Dynamics of jet formation and collapse for axisymmetric surface gravity waves: coupled 3D potential flow and SPH simulations

TL;DR

The paper addresses axisymmetric gravity-wave breaking with large jet heights by coupling a fully nonlinear potential-flow solver (OceanWave3D) with a GPU-accelerated SPH solver (DualSPHysics) in OceanSPHysics3D. This fully three-dimensional framework resolves the entire sequence from directional focusing and primary curvature-collapse jet formation to inertial cavity collapse and secondary-jet generation, validated against McAllister et al. experiments and Longuet-Higgins theory. A key finding is that the primary jet arises from curvature collapse while the secondary jet results from an inertial cavity collapse, with distinct self-similarity properties and extreme local accelerations; the study also demonstrates substantial computational savings (~90%) versus full-basin SPH. The work establishes a robust platform for exploring extreme free-surface phenomena and lays groundwork for future two-phase, air-entrainment, and multi-resolution extensions to capture coupled gas–liquid dynamics in 3D.

Abstract

Axisymmetric waves occur across a wide range of scales. This study analyses large-scale gravity-dominated axisymmetric waves, with jet heights of up to 6 m, for which surface-tension effects are negligible. The Bond number is O(10^5) and the Weber number ranges from O(10^4) to O(10^6). Our aim is to clarify the dynamics of highly nonlinear axisymmetric jet formation, cavity collapse and the consequent generation of secondary jets. The newly developed three-dimensional framework OceanSPHysics3D, combining unsteady potential flow with smoothed particle hydrodynamics, enables full simulation of jet initiation and collapse. The computed free-surface elevations and jet evolution agree well with the experiments of McAllister et al. (Journal of Fluid Mechanics, 2022) and with an analytical jet-tip-angle formulation by Longuet-Higgins (Journal of Fluid Mechanics, 1983). The simulations elucidate how the falling primary jet induces a secondary jet. The mechanisms forming the pre-jet trough and the post-jet cavity are fundamentally different. The pre-jet trough arises geometrically from directional focusing of the constituent waves, yielding a self-similar shape when appropriately scaled. In contrast, the post-jet cavity is formed inertially by the falling continuous jet and lacks both spatial and temporal self-similarity. Its collapse also differs: the cavity pinches off at the neck to generate upward and downward secondary jets, with local accelerations reaching approximately 150 times gravity. The primary jet scale governs the ensuing secondary-jet dynamics, including vortex-ring formation and strong vertical mixing. These findings illustrate the complexity of axisymmetric jet dynamics and demonstrate the ability of the present framework to reproduce the key coupled processes in such extreme free-surface events.

Figures (24)

• Figure 1: (a) Comparison between the experimental axisymmetric jet profile and the circular-basin SPH simulation Kanehira19, showing that SPH underestimates the jet height and fails to reproduce the sharp jet tip. (b) Conceptual diagram of the OceanSPHysics3D framework, in which a fully nonlinear potential-flow (FNPF) domain solves the large-scale wave propagation, while an embedded local SPH subdomain resolves wave breaking. The two solvers exchange information through the overlap region $\varOmega_{\mathrm{overlap}}$.
• Figure 2: Numerical domain in OceanWave3D (a) and the SPH subdomain (b). The SPH subdomain is highlighted by the blue rectangle in panel (a).
• Figure 3: Sequence of three-dimensional water free surface profiles simulated in the SPH for Exp. 50. Panels (a)--(c) represent curvature collapse to vertical jet evolution, whereas (d)--(f) show falling jet, secondary cavity formation and collapse.
• Figure 4: Sequence of three-dimensional water free surface profiles simulated in the SPH for Exp. 75. Panels (a)--(c) represent curvature collapse to vertical jet evolution, whereas (d)--(f) show falling jet, secondary cavity formation and collapse.
• Figure 5: Time series comparison of water surface elevation. Panels (a--c) show the surface elevations measured near the open boundary ($x=-1.37$), while (d--f) show the surface elevations at the centre of the simulation domain ($x,y=0$). WG denotes the wave-gauge measurements and Q denotes the Qualisys floating-marker measurements.