[Paper Review] Macroscopic Lattice Boltzmann Method (MacLAB)

Jian Zhou|arXiv (Cornell University)|Jan 5, 2019

Lattice Boltzmann Simulation Studies30 references4 citations

TL;DR

This paper introduces the Macroscopic Lattice Boltzmann Method (MacLAB), a reformulation of the standard lattice Boltzmann method that eliminates the collision step by relying solely on the lattice size $\delta x$ as the single fundamental parameter. By embedding viscosity through particle speed $e = \delta x / \delta t$, MacLAB achieves unconditional stability, reduced memory usage, and direct use of macroscopic boundary conditions, demonstrating accurate simulation of viscous and inviscid flows in 2D and 3D benchmarks including cavity and Taylor-Green vortex flows.

ABSTRACT

The birth of the lattice Boltzmann method (LBM) fulfils a dream that simple arithmetic calculations can simulate complex fluid flows without solving complicated partial differential flow equations. Its power and potential of resolving more and more challenging physical problems have been and are being demonstrated in science and engineering covering a wide range of disciplines such as physics, chemistry, biology, material science and image analysis. The method is a highly simplified model for fluid flows using a few limited fictitious particles that move one grid at a constant time interval and collide each other at a grid point on uniform lattices, which are the two routine steps for implementation of the method to simulate fluid flows. As such, a real complex particle dynamics is approximated as a regular particle model using three parameters of lattice size, particle speed and collision operator. A fundamental question is "Are the two steps integral to the method or can the three parameters be reduced to one for a minimal lattice Boltzmann method?". Here, I show 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 lattice size dx, 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. The findings have been 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.

Motivation & Objective

• To investigate whether the collision step in the lattice Boltzmann method can be removed without sacrificing accuracy or stability.
• To develop a minimal LBM formulation based solely on macroscopic physical variables and a single parameter, lattice size $\delta x$.
• To achieve unconditional stability, reduced memory usage, and direct macroscopic boundary condition implementation.
• To validate the method on viscous and inviscid flows across 2D and 3D configurations.

Proposed method

• Reformulates the standard LBM by removing the collision step, retaining only the streaming step with particle speed $e = \delta x / \delta t$.
• Defines the particle distribution function using only the equilibrium distribution $f_{\alpha}^{eq}$, eliminating the need for a collision operator.
• Embeds fluid viscosity naturally through the particle speed $e$, making $\nu = \frac{1}{6}(2\tau - 1)e\delta x$ implicit in the lattice dynamics.
• Uses the standard D2Q9 and D3Q19 lattice structures but removes the relaxation time $\tau$ and collision matrix $\Omega_{\alpha\beta}$.
• Applies the Chapman-Enskog analysis to confirm that the resulting scheme recovers the Navier-Stokes equations at second-order accuracy.
• Employs macroscopic variables (density $\rho$, velocity $u_i$) directly as boundary conditions, simplifying implementation.

Experimental results

Research questions

• RQ1 Can the lattice Boltzmann method be simplified to a single-parameter model by eliminating the collision step?
• RQ2 Does a collision-free LBM based only on lattice size $\delta x$ still recover the Navier-Stokes equations accurately?
• RQ3 Can such a method achieve unconditional stability and reduced memory usage compared to standard LBM?
• RQ4 How does the method perform on benchmark viscous and inviscid flow problems in 2D and 3D?
• RQ5 Can macroscopic boundary conditions be directly applied without reconstructing distribution functions?

Key findings

• The MacLAB model successfully recovers the Navier-Stokes equations at second-order accuracy through Chapman-Enskog analysis, confirming its theoretical validity.
• The method achieves unconditional stability, eliminating the numerical instability issues common in single-relaxation-time LBM.
• Memory usage is significantly reduced because the collision step and associated storage of distribution functions are no longer required.
• Boundary conditions can be applied directly using macroscopic variables such as density and velocity, simplifying implementation.
• Numerical tests confirm accurate simulation of 2D and 3D cavity flows, as well as unsteady Taylor-Green vortex flows, for both viscous and inviscid cases.
• The particle speed $e = \delta x / \delta t$ implicitly encodes fluid viscosity, making $\nu = \frac{1}{6}(2\tau - 1)e\delta x$ a natural consequence of the lattice structure.