  Shallow flow or thin liquid film models are used for a wide range of physical
and engineering problems. Shallow flow models allow capturing the free surface
of the fluid with little effort and reducing the three-dimensional problem to a
quasi two-dimensional problem through depth-integrating the flow fields.
Despite remarkable progress of such models in the last decade, accurate
description of complex topography remains a challenge. Interaction with
topography is particularly critical for granular flows, because their rheology
requires modeling of the pressure field, which is strongly linked to surface
curvature and associated centrifugal forces. Shallow granular flow models are
usually set up in surface-aligned curvilinear coordinates, and velocity is
represented as a two-dimensional surface-aligned vector field. The
transformation from Cartesian to curvilinear coordinates introduces fictitious
forces, however, which result in complex governing equations. In this paper, we
set up the shallow flow model in three-dimensional Cartesian coordinates and
preserve three-dimensional velocity in the depth-integrated model. Topography
is taken into account with a constraint on velocity. This approach is commonly
applied by the thin liquid film community. The advantage is a curvature-free
mathematical description that is convenient for complex topographies. The
constraint on velocity yields a solution for the pressure field, which is
required for the pressure-dependent rheology of granular materials. The model
is therefore well-suited for granular flows on three-dimensional terrain, e.g.,
avalanches.
