Applicability of kinematic wave approximation to shallow mud-flows with a yield stress

The study of unsteady shallow-layer flows, as those occurring during floods, may be performed through either fully dynamic models or using simplified momentum equations (e.g. kinematic, diffusion and quasi-steady approximations). Among the latter, kinematic approximation may provide a significant reduction of the computational effort, which represents a key advantage in time-critical applications. The proper application of this simplified model is however generally subjected to some limitations, which the modeler has to know and accurately verify to avoid errors. The present paper aims to investigate the applicability range of the kinematic approximation to shallow flows, with special concern to unsteady mud-flows, such as the hyper-concentrated floods. In the present analysis the rheological model proposed by O'Brien et al. (1993), broadly appreciated and widespread in the technical community, is considered. This model relies on a quadratic relationship linking the shear rate to the total shear stress, accounting for turbulent, dispersive and cohesive yield stresses. To provide relatively simple applicability criteria, linear analysis is applied to compare the celerity of a small perturbation of an initial steady uniform flow as predicted by the simplified model with those of the full dynamic model. Based on this comparison, applicability criteria for the kinematic approximation for mud-flows are derived. In particular, the proposed criteria rely on the comparison between the flood rising-time and a threshold time-scale: whenever the former is larger than the latter, the estimation of the propagation celerity of the simplified model is guaranteed with a prescribed accuracy. These criteria are presented considering the effect of all model parameters, based on the particular case of a pyroclastic mud, typically encountered in volcanic areas of southern Italy. The achieved results are discussed, aiming to guide the choice of the appropriate simplified model for mud flood analysis