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Air entrainment in transient flows in closed water pipes : A two-layer approach

Published online by Cambridge University Press:  11 January 2013

C. Bourdarias
Laboratoire de Mathématiques, UMR 5127 – CNRS and Université de Savoie, 73376 Le Bourget-du-Lac Cedex, France..;
M. Ersoy
BCAM–Basque Center for Applied Mathematics, Bizkaia Technology Park 500, 48160 Derio, Basque Country, Spain. Present address : IMATH–Institut de Mathématiques de Toulon et du Var, Université du sud Toulon-Var, Bâtiment U, BP 20132 – 83957 La Garde Cedex, France.;
Stéphane Gerbi
Laboratoire de Mathématiques, UMR 5127 – CNRS and Université de Savoie, 73376 Le Bourget-du-Lac Cedex, France..;
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In this paper, we first construct a model for free surface flows that takes into account the air entrainment by a system of four partial differential equations. We derive it by taking averaged values of gas and fluid velocities on the cross surface flow in the Euler equations (incompressible for the fluid and compressible for the gas). The obtained system is conditionally hyperbolic. Then, we propose a mathematical kinetic interpretation of this system to finally construct a two-layer kinetic scheme in which a special treatment for the “missing” boundary condition is performed. Several numerical tests on closed water pipes are performed and the impact of the loss of hyperbolicity is discussed and illustrated. Finally, we make a numerical study of the order of the kinetic method in the case where the system is mainly non hyperbolic. This provides a useful stability result when the spatial mesh size goes to zero.

