Fully nonlinear simulations of interactions between solitary waves and structures based on the finite element method

doi:10.1016/j.oceaneng.2015.08.007

Ocean Engineering

Volume 108, 1 November 2015, Pages 202-215

https://doi.org/10.1016/j.oceaneng.2015.08.007 Get rights and content

Highlights

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Wave reflection and separation are observed in the case of solitary waves propagate over a step.
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The wave peak increases at front of single cylinder but decreases at back with incoming wave amplitude.
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The diffracted wave by single cylinder is a traveling wave and its peaks generally increase with incoming wave amplitude.
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Wave runup at the front increases with initial draught and breadth and decreases at the back.
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In twin-cylinder cases, the wave at the back of cylinder one is strongly affected in smaller spacing.

Abstract

Fully nonlinear interactions between solitary waves and structures are studied based on a finite element method (FEM) in two-dimensions. A mesh with higher order 8-node quadrilateral isoperimetric element is used in the simulation. The velocity potential is obtained by solving a linear matrix system using the conjugate gradient (CG) method with a symmetric successive overelaxlation (SSOR) preconditioner, and the velocity in the fluid domain is calculated through differentiating shape functions directly. The wave elevation and potential on the free surface are updated by the fourth order Runge–Kutta method. Waves and hydrodynamic forces are obtained for single solitary waves acting on a wall and propagating on a tank with a step, collision between two solitary waves, interactions between a solitary wave and a single and twin rectangular cylinders on free surface. Some results are compared with previous studies of analytical solutions and those by experiment. The simulations have provided many results to show the nonlinearities of waves and hydrodynamic forces with different incoming wave amplitudes, and the effect of dimensions of the single cylinder and the spacing between twin cylinders on the wave and forces are also discussed.

Introduction

Solitary wave is a single peaked wave propagating with constant speed. It appears on the ocean surface in the coastal region or down below the ocean surface in deep water in the form of internal wave. In the former case, the wave will depend on the ratios of the wave amplitude to water depth and to wave length. In the latter case, when the length scale of the water depth is missing, it is replaced by the scale over the liquid density changes. It therefore has a wide range of applications in coastal engineering, ocean engineering and naval architecture. Our focus here is the surface solitary wave which primarily appears in shallow water. It was first observed by Russell in 1834 and has been studied extensively even since based on the mathematical, numerical and experimental approaches. Su and Mirie (1980) and Mirie and Su (1982) considered the two colliding solitary waves passing through each other based on the perturbation method and numerical simulation. Chen et al. (2015) studied a solitary wave impacting on a vertical wall through experiment. Seabra-Santos et al. (1987) undertook numerical and experimental study of the transformation of a solitary wave over a shelf and an obstacle. Losada et al. (1989) did similar numerical and experimental study on solitary waves with moderate amplitude propagating over a step. Huang and Dong (2001) and Lin (2004) simulated solitary wave propagating over a submerged rectangular cylinder, respectively. Seiffert et al. (2014) undertook experimental and numerical study of solitary wave interactions with coastal-bridge, and a further experimental study on coastal-bridge with girders was made by them (Hayatdavoodi et al., 2014).

There is also a large volume of work on interactions of solitary waves with structures based on the Boussinesq or KdV equation. Sibley et al. (1982) studied forces on horizontal cylinders due to solitary waves. Wang et al. (1992) examined the scattering of solitary waves around a vertical cylinder by solving the generalized Boussinesq equations. Yates and Wang (1994) also studied the scattering of solitary wave by a vertical circular cylinder both experimentally and numerically. Zhao et al. (2007) considered the solitary wave scattering by an array of vertical circular cylinders based on the generalized Boussinesq equations. On the other hand, Lu (1991) developed a matched asymptotic expansion and boundary element based numerical method to study weakly nonlinear shallow waves and its interactions with structures on free surface. The whole fluid domain is divided into two subfields: the governing equation is the Boussinesq equation in the outer domain and is Laplace equation in the inner.

