In this work we present a deduction of the Saint-Venant–Exner model through an asymptotic analysis of the Navier–Stokes equations. A multi-scale analysis is performed in order to take into account that the velocity of the sediment layer is smaller than the one of the fluid layer. This leads us to consider a shallow water type system for the fluid layer and a lubrication Reynolds equation for the sediment one. This deduction provides some improvements with respect to the classic Saint-Venant–Exner model: (i) the deduced model has an associated energy. Moreover, it allows us to explain why classic models do not have an associated energy and how they can be modified in order to recover a model with this property. (ii) The model incorporates naturally a necessary modification that must be taken into account in order to be applied to arbitrarily sloping beds. Furthermore, we show that in general this modification is different from the ones considered classically. Nevertheless, it coincides with a classic one in the case of constant free surface. (iii) The deduced solid transport discharge naturally depends on the thickness of the moving sediment layer, which allows to ensure sediment mass conservation. Moreover, we include a simplified version of the model for the case of quasi-stationary regimes. Some of these simplified models correspond to a generalization of classic ones such as Meyer-Peter and Müller and Ashida–Michiue models. Three numerical tests are presented to study the evolution of a dune for several definition of the repose angle, to see the influence of the proposed definition of the effective shear stress in comparison with the classic one, and by comparing with experimental data.
Mots-clés : Saint-Venant–Exner, bedload, Reynolds equation
@article{M2AN_2017__51_1_115_0,
author = {Fern\'andez-Nieto, Enrique D. and Luna, Tom\'as Morales de and Narbona-Reina, Gladys and Zabsonr\'e, Jean de Dieu},
title = {Formal deduction of the {Saint-Venant{\textendash}Exner} model including arbitrarily sloping sediment beds and associated energy},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {115--145},
publisher = {EDP-Sciences},
volume = {51},
number = {1},
year = {2017},
doi = {10.1051/m2an/2016018},
mrnumber = {3601003},
zbl = {1360.35176},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2016018/}
}
TY - JOUR AU - Fernández-Nieto, Enrique D. AU - Luna, Tomás Morales de AU - Narbona-Reina, Gladys AU - Zabsonré, Jean de Dieu TI - Formal deduction of the Saint-Venant–Exner model including arbitrarily sloping sediment beds and associated energy JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 115 EP - 145 VL - 51 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2016018/ DO - 10.1051/m2an/2016018 LA - en ID - M2AN_2017__51_1_115_0 ER -
%0 Journal Article %A Fernández-Nieto, Enrique D. %A Luna, Tomás Morales de %A Narbona-Reina, Gladys %A Zabsonré, Jean de Dieu %T Formal deduction of the Saint-Venant–Exner model including arbitrarily sloping sediment beds and associated energy %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 115-145 %V 51 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2016018/ %R 10.1051/m2an/2016018 %G en %F M2AN_2017__51_1_115_0
Fernández-Nieto, Enrique D.; Luna, Tomás Morales de; Narbona-Reina, Gladys; Zabsonré, Jean de Dieu. Formal deduction of the Saint-Venant–Exner model including arbitrarily sloping sediment beds and associated energy. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 1, pp. 115-145. doi : 10.1051/m2an/2016018. http://archive.numdam.org/articles/10.1051/m2an/2016018/
and , Study on hydraulic resistance and bedload transport rate in alluvial streams. JSCE Tokyo 206 (1972) 59–69.
, , , and , Investigation of the mobile granular layer in bed-load transport. J. Fluid Mech. 736 (2013) 594–615. | DOI | Zbl
, The Flow of Cohesionless Grains in Fluids. Roy. Soc. London Philos. Trans. Ser. A: Math. Phys. Sci. 249 (1956) 235–297. | DOI | MR | Zbl
R.A. Bagnold, An approach to the sediment transport problem from general physics. In: Physical of Sediments and Transport by Wind and Water. Amer. Society of Civil Engineers (1988).