Research Article
© EDP Sciences, SMAI, 2013

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Abgrall, R. and Karni, S., Two-layer shallow water system : a relaxation approach. SIAM J. Sci. Comput. 31 (2009) 16031627. Google Scholar
Audusse, E., A multilayer Saint-Venant model : derivation and numerical validation. Discrete Contin. Dyn. Syst. Ser. B 5 (2005) 189214. Google Scholar
R. Barros and W. Choi, On the hyperbolicity of two-layer flows, in Frontiers of applied and computational mathematics. World Sci. Publ., Hackensack, NJ (2008) 95–103.
F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, in Frontiers in Mathematics. Birkhäuser Verlag, Basel (2004)
Bouchut, F. and Morales, T., An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment. ESAIM : M2AN 42 (2008) 683689. Google Scholar
Bourdarias, C. and Gerbi, S., A finite volume scheme for a model coupling free surface and pressurised flows in pipes. J. Comput. Appl. Math. 209 (2007) 109131. Google Scholar
Bourdarias, C., Ersoy, M. and Gerbi, S., A kinetic scheme for pressurised flows in non uniform closed water pipes. Monografias de la Real Academia de Ciencias de Zaragoza 31 (2009) 120. Google Scholar
Bourdarias, C., Ersoy, M. and Gerbi, S., A model for unsteady mixed flows in non uniform closed water pipes and a well-balanced finite volume scheme. International Journal on Finite Volumes 6 (2009) 147. Google Scholar
C. Bourdarias, M. Ersoy and S. Gerbi, A kinetic scheme for transient mixed flows in non uniform closed pipes : a global manner to upwind all the source terms. J. Sci. Comput. (2011) 1–16.
Bourdarias, C., Ersoy, M. and Gerbi, S., A mathematical model for unsteady mixed flows in closed water pipes. Science China Math. 55 (2012) 221244. Google Scholar
C. Bourdarias, M. Ersoy and S. Gerbi, Unsteady mixed flows in non uniform closed water pipes : a full kinetic approach (2011). Submitted.
Castro, M., Macías, J. and Parés, C., A Q-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system. ESAIM : M2AN 35 (2001) 107127. Google Scholar
Cerne, S., Petelin, S. and Tiselj, I., Coupling of the interface tracking and the two-fluid models for the simulation of incompressible two-phase flow. J. Comput. Phys. 171 (2001) 776804. Google Scholar
Chaudhry, M.H., Bhallamudi, S.M., Martin, C.S. and Naghash, M., Analysis of transient pressures in bubbly, homogeneous, gas-liquid mixtures. J. Fluids Eng. 112 (1990) 225231. Google Scholar
Dafermos, C.M., Generalized characteristics in hyperbolic systems of conservation laws. Arch. Ration. Mech. Anal. 107 (1989) 127155. Google Scholar
Dal Maso, G., Lefloch, P.G. and Murat, F., Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74 (1995) 483548. Google Scholar
M. Ersoy, Modélisation, analyse mathématique et numérique de divers écoulements compressibles ou incompressibles en couche mince. Ph.D. thesis, Université de Savoie, Chambéry (2010).
Faille, I. and Heintze, E., A rough finite volume scheme for modeling two-phase flow in a pipeline. Comput. Fluids 28 (1999) 213241. Google Scholar
Fuller, A.T., Root location criteria for quartic equations. IEEE Trans. Autom. Control 26 (1981) 777782. Google Scholar
Gerbeau, J.-F. and Perthame, B., Derivation of viscous Saint-Venant system for laminar shallow water numerical validation. Discrete Contin. Dyn. Syst. Ser. B 1 (2001) 89102. Google Scholar
Hamam, M.A. and McCorquodale, A., Transient conditions in the transition from gravity to surcharged sewer flow. Can. J. Civ. Eng. 9 (1982) 189196. Google Scholar
Hibiki, T. and Ishii, M., One-dimensional drift-flux model and constitutive equations for relative motion between phases in various two-phase flow regimesaa. Int. J. Heat Mass Transfer 46 (2003) 49354948. Google Scholar
T. Hibiki and M. Ishii, Thermo-fluid dynamics of two-phase flow. With a foreword by Lefteri H. Tsoukalas. Springer, New York (2006).
Ovsjannikov, L.V., Models of two-layered “shallow water”. Zh. Prikl. Mekh. i Tekhn. Fiz. 180 (1979) 314. Google Scholar
Parés, C., Numerical methods for nonconservative hyperbolic systems : a theoretical framework. SIAM J. Numer. Anal. 44 (2006) 300321. Google Scholar
Perthame, B. and Simeoni, C., A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201231. Google Scholar
L. Sainsaulieu, An Euler system modeling vaporizing sprays, in Dynamics of Hetergeneous Combustion and Reacting Systems, Progress in Astronautics and Aeronautics, AIAA, Washington, DC 152 (1993).
Sainsaulieu, L., Finite volume approximate of two-phase fluid flows based on an approximate Roe-type Riemann solver. J. Comput. Phys. 121 (1995) 128. Google Scholar
Savage, S.B. and Hutter, K., The motion of a finite mass of granular material down a rough incline. J. Fluid Mech. 199 (1989) 177215. Google Scholar
C. Savary, Transcritical transient flow over mobile beds, boundary conditions treatment in a two-layer shallow water model. Ph.D. thesis, Louvain (2007).
J.B. Schijf and J.C. Schönfled, Theoretical considerations on the motion of salt and fresh water, in Proc. of Minnesota International Hydraulic Convention. IAHR (1953) 322–333.
C.S.S. Song, Two-phase flow hydraulic transient model for storm sewer systems, in Second international conference on pressure surges, BHRA Fluid engineering. Bedford, England (1976) 17–34.
C.S.S. Song, Interfacial boundary condition in transient flows, in Proc. of Eng. Mech. Div. ASCE, on advances in civil engineering through engineering mechanics (1977) 532–534.
Song, C.S.S., Cardle, J.A. and Leung, K.S., Transient mixed-flow models for storm sewers. J. Hydraul. Eng. 109 (1983) 14871503. Google Scholar
Stewart, H.B. and Wendroff, B., Two-phase flow : models and methods. J. Comput. Phys. 56 (1984) 363409. Google Scholar
Tiselj, I. and Petelin, S., Modelling of two-phase flow with second-order accurate scheme. J. Comput. Phys. 136 (1997) 503521. Google Scholar
Wiggert, D.C. and Sundquist, M.J., The effects of gaseous cavitation on fluid transients. J. Fluids Eng. 101 (1979) 7986. Google Scholar
E.B. Wylie and V.L. Streeter, Fluid transients in systems. Prentice Hall, Englewood Cliffs, NJ (1993).