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A Quadrature-Based Kinetic Model for Dilute Non-Isothermal Granular Flows

Published online by Cambridge University Press:  20 August 2015

Alberto Passalacqua ,
Janine E. Galvin ,
Prakash Vedula ,
Christine M. Hrenya  and
Rodney O. Fox
Alberto Passalacqua*
Affiliation:
Department of Chemical and Biological Engineering, Iowa State University, Ames, IA 50011-2230, USA
Janine E. Galvin*
Affiliation:
United States Department of Energy National Energy Technology Laboratory, Morgantown, WV 26507-0880, USA
Prakash Vedula*
Affiliation:
School of Aerospace and Mechanical Engineering, University of Oklahoma, Norman, OK 73019-0601, USA
Christine M. Hrenya*
Affiliation:
Department of Chemical and Biological Engineering, University of Colorado, Boulder, CO 80309-0424, USA
Rodney O. Fox*
Affiliation:
Department of Chemical and Biological Engineering, Iowa State University, Ames, IA 50011-2230, USA
*
Corresponding author.Email:albertop@iastate.edu
Email:janine.galvin@netl.doe.gov
Email:pvedula@ou.edu
Email:hrenya@colorado.edu
Email:rofox@iastate.edu
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Abstract

A moment method with closures based on Gaussian quadrature formulas is proposed to solve the Boltzmann kinetic equation with a hard-sphere collision kernel for mono-dispersed particles. Different orders of accuracy in terms of the moments of the velocity distribution function are considered, accounting for moments up to seventh order. Quadrature-based closures for four different models for inelastic collision-the Bhatnagar-Gross-Krook, ES-BGK, the Maxwell model for hard-sphere collisions, and the full Boltzmann hard-sphere collision integral-are derived and compared. The approach is validated studying a dilute non-isothermal granular flow of inelastic particles between two stationary Maxwellian walls. Results obtained from the kinetic models are compared with the predictions of molecular dynamics (MD) simulations of a nearly equivalent system with finite-size particles. The influence of the number of quadrature nodes used to approximate the velocity distribution function on the accuracy of the predictions is assessed. Results for constitutive quantities such as the stress tensor and the heat flux are provided, and show the capability of the quadrature-based approach to predict them in agreement with the MD simulations under dilute conditions.

