June  2015, 20(4): 961-988. doi: 10.3934/dcdsb.2015.20.961

An energy-consistent depth-averaged Euler system: Derivation and properties

1. 

INRIA Roquencourt, B.P. 105, 78153 Le Chesnay Cedex

2. 

University Paris Diderot, Sorbonne Paris Cité, Institut de Physique du Globe de Paris, Seismology group, 1 rue Jussieu, 75005 Paris, France

3. 

CEREMA, 134 rue de Beauvais, F-60280 Margny-Lès-Compiègne, France

4. 

Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

Received  April 2014 Revised  September 2014 Published  February 2015

In this paper, we present an original derivation process of a non-hydrostatic shallow water-type model which aims at approximating the incompressible Euler and Navier-Stokes systems with free surface. The closure relations are obtained by a minimal energy constraint instead of an asymptotic expansion. The model slightly differs from the well-known Green-Naghdi model and is confronted with stationary and analytical solutions of the Euler system corresponding to rotational flows. At the end of the paper, we give time-dependent analytical solutions for the Euler system that are also analytical solutions for the proposed model but that are not solutions of the Green-Naghdi model. We also give and compare analytical solutions of the two non-hydrostatic shallow water models.
Citation: Marie-Odile Bristeau, Anne Mangeney, Jacques Sainte-Marie, Nicolas Seguin. An energy-consistent depth-averaged Euler system: Derivation and properties. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 961-988. doi: 10.3934/dcdsb.2015.20.961
References:
[1]

B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics,, Invent. Math., 171 (2008), 485.  doi: 10.1007/s00222-007-0088-4.  Google Scholar

[2]

B. Alvarez-Samaniego and D. Lannes, A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations,, Indiana Univ. Math. J., 57 (2008), 97.  doi: 10.1512/iumj.2008.57.3200.  Google Scholar

[3]

A.-J.-C. Barré de Saint-Venant, Théorie du mouvement non permanent des eaux avec applications aux crues des rivières et à l'introduction des marées dans leur lit,, C. R. Acad. Sci. Paris, 73 (1871), 147.   Google Scholar

[4]

J.-L. Bona, T.-B. Benjamin and J.-J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Philos. Trans. Royal Soc. London Series A, 272 (1972), 47.  doi: 10.1098/rsta.1972.0032.  Google Scholar

[5]

P. Bonneton, E. Barthelemy, F. Chazel, R. Cienfuegos, D. Lannes, F. Marche and M. Tissier, Recent advances in Serre-Green Naghdi modelling for wave transformation, breaking and runup processes,, European Journal of Mechanics - B/Fluids, 30 (2011), 589.  doi: 10.1016/j.euromechflu.2011.02.005.  Google Scholar

[6]

F. Bouchut, A. Mangeney-Castelnau, B. Perthame and J.-P. Vilotte, A new model of Saint-Venant and Savage-Hutter type for gravity driven shallow water flows,, Comptes Rendus Mathematique, 336 (2003), 531.  doi: 10.1016/S1631-073X(03)00117-1.  Google Scholar

[7]

F. Bouchut and M. Westdickenberg, Gravity driven shallow water models for arbitrary topography,, Comm. in Math. Sci., 2 (2004), 359.  doi: 10.4310/CMS.2004.v2.n3.a2.  Google Scholar

[8]

Y. Brenier, Homogeneous hydrostatic flows with convex velocity profiles,, Nonlinearity, 12 (1999), 495.  doi: 10.1088/0951-7715/12/3/004.  Google Scholar

[9]

M.-O. Bristeau, N. Goutal and J. Sainte-Marie, Numerical simulations of a non-hydrostatic Shallow Water model,, Computers & Fluids, 47 (2011), 51.  doi: 10.1016/j.compfluid.2011.02.013.  Google Scholar

[10]

M.-O. Bristeau and J. Sainte-Marie, Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 733.  doi: 10.3934/dcdsb.2008.10.733.  Google Scholar

[11]

