Propagation Of Nonlinear Water Waves Research Articles

Meshless Generalized Finite Difference Method for the Propagation of Nonlinear Water Waves under Complex Wave Conditions

The propagation of nonlinear water waves under complex wave conditions is the key issue of hydrodynamics both in coastal and ocean engineering, which is significant in the prediction of strongly nonlinear phenomena regarding wave–structure interactions. In the present study, the meshless generalized finite difference method (GFDM) together with the second-order Runge–Kutta method (RKM2) is employed to construct a fully three-dimensional (3D) meshless numerical wave flume (NWF). Three numerical examples, i.e., the propagation of freak waves, irregular waves and focused waves, are implemented to verify the accuracy and stability of the developed 3D GFDM model. The results show that the present numerical model possesses good performance in the simulation of nonlinear water waves and suggest that the 3D “RKM2-GFDM” meshless scheme can be adopted to further simulate more complex nonlinear problems regarding wave–structure interactions in ocean engineering.

Simulation of fully nonlinear water wave propagation over the flat bottom and uneven bottom by meshless numerical wave tank

Simulation of fully nonlinear water wave propagation over the flat bottom and uneven bottom by meshless numerical wave tank

Global attractor for weakly damped, forced mKdV equation in low regularity spaces

The modified Korteweg-de Vries equation (for short, mKdV) models the propagation of nonlinear water waves in the shallow water approximation. We consider the weakly damped and forced mKdV under the periodic boundary condition. We often study mKdV equation by using the Miura transformation, which converts solutions of mKdV to solutions of the Korteweg-de Vries equation (KdV). However, if mKdV has damping and external forcing terms, the Miura transformation does not work well. To see the asymptotic behaviour of the solutions of mKdV equation, the study of global attractor is important. We prove the existence of the global attractor for $$ {s > 9/10} $$ in $$\dot{H}^{s}$$ for mKdV equation by applying the modified energies and almost conserved quantities.

On the propagation of nonlinear water waves in a three-dimensional numerical wave flume using the generalized finite difference method

Nonlinear water waves are common physical phenomena in the field of coastal and ocean engineering, which plays a critical role in the investigation of hydrodynamics regarding offshore and deep-water structures. In the present study, a three-dimensional (3D) numerical wave flume (NWF) is constructed to simulate the propagation of nonlinear water waves. On the basis of potential flow theory, the second-order Runge-Kutta method (RKM2) combining with a semi-Lagrangian approach is carried out to discretize the temporal variable of the 3D Laplace’s equation. For the spatial variables, the generalized finite difference method (GFDM) is adopted to solve the governing equations for the deformable computational domain at each time step. The upstream condition is considered as a wave-making boundary with imposing horizontal velocity while the downstream condition as a wave-absorbing boundary with a pre-defined sponge layer to deal with the phenomenon of wave reflection. Three numerical examples are investigated and discussed in detail to validate the accuracy and stability of the developed 3D GFDM-based NWF. The results show that the newly-proposed numerical method has good performance in the prediction of the dynamic evolution of nonlinear water waves, and suggests that the novel 3D “RKM2-GFDM” meshless scheme can be employed to further investigate more complicated hydrodynamic problems in practical applications.

Well-posedness of the classical solutions for a Kawahara–Korteweg–de Vries-type equation

The Kawahara–Korteweg–de Vries-type equation occurs in the modelization of magneto-acoustic waves in plasmas and propagation of nonlinear water waves in the long-wavelength region as in the case of Korteweg–de Vries equation. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem associated with this equation.

Application of generalized finite difference method to propagation of nonlinear water waves in numerical wave flume

In this paper, a numerical wave flume is formed by combining the generalized finite difference method (GFDM), the Runge–Kutta method, the semi-Lagrangian technique, the ramping function and the sponge layer to efficiently and accurately analyze the propagation of nonlinear water waves. On the basis of potential flow, the mathematical description of wave propagation is a time-dependent boundary value problem, governed by a Laplace equation for velocity potential and two nonlinear free-surface boundary conditions. The incident waves are introduced by imposing horizontal velocity along upstream boundary, as a sponge layer is placed at the end of flume to absorb wave energy and avoid any reflection of waves. The GFDM, a newly-developed meshless numerical method, and the second-order Runge–Kutta method were, respectively, adopted for spatial and temporal discretizations of the moving-boundary problems. The GFDM, which is truly free from mesh generation and numerical quadrature, is easy-to-program, straightforward and efficient, especially for moving-boundary problems. Four numerical examples are adopted in this paper to validate the stability, the efficiency and the accuracy of the proposed meshless numerical wave flume. The GFDM results were compared with other numerical solutions and experimental data to verify the merits and robustness of the proposed meshless numerical model.

