Effect of Density Ratio and Surface Tension on Vortex–Interface Interactions: A Numerical Study

Abstract

In two-phase flow, the interaction between multi-scale vortex structures and interfaces (bubbles or free surfaces) triggers a range of complex physical phenomena. This study employs numerical simulations to investigate the interaction between a horizontal vortex and the interface separating two layers of immiscible fluids with different densities (e.g., water and air). The vortex is initialized as an internal motion within the heavier phase. We focus specifically on the impact of the phase density ratio and surface tension. Numerical simulations reveal that when the density ratio is near unity, interface rupture occurs only at high Weber numbers ($\mathit{We}$), where low surface tension enables the rupture of sharp interface points. Conversely, at high surface tension (low $\mathit{We}$), these sharp points stretch into thin liquid films, significantly increasing the surface area without causing breakage. As the density ratio increases, interface rupture at sharp points accelerates, even under high surface tension, leading to faster dissipation of the initial vortex. In high-$\mathit{We}$ scenarios, an increased density ratio promotes the faster formation and greater intensity of new vortex layers at the interface. However, increasing surface tension enhances the vorticity of these layers but simultaneously slows their generation rate. The findings highlight the critical interplay between surface tension and density differences in vortex–interface interactions, with surface tension stabilizing the interface and density differences driving more intense vortex shedding and deformation. These insights offer valuable guidance for understanding two-phase flow behavior and improving the design of systems involving multiphase fluids.

1. Introduction

The study of vortex–interface interactions in immiscible two-phase flows is a critical area of research with wide-ranging applications in marine engineering, offshore structures, and multiphase flow systems [1,2]. Although often studied as a fundamental unit process, this interaction serves as a canonical model for complex hydrodynamic phenomena. For instance, in the turbulent wake of a transom stern vessel, large-scale vortical structures interact violently with the free surface, leading to surface ‘scarring’ and significant air entrainment. Similarly, in plunging breaking waves, horizontal ‘roller’ vortices dictate the onset of ‘white water’ and bubble generation. Consequently, understanding the fundamental mechanisms underlying these vortex-induced interface deformations is essential for improving our ability to model and predict complex flow behaviors in environments such as ocean waves, fuel injection systems, and industrial fluid processes.

The interaction between vortices and interfaces is a dynamic and multifaceted process, influenced by a variety of factors, including the phase density ratio, surface tension, and the intrinsic properties of the fluids involved. Previous research has established that vortex behavior is significantly altered by the presence of interfaces, particularly when the two fluids have different densities or viscosities. Early studies by Yih (1967) [3] and Hooper & Boyd (1983) [4] examined the effects of viscosity contrasts on the stability of interfaces, highlighting how Kelvin-Helmholtz instabilities can be modified by the presence of a fluid interface. These studies laid the groundwork for understanding how instabilities can be driven by shear forces at the interface, even in low Reynolds number flows. Moreover, the role of surface tension has been found to play a crucial role in stabilizing or destabilizing the interface, influencing the growth of instabilities.

On the other hand, vortices can also induce significant changes in the interface. As vortical structures evolve, they generate vorticity layers at the interface, which can lead to complex deformations and even interface breakage. Sharp’s (1982) [5] work on Rayleigh-Taylor instability demonstrated how vortex-induced vorticity at the interface between two fluids of different densities can lead to the formation of droplets and film break-up. This phenomenon has been further investigated by Villermaux & Clanet (2002) and Ling et al. (2019) [6,7], who showed how shear instability at the interface leads to the fragmentation of fluid sheets and the generation of droplets under the influence of vortices. Their studies suggest that vortex-induced deformation is a fundamental process in droplet formation during wave breaking and other multiphase flow phenomena.

The generation of vortices by interface evolution is another critical aspect of these interactions. Longuet-Higgins (1953) [8] was one of the first to explore the vorticity generation at free surfaces in viscous fluids, demonstrating that the motion of the free surface could induce vorticity due to the pressure gradient across the interface. Later works by Lundgren & Koumoutsakos (1999) [9] and Wu et al. (1998) [10] extended this theory to incorporate the effects of surface tension and fluid density differences, providing more comprehensive models for the vorticity generation at fluid-fluid interfaces. These findings suggest that the interaction between the vortex structures and the interface can be reciprocal, where vortices drive interface deformation and the interface evolution, in turn, modifies the vorticity field, creating a complex feedback loop.

To better understand these interactions, numerical simulations, particularly Direct Numerical Simulation (DNS), have proven to be an invaluable tool. DNS enables a detailed, high-fidelity representation of fluid dynamics, capturing both the small-scale and large-scale features of vortex–interface interactions. Iafrati & Campana (2005) [11] utilized DNS to investigate wave breaking and the onset of instability in two-phase flows, while Deike et al. (2015, 2016) [12,13] used DNS to study the role of vorticity layers in the fine-scale structures that develop during wave breaking. Their work demonstrated that vortex–interface interactions are central to understanding the fine-scale dynamics in wave-breaking processes. More recently, Loisy & Naso (2017) [14] applied DNS to study the interactions between bubbles and turbulence, further revealing how vortices influence the deformation and rise of bubbles in a turbulent flow.

