Experiments and numerical methods are developed to investigate the water entry of a freefall wedge with a focus on the evolution of the pressure on the impact sides (the side contacting water) and the top side (the dry side on the top of the wedge), evolution of the global hydrodynamic loads, evolution of the air-water interface, and wedge motion. It is found that a typical water entry of a freefall wedge can be divided into slamming, transition, collapse and post-closure stages. A single-fluid numerical model is presented to simulate the first three stages. The results are compared to experiments and good agreements are obtained. A two-fluid BEM is proposed to investigate the influence of the air flow before the closure of the cavity created on the top of the wedge. It is found that for the closure of the 2D cavity, the air flow starts to play an important role just before closure but due to the short duration, the influence of air flow on the body velocity and configuration of the air-water interface is limited.

We investigate the unsteady hydrodynamic force of solid objects vertically entering water with an air cavity behind the falling body. Physical models are proposed to represent the force components corresponding to the body acceleration, the gravity and the velocity of the body and the fluid particles. The theoretical or numerical solutions of the physical models are presented to understand the evolution of the force components. The body-acceleration force component is expressed as the high-frequency added mass times the body acceleration. Near the undisturbed free surface, the added mass grows strongly with increasing the submerged depth. It tends to be steady after the submerged depth is greater than a few characteristic lengths. The gravity force component consists of an upward hydrostatic term and a downward dynamic term. Generally, the hydrostatic term, which is obtained by integrating the gravity term in the Bernoulli's equation over the wetted body surface, is much larger than the gravity force component. For the threedimensional bodies, the gravity force component is found to vary as a power of the submerged depth, where the exponent is about 0.83. The velocity force component is represented as the drag coefficient defined by the V-squared law, which is characterized by the body geometry. The drag coefficient may experience three successive stages with increasing the submerged depth.

A high-order Harmonic Polynomial Method (HPM) is developed for solving the Laplace equation with complex boundaries. The "irregular cell" is proposed for the accurate discretization of the Laplace equation, where it is difficult to construct a high-quality stencil. An advanced discretization scheme is also developed for the accurate evaluation of the normal derivative of potential functions on complex boundaries. Thanks to the irregular cell and the discretization scheme for the normal derivative of the potential functions, the present method can avoid the drawback of distorted stencils, i.e., the possible numerical inaccuracy/instability. Furthermore, it can involve stationary or moving bodies on the Cartesian grid in an accurate and simple way. With the proper free-surface tracking methods, the harmonic polynomial method has been successfully applied to the accurate and stable modeling of highly-nonlinear free-surface potential flows with and without moving bodies, i.e., sloshing, water entry and plunging breaker.

In this paper, a nonlinear boundary element method (BEM) is developed for investigating air cavity formation during the high-speed water entry of wedges. A technique is proposed for dynamic re-gridding of free surface boundaries. This technique applies to both equally and nonequally spaced grids, and it is able to suppress the numerical instabilities encountered using a BEM for simulating free surface flows. The authors also develop a purely numerical method to simulate nonviscous flow separation, which occurs when the flow reaches the knuckle of the wedge. The present nonlinear BEM has been verified by comparisons with similarity solutions. We also compare numerical results with experimental results. Finally, we give a numerical prediction of the evolution of the cavity until the closure of the cavity, and the influence of the initial entry velocity, wedge mass, and deadrise angle on the characteristics of the transient cavities is investigated.