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Modeling of Turbulence in Rapid Granular Flows

  • Kolumban HutterEmail author
  • Yongqi Wang
Chapter
Part of the Advances in Geophysical and Environmental Mechanics and Mathematics book series (AGEM)

Abstract

This chapter is devoted to a phenomenological theory of granular materials subjected to slow frictional as well as rapid flows with intense collisional interactions. The microstructure of the material is taken into account by considering the solid volume fraction as a basic field. This variable enters the formulation via the balance law of configurational momentum, including corresponding contributions to the energy balance, as originally proposed by Goodman and Cowin (Arch Rational Mech Anal 44:249–266, 1972, [25]), but modified here by adequately introducing an internal length. The subgrid motion is interpreted as volume fraction variation in relatively moderate laminar variation and rapid fluctuations, which manifest themselves in correspondingly filtered equations in terms of correlation products as in turbulence theories. We apply an ergodic (Reynolds ) filter to these equations as in classical turbulent RANS-modeling and deduce averaged balances of mass, linear and configurational momenta, energy, turbulent and configurational kinetic energy. Moreover, we postulate balance laws for the dissipation rates of the turbulent kinetic energy. All these comprise 10 evolution equations for a larger number of field variables. Closure relations are formulated for the laminar constitutive quantities and the correlation terms, all postulated to obey the material objectivity rules. To apply the entropy principle, three coldness measures are introduced for capturing material, configurational and turbulent dissipative quantities, they simplify the analysis of müller’s entropy principle. The thermodynamic analysis delivers equilibrium properties of the constitutive quantities and linear expressions for the non-equilibrium closure relations.

Keywords

Granular materials Extended GoodmanCowin type microstructure Turbulent motions Reynolds filtering Laminar and turbulent constitutive modeling Entropy principle for laminar and turbulent motions Material turbulent and granular temperature 

