[论文解读] A fluctuating lattice Boltzmann formulation based on orthogonal central moments

Alessandro De Rosis, Yang Zhou|arXiv (Cornell University)|Feb 18, 2026

Lattice Boltzmann Simulation Studies被引用 0

一句话总结

该论文开发了一种使用正交中心矩的涨落格子玻尔兹曼方法，在 moment 空间推导噪声，并在 D2Q9 和 D3Q27 晶格上实现格子层面的精确涨落-耗散平衡。

ABSTRACT

Thermal fluctuations play a central role in fluid dynamics at mesoscopic scales and must be incorporated into numerical schemes in a manner consistent with statistical mechanics. In this work, we develop a fluctuating lattice Boltzmann formulation based on an orthogonal central-moments-based representation. Stochastic forcing is introduced directly in the space of central moments (CMs) and consistently paired with mode-dependent relaxation, yielding a discrete kinetic model that satisfies the fluctuation-dissipation theorem exactly at the lattice level. Owing to the orthogonality of the basis, the equilibrium covariance matrix of the central moments is diagonal, and each non-conserved mode can be interpreted as an independent discrete Ornstein-Uhlenbeck process with variance fixed by equilibrium thermodynamics. The resulting formulation guarantees exact equipartition of kinetic energy at equilibrium, preserves Galilean invariance, and retains the enhanced numerical stability characteristic of CMs-based collision operators. Explicit fluctuating schemes are constructed for the D2Q9 and D3Q27 lattices. The extension to reduced-velocity discretisation is discussed too. A comprehensive set of numerical tests verifies correct thermalisation, isotropy of equilibrium statistics, and the expected scaling of velocity fluctuations with thermal energy, density, and relaxation time. In contrast to fluctuating BGK formulations, the present method remains stable and well posed in the over-relaxation regime, including in the immediate vicinity of the stability limit. These results demonstrate that CMs-based lattice Boltzmann methods provide a natural and robust framework for fluctuating hydrodynamics, in which dissipation, noise, and kinetic mode structure are consistently aligned at the discrete level.

研究动机与目标

• 在格子玻尔兹曼方法中以符合统计力学与守恒定律的方式引入热涨落的动机。
• 开发基于中心矩的涨落 LBM (CM-FLBM)，实现伽利金不变量性与稳定性。
• 推导 D2Q9 与 D3Q27 的显式涨落碰撞算子并讨论降速扩展。
• 确保在平衡时动能的完全等分以及中心矩空间中的对角线平衡协方差。

提出的方法

• 直接在中心矩中进行带模式相关松弛的随机强迫的表述。
• 使用正交的中心矩基底获得对角线平衡协方差并实现每个模式的独立噪声。
• 推导碰撞后中心矩并回变换到 D2Q9 与 D3Q27 的群体分布。
• 证明该方案满足涨落-耗散平衡，并在流体极限下恢复 Landau–Lifshitz 的涨落流体力学。
• 给出噪声振幅和碰撞步骤的显式表达式，确保在过松弛状态下的伽利金不变量性与稳定性。

Figure 1: Test 2: grid-convergence of the Taylor–Green vortex. Log–log plot of the $L_{2}$ -norm of the velocity error versus grid resolution, showing a fitted slope of $-2$ , consistent with second-order accuracy of the scheme in space and time. Findings obtained by the adoption of an orthogonal ba

实验结果

研究问题

• RQ1 如何在格子层面以符合 FDT 的方式将热涨落并入 LBM？
• RQ2 正交中心矩基是否允许对非守恒模式进行独立的随机强迫并保持对角线平衡协方差？
• RQ3 D2Q9 与 D3Q27 的显式 CM-FLBM 方案是否能再现正确的平衡涨落与能量等分？
• RQ4 在 CM-LBM 中使用四阶 Hermite 一致性平衡对确定性与随机贡献的分离有何影响？
• RQ5 在广泛的松弛参数范围（包括过松弛区间）下 CM-FLBM 的性能表现如何？

主要发现

• CM-FLBM 在晶格层面实现了动能的精确平衡等分。
• 在正交 CM 基中，平衡协方差为对角线，使得每个模式的独立松弛与噪声成为可能。
• 推导并展示了 D2Q9 与 D3Q27 的显式涨落算子，保留伽利金不变量性与稳定性。
• 该方法在过松弛区间仍然稳健，在许多情形下优于涨落 BGK 形式。
• 数值测试验证了正确的热化、平衡统计量的各向同性，以及速度涨落随热能、密度和松弛时间的期望尺度。

Figure 2: Test 3. Time evolution of the domain-averaged velocity variances $\langle u_{x}^{2}\rangle$ (dark magenta line with triangles) and $\langle u_{y}^{2}\rangle$ (dark cyan line with inverted triangles) for a homogeneous system with thermal fluctuations enabled ( $k_{B}T=1/3000$ , $\rho_{0}=1$