rebound, a feature absent in the infinite-wall case.

Finite plate interactions (figure 11 c,f,i) exhibit the strongest departures from canonical behaviour. The PVR maintains higher total velocities for longer durations, consistent with reduced flow confinement, yet the eventual velocity decay confirms that even small plates act as significant obstructions. The wall-tangential component shows extended stretching (prolonged VSI phase), while the wall-normal component highlights asymmetric rebound dynamics dominated by edge-generated vorticity. The $1.5\,D_n$ plate displays only a single rebound (VRI) followed by steady descent, whereas the $2\,D_n$ plate exhibits a larger initial rebound loop before crossing the edge of the plate and decaying. These trends indicate that the edge interaction weakens the classical secondary rebound and redistributes the momentum into lateral motion along the edge rather than the normal ejection.

In summary, the vortex-core velocity analysis reinforces the regime classification proposed earlier. The wall-tangential and wall-normal components distinctly separate the phases of stretching and rebound, while the total velocity reveals the degree of momentum arrest imposed by the plate. Collectively, these metrics quantify how the finite-surface extent modifies the sequence and coherence of the vortex rebound dynamics. The finite-edge effect on vortex-core velocity is shown to decrease the arresting of momentum (figure 11 c), increase the stretching velocity (figure 11 f) and only reveals one rebound event (figure 11 i).

3.3.2. Vortex-core deformation and morphology

The evolution of the PVR core shape provides complementary insight into how finite-surface extent modifies near-wall strain and vorticity redistribution (figure 12). Here, the core eccentricity serves as a proxy for the degree of deformation: a perfectly circular core has an eccentricity of zero, while higher values correspond to increasing elongation and shear-induced distortion. Although this scalar measure cannot directly quantify local shear or three-dimensional stretching, it captures the integrated imprint of these effects on the coherent structure. Furthermore, core eccentricity is found to drive short-wave instabilities in vortices (Pierrehumbert Reference Pierrehumbert1986; Leweke & Williamson Reference Leweke and Williamson1998; Eloy & Dizès Reference Eloy and Dizès1999). The measure of eccentricity could therefore have implications on PVR stability through the surface collision.

Figure 12. Temporal evolution of PVR core eccentricity for (a) infinite, (b) quasi-infinite and (c) finite plate interactions at ${\textit{Re}}_{\varGamma }=1280$ . Eccentricity quantifies the deformation of the vortex core during the stretching and rebound phases.

For the infinite plate regime (figure 12 a), the PVR enters the field of view with an approximately constant eccentricity ( $-1 \leqslant t^* \lessapprox 0$ ), consistent with an elliptical cross-section typical of isolated rings (Shariff, Leonard & Ferziger Reference Shariff, Leonard and Ferziger1989). Upon nearing the wall ( $t^* \approx 0$ ) the PVR has maintained its initial eccentricity, indicating the FT phase of the vortex interaction. From ( $0 \leqslant t^* \lessapprox 2.5$ ), the eccentricity steadily increases as the PVR core elongates and deforms along the surface of the plate. The eccentricity peaks and remains steady as the interaction transitions to the VRI phase. Interestingly, the eccentricity is nearly constant at this peak as it is effectively pressed against the separated boundary layer and SVR. As the PVR develops through the VRI phase, the subsequent relaxation of the core shape ( $t^* \gtrsim 4$ ) marked by decreasing eccentricity reflects reduced wall influence and partial shape recovery before the onset of a weaker second stretching phase. Within this regime, variations across plate sizes are negligible, confirming that edge effects are absent.

In the quasi-infinite regime (figure 12 b), the eccentricity evolution closely parallels that of the infinite case but with subtly stronger deformation and delayed relaxation. The initial rise in eccentricity again marks the onset of the VSI phase, followed by a peak during the rebound as the PVR interacts with the rolled-up boundary layer and weak edge-generated vorticity. The enhanced SVR strength in this regime steepens local vorticity gradients, producing higher core elongation and greater cross-core annihilation, consistent with the stronger circulation decay observed in figure 9 a. The extended relaxation period for the $2.5\,D_n$ plate suggests a slower rebound progression, in agreement with the phase lag identified in the core velocity data (figure 11 e,f).

