A model for unsteady mixed flows in non uniform closed water pipes and a well-balanced finite volume scheme

TL;DR

This paper introduces a new coupled model for unsteady mixed flows in non-uniform closed water pipes, incorporating changes in section and slope, and presents a well-balanced finite volume scheme for numerical simulation.

Contribution

It develops a coupled PFS-model for free surface and pressurized flows with a novel treatment of transition points and a Roe-like scheme for accurate numerical solutions.

Findings

The model accurately captures flow transitions at pipe changes.

The finite volume scheme effectively handles geometrical source terms.

Numerical tests validate the model's robustness and accuracy.

Abstract

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.