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

Axisymmetric waves occur across a wide range of scales. This study analyses large-scale (with a jet height of up to 6 m) gravity-dominated axisymmetric waves (Bond number (Formula presented) and Weber number (Formula presented)), for which surface-tension effects are negligible. 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 SPH, provides an integrated platform capable of handling multidirectional and axisymmetric waves, thereby enabling the full simulation of axisymmetric jet initiation and collapse. The computed free-surface elevations and jet evolution agree well with the experiments of McAllister et al. (2022 J. Fluid Mech. 935, A5) and an analytical jet-tip-angle formulation of Longuet-Higgins (1983 J. Fluid Mech. 127, 103–121). The simulations elucidate how the falling primary jet induces a secondary jet. The mechanisms forming the pre-jet trough and 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 (Formula presented). 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.