在格子玻尔兹曼方法中，矩基底何时重要？

中文摘要 AI 辅助

是否对矩基底进行正交化会改变中心矩格子玻尔兹曼方案，这取决于速度集的几何结构。基底仅通过共轭矩阵 $\mathit{T}^{-1} \Lambda \mathit{T}$ 进入碰撞项，因此两个由常数矩阵 $\mathit{A}$ 关联的基底定义完全相同的方案，当且仅当 $[\Lambda,\mathit{A}]_{ij}=\mathit{A}_{ij}(\lambda_i-\lambda_j)$ 为零；并且在物理状态上，只有非守恒矩对应的列才起作用。我们证明，在速度属于 $\{0,\pm1\}^d$ 且权重对称的格子上，正交化仅通过面对角线和体对角线达到剪切矩：无论正交化方式如何，在 D2Q9 和 D3Q15 上剪切率是自由的，而在 D3Q19 和 D3Q27 上，当四阶矩不以剪切率松弛时，方案会发生变化。在对称性被破坏的矩形格子上，独立的体积率产生一个正交对应方案，该方案与已发表方案的输运系数不同，而已发表方案本身在静止平衡的内积下是正交的，除了一致性要求的一个松弛项。在公式不同但流体动力学相同的场合，在静止平衡内积下正交的基底（这保证了静止时的线性稳定性）在最不稳定的流动方向上从未比其它方案差超过几个百分点，通常更稳健，并且相对于 Taylor-Green 直接数值模拟（DNS）更精确。通过非正交变换和共轭松弛矩阵计算，在 D3Q27 上其成本最多增加 $2\\%$。

英文摘要

Whether orthogonalising the moment basis changes a central-moment lattice Boltzmann scheme is decided by the geometry of the velocity set. The basis enters the collision only through the conjugate $\mathit{T}^{-1} Λ\mathit{T}$, so two bases related by a constant matrix $\mathit{A}$ define exactly the same scheme if and only if $[Λ,\mathit{A}]_{ij}=\mathit{A}_{ij}(λ_i-λ_j)$ vanishes, and on physical states only its columns for the non-conserved moments matter. We prove that on lattices with velocities in $\{0,\pm1\}^d$ and symmetric weights orthogonalisation reaches the shear moments only through the face and body diagonals: whatever the orthogonalisation, the shear rate is free on D2Q9 and D3Q15, whereas on D3Q19 and D3Q27 the scheme changes whenever the fourth-order moments do not relax at the shear rate. On rectangular lattices, where the symmetry is broken, an independent bulk rate yields an orthogonal counterpart that does not share the transport coefficients of the published scheme, which is itself orthogonal in the inner product of the rest equilibrium up to the one relaxation entry that consistency requires. Where the formulations differ but share their hydrodynamics, a basis orthogonal in the inner product of the rest equilibrium, which guarantees linear stability at rest, was never less robust in the least stable flow direction by more than a few per cent, often much more robust, and the more accurate against a Taylor-Green DNS. Computed through the non-orthogonal transforms with a conjugated relaxation matrix, it costs at most $2\%$ more on D3Q27.

发表机构

• The University of Manchester（曼彻斯特大学）

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