Hostname: page-component-848d4c4894-p2v8j Total loading time: 0 Render date: 2024-06-11T23:59:34.491Z Has data issue: false hasContentIssue false
  • English
  • Français

A theoretical formulation of dilatation/contraction for continuum modelling of granular flows

Published online by Cambridge University Press:  20 April 2021

Huabin Shi Open the ORCID record for Huabin Shi [Opens in a new window] ,
Ping Dong Open the ORCID record for Ping Dong [Opens in a new window] ,
Xiping Yu Open the ORCID record for Xiping Yu [Opens in a new window]  and
Yan Zhou Open the ORCID record for Yan Zhou [Opens in a new window]
Huabin Shi*
Affiliation:
State Key Laboratory of Internet of Things for Smart City and Department of Civil and Environmental Engineering, University of Macau, Avenida da Universidade, Taipa, Macao, China
Ping Dong
Affiliation:
School of Engineering, University of Liverpool, Brownlow Hill, LiverpoolL69 3GH, United Kingdom
Xiping Yu
Affiliation:
State Key Laboratory of Hydroscience and Engineering, Department of Hydraulic Engineering, Tsinghua University, Beijing100084, China
Yan Zhou
Affiliation:
School of Engineering, University of Liverpool, Brownlow Hill, LiverpoolL69 3GH, United Kingdom
*
†Email address for correspondence: huabinshi@um.edu.mo
Get access Rights & Permissions [Opens in a new window]

Abstract

Shear dilatation/contraction of granular materials has long been recognized as an important process in granular flows but a comprehensive theoretical description of this process for a wide range of shear rates is not yet available. In this paper, a theoretical formulation of dilatation/contraction is proposed for continuum modelling of granular flows, in which the dilatation/contraction effects consist of a frictional component, which results from the rearrangement of enduring-contact force chains among particles, and a collisional component, which arises from inter-grain collisions. In this formulation, a frictional solid pressure, which considers the rearrangement of contact force chains under shear deformation, is proposed for the frictional dilatation/contraction, while well-established rheological laws are adopted for the collisional inter-grain pressure to account for the collisional dilatancy effect. The proposed formulation is first verified analytically by describing the shear-weakening behaviour of granular samples in a torsional shear rheometer and by capturing the incipient failure of both dry and immersed granular slopes. The proposed dilatation/contraction formulation is then further validated numerically by integrating it into a two-fluid continuum model and applying the model to study the collapse of submerged granular columns, in which the dilatation/contraction plays a critical role.

