The three-dimensional (3D) Taylor–Green vortex (TGV) flow is one of the simplest systems to study the generation of different scales of vortices due to the growth of disturbances via effects of different physical mechanisms, including vortex-stretching as an additional source for instability, showing not only the creation of turbulence but also turbulent decay. The strong anisotropic and well-organized flow becomes unstable at early time due to transfer of energy to small scales. The analysis of instability of the periodic 3D TGV flow for the Reynolds number of Re = 2000 is reported here. The direct numerical simulation of the periodic 3D TGV flow is carried out using high accuracy numerical methods for the (vector-potential, vorticity)-formulation, which exactly satisfy the solenoidality condition for vector-potential and vorticity in the computational domain. The evolution of disturbances is examined using the instability theories of the disturbance mechanical energy of the Navier–Stokes equation and the role of rotationality by the disturbance enstrophy transport equation (DETE), which is derived from the enstrophy transport equation. The 3D TGV flow exhibits a tornado-type structure at the center of the domain at intermediate stages of transition to turbulence, which is analyzed using the vortex-identification method of λ₂-criteria and the DETE method, as described by Sengupta et al. [“Tracking disturbances in transitional and turbulent flows: Coherent structures,” Phys. Fluids 31(12), 124106 (2019)]. Here, it is observed that the coherent structure is diffused in the λ₂-contours. Third generation vortex-identification methods are analyzed for capturing the transient, rotating vortex. The combination of new Omega- and the Liutex/Rortex-methods, as reviewed by C. Liu et al. [“Third generation of vortex identification methods: Omega and Liutex/Rortex based systems,” J. Hydrodyn. 31(2), 205–223 (2019)], captures the evolution of the transient vortex, but the structure identified by these methods appears to be diffused, while the DETE method clearly captures the vortex geometry and highlights the formation of the vortex at early times to aid in predicting the flow evolution.

中文翻译：

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三维Navier-Stokes方程的非线性不稳定性分析：泰勒-格林涡旋问题

三维（3D）泰勒-格林涡流（TGV）流是研究由于扰动的增长而产生的不同尺度涡旋的最简单系统之一，扰动是通过不同物理机制的影响而产生的，包括涡旋拉伸作为附加不稳定的根源，不仅显示了湍流的产生，而且还显示了湍流的衰减。由于能量转移到小规模，强烈的各向异性和组织良好的流动在早期变得不稳定。Re的雷诺数的周期性3D TGV流的不稳定性分析=此处报告为2000。使用（矢量势，涡度）公式的高精度数值方法对3D TGV周期性流动进行直接数值模拟，该方法在计算域中完全满足矢量势和涡度的螺线管条件。使用Navier–Stokes方程的扰动机械能的不稳定性理论以及通过从涡旋迁移方程推导的扰动涡旋迁移方程（DETE）来研究旋转的作用，从而检验了扰动的演变。所述3D TGV流在域的在过渡到湍流，这是使用的涡识别方法分析的中间阶段的中心显示出龙卷风型结构λ ₂-标准和DETE方法，如Sengupta等人所述。[“在过渡流和湍流中的跟踪扰动：相干结构，”物理。流体31（12），124106（2019）]。在这里，可以观察到相干结构在扩散λ _2个-contours。分析了第三代涡流识别方法，以捕获瞬态旋转涡流。新的Omega方法和Liutex / Rortex方法的组合，如C. Liu等人所综述。[“涡流识别方法的第三代：基于Omega和Liutex / Rortex的系统，” J。Hydrodyn。31（2），205–223（2019）]，捕获了瞬态涡旋的演化，但是通过这些方法识别的结构似乎是分散的，而DETE方法清楚地捕获了涡旋的几何形状并突出了早期涡旋的形成。有助于预测流量演变的时间。