Novel discretization strategies for the 2D non-Newtonian resistance term in geophysical shallow flows

In the context of two-dimensional models for complex geophysical surface flows such as debris flows, muddy slurries, oil spills over land, hyperconcentrated floods, lava flows, etc, depth-averaged rheological models relate the shear stress state within the fluid column to the depth-averaged local flow features. Despite it is the most influencing term on the mobility of complex shallow flows, the numerical treatment of the resistance contribution to the flow momentum is still a challenging topic, especially when dealing with 2D large-scale applications. In this work, two novel strategies for the explicit upwind discretization of generalized non-Newtonian resistance terms in two-dimensional numerical models are proposed, called integral and differential approaches. These new strategies are applicable to generalized rheological formulations in any type of mesh topology. Results from benchmark tests running in orthogonal, triangle structured and triangle unstructured meshes demonstrate that both approaches represent an improvement for the explicit upwind integration of the 2D resistance force compared with previous procedures. It is shown that the alignment of the flow with the mesh main-axis, which has been previously attributed to faults of 2D FV numerical methods and insufficient mesh refinements, is directly related to the loss of the rotational invariance of the integrated resistance force. This is caused by the erroneous procedure for including the 2D resistance term into the local flux balance at the cell edges. Furthermore, a novel implicit centered method for the integration of the 2D resistance force has also been derived for the quadratic frictional non-linear resistance formulation. Despite the implicit procedure fails to converge to steady uniform flow states, the differential explicit upwind and the implicit centered methods show similar level of accuracy, robustness and computational efficiency for transient 2D frictional visco-plastic flows.