This paper proposes a thin layer depth-averaged two-phase model provided by a dissipative energy balance to describe avalanches of solid-fluid mixtures. This model is derived from a 3D two-phase model based on the equations proposed by Jackson [The Dynamics of Fluidized Particles. Cambridges Monographs on Mechanics (2000)] which takes into account the force of buoyancy and the forces of interaction between the solid and fluid phases. Jackson’s model is based on mass and momentum conservation within the two phases, i.e. two vector and two scalar equations. This system has five unknowns: the solid volume fraction, the solid and fluid pressures and the solid and fluid velocities, i.e. three scalars and two vectors. As a result, an additional equation 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. In particular, Pitman and Le [Philos. Trans. R. Soc. A 363 (2005) 799–819] replaced this closure simply by imposing an extra boundary condition. If the pressure is assumed to be hydrostatic, this condition can be considered as a closure condition. However, the corresponding model cannot account for a dissipative energy balance. We propose here a closure equation to complete Jackson’s model, imposing incompressibility of the solid phase. We prove that the resulting whole 3D model is compatible with a dissipative energy balance. From this model, we deduce a 2D depth-averaged model and we also prove that the energy balance associated with this model is dissipative. Finally, we propose a numerical scheme to approximate the depth-averaged model. We present several numerical tests for the 1D case that are compared to the results of the model proposed by Pitman and Le.
DOI : 10.1051/m2an/2014026
Mots-clés : Granular flows, two-phase flows, thin layer approximation, energy balance, non-conservative systems, projection method, finite volume schemes
@article{M2AN_2015__49_1_101_0,
author = {Bouchut, F. and Fern\'andez-Nieto, E.D. and Mangeney, A. and Narbona-Reina, G.},
title = {A two-phase shallow debris flow model with energy balance},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {101--140},
publisher = {EDP-Sciences},
volume = {49},
number = {1},
year = {2015},
doi = {10.1051/m2an/2014026},
mrnumber = {3342194},
zbl = {1310.76172},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2014026/}
}
TY - JOUR AU - Bouchut, F. AU - Fernández-Nieto, E.D. AU - Mangeney, A. AU - Narbona-Reina, G. TI - A two-phase shallow debris flow model with energy balance JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 101 EP - 140 VL - 49 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2014026/ DO - 10.1051/m2an/2014026 LA - en ID - M2AN_2015__49_1_101_0 ER -
%0 Journal Article %A Bouchut, F. %A Fernández-Nieto, E.D. %A Mangeney, A. %A Narbona-Reina, G. %T A two-phase shallow debris flow model with energy balance %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 101-140 %V 49 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2014026/ %R 10.1051/m2an/2014026 %G en %F M2AN_2015__49_1_101_0
Bouchut, F.; Fernández-Nieto, E.D.; Mangeney, A.; Narbona-Reina, G. A two-phase shallow debris flow model with energy balance. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 1, pp. 101-140. doi : 10.1051/m2an/2014026. http://archive.numdam.org/articles/10.1051/m2an/2014026/
and , A fluid mechanical description of fluidized beds. Ind. Eng. Chem. Fundam. 6 (1967) 527–539. | DOI
, and , Instabilities and the formation of bubbles in fluidized beds. J. Fluid Mech. 303 (1995) 327–366. | DOI | Zbl
, Circle, sphere, and drop packings. Phys. Rev. E 53 (1996) 2571. | DOI | MR
, and , Dense suspensions in rotating-rod flows: normal stresses and particle migration. J. Fluid Mech. 686 (2011) 5–25. | DOI | Zbl
and , Gravity driven shallow water models for arbitrary topography. Commun. Math. Sci. 2 (2004) 359–389. | DOI | MR | Zbl
, , and , A new model of Saint Venant and Savage-Hutter type for gravity driven shallow water flows. C.R. Acad. Sci. Paris, Sér. I 336 (2003) 531–536. | DOI | MR | Zbl
, , and , On new erosion models of Savage-Hutter type for avalanches. Acta Mech. 199 (2008) 181–208. | DOI | Zbl
, and , Dynamics, deposit and erosion processes in granular collapse over sloping beds. J. Geophys. Res. 119 (2013) 504–532.
, , , and , Numerical modeling of landquakes. Geophys. Res. Lett. 37 (2010) L15305. | DOI
D.L. George and R.M. Iverson, A two-phase debris-flow model that includes coupled evolution of volume fractions, granular dilatancy, and pore-fluid pressure. Italian J. Engrg. Geol. Environ. (2011) DOI:10.4408/IJEGE.2011-03.B-047.
and , Two problems in the gravity flow of granular materials. J. Fluid Mech. 45 (1971) 321–339. | DOI | Zbl
J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics. Kluwer Academic Publishers (1983).
