Comptes Rendus
no. 9-10
Boussinesq equation, surface waves in fluids
Dissipative Boussinesq equations
[Équations de Boussinesq dissipatives]
Denys Dutykh1 ; Frédéric Dias1 
1 CMLA, ENS Cachan, CNRS, PRES UniverSud, 61, avenue President Wilson, 94235 Cachan cedex, France
Comptes Rendus. Mécanique, Joseph Boussinesq, a Scientist of bygone days and present times, Volume 335 (2007) no. 9-10, pp. 559-583.

La théorie classique des vagues repose sur la théorie des écoulements non-visqueux. Cependant il est important d'inclure les effets visqueux dans certaines applications. Deux modèles sont ainsi proposés pour rajouter des effets dissipatifs dans le contexte des équations de Boussinesq, qui incluent les effets de faible dispersion et faible nonlinéarité dans le cadre des équations en eau peu profonde. Les équations de Boussinesq dissipatives sont ensuite intégrées numériquement.

The classical theory of water waves is based on the theory of inviscid flows. However it is important to include viscous effects in some applications. Two models are proposed to add dissipative effects in the context of the Boussinesq equations, which include the effects of weak dispersion and nonlinearity in a shallow water framework. The dissipative Boussinesq equations are then integrated numerically.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2007.08.003
Keywords: Computational fluid mechanics, Boussinesq equations, Dissipation, Viscosity, Long wave approximation, Tsunami propagation
Mots-clés : Mécanique des fluides numérique, Équations de Boussinesq, Dissipation, Viscosité, Approximation onde longue, Propagation de tsunami

Denys Dutykh 1 ; Frédéric Dias 1

1 CMLA, ENS Cachan, CNRS, PRES UniverSud, 61, avenue President Wilson, 94235 Cachan cedex, France 
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Denys Dutykh; Frédéric Dias. Dissipative Boussinesq equations. Comptes Rendus. Mécanique, Joseph Boussinesq, a Scientist of bygone days and present times, Volume 335 (2007) no. 9-10, pp. 559-583. doi : 10.1016/j.crme.2007.08.003. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2007.08.003/

[1] J.V. Boussinesq Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal, C. R. Acad. Sci. Paris, Volume 73 (1871), pp. 256-260

[2] J. Boussinesq 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, J. Math. Pures Appl., Volume 17 (1872), pp. 55-108

[3] A.J.C. de Saint-Venant Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l'introduction des marées dans leur lit, C. R. Acad. Sci. Paris, Volume 73 (1871), pp. 147-154

[4] D.H. Peregrine Long waves on a beach, J. Fluid Mech., Volume 27 (1967), pp. 815-827

[5] O. Nwogu Alternative form of Boussinesq equations for nearshore wave propagation, J. Waterway, Port, Coastal and Ocean Engrg., Volume 119 (1993), pp. 618-638

[6] G. Wei; J.T. Kirby; S.T. Grilli; R. Subramanya A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves, J. Fluid Mech., Volume 294 (1995), pp. 71-92

[7] P.A. Madsen; H.A. Schaffer Higher-order Boussinesq-type equations for surface gravity waves: Derivation and analysis, Phil. Trans. R. Soc. Lond. A, Volume 356 (1998), pp. 3123-3184

[8] J.T. Kirby Boussinesq models and applications to nearshore wave propagation, surfzone processes and wave-induced currents (V.C. Lakhan, ed.), Advances in Coastal Modeling, Elsevier, 2003, pp. 1-41

[9] J.L. Bona; M. Chen; J.-C. Saut Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: Derivation and linear theory, J. Nonlinear Sci., Volume 12 (2002), pp. 283-318

[10] H. Lamb Hydrodynamics, Cambridge University Press, 1932

[11] J. Boussinesq Lois de l'extinction de la houle en haute mer, C. R. Acad. Sci. Paris, Volume 121 (1895), pp. 15-20

[12] J.L. Bona; W.G. Pritchard; L.R. Scott An evaluation of a model equation for water waves, Phil. Trans. R. Soc. Lond. A, Volume 302 (1981), pp. 457-510

[13] J. Boussinesq Essai sur la théorie des eaux courantes, Mémoires présentés par divers Savants à l'Académie des Sciences, Volume 23 (1877), pp. 1-680

