Journées autour des écoulements granulaires

A well-balanced entropy scheme for a shallow water type system describing two-phase debris flows

3 juil. 2023, 16:50

50m

Amphi Hennequin (Campus des Cézeaux - Aubière)

Présentation Présentation orale Lundi après-midi

Orateur

Elias Drach (Université Gustave Eiffel)

Description

In the context of modeling two-phase debris flows involving grains and fluid, some shallow water systems arise with internal variables.
Our work focus on such a shallow water system with two internal variables and a topography $b$ which adds a nonconservative term. \

For numerical purposes, it is desirable to deal with a system where the mathematical entropy (the physical energy of the system) is convex with respect to the chosen conservative variables. Then at the numerical level, we can look for a scheme satisfying a semi-discrete entropy inequality. It also preserves the steady state at rest, so-called "well-balanced".

Our system is written as

$${\partial }_{t}h+{\mathrm{\nabla }}_x(hv)=0,$$
$${\partial }_t(hv)+{\mathrm{\nabla }}_x(hv\otimes v)+g_c{\mathrm{\nabla }}_x(r\frac{h^2}{2})+g_{c}h{\mathrm{\nabla }}_x(b+\tilde{b})=T,$$
$${\partial }_t\rho +v\cdot {\mathrm{\nabla }}_x\rho ={\mathrm{\Phi }}_1,$$
$${\partial }_{t}r+v\cdot {\mathrm{\nabla }}_{x}r={\mathrm{\Phi }}_2,$$
with the energy
$$E=h\frac{|v{|}^2}{2}+g_{c}h(b+\tilde{b})+g_{c}r\frac{h^2}{2}.$$

The physical unknowns of the system are the total mass $h$, the velocity $v$, the density of the mixture layer $\rho$ and a variable $r$ depending on the proportion of fluid between the layers.
Sources terms ${\mathrm{\Phi }}_1$, ${\mathrm{\Phi }}_2$ and $T$ contains multivalued friction and dilatancy effects. \

Writing the system with conservative variables for which the energy is convex, we derive a well-balanced scheme satisfying a semi-discrete entropy inequality.
A numerical test case of injection of some mixture and fluid into a box will be discussed to illustrate the importance of the dilatancy effect.

Documents de présentation

drach_elias_mathgeophy_2023.pdf