On the weak turbulent motions of an isothermal dry granular dense flow with incompressible grains: part I. Equilibrium turbulent closure models
Tóm tắt
The weak turbulent motions of a dry granular dense flow and the influence of the turbulent fluctuations caused by the minor short-term elastic/inelastic instantaneous collisions and the major long-term enduring frictional contacts among the grains on the mean flow characteristics are investigated. To this end, the conventional Reynolds-averaging process is applied to obtain the balance equations for the mean primitive fields associated with turbulent closure models. The thermodynamic analysis, based on the Mueller–Liu entropy principle, is carried out to derive the equilibrium formulations of the closure models. It shows that the effect of the turbulent fluctuations on the mean flow characteristics as well as the turbulent kinetic energy and dissipation can be taken into account by the granular coldness: a phenomenological measure of the fluctuating kinetic energy intensity. The implementation of the complete thermodynamically consistent turbulent closure models and the simulation of a gravity-driven stationary flow down an inclined moving plane compared with the experimental outcomes are provided in Part II of the present study.
Tài liệu tham khảo
Ahmadi G (1985) A turbulence model for rapid flows of granular materials. Part I. Basic theory. Powder Technol 44:261–268
Ahmadi G, Shahinpoor M (1983) Towards a turbulent modeling of rapid flow of granular materials. Powder Technol 35:241–248
Andò E, Hall SA, Viggiani G, Desrues J, Bèsuelle P (2012) Grain-scale experimental investigation of localised deformation in sand: a discrete particle tracking approach. Acta Geotech 7:1–13
Aranson IS, Tsimring LS (2009) Granular patterns. Oxford University Press, Oxford
Aste T, Matteo TD, Tordesillas A (eds) (2007) Granular and complex materials. World Scientific, Singapore
Batchelor GK (1993) The theory of homogeneous turbulence. Cambridge University Press, Cambridge
Bradshaw P (1971) An introduction to turbulence and its measurement. Pergamon Press, Oxford
Campell CS (1990) Rapid granular flows. Annu Rev Fluid Mech 22:57–92
Campbell CS (2005) Stress-controlled elastic granular shear flows. J Fluid Mech 539:273–297
Coussot P (1997) Mudflow rheology and dynamics. AA Balkema, Rotterdam
da Cruz F, Emam S, Prochnow M, Roux J-N, Chevoir F (2005) Rheophysics of dense granular materials: discrete simulation of plane shear flows. Phys Rev E 72:021309
Daniel RC, Poloski AP, Sáez AE (2007) A continuum constitutive model for cohesionless granular flows. Chem Eng Sci 62:1343–1350
Ding J, Gidaspow D (1990) A bubbling fluidization model using kinetic theory of granular flow. AIChE J 36(4):523–538
du Vachat R (1977) Realizability inequalities in turbulent flows. Phys Fluids 20:551–556
Duran J (2000) Sands, powders and grains. Springer, Wien
Eringen AC, Kadafar CB (1976) Polar field theories. In: Eringen AC (ed) Continuum physics IV. Academic Press, New York
Faccanoni G, Mangeney A (2013) Exact solution for granular flows. Int J Numer Anal Methods Geomech 37:1408–1433
Fang C (2009) Gravity-driven dry granular slow flows down an inclined moving plane: a comparative study between two concepts of the evolution of porosity. Rheol Acta 48:971–992
Fang C (2010) Rheological characteristics of solid–fluid transition in dry granular dense flows: a thermodynamically consistent constitutive model with a pressure-ratio order parameter. Int J Numer Anal Methods Geomech 34:881–905
Fang C, Wang Y, Hutter K (2006) A thermo-mechanical continuum theory with internal length for cohesionless granular materials. Part I. A class of constitutive models. Continuum Mech Thermodyn 17(8):545–576
Fang C, Wang Y, Hutter K (2006) Shearing flows of a dry granular material—hypoplastic constitutive theory and numerical simulations. Int J Numer Anal Methods Geomech 30:1409–1437
Fang C, Wu W (2014) On the weak turbulent motions of an isothermal dry granular dense flow with incompressible grains: part II. Complete closure models and numerical simulations. Acta Geotech. doi:10.1007/s11440-014-0314-3
MiDi GDR (2004) On dense granular flows. Eur Phys J E 14:341–365
Goencue F, Luding S (2013) Effect of particle friction and polydispersity on the macroscopic stress–strain relations of granular materials. Acta Geotech 8(6):629–643
Goldhirsch I (2008) Introduction to granular temperature. Powder Technol 182:130–136
Goodman MA, Cowin SC (1971) Two problems in the gravity flow of granular materials. J Fluid Mech 45:321–339
Goodman MA, Cowin SC (1972) A continuum theory for granular materials. Arch Ration Mech Anal 44:249–266
