Macroscopic Lattice Boltzmann Method

Abstract

The lattice Boltzmann method (LBM) is a highly simplified model for fluid flows using a few limited fictitious particles. It has been developed into a very efficient and flexible alternative numerical method in computational physics, demonstrating its great power and potential for resolving more and more challenging physical problems in science and engineering covering a wide range of disciplines such as physics, chemistry, biology, material science and image analysis. The LBM is implemented through the two routine steps of streaming and collision using the three parameters of the lattice size, particle speed and collision operator. A fundamental question is if the two steps are integral to the method or if the three parameters can be reduced to one for a minimal lattice Boltzmann method. In this paper, it is shown that the collision step can be removed and the standard LBM can be reformulated into a simple macroscopic lattice Boltzmann method (MacLAB). This model relies on macroscopic physical variables only and is completely defined by one basic parameter of the lattice size δx, bringing the LBM into a precise “lattice” Boltzmann method. The viscous effect on flows is naturally embedded through the particle speed, making it an ideal automatic simulator for fluid flows. Three additional advantages compared to the existing LBMs are that: (i) physical variables can directly be retained as the boundary conditions; (ii) much less computational memory is required; and (iii) the model is unconditionally stable. The findings are demonstrated and confirmed with numerical tests including flows that are independent of and dependent on fluid viscosity, 2D and 3D cavity flows and an unsteady Taylor–Green vortex flow. This provides an efficient and powerful model for resolving physical problems in various disciplines of science and engineering.

Quantum algorithm for the advection–diffusion equation simulated with the lattice Boltzmann method

“…The advection velocity vector for 2D case is − → u = ui + v j, where i and j are unit vectors along x and y direction. It should be noted that following the work of Zhou [47] parameter ω is set to 1.0 in both cases.…”

“…( 5)). To simplify the collision step without destroying the very core of the lattice Boltzmann method, according to Zhou [47], parameter ω is set to 1.0. As a result, the left side of Eq.…”

Macroscopic axisymmetric lattice Boltzmann method (MacAxLAB)

“…After substitution of τ = 1 into the above equation following Zhou's idea in MacLAB [21], it can be simplified to…”

“…Setting τ = 1 and taking Eq. ( 32) following Zhou's idea [21] lead to fα (x, t) = f eq α (x − e α δt, t − δt)]…”

“…Also, the no slip boundary condition cannot be exactly achieved through application of the most popular and efficient bounce-back scheme. These drawbacks have been removed by Zhou [21] in his newly developed macroscopic lattice Boltzmann method (MacLAB) for Navier-Stokes equations for fluid flows. In this paper, the MacLAB is extended to solve the axisymmetric flow equations.…”