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Hydrodynamic modeling of dilute and dense granular flow

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Abstract

We study numerically a continuum model for granular flow, which covers the regime of fast dilute flow as well as slow dense flow up to vanishing velocity. The constitutive relations at small and intermediate densities are equivalent to those derived from kinetic theory of granular flow. The existence of an inherent instability due to the vanishing kinetic or collisional pressure for small granular temperatures requires a cross over from a collisional pressure to an a thermal yield pressure at densities close to random close packing. Contrary to a kinetic viscosity, the viscosity turns into a function diverging for small temperatures analogous to the diverging viscosities of liquids close to the glass transition. In this respect the presented model is a simplified version of a model of Savage (J Fluid Mech 377:1–26, 1998), which nevertheless recovers many aspects of dense granular flow. As examples we show simulations of sandpiles with predictable slopes, hopper simulations with mass and core flow and angle dependent critical sand heights in flows down an inclined plane. We solve the system of the strongly nonlinear singular hydrodynamic equations with the help of a newly developed nonlinear time stepping algorithm together with a finite volume space discretization. The numerical algorithm is implemented using a finite volume solver framework developed by the authors which allows discretization on cell-centred bricks in arbitrary domains.

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References

  1. Savage S.B.: Analysis of slow high-concentration flows of granular materials. J. Fluid Mech. 377, 1–26 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. Kadanoff, L.P.: Built upon sand: theoretical ideas inspired by granular flows. Rev. Mod. Phys. 71(1) (1999)

  3. Brilliantov N.V., Pöschel T.: Kinetic Theory of Granular Gases. Oxford Graduate Texts. Oxford University Press, Berlin (2003)

    Google Scholar 

  4. Luding S., McNamara S.: How to handle the inelastic collapse of a dissipative hard-sphere gas with the tc model. Granul. Matter 1(3), 113–128 (1998)

    Article  Google Scholar 

  5. Luding S.: On the relevance of molecular chaos for granular flows. ZAMM 80, 9–12 (2000)

    Article  Google Scholar 

  6. Gidaspow D.: Multiphase Flow and Fluidization. Academic Press, New York (1994)

    MATH  Google Scholar 

  7. Khain E., Meerson B.: Onset of thermal convection in a horizontal layer of granular gas. Phys. Rev. E 67(2), 021,306 (2003)

    Article  MathSciNet  Google Scholar 

  8. Carrillo J.A., Pöschel T., Saluena C.: Granular hydrodynamics and pattern formation in vertically oscillated granular disk layers. J. Fluid Mech. 597(1), 119–144 (2008)

    MATH  MathSciNet  ADS  Google Scholar 

  9. Meerson B., Pöschel T., Bromberg Y.: Close-packed floating clusters: granular hydrodynamics beyond the freezing point?. Phys. Rev. Lett. 91(2), 024301 (2003)

    Article  ADS  Google Scholar 

  10. Meerson, B., Díez-Minguito, M., Schwager, T., Pöschel, T.: Close-packed granular clusters: hydrostatics and persistent gaussian fluctuations. Granul. Matter (10), 21–27 (2007)

  11. Bocquet L., Losert W., Schalk D., Lubensky T.C., Gollub J.P.: Granular shear flow dynamics and forces: experiment and continuum theory. Phys. Rev. E 65(1), 011307 (2001)

    Article  ADS  Google Scholar 

  12. Bocquet L., Errami J., Lubensky T.: Hydrodynamic model for a dynamical jammed-toflowing transition in gravity driven granular media. Phys. Rev. Lett. 89, 184301 (2002)

    Article  ADS  Google Scholar 

  13. Schmidt, S.: On numerical simulation of granular flow. Ph.D. thesis, Technische Universität Kaiserslautern http://kluedo.ub.uni-kl.de/volltexte/2009/2364/ (2009)

  14. Landau L., Lifshitz E.M.: Fluid Mechanics, Course of Theoretical Physics, vol. 6. Pergamon Press, Oxford (1987)

    Google Scholar 

  15. Garzo V., Dufty J.W.: Dense fluid transport for inelastic hard spheres. Phys. Rev. E 59, 5895–5911 (1999)

    Article  ADS  Google Scholar 

  16. Luding S.: Global equation of state of two-dimensional hard sphere systems. Phys. Rev. E 63(4), 042201 (2001)

    Article  ADS  Google Scholar 

  17. Garcia-Rojo R., Luding S., Brey J.J.: Transport coefficients for dense hard-disk systems. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 74(6), 061305 (2006)

