Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T07:13:59.719Z Has data issue: false hasContentIssue false
  • English
  • Français

Lattice ellipsoidal statistical BGK model for thermal non-equilibrium flows

Published online by Cambridge University Press:  08 February 2013

Jianping Meng ,
Yonghao Zhang ,
Nicolas G. Hadjiconstantinou ,
Gregg A. Radtke  and
Xiaowen Shan
Jianping Meng
Affiliation:
Department of Mechanical & Aerospace Engineering, University of Strathclyde, Glasgow G1 1XJ, UK
Yonghao Zhang*
Affiliation:
Department of Mechanical & Aerospace Engineering, University of Strathclyde, Glasgow G1 1XJ, UK
Nicolas G. Hadjiconstantinou
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Gregg A. Radtke
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Xiaowen Shan
Affiliation:
Beijing Aeronautical Science & Technology Research Institute, Commercial Aircraft Corporation of China, Ltd, Beijing, 102211, China
*
†Email address for correspondence: yonghao.zhang@strath.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

A thermal lattice Boltzmann model is constructed on the basis of the ellipsoidal statistical Bhatnagar–Gross–Krook (ES-BGK) collision operator via the Hermite moment representation. The resulting lattice ES-BGK model uses a single distribution function and features an adjustable Prandtl number. Numerical simulations show that using a moderate discrete velocity set, this model can accurately recover steady and transient solutions of the ES-BGK equation in the slip-flow and early transition regimes in the small-Mach-number limit that is typical of microscale problems of practical interest. In the transition regime in particular, comparisons with numerical solutions of the ES-BGK model, direct and low-variance deviational Monte Carlo simulations show good accuracy for values of the Knudsen number up to approximately $0. 5$. On the other hand, highly non-equilibrium phenomena characterized by high Mach numbers, such as viscous heating and force-driven Poiseuille flow for large values of the driving force, are more difficult to capture quantitatively in the transition regime using discretizations chosen with computational efficiency in mind such as the one used here, although improved accuracy is observed as the number of discrete velocities is increased.

JFM classification

Mathematical Foundations: Computational methods Rarefied Gas Flow: Kinetic theory Micro-/Nano-fluid dynamics: Micro-/Nano-fluid dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 718 , 10 March 2013 , pp. 347 - 370
Copyright
©2013 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andries, P., Bourgat, J.-F., le Tallec, P. & Perthame, B. 2002 Numerical comparison between the Boltzmann and ES-BGK models for rarefied gases. Comput. Meth. Appl. Mech. Engng 191 (31), 3369–3390.Google Scholar
Andries, P., Tallec, P. Le., Perlat, J.-P. & Perthame, B. 2000 The Gaussian-BGK model of Boltzmann equation with small Prandtl numbe. Eur. J. Mech. (B/Fluids) 19 (6), 813–830.Google Scholar
Ansumali, S. & Karlin, I. V. 2002 Kinetic boundary conditions in the lattice Boltzmann method. Phys. Rev. E 66 (2), 26311.Google Scholar
Ansumali, S., Karlin, I. V., Arcidiacono, S., Abbas, A. & Prasianakis, N. I. 2007 Hydrodynamics beyond Navier–Stokes: exact solution to the lattice Boltzmann hierarchy. Phys. Rev. Lett. 98 (12), 124502.CrossRefGoogle Scholar
Aoki, K., Takata, S. & Nakanishi, T. 2002 Poiseuille-type flow of a rarefied gas between two parallel plates driven by a uniform external force. Phys. Rev. E 65 (2), 26315.Google Scholar
Cercignani, C. 2000 Rarefied Gas Dynamics: From Basic Concepts to Actual Calculations. Cambridge University Press.Google Scholar
