Scientists developed many vortex identification methods and Liu classified the traditional vortex identification methods into two generations. Vorticity and vorticity-based methods belong to the first generation while eigenvalues-based methods are considered as the second generation, such as Q criterion and λci method. However, although these methods are widely used, there is still room for improvements. People used to consider direction of vorticity is the swirