Skip to main content Accessibility help
×
×
Home

A two-phase two-layer model for fluidized granular flows with dilatancy effects

  • François Bouchut (a1), Enrique D. Fernández-Nieto (a2), Anne Mangeney (a3) (a4) and Gladys Narbona-Reina (a2)
Abstract

We propose a two-phase two-thin-layer model for fluidized debris flows that takes into account dilatancy effects, based on the closure relation proposed by Roux & Radjai (Physics of Dry Granular Media, 1998, Springer, pp. 229–236). This relation implies that the occurrence of dilation or contraction of the granular material 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, with accurate asymptotic expansions. 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. For an appropriate form of dilatancy law we obtain a depth-averaged model with a dissipative energy balance in accordance with the corresponding three-dimensional initial system.

Copyright
Corresponding author
Email address for correspondence: francois.bouchut@u-pem.fr
References
Hide All
Anderson, T. B. & Jackson, R. 1967 A fluid mechanical description of fluidized beds. Ind. Engng Chem. Fundam. 6, 527539.
Andreini, N., Ancey, C. & Epely-Chauvin, G. 2013 Granular suspension avalanches. II: plastic regime. Phys. Fluids 25, 033302.
Andreotti, B., Forterre, Y. & Pouliquen, O. 2011 Les milieux granulaires. In Physique Savoirs Actuels. EDP Sciences.
Beavers, G. S. & Joseph, D. D. 1967 Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197207.
Bolton, M. D. 1986 The strength and dilatancy of sands. Gèotechnique 36, 6578.
Bouchut, F. & Boyaval, S. 2016 Unified derivation of thin-layer reduced models for shallow free-surface gravity flows of viscous fluids. Eur. J. Mech. (B/Fluids) 55, 116131.
Bouchut, F., Fernández-Nieto, E. D., Mangeney, A. & Narbona-Reina, G. 2015 A two-phase shallow debris flow model with energy balance. ESAIM: Math. Modelling Numer. Anal. 49, 101140.
Bouchut, F., Mangeney-Castelnau, A., Perthame, B. & Vilotte, J.-P. 2003 A new model of Saint Venant and Savage–Hutter type for gravity driven shallow water flows. C. R. Acad. Sci. Paris série I 336, 531536.
Bouchut, F. & Westdickenberg, M. 2004 Gravity driven shallow water models for arbitrary topography. Commun. Math. Sci. 2, 359389.
Brenner, H. 2009 Bi-velocity hydrodynamics, multicomponent fluids. Intern. J. Engng Sci. 47, 902929.
Brenner, H. 2010 Diffuse volume transport in fluids. Phys. A 389, 40264045.
Cassar, C., Nicolas, M. & Pouliquen, O. 2005 Submarine granular flows down inclined planes. Phys. Fluids 17, 103301.
Da Cruz, F., Emam, S., Prochnow, M., Roux, J.-N. & Chevoir, F. 2005 Rheophysics of dense granular materials: discrete simulation of plane shear flows. Phys. Rev. E 72, 021309.
Fernández-Nieto, E. D., Bouchut, F., Bresch, D., Castro Díàz, M. J. & Mangeney, A. 2008 A new Savage–Hutter type model for submarine avalanches and generated tsunami. J. Comput. Phys. 227, 77207754.
Forterre, Y. & Pouliquen, O. 2008 Flows of dense granular media. Annu. Rev. Fluid Mech. 40, 124.
GDR MiDi group 2004 On dense granular flows. Eur. Phys. J. E 14, 341365.
George, D. L. & Iverson, R. M.2011 A two-phase debris-flow model that includes coupled evolution of volume fractions, granular dilatancy, and pore-fluid pressure. In 5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy, 14–17 June 2011: Italian J. Engng Geology Environ, pp. 415–424. Universita La Sapienza.
George, D. L. & Iverson, R. M. 2014 A depth-averaged debris-flow model that includes the effects of evolving dilatancy. II: numerical predictions and experimental tests. Proc. R. Soc. Lond. A 470, 20130820.
Gray, J. M. N. T. & Edwards, A. N. 2014 A depth-averaged 𝜇(I)-rheology for shallow granular free-surface flows. J. Fluid Mech. 755, 503534.
Iverson, R. M. 1997 The physics of debris flows. Rev. Geophys. 35, 245296.
Iverson, R. M. 2000 Landslide triggering by rain infiltration. Water Resour. Res. 36, 18971910.
Iverson, R. M. 2005 Regulation of landslide motion by dilatancy and pore pressure feedback. J. Geophys. Res. 110, F02015.
