Elsevier

Coastal Engineering

Volume 53, Issue 11, November 2006, Pages 929-945
Coastal Engineering

A numerical study of nonlinear wave run-up on a vertical plate

https://doi.org/10.1016/j.coastaleng.2006.06.004Get rights and content

Abstract

A finite difference model based on a recently derived highly-accurate Boussinesq-type formulation is presented. Up to the third-order space derivatives in terms of the velocity variables are retained, and the horizontal velocity variables are re-formulated in terms of a velocity potential. This decreases the total number of unknowns in two horizontal dimensions from seven to five, simplifying the implementation, and leading to increased computational efficiency. Analysis of the embedded properties demonstrates that the resulting model has applications with errors of 2 to 3% for (wavenumber times depth) kh  10 in terms of dispersion and kh  4 in terms of internal kinematics. The stability and accuracy of the discrete linearised systems are also analysed for both potential and velocity formulations and the advantages and disadvantages of each are discussed. The velocity potential model is then used to study physically demanding problems involving highly nonlinear wave run-up on a bottom-mounted (surface-piercing) plate. New cases involving oblique incidence are considered. In all cases, comparisons with recent physical experiments demonstrate good quantitative accuracy, even in the most demanding cases, where the local wave steepness can exceed (waveheight divided by wavelength) H / L = 0.20. The velocity potential model is additionally shown to have numerical advantages when dealing with wave–structure interactions, requiring less smoothing around exterior structural corners.

Introduction

Boussinesq-type equations have been widely studied and extended in recent years. They represent an attractive tool for coastal applications and engineering purposes. They are traditionally formulated in terms of horizontal velocities (u, v) and the free surface elevation η, with the vertical velocity variable w explicitly eliminated from the equations. The traditional approach results in a system of two governing equations: the mass and momentum conservation equations. Nevertheless, such formulations clearly show some trade off in their performance regarding nonlinearity, shoaling and dispersion. Agnon et al. (1999) have shown that by retaining the vertical velocity variable as an unknown, far better accuracy can be obtained through truncated series expansions, leading to a potentially infinite radius of convergence (Madsen and Agnon, 2003). This original procedure is based on an exact formulation of the boundary conditions at the free surface and at the sea bottom combined with an approximate solution to the Laplace equation given in terms of truncated series expansions. As a result, the six-equation model proposed shows roughly equivalent accuracy in linear and nonlinear embedded properties.

The most accurate Boussinesq-type formulation yet derived corresponds to the equations of Madsen et al. (2003) (see also Madsen et al., 2002). This model combines the Agnon et al. (1999) idea of decoupling the linear and the nonlinear part of the fluid problem and the inspiration of Nwogu (1993) to use a mid-depth expansion point of the truncated operators. Limitations regarding the accuracy of the vertical velocity profiles are thus largely removed. The new approach proposed here considers the equations originally derived by Madsen et al. (2003) (their method III), utilising Padé [2,2] approximants introduced in operators at z = 0 and z =  h, and rewritten in terms of a velocity potential instead of the commonly used horizontal velocity variables. These are shown to give an accurate description of dispersive nonlinear waves up to dimensionless water depths of kh  10. Accurate nonlinear internal wave kinematics are restricted to kh  4. The advantage of using the velocity potential is that it decreases the total number of equations to be solved numerically thus simplifying the implementation and reducing the computational costs in terms of storage and time. A possible disadvantage is that rotation in the horizontal plane (corresponding to the vertical component of the vorticity vector) is not allowed. This has been commonly removed within the present high-order Boussinesq-type approach, however, even when velocity variables have been utilised (see e.g. Fuhrman and Bingham, 2004, Fuhrman et al., 2005) (hence also resulting in an irrotational formulation), since the resulting equations are easier to solve numerically.

The focus of the present paper is on the extension of this Boussinesq-type model to allow fluid domains with rectangular bottom-mounted (surface-piercing) structures. In potential flow, it is well known that velocities are singular at an exterior structural corner. This can introduce several practical numerical problems, which can affect both the stability and the accuracy of the model.

The Boussinesq-type equations to be solved are reviewed in Section 2. The embedded linear and nonlinear properties of the formulation are quantified in Section 3. The finite difference based numerical model for solving these equations is described in Section 4. A numerical stability and accuracy analysis comparing the potential formulation to an equivalent velocity formulation is given in Section 5. Section 6 focuses on the insertion of structures into the fluid domain and underlines the advantages of using a velocity potential formulation in the context of rectangular box-shaped structures in a potential flow. Several applications dealing with nonlinear wave–structure interactions in deep water are described in 7 Nonlinear wave run-up on a vertical rigid plate, 8 Oblique waves. Experimental and numerical results are compared, demonstrating the ability of the model to accurately deal with complicated, but practical problems involving highly nonlinear waves and rectangular box-shaped structures. An exhaustive study of the experiments conducted at the BGO-FIRST offshore wave tank by Molin et al. (2005) is presented. A similar vertical rigid plate model is then submitted to oblique incident wave fields. Section 9 gives some conclusions including the suggestion that the observed highly nonlinear run-up phenomenon occurs only at incidence angles of less than 20°.

