Numerical analysis of the axisymmetric lattice Boltzmann method for steady and oscillatory flows in periodic geometries

Abstract

Compared to more typical computational fluid dynamics techniques, the lattice Boltzmann method (LBM) is relatively new and unexplored. In recent years, axisymmetric LBM formulations, which can simulate flow in rotationally symmetric 3D geometries, have been published. Here we verify a novel axisymmetric LBM implementation using numerical criteria. Hagen–Poiseuille and Womersley flow are considered within a straight tube where analytic solutions are available. With this, we establish sufficient accuracy of the approximated flow and study the effects of changing simulation parameters (e.g. Reynolds number, Womersley number) and spatial/temporal parameters (e.g. relaxation time, mesh nodes, time steps). Furthermore, steady and oscillatory flows within a periodically-varying, longitudinally asymmetric geometry are considered. Analytic solutions are not available in these cases; however, the validity of the axisymmetric LBM for curved boundaries is ensured through convergence, mesh independence and qualitative observations. Guaranteeing reasonable flow field determination for the aformentioned geometry is relevant to a larger problem where particulate suspension is pumped back and forth through a membrane of axisymmetric micropores. In these circumstances, experiments have induced directed particle transport even though there is no net flow of the carrier fluid. Hence, our work aims to improve current numerical simulations of these flow problems to better understand the factors that facilitate particle transport. References R. D. Astumian and P. Hänggi. Brownian motors. Phys. Today 55.11 (2002), pp. 33–39. doi: 10.1063/1.1535005. (Cit. on p. C214). W. R. Bowen and F. Jenner. Theoretical descriptions of membrane filtration of colloids and fine particles: An assessment and review. Adv. Colloid Interface Sci. 56 (1995), pp. 141–200. doi: 10.1016/0001-8686(94)00232-2 N. Islam. Fluid flow and particle transport through periodic capillaries. Bull. Aust. Math. Soc. 96.3 (2017), pp. 521–522. doi: 10.1017/S0004972717000739 C. Kettner, P. Reimann, P. Hänggi, and F. Müller. Drift ratchet. Phys. Rev. E 61.1 (2000), pp. 312–323. doi: 10.1103/PhysRevE.61.312 S. H. Kim and H. Pitsch. A generalized periodic boundary condition for lattice Boltzmann method simulation of a pressure driven flow in a periodic geometry. Phys. Fluids 19.10, 108101 (2007). doi: 10.1063/1.2780194 T. Krüger, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G. Silva, and E. M. Viggen. The lattice Boltzmann method: Principles and practice. Vol. 10. Graduate Texts in Physics. Springer, 2017, pp. 978–3. doi: 10.1007/978-3-319-44649-3 S. Matthias and F. Müller. Asymmetric pores in a silicon membrane acting as massively parallel brownian ratchets. Nature 424 (2003), pp. 53–57. doi: 10.1038/nature01736 S. J. Stephen, B. M. Johnston, and P. R. Johnston. Comparing lattice Boltzmann simulations of periodic fluid flow in repeated micropore structures with longitudinal symmetry and asymmetry. Proceedings of the 15th Biennial Engineering Mathematics and Applications Conference, EMAC-2021. Ed. by A. Clark, Z. Jovanoski, and J. Bunder. Vol. 63. ANZIAM J. 2022, pp. C69–C83. doi: 10.21914/anziamj.v63.17158 W. Wang and J. Zhou. Enhanced Lattice Boltzmann modelling of axisymmetric flows. Proceedings of the Institution of Civil Engineers—Engineering and Computational Mechanics 167.4 (2014), pp. 156–166. doi: 10.1680/eacm.14.00005 J. G. Zhou. Axisymmetric lattice Boltzmann method revised. Phys. Rev. E 84.3, 036704 (2011). doi: 10.1103/PhysRevE.84.036704