Iverson, R. M. 2009 Elements of an improved model of debris-flow motion. In Powders and Grains, American Institute of Physics, Proceedings, vol. 1145, pp. 916.
Iverson, R. M. & George, D. L. 2014 A depth-averaged debris-flow model that includes the effects of evolving dilatancy. I: physical basis. Proc. R. Soc. Lond. A 470, 20130819.
Iverson, R. M. & George, D. L. 2016 Modelling landslide liquefaction, mobility bifurcation and the dynamics of the 2014 Oso disaster. Gèotechnique 66, 175187.
Iverson, R. M., Logan, M., LaHusen, R. G. & Berti, M. 2010 The perfect debris flow? Aggregated results from 28 large-scale experiments. J. Geophys. Res. 115, F03005.
Jackson, R. 1983 Some Mathematical and Physical Aspects of Continuum Models for the Motion of Granular Materials, in Theory of Dispersed Multiphase Flow. Proceedings of an Advanced Seminar, Conducted by the Mathematics Research Center, the University of Wisconsin–Madison, 26–28 May 1982 (ed. Meyer, R. E.), pp. 291337. Academic.
Jackson, R. 2000 The Dynamics of Fluidized Particles. (Cambridge Monographs on Mechanics) , Cambridge University Press.
Kowalski, J. & McElwaine, J. N. 2013 Shallow two-component gravity-driven flows with vertical variation. J. Fluid Mech. 714, 434462.
Lee, C. H., Huang, C. J. & Chiew, Y. M. 2015 A three-dimensional continuum model incorporating static and kinetic effects for granular flows with applications to collapse of a two-dimensional granular column. Phys. Fluids 27, 113303.
Lhuillier, D. 2009 Migration of rigid particles in non-Brownian viscous suspensions. Phys. Fluids 21, 023302.
Mitchell, J. K. 1993 Fundamentals of Soil Behaviours. Wiley.
Montserrat, S., Tamburrino, A., Roche, O. & Niño, Y. 2012 Pore fluid pressure diffusion in defluidizing granular columns. J. Geophys. Res. 117, F02034.
Morales de Luna, T. 2008 A Saint Venant model for gravity driven shallow water flows with variable density and compressibility effects. Math. Comput. Model. 47, 436444.
Morris, J. F. & Boulay, F. 1999 Curvilinear flows of non-colloidal suspensions: the role of normal stresses. J. Rheol. 43, 12131237.
Nott, P. R., Guazzelli, E. & Pouliquen, O. 2011 The suspension balance model revisited. Phys. Fluids 23, 043304.
Pailha, M., Nicolas, M. & Pouliquen, O. 2008 Initiation of underwater granular avalanches: influence of the initial volume fraction. Phys. Fluids 20, 111701.
Pailha, M. & Pouliquen, O. 2009 A two-phase flow description of the initiation of underwater granular avalanches. J. Fluid Mech. 633, 115135.
Pelanti, M., Bouchut, F. & Mangeney, A. 2008 A Roe-type scheme for two-phase shallow granular flows over variable topography. ESAIM: Math. Modelling Numer. Anal. 42, 851885.
Penel, Y., Dellacherie, S. & Després, B. 2015 Coupling strategies for compressible-low Mach number flows. Math. Models Meth. Appl. Sci. 25, 10451089.
Pitman, E. B. & Le, L. 2005 A two-fluid model for avalanche and debris flows. Phil. Trans. R. Soc. Lond. A 363, 15731601.
Reynolds, O. 1885 On the dilatancy of media composed of rigid particles in contact. Phil. Mag. 5 20, 469481.
Richardson, J. F. & Zaki, W. N. 1954 Sedimentation and fluidization: Part I. Trans. Inst. Chem. Engrs 32, 3553.
Rondon, L., Pouliquen, O. & Aussillous, P. 2011 Granular collapse in a fluid: role of the initial volume fraction. Phys. Fluids 23, 073301.
Roux, S. & Radjai, F. 1998 Texture-dependent rigid plastic behaviour. In Physics of Dry Granular Media (ed. Herrmann, H. J. et al. ), NATO ASI Series, vol. 350, pp. 229236. Springer.
Savage, S. B. & Hutter, K. 1989 The motion of a finite mass of granular material down a rough incline. J. Fluid Mech. 199, 177215.
Schaeffer, D. G. & Iverson, R. 2008 Steady and intermittent slipping in a model of landslide motion regulated by pore-pressure feedback. SIAM J. Appl. Maths 69, 768786.
Schofield, A. N. & Wroth, C. P. 1968 Critical State Soil Mechanics. McGraw-Hill Inc.
Vardoulakis, I. 1986 Dynamic stability analysis of undrained simple shear on water-saturated granular soils. Intl J. Numer. Anal. Mech. Geomech. 10, 177190.
Wood, D. M. 1990 Soil Behavior and Critical State Soil Mechanics. Cambridge University Press.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed