Slip velocity boundary conditions for the lattice Boltzmann modeling of microchannel flows

Abstract

Slip flows in ducts are important in numerous engineering applications, most notably in microchannel flows. Compared to the standard no-slip Dirichlet condition, the case of slip formulates as a Robin-type condition for the fluid tangential velocity. Such an increase in mathematical complexity is accompanied by a more challenging numerical transcription. The present work concerns with this topic, addressing the modeling of the slip velocity boundary condition in the lattice Boltzmann method (LBM) applied to steady slow viscous flows inside ducts of nontrivial shapes. As novelty, we extend the newly revised local second-order boundary (LSOB) Dirichlet fluid flow method [Philos. Trans. R. Soc. A 378, 20190404 (2020)] to implement the slip velocity condition within the two-relaxation-time (TRT) framework. The LSOB follows an in-node philosophy where its operation principle seeks to explicitly reconstruct the unknown boundary populations in the form of a third-order accurate Chapman-Enskog expansion, where the wall slip condition is built-in as a normal Taylor-type condition. The key point of this approach is that the required first-and second-order momentum derivatives, rather than computed through nonlocal finite difference approximations, are locally determined through a simple local linear algebra procedure, whose formulation is particularly aided by the TRT symmetry argument. To express the obtained derivatives, two approaches are considered, called Lnode and Lwall, which operate with node and wall variables, respectively. These two formulations are developed to prescribe the physical slip condition over plane and curved walls, including the corners. Their consistency and accuracy characteristics are examined against alternative linkwise strategies to impose the wall slip velocity, such as the kinetic-based diffusive bounce-back scheme, the central linear interpolation slip scheme, and the multireflection slip scheme. The several slip schemes are tested over different 3D microchannel configurations, with walls not conforming with the LBM uniform mesh. Numerical tests confirm the advanced accuracy characteristics of the proposed LSOB slip boundary scheme, revealing the added challenge of the wall slip modeling, and that parabolic accuracy is a necessary requirement to reach second-order accuracy within this problem class.

Unified directional parabolic-accurate lattice Boltzmann boundary schemes for grid-rotated narrow gaps and curved walls in creeping and inertial fluid flows

“…In principle, Eq. (40) gives some advantage to CELI-IP over IPLI, whereas numerically the two schemes perform very similarly.…”

“…Accordingly, SSN 0 = SSN and SSN k , k = {1, 4}, satisfy the stability conditions in Eq. (40) with Eqs. (62a) and (62b).…”

“…Considering straight-wall modeling, we find either first-order [32][33][34][35] or second-order [36] finite-difference shear-stress nonequilibrium reconstruction and on-grid moment-based method [37]. Regarding off-lattice boundary conditions, it is worth mentioning the single-node projection method for curved walls called Local Second-Order Boundary (LSOB) [28] and its recent variants, namely, the simplified degraded [38], no-slip [39], and slip [40] reformulations for pipe flows, and the A-LSOB extensions [41,42] for Dirichlet and Neumann conditions in advection-diffusion scalar transport. The distinctive key point is that LSOB introduces boundary condition through a wall-normal Taylor closure relation, whereas the one-point approach [43] combines the matrix projections with the lower-order directional Dirichlet prescription.…”