Accurate modelling of uni-directional surface waves (original) (raw)
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Variational derivation of KdV-type models for surface water waves
Physics Letters A, 2007
Using the Hamiltonian formulation of surface waves, we approximate the kinetic energy and restrict the governing generalized action principle to a submanifold of uni-directional waves. Different from the usual method of using a series expansion in parameters related to wave height and wavelength, the variational methods retains the Hamiltonian structure (with consequent energy and momentum conservation) and makes it possible to derive equations for any dispersive approximation. Consequentially, the procedure is valid for waves above finite and above infinite depth, and for any approximation of dispersion, while quadratic terms in the wave height are modeled correctly. For finite depth this leads to higher-order KdV type of equations with terms of different spatial order. For waves above infinite depth, the pseudo-differential operators cannot be approximated by finite differential operators and all quadratic terms are of the same spatial order.
Practical use of variational principles for modeling water waves
Physica D: Nonlinear Phenomena, 2012
This paper describes a method for deriving approximate equations for irrotational water waves. The method is based on a 'relaxed' variational principle, i.e., on a Lagrangian involving as many variables as possible. This formulation is particularly suitable for the construction of approximate water wave models, since it allows more freedom while preserving the variational structure. The advantages of this relaxed formulation are illustrated with various examples in shallow and deep waters, as well as arbitrary depths. Using subordinate constraints (e.g., irrotationality or free surface impermeability) in various combinations, several model equations are derived, some being well-known, other being new. The models obtained are studied analytically and exact travelling wave solutions are constructed when possible.
On the Efficient Numerical Simulation of Directionally Spread Surface Water Waves
Journal of Computational Physics, 2001
This paper concerns the description of transient and highly nonlinear, near-breaking, surface water waves that are characterized by a spread of wave energy in both frequency and direction. A new spectral wave model is described that allows both the unsteadiness and the directionality of a wave field to be described in a fully nonlinear sense. The methodology underlying the scheme is similar to the unidirectional model developed previously by Craig and Sulem . An approximation of the Dirichlet-Neumann operator is made that transforms the boundary values of the velocity potential, φ, at the water surface into values of φ z . This allows an initial spatial representation of the water surface elevation and the velocity potential on this surface to be time marched using fast Fourier transforms. The advantages of this technique lie in both its efficiency and its robustness. These are of fundamental importance when seeking to model extreme ocean waves, involving broad-banded frequency spectra and realistic directional spreads, since they incorporate a large range of horizontal length scales. In its present form, the model is appropriate to waves propagating on water of constant depth; it runs on a PC and is sufficiently stable to predict the evolution of nearbreaking waves. Indeed, the only significant restriction arises due to the Fourier series representation. This requires the water surface elevation to be a single-valued function of the horizontal coordinates and therefore limits the model to non-overturning waves. The new numerical scheme is validated against a fifth-order Stokes solution for regular waves and the recent experimental observations provided by Johannessen and Swan . These latter comparisons are particularly important, confirming that the model is able to describe the rapid and highly significant energy transfers that occur across the wavenumber spectrum in the vicinity of an extreme event. These are strongly dependent upon the directionality of the wavefield and critically important when seeking to define the characteristics of an extreme, near-breaking, wave. The paper concludes with an example of the formation of a realistic, fully nonlinear and directionally spread wave group in the open ocean. c 2001 Elsevier Science
A water wave model with horizontal circulation and accurate dispersion
2009
We describe a new water wave model which is variational, and combines a depth-averaged vertical (component of) vorticity with depth-dependent potential flow. The model facilitates the further restriction of the vertical profile of the velocity potential to n-th order polynomials or a finite element profile with a small number of elements (say), leading to a framework for efficient modelling of the interaction of steepening and breaking waves near the shore with a large-scale horizontal flow. The equations are derived from a constrained variational formulation which leads to conservation laws for energy, mass, momentum and vertical vorticity (or circulation). We show that the potential flow water wave equations and the shallow-water equations are recovered in the relevant limits, and provide approximate shock relations for the model which can be used in numerical schemes to model breaking waves.