Previous DNS studies have primarily focused on the statistical properties of droplets and bubbles in fully developed turbulence. However, the fundamental mechanisms triggering the initial interface breakup are often obscured in such complex flows. In contrast, this study targets the deterministic onset mechanisms by employing a simplified numerical model. This study uses numerical simulations to investigate the combined effects of density ratio and surface tension. We adopt a two-dimensional (2D) model representing a horizontal vortex parallel to the interface. We acknowledge that three-dimensional (3D) effects, such as vortex stretching, are important in real flows. However, we chose this 2D approach to clearly isolate the fundamental roles of density difference and surface tension, avoiding the extra complexity of 3D turbulence. This setup is relevant to typical flows like the wake behind a transom stern or rollers in breaking waves. Full 3D effects are beyond the scope of this baseline study and will be addressed in our future work.

In this paper, a high-fidelity simulation of interface evolution induced by a Lamb–Oseen vortex is conducted. During the interface-vortex interaction, the interface undergoes significant deformation and may even shatter under the influence of the vortices. We focus on examining the effects of phase density differences and surface tension on the interaction mechanisms. Through comprehensive investigation and statistical analysis of the entrapped bubbles, this study enhances our understanding of interface-vortex interactions in ocean dynamics. Section 2 presents the numerical setup and methods used in the simulation. In Section 3, we first describe the idealized numerical experiments, then discuss the roles of the density ratio and surface tension in the interface evolution process. Additionally, we analyze the combined effects of density differences and surface tension. Concluding remarks are provided in Section 4.

2. Numerical Setup and Methods

In this study, we employ the self-developed solver BAMR-SJTU, which has been rigorously validated through three-dimensional numerical simulations of various fluid phenomena, including bow wave breaking [15,16], and hydraulic jumps (Li et al., 2021, 2022) [17,18]. This solver is specifically designed to model the complex interactions between the air and liquid phases, ensuring accurate representation of two-phase flow dynamics. While a brief overview of the numerical methods used is provided here, more detailed information can be found in the referenced works.

To accurately capture complex interface topologies, we implement the CLSVOF (Coupled Level-Set and Volume-of-Fluid) method, which is renowned for its mass-preserving properties [19]. This method enables precise tracking of the interface, a critical aspect for simulations involving large deformations. Additionally, a mass and momentum-consistent advection method [18,20] has been developed to enhance the numerical stability and robustness of simulations, particularly in high-density-ratio two-phase flows. In order to account for the role of surface tension in the formation and evolution of bubbles and droplets, we employ a sharp surface tension (SSF) model [19], which is based on the ghost fluid method [21]. This model ensures an accurate representation of surface forces, which is essential for capturing the fine details of bubble dynamics and droplet formation.

2.1. Governing Equations

The governing equations for incompressible two-phase flows in the present study are based on the continuity and momentum equations, which are expressed in conservative form as follows:

$${\displaystyle \frac{\partial \rho }{\partial t}}+\nabla ·\left(\rho \mathbf{u}\right)=0,$$

(1a)

$${\displaystyle \frac{\partial \left(\rho \mathbf{u}\right)}{\partial t}}+\nabla ·\left(\rho \mathbf{u}\mathbf{u}\right)=-\nabla p+\nabla ·\mathsf{\tau }+\rho \mathbf{g}+{\mathbf{T}}_{\mathsf{\sigma }},$$

(1b)

where the velocity vector is denoted by u, and $p$ represents the pressure. The shear stress tensor, τ, is defined as $\tau =\mu [\nabla \mathbf{u}+(\nabla \mathbf{u}{) }^T]$, where $\mu$ is the dynamic viscosity and $\rho$ is the density. These properties can be computed using the volume fraction for immiscible two-phase flow. The term g represents the gravitational acceleration, and ${\mathbf{T}}_{\mathsf{\sigma }}$ denotes the surface tension, which is non-zero only at the interface.

The equation governing interface advection is expressed as follows:

$${\displaystyle \frac{\partial C}{\partial t}}+\nabla ·\left(C\mathbf{u}\right)-C\nabla ·\mathbf{u}=0$$

(2)

where $C$ is the fractional volume function.

The density $\rho$ and viscosity $\mu$ in Equation (1) can be determined by utilizing the volume fraction $C$ in the following manner:

$$\rho =C{\rho }_l+\left(1-C\right){\rho }_a$$

(3a)

$$\mu =C{\mu }_l+\left(1-C\right){\mu }_a$$

(3b)

where the subscripts $l$ and $a$ correspond to the liquid and air phases, respectively.

