Shock Waves

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Limitation principle for computational fluid dynamics

  • C. Liu
  • G. Zhou
  • W. Shyy
  • K. XuEmail author
Original Article


Theoretical gas dynamics uses the physical Knudsen number \(\mathrm {Kn}_\mathrm{p}\), which is defined as the ratio of the particle mean free path \(\lambda \) to the characteristic length scale L, to categorize the flow into different regimes. The Boltzmann equation is the fundamental equation for dilute gases, while the Navier–Stokes (NS) equations are used for the description of continuum flow at \(\mathrm {Kn}_\mathrm{p}\le 10^{-3}\). For computational fluid dynamics (CFD), the numerical resolution is limited by the discrete cell size and time step. Therefore, we can define a cell Knudsen number \(\mathrm {Kn}_\mathrm{c}\) as the ratio of the particle mean free path \(\lambda \) to the cell size \(\Delta x\). In CFD, the numerical solution and the corresponding numerical flow regime are fully controlled by a numerical Knudsen number \(\mathrm {Kn}_\mathrm{n}\), which is a function of the physical Knudsen number \(\mathrm {Kn}_\mathrm{p}\) and the cell Knudsen number \(\mathrm {Kn}_\mathrm{c}\). The limitation principle relates to the connections between \(\mathrm {Kn}_\mathrm{n}\), \(\mathrm {Kn}_\mathrm{p}\), and \(\mathrm {Kn}_\mathrm{c}\). In this paper, based on the relationship between the modeling equation, cell resolution, and the physical structure thickness, we propose the division of numerical flow regimes. According to the limitation principle, the range of validity of the NS equations is extended to \(\max (\mathrm {Kn}_\mathrm{p},\mathrm {Kn}_\mathrm{c})\le 10^{-3}\). During a mesh refinement process, in some cases the NS equations alone may not be able to capture the flow physics once the large gradients and high-frequency modes are resolved by numerical mesh size and time step. In order to obtain a physical solution in the corresponding numerical scale efficiently, a multiscale method is preferred to identify the flow physics in the corresponding cell Knudsen number \(\mathrm {Kn}_\mathrm{c}\), such as capturing hydrodynamic wave propagation in the coarse mesh resolution case and the kinetic particle transport in the fine mesh case. The unified gas-kinetic scheme (UGKS) is such a multiscale method for providing continuum, near-continuum, and non-equilibrium solutions with a variation of cell Knudsen number. Numerical examples with different physical Knudsen numbers are calculated under different cell Knudsen numbers. These results show the mesh size effect on the numerical representation of a physical solution. In comparison with the NS and direct Boltzmann solvers, the multiscale UGKS is able to capture flow physics in different regimes seamlessly with a variation of numerical resolution.


Cell Knudsen number Grid refinement Multiscale modeling Non-equilibrium flow 



The current research was supported by the Hong Kong Research Grant Council (16206617, 16207715) and the National Science Foundation of China (11772281, 91530319).


  1. 1.
    Aboulhasanzadeh, B., Mohseni, K.: An observable regularization of compressible two-phase flow. Proc. Comput. Sci. 108, 1943–1952 (2017). CrossRefGoogle Scholar
  2. 2.
    Xu, K., Liu, C.: A paradigm for modeling and computation of gas dynamics. Phys. Fluids 29, 026101 (2017). CrossRefGoogle Scholar
  3. 3.
    Li, J., Li, Q., Xu, K.: Comparison of the generalized Riemann solver and the gas-kinetic scheme for inviscid compressible flow simulations. J. Comput. Phys. 230(12), 5080–5099 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Xu, K.: Direct Modeling for Computational Fluid Dynamics: Construction and Application of Unified Gas-Kinetic Scheme. World Scientific, Singapore (2015). CrossRefzbMATHGoogle Scholar
  5. 5.
    Chapman, S., Cowling, T.G., Burnett, D.: The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases. Cambridge University Press, Cambridge (1990)Google Scholar
  6. 6.
    Chen, S., Xu, K.: A comparative study of an asymptotic preserving scheme and unified gas-kinetic scheme in continuum flow limit. J. Comput. Phys. 288, 52–65 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Nasrabadi, N.M.: Pattern recognition and machine learning. J. Electron. Imaging 16(4), 049901 (2007). MathSciNetCrossRefGoogle Scholar
  8. 8.
    Liu, C., Xu, K., Sun, Q., Cai, Q.: A unified gas-kinetic scheme for continuum and rarefied flows IV: Full Boltzmann and model equations. J. Comput. Phys. 314, 305–340 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Wu, L., White, C., Scanlon, T.J., Reese, J.M., Zhang, Y.: Deterministic numerical solutions of the Boltzmann equation using the fast spectral method. J. Comput. Phys. 250, 27–52 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Wang, R.-J., Xu, K.: The study of sound wave propagation in rarefied gases using unified gas-kinetic scheme. Acta Mech. Sin. 28(4), 1022–1029 (2012). CrossRefzbMATHGoogle Scholar
  11. 11.
    Zheng, Y., Garcia, A.L., Alder, B.J.: Comparison of kinetic theory and hydrodynamics for Poiseuille flow. J. Stat. Phys. 109(3–4), 495–505 (2002). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Aoki, K., Takata, S., Nakanishi, T.: Poiseuille-type flow of a rarefied gas between two parallel plates driven by a uniform external force. Phys. Rev. E 65(2), 026315 (2002). CrossRefGoogle Scholar
  13. 13.
    Xiao, T., Xu, K., Cai, Q., Qian, T.: An investigation of non-equilibrium heat transport in a gas system under external force field. Int. J. Heat Mass Transf. 126, 362–379 (2018). CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsHong Kong University of Science and TechnologyKowloonChina
  2. 2.Department of Mechanical and Aerospace EngineeringHong Kong University of Science and TechnologyKowloonChina
  3. 3.HKUST Shenzhen Research InstituteShenzhenChina
  4. 4.College of EngineeringPeking UniversityBeijingChina

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