For a long time, people hold an idea that vorticity is equivalent to vortex for fluid flow since vorticity represents rotation for rigid body. Nevertheless, many experimental results do not support this opinion. So, several improved methods have been proposed, including Q, ${\lambda }_{ci}$, ${\lambda }_2$, $\Delta$ methods and etc, which are all based on the eigenvalues of the velocity gradient tensor. These methods share a common drawback that they are all scalars and, as a result, are unable to locate the swirling axis which is an important information of rotation. To overcome this shortage, Liutex was proposed as a vector indicator of vortex. The direction of Liutex represents the swirling axis and the magnitude of Liutex is defined as twice the angular speed of rotation. After the introduction of Liutex, many experiments and numerical simulations have shown that Liutex can accurately and correctly capture both big and small vortex, which is better than the existing methods. In this paper, the explicit formulae of vorticity, Q, ${\lambda }_{ci}$ and $\Delta$ in terms of Liutex are derived, followed by correlation analysis based on a direct numerical simulation (DNS) result of boundary layer transition and a large eddy simulation (LES) result of supersonic ramp flow with a fully developed turbulent boundary layer. The results show that correlation between vorticity and Liutex is very small and even negative in strong shear regions. Although the correlations of Q, ${\lambda }_{ci}$, ${\lambda }_2$ and $\Delta$ are better than vorticity, they are still small in strong shear regions. The expression of relation between vorticity, Q, ${\lambda }_{ci}$, $\Delta$ and Liutex reveals how these methods are contaminated by shear or stretching or both.

长期以来，人们一直认为涡旋等于流体流动的涡旋，因为涡旋代表刚体的旋转。尽管如此，许多实验结果并不支持这一观点。因此，提出了几种改进的方法，包括Q，${\lambda }_{C\mathrm{一世}}$， ${\lambda }_{\mathrm{2个}}$， $\Delta$方法等，这些方法均基于速度梯度张量的特征值。这些方法有一个共同的缺点，即它们都是标量，因此无法定位作为旋转的重要信息的旋转轴。为了克服这种不足，建议使用Liutex作为涡流的矢量指标。Liutex的方向代表旋转轴，Liutex的大小定义为旋转角速度的两倍。在引入Liutex之后，许多实验和数值模拟表明Liutex可以准确而正确地捕获大涡和小涡，这比现有方法要好。在本文中，涡度的显式公式Q${\lambda }_{C\mathrm{一世}}$ 和 $\Delta$得出有关Liutex的信息，然后基于边界层过渡的直接数值模拟（DNS）结果和具有完全展开的湍流边界层的超音速斜坡流的大涡模拟（LES）结果进行相关分析。结果表明，在强剪切区域，涡度与Liutex之间的相关性很小，甚至为负相关。虽然Q的相关性${\lambda }_{C\mathrm{一世}}$， ${\lambda }_{\mathrm{2个}}$ 和 $\Delta$比涡旋好，但在强剪切区域它们仍然很小。涡度Q之间的关系表达式${\lambda }_{C\mathrm{一世}}$， $\Delta$ Liutex揭示了这些方法是如何受到剪切或拉伸或两者同时污染的。