A mathematical model for unsteady mixed flows in closed water pipes. (English) Zbl 1387.35484

Summary: We present the formal derivation of a new unidirectional model for unsteady mixed flows in nonuniform closed water pipes. In the case of free surface incompressible flows, the FS-model is formally obtained, using formal asymptotic analysis, which is an extension to more classical shallow water models. In the same way, when the pipe is full, we propose the P-model, which describes the evolution of a compressible inviscid flow, close to gas dynamics equations in a nozzle. In order to cope with the transition between a free surface state and a pressured (i.e., compressible) state, we propose a mixed model, the PFS-model, taking into account changes of section and slope variation.


35Q35 PDEs in connection with fluid mechanics
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Full Text: DOI arXiv Link


[1] Alvarez-Samaniego B, Lannes D. Large time existence for 3D water-waves and asymptotics. Invent Math, 2008, 171: 485–541 · Zbl 1131.76012
[2] Blommaert G. Étude du comportement dynamique des turbines francis: contrôle actif de leur stabilité de fonctionnement. PhD Thesis, EPFL, 2000
[3] Bouchut F, Fernández-Nieto E D, Mangeney A, et al. On new erosion models of Savage-Hutter type for avalanches. Acta Mech, 2008, 199: 181–208 · Zbl 1152.76053
[4] Bouchut F, Mangeney-Castelnau A, Perthame B, et al. A new model of Saint Venant and Savage-Hutter type for gravity driven shallow water flows. C R Math Acad Sci Paris, 2003, 336: 531–536 · Zbl 1044.35056
[5] Bourdarias C, Ersoy M, Gerbi S. A model for unsteady mixed flows in non uniform closed water pipes and a well-balanced finite volume scheme. Internat J Finite Volumes, 2009, 6: 1–47 · Zbl 1325.76126
[6] Bourdarias C, Gerbi S. A finite volume scheme for a model coupling free surface and pressurised flows in pipes. J Comp Appl Math, 2007, 209: 109–131 · Zbl 1135.76036
[7] Bourdarias C, Gerbi S. A conservative model for unsteady flows in deformable closed pipe and its implicit second order finite volume discretisation. Computers & Fluids, 2008, 37: 1225–1237 · Zbl 1237.76081
[8] Bourdarias C, Gerbi S, Gisclon M. A kinetic formulation for a model coupling free surface and pressurised flows in closed pipes. J Comp Appl Math, 2008, 218: 522–531 · Zbl 1152.76020
[9] Boutin B. Étude mathématique et numérique d’équations hyperboliques non-linéaires: couplage de modèles et chocs non classiques. PhD Thesis, CEA de Saclay et Laboratoire J.-L. Lions, 2009
[10] Boutounet M, Chupin L, Noble P, et al. Shallow water viscous flows for arbitrary topopgraphy. Commun Math Sci, 2008, 6: 29–55 · Zbl 1136.76020
[11] Bresch D, Noble P. Mathematical justification of a shallow water model. Methods Appl Anal, 2007, 14: 87–117 · Zbl 1158.35401
[12] Capart H, Sillen X, Zech Y. Numerical and experimental water transients in sewer pipes. J Hydraulic Res, 1997, 35: 659–672
[13] Cunge J A. Modèle pour le calcul de la propagation des crues. La Houille Blanche, 1971, 3: 219–223
[14] Decoene A, Bonaventura L, Miglio E, et al. Asymptotic derivation of the section-averaged shallow water equations for natural river hydraulics. Methods Appl Anal, 2007, 14: 87–117 · Zbl 1207.35092
[15] Dong N T. Sur une méthode numérique de calcul des écoulements non permanents soit à surface libre, soit en charge, soit partiellement à surface libre et partiellement en charge. La Houille Blanche, 1990, 2: 149–158
[16] Ersoy M. Modélisation, analyse mathématique et numérique de divers écoulements compressibles ou incompressibles en couche mince. PhD Thesis, Université de Savoie (France), 2010
[17] Fuamba M. Contribution on transient flow modelling in storm sewers. J Hydraulic Res, 2002, 40: 685–693
[18] Gerbeau J F, Perthame B. Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Contin Dyn Syst Ser B, 2001, 1: 89–102 · Zbl 0997.76023
[19] Levermore C D, Oliver M, Titi E S. Global well-posedness for models of shallow water in a basin with a varying bottom. Indiana Univ Math J, 1996, 45: 479–510 · Zbl 0953.76011
[20] Lighthill M J, Whitham G B. On kinematic waves, II: A theory of traffic flow on long crowded roads. Proc R Soc Lond A, 1955, 229: 317–345 · Zbl 0064.20906
[21] Marche F. Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects. European J Mech Ser B Fluids, 2007, 26: 49–63 · Zbl 1105.76021
[22] Mochon S. An analysis of the traffic on highways with changing surface conditions. Math Model, 1987, 9: 1–11
[23] Richards P I. Shock waves on the highway. Oper Res, 1956, 4: 42–51
[24] Roe P L. Some contributions to the modelling of discontinuous flows. In: Large-scale Computations in FluidMechanics, Part 2. Lectures in Appl Math, vol. 22. Providence, RI: Amer Math Soc, 1985, 163–193
[25] Streeter V L, Wylie E B, Bedford K W. Fluid Mechanics. New York: McGraw-Hill, 1998
[26] Toro E F. Riemann problems and the WAF method for solving the two-dimensional shallow water equations. Philos Trans Roy Soc London Ser A, 1992, 338: 43–68 · Zbl 0747.76027
[27] Wylie E B, Streeter V L. Fluid Transients. New York: McGraw-Hill, 1978
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.