A three-dimensional arbitrary Lagrangian–Eulerian Petrov–Galerkin finite element model for fully nonlinear free-surface waves

A three-dimensional numerical model based on the incompressible Navier–Stokes equations is developed to simulate fully nonlinear water wave propagation problems. The Navier–Stokes equations are solved using the Arbitrary Lagrangian–Eulerian scheme (ALE) and the Petrov–Galerkin Finite Element Method (PG-FEM). A fractional step method is used in the time discretization, where the convection and diffusion terms are separated from the pressure terms when solving the momentum equations. A damping layer is set in front of the outgoing boundary for absorbing the outgoing waves. The model is validated against a series of experimental data, including wave propagation over a submerged bar, wave propagation over a semicircular shoal and wave propagation over a submerged vertical breakwater. The good agreement between the numerical results and experimental data demonstrates the capacity and the accuracy of the model for simulating wave propagation over uneven seabed and wave interaction with submerged structures.

Ostéotomies métatarsiennes distales percutanées : suivi rétrospectif de 50 cas

Modeling the propagation of nonlinear free surface waves is a challenging task but inevitable in studying nearshore wave propagation processes. More recently, meshless methods are emerging as potential tools for studying water wave related problems, in particular when there is a flow separation. In this study, smoothed particle hydrodynamics (SPH) has been shown to be useful to study the propagation of nonlinear water waves generated in a numerical flume. The problems considered address various aspects associated with long time simulation and highly nonlinear waves such as bore formation and wave breaking. An explicit dynamic free surface boundary condition has been imposed in the SPH model for steady long time simulation. A correction has been prescribed for the mass of the ghost particles at the geometric singular zones for the present application. The developed model is validated with FEM based simulation and other meshless methods. The split-up of a single solitary wave as it crosses continental shelf is simulated by the present method.

Experimental and Lagrangian modeling of nonlinear water waves propagation on a sloping bottom

This paper presents an experimental and theoretical investigation of nonlinear water wave propagation over a sloping bed. Firstly, a series of monochromatic wave laboratory experiments were performed in order to measure the particle trajectories, evolution of wave profile, and wave phase velocity as wave propagates on a sloping bottom. The particle trajectories are quantified by means of images from a high speed camera, whereas the evolution of wave profile and variation of wave phase velocity are measured by a wave gauge array. Subsequently, the free-surface elevation, phase velocity, particle trajectories, and breaking wave height are estimated using a Lagrangian nonlinear wave transformation model. Model predictions show a reasonable agreement with experimental data.

SPH Simulation of shallow water wavepropagation

Modeling the propagation of nonlinear free surface waves is a challenging task but inevitable in studying nearshore wave propagation processes. More recently, meshless methods are emerging as potential tools for studying water wave related problems, in particular when there is a flow separation. In this study, smoothed particle hydrodynamics (SPH) has been shown to be useful to study the propagation of nonlinear water waves generated in a numerical flume. The problems considered address various aspects associated with long time simulation and highly nonlinear waves such as bore formation and wave breaking. An explicit dynamic free surface boundary condition has been imposed in the SPH model for steady long time simulation. A correction has been prescribed for the mass of the ghost particles at the geometric singular zones for the present application. The developed model is validated with FEM based simulation and other meshless methods. The split-up of a single solitary wave as it crosses continental shelf is simulated by the present method.