References

  1. 1.
    Ahmadi, G.: Thermodynamics of multi-temperature fluids with applications to turbulence. Appl. Math. Model. 9, 271–274 (1985)CrossRefGoogle Scholar
  2. 2.
    Ahmadi, G.: Thermodynamically consistent \(k-Z\) models for compressible turbulent flows. Appl. Math. Model. 12, 391–398 (1988)Google Scholar
  3. 3.
    Ahmadi, G.: A two-equation turbulence model for compressible flows based on the second law of thermodynamics. J. Non-equilib. Thermodyn. 14, 45–49 (1989)CrossRefGoogle Scholar
  4. 4.
    Ahmadi, G.: A thermodynamically consistent rate-dependent model for turbulence. Part I: Formulation. Int. J. Non-linear Mech. 26, 595–606 (1991)CrossRefGoogle Scholar
  5. 5.
    Ahmadi, G., Abu-Zaid, S.: A thermodynamically consistent stress transport model for rotating turbulent flows. Geophys. Astrophys. Fluid Dyn. 61, 109–125 (1991)CrossRefGoogle Scholar
  6. 6.
    Aranson, I.S., Tsimring, L.S.: Granular Patterns. Oxford University Press, Oxford (2009)Google Scholar
  7. 7.
    Ausloos, M., Lambiotte, R., Trojan, K., Koza, Z., Pekala, M.: Granular matter: a wunderfu lworld of clusters in far-from equilibrium systems. Phys. A 357, 337–349 (2005)CrossRefGoogle Scholar
  8. 8.
    Bagnold, R.A.: The Physics of Blown Sand and Desert Dunes, p. 265. Dover Earth Science, USA (2005). ISBN: 9780486439310Google Scholar
  9. 9.
    Campbell, C.S.: Computer simulation of rapid granular flows. In: Proceedings of the 10th National Congress on Applied Mechanics, Austin, Texas. ASME, New York (1986)Google Scholar
  10. 10.
    Campbell, C.S.: Rapid granular flows. Ann. Rev. Fluid Mech. 22, 57–92 (1990)CrossRefGoogle Scholar
  11. 11.
    Chowdhury, S.J., Ahmadi, G.: A thermodynamically consistent rate-dependent model for turbulence. Part I: Theory II: Computational results. Int. J. Non-lin. Mech. 27, 705–718 (1992)Google Scholar
  12. 12.
    du Vachat, R.: Realizability inequalities in turbulent flows. Phys. Fluids 20, 551 (1977)CrossRefGoogle Scholar
  13. 13.
    Egolf, P., Hutter, K.: The solution of elementary turbulent shear flows. Forthcoming (2017)Google Scholar
  14. 14.
    Fang, C., Wang, Y., Hutter, K.: A thermo-mechanical continuum theory with internal length for cohesionless granular materials. Part I: a class of constitutive models. Conitn. Mech. Thermodyn. 17(8), 545–576 (2006)Google Scholar
  15. 15.
    Fang, C., Wang, Y., Hutter, K.: A thermo-mechanical continuum theory with internal length for cohesionless granular materials. Part II: non-equilibrium postulates and numerical simulations of simple shear, plane Poiseuille and gravity driven problems. Conitn. Mech. Thermodyn. 17(8), 577–613 (2006)Google Scholar
  16. 16.
    Fang, C.: A \(k-\varepsilon \) turbulence closure model of an isothermal dry granular dense matter. Contin. Mech. Thermodyn. 28, 1049–1069 (2016)Google Scholar
  17. 17.
    Farkas, Gy.: A Fourier-féle mechanikai elv alkalmazái. Mathematikai és Természettundományi Értesitö, 12, 457–472 (1894)Google Scholar
  18. 18.
    Favre, A.: Contribution à l’étude expérimentale des mouvements hydrodynamiques à deux dimensions. Gautier Villars, Paris (1938)Google Scholar
  19. 19.
    Favre, A.: Equations des gaz turbulents compressible. II. Methode de vitesses moyennes; Méthode des vitesses macroscopiques pondérées par la masse volumique. J. Mec. 4, 391 (1965)Google Scholar
  20. 20.
    Favre, A.: La turbulence en mécanique des fluides: Bases théoriques et expérimentales, méthodes statistiques. Gautier Villars, Paris (1976)Google Scholar
  21. 21.
    Favre, A.: Mesures statistiques de la correlation dans letemps, premires applications à l’ études de mouvements turbulentes en suffleries, p. 67. ONERA Publication (1954)Google Scholar
  22. 22.
    Favre, A.: Turbulent Fluxes through the Sea Surface, Wave Dynamics. Plenum Press, New York (1978)Google Scholar
  23. 23.
