This paper summarizes and extends some mathematical results for a model for a class of water-entry problems characterized by the geometrical property that the impacting body is nearly parallel to the undisturbed water surface and that the impact is so rapid that gravity can be neglected. Explicit solutions for the pressure distributions are given in the case of two-dimensional flow and a variational formulation is described which provides a simple numerical algorithm for three-dimensional flows. We also pose some open questions concerning the well-posedness and physical relevance of the model for exit problems or when there is an air gap between the impacting body and the water.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112091001076

Incompressible water-entry problems at small deadrise angles

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

A jamming mechanism for the observed phenomenon of the sudden hardening of suspensions of micrometre-sized particles on impact (enough to enable a person to run over them) is described and quantified. If you walk fast enough, they say, you can walk over quicksand that would quickly trap the immobile and unwary. The forces involved in that feat are explained in this study. Liquids typically flow around an intruding object, but dense aqueous suspensions of micrometre-sized particles can harden under impact. Shear thickening — a tendency of the sheared suspension to dilate — is often invoked to explain the temporary hardening of such liquids, but is difficult to reconcile with the magnitude of such effects. Here, Scott Waitukaitis and Heinrich Jaeger demonstrate that the remarkable impact resistance is produced by a different mechanism. Using detailed imaging to capture the dynamics of the process — modelled by an aluminium rod striking a dense suspension of cornflour and water — they find that the stresses originate from an impact-generated solidification front that transforms an initially compressible particle matrix into a rapidly growing jammed region. Although liquids typically flow around intruding objects, a counterintuitive phenomenon occurs in dense suspensions of micrometre-sized particles: they become liquid-like when perturbed lightly, but harden when driven strongly1,2,3,4,5. Rheological experiments have investigated how such thickening arises under shear, and linked it to hydrodynamic interactions1,3 or granular dilation2,4. However, neither of these mechanisms alone can explain the ability of suspensions to generate very large, positive normal stresses under impact. To illustrate the phenomenon, such stresses can be large enough to allow a person to run across a suspension without sinking, and far exceed the upper limit observed under shear or extension2,4,6,7. Here we show that these stresses originate from an impact-generated solidification front that transforms an initially compressible particle matrix into a rapidly growing jammed region, ultimately leading to extraordinary amounts of momentum absorption. Using high-speed videography, embedded force sensing and X-ray imaging, we capture the detailed dynamics of this process as it decelerates a metal rod hitting a suspension of cornflour (cornstarch) in water. We develop a model for the dynamic solidification and its effect on the surrounding suspension that reproduces the observed behaviour quantitatively. Our findings suggest that prior interpretations of the impact resistance as dominated by shear thickening need to be revisited.

https://jfi.uchicago.edu/~jaeger/group/JaegerGroupPapers/suspensions/nature11187.pdf

Impact-activated solidification of dense suspensions via dynamic jamming fronts

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.

/pdf/water-entry-of-spinning-spheres-1hghk0kzsb.pdf

Water entry of spinning spheres

We present a study of the forces during free-surface water entry of spheres of varying masses, diameters, and surface treatments. Previous studies have shown that the formation of a subsurface air cavity by a falling sphere is conditional upon impact speed and surface treatment. This study focuses on the forces experienced by the sphere in both cavity-forming and non-cavity-forming cases. Unsteady force estimates require accurate determination of the deceleration for both high and low mass ratios, especially as inertial and hydrodynamic effects approach equality. Using high-speed imaging, high-speed particle image velocimetry, and numerical simulation, we examine the nature of the forces in each case. The effect of mass ratio is shown, where a lighter sphere undergoes larger decelerations and more dramatic trajectory changes. In the non-cavity-forming cases, the forces are modulated by the growth and shedding of a strong, ring-like vortex structure. In the cavity-forming cases, little vorticity is shed by the sphere, and the forces are modulated by the unsteady pressure required for the opening and closing of the air cavity. A data-driven boundary-element-type method is developed to accurately describe the unsteady forces using cavity shape data from experiments.

Unsteady forces on spheres during free-surface water entry

Detailed experimental results are presented for the initial impact force on a sphere striking a horizontal liquid surface vertically at speeds in the range 1-3 m s−1. Results are discussed in terms of an impact drag coefficient. Liquids having viscosities in the range 10−3−102 Pa s have been studied. For low viscosities the results have been compared with the theoretical calculations of Shiffman & Spencer. Good agreement has been found in most respects; in particular the impact force varies as the square root of the depth for depths less than a tenth of the radius. The impact drag coefficient has also been studied through the transition from inertia to viscosity-dominated conditions. The variation of the impact drag coefficient is presented as a function of Reynolds number, and its variation in the range 5 × 10−2 < Re < 5 × 103 is shown to resemble that of a fully immersed sphere moving steadily in a homogeneous fluid.

M. Moghisi

Papers

An experimental investigation of the initial force of impact on a sphere striking a liquid surface