, and , Searching for the perfect packing. Phys. A 330 (2003) 77–82. | DOI | MR | Zbl
and , Entrainment of debris in rock avalanches: An analysis of a long run-out mechanism. Bull. Geol. Soc. Am. 116 (2004) 1240–1252. | DOI
, The physics of debris flows, Rev. Geophys. 35 (1997) 245–296. | DOI
, , Flow of variably fluidized granular masses across three-dimensional terrain 1: Coulomb mixture theory. J. Geophys. Res. 106 (2001) 537–552. | DOI
, , and , The perfect debris flow? aggregated results from 28 large-scale experiments. J. Geophys. Res. 115 (2010) F03005. DOI:10.1029/2009JF001514.
, , , , and , Positive feedback and momentum growth during debris-flow entrainment of wet bed sediment. Nature Geoscience 4 (2011) 116–121. | DOI
R. Jackson, The Dynamics of Fluidized Particles. Cambridges Monographs on Mechanics (2000). | MR | Zbl
G. Jhonson, M. Massoudi and K.R. Rajagopal, A review of interaction mechanisms in fluidsolid flows. Technical Report. DOE/PETC/TR-90/9, U.S. Dep. of Energy, Pittsburgh Energy Tech. Ctr., Pittsburgh, USA (1990).
, The mobility of long-runout landslides. Eng. Geol. 63 (2002) 301–331. | DOI
and , Mobility and topographic effects for large Valles Marineris landslides on Mars. Geophys. Res. Lett. 34 (2007) L10201. | DOI
, and , Frictional weakening in landslides on Earth and on other planetary bodies. Nature Commun. 5 (2014) 3417. | DOI
, , , , , and , Numerical modeling of avalanches based on Saint-Venant equations using a kinetic scheme. J. Geophys. Res. 108 (2003) 2527. | DOI
, , , , and , On the use of Saint-Venant equations for simulating the spreading of a granular mass. J. Geophys. Res. 110 (2005) B09103.
, , , and , Avalanche mobility induced by the presence of an erodible bed and associated entrainment. Geophys. Res. Lett. 34 (2007) L22401. | DOI
, , , , and , Erosion and mobility in granular collapse over sloping beds. J. Geophys. Res.-Earth Surf. 115 (2010) F03040. | DOI
, Landslide boost from entrainment. Nature Geosci. 4 (2011) 77–78. | DOI
, , , , , and , Sinuous gullies on Mars: Frequency, distribution, and implications for flow properties. J. Geophys. Res. Planets 115 (2010) E11001. | DOI
, , , On the role of the ambient fluid on gravitational granular flow dynamics. J. Fluid. Mech. 648 (2010) 381–404. | DOI | MR | Zbl
, , , , , and , Numerical modeling of the Mount Steller landslide flow history and of the generated long period seismic waves. Geophys. Res. Lett. 39 (2012) L16402. | DOI
, , and , Mixing of a granular layer falling through a fluid. Phys. Rev. E 82 (2010) 011301. | DOI
, and , Sediment dynamics. Part I: Bed-load transport by shearing flows. J. Fluid Mech. 636 (2009) 295–319. | DOI | MR | Zbl
and , On the well-balance property of Roe’s method for nonconservative hyperbolic systems. Applications to shallow-water systems. ESAIM: M2AN 38 (2004) 821–852. | DOI | Numdam | MR | Zbl
and , A two-phase flow description of the initiation of underwater granular avalanches. J. Fluid Mech. 633 (2009) 115–135. | DOI | MR | Zbl
, and , A Roe-type scheme for two-phase shallow granular flows over variable topography. ESAIM: M2AN 42 (2008) 851–885. | DOI | Numdam | MR | Zbl
, and , A Riemann solver for single-phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers. J. Comput. Phys. 230 (2011) 515–550. | DOI | MR | Zbl
and , A two-fluid model for avalanche and debris flows. Philos. Trans. R. Soc. A 363 (2005) 1573–1601. | DOI | MR | Zbl
, and , Modelling debris flows down general channels. Natural Hazards Earth System Sci. 5 (2005) 799–819. | DOI
, and , Granular collapse in a fluid: role of the initial volume fraction. Phys. Fluids 23 (2011) 073301. | DOI
and , Sedimentation and Fluuidization: part I. Trans. Inst. Chem. Eng. 32 (1954) 35–53.
, , and , On the run out distance of geophysical gravitational flows: insight from fluidized granular collapse experiments. Earth Planet. Sci. Lett. 311 (2011) 375–385. | DOI
, , , , and , Dynamic pore pressure variations induce substrate erosion by pyroclastic flows, Geology 41 (2013) 1107–1110. | DOI
S. Roux and F. Radjai, Texture-dependent rigid plastic behavior. In Proc. of Physics of Dry Granular Media 1997. Edited by H.J. Herrmann. Kluwer. Cargèse, France (1998) 305–311.
and , The motion of a finite mass of granular material down a rough incline. J. Fluid Mach. 199 (1989) 177–215. | DOI | MR | Zbl
and , Coupled continuum discrete model for saturated granular materials. J. Engrg. Mech. 131 (2005) 413–426. | DOI
, , , , A. Wachs: Micro-rheology of dense particulate flows: application to immersed avalanches. J. Non-Newtonian Fluid 166 (2011) 63–72. | DOI | Zbl
, and , , Space-filling properties of polydisperse granular media. Phys. Rev. E 76 (2007) 021301. | DOI
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