    Favre, A.: Equations des gaz turbulents compressible I. Formes generales. J. Mec. 4, 361–390 (1965)Google Scholar
  24. 24.
    Gilbert, G.T.: Positive definite matrices and Sivester’s criterion. Am. Math. Mon. 98(1), 44–46 (1991)CrossRefGoogle Scholar
  25. 25.
    Goodman, M.A., Cowin, S.C.: A continuum theory for granular materials. Arch. Rational Mech. Anal. 44, 249–266 (1972)CrossRefGoogle Scholar
  26. 26.
    Hager, W.H.: Hydraulicians in Europe 1800-2000, Volume 2. CRC Press, Boca Raton (2009)Google Scholar
  27. 27.
    Hauser, R.A., Kirchner, N.: A historical note on the entropy principle of Müller and Liu. Conitn. Mech. Thermodyn. 14(2), 223–226 (2002)CrossRefGoogle Scholar
  28. 28.
    Hutter, K., Jöhnk, K.: Continuum Methods of Physical Modeling, p. 635. Springer, Berlin (2004)Google Scholar
  29. 29.
    Hutter, K., Wang, Y.: Fluid and Thermodynamics, vol. 2, p. 633. Springer, Switzerland (2016)Google Scholar
  30. 30.
    Hutter, K., Rajagopal, K.R.: On flows of granular materials. Contin. Mech. Thermodyn. 6, 81–139 (1994)CrossRefGoogle Scholar
  31. 31.
    Hutter, K., Schneider, L.: Important aspects in the formulation of solid-fluid debris-flow models. Part I. Thermodynamic implications. Contin. Mech. Thermodyn. 22(5), 363–390 (2010)CrossRefGoogle Scholar
  32. 32.
    Hutter, K., Schneider, L.: Important aspects in the formulation of solid-fluid debris-flow models. Part II. Constitutive modelling. Contin. Mech. Thermodyn. 22(5), 391–411 (2010)CrossRefGoogle Scholar
  33. 33.
    Jenkins, J.T., Savage, S.B.: The mean stress resulting from interparticle collisions in a rapid granular shear flow. In: Brulin, B. Hsien, R.K.T. (eds.) Proceedings of the Fourth International Conference on Continuum Models of Discrete Systems. North Holland Publishing Company, Amsterdam (1981)Google Scholar
  34. 34.
    Jenkins, J.T., Savage, S.B.: A theory for the rapid flow of identical, smooth, nearly elastic spherical particles. J. Fluid Mech. 130, 187–202 (1981)CrossRefGoogle Scholar
  35. 35.
    Johnson, P.C., Jackson, R.: Frictional-collisional constitutive relations for granular materials, with application to plane shearing. J. Fluid Mech. 1(76), 67–93 (1987)CrossRefGoogle Scholar
  36. 36.
    Kirchner, N.P., Hutter, K.: Modelling particle size segregation in granular mixtures. In: Hutter, K., Kirchner, N. (eds.) Dynamic Response of Granular and Porous Materials under Large and Catastrophic Deformation. Lecture Notes in Applied and Computational Mechanics, vol. 11, pp. 367–391. Spriner, Berlin (2003)Google Scholar
  37. 37.
    Kirchner, N.P., Teufel, T.: Thermodynamically consistent modeling of abrsasive granular materials. II: thermodynamic equilibrium and applications to steady shear flows. Proc. Royal Soc. Lond. A458 3053–3077 (2002)Google Scholar
  38. 38.
    Kirchner, N.: Thermodynamically consistent modeling of abrsasive granular materials. I non-equilibrium theory. I: non-equilibrium theory. Proc. Royal Soc. Lond. A458, 2153–2176 (2002)Google Scholar
  39. 39.
    Liu, I-Shih: Method of Lagrange multipliers for exploitation of the entropy principle. Arch. Rational Mech. Anal. 46, 131–148 (1972)Google Scholar
  40. 40.
    Luca, I., Fang, C., Hutter, K.: A thermodynamic model of turbulent motions in a granular material. Contin. Mech. Thermodyn 16, 363–390 (2004)CrossRefGoogle Scholar
  41. 41.
    Lun, C.K.K., Savage, S.B., Jeffrey, D.J., Chepurny, N.: Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flow field. J. Fluid. Mech 140, 223–256 (1984)CrossRefGoogle Scholar
  42. 42.
    Marshall, J.S., Naghdi, P.M.: Thermodynamical theory of turbulence I. Basic developments. Philos. Trans. R. Soc. Lond. A327, 415–488 (1989)Google Scholar
  43. 43.
    Minkowski, H.: Geometrie der Zahlen. Teubner, Leipzig (1896)Google Scholar
  44. 44.
    Müller, I.: Thermodynamics. Pitman, New York (1985)Google Scholar
