Skip to main content
SpringerLink
Log in

Gas-kinetic numerical schemes for one- and two-dimensional inner flows

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

Several kinds of explicit and implicit finite-difference schemes directly solving the discretized velocity distribution functions are designed with precision of different orders by analyzing the inner characteristics of the gas-kinetic numerical algorithm for Boltzmann model equation. The peculiar flow phenomena and mechanism from various flow regimes are revealed in the numerical simulations of the unsteady Sod shock-tube problems and the two-dimensional channel flows with different Knudsen numbers. The numerical remainder-effects of the difference schemes are investigated and analyzed based on the computed results. The ways of improving the computational efficiency of the gas-kinetic numerical method and the computing principles of difference discretization are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Sod, G. A. A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. Journal of Computational Physics 27, 1–31 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chu, C. K. Kinetic-theoretic description of the formation of a shock wave. Physics of Fluids 8(1), 12–22 (1965)

    Article  Google Scholar 

  3. Reitz, R. D. One-dimensional compressible gas dynamics calculations using the Boltzmann equations. Journal of Computational Physics 42, 108–123 (1981)

    Article  MATH  Google Scholar 

  4. Shu, C. W. and Osher, S. Efficient implementation of essentially non-oscillatory shock capturing schemes I. Journal of Computational Physics 77, 439–471 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  5. Gorelov, S. L. and Kogan, M. N. Solution of linear problem of rarefied gasdynamics by the Monte Carlo method. Fluid Dynamics 3, 96–98 (1968)

    Article  Google Scholar 

  6. Huang, A. B. and Hwang, P. F. Test of statistical models for gases with and without internal energy states. Physics of Fluids 16(4), 466–575 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  7. Alofs, D. J., Flagan, R. C., and Springer, G. S. Density distribution measurements in rarefied gases contained between parallel plates at high temperature differences. Physics of Fluids 14(3), 529–533 (1971)

    Article  Google Scholar 

  8. Tritton, D. J. Physical Fluid Dynamics, Oxford University Press, Oxford (1988)

    Google Scholar 

  9. Prendergast, K. H. and Xu, K. Numerical hydrodynamics from gas-kinetic theory. Journal of Computational Physics 109, 53–64 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  10. Yang, J. Y. and Huang, J. C. Rarefied flow computations using nonlinear model Boltzmann equations. Journal of Computational Physics 120, 323–339 (1995)

    Article  MATH  Google Scholar 

  11. Zheng, Y., Garcia, A. L., and Alder, B. J. Comparison of kinetic theory and hydrodynamics for Poiseuille flow. Journal of Statistical Physics 109, 495–505 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. Luo, L. S., Chen, H., Chen, S., Doolen, G. D., and Lee, Y. C. Generalized hydrodynamic transport in lattice gas automata. Physical Review A 43, 7097–7100 (1991)

    Article  Google Scholar 

  13. Chen, S. and Doolen, G. D. Lattice Boltzmann method for fluid flows. Annual Review of Fluid Mechanics 30, 329–364 (1998)

    Article  MathSciNet  Google Scholar 

  14. 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. European Journal of Mechanics-B/Fluids 9(3), 449–456 (1990)

    Google Scholar 

  15. Xu, K. and Li, Z. H. Microchannel flow in the slip regime: gas-kinetic BGK-Burnett solutions. Journal of Fluid Mechanics 513, 87–110 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Li, Z. H. Study on the Unified Algorithm from Rarefied Flow to Continuum, Ph. D. dissertation, China Aerodynamics Researchment and Development Center, Mianyang (2001)

    Google Scholar 

  17. Li, Z. H. and Zhang, H. X. Numerical investigation from rarefied flow to continuum by solving the Boltzmann model equation. International Journal of Numerical Methods in Fluids 42, 361–382 (2003)

    Article  MATH  Google Scholar 

  18. Li, Z. H. and Zhang, H. X. Study on gas kinetic unified algorithm for flows from rarefied transition to continuum. Journal of Computational Physics 193, 708–738 (2004)

    Article  MATH  Google Scholar 

  19. Li, Z. H. and Zhang, H. X. Gas-kinetic description of shock wave structures by solving Boltzmann model equation. International Journal of Computational Fluid Dynamics 22(9), 623–638 (2008)

    Article  MathSciNet  Google Scholar 

  20. Zhang, H. X. and Zhuang, F. G. NND schemes and their applications to numerical simulation of two- and three-dimensional flows. Advances in Applied Mechanics 29, 193–256 (1992)

    Article  MATH  Google Scholar 

  21. Shizgal, B. A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems. Journal of Computational Physics 41, 309–328 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  22. Yee, H. C. Construction of explicit and implicit symmetric TVD schemes and their applications. Journal of Computational Physics 68, 151–179 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  23. Zhang, H. X. and Shen, M. Y. Computational Fluid Dynamics—Fundamentals and Applications of Finite Difference Methods, National Industry Press, Beijing (2003)

    Google Scholar 

  24. Liu, R. X. and Shu, Q. W. Some New Methods on Computational Fluid Dynamics, Science Press, Beijing (2003)

    Google Scholar 

  25. Schlichting, H. Boundary-Layer Theory, Chap. XII, McGraw-Hill, New York (1968)

    Google Scholar 

  26. Nance, R. P., Hash, D. B., and Hassan, H. A. Role of boundary conditions in Monte Carlo simulation of microelectromechanical systems. Journal of Thermophysics and Heat Transfer 12(3), 447–449 (1998)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Hypervelocity Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang, 621000, Sichuan Province, P. R. China

    Zhi-hui Li  (李志辉), Lin Bi  (毕林) & Zhi-gong Tang  (唐志共)

  2. National Laboratory for Computational Fluid Dynamics, Beijing, 100191, P. R. China

    Zhi-hui Li  (李志辉)

Authors
  1. Zhi-hui Li  (李志辉)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lin Bi  (毕林)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhi-gong Tang  (唐志共)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhi-hui Li  (李志辉).

Additional information

Communicated by Zhe-wei ZHOU

Project supported by the National Natural Science Foundation of China (No. 10621062) and the Research Fund for Next Generation of General Armament Department (No. 9140A13050207KG29)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, Zh., Bi, L. & Tang, Zg. Gas-kinetic numerical schemes for one- and two-dimensional inner flows. Appl. Math. Mech.-Engl. Ed. 30, 889–904 (2009). https://doi.org/10.1007/s10483-009-0708-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-009-0708-x

Key words

Chinese Library Classification

2000 Mathematics Subject Classification

Search

Navigation