Cercignani, C. 2006 Slow Rarefied Flows: Theory and Application to Micro-Electro-Mechanical Systems. Birkhauser.CrossRefGoogle Scholar
Chen, S. Y. & Doolen, G. D. 1998 Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30, 329–364.CrossRefGoogle Scholar
Chikatamarla, S. S. & Karlin, I. V. 2006 Entropy and Galilean invariance of lattice Boltzmann theories. Phys. Rev. Lett. 97 (19), 190601.Google Scholar
Chikatamarla, S. S. & Karlin, I. V. 2009 Lattices for the lattice Boltzmann method. Phys. Rev. E 79 (4), 46701.Google Scholar
Clerk Maxwell, J. 1879 On stresses in rarified gases arising from inequalities of temperature. Phil. Trans. R. Soc. Lond. 170, 231–256.Google Scholar
Gallis, M. A. & Torczynski, J. R. 2011 Investigation of the ellipsoidal-statistical Bhatnagar–Gross–Brook kinetic model applied to gas-phase transport of heat and tangential momentum between parallel walls. Phys. Fluids 23 (3), 030601.Google Scholar
Garzó, V. & Santos, A. 1995 Comparison between the Boltzmann and BGK equations for uniform shear flow. Physica A 213, 426–434.CrossRefGoogle Scholar
Gonnella, G., Lamura, A. & Sofonea, V. 2007 Lattice Boltzmann simulation of thermal nonideal fluids. Phys. Rev. E 76 (3), 36703.Google Scholar
Grad, H. 1949 On the kinetic theory of rarefied gases. Commun. Pure Appl. Maths 2, 331–407.Google Scholar
Grad, H. 1958 Principles of the Kinetic Theory of Gases, Handbuch der Physik XII: Thermodynamik der Gase Edn. Springer.CrossRefGoogle Scholar
Graur, I. & Polikarpov, A. 2009 Comparison of different kinetic models for the heat transfer problem. Heat Mass Transfer 46 (2), 237–244.Google Scholar
Gu, X. J. & Emerson, D. R. 2007 A computational strategy for the regularized 13 moment equations with enhanced wall-boundary conditions. J. Comput. Phys. 225 (1), 263–283.Google Scholar
Gu, X.-J. & Emerson, D. R 2009 A high-order moment approach for capturing non-equilibrium phenomena in the transition regime. J. Fluid Mech. 636, 177–216.Google Scholar
Guo, Z., Zhao, T. S. & Shi, Y. 2006 Physical symmetry, spatial accuracy, and relaxation time of the lattice Boltzmann equation for microgas flows. J. Appl. Phys. 99 (7), 74903.CrossRefGoogle Scholar
Guo, Z., Zheng, C., Shi, B. & Zhao, T. S. 2007 Thermal lattice Boltzmann equation for low Mach number flows: decoupling model. Phys. Rev. E 75 (3), 36704.Google Scholar
Hadjiconstantinou, N. G. 2006 The limits of Navier–Stokes theory and kinetic extensions for describing small-scale gaseous hydrodynamics. Phys. Fluids 18 (11), 111301.Google Scholar
Han, Y.-L., Muntz, E. P. & Alexeenko, A. 2011 Nanoscale and microscale thermophysical engineering experimental and computational studies of temperature gradient–driven molecular transport in gas flows through nano/microscale Channels. Nanoscale Microscale Thermophys. Engng 37–41.Google Scholar
Han, Y.-L., Muntz, E. P., Alexeenko, A. & Young, M. 2007 Experimental and computational studies of temperature gradient-driven molecular transport in gas flows through nano/microscale channels. Nanoscale Microscale Thermophys. Engng 11 (1–2), 151–175.Google Scholar
He, X., Chen, S. & Doolen, G. D 1998 A novel thermal model for the lattice Boltzmann method in incompressible limit. J. Comput. Phys. 146 (1), 282–300.Google Scholar
Holway, L. H. 1966 New statistical models for kinetic theory: methods of construction. Phys. Fluids 9 (9), 1658–1673.Google Scholar
Homolle, T. M. M. & Hadjiconstantinou, N. G. 2007 A low-variance deviational simulation Monte Carlo for the Boltzmann equation. J. Comput. Phys. 226 (2), 2341–2358.Google Scholar
Kim, S. H., Pitsch, H. & Boyd, I. D. 2008a Accuracy of higher-order lattice Boltzmann methods for microscale flows with finite Knudsen numbers. J. Comput. Phys. 227 (19), 8655–8671.Google Scholar
