基于速度梯度张量四重分解的涡旋判据评估
Evaluation of Vortex Criteria by Virtue of the Quadruple Decomposition of Velocity Gradient Tensor
AI总结：
本文基于速度梯度张量的四重分解，研究了$ω$、$Q$、$\varDelta$和$λ_{ci}$四种涡旋判据的物理含义与局限性，明确了净旋转的运动学机制，增进了对流体涡旋运动的理解。
摘要：
基于对速度梯度张量的分析，本文研究了四种涡旋判据：$ω$、$Q$、$\varDelta$和$λ_{ci}$的物理含义及局限性，并揭示了通常由各种涡旋判据的水平集所图示的涡旋模式的实际物理意义。首次提出了一种基于速度梯度张量正规性的四重分解，将流体微元的运动分解为膨胀、沿正规标架的轴向拉伸、面内畸变和简单剪切，以阐明各种涡旋判据的运动学解释。由涡量$ω$表征的平均旋转始终包含简单剪切；$Q$判据能反映不变平面内相对于流体微元轴向拉伸的净旋转强度，且它是净旋转存在的充分非必要条件；$\varDelta$判据能精确识别净旋转的存在，但它并不代表净旋转的强度；在存在净旋转的情况下，$λ_{ci}$表征其绝对强度。净旋转是不变平面内正规旋转与简单剪切的总效应，其中前者是最基本的旋转。新引入的四重分解可增进我们对流体中涡旋及其运动的理解。
英文摘要：
Based on the analysis of the velocity gradient tensor, we investigate in this paper the physical interpretation and limitations of four vortex criteria: $ω$, $Q$, $\varDelta$ and $λ_{ci}$, and reveal the actual physical meaning of vortex patterns which are usually illustrated by level sets of various vortex criteria. A quadruple decomposition based on the normality of the velocity gradient tensor is proposed for the first time, which resolves the motion of a fluid element into dilation, axial stretch along the normal frame, in-plane distortion, and simple shear, in order to clarify the kinematical interpretation of various vortex criteria. The mean rotation characterized by the vorticity $ω$ always consists of simple shear; the $Q$-criterion can reflect the strength of net rotation within the invariant plane relative to the axial stretch of a fluid element, and it is a sufficient but unnecessary condition for the existence of net rotation; the $\varDelta$-criterion can exactly identify the existence of net rotation, but it is not the strength of net rotation; in the case that net rotation exists, $λ_{ci}$ characterize its absolute strength. net rotation is the total effect of the normal rotation within the invariant plane and the simple shear, where the former is the most basic rotation. The newly introduced quadruple decomposition can improve our understanding of vortices in fluids and their motions.
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