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A mathematical model for unsteady mixed flows in closed water pipes

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Abstract

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.

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References

  1. Alvarez-Samaniego B, Lannes D. Large time existence for 3D water-waves and asymptotics. Invent Math, 2008, 171: 485–541

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MATH  Google Scholar 

  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

    MathSciNet  MATH  Google Scholar 

  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

    MathSciNet  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    MathSciNet  MATH  Google Scholar 

  11. Bresch D, Noble P. Mathematical justification of a shallow water model. Methods Appl Anal, 2007, 14: 87–117

    MathSciNet  MATH  Google Scholar 

  12. Capart H, Sillen X, Zech Y. Numerical and experimental water transients in sewer pipes. J Hydraulic Res, 1997, 35: 659–672

    Article  Google Scholar 

  13. Cunge J A. Modèle pour le calcul de la propagation des crues. La Houille Blanche, 1971, 3: 219–223

    Google Scholar 

  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

    MathSciNet  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  22. Mochon S. An analysis of the traffic on highways with changing surface conditions. Math Model, 1987, 9: 1–11

    Article  MathSciNet  Google Scholar 

  23. Richards P I. Shock waves on the highway. Oper Res, 1956, 4: 42–51

    Article  MathSciNet  Google Scholar 

  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

    Google Scholar 

  25. Streeter V L, Wylie E B, Bedford K W. Fluid Mechanics. New York: McGraw-Hill, 1998

    Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  27. Wylie E B, Streeter V L. Fluid Transients. New York: McGraw-Hill, 1978

    Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Mathématiques, Université de Savoie, Le Bourget du Lac, 73376, France

    Christian Bourdarias & Stéphane Gerbi

  2. Basque Center for Applied Mathematics, Bizkaia Technology Park, Derio Basque Country, 48160, Spain

    Mehmet Ersoy

  3. Institut des Mathématiques de Toulon et du Var, Université du Sud Toulon-Var, La Garde Cedex, 83957, France

    Mehmet Ersoy

Authors
  1. Christian Bourdarias
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  2. Mehmet Ersoy
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  3. Stéphane Gerbi
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Correspondence to Stéphane Gerbi.

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Dedicated to the NSFC-CNRS Chinese-French summer institute on fluid mechanics in 2010

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Bourdarias, C., Ersoy, M. & Gerbi, S. A mathematical model for unsteady mixed flows in closed water pipes. Sci. China Math. 55, 221–244 (2012). https://doi.org/10.1007/s11425-011-4353-z

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