The project activities have been declined via 4 Work Packages (WPs):
Work Package 1:
The goal of WP1 was to develop a well-posed multi-layer model for granular flows in the dense-collisional regime. A first activity was to study the simple case of two layers, in order to overcome the issues due to the hyperbolicity loss and, possibly, extend the method of the extra momentum exchanges also to the multilayer case. A theoretical study on the eigenstructure of the two-layer model has been performed by asymptotic analysis. This study increased the insight into the mathematical structure of two- and multi-layer models and revealed a new family of stability criteria, valid for any flow condition.
The second part of WP1 was devoted to the development of the multi-layer model. The viscoplastic µ(I) rheology has been employed as a constitutive law, while the evolution of the volume fraction has been considered through an advection-diffusion transport equation, implementing a dilatancy law, depending on the inertial number, and also designed to account for possible non-local effects. To regularize the ill-posed first-order multi-layer model, the structure of the µ(I) rheology has been successfully exploited, by considering an approximation of the in-plane stress gradients.
Work Package 2:
In WP2 the multi-layer model has been extended to the case of irregular basal topographies. A fixed Cartesian frame of reference with one axis parallel to the mean bed slope is considered, while the topography irregularities are treated by introducing a superimposed bed function. For the closure of the multi-layer model, suitable discretized expressions, consistent with the chosen constitutive and dilatancy laws, have been developed. As well, different closures for the mass exchanges among the layers have been investigated.
Work Package 3:
In WP3 the model has been numerically integrated. A multi-stage time-splitting approach, with the advection part of the partial differential equation system being solved through a lateralized LHLL approximate Riemann solver, has been used. High-resolution in space has been obtained by a MUSCL predictor-corrector treatment. For the numerical treatment of the diffusion-like term an explicit scheme has been implemented. To guarantee the numerical stability of this stage of the numerical scheme, a multi-step time-marching approach has been considered. An advantage of the proposed multi-layer model, over fully 3D models, is represented by its higher computational efficiency. Indeed, the computational costs are found to increase at most linearly with the number of the layers. Moreover, thanks to its well-posedness for any value of the inertial number, the multi-layer model can safely be used on irregular topographies with arbitrary slopes. Various numerical simulations have been performed in order to investigate the main properties of the model.
Work Package 4:
In WP4 the MSCA researcher compared the multi-layer model with experimental data. A laboratory dataset, previously gathered at the LIDAM, has been employed for comparisons of both velocity and volume fraction profiles in the steady state. Additionally, two experimental datasets from the literature have been used for the comparisons in the unsteady state. For a reliable comparison of the multi-layer model with experimental data, a crucial task was to consider the effects of the sidewall resistances that are typically non-negligible in the majority of experimental setups. To this end, a rate-independent Coulomb-friction law has been introduced at the sidewall surfaces of each layer. The comparisons with experimental data confirmed the good capabilities of the multi-layer model in capturing the essential physics of granular flows in both steady and transient states. A relatively worse agreement with the experimental volume fraction was observed in the upper more dilute flow region, which suggests that a more sophisticated dilatancy law would be advisable in a future development of the model.
