Section: New Results

Mathematical Modelling

Mathematical and numerical model for nonlinear viscoplasticity

Participants : Nicolas Favrie, Sergey Gavrilyuk.

A macroscopic model describing elastic plastic solids is derived in a special case of the internal specificc energy taken in separable form: it is the sum of a hydrodynamic part depending only on the density and entropy, and a shear part depending on other invariants of the Finger tensor. In particular, the relaxation terms are constructed compatible with the von Mises yield criteria. In addition, Maxwell-type material behaviour is shown up: the deviatoric part of the stress tensor decays during plastic deformations. Numerical examples show the ability of this model to deal with real physical phenomena [15] .

A discrete model for compressible flows in heterogeneous media

Participants : Olivier Le Métayer, Alexandre Massol, Nicolas Favrie, Sarah Hank.

This work deals with the building of a discrete model able to describe and to predict the evolution of complex gas flows in heterogeneous media. In many physical applications, large scales numerical simulation is no longer possible because of a lack of computing resources. Indeed the medium topology may be complex due to the presence of many obstacles (walls, pipes, equipments, geometric singularities etc.). Aircraft powerplant compartments are examples where topology is complex due to the presence of pipes, ducts, coolers and other equipment. Other important examples are gas explosions and large scale dispersion of hazardous materials in urban places, cities or underground involving obstacles such as buildings and various infrastructures. In all cases efficient safety responses are required. Then a new discrete model is built and solved in reasonable execution times for large cell volumes including such obstacles. Quantitative comparisons between experimental and numerical results are shown for different significant test cases, showing excellent agreement [18] .

A hyperbolic Eulerian model for dilute two-phase suspensions

Participants : Sarah Hank, Richard Saurel, Olivier Le Métayer.

Conventional modeling of two-phase dilute suspensions is achieved with the Euler equations for the gas phase and gas dynamics pressureless equations for the dispersed phase, the two systems being coupled by various relaxation terms. The gas phase equations form a hyperbolic system but the particle phase corresponds to a hyperbolic degenerated one. Numerical difficulties are thus present when dealing with the dilute phase system. In the present work, we consider the addition of turbulent effects in both phases in a thermodynamically consistent way. It results in two strictly hyperbolic systems describing phase's dynamics. Another important feature is that the new model has improved physical capabilities. It is able, for example, to predict particle dispersion, while the conventional approach fails. These features are highlighted on several test problems involving particles jets dispersion and are compared against experimental data. With the help of a single parameter (a turbulent viscosity), excellent agreement is obtained for various experimental configurations studied by different authors [17] .

Diffuse interface model for compressible fluid - Compressible elastic-plastic solid interaction

Participants : Nicolas Favrie, Sergey Gavrilyuk.

An Eulerian hyperbolic diffuse interface model for elastic plastic solid fluid interaction is constructed. The system of governing equations couples Euler equations of compressible fluids and a visco-plastic model of Maxwell type materials (the deviatoric part of the stress tensor decreases during plastic deformations) in the same manner as models of multicomponent fluids. In particular, the model is able to create interfaces which were not present initially.

The model is thermodynamically compatible: it verifies the entropy inequality. However, a numerical treatment of the model is particularly challenging. Indeed, the model is non-conservative, so a special numerical splitting is proposed to overcome this difficulty. The numerical algorithm contains two relaxation procedures. One of them is physical and is related to the plastic relaxation mechanism (relaxation toward the yield surface). The second one is numerical. It consists in replacing the algebraic equation expressing a mechanical equilibrium between components by a partial differential equation with a short relaxation time. The numerical method was tested in one dimensional case (Wilkins' flying plate problem), two-dimensional plane case (impact of a projectile on a plate) and axisymmetrical case (Taylor test problem, impact with penetration effects, etc.). Numerical examples show the ability of the model to deal with real physical phenomena [13] .

Criterion of hyperbolicity for non-conservative quasilinear systems admitting a partially convex conservation law

Participants : Alain Forestier, Sergey Gavrilyuk.

A system of conservation laws admitting an additional convex conservation law can be written as a symmetric $t$-hyperbolic in the sense of Friedrichs system. However, in mathematical modeling of complex physical phenomena, it is customary to use non-conservative hyperbolic models. We generalize the Godunov Friedrichs Lax approach to this new class of models [16] .

A new model of roll waves: comparison with Brock's experiments

Participants : Gaël Richard, Sergey Gavrilyuk.

We derive a mathematical model of shear flows of shallow water down an inclined plane. Periodic stationary solutions to this model describing roll waves were obtained. The solutions are in good agreement with experimental profiles of roll waves measured in Brock's experiments (1967). In particular, the height of the vertical front of the waves, the shock thickness and the wave amplitude are well captured by the model [21] .

Modelling gas dynamics in one-dimensional ducts with abrupt area change

Participants : R Menina, Richard Saurel, M Zereg, Lazhar Houas.

Most gas dynamic computations in industrial ducts are done in one dimension with cross-section-averaged Euler equations. This poses a fundamental difficulty as soon as geometrical discontinuities are present. The momentum equation contains a non-conservative term involving a surface pressure integral, responsible for momentum loss. Definition of this integral is very difficult from a mathematical standpoint as the flow may contain other discontinuities (shocks, contact discontinuities). From a physical standpoint, geometrical discontinuities induce multidimensional vortices that modify the surface pressure integral. In the present paper, an improved one-dimensional flow model is proposed. An extra energy (or entropy) equation is added to the Euler equations expressing the energy and turbulent pressure stored in the vortices generated by the abrupt area variation. The turbulent energy created by the flow area change interaction is determined by a specific estimate of the surface pressure integral. Model's predictions are compared with two-dimensional averaged results from numerical solution of the Euler equations. Comparison with shock tube experiments is also presented. The new one dimensional averaged model improves the conventional cross-section-averaged Euler equations and is able to reproduce the main flow features [19] .