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

TL;DR

This paper develops a two-layer kinetic model for air entrainment in transient free surface flows within closed water pipes, addressing hyperbolicity issues and providing numerical stability analysis.

Contribution

It introduces a novel two-layer kinetic scheme for modeling air entrainment in pipe flows, incorporating a treatment for boundary conditions and hyperbolicity challenges.

Findings

The model captures air entrainment effects in transient pipe flows.

Numerical tests demonstrate the scheme's stability despite non-hyperbolicity.

The kinetic method's order of accuracy is analyzed for non-hyperbolic systems.

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.