Title: 2D granular flows with the $μ(I)$ rheology and side walls friction: a well balanced multilayer discretization,
Abstract: We present here numerical modelling of granular flows with the $\mu(I)$
rheology in confined channels. The contribution is twofold: (i) a model to
approximate the Navier-Stokes equations with the $\mu(I)$ rheology through an
asymptotic analysis. Under the hypothesis of a one-dimensional flow, this model
takes into account side walls friction; (ii) a multilayer discretization
following Fernández-Nieto et al. (J. Fluid Mech., vol. 798, 2016, pp.
643-681). In this new numerical scheme, we propose an appropriate treatment of
the rheological terms through a hydrostatic reconstruction which allows this
scheme to be well-balanced and therefore to deal with dry areas. Based on
academic tests, we first evaluate the influence of the width of the channel on
the normal profiles of the downslope velocity thanks to the multilayer approach
that is intrinsically able to describe changes from Bagnold to S-shaped (and
vice versa) velocity profiles. We also check the well balance property of the
proposed numerical scheme. We show that approximating side walls friction using
single-layer models may lead to strong errors. Secondly, we compare the
numerical results with experimental data on granular collapses. We show that
the proposed scheme allows us to qualitatively reproduce the deposit in the
case of a rigid bed (i. e. dry area) and that the error made by replacing the
dry area by a small layer of material may be large if this layer is not thin
enough. The proposed model is also able to reproduce the time evolution of the
free surface and of the flow/no-flow interface. In addition, it reproduces the
effect of erosion for granular flows over initially static material lying on
the bed. This is possible when using a variable friction coefficient $\mu(I)$
but not with a constant friction coefficient.