Air-water interface dynamics and energy transition in air of a sphere passed vertically upward through the interface

The water exit problem is a fundamental problem in fluid dynamics, and the behavior of the air-water interface and the energy transition of an object exiting water have not been thoroughly investigated. In this study, a solid sphere with a density of 2.64 x 10(3) kg/m(3) and diameter of d = 25.4 mm was launched vertically upward in water toward the air-water interface. The motion of the sphere and the behavior of the interface were investigated for varying submergence depths H from the launch position of the sphere to the interface. The launch velocity was set so that the Reynolds number immediately after the sphere passed the air-water interface was about 3000 for all cases of H. The kinetic and potential energies of the sphere and the energy lost because of the air-water interface (i.e., interfacial containing energy) were estimated based on the classical law of energy conservation. For H/d <= 3, the ratio between the kinetic energy immediately after passing through the air-water interface and the potential energy at the maximum displacement position decreases with increasing H, but this energy ratio takes a constant value of 0.57 for H/d >= 4. Additionally, for H/d >= 4, kinetic energy is transformed to potential energy and interfacial containing energy at a fixed ratio for each vertical position. The spreading characteristics of the water mass after the sphere has passed the air-water interface and the thickness and width of the interfacial water sheet when the top of the sphere reached the calm free surface were investigated by focusing on their relationship to the energy distribution. For H/d >= 4, where the energy ratio takes a constant value, the increase in the rate of spread of the water mass with increasing H/d is clearly smaller than that of H/d <= 4. Furthermore, for H/d >= 4, the ratio between the thickness and width of the interfacial water sheet is constant. In other words, in the region where the ratio between the kinetic energy of the sphere immediately after passing through the air-water interface and the potential energy at the maximum displacement position has a constant value, the shape of the interfacial water sheet is self-similar. These findings contribute to the determination of variable parameters when modeling the water exit problem.