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Solving the Boltzmann equation deterministically by the fast spectral method: application to gas microflows

Published online by Cambridge University Press:  28 March 2014

Lei Wu ,
Jason M. Reese  and
Yonghao Zhang
Lei Wu
Affiliation:
James Weir Fluids Laboratory, Department of Mechanical and Aerospace Engineering, University of Strathclyde, Glasgow G1 1XJ, UK
Jason M. Reese
Affiliation:
School of Engineering, University of Edinburgh, Edinburgh EH9 3JL, UK
Yonghao Zhang*
Affiliation:
James Weir Fluids Laboratory, Department of Mechanical and Aerospace Engineering, University of Strathclyde, Glasgow G1 1XJ, UK
*
†Email address for correspondence: yonghao.zhang@strath.ac.uk
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Abstract

Based on the fast spectral approximation to the Boltzmann collision operator, we present an accurate and efficient deterministic numerical method for solving the Boltzmann equation. First, the linearized Boltzmann equation is solved for Poiseuille and thermal creep flows, where the influence of different molecular models on the mass and heat flow rates is assessed, and the Onsager–Casimir relation at the microscopic level for large Knudsen numbers is demonstrated. Recent experimental measurements of mass flow rates along a rectangular tube with large aspect ratio are compared with numerical results for the linearized Boltzmann equation. Then, a number of two-dimensional microflows in the transition and free-molecular flow regimes are simulated using the nonlinear Boltzmann equation. The influence of the molecular model is discussed, as well as the applicability of the linearized Boltzmann equation. For thermally driven flows in the free-molecular regime, it is found that the magnitudes of the flow velocity are inversely proportional to the Knudsen number. The streamline patterns of thermal creep flow inside a closed rectangular channel are analysed in detail: when the Knudsen number is smaller than a critical value, the flow pattern can be predicted based on a linear superposition of the velocity profiles of linearized Poiseuille and thermal creep flows between parallel plates. For large Knudsen numbers, the flow pattern can be determined using the linearized Poiseuille and thermal creep velocity profiles at the critical Knudsen number. The critical Knudsen number is found to be related to the aspect ratio of the rectangular channel.

JFM classification

Mathematical Foundations: Computational methods Micro-/Nano-fluid dynamics: Micro-/Nano-fluid dynamics Rarefied Gas Flow: Rarefied Gas Flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 746 , 10 May 2014 , pp. 53 - 84
Copyright
© 2014 Cambridge University Press 