DEPTH-INTEGRATED NUMERICAL MODELING OF TURBULENT TRANSPORT BY LONG WAVES AND CURRENTS

In nature, flows are 3D phenomenon, but, in many geophysical settings, the water depth scale is smaller relative to the horizontal scale, such that horizontal 2D (H2D) motions dominate the flow structure. In those cases, especially in large domains, the H2D numerical model can be a practical and accurate tool - if the 3D physical properties can be included properly into the H2D model. Some of the H2D approaches in widespread use are the Boussinesq-type equations (BE) and shallow water equations (SWE) derived by a perturbation approach or depth averaging. The BE can account for some of the dispersive, turbulent and rotational flow properties frequently observed in nature (Kim et al., 2009). Also it has the ability of coupling currents and waves and can predict nonlinear water wave propagation over an uneven bottom from deep (or intermediate) water to the shallow water area. However, during the derivation of a H2D equation set, BE or SWE, some of the 3D flow properties like the dispersive stresses (Kuipers and Vreugdenhill, 1973) and the effects of the unresolved small scale 3D turbulence are excluded. Subsequently, there must be some limitations for predicting horizontal flow structures which can be generated through these neglected 3D effects. Naturally, any inaccuracy of the hydrodynamic flow model is reflected in the results of a coupled scalar transport model. In order to incorporate 3D turbulence effects into H2D flow models, various approaches have been proposed. Among many others, the stochastic backscatter model (BSM) proposed by Hinterberger et al. (2007) can account for the mechanism of inverse energy transfer from unresolved 3D turbulence to resolved 2D flow motions. Reasonable results were obtained by the proposed methods. Similar to the flow model, for scalar transport it is desired to develop a H2D model that can approximately account for the vertical deviations of concentration and velocity, and the associated mixing. For the accurate prediction of transport, an accurate numerical solver which can minimize numerical dispersion, dissipation and diffusion should be developed. Recently, the finite volume method (FVM) using approximate Riemann solvers has been developed and applied successfully. In this study, a depth-integrated model including subgrid scale mixing effects for turbulent transport by long waves and currents is presented. A fully-nonlinear, depth-integrated set of equations for weakly dispersive and rotational flow are derived by the long wave perturbation approach. The same approach is applied to derive a depth-integrated scalar transport model. The proposed equations are solved by a fourth-order accurate FVM. The depth-integrated flow and transport models are applied to typical problems which have different mixing mechanisms. Several important conclusions are obtained from the simulations: (i) From a mixing layer simulation it is revealed that the dispersive stress implemented with a stochastic BSM plays an important role for energy transfer. (ii) The proposed transport model coupled with the depth-integrated flow model can predict the passive scalar transport based on the turbulent intensity - not by relying on empirical constants. (iii) For near field transport simulations, the inherent limitation of the two-dimensional horizontal model to capture vertical structure is recognized. (iv) If the main mechanism of flow instability originates from relatively large-scale bottom topography features, then the effects of the dispersive stresses are less important.

LAGRANGIAN BREAKER CHARACTERISTICS FOR NONLINEAR WATER WAVES PROPAGATING ON SLOPING BOTTOMS

In this paper, a new third-order Lagrangian asymptotic solution describing nonlinear water wave propagation on the surface of a uniform sloping bottom is presented. The model is formulated in the Lagrangian variables and we use a two-parameter perturbation method to develop a new mathematical derivation. The particle trajectories, wave pressure and Lagrangian velocity potential are obtained as a function of the nonlinear wave steepness ï¥ and the bottom slope ï¡ perturbed to third order. The analytical solution in Lagrangian form satisfies state of the normal pressure at the free surface. The condition of the conservation of mass flux is examined in detail for the first time. The two important properties in Lagrangian coordinates, Lagrangian wave frequency and Lagrangian mean level, are included in the third-order solution. The solution can also be used to estimate the mean return current for waves progressing over the sloping bottom. The Lagrangian solution untangle the description of the features of wave shoaling in the direction of wave propagation from deep to shallow water, as well as the process of successive deformation of a wave profile and water particle trajectories leading to wave breaking. The proposed model has proved to be capable of a better description of non-linear wave effects than the corresponding approximation of the same order derived by using the Eulerian description. 
 The proposed solution has also been used to determine the wave shoaling process, and the comparisons between the experimental and theoretical results are presented in Fig.1a~1b. In addition, the basic wave-breaking criterion, namely the kinematical Stokes stability condition, has been investigated. The comparisons between the present theory, empirical formula of Goda (2004) and the experiments made by Iwagali et al.(1974), Deo et al.(2003) and Tsai et al.(2005) for the breaking index(Hb/L0) versus the relative water depth(d0/L0) under two different bottom slopes are depicted in Figs 2a~2b. It is found that the theoretical breaking index is well agreement with the experimental results for three bottom slopes. However,for steep slope of 1/3 shown in Fig 2b, the result of Goda's empirical formula gives a larger value in comparison with the experimental data and the present theory. Some of empirical formulas presented the breaking wave height in terms of deepwater wave condition, such as in Sunamura (1983) and in Rattanapitikon and Shibayama(2000). Base on the results depicted in Fig. 3a~3b, it showed that the theoretical results are in good agreement with the experimental data (Iwagali et al. 1974, Deo et al.2003 and Tsai et al. 2005) than the empirical formulas. The empirical formula of Sunamura (1983) always predicts an overestimation value.