The work mentioned above is based on the shallow water equations such as Boussinesq or KdV equation. They are obtained through approximation from the governing Laplace equation and dynamic and kinematic boundary conditions on the free surface. These equations are valid within certain limits of the ratios of the wave amplitude to water depth and wave length to water depths, and the relative magnitudes of these two ratios. While these assumptions are valid for the wave on its own, the local waves generated through the interactions of the solitary wave and the structure may not follow these assumptions. It would therefore be more appropriate to use the original governing Laplace equations with the fully nonlinear free surface boundary conditions. Based on this approach, Isaacson (1982) investigated the diffraction of solitary waves by a circular cylinder using an integral equation method. Kim et al. (1983) studied solitary wave generation and propagation, including a single solitary wave and two successive solitary waves rushing up a sloping beach in two-dimensions. Chian and Ertekin (1992) further investigated the diffraction of solitary waves by horizontal submerged circular cylinders, and provided the free-surface profiles, particle velocities and wave forces. Cao et al. (1993) employed a desingularized boundary integral method to simulate interactions between solitary waves and submerged two dimensional circular cylinders. Cooker et al. (1997) and Chambarel et al. (2009) simulated reflection of solitary wave in large amplitudes at a vertical wall and head-on collision of two solitary waves, respectively, based on the boundary element method.

It is known from mentioned above that various theories and experiments has be widely used in studying solitary wave propagating in a tank without anything or with a submerged body inside. Very little has, however, been done for interactions between solitary waves and structures on free surface, which has important application in coastal engineering and ship engineering such as pontoon breakwater and floating boats in shallow water. In the present paper, we employ a higher order finite element based numerical method to investigate the interactions between solitary waves and two-dimensional (2D) structures on free surface based on the fully nonlinear potential theory. Various cases are considered in the paper. We first simulate some cases of a single solitary wave in a tank, interactions between two solitary waves and solitary waves propagating over a step as validation, and we then consider solitary wave acting with a single- or twin-cylinder rectangular cylinder on free surface, and the effect of various parameters (e.g., wave amplitude, initial draught and breadth of cylinder) on waves and hydrodynamics forces has been discussed.

Section snippets

Governing equation and boundary conditions

We consider the problem of a body on free surface in an incoming solitary wave as shown in Fig. 1. A Cartesian coordinate system $o x y$ is defined in which the x-axis coincides with the undisturbed free surface, and the y-axis is positive upward and is at the left side of the domain. The free surface and the body surface are denoted as $S_{f}$ and $S_{b}$ respectively. The bottom of the liquid is assumed to be a horizontal plane at y=−h and is denoted by S_bot. The fluid is assumed incompressible and

Finite element discritazation and numerical procedures

The wave velocity potential problem may be solved using the time stepping method. At each time step, we adopt the higher order finite element method with 8-node quadrilateral iosparametric elements. The coefficient matrix in the finite element linear system is symmetric, positive definite and even sparse. Once the coefficients are found, the linear system is then solved through an iteration based on the conjugate gradient method with a symmetric successive over relaxation (SSOR) preconditioner.

Solitary waves propagate in a tank

We first consider a single solitary wave in a fluid domain, whose length is L and calm water depth is h. A vertical wall is located at the right side of the domain. The solitary wave propagates towards the wall. In the simulation, the calm water depth is chosen as h=1.0 m, the wave height H=0.4h and x_p=L/2. The length of the domain is chosen to be L=3L_eff, where L_eff is the efficient wave length and is given by Nakayama (1983) as $L_{e f f} = 9.5776 h \sqrt{\frac{h}{H}}$

The finite element nodes are uniformly distributed

Conclusions

A higher order finite element method has been utilized to analyze the fully nonlinear interactions between solitary waves and structures. The velocity potential at each time step is obtained through solving matrix equations based on an iteration method. The fourth order Runge–Kutta method is used to update the wave elevation and potential on the free surface at each time step. The radiation condition is imposed through placing a damping zone at one or both ends of the tank.

Extensive simulations

Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant no. 51279179) and State Key Laboratory of Ocean Engineering (Shanghai Jiao Tong University) (Grant no. 1111), to which the authors are most grateful.

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Fully nonlinear simulations of interactions between solitary waves and structures based on the finite element method

Highlights

Abstract

Introduction

Section snippets

Governing equation and boundary conditions

Finite element discritazation and numerical procedures

Solitary waves propagate in a tank

Conclusions

Acknowledgment

Eur. J. Mech. B/Fluids

Wave Motion

Coast. Eng.

Coast. Eng.

Coast. Eng.

Coast. Eng.

Coast. Eng.

Appl. Ocean Res.

Comput. Fluids

Ocean Eng.

J. Fluids Struct.

Comput. Fluids

Ocean Eng.

Ocean Eng.

An integral equation for unsteady surface waves and a comment on the Boussinesq equation

J. Fluid Mech.

Numerical computations of two-dimensional solitary waves generated by moving disturbances

Int. J. Numer. Methods Fluids

Head-on collision of two solitary waves and residual falling jet formation

Nonlinear Process. Geophys.