Variational Principles and Hamiltonian Formulation for Nonlinear Water Waves

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Numerical Simulation for Nonlinear Water Waves Propagating along the Free Surface

The main aim of this work is to introduce the analytical approximate solutions of the water wave problem for a fluid layer of finite depth in the presence of gravity. To achieve this aim, we begun with the derivation of the Korteweg-de Vries equations for solitons by using the method of multiple scale expansion. The proposed problem describes the behavior of the system for free surface between air and water in a nonlinear approach. To solve this problem, we use the well-known analytical method, namely, variational iteration method (VIM). The proposed method is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. The proposed method provides a sequence of functions which may converge to the exact solution of the proposed problem. Finally, we observe that the elevation of the water waves is in form of traveling solitary waves.

A short comment on the effect of a shear layer on nonlinear water waves

In this paper we study nonlinear periodic deep water waves propagating on a background shear current,which decays exponentially with depth.We extend the study of Cheng,Cang and Liao(2009) by introducing a second parameter which measures the depth of the shear current.A high-order convergent analytical series solution is obtained by the homotopy analysis method(HAM).A detailed analysis of the impact of the depth parameter is given.We find that increasing this parameter so that the shear current is thinner reduces the wave phase speed,smoothes the wave crest,sharpens the trough,and enlarges the maximum wave height for the case of propagating waves on an opposing current;while it produces the opposite effect on an aiding current.

HAMILTONIAN FORMULATION OF NONLINEAR WATER WAVES IN A TWO-FLUID SYSTEM

In this paper, it is dealt with that the Hamiltonian formulation of nonlinear water waves in a two_fluid system,which consists of two layers of constant_density incompressible inviscid fluid with a horizontal bottom,an interface and a free surface. The velocity potentials are expanded in power series of the vertical coordinate. By taking the kinetic thickness of lower fluid_layer and the reduced kinetic thickness of upper fluid_layer as the generalized displacements, choosing the velocity potentials at the interface and free surface as the generalized momenta and using Hamilton's principle, the Hamiltonian canonical equations for the system are derived with the Legendre transformation under the shallow water assumption. Hence the results for single_layer fluid are extended to the case of stratified fluid.

Fully Nonlinear Shallow Water Waves Simulation Using Green-Naghdi Theory

Green-Naghdi （G-N） theory is a fully nonlinear theory for water waves. Some researchers call it a fully nonlinear Boussinesq model. Different degrees of complexity of G-N theory are distinguished by ＂levels＂ where the higher the level, the more complicated and presumably more accurate the theory is. In the research presented here a comparison was made between two different levels of G-N theory, specifically level II and level III G-N restricted theories. A linear analytical solution for level III G-N restricted theory was given. Waves on a planar beach and shoaling waves were both simulated with these two G-N theories. It was shown for the first time that level III G-N restricted theory can also be used to predict fluid velocity in shallow water. A level III G-N restricted theory is recommended instead of a level II G-N restricted theory when simulating fullv nonlinear shallow water waves.

Numerical simulation for the two-dimensional nonlinear shallow water waves

This study deals with the general numerical model to simulate the two-dimensional tidal flow, flooding wave (long wave) and shallow water waves (short wave). The foundational model is based on nonlinear Boussinesq equations. Numerical method for modelling the short waves is investigated in detail. The forces, such as Coriolis forces, wind stress, atmosphere and bottom friction, are considered. A two-dimensional implicit difference scheme of Boussinesq equations is proposed. The low-reflection outflow open boundary is suggested. By means of this model,both velocity fields of circulation current in a channel with step expansion and the wave diffraction behind a semi-infinite breakwater are computed, and the results are satisfactory.

Interactions of nonlinear gravity waves and uniform current in Lagrangian system

The particle trajectory on a weakly nonlinear progressive surface wave obliquely interacting with a uniform current is studied by using an Euler-Lagrange transformation.The third-order asymptotic solution is a periodic bounded function of Lagrangian labels and time,which imply that the entire solution is uniformly-valid.The explicit parametric solution highlights the trajectory of a water particle and mass transport associated with a particle displacement can now be obtained directly in Lagrangian form.The angular frequency and Lagrangian mean level of the particle motion in Lagrangian form differ from those of the Eulerian.The variations in the water particle orbits resulting from the oblique interaction with a steady uniform current of different magnitudes are also investigated.

A Brief Summary of Finite Element Method Applications to Nonlinear Wave-structure Interactions

We review recent advances in the finite element method (FEM) simulations of interactions between waves and structures. Our focus is on the potential theory with the fully nonlinear or second-order boundary condition. The present paper has six sections. A review of previous work on interactions between waves and ocean structures is presented in Section one. Section two gives the mathematical formulation. In Section three, the finite element discretization, mesh generation and the finite element linear system solution methods are described. Section four presents numerical methods including time marching schemes, computation of velocity, remeshing and smoothing techniques and numerical radiation conditions. The application of the FEM to the wave-structure interactions are presented in Section five followed by the concluding remarks in Section six.

Numerical solutions for two nonlinear wave equations

The split-step pseudo-spectral method is a useful method for solving nonlinear wave equations. However, it is not widely used because of the limitation of the periodic boundary condition. In this paper, the method is modified at its second step by avoiding transforming the wave height function into a frequency domain function. Thus, the periodic boundary condition is not required, and the new method is easy to implement. In order to validate its performance, the proposed method was used to solve the nonlinear parabolic mild-slope equation and the spatial modified nonlinear Schrodinger （MNLS） equation, which were used to model the wave propagation under different bathymetric conditions. Good agreement between the numerical and experimental results shows that the present method is effective and efficient in solving nonlinear wave eouations.

