The granular monoclinal wave

Abstract

We study granular chute flow using the classic Saint-Venant approach, with the shear stresses within the granular sheet being incorporated via a friction law due to Pouliquen & Forterre (J. Fluid Mech., vol. 453, 2002, pp. 113–151) and with the in-plane stresses (which are ignored in the traditional formulation for normal fluids) being represented by a viscous-like term recently derived by Gray & Edwards (J. Fluid Mech., vol. 755, 2014, pp. 503–534). On the basis of this model, we predict that the granular sheet is able to sustain monoclinal waves, i.e. travelling shock structures that monotonically connect a thick region of uniform flow to a thinner one. We examine the balance of forces that determine the shape of this particular waveform and give the precise window of system parameters for which monoclinal waves are expected to appear in experiments.

On the structure of granular jumps: the dynamical systems approach

“…In the second snapshot (at t = 0.32 s) the dimple has passed the critical level h = h c and is now travelling along the upper, subcritical flank. The evolution of the system has been evaluated using the method of lines (Schiesser 1991;Razis et al 2018), with a computational space step Δx = 0.6 × 10 −4 m. The grid independence of the result has been checked by using smaller space steps.…”

“…Also in granular flows the diffusive term plays a similar role. It always remains very small compared to the other terms in the momentum balance (Razis, Kanellopoulos & van der Weele 2018), yet its contribution is crucial for two reasons: (a) It comes into action when there are variations inū(x, t) (implying also variations in h(x, t)), flattening them out and averting the build-up of discontinuities. (b) Being the only term in the equation of motion that involves a second-order derivative, it adds a second dimension to the phase space of the associated dynamical system (see § 3), thus paving the way for a host of two-dimensional structures -e.g.…”

The granular monoclinal wave: a dynamical systems survey

“…This condition expresses the balance between the forces of gravity and friction (Razis et al. 2018). Now, we can express the velocity of the uniform plateaus as follows: and the corresponding Froude number takes the form The velocity of the shock front is given by the well-known formula (Whitham 1974) which reflects the fact that in the steady state, for an observer in the co-moving frame, the flux of material entering the shock structure must be equal to the flux leaving the shock, i.e.…”

“…Although in the shock region the inertial terms are not exactly zero, they are sufficiently small to justify this expression which is based on kinematic considerations alone, namely the mass conservation across the shock (Razis et al. 2018).…”

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Figure 1.Granular monoclinal flood wave on a chute (Razis et al. 2018): a flowing sheet of uniform height and depth-averaged velocity overtakes a shallower and slower flow of height and velocity . The monoclinal wave, being the travelling shock structure connecting these two regions, is stable as long as the Froude number of the upper plateau does not exceed a critical value given by (3.18).

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