Momentum transfer of a Boltzmann-lattice fluid with boundaries

TLDR

The study proposes a lattice Boltzmann boundary condition for moving curved boundaries that blends bounce‑back with first‑ or second‑order spatial interpolation. The authors validate the boundary condition by comparing lattice Boltzmann results with analytic Navier–Stokes solutions for circular Couette flow and with finite‑element solutions for flow past a periodic array of cylinders, and they analyze torque from momentum transfer for impulsively started and rotating fluid–cylinder configurations. The boundary condition is simple, robust, efficient, and achieves second‑order accuracy, with lattice Boltzmann solutions closely matching Navier–Stokes results for both circular Couette flow and flow past cylindrical arrays.

Abstract

We study the velocity boundary condition for curved boundaries in the lattice Boltzmann equation (LBE). We propose a LBE boundary condition for moving boundaries by combination of the “bounce-back” scheme and spatial interpolations of first or second order. The proposed boundary condition is a simple, robust, efficient, and accurate scheme. Second-order accuracy of the boundary condition is demonstrated for two cases: (1) time-dependent two-dimensional circular Couette flow and (2) two-dimensional steady flow past a periodic array of circular cylinders (flow through the porous media of cylinders). For the former case, the lattice Boltzmann solution is compared with the analytic solution of the Navier–Stokes equation. For the latter case, the lattice Boltzmann solution is compared with a finite-element solution of the Navier–Stokes equation. The lattice Boltzmann solutions for both flows agree very well with the solutions of the Navier–Stokes equations. We also analyze the torque due to the momentum transfer between the fluid and the boundary for two initial conditions: (a) impulsively started cylinder and the fluid at rest, and (b) uniformly rotating fluid and the cylinder at rest.