Vortex ring connection to a free surface

Abstract We perform numerical simulations of the interaction between a vortex ring and a free surface, and provide a new interpretation of the mechanism by which the vortex ring connects to the free surface. Large vorticity gradients at the free surface result in the diffusion of surface-tangential vorticity out of the fluid. This is accompanied by the diffusion of opposite-signed surface-normal vorticity away from the connection line, along the free surface, which results in the attachment of the vortex ring to the free surface. Compared to existing descriptions, this interpretation explains better how the solenoidal property that vortex lines do not end in the fluid is maintained. By including an interface vortex sheet at the free surface, the important property of vorticity conservation is maintained throughout the interaction. The upper part of the vortex ring simply diffuses out of the fluid, and into the interface vortex sheet, with the total circulation remaining constant.

Interface Fluid 2 Fluid 1 Figure 1.Flow configuration considered in this study.A vortex ring of circulation Γ 0 , radius R 0 and core radius a is positioned at depth H beneath the interface (or free surface), and approaches the interface at angle α.The computational domain is a box with side lengths L x , L y , and L z,1 (in fluid 1) and L z,2 (in fluid 2).For free-surface flows, the upper fluid is ignored, and the interface is replaced with the free-surface boundary.
methods used in this study, while in § 2.5, we present the mesh used for numerical computations.Finally, in § § 2.6 and 2.7, we present a validation study for the numerical methods used in this study.

Problem description
The flow configuration studied in this article is shown in figure 1, and is similar to the set-up used by Zhang et al. (1999).We consider a vortex ring with initial circulation Γ 0 , ring radius R 0 and core radius a, positioned at an initial depth H beneath a free surface.The vortex ring is inclined, and approaches the free surface at an angle α.While the main focus of this article is on free-surface flows, we also consider the interaction between a vortex ring and a fluid-fluid interface.For the two-fluid case, we must consider the fluid domains below (fluid 1) and above (fluid 2) the interface, while we consider only the lower fluid (fluid 1) in the free-surface case.The free surface is an approximation for the fluid-fluid interface, in the limit that density and viscosity of the upper fluid approach zero.
Assuming incompressible flow, the continuity and momentum equations in each fluid are written as where subscript i = 1, 2 indicates quantities defined in fluids 1 and 2, respectively.Density is denoted by ρ, and the dynamic and kinematic viscosities are denoted μ and ν, respectively.Finally, g is the acceleration due to gravity.Following Zhang et al. (1999), flow parameters are non-dimensionalised by the initial circulation (Γ 0 ) and radius (R 0 ) of the vortex ring, and the density of the lower fluid (ρ 1 ).The following dimensionless parameters are then used to characterise this flow: the Reynolds number Re = Γ 0 /ν 1 , the Froude number Fr = Γ 0 /g 1/2 R We = R 0 σ/(ρ 1 Γ 2 0 ), the initial depth to radius ratio H/R 0 , and the initial vortex core to ring diameter ratio, a/R 0 .The effects of surface tension, σ , are not considered in this article, thus the Weber number is set to We = 0.For two-fluid flows, the ratios of density (ρ 1 /ρ 2 ) and dynamic viscosity (μ 1 /μ 2 ) across the interface must also be considered.
We generate the initial velocity field using the approach outlined in Zhang et al. (1999).First, we assume a Gaussian profile for the initial vorticity distribution in the vortex core, where ω axial is the vorticity component aligned with the vortex core axis, and r is the distance from the vortex axis.Then a streamfunction is obtained by solving the Poisson equation with boundary conditions Ψ x = Ψ y = ∂Ψ z /∂z = 0 on the free surface, and Ψ = 0 for all remaining boundaries.Then the initial velocity field is obtained from the streamfunction: (2.5) Note that the Gaussian profile is not a steady-state solution, and the ring undergoes an adjustment to a more stable profile at the beginning of the simulation (Zhang et al. 1999).

Interfacial boundary conditions
The following boundary conditions are used for a fluid-fluid interface.First is the continuity of velocity, u 1 = u 2 , (2.6) which is due to both the no-slip and no-penetration conditions.The remaining boundary conditions are due to the balance of tangential and normal stresses on the interface (Tuković & Jasak 2012): (2.7) (2.8) In (2.7) and (2.8), ŝ is the unit normal vector to the surface, ∇ is the surface gradient operator, and κ = −∇ • ŝ is the mean curvature of the interface.Also, u = u − (u • ŝ)ŝ is the surface-parallel velocity.For a free surface, the upper fluid exerts no stresses on the lower fluid, apart from a constant pressure (Lundgren & Koumoutsakos 1999).Then the tangential and normal shear-stress balances in (2.7) and (2.8) reduce to the boundary conditions (2.9) (2.10) they allow vorticity gradients to be computed using grid points that lie on the interface.The primary numerical method used in this study is a moving-mesh interface tracking scheme, interTrackFoam (Tuković & Jasak 2012), from the open-source software package, foam-extend 4.1 (a fork of the OpenFOAM software).The interTrackFoam solver uses an arbitrary Lagrangian-Eulerian (ALE) scheme to track motion of the interface, and can be applied to both single-fluid (free surface) or two-fluid (interfacial) flows.The interTrackFoam solver is described in Tuković & Jasak (2012), and was validated against several test cases in that article.It has also been used (sometimes in a modified form) to study the motion of free-rising bubbles (Pesci et al. 2018;Charin et al. 2019), Taylor bubbles (Marschall et al. 2014) and transient capillary rise (Gründing et al. 2020).It has been validated against other numerical methods (Gründing et al. 2020;Marschall et al. 2014), as well as experimental measurements (Marschall et al. 2014), and good agreement has been obtained.
The interTrackFoam solver uses a finite volume-method, where the incompressible continuity and momentum equations are integrated across a set of control volumes, where V is the control volume, and n is the outward-oriented unit vector to the control-volume boundary.Also, u is the fluid velocity, ν is the dynamic viscosity, and pi is the static pressure minus the hydrostatic pressure (p i = p i − ρ i gz).Finally, v is the velocity of the control-volume boundary, which may be different from the fluid velocity.
Once again, subscript i = 1, 2 indicates quantities defined in the lower and upper fluids, respectively.The advantage of this scheme is that the computational mesh is allowed to move, with velocity v, so that grid points track the interface as it moves.This requires that on the free surface, the surface-normal velocity of the mesh matches the fluid velocity: (2.13) To ensure that the internal mesh remains smooth, the velocity of internal grid points is obtained from the velocity on the boundary using a mesh-motion solver (Jasak & Tukovic 2006).Since the numerical mesh moves with the interface, the boundary conditions (2.6)-(2.10)can be applied directly to the interface or free surface.Details on how these boundary conditions are implemented in the interTrackFoam solver can be found in Tuković & Jasak (2012).
In the interTrackFoam solver, (2.11) and (2.12) are discretised using the finite-volume method.Spatial derivatives are converted to boundary fluxes on control-volume faces using Gauss' theorem, and linear interpolation was used to construct the fluxes on cell faces.Time is divided into discrete time steps, and the temporal derivative is discretised using a second-order backwards time scheme, where n is the current time step.Finally, the discretised equations are solved using an iterative method, based on the PISO algorithm (Issa 1986).For further details on how this method is implemented, refer to Tuković & Jasak (2012).

