Roll wave prediction model of Herschel–Bulkley fluids evolving on porous bottom

Abstract

The media have been reporting natural disasters frequently associated with climate change. When it comes to mudflows, the impact is substantial, especially on highly vulnerable communities. Under appropriate conditions (inclination, discharge and rheology), waves with constant amplitude, length and celerity can occur on the free surface of the flow. Such phenomenon is called roll wave, and it generally intensifies the disaster. Researchers usually represent the phenomenon through mathematical models. The present work aims to implement a new roll wave model, by taking into account three situations: the employment of Cauchy equations in a shallow water regime; the study of a Herschel–Bulkley fluid flowing on a channel with porous bottom and non-zero velocity at the bottom, and the analysis of a Darcian flow evolving on a porous medium. Numerical simulations were conducted in order to determine new criteria for roll wave generation and to describe how these parameters interfere with the characteristics of the phenomenon. As an example of the result reached, for a Froude number Fr<1 (with n=0.6 and C=0.2), the presence of a porous bottom was able to increase the wave amplitude by 20% and the bottom shear stress by 10%, when compared to an impermeable bottom. For a Fr⟶∞, the variation in the wave amplitude reached 80% and the bottom shear stress presented an even more significant increase (¿ 100%), indicating high sensitivity to the porous effect. In non-Newtonian fluids, the domain for roll wave generation is extended and the porous bottom acts basically as an additive element, altering wave properties and its effects on the bottom. The resulting data provided trend curves for the amplitude, the celerity and the wavelength of roll waves, building the basis for a prediction model with potential application in engineering, especially in determining shear stresses on the bottom that allow to infer erosion rates produced by the phenomenon on channels with porous bottom or on wetlands with the presence of cohesive material.