At low Froude numbers, vertical and oblique impacts of deformable spheres into water are performed. The dependence levels of the hydrodynamics, cavity formations, and stress responses on the oblique angle are investigated. Based on theoretical and numerical analyses, the differences between the vertical and oblique impacts in sphere deformations and cavity formations are clarified. Elastic forces inside the sphere induce different models of the deformation behaviors during oblique water entry, and the sphere with a smaller oblique angle extends the formation time of the oblate deformation and nested cavity. Over a single deformation period, the vertical hydrodynamic forces do not continuously increase with the oblique angle, and they depend on the deformation behavior. Additionally, the evolution in the high-stress area at the bottom of the sphere caused by the oblique motion is similar to a solar eclipse. Over the first deformation period, the energy change in the sphere with different oblique angles can be predicted from the dimensionless ratio of the material shear modulus to the impact hydrodynamic pressure.

This paper presents a comparative study on water entry and water walking events of elastic spheres using a fluid–structure interaction method. The sphere displacements, cavity formations, and hydrodynamic characteristics during the water entry and water walking events are investigated. When the impact angle decreases, an obliquely impacting elastic sphere on the water surface can undergo a transition from water entry events to water walking events. Based on the morphological classification of the cavity opening, the shapes of cavity openings formed by elastic spheres can be roughly predicted from material properties and impact conditions. Furthermore, the water walking event is characterized by three typical stages: impact below the surface, walking across the surface, and skipping from the surface. Each stage is analyzed in terms of sphere displacements and hydrodynamic characteristics. The results also show that the vertical hydrodynamic forces acting on elastic spheres are not always smaller than those acting on rigid spheres at all impact angles. The critical impact angle that results in an equivalent vertical hydrodynamic force acting on both elastic and rigid spheres can be predicted from the material properties and impact conditions of the elastic sphere.

The hollow cylinder is a special cylindrical body with a through structure in the middle position. A collaborative method based on the finite volume method and the finite element method is used to numerically calculate high-speed vertical water entry of the hollow cylinder. The results show that the hollow cylinder exhibits a unique internal jet phenomenon, and the height of the jet shows an almost linear growth trend. Compared to the completely sealed hollow cylinder, the displacement and velocity of the hollow cylinder after entering into water are greater, but the volume of generated cavities is smaller. The force and stress at the bottom of the hollow cylinder are smaller than those at the bottom of the completely sealed hollow cylinder. The force and stress at the bottom of the hollow cylinder are greater than those at its top position. The greater thickness of the hollow cylinder has greater velocity and displacement, as well as greater peak values of the force and stress at its bottom position. The stress fluctuation at the bottom of the hollow cylinder with a smaller thickness is more pronounced.

The importance of high-speed water entry is acknowledged within the defense industry. This study numerically investigates the water entry of a high-speed rectangle projectile, focusing on cavity dynamics and shock wave generation. A computational model is employed to accurately simulate the intricate fluid dynamics of compressible multiphase flows. This model integrates a dual-phase flow algorithm with a thermally sensitive Tait equation of state for the liquid phase. The primary focus lies in understanding the effects of fluid compressibility on cavity evolution and shock wave propagation across different Froude numbers. The findings reveal that compressibility induces changes in cavity formation size, leading to significant variations in phase composition within the cavity. Furthermore, compressibility enhances the air cushion effect upon surface impact, resulting in delayed water entry and concurrent reduction in projectile drag. Moreover, a prognostic model is proposed, correlating shock pressure with propagation distance, thereby validating theoretical hypotheses advanced by Lee et al. [J. Fluid Struct., 11, 819–844 (1997)].

A two-way fluid–structure interaction method is used to investigate the cavity evolution, trajectory, and motion characteristics of a hollow cylinder during oblique high-speed water entry. The finite volume method is used to handle the fluid domain, and the finite element method is used to handle the structural domain. Meanwhile, key data from the two computational domains are exchanged during each iteration process to achieve fluid–structure coupling simulation. The results indicate that a jet forms in the middle of the hollow cylinder, which will deflect and eventually collide toward the upstream side of the cavity. Due to the obstruction of the internal jet, the water entry angle of the hollow cylinder changes less than that of a completely sealed hollow cylinder. Compared to the completely sealed hollow cylinder, the completely open hollow cylinder has significantly smaller deformation and strain energy during water entry, resulting in better stability and durability. As the initial velocity increases, the displacement, fluid force, and stress of the completely open hollow cylinder all increase. As the initial water entry angle increases, the stress and strain energy at a specific angle (45°) are smaller than those of a completely sealed hollow cylinder. As the diameter increases, the stress and strain energy experienced by the completely open hollow cylinder gradually increase. As the thickness increases, the force and stress acting on the completely open hollow cylinder increase. As the length increases, the trajectory of the hollow cylinder becomes closer to a straight line.

Porous hollow sphere is a hollow sphere with holes arranged in a certain pattern inside. This article presents the results of the porous hollow sphere during water entry using high-speed photography technology. The main consideration is the influence of initial velocity and hole size on the evolution characteristics of cavity flow, splash crown, and jet. Due to the presence of obvious non-chamfered holes, the porous hollow sphere still generates significant cavities when vertically penetrating into water at a lower initial velocity. Meanwhile, it generates unique cavity and the crown which has obvious columns, with the same number of cavity columns as the holes in the middle of the porous hollow sphere. During water entry, there is a jet in the middle position at the top of the sphere. When the pinch-off phenomenon occurs, there is still a jet at this position, which keeps the upper and lower cavities in contact. The shape of the cavity changes from hyperbolic to triangular and then to W-shaped. As the initial velocity increases, the contour of cavity column becomes less distinct, the surface of cavity becomes smoother, and the number of cavities at the top of the porous hollow sphere increases after cavity closed.