[Paper Review] Chapman-Enskog Analysis of Finite Volume Lattice Boltzmann Schemes

Nima H. Siboni, Dierk Raabe+1 more|arXiv (Cornell University)|Jul 20, 2014

Lattice Boltzmann Simulation Studies48 references3 citations

TL;DR

This paper presents a systematic Chapman-Enskog analysis of finite volume lattice Boltzmann schemes in two dimensions, deriving the relationship between the lattice Boltzmann relaxation time and fluid kinematic viscosity for different flux evaluation schemes. It shows that the constant upwind scheme introduces artificial numerical viscosity, while central and linear upwind schemes avoid this artifact, enabling accurate recovery of the Navier-Stokes equations with consistent viscosity scaling.

ABSTRACT

In this paper, we provide a systematic analysis of some finite volume lattice Boltzmann schemes in two dimensions. A complete iteration cycle in time evolution of discretized distribution functions is formally divided into collision and propagation (streaming) steps. Considering mass and momentum conserving properties of the collision step, it becomes obvious that changes in the momentum of finite volume cells is just due to the propagation step. Details of the propagation step are discussed for different approximate schemes for the evaluation of fluxes at the boundaries of the finite volume cells. Moreover, a full Chapman-Enskog analysis is conducted allowing to recover the Navier-Stokes equation. As an important result of this analysis, the relation between the lattice Boltzmann relaxation time and the kinematic viscosity of the fluid is derived for each approximate flux evaluation scheme. In particular, it is found that the constant upwind scheme leads to a positive numerical viscosity while the central scheme as well as the linear upwind scheme are free of this artifact.

Motivation & Objective

• To provide a systematic analytical framework for finite volume lattice Boltzmann methods, which are widely used but lack comprehensive theoretical analysis.
• To investigate how different flux evaluation schemes in finite volume LBM affect the macroscopic fluid behavior, particularly viscosity.
• To derive the effective kinematic viscosity in terms of the lattice Boltzmann relaxation time for various schemes using Chapman-Enskog expansion.
• To clarify the origin of numerical viscosity in upwind-type schemes and distinguish it from physical viscosity.
• To validate the consistency of finite volume LBM with the Navier-Stokes equations under multiscale asymptotic analysis.

Proposed method

• The time evolution of the distribution function is decomposed into collision and propagation (streaming) steps, with mass and momentum conservation enforced in the collision step.
• Finite volume cells are defined around lattice nodes, with fluxes computed at cell boundaries using different approximate schemes: central, linear upwind, and constant upwind.
• A multiscale Chapman-Enskog expansion is applied, introducing a small parameter ε to separate fast (collision) and slow (macroscopic) time scales.
• The distribution function is expanded as F_i = F_i^0 + εF_i^1 + ε²F_i^2 + ..., and the equations are solved order-by-order in ε to derive the macroscopic equations.
• The Navier-Stokes equations are recovered in the incompressible limit, with viscous terms derived from second-order contributions in the expansion.
• The effective viscosity is computed by comparing the derived viscous terms with the standard Navier-Stokes form, yielding explicit relations between τ and μ for each scheme.

Experimental results

Research questions

• RQ1 How do different flux evaluation schemes in finite volume lattice Boltzmann methods affect the effective viscosity of the simulated fluid?
• RQ2 What is the relationship between the lattice Boltzmann relaxation time τ and the kinematic viscosity ν in finite volume schemes?
• RQ3 Why does the constant upwind scheme introduce an additional numerical viscosity not present in central or linear upwind schemes?
• RQ4 Can the Chapman-Enskog analysis consistently recover the Navier-Stokes equations for finite volume LBM with non-uniform flux approximations?
• RQ5 What is the origin of the extra viscous term in the constant upwind scheme, and how does it modify the physical viscosity?

Key findings

• The constant upwind scheme introduces an additional numerical viscosity term proportional to 1/2, resulting in an effective viscosity μ = ρc_s²(τ + 1/2), which exceeds the physical viscosity.
• The central and linear upwind schemes do not introduce this spurious viscosity, leading to a clean recovery of the physical viscosity μ = ρc_s²τ.
• The second-order Chapman-Enskog expansion shows that the viscous term in central and linear upwind schemes arises from the collision and streaming contributions, consistent with the standard Navier-Stokes form.
• The viscous term in the constant upwind scheme is attributed to the second-order term in the flux approximation, specifically from the ∂²F₀/∂x² and ∂²F₀/∂y² contributions.
• The analysis confirms that central and linear upwind schemes are free from artificial numerical viscosity, making them more accurate for simulating physical fluid dynamics.
• The derivation establishes a clear analytical link between the lattice Boltzmann relaxation time τ and the kinematic viscosity ν for each flux evaluation method, enabling proper parameter calibration.