R. Camassa, D. D. Holm and C. D. Levermore, Long-time effects of bottom topography in shallow water,, Phys. D, 98 (1996), 258.  doi: 10.1016/0167-2789(96)00117-0.  Google Scholar

[12]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation,, Adv. Appl. Math., 31 (1994), 1.  doi: 10.1016/S0065-2156(08)70254-0.  Google Scholar

[13]

F. Chazel, D. Lannes and F. Marche, Numerical simulation of strongly nonlinear and dispersive waves using a Green-Naghdi model,, J. Sci. Comput., 48 (2011), 105.  doi: 10.1007/s10915-010-9395-9.  Google Scholar

[14]

A. J. Chorin, Numerical solution of the Navier-Stokes equations,, Math. Comp., 22 (1968), 745.  doi: 10.1090/S0025-5718-1968-0242392-2.  Google Scholar

[15]

A. Decoene, L. Bonaventura, E. Miglio and F. Saleri, Asymptotic derivation of the section-averaged shallow water equations for river hydraulics,, M3AS, 19 (2009), 387.  doi: 10.1142/S0218202509003474.  Google Scholar

[16]

D. Dutykh, T. Katsaounis and D. Mitsotakis, Finite volume methods for unidirectional dispersive wave models,, Internat. J. Numer. Methods Fluids, 71 (2013), 717.  doi: 10.1002/fld.3681.  Google Scholar

[17]

S. Ferrari and F. Saleri, A new two-dimensional Shallow Water model including pressure effects and slow varying bottom topography,, M2AN Math. Model. Numer. Anal., 38 (2004), 211.  doi: 10.1051/m2an:2004010.  Google Scholar

[18]

J.-F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation,, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 89.  doi: 10.3934/dcdsb.2001.1.89.  Google Scholar

[19]

A. E. Green, N. Laws and P. M. Naghdi, On the theory of water waves,, Proc. Roy. Soc. (London) Ser. A, 338 (1974), 43.  doi: 10.1098/rspa.1974.0072.  Google Scholar

[20]

A. Green and P. Naghdi, A derivation of equations for wave propagation in water of variable depth,, J. Fluid Mech., 78 (1976), 237.  doi: 10.1017/S0022112076002425.  Google Scholar

[21]

E. Grenier, On the derivation of homogeneous hydrostatic equations,, ESAIM: M2AN, 33 (1999), 965.  doi: 10.1051/m2an:1999128.  Google Scholar

[22]

D. Lannes and P. Bonneton, Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation,, Physics of Fluids, 21 (2009).  doi: 10.1063/1.3053183.  Google Scholar

[23]

O. Le Métayer, S. Gavrilyuk and S. Hank, A numerical scheme for the Green-Naghdi model,, J. Comp. Phys., 229 (2010), 2034.  doi: 10.1016/j.jcp.2009.11.021.  Google Scholar

[24]

C. D. Levermore, Entropy-based moment closures for kinetic equations,, in Proceedings of the International Conference on Latest Developments and Fundamental Advances in Radiative Transfer (Los Angeles, 26 (1997), 591.  doi: 10.1080/00411459708017931.  Google Scholar

[25]

C. Levermore and M. Sammartino, A shallow water model with eddy viscosity for basins with varying bottom topography,, Nonlinearity, 14 (2001), 1493.  doi: 10.1088/0951-7715/14/6/305.  Google Scholar

[26]

Y. A. Li, A shallow-water approximation to the full water wave problem,, Comm. Pure Appl. Math., 59 (2006), 1225.  doi: 10.1002/cpa.20148.  Google Scholar

[27]

P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1: Incompressible models,, Oxford University Press, (1996).   Google Scholar

[28]

A. Lucas, A. Mangeney and J. P. Ampuero, Frictional weakening in landslides on earth and on other planetary bodies,, Nature Communication, 5 (2014).  doi: 10.1038/ncomms4417.  Google Scholar

[29]

N. Makarenko, A second long-wave approximation in the Cauchy-Poisson problem(in russian),, Dyn. Contin. Media, 77 (1986), 56.   Google Scholar

[30]

