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

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. 1991 Mathematics Subject Classification. 74S10, 35L60, 74G15. The dates will be set by the publisher. Notations concerning geometrical quantities θ(x) angle of the inclination of the main pipe axis z = Z(x) at position x Z(x) slope Ωw(t, x) cross-section area of the pipe orthogonal to the axis z = Z(x) filled with water Ωa(t, x) cross-section area of the pipe orthogonal to the axis z = Z(x) filled with air Ω(x) Ω(x) = Ωw(t, x) ∪ Ωa(t, x) S(x) area of Ω(x) R(x) radius of the cross-section Ω(x) σ(x, z) σ(x, z) = βr(x, z) − βl(x, z) width of the cross-section Ω(x) at altitude z with βr(x, z) (resp. βl(x, z)) is the right (resp. left) boundary point at altitude z.