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We present the derivation of a new unidirectional model for We present thederivation of a new unidirectional model for unsteady mixed flows in nonuniform closed water pipes. We introduce a local reference frame to take intoaccount the local perturbation caused by the changes of section and slope. Thenan asymptotic analysis is performed to obtain a model for free surface flowsand another one for pressurized flows. By coupling these models through thetransition points by the use of a common set of variables and a suitablepressure law, we obtain a simple formulation called PFS-model close to theshallow water equations with source terms. It takes into account the changes ofsection and the slope variation in a continuous way through transition points.Transition point between the two types of flows is treated as a free boundaryassociated to a discontinuity of the gradient of pressure. The numericalsimulation is performed by making use of a Roe-like finite volume scheme thatwe adapted to take into account geometrical source terms in the convectionmatrix. Finally some numerical tests are presented.

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