, and , Submarine granular flows down inclined planes. Phys. Fluids 17 (2005) 103301. | DOI | Zbl
, and , Sediment transport models in Shallow Water equations and numerical approach by high order finite volume methods. Comput. Fluids 37 (2008) 299–316. | DOI | MR | Zbl
and , A 3D numerical model for incompress-ible two-phase flow of a granular bed submitted to a laminar shearing flow. Comput. Meth. Appl. Mech. Eng. 199 (2010) 439–449. | DOI | MR | Zbl
, Interface boundary value problem for the Navier–Stokes equations in thin domains. J. Differ. Eq. 208 (2005) 449–493. | MR | Zbl
, Selection of the ripple length on a granular bed sheared by a liquid flow. Phys. Fluids 18 (2006) 121508. | DOI | Zbl
, and , Selection of ripple length on a granular bed sheared by a liquid flow. Phys. Fluids 18 (2006) 121508. | Zbl
M. Church, P. Biron and A. Roy, Gravel Bed Rivers: Processes, Tools, Environments. John Wiley & Sons (2012).
, A Rigorous Justification of the Reynolds Equation. Quart. Appl. Math. XLV 4 (1987) 627–644. | DOI | MR | Zbl
, Revisiting the linear theory of sand dune formation. J. Fluid Mech. 502 1–6 (2004). | DOI | MR | Zbl
, The hydrodynamic theory of film lubrication. Proc. R. Soc. London, Ser. A 197 (1949) 201–217. | DOI | Zbl
, and , A two time-scale model for tidal bed-load transport. Commun. Math. Sci. 10 (2012) 875–888. | DOI | MR | Zbl
, Formulas for the transportation of bedload. ASCE 107 (1942) 561–575.
, Uber die wechselwirkung zwischen wasser und geschiebe in flussen, Sitzungsber. Akad. Wissenschaften pt. IIa. Bd. 134 (1925). | JFM
and . A sediment transport model for straight alluvial channels. Nordic Hydrol. 7 (1976) 293–306. | DOI
F. Engelund and E. Hansen, A monograph on sediment transport in alluvial streams. Danish Technical Press, Copenhagen (1967).
and , Erosion and transport of bedload sediment. J. Hydraulaul. Res. 14 (1976) 127–144. | DOI
, Modelling and numerical simulation of submarine sediment shallow flows: transport and avalanches. SeMA J. 49 (2009) 83–103. | MR | Zbl
, , , and , A new-Savage-Hutter type model for submarine avalanches and genrated tsunami. J. Comput. Phys. 227 (2008) 7720–7754. | DOI | MR | Zbl
, and , Formal derivation of a bilayer model coupling Shallow Water and Reynolds equations: evolution of a thin pollutant layer over the water. Eur. J. Appl. Math. 24 (2013) 803–833. | DOI | MR | Zbl
, , and , On the influence of the thickness of the sediment moving layer in the definition of the bedload trasport formula in Exner systems. Comp. Fluids 91 (2014) 87–106. | DOI | MR | Zbl
and , Flows of dense granular media. Ann. Rev. Fluid Mech. 40 (2008) 1–24. | DOI | MR | Zbl
, and , The formation of river channel. SIAM J. Appl. Math. 67 (2007) 1016–1040. | DOI | MR | Zbl
, On the development of dunes in erodible channel. J. Fluid Mech. 64 (1974) 1–16. | DOI | Zbl
, and , Free surface flows over mobile bed: mathematical analysis and numerical modeling of coupled and decoupled approaches. Commun. Appl. Ind. Math. 2 (2011). | MR | Zbl
A.J. Grass, Sediments transport by waves and currents. SERC London Cent. Mar. Technol., Report No. FL29 (1981).
and , Derivation of viscous Saint-Venant system for laminar Shallow-Water; Numerical validation. Disc. Cont. Dynam. Syst. Series B 1 (2001) 89–102. | MR | Zbl
R. Jackson, The Dynamics of Fluidized Particles. Cambridge University Press, Cambridge, (2000). | MR | Zbl
, Criteria for determining sand transport by surface creep and saltation. Trans. AGU. 23 (1942) 639–643. | DOI
and . A new vectorial bedload formulation and its application to the time evolution of straight river channels. J. Fluid Mech. 267 (1994) 153–183. | DOI | Zbl
and , Evolution of a model dune in a shear flow. Eur. J. Mech. B/Fluids 25 (2006) 348–359. | DOI | MR | Zbl
, and The granular column collapse as a continuum: validity of a two-dimensional Navier–Stokes model with a -rheology. J. Fluid. Mech. 686 (2011) 378–408. | DOI | MR | Zbl
, Movement of sand in tunnels. Proc. A.S.C.E. 95 (1969) 1835–1846.
