We present the results of a combined experimental and theoretical investigation of the normal impact of hydrophobic spheres on a water surface. Particular attention is given to characterizing the shape of the resulting air cavity in the low Bond number limit, where cavity collapse is driven principally by surface tension rather than gravity. A parameter study reveals the dependence of the cavity structure on the governing dimensionless groups. A theoretical description based on the solution to the Rayleigh–Besant problem is developed to describe the evolution of the cavity shape and yields an analytical solution for the pinch-off time in the zero Bond number limit. The sphere's depth at cavity pinch-off is also computed in the low Weber number, quasi-static limit. Theoretical predictions compare favourably with our experimental observations in the low Bond number regime, and also yield new insight into the high Bond number regime considered by previous investigators. Discrepancies are rationalized in terms of the assumed form of the velocity field and neglect of the longitudinal component of curvature, which together preclude an accurate description of the cavity for depths less than the capillary length. Finally, we present a theoretical model for the evolution of the splash curtain formed at high Weber number and couple it with the underlying cavity dynamics.

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Water entry of small hydrophobic spheres

The free-surface impact of solid objects has been investigated for well over a century. This canonical problem is influenced by many physical parameters, including projectile geometry, material properties, fluid properties, and impact parameters. Through advances in high-speed imaging and visualization techniques, discoveries about the underlying physics have improved our understanding of these phenomena. Improvements to analytical and numerical models have led to critical insights into cavity formation, the depth and time of pinch-off, forces, and trajectories for myriad different impact parameters. This topic spans a wide range of regimes, from low-speed entry phenomena dominated by surface tension to high-speed ballistics, for which cavitation is important. This review surveys experimental, theoretical, and numerical studies over this broad range, utilizing canonical images where possible to enhance intuition and insight into the rich phenomena.

Water Entry of Projectiles

The complex hydrodynamics of water entry by a spinning sphere are investigated experimentally for low Froude numbers. Standard billiard balls are shot down at the free surface with controlled spin around one horizontal axis. High-speed digital video sequences reveal unique hydrodynamic phenomena which vary with spin rate and impact velocity. As anticipated, the spinning motion induces a lift force on the sphere and thus causes significant curvature in the trajectory of the object along its descent, similar to a curveball pitch in baseball. However, the splash and cavity dynamics are highly altered for the spinning case compared to impact of a sphere without spin. As spin rate increases, the splash curtain and cavity form and collapse asymmetrically with a persistent wedge of fluid emerging across the centre of the cavity. The wedge is formed as the sphere drags fluid along the surface, due to the no-slip condition; the wedge crosses the cavity in the same time it takes the sphere to rotate one-half a revolution. The spin rate relaxation time plateaus to a constant for tangential velocities above half the translational velocity of the sphere. Non-dimensional time to pinch off scales with Froude number as does the depth of pinch-off; however, a clear mass ratio dependence is noted in the depth to pinch off data. A force model is used to evaluate the lift and drag forces on the sphere after impact; resulting forces follow similar trends to those found for spinning spheres in oncoming flow, but are altered as a result of the subsurface air cavity. Images of the cavity and splash evolution, as well as force data, are presented for a range of spin rates and impact speeds; the influence of sphere density and diameter are also considered.

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Water entry of spinning spheres

Substantial discrepancy in the conditions for attainment of different closure modes of a ventilated supercavity has existed widely in the published literature. In this study, supercavity closure is investigated with an objective to understand the physical mechanisms determining closure formation and transition between different closure modes and to reconcile the observations from prior studies under various flow settings. The experiments are conducted in a closed-wall recirculating water tunnel to image ventilated supercavity closure using high speed and high-resolution photography and simultaneously measure pressure inside the cavity. The flow conditions are varied systematically to cover a broad range of water velocity, ventilation flow rate and cavitator size, which correspond to different Froude numbers, air entrainment coefficients and blockage ratios, respectively. In addition to the classical closure modes reported in the literature (e.g. re-entrant jet, twin vortex, quad vortex, etc.), the study has revealed a number of new closure modes that occur during the transition between classical modes, or under very specific flow conditions. Closure maps are constructed to depict the flow regimes, i.e. the range of Froude number and air entrainment coefficient, for various closure modes at different blockage ratios. From the closure map at each blockage ratio, a critical ventilation flow rate, below which the supercavity collapses into foamy cavity upon reduction of Froude number, is identified. The air entrainment coefficients corresponding to such critical ventilation rate are found to be independent of blockage ratio. It has been observed that in the process of generating a supercavity by increasing ventilation flow rate, the cavitation number gradually reduces to a minimum value and stays fixed upon further increments in the ventilation rate. Once a supercavity is formed, the ventilation rate can be decreased to a much lower value with no change in cavitation number while still maintaining a supercavity. This process is accompanied by a change in closure modes, which generally goes from twin vortex, to quad vortex, and then to re-entrant jet. In addition, the blockage effect is shown to play an important role in promoting the occurrence of twin-vortex closure modes. Subsequently, a physical framework governing the variation of different closure modes is proposed, and is used to explain mode transition upon the change of flow conditions, including the blockage effect. This framework is further extended to shed light on the occurrence of closure modes for ventilated supercavitation experiments across different types of flow facilities, the natural supercavity closure and the pulsating supercavity reported in the literature. Finally, in combination with a recent numerical study, our research discusses the role of the internal flow physics on the observed features during supercavity formation and closure-mode transition, paving the way for future investigations in this direction.

An experimental investigation into supercavity closure mechanisms

We report the experimental observation of a well-defined rippling of the air cavity entrained by a rapidly moving solid object entering the free surface of a liquid (water or ethanol). The ripples are fixed in the lab frame, and begin just after the pinch-off (deep seal) of the cavity, simultaneous with the acoustic emission. This acoustic resonance corresponds approximately to the Minnaert frequency for volume oscillations of the bubble. We present an irrotational model which explains the ripples as a spatial rectification of these volume oscillations by the surface of the moving object.

Cavity ripples observed during the impact of solid objects into liquids

When a high-speed projectile penetrates into water, a cavity is formed behind the projectile. The gas enclosed in the cavity experiences a nonequilibrium process, i.e., the gas pressure decreases as the projectile moves more deeply into water. As a result, the cavity is sealed near the free surface (surface closure) and subsequently the cavity breaks up in water (deep closure). Accompanying the break-up of the cavity, secondary shock waves appear. This is the so-called supercavitation in water entry. We describe an experimental investigation into the water entry phenomenon. Projectiles of 342 m/s were generated from a small-bore rifle that was fixed vertically in the experimental facility. The projectiles were fired into a windowed water tank. A shadowgraph optical observation was performed to observe the entry process of the projectile and the formation and collapse of the cavity behind the projectile. A number of interesting observations relating to the motion of the free surface, the splash, the underwater bubbly flow and so on were found

Takuya Takami

Papers

Optical Observation of the Supercavitation Induced by High-Speed Water Entry