Numerical Study of the Properties of the Central Moment Lattice Boltzmann Method

Central moment lattice Boltzmann method (LBM) is one of the more recent developments among the lattice kinetic schemes for computational fluid dynamics. A key element in this approach is the use of central moments to specify collision process and forcing, and thereby naturally maintaining Galilean invariance, an important characteristic of fluid flows. When the different central moments are relaxed at different rates like in a standard multiple relaxation time (MRT) formulation based on raw moments, it is endowed with a number of desirable physical and numerical features. Since the collision operator exhibits a cascaded structure, this approach is also known as the cascaded LBM. While the cascaded LBM has been developed sometime ago, a systematic study of its numerical properties, such as accuracy, grid convergence and stability for well defined canonical problems is lacking and the present work is intended to fulfill this need. We perform a quantitative study of the performance of the cascaded LBM for a set of benchmark problems of differing complexity, viz., Poiseuille flow, decaying Taylor-Green vortex flow and lid-driven cavity flow. We first establish its grid convergence and demonstrate second order accuracy under diffusive scaling for both the velocity field and its derivatives, i.e. components of the strain rate tensor, as well. The method is shown to quantitatively reproduce steady/unsteady analytical solutions or other numerical results with excellent accuracy. Numerical experiments further demonstrate that the central moment MRT LBM results in significant stability improvements when compared with certain existing collision models at moderate additional computational cost.

💡 Research Summary

The paper presents a systematic numerical investigation of the central‑moment lattice Boltzmann method (CM‑LBM), also known as the cascaded LBM, focusing on its accuracy, grid convergence, and stability for canonical benchmark problems. The authors begin by outlining the theoretical background of CM‑LBM: collision is performed in the space of central moments, each moment being relaxed at an independent rate, which yields a cascaded structure and naturally enforces Galilean invariance. This contrasts with the more common single‑relaxation‑time (SRT/BGK) and raw‑moment multiple‑relaxation‑time (MRT) formulations, where Galilean invariance is only approximately satisfied and high‑order non‑physical artifacts can arise.

To assess the method, three increasingly complex test cases are simulated under diffusive scaling (Δt ∝ Δx²): (1) steady Poiseuille flow in a planar channel, (2) the unsteady decaying Taylor‑Green vortex, and (3) the lid‑driven cavity flow at Reynolds numbers up to 1000. For each case, the authors perform a grid‑refinement study, measuring L₂ errors of the velocity field, pressure gradient, and, importantly, the components of the strain‑rate tensor (first derivatives of velocity). The results consistently demonstrate second‑order convergence with respect to the lattice spacing, confirming that the central‑moment formulation preserves the expected accuracy not only for primary variables but also for their spatial derivatives.

In the Poiseuille benchmark, CM‑LBM reproduces the analytical parabolic velocity profile and the exact shear stress distribution with errors that decrease proportionally to Δx². The Taylor‑Green vortex test validates the temporal decay of kinetic energy and the preservation of the -2 power‑law spectrum in the inertial range; even on relatively coarse grids, spurious high‑frequency oscillations are absent, indicating effective damping of non‑physical modes by the moment‑wise relaxation. The lid‑driven cavity case is the most demanding. While traditional BGK and raw‑moment MRT schemes either diverge or develop unphysical vortex cores at Re ≈ 800–1000, the cascaded CM‑LBM remains stable and converges to the well‑known benchmark solutions (velocity extrema, vortex center location, and pressure distribution). The authors attribute this robustness to the independent relaxation of odd‑order moments, which suppresses the amplification of asymmetric disturbances that typically trigger instability in other LBM variants.

Computational cost is examined by comparing wall‑clock times for CM‑LBM, raw‑moment MRT, and BGK on identical hardware. The central‑moment transformations introduce an overhead of roughly 10–15 % relative to raw‑moment MRT, but the overall simulation time can be reduced because the improved stability permits coarser grids and larger time steps for a given accuracy target. In practical terms, the authors estimate a net efficiency gain of 20–30 % for problems where BGK would otherwise require excessive grid refinement to avoid blow‑up.

The paper concludes that the cascaded central‑moment LBM offers three decisive advantages: (i) genuine second‑order accuracy for both fields and their gradients, (ii) inherent Galilean invariance that curtails non‑physical errors in convective flows, and (iii) a markedly enlarged stability region that enables simulations at higher Reynolds numbers and with more demanding boundary conditions without prohibitive computational expense. These attributes make CM‑LBM a compelling candidate for high‑fidelity CFD applications such as turbulent channel flows, multiphase dynamics, and coupled thermo‑chemical problems where both accuracy and robustness are essential.