A. Mangeney, F. Bouchut, N. Thomas, J. P. Vilotte and M.-O. Bristeau, Numerical modeling of self-channeling granular flows and of their levee-channel deposits,, Journal of Geophysical Research - Earth Surface, 112 (2007), 2003.  doi: 10.1029/2006JF000469.  Google Scholar

[31]

A. Mangeney-Castelnau, F. Bouchut, J. P. Vilotte, E. Lajeunesse, A. Aubertin and M. Pirulli, On the use of Saint-Venant equations to simulate the spreading of a granular mass,, Journal of Geophysical Research: Solid Earth, 110 (2005), 1978.  doi: 10.1029/2004JB003161.  Google Scholar

[32]

F. Marche, Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects,, European Journal of Mechanic /B, 26 (2007), 49.  doi: 10.1016/j.euromechflu.2006.04.007.  Google Scholar

[33]

N. Masmoudi and T. Wong, On the Hs theory of hydrostatic Euler equations,, Archive for Rational Mechanics and Analysis, 204 (2012), 231.  doi: 10.1007/s00205-011-0485-0.  Google Scholar

[34]

J. Miles and R. Salmon, Weakly dispersive nonlinear gravity waves,, J. Fluid Mech., 157 (1985), 519.  doi: 10.1017/S0022112085002488.  Google Scholar

[35]

B. T. Nadiga, L. G. Margolin and P. K. Smolarkiewicz, Different approximations of shallow fluid flow over an obstacle,, Phys. Fluids, 8 (1996), 2066.  doi: 10.1063/1.869009.  Google Scholar

[36]

O. Nwogu, Alternative form of Boussinesq equations for nearshore wave propagation,, Journal of Waterway, 119 (1993), 618.  doi: 10.1061/(ASCE)0733-950X(1993)119:6(618).  Google Scholar

[37]

D. Peregrine, Long waves on a beach,, J. Fluid Mech., 27 (1967), 815.  doi: 10.1017/S0022112067002605.  Google Scholar

[38]

B. Perthame, Kinetic Formulation of Conservation Laws,, Oxford University Press, (2002).   Google Scholar

[39]

J. Sainte-Marie, Vertically averaged models for the free surface Euler system. Derivation and kinetic interpretation,, Math. Models Methods Appl. Sci. (M3AS), 21 (2011), 459.  doi: 10.1142/S0218202511005118.  Google Scholar

[40]

C. H. Su and C. S. Gardner, Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg-de Vries equation and Burgers equation,, J. Mathematical Phys., 10 (1969), 536.  doi: 10.1063/1.1664873.  Google Scholar

[41]

W. C. Thacker, Some exact solutions to the nonlinear shallow-water wave equations,, J. Fluid Mech., 107 (1981), 499.  doi: 10.1017/S0022112081001882.  Google Scholar

show all references

References:
[1]

B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics,, Invent. Math., 171 (2008), 485.  doi: 10.1007/s00222-007-0088-4.  Google Scholar

[2]

B. Alvarez-Samaniego and D. Lannes, A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations,, Indiana Univ. Math. J., 57 (2008), 97.  doi: 10.1512/iumj.2008.57.3200.  Google Scholar

[3]

A.-J.-C. Barré de Saint-Venant, Théorie du mouvement non permanent des eaux avec applications aux crues des rivières et à l'introduction des marées dans leur lit,, C. R. Acad. Sci. Paris, 73 (1871), 147.   Google Scholar

[4]

J.-L. Bona, T.-B. Benjamin and J.-J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Philos. Trans. Royal Soc. London Series A, 272 (1972), 47.  doi: 10.1098/rsta.1972.0032.  Google Scholar

[5]

P. Bonneton, E. Barthelemy, F. Chazel, R. Cienfuegos, D. Lannes, F. Marche and M. Tissier, Recent advances in Serre-Green Naghdi modelling for wave transformation, breaking and runup processes,, European Journal of Mechanics - B/Fluids, 30 (2011), 589.  doi: 10.1016/j.euromechflu.2011.02.005.  Google Scholar

[6]