Kim, S. H., Pitsch, H. & Boyd, I. D. 2008b Slip velocity and Knudsen layer in the lattice Boltzmann method for microscale flows. Phys. Rev. E 77 (4), 26704.Google Scholar
Lallemand, P. & Luo, L.-S. 2003 Theory of the lattice Boltzmann method: acoustic and thermal properties in two and three dimensions. Phys. Rev. E 68 (3), 36706.Google Scholar
Manela, A. & Hadjiconstantinou, N. G. 2007 On the motion induced in a gas confined in a small-scale gap due to instantaneous heating. J. Fluid Mech. 593, 453–462.Google Scholar
Manela, A. & Hadjiconstantinou, N. G. 2008 Gas motion induced by unsteady boundary heating in a small-scale slab. Phys. Fluids 20 (11), 117104.Google Scholar
Manela, A. & Hadjiconstantinou, N. G. 2010 Gas-flow animation by unsteady heating in a microchannel. Phys. Fluids 22 (6), 62001.CrossRefGoogle Scholar
Mansour, M. M., Baras, F. & Garcia, A. L. 1997 On the validity of hydrodynamics in plane Poiseuille flows. Physica A 240 (1–2), 255–267.Google Scholar
Meng, J. & Zhang, Y. 2011a Accuracy analysis of high-order lattice Boltzmann models for rarefied gas flows. J. Comput. Phys. 230 (3), 835–849.Google Scholar
Meng, J. & Zhang, Y. 2011b Gauss–Hermite quadratures and accuracy of lattice Boltzmann models for nonequilibrium gas flows. Phys. Rev. E 83 (3).Google Scholar
Mieussens, L. & Struchtrup, H. 2004 Numerical comparison of Bhatnagar–Gross–Krook models with proper Prandtl number. Phys. Fluids 16 (8), 2797–2813.Google Scholar
Prasianakis, N. I. & Karlin, I. V. 2007 Lattice Boltzmann method for thermal flow simulation on standard lattices. Phys. Rev. E 76 (1), 16702.Google Scholar
Prasianakis, N. I. & Karlin, I. V. 2008 Lattice Boltzmann method for simulation of compressible flows on standard lattices. Phys. Rev. E 78 (1), 16704.Google Scholar
Prasianakis, N. I., Karlin, I. V., Mantzaras, J. & Boulouchos, K. B. 2009 Lattice Boltzmann method with restored Galilean invariance. Phys. Rev. E 79 (6), 66702.CrossRefGoogle ScholarPubMed
Radtke, G. A. & Hadjiconstantinou, N. G. 2009 Variance-reduced particle simulation of the Boltzmann transport equation in the relaxation-time approximation. Phys. Rev. E 79 (5), 56711.Google Scholar
Radtke, G. A., Hadjiconstantinou, N. G. & Wagner, W. 2011 Low-noise Monte Carlo simulation of the variable hard-sphere gas. Phys. Fluids 23, 030606.Google Scholar
Sbragaglia, M., Benzi, R., Biferale, L., Chen, H., Shan, X. & Succi, S. 2009 Lattice Boltzmann method with self-consistent thermo-hydrodynamic equilibria. J. Fluid Mech. 628, 299–309.Google Scholar
Sbragaglia, M. & Succi, S. 2005 Analytical calculation of slip flow in lattice Boltzmann models with kinetic boundary conditions. Phys. Fluids 17 (9), 93602.Google Scholar
Sbragaglia, M. & Succi, S. 2006 A note on the lattice Boltzmann method beyond the Chapman–Enskog limits. Europhys. Lett. 73 (3), 370–376.Google Scholar
Scagliarini, A., Biferale, L., Sbragaglia, M., Sugiyama, K. & Toschi, F. 2010 Lattice Boltzmann methods for thermal flows: continuum limit and applications to compressible Rayleigh–Taylor systems. Phys. Fluids 22 (5), 55101.Google Scholar
Shan, X. 1997 Simulation of Rayleigh–Bénard convection using a lattice Boltzmann method. Phys. Rev. E 55 (3), 2780–2788.CrossRefGoogle Scholar
Shan, X. 2010 General solution of lattices for Cartesian lattice Bhatanagar–Gross–Krook models. Phys. Rev. E 81 (3), 36702.Google Scholar
Shan, X. & Chen, H. 2007 A general multiple-relaxation Boltzmann collision model. Intl J. Mod. Phys. C 18, 635–643.CrossRefGoogle Scholar
Shan, X. W. & He, X. Y. 1998 Discretization of the velocity space in the solution of the Boltzmann equation. Phys. Rev. Lett. 80 (1), 65–68.CrossRefGoogle Scholar
