Mehmet Ersoy

A mathematical model for unsteady mixed flows in closed water pipes

We present the formal derivation of a new unidirectional model for unsteady mixed flows in nonuniform closed water pipes. 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 P-model, which describes the evolution of a compressible inviscid flow, close to gas dynamics equations in a nozzle. In order to cope with 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.

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

In this paper, we first construct a model for transient free surface flows that takes into account the air entrainment by a sytem of 4 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). Then, we propose a mathematical kinetic interpretation of this system to finally construct a well-balanced kinetic scheme having the properties of conserving the still water steady state and possesing an energy. Finally, numerical tests on closed uniforms water pipes are performed and discussed.

Unsteady mixed flows in non uniform closed water pipes: a Full Kinetic Approach

We recall the Pressurized and Free Surface 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 flooding zones and most of all it is very easy to implement it. Finally numerical experiments versus laboratory experiments are presented and the scheme produces results that are in a very good agreement. We also present a numerical comparison with analytic solutions for free surface flows in non uniform pipes: the numerical scheme has a very good behavior. A code to code comparison for pressurized flows is also conducted and leads to a very good agreement. We also perform a numerical experiment when flooding and drying flows may occur and finally make a numerical study of the order of the kinetic method.

A Kinetic Scheme for Transient Mixed Flows in Non Uniform Closed Pipes: A Global Manner to Upwind All the Source Terms

We present a numerical kinetic scheme for an unsteady mixed pressurized and free surface model. This model has a source term depending on both the space variable and the unknown U of the system. Using the Finite Volume and Kinetic (FVK) framework, we propose an approximation of the source terms following the principle of interfacial upwind with a kinetic interpretation. Then, several numerical tests are presented.