Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems

From the free surface Navier-Stokes system, we derive the non-hydrostatic Saint-Venant system for the shallow waters including friction and viscosity. The derivation leads to two formulations of growing complexity depending on the level of approximation chosen for the fluid pressure. The obtained models are compared with the Boussinesq models.

[1]  Sandra Soares Frazao,et al.  Undular bores and secondary waves -Experiments and hybrid finite-volume modelling , 2002 .

[2]  J. Bona,et al.  Model equations for long waves in nonlinear dispersive systems , 1972, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[3]  Min Chen,et al.  Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media. I: Derivation and Linear Theory , 2002, J. Nonlinear Sci..

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

[5]  Philippe Bonneton,et al.  A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. Part II : boundary conditions and validation , 2007 .

[6]  F. Ursell,et al.  The long-wave paradox in the theory of gravity waves , 1953, Mathematical Proceedings of the Cambridge Philosophical Society.

[7]  D. Peregrine.  Long waves on a beach , 1967, Journal of Fluid Mechanics.

[8]  F. Marche.  Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects , 2007 .

[9]  Jim Douglas,et al.  Superconvergence of a finite element approximation to the solution of a Sobolev equation in a single space variable , 1981 .

[10]  M. Walkley,et al.  A numerical method for extended Boussinesq shallow-water wave equations , 1999 .

[11]  J. Boussinesq,et al.  Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. , 1872 .

[12]  Jean-Frédéric Gerbeau,et al.  Derivation of viscous Saint-Venant system for laminar shallow water , 2001 .

[13]  C. D. Levermore,et al.  A shallow water model with eddy viscosity for basins with varying bottom topography , 2001 .

[14]  F. Saleri,et al.  A new two-dimensional shallow water model including pressure effects and slow varying bottom topography , 2004 .

[15]  P. Lions.  Mathematical topics in fluid mechanics , 1996 .

[16]  Michael Westdickenberg,et al.  Gravity driven shallow water models for arbitrary topography , 2004 .

[17]  E. Audusse,et al.  A multilayer Saint-Venant model: Derivation and numerical validation , 2005 .

[18]  O. Pironneau,et al.  ROUGH BOUNDARIES AND WALL LAWS , 1998 .

[19]  S. Perotto,et al.  Adaptive finite element methods for Boussinesq equations , 2000 .

[20]  Philippe Bonneton,et al.  A fourth‐order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq‐type equations. Part I: model development and analysis , 2006 .

[21]  J. Bona,et al.  Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: II. The nonlinear theory , 2004 .