Skip to main content Accessibility help
×
Home
Hostname: page-component-7ccbd9845f-mpxzb Total loading time: 0.447 Render date: 2023-01-28T01:22:19.640Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

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*
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:jchuang@mail.ntou.edu.tw
Get access

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.

Type
Research Article
Copyright
Copyright © Global Science Press Limited 2013

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

[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 2005963 (2005).Google Scholar
[2]Annis, B. K., Thermal creep in gases, J. Chem. Phys., 57 (1972), 28982905.CrossRefGoogle 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), 26452661.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), 919954.CrossRefGoogle Scholar
[5]Barisik, Murat and Beskok, Ali, Molecular dynamics simulations of shear-driven gas flows in nano-channels, Microfluid Nanofluid, 11 (2011), 611622.CrossRefGoogle 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.CrossRefGoogle 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), 511525.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.Google 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), 807822.CrossRefGoogle 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), 66436664.CrossRefGoogle Scholar
[14]Fan, J. and Shen, C., Statistical simulation of low-speed rarefied gas flows, J. Comput. Phys., 167 (2001), 393412.CrossRefGoogle 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), 123.CrossRefGoogle 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), 662690.CrossRefGoogle Scholar
[18]Kogan, M. N., Rarefied Gas Dynamics, Plenum Press, New York, 1969.CrossRefGoogle 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), 20442062.CrossRefGoogle Scholar
[21]Morinishi, Koji, Numerical simulation for gas microflows using Boltzmann equation, Comput. Fluids, 35 (2006), 978985.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.CrossRefGoogle Scholar
[24]Sharipov, F., Non-isothermal gas flow through rectangular microchannels, J. Micromech. Microeng., 9 (1999), 394401.CrossRefGoogle 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), 400425.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), 564575.CrossRefGoogle Scholar
[29]Titarev, V. A., Implicit unstructured-mesh method for calculating Poiseuille flows of rarefied gas, Commun. Comput. Phys., 8 (2010), 427444.Google Scholar
[30]Titarev, V. A., Efficient deterministic modelling of three-dimensional rarefied gas flows, Commun. Comput. Phys., 12 (2012), 162192.CrossRefGoogle 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), 23082313.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), 10221029.CrossRefGoogle 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), 289335.CrossRefGoogle Scholar
[35]Xu, K., Regularization of the Chapman-Enskog expansion and its description of shock structure, Phys. Fluids, 14(4) (2002), L17L20. 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), 77477764.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), 698711.CrossRefGoogle Scholar
[38]Xu, K. and Josyula, E., Continuum formulation for non-equilibrium shock structure calculation, Commun. Comput. Phys., 1(3) (2006), 425448.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), 72507256.CrossRefGoogle Scholar
68
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

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

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

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

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

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

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *