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Air entrainment in transient flows in closed water pipes: a two-layer approach. (English) Zbl 1267.76009

Summary: 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.

MSC:

76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76N15 Gas dynamics (general theory)
76S05 Flows in porous media; filtration; seepage
76T99 Multiphase and multicomponent flows
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74S10 Finite volume methods applied to problems in solid mechanics
35L60 First-order nonlinear hyperbolic equations
74G15 Numerical approximation of solutions of equilibrium problems in solid mechanics
35Q35 PDEs in connection with fluid mechanics
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