A Numerical Study of the Exact Evolution Equations for Surface Waves in Water of Finite Depth
Studies in Applied Mathematics, 2004
We describe a pseudo-spectral numerical method to solve the systems of one-dimensional evolution equations for free surface waves in a homogeneous layer of an ideal fluid. We use the method to solve a system of one-dimensional integro-differential equations, first proposed by Ovsjannikov and later derived by Dyachenko, Zakharov, and Kuznetsov, to simulate the exact evolution of nonlinear free surface waves governed by the two-dimensional Euler equations. These equations are written in the transformed plane where the free surface is mapped onto a flat surface and do not require the common assumption that the waves have small amplitude used in deriving the weakly nonlinear Korteweg-de Vries and Boussinesq long-wave equations. We compare the solution of the exact reduced equations with these weakly nonlinear long-wave models and with the nonlinear long-wave equations of Su and Gardner that do not assume the waves have small amplitude. The Su and Gardner solutions are in remarkably close agreement with the exact Euler solutions for large amplitude solitary wave interactions while the interactions of low-amplitude solitary waves of all four models agree. The simulations demonstrate that our method is an efficient and accurate approach to integrate all of these equations and conserves the mass, momentum, and energy of the Euler equations over very long simulations.
Modelling and simulation of surface water waves
Mathematics and Computers in Simulation, 2002
The evolution of waves on the surface of a layer of fluid is governed by non-linear effects from surface deformations and dispersive effects from the interaction with the interior fluid motion. Several simulation tools are described in this paper and compared with real life experiments in large tanks of a hydrodynamic laboratory. For the full surface wave equations a numerical FEM/FD program is described that solves both the interior flow and the surface evolution; apart from being very efficient, the program performs remarkably well. For theoretical analysis, simplified equations are desired. These can be obtained by modelling the interior flow to some degree of accuracy, leading to a single equation for the surface elevation. As representatives of this class, we discuss KdV-type of equations and, for wave packets, the NLS equation. We show that even this last equation describes quite well the large, non-symmetric deformations of the envelope of bi-harmonic waves when formulated as a signalling problem.
Wave Motion, 2011
In this paper we formulate relatively simple models to describe the propagation of coastal waves from deep parts in the ocean to shallow parts near the coast. The models have good dispersive properties that are based on smooth quasi-homogeneous interpolation of the exact dispersion above flat bottom. This dispersive quality is then maintained in the second order nonlinear terms of uni-directional equations as known from the AB-equation. A linear coupling is employed to obtain bi-directional propagation which includes (interactions with) reflected waves. The derivation of the models is consistent with the basic variational formulation of surface waves without rotation. A subsequent spatial discretization that takes this variational structure into account leads to efficient and accurate codes, as will be shown in Part 2.
On direct methods in water-wave theory
Model equations for three-dimensional, inviscid flow between two arbitrary, timevarying material surfaces are derived using a 'direct ' or variational approach due to Kantorovich. This approach results in a hierarchy of approximate theories, each of a higher level of spatial approximation and complexity. It can be shown that the equations are equivalent in substance to 'the theory of directed fluid sheets ' of Green & Naghdi (1974& Naghdi ( , 1976.
Variational water-wave model with accurate dispersion and vertical vorticity
Journal of Engineering Mathematics, 2009
A new water-wave model has been derived which is based on variational techniques and combines a depth-averaged vertical (component of) vorticity with depth-dependent potential flow. The model facilitates the further restriction of the vertical profile of the velocity potential to n-th order polynomials or a finite-element profile with a small number of elements (say), leading to a framework for efficient modelling of the interaction of steepening and breaking waves near the shore with a large-scale horizontal flow. The equations are derived from a constrained variational formulation which leads to conservation laws for energy, mass, momentum and vertical vorticity. It is shown that the potential-flow water-wave equations and the shallow-water equations are recovered in the relevant limits. Approximate shock relations are provided, which can be used in numerical schemes to model breaking waves.
A variational model for fully non-linear water waves of Boussinesq type
Physical Review E, 2005
Using a variational principle and a parabolic approximation to the vertical structure of the velocity potential, the equations of motion for surface gravity waves over mildly sloping bathymetry are derived. No approximations are made concerning the non-linearity of the waves. The resulting model equations conserve mass, momentum and positive-definite energy. They are shown to have improved frequency-dispersion characteristics, as compared