Abstract 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 × 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.

中文翻译：

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垂直向上通过界面的球体的空气-水界面动力学和能量转换

摘要 出水问题是流体动力学中的一个基本问题，空气-水界面的行为和物体出水的能量转移尚未得到彻底研究。在这项研究中，一个密度为 2.64 × 10 3 kg / m 3 且直径为 d = 25.4 mm 的实心球体在水中垂直向上发射到空气 - 水界面。针对从球体发射位置到界面的不同浸没深度 H，研究了球体的运动和界面的行为。发射速度设置为使球体通过空气-水界面后的雷诺数在所有 H 情况下立即约为 3000。球体的动能和势能以及由于空气-水界面而损失的能量（即, 界面包含能量）是根据经典能量守恒定律估计的。对于 H / d ⩽ 3 ，刚通过气水界面后的动能与最大位移位置的势能之比随着 H 的增加而减小，但该能量比对于 H / d 取恒定值 0.57 ⩾ 4 . 此外，对于 H / d ⩾ 4 ，对于每个垂直位置，动能以固定比率转换为势能和界面包含能量。重点研究了球体通过气水界面后水团的扩散特性以及球体顶部到达平静自由面时界面水片的厚度和宽度与能量分布的关系。对于 H / d ⩾ 4 ，在能量比取恒定值的情况下，随着 H / d 的增加，水团扩散速率的增加明显小于 H / d ⩽ 4 。此外，对于 H / d ⩾ 4 ，界面水层的厚度和宽度之间的比率是恒定的。换言之，在球体刚通过气水界面后的动能与最大位移位置的势能之比为恒定值的区域内，界面水层的形状为自相似的。这些发现有助于在模拟出水问题时确定可变参数。界面水层的厚度与宽度之比是恒定的。换言之，在球体刚通过气水界面后的动能与最大位移位置的势能之比为恒定值的区域内，界面水层的形状为自相似的。这些发现有助于在模拟出水问题时确定可变参数。界面水层的厚度与宽度之比是恒定的。换言之，在球体刚通过气水界面后的动能与最大位移位置的势能之比为恒定值的区域内，界面水层的形状为自相似的。这些发现有助于在模拟出水问题时确定可变参数。