Integral method for flows down an incline: viscous, turbulent and granular cases

TL;DR

This paper extends the integral method for modeling inclined flows to viscous, turbulent, and granular cases, providing a versatile approach that captures key dynamics with reduced complexity.

Contribution

The paper generalizes the integral method to non-Newtonian flows, including turbulent and granular rheologies, demonstrating its applicability across different flow regimes.

Findings

Successfully modeled anti-dunes in viscous flows

Analyzed transverse velocity profiles in turbulent channels

Explored Kapitza instability in granular flows

Abstract

The integral method can be used to model accurately flows down an inclined plane. Such a method consists in projecting the full 3D equations on a lower dimensional representation. The vertical velocity profiles have their functional form fixed, based from the exact solution of homogeneous steady flows. This projection is achieved by integration of the momentum equation over the flow depth -- Saint-Venant approach. Here we generalize the viscous case to two non-newtonian constitutive relations: a Prandtl-like turbulent closure and a local granular rheology. We discuss one application in each case: the formation of anti-dunes in viscous streams, the transverse velocity profile in turbulent channels and the Kapitza instability in dense granular flows. They demonstrate the usefulness of this approach to get a model qualitatively correct, quantitatively reasonable and in which the dynamical mechanisms at work can be easily identified.