Journées autour des écoulements granulaires

Jul 3 – 5, 2023
Campus des Cézeaux - Aubière
Europe/Paris timezone

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

Jul 3, 2023, 4:50 PM
50m
Amphi Hennequin (Campus des Cézeaux - Aubière)

Amphi Hennequin

Campus des Cézeaux - Aubière

3, place Vasarely 63 178 Aubière
Présentation Présentation orale

Speaker

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+\nabla_x(hv)=0,
%\label{eq:h}

\partial_t\bigl(hv\bigr)+\nabla_x\Bigl(h v\otimes v \Bigr)+g_c\nabla_x\Bigl(r
\frac{h^2}{2}\Bigr)+g_c h\nabla_x(b+\tilde b)=T,
%\label{eq:hv}

\partial_t\rho+v\cdot\nabla_x\rho=\Phi_1,
%\label{eq:rho}

\partial_t r +v\cdot\nabla_x r = \Phi_2,
%\label{eq:r}

with the energy

E = h\frac{|v|^2}{2}+g_ch(b+\tilde b)
+g_cr\frac{h^2}{2}.
%\label{entropy}

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 $\Phi_1$, $\Phi_2$ 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.

Primary authors

Elias Drach (Université Gustave Eiffel) Francois Bouchut (CNRS & Université Gustave Eiffel)