Flow Dynamics and Bed Resistance of Wave Propagation over Bed Ripples

The viscous, two-dimensional, free-surface flow induced by the propagation of nonlinear water waves over a rigid rippled bed was simulated numerically. The simulations were based on the numerical solution of the Navier-Stokes equations subject to fully nonlinear free-surface boundary conditions and appropriate bottom, inflow, and outflow boundary conditions. The equations were properly transformed so that the computational domain became time-independent. A hybrid scheme was used for the spatial discretization with finite differences in the streamwise direction and a pseudospectral approximation with Chebyshev polynomials in the vertical direction. A fractional time-step scheme was used for the temporal discretization. Over the rippled bed, the wave boundary layer thickness increased significantly, while vortex shedding at the ripple crest generated alternating circulation regions over the ripple trough. The velocity of the Eulerian drift profile was opposite to the direction of wave propagation far above the ripples, whereas close to the bed, its magnitude was influenced by the ripples up to a height of about six times the ripple height above the ripple crest. The amplitude of the wall shear stress on the ripples increased with increasing ripple steepness, whereas the amplitude of the corresponding friction drag force on a ripple was insensitive to this increase. The amplitude of the form drag force attributable to the dynamic pressure increased with increasing ripple steepness; therefore, the percentage of friction in the total drag force decreased with increasing ripple steepness. The period-averaged drag forces on a ripple were very weak, while the influence of form drag increased with increasing ripple steepness.

A mathematical model for weakly nonlinear water wave propagation

Nonlinear wave equations are derived in this paper based on combining the Stokes second-order wave theory with the Boussinesq theory. In shallow water the new equations reduce to the improved Boussinesq equations and in deep water the second-order Stokes waves can be simulated directly without water depth restriction. There is no contradiction between the Boussinesq theory and the second-order Stokes theory for the new equations. Numerical results of the wave equations are compared with analytical solutions of nonlinear Cnoidal waves and second-order Stokes waves. Good agreement has been achieved. The new equations can be easily and directly applied for random wave modelling in finite water depth.

Numerical simulations of inviscid and viscous free‐surface flows with application to nonlinear water waves

AbstractThe purpose of the present study is to establish a numerical model appropriate for solving inviscid/viscous free‐surface flows related to nonlinear water wave propagation. The viscous model presented herein is based on the Navier–Stokes equations, and the free‐surface is calculated through an arbitrary Lagrangian–Eulerian streamfunction‐vorticity formulation. The streamfunction field is governed by the Poisson equation, and the vorticity is obtained on the basis of the vorticity transport equation. For computing the inviscid flow the Laplace streamfunction equation is used. These equations together with the respective (appropriate) fully nonlinear free‐surface boundary conditions are solved using a finite difference method. To demonstrate the model feasibility, in the present study we first simulate collision processes of two solitary waves of different amplitudes, and compute the phenomenon of overtaking of such solitary waves. The developed model is subsequently applied to calculate (both inviscid and the viscous) flow field, as induced by passing of a solitary wave over submerged rectangular structures and rigid ripple beds. Our study provides a reasonably good understanding of the behavior of (inviscid/viscous) free‐surface flows, within the framework of streamfunction‐vorticity formulation. The successful simulation of the above‐mentioned test cases seems to suggest that the arbitrary Lagrangian–Eulerian/streamfunction‐vorticity formulation is a potentially powerful approach, capable of effectively solving the fully nonlinear inviscid/viscous free‐surface flow interactions. Copyright © 2009 John Wiley & Sons, Ltd.

Propagation of axi-symmetric nonlinear shallow water waves over slowly varying depth

A problem in nonlinear water-wave propagation on the surface of an inviscid, stationary fluid is presented. The primary surface wave, suitably initiated at some radius, is taken to be a slowly evolving nonlinear cylindrical wave (governed by an appropriate Korteweg–de Vries equation); the depth is assumed to be varying in a purely radial direction. We consider a sech 2 profile at an initial radius (which is, following our scalings, rather large), and we describe the evolution as it propagates radially outwards. This initial profile was chosen because its evolution over constant depth is understood both analytically and numerically, even though it is not an exact solitary-wave solution of the cylindrical KdV equation. The propagation process will introduce reflected and re-reflected components which will also be described. The precise nature of these reflections is fixed by the requirements of mass conservation. The asymptotic results presented describe the evolution of the primary wave, the development of an outward shelf and also an inward (reflected) shelf. These results make use of specific depth variations (which were chosen to simplify the solution of the relevant equations), and mirror those obtained for the problem of 1D plane-waves over variable depth, although the details here are more complex due to the axi-symmetry.

A three dimensional multigrid model for fully nonlinear water waves

In this paper a three dimensional multigrid model is developed for the linear and fully nonlinear water wave propagation. The Laplace equation is transformed from an irregular calculation domain to a regular one and the boundary conditions on water surface and sea bottom can be implemented precisely. The multigrid method is used to solve the governing equation and the requirement of the computer storage is very small. The difference in computer time for running the linear model and the fully nonlinear model is not significant. The present model is valid over the complete range of water depths. For fully nonlinear water wave problems the present model is particularly efficient. The model is used to investigate the validity of the mild-slope equation for the case of strong wave focusing behind an elliptical shoal and also applied to Whalin's experiment. Simulation of wave breaking is not included in the present model.