The underlying physical mechanism of the residual-based large eddy simulation (LES) based on the variational multiscale (VMS) method is clarified. Resolved large-scale energy transportation equation is mathematically derived for turbulent kinetic energy budget analysis. Firstly, statistical results of benchmark turbulent channel flow at $R{e}_{\tau }=180$ obtained using a coarse mesh are compared with the results obtained by the classical LES with the Smagorinsky and dynamic subgrid stress (SGS) model. The present LES shows an advantage in predicting the statistical results of the incompressible turbulent flows. Secondly, the contributions of the unresolved small-scale presentation terms (Term $\text{I}$-$\text{IV}$ in Eq. (10)) to the turbulent kinetic dissipation are analysed for the VMS method. The results show that the turbulent kinetic dissipation provided by the numerical diffusion in the VMS method is smaller in the inner layer, larger in the outer layer of the channel flow than those by the Smagorinsky and dynamic SGS model. The turbulent kinetic dissipation in the VMS method is mainly given by the numerical diffusion provided by one of the “cross-stress” terms (Term $\text{I}$, same as the stabilization term in the SUPG method) and LSIC term (Term $\text{IV}$). The other one of the “cross-stress” terms (Term $\text{II}$) gives rise to the positive turbulent kinetic energy budget, and does not dissipate the turbulent kinetic energy. The so-called “Reynolds stress” term (Term $\text{III}$) dissipates the turbulent energy but provides a very small numerical diffusion. Finally, on the basis of the turbulent kinetic energy dissipation analysis, a new residual-based stabilized finite element formulation is proposed by modifying the large-scale equation in the VMS method. Numerical experiments of 2D lid-driven cavity flow and 3D incompressible turbulent channel flow are tested to validate the proposed formulation. It is shown that all the stabilization terms in the proposed formulation produce additional numerical diffusions and physically increase the total turbulent kinetic dissipation. Consequently, an apparent improvement in both the first-order and second-order statistical quantities are pursued by the new stabilized finite element formulation.

阐明了基于变分多尺度 (VMS) 方法的基于残差的大涡模拟 (LES) 的潜在物理机制。求解的大规模能量传输方程是数学推导的湍流动能收支分析。首先，基准湍流通道流的统计结果$\mathrm{电阻}{\mathrm{电子}}_{\tau }=180$将使用粗网格获得的结果与使用 Smagorinsky 和动态亚网格应力 (SGS) 模型的经典 LES 获得的结果进行比较。目前的 LES 在预测不可压缩湍流的统计结果方面具有优势。其次，未解决的小规模表示项（Term$\text{一世}$——$\text{四}$在方程式中 (10)) 对湍流动力学耗散进行了分析，用于 VMS 方法。结果表明，与 Smagorinsky 和动态 SGS 模型相比，VMS 方法中数值扩散提供的湍流动力学耗散在内层较小，在外层较大。VMS 方法中的湍流动力学耗散主要由“交叉应力”项（Term$\text{一世}$, 与 SUPG 方法中的稳定项相同）和 LSIC 项（Term $\text{四}$）。另一个“交叉应力”术语（Term$\text{二}$) 产生正的湍动能预算，并且不耗散湍动能。所谓的“雷诺应力”项（Term$\text{三}$) 耗散湍流能量，但提供非常小的数值扩散。最后，在湍动能耗散分析的基础上，通过修改VMS方法中的大尺度方程，提出了一种新的基于残差的稳定有限元公式。对 2D 盖驱动腔流和 3D 不可压缩湍流通道流的数值实验进行了测试，以验证所提出的公式。结果表明，所提出的公式中的所有稳定项都会产生额外的数值扩散，并在物理上增加了总湍流动力学耗散。因此，新的稳定有限元公式追求一阶和二阶统计量的明显改进。