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A Unified Gas-Kinetic Scheme for Continuum and Rarefied Flows III: Microflow Simulations

Published online by Cambridge University Press:  03 June 2015

Juan-Chen Huang ,
Kun Xu  and
Pubing Yu
Juan-Chen Huang*
Affiliation:
Department of Merchant Marine, National Taiwan Ocean University, Keelung 20224, Taiwan
Kun Xu*
Affiliation:
Mathematics Department, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
Pubing Yu*
Affiliation:
Mathematics Department, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
*
Corresponding author.Email:[email protected]
Email:[email protected]
Email:[email protected]
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Abstract

Due to the rapid advances in micro-electro-mechanical systems (MEMS), the study of microflows becomes increasingly important. Currently, the molecular-based simulation techniques are the most reliable methods for rarefied flow computation, even though these methods face statistical scattering problem in the low speed limit. With discretized particle velocity space, a unified gas-kinetic scheme (UGKS) for entire Knudsen number flow has been constructed recently for flow computation. Contrary to the particle-based direct simulation Monte Carlo (DSMC) method, the unified scheme is a partial differential equation-based modeling method, where the statistical noise is totally removed. But, the common point between the DSMC and UGKS is that both methods are constructed through direct modeling in the discretized space. Due to the multiscale modeling in the unified method, i.e., the update of both macroscopic flow variables and microscopic gas distribution function, the conventional constraint of time step being less than the particle collision time in many direct Boltzmann solvers is released here. The numerical tests show that the unified scheme is more efficient than the particle-based methods in the low speed rarefied flow computation. The main purpose of the current study is to validate the accuracy of the unified scheme in the capturing of non-equilibrium flow phenomena. In the continuum and free molecular limits, the gas distribution function used in the unified scheme for the flux evaluation at a cell interface goes to the corresponding Navier-Stokes and free molecular solutions. In the transition regime, the DSMC solution will be used for the validation of UGKS results. This study shows that the unified scheme is indeed a reliable and accurate flow solver for low speed non-equilibrium flows. It not only recovers the DSMC results whenever available, but also provides high resolution results in cases where the DSMC can hardly afford the computational cost. In thermal creep flow simulation, surprising solution, such as the gas flowing from hot to cold regions along the wall surface, is observed for the first time by the unified scheme, which is confirmed later through intensive DSMC computation.