F. Bouchut, A. Mangeney-Castelnau, B. Perthame and J.-P. Vilotte, A new model of Saint-Venant and Savage-Hutter type for gravity driven shallow water flows,, Comptes Rendus Mathematique, 336 (2003), 531.  doi: 10.1016/S1631-073X(03)00117-1.  Google Scholar

[7]

F. Bouchut and M. Westdickenberg, Gravity driven shallow water models for arbitrary topography,, Comm. in Math. Sci., 2 (2004), 359.  doi: 10.4310/CMS.2004.v2.n3.a2.  Google Scholar

[8]

Y. Brenier, Homogeneous hydrostatic flows with convex velocity profiles,, Nonlinearity, 12 (1999), 495.  doi: 10.1088/0951-7715/12/3/004.  Google Scholar

[9]

M.-O. Bristeau, N. Goutal and J. Sainte-Marie, Numerical simulations of a non-hydrostatic Shallow Water model,, Computers & Fluids, 47 (2011), 51.  doi: 10.1016/j.compfluid.2011.02.013.  Google Scholar

[10]

M.-O. Bristeau and J. Sainte-Marie, Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 733.  doi: 10.3934/dcdsb.2008.10.733.  Google Scholar

[11]

R. Camassa, D. D. Holm and C. D. Levermore, Long-time effects of bottom topography in shallow water,, Phys. D, 98 (1996), 258.  doi: 10.1016/0167-2789(96)00117-0.  Google Scholar

[12]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation,, Adv. Appl. Math., 31 (1994), 1.  doi: 10.1016/S0065-2156(08)70254-0.  Google Scholar

[13]

F. Chazel, D. Lannes and F. Marche, Numerical simulation of strongly nonlinear and dispersive waves using a Green-Naghdi model,, J. Sci. Comput., 48 (2011), 105.  doi: 10.1007/s10915-010-9395-9.  Google Scholar

[14]

A. J. Chorin, Numerical solution of the Navier-Stokes equations,, Math. Comp., 22 (1968), 745.  doi: 10.1090/S0025-5718-1968-0242392-2.  Google Scholar

[15]

A. Decoene, L. Bonaventura, E. Miglio and F. Saleri, Asymptotic derivation of the section-averaged shallow water equations for river hydraulics,, M3AS, 19 (2009), 387.  doi: 10.1142/S0218202509003474.  Google Scholar

[16]

D. Dutykh, T. Katsaounis and D. Mitsotakis, Finite volume methods for unidirectional dispersive wave models,, Internat. J. Numer. Methods Fluids, 71 (2013), 717.  doi: 10.1002/fld.3681.  Google Scholar

[17]

S. Ferrari and F. Saleri, A new two-dimensional Shallow Water model including pressure effects and slow varying bottom topography,, M2AN Math. Model. Numer. Anal., 38 (2004), 211.  doi: 10.1051/m2an:2004010.  Google Scholar

[18]

J.-F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation,, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 89.  doi: 10.3934/dcdsb.2001.1.89.  Google Scholar

[19]

A. E. Green, N. Laws and P. M. Naghdi, On the theory of water waves,, Proc. Roy. Soc. (London) Ser. A, 338 (1974), 43.  doi: 10.1098/rspa.1974.0072.  Google Scholar

[20]

A. Green and P. Naghdi, A derivation of equations for wave propagation in water of variable depth,, J. Fluid Mech., 78 (1976), 237.  doi: 10.1017/S0022112076002425.  Google Scholar

[21]

E. Grenier, On the derivation of homogeneous hydrostatic equations,, ESAIM: M2AN, 33 (1999), 965.  doi: 10.1051/m2an:1999128.  Google Scholar

[22]

D. Lannes and P. Bonneton, Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation,, Physics of Fluids, 21 (2009).  doi: 10.1063/1.3053183.  Google Scholar

[23]

O. Le Métayer, S. Gavrilyuk and S. Hank, A numerical scheme for the Green-Naghdi model,, J. Comp. Phys., 229 (2010), 2034.  doi: 10.1016/j.jcp.2009.11.021.  Google Scholar

