A σ‐coordinate non‐hydrostatic model with embedded Boussinesq‐type‐like equations for modeling deep‐water waves

A σ-coordinate non-hydrostatic model, combined with the embedded Boussinesq-type-like equations, a reference velocity, and an adapted top-layer control, is developed to study the evolution of deep-water waves. The advantage of using the Boussinesq-type-like equations with the reference velocity is to provide an analytical-based non-hydrostatic pressure distribution at the top-layer and to optimize wave dispersion property. The σ-based non-hydrostatic model naturally tackles the so-called overshooting issue in the case of non-linear steep waves. Efficiency and accuracy of this non-hydrostatic model in terms of wave dispersion and nonlinearity are critically examined. Overall results show that the newly developed model using a few layers is capable of resolving the evolution of non-linear deep-water wave groups. Copyright © 2009 John Wiley & Sons, Ltd.

[1]  Chin H. Wu,et al.  An efficient and accurate non‐hydrostatic model with embedded Boussinesq‐type like equations for surface wave modeling , 2009 .

[2]  Measurements of wave modulation and breaking , 2005 .

[3]  Marcel Zijlema,et al.  Efficient computation of surf zone waves using the nonlinear shallow water equations with non-hydrostatic pressure , 2008 .

[4]  Scott F. Bradford.  Godunov-Based Model for Nonhydrostatic Wave Dynamics , 2005 .

[5]  Chin H. Wu,et al.  Efficient non-hydrostatic modelling of surface waves interacting with structures , 2007 .

[6]  O. Phillips.  The dynamics of the upper ocean , 1966 .

[7]  Pengzhi Lin,et al.  A σ‐coordinate three‐dimensional numerical model for surface wave propagation , 2002 .

[8]  Karsten Trulsen,et al.  Spatial Evolution of Water Surface Waves: Numerical Simulation and Experiment of Bichromatic Waves , 2001 .

[9]  R. Falconer,et al.  Three‐dimensional numerical modelling of free surface flows with non‐hydrostatic pressure , 2002 .

[10]  Aifeng Yao,et al.  Laboratory measurements of limiting freak waves on currents , 2004 .

[11]  John Grue,et al.  An efficient model for three-dimensional surface wave simulations , 2005 .

[12]  Marcel Zijlema,et al.  An accurate and efficient finite‐difference algorithm for non‐hydrostatic free‐surface flow with application to wave propagation , 2003 .

[13]  E. van Groesen,et al.  Experiments and Numerics of Bichromatic Wave Groups , 2001 .

[14]  P. Liu,et al.  A two-layer approach to wave modelling , 2004, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[15]  T. Y. Wu,et al.  A unified theory for modeling water waves , 2001 .

[16]  C. Mei.  The applied dynamics of ocean surface waves , 1983 .

[17]  Xinjian Chen.  A fully hydrodynamic model for three‐dimensional, free‐surface flows , 2003 .

[18]  Chin H. Wu,et al.  A two‐dimensional vertical non‐hydrostatic σ model with an implicit method for free‐surface flows , 2004 .

[19]  John D. Fenton,et al.  Numerical methods for nonlinear waves , 2003 .

[20]  Chin H. Wu,et al.  Breaking criteria and energy losses for three‐dimensional wave breaking , 2002 .

[21]  Chin H. Wu,et al.  A new efficient 3D non-hydrostatic free-surface flow model for simulating water wave motions , 2006 .

[22]  Evolution of sidebands in deep-water bichromatic wave trains , 2007 .

[23]  R. G. Dean,et al.  FREAK WAVES: A POSSIBLE EXPLANATION , 1990 .

[24]  O. Nwogu.  Alternative form of Boussinesq equations for nearshore wave propagation , 1993 .

[25]  K. Hall,et al.  High‐order compact numerical schemes for non‐hydrostatic free surface flows , 2006 .

[26]  D. Yue,et al.  COMPUTATION OF NONLINEAR FREE-SURFACE FLOWS , 1996 .

[27]  A three‐dimensional non‐hydrostatic vertical boundary fitted model for free‐surface flows , 2008 .

[28]  Stefan Mayer,et al.  A FRACTIONAL STEP METHOD FOR UNSTEADY FREE-SURFACE FLOW WITH APPLICATIONS TO NON-LINEAR WAVE DYNAMICS , 1998 .

[29]  W. Melville,et al.  Laboratory measurements of deep-water breaking waves , 1990, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[30]  P. Liu,et al.  Numerical Simulations of Nonlinear Short Waves Using a Multilayer Model , 2005 .

[31]  John D. Fenton,et al.  A Fifth‐Order Stokes Theory for Steady Waves , 1985 .

[32]  Efim Pelinovsky,et al.  Physical Mechanisms of the Rogue Wave Phenomenon , 2003 .

[33]  A 3‐D non‐hydrostatic pressure model for small amplitude free surface flows , 2006 .

[34]  Marcel Zijlema,et al.  Non‐hydrostatic 3D free surface layer‐structured finite volume model for short wave propagation , 2009 .

[35]  Aifeng Yao,et al.  Incipient breaking of unsteady waves on sheared currents , 2005 .

[36]  Hengliang Yuan,et al.  An implicit three‐dimensional fully non‐hydrostatic model for free‐surface flows , 2004 .

[37]  Jan-Tai Kuo,et al.  A higher-order non-hydrostatic σ model for simulating non-linear refraction–diffraction of water waves , 2009 .

[38]  Bin Li.  A 3-D model based on Navier–Stokes equations for regular and irregular water wave propagation , 2008 .

[39]  Eng Soon Chan,et al.  A Comparison of Two- and Three-Dimensional Wave Breaking , 1998 .

[40]  Harry B. Bingham,et al.  A new Boussinesq method for fully nonlinear waves from shallow to deep water , 2002, Journal of Fluid Mechanics.

[41]  Paul C. Liu,et al.  Freak waves - more frequent than rare! , 2004 .

[42]  Eliezer Kit,et al.  Evolution of a nonlinear wave field along a tank: experiments and numerical simulations based on the spatial Zakharov equation , 2001, Journal of Fluid Mechanics.

[43]  Bin Li,et al.  Three-Dimensional Model of Navier-Stokes Equations for Water Waves , 2001 .

[44]  Vincenzo Casulli,et al.  A SEMI-IMPLICIT FINITE DIFFERENCE METHOD FOR NON-HYDROSTATIC, FREE-SURFACE FLOWS , 1999 .

[45]  Roger Alexander Falconer,et al.  An implicit numerical algorithm for solving non-hydrostatic free-surface flow problems , 2001 .

[46]  J. A. Battjes,et al.  Generalized evolution equations for nonlinear surface gravity waves over two-dimensional topography , 2006, Journal of Fluid Mechanics.

[47]  Marcel Zijlema,et al.  Further experiences with computing non‐hydrostatic free‐surface flows involving water waves , 2005 .

[48]  Hengliang Yuan,et al.  Fully Nonhydrostatic Modeling of Surface Waves , 2006 .

[49]  Chin H. Wu,et al.  Energy Dissipation of Unsteady Wave Breaking on Currents , 2004 .

[50]  Roy A. Walters,et al.  A semi‐implicit finite element model for non‐hydrostatic (dispersive) surface waves , 2005 .

[51]  Jan-Tai Kuo,et al.  A higher-order σ-coordinate non-hydrostatic model for nonlinear surface waves , 2007 .