Volume-of-fluid method
To validate the results of the interTrackFoam solver, additional simulations are performed using the volume-of-fluid (VOF) method (Hirt & Nichols 1981).Two different implementations of the VOF method were used: the interFoam solver from foam-extend 4.1, and the VOF method implemented in the commercial software package ANSYS FLUENT.
Instead of tracking the interface directly, the VOF method introduces a volume fraction F that indicates the fraction of each fluid contained in a given control volume.For two-fluid flows, F = 1 for cells that contain only the lower fluid (fluid 1), while F = 0 for cells containing only the upper fluid (fluid 2).For cells that contain the interface between two fluids, the volume fraction lies in the interval 0 < F < 1.The interface is therefore captured indirectly, via the evolution of the volume fraction field.
For incompressible flows, with no mass transfer between phases, the volume fraction satisfies the conservation law (Hirt & Nichols 1981) (2.15) However, care must be taken to ensure that the interface remains sharp when solving this equation (Hirt & Nichols 1981).In the interFoam solver, this is achieved by modifying the advective fluxes near the interface (Deshpande, Anumolu & Trujillo 2012), by incorporating an interface compression term.In ANSYS FLUENT, the geometric reconstruction scheme was used (ANSYS 2019) to maintain a sharp interface.
In the VOF method, a single set of momentum equations is solved for both fluids, across the entire computational domain (Deshpande et al. 2012): (2.16) In this equation, fluid properties, such as density and viscosity, are taken to be average values based on the volume fraction: Equations (2.15) and (2.16) are discretised using the finite-volume approach.For details of the numerical implementation, refer to Deshpande et al. (2012) for the implementation in interFoam, and to ANSYS (2019) for the implementation in ANSYS FLUENT.
We remark that typically, vorticity and vorticity gradients exhibit a discontinuity across the interface (Terrington et al. 2022).However, under the VOF method, this discontinuity is spread over a region near the interface where 0 < F < 1, which leads to difficulty in determining the boundary vorticity flux at the interface.Therefore, the interface tracking interTrackFoam solver is preferred for this study, and the VOF method is used mainly to validate the results of the interTrackFoam solver.For free-surface simulations, the upper boundary of the computational domain is located on the free surface, while for two-fluid simulations, the computational domain extends to elevation L z,2 above the interface.Unless otherwise stated, the computational domain used in this study has side lengths When using the interTrackFoam solver, the interface (or free surface) forms part of the computational domain boundary, and the numerical mesh will deform with the surface.For two-fluid simulations, separate meshes are used for the upper and lower fluids, with the interface forming the boundary between these meshes.For free-surface flows, only a single mesh is used, with a boundary on the free surface.When the VOF method is used, a single stationary grid is used for the entire computational domain, and the interface is not part of the computational domain boundary.
The computational domain was meshed using a block-structured approach, as illustrated in figure 2. A fine grid was used in the region near the vortex ring, bounded by −2 x 2, −3 y 3 and −3.5 z 0, with a lower grid resolution used outside this region.Additionally, the cell size in the vertical (z) direction was reduced near the interface, to capture accurately vorticity gradients near the surface.For simulations run in foam-extend 4.1, meshes were generated using the blockMesh application, while for simulations run in ANSYS FLUENT, the mesh was generated using ICEM CFD.
Simulations were performed in a Galilean reference frame, translating with a velocity approximately equal to the vortex ring velocity, to ensure that the vortex ring remained within the fine-mesh region.This was achieved by applying a constant velocity inlet to the upstream boundary (at x = +L x ), with inlet velocity U ∞ /(Γ 0 /R 0 ) = (−0.1,0, 0).Outlet boundary conditions were applied to the downstream boundary (at x = −L x ), while the remaining boundaries, aside from the interface, were set to free-slip walls.

Validation study
A grid resolution study was performed using the meshes outlined in table 1, where mesh 1 has the coarsest resolution, and mesh 3 has the finest resolution.Simulations were run using interTrackFoam, for a single-fluid free-surface flow.Physical parameters were chosen to match Case 2 from Zhang et al. (1999): Re = 1570, Fr = 0.47, α = 80 • , H/R 0 = 1.57and a/R 0 = 0.35.  1. Numerical grids used in the convergence study, where N is the total number of cells, while x, y and z indicate the cell spacing in the fine-mesh region (figure 2).Spacing in the z direction is provided at both z = −3.5 ( z (3.5) ), and at the interface ( z (0) ).In figure 3, we plot the maximum spanwise vorticity (ω y ) in the symmetry plane (y = 0), normalised by the initial peak vorticity (ω 0 ), for each mesh resolution.The maximum vorticity in the symmetry plane is nearly identical between meshes 2 and 3, demonstrating that adequate resolution has been obtained in this region.The maximum vertical vorticity (ω z ) in the free surface is also provided in figure 3. Noting convergence of the predictions between resolutions and that the differences in the maximum vorticity between meshes 2 and 3 are relatively small, the grid resolution is assumed satisfactory.The finest mesh (mesh 3) is used for all subsequent simulations in this study.
The grid resolution study was performed using identical physical parameters to Case 2 from Zhang et al. (1999), in order to validate our results against their simulations.However, as shown in figure 4, the profiles of maximum vorticity magnitude, in both the symmetry plane and the free surface, are vastly different.In our simulations, the maximum vorticity in the symmetry plane increases to over 2.8 times the initial peak vorticity, as the vortex ring interacts with the free surface, while Zhang et al. (1999) report a monotonic decrease in the vorticity magnitude.In both cases, the maximum value of vertical vorticity in the free surface is of a magnitude similar to the maximum vorticity in the symmetry plane,  Zhang et al. (1999), showing the maximum spanwise vorticity in the symmetry plane (solid lines), and the maximum vertical vorticity (ω z ) in the free surface (dashed lines).The physical parameters are Re = 1570, Fr = 0.47, α = 80 • , H/R 0 = 1.57and a/R 0 = 0.35. Figure 5.Comparison between the current numerical results (interTrackFoam), the numerical simulations of Zhang et al. (1999), and the experimental measurements of Gharib et al. (1994), showing the maximum spanwise vorticity in the symmetry plane (solid lines), and vertical vorticity in the free surface (dashed lines).The physical parameters are Re = 1150, Fr = 0.19, α = 83 • , H/R 0 = 2 and a/R 0 = 0.3.Results from the interTrackFoam solver when averaged over the interrogation window from the experimental particle image velocimetry (PIV) measurements are also presented.
and is therefore much higher in our simulations (≈ 2.5ω 0 ), compared to  Zhang et al. (1999) validate their numerical method against the experimental measurements of Gharib et al. (1994).We performed an additional validation simulation using physical parameters matching the experimental conditions: Re = 1150, Fr = 0.19, α = 83 • , H/R 0 = 2, and a/R 0 = 0.3, and the results are presented in figure 5.The Zhang et al. simulations are in reasonable agreement with the experimental measurements of Gharib et al., whereas our simulations are remarkably different.Once again, we find a sharp increase in the maximum spanwise vorticity during the vortex reconnection process, while Zhang et al. (1999) and Gharib et al. (1994) report no increase in the maximum spanwise vorticity.
Since numerical simulations performed using the interTrackFoam solver disagree with existing numerical and experimental results, we performed simulations using the VOF method, for the same physical parameters.The VOF method requires two fluids, and the density and viscosity ratios were set to ρ 1 /ρ 2 = 1000 and μ 1 /μ 2 = 100, so the influence of the upper fluid is small.A comparison between the different numerical methods used (interTrackFoam, for both free-surface and two-fluid flows, and the VOF method, using both interFoam, and ANSYS FLUENT) is shown in figure 6, for physical parameters matching Case 2 from Zhang et al. (1999).The maximum vorticities in the symmetry plane are nearly identical for each method, providing support for the interTrackFoam solver.
Moreover, the sharp rise in spanwise vorticity seen in our simulations is also observed in symmetrical vortex ring connections (Kida, Takaoka & Hussain 1991), which are equivalent to a perfectly flat (zero Froude number) free surface.To simulate a flat free surface, simulations were performed using the pimpleFoam solver in foam-extend 4.1, with the free surface replaced by a free-slip wall.In figure 7(a), we compare our results to the numerical simulations of Kida et al. (1991) and Balakrishnan (2013), at Re = 1153, α = 90 • , a/R 0 = 0.4 and H/R 0 = 1.858, in a periodic (in the x and y directions) box with side lengths L x /R 0 = 6.4,L y /R 0 = 6.4 and L z /R 0 = 4.5.Results are in good agreement, and the slight disagreement can be attributed to the orientation of our periodic box (which differs from theirs by 45 • ).Importantly, we observe a sharp rise in the maximum spanwise vorticity in the initial stages of the connection process.Having validated our flat-surface method against existing results, in figure 7(b) we compare a flat-surface simulation obtained using pimpleFoam to the free-surface simulation obtained using interTrackFoam, for Fr = 0.47, Re = 1570, α = 80 • , H/R 0 = 1.57and a/R 0 = 0.35.The maximum vorticities in the symmetry plane and the free surface are nearly identical for both cases.This is reasonable, given that surface deformations are small at Fr = 0.47.In § 6.2, we perform simulations at Fr = 1, where the surface deformation is not negligible.We find good agreement between the moving-mesh solver (interTrackFoam) and the VOF method (interFoam), which demonstrates that both methods are capable of capturing accurately the free-surface deformation.Therefore, we can assume that these methods are also reliable at Fr = 0.47 (figure 6), where the evolution of the maximum spanwise vorticity is not significantly different from that at a flat shear-free surface.Importantly, the sharp rise in spanwise vorticity magnitude seen in our free-surface simulations is also observed for flat shear-free surfaces, and is therefore a physically reasonable result at low Froude numbers.
To explain the discrepancy between our results and previous experimental and numerical measurements, we remark that neither the experimental measurements of Gharib et al. (1994) nor the numerical simulations of Zhang et al. (1999) possess sufficient resolution to capture the peak vorticity magnitude observed in our simulations.Figure 8 displays contours of spanwise vorticity in the symmetry plane, which shows that the region responsible for large vorticity magnitudes is a thin layer near the free surface.Overlaid on this figure is a square outline showing the approximate size of the 32 × 32 pixel interrogation window used to compute each velocity vector in Gharib & Weigand (1996), which we assume is similar to the interrogation window used in Gharib et al. (1994).It is clear that the PIV measurements are not able to resolve the thin layer of elevated vorticity seen in our simulations, and the corresponding PIV vorticity field would be quite smeared.In figure 5, we present the maximum spanwise vorticity when averaged over the PIV interrogation window, which compares quite favourably with the existing Overlaid is a square outline showing the approximate size of the 32 × 32 pixel windows used to obtain each velocity vector for PIV by Gharib & Weigand (1996).The horizontal lines show the vertical grid spacing used by Zhang et al. (1999) at their highest resolution.The location of maximum vorticity magnitude is indicated by an asterisk.
experimental and numerical results.Therefore, our simulation results are consistent with the experimental measurements, once the vorticity is downsampled to match the resolution of the PIV measurements.Zhang et al. (1999) use central finite differencing in the vertical direction, with their finest grid size (128 points distributed uniformly in the vertical direction) shown by horizontal lines in figure 8.For this grid spacing, the location of maximum spanwise vorticity predicted by our simulations is less than two cell heights from the free surface.This is consistent with figures 11 and 13 of Zhang et al. (1999), where the maximum spanwise vorticity occurs only two cell heights below the free surface.Moreover, this corresponds to a single cell height on their 64 × 64 × 64 grid, and it is quite surprising that Zhang et al. (1999) find such good agreement between the maximum vorticity magnitude on each grid.Given that the maximum value of vorticity magnitude occurs only two cell heights beneath the free surface on their finest grid, it is unlikely that Zhang et al. (1999) can resolve accurately the vorticity magnitude near the free surface using central finite differencing.Our finest mesh, on the other hand, has an order of magnitude lower cell height near the free surface, and is therefore sufficient to resolve accurately the maximum vorticity near the free surface.

