Dynamics of submerged gravitational granular flows

From a mathematical point of view, debris flows can be treated as gravitational granular flows in a water matrix. In absence of cohesive materials, the rheology of the liquid phase is Newtonian, while the rheology of the granular phase is more complex. In the gravitational flows analyzed in this paper, we are considering the motion of a granular flow flowing on an immobile bed composed by the same material, in equilibrium state. In this condition the presence of a stratification of the rheological mechanisms across the flow depth has been observed [3]. Near the free surface the granular flow is characterized by instantaneous contact among the particles (collisional regime), while near the immobile bed the granular flow is dominated by prolonged contacts among particles (frictional regime). In general, the tensor of the granular phase can be assumed to be the composition of two tensors, one, T g−coll ij , representative of the stresses exchanged with a collisional mechanism, one, T g−fric ij , representative of the stresses expressed by a frictional mechanism. The rheology of the collisional regime is well described by means of the kinetic theory, in which the granular flow is assimilated to a dense gas [4]. The rheology of the frictional regime, however, is still under debate and a persuasive formulation of this regime does not exist yet. Often, one assumes that frictional stress is Coulombian, but experimental evidence shows that this hypothesis is rather limitative [2]. A successful scheme of this kind was recently proposed by the GDR-Midi group [6], but this model is based on some parameters, the determination of which is doubtful and it does not contain a suitable formulation for the granular pressure. Armanini [1] proposes a reinterpretation of the model, as weighted average of a Coulombian stress (dependent on static friction angle at the bottom), and of a dynamic stress, represented by a dynamic friction angle. Besides in the same work also a relation for the granular pressure is given. In the present paper, we are presenting a generalization of this model its numerical solution for two-dimensional, uniform flows. Under this hypothesis the system of differential equations that describes the flow is: ∂τ 12 ∂x2 = −(1 + C∆)ρwg ∂z ∂x1 ∂p ∂x2 = −C∆ρwg ∂z ∂x2 ∂ ∂x2 ( −f4 √ Θdp ∂Θ ∂y ) = f2 √ Θdp ( ∂u1 ∂y )2 − f5ρs Θ dp (1) where dp is the size of the particles; Θ the granular temperature; u the velocity of granular phase; ∆ = (ρs − ρw)/ρw; z is the vertical direction; c is the volume concentration of the granular phase; x1 and x2 are the longitudinal and the normal directions respectively, and f1,f2 and f4 are functions of the concentration. With the following relationships: τ 12 = f2 √ Θdp ∂ug1 ∂y + p I 0 I2 0 + I2 tanφ and pg12 = f1ρsΘ+ p g z=0e − (c−c ∗)2 A (2) where C∗ is the solid concentration in the bed, and I = dp(∂u g 1/∂y)/ √ pg12/ρs, while I0 and A are costants. The system (1)-(2) can be written as a differential algebraic system of the general form dE(Q) dy = S(y,Q), (3) with two nonlinear vector functions E(Q) and S(Q). The boundary value problem (1)(2) is solved on the basis of a shooting method which resolves the DAE (3) numerically. We studied two options: first, using a simple Crank–Nicholson type scheme of the form E(Qn+1)− E(Qn) ∆y − 1 2 (S(yn, Qn) + S(yn+1, Qn+1)) = 0, (4) which is solved for Qn+1 using a Newton method with an appropriate globalization strategy based on linesearch. As a second option we use a high order continuous Galerkin finite element method for the solution of (3), similar to the discontinuous Galerkin finite element scheme proposed in [7] for the solution of the ODE systems arising in boundary layer theory. The numerical results obtained from the model (1)-(2) are satisfactory compared with the experimental data.