The perceived velocity gradient tensor (PVGT), constructed from four fluid tracers forming a tetrahedron, provides a natural way to study the structure of velocity fluctuations and its dependence on spatial scales. It generalizes and shares qualitatively many properties with the true velocity gradient tensor. Here, we establish the evolution equation for the PVGT, and, for homogeneous and isotropic incompressible turbulent flows, we analyse the dynamics of the PVGT in particular using its second- and third-order invariants. We show that, for PVGT based on regular tetrads with lateral size $R_{0}$ , the second-order invariants can be expressed solely in terms of the usual second-order velocity structure functions, while the third-order invariants involve the usual third-order longitudinal velocity structure function and a less well known three-point velocity correlation function. For homogeneous and isotropic turbulence, exact relations between the second moments of strain and vorticity, as well as enstrophy production and the third moments of the strain, are derived. These generalized relations are valid for all ranges of $R_{0}$ , and reduce to classical results for the velocity gradient tensor when $R_{0}$ is in the dissipative range. With the help of these relations, we quantify the importance of the various terms, such as vortex stretching, as a function of the scale $R_{0}$ . Our analysis, which is supported by the results of direct numerical simulations of turbulent flows in the Reynolds-number range $100\leqslant R_{\unicode[STIX]{x1D706}}\leqslant 610$ , allows us to demonstrate that strain prevails over vorticity when $R_{0}$ is in the inertial range.

中文翻译：

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均匀和各向同性湍流中感知速度梯度张量的动力学和不变量

感知速度梯度张量 (PVGT) 由形成四面体的四个流体示踪剂构成，为研究速度波动的结构及其对空间尺度的依赖性提供了一种自然的方法。它概括并定性地与真实速度梯度张量共享许多属性。在这里，我们建立了 PVGT 的演化方程，并且对于均匀和各向同性的不可压缩湍流，我们特别使用其二阶和三阶不变量来分析 PVGT 的动力学。我们表明，对于基于横向尺寸为 $R_{0}$ 的规则四分体的 PVGT，二阶不变量可以仅用通常的二阶速度结构函数表示，而三阶不变量涉及通常的三阶纵向速度结构函数和鲜为人知的三点速度相关函数。对于均匀和各向同性的湍流，可以推导出应变和涡度的二阶矩以及熵产生和应变的三阶矩之间的精确关系。这些广义关系对 $R_{0}$ 的所有范围都有效，并且当 $R_{0}$ 在耗散范围内时，可以简化为速度梯度张量的经典结果。在这些关系的帮助下，我们将各种术语（例如涡旋拉伸）的重要性量化为尺度 $R_{0}$ 的函数。我们的分析，