The derivation of shallow water models from Navier–Stokes equations is revisited yielding a class of two-layer shallow water models. An improved velocity profile is proposed, based on the superposition of an inviscid fluid and a viscous layer inspired by the Interactive Boundary Layer interaction used in aeronautics. This leads to a new friction law which depends not only on velocity and depth but also on the variations of velocity and thickness of the viscous layer. The resulting system is an extended shallow water model consisting of three depth-integrated equations: the first two are mass and momentum conservation in which a slight correction on hydrostatic pressure has been made; the third one, known as von Kármán equation, describes the evolution of the viscous layer. This coupled model is shown to be conditionally hyperbolic, and a Godunov-type finite volume scheme is also proposed. Several numerical examples are provided and compared to the Multi-Layer Saint-Venant model. They emphasize the ability of the model to deal with unsteady viscous effects. They illustrate also the phase-lag between friction and topography, and even recover possible reverse flows.
Mots-clés : Shallow water, viscous layer, friction law, Prandtl equation, von Kármán equation
@article{M2AN_2019__53_1_269_0,
author = {James, Fran\c{c}ois and Lagr\'ee, Pierre-Yves and Le, Minh H. and Legrand, Mathilde},
title = {Towards a new friction model for shallow water equations through an interactive viscous layer},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {269--299},
publisher = {EDP-Sciences},
volume = {53},
number = {1},
year = {2019},
doi = {10.1051/m2an/2018076},
zbl = {1426.35151},
mrnumber = {3938846},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2018076/}
}
TY - JOUR AU - James, François AU - Lagrée, Pierre-Yves AU - Le, Minh H. AU - Legrand, Mathilde TI - Towards a new friction model for shallow water equations through an interactive viscous layer JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2019 SP - 269 EP - 299 VL - 53 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2018076/ DO - 10.1051/m2an/2018076 LA - en ID - M2AN_2019__53_1_269_0 ER -
%0 Journal Article %A James, François %A Lagrée, Pierre-Yves %A Le, Minh H. %A Legrand, Mathilde %T Towards a new friction model for shallow water equations through an interactive viscous layer %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2019 %P 269-299 %V 53 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2018076/ %R 10.1051/m2an/2018076 %G en %F M2AN_2019__53_1_269_0
James, François; Lagrée, Pierre-Yves; Le, Minh H.; Legrand, Mathilde. Towards a new friction model for shallow water equations through an interactive viscous layer. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 1, pp. 269-299. doi : 10.1051/m2an/2018076. http://www.numdam.org/articles/10.1051/m2an/2018076/
[1] and , Exact solutions to the Riemann problem of the shallow water equations with a bottom step. Comput. Fluids 30 (2001) 643–671. | DOI | MR | Zbl
[2] , , and , A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation. ESAIM: M2AN 45 (2011) 169–200. | DOI | Numdam | MR | Zbl
[3] , The fluid dynamics of river dunes: a review and some future research directions. J. Geophys. Res. Earth Surf. 110 (2005) F4. | DOI
[4] , Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources. Frontiers in Mathematics Birkhäuser Verlag Basel 2004 (2004). | DOI | MR | Zbl
[5] , and , Frontal geostrophic adjustment and nonlinear wave phenomena in one-dimensional rotating shallow water. Part 2. High-resolution numerical simulations. J. Fluid Mech. 514 (2004) 35–63. | DOI | MR | Zbl
[6] and , Implicit schemes with large time step for non-linear equations: application to river flow hydraulics. Int. J. Numer. Meth. Fluids 46 (2004) 607–636. | DOI | Zbl
[7] , and , Finite volume method for simulating extreme flood events in natural channels, J. Hydraul. Res. 41 (2003) 167–177. | DOI
[8] , 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
[9] and , Shooting and parallel shooting methods for solving the falkner-skan boundary-layer equation. J. Comput. Phys. 7 (1971) 289–300. | DOI | Zbl
[10] , and , Sand ripples and dunes. Annu. Rev. Fluid Mech. 45 (2013) 469–493. | DOI | MR | Zbl
[11] , Open-Channel Hydraulics. McGraw-Hill, New York, NY (1959).
[12] , 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. 73 (1871) 147–154. | JFM
[13] , and , Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74 (1995) 483–548. | MR | Zbl
[14] , , and , Simulation of rain-water overland-flow. In: Proceedings of the 12th International Conference on Hyperbolic Problems. University of Maryland, College Park (USA) (2008). | Zbl
[15] , and Eds., Proceedings of symposia in applied mathematics. Amer. Math. Soc. 67 (2009) 537–546.
[16] , , and , Numerical modeling of subaqueous sand dune morphodynamics. J. Geophys. Res. Earth Surf. 121 (2016) 565–587. | DOI
[17] and , On an evolution equation for sand dunes. SIAM J. Appl. Math. (2010).
[18] , , and , Overland flow and infiltration modelling for small plots during unsteady rain: numerical results versus observed values. J. Hydrol. 228 (2000) 265–282. | DOI
[19] , Uber die wechselwirkung zwischen wasser und geschiebe in flussen. Akad. Wiss. Wien Math. Naturwiss. Klasse 134 (1925) 165–204.
