Block iterative frequency-based lattice Boltzmann algorithm for microscale oscillatory flow

A recent article (Yong Shi and John E. Sader, Phys. Rev. E 81, 036706 (2010)) developed a linearized frequency-based lattice Boltzmann (LB) model to describe oscillatory Stokes flow in microelectromechanical systems (MEMS). Nonetheless, its numerical algorithm is formulated in the conventional time-marching form with addition of virtual time. In this article, we propose a different algorithm to solve the linearized LB model in the frequency domain using the block iteration scheme integrating the tri-diagonal matrix method, Gaussian Seidel-based alternative direction iteration and over relaxing technique. This change in the LB algorithm leads to direct modeling of microscale oscillatory flow in the frequency domain without mimicking a numerical time evolution. Through simulating the one-dimensional oscillatory Couette flow and two-dimensional flow around an oscillating circular cylinder, we examined numerical accuracy of this block iterative LB algorithm proposed in this article. Importantly, we also explored its computational efficiency in comparison to that of the conventional time-marching LB algorithm and several simplified block-iterative LB algorithms. Our results demonstrate the block iterative LB algorithm in this article is a useful alternative numerical solver exhibiting nearly 2nd order accuracy for simulating frequency-dependent oscillatory flow in MEMS. The simulations also reveal rich possibilities in construction of the block iterative LB algorithm through combining the LB theory in the frequency domain with advanced CFD numerical techniques.