Modelling and simulation of the three-dimensional hydrodynamic problem

This paper deals with slamming phenomenon (impact between bow ship and water free surface). Slamming loads on ship may be sufficiently important so as to create plastic deformations of the hull external structure. In extreme cases, they have been recognised for being responsible for the loss of ships. The problem to solve is transient and highly non-linear due to the character of the flow. In the present paper, the three-dimensional Wagner problem is solved numerically using a variational formulation together with a Finite Element Method. Three-dimensional results for simple rigid bodies such as a cone and an ellipsoid are successfully compared with analytical results. Results for deformable structure will be presented.


INTRODUCTION
In rough seas where large relative amplitude ship motions occur, the bow may emerge out of the water. In these conditions of the state of the sea, the ship hull can experience impact loads as the bow reenters the incoming wave. This phenomenon is commonly referred as slamming. Despite the pressure field due to the impact remains localized in space and time, slamming loads can locally create plastic deformations to the hull external structure. In extreme cases, they have been recognized for being responsible for the loss of ships. The purpose of this paper is to give some insight about this subject.
There are several approaches to study such a problem. On one hand, the Navier Stokes equations can be used to solve in velocity formulation, and on the other hand a velocity potential theory is introduced. In this paper, the second approach is going to be applied. The velocity potential equations are studied by an asymptotic analysis. The two dimensional problem has been widely studied and the theoretical analysis is now well established in case of a rigid body. First order analytical solutions have been found for a wedge with small deadrise angles (Wagner 1932), a cylinder (Cointe 1987), and arbitrary two dimensional blunt body shapes (Cointe 1989, Howinson, et al 1991. The latest consider an axisymetric body yet the results are closed to two-dimensional ones. In above mentioned studies, the body is assumed to be rigid. The two dimensional asymptotic solutions have been well compared to drop tests of different shapes (Cointe and Armand 1987, Fontaine and Cointe 1997, Zhao and al 1996, Magee and Fontaine 1998 and for rigid cone in three-dimensional (Faltinsen, 1997). For three-dimensional case, analytical solutions are far from possible expect in case of particular geometry (Scolan, Korobkin, 2001). Korobkin (1995), Kvalsvold (1995), andFaltinsen (1997), studied ship bow behaviour modelling the structure as a beam so that the coupled fluid-structure mathematical problem can be solved simultaneously. It results that hydroelasticity have not to be neglected. More recently, Donguy, Peseux and Fontalne (2000) have 1 shown that structural deformations present an important threedimensional character. So three-dimensional effects have to be taken into account.
This study takes place within this framework. We try to develop a purely numerical approach to solve general three dimensional problem of slamming, for rigid and deformable body, considering fully coupled fluid-structure interaction problem.
In a first part, the three dimensional governing equations for the fluid are presented. The setting equation relies on the asymptotic theory. For any geometry of body, the velocity potential formulation can not be solved analytically. So, a numerical resolution is performed by the finite element method. In a second part, the structure problem is considered and is classically solved by the finite element method. Then, we will focus on the coupled problem. The fluid structure interaction introduces a coupling matrix that we validate in two or three dimensional cases. Finally, we present solution of the coupled problem, which is iteratively solved by the Newmark time integration scheme. We consider the problem of the impact of a three-dimensional body on the free surface (Fig. 1). The water entry problem is expressed within the framework of potential flow theory. Indeed, the fluid is assumed to be perfect and incompressible and the flow is irrotational. The velocity field can therefore be evaluated according to ~ = gr?tdO, where ~=~x,y,z,t) is the velocity potential. Under these assumptions, the velocity potential must satisfy the following boundary value problem: -~n = u, .n on the body (2) 9¢p 1 _ 2 + P + g z =0 on the free surface (3) "~t +-2 ( grad~ ) 9 Oh 94 "7-+ g rdd~,g rddh -7 -= 0 dt OZ on the free surface (4) where the displacement of the body is given by its ff, (x,y,t) and the free surface elevation is written as z = h(x,y,t).
The body boundary condition, Eq. 2, expresses the continuity of the normal velocity component on the fluid-structure interface. The kinematic and dynamic free surface conditions, Eq. 3 and Eq. 4 respectively, state that the free surface is a material surface along which the pressure is constant. Moreover, the fluid is assumed to be initially at rest: and the flow to be unperturbed far from the body : Once the velocity potential is known, the pressure on the body is calculated by Bemoulli's equation: p Ot z (7)

