Numerical Simulation of Granular Column Collapses with Pressure-Dependent Viscoplastic Model using the Smoothed Particle Hydrodynamic Method
Authors
Abstract
This paper presents a numerical analysis of granular column collapse phenomenon using a two-dimensional smoothed particle hydrodynamics model and a local constitutive law proposed by Jop et al. This constitutive law, which is based on the viscoplastic behaviour of dense granular material flows, is characterized by an apparent viscosity depending both on the local strain rate and the local pressure. The rheological parameters are directly derived from the experiments. A simple proposed regularization method used in the viscosity relation to reproduce the stopping condition and the free surface of a granular flow where the pressure is disappeared. Pressure oscillation, as the main disadvantage of the weakly compressible SPH method, leads to an inaccurate pressure distribution. In this research, a new algorithm is proposed to remove the nonphysical oscillations by relating the divergence of velocity to the Laplacian of pressure. The simulations based on the proposed SPH algorithm satisfactorily capture the dynamics of gravity-driven granular flows observed in the experiments. The maximum thickness of a granular flowing on a rough inclined plane is obtained based on the local rheology model and compared with the experimental results. The run-out distances and the slopes of the deposits in the simulations showed a good agreement with the values found in the experiments. The results of the simulation proved that the initial column ratio played an important role in spreading the granular mass
Keywords
1. Lajeunesse, E., Mangeney-Castelnau, A., and Vilotte, J. P., “Spreading of a Granular Mass on a Horizontal Plane”, Physics of Fluids, Vol. 16, pp. 2371-2381, 2004.
2. Lajeunesse, E., Monnier, J., and Homsy, G., “Granular Slumping on a Horizontal Surface”, Physics of Fluids, Vol. 17, p. 103302, 2005.
3. Lube, G., Huppert, H. E., Sparks, R. S. J., and Hallworth, M. A., “Axisymmetric Collapses of Granular Columns”, Journal of Fluid Mechanics, Vol. 508, pp. 175-199, 2004.
4. Lube, G., Huppert, H. E., Sparks, R. S. J., and Freundt, A., “Collapses of Two-Dimensional Granular Columns”, Physical Review E, Vol. 72, p. 041301, 2005.
5. Lube, G., Huppert, H. E., Sparks, R. S. J., and Freundt, A., “Static and Flowing Regions in Granular Collapses Down Channels”, Physics of Fluids, Vol. 19, p. 043301, 2007.
6. Pouliquen, O., “Scaling Laws in Granular Flows Down Rough Inclined Planes”, Physics of Fluids, Vol. 11, pp. 542-548, 1999.
7. Mangeney, A., Roche, O., Hungr, O., Mangold, N., Faccanoni, G., and Lucas, A., “Erosion and Mobility in Granular Collapse Over Sloping Beds”, Journal of Geophysical Research: Earth Surface, Vol. 115, 2010.
8. Da Cruz, F., Emam, S., Prochnow, M., Roux, J. -N., and Chevoir, F., “Rheophysics of Dense Granular Materials: Discrete Simulation of Plane Shear Flows”, Physical Review E, Vol. 72, p. 021309, 2005.
9. Jop, P., Forterre, Y., and Pouliquen, O., “A Constitutive Law for Dense Granular Flows”, Nature, Vol. 441, pp. 727-730, 2006.
10. Kamrin, K., and Koval, G., “Nonlocal Constitutive Relation for Steady Granular Flow”, Physical Review Letters, Vol. 108, p. 178301, 2012.
11. Lagrée, P. -Y., Staron, L., and Popinet, S., “The Granular Column Collapse as a Continuum: Validity of a Two-Dimensional Navier-Stokes Model with a μ (I)-Rheology”, Journal of Fluid Mechanics, Vol. 686, pp. 378-408, 2011.
12. Kamrin, K., “Nonlinear Elasto-Plastic Model for Dense Granular Flow”, International Journal of Plasticity, Vol. 26, pp. 167-188, 2010.
13. Cundall, P. A., and Strack, O. D., “A Discrete Numerical Model for Granular Assemblies”, Geotechnique, Vol. 29, pp. 47-65, 1979.
14. Lucy, L. B., “A Numerical Approach to the Testing of the Fission Hypothesis”, The Astronomical Journal, Vol. 82, pp. 1013-1024, 1977.
15. Gingold, R. A., and Monaghan, J. J., “Smoothed Particle Hydrodynamics: Theory and Application to Non-Spherical Stars”, Monthly Notices of the Royal Astronomical Society, Vol. 181, pp. 375-389, 1977.
16. Randles, P., and Libersky, L., “Smoothed Particle Hydrodynamics: Some Recent Improvements and Applications”, Computer Methods in Applied Mechanics and Engineering, Vol. 139, pp. 375-408, 1996.
