A Nonlinear Coupled-Mode Model for Water Waves Over a General Bathymetry

2002 ◽  
Author(s):  
Gerassimos A. Athanassoulis ◽  
Konstandinos A. Belibassakis
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Vertical Structure ◽  
Local Mode ◽  
Surface Mode ◽  
Nonlinear Water Waves ◽  
Intermediate Depth ◽  
Coupled Mode ◽  
End Conditions ◽  
Sloping Bottom

A non-linear coupled-mode system of horizontal equations is derived with the aid of Luke’s (1967) variational principle, which models the evolution of nonlinear water waves in intermediate depth over a general bathymetry. The vertical structure of the wave field is exactly represented by means of a local-mode series expansion of the wave potential, Athanassoulis & Belibassakis (2000). This series contains the usual propagating and evanescent modes, plus two additional modes, the free-surface mode and the sloping-bottom mode, enabling to consistently treat the non-vertical end-conditions at the free-surface and the bottom boundaries. The system fully accounts for the effects of non-linearity and dispersion.

A Unified Coupled-Mode Approach to Nonlinear Waves in Finite Depth: Potential Flow

2008 ◽  
Author(s):  
G. A. Athanassoulis ◽  
K. A. Belibassakis
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Water Wave ◽  
Bottom Topography ◽  
Local Mode ◽  
Surface Mode ◽  
Nonlinear Water Waves ◽  
Coupled Mode ◽  
Wide Range ◽  
Sloping Bottom

A non-linear coupled-mode system of horizontal equations is presented, as derived from Luke’s (1967) variational principle, which models the evolution of nonlinear water waves in intermediate depth over a general bottom topography. The vertical structure of the wave field is represented by means of a complete local-mode series expansion of the wave potential. This series contains the usual propagating and evanescent modes, plus two additional terms, the free-surface mode and the sloping-bottom mode, enabling to consistently treat the non-vertical end-conditions at the free-surface and the bottom boundaries. The present coupled-mode system fully accounts for the effects of non-linearity and dispersion, and has the following main features: (i) various standard models of water-wave propagation are recovered by appropriate simplifications, and (ii) it exhibits fast convergenge, and thus, a small number of modes (up to 5) are usually enough for the precise numerical solution, provided that the two new modes (the free-surface and the sloping-bottom ones) are included in the local-mode series. In the present work, the coupled-mode system is applied to the numerical investigation of families of steady traveling wave solutions in constant depth, corresponding to a wide range of water depths, ranging from intermediate to shallow-water wave conditions and its results are compared vs. Stokes and cnoidal wave theories, respectively. Also, numerical results are presented for waves propagating over variable bathymetry regions and compared with second-order Stokes theory and experimental data.

A Fast Convergent Modal-Expansion of the Wave Potential With Application to the Hydrodynamic and Hydroelastic Analysis of Floating Bodies in General Bathymetry

2009 ◽  
Author(s):  
K. A. Belibassakis ◽  
G. A. Athanassoulis
Keyword(s):  
Free Surface ◽  
Variable Thickness ◽  
Water Waves ◽  
Local Mode ◽  
Floating Bodies ◽  
Coupled Mode ◽  
Wave Potential ◽  
Non Linear ◽  
Hydroelastic Analysis ◽  
Sloping Bottom

