Numerical simulation of solitary waves on plane slopes

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Abstract

In this paper, we present a numerical method for the computation of surface water waves over bottom topography. It is based on a series expansion representation of the Dirichlet–Neumann operator in terms of the surface and bottom variations. This method is computationally very efficient using the fast Fourier transform. As an application, we perform computations of solitary waves propagating over plane slopes and compare the results with those obtained from a boundary element method. A good agreement is found between the two methods.

Introduction

Surface water wave propagation over variable depth has been studied for many years in applications to coastal engineering. This problem can be very complex due to the richness of coastal wave dynamics: when entering shallow water, waves are strongly affected by the bottom through shoaling, refraction, diffraction and reflection. Nonlinear effects related to wave–wave and wave–bottom interactions can cause wave scattering and depth-induced wave breaking. In turn, nonlinear waves can have a great influence on sediment transport and the formation of sandbars in nearshore regions.

Traditionally, the modeling of water waves over variable depth has been tackled using both the full Euler equations and long wave models such as shallow water equations or Boussinesq equations. Much work has been devoted to develop numerical methods for the solution to the full equations. Two main classes of numerical methods are boundary element methods [18], [10], [15] and spectral methods. In particular, spectral methods based on perturbative expansions have been developed by Dommermuth and Yue [11], West et al. [25], Craig and Sulem [9], Nicholls [19] and Bateman et al. [2] for the modeling of water waves over constant or infinite depth. Similar methods have been proposed by Liu and Yue [17] and Smith [23] to include the effects of a bottom topography.

In this paper, we present a numerical method for the simulation of water waves over variable depth, based on an extension of the work of Craig and Sulem [9]. We use a series expansion formulation of the Dirichlet–Neumann operator on the free surface in terms of the surface and bottom variations. The spatial discretization is efficiently performed by a pseudospectral method using the fast Fourier transform. To test the accuracy of our method, we perform computations of solitary waves propagating over plane slopes and compare the results with those obtained from a boundary element method. For convenience, we restrict ourselves to a two-dimensional configuration.

The paper is organized as follows. In Section 2, we present the mathematical formulation of the problem and introduce the Dirichlet–Neumann operator. In Section 3, we give a description of the numerical method for the discretization of the equations. Section 4 describes the boundary element method, and in Section 5, we present our numerical results of solitary waves propagating over a sloping bottom.

Section snippets

Equations of motion and Dirichlet–Neumann operator

We consider the motion of a free surface on top of a two-dimensional fluid domain defined by Ω(β,η)={(x,y):x∈R,−h+β<y<η}, where β(x) denotes the bottom perturbation and η(x,t) denotes the free surface. The mean water level is located at y=0 and the constant reference depth is represented by h. The fluid is assumed to be incompressible, inviscid and irrotational, so that the fluid velocity is given by u=∇ϕ and the velocity potential ϕ(x,y,t) satisfiesΔϕ=0inΩ(β,η).On the bottom boundary y=−h+β,

Numerical method

We assume periodic boundary conditions in the x-direction and, as in [9], [19], we use a pseudospectral method for the space discretization of the problem. This is a natural choice for the computations of G(β,η) and L(β) since each term in (11), (14) consists of concatenations of Fourier multipliers with powers of β and η. Both operators are approximated by a finite number of termsG(β,η)≃∑l=0MGG(l)(β,η),L(β)≃∑j=0MLLj(β),where the orders MG and ML are independently chosen according to the

Boundary integral formulation and boundary element method

In this section, we briefly describe the numerical model developed by Grilli et al. [15] for the simulation of two-dimensional nonlinear water waves over bottom topography. This model has been recently extended to three dimensions by Grilli et al. [13].

The governing equations are still given for a potential flow. However, the kinematic and dynamic boundary conditions (3), (4) are expressed in the mixed Eulerian–Lagrangian formulation asdtx=u=∇ϕ,dtϕ=12|∇ϕ|2−gy,where d/dt=∂/∂t+∇ϕ⋅∇ denotes the

Numerical results

The numerical method described in Section 3 is used to compute the evolution of solitary waves traveling up plane slopes. The accuracy of the computations are assessed by direct comparison with results obtained from the BE method described in Section 4.

The geometry of the bottom is specified as follows: a flat bottom of depth h over a short distance d from the left extremity of the domain, and a slope s for x≥d. All physical quantities are non-dimensionalized according to long wave theory, i.e.

Conclusions

We have presented a numerical method for the computation of unsteady nonlinear surface waves over bottom topography. As an extension of the work of Craig and Sulem [9], the model is based on the expansion of the Dirichlet–Neumann operator in terms of the surface and bottom variations and a pseudospectral method is used for the spatial discretization of the problem. In this paper, we have applied the model to the computation of solitary waves propagating over plane slopes and direct comparisons

Acknowledgements

The authors thank W. Craig, C. Sulem and S.T. Grilli for helpful advice. P.G. acknowledges financial support and disposal of computing resources from SHARCNET. D.P.N. gratefully acknowledges support from NSF through grant nos. DMS-0196452 and DMS-0139822.

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