[20] and , Solutions of the boundary-layer equations. London, Edinburgh, and Dublin Philos. Mag. J. Sci. 12 (1931) 865–896. | JFM | Zbl
[21] , Dunes and drumlins. In: Geomorphological Fluid Mechanics. Springer, Berlin (2001) 430–454. | DOI | Zbl
[22] , Finite volume methods and adaptive refinement for tsunami propagation and inundation. Ph.D. thesis, University of Washington (2006). | MR
[23] and , Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Contin. Dyn. Syst. Ser. B 1 (2001) 89–102. | MR | Zbl
[24] , and , A 2D nonlinear multiring model for blood flow in large elastic arteries. J. Comput. Phys. 350 (2017) 136–165. | DOI | MR | Zbl
[25] , , Numerical approximations of hyperbolic systems of conservation laws. Applied Mathematical Sciences Springer-Verlag, New York, NY 118 (1996). | DOI | MR | Zbl
[26] , , , A Godunov-type scheme for shallow water equations dedicated to simulations of overland flows on steep slopes. In: International Conference on Finite Volumes for Complex Applications, Springer, Berlin (2017) 275–283. | MR | Zbl
[27] and , Proceedings of the 2nd workshop on dam-break wave simulation. Technical Report, EDF-DER (1997).
[28] , Modeling overland flow contamination by chemicals mixed in shallow soil horizons under variable source area hydrology. Water Resour. Res. 32 (1996) 753–758. | DOI
[29] , and , On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1983) 35–61. | DOI | MR | Zbl
[30] and , The effects of hydraulic resistance on dam-break and other shallow inertial flows. J. Fluid Mech. 501 (2004) 179–212. | DOI | MR | Zbl
[31] , A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press, Cambridge (1997). | DOI | MR | Zbl
[32] , , and , Falling Liquid Films. Springer Science & Business Media 176 (2011). | MR | Zbl
[33] , The mechanics of dunes and antidunes in erodible-bed channels. J. Fluid Mech. 16 (1963) 521–544. | DOI | Zbl
[34] and , Numerical solution for the falkner–skan boundary layer viscous flow over a wedge. Int. J. Eng. Sci. Technol. 3 (2013) 18–36.
[35] , and , Propagation and run-up of nearshore tsunamis with hllc approximate riemann solver. Ocean Eng. 34 (2007) 1164–1173. | DOI
[36] , , , , , , and , Modeling rain-driven overland flow: empirical versus analytical friction terms in the shallow water approximation. J. Hydrol. 536 (2016) 1–9. | DOI
[37] and , Evolution of a model dune in a shear flow. Eur. J. Mech.-B/Fluids 25 (2006) 348–359. | DOI | MR | Zbl
[38] , A triple deck model of ripple formation and evolution. Phys. Fluids 15 (2003) 2355–2368. | DOI | MR | Zbl
[39] , Interactive boundary layer (IBL). In: Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances. Springer, Berlin (2010) 247–286. | MR | Zbl
[40] , , , and , Characterization of the pressure drop in a 2D symmetrical pipe: some asymptotical, numerical, and experimental comparisons. ZAMM-J. Appl. Math. Mech./Z. Angew. Math. Mech. 85 (2005) 141–146. | DOI | Zbl
[41] and , The RNS/Prandtl equations and their link with other asymptotic descriptions: application to the wall shear stress scaling in a constricted pipe. Int. J. Eng. Sci. 43 (2005) 352–378. | DOI | MR | Zbl
[42] , and , Asymmetrical effects in a 2D stenosis. Eur. J. Mech.-B/Fluids 26 (2007) 83–92. | DOI | MR | Zbl
[43] , , and , A faster numerical scheme for a coupled system modeling soil erosion and sediment transport. Water Resour. Res. 51 (2015) 987–1005. | DOI
[44] and , Godunov method for nonconservative hyperbolic systems. ESAIM: M2AN 41 (2007) 169–185. | DOI | Numdam | MR | Zbl
[45] , , , and , Detailed simulation of morphodynamics: 3. Ripples and dunes. Water Resour. Res. 49 (2013) 5930–5943. | DOI
[46] , , and , Modeling river dune development and dune transition to upper stage plane bed. Earth Surf. Process. Landf. 15 (2015) 323–335.
[47] , 95.60 a new bound for polynomials when all the roots are real. Math. Gazette 95 (2011) 520–526. | DOI
[48] , , and , Modeling river dune evolution using a parameterization of flow separation. J. Geophys. Res. Earth Surf. 114 (2009) F1. | DOI
[49] , Quadtree-adaptative tsunami modelling. Ocean Dyn. 61 (2011) 1261–1285. | DOI
[50] , Motion of fluids with very little viscosity. NACA Transl. 452 (1928) 1–8.
[51] and , A new model of roll waves: comparison with brock’s experiments. J. Fluid Mech. 698 (2012) 374–405. | DOI | MR | Zbl
[52] and , An analytical solution for the lateral transport of dissolved chemicals in overland flow. Water Resour. Res. 31 (1995) 1031–1040. | DOI
[53] , Boundary-layer Theory. McGraw-Hill, New York, NY (1968). | Zbl
[54] , On the impulsive motion of a flat plate in a viscous fluid. Quart. J. Mech. Appl. Math. 4 (1951) 182–198. | DOI | MR | Zbl
[55] , On the impulsive motion of a flat plate in a viscous fluid. II. Quart. J. Mech. Appl. Math. 26 (1973) 143–152. | DOI | Zbl
[56] , , , , , , , and , Measurement and modelling of high-resolution flow-velocity data under simulated rainfall on a low-slope sandy soil. J. Hydrol. 348 (2008) 1–12. | DOI
[57] , and , Case study: malpasset dam-break simulation using a two-dimensional finite volume methods. J. Hydraul. Eng. 128 (2002) 460–472. | DOI
[58] , Perturbation Methods in Fluid Mechanics. Parabolic Press, Incorporated, Hackensack, NY (1975). | MR | Zbl
[59] , The spaces BV and quasilinear equations. Matematicheskii Sbornik 115 (1967) 255–302. | Zbl
[60] and , An iterative method for solving the falkner–skan equation. Appl. Math. Comput. 210 (2009) 215–222. | MR | Zbl
Cité par Sources :