2.__22 Asymptotic formulation
The above mentioned boundaries value problem cannot easily be solved in 3D, even numerically, due to the local and moving flow character. Classically, the hydrodynamic problem is simplified into: = 0 on FL (10) Oh 00 Ot = 9-"z on FL (11) where FB is the projection on z = 0 of the wetted body surface and FL the position of the undisturbed free surface. Of represent the linearized fluid domain (see Fig. 1). Cointe (1989), Wilson (1989) used the asymptotic expansion method to justify the mixte Neuman-Dirichlet problem. The perturbation procedure strongly relies on the blunt body assumption since the small parameter used in the asymptotic expansion is the ratio between the immersion parameter and the characteristic length scale of the body wetted width. Within a far field point of view, the body boundary condition, Eq. 2, can be written on z = 0 without introducing significant error, therefore justifying Eq. 9. The exact dynamic condition, Eq. 3, has been replaced by a Dirichlet condition for the potential on the undisturbed position of the free surface, Eq. 10. Physically the acceleration in the fluid is assumed to be large compared to gravity which can be neglected during the first instants. Finally, the simplified kinematical free surface condition, Eq. 11, states that the vertical displacement of the free surface is equal to the fluid vertical motion, evaluated on the undisturbed position of the free surface. The quadratic terms have to remain small compared to the linear ones in order to keep valid this approximation. The resulting problem, Eqs. 8-11 is often referred to the generalised Wagner (1932) problem. In the present formulation, three-dimensional effects are retained through the Laplace's equation, and structural deformations of the body surface are taken into account in the body boundary condition.

Numerical resolution
The resolution procedure for the velocity and displacement potentials is based on the Finite Element Method. The weighted residual method is applied considering the weighting functions ~0 which verifies the boundary condition ~o = 0 on FL • The problem comes to minimize the integral quantity: (12) where the residue R(~) is set equal to Aq~. Then, applying Green's identity and taking into account of boundary conditions, Eq. 13, leads to the weak formulation: The potential ¢ and weight functions q~ are approximated using Galerkin's method leading to: where the shape functions ivy depend on the types of elements, and (0} e denotes the nodal potential vector of the finite element (e).
Finally, the following linear system is deduced from the discretization of the bilinear forms, Eq. 13: To evaluate the right hand side of eq. (17), it is necessary to know the wetted portion of the body Fn or, equivalently, the position of the contact line between the body and the fluid. Due to the deformations of the free surface, the wetted surface is part of the unknown of the problem. An additional equation is therefore needed to close the problem. Physically, the contact line is determined by imposing to the solution to satisfy volume of the fluid conservation (Wilson, 1989, Fontaine & Cointe, 1992 or equivalently, by imposing the existence of an intersection point between the elevation of the free surface and the body as intuitively done by Wagner (1932). Volume conservation should be automatically satisfied since Laplace's equation is solved, the wetted surface beeing determined through an iterative procedure. (Donguy, Peseux and Fontaine, 2000) Once convergence has been reached, the position of the contact line is known. The linear system for the velocity potential is then solved. As a result, a simple first order backward finite difference scheme gives an accurate estimation of the pressure. This last solution represents only the fluid flow far from the body. Consequently, pressure distribution is singular close to the boundary wetted surface. This singularity is corrected by matching numerical solution with the analytical solution of the flow near the body.

HYDROELASTICITY
In this section, the previous analysis is extended to treat the case of deformable body. The equations governing the structural deformations are first recalled. A coupled method to solve the fluidstructure interaction problem is then proposed.

3.__! Governincj structural equations
The usual assumptions of small perturbations is made. The equilibrium equation reads: where X denotes the stress tensor, ff, corresponds to the structural displacement, and f corresponds to volume force associated for example with the gravity field. If gravity effects do not contribute to the local deformations here, they nevertheless influence the global motion of the body before impact. Boundary condition on the structure is: where the subscript f and s refer to the fluid and the structure, respectively, The structural modeling is performed within the framework of linear elasticity. Hooke's law is used to express the relation between the stresses and the=strains which are given by the linearised Green-Lagrange tensor E :

3.__22 Coupled problem
The structural problem is also solved using the Finite Element Method. The principle of virtual works is applied to Eq. 18 together with boundary condition, Eq. 19, leading classically to the discretised form: By definition, hydroelastic effects arise as a result of a strong interaction between the fluid flow and the structural response. From Eq. 22, it can be seen that the pressure imposed by the fluid will drive the structural deformations.The structural deformations influence the pressure field through the body boundary condition as shown by Eq. 17. In order to obtain a robust method to solve the coupled fluidstructure interaction problem, all the evolution equations are integrated simultaneously. The structural deformations are included in the fluid flow discrete formulation, while the pressure is expressed as the time derivative of the velocity potential in the discrete structural analysis. The following problem is then obtained: where the coupling terms [FS] representing the fluid -structure interaction appear explicitly. Eq. 26 shows that the coupling term involves only the values of the structural displacement g, and the velocity potential ~ on the wetted surface of the body, FB.

