A shock-capturing wave-propagation method for dry and saturated granular flows

The Savage–Hutter (SH) equations for dry granular flows are a system of hyperbolic balance laws which is based on a Coulomb friction approach for the description of internal failure and basal sliding and determines the time-dependent behaviour of depth and depth-integrated velocity components in a terrain following coordinate system (tangential to the sliding bed). Alternatively the Iverson–Denlinger (ID) equations are a system of hyperbolic balance laws for the determination of the time-dependent behaviour of fluid-saturated granular flows. They are based on the SH-theory, explicitly consider the fluid phase using a two-phase approach, but do not correspond with the SH-theory in the cases of a vanishing fluid phase. Important terms originating from the kinematic bottom boundary condition and taking care of the variable bed slope are neglected and a term taking the internal failure into account was added. In this paper I present a new numerical method, a wave-propagation method for the solution of the SH- and the ID-equations. It works in the finite volume context and uses Godunov-type schemes with spatially discretized flux functions. Since the SH-equation as well as the ID-equations are balance laws, the source terms are taken into account in form of adapted flux differences before the wave decomposition is performed. A first order as well as a second order version are derived. They are compared with the classical fractional-step or operator-splitting method for the solution of balance laws, which serves as a reference method. Both methods are applied on several test problems: (1) a dry granular flow in a rectangular flume with a bed surface inclination of Θ=31.4°, (2) a dry granular flow in a rectangular flume (Θ=40°), (3) a dry granular flow down an inclined plane (Θ=31.4°), (4) a dry granular flow down an inclined plane diverted by an obstacle (Θ=40°) and (5) a fluid-saturated granular flow down an inclined plane (Θ=31.4°).