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A general fluid–sediment mixture model and constitutive theory validated in many flow regimes

Published online by Cambridge University Press:  28 December 2018

Aaron S. Baumgarten  and
Ken Kamrin Open the ORCID record for Ken Kamrin [Opens in a new window]
Aaron S. Baumgarten
Affiliation:
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Ken Kamrin*
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: [email protected]
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Abstract

We present a thermodynamically consistent constitutive model for fluid-saturated sediments, spanning dense to dilute regimes, developed from the basic balance laws for two-phase mixtures. The model can represent various limiting cases, such as pure fluid and dry grains. It is formulated to capture a number of key behaviours such as: (i) viscous inertial rheology of submerged wet grains under steady shearing flows, (ii) the critical state behaviour of grains, which causes granular Reynolds dilation/contraction due to shear, (iii) the change in the effective viscosity of the fluid due to the presence of suspended grains and (iv) the Darcy-like drag interaction observed in both dense and dilute mixtures, which gives rise to complex fluid–grain interactions under dilation and flow. The full constitutive model is combined with the basic equations of motion for each mixture phase and implemented in the material point method (MPM) to accurately model the coupled dynamics of the mixed system. Qualitative results show the breadth of problems which this model can address. Quantitative results demonstrate the accuracy of this model as compared with analytical limits and experimental observations of fluid and grain behaviours in inhomogeneous geometries.

JFM classification

Granular media: Granular media Multiphase and Particle-laden Flows: Multiphase and Particle-laden Flows Complex Fluids: Suspensions
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 861 , 25 February 2019 , pp. 721 - 764
Copyright
© 2018 Cambridge University Press 

