Large-eddy simulation (LES) of turbulence in complex geometries is often conducted using strongly inhomogeneous resolution. The issues associated with resolution inhomogeneity are related to the noncommutativity of the filtering and differentiation operators, which introduces a commutation term into the governing equations. Neglect of this commutation term gives rise to commutation error. While the commutation error is well recognized, it is often ignored in practice. Moreover, the commutation error arising from the implicit part of the filter (i.e., projection onto the underlying discretization) has not been well investigated. Modeling the commutator between numerical projection and differentiation is crucial for correcting errors induced by resolution inhomogeneity in practical LES settings, which typically rely solely on implicit filtering. Here we employ a multiscale asymptotic analysis to investigate the characteristics of the commutator. This provides a statistical description of the commutator, which can serve as a target for the statistical characteristics of a commutator model. Further, we investigate how commutation error manifests in simulation and demonstrate its impact on the convection of a packet of homogeneous isotropic turbulence through an inhomogeneous grid. A connection is made between the commutation error and the propagation properties of the underlying numerics. A modeling approach for the commutator is proposed that is applicable to LES with filters that include projections to the discrete solution space and that respects the numerical properties of the LES evolution equation. It may also be useful in addressing other LES modeling issues such as discretization error.

中文翻译：

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大涡模拟中分辨率不均匀性的影响

复杂几何形状中湍流的大涡模拟 (LES) 通常使用强烈的非均匀分辨率进行。与分辨率不均匀性相关的问题与滤波和微分算子的非对易性有关，这将交换项引入控制方程。忽略该换向项会导致换向误差。虽然换向误差是公认的，但在实践中经常被忽略。此外，由滤波器的隐式部分（即投影到底层离散化）引起的换向误差还没有得到很好的研究。对数值投影和微分之间的换向器进行建模对于纠正实际 LES 设置中分辨率不均匀引起的错误至关重要，这通常仅依赖于隐式滤波。在这里，我们采用多尺度渐近分析来研究换向器的特性。这提供了换向器的统计描述，可以作为换向器模型统计特性的目标。此外，我们研究了换向误差在模拟中的表现，并证明了它对通过非均匀网格的均匀各向同性湍流包的对流的影响。在换向误差和基础数值的传播特性之间建立了联系。提出了一种换向器建模方法，该方法适用于带有滤波器的 LES，该滤波器包括对离散解空间的投影，并尊重 LES 演化方程的数值特性。