The compressible granular collapse in a fluid as a continuum: validity of a Navier-Stokes model with $\mu(J)$-$\phi(J)$-rheology

The incompressible $\mu(I)$-rheology has been used to study subaerial granular flows with remarkable success. For subaquatic granular flows, drag between grains and the pore fluid is substantially higher and the physical behaviour is more complex. High drag forces constrain the rearrangement of grains and dilatancy, leading to a considerable build-up of pore pressure. Its transient and dynamic description is the key to modelling subaquatic granular flows but out of the scope of incompressible models. In this work, we advance from the incompressible $\mu(I)$-rheology to the compressible $\mu(J)$-$\phi(J)$-rheology to account for pore pressure, dilatancy, and the scaling laws under subaquatic conditions. The model is supplemented with critical state theory to yield the correct properties in the quasi-static limit. The pore fluid is described by an additional set of conservation equations and the interaction with grains is described by a drag model. This new implementation enables us to include most of the physical processes relevant for submerged granular flows in a highly transparent manner. Both, the incompressible and compressible rheologies are implemented into OpenFOAM and various simulations at low and high Stokes numbers are conducted with both frameworks. We found a good agreement of the $\mu(J)$-$\phi(J)$-rheology with low Stokes number experiments, that incompressible models fail to describe. The combination of granular rheology, pore pressure, and drag model leads to complex phenomena such as apparent cohesion, remoulding, hydroplaning, and turbidity currents. The simulations give remarkable insights into these phenomena and increase our understanding of subaquatic mass transports.