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References

Abe, K., Soga, K. & Bandara, S. 2013 Material point method for coupled hydromechanical problems. J. Geotech. Geoenviron. Engng 140 (3), 04013033.Google Scholar
Allen, B. & Kudrolli, A. 2017 Depth resolved granular transport driven by shearing fluid flow. Phys. Rev. Fluids 2 (2), 024304.Google Scholar
Amarsid, L., Delenne, J. Y., Mutabaruka, P., Monerie, Y., Perales, F. & Radjai, F. 2017 Viscoinertial regime of immersed granular flows. Phys. Rev. E 96, 012901.Google Scholar
Anand, L. & Su, C. 2005 A theory for amorphous viscoplastic materials undergoing finite deformations, with applications to metallic glasses. J. Mech. Phys. Solids 53 (6), 1362–1396.Google Scholar
Bandara, S. & Soga, K. 2015 Coupling of soil deformation and pore fluid flow using material point method. Comput. Geotech. 63, 199–214.Google Scholar
Bardenhagen, S. G. & Kober, E. M. 2004 The generalized interpolation material point method. Comput. Model. Engng Sci. 5 (6), 477–496.Google Scholar
Beetstra, R., van der Hoef, M. A. & Kuipers, J. A. M. 2007 Drag force of intermediate Reynolds number flow past mono- and bidisperse arrays of spheres. AIChE J. 53 (2), 489–501.Google Scholar
Boyer, F., Gauzelli, E. & Pouliquen, O. 2011 Unifying suspension and granular rheology. Phys. Rev. Lett. 107 (18), 188301.Google Scholar
Brackbill, J. U. & Ruppel, H. M. 1986 Flip: a method for adaptively zoned, particle-in-cell calculations of fluid flows in two dimensions. J. Comput. Phys. 65 (2), 314–343.Google Scholar
Brackbill, J. U., Kothe, D. B. & Ruppel, H. M. 1988 Flip: a low-dissipation, particle-in-cell method for fluid flow. Comput. Phys. Commun. 48 (1), 25–38.Google Scholar
Carman, P. C. 1937 Fluid flow through granular beds. Trans. Inst. Chem. Engrs 15, 150–166.Google Scholar
Cassar, C., Nicolas, M. & Pouliquen, O. 2005 Submarine granular flows down inclined planes. Phys. Fluids 17 (10), 103301.Google Scholar
Ceccato, F., Beuth, L., Vermeer, P. A. & Simonini, P. 2016 Two-phase material point method applied to the study of cone penetration. Comput. Geotech. 80, 440–452.Google Scholar
Ceccato, F. & Simonini, P. 2016 Granular flow impact forces on protection structures: Mpm numerical simulations with different constitutive models. Proc. Engng 158, 164–169.Google Scholar
Chang, C. & Powell, R. L. 1993 Dynamic simulation of bimodal suspensions of hydrodynamically interacting spherical particles. J. Fluid Mech. 253, 1–25.Google Scholar
Chang, C. & Powell, R. L. 1994 Effect of particle size distributions on the rheology of concentrated bimodal suspensions. J. Rheol. 38 (1), 85–98.Google Scholar
Chong, J. S., Christiansen, E. B. & Baer, A. D. 1971 Rheology of concentrated suspensions. J. Appl. Polym. Sci. 15 (8), 2007–2021.Google Scholar
Clift, R., Grace, J. R. & Weber, M. E. 2005 Bubbles, Drops, and Particles. Courier Corporation.Google Scholar
Cook, B. K., Noble, D. R. & Williams, J. R. 2004 A direct simulation method for particle-fluid systems. Engng Comput. 21 (2/3/4), 151–168.Google Scholar
Coussy, O. 2004 Poromechanics. Wiley.Google Scholar
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 (2), 021309.Google Scholar
Drumheller, D. S. 2000 On theories for reacting immiscible mixtures. Intl J. Engng Sci. 38, 347–382.Google Scholar
Dunatunga, S. & Kamrin, K. 2015 Continuum modelling and simulation of granular flows through their many phases. J. Fluid Mech. 779, 483–513.Google Scholar
Dupuit, J. É. J. 1863 Études théoriques et pratiques sur le mouvement des eaux dans les canaux découverts et à travers les terrains perméables: avec des considérations relatives au régime des grandes eaux, au débouché à leur donner, et à la marche des alluvions dans les rivières à fond mobile. Dunod.Google Scholar
Einstein, A. 1906 Calculation of the viscosity-coefficient of a liquid in which a large number of small spheres are suspended in irregular distribution. Ann. Phys. Leipzig 19, 286–306.Google Scholar
Fern, E. J. & Soga, K. 2016 The role of constitutive models in mpm simulations of granular column collapses. Acta Geotech. 11 (3), 659–678.Google Scholar
Gurtin, M. E., Fried, E. & Anand, L. 2010 The Mechanics and Thermodynamics of Continua. Cambridge University Press.Google Scholar
Henann, D. L. & Kamrin, K. 2013 A predictive, size-dependent continuum model for dense granular flows. Proc. Natl Acad. Sci. 110 (17), 6730–6735.Google Scholar
van der Hoef, M. A., Beetstra, R. & Kuipers, J. A. M. 2005 Lattice-Boltzmann simulations of low-Reynolds-number flow past mono- and bidisperse arrays of spheres: results for the permeability and drag force. J. Fluid Mech. 528, 233–254.Google Scholar
Houssais, M., Ortiz, C. P., Durian, D. J. & Jerolmack, D. J. 2015 Onset of sediment transport is a continuous transition driven by fluid shear and granular creep. Nat. Commun. 6, 6527.Google Scholar
Huang, P., Zhang, X., Ma, S. & Huang, X. 2011 Contact algorithms for the material point method in impact and penetration simulation. Intl J. Numer. Meth. Engng 85 (4), 498–517.Google Scholar
Jackson, R. 2000 The Dynamics of Fluidized Particles. Cambridge University Press.Google Scholar
Jop, P., Forterre, Y. & Pouliquen, O. 2006 A constitutive law for dense granular flows. Nature 441 (7094).Google Scholar
Kamrin, K. & Henann, D. L. 2015 Nonlocal modeling of granular flows down inclines. Soft Matt. 11 (1), 179–185.Google Scholar
Kamrin, K. & Koval, G. 2012 Nonlocal constitutive relation for steady granular flow. Phys. Rev. Lett. 108 (17), 178301.Google Scholar
Klika, V. 2014 A guide through available mixture theories for applications. Critical Rev. Solid State Mater. Sci. 39 (2), 154–174.Google Scholar
Koh, C. G., Gao, M. & Luo, C. 2012 A new particle method for simulation of incompressible free surface flow problems. Intl J. Numer. Methods Engng 89 (12), 1582–1604.Google Scholar
Maljaars, J. M.2016 A hybrid particle-mesh method for simulating free surface flows. Delft University of Technology.Google Scholar
Mast, C. M., Mackenzie-Helnwein, P., Arduino, P., Miller, G. R. & Shin, W. 2012 Mitigating kinematic locking in the material point method. J. Comput. Phys. 231 (16), 5351–5373.Google Scholar
Pailha, M. & Pouliquen, O. 2009 A two-phase flow description of the initiation of underwater granular avalanches. J. Fluid Mech. 633, 115–135.Google Scholar
Poslinski, A. J., Ryan, M. E., Gupta, R. K., Seshadri, S. G. & Frechette, F. J. 1988 Rheological behavior of filled polymeric systems ii. The effect of a bimodal size distribution of particulates. J. Rheol. 32 (8), 751–771.Google Scholar
Rondon, L., Pouliquen, O. & Aussillous, P. 2011 Granular collapse in a fluid: role of the initial volume fraction. Phys. Fluids 23 (7), 073301.Google Scholar
Roux, S. & Radjai, F. 1998 Texture-dependent rigid-plastic behavior. In Physics of Dry Granular Media (ed. Herrmann, H. J., Hovi, J. P. & Luding, S.), pp. 229–236. Springer.Google Scholar
Roux, S. & Radjai, F. 2001 Statistical approach to the mechanical behavior of granular media. In Mechanics for a New Mellennium (ed. Aref, H. & Phillips, J. W.), pp. 181–196. Kluwer.Google Scholar
Rudnicki, J. W. & Rice, J. R. 1975 Conditions for the localization of deformation in pressure-sensitive dilatant materials. J. Mech. Phys. Solids 23 (6), 371–394.Google Scholar
Shapiro, A. P. & Probstein, R. F. 1992 Random packings of spheres and fluidity limits of monodisperse and bidisperse suspensions. Phys. Rev. Lett. 68 (9), 1422–1425.Google Scholar
Soga, K., Alonso, E., Yerro, A., Kumar, K. & Bandara, S. 2015 Trends in large-deformation analysis of landslide mass movements with particular emphasis on the material point method. Geotechnique 66 (3), 248–273.Google Scholar
Steffen, M., Kirby, R. M. & Berzins, M. 2008 Analysis and reduction of quadrature errors in the material point method (mpm). Intl J. Numer. Meth. Engng 76 (6), 922–948.Google Scholar
Stickel, J. J. & Powell, R. L. 2005 Fluid mechanics and rheology of dense suspensions. Annu. Rev. Fluid Mech. 37, 129–149.Google Scholar
Storms, R. F., Ramarao, B. V. & Weiland, R. H. 1990 Low shear rate viscosity of bimodally dispersed suspensions. Powder Technol. 63 (3), 247–259.Google Scholar
Sulsky, D., Chen, Z. & Schreyer, H. L. 1994 A particle method for history-dependent materials. Comput. Meth. Appl. Mech. Engng 118 (1–2), 179–196.Google Scholar
Truesdell, C. & Noll, W. 1965 The Non-Linear Field Theories of Mechanics. Springer.Google Scholar
Turian, R. M. & Yuan, T.-F. 1977 Flow of slurries in pipelines. AIChE J. 23 (3), 232–243.Google Scholar
Wilmanski, K. 2008 Continuum Thermodynamics – Part 1: Foundations. World Scientific.Google Scholar
Zhao, Y. & Davis, R. H. 2002 Interaction of two touching spheres in a viscous fluid. Chem. Engng Sci. 57 (11), 1997–2006.Google Scholar