Keywords

76P0576T2578M0582B4082C80Kinetic equationquadrature-based moment methodgranular gascollision integralBGK-type model
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 1 , July 2011 , pp. 216 - 252
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Adler, B. J., and Wainwright, T. E., Studies in molecular dynamics, II, behavior of small number of elastic spheres, J. Chem. Phys., 33 (1960), 2363–2382.Google Scholar
[2]Beylich, A. E., Solving the kinetic equation for all Knudsen numbers, Phys. Fluids., 12 (2000), 444–465.Google Scholar
[3]Bhatnagar, P. L., Gross, E. P., and Krook, M., A model for collisional processes in gases, I, small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511–525.CrossRefGoogle Scholar
[4]Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford University Press, 1994.Google Scholar
[5]Bobylev, A. V., Carrillo, J. A., and Gamba, I. M., On some properties of kinetic and hydrody-namic equations for inelastic interactions, J. Stat. Phys., 98(3/4) (2000), 743–773.Google Scholar
[6]Brey, J. J., Dufty, J. W., Kim, C.-S., and Santos, A., Hydrodynamics for granular flow at low density, Phys. Rev. E., 58 (1998), 4638–4653.Google Scholar
[7]Brey, J. J., Moreno, F., and Dufty, J. W., Reply to “Comment on Model kinetic equation for low-density granular flow”, Phys. Rev. E., 57(5) (1998), 6212–6213.Google Scholar
[8]Brey, J. J., Ruiz-Montero, M. J., and Cubero, D., Homogeneous cooling state of a low-density granular flow, Phys. Rev. E., 54 (1996), 3664–3671.CrossRefGoogle ScholarPubMed
[9]Carnahan, N. F., and Starling, K. E., Equation of state for nonattracting rigid spheres, J. Chem. Phys., 51 (1969), 635–636.Google Scholar
[10]Carrillo, J. A., Cercignani, C., and Gamba, I. M., Steady states of a Boltzmann equation for driven granular media, Phys. Rev. E., 62(6) (2000), 7700–7707.Google Scholar
[11]Carrillo, J.A., Majorana, A., and Vecil, F., A semi-Lagrangian deterministic solver for the semiconductor Boltzmann-Poisson system, Commun. Comput. Phys., 2(5) (2007), 1027–1054.Google Scholar
[12]Cercignani, C., The Boltzmann Equation and Its Applications, Springer-Verlag, 1988.Google Scholar
[13]Cercignani, C., Rarefied Gas Dynamics, Cambridge University Press, 2000.Google Scholar
[14]Chapman, S., and Cowling, T. G., The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, 2 edition, 1961.Google Scholar
[15]Desjardin, O., Fox, R. O., and Villedieu, P., A quadrature-based moment method for dilute fluid-particle flows, J. Comput. Phys., 227 (2008), 2524–2539.Google Scholar
[16]Dufty, J. W., Baskaran, A., and Zogaib, L., Gaussian kinetic model for granular gases, Phys. Rev. E., 69 (2004), 051301.Google Scholar
[17]Ferziger, J. H, and Peric, M., Computational Methods for Fluid Dynamics, Springer, 2002.Google Scholar
[18]Fox, R. O., A quadrature based third-order moment method for dilute gas-particle flows, J. Comput. Phys., 227 (2008), 6313–6350.CrossRefGoogle Scholar
[19]Fox, R. O., Higher-order quadrature-based moment methods for kinetic equations, J. Com-put. Phys., 228 (2009), 7771–7791.Google Scholar
[20]Fox, R. O., Optimal moment sets for multivariate direct quadrature method of moments, Ind. Eng. Chem. Res., 48 (2009), 9686–9696.Google Scholar
[21]Fox, R. O., Laurent, F., and Massot, M., Numerical simulation of spray coalescence in an Eule-rian framework: direct quadrature method of moments and multi-fluid method, J. Comput. Phys., 227(6) (2008), 3058–3088.Google Scholar
[22]Fox, R. O., and Vedula, P., Quadrature-based moment model for moderately dense polydisperse gas-particle flows, Ind. Eng. Chem. Res., 49(11) (2010), 5174–5187.CrossRefGoogle Scholar
[23]Galvin, J. E., Hrenya, C. M., and Wildman, R. D., On the role of the Knudsen layer in rapid granular flows, J. Fluid. Mech., 585 (2007), 73–92.CrossRefGoogle Scholar
[24]Garzo, V., and Dufty, J. W., Dense fluid transport for inelastic hard spheres, Phys. Rev. E., (59) (1999), 5895–5911.Google Scholar
[25]Goldhirsch, I., Rapid granular flows, Ann. Rev. Fluid. Mech., 35 (2003), 267–293.CrossRefGoogle Scholar
[26]Goldshtein, A., and Shapiro, M., Mechanics of collisional motion of granular materials, I. general hydrodynamic equations, J. Fluid. Mech., 282 (1995), 75–114.Google Scholar
[27]Gordon, R. G., Error bounds in equilibrium statistical mechanics, J. Math. Phys., 9 (1968), 655–662.Google Scholar
[28]Holway, L. H., New statistical models for kinetic theory, methods of construction, Phys. Fluids., 9(9) (1966), 1658–1673.Google Scholar
[29]Jenkins, J. T., Kinetic theory for nearly elastic spheres, In Hermann, H. J., Hovi, J. P., and Luding, S., editors, Physics of Dry Granular Media, Kluwer, 1998.CrossRefGoogle Scholar
[30]Jenkins, J. T., and Savage, S. B., A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles, J. Fluid. Mech., 130 (1983), 187–202.Google Scholar
[31]Marchisio, D. L., Vigil, R. D., and Fox, R. O., Quadrature method of moments for aggregation-breakage processes, Colloid, J.. Interface. Sci., 258(2) (2003), 322–334.Google Scholar
[32]Maxwell, J. C., On stresses in rarefied gases arising from inequalities of temperature, Philos. Trans. R. Soc., 170 (1879), 231–256.Google Scholar
[33]McGraw, R., Description of aerosol dynamics by the quadrature method of moments, Aero. Sci. Technol., 27 (1997), 255–265.Google Scholar
[34]Perthame, B., Boltzmann type schemes for compressible Euler equations in one and two space dimensions, SIAM J. Numer. Anal., 29(1) (1990), 1–19.Google Scholar
[35]Struchtrup, H., Macroscopic Transport Equations for Rarefied Gas Flows, Springer, 2005.Google Scholar
[36]Vikas, V., Wang, Z. J., Passalacqua, A., and Fox, R. O., Development of high-order realizable finite-volume schemes for quadrature-based moment method, 4 January 2010.Google Scholar
[37]Wheeler, J. C., Modified moments and Gaussian quadratures, Rocky. Mount. J. Math., 4 (1974), 287–296.Google Scholar
[38]Wright, D., McGraw, R., and Rosner, D. E., Bivariate extension of the quadrature method of moments for modeling simultaneous coagulation and sintering particle populations, J. Colloid. Interface. Sci., 236(2) (2001), 242–251.Google Scholar
[39]Yuan, C., and Fox, R. O., Conditional quadrature method of moments for kinetic equations, J. Comput>. Phys., submitted, 2010.Google Scholar