The surface tension in Equation (1b) can be expressed as:

$${\mathit{T}}_{\sigma }=\sigma \kappa \delta \mathbf{n}$$

(4)

The surface tension is defined by the surface tension coefficient σ and the local curvature κ. The vector n represents the normal vector to the interface, while δ denotes the Dirac delta function. To estimate the curvature, we use the height function (HF) method, enhanced with a curvature smoothing strategy, which ensures second-order accuracy in spatial convergence. Numerical experiments demonstrate that the surface tension model used in this study remains stable even with large computational time steps.

2.2. Computational Set-Up

In this study, we investigate the interaction between the interface and a Lamb–Oseen vortex [22] with total circulation ${\mathsf{\Gamma }}_0$ and core radius $r_o$, which serves as a model for an evolving vortex in turbulent flow. The simulation accounts for the radial expansion of the vortex over time due to viscous diffusion. Initially, the flow field is initialized with a velocity corresponding to a Lamb–Oseen vortex, expressed in Cartesian coordinates as follows:

$$v_x=-{\displaystyle \frac{{\mathsf{\Gamma }}_{0}y}{2\pi \left(x^2+y^2\right)}}\left[1-\exp \left(-{\displaystyle \frac{x^2+y^2}{{r_o}^2}}\right)\right]$$

(5a)

$$v_y={\displaystyle \frac{{\mathsf{\Gamma }}_{0}x}{2\pi (x^2+y^2)}}\left[1-\mathrm{e}\mathrm{x}\mathrm{p}(-{\displaystyle \frac{x^2+y^2}{{r_o}^2}})\right]$$

(5b)

and the corresponding vorticity is given by

$$w_z={\displaystyle \frac{{\mathsf{\Gamma }}_0}{\pi \left(x^2+y^2\right)}}\exp \left(-{\displaystyle \frac{x^2+y^2}{{r_o}^2}}\right)$$

(5c)

where ${\Gamma }_0$ and $r_o$ represent the initial circulation and the initial core size, respectively.

The initial conditions are depicted in Figure 1, where the initial length of the interface corresponds to the extent of the computational domain. In this study, the evolution of the interface is driven by a Lamb–Oseen vortex, with a distinct separation between the regions of initial vorticity and the initial interface. The physical model consists of two layers of immiscible fluids. Specifically, Phase 1 (${\rho }^1$) is located at the top, while Phase 2 (${\rho }^2$) occupies the bottom region, as shown in Figure 1a. An internal Lamb–Oseen vortex is initialized entirely within the heavier phase (Phase 2), centered at the origin, governed by a circulation ${\mathsf{\Gamma }}_0$ and a core radius $r_o$, which is smaller than the distance between the vortex core and the interface. This configuration allows for a clear distinction between the vortex and the interface, facilitating an in-depth examination of their interactions. For the presentation of results, a mixed unit convention is adopted to best illustrate the flow physics. Geometric quantities, such as the interface length, as well as the time variable ($t$), are normalized by the computational domain width ($\lambda =1 \mathrm{m}$) and the characteristic time scale ($T=1 \mathrm{s}$), respectively, and are therefore non-dimensional. However, to retain the physical magnitude of the flow field, dynamic quantities such as velocity and vorticity are presented in dimensional units, specifically $\mathrm{m}/\mathrm{s}$ and $\mathrm{s}⁻¹$, respectively.

To study the resulting flow behavior, the present investigation explores various intensities of vortex-induced breaking by varying the circulation of the vortex. By adjusting ${\mathsf{\Gamma }}_0$, the intensity of the vortex’s influence on the interface is modified, enabling the study of different flow regimes. The simulation is conducted under free-slip boundary conditions on both the top and bottom walls, ensuring the absence of normal velocity gradients at these boundaries. Figure 1b,c illustrate the distribution of vortex strength and velocity in the radial direction for the Lamb–Oseen vortex, providing valuable insights into the evolution of the vortex and its interaction with the interface.

In this study, the distance between the vortex and the interface, $d_o$, and the vortex radius, $r_o$, are held constant, with $d_o=0.1$ and $r_o=0.02$, ensuring a clear separation between the interface and the vortex. Additionally, the average density ${\rho }_m$ on both sides of the interface is defined as the arithmetic average of ${\rho }^1$ and ${\rho }^2$, where ${\rho }^1$ and ${\rho }^2$ represent the densities on either side of the interface, i.e.,

$${\rho }_m=({\rho }^1+{\rho }^2) /2$$

(6)

The flow dynamics are governed by three key non-dimensional control parameters: the density ratio ($r_p$), the Reynolds number ($Re$), and the Weber number ($We$). These are defined as follows:

$$r_p={\displaystyle \frac{{\rho }^2}{{\rho }^1}}$$

(7)

$$Re={\displaystyle \frac{{\rho }_m{\Gamma }_o}{2\pi \mu }}$$

(8)

$$We={\displaystyle \frac{{\rho }_m{{\Gamma }_o}^2}{{\left(2\pi \right) }^2{d}_o\sigma }}$$

(9)