17. Eghtesad, A., Shafiei, A., and Mahzoon, M., “Study of Dynamic Behavior of Ceramic-Metal FGM under High Velocity Impact Conditions using CSPM Method”, Applied Mathematical Modelling, Vol. 36, pp. 2724-2738, 2012.
18. Shao, S., and Gotoh, H., “Turbulence Particle Models for Tracking free Surfaces”, Journal of Hydraulic Research, Vol. 43, pp. 276-289, 2005.
19. Monaghan, J. J., “Simulating Free Surface Flows with SPH”, Journal of Computational Physics, Vol. 110, pp. 399-406, 1994.
20. Laigle, D., Lachamp, P., and Naaim, M., “SPH-Based Numerical Investigation of Mudflow and other Complex Fluid Flow Interactions with Structures”, Computational Geosciences, Vol. 11, pp. 297-306, 2007.
21. Pasculli, A., Minatti, L., Sciarra, N., and Paris, E., “SPH Modeling of Fast Muddy Debris Flow: Numerical and Experimental Comparison of Certain Commonly Utilized Approaches”, Italian Journal of Geosciences, Vol. 132, pp. 350-365, 2013.
22. Bui, H. H., Fukagawa, R., Sako, K., and Ohno, S., “Lagrangian Meshfree Particles Method (SPH) for Large Deformation and Failure Flows of Geomaterial using Elastic–Plastic Soil Constitutive Model”, International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 32, pp. 1537-1570, 2008.
23. Nguyen, C. T., Nguyen, C. T., Bui, H. H., Nguyen, G. D., and Fukagawa, R., “A New SPH-Based Approach to Simulation of Granular Flows using Viscous Damping and Stress Regularisation”, Landslides, Vol. 14, pp. 69-81, 2017.
24. Pouliquen, O., Cassar, C., Jop, P., Forterre, Y., and Nicolas, M., “Flow of Dense Granular Material: Towards Simple Constitutive Laws”, Journal of Statistical Mechanics: Theory and Experiment, Vol. 2006, p. P07020, 2006.
25. Liu, G. -R. and Liu, M. B., Smoothed Particle Hydrodynamics: a Meshfree Particle Method, World Scientific, 2003.
26. Gotoh, H., Shibahara, T., and Sakai, T., “Sub-Particle-Scale Turbulence Model for the {MPS} Method - {L}Agrangian Flow Model for Hydraulic Engineering”, Advanced Methods for Computational Fluid Dynamics, Vol. 4-9, pp. 339-347, 2001.
27. Abdelrazek, A. M., Kimura, I., and Shimizu, Y., “Simulation of Three-Dimensional Rapid Free-Surface Granular Flow Past Different Types of Obstructions using the SPH Method”, Journal of Glaciology, Vol. 62, pp. 335-347, 2016.
28. Oger, G., Doring, M., Alessandrini, B., and Ferrant, P., “An Improved SPH Method: Towards Higher Order Convergence”, Journal of Computational Physics, Vol. 225, pp. 1472-1492, 2007.
29. Cleary, P. W., and Monaghan, J. J., “Conduction Modelling using Smoothed Particle Hydrodynamics”, Journal of Computational Physics, Vol. 148, pp. 227-264, 1999.
30. Schwaiger, H. F., “An Implicit Corrected SPH Formulation for Thermal Diffusion with Linear Free surface Boundary Conditions”, International Journal for Numerical Methods in Engineering, Vol. 75, pp. 647-671, 2008.
31. Monaghan, J., and Gingold, R. A., “Shock Simulation by the Particle Method SPH”, Journal of Computational Physics, Vol. 52, pp. 374-389, 1983.
32. Morris, J., and Monaghan, J., “A Switch to Reduce SPH Viscosity”, Journal of Computational Physics, Vol. 136, pp. 41-50, 1997.
33. Morris, J. P., Fox, P. J., and Zhu, Y., “Modeling Low Reynolds Number Incompressible Flows using SPH”, Journal of Computational Physics, Vol. 136, pp. 214-226, 1997.
34. Adami, S., Hu, X., and Adams, N., “A Generalized Wall Boundary Condition for Smoothed Particle Hydrodynamics”, Journal of Computational Physics, Vol. 231, pp. 7057-7075, 2012.
35. Fatehi, R., and Manzari, M., “A Consistent and Fast Weakly Compressible Smoothed Particle Hydrodynamics with a New Wall Boundary Condition”, International Journal for Numerical Methods in Fluids, Vol. 68, pp. 905-921, 2012.
