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Published May 2024 | Published
Journal Article Open

Variational solution to the lattice Boltzmann method for Couette flow

Abstract

Literature studies of the lattice Boltzmann method (LBM) demonstrate hydrodynamics beyond the continuum limit. This includes exact analytical solutions to the LBM, for the bulk velocity and shear stress of Couette flow under diffuse reflection at the walls through the solution of equivalent moment equations. We prove that the bulk velocity and shear stress of Couette flow with Maxwell-type boundary conditions at the walls, as specified by two-dimensional isothermal lattice Boltzmann models, are inherently linear in Mach number. Our finding enables a systematic variational approach to be formulated that exhibits superior computational efficiency than the previously reported moment method. Specifically, the number of partial differential equations (PDEs) in the variational method grows linearly with quadrature order while the number of moment method PDEs grows quadratically. The variational method directly yields a system of linear PDEs that provide exact analytical solutions to the LBM bulk velocity field and shear stress for Couette flow with Maxwell-type boundary conditions. It is anticipated that this variational approach will find utility in calculating analytical solutions for novel lattice Boltzmann quadrature schemes and other flows.

Copyright and License

© 2024 American Physical Society.

Acknowledgement

This research is supported by an Australian Government Research Training Program (RTP) Scholarship. Y.S. acknowledges support from Ningbo Natural Science Foundation (Project ID 2022J175).

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Additional details

Created:
May 30, 2024
Modified:
May 30, 2024