    Google Scholar 

  18. Khain E.: Hydrodynamics of fluid-solid coexistence in dense shear granular flow. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 75(5), 051310 (2007)

    ADS  Google Scholar 

  19. Luding S.: Towards dense, realistic granular media in 2d. Nonlinearity 22(12), R101–R146 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  20. Fingerle A., Herminghaus S.: Equation of state of wet granular matter. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 77(1), 011306 (2008)

    ADS  Google Scholar 

  21. Dahler J.S.: Transport phenomena in a fluid composed of diatomic molecules. J. Chem. Phys. 30, 1447–1475 (1959)

    Article  ADS  Google Scholar 

  22. Campbell C.S.: Boundary interaction for two dimensional granular flows. part1. flat boundaries, asymmetric stresses and couple stress. J. Fluid Mech. 247, 111–136 (1993)

    Article  ADS  Google Scholar 

  23. Mitarai N., Hayakawa H., Nakanishi H.: Collisional granular flow as a micropolar fluid. Phys. Rev. Lett. 88(17), 174301 (2002)

    Article  ADS  Google Scholar 

  24. Landau L., Lifshitz E.M.: Fluid Mechanics, Course of Theoretical Physics, vol. 7. Pergamon Press, Oxford (1986)

    Google Scholar 

  25. Hansen J.P., McDonald I.R.: Theory of Simple Liquids. Academic Press, New York (1986)

    Google Scholar 

  26. Khain E., Meerson B.: Shear-induced crystallization of a dense rapid granular flow: hydrodynamics beyond the melting point. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 73(6), 061301 (2006)

    Google Scholar 

  27. To be more precise for constant volume the specific heat should be replaced by the specific heat at constant volume

  28. McNamara S., Young W.R.: Inelastic collapse in two dimensions. Phys. Rev. E 50(1), R28–R31 (1994)

    Article  ADS  Google Scholar 

  29. Khain E., Meerson B.: Symmetry-breaking instability in a prototypical driven granular gas. Phys. Rev. E 66(2), 021306 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  30. D’Anna G., Mayor P., Barrat A., Loretto V., Nori F.: Observing brownian motion in vibration-fluidized granular matter. Nature 424, 909–912 (2003)

    Article  ADS  Google Scholar 

  31. Jiang Y., Liu M.: Granular solid hydrodynamics. Granul. Matter 11(3), 139–156 (2009)

    Article  Google Scholar 

  32. Churbanov A.: A Unified Algorithm to Predict Compressible and Incompressible Flows and Incompressible Flows. Lecture Notes in Computer Science, vol. 2542/2003, pp. 412–419. Springer, Berlin (2003)

    Google Scholar 

  33. Van Heul D., Vuik C., Wesseling P.: A conservative pressure-correction method for flow at all speeds. Comput. Fluids 32(8), 1113–1132 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  34. Gastaldo, L., Babik, F., Herbin, R., Latche, J.C.: An unconditionally stable pressure correction scheme for barotropic compressible navier-stokes equations. In: ECCOMAS CFD (2006)

  35. Daerr A., Douady S.: Two types of avalanche behaviour in granular media. Nature 399, 241–243 (1999)

    Article  ADS  Google Scholar 

  36. GDR-MiDi: On dense granular flow. Eur. Phys. J. E 14, 341–365 (2004)

    Google Scholar 

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Authors and Affiliations

  1. Fraunhofer ITWM, Kaiserslautern, Germany

    Arnulf Latz & Sebastian Schmidt

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  1. Arnulf Latz
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  2. Sebastian Schmidt
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Correspondence to Sebastian Schmidt.

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Latz, A., Schmidt, S. Hydrodynamic modeling of dilute and dense granular flow. Granular Matter 12, 387–397 (2010). https://doi.org/10.1007/s10035-010-0187-6

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  • DOI: https://doi.org/10.1007/s10035-010-0187-6

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