Basal boundary conditions for granular surface flows over fragile and brittle erodible beds

Abstract

Many granular surface flows occur as shear flows of finite thickness, over erodible beds composed of the same granular material. Such beds may be fragile, and offer no more resistance to erosion than to sustained shear. Or they may be brittle, and offer instead an excess resistance to erosion. To take this contrast into account, new basal boundary conditions are proposed. Their implications for parallel flows down infinite slopes are then examined for three different cases: stationary flows; starting; and stopping transients. For all three cases, flow behaviour is altered significantly when beds present an excess resistance to erosion. For stationary flows, non-unique velocity profiles are obtained, implying hysteresis or history-dependence. For starting transients, a power law growth of the flow thickness is predicted, instead of the jump to finite or infinite depth that would otherwise occur. For stopping transients, flows start to decelerate with a finite basal shear rate, even over erodible substrates. Analytical solutions to the corresponding free and moving boundary problems are obtained, and checked against numerical results. Model predictions are then compared with experimental measurements. Overall, good agreement is obtained. In particular, the model describes well the very different erosional responses observed for fragile and brittle beds.

A continuum model of discrete granular avalanches

“…By analogy with the constant slope case treated in Capart (2023), we seek an exact similarity solution whereby the flowing layer simultaneously thickens and accelerates, while maintaining a velocity profile of Bagnold shape. The time-evolving velocity profile satisfying the boundary conditions at the surface and at the base is then…”

“…For the entrainment stage, the time-evolving Bagnold profile (3.2) is the exact solution to the assumed governing equations, boundary and initial conditions, provided that the two ODEs (3.3) and (3.4) are satisfied. This was derived step by step in Capart (2023), and can be checked by direct substitution into (2.33) to (2.37). During bypass and detrainment, the Bagnold shape no longer applies and the velocity profile deforms into a sigmoidal shape.…”

“…Moreover, their behaviour depends on the single dimensionless number h * e = ( 5 8 ) 1/4 (S * f − 1) 1/2 , which itself depends only on the oversteepening at failure, S * f . Generalizing an approach applied earlier by Parez et al (2016) and Capart (2023) to the constant slope case, we seek series solutions for S * (t) and u * (η, t * ) of the form (3.23) where the roots λ n and coefficients C n can be real or complex, and the infinite series is truncated to a finite number of terms N = 8 (in practice, N = 3 is sufficient for accuracy to at least three significant figures). The eigenfunctions f n (η) that satisfy the partial differential equation (3.19) and boundary conditions (3.20a,b) are obtained as…”

“…Granular motion is often limited to a surface layer, but the thickness of this layer can grow and decay over time as the avalanche entrains and detrains grains at the base. The basal boundary is therefore a moving non-material interface (Jop et al 2007;Lusso et al 2021;Capart 2023). As for other geomorphic flows (Hungr 1995), bulking and debulking may significantly affect the flow dynamics, and in particular control the peak flow rate.…”

“…The first is a local flow rheology, provided by the linearized μ(I) model relating the stress ratio μ to the inertia number I (da Cruz et al 2005;Jop et al 2006). The second is a new set of basal boundary conditions, proposed recently for granular flows over brittle erodible beds (Capart 2023). These assert that, for entrainment to occur, the basal shear stress must overcome the erosion resistance of the jammed deposit, while for detrainment to take place the basal shear rate must vanish.…”