Study of initial stage of entry of a solid sphere into shallow liquid with Synthetic Schlieren technique

Abstract

Vertical entry of a 213 mm diameter solid sphere moving with a low speed of 16–40 mm/s onto a shallow liquid layer is investigated experimentally. The focus of the study is on deformation of the free surface before the actual contact between solid and liquid in presence of air cushioning. Owing to local character of air cushioning, the sphere is modeled by a spherical segment. The measurements were conducted with Synthetic-Schlieren technique, adapted to the present experimental configuration. In the studied range of parameters, the value of the Weber number is order O(1), giving rise to capillary oscillations of the free surface below the sphere. The primary solid-liquid contact at higher impact speeds (higher Weber number) occurs due to growth of a circular rim; the radius of contact is predicted satisfactory by existing theories assuming the lubrication/inviscid balance as a key mechanism. At lower speeds (lower Weber number), the contact is significantly delayed, being related with the effects due to surface tension and final horizontal extent of the spherical segment. In case of delayed contact, the evolution of free surface can be studied over a significant time span: the radial profile of the rim evolves into a flat-top structure, reminiscent of axisymmetric bore wave. The front of this wave propagates with constant speed, defined (with a good accuracy) by the condition that group velocity is equal to phase velocity of the waves. This condition is satisfied in i) the long-wave limit and ii) for the capillary waves of lowest admissible phase velocity. The second case is observed in the present experiments. The capillary waves generated by the impact “ride” on the top of the bore wave. The bore wave itself generates capillary waves in front of it, thereby producing a structure reminiscent of dispersive shock waves. Phase velocity and wavenumber of capillary waves are described well by linear theory of gravity-capillary waves in a liquid of finite depth.