Abstract
Using a continuum Navier-Stokes solver with the μ(I) flow law implemented to model the viscous behavior, and the discrete Contact Dynamics algorithm, the discharge of granular silos is simulated in two dimensions from the early stages of the discharge until complete release of the material. In both cases, the Beverloo scaling is recovered. We first do not attempt a quantitative comparison, but focus on the qualitative behavior of velocity and pressure at different locations in the flow. A good agreement for the velocity is obtained in the regions of rapid flows, while areas of slow creep are not entirely captured by the continuum model. The pressure field shows a general good agreement, while bulk deformations are found to be similar in both approaches. The influence of the parameters of the μ(I) flow law is systematically investigated, showing the importance of the dependence on the inertial number I to achieve quantitative agreement between continuum and discrete discharge. However, potential problems involving the systems size, the configuration and “non-local” effects, are suggested. Yet the general ability of the continuum model to reproduce qualitatively the granular behavior is found to be very encouraging.
Graphical abstract
Similar content being viewed by others
References
T. Le Pennec, K.J. Måløy, E.G. Flekkøy, J.C. Messager, M. Ammi, Phys. Fluids 10, 3072 (1998).
A. Janda, R. Harich, I. Zuriguel, D. Maza, P. Cixous, A. Garcimartí n, Phys. Rev. E 79, 031302 (2009).
D.M. Walker, Chem. Engng. Sci. 21, 975 (1966).
C.E. Davies, V. Chew, New Zealand J. Dairy Sci. Techol. 18, 47 (1983).
W.A. Beverloo, H.A. Leniger, J. van de Velde, Chem. Eng. Sci. 15, 260 (1961).
X.L. Wu, K.J. Maløy, A. Hansen, M. Ammi, D. Bideau, Phys. Rev. Lett. 71, 1363 (1993).
J.M.N.T. Gray, K. Hutter, Contin. Mech. Thermodyn. 9, 341 (1997).
A. Samadani, L. Mahadevan, A. Kudrolli, J. Fluid Mech. 452, 293 (2002).
J.M. Rotter, J.M.F.G. Holst, J.Y. Ooi, A. M. Sanad, Philos. Trans. R. Soc. London, Ser. A 356, 2685 (1998).
K. Kamrin, Int. J. Plasticity 26, 167 (2010).
L. Staron, P.-Y. Lagrée, S. Popinet, Phys. Fluids 24, 113303 (2012).
J. Sun, S. Sundaresan, arXiv:1207.1751v1 [cond-mat.soft] (2012).
C. Mankoc, A. Janda, R. Arévalo, J.M. Pastor, I. Zuriguel, A. Garcimartí n, D. Maza, Granular Matter 7, 407 (2007).
C.H. Rycroft, K. Kamrin, M.Z. Bazant, J. Mech. Phys. Solids 57, 828 (2009).
P. Claudin, J.-P. Bouchaud, M.E. Cates, J.P. Wittmer, Phys. Rev. E 57, 4441 (1998).
J.-N. Roux, Phys. Rev. E 61, 6802 (2000).
GDR MiDi, Eur. Phys. J. E 14, 341 (2004).
F. da Cruz, S. Emam, M. Prochnow, J.-N. Roux, F. Chevoir, Phys. Rev. E 72, 021309 (2005).
P. Jop, Y. Forterre, O. Pouliquen, Nature 441, 727 (2006).
G.B. Crosta, S. Imposimato, D. Roddeman, J. Geophys. Res. 114, F03020 (2009).
P.-Y. Lagrée, L. Staron, S. Popinet, J. Fluid Mech. 686, 378 (2011).
L. Lacaze, R.R. Kerswell, Phys. Rev. Lett. 102, 108305 (2009).
J. Chauchat, M. Médale, Comput. Methods Appl. Mech. Engrg. 199, 439 (2010).
C. Ancey, P. Coussot, P. Evesque, J. Rheol. 43, 1673 (1999).
T.J. Hatano, Phys. Rev. E 75, 060301 (2007).
O. Pouliquen, Y. Forterre, Philos. Trans. R. Soc. A 367, 5091 (2009).
K. Kamrin, G. Koval, Phys. Rev. Lett. 108, 178301 (2012).
J. Sun, S. Sundaresan, J. Fluid Mech. 682, 590 (2011).
S. Popinet, J. Comput. Phys. 190, 572 (2003).
S. Popinet J. Comput. Phys.2285838(2009.
M. Jean, J.-J. Moreau, Unilaterality and dry friction in the dynamics of rigid bodies collection, in Proceedings of the Contact Mechanics International Symposium, edited by A. Curnier (Presses Polytechniques et Universitaires Romandes, 1992) pp. 31-48.
F. Radjai, V. Richefeu, Mech. Mater. 41, 715 (2009).
F. Radjai, L. Brendel, S. Roux, Phys. Rev. E 54, 861 (1996).
I. Zuriguel, A. Garcimartín, D. Maza, L.A. Pugnaloni, J.M. Pastor, Phys. Rev. E 71, 051303 (2005).
A. Janda, I. Zuriguel, D. Maza, Phys. Rev. Lett. 108, 248001 (2013).
A.V. Potapov, C.S. Campbell, Phys. Fluids 8, 2884 (1996).
J. Choi, A. Kudrolli, M.Z. Bazant, J. Phys: Condens. Matter, 17, S2533 (2005).
I. Bartos, I.M. Jánosi, Granular Matter 9, 81 (2006).
H.G. Sheldon, D.J. Durian, Granular Matter 12, 579 (2010).
M.A. Aguirre, J.G. Grande, A. Calvo, L.A. Pugnaloni, J.-C. Géminard, Phys. Rev. E 83, 061305 (2011).
C. González-Montellano, F. Ayuga, J.Y. Ooi, Granular Matter 13, 149 (2011).
J.E. Hilton, P.W. Cleary, Phys. Rev. E 84, 011307 (2011).
H.A. Janssen, Zeitschr. Vereines Deutsch. Ing. 39, 1045 (1895).
M. Sperl, Granular Matter 8, 59 (2006).
G. Ovarlez, C. Fond, E. Clément, Phys. Rev. E 67, 060302 (2003).
C. Perge, M.A. Aguirre, P.A. Gago, L.A. Pugnaloni, D. Le Tourneau, J.-C. Géminard, Phys. Rev. E 85, 021303 (2012).
R.M. Nedderman, Static and Kinematics of Granular Materials (Cambridge University Press, 1992).
D. Hirshfeld, Y. Radzyner, D.C. Rapaport, Phys. Rev. E 56, 4404 (1997).
P.A. Langston, U. Tüzün, D.M. Heyes, Chem. Engin. Sci. 50, 967 (1995).
M.F. Djouwe Meffeja, PhD Thesis, Université de Rennes 1 (2012).
A. Garcimartí n, I. Zuriguel, A. Janda, D. Maza, Phys. Rev. E 84, 031309 (2011).
R. Artoni, A.C. Santomaso, M. Go, P. Canu, Phys. Rev. Lett. 108, 238002 (2012).
P. Jop, Y. Forterre, O. Pouliquen, J. Fluid Mech. 541, 167 (2005).
O. Pouliquen, Phys. Fluids 11, 542 (1999).
D.L. Henann, K. Kamrin, Proc. Natl. Acad. Sci. U.S.A. 110, 6730 (2013).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Staron, L., Lagrée, P.Y. & Popinet, S. Continuum simulation of the discharge of the granular silo. Eur. Phys. J. E 37, 5 (2014). https://doi.org/10.1140/epje/i2014-14005-6
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1140/epje/i2014-14005-6