Effect of initial vortex ring depth
The numerical study of Zhang et al. (1999) used initial vortex ring depth H/R 0 = 1.57, which means that the upper core of the vortex ring is located initially at depth ≈ 0.5R 0 , which is similar to the core radius, a/R 0 = 0.35.This means the upper part of the vortex ring is located initially very close to the free surface, and may not be representative of a vortex ring that has approached the free surface from a distance.
To investigate the effect of the initial vortex ring depth, a set of numerical simulations with different initial depth ratios was performed.All other physical parameters were identical to Case 2 from Zhang et al. (1999).The time history of the magnitude of spanwise vorticity in the symmetry plane is plotted in figure 9(a), for each depth ratio.At the beginning of the simulation, the vorticity distribution in the vortex ring core adjusts from the initial Gaussian profile to a more stable distribution (Zhang et al. 1999).This is accompanied by a small increase in the maximum vorticity magnitude over duration approximately t/(R 2 0 /Γ 0 ) = 2. Following the initial adjustment of the vortex core, the maximum vorticity in the vortex ring decreases gradually, due to viscous diffusion.For depth ratios greater than H/R 0 = 2, the maximum vorticities in the upper and lower portions of the vortex ring decrease at the same rate, which implies that the vortex ring does not experience significant influence from the free surface.However, at approximately t/(R 2 0 /Γ 0 ) = −20, the vorticity in the upper vortex core begins to increase, while the vorticity in the lower core continues to decrease.This indicates that the upper part of the vortex ring has begun to experience significant influence from the free surface.
In figure 9(b), we plot the vertical position of the maximum vorticity magnitude in the symmetry plane, for each depth ratio.At the time when the upper core begins to experience significant influence from the free surface (t/(R 2 0 Γ 0 ) ≈ −20), the upper vortex core is situated at depth approximately one ring radius beneath the free surface (z c /R 0 = −1).This suggests that the blockage layer thickness -where the vortex ring experiences significant influence from the free surface -is approximately equal to the ring radius.For vortex ring depths less than or equal to H/R 0 = 2, the upper part of the vortex ring is located initially inside the blockage layer.To ensure that the vortex ring begins outside the blockage layer -and therefore our simulations capture the dynamics of the vortex ring as it passes through the entire blockage layer -depth ratio H/R 0 = 2.5 is used for the remaining simulations in this study.

Overview of vortex connection
In this section, we provide an overview of the interaction between a vortex ring and a free surface, showing the main features of the reconnection process.We focus on a single  simulation, with parameters Re = 1570, Fr = 0.47, α = 80 • , H/R 0 = 2.5, a/R 0 = 0.35 and We = 0.The effect of changes to these parameters is considered later, in § 6.

Free-surface interaction
Figure 10 presents isosurfaces of vorticity magnitude at several time instances, revealing the evolution of the vortex ring as it interacts with the free surface.(A transient animation of this figure is provided as movie 1 in the supplementary material available at https:// doi.org/10.1017/jfm.2022.529.)Initially, the vortex ring approaches the free surface under its own self-induced velocity (figure 10a).Then the upper portion of the vortex ring is flattened and stretched as it impinges upon the surface (figure 10b).This leads to large vorticity gradients at the free surface, so that vorticity from the upper portion of the vortex ring is diffused out of the fluid by viscous forces.As a result, the vortex ring becomes broken open near the free surface, leaving the open ends of the vortex ring attached to the free surface (figures 10c,d).
While the isosurfaces of vorticity magnitude (figure 10) suggest that vortex connection has occurred, a change in topology of vorticity isosurfaces does not necessarily imply a change in the topology of vortex lines (Kida & Takaoka 1994).In figure 11, vortex lines are plotted at two different times, before and after connection to the free surface has occurred.(An animation of this figure is provided in supplementary movie 2.) Initially, all vortex lines are closed loops (figure 11a,b); however, during the interaction, vortex lines are broken open near the free surface and become attached to the free surface (figure 11c,d).Isosurfaces of vorticity magnitude are also plotted in figure 11, demonstrating that, for the present flow, the geometries of vortex lines and vorticity isosurfaces are closely related.
A colour plot of the free-surface displacement, η, is also provided in figure 10.As the vortex ring approaches the free surface, the surface is elevated ahead of the vortex ring, with a depression formed behind the vortex ring (figures 10a,b).As the vortex ring connects to the free surface, the free-surface depression splits into two surface dimples, which are located where the vortex ring is attached to the surface (figures 10c,d).
The qualitative aspects of vortex reconnection observed here compare fairly well to Zhang et al. (1999); however, there are some key differences.First, in our results, the upper vortex core is flattened and stretched far more than in the Zhang et al. results.Second, in our simulations, some vorticity remains unattached to the free surface, which was not observed by Zhang et al. (1999).

Two-fluid flow
The free-surface boundary conditions (2.9) and (2.10) are often used as an approximation for the air-water interface, where, due to the low density and viscosity of air, motion in the upper fluid (air) has little influence on the dynamics of the lower fluid (water).In this subsection, we consider the interaction between a vortex ring and a fluid-fluid interface, with density ratio ρ 1 /ρ 2 = 1000 and viscosity ratio μ 1 /μ 2 = 100, which are comparable to a water-air interface.Motion in the lower fluid is nearly identical to the free-surface case; the main difference between the free surface and the fluid-fluid interface is the appearance of vorticity in the upper fluid.This is seen more clearly in figure 13, which plots vortex lines in both fluids, both before and after connection to the interface has occurred.(A transient animation of this figure is provided in supplementary movie 4.) Before the interaction, vortex lines in both fluids are closed loops, which lie entirely within a single fluid.After the interaction, vortex lines that attach to the interface are closed loops, passing through both fluids.Effectively, the upper part of the vortex ring has diffused into the upper fluid, and the vortex ring is contained in both fluids.