Large-scale edge waves generated by a moving atmospheric pressure

Long waves generated by a moving atmospheric pressure distribution, associated with a storm, in coastal region are investigated numerically. For simplicity the moving atmospheric pressure is assumed to be moving only in the alongshore direction and the beach slope is assumed to be a constant in the on-offshore direction. By solving the linear shallow water equations we obtain numerical solutions for a wide range of physical parameters, including storm size （2a）, storm speed （U）, and beach slope （a）. Based on the numerical results, it is determined that edge wave packets are generated if the storm speed is equal to or greater than the critical velocity, Ucr, which is defined as the phase speed of the fundamental edge wave mode whose wavelength is scaled by the width of the storm size. The length and the location of the positively moving edge wave packet is roughly Ut/2 〈 y 〈 Ut, where y is in the alongshore direction and t is the time. Once the edge wave packet is generated, the wavelength is the same as that of the fundamental edge wave mode corresponding to the storm speed and is independent of the storm size, which can, however, affect the wave amplitude. When the storm speed is less than the critical velocity, the primary surface signature is a depression directly correlated to the atmospheric pressure distribution.

Rogue Waves and Lump Solitons of the(3+1)-Dimensional Generalized B-type Kadomtsev–Petviashvili Equation for Water Waves

In this paper, the(3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation for water waves is investigated. Through the Hirota method and Kadomtsev–Petviashvili hierarchy reduction, we obtain the first-order,higher-order, multiple rogue waves and lump solitons based on the solutions in terms of the Gramian. The first-order rogue waves are the line rogue waves which arise from the constant background and then disappear into the constant background again, while the first-order lump solitons propagate stably. Interactions among several first-order rogue waves which are described by the multiple rogue waves are presented. Elastic interactions of several first-order lump solitons are also presented. We find that the higher-order rogue waves and lump solitons can be treated as the superpositions of several first-order ones, while the interaction between the second-order lump solitons is inelastic.

Invariant subspaces,exact solutions and stability analysis of nonlinear water wave equations

The key purpose of the present research is to derive the exact solutions of nonlinear water wave equations(NLWWEs)in oceans through the invariant subspace scheme(ISS).In this respect,the NLWWEs which describe specific nonlinear waves are converted to a number of systems of ordinary differential equations(ODEs)such that the resulting systems can be efficiently handled by computer algebra systems.As an accomplishment,the performance of the well-designed ISS in extracting a group of exact solutions is formally confirmed.In the end,the stability analysis for the NLWWE is investigated through the linear stability scheme.

Real-time flood forecasting of Huai River with flood diversion and retarding areas

A combination of the rainfall-runoff module of the Xin’anjiang model, the Muskingum routing method, the water stage simulating hydrologic method, the diffusion wave nonlinear water stage method, and the real-time error correction method is applied to the real-time flood forecasting and regulation of the Huai River with flood diversion and retarding areas. The Xin’anjiang model is used to forecast the flood discharge hydrograph of the upstream and tributary. The flood routing of the main channel and flood diversion areas is based on the Muskingum method. The water stage of the downstream boundary condition is calculated with the water stage simulating hydrologic method and the water stages of each cross section are calculated from downstream to upstream with the diffusion wave nonlinear water stage method. The input flood discharge hydrograph from the main channel to the flood diversion area is estimated with the fixed split ratio of the main channel discharge. The flood flow inside the flood retarding area is calculated as a reservoir with the water balance method. The faded-memory forgetting factor least square of error series is used as the real-time error correction method for forecasting discharge and water stage. As an example, the combined models were applied to flood forecasting and regulation of the upper reaches of the Huai River above Lutaizi during the 2007 flood season. The forecast achieves a high accuracy and the results show that the combined models provide a scientific way of flood forecasting and regulation for a complex watershed with flood diversion and retarding areas.

IMPROVED MODEL FOR THREE DIMENSIONAL NONLINEAR WATER WAVE FORCE PREDICTION

An improved model for numerically predicting nonlinear wave forces exerted on an offshore structure is pro- posed.In a previous work[9],the authors presented a model for the same purpose with an open boundary condi- tion imposed,where the wave celerity has been defined constant.Generally,the value of wave celerity is time-de- pendent and varying with spatial location.With the present model the wave celerity is evaluated by an upwind dif- ference scheme,which enables the method to be extended to conditions of variable finite water depth,where the value of wave celerity varies with time as the wave approaches the offshore structure.The finite difference method incorporated with the time-stepping technique in time domain developed here makes the numerical evolution effec- tive and stable.Computational examples on interactions between a surface-piercing vertical cylinder and a solitary wave or a cnoidal wave train demonstrates the validity of this program.

Specific wave structures of a fifth-order nonlinear water wave equation

Investigated in the present paper is a fifth-order nonlinear evolution(FONLE)equation,known as a nonlinear water wave(NLWW)equation,with applications in the applied sciences.More precisely,a traveling wave hypothesis is firstly applied that reduces the FONLE equation to a 1D domain.The Kudryashov methods(KMs)are then adopted as leading techniques to construct specific wave structures of the governing model which are classified as W-shaped and other solitons.In the end,the effect of changing the coefficients of nonlinear terms on the dynamical features of W-shaped and other solitons is investigated in detail for diverse groups of the involved parameters.