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References

Baker, L. L. & Hadjiconstantinou, N. G. 2005 Variance reduction for Monte Carlo solutions of the Boltzmann equation. Phys. Fluids 17 (5), 051703.Google Scholar
Bird, G. A. 1994 Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford University Press.Google Scholar
Cercignani, C. 1990 Mathematical Methods in Kinetic Theory. Plenum.CrossRefGoogle Scholar
Cercignani, C. & Daneri, A. 1963 Flow of a rarefied gas between two parallel plates. J. Appl. Phys. 34, 3509–3513.Google Scholar
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Non-uniform Gases. Cambridge University Press.Google Scholar
Doi, T. 2010 Numerical analysis of the Poiseuille flow and thermal transpiration of a rarefied gas through a pipe with a rectangular cross-section based on the linearized Boltzmann equation for a hard-sphere molecular gas. J. Vac. Sci. Technol. A 28, 603–612.Google Scholar
Doi, T. 2012a Effect of weak gravitation on the plane Poiseuille flow of a highly rarefied gas. Z. Angew. Math. Phys. 63, 1091–1102.Google Scholar
Doi, T. 2012b Plane thermal transpiration of a rarefied gas in the presence of gravitation. Vacuum 86, 1541–1546.Google Scholar
Ewart, T., Perrier, P., Graur, I. A. & Méolans, J. G. 2007 Mass flow rate measurements in a microchannel, from hydrodynamic to near free molecular regimes. J. Fluid Mech. 584, 337–356.Google Scholar
Fan, J. & Shen, C. 2001 Statistical simulation of low-speed rarefied gas flows. J. Comput. Phys. 167, 393–412.CrossRefGoogle Scholar
Frangi, A., Ghisi, A. & Frezzotti, A. 2007 Analysis of gas flow in MEMS by a deterministic 3D BGK kinetic model. Sensor Lett. 6 (1), 1–7.Google Scholar
Funagane, H. & Takata, S. 2012 Hagen-Poiseuille and thermal transpiration flows of a highly rarefied gas through a circular pipe. Fluid Dyn. Res. 44, 055506.Google Scholar
Gamba, I. M. & Tharkabhushanam, S. H. 2009 Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states. J. Comput. Phys. 228 (6), 2012–2036.Google Scholar
Garcia-Colin, L. S., Velasco, R. M. & Uribe, F. J. 2008 Beyond the Navier–Stokes equations: Burnett hydrodynamics. Phys. Rep. 465, 149–189.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
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.CrossRefGoogle Scholar
Huang, J. C., Xu, K. & Yu, P. B. 2012 A unified gas-kinetic scheme for continuum and rarefied flows II: multi-dimensional cases. Commun. Comput. Phys. 12, 662–690.Google Scholar
Huang, J. C., Xu, K. & Yu, P. B. 2013 A unified gas-kinetic scheme for continuum and rarefied flows III: microflow simulations. Commun. Comput. Phys. 14, 1147–1173.Google Scholar
John, B., Gu, X. J. & Emerson, D. R. 2010 Investigation of heat and mass transfer in a lid-driven cavity under nonequilibrium flow conditions. Numer. Heat Transfer B 52, 287–303.CrossRefGoogle Scholar
John, B., Gu, X. J. & Emerson, D. R. 2011 Effects of incomplete surface accommodation on non-equilibrium heat transfer in cavity flow: a parallel DSMC study. Comput. Fluids 45, 197–201.Google Scholar
Loyalka, S. K. 1971 Kinetic theory of thermal transpiration and mechanocaloric effect. I. J. Chem. Phys. 55, 4497.Google Scholar
Masters, N. D. & Ye, W. J. 2007 Octant flux splitting information preservation DSMC method for thermally driven flows. J. Comput. Phys. 226 (2), 2044–2062.Google Scholar
Mieussens, L. & Struchtrup, H. 2004 Numerical comparison of Bhatnagar–Gross–Krook models with proper Prandtl number. Phys. Fluids 16 (8), 2297–2813.Google Scholar
Mouhot, C. & Pareschi, L. 2006 Fast algorithms for computing the Boltzmann collision operator. Maths Comput. 75 (256), 1833–1852.Google Scholar
Ohwada, T., Sone, Y. & Aoki, K 1989 Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard-sphere molecules. Phys. Fluids A 1, 2042–2049.Google Scholar
Pareschi, L. & Russo, G. 2000 Numerical solution of the Boltzmann equation I: spectrally accurate approximation of the collision operator. SIAM J. Numer. Anal. 37 (4), 1217–1245.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 (3), 030606.CrossRefGoogle Scholar
Rana, A., Torrilhon, M. & Struchtrup, H. 2013 A robust numerical method for the R13 equations of rarefied gas dynamics: application to lid driven cavity. J. Comput. Phys. 236, 169–186.Google Scholar
Sharipov, F. 1994a Onsager-Casimir reciprocity relations for open gaseous systems at arbitrary rarefaction. I. General theory for single gas. Physica A 203, 437–456.Google Scholar
Sharipov, F. 1994b Onsager–Casimir reciprocity relations for open gaseous systems at arbitrary rarefaction. II. Application of the theory for single gas. Physica A 203, 457–485.Google Scholar
Sharipov, F. & Bertoldo, G. 2009a Numerical solution of the linearized Boltzmann equation for an arbitrary intermolecular potential. J. Comput. Phys. 228, 3345–3357.CrossRefGoogle Scholar
Sharipov, F. & Bertoldo, G. 2009b Poiseuille flow and thermal creep based on the Boltzmann equation with the Lennard-Jones potential over a wide range of the Knudsen number. Phys. Fluids 21, 067101.Google Scholar
Sharipov, F. & Seleznev, V. 1994 Rarefied flow through a long tube at any pressure ratio. J. Vac. Sci. Technol. A 12 (5), 2933–2935.Google Scholar
Sharipov, F. & Seleznev, V. 1998 Data on internal rarefied gas flows. J. Phys. Chem. Ref. Data 27 (3), 657–706.Google Scholar
Sharipov, F. & Strapasson, J. L. 2012 Ab initio simulation of transport phenomena in rarefied gases. Phys. Rev. E 86, 031130.Google Scholar
Sharipov, F. & Strapasson, J. L. 2013 Benchmark problems for mixtures of rarefied gases. I. Couette flow. Phys. Fluids 25, 027101.Google Scholar
Sone, Y. 2002 Kinetic Theory and Fluid Dynamics. Birkhauser.Google Scholar
Takata, S. 2009a Note on the relation between thermophoresis and slow uniform flow problems for a rarefied gas. Phys. Fluids 21, 112001.Google Scholar
Takata, S. 2009b Symmetry of the linearized Boltzmann equation and its application. J. Stat. Phys. 136, 751.Google Scholar
Takata, S. & Funagane, H. 2011 Poiseuille and thermal transpiration flows of a highly rarefied gas: over-concentration in the velocity distribution function. J. Fluid Mech. 669, 242–259.Google Scholar
Venkattraman, A. & Alexeenko, A. A. 2012 Binary scattering model for Lennard-Jones potential: transport coefficients and collision integrals for non-equilibrium gas flow simulations. Phys. Fluids 24, 027101.Google Scholar
Wu, L., White, C., Scanlon, T. J., Reese, J. M. & Zhang, Y. H. 2013 Deterministic numerical solutions of the Boltzmann equation using the fast spectral method. J. Comput. Phys. 250, 27–52.Google Scholar