Interface vortex sheet
In several formulations of vorticity dynamics (Lundgren & Koumoutsakos 1999;Brøns et al. 2014;Terrington et al. 2020Terrington et al. , 2022)), a vortex sheet is included at the free surface, so that the total circulation is conserved.In three dimensions, Terrington et al. (2022) define the local density of vector circulation in the interface vortex sheet above a free surface as By including the interface circulation in this manner, the total circulation is conserved (Terrington et al. 2022).During the connection to the free surface, vorticity lost from the lower fluid is transferred into the interface vortex sheet ( § 4.2).The interface vortex sheet also satisfies a generalised solenoidal property (Terrington et al. 2022) where ∇ is the surface gradient operator (Wu 1995).Equation (3.2) shows that vortex lines do not simply end on the free surface, but instead continue as circulation in the interface vortex sheet.This relationship is illustrated in figure 14, which plots vortex lines in the interface vortex sheet, as well as colour contours of surface-normal vorticity on the free surface.Before the connection has occurred, the surface-normal vorticity is zero, and the interface vortex sheet is divergence-free (∇ • γ = 0), so that vortex lines in the interface vortex sheet are all closed loops (figure 14a).However, after connection to the free surface has occurred, vortex lines in the interface vortex sheet are generated from regions of positive surface-normal vorticity, and end in regions of negative surface-normal vorticity (figure 14b).
Regions of high surface-normal vorticity, which act as sources and sinks for the interface vortex sheet, indicate regions where the vortex ring is attached to the free surface.This is shown in figure 15  of circulation in the free-surface vortex sheet.In this sense, vortex lines do not end on the free surface, but continue as vortex lines in the interface vortex sheet.After the connection to the free surface, the vortex ring remains a closed loop, but with part of the vortex ring attached to the free surface.Essentially, the upper part of the vortex ring has diffused out of the fluid, and is now contained in the interface vortex sheet.
Comparing figures 15 and 13 reveals several similarities between the interface vortex sheet above a free surface, and the vorticity field above a fluid-fluid interface.Before the interaction, vortex lines in the upper fluid are closed loops, as are the vortex lines in the interface vortex sheet.After the interaction, vortex lines in the interface vortex sheet begin and end at points where the vortex ring attaches to the free surface, as do vortex lines in the upper fluid for the two-fluid case.In this sense, the free-surface vortex sheet can be interpreted as an approximation for the vorticity that would be found above an air-water interface.

The mechanism of vortex connection to a free surface
In this section, we consider the dynamical mechanisms that drive the vortex ring connection to a free surface.First, in § 4.1, we provide a summary of the necessary vorticity dynamics.In § 4.2, a control-volume analysis is used to investigate the transfer of vorticity between the fluid and the interface vortex sheet.In § 4.3, we use a control-surface analysis to describe the appearance of surface-normal vorticity in the free surface.Then, in § 4.4, we consider the dynamics of a vortex ring connection to a fluid-fluid interface.Finally, in § 4.5, we show that the analysis considered in this section leads to a novel interpretation of the vortex connection process, and compare to existing interpretations.

Vorticity dynamics
Taking the curl of the momentum equation yields a transport equation for vorticity (the Helmholtz equation).For an incompressible Newtonian fluid, this equation is expressed as If this expression is integrated across a control volume (V), then these dynamical processes can be interpreted as boundary fluxes (Terrington et al. 2022): where ∂V is the boundary surface of V, n is the outward-oriented unit normal to ∂V, and v b is the velocity of the control-volume boundary.From left to right, terms on the right-hand side of (4.2) represent the effects of advection, vortex stretching/tilting and viscous diffusion as boundary fluxes.Now, the form of the viscous boundary flux in (4.2) is not unique.Here, we have used the Lyman (1990) definition, σ = ν n × (∇ × ω), rather than the original definition proposed by Lighthill (1963), σ = −ν n • ∇ω.When integrated across a control volume, both definitions yield the same rate of change of vorticity, so either definition may be used in (4.2).Terrington, Hourigan & Thompson (2021) have demonstrated several advantages to using Lyman's definition.Of particular relevance to the present work, Lyman's definition was found to provide a more elegant interpretation of the connection between antiparallel vortex filaments in the fluid interior.In the present article, we also show that Lyman's definition also provides a more elegant interpretation of the mechanism behind the attachment of vortex filaments to the free surface during the connection process.
The shear-free boundary condition (2.9) can be written as a boundary condition for the surface-tangential vorticity on the free surface (Longuet-Higgins 1998; Peck & Sigurdson 1998;Terrington et al. 2022): where ω = ω − (ŝ • ω)ŝ is the surface-parallel vorticity on the free surface, ŝ is the unit normal to the interface, directed out of fluid 1, ∇ is the surface gradient operator (Wu 1995), and K = ∇ ŝ is the surface curvature tensor.The viscous flux of vorticity at the free surface results in the diffusion of vorticity either into or out of the fluid, in order to satisfy this boundary condition (Terrington et al. 2022).At low Froude numbers, the right-hand side of (4.3) is small, and the vorticity on the free surface is close to zero.However, at higher Froude numbers, distortion of the free surface leads to the appearance of secondary vorticity on the free surface (Gharib & Weigand 1996;Zhang et al. 1999).

Control-volume analysis
To examine the mechanisms leading to the loss of vorticity from the fluid, we consider the control volume depicted in figure 16.The outer boundary of V 1 is divided into two portions: I is the part that lies on the free surface, while ∂V 1 is the part in the fluid interior.The vector n is the unit normal to ∂V 1 , while ŝ is the unit normal to the free surface.
In this subsection, we consider the volume integral of vorticity which we have termed the 'vector circulation' (Terrington et al. 2021(Terrington et al. , 2022)).We also consider the total vector circulation contained in the interface vortex sheet, t/(R 0 2 /Γ 0 ) quantities are plotted in figure 17(a).The net spanwise vorticity in the fluid is zero initially, due to the symmetry of the vortex ring.However, Γ 1,y increases throughout the interaction, as negative vorticity from the upper part of the vortex ring diffuses out of the fluid.The loss of negative spanwise vorticity from the fluid is balanced by an approximately equal increase in negative circulation found in the vortex sheet, so that the total circulation (Γ 1,y + Γ γ,y ) remains approximately constant.Using (4.2), we construct the following expression for the rate of change of vorticity in The rate of change of circulation above the free surface is given by the expression (Terrington et al. 2022) dΓ γ dt = The first two terms are equal and opposite to the free-surface flux terms in (4.6), so that the effects of viscous diffusion and vortex stretching/tilting act to transfer circulation between the fluid interior and the interface vortex sheet.Any loss of negative circulation is balanced by an equal gain of negative circulation in the interface vortex sheet.Combining (4.7) and (4.6), we have which includes several boundary source terms that represent the generation of interface circulation by tangential pressure gradients (p 1 /ρ 1 ) or gravitational forces (φ g ), as well as the advection of circulation along the interface (u 1 • u 1 ).For the control volume used in this analysis, these boundary terms are small but non-zero, leading to small oscillations in the total circulation.However, these terms will approach zero if ∂I is positioned sufficiently far from the vortex ring, and the total circulation will be conserved.
The time histories of the viscous and vortex stretching/tilting fluxes, as well as the rate of change of spanwise vorticity, are plotted in figure 17(b).Initially, the increase in Γ 1,y (and decrease in Γ γ,y ) is due to the viscous flux of vorticity out of the fluid.During the initial stage of the connection process, negative spanwise vorticity from the upper part of the vortex ring is brought near the free surface.At low Froude numbers, (4.3) requires the tangential vorticity at the free surface to be small, leading to a large gradient of spanwise vorticity (∂ω y /∂z) on the free surface.This vorticity gradient drives the viscous flux of vorticity out of the fluid and into the free-surface vortex sheet.
While the viscous flux dominates the initial loss of negative circulation from the fluid, the vortex stretching term (F st ) becomes the dominant mechanism in the later portion of this interaction.An interpretation of the vortex stretching/tilting flux was presented in (Terrington et al. 2022).Eyink (2008) gives a simple proof for the expression where x is the position vector, which (assuming that ω • n = 0 on ∂V 1 ) shows that the total vorticity in V 1 is determined completely by the locations where vortex filaments are attached to the free surface.The vortex stretching flux (u(ω • ŝ)) can be interpreted as representing the advection of surface-normal vorticity in the free surface.This changes the locations where vortex filaments attach to the surface, therefore requiring a change to the total vorticity in the fluid.This is illustrated in figure 18, where a positive F st requires surface-attached vorticity to be carried away from the symmetry plane by advection.As a result, new spanwise vorticity must be generated by vortex stretching in the fluid interior, since vortex lines cannot end inside the fluid.Moreover, an equal quantity of opposite-signed circulation must be created in the interface vortex sheet, in order to satisfy the generalised solenoidal condition (3.2).