A non-linear coupled-mode system of horizontal equations has been derived with the aid of Luke’s (1967) variational principle, modelling the evolution of nonlinear water waves in intermediate depth and over a general bathymetry Athanassoulis & Belibassakis (2002, 2008). Following previous work by the authors in the case of linearised water waves (Athanassoulis & Belibassakis 1999), the vertical structure of the wave field is exactly represented by means of a local-mode series expansion of the wave potential. This series contains the usual propagating and evanescent modes, plus two additional modes, the free-surface mode and the sloping-bottom mode, enabling to consistently treat the non-vertical end-conditions at the free-surface and the bottom boundaries. The coupled-mode system fully accounts for the effects of non-linearity and dispersion. The main feature of this approach that a small number of modes (of the order of 5–6) are enough for the precise numerical solution, provided that the two new modes (the free-surface and the sloping-bottom ones) are included in the local-mode series. The consistent coupled-mode system has been applied to numerical investigation of families of steady nonlinear travelling wave solutions in constant depth (Athanassoulis & Belibassakis 2007) showing good agreement with known solutions both in the Stokes and the cnoidal wave regimes. In the present work we focus on the hydroelastic analysis of floating bodies lying over variable bathymetry regions, with application to the non-linear scattering of water waves by large floating structures (of VLFS type or ice sheets) characterised by variable thickness (draft), flexural rigidity and mass distributions, modelled as thin plates of variable thickness, extending previous approaches (see, e.g., Porter & Porter 2004, Belibassakis & Athanassoulis 2005, 2006, Bennets et al 2007). Numerical examples are presented, showing that useful results can be obtained for the analysis of large floating elastic bodies or structures very efficiently by keeping only a few terms in the expansion. Ideas for extending our approach to 3D are also discussed.

scholarly journals New exact relations for steady irrotational two-dimensional gravity and capillary surface waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170220 ◽  
Author(s):  
Didier Clamond
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Water Waves ◽  
Two Dimensional ◽  
Physical Plane ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Dimensional Gravity ◽  
Irrotational Motion ◽  
Exact Relations

Steady two-dimensional surface capillary–gravity waves in irrotational motion are considered on constant depth. By exploiting the holomorphic properties in the physical plane and introducing some transformations of the boundary conditions at the free surface, new exact relations and equations for the free surface only are derived. In particular, a physical plane counterpart of the Babenko equation is obtained. This article is part of the theme issue ‘Nonlinear water waves’.

Nonlinear Water Waves in the Presence of Submerged Elliptic Cylinder

2004 ◽  
Author(s):  
Nikolai I. Makarenko
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Boundary Integral Equations ◽  
Fluid Velocity ◽  
Elliptic Cylinder ◽  
Small Time ◽  
Boundary Integral ◽  
Nonlinear Water Waves ◽  
Wave Elevation ◽  
Boundary Equations

The fully nonlinear problem on unsteady two-dimensional water waves generated by elliptic cylinder, that is horizontally submerged beneath a free surface, is considered. An analytical boundary integral equations method using a version of Milne-Thomson transformation is developed. Boundary equations (the BEq system) determine immediately exact wave elevation and fluid velocity at free surface. Small-time solution expansion is obtained in the case of accelerated cylinder starting from rest.

scholarly journals Theoretical and experimental study of particle trajectories for nonlinear water waves propagating on a sloping bottom

2012 ◽  
Vol 370 (1964) ◽  
pp. 1543-1571 ◽  
Author(s):  
Yang-Yih Chen ◽  
Meng-Syue Li ◽  
Hung-Chu Hsu ◽  
Chiu-On Ng
Keyword(s):  
Asymptotic Solution ◽  
Water Waves ◽  
Surface Profile ◽  
Lagrangian Description ◽  
Nonlinear Water Waves ◽  
Third Order ◽  
Particle Trajectories ◽  
Free Surface Profile ◽  
Sloping Beach ◽  
Sloping Bottom

A third-order asymptotic solution in Lagrangian description for nonlinear water waves propagating over a sloping beach is derived. The particle trajectories are obtained as a function of the nonlinear ordering parameter ε and the bottom slope α to the third order of perturbation. A new relationship between the wave velocity and the motions of particles at the free surface profile in the waves propagating on the sloping bottom is also determined directly in the complete Lagrangian framework. This solution enables the description of wave shoaling in the direction of wave propagation from deep to shallow water, as well as the successive deformation of wave profiles and water particle trajectories prior to breaking. A series of experiments are conducted to investigate the particle trajectories of nonlinear water waves propagating over a sloping bottom. It is shown that the present third-order asymptotic solution agrees very well with the experiments.

