Christian Bourdarias

We recall the PFS model constructed for the modeling of unsteady mixed flows in closed water pipes where transition points between the free surface and pressurized flow are treated as a free boundary associated to a discontinuity of the gradient of pressure. Then we present a numerical kinetic scheme for the computations of unsteady mixed flows in closed water pipes. This kinetic method that we call FKA for "Full Kinetic Approach" is an easy and mathematically elegant way to deal with multiple transition points when the changes of state between free surface and pressurized flow occur. We use two approaches namely the "ghost waves approach" and the "Full Kinetic Approach" to treat these transition points. We show that this kinetic numerical scheme has the following properties: it is wet area conservative, under a CFL condition it preserves the wet area positive, it treats "naturally" the drying and flooding area and most of all it preserves every stationary flow. Finally numerical experiment versus laboratory experiment is presented and the scheme produces results that are in a very good agreement. We also present a numerical experiment when flooding and drying flows may occur and finally make a numerical study of the order of the kinetic method.

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.

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.

We present the formal derivation of a new unidirectional model for unsteady mixed flows in non uniform closed water pipe. In the case of free surface incompressible flows, the \FS-model is formally obtained, using formal asymptotic analysis, which is an extension to more classical shallow water models. In the same way, when the pipe is full, we propose the \Pres-model, which describes the evolution of a compressible inviscid flow, close to gas dynamics equations in a nozzle. In order to cope the transition between a free surface state and a pressured (i.e. compressible) state, we propose a mixed model, the \PFS-model, taking into account changes of section and slope variation.

We present the derivation of a new unidirectional model for We present the derivation of a new unidirectional model for unsteady mixed flows in non uniform closed water pipes. We introduce a local reference frame to take into account the local perturbation caused by the changes of section and slope. Then an asymptotic analysis is performed to obtain a model for free surface flows and another one for pressurized flows. By coupling these models through the transition points by the use of a common set of variables and a suitable pressure law, we obtain a simple formulation called PFS-model close to the shallow water equations with source terms. It takes into account the changes of section and the slope variation in a continuous way through transition points. Transition point between the two types of flows is treated as a free boundary associated to a discontinuity of the gradient of pressure. The numerical simulation is performed by making use of a Roe-like finite volume scheme that we adapted to take into account geometrical source terms in the convection matrix. Finally some numerical tests are presented.

The aim of this paper is to present a kinetic numerical scheme for the computations of transient pressurised flows in closed water pipes. Firstly, we detail the mathematical model written as a conservative hyperbolic partial differentiel system of equations, and the we recall how to obtain the corresponding kinetic formulation. Then we build the kinetic scheme ensuring an upwinding of the source term due to the topography performed in a close manner described by Perthame et al. using an energetic balance at microscopic level for the Shallow Water equations. The validation is lastly performed in the case of a water hammer in a uniform pipe: we compare the numerical results provided by an industrial code used at EDF-CIH (France), which solves the Allievi equation (the commonly used equation for pressurised flows in pipes) by the method of characteristics, with those of the kinetic scheme. It appears that they are in a very good agreement.