[24]

C. D. Levermore, Entropy-based moment closures for kinetic equations,, in Proceedings of the International Conference on Latest Developments and Fundamental Advances in Radiative Transfer (Los Angeles, 26 (1997), 591.  doi: 10.1080/00411459708017931.  Google Scholar

[25]

C. Levermore and M. Sammartino, A shallow water model with eddy viscosity for basins with varying bottom topography,, Nonlinearity, 14 (2001), 1493.  doi: 10.1088/0951-7715/14/6/305.  Google Scholar

[26]

Y. A. Li, A shallow-water approximation to the full water wave problem,, Comm. Pure Appl. Math., 59 (2006), 1225.  doi: 10.1002/cpa.20148.  Google Scholar

[27]

P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1: Incompressible models,, Oxford University Press, (1996).   Google Scholar

[28]

A. Lucas, A. Mangeney and J. P. Ampuero, Frictional weakening in landslides on earth and on other planetary bodies,, Nature Communication, 5 (2014).  doi: 10.1038/ncomms4417.  Google Scholar

[29]

N. Makarenko, A second long-wave approximation in the Cauchy-Poisson problem(in russian),, Dyn. Contin. Media, 77 (1986), 56.   Google Scholar

[30]

A. Mangeney, F. Bouchut, N. Thomas, J. P. Vilotte and M.-O. Bristeau, Numerical modeling of self-channeling granular flows and of their levee-channel deposits,, Journal of Geophysical Research - Earth Surface, 112 (2007), 2003.  doi: 10.1029/2006JF000469.  Google Scholar

[31]

A. Mangeney-Castelnau, F. Bouchut, J. P. Vilotte, E. Lajeunesse, A. Aubertin and M. Pirulli, On the use of Saint-Venant equations to simulate the spreading of a granular mass,, Journal of Geophysical Research: Solid Earth, 110 (2005), 1978.  doi: 10.1029/2004JB003161.  Google Scholar

[32]

F. Marche, Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects,, European Journal of Mechanic /B, 26 (2007), 49.  doi: 10.1016/j.euromechflu.2006.04.007.  Google Scholar

[33]

N. Masmoudi and T. Wong, On the Hs theory of hydrostatic Euler equations,, Archive for Rational Mechanics and Analysis, 204 (2012), 231.  doi: 10.1007/s00205-011-0485-0.  Google Scholar

[34]

J. Miles and R. Salmon, Weakly dispersive nonlinear gravity waves,, J. Fluid Mech., 157 (1985), 519.  doi: 10.1017/S0022112085002488.  Google Scholar

[35]

B. T. Nadiga, L. G. Margolin and P. K. Smolarkiewicz, Different approximations of shallow fluid flow over an obstacle,, Phys. Fluids, 8 (1996), 2066.  doi: 10.1063/1.869009.  Google Scholar

[36]

O. Nwogu, Alternative form of Boussinesq equations for nearshore wave propagation,, Journal of Waterway, 119 (1993), 618.  doi: 10.1061/(ASCE)0733-950X(1993)119:6(618).  Google Scholar

[37]

D. Peregrine, Long waves on a beach,, J. Fluid Mech., 27 (1967), 815.  doi: 10.1017/S0022112067002605.  Google Scholar

[38]

B. Perthame, Kinetic Formulation of Conservation Laws,, Oxford University Press, (2002).   Google Scholar

[39]

J. Sainte-Marie, Vertically averaged models for the free surface Euler system. Derivation and kinetic interpretation,, Math. Models Methods Appl. Sci. (M3AS), 21 (2011), 459.  doi: 10.1142/S0218202511005118.  Google Scholar

[40]

C. H. Su and C. S. Gardner, Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg-de Vries equation and Burgers equation,, J. Mathematical Phys., 10 (1969), 536.  doi: 10.1063/1.1664873.  Google Scholar

[41]

W. C. Thacker, Some exact solutions to the nonlinear shallow-water wave equations,, J. Fluid Mech., 107 (1981), 499.  doi: 10.1017/S0022112081001882.  Google Scholar

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