Section snippets

Boussinesq-type approach

Consider the irrotational flow of an incompressible inviscid fluid with a free surface. A Cartesian coordinate system is adopted, with the x-axis and y-axis located on the still-water plane and with the z-axis pointing vertically upwards. The fluid domain is bounded by the sea bed at z =  h(x, y) and the free surface at z = η(x, y, t). We introduce the velocity potential ϕ(x, y, z, t), which is related to the velocity components through:ϕu,ϕzw,(x,y).

Following Zakharov (1968) (see also

Analysis of embedded properties

By applying the linear Fourier analysis described in detail by Madsen et al. (2003) Section 3, it is straightforward to show that this method obeys the following linear dispersion relation:c2gh=1+760k2h2+1600k4h41+920k2h2+11600k4h4+114400k6h6,where c = ω / k is the wave celerity, ω the radian frequency and k the wavenumber. Comparing this to the exact value of c2Stokes / (gh) = tanh (kh) / (kh) gives the curve marked “Linear” in Fig. 1.

In order to quantify the accuracy of the method for nonlinear waves

Numerical model

In this section the numerical solutions of the formulation described above will be considered. Compared to a formulation in terms of velocity variables where A corresponds to a square matrix of size 3 Nx × Ny (where Nx, Ny are the number of grid points along the x- and the y-directions respectively), the square matrix A obtained from the velocity potential formulation is of rank 2 Nx × Ny. This represents a major advantage in terms of computational efficiency when dealing with numerical solutions

Linear stability and accuracy analysis

We will now analyse the stability and accuracy of the linearised numerical model. For simplicity, the analysis is restricted to flat-bottom problems with a single horizontal dimension. As it is more usual to formulate Boussinesq-type equations in terms of velocity variables, rather than a velocity potential, it is of interest to compare the numerical behaviour of both formulations. Both will therefore be analysed in the following subsections. Throughout this section, standard von Neumann

Inclusion of rectangular box-shaped structures

This section focuses on the inclusion of rectangular bottom-mounted (surface-piercing) structures into the Boussinesq-type model. As is well known, the linear potential flow solution around an exterior corner is weakly singular at the corner itself. To avoid explicit evaluation of quantities at the corner, all structural walls are placed half way between grid points. Boundary conditions at structural walls are then imposed in the same way as for the surrounding walls of the basin, i.e. by

Experimental set-up

The physical experiments involving highly nonlinear wave–structure interaction in deep water conducted by Molin et al. (2005) will now be considered as a first practical validation of the model. The tests were performed at the BGO-FIRST offshore wave tank on a rigid vertical plate model submitted to regular waves of varying periods and steepnesses. During the experiments, at wavelengths comparable with the width of the plate, a strong run-up occurring at the plate-wall intersection on the

Generation of oblique incident wave fields

This section deals with the generation and absorption of oblique waves in the numerical model. To accomplish this, pure wave generation zones, immediately followed by relaxation zones are placed along two walls of the domain, and damping zones (or sponge layers) are placed along the opposing walls, in the arrangement shown in Fig. 13. The areas with the arrows represent the pure generation zones. This technique prevents diffraction effects from disturbing the incident wave and allows reflected

Conclusions

A high-order Boussinesq-type model has been developed and used to investigate nonlinear wave interaction with a piecewise rectangular bottom-mounted (surface-piercing) structure. The method used is based on the Padé [2,2] variant of the method originally derived by Madsen et al. (2003). This has been re-formulated in terms of a velocity potential (thus requiring up to fourth-order horizontal derivatives), and discretised using finite differences. An analysis of the embedded properties

Acknowledgements

E. Jamois thanks Dr O. Kimmoun and Dr F. Remy (Ecole Généraliste d'Ingénieurs de Marseille) for their contribution to this project and Prof. P.A. Madsen (Technical University of Denmark) for providing the variable depth coefficients of the formulation and helpful discussions. D.R. Fuhrman and H.B. Bingham wish to thank the Danish Technical Research Council (STVF grant no. 9801635) for financial support, and the Danish Center for Scientific Computing for supercomputing resources.

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