Hinze JO (1975) Turbulence. McGraw-Hill, New York
Hutter K, Joehnk K (2004) Continuum methods of physical modeling. Springer, Berlin
Hutter K, Laloui L, Vulliet L (1999) Thermodynamically based mixture models of saturated and unsaturated soils. Mech Cohes Frict Mater 4:295–338
Hutter K, Rajagopal KR (1994) On flows of granular materials. Continuum Mech Thermodyn 6:81–139
Jaeger HM, Nagel SR (1992) Physics of the granular state. Science 255:1523–1531
Jakob M, Hungr O (2005) Debris-flow hazards and related phenomena. Springer, Berlin
Jenkins JT, Savage SB (1983) A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. J Fluid Mech 130:187–202
Johnson PC, Jackson R (1987) Frictional–collisional constitutive relations for granular materials, with applications to plane shearing. J Fluid Mech 176:67–93
Jop P (2008) Hydrodynamic modeling of granular flows in a modified Couette cell. Phys Rev E 77:032301
Jop P, Forterre Y, Pouliquen O (2006) A constitutive law for dense granular flows. Nature 411:727–730
Kirchner N (2002) Thermodynamically consistent modeling of abrasive granular materials. I: Non-equilibrium theory. Proc R Soc Lond A 458:2153–2176
Ko HY, Scott RF (1967) Deformation of sand in hydrostatic compression. J Soil Mech Found Div Proc ASCE 93:137–156
Liu I (1972) Method of lagrange multipliers for exploitation of the entropy principle. Arch Ration Mech Anal 46:131–148
Luca I, Fang C, Hutter K (2004) A thermodynamic model of turbulent motions in a granular material. Continuum Mech Thermodyn 16:363–390
Lumley JL, Tennekes H (1972) A first course in turbulence. MIT Press, Cambridge
Lun KK, Savage SB, Jeffery DJ, Chepurny N (1984) 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
Ma D, Ahmadi G (1985) A turbulence model for rapid flows of granular materials. Part II simple shear flows. Powder Technol 44:269–279
Mehta A (2007) Granular physics. Cambridge University Press, New York
Mindlin RD (1964) Microstructure in linear elasticity. Arch Ration Mech Anal 16:51–78
Muehlhaus HB, Shi J, Olsen-Kettle L, Moresi L (2011) A double slip non-coaxial flow rule for viscous-plastic Cosserat materials. Acta Geotech 6(4):219–229
Mueller I (1985) Thermodynamics. Pitmann, New York
Ogawa S (1978) Multitemperature theory of granular materials. In: Cowin SC, Satake M (eds) Proceedings of continuum mechanical and statistical approaches in the mechanics of granular materials, US-Japan Seminar, Sendai, Japan
Pudasaini S, Hutter K (2007) Avalanche dynamics. Springer, Berlin
Rao KK, Nott PR (2008) Introduction to granular flows. Cambridge University Press, London
Richman MW (1988) Boundary conditions based upon a modified Maxwellian velocity distribution for flows if identical, smooth, nearly elastic spheres. Acta Mech 75:227–240
Richman MW, Marciniec RP (1990) Gravity-driven granular flows of smooth, inelastic spheres down bumpy inclines. J Appl Mech 57:1036–1043
Rung T, Thiele F, Fu S (1999) On the realizability of non-linear stress–strain relationships for Reynolds’ stress closures. Flow Turbul Combust 60:333–359
Sadiki A, Hutter K (1996) 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 modeling. Continuum Mech Thermodyn 8:341–349
Sadiki A, Hutter K (2000) On thermodynamics of turbulence. J Non Equilib Thermodyn 25:131–160
Savage SB, Jeffery DJ (1981) The stress tensor in a granular flow at high shear rates. J Fluid Mech 110:255–272
Shih TH, Zhu J, Lumley JL (1993) A realizable Reynolds’ stress algebraic equation model. NASA TM-105993
Svendsen B, Hutter K (1999) On the thermodynamics of a mixture of isotropic materials with constraints. Int J Eng Sci 33(14):2021–2054
Takahashi T (2007) Debris flow: mechanics, prediction and countermeasures. Taylor & Francis, London
Truesdell C, Muncaster RG (1980) Fundamentals of Maxwell’s kinetic theory of a simple monatomic gas: treated as a branch of rational mechanics. Academic Press, New York
Tsinober A (2009) An informal conceptual introduction to turbulence. Springer, Heidelberg
Vescovi D, di Prisco C, Berzi D (2013) From solid to granular gases: the steady state for granular materials. Int J Numer Anal Methods Geomech 37:2937–2951
Volfson D, Tsimging LD, Aranson IS (2003) Partially fluidized shear granular flow: continuum theory and molecular dynamics simulations. Phys Rev E 68:021301
Wang Y, Hutter K (1999) A constitutive theory of fluid-saturated granular materials and its application in gravitational flows. Rheol Acta 38:214–223
Wang Y, Hutter K (2001) Granular material theories revisited. In: Balmforth NJ, Provenzale A (eds) Geomorphological fluid mechanics. Springer, Heidelberg, pp 79–107