Control-surface analysis
The dynamical mechanisms leading to the connection of vorticity to the free surface can be understood by considering the behaviour of circulation in both the symmetry plane ).This term can be interpreted as representing the advection of vortex filaments in the free surface, which must correspond to the creation of spanwise vorticity by vortex stretching in the fluid interior.An equal and opposite quantity of circulation is also created in the interface vortex sheet, so that the total spanwise circulation is conserved.To quantify the reconnection process, we consider the circulation contained in the system of control surfaces illustrated in figure 20: S 1 is the symmetry plane; I 1 is the portion of the free surface where y < 0; and I 2 is the portion of the free surface where y > 0. The time history of circulation in each reference surface is plotted in figure 21(a), where Γ I 1 , Γ I 2 and Γ S 1 are the circulations contained in surfaces I 1 , I 2 and S 1 , respectively.As the upper part of the vortex ring diffuses out of the free surface, the circulation in t/(R 0 2 /Γ 0 ) S 1 increases (becomes more positive), and this is balanced by an equal gain of negative circulation in I 1 , and of positive circulation in I 2 .
The transfer of circulation between the symmetry plane and the free surface is related closely to the top topological changes to vortex lines observed in figure 11.Two kinds of vortex line topology are present in this flow: closed loops and open filaments.Loop-type vortex filaments pass through the symmetry plane twice, so the net spanwise circulation in the symmetry plane due to a closed vortex filament is zero.As the initial vortex ring contains only closed filaments, the initial circulations in S 1 , I 1 and I 2 are all zero.Open filaments, however, begin on I 2 , pass through S 1 once, and end on I 1 , providing an equal magnitude of positive circulation in I 2 , and negative circulation in S 1 and I 1 .The changes in circulation shown in figure 21 therefore describe the opening of vortex loops, and the attachment of open vortex lines to the free surface.
In figure 21 we also plot the interface circulation contained in the connection line (C): The loss of negative circulation from S 1 is also balanced by an equal gain of negative circulation in C, so that the total circulation Γ S 1 + Γ γ is constant.Effectively, the upper part of the vortex ring has been transferred into the interface vortex sheet, and vortex lines intersect the symmetry plane twice: once as vorticity in the fluid interior, and once as circulation in the interface vortex sheet.The rate of change of circulation in each surface is given by the fluxes of vorticity across the boundary of each surface.Assuming that the outer boundary curves (∂S 1 , ∂I 1 and ∂I 2 ) are sufficiently far from the vortex ring, fluxes across these surfaces are negligible, and the circulation in each surface can change only due to vorticity fluxes along the connection line C. Then the rate of change of circulation in each surface is given by (Terrington et al. 2022) Equations (4.11) attribute the loss of circulation from the symmetry plane, and the gain of circulation in I 1 and I 2 , to the viscous flux of vorticity across the connection line.Importantly, the loss of circulation from the symmetry plane represents the breaking open of vortex filaments, while the appearance of vorticity in the free surface indicates the attachment of these vortex lines to the free surface.Equations (4.11) show that these processes always occur simultaneously -which must be the case, since vortex filaments cannot end in the fluid interior -and are driven by the same physical process: the viscous boundary flux.
First, consider the loss of spanwise vorticity from the symmetry plane, which is illustrated by the contour plots of spanwise vorticity presented in figure 22.Initially, the vortex ring approaches the free surface under its own self-induced velocity.As the upper vortex core interacts with the free surface, it is flattened and stretched (figures 22b,c), resulting in a strong gradient of spanwise vorticity at the free surface (∂ω y /∂z).According to (4.11), this results in a viscous flux of spanwise vorticity out of the symmetry plane, so that most of the negative vorticity in the upper vortex core is diffused out of the fluid (figure 22d).
According to (4.11d), F ν also contributes to the rate of change of interface circulation, so the circulation lost from S 1 is transferred into the interface vortex sheet.Additional boundary terms also appear in this equation, representing the generation of interface circulation by tangential pressure gradients and body forces, as well as the advection of interface circulation along the surface.If the boundary points, a and b, are sufficiently far from the vortex ring, these terms approach zero, and the total circulation in the symmetry  plane -including vorticity in the fluid, and circulation in the interface vortex sheet -is constant.Now we consider the appearance of vorticity in the free surface, which is illustrated by the contours of surface-normal vorticity presented in figure 23.Initially, there is almost no vorticity in the free surface (figure 23a).However, during the interaction, positive vorticity appears in I 2 , while negative vorticity appears in I 1 .Equations (4.11) attribute this to the viscous vorticity flux at the connection line (y = 0), which is due to the same vorticity gradients that lead to the loss of circulation from the symmetry plane (primarily the gradient of spanwise vorticity, ∂ω y /∂z).In other words, the flux of negative spanwise vorticity out of the symmetry plane is associated with an equal flux of negative surface-normal vorticity into I 1 , and positive surface-normal vorticity into I 2 .This is necessary to ensure that the kinematic condition that vortex lines do not end in the fluid interior is satisfied.

Two-fluid flows
For two-fluid flows, the vorticity balance is similar to the free-surface case considered in § § 4.2 and 4.3, although vorticity is transferred into the upper fluid, rather than into the interface vortex sheet.For the control-volume analysis, we use the system of control volumes shown in figure 24(a), where V 1 and V 2 are control volumes in fluids 1 and 2, respectively.The two volumes share a common boundary, I, on the interface, while ∂V 1 (a) 1.0 sufficiently far from the vortex ring, then the net vorticity creation will be zero.Generation of vorticity by local pressure gradients on some section of the interface may still occur, but this will be balanced by the creation of opposite-signed vorticity elsewhere.
The time history of total spanwise vorticity in each fluid is plotted in figure 25(b), which can be compared to the free-surface case in figure 17(a).As the upper portion of the vortex ring diffuses out of the fluid, there is a loss of negative spanwise vorticity from the lower fluid.This is balanced approximately by an equal gain of negative vorticity in the upper fluid, so that the total vorticity in both fluids remains approximately constant.Effectively, negative vorticity lost from the lower fluid is transferred into the upper fluid.This is similar to the free-surface case, where vorticity lost from the lower fluid is transferred into the interface vortex sheet.
Note, however, that significant oscillations in the total vorticity are present in figure 25(b), which are due to the pressure generation term.This is confirmed in figure 26(b), which plots the various terms in (4.12) and demonstrates clearly that the oscillations in total circulation are due to pressure gradients on the interface.The pressure term will decrease to zero as the size of the control volume is increased, and the total circulation will be constant.
For the control-surface analysis, we use the system of control surfaces in figure 24(b), which includes the parts of the symmetry plane in fluid 1 (S 1 ) and fluid 2 (S 2 ), as well as the surfaces I 1 and I 2 on the interface.The rate of change of circulation in each surface is given by the viscous flux of vorticity along the connection line: (dΓ dΓ where F ν,1 is the flux of vorticity into fluid 1, and F ν,2 is the flux of vorticity into fluid 2.Moreover, the net rate of change of circulation in the symmetry plane is given by which changes only due to the net generation of vorticity by pressure gradients on the interface.If the boundary points a and b are positioned sufficiently far from the vortex ring, then the pressure term will be zero, and the total circulation will be conserved. The time history of circulation in each reference surface is plotted in figure 25(a).As in the free-surface case (figure 21a), the viscous flux (F ν,1 ) is responsible for the loss of circulation from the lower fluid (S 1 ), as well as the attachment of surface-normal vorticity to the free surface.Moreover, the loss of circulation from S 1 is balanced by an equivalent flux of circulation into S 2 (F ν,2 ), so the total circulation in the symmetry plane (Γ S 1 + Γ S 2 ) is approximately constant.This resembles the free-surface case, where the loss of circulation from the fluid is balanced by the increase in circulation contained in the interface vortex sheet.
As with the control-volume analysis, significant oscillations are observed in the total circulation in the symmetry plane (Γ S 1 + Γ S 2 ), due to the creation of vorticity by pressure gradients.The various terms from (4.14) are plotted in figure 26(a), which confirms that the oscillations in total circulation are due to the pressure generation term.Once again, as the control-surface boundary is moved further from the vortex ring, these oscillations will decrease, and the total circulation will be constant.(1999).In our description, the attachment of vorticity to the free surface is attributed to the diffusion of surface-normal vorticity along the free surface, and away from the connection line, while in the Zhang et al. (1999) interpretation, this is attributed to the diffusion of surface-normal vorticity from the fluid interior.
The similarities between the circulation balances for free-surface and two-fluid flows suggests that the interface vortex sheet above a free surface plays a similar role to the vorticity that would be found above an air-water interface.For the two-fluid case, the loss of vorticity from the lower fluid is balanced by an equal increase in the vorticity contained in the upper fluid.For the free-surface case, the loss of vorticity from the lower fluid is instead balanced by an increase in the circulation contained in the interface vortex sheet.

