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Dynamic responses of the closure relations, specific turbulent Helmholtz free energy and turbulent viscosity are postulated followed by experimental calibrations. The established closure model is applied to analyses of a gravity-driven stationary avalanche with incompressible grains down an incline. While the mean velocity and volume fraction increase from their minimum values on the plane toward maximum values on the free surface exponentially, two-fold turbulent kinetic energies and dissipations evolve in a reverse manner. Most two-fold turbulent kinetic energies and dissipations are confined within the thin turbulent boundary layer immediately above the plane, with nearly vanishing two-fold turbulent kinetic energies and finite two-fold turbulent dissipations in the passive layer. The two layers are similar to those of Newtonian fluids in turbulent boundary layer flows, and are preferable recognized by the distributions of turbulent kinetic energies and dissipations.

On the turbulent boundary layer of a dry granular avalanche down an incline. II – Closure model and numerical simulations