Finite-plate interactions (figure 12 c) exhibit markedly stronger and more persistent deformation. This stronger deformation can be reasonably explained by the indication that the PVR cores interact more closely to the surface of the plate as shown in figure 10(b). Both plate cases show eccentricities exceeding 0.9, signifying extreme core elongation and deformation upon the surface of the plate. For the $1.5\,D_n$ plate, the core fails to recover its shape, having undergone a more extreme clipping at the edge of the plate as indicated by its rapid circulation decay and trajectory (see figure 10). The $2\,D_n$ plate experiences partial recovery after the first rebound, but the phase lag introduced by a stronger SVR prevents complete relaxation within the observed duration. These high eccentricities reveal the more intense plate interaction experienced by plates of this size regime and are consistent with the attenuated rebound loops and accelerated PVR decay characteristics explored previously.

Overall, the evolution of PVR eccentricity delineates how finite-surface extent alters the balance between wall-induced compression, tangential stretching and rebound relaxation. While eccentricity alone does not quantify shear production, its temporal signature robustly reflects the strength and persistence of deformation, offering a concise kinematic measure of interaction intensity across regimes. Although this work does not explore the three-dimensional growth of such instabilities, it is worth noting that increased eccentricity of the finite plate regime would impact the stability of the PVR and perhaps play a role in the enhanced PVR decay revealed in figure 10(a).

3.3.3. Secondary vorticity separation

The evolution of the boundary-layer separation point provides a direct kinematic measure of where and when secondary vorticity detaches from the surface, offering a bridge between qualitative flow visualisations and quantitative circulation dynamics (figure 13). Although the separation behaviour is already implicit in the vorticity and velocity fields, tracking its position clarifies how finite-surface extent modifies the onset and persistence of secondary structures. Measurements are restricted to the first rebound phase, since subsequent interactions yield weak near-wall velocities that fall below measurement accuracy.

Figure 13. Temporal evolution of the boundary-layer separation point for (a) infinite, (b) quasi-infinite and (c) finite plate interactions at ${\textit{Re}}_{\varGamma }=1280$ . The separation location is expressed in the plate-based coordinate system and tracked through the initial rebound phase.

For the infinite plate regime (figure 13 a), the separation point initially forms on the upwash side of the impinging vortex around $t^* \approx 1$ , coinciding with the transition from free translation to vortex stretching. As the PVR elongates along the wall ( $0 \lessapprox t^* \lessapprox 2$ ), the separation location migrates radially outward, mirroring the expanding footprint of the induced wall jet. Once the PVR begins to rebound and the local streamwise velocity decreases, the separation point reaches a quasi-steady position that fluctuates weakly due to unsteady boundary-layer roll-up. This behaviour reproduces the canonical infinite-wall response, with distinct phases of unsteady growth followed by steady persistence.

In the quasi-infinite regime (figure 13 b), the timing of initial separation remains nearly identical to the infinite case, but the subsequent evolution differs markedly. The separation front continues to propagate outward until it reaches the edge of the plate, where the wall jet detaches laterally rather than normally. This transition from wall-normal to edge-directed separation defines the onset of mixed boundary layer and edge vorticity generation. The earlier arrival of the separation point at the edge corresponds to stronger induced flow and higher retention of the PVR circulation, consistent with the increased stretching velocities and rebound amplitudes documented in figure 11(b,e,h). Thus, even without large-scale changes in timing, the topology of separation shifts fundamentally as the plate extent decreases.

Finite plate interactions (figure 13 c) exhibit a qualitatively distinct separation pattern. Here, the detachment originates at the edge of the plate rather than on the surface, preceding the lateral expansion of the PVR. The $1.5\,D_n$ plate shows early and persistent edge separation that eventually disappears around $t^* \approx 4.8$ , when the weakened PVR core moves sufficiently away that the induced flow along the edge drops below the detection threshold. This disappearance marks the loss of near-wall influence, whether due to vortex weakening, geometric shielding or both, and coincides with the cessation of rebound activity seen in trajectory and circulation data (figure 10 a,b).

Overall, tracking the separation point quantitatively substantiates the regime classification proposed earlier. The infinite regime is characterised by classical wall-normal detachment; the quasi-infinite regime introduces a hybrid wall-to-edge transition; and the finite regime is dominated by purely edge-driven separation. This transition in separation topology underpins the observed differences in secondary vorticity strength and rebound dynamics across regimes, reinforcing that the dominant control parameter is the plate extent relative to the vortex ring diameter rather than Reynolds number or nominal circulation.