JFM classification

Granular media: Granular media Multiphase and Particle-laden Flows: Particle/fluid flow Geophysical and Geological Flows: Sediment transport
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 916 , 10 June 2021 , A56
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aussillous, P., Chauchat, J., Pailha, M., Médale, M. & Guazzelli, É 2013 Investigation of the mobile granular layer in bedload transport by laminar shearing flows. J. Fluid Mech. 736, 594–615.CrossRefGoogle Scholar
Barker, T., Schaeffer, D.G., Bohorquez, P. & Gray, J.M.N.T. 2015 Well-posed and ill-posed behaviour of the μ (I)-rheology for granular flow. J. Fluid Mech. 779, 794–818.CrossRefGoogle Scholar
Barker, T., Schaeffer, D.G., Shearer, M. & Gray, J.M.N.T. 2017 Well-posed continuum equations for granular flow with compressibility and μ (I)-rheology. Proc. R. Soc. A 473, 20160846.CrossRefGoogle ScholarPubMed
Baumgarten, A.S. & Kamrin, K. 2019 A general fluid-sediment mixture model and constitutive theory validated in many flow regimes. J. Fluid Mech. 861, 721–764.CrossRefGoogle Scholar
Bonnet, F., Richard, T. & Philippe, P. 2010 Sensitivity to solid volume fraction of gravitational instability in a granular medium. Granul. Matt. 12, 317–325.CrossRefGoogle Scholar
Bouchut, F., Fernández-Nieto, E.D., Mangeney, A. & Narbona-Reina, G. 2016 A two-phase two-layer model for fluidized granualr flows with dilatancy effects. J. Fluid Mech. 801, 166–221.CrossRefGoogle Scholar
Boyer, F., Guazzelli, É & Pouliquen, O. 2011 Unifying suspension and granular rheology. Phys. Rev. Lett. 107, 188301.CrossRefGoogle ScholarPubMed
Chauchat, J. 2018 A comprehensive two-phase flow model for unidirectional sheet-flows. J. Hydraul. Res. 56 (1), 15–28.CrossRefGoogle Scholar
Chen, X., Li, Y., Niu, X., Chen, D. & Yu, X. 2011 A two-phase approach to wave-induced sediment transport under sheet flow conditions. Coastal Engng 58 (11), 1072–1088.CrossRefGoogle Scholar
Chialvo, S., Sun, J. & Sundaresan, S. 2012 Bridging the rheology of granular flows in three regimes. Phys. Rev. E 85, 021305.CrossRefGoogle ScholarPubMed
Dalrymple, R.A. & Rogers, B.D. 2006 Numerical modeling of water waves with the SPH method. Coastal Engng 53 (2-3), 141–147.CrossRefGoogle Scholar
Delannay, R., Valance, A., Mangeney, A., Roche, O. & Richard, P. 2017 Granular and particle-laden flows: from laboratory experiments to field observations. J. Phys. D: Appl. Phys. 50, 053001.CrossRefGoogle Scholar
Drew, D.A. 1983 Mathematical modeling of two-phase flow. Annu. Rev. Fluid Mech. 15 (1), 261–291.CrossRefGoogle Scholar
Dsouza, P.V. & Nott, P.R. 2020 A non-local constitutive model for slow granular flow that incorporates dilatancy. J. Fluid Mech. 888, R3.CrossRefGoogle Scholar
Forterre, Y. & Pouliquen, O. 2008 Flows of dense granular media. Annu. Rev. Fluid Mech. 40, 1–24.CrossRefGoogle Scholar
Gidaspow, D. 1994 Multiphase Flow and Fluidization, pp. 35–37. Academic.Google Scholar
Gonzalez-Ondina, J.M., Fraccarollo, L. & Liu, P.L.F. 2018 Two-level, two-phase model for intense, turbulent sediment transport. J. Fluid Mech. 839, 198–238.CrossRefGoogle Scholar
Gravish, N. & Goldman, D.I. 2014 Effect of volume fraction on granular avalanche dynamics. Phys. Rev. E 90, 032202.CrossRefGoogle ScholarPubMed
Guo, X., Peng, C., Wu, W. & Wang, Y. 2016 A hypoplastic constitutive model for debris materials. Acta Geotech. 11, 1217–1229.CrossRefGoogle Scholar
Hérault, A., Bilotta, G. & Dalrymple, R.A. 2010 SPH on GPU with CUDA. J. Hydraul Res. 48 (S1), 74–79.CrossRefGoogle Scholar
Heyman, J., Delannay, R., Tabuteau, H. & Valance, A. 2017 Compressibility regularizes the μ(I)-rheology for dense granular flows. J. Fluid Mech. 830, 553–568.CrossRefGoogle Scholar
Houssais, M. & Jerolmack, D.J. 2017 Toward a unifying constitutive relation for sediment transport across environments. Geomorphology 277, 251–264.CrossRefGoogle Scholar
Hsu, T.J., Jenkins, J.T. & Liu, P.L.F. 2004 On two-phase sediment transport: sheet flow of massive particles. Proc. R. Soc. A: Math. Phys. 460 (2048), 2223–2250.CrossRefGoogle Scholar
