Fully nonlinear investigation on water entry of a rigid paraboloid

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Abstract

Oblique water entry of a rigid paraboloid is investigated by fully nonlinear boundary element method. The deadrise angle of the paraboloid increases gradually from zero at the bottom tip and a slenderness coefficient is introduced to symbolize the shape of the paraboloid. Convergence study with respect to time step and element size is carried on to assure the numerical procedure. The free surface elevation and pressure distribution are depicted in time domain. The present solution is compared with the linearized Wagner method and the composite solution with spray jets to investigate the influence of exact fully nonlinear boundary conditions. As there is no sharp edge or tip around the paraboloid, the water flows over the body both from lateral side and down side in oblique entry, the stagnation point relative to the body moves towards the flow-coming direction, and the negative pressure can be observed on both lateral side and down side of the paraboloid. The horizontal speed has great influence on horizontal total force and little influence on vertical total force, while the effect of the slenderness coefficient just has the opposite effect.

Introduction

The determination of the impact force between marine structures and water is crucial in the process of structural design. Due to the impact, large free surface motion is induced and high hydrodynamic pressure is generated. For example, when a missile crashes into water, very short duration of impact gives a remarkable free surface motion and high magnitude of local pressure. The large local pressure acts on the slender body of the missile and a huge momentum as well as resultant sectional force is formed to alter its forward motion or even break off the structure. The methods for predicting the impact loads due to water entry have been extensively developed and applied in various industrial areas, such as slamming pressure prediction on ship hull, interaction between incident wave and offshore structures, water entry and exit of underwater vehicles, landing of water planes, etc [1], [2], [3]. Water entry as one typical form of the fluid-structure interaction problems poses great technical challenges when tracing the evolution of the free surface and coupled motion. Studies on water entry are attractive for both engineering and research purposes.

The earliest study on water entry can be traced back to Von Karman [4] who ignored the influence of free surface disturbance on wetted area of the impactor. While the ignored free surface deformation was considered by Wagner [5] on the basis of Karman's theory and then the Wagner theory was extensively used by latter researchers. As linearized boundary conditions and assumption of small deadrise angles were used in Wagner theory, the method was usually applicable for water entry of those blunt bodies at the initial stage. Cointe and Armand [6] and Howison et al. [7] employed the exact body shape and kinematic boundary condition to compute the fluid flow. Scolan and Korobkin [8] presented exact analytical Wagner solution of water entry of an elliptic paraboloid. They proved that the interaction line of the elliptic paraboloid and the water is also elliptic. Korobkin and Scolan [9] performed an additional linearization on the basis of an axisymmetric solution. The additional linearization can be used to study those bodies whose shapes are close to axisymmetric bodies such as elliptic paraboloid, inclined cone and pyramid. Gazzola et al. [10] carried research on water entry of asymmetric bodies. Qin et al. [11] investigated on the impact between a wedge with angular momentum and calm water. Moore et al.[12] concentrated on the oblique water entry problem of three-dimensional bodies. Khabakhpasheva and Korobkin [13] and Shams and Porfiri [14] studied water entry of elastic wedges by applying Wagner theory. Although Wagner theory was proposed based on assumption of small deadrise angle, significant progress has been made toward bodies with large deadrise angles entering into calm water [15], [16], [17].

In addition to Wagner's approximation, there has been some other analytical work on water entry. Cumberbatch [18] gave the mathematical solution of the vertical impact between a liquid wedge and a flat surface. Dobrovol'skaya [19] converted the water entry problem of a wedge into an integro-differential equation. Shu [20] used Taylor expansion to study the oblique collision between a water wedge and a flat wall. Semenov and Iafrati [21] simulated water entry of an asymmetric wedge into initially calm water by the integral hodograph method. Semenov et al. [22] used the same methods to study the collision between two liquid wedges. Other theoretical models for the prediction of slamming loads were proposed by Vorus [23] and Oliver [24].

