Closure relations for mobile bed debris flows in a wide range of slopes and concentrations

• Closure relations for depth integrated debris flows mathematical models.
• Two relations, concerning the bed shear stress and the solid concentration.
• The relations are valid in a wide range of slopes and concentrations.
• Comparison between theory and laboratory data shows good agreement.

Debris flows are flows of water and sediment driven by gravity that initiate in the upper part of a stream, where the slope is very steep, allowing high values of solid concentration (hyperconcentrated flows), and that stop in the lower part of the basin, which is characterized by much lower slopes and reduced speeds and concentrations. Modelling these flows requires mathematical and numerical tools capable of simulating the behavior of a fluid in a wide range of concentrations of the solid phase, spanning from hyperconcentrated flows to flows in the fluvial regime. According to a two-phase approach, the depth integrated equations of mass and momentum conservation for water and sediments, under the shallow water hypothesis, are employed to solve field problems related to debris flows. These equations require suitable closure relations that in this case should be valid in a very wide range of slopes. In the hypothesis of absence of cohesive material, we derived these closure relations properly combining the relative relations valid separately in the fluvial and in the hyperconcentrated regimes. In the intermediate regime, the shear stress is due to the combined effect of the deformation of the liquid phase (grain roughness turbulence) and of inter-particle collisions. Therefore, an approach based on the sum of the effects of the two causes has been proposed, combining the Darcy–Weisbach equation and the Bagnoldian grain-inertia theory.A similar treatment has been made for the transport capacity relations, combing the Bagnold expression of the collisional regime with a transport capacity monomial formula valid in the fluvial regime.The closure relations are expressed in non-dimensional form as a function of the Froude number, of the solid concentration, of the relative submergence, and of the slope.In order to test the closure relation, a set of experiments with mixtures of non-cohesive sediments and water have been carried out in a laboratory flume under steady uniform flow conditions, with different solid and liquid discharges and different grain size distributions. The closure equations are satisfactorily tested against experimental investigation.