# An overview of layer-averaged models for complex rheologies and well-balanced approximations

**Speaker:**

Fernandez-Nieto

**Affiliation:**

Universidad de Sevilla

**When:**

Tuesday 6th February 2024

**Time:**

10:30:00

**Where:**

Aula Saleri

**Abstract:**

In this talk first an overview on layer-averaged approximations, also named multilayer models (see [1]-[3]), will be presented. Firstly, we focus on the derivation of this approach for Navier-Stokes equations with hydrostatic pressure and constant viscosity introduced in [3]. In this paper the model presented in [1] is derived as a discontinuous Garlekin method. Secondly, several generalizations for complex reologies are considered, where we focus mainly the simulation of granular avalanches with a mu(I)-rheology and viscoplastic avalanches by a Herschel-Bulkley type model. These two rheologies have some similarities and some hugh differences. For example, in both cases we can observe in the simulations and in experimental measurements an important vertical structure of the velocity profile. But we can observe very different profiles of the velocity. In granular avalanches the particles near the free surge present a great movility and we observe zero velocity near the bottom. Contrariously, in Hersche-Bulkley models we can observe a bigger gradient of the velocity near the bottom and a plug or pseudo-plug area, moving at constant velocity but not zero, near the free surface. This implies that both models have different forms of the stationary solutions. Thus, different strategies are proposed to obtain well-balanced finite volume approximations.
For the case of mu(I)-rheology, several depth-averaged and laver-averaged models can be found in the bibliography that approximates Navier-Stokes system by asymptotic approximations (see [2,4,7]), but in no case all components of the stress tensor are considered, even in the case of weakly non-hydrostatic models. In [5] a layer-averaged model to approximate Herschel-Bulkley model is proposed, in this case by considering a hydrostatic pressure and the main order of the stress tensor component, by introducing and asymptotic analysis.
To finish this talk, we will present briefly the results of [6]. A layer-averaged approximation of Navier- Stokes system with complex rheologies, taking into account all components of the stress tensor, is introduced.
References:
[1] E. Audusse, M. Bristeau, B. Perthame, J. Sainte-Marie. A multilayer Saint-Venant system with mass exchanges for shallow water flows, derivation and numerical validation. ESAIM: Mathematical Modelling and Numerical Analysis, 45:169–200, 2011.
[2] C. Escalante, E.D. Fernandez-Nieto, J Garres-Diaz, A. Mangeney. Multilayer Shallow Model for Dry Granular Flows with a Weakly Non-hydrostatic Pressure. Journal of Scientific Computing, 96(3), 88, 2023.
[3] E.D. Fernandez-Nieto, E.H. Kone, T. Chacon. A Multilayer Method for the Hydrostatic Navier-Stokes Equations: A Particular Weak Solution. Journal Scientific Computing, 60:408–437, 2014.
[4] E.D. Fernandez-Nieto, J. Garres-Diaz, A. Mangeney, G. Narbona-Reina. 2D granular flows with the µ(I) rheology and side walls friction: A well-balanced multilayer discretization. Journal of Computational Physics, 356: 192–219, 2018.
[5] E.D. Fernandez-Nieto, J Garres-Diaz, P. Vigneaux. Multilayer models for hydrostatic Herschel-Bulkley viscoplastic flows. Computers & Mathematics with Applications, 139: 99–117, 2023.
[6] E.D. Fernandez-Nieto, J Garres-Diaz. Layer-averaged approximation of Navier-Stokes system with complex rheologies. ESAIM: Mathematical Modelling and Numerical Analysis, 57(5):2735–2774, 2023.
[7] J. Garres-Diaz, E.D. Fernandez-Nieto, T. Morales de Luna, A. Mangeney. A weakly non-hydrostatic shallow model for dry granular flows. Journal of Scientific Computing, 86(2), 25, 2021.
Contatti:
luca.formaggia@polimi.it
carlo.defalco@polimi.it