Mechanism of vortex connection
The analysis considered in this section suggests a novel interpretation of the vortex connection to a free surface.In this subsection, we compare this interpretation to the previous mechanism proposed by Gharib & Weigand (1996) and Zhang et al. (1999).
A sketch of the vortex connection mechanism is presented in figure 27(a).As the upper portion of the vortex ring interacts with the free surface, the gradient of spanwise vorticity at the free surface (∂ω y /∂z) results in a viscous flux of surface-tangential vorticity out of the fluid.The upper part of the vortex ring diffuses out of the fluid, and into the interface vortex sheet.The flux of spanwise vorticity out of the fluid is accompanied by the diffusion of opposite-signed surface-normal vorticity away from the connection line, along the free surface, which results in the attachment of vortex lines to the free surface.
Under this interpretation, the attachment of vortex filaments to the free surface is attributed directly to the viscous flux of spanwise vorticity out of the fluid.Therefore, the mechanism that drives the loss of spanwise vorticity from the fluid is also responsible for the attachment of vortex filaments to the free surface.This is beneficial, since the divergence-free property of the vorticity field -where vortex lines cannot end in the fluid interior -requires that the circulation lost from the symmetry plane must be gained in the free surface.Our interpretation illustrates clearly how this condition is enforced.
This interpretation of the vortex attachment mechanism differs substantially from previous interpretations (Rood 1994a,b;Gharib & Weigand 1996;Zhang et al. 1999).While Rood (1994a) describes in great detail the mechanisms by which surface-tangential vorticity diffuses out of the fluid, he does not ascribe the attachment of vorticity to the surface to any particular physical process.Instead, he states that 'a portion of the closed vortex loop fluxes out of the fluid, leaving the remainder of the loop with ends terminating from the fluid and the gain of circulation in the free surface as a single process, which more clearly explains how the kinematic condition that vortex lines do not end in the fluid is enforced.

Vorticity dynamics in the fluid interior
In this section, we consider the dynamics of vorticity in the fluid interior, beneath the free surface.To understand the dynamics of the vortex ring connection, Zhang et al. (1999) introduce two dynamical regions: the blockage layer, and the viscous sublayer.The blockage layer is due to the constraint on surface-normal velocity at the free surface, and leads to elevated vortex stretching and tilting.The viscous layer is due to the shear-free boundary condition (4.3), which results in a region of elevated viscous diffusion near the free surface.In addition to vortex connection to a free surface, the viscous and blockage layers also play an important role in the dynamics of free-surface turbulence (Shen et al. 1999;Bodart, Cazalbou & Joly 2010).
To examine the viscous and blockage layers, a control-surface analysis is used.We consider the system of control surfaces shown in figure 29: we introduce a line segment C z (z), at depth z beneath the surface.This line segment divides the symmetry plane into portions below (S a ) and above (S b ) the curve C z .We also consider the horizontal surface S z (z), which is at the same depth as z.
Using (2.2) from Terrington et al. (2021), the vorticity flux across the line C z (z) gives the rate of transfer of vorticity between S a and S b as u z ω y dx, (5.2) ν ∂ω z /∂y − ∂ω y /∂z dx. (5.

Viscous layer
The viscous layer is a thin region near the free surface where viscous diffusion is dominant, due to large vorticity gradients arising from the shear-free boundary condition (4.3).The viscous layer can be identified by considering the instantaneous profiles of the advective (F u ) and viscous (F ν ) fluxes, which are plotted in figure 31 for two different flow times.When the vortex ring is far from the free surface (figure 31a), the viscous flux is small, and vorticity transport is driven predominantly by advection.During the vortex reconnection process, when the vortex ring is close to the free surface (figure 31b), the advection term is dominant for depths greater than approximately z/R 0 = 0.1.However, the advection term approaches zero near the free surface, and the viscous term dominates the vorticity transport in a thin layer (the viscous sublayer), with thickness of order δ ν ≈ 0.1R 0 .This is in agreement with Zhang et al. (1999), who also find the viscous layer thickness to be approximately δ ν ≈ 0.1R 0 .
Instead of using the instantaneous flux profiles, we can instead consider the time averaged fluxes (5.8) In this expression, H is the Heaviside step function, which is used so that only fluxes corresponding to the upper vortex ring core are considered (i.e.contributions from regions containing positive spanwise vorticity are ignored).Profiles of the time-averaged viscous and advective fluxes are provided in figure 32(a).Outside the viscous sublayer (z < −0.1R 0 ), the advection term is dominant, and remains close to unity, while the time-averaged viscous flux is negligible.Inside the viscous layer, however, the advective flux decreases rapidly to zero, and the time-averaged viscous flux is dominant.

Blockage layer
The blockage layer is due to the blockage effect of the free surface, where, at low Froude numbers, the surface-normal velocity approaches zero at the free surface.This leads to a region near the free surface with elevated gradients of surface-normal velocity (∂u z /∂z), resulting in an increase in vortex stretching and tilting within the blockage layer.Since the blockage layer is due to a constrained surface-normal velocity at the free surface, the blockage effect is reduced, or even entirely absent, at high Froude numbers (Brocchini & Peregrine 2001).
The advective flux F u (z) includes the effects of both advection and vortex stretching/tilting, so the blockage layer cannot be identified using the time-averaged fluxes in figure 32(a).To identify the effects of vortex stretching and tilting, we first recognise the following relationship between I ω y and F u : which is a pseudo-one-dimensional transport equation for the spanwise vorticity in the symmetry plane.The gradient in vertical flux can then be related to the conventional advection and vortex stretching terms: The profiles of time-averaged advection (d Fav ) and vortex stretching (d Fst ) are plotted in figure 32(b), along with the gradients of the total vorticity fluxes.Outside the blockage layer (which is of order R 0 ), the time-average effect of both advection and vortex stretching is zero.However, for depths shallower than z ≈ R 0 , the time-averaged vortex stretching term becomes significant.The sign of d Fst is negative, indicating a net increase in negative spanwise vorticity due to vortex stretching within the blockage layer.Outside the viscous layer, the increase in vorticity due to vortex stretching is balanced by a decrease in vorticity magnitude due to the advection terms.Within the viscous layer, the strength of the advection term decreases, and the effects of vortex stretching are balanced by the viscous term.
From figure 32(b), the blockage effect is negligible for depths greater than the ring radius, suggesting a blockage layer thickness of approximately δ b ≈ R 0 .This is consistent with figure 9, where the influence the free surface on the evolution of the maximum vorticity first becomes noticeable when the upper vortex core is at depth z = −R 0 .However, the effect of the blockage layer remains small until a depth of approximately 0.4R 0 , which is close to the blockage layer depth suggested by Zhang et al. (1999).
Using (5.4), the vertical vorticity fluxes also give the rate of change of vertical vorticity in S z .This can be related to the advection and tilting of ω z as where we have assumed that ω is zero on all boundaries of S z (z), aside from C z (z).The right-hand size of (5.13) includes terms related to both vortex tilting, (5.14) and advection, (5.15) The time-averaged contributions of vortex tilting ( Ftl,z ) and advection ( Fav,z ) to the rate of change of vertical vorticity are plotted in figure 33.Outside the blockage layer, the time-averaged advection term is close to zero, and vortex tilting drives the rate of change of vertical vorticity.Within the blockage layer, however, the advection term and the vortex tilting term provide approximately equal contributions.This disagrees with (Zhang et al. 1999), who suggest that vortex tilting increases inside the blockage layer.Note that Zhang et al. consider instantaneous profiles of the vortex stretching and tilting terms -rather than time-averaged -which does not necessarily capture the change in dynamics as the vortex ring traverses the blockage layer.

