Emerging, ripening, and attenuating stages of granular roll waves

Abstract

Abstract Adopting the form of the dynamic μ(Fr,h) basal friction law and introducing a second-order viscous term into the Saint Venant-type equations, we simulated two-dimensional granular roll waves generated in a rectangular chute applying the second-order-accurate total-variation-diminishing MacCormack scheme. Consistent with previous findings, we found that the amplitude and wavelength of the wave increased with transit distance. Although the μ(Fr,h) dynamic basal friction law is the more influential factor in shaping the waves, introducing a viscous term slightly reduces the wave peaks. A sensitive analysis indicates a quadratic relationship between wave peak height and the coefficient in the depth-averaged viscosity with ν≤2.4 × 10−2 m1.5/s. In the numerical simulation of roll waves subject to initial perturbations of different magnitudes, three stages in their evolution (namely, emerging, ripening, and attenuating) are evident from their difference in transit behavior. In the emerging stage, the normalized maximum height of the roll waves increases exponentially during propagation. In the ripening stage, this height grows linearly with transit distance whereas the growth rate decreases with the strength of the initial perturbation. During the attenuating stage, the maximum roll-wave height forms a hump in the upstream region where the waves attenuate. Roll-wave peaks and troughs for all three stages tend to approach equilibrium heights further down the chute. Subcritical flow roll-wave simulations show that as flow speed decreases, a reverse transformation pattern is found; specifically, the amplitude and wavelength of the wave decrease further down the chute.