The expression for the vorticity generated at the interface is given by:

$$\mathsf{\Sigma }={\displaystyle \frac{r_p+1}{2r_p}}(\left(r_p-1\right){\displaystyle \frac{\partial p_m}{\partial s}}-{\displaystyle \frac{(r_p+1) }{2}}{\displaystyle \frac{1}{We}}{\displaystyle \frac{\partial k}{\partial s}})$$

(10)

According to Equations (7)–(9), the generation of vorticity at the interface is primarily influenced by three key factors: first, the density difference between the two phases, which is controlled by the parameter $r_p$; second, the characteristics of the initial vortex, which are governed by the Reynolds number $\mathrm{Re}$; and third, the effects of surface tension, which are controlled by the Weber number $\mathrm{We}$. Each of these factors plays a crucial role in determining the vorticity dynamics near the interface. The density ratio $r_p$ reflects the relative densities of the two phases and directly influences the vorticity generation through its impact on the flow behavior. The Reynolds number $\mathrm{Re}$, which characterizes the relative importance of inertial forces versus viscous forces, is crucial for understanding the behavior of the initial vortex and its interaction with the interface. The Weber number $\mathit{We}$, which quantifies the ratio of inertial forces to surface tension forces, governs the influence of surface tension in shaping the interface and controlling vorticity generation.

This study will explore how these three parameters—$r_p$, $\mathrm{Re}$, and $\mathit{We}$—affect the generation of vorticity near the interface under varying conditions. Specifically, we will examine the individual and combined effects of these parameters on vorticity generation, shedding light on the complex interactions between density differences, vortex dynamics, and surface tension. By varying these parameters, we aim to provide a comprehensive understanding of how the interplay of these factors contributes to the formation and evolution of vorticity at the interface in two-phase flows.

2.3. Mesh Convergence Verification

To evaluate the impact of grid resolution on the numerical results, this study conducted a convergence analysis using three different levels of grid refinement. Figure 2 illustrates the distribution of the initial grid configurations at three different refinement levels, while Figure 3 demonstrates the evolution of the dynamic mesh, with refinement levels ranging from 3 to 6. These refinement levels correspond to three types of grids: coarse grids (labeled 3–5, indicating a minimum refinement level of 3 and a maximum of 5), medium grids (3–6), and fine grids (3–7), with minimum grid spacings of $\mathsf{\Delta }=8\times {10}^{-3}$, $\mathsf{\Delta }=4\times {10}^{-3}$, and $\mathsf{\Delta }=2\times {10}^{-3}$ respectively, as summarized in

Table 1. Comparison of three different grid types.

Grid Type	Refinement Level	Minimum Grid Spacing	Initial Grid Number
Coarse grids	3–5	$8\times {10}^{-3}$	8586
Medium grids	3–6	$4\times {10}^{-3}$	13,446
Fine grids	3–7	$2\times {10}^{-3}$	43,578

. Figure 4 presents a comparison of the numerical results for the interface-vortex profiles obtained using these three different grid configurations. On the refined grids (3–6 and 3–7), the interface-vortex profiles show a high degree of consistency, suggesting that the grid resolution in this study converges well during the computational process.

Additionally, as shown in Figure 5, the time evolution of key physical quantities such as interface length, maximum velocity, and maximum vorticity is compared across the three different grid types. The analysis revealed that the coarse grid yields less accurate results, while the fine and medium grids demonstrate strong consistency. This further confirms the convergence of the grid resolution used in the current study. Based on a balance between computational efficiency and accuracy, the medium-resolution grid (3–6) is chosen for the majority of the numerical simulations in this chapter. However, to more accurately capture small-scale flow structures in specific regions, the finer grid (3–7) is selectively employed. This approach ensures both computational efficiency and the ability to resolve finer details of the flow dynamics.

2.4. Numerical Methods

As shown in Figure 2, to accurately capture small-scale flow structures, we employ a block-structured Adaptive Mesh Refinement (AMR) strategy [23,24]. This approach enables selective application of the finest mesh refinement in critical regions, such as the free surface and areas with high vorticity, where the flow exhibits significant gradients or complex features. Figure 3 illustrates how the finest meshes are concentrated in these critical zones, ensuring that small-scale flow structures are resolved with high precision, while minimizing computational effort in less critical regions. This targeted refinement significantly enhances computational efficiency, enabling the simulation of multiscale processes across a broad range of spatial scales.

The entire computational domain is divided into blocks, each serving as a basic manipulation unit. This block-structured method facilitates the efficient implementation of high-order numerical schemes, such as the Weighted Essentially Non-Oscillatory (WENO) method [25], which is particularly effective at capturing sharp gradients and discontinuities in the flow field. The use of wide stencils in these high-order schemes ensures accurate representation of the flow dynamics, even in regions with complex interface behavior, such as the free surface and vorticity layers. By combining AMR with high-order numerical schemes, we achieve a balance between high accuracy and computational efficiency, enabling the resolution of intricate flow structures without excessive computational costs. This approach allows for a more comprehensive understanding of complex fluid dynamics while maintaining the practicality of the simulation in terms of resource usage.