Wagner theory or other analytical methods usually have strict limitations on body shape, entry speed and boundary conditions. From this aspect, numerical methods are more robust to predict impact pressure and loads in more complex conditions. Boundary element method (BEM) is one commonly used method for solving water entry problems. Due to large distortion of free surface in the process of water entry, fully nonlinear boundary conditions are adopted. The body surface, the free surface as well as the far-away control surface are meshed in time domain. To present consecutive mesh, the calculation by BEM is usually started with a contact zone and a small part of the body was put beneath the water surface [25], [26], [27], [28]. While Wu et al. [29] introduced a stretched coordinate system in water entry problems and the distance that the body has travelled into the water was chosen to nondimensionalize the physical problem which allows us to adopt the similar element size and computational domain in the space when the time step is marching forward. This stretched coordinate system method was used in a series of water entry problems [30], [31], [32], [33]. Then boundary element method was further extended to solve 3D axisymmetric and non-axisymmetric problems. Xu et al. [34] simulated water entry of a cone in free fall motion, and Sun and Wu [35,36] solved oblique water entry problem of conical bodies at constant and varying speed. The above work on water entry by BEM all considered semi-infinite flow field, it is also convenient to deal with problems of finite boundary or incident waves by using BEM. Water entry of a wedge in shallow water was investigated by Lin and Ho [37]. Recently, two dimensional water entry in presence of wave and current was studied by Cheng et al. [38] and Cheng et al. [39], and three dimensional water entry in Stokes wave was solved by Sun et al. [40].

Apart from BEM, other commonly used CFD methods for solving water entry problem include SPH and FVM, which have advantages of simulating large distortion of free surface. Oger et al. [41] considered the coupling of a falling wedge and water based on SPH method. Skillen et al. [42] used incompressible SPH (ISPH) method to study water entry of a cylinder and a wedge. Facci et al. [43] analyzed 3D water entry problem by using FVM. Marrone et al. [3] conducted 3D SPH simulations of water entry of complex bodies. Sun et al. [44] applied the recently developed δ+-SPH method for 2D and 3D water entry problems. Wang et al. [45] adopted SPH scheme to simulate water entry of elastomer.

As paraboloids have flat bottoms, water entry of paraboloids is most frequently studied by Wagner method based on the framework of linear approximation [8,17,46]. And self-similar solution of water entry of an expanding paraboloid is given by Wu and Sun [47]. It should be noticed that the increase of the deadrise angle of the paraboloid during water entry will bring body nonlinearity and thus it is necessary that nonlinear boundary conditions are included to see how it works in time domain. For the above reasons, the paper presents an efficient prediction method of slamming loads on three dimensional rigid bodies with curvature under fully nonlinear boundary conditions. The paper dissects this problem by analyzing water entry of a rigid paraboloid. Our framework focuses on variation of the physical behaviors during slamming of the paraboloid. The influence of fully nonlinear boundary conditions on impact pressure is studied through the comparison with linear results by Wagner method and the composite solution with spray jets. The distribution behavior of negative pressure and the movement of the stagnation point is investigated by comparison with that of cone.

Section snippets

Mathematical model and numerical procedures

A three-dimensional mathematical model of a rigid paraboloid entering into water at constant speed is described in Fig. 1. A space-fixed Cartesian coordinate system O-xyz is defined, in which O − xy plane lies on the undisturbed free surface, and the z-axis points upward. At t = 0, the tip of the paraboloid is at the origin of the coordinate system. The velocity components U and W are constants in x and z direction respectively. W is assumed to be positive when the paraboloid enters water

Convergence study

We first verify our methodology and numerical procedure through convergence study. We consider the case of a paraboloid of λ = 0.2 entering vertically in water at constant speed of W. The smallest element is set near the jet root where the curvature of the free surface is largest. The element size increases gradually away from the jet root at a given ratio, which are chosen as 1.03. From the analysis in Section 2, it can be seen that the non-dimensional free surface elevation β and the

Conclusions

The oblique water entry problem of a paraboloid at constant speed is investigated and the 3D flow behaviors are analyzed in the article, based on the convergence study with respect to free surface elevation, pressure distribution as well as the resultant force. The present results are compared with linearized Wagner method with no spray jets and asymptotic composite solution with spray jets. The comparison shows that the three methods conform better during initial impact stage with small

Acknowledgements

This work is supported by National Key R&D Program of China (2018YFC0310500), the National Natural Science Foundation of China (grant no. 51679045, 51579052, 51679055) and the Fundamental Research Funds for the Central Universities (HEUCFP201742). This work is also supported by Lloyd’s Register Foundation (LRF) through the joint center involving University College London, Shanghai Jiaotong University and Harbin Engineering University, to which the authors are most grateful. LRF supports the

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      Citation Excerpt :

      The predictions of the proposed model were in good agreement with respect to the experimental data. Sun et al. [18] investigated the water entry problem of a rigid paraboloid by fully nonlinear boundary element method. The results were compared with linearized Wagner method with no spray jets and asymptotic composite solution with spray jets.

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