A variational approach to Boussinesq modelling of fully nonlinear water waves

2010 ◽  
Vol 657 ◽  
pp. 36-63 ◽  
Author(s):  
GERT KLOPMAN ◽  
BRENNY VAN GROESEN ◽  
MAARTEN W. DINGEMANS
Keyword(s):  
Water Waves ◽  
Vertical Structure ◽  
Shape Function ◽  
Frequency Dispersion ◽  
Nonlinear Water Waves ◽  
Long Distance ◽  
Linear Dispersion ◽  
Flow Equations ◽  
Time Space ◽  
Hyperbolic Cosine

In this paper we present a new method to derive Boussinesq-type equations from a variational principle. These equations are valid for nonlinear surface-water waves propagating over bathymetry. The vertical structure of the flow, required in the Hamiltonian, is approximated by a (series of) vertical shape functions associated with unknown parameter(s). It is not necessary to make approximations with respect to the nonlinearity of the waves. The resulting approximate Hamiltonian is positive definite, contributing to the good dynamical behaviour of the resulting equations. The resulting flow equations consist of temporal equations for the surface elevation and potential, as well as a (set of) elliptic equations for some auxiliary parameter(s). All equations only contain low-order spatial derivatives and no mixed time–space derivatives. Since one of the parameters, the surface potential, can be associated with a uniform shape function, the resulting equations are very well suited for wave–current interacting flows.The variational method is applied to two simple models, one with a parabolic vertical shape function and the other with a hyperbolic-cosine vertical structure. For both, as well as the general series model, the flow equations are derived. Linear dispersion and shoaling are studied using the average Lagrangian. The model with a parabolic vertical shape function has improved frequency dispersion, as compared to classical Boussinesq models. The model with a hyperbolic-cosine vertical structure can be made to have exact phase and group velocity, as well as shoaling, for a certain frequency.For the model with a parabolic vertical structure, numerical computations are done with a one-dimensional pseudo-spectral code. These show the nonlinear capabilities for periodic waves over a horizontal bed and an underwater bar. Further some long-distance computations for soliton wave groups over bathymetry are presented.

scholarly journals Prediction of the free-surface elevation for rotational water waves using the recovery of pressure at the bed

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170102 ◽  
Author(s):  
D. Henry ◽  
G. P. Thomas
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Surface Profile ◽  
Two Dimensions ◽  
Arbitrary Distribution ◽  
Nonlinear Water Waves ◽  
Steady Current ◽  
Linear Waves ◽  
Free Surface Profile ◽  
Near Shore

This paper considers the pressure–streamfunction relationship for a train of regular water waves propagating on a steady current, which may possess an arbitrary distribution of vorticity, in two dimensions. The application of such work is to both near shore and offshore environments, and in particular, for linear waves we provide a description of the role which the pressure function on the seabed plays in determining the free-surface profile elevation. Our approach is shown to provide a good approximation for a range of current conditions. This article is part of the theme issue ‘Nonlinear water waves’.

Nonlinear water waves generated by an accelerated circular cylinder

1979 ◽  
Vol 92 (4) ◽  
pp. 767-781 ◽  
Author(s):  
H. J. Haussling ◽  
R. M. Coleman
Keyword(s):  
Free Surface ◽  
Circular Cylinder ◽  
Water Waves ◽  
Numerical Solutions ◽  
Nonlinear Effects ◽  
Surface Boundary ◽  
Nonlinear Water Waves ◽  
Pressure Distributions ◽  
Surface Boundary Conditions ◽  
Surface Profiles

Numerical solutions for the irrotational flow of an incompressible fluid about a circular cylinder accelerated from rest below a free surface are presented. The usual restriction to linearized free-surface boundary conditions has been avoided. The transient period from the start to a local steady state or to the development of a very steep wave slope is investigated in terms of free-surface profiles and body-surface pressure distributions. Linear and nonlinear results are used to illustrate the transition from deep submergence when nonlinear effects are small to shallow submergence when linearized analysis is inaccurate.

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