3. Results and Discussion

3.1. Idealized Numerical Experiments

Initially, we conduct simulations under idealized conditions to understand the fundamental behavior of vortex–interface interactions. Specifically, when the densities of the two phases are equal (i.e., $r_p=1$) and surface tension is neglected (i.e., $We=\mathrm{\infty }$), the baroclinic source term in Equation (10) vanishes. Under these conditions, rather than being entirely passive, the interface dynamics are kinematically governed by the strain field imposed by the primary vortex. Although no new vorticity is generated via the baroclinic mechanism, the interface still undergoes significant stretching and deformation driven by viscous coupling and the non-uniform velocity distribution. Consequently, in this regime, the interface essentially acts as a material line tracking the flow kinematics without exerting feedback (such as capillary forces) on the primary vortex, as shown in Figure 6, Figure 7 and Figure 8.

Under these ideal conditions, we systematically vary the Reynolds number ($Re$) to three distinct values: $Re=\mathrm{13,997}$, $Re=\mathrm{34,994}$, and $Re=\mathrm{55,991}$. These different Reynolds numbers represent flows with varying inertial forces relative to viscous forces, with higher $Re$ values corresponding to stronger inertial effects and more intense vortex activity. The resulting interface evolution is shown in Figure 6, Figure 7 and Figure 8. As expected, the simulations are in agreement with hydrodynamic theory, which predicts that at higher $Re$, vortices become stronger and their impact on the interface increases. The higher Reynolds numbers correspond to more energetic flow, leading to the amplification of vortex strength and a greater ability to perturb the interface. Consequently, at high Reynolds numbers, the interface undergoes more significant breaking. In the case of $Re=\mathrm{55,991}$, the interface becomes highly curved, with extensive breaking occurring near the liquid film.

The increase in vortex strength at higher Reynolds numbers is due to the growing influence of inertial forces, which dominate over viscous forces. As these vortices interact with the interface, they induce shear and circulation at the interface, causing it to deform. These deformations, while passive in the absence of surface tension, still reveal how vortices influence the interface dynamics. This effect becomes more pronounced as $Re$ increases, where the vortices become more chaotic and lead to larger-scale perturbations of the interface.

For a more intuitive comparison, we overlay the results for the three Reynolds numbers in a single figure, showing the interface at four different time steps. This comparison, presented in Figure 9, allows for a visual assessment of the differences in interface evolution under varying vortex strengths. As the Reynolds number increases, the vortices cause greater and more rapid distortion of the interface, particularly in regions with high vorticity. To further quantify the results, we use the total length of the interface as an indicator of the degree of deformation. A more deformed interface corresponds to a larger total length, due to the increased surface area created by the vortical motion. The time evolution of this total interface length, corresponding to different vortex strengths, is presented in Figure 10. This analysis provides a clear and quantitative measure of how vortex strength influences the interface over time, offering deeper insights into the physical mechanisms underlying vortex–interface interactions.

3.2. Consideration of Surface Tension

This section focuses on investigating the role of surface tension in vortex–interface interactions, simplifying the problem by neglecting the density differences between the two phases, i.e., setting $r_p=1$ and varying the Weber number ($\mathit{We}$). The Reynolds number ($\mathit{Re}$) is fixed at 34,994 for the study. We explore three distinct values of $\mathrm{We}$, specifically $\mathit{We}=3.16$, $\mathit{We}=0.633$, and $\mathit{We}=0.316$, with the corresponding interface evolution shown in Figure 11, Figure 12 and Figure 13. As in Section 3.2, the total length of the interface is used as a quantitative indicator to assess the degree of interface deformation, and the time evolution of the interface length is analyzed, with results presented in Figure 14.

When $\mathrm{We}$ is large, the dynamics of the interface primarily change in regions with sharp interface curvature, where the curvature is more pronounced. This behavior can be explained by Equation (10) for $r_p=1$, which suggests that vorticity generation at the interface is influenced by the curvature gradient:

$$\mathsf{\Sigma }=-{\displaystyle \frac{1}{We}}{\displaystyle \frac{\partial \kappa }{\partial s}}$$

(11)

Here, the source term is directly proportional to the curvature gradient along the interface, implying that stronger curvature leads to greater vorticity generation. Even for relatively high $\mathit{We}$, the formation of a distinct liquid film edge can be observed, as shown in Figure 11 and Figure 12. As surface tension increases (i.e., $\mathit{We}$ decreases), the size of the liquid film edge grows. This is a direct consequence of increased surface tension, which stabilizes the interface and promotes the formation of a sharper, more distinct liquid film. This effect is particularly evident when comparing Figure 11 and Figure 12, where a more pronounced liquid film edge is observed with lower $\mathit{We}$.