F. Marche, Theoretical and numerical study of Shallow Water models. Applications to Nearshore hydrodynamics. Ph.D. thesis, University of Bordeaux (2005).
and , Formulas for bed-load transport. Rep. 2nd Meet. Int. Assoc. Hydraul. Struct. Res., Stockolm 1948. ASCE. 107 (1942) 561–575.
, and , A Duality Method for Sediment Transport Based on a Modified Meyer-Peter and Müller Model. J. Sci. Comput. 48 (2009) 258–273. | DOI | MR | Zbl
, , and , On a Shallow Water Model for the Simulation of Turbidity Currents. Commun. Comput. Phys. 6 (2011) 848–882. | DOI | MR | Zbl
, , and , Derivation of a bi-layer Shallow–Water model with viscosity. Numerical validation. CMES 43 (2009) 27–71. | MR | Zbl
, Coastal Bottom Boundary Layers and Sediment Transport. World Scientific Publishing, Singapore. Adv. Ser. Ocean Eng. 4 (1992). | DOI
, and , Long scale evolution of thin films. Rev. Mod. Phys. 69 (1997). | DOI
, and , Sediment dynamics. Part I: Bed-load transport by shearing flows. J. Fluid Mech. 636 (2009) 295–319. | DOI | MR | Zbl
M. Pailha, J. Chauchat, P. Aussillous, M. Médale and E. Guazzelli. Bed-load Transport. Part 2: The Mobile Granular Layer. Ph.D. thesis (2011).
, and , Bedload at low Shields stress on arbitrarily sloping beds: Alternative entrainment formulation. Water Resources Res. 29 (2003) 7.
and , On the well-balance property of Roe’s method for nonconservative hyperbolic systems. Applications to shallow-water systems. ESAIM: M2AN 38 (2004) 821–852. | DOI | Numdam | MR | Zbl
E. Peña González, Estudio numérico y experimental del transporte de sedimentos en cauces aluviales. Ph.D. thesis, Universidade da Coruña, Grupo de Ingeniería del agua y del medio ambiente (2002).
and , A two-phase model for sheet flow regime based on dense granular flow rheology. J. Geophys. Res. Oceans 118 (2013) 619–634. | DOI
M. Peybernes, Analyse de problème mathématiques de la mécanique des fluides de type bi-couche et à frontière libre. Ph.D. thesis, University of Pascal Paoli (2006).
O. Reynolds, On the theory of lubrication and its application to Mr. Beauchamp Tower’s experiment. Phil. Trans. R. Soc. London Part I (1886) 228–310.
, Sediment transport (I): bedload transport. J. Hydraul. Div. Proc. ASCE 110 (1984) 1431–56. | DOI
P.C. Roos, Seabed Pattern Dynamics and Offshore Sand Extraction. Ph.D. thesis, University of Twente (2004).
H. Schuttelaars, Evolution and Stability Analysis of Bottom Patterns in Tidal Embayements. Ph.D. thesis, Utrecht University (1997).
, and , Bedload at low Shields stress on arbitrarily sloping beds: Failure of the Bagnold hypothesis. Water Resources Res. 38 (2002) 11. | DOI
. . and , Bed deformation in curved alluvial channels. J. Hudraulic Res. 23 (1985) 57–79. | DOI
, , and , A discontinuous Galerkin finite element model for morphodynamical evolution in shallow flows. Comp. Meth. App. Mech. Eng. 197 (2008) 2930–2947. | DOI | MR | Zbl
, , , , and , Medium-term 2DH coastal area modelling. Coastal Engineering 21 (1993) 193–224. | DOI
J.D. Zabsonré, Modèles visqueux en sédimentation et stratification: obtention formelle, stabilité théorique et schémas volumes finis bien équilibrés. Ph.D. thesis, Université de Savoie (2008).
, and , An energetically consistent viscous sedimentation model. Math. Models Methods Appl. Sci. 19 (2009) 477–499. | DOI | MR | Zbl
Cité par Sources :
