ON THE ACCURACY OF LOW- AND HIGH-ORDER LATTICE BOLTZMANN EQUATIONS IN APPLICATIONS TO SLOW ISOTHERMAL MICROFLOWS

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

The question of the applicability of two-dimensional lattice Boltzmann equations of different orders in standard lattices to the description of slow isothermal sparse flows is considered. The twodimensional Poiseuille flow at different Knudsen numbers is used as a reference problem. This problem is numerically solved using several lattice Boltzmann equations of low and high orders having from 9 to 53 discrete velocities. The results are compared with solutions of the linearized Boltzmann, Bhatnagar-GrossKrook equations, which are used as reference ones. Numerical experiments have shown that an increase in the order of the Boltzmann lattice equation (i.e., the number of first moments of the local-Maxwell distribution reproduced by the discrete local equilibrium of the Boltzmann lattice equation) does not always lead to an increase in accuracy. In particular, a new low-order model for 16 velocities is proposed, which correctly describes the diffuse reflection at solid boundaries at a qualitative level. It is shown that for this model, it is possible to obtain sufficiently accurate values of the volumetric flow rate of sliding velocities for a wide range of Knudsen numbers in comparison with other models under consideration.

About the authors

O. V Ilyin

Federal Research Center Computer Science and Control, RAS

Email: oilyin@gmail.com
Moscow, Russia

References

  1. Kruger T., Kusumaatmaja H., Kuzmin A., Shardt O., Silva G., Viggen E. The Lattice Boltzmann Method. Principles and Practice. Springer, 2017.
  2. McNamara G., Zanetti G. Use of the Boltzmann equation to simulate lattice gas automata // Phys. Rev. Lett. 1988. V 61. P. 2332-2335.
  3. Karniadakis G., Beskok A., Aluru N. Microflows and Nanoflows. Fundamentals and Simulation. Springer, 2005.
  4. Wang J., Chen L., Kang Q., Rahman S. The lattice Boltzmann method for isothermal micro-gaseous flow and its application in shale gas flow: A review // Int. J. Heat Mass Transf. 2016. V. 95. P. 94-108.
  5. Shan X., Yuan X., Chen H. Kinetic theory representation of hydrodynamics: a way beyond the Navier-Stokes equation // J. Fluid Mech. 2006. V. 550. P. 413—441.
  6. Shan X., He X. Discretization of the Velocity Space in the Solution of the Boltzmann Equation // Phys. Rev. Lett. 1998. V. 80. P. 65-67.
  7. Philippi P., Hegele Jr. L., dos Santos L., Surmas R. From the continuous to the lattice Boltzmann equation: The discretization problem and thermal models // Phys. Rev. E. 2006. V. 73. P. 056702.
  8. Surmas R., Ortiz C., Philippi P. Simulating thermohydrodynamics by finite difference solutions of the Boltzmann equation // Eur. Phys. J. 2009. V. 90. P. 81-90.
  9. Chikatamarla S., Karlin I. Lattices for the lattice Boltzmann method // Phys. Rev. E. 2009. V. 79. P. 046701.
  10. Shan X. General solution of lattices for Cartesian lattice Bhatanagar—Gross—Krook models// Phys. Rev E. 2010. V. 81. P. 036702.
  11. Shan X. The mathematical structure of the lattices of the lattice Boltzmann method //J. Comput. Sci. V 17. P. 475— 481.
  12. Feuchter C., Schleifenbaum W. High-order lattice Boltzmann models for wall-bounded flows at finite Knudsen numbers // Phys. Rev. E. 2016. V. 94. P. 013304.
  13. Spiller D., Dunweg B. Semiautomatic construction of lattice Boltzmann models // Phys. Rev. E. 2020. V 101. P. 043310.
  14. Meng J.-P., Zhang Y. Gauss-Hermite quadratures and accuracy of lattice Boltzmann models for non-equilibrium gas flows // Phys. Rev. E. 2011. V. 83. P. 036704.
  15. Shi Y., Wu L., Shan X. Accuracy of high-order lattice Boltzmann method for non-equilibrium gas flow // J. Fluid Mech. 2020. V. 907. P. A25.
  16. Gross E. Ziering S. Kinetic theory of Linear Shear flow // Phys. Fluid. 1958. V 1. P 215—223.
  17. Ambrus V., Sofonea V. Implementation of diffuse-reflection boundary conditions using lattice Boltzmann models based on half-space Gauss-Laguerre quadratures // Phys. Rev E. 2014. V. 89. P. 041301(R).
  18. Ambrus V., Sofonea V. Application of mixed quadrature lattice Boltzmann models for the simulation of Poiseuille flow at non-negligible values of the Knudsen number // J. Comput. Sci. 2016. V. 17. P. 403.
  19. Ambrus V., Sofonea V. Lattice Boltzmann models based on half-range Gauss-Hermite quadratures //J. Comput. Phys. 2016. V. 316. P. 760.
  20. Shi Y. Velocity discretization for lattice Boltzmann method for noncontinuum bounded gas flows at the micro- and nanoscale // Phys. Fluid. 2022. V. 34. P. 082013.
