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.
Mots-clés : two-layer vertically averaged flow, free surface water flows, loss of hyperbolicity, nonconservative product, two-layer kinetic scheme, real boundary conditions
@article{M2AN_2013__47_2_507_0,
author = {Bourdarias, C. and Ersoy, M. and Gerbi, St\'ephane},
title = {Air entrainment in transient flows in closed water pipes : {A} two-layer approach},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {507--538},
publisher = {EDP-Sciences},
volume = {47},
number = {2},
year = {2013},
doi = {10.1051/m2an/2012036},
mrnumber = {3021696},
zbl = {1267.76009},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2012036/}
}
TY - JOUR AU - Bourdarias, C. AU - Ersoy, M. AU - Gerbi, Stéphane TI - Air entrainment in transient flows in closed water pipes : A two-layer approach JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 507 EP - 538 VL - 47 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2012036/ DO - 10.1051/m2an/2012036 LA - en ID - M2AN_2013__47_2_507_0 ER -
%0 Journal Article %A Bourdarias, C. %A Ersoy, M. %A Gerbi, Stéphane %T Air entrainment in transient flows in closed water pipes : A two-layer approach %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 507-538 %V 47 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2012036/ %R 10.1051/m2an/2012036 %G en %F M2AN_2013__47_2_507_0
Bourdarias, C.; Ersoy, M.; Gerbi, Stéphane. Air entrainment in transient flows in closed water pipes : A two-layer approach. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 2, pp. 507-538. doi : 10.1051/m2an/2012036. http://www.numdam.org/articles/10.1051/m2an/2012036/
[1] and , Two-layer shallow water system : a relaxation approach. SIAM J. Sci. Comput. 31 (2009) 1603-1627. | MR | Zbl
[2] , A multilayer Saint-Venant model : derivation and numerical validation. Discrete Contin. Dyn. Syst. Ser. B 5 (2005) 189-214. | MR | Zbl
[3] and , On the hyperbolicity of two-layer flows, in Frontiers of applied and computational mathematics. World Sci. Publ., Hackensack, NJ (2008) 95-103. | MR
[4] , Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, in Frontiers in Mathematics. Birkhäuser Verlag, Basel (2004) | MR | Zbl
[5] and , An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment. ESAIM : M2AN 42 (2008) 683-689. | Numdam | MR | Zbl
[6] and , A finite volume scheme for a model coupling free surface and pressurised flows in pipes. J. Comput. Appl. Math. 209 (2007) 109-131. | MR | Zbl
[7] , and , A kinetic scheme for pressurised flows in non uniform closed water pipes. Monografias de la Real Academia de Ciencias de Zaragoza 31 (2009) 1-20. | MR
[8] , and , 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) 1-47. | MR | Zbl
[9] , and , 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. | MR
[10] , and , A mathematical model for unsteady mixed flows in closed water pipes. Science China Math. 55 (2012) 221-244. | MR
[11] , and , Unsteady mixed flows in non uniform closed water pipes : a full kinetic approach (2011). Submitted. | Zbl
[12] , and , 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) 107-127. | Numdam | MR | Zbl
[13] , and , Coupling of the interface tracking and the two-fluid models for the simulation of incompressible two-phase flow. J. Comput. Phys. 171 (2001) 776-804. | Zbl
[14] , , and , Analysis of transient pressures in bubbly, homogeneous, gas-liquid mixtures. J. Fluids Eng. 112 (1990) 225-231.
[15] , Generalized characteristics in hyperbolic systems of conservation laws. Arch. Ration. Mech. Anal. 107 (1989) 127-155. | MR | Zbl
[16] , and , Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74 (1995) 483-548. | MR | Zbl
[17] , Modélisation, analyse mathématique et numérique de divers écoulements compressibles ou incompressibles en couche mince. Ph.D. thesis, Université de Savoie, Chambéry (2010).
[18] and , A rough finite volume scheme for modeling two-phase flow in a pipeline. Comput. Fluids 28 (1999) 213-241. | Zbl
[19] , Root location criteria for quartic equations. IEEE Trans. Autom. Control 26 (1981) 777-782. | MR | Zbl
[20] 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
[21] and , Transient conditions in the transition from gravity to surcharged sewer flow. Can. J. Civ. Eng. 9 (1982) 189-196.
[22] and , 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) 4935-4948. | Zbl
[23] and , Thermo-fluid dynamics of two-phase flow. With a foreword by Lefteri H. Tsoukalas. Springer, New York (2006). | MR | Zbl
[24] , Models of two-layered “shallow water”. Zh. Prikl. Mekh. i Tekhn. Fiz. 180 (1979) 3-14. | MR
[25] , Numerical methods for nonconservative hyperbolic systems : a theoretical framework. SIAM J. Numer. Anal. 44 (2006) 300-321. | MR | Zbl
[26] and , A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201-231. | MR | Zbl
[27] , An Euler system modeling vaporizing sprays, in Dynamics of Hetergeneous Combustion and Reacting Systems, Progress in Astronautics and Aeronautics, AIAA, Washington, DC 152 (1993).
[28] , Finite volume approximate of two-phase fluid flows based on an approximate Roe-type Riemann solver. J. Comput. Phys. 121 (1995) 1-28. | MR | Zbl
[29] and , The motion of a finite mass of granular material down a rough incline. J. Fluid Mech. 199 (1989) 177-215. | MR | Zbl
[30] , Transcritical transient flow over mobile beds, boundary conditions treatment in a two-layer shallow water model. Ph.D. thesis, Louvain (2007).
[31] and , Theoretical considerations on the motion of salt and fresh water, in Proc. of Minnesota International Hydraulic Convention. IAHR (1953) 322-333.
[32] , 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.
[33] , Interfacial boundary condition in transient flows, in Proc. of Eng. Mech. Div. ASCE, on advances in civil engineering through engineering mechanics (1977) 532-534.
[34] , and , Transient mixed-flow models for storm sewers. J. Hydraul. Eng. 109 (1983) 1487-1503.
[35] and , Two-phase flow : models and methods. J. Comput. Phys. 56 (1984) 363-409. | MR | Zbl
[36] and , Modelling of two-phase flow with second-order accurate scheme. J. Comput. Phys. 136 (1997) 503-521. | Zbl
[37] and , The effects of gaseous cavitation on fluid transients. J. Fluids Eng. 101 (1979) 79-86.
[38] and , Fluid transients in systems. Prentice Hall, Englewood Cliffs, NJ (1993).
Cité par Sources :
