Invariants of the reduced velocity gradient tensor in turbulent flows
Invariants of the reduced velocity gradient tensor in turbulent flows
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DOI：
10.1017/jfm.2012.558
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发表时间：
2012-11
影响因子：
3.7
通讯作者：
J. Cardesa;Dhiren Mistry;Lian Gan;James R. Dawson
中科院分区：
文献类型：
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作者：
J. Cardesa;Dhiren Mistry;Lian Gan;James R. Dawson
摘要：在本文中，我们研究了由不可压缩三维 (3D) 流的二维 (2D) 切片形成的简化 $2\times 2$ 速度梯度张量 (VGT) 的不变量 $p$ 和 $q$。使用来自 2D 粒子图像测速 (PIV) 测量和各种湍流的 3D 直接数值模拟的数据，我们表明 $p$ 和 $q$ 的联合概率密度函数 (p.d.f.s) 表现出与 $\langle pq\rangle \lt 0$ 一致的共同特征不对称形状。提出了对这种不等式的解释。假设局部同质性，我们得出 $\langle p\rangle = 0$ 和 $\langle q\rangle = 0$ 。加上局部各向同性，$\langle pq\rangle $ 的符号被证明与 $\partial {u}_{1} / \partial {x}_{1} $ 的偏度符号相同，因此为负。这表明，在 $p{{\ndash}}q$ 的联合 p.d.f.s 中观察到的不对称性源于最小尺度上涡旋拉伸的普遍优势。该联合 PDF 的一些优点讨论了与从完整的$3\times 3$ VGT 获得的$Q{{\ndash}}R$ 的比较。通过分析与降低的 VGT 相关的降低的应变率矩阵的特征值，我们证明在某些情况下，2D 数据可以明确区分完整的 3\times 3$ 应变率张量的双轴（板材成型）和轴向（管材成型）应变率配置。
Abstract In this paper we examine the invariants $p$ and $q$ of the reduced $2\times 2$ velocity gradient tensor (VGT) formed from a two-dimensional (2D) slice of an incompressible three-dimensional (3D) flow. Using data from both 2D particle image velocimetry (PIV) measurements and 3D direct numerical simulations of various turbulent flows, we show that the joint probability density functions (p.d.f.s) of $p$ and $q$ exhibit a common characteristic asymmetric shape consistent with $\langle pq\rangle \lt 0$ . An explanation for this inequality is proposed. Assuming local homogeneity we derive $\langle p\rangle = 0$ and $\langle q\rangle = 0$ . With the addition of local isotropy the sign of $\langle pq\rangle $ is proved to be the same as that of the skewness of $\partial {u}_{1} / \partial {x}_{1} $ , hence negative. This suggests that the observed asymmetry in the joint p.d.f.s of $p{{\ndash}}q$ stems from the universal predominance of vortex stretching at the smallest scales. Some advantages of this joint p.d.f. compared with that of $Q{{\ndash}}R$ obtained from the full $3\times 3$ VGT are discussed. Analysing the eigenvalues of the reduced strain-rate matrix associated with the reduced VGT, we prove that in some cases the 2D data can unambiguously discriminate between the bi-axial (sheet-forming) and axial (tube-forming) strain-rate configurations of the full $3\times 3$ strain-rate tensor.