Baumgarten and Kamrin supplementary movie 1

Simulated collapses for the loose initial packing (bottom) and the dense initial packing (top) described in section 5.1.2. Solid phase material points are colored by equivalent plastic shearing-rate according to the scale. Fluid phase material points are colored light gray.

Download Baumgarten and Kamrin supplementary movie 1(Video)
Video 3.7 MB

Baumgarten and Kamrin supplementary movie 2

Simulated erosion flows described in section 5.1.3 and table 5. Solid phase material points are colored by packing fraction according to the scale. Fluid phase material points are colored light gray.

Download Baumgarten and Kamrin supplementary movie 2(Video)
Video 9.4 MB

Baumgarten and Kamrin supplementary movie 3

Simulated intruder problem described in section 5.2.1. Solid phase material points are colored by packing fraction according to the scale. Fluid phase material points are colored dark gray. The intruder material points are colored dark gray.

Download Baumgarten and Kamrin supplementary movie 3(Video)
Video 3 MB

Baumgarten and Kamrin supplementary movie 4

Simulated slope failures for dry slope (top) and partially submerged slope (bottom) as described in section 5.2.2. Solid phase material points are colored by equivalent plastic shearing-rate according to the scale. Fluid phase material points are colored light gray.

Download Baumgarten and Kamrin supplementary movie 4(Video)
Video 10 MB