[14] E.O. Tuck The effect of a surface layer of viscous fluid on the wave resistance of a thin ship, J. Ship Res., Volume 18 (1974), pp. 265-271

[15] M.S. Longuet-Higgins Theory of weakly damped stokes waves: a new formulation and its physical interpretation, J. Fluid Mech., Volume 235 (1992), pp. 319-324

[16] B. Spivak; J.-M. Vanden-Broeck; T. Miloh Free-surface wave damping due to viscosity and surfactants, Eur. J. Mech. B Fluids, Volume 21 (2002), pp. 207-224

[17] C. Skandrani; C. Kharif; J. Poitevin Nonlinear evolution of water surface waves: the frequency down-shift phenomenon, Contemp. Math., Volume 200 (1996), pp. 157-171

[18] F. Dias, A.I. Dyachenko, V.E. Zakharov, Theory of weakly damped free-surface flows: a new formulation based on potential flow solutions, Phys. Lett. A (2007), in press

[19] K.D. Ruvinsky; F.I. Feldstein; G.I. Freidman Numerical simulations of the quasistationary stage of ripple excitation by steep-capillary waves, J. Fluid Mech., Volume 230 (1991), pp. 339-353

[20] A.B. Kennedy; Q. Chen; J.T. Kirby; R.A. Dalrymple Boussinesq modelling of wave transformation, breaking, and runup, J. Waterway, Port, Coastal and Ocean Engrg., Volume 126 (2000), pp. 39-47

[21] J.A. Zelt The run-up of nonbreaking and breaking solitary waves, Coastal Engrg., Volume 15 (1991), pp. 205-246

[22] K.L. Heitner; G.W. Housner Numerical model for tsunami runup, J. Waterway, Port, Coastal and Ocean Engrg., Volume 96 (1970), pp. 701-719

[23] D. Dutykh; F. Dias Viscous potential free-surface flows in a fluid layer of finite depth, C. R. Acad. Sci. Paris, Ser. I, Volume 345 (2007), pp. 113-118

[24] L. Jiang; C.-L. Ting; M. Perlin; W.W. Schultz Moderate and steep Faraday waves: instabilities, modulation and temporal asymmetries, J. Fluid Mech., Volume 329 (1996), pp. 275-307

[25] W. Zhang; J. Vinals Pattern formation in weakly damped parametric surface waves, J. Fluid Mech., Volume 336 (1997), pp. 301-330

[26] D. Dutykh; F. Dias Water waves generated by a moving bottom (A. Kundu, ed.), Tsunami and Nonlinear Waves, Springer Verlag (Geo Sci.), 2007, pp. 63-94

[27] F. Ursell The long-wave paradox in the theory of gravity waves, Proc. Camb. Phil. Soc., Volume 49 (1953), pp. 685-694

[28] M. Benoit, Contribution à l'étude des états de mer et des vagues, depuis l'océan jusqu'aux ouvrages cotiers, 2006, Mémoire d'habilitation à diriger des recherches

[29] G.B. Whitham Linear and Nonlinear Waves, John Wiley & Sons Inc., New York, 1999

[30] T.B. Benjamin Lectures on nonlinear wave motion, Lectures in Appl. Math., vol. 15, Amer. Math. Soc., Providence, RI, 1974, pp. 3-47

[31] D.H. Peregrine Equations for water waves and the approximation behind them, Waves on Beaches and Resulting Sediment Transport, Academic Press, New York, 1972, pp. 95-121

[32] R.J. Murray, Short wave modelling using new equations of Boussinesq type, in: Proc. 9th Australian Conf. on Coast. and Oc. Engrg., 1989

[33] P.A. Madsen; R. Murray; O.R. Sorensen A new form of the Boussinesq equations with improved linear dispersion characteristics, Coastal Engrg., Volume 15 (1991), pp. 371-388

[34] J.M. Witting A unified model for the evolution of nonlinear water waves, J. Comput. Phys., Volume 56 (1984), pp. 203-236

[35] M. Chen Exact traveling-wave solutions to bidirectional wave equations, Intern. J. Theoret. Phys., Volume 37 (1998), pp. 1547-1567

[36] A.C. Newell Finite amplitude instabilities of partial difference equations, SIAM J. Appl. Math., Volume 33 (1977), pp. 133-160

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