3.3.4. Reynolds number effects

As shown in previous sections, the finite-wall effects are primarily governed by the separation behaviour of the boundary layer and the corresponding generation of secondary vorticity. In this section we investigate whether Reynolds number alters these finite-edge effects in any significant way. Prior work indicates that vortex ring–wall collisions on flat plates evolve predictably with Reynolds number, where a higher ${\textit{Re}}_{\varGamma }$ enhances the induced secondary structures but also accelerates the decay of the PVR (Chu et al. Reference Chu, Wang and Hsieh1993; Harris & Williamson Reference Harris and Williamson2012; Mishra, Pumir & Ostilla-Mónico Reference Mishra, Pumir and Ostilla-Mónico2021). To evaluate whether similar trends occur for finite surfaces, we first assess whether the underlying boundary-layer separation mechanics, the driving mechanism for finite-edge effects, change within a broad range of ${\textit{Re}}_{\varGamma }\approx 600$ – $2800$ . We then perform a more quantitative analysis across a narrower window of ${\textit{Re}}_{\varGamma }\approx 1280$ – $1925$ to examine subtler variations, if any, that might manifest beyond the dominant geometric effects.

The $2.5\,D_n$ plate, representative of the quasi-infinite regime, demonstrates that the Reynolds number has a negligible effect on the fundamental separation mechanism of the boundary layer (figure 14). The first row (panels a,e,i,m) shows the approaching PVR, which enters the field of view with comparable strength and structure across all Reynolds numbers. As the vortex impacts the wall, boundary-layer fluid is drawn outward and shed over the edge of the plate (panels b,f,j,n). The critical phase occurs when the boundary layer erupts from the surface (panels c,g,k,o), forming the SVR. This eruption consistently initiates along the upwash side of the PVR, as described by Jabbar & Naguib (Reference Jabbar and Naguib2019), and the subsequent roll-up (panels d,h,l,p) proceeds in nearly identical fashion regardless of ${\textit{Re}}_{\varGamma }$ . Although higher-Reynolds-number rings exhibit slightly sharper gradients in the visualised dye, the timing, location and geometry of separation remain unchanged. Because the impinging vortex diameter and induced upwash region are independent of ${\textit{Re}}_{\varGamma }$ , the finite-edge interaction maintains the same structure and timing across the entire range examined. Hence, no regime shift arises as a result of Reynolds number variation.

Figure 14. Asynchronous flow visualisation data feature matched in an ad hoc matter (due to camera limitations) of the $2.5\,D_n$ plate for ${\textit{Re}}_{\varGamma }=$ 600 (a–d), 1200 (e–h), 1800 (i–l) and 2800 (m–p). The PVR is shown in green while the boundary layer and SVR are shown in orange.

A similar invariance is observed for the finite-plate regime, represented by the $2.0\,D_n$ plate (figure 15). As the PVR approaches the wall (panels a,e,i,m), induced motion along the surface generates secondary vorticity that rolls up along the edge (panels b,f,j,n). This vorticity subsequently wraps into a coherent SVR (panels c,g,k,o), which is responsible for the rebound motion beyond the plate’s edge (panels d,h,l,p). While subtle differences in dye intensity and the apparent sharpness of the rebound appear at higher ${\textit{Re}}_{\varGamma }$ , the global flow topology and sequence of events remain unchanged. There is no evidence of a regime transition or any modification in the relative timing of the VSI or VRI phases. Thus, for both quasi-infinite and finite plates, the flow visualisations confirm that the separation behaviour and finite-edge mechanism are essentially Reynolds number invariant over ${\textit{Re}}_{\varGamma }\approx 600$ – $2800$ .

Figure 15. Asynchronous flow visualisation data feature matched in an ad hoc matter (due to camera limitations) of the $2\,D_n$ plate for ${\textit{Re}}_{\varGamma }=$ 600 (a–d), 1200 (e–h), 1800 (i–l) and 2800 (m–p). The PVR is shown in green while the boundary layer and SVR are shown in orange.

