Water entry of a floating body into waves with air cavity effect

Highlights

• A fully nonlinear time-domain HOBEM is developed to investigate the impact problem between semi-circular floating body and Stokes wave with asymmetric air pocket.

• Two stretch coordinate systems are adopted to keep the mesh accurate.

• Some auxiliary functions are adopted to decouple the interaction of the floating body with degree of freedom and Stokes wave.

• Relatively complete parametric studies, such as the wave amplitude, wave length, initial wave phase, initial air pressure, prescribe floating body velocity, are conducted to investigate the problem.

Abstract

Based on a fully nonlinear time-domain higher-order boundary element method (HOBEM), the present paper investigates a semi-circular floating body entering obliquely into Stokes wave with air entrapment. In this model, a fifth-order analytical solution is used to simulate the Stokes wave as incident boundary condition. A long and thin jet layer is simulated and assumed to separate from wetted body surface along its tangential direction, which avoids fluid particular leaving or invading body surface. When inner free surfaces on both sides of body collide with each other, a closured air cavity is formed based on the assumption of adiabatic. Dual local stretched coordinate systems are applied to resolve the impact with air cavity effect and downward motion of inner jet due to the local impacts between two water surface which is mainly dynamic effects. Some auxiliary functions are employed to obtain the pressure distribution induced by wave impact. The developed model is verified against the published numerical results for water entry with air cavity effect in the absence of waves. Then, numerical simulations are undertaken to investigate the mechanism of wave entry with air cavity formation through designed parameters, i.e. wave amplitude, wave length, initial wave phase, initial air pressure, prescribed floating body velocity. Numerical results indicate that the presence of wave delays the formation of air pocket and leads to more asymmetric shape of air pocket with the increase of wave nonlinearity when the body impacts the wave peak. After the occurrence of air pocket, the fluid pressure sharply increases.

Introduction

Air cavity phenomenon is usually generated during the physical process of fluid/fluid and fluid/structure impact problems. For example, a floating body enters into water, leading to the large deformation of free surface with a formed gap. As time progresses, the inner surfaces beside the gap move closer to each other and then an air cavity is developed. Water entry also has extensive engineering application, such as the launching of lifeboat from a ship or a platform. Another practical example, when waves hit the coastal structures or offshore platforms, their path are suddenly blocked and then the air often is trapped to form an air cavity. These impacts with air cavity effects would lead to the dramatic increase of fluid pressure along the body surface with the jet root very rapidly. In addition, it was found that the major contribution to water entry with air cavity came from the pulsations of the air pocket formed behind the body (Richardson, 1955). The secondary water impact on the top surface of the body may cause damage at the upper surface of the body. However, the mechanism of air cavity occurrence and its effect on wave impact still remain not well-understood due to various factors, such as wave nonlinearity, compressibility of air, air cavity deformation and oscillation of air pressure.

Previous theoretical and numerical simulations of the impact problem induced by water entry are largely based on cases of a two-dimensional wedge with finite or infinite length. When infinite length of the wedge is considered, fluid always climbs up along the body surface to form a jet layer which needs some particular treatments to ensure sufficient numerical accuracy. At earliest studied stage, the jet layer was kept to be static and its effect on impact pressure was ignored because the pressure of jet surface was equal to the atmospheric pressure (Kármàn, 1929). Later, jet tip cutting method was adopted when the jet starts to overturn due to gravity effect. The distribution of fluid velocity and pressure within the thin jet layer is assumed to increase linearly, which avoids large number of small elements on jet surface. The proponents of this method are Battistin and Iafrati, 2003, Xu et al., 2010, Sun and Faltinsen, 2007, Sun et al., 2015a, Sun et al., 2015b, and Cheng et al., 2018a, Cheng et al., 2018b, Cheng et al., 2019a, Cheng et al., 2019b. However, the jet cutting technique would affect the development of water surface shape when the jet detachment along body surface occurs for the wedge with finite length, which further causes some numerical error as the gravity effect becomes dominant. To solve this problem, Bao et al. (2016) kept the motion of free jet in their simulations by imposing intersection condition between free surface and body surface, and thus the fluid velocity and pressure can be obtained directly without any numerical treatment. Tassin et al., 2014, Wang et al., 2015a, Wang et al., 2015b, Semenov and Wu, 2016 simulated two jet layers respectively to moves toward the left and right sides of the wedge and the performance is stopped before jet layer falls into water surface. Dias and Vanden-Broeck, 2011, Sun et al., 2015a, Sun et al., 2015b and Cheng et al., 2019a, Cheng et al., 2019b employed the domain decomposition method in boundary element method (BEM) to investigate the overlapping phenomenon of a jet falling into main fluid domain.

