Abstract
Many continuum theories for granular flow produce an equation of motion for the fluctuating kinetic energy density (“granular temperature”) that accounts for the energy lost in inelastic collisions. Apart from the presence of an extra dissipative term, this equation is very similar in form to the usual temperature equation in hydrodynamics. It is shown how a lattice-kinetic model based on the Bhatnagar-Gross-Krook (BGK) equation that was previously derived for a miscible two-component fluid may be modified to model the continuum equations for granular flow. This is done by noting that the variable corresponding to the concentration of one species follows an equation that is essentially analogous to the granular temperature equation. A simulation of an unforced granular fluid using the modified model reproduces the phenomenon of “clustering instability,” namely the spontaneous agglomeration of particles into dense clusters, which occurs generically in all granular flows. The success of the continuum theory in capturing the gross features of this basic phenomenon is discussed. Some shear flow simulations are also presented.
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References
P. Bhatnagar, E. P. Gross, and M. K. Krook,Phys. Rev. 94:511 (1954).
C. S. Campbell,J. Fluid Mech. 203:449–473 (1989).
C. S. Campbell and C. E. Brennan,Trans. ASME: J. Appl. Mech. 52:172–178 (1985).
S. Chapman and T. G. Cowling,The Mathematical Theory of Nonuniform Gases, 3rd ed. (Cambridge University Press, Cambridge, 1970).
D. d'Humières, P. Lallemand, and U. Frisch,Europhys. Lett. 2:291 (1980).
G. D. Doolen, ed.,Lattice Gas Methods for Partial Differential Equations Addison-Wesley, Reading, Massachusetts, 1990).
U. Frisch, D. d'Humières, B. Hasslacher, P. Lallemand, Y. Pomeau, and J.-P. Rivet,Complex Systems 1:649 (1987).
U. Frisch, B. Hasslacher, and Y. Pomeau,Phys. Rev. Lett. 56:1505 (1986).
I. Goldhirsch, M.-L. Tan, and G. Zanetti,J. Sci. Comp. 8(1):1 (1993).
I. Goldhirsch and G. Zanetti,Phys. Rev. Lett. 70:1619–1662 (1993).
P. K. Haff,J. Fluid Mech. 134:401–430 (1983).
H. J. Herrmann,Physica A 191:263–276 (1992).
M. A. Hopkins and M. Y. Louge,Phys. Fluids A 3(1):47–57 (1991).
J. T. Jenkins and M. W. Richman,Arch. Rat. Mech. Anal 87:355–377 (1985).
J. T. Jenkins and M. W. Richman,Phys. Fluids 28:3485–3494 (1985).
J. T. Jenkins and S. B. Savage,J. Fluid Mech. 130:187–202 (1983), and references therein.
C. K. K. Lun,J. Fluid Mech. 223:539–559 (1991).
C. K. K. Lun and S. B. Savage,J. Appl. Mech. 154:47–53 (1987), and references therein.
S. McNamara,Phys. Fluids A 5:3056–3070 (1993).
S. McNamara and W. R. Young,Phys. Fluids A 4:496–504 (1992).
S. McNamara and W. R. Young,Phys. Fluids A 5:34–45 (1993).
S. Ogawa, Multitemperature theory of granular materials inProceedings of US-Japan Seminar on Continuum Mechanics and Statistical Approach to the Mechanics of Granular Matter (Gukujutsu Bunken Fukyukai, Tokyo, 1978), Vol. 31, pp. 208–217.
T. Pöschel,J. Phys. II France 3:27–40 (1993).
Y. H. Qian, Lattice Gas and lattice kinetic theory applied to Navier-Stokes equation, Ph.D. thesis, University of Paris 6 (1990).
Y. H. Qian, D. d'Humières, and P. Lallemand,Europhys. Lett. 17(6):479–484, (1992).
Y. H. Qian and S. A. Orszag,Europhys. Lett 21(3):255–259 (1993).
Y. H. Qian, M.-L. Tan, and S. A. Orszag, A lattice-BGK model for miscible fluids, inProceedings 5th International Symposium on Computational Fluid Dynamics (Sendai, Japan, 1993).
S. B. Savage,J. Fluid Mech. 241:109–123 (1992).
P. J. Schmid and H. K. Kytomaa,J. Fluid Mech. 264:255–275 (1994).
M.-L. Tan, Microstructures and macrostructures in rapid granular flows, Ph.D. thesis, Princeton University, (1995).
O. R. Walton, H. Kim, and A. D. Rosato, Microstructure and stress differences in shearing flows, inProceedings of Mechanics Computing in 1990's and Beyond, Adeli, H. and Sierakowski, eds. (ASCE, New York, 1991), Vol. 2, pp. 1249–1253.
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Tan, M.L., Qian, Y.H., Goldhirsch, I. et al. Lattice-BGK approach to simulating granular flows. J Stat Phys 81, 87–103 (1995). https://doi.org/10.1007/BF02179970
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DOI: https://doi.org/10.1007/BF02179970