Christian Bourdarias

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.

A kinetic scheme for unsteady pressurised flows in closed water pipes

This paper is concerned with decay estimate of solutions to the semilinear wave equation with strong damping in a bounded domain. Introducing an appropriate Lyapunov function, we prove that when the damping is linear, we can find initial data, for which the solution decays exponentially. This result improves an early one in [4].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 then 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 and Simeoni (2001) [1] and Botchorishvili et al. (2003) [2] using an energetic balance at microscopic level. The validation is lastly performed in the case of a water hammer in an 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.

A pseudo active kinematic constraint for a biological living soft tissue: An effect of the collagen network

Recent studies in mammalian hearts show that left ventricular wall thickening is an important mechanism for systolic ejection and that during contraction the cardiac muscle develops significant stresses in the muscular cross-fiber direction. We suggested that the collagen network surrounding the muscular fibers could account for these mechanical behaviors. To test this hypothesis we develop a model for large deformation response of active, incompressible, nonlinear elastic and transversely isotropic living soft tissue (such as cardiac or arteries tissues) in which we include a coupling effect between the connective tissue and the muscular fibers. Then, a three-dimensional finite element formulation including this internal pseudo-active kinematic constraint is derived. Analytical and finite element solutions are in a very good agreement. The numerical results show this wall thickening effect with an order of magnitude compatible with the experimental observations.

Convergence of fluctuation-splitting schemes for two dimensional scalar conservation laws with a kinetic solver

Summary. The “fluctuation-splitting schemes” (FSS in short) have been introduced by Roe and Sildikover to solve advection equations on rectangular grids and then extended to triangular grids by Roe, Deconinck, Struij... For a two dimensional nonlinear scalar conservation law, we consider the case of a triangular grid and of a kinetic approach to reduce the discretization of the nonlinear equation to a linear equation and apply a particular FSS called N-scheme. We show that the resulting scheme converges strongly in

Numerical simulation of a compressible two-layer model: a first attempt with an implicit-explicit splitting scheme

L^2_{loc}

Numerical Simulation of Mixed Flows in Hydroelectric Circuits with Temporary Flows Flowmix Software

in a finite volume sense.

A Fractional Step Method to Simulate Mixed Flows in Pipes with a Compressible Two-Layer Model

This work is devoted to the numerical simulation of the compressible two-layer model developed in [16]. The latter is an hyperbolic two-fluid two-pressure model dedicated to gas-liquid flows in pipes, especially stratified air-water flows. Using explicit schemes, one obtains a CFL condition based on the celerity of (fast) acoustic waves which typically brings large numerical diffusivity for the (slow) material waves and small time steps. In order to overcome these drawbacks, the proposed scheme involves an operator splitting and an implicit-explicit time discretization. Thus, the full system is split into two hyperbolic subsystems. The first one deals with the transport equation on the liquid height using an explicit scheme and upwind fluxes. The second one deals with the averaged mass and momentum conservation equations of both phases using an implicit scheme which handles the propagation of acoustic waves. At last, the positivity of heights and densities is ensured under a CFL condition which involves material velocities. Numerical experiments are performed using acoustic as well as material time steps. Adding the Rusanov scheme for comparison, the best accuracy is obtained with the proposed scheme used with acoustic time steps. When used with material time steps, efficiency on the slow waves and stability are obtained regarding analytical solutions of the convective part.