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Peano
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Peano
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The physical model used to simulate avalanches is based on the Shallow Water Equations (SWE), extended to incorporate the Voellmy-Salm friction model, where the friction resistance of the granular material is represented as a source term. The Voellmy-Salm model accounts for both velocity-independent Coulomb friction and turbulent-like drag, which are critical for accurately simulating granular flows.
The Voellmy-Salm rheological law, initially developed by Voellmy and later extended by Salm for dense granular flows, is widely adopted in avalanche and landslide modeling. This semi-empirical framework decomposes the basal shear stress \( \tau \) into two physically distinct components:
\[ \tau = \mu \sigma + \frac{\rho g V^2}{\xi} \]
where \( \sigma \) is the normal stress at the flow base; \( \mu \) is the basal friction coefficient; \( \rho \) is the bulk density of the granular material; \( g \) is the gravitational acceleration; \( \xi \) is the Voellmy turbulence coefficient; and \( V \) is the velocity vector.
The SWE augmented with the Friction Law as a source term, forms the governing equations for gravity-driven granular avalanches:
\begin{align*} \begin{cases} \frac{\partial h}{\partial t} + \frac{\partial (hu)}{\partial x} + \frac{\partial (hv)}{\partial y} = 0 \\ \frac{\partial (hu)}{\partial t} + \frac{\partial}{\partial x}(hu^2 + \frac{gh^2}{2}) + \frac{\partial (huv)}{\partial y} = -gh \frac{\partial z}{\partial x} - \frac{u}{V}(\mu gh \cos(\alpha)+ g \frac{V^2}{\xi}) \\ \frac{\partial (hv)}{\partial t} + \frac{\partial (hvu)}{\partial x}+ \frac{\partial}{\partial y}(hv^2 + \frac{gh^2}{2}) = -gh \frac{\partial z}{\partial y} - \frac{v}{V}(\mu gh \cos(\alpha)+ g \frac{V^2}{\xi}) \\ \end{cases}, \end{align*}
where \( V \) is the magnitude of the velocity vector. \( \alpha \) is the local slope angle, \( \mu \) is the basal friction coefficient, and \( \xi \) is the Voellmy turbulence coefficient. \( \mu \) and \( \xi \) are asummed constant here.
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