  21. Shi Y. Comparison of different Gaussian quadrature rules for lattice Boltzmann simulations of noncontinuum Couette flows: From the slip to free molecular flow regimes // Phys. Fluid. 2023. V. 35. P. 072015.
  22. Ilyin O. Intermediate Lattice Boltzmann-BGK method and its application to micro-flows // J. Phys.: Conf. Ser. 2019. V. 1163. P. 012030.
  23. Ilyin O. Gaussian Lattice Boltzmann method and its applications to rarefied flows // Phys. Fluid. 2020. V. 32. P. 012007.
  24. Di Staso G., Clercx H., Succi S., Toschi F. Lattice Boltzmann accelerated direct simulation Monte Carlo for dilute gas flow simulations // Phil. Trans. R. Soc. A. 2016. V. 374. P. 20160226.
  25. Di Staso G., Clercx H., Succi S., Toschi F. DSMC-LBM mapping scheme for rarefied and non-rarefied gas flows // J. Comput. Sci. 2016. V. 17. P. 357—369.
  26. Di Staso G., Srivastava S., Arlemark E., Clercx H. Toschi F. Hybrid lattice Boltzmann-direct simulation Monte Carlo approach for flows in three-dimensional geometries // Comput. Fluid. 2018. V. 172. P. 492-509.
  27. Aristov V., Ilyin O., Rogozin O. A hybrid numerical scheme based on coupling discrete-velocities models for the BGK and LBGK equations // AIP Conf. Proc. 2019. V. 2132. P. 060007.
  28. Aristov V., Ilyin O., Rogozin O. Kinetic multiscale scheme based on the discrete-velocity and lattice-Boltzmann methods // J. Comput. Sci. 2020. V. 40. P. 101064.
  29. Kim S., Pitsch H., Boyd I. Accuracy of higher-order lattice Boltzmann methods for microscale flows with finite Knudsen numbers // J. Comput. Phys. 2008. V. 227. P. 8655-8671.
  30. Tang G., Zhang Y., Emerson D. Lattice Boltzmann models for nonequilibrium gas flows // Phys. Rev. E. 2008. V. 77. P. 046701.
  31. Shi Y., Brookes P., Yap Y., Sader J. Accuracy of the lattice Boltzmann method for low-speed noncontinuum flows // Phys. Rev. E. 2011. V. 83. P. 045701(R).
  32. de Izarra L., Rouet J.-L., Izrar B. High-order lattice Boltzmann models for gas flow for a wide range of Knudsen numbers // Phys. Rev. E. 2011. V. 84. P. 066705.
  33. Chikatamarla S., Karlin I. Entropy and Galilean invariance of lattice Boltzmann theories// Phys. Rev. Lett. 2006. V. 97. P. 190601.
  34. Bardow A., Karlin I., Gusev A. Multispeed models in off-lattice Boltzmann simulations // Phys. Rev. E. 2008. V. 77. P. 025701(R).
  35. Ilyin O. Nonclassical Heat Transfer in a Microchannel and a Problem for Lattice Boltzmann Equations // Comp. Math. Math. Phys. 2023. V 63. P 2297-2305.
  36. Ansumali S., Karlin I. Kinetic boundary conditions in the lattice Boltzmann method // Phys. Rev. E. 2002. V. 66. P. 026311.
  37. Meng J., Zhang Y. Diffuse reflection boundary condition for high-order lattice Boltzmann models with streamingcollision mechanism // J. Comput. Phys. 2014. V. 258. P. 601-612.
  38. Cercignani C., Lampis M., Lorenzani S. Variational approach to gas flows in microchannels // Phys. Fluid. 2004. V. 16. P. 3426-3437.
  39. Ohwada T., Sone Y., Aoki K. 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. Fluid. 1989. V. 1. P. 2042-2049.
  40. Toschi F., Succi S. Lattice Boltzmann method at finite Knudsen numbers // Europhys. Lett. 2005. V. 69. P. 549.
  41. Montessori A., Prestininzi P., La Rocca M., Succi S. Lattice Boltzmann approach for complex nonequilibrium flows // Phys. Rev. E. 2015. V. 92. P. 043308.
  42. Kandhai D., Koponen A., Hoekstra A., Kataja M., Timonen J., Sloot P.M.A. Implementation Aspects of 3D Lattice-BGK: Boundaries, Accuracy, and a New Fast Relaxation Method // J. Comput. Phys. 1999. V. 150. P. 482-501.
  43. Bobylev A., Cercignani C. Discrete Velocity Models Without Nonphysical Invariants // J. Stat. Phys. 1999. V. 97. P. 677-686.
  44. Веденяпин В., Орлов Ю. О законах сохранения для полиномиальных гамильтонианов и для дискретных моделей уравнения Больцмана // ТМФ. 1999. Т. 121. C. 307-315.
  45. Vedenyapin V. Velocity inductive construction for mixtures // Transp. Theor. Stat. Phys. 1999. V. 28. P. 727-742.
  46. Bobylev A., Vinerean M. Construction of Discrete Kinetic Models with Given Invariants // J. Stat. Phys. 2008. V. 132. P. 153-170.
  47. Guo Z., Shi B., Zheng C. Chequerboard effects on spurious currents in the lattice Boltzmann equation for two-phase flows // Phil. Trans. R. Soc. A. 2011. V. 369. P. 2283-2291.
  48. Su W., Lindsay S., Liu H., Wu L. Comparative study of the discrete velocity and lattice Boltzmann methods for rarefied gas flows through irregular channels // Phys. Rev E. 2017. V. 96. P. 023309.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2024 Russian Academy of Sciences