Some exact solutions for debris and avalanche flows

Abstract

Exact analytical solutions to simplified cases of nonlinear debris avalanche model equations are necessary to calibrate numerical simulations of flow depth and velocity profiles on inclined surfaces. These problem-specific solutions provide important insight into the full behavior of the system. In this paper, we present some new analytical solutions for debris and avalanche flows and then compare these solutions with experimental data to measure their performance and determine their relevance. First, by combining the mass and momentum balance equations with a Bagnold rheology, a new and special kinematic wave equation is constructed in which the flux and the wave celerity are complex nonlinear functions of the pressure gradient and the flow depth itself. The new model can explain the mechanisms of wave advection and distortion, and the quasiasymptotic front bore observed in many natural and laboratory debris and granular flows. Exact time-dependent solutions for debris flow fronts and associated velocity profiles are then constructed. We also present a novel semiexact two-dimensional plane velocity field through the flow depth. Second, starting with the force balance between gravity, the pressure gradient, and Bagnold’s grain-inertia or macroviscous forces, we construct a simple and very special nonlinear ordinary differential equation to model the steady state debris front profile. An empirical pressure gradient enhancement factor is introduced to adequately stretch the flow front and properly model nonhydrostatic pressure in granular and debris avalanches. An exact solution in explicit form is constructed, and is expressed in terms of the Lambert–Euler omega function. Third, we consider rapid flows of frictional granular materials down a channel. The steady state mass and the momentum balance equations are combined together with the Coulomb friction law. The Chebyshev radicals are employed and the exact solutions are developed for the velocity profile and the debris depth. Similarly, Bagnold’s fluids are also used to construct alternative exact solutions. Many interesting and important aspects of all these exact solutions, their applications to real-flow situations, and the influence of model parameters are discussed in detail. These analytical solutions, although simple, compare very well with experimental data of debris flows, granular avalanches, and the wave tips of dam break flows. A new scaling law for Bagnold’s fluids is established to relate the settlement time of debris deposition. It is found analytically that the macroviscous fluid settles (comes to a standstill) considerably faster than the grain-inertia fluid, as manifested by dispersive pressure.