All these above studies are, however, for water entry without considering air cavity. As long as the entering speed is sufficiently large and time is enough long, an air cavity will be formed due to the collision of free surfaces, which makes impact problem becomes more complicate due to pressure oscillation and compressibility of the air. The problem of water entry with air cavity attracts plenty of researchers. For example, Gaudet (1998) applied a three-dimensional circular disks entering calm water based on the boundary element method to investigate the air cavity seal depth related to low Froude number. Duez et al. (2007) investigated the response of a superhydrophobic sphere entering into calm water under different entering velocity. Do-Quang and Amberg (2009) developed a 2D numerical model based on the Navier–Stokes equations to study the effect of spere surface wetting to splash. Iranmanesh and Passandideh-Fard (2017) employed a three-dimensional numerical approach based on the volume-of-fluid technique to solve horizontal circular cylinder entering water with air pocket. Wang and Faltinsen (2013) adopted a nonlinear boundary element method (BEM) to investigate air cavity formation during the high-speed water entry of wedge. Sun et al. (2019) used BEM to explore a floating body sinking into calm water with open cavity and closed bubble, which good simulate the whole process of air pocket formation including closure of the pocket and collision of the volume. Wang et al. (2019) modeled the unsteady hydrodynamic forces under different cross sections, i.e. circular cylinder and wedge, freely entering calm water surface with air cavity within the framework of potential flow. Besides, experimental method has been utilized to investigate cavity trajectory. Among typical work, Watson et al. (2018) experimentally investigated smooth, hydrophilic sphere freely into liquid including a single layer of fabric on the surface and found that the existence of fabric promoted formation of air cavity. Speirs et al. (2018) presented a sphere entering into a water–surfactant​ mixture, which showed that surfactant can change the surface tension and further alter the critical velocity for air pocket formation. Shepard et al. (2019) studied the dependent relation between body velocity and the submerging depth of cavity pinch-off using high-speed camera. Furthermore, Tan (2019) carried out experiments to investigate solid sphere entering into an oil–water system. Compared above experimental studies, it can be found that the flow field around the body becomes more complex, which is worth to conduct a more systematic investigation of the influence of air cavity on the pressure distribution and free surface.

The foregoing studies are limited to the structure entering into calm water without taking into account wave effects. In realistic conditions, waves generally change the fluid pressure distribution on body surface and play an important role in hydrodynamic interaction between structure and fluid when the wave length and wave height are longer related to the body dimension. Water entry of a floating body into waves with air cavity would lead to some new physical features which brings new challenges during simulation. Firstly, Contact location of the two depart flow is difficult to be determined due to the propagation characteristic of wave. In previous clam water entry studies, the inward two depart flows touch exactly on the axis of the structure symmetry and the shape of an air cavity is easily calculated. When waves exist, the impact location may be anywhere, leading to an uncertainty in the shape and volume of generated air cavity. Thus, the free surface must be tracked precisely at each time step to update the accurate time point and location of liquid–liquid impact. In such situation, the pressure distribution on floating body becomes asymmetric due to wave parameters i.e. initial wave phase and horizontal wave velocity, and the floating body moves with multi-degree of freedom. Thus, the merging of cavity surface cannot be treated as the same with the methods adopted in the studies of breaking waves hitting on a vertical wall or the vertical clam water entry, such as the works of Chan and Melville, 1988, Song and Zhang, 2018, Sun et al., 2018. Secondly, the presence of waves changes the deadrise angle between the body surface and free surface, which affect the development of the jet and leads to large deformation of free surface with gravity effect due to wave nonlinearity. This non-self-similar problem requires special attention on simulation of local impact region in time domain. Therefore, it is worth to conduct a more system and comprehensive study of the wave effects on water entry with air cavity.

The primary aim of the present work is to solve the impact problem induced by a semi-circular floating body obliquely entering into waves with air cavity effect. The whole numerical model is developed using a time-domain higher-order element method based on the potential flow theory with fully nonlinear conditions. The fifth-order stokes wave at infinite water depth is adopted as the incident wave. The formation of air cavity is assumed to be an adiabatic process. When the depart flow impact happens, dual stretched coordinate system is used to control the mesh quantity. This paper is organized as follows. In Section 2, the mathematical model is established based on the incompressible potential flow theory. Next, the model is verified against the previous numerical results in Section 3. In Section 4, effects of extensive parameters are studied, such as the wave height, wave length and the entering water velocity. And the detailed results are provided through free surface elevation, pressure distribution on floating body. In Section 5, the conclusions of this study are presented.