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. A: Math. Phys. 470, 20130819.CrossRefGoogle Scholar
Jenkins, J.T. & Savage, S.B. 1983 A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. J. Fluid Mech. 130, 187–202.CrossRefGoogle Scholar
Johnson, P.C. & Jackson, R. 1987 Frictional-collisional constitutive relations for granular materials, with application to plane shearing. J. Fluid Mech. 176, 67–93.CrossRefGoogle Scholar
Johnson, P.C., Nott, P. & Jackson, R. 1990 Frictional-collisional equations of motion for particulate flows and their application to chutes. J. Fluid Mech. 210, 501–535.CrossRefGoogle Scholar
Jop, P., Forterre, Y. & Pouliquen, O. 2006 A constitutive law for dense granular flows. Nature 441, 727–730.CrossRefGoogle ScholarPubMed
Kaitna, R., Dietrich, W.E. & Hsu, L. 2014 Surface slopes, velocity profiles and fluid pressure in coarse-grained debris flows saturated with water and mud. J. Fluid Mech. 741, 377–403.CrossRefGoogle Scholar
Lee, C.H. & Huang, Z. 2018 A two-phase flow model for submarine granular flows: With an application to collapse of deeply-submerged granular columns. Adv. Water Resour. 115, 286–300.CrossRefGoogle Scholar
Lu, K., Brodsky, E.E. & Kavehpour, H.P. 2007 Shear-weakening of the transitional regime for granular flow. J. Fluid Mech. 587, 347–372.CrossRefGoogle Scholar
Maurin, R., Chauchat, J. & Frey, P. 2016 Dense granular flow rheology in turbulent bedload transport. J. Fluid Mech. 804, 490–512.CrossRefGoogle Scholar
Meruane, C., Tamburrino, A. & Roche, O. 2010 On the role of the ambient fluid on gravitational granular flow dynamics. J. Fluid Mech. 648, 381–404.CrossRefGoogle Scholar
Ouriemi, M., Aussillous, P. & Guazzelli, É 2009 Sediment dynamics. Part 1, Bed-load transport by laminar shearing flows. J. Fluid Mech. 636, 295–319.CrossRefGoogle Scholar
Pailha, M., Nicolas, M. & Pouliquen, O. 2008 Initiation of underwater granular avalanches: Influence of the initial volume fraction. Phys. Fluids 20, 111701.CrossRefGoogle Scholar
Pailha, M. & Pouliquen, O. 2009 A two-phase flow description of the initiation of underwater granular avalanches. J. Fluid Mech. 633, 115–135.CrossRefGoogle Scholar
Reynolds, O. 1885 On the dilatancy of media composed of rigid particles in contact. With experimental illustrations. Lond. Edinb. Dubl. Phil. Mag. 20 (127), 469–481.CrossRefGoogle Scholar
Rondon, L., Pouliquen, O. & Aussillous, P. 2011 Granular collapse in a fluid: Role of the initial volume fraction. Phys. Fluids 23, 073301.CrossRefGoogle Scholar
Roux, S. & Radjai, F. 1998 Texture-dependent rigid plastic behaviour. In Proceedings: Physics of Dry Granular Media (ed. H.J. Herrmann, J.P. Hovi & S. Luding), Sep. 1997, pp. 229–235. Springer.CrossRefGoogle Scholar
Rowe, P.W. 1962 The stress-dilatancy relation for static equilibrium of an assembly of particles in contact. P. Roy. Soc. A: Math. Phys. 269 (1339), 500–527.Google Scholar
Shi, H., Si, P., Dong, P. & Yu, X. 2019 A two-phase SPH model for massive sediment motion in free surface flows. Adv. Water Resour. 129, 80–98.CrossRefGoogle Scholar
Shi, H., Yu, X. & Dalrymple, R.A. 2017 Development of a two-phase SPH model for sediment laden flows. Comput. Phys. Commun. 221, 259–272.CrossRefGoogle Scholar
Si, P., Shi, H. & Yu, X. 2018 Development of a mathematical model for submarine granular flows. Phys. Fluids 30, 083302.CrossRefGoogle Scholar
Trulsson, M., Andreotti, B. & Claudin, P. 2012 Transition from the viscous to inertial regime in dense suspensions. Phys. Rev. Lett. 109 (11), 118305.CrossRefGoogle ScholarPubMed
van Wachem, B.G.M., Schouten, J.C., van den Bleek, C.M., Krishna, R. & Sinclair, J.L. 2001 Comparative analysis of CFD models for dense gas-solid systems. AIChE J. 47 (5), 1035–1051.CrossRefGoogle Scholar
Wang, C., Wang, Y., Peng, C. & Meng, X. 2017 Dilatancy and compaction effects on the submerged granular column collapse. Phys. Fluids 29, 103307.CrossRefGoogle Scholar
Wood, D.M. 1990 Soil Behaviour and Critical State Soil Mechanics. Cambridge University Press.Google Scholar