Effect of parameters
In this section, we consider the effects of Reynolds and Froude numbers on the evolution of the vortex ring.As the Reynolds number is increased, the thickness of the viscous layer is reduced, leading to a larger peak in maximum vorticity.As the Froude number is increased, distortion of the free surface becomes more significant, leading to the creation of secondary vorticity, which interacts with the primary vortex ring and delays the connection to the free surface.The time history of maximum magnitude of spanwise vorticity (ω y ) in the symmetry plane, as well as maximum magnitude of vertical vorticity (ω z ) in the free surface, is shown in figure 34.This figure reveals several aspects of the vortex ring evolution that are affected by the Reynolds number.First, as the Reynolds number is reduced, the maximum vorticity in the vortex ring decays more rapidly during the initial approach to the free surface.Due to elevated levels of viscous diffusion, the vortex ring core is spread out more rapidly, leading to a lower peak vorticity and a larger core radius.Figure 34 also shows that the increase in maximum vorticity during the reconnection process is stronger at high Reynolds numbers, and is almost entirely absent at low Reynolds numbers.This can be understood by considering the thickness of the viscous layer ( § 5.1).As the Reynolds number is increased, the thickness of the viscous layer decreases (Zhang et al. 1999), leading to enhanced vortex stretching in the blockage layer.
This can be seen in figure 35, which shows contours of spanwise vorticity at t/(R 2 0 /Γ 0 ) = 35, for Re = 1000 and 2000.(A transient animation of this figure is provided in supplementary movie 7.) In both cases, the upper vortex core is contained within a thin layer near the surface.However, this layer is much thinner for Re = 2000 compared to Re = 1000, indicating a thinner viscous layer.We also consider the time-averaged viscous and advective fluxes, which are plotted in figure 36(a) for Re = 500, 1000 and 2000, which clearly illustrate that as the Reynolds number is increased, the thickness of the viscous layer decreases.As the viscous layer becomes thinner at higher Reynolds numbers, increased vortex stretching occurs in the blockage layer, leading to a higher maximum vorticity in the upper vortex core.In figure 36(b), we consider the time-averaged vortex stretching (d Fst ) and advection (d Fav ) terms, for Re = 500, 1000 and 2000.Outside of the viscous layer, the vortex stretching and advection terms are nearly independent of Reynolds number, since the blockage effect of the free surface does not depend on viscosity.The strength of the blockage effect increases as the free surface is approached, until it is impeded by viscous diffusion within the viscous layer.As the thickness of the viscous layer is reduced, the blockage effect is allowed to occur closer to the free surface, leading to a much greater maximum magnitude of vortex stretching.
The increase in vortex stretching that occurs at higher Reynolds numbers leads to larger vorticity gradients at the free surface.Therefore, although the viscosity is reduced, the boundary vorticity flux (νŝ × (∇ × ω)) remains significant.This explains why the time scale for vortex connection in figure 34 is relatively insensitive to Reynolds number, despite vortex connection being a purely viscous process.

Froude number
To investigate the effects of Froude number, a series of simulations was performed with the moving-mesh solver (interTrackFoam), using Froude numbers from Fr = 0.01 to 1.The moving-mesh solver is not capable of handling breaking of the free surface and air entrainment, so could not be used above Fr = 1.To investigate higher Froude numbers, a series of simulations were run using the VOF solver (interFoam), using Froude numbers from Fr = 1 to 2. All other physical parameters were held constant: Re = 1570, α = 80 • , a/R 0 = 0.35 and H/R 0 = 2.5.In figure 37 Time histories of the maximum magnitude of spanwise vorticity in the symmetry plane for each Froude number are plotted in figure 37.At high Froude numbers, the magnitude of secondary vorticity surpasses that of the primary vortex.The profiles in figure 37 for Fr 1 correspond to the primary vortex core, not the secondary vorticity.There is little change in the profiles between Fr = 0.01 and Fr = 0.75, indicating that at these Froude numbers, deformation of the free surface has little impact on the evolution of the vortex ring.The influence of free-surface deformations becomes significant at Fr = 1.As the Froude number is increased beyond this, the maximum value of spanwise vorticity in the vortex ring is reduced; however, this peak value of spanwise vorticity in the primary vortex core persists for a longer duration.
In figure 38, we show contours of spanwise vorticity in the symmetry plane at Fr = 0.75 and 1, obtained using the moving-mesh solver (interTrackFoam).(A transient animation of this figure is provided in supplementary movie 8.) At Fr = 0.75, slight deformations in the free surface are observed, and a small amount of secondary vorticity is present on the free surface.However, the free-surface deformations and secondary vorticity are not strong enough to alter significantly the vortex ring evolution.At Fr = 1, however, larger deformations in the free surface are observed, leading to increased generation of secondary vorticity.Comparing figures 39(b) and 39(d), we see that the secondary vorticity for Fr = 1 has deflected the head of the primary vortex away from the free surface.As a result, the primary vortex core does not diffuse into the free surface as rapidly, and persists in the fluid for a longer duration.
To understand the appearance of secondary vorticity, consider the free-surface boundary condition (4.3).The first term, −ŝ × ∇ (u 1 • ŝ), is due to the normal velocity of the free surface, and represents the rotation of the free surface, while the second term, −ŝ × (u 1 • K ), is due to the curvature of the free surface, and represents the rotation of fluid elements as they follow the curvature of the free surface (Terrington et al. 2020(Terrington et al. , 2022)).Both of these effects are larger as the Froude number is increased, resulting in a greater quantity of secondary vorticity appearing at the free surface.
In figure 39, we plot contours of spanwise vorticity for Fr = 1, 1.5 and 2, obtained using the VOF solver (interFoam).(A transient animation is provided in supplementary movie 9.) As the Froude number is increased, larger surface deformations are observed, leading to increased generation of secondary vorticity.The secondary vorticity detaches from the free surface and forms a counter-rotating vortex pair, which pulls the primary vortex core away from the free surface.This interaction prevents the primary vortex ring from diffusing out of the free surface, so the upper vortex core persists for a longer duration.
For Fr = 1.5 and 2, we also see evidence of air entrainment in figures 39(c) and 39(e).The air entrainment process appears to be similar to that observed in André & Bardet (2017), where the primary vortex creates a thin air ligament near the free surface, which pinches off and becomes an air bubble.Large surface curvature is observed at the tip of the air ligament, which is where the secondary vortex is generated.The air entrainment process for a vortex ring interacting with a free surface has been discussed in greater detail by Yu et al. (2018).

Conclusions
We have provided a new interpretation of the mechanism by which a vortex ring attaches to a free surface.As the vortex ring approaches the free surface, vortex stretching in the blockage layer leads to large vorticity gradients at the free surface.This Vortex ring connection to a free surface accompanied by the viscous flux of surface-normal vorticity along the free surface and away from the connection line, which results in the attachment of the ends of the vortex ring to the free surface.This description explains clearly how the loss of circulation from the symmetry plane is balanced by an equal increase in circulation in the free surface, which is a consequence of the kinematic condition that vortex lines do not end inside the fluid.
By including an interface vortex sheet at the free surface, the total circulation in this flow is conserved.The loss of vorticity from the fluid is balanced by an increase in the circulation found in the interface vortex sheet.Moreover, the interface vortex sheet satisfies a generalised solenoidal condition, so that after the connection, the vortex ring remains a closed loop, with the upper portion of the vortex ring passing through the interface vortex sheet.During the reconnection process, the upper part of the vortex ring simply diffuses out of the fluid and into the interface vortex sheet.This is similar to the interaction between a vortex ring and an air-water interface, where the upper part of the vortex ring diffuses out of the lower fluid and into the upper fluid.
Finally, we have considered the effects of Reynolds and Froude numbers on the vortex connection process.As the Reynolds number is increased, the thickness of the viscous layer decreases, leading to increased vortex stretching within the blockage layer.This leads to larger vorticity gradients at the free surface, which produce the boundary vorticity flux.
As the Froude number is increased, larger deformation of the free surface is observed, leading to the creation of secondary vorticity at the free surface.The secondary vorticity pulls the primary vortex ring away from the free surface, prolonging the vortex connection process.
Funding.This work was supported by computational resources provided by the Australian Government through the National Computational Infrastructure and Pawsey Supercomputer Centre (Merit Grants n67 and d71) under the National Computational Merit Allocation Scheme, and was supported by the Australian Government through the Australian Research Council's Discovery Projects funding scheme (project DP210100990).S.J.T. acknowledges the support of a Sir James McNeil Scholarship and Monash Graduate Excellence Scholarship.
Declaration of interests.The authors report no conflict of interest.

Figure 2 .
Figure 2.An example of the meshing scheme used in this study, viewed from (a) the x-z plane, and (b) the y-z plane.Dark lines divide regions with different mesh grading.The locations of the representative cell sizes in table1are also shown.For clarity, the grid shown here is much coarser than those used for numerical simulations.For single-fluid simulations, the mesh in the upper fluid is removed, and the free surface (z = 0) is the upper boundary of the computational domain.

944Figure 8 .
Figure 8. Contours of spanwise vorticity (ω y ) in the symmetry plane at t/(R 2 0 /Γ 0 ) = 23, obtained using the interTrackFoam solver with physical parameters Re = 1190, Fr= 0.19, α = 83 • , a/R 0 = 0.3 and H/R 0 = 2.Overlaid is a square outline showing the approximate size of the 32 × 32 pixel windows used to obtain each velocity vector for PIV byGharib & Weigand (1996).The horizontal lines show the vertical grid spacing used byZhang et al. (1999) at their highest resolution.The location of maximum vorticity magnitude is indicated by an asterisk.

Figure 9 .
Figure 9. (a) Time history of the maximum magnitude of spanwise vorticity (ω y ) in the symmetry plane.(b) Vertical position of the maximum vorticity in the upper vortex ring core, for a range of initial depth ratios (H/R 0 ).Solid lines indicate the maximum vorticity in the upper core, while dashed lines indicate vorticity in the lower core.The flow times have been shifted so that the peak vorticity in each case occurs at t = 0.The physical parameters are Re = 1570, Fr = 0.47, α = 80 • and a/R 0 = 0.35.