3.__33 Validation of the couplinq matrix in 2D
The numerical resolution of Eq. 24 or Eq. 25 is performed using commercial Finite Element Method numerical code Castem. An external procedure is implemented for the evaluation of the coupling matrix [FS]. The correct implementation of the approach is checked for the test case problem of sloshing in a rectangular tank including an elastic beam in its center, as shown in Figure 2 for twodimensional case. This problem has been studied experimentally by Cha'f (Cha~, 1996).
For this problem, gravity cannot be neglected and so the formulation introduces a mass matrix. We obtained the following problem (Morand and Ohayon, 1992): (27) (29) The mechanical excitation consists in a harmonic load applied to the beam or the plate extremity. A frequency scanning is performed, see Fig. 4, to evaluate the eigenfrequencies which correspond to the peaks of the beam displacement. The free surface elevation for the first four modes of the coupled problem is shown in Fig. 3, together with the modal shape of the elastic beam. The numerical values of the eigenfrequencies are presented in table 1, column C1. They are compared to the results by Peseux et al (1999), column entitled "Ref'. A toolbox that allows the direct computation of the resonant frequencies was also applied, see column C2, All the results are relatively close to each other, especially, the two last columns. The computation by Peseux was performed using another f.e.m, code, which may explains the small differences in the results. The agreement between the three methods is very satisfactory.   To validate the coupling matrix in three-dimensional case, we are interested in the test case problem of sloshing in a rectangular tank including an elastic plate in its center, as shown in Figure 5a. The figure 5b shows the finite elements mesh used, which are hexaedron elements for the fluid and quadrilateral elements for the structure. The equations of the problem are the same as the Eq. 27 -Eq. 29.

[FSff = Ires {N/]{n }r [N s Iris
Once again, the mechanical excitation consists in a harmonic load applied to the plate extremity. A frequency scanning is performed, see Fig. 6, to evaluate the eigenfrequencies which correspond to the peaks of the plate displacement. The free surface elevation with the modal shape of the plate beam for the first four modes of the coupled problem is shown in Fig. 7. In table 2, we reported the numerical values of the eigenfrequencies are presented in table 2. Morever, experimental results for plate vibration (Peseux, 1989) are given in the column "Exp". Comparison of frequencies leads to good agreement, which give confidence into the external numerical procedure that has been developed in the present study. Developing this procedure, and controlling precisely the coupling terms resulting from the fluid -structure interaction is necessary to treat the water impact problem of an elastic body.  (31) where: The fluid -structure interaction problem, Eq. 31, is then integrated starting from the initial conditions: in a numerical temporal explicit or implicit scheme with a time step dr. Nevertheless, studying for rigid body slamming, it appears that time step dt for the fluid problem has to be less than the fiftieth part of the instant to calculate accurately pressure distribution. Choosing the same time step for the coupled problem simulation leads to generate a high costly time procedure.
Consequently, we solved the fluid-structure interaction problem with an iterative procedure. The fluid problem is first solved with a step time S t. After determination of the force due to the pressure, the structural resolution is within the Newmark temporal integrated sheme with the step time At. The time schema is represented by In the present paper, a numerical method is proposed to solve the three-dimensional hydrodynamic impact problem for blunt bodies.
In a first part, we have presented the fluid problem formulated in velocity potential and relying on asymptotic theory. The equations deduced from such formulation have been solved through the FEM. In a second part, we have examined the structural governing equations, which have been classically resolved by FEM as well. Finally the two previous methods had been used to treat the case of deformable bodies. Yet this formulation introduces a coupling matrix. The correct evaluation of the terms has been checked for the sloshing problem of a tank with an elastic beam in two-dimensional case and elastic plate in three-dimensional case. The coupling problem has been integrated in time by a Newmark schema with an iterative coupling. The results that we have obtained under these assumptions are rather good.
This formulation entails the boundary conditions projection. To avoid this assumption, and follow the evolution of the free surface, other method can be used like the ALE (Arbitrary Lagrangian Eulerian) method.