Keywords

51.10.+y47.11.-j47.11.St47.45.-nUnified schemenon-equilibrium microflowthermal creep flows
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 5 , November 2013 , pp. 1147 - 1173
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Alexeenko, A. A., Gimelshein, S. F., Muntz, E. P. and Ketsdever, A. D., Modeling of thermal transpiration flows for Knudsen compressor optimization meeting, in: Presented at 43rd Aerospace Sciences Meeting, Reno, NV, AIAA Paper 2005–963 (2005).Google Scholar
[2]Annis, B. K., Thermal creep in gases, J. Chem. Phys., 57 (1972), 2898–2905.Google Scholar
[3]Aoki, K., Takata, S., Hidefumi, A. and Golse, F., A rarefied gas flow caused by a discontinuous wall temperature, Phys. Fluids, 13 (2001), 2645–2661.CrossRefGoogle Scholar
[4]Aoki, Kazuo, Degond, Pierre and Mieussens, Luc, Numerical simulations of rarefied gases in curved channels: thermal creep, circulating flow and pumping effect, Commun. Comput. Phys., 6(5), 919–954.CrossRefGoogle Scholar
[5]Barisik, Murat and Beskok, Ali, Molecular dynamics simulations of shear-driven gas flows in nano-channels, Microfluid Nanofluid, 11 (2011), 611–622.Google Scholar
[6]Belotserkovskii, O. M. and Khlopkov, Y. I., Monte Carlo Methods in Mechanics of Fluid and Gas, World Scientific, 2010.CrossRefGoogle Scholar
[7]Beylich, Alfred E., Solving the kinetic equation for all Knudsen numbers, Phys. Fluids, 12 (2000), 160;444.Google Scholar
[8]Bhatnagar, P. L., Gross, E. P. and Krook, M., A Model for collision processes in gases I: small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511–525.CrossRefGoogle Scholar
[9]Bielenberg, J. R. and Brenner, H., A continuum model of thermal transpiration, J. Fluid Mech., 546 (2006), 123.Google Scholar
[10]Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford Science Publications, 1994.CrossRefGoogle Scholar
[11]Burt, J. M. and Boyd, I. D., Convergence detection in direct simulation Monte Carlo calculations for steady state flows, Commun. Comput. Phys., 10 (2011), 807–822.Google Scholar
[12]Chapman, S. and Cowling, T. G., The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, 1990.Google Scholar
[13]Chen, S. Z., Xu, K., Li, C. B. and Cai, Q. D., A unified gas kinetic scheme with moving mesh and velocity space adaptation, J. Comput. Phys., 231 (20) (2012), 6643–6664.Google Scholar
[14]Fan, J. and Shen, C., Statistical simulation of low-speed rarefied gas flows, J. Comput. Phys., 167 (2001), 393–412.Google Scholar
[15]Karniadakis, G., Beskok, A. and Aluru, N., Microflows and Nanoflows, Springer Sci-ence+Business Media, Inc., 2005.Google Scholar
[16]Han, Y. L., Working gas temperature and pressure change for microscale thermal creep-driven flow caused by discontinuous wall temperature, Fluid Dyn. Res., 42 (2010), 1–23.Google Scholar
[17]Huang, J. C., Xu, K. and Yu, P. B., A unified gas-kinetic scheme for continuum and rarefied flows II: multidimensional cases, Commun. Comput. Phys., 3(3) (2012), 662–690.Google Scholar
[18]Kogan, M. N., Rarefied Gas Dynamics, Plenum Press, New York, 1969.Google Scholar
[19]Knudsen, M., The Kinetic Theory of Gases, third ed., Wiley, New York, 1950.Google Scholar
[20]Masters, N. D. and Ye, W., Octant flux splitting information preserving DSMC method for thermally driven flows, J. Comput. Phys., 226 (2007), 2044–2062.Google Scholar
[21]Morinishi, Koji, Numerical simulation for gas microflows using Boltzmann equation, Comput. Fluids, 35 (2006), 978–985.CrossRefGoogle Scholar
[22]Radtke, G. A., Hadjiconstantinou, N. G. and Wagner, W., Low-noise Monte Carlo simulation of the variable hard sphere gas, Phys. Fluids, 23 (2011), 030606.CrossRefGoogle Scholar
[23]Shakhov, E. M., Generalization of the Krook kinetic equation, Fluid Dyn., 3(1968), 95.Google Scholar
[24]Sharipov, F., Non-isothermal gas flow through rectangular microchannels, J. Micromech. Microeng., 9 (1999), 394–401.Google Scholar
[25]Sone, Y., Takata, S. and Ohwada, T., Numerical analysis of the plane Couette flow of a rarefied gas on the basis of the linearized Boltzmann equation for hard-sphere molecules, Euro. J. Mech. B/Fluids, 9 (1990), 273.Google Scholar
[26]Sun, Q. and Boyd, I. D., A direct simulation method for subsonic microscale gas flows, J. Comput. Phys., 179 (2002), 400–425.CrossRefGoogle Scholar
[27]Sun, Q., Information Preservation Methods for Modeling Micro-Scale Gas Flows, Ph.D. thesis, 2003, The University of Michigan.Google Scholar
[28]Tcheremissine, F. G., Solution of the Boltzmann kinetic equation for low speed flows, Transport Theory Statistical Phys., 37 (2008), 564–575.Google Scholar
[29]Titarev, V. A., Implicit unstructured-mesh method for calculating Poiseuille flows of rarefied gas, Commun. Comput. Phys., 8 (2010), 427–444.CrossRefGoogle Scholar
[30]Titarev, V. A., Efficient deterministic modelling of three-dimensional rarefied gas flows, Commun. Comput. Phys., 12 (2012), 162–192.Google Scholar
[31]Vargo, S. E., Muntz, E. P., Shiflett, G. R. and Tang, W. C., Knudsen compressor as a micro- and macroscale vacuum pump without moving parts or fluids, J. Vacuum Sci. Tech. A, 7 (1999), 2308–2313.CrossRefGoogle Scholar
[32]Wang, R. J. and Xu, K., The study of sound wave propagation in rarefied gases using unified gas-kinetic scheme, Acta Mech. Sinica, 28 (2012), 1022–1029.Google Scholar
[33]Xu, K., Numerical Hydrodynamics from Gas-Kinetic Theory, Ph.D. thesis, 1993, Columbia University.Google Scholar
[34]Xu, K., A Gas-Kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method, J. Comput. Phys., 171 (2001), 289–335.Google Scholar
[35]Xu, K., Regularization of the Chapman-Enskog expansion and its description of shock structure, Phys. Fluids, 14(4) (2002), L17–L20. DOI: 10.1063/1.1453467CrossRefGoogle Scholar
[36]Xu, K. and Huang, J. C., A unified gas-kinetic scheme for continuum and rarefied flows J. Comput. Phys., 229 (2010), 7747–7764.CrossRefGoogle Scholar
[37]Xu, K. and Huang, J. C., An improved unified gas-kinetic scheme and the study of shock structures, IMA J. Appl. Math., 76(5) (2011), 698–711.CrossRefGoogle Scholar
[38]Xu, K. and Josyula, E., Continuum formulation for non-equilibrium shock structure calculation, Commun. Comput. Phys., 1(3) (2006), 425–448.Google Scholar
[39]Xu, K., Liu, H. W. and Jiang, J. Z., Multiple-temperature kinetic model for continuum and near continuum flows, Phys. Fluids, 19 (2007), 016101. DOI: 10.1063/1.2429037.CrossRefGoogle Scholar
[40]Zhang, J., Fan, J. and Jiang, J. Z., Multiple temperature model for the information preservation method and its application to nonequilibrium gas flows, J. Comput. Phys., 230 (2011), 7250–7256.Google Scholar