36. Goldhirsch, I., “Rapid Granular Flows”, Annual Review of Fluid Mechanics, Vol. 35, pp. 267-293, 2003.
37. Papanastasiou, T. C., “Flows of Materials with Yield”, Journal of Rheology, Vol. 31, pp. 385-404, 1987.
38. MiDi, G., “On Dense Granular Flows”, The European Physical Journal E, Vol. 14, pp. 341-365, 2004.
39. Jop, P., Forterre, Y., and Pouliquen, O., “Crucial Role of Sidewalls in Granular Surface Flows: Consequences for the Rheology”, Journal of Fluid Mechanics, Vol. 541, pp. 167-192, 2005.
40. Forterre, Y. and Pouliquen, O., “Long-Surface-Wave Instability in Dense Granular Flows”, Journal of Fluid Mechanics, Vol. 486, pp. 21-50, 2003
2. Lajeunesse, E., Monnier, J., and Homsy, G., “Granular Slumping on a Horizontal Surface”, Physics of Fluids, Vol. 17, p. 103302, 2005.
3. Lube, G., Huppert, H. E., Sparks, R. S. J., and Hallworth, M. A., “Axisymmetric Collapses of Granular Columns”, Journal of Fluid Mechanics, Vol. 508, pp. 175-199, 2004.
4. Lube, G., Huppert, H. E., Sparks, R. S. J., and Freundt, A., “Collapses of Two-Dimensional Granular Columns”, Physical Review E, Vol. 72, p. 041301, 2005.
5. Lube, G., Huppert, H. E., Sparks, R. S. J., and Freundt, A., “Static and Flowing Regions in Granular Collapses Down Channels”, Physics of Fluids, Vol. 19, p. 043301, 2007.
6. Pouliquen, O., “Scaling Laws in Granular Flows Down Rough Inclined Planes”, Physics of Fluids, Vol. 11, pp. 542-548, 1999.
7. Mangeney, A., Roche, O., Hungr, O., Mangold, N., Faccanoni, G., and Lucas, A., “Erosion and Mobility in Granular Collapse Over Sloping Beds”, Journal of Geophysical Research: Earth Surface, Vol. 115, 2010.
8. Da Cruz, F., Emam, S., Prochnow, M., Roux, J. -N., and Chevoir, F., “Rheophysics of Dense Granular Materials: Discrete Simulation of Plane Shear Flows”, Physical Review E, Vol. 72, p. 021309, 2005.
9. Jop, P., Forterre, Y., and Pouliquen, O., “A Constitutive Law for Dense Granular Flows”, Nature, Vol. 441, pp. 727-730, 2006.
10. Kamrin, K., and Koval, G., “Nonlocal Constitutive Relation for Steady Granular Flow”, Physical Review Letters, Vol. 108, p. 178301, 2012.
11. Lagrée, P. -Y., Staron, L., and Popinet, S., “The Granular Column Collapse as a Continuum: Validity of a Two-Dimensional Navier-Stokes Model with a μ (I)-Rheology”, Journal of Fluid Mechanics, Vol. 686, pp. 378-408, 2011.
12. Kamrin, K., “Nonlinear Elasto-Plastic Model for Dense Granular Flow”, International Journal of Plasticity, Vol. 26, pp. 167-188, 2010.
13. Cundall, P. A., and Strack, O. D., “A Discrete Numerical Model for Granular Assemblies”, Geotechnique, Vol. 29, pp. 47-65, 1979.
14. Lucy, L. B., “A Numerical Approach to the Testing of the Fission Hypothesis”, The Astronomical Journal, Vol. 82, pp. 1013-1024, 1977.
15. Gingold, R. A., and Monaghan, J. J., “Smoothed Particle Hydrodynamics: Theory and Application to Non-Spherical Stars”, Monthly Notices of the Royal Astronomical Society, Vol. 181, pp. 375-389, 1977.
16. Randles, P., and Libersky, L., “Smoothed Particle Hydrodynamics: Some Recent Improvements and Applications”, Computer Methods in Applied Mechanics and Engineering, Vol. 139, pp. 375-408, 1996.
17. Eghtesad, A., Shafiei, A., and Mahzoon, M., “Study of Dynamic Behavior of Ceramic-Metal FGM under High Velocity Impact Conditions using CSPM Method”, Applied Mathematical Modelling, Vol. 36, pp. 2724-2738, 2012.
18. Shao, S., and Gotoh, H., “Turbulence Particle Models for Tracking free Surfaces”, Journal of Hydraulic Research, Vol. 43, pp. 276-289, 2005.
19. Monaghan, J. J., “Simulating Free Surface Flows with SPH”, Journal of Computational Physics, Vol. 110, pp. 399-406, 1994.