Figure 11 .
Figure 11.Vortex lines, as well as isosurfaces of vorticity magnitude, (a,b) before and (c,d) after the free-surface connection.Vortex lines are colour coded, with red indicating closed loops, and blue indicating vortex lines that begin and terminate on the free surface.Both front (a,c) and side (b,d) views are provided.The dashed line in (a,c) indicates the symmetry plane (y = 0).

Figure 13 .
Figure 13.Vortex lines and isosurfaces of vorticity magnitude, (a) before and (b) after vortex connection to a fluid-fluid interface, at ρ 1 /ρ 2 = 1000 and μ 1 /μ 2 = 100.Vortex lines are colour coded, with red indicating closed loops that lie entirely within fluid 1, green indicating closed loops that lie entirely in fluid 2, and blue indicating vortex lines that pass through both fluids.

Figure 14 .
Figure 14.Vortex lines in the interface vortex sheet (i.e.curves tangent to the interface circulation density, γ ), as well as colour contours of the surface-normal vorticity, at two different flow times.
, which plots vortex lines in the fluid interior as well as in the interface vortex sheet.(A transient animation of this figure is provided in supplementary movie 5.) As vortex lines in the fluid interior connect to the free surface, they act as sources or sinks 944 A56-17 https://doi.org/10.1017/jfm.2022.

Figure 15 .
Figure 15.Vortex lines in the fluid as well as in the interface vortex sheet, both (a) before and (b) after vortex reconnection has occurred.Vortex lines are colour coded, with red indicating closed loops, while blue indicates vortex lines that begin and terminate on the free surface.

Figure 17
Figure 17.(a) Time history of the spanwise component of vector circulation in the fluid (Γ 1,y ) and in the interface vortex sheet (Γ γ,y ).(b) Comparison between the rate of change of spanwise vorticity, and the viscous and vortex stretching terms from (4.6).
.6) where we have assumed that vorticity fluxes across ∂V 1 are negligible.This equation indicates the two mechanisms that contribute to the rate of change of vorticity in the fluid: the viscous boundary flux F ν = I νŝ × (∇ × ω) dS, and the vortex stretching/tilting flux F st = I (ω • ŝ)u dS.944 A56-20 https://doi.org/10.1017/jfm.2022.

944Figure 18 .
Figure18.Illustration of the vortex stretching/tilting flux u(ω • ŝ) from (4.6).This term can be interpreted as representing the advection of vortex filaments in the free surface, which must correspond to the creation of spanwise vorticity by vortex stretching in the fluid interior.An equal and opposite quantity of circulation is also created in the interface vortex sheet, so that the total spanwise circulation is conserved.

Figure 19 .
Figure 19.Contours of spanwise vorticity (ω y ) in the symmetry plane (y = 0), and of surface-normal vorticity (ω • ŝ) in the free surface, both (a) before and (b) after the free surface connection has occurred.Contours of dimensionless vorticity are plotted for ω/(Γ 0 /R 2 0 ) = . . ., −1.25, −0.75, −0.25, 0.25, 0.75, 1.25, . ... (y = 0) and the free surface.Contours of the spanwise vorticity (ω y ) in the symmetry plane, and of surface-normal vorticity (ω • ŝ) in the free surface, are presented in figure 19, both before and after reconnection has occurred.(A transient animation of this figure is provided in supplementary movie 6.)This figure demonstrates one of the main features of vortex connection to the free surface: the loss of negative spanwise vorticity from the symmetry plane (which signifies the breaking open of vortex filaments near the free surface), and the appearance of new vorticity in the free surface (indicating the attachment of vortex lines to the free surface).To quantify the reconnection process, we consider the circulation contained in the system of control surfaces illustrated in figure20: S 1 is the symmetry plane; I 1 is the portion of the free surface where y < 0; and I 2 is the portion of the free surface where y > 0. The time history of circulation in each reference surface is plotted in figure21(a), where Γ I 1 , Γ I 2 and Γ S 1 are the circulations contained in surfaces I 1 , I 2 and S 1 , respectively.As the upper part of the vortex ring diffuses out of the free surface, the circulation in

Figure 21 .
Figure 21.(a) Time history of the circulation in the surfaces I 1 , I 2 and S 1 , as well as of the interface circulation Γ γ .(b) A comparison between the rate of change of circulations and the viscous boundary flux F ν , defined in (4.11).
.11e) where F ν is the viscous flux of vorticity across the connection line (C), and subscript (b − a) indicates the difference in some quantity between the endpoints of C (θ (b−a) = θ b − θ a ).

Figure 22 .
Figure 22.Contours of spanwise vorticity (ω y ) in the symmetry plane (y = 0) at several times throughout the interaction.The colourmap and contour levels are the same as in figure 19.

Figure 23 .
Figure 23.Contours of surface-normal vorticity (ω • ŝ) in the free surface at several times throughout the interaction.The colourmap and contour levels are the same as in figure 19.The dashed line indicates the connection line (y = 0), which is the shared boundary of the surfaces I 1 , I 2 and S 1 .

Figure 24 .
Figure24.System of (a) control volumes, and (b) control surfaces used to analyse the vortex ring connection to a fluid-fluid interface.The volumes V 1 and V 2 are control volumes in the lower and upper fluids, respectively.Also, S 1 and S 2 are the portions of the symmetry plane in the upper and lower fluids, respectively, while I 1 and I 2 are surfaces that lie in the interface.

Figure 25 .
Figure25.Time histories of (a) the circulation in both the symmetry plane (S 1 and S 2 ) and the interface (I 1 and I 2 ).(b) The spanwise component of vector circulation in each fluid (Γ 1,y and Γ 2,y ), for the interaction between a vortex ring and a fluid-fluid interface.

Figure 26 .
Figure26.The net rate of change of (a) total circulation in the symmetry plane (Γ S 1 + Γ S 2 ), and (b) spanwise component of vector circulation in both fluids (Γ 1,y + Γ 2,y ), as well as the net vorticity generation by pressure gradients (S p and S p,y ) and net viscous flux of vorticity out of the interface (F ν,1 + F ν,2 ).

944Figure 27
Figure27.A comparison between (a) our interpretation of the vortex attachment process, and (b) the mechanism proposed byZhang et al. (1999).In our description, the attachment of vorticity to the free surface is attributed to the diffusion of surface-normal vorticity along the free surface, and away from the connection line, while in theZhang et al. (1999) interpretation, this is attributed to the diffusion of surface-normal vorticity from the fluid interior.

944Figure 29 .
Figure 29.System of control surfaces used to investigate the dynamics of vorticity in the fluid interior.The line segment C z (z) divides the symmetry plane into portions below (S a ) and above (S b ) the curve C z .The horizontal surface S z (z) is situated at the same depth as C z (z).

Figure 30 .
Figure 30.Profiles of the integrals of spanwise (I ωy ) and vertical vorticity (I ωz ), with depth beneath the free surface, both (a) before and (b) after the connection to the free surface.The physical parameters are Re = 1570, Fr = 0.47, α = 80 • , H/R 0 = 3.00 and a/R 0 = 0.35.

Figure 31 .
Figure 31.Profiles of the vertical flux of spanwise vorticity (F(z)) with depth beneath the free surface, at flow times (a) before and (b) during the connection to the free surface.

944Figure 32
Figure 32.(a) Profiles of the time-averaged vertical vorticity flux ( F(z)), with distance below the free surface.(b) Gradients of the time-averaged vorticity flux, and the contributions from advection (d Fav ) and vortex stretching (d Fst ).

Figure 33 .Figure 34 .
Figure 33.Profiles of the time-averaged vertical vorticity flux, and the integrated rate of change of vertical vorticity due to advection ( Fav,z ) and vortex tilting ( Ftl,z )

Figure 36 .
Figure 36.(a) Profiles of the time-averaged advective ( Fu ) and viscous ( Fν ) vorticity fluxes in the symmetry plane, and (b) profiles of the vortex stretching (d Fst ) and advection (d Fav ) terms, for a selection of Reynolds numbers.

944Figure 37 .
Figure37.Time histories of the maximum magnitude of spanwise vorticity in the symmetry plane associated with the primary vortex ring core for a range of Froude numbers, for a vortex ring with physical parameters Re = 1570, α = 80 • , a/R 0 = 0.35 and H/R 0 = 2.5.Results obtained on both the moving-mesh solver (interTrackFoam) and the VOF solver (interFoam) are presented.