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

¹ Laboratoire de Mathématiques, UMR 5127 – CNRS and Université de Savoie, 73376 Le Bourget-du-Lac Cedex, France.
Christian.Bourdarias@univ-savoie.fr; Stephane.Gerbi@univ-savoie.fr
² BCAM–Basque Center for Applied Mathematics, Bizkaia Technology Park 500, 48160 Derio, Basque Country, Spain.
³ 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.
Mehmet.Ersoy@univ-tln.fr

Revised: 29 May 2012

Abstract

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.