20. Laigle, D., Lachamp, P., and Naaim, M., “SPH-Based Numerical Investigation of Mudflow and other Complex Fluid Flow Interactions with Structures”, Computational Geosciences, Vol. 11, pp. 297-306, 2007.
21. Pasculli, A., Minatti, L., Sciarra, N., and Paris, E., “SPH Modeling of Fast Muddy Debris Flow: Numerical and Experimental Comparison of Certain Commonly Utilized Approaches”, Italian Journal of Geosciences, Vol. 132, pp. 350-365, 2013.
22. Bui, H. H., Fukagawa, R., Sako, K., and Ohno, S., “Lagrangian Meshfree Particles Method (SPH) for Large Deformation and Failure Flows of Geomaterial using Elastic–Plastic Soil Constitutive Model”, International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 32, pp. 1537-1570, 2008.
23. Nguyen, C. T., Nguyen, C. T., Bui, H. H., Nguyen, G. D., and Fukagawa, R., “A New SPH-Based Approach to Simulation of Granular Flows using Viscous Damping and Stress Regularisation”, Landslides, Vol. 14, pp. 69-81, 2017.
24. Pouliquen, O., Cassar, C., Jop, P., Forterre, Y., and Nicolas, M., “Flow of Dense Granular Material: Towards Simple Constitutive Laws”, Journal of Statistical Mechanics: Theory and Experiment, Vol. 2006, p. P07020, 2006.
25. Liu, G. -R. and Liu, M. B., Smoothed Particle Hydrodynamics: a Meshfree Particle Method, World Scientific, 2003.
26. Gotoh, H., Shibahara, T., and Sakai, T., “Sub-Particle-Scale Turbulence Model for the {MPS} Method - {L}Agrangian Flow Model for Hydraulic Engineering”, Advanced Methods for Computational Fluid Dynamics, Vol. 4-9, pp. 339-347, 2001.
27. Abdelrazek, A. M., Kimura, I., and Shimizu, Y., “Simulation of Three-Dimensional Rapid Free-Surface Granular Flow Past Different Types of Obstructions using the SPH Method”, Journal of Glaciology, Vol. 62, pp. 335-347, 2016.
28. Oger, G., Doring, M., Alessandrini, B., and Ferrant, P., “An Improved SPH Method: Towards Higher Order Convergence”, Journal of Computational Physics, Vol. 225, pp. 1472-1492, 2007.
29. Cleary, P. W., and Monaghan, J. J., “Conduction Modelling using Smoothed Particle Hydrodynamics”, Journal of Computational Physics, Vol. 148, pp. 227-264, 1999.
30. Schwaiger, H. F., “An Implicit Corrected SPH Formulation for Thermal Diffusion with Linear Free surface Boundary Conditions”, International Journal for Numerical Methods in Engineering, Vol. 75, pp. 647-671, 2008.
31. Monaghan, J., and Gingold, R. A., “Shock Simulation by the Particle Method SPH”, Journal of Computational Physics, Vol. 52, pp. 374-389, 1983.
32. Morris, J., and Monaghan, J., “A Switch to Reduce SPH Viscosity”, Journal of Computational Physics, Vol. 136, pp. 41-50, 1997.
33. Morris, J. P., Fox, P. J., and Zhu, Y., “Modeling Low Reynolds Number Incompressible Flows using SPH”, Journal of Computational Physics, Vol. 136, pp. 214-226, 1997.
34. Adami, S., Hu, X., and Adams, N., “A Generalized Wall Boundary Condition for Smoothed Particle Hydrodynamics”, Journal of Computational Physics, Vol. 231, pp. 7057-7075, 2012.
35. Fatehi, R., and Manzari, M., “A Consistent and Fast Weakly Compressible Smoothed Particle Hydrodynamics with a New Wall Boundary Condition”, International Journal for Numerical Methods in Fluids, Vol. 68, pp. 905-921, 2012.
36. Goldhirsch, I., “Rapid Granular Flows”, Annual Review of Fluid Mechanics, Vol. 35, pp. 267-293, 2003.
37. Papanastasiou, T. C., “Flows of Materials with Yield”, Journal of Rheology, Vol. 31, pp. 385-404, 1987.
38. MiDi, G., “On Dense Granular Flows”, The European Physical Journal E, Vol. 14, pp. 341-365, 2004.
39. Jop, P., Forterre, Y., and Pouliquen, O., “Crucial Role of Sidewalls in Granular Surface Flows: Consequences for the Rheology”, Journal of Fluid Mechanics, Vol. 541, pp. 167-192, 2005.
40. Forterre, Y. and Pouliquen, O., “Long-Surface-Wave Instability in Dense Granular Flows”, Journal of Fluid Mechanics, Vol. 486, pp. 21-50, 2003
)