Although the qualitative behaviour is consistent, a closer look at the quantitative metrics reveals predictable variations. For the large-plate regime ( $5\,D_n$ ), the circulation data (figure 16 a) show that higher ${\textit{Re}}_{\varGamma }$ increases the rate of PVR decay during the VSI phase because of the heightened vorticity gradients, as expected from prior studies (Chu et al. Reference Chu, Wang and Hsieh1993; Harris & Williamson Reference Harris and Williamson2012; Mishra et al. Reference Mishra, Pumir and Ostilla-Mónico2021). This results from stronger induced boundary layers at higher Reynolds numbers, which amplify vorticity gradients and enhance cross-core annihilation. The strengthened boundary layer also generates a more robust SVR, increasing its peak circulation by roughly $\Delta \varGamma \approx 0.05\,\varGamma _0$ . This stronger secondary vorticity is mirrored in the trajectories (figure 16 b), where higher- ${\textit{Re}}_{\varGamma }$ PVRs rebound slightly later and at greater height due to higher momentum and enhanced coupling between the primary and secondary vortices. However, the subsequent rebound loops become less coherent as diffusion accelerates, consistent with faster loss of vortex integrity. These effects are minor and do not alter the overall structure of the interaction.

Figure 16. (a) Normalised circulation data as a function of non-dimensional time for the PVR and SVR for interactions with the $5\,D_n$ plate at all ${\textit{Re}}_{\varGamma }$ . (b) Vortex-core trajectories for the PVR (circles) and SVR (triangles) of the $5\,D_n$ plate interaction at all ${\textit{Re}}_{\varGamma }$ from $t^*\approx 0$ to $t^*=11$ for the PVR and from $t^*=2.5$ to $t^*=5$ for the SVR, where $x_i$ is the x value of the PVR at the first instant of tracking.

In the quasi-infinite plate regime, the Reynolds number effects on circulation follow similar patterns (figure 17 a). Higher ${\textit{Re}}_{\varGamma }$ enhances SVR strength during the first rebound, but at the cost of faster PVR decay. The resulting trajectories (figure 17 b) reveal that higher- ${\textit{Re}}_{\varGamma }$ PVRs approach the plate more closely before rebound, and the second rebound phase becomes slightly delayed. This phase shift stems from differences in the circulation retained at the instant of edge interaction. At later times, weaker PVRs at higher ${\textit{Re}}_{\varGamma }$ generate a less coherent SVR, which limits the rebound height and leads to premature dissipation. Although subtle, these effects highlight the influence of Reynolds number on timing and coherence, but not on the fundamental regime classification. The observed phase lag is also consistent with increased stretching velocities seen in quasi-infinite plates, which delay the onset of the secondary rebound.

Figure 17. (a) Normalised circulation data as a function of non-dimensional time for the PVR and SVR for interactions with the $2.5\,D_n$ plate at all ${\textit{Re}}_{\varGamma }$ . (b) Vortex-core trajectories for the PVR (circles) and SVR (triangles) of the $2.5\,D_n$ plate interaction at all ${\textit{Re}}_{\varGamma }$ from $t^*\approx 0$ to $t^*=11$ for the PVR and from $t^*=2.5$ to $t^*=5$ for the SVR, where $x_i$ is the x value of the PVR at the first instant of tracking.

Figure 18. (a) Normalised circulation data as a function of non-dimensional time for the PVR and SVR for interactions with the $2\,D_n$ plate at all ${\textit{Re}}_{\varGamma }$ . (b) Vortex-core trajectories for the PVR (circles) and SVR (triangles) of the $2\,D_n$ plate interaction at all ${\textit{Re}}_{\varGamma }$ from $t^*\approx 0$ to $t^*=11$ for the PVR and from $t^*=0$ to $t^*=5$ for the SVR, where $x_i$ is the x value of the PVR at the first instant of tracking.

In the finite-plate regime (figure 18), the circulation decay and rebound behaviour again mirror those of the large-plate case. The boundary layer separates primarily from the edge even at the lowest ${\textit{Re}}_{\varGamma }$ , and its onset timing, SVR roll-up and rebound trajectory remain effectively unchanged. Although higher- ${\textit{Re}}_{\varGamma }$ interactions exhibit slightly faster initial stretching and decay, these variations are minor and within the uncertainty bounds of the measurement. Importantly, the finite-wall interactions show no emergence of new flow features, confirming that Reynolds number variations do not modify the underlying mechanism that differentiates finite and quasi-infinite regimes.

Overall, these results demonstrate that geometric extent, not Reynolds number, governs the interaction physics across the investigated range. The boundary-layer eruption, secondary vortex formation and rebound dynamics remain invariant to ${\textit{Re}}_{\varGamma }$ , with only minor quantitative differences attributable to known viscous-scaling trends. Thus, the proposed regime classification framework remains robust and independent of Reynolds number within the tested range. Future work may refine these findings by exploring transitional plate sizes between the quasi-infinite and finite regimes, where a slight shift in separation mechanism could occur during the VSI phase.