Additionally, there exists a critical range for $\mathit{We}$, below which surface tension prevents significant curling phenomena from occurring. This is illustrated in Figure 13, where no notable interface deformation occurs for $\mathit{We}$ values below this threshold. In this regime, a vortex layer can detach from the sharp interface, leading to vortex shedding and the formation of a dipole, accompanied by the generation of capillary waves. This mechanism can be understood as the transition of the interface from a stable state to one where surface tension induces more complex dynamic effects, such as vortex formation and wave propagation. The observed behavior highlights the critical role of surface tension in shaping the interface dynamics, driving the formation of intricate fluid structures that evolve over time.

As surface tension increases, the size of the liquid film edge continues to grow, and the vorticity generated at the interface also increases, as shown in Figure 15. This figure provides an intuitive comparison of the changes in vorticity for different $We$ values. From Figure 15, we can find that the vorticity generated at the interface could exceed the peak vorticity of the initial vortex. To quantify this effect, we analyze the total vorticity generated near the interface and present a comparison in Figure 16. However, the differences between $We=0.633$ and $We=0.316$ are not particularly significant, as shown in Figure 17, where the maximum vorticity strengths over time are compared. The similar vorticity field structures lead to comparable dimensionless vorticity values.

Despite this, as $We$ decreases, the size of the vorticity field grows. However, the reduction in curvature is compensated by the increase in surface tension, which keeps the peak vorticity relatively constant. This indicates that the effect of surface tension becomes more significant as $We$ decreases, with the interface dynamics increasingly governed by surface forces rather than inertial forces. The vorticity generated at the interface leads to the creation of a velocity field, which can be estimated using the following approach: First, the vorticity from the deformation vortices in the total vorticity field is removed to obtain the filtered vorticity field $\tilde{\omega }$. Then, the velocity potential $\psi$ is determined by solving the following equation:

$${\nabla }^2\psi =-\tilde{\omega }$$

(12)

Subsequently, the velocity field is computed using $u_x={\displaystyle \frac{\partial \psi }{\partial y}}$ and $u_y=-{\displaystyle \frac{\partial \psi }{\partial x}}$. Figure 18 shows that the velocity field is predominantly dipolar in nature, with the situation at $t=2$ being particularly pronounced. This velocity field originates from the formation of the liquid film edge, and the size and formation of this edge depend on $We$. As $We$ increases, the size of the liquid film edge decreases, and the formation time increases. This suggests that as surface tension rises, the interface dynamics become more stable and evolve more slowly.

The numerical results in this section also reveal that, regardless of the $\mathit{We}$ value, the dynamic properties of the interface remain relatively unchanged over a broad range of Reynolds numbers ($\mathrm{Re}$). Notably, for higher values of $\mathit{We}$, the effect of $\mathrm{Re}$ on the interface shape becomes negligible, even over extended periods. This observation suggests that, at high surface tension, the primary factors influencing interface dynamics are surface tension and curvature, rather than the Reynolds number. As a result, the interface dynamics stabilize under these conditions, with surface tension playing a dominant role in determining the behavior of the interface.

3.3. The Effect of Density Ratio on the Interface Evolution

Next, we focus on studying the impact of the density ratio between the two phases on the interaction between the vortex and the interface, while neglecting the effects of surface tension, i.e., setting $We=\infty$ and varying the phase density ratio $r_p$. The Reynolds number ($Re$) is fixed at 34,994 for all simulations. Under the influence of the existing vortex, the interface begins to curl. However, unlike the passive interface behavior observed in Section 3.2, Figure 19 and Figure 20 show that even when the density ratio $r_p$ is close to 1, the sharp interface undergoes rapid and significant changes. These deformations are much more pronounced compared to the passive interface behavior described in Section 3.1. In this case, the source term generates a vorticity layer at the interface, rather than a localized dipole near high-curvature points. This can be explained by Equation (10) for $We=1$:

$$\mathsf{\Sigma }={\displaystyle \frac{r_p-1}{2r_p}}{\displaystyle \frac{\partial p_m}{\partial s}}$$

(13)

Here, the source term is proportional to the pressure gradient along the interface. As the density difference between the phases increases, the pressure differential across the interface becomes more pronounced, leading to greater vorticity generation. This explains why, with increasing density differences, the interface exhibits more intense dynamic interactions. The increased pressure differential drives stronger vortex formation and more significant deformation of the interface. This study emphasizes the critical role of density differences in vortex–interface interactions, demonstrating that even small variations in the density ratio can significantly influence the interface’s response to vortex dynamics.

To explore the physical mechanisms behind these observations, we consider a rough approximation, assuming that the observed vortex behaves similarly to the passive spiral interface described in Section 3.1, with the pressure field primarily driven by the undisturbed Lamb–Oseen vortex. This approximation evaluates the vorticity generated along the spiral as follows:

$${\displaystyle \frac{\partial p_m}{\partial r}}{\displaystyle \frac{\partial r}{\partial s}}={\displaystyle \frac{u_{\theta }^2}{r}}{\displaystyle \frac{\partial r}{\partial s}}={\displaystyle \frac{u_{\theta }^2}{r}}{\displaystyle \frac{\partial r}{\partial {\Phi }_0}}{\displaystyle \frac{1}{\frac{\partial s}{\partial {\Phi }_0}}}$$

(14)

Unlike in Section 3.2, where the source term is concentrated near high-curvature points, here the vorticity source is more evenly distributed along the interface, resulting in a broader vorticity layer. This more uniform distribution of vorticity contributes to the observed dynamics and highlights the differences in vortex–interface interactions between the two scenarios.