  45. 45.
    Ogawa, S.: Multitemperature theory of granular materials. Proceedings of the Continuum Mechanical and Statistical Approaches in the Mechanics of Granular Materials. In: Cowin, S.C., Satake, M. (eds) Proceedings of the Continuum Mechanical and Statistical Approaches in the Mechanics of Granular Materials, US-Japan Seminar, Sendai, Japan (1978)Google Scholar
  46. 46.
    Pöschel, T., Brillantov, N.V.: Granular Gas Dynamics. Lecture notes in Physics, vol. 624. Springer, New York (2013)Google Scholar
  47. 47.
    Rao, K.K., Nott, P.R.: Introduction to Granular Flows. Cambridge University Press, London (2008)Google Scholar
  48. 48.
    Richman, M.W.: Boundary conditions based upon a modified Maxwellian velocity distribution for flows of identical, smooth, nearly elastic spheres. Acta Mech. 75, 227–240 (1988)CrossRefGoogle Scholar
  49. 49.
    Richman, M.W., Marciniec, R.P.: Gravity-driven granular flows of smooth, inelastic spheres down bumpy inclines. J. Appl. Mech. 57, 1036–1043 (1990)CrossRefGoogle Scholar
  50. 50.
    Rung, T., Thiele, F., Fu, S.: On the realizability of non-linear stress-strain relationships for Reynolds-stress closures. Flow Turbul. Combust. 60, 333–359 (1999)CrossRefGoogle Scholar
  51. 51.
    Sadiki. A.: Turbulentzmodellierung und Thermodynamik. Habilitationsschrift. TU Darmstadt (1998)Google Scholar
  52. 52.
    Sadiki, A., Hutter, K.: On the frame dependence and form invariance of the transport equations for the Reynolds stress tensor and the turbulent heat flux vector: its consequences on closure models in turbulence modelling. Contin. Mech. Thermodyn. 8, 341–349 (1996)CrossRefGoogle Scholar
  53. 53.
    Sadiki, A., Hutter, K.: On thermodynamics of turbulence. J. Non-equilib. Thermodyn. 25, 130–160 (2000)CrossRefGoogle Scholar
  54. 54.
    Savage, S.B.: The mechanics of rapid granular flows. Adv. Appl. Mech. 24, 289–366 (1984)CrossRefGoogle Scholar
  55. 55.
    Savage, S.B., Jeffrey, D.J.: The stress tensor in a granular flow at high shear rates. J. Fluid Mech. 110, 255–272 (1981)CrossRefGoogle Scholar
  56. 56.
    Schneider, L., Hutter, K.: Solid Fluid Mixtures of Frictional Materials in Geophysical and Geotechnical Context Based on a Concise Thermodynamic Analysis. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Berlin (2009)Google Scholar
  57. 57.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete Mathematics. Wiley, Chichester (1996)Google Scholar
  58. 58.
    Schumann, V.: Realizability of Reynolds stress turbulence models. Phys. Fluids 20, 721–725 (1977)CrossRefGoogle Scholar
  59. 59.
    Shih, T.H., Zhu, J., Lumley, J.L.: A realizable Reynolds stress algebraic equation model. NASA TM-105993 (1993)Google Scholar
  60. 60.
    Shih, T.H.: Constitutive relations and realizability of single-point turbulence closures. In: Hallbäck, M., et al. (eds.) Turbulence and Transition Modeling, pp. 155–192. Grademic Holland (1996)Google Scholar
  61. 61.
    Svendsen, B., Hutter, K., Laloui, L.: Constitutive models for granular materials including quasi-static frictional behaviour: toward a thermodynamic formulation of plasticity. Contin. Mech. Thermodyn. 11, 263–275 (1999)CrossRefGoogle Scholar
  62. 62.
    Wang, Y., Hutter, K.: A constitutive theory of fluid saturated granular materials and its application in gravitational flows. Rheol. Acta 38, 214–223 (1999)CrossRefGoogle Scholar
  63. 63.
    Wang, Y., Hutter, K.: Shearing flows in a Goodman–Cowin type granular material theory and numerical results. Part. Sci. Technol. 17, 97–124 (1999)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.c/o Laboratory of Hydraulics, Hydrology, GlaciologyETH ZürichZürichSwitzerland
  2. 2.Department of Mechanical EngineeringDarmstadt University of TechnologyDarmstadtGermany

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