A General formulation for the slip velocity boundary condition in lattice Boltzmann methods

Abstract

Numerous fluid flow problems employ the slip velocity boundary condition, which due to their complexity are often only accessible through CFD. While the lattice Boltzmann method (LBM) [1] has been credited as a natural CFD strategy to model the wall slip phenomenon, it is now recognized that numerical artefacts affect the LBM solution [2] if the LBM slip boundary scheme is not properly calibrated. Unfortunately, this calibration uses an ad hoc procedure, and it does not work in general [2]. Currently, the development of consistent LBM slip boundary schemes tends to relate the slip velocity boundary condition with the closure relation of the LBM boundary scheme [2, 3, 4]. In this context, both linkwise or wet-node operation principles can be explored, though none is optimal. In linkwise philosophy [2, 3], boundary populations are constructed along lattice links, making it difficult to handle directional information, like simultaneous satisfaction of the slip (Robin-type) condition for the velocity tangential component and the no-penetration (Dirichlet-type) condition for the velocity normal component. In wet-node philosophy [4], the directional information is naturally handled, but the construction of the boundary populations is more evolving and harder to cope with. This work proposes a scheme that combines these two philosophies. Here, any standard linkwise scheme, e.g. a multireflection scheme [5], can be used to prescribe the no-slip condition over normal and tangential directions, which is amended with a correction term to enforce the conditions for the velocity slip only along the wall tangential direction(s). This correction brings in the information from the first- and second-order velocity derivatives along the pertinent wall directions, which are found in a simple and local way through the LSOB wet-node operation principle [4]. Our theoretical analysis shows that, the proposed mixed linkwise/wet-node scheme, reaches parabolic accuracy for both no-penetration and slip velocity conditions. This high level of accuracy is also confirmed through a series of numerical simulations performed in well-established benchmark test cases of slip flow over planar and curved walls. The comparison of the numerical results here obtained against previously published ones [2, 3, 4] pinpoints the superiority of the present strategy in modelling the slip velocity boundary condition in LBM offering supremacy both in terms of accuracy and simplicity of implementation.