Next, we discuss two cases with different density ratios: Case 1, where $r_p>1$, and Case 2, where $r_p<1$. For $r_p>1$, the dynamics at the sharp interface change rapidly, with small-scale structures appearing at the interface tip, as shown in Figure 20, Figure 21 and Figure 22. The vorticity source term, as approximated by Equation (14), shows that the vorticity at the upper part of the interface is positive (${\mathsf{\Phi }}_0>0$), while at the lower part, it is negative (${\mathsf{\Phi }}_0<0$). Figure 23 illustrates the velocity field generated by the interface vorticity, calculated using Equation (12). This additional velocity field enhances the effect of the existing vortex velocity field, leading to greater stretching of the liquid film and ultimately resulting in fluid fragmentation near the interface tip, as seen in the formation of small fluid droplets. This effect is more pronounced for higher $r_p$, as larger density differences promote faster vortex shedding and interface deformation.

The enhanced stretching of the liquid film is a direct result of the self-amplifying mechanism associated with the vorticity layer generated at the interface. As surface tension increases, the liquid film edge grows larger, and the vorticity generated at the interface intensifies. This stretching leads to exponential growth, which explains the observation of the sharp interface evolving with exponential thinning behavior, as the vorticity layer amplifies the deformation process.

For $r_p>1$, the interface becomes unstable on the outer side due to the existing vortex velocity field. The initial vorticity generated at the interface also induces shear effects, which further intensify the generalized Rayleigh-Taylor (RT) instability driven by the vortex dynamics. This shear effect accelerates the breakup of the fluid near the interface tip, as illustrated by the comparison between Figure 7 and Figure 22. As the vorticity generated at the interface increases, the velocity induced by this additional vorticity eventually surpasses the velocity induced by the existing vortex, causing the vorticity to roll up and shed from the tip, forming a new vortex (see Figure 22).

This new vortex, in turn, influences the interface’s dynamic response, triggering the formation of a heavy liquid (liquid 2) film. The rate of vorticity generation and the intensity of this effect strongly depend on the density ratio $r_p$, as the differential momentum between the phases governs the overall vorticity generation at the interface.

In contrast, for the case where $r_p<1$, the sign of the vorticity generated in Equation (13) changes. This change is clearly visible when comparing Figure 19 with Figure 18, Figure 24 and Figure 25. The reversal of the vorticity sign results in a relative velocity that opposes the existing vortex velocity field, as seen in Figure 26. This reversed vorticity induces a distinct interface evolution: compared to the single-phase case, the liquid film in this condition becomes thicker. Similar to the case where ${\mathrm{r}}_{\mathrm{p}}>1$, a localized centrifugal Rayleigh-Taylor instability is still observed, confined to the inner side of the interface (see Figure 18, Figure 24, and Figure 25) where the local density gradient opposes the effective centrifugal gravity.

When comparing the results of $r_p<1$ and $r_p>1$, the interface evolution comparison is shown in Figure 27. From Figure 27, it is evident that the liquid film in the $r_p<1$ case breaks more quickly (is more prone to shedding) than in the $r_p>1$ case. However, the overall deformation of the interface at the sharp point is slower and less intense than in the $r_p>1$ case. As the density ratio $r_p$ increases, the liquid film breaks more quickly, and the deformation becomes more violent. Quantitative analysis of these effects, considering the total vorticity generated near the interface, is shown in Figure 28. It is observed that the total vorticity generated decreases as the density ratio approaches 1, which is corroborated by the comparison in Figure 27. For the case with $r_p=2$, the vorticity grows more rapidly, and this quantitative result supports the observation in Figure 27, where the interface deformation near the sharp point is faster and more violent for higher $r_p$.

Further, two additional quantitative analysis plots focusing on the maximum vortex strength and the formed interface length are shown in Figure 29 and Figure 30. From these plots, it is clear that the dissipation of the initial vortex in the $r_p<1$ case occurs more quickly (Figure 29), while Figure 30 visually shows that as $r_p$ increases, the overall interface change becomes more intense.

3.4. The Together Effects of Density and Surface Tension

Building upon the work presented in the previous two sections, this section explores the combined effects of surface tension and phase density ratio on vortex–interface interactions. Specifically, we set the phase density ratio $r_p\ne 1$ and vary the Weber number $We$ to analyze how these two factors jointly influence the dissipation of the initial vortex and the generation of vorticity at the interface.

