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

Published online by Cambridge University Press:  11 January 2013

C. Bourdarias ,
M. Ersoy  and
Stéphane Gerbi
C. Bourdarias
Affiliation:
Laboratoire de Mathématiques, UMR 5127 – CNRS and Université de Savoie, 73376 Le Bourget-du-Lac Cedex, France.. [email protected]; [email protected]
M. Ersoy
Affiliation:
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.; [email protected]
Stéphane Gerbi
Affiliation:
Laboratoire de Mathématiques, UMR 5127 – CNRS and Université de Savoie, 73376 Le Bourget-du-Lac Cedex, France.. [email protected]; [email protected]
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Abstract

In this paper, we first construct a model for free surface flows that takes into accountthe air entrainment by a system of four partial differential equations. We derive it bytaking averaged values of gas and fluid velocities on the cross surface flow in the Eulerequations (incompressible for the fluid and compressible for the gas). The obtained systemis conditionally hyperbolic. Then, we propose a mathematical kinetic interpretation ofthis system to finally construct a two-layer kinetic scheme in which a special treatmentfor the “missing” boundary condition is performed. Several numerical tests on closed waterpipes are performed and the impact of the loss of hyperbolicity is discussed andillustrated. Finally, we make a numerical study of the order of the kinetic method in thecase where the system is mainly non hyperbolic. This provides a useful stability resultwhen the spatial mesh size goes to zero.

Keywords

Two-layer vertically averaged flowfree surface water flowsloss of hyperbolicitynonconservative producttwo-layer kinetic schemereal boundary conditions
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 2 , March 2013 , pp. 507 - 538
Copyright
© EDP Sciences, SMAI, 2013

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