**Abstract:** | * We present in this talk a two-phase two-thin-layer model for fluidized debris flows that takes into account dilatancy effects. It describes the velocity of both the solid and the fluid phases, the compression/dilatation of the granular media and its interaction with the pore fluid pressure (see [Bouchut et al., 2016]).
The model is derived from a 3D two-phase model proposed by [Jackson, 2000]. This system has 5 unknowns: the solid and fluid velocities, the solid and fluid pressures and the solid volume fraction. As a result, an additional equation inside the mixture is necessary to close the system. Surprisingly, this issue is inadequately accounted for in the models that have been developed on the basis of Jackson's work [Bouchut et al., 2015]. In particular, in [Pitman-Le, 2005] authors replaced this closure simply by imposing an extra boundary condition at the surface of the flow. When making a shallow expansion, this condition can be considered as a closure condition. However, the corresponding model cannot account for a dissipative energy balance. We present in this talk (see [Bouchut et al., 2016]), an approach to correctly deal with the thermodynamics of Jackson's model by closing the mixture equations by a weak compressibility relation following [Roux-Radjai, 1998]. This relation implies that the occurrence of dilation or contraction of the granular material in the model depends on whether the solid volume fraction is respectively higher or lower than a critical value. When dilation occurs, the fluid is sucked into the granular material, the pore pressure decreases and the friction force on the granular phase increases. On the contrary, in the case of contraction, the fluid is expelled from the mixture, the pore pressure increases and the friction force diminishes. To account for this transfer of fluid into and out of the mixture, a two-layer model is proposed with a fluid layer on top of the two-phase mixture layer. Mass and momentum conservation are satisfied for the two phases, and mass and momentum are transferred between the two layers. A thin-layer approximation is used to derive average equations. Special attention is paid to the drag friction terms that are responsible for the transfer of momentum between the two phases and for the appearance of an excess pore pressure with respect to the hydrostatic pressure.
Finally, we present several numerical tests by comparing with the models proposed in [Pailha-Pouliquen, 2009] and [Iverson-George, 2014], and with experimental data for the case of uniform flows.
References:
[Bouchut et al., 2015] F. Bouchut, E.D. Fernandez-Nieto, A. Mangeney, G. Narbona-Reina, A two-phase shallow debris flow model with energy balance, ESAIM: Math. Modelling Num. Anal., 49, 101-140 (2015).
[Bouchut et al., 2016] F. Bouchut, E. D. Fernandez-Nieto, A. Mangeney, G. Narbona-Reina, A two-phase two-layer model for fluidized granular flows with dilatancy effects, J. Fluid Mech., submitted (2016).
[Iverson-George, 2014] R. M. Iverson and D. L. George. A depth-averaged debris-flow model that includes the effects of evolving dilatancy. I. Physical basis. Proc. R. Soc. A, 470:20130819, (2014).
[Jackson, 2000] R. Jackson, The Dynamics of Fluidized Particles, Cambridges Monographs on Mechanics (2000).
[Pailha-Pouliquen, 2009] M. Pailha and O. Pouliquen. A two-phase flow description of the initiation of underwater granular avalanches. J. Fluid Mech., 633:115–135, (2009)
[Pitman-Le, 2005] E.B. Pitman, L. Le, A two-fluid model for avalanche and debris flows, Phil.Trans. R. Soc. A, 363, 1573-1601 (2005).
[Roux-Radjai, 1998] S. Roux, F. Radjai, Texture-dependent rigid plastic behaviour, Proceedings: Physics of Dry Granular Media, September 1997. (eds. H. J. Herrmann et al.). Kluwer. Carge?se, France, 305-311 (1998).
contatto: luca.bonaventura@polimi.it
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