For this study, we select values for the phase density ratio $r_p$ as 1.1, 1.5, and 2, and values for $We$ as 3.166, 0.633, and 0.316, with $Re=\mathrm{34,994}$. The corresponding interface evolution for these different conditions is shown in Figure 31, Figure 32 and Figure 33, where the vertical axis corresponds to the values of $r_p$ (1.1, 1.5, and 2), and the horizontal axis corresponds to the values of $We$ (3.166, 0.633, and 0.316). The comparison is made at three different time steps, $t=0.5$, 1, and 1.5.

For a more intuitive comparison of how the interface evolves over time with varying $r_p$ and $We$, we present Figure 34, which clearly illustrates the effects of these two parameters on vortex–interface interactions, as described in Equation (10). From Figure 34, we can observe that when the density ratio $r_p$ is close to 1, interface breakage occurs only under high $We$ (low surface tension). Under low $We$ (high surface tension), the sharp interface becomes stretched into a liquid film, increasing the surface area without breakage.

However, as the density ratio $r_p$ increases, even under low $We$ and high-surface-tension conditions, the interface becomes more prone to breakage at the sharp points. This can be explained by the fact that increasing the density contrast between the two phases creates a larger differential momentum between the phases, enhancing the vorticity generation at the interface. The larger the density ratio, the greater the momentum transfer between the phases, which leads to faster deformation of the interface and more rapid vortex dissipation. As a result, the initial vortex dissipates more quickly when the density contrast is higher, even in the presence of high surface tension.

On the other hand, surface tension plays a stabilizing role at the interface. When $We$ is large (low surface tension), the interface is more susceptible to deformation and breakage because surface forces are weaker, allowing the vortices to cause more pronounced interface instabilities. As $We$ decreases (increasing surface tension), the interface becomes more stable, and the vortex-induced deformation is slower. This effect is clearly visible in the comparison between Figure 11 and Figure 12, where lower $We$ values lead to the formation of larger liquid film edges, while higher $We$ values result in sharper, more unstable interfaces.

For high $We$ (low surface tension), we observe that as the density ratio $r_p$ increases, the rate at which new vorticity layers form at the interface also increases, and the vorticity generated becomes more intense. This is due to the larger density contrast between the phases, which promotes faster momentum transfer and strengthens vortex activity at the interface. However, when surface tension increases (i.e., when $We$ decreases), the vorticity generated at the interface becomes larger, but the rate of generation of new vorticity layers slows down. This is due to the stabilizing effect of surface tension, which reduces the intensity of vortex formation while allowing for a more gradual deformation of the interface.

In summary, the combined effects of surface tension and density ratio significantly influence the vortex–interface interaction. As the density ratio $r_p$ increases, the interface becomes more prone to breakage due to the stronger momentum differential between the phases. Surface tension moderates this effect, with high surface tension (low $We$) stabilizing the interface and reducing vortex-induced deformation, while low surface tension (high $We$) allows for more pronounced interface breakage and vortex shedding. The balance between these two factors dictates the dynamics of the interface and the vortex behavior.

4. Conclusions

This study primarily uses numerical simulations to investigate the interaction between vortices and the interface, focusing on the effects of phase density differences and surface tension on the interaction mechanisms from a two-dimensional perspective. The main emphasis of the research is on the generation and evolution of the vorticity layer at the interface, which serves as the basis for comparison in vortex–interface interaction simulations.

The main conclusions drawn from the numerical simulations are as follows:

First, in the absence of density differences, surface tension acts as the primary stabilizing force. As surface tension increases (i.e., decreasing Weber number), the interface forms a distinct, expanding liquid film edge rather than breaking up. A critical Weber number exists, below which significant curling is suppressed, leading to the formation of dipoles and capillary waves. Notably, while higher surface tension results in a larger total integrated vorticity due to the extended interface length, the peak vorticity magnitude remains relatively constant. This indicates that surface tension regulates the spatial distribution and coherence of vorticity rather than its maximum intensity.

Second, the introduction of a density difference alters the dynamics through the baroclinic torque mechanism. This effect generates intense secondary vorticity along the interface, which induces an additional velocity field that enhances film stretching and thinning. Crucially, when this baroclinic vorticity becomes sufficiently strong, its self-induced velocity surpasses that of the primary vortex, triggering the roll-up and shedding of secondary vortices from the interface tip. This process accelerates ligament breakup and droplet formation, highlighting that the density gradient is the primary driver for topological instabilities and vortex shedding.

Finally, the combined analysis reveals that the interface fate is determined by the competition between baroclinic destabilization and capillary stabilization. For density ratios near unity, the interface remains stable and forms a liquid film unless surface tension is very low. However, as the density ratio increases, the enhanced baroclinic torque can overcome even strong capillary forces, rendering the interface prone to rupture and facilitating vortex shedding even at low Weber numbers. Thus, while surface tension governs the accumulation of vorticity, the density difference serves as the catalyst for accelerating vorticity generation and triggering the transition from stable deformation to breakup. While the present study focuses on deterministic interactions, future work will extend this analysis to stochastic flow fields to assess the statistical properties of the breakup process under random perturbations.