Shan, X. W., Yuan, X. F. & Chen, H. D. 2006 Kinetic theory representation of hydrodynamics: a way beyond the Navier Stokes equation. J. Fluid Mech. 550, 413–441.Google Scholar
Shim, J. & Gatignol, R. 2011 Thermal lattice Boltzmann method based on a theoretically simple derivation of the Taylor expansion. Phys. Rev. E 83 (4), 2–6.Google Scholar
Sofonea, V. 2009 Implementation of diffuse reflection boundary conditions in a thermal lattice Boltzmann model with flux limiters. J. Comput. Phys. 228 (17), 6107–6118.Google Scholar
Struchtrup, H. 2004 Stable transport equations for rarefied gases at high orders in the knudsen number. Phys. Fluids 16 (11), 3921–3934.CrossRefGoogle Scholar
Struchtrup, H. 2005a Derivation of 13 moment equations for rarefied gas flow to second-order accuracy for arbitrary interaction potentials. Multiscale Model. Simul. 3 (1), 221–243.Google Scholar
Struchtrup, H. 2005b Macroscopic Transport Equations for Rarefied Gas Flows. Springer.Google Scholar
Struchtrup, H. & Torrilhon, M. 2003 Regularization of Grad’s 13 moment equations: derivation and linear analysis. Phys. Fluids 15 (9), 2668–2680.Google Scholar
Succi, S. 2002 Mesoscopic modeling of slip motion at fluid-solid interfaces with heterogeneous catalysis. Phys. Rev. Lett. 89 (6).CrossRefGoogle ScholarPubMed
Taheri, P. & Struchtrup, H. 2010 An extended macroscopic transport model for rarefied gas flows in long capillaries with circular cross section. Phys. Fluids 22 (11), 112004.Google Scholar
Taheri, P., Torrilhon, M. & Struchtrup, H. 2009 Couette and poiseuille microflows: analytical solutions for regularized 13-moment equations. Phys. Fluids 21 (1), 017102.CrossRefGoogle Scholar
Tang, G. H., Gu, X. J., Barber, R. W., Emerson, D. R. & Zhang, Y. H. 2008a Lattice Boltzmann model for thermal transpiration. Phys. Rev. E 78 (2), 27701.Google Scholar
Tang, G., Zhang, Y. & Emerson, D. R. 2008b Lattice Boltzmann models for nonequilibrium gas flows. Phys. Rev. E 77 (4), 46701.Google Scholar
Tian, Z.-W., Zou, C., Liu, H.-J., Guo, Z.-L., Liu, Z.-H. & Zheng, C.-G. 2007 Lattice Boltzmann scheme for simulating thermal micro-flow. Physica A 385 (1), 59–68.Google Scholar
Torrilhon, M. & Struchtrup, H. 2008 Boundary conditions for regularized 13-moment-equations for micro-channel-flows. J. Comput. Phys. 227 (3), 1982–2011.Google Scholar
Toro, E. F. 2009 Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer.Google Scholar
Toschi, F. & Succi, S. 2005 Lattice Boltzmann method at finite Knudsen numbers. Europhys. Lett. 69 (4), 549–555.Google Scholar
Wagner, W. 2008 Deviational particle Monte Carlo for the Boltzmann equation. Monte Carlo Meth. Applic. 14 (3), 191–268.Google Scholar
Watari, M. 2009 Velocity slip and temperature jump simulations by the three-dimensional thermal finite-difference lattice Boltzmann method. Phys. Rev. E 79 (6), 66706.CrossRefGoogle ScholarPubMed
Watari, M. & Tsutahara, M. 2003 Two-dimensional thermal model of the finite-difference lattice Boltzmann method with high spatial isotropy. Phys. Rev. E 67 (3), 36306.Google Scholar
Yudistiawan, W. P., Ansumali, S. & Karlin, I. V. 2008 Hydrodynamics beyond Navier–Stokes: the slip flow model. Phys. Rev. E 78 (1), 16705.Google Scholar
Zhang, Y., Gu, X., Barber, R. W. & Emerson, D. R. 2006 Capturing Knudsen layer phenomena using a lattice Boltzmann model. Phys. Rev. E 74 (4), 46704.Google Scholar
Zhang, Y., Qin, R. & Emerson, D. R. 2005 Lattice Boltzmann simulation of rarefied gas flows in microchannels. Phys. Rev. E 71 (4), 47702.Google Scholar
Zheng, L., Guo, Z., Shi, B. & Zheng, C. 2010 Kinetic theory based lattice Boltzmann equation with viscous dissipation and pressure work for axisymmetric thermal flows. J. Comput. Phys. 229 (16), 5843–5856.Google Scholar