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Closure Terms

Momentum balance

The connection between the fine scale representation of the grains and the coarse scale representation of the fluid flow require an empirical force to model the momentum transfer between the discrete and the continuous phases. This closure term is a viscous force taking into account the relative velocity between the grains and the fluid, the presence of neighbors and many other physical parameters.

In the continuous representation of the mixture, this fluid-grains interaction force is applied as a source term that is often dominant. A semi-implicit scheme is used to stabilize the numerical resolution of the average Navier-Stokes equations at sufficiently high time-step require to benefit from the implicit model we use.

The empirical closure terms have been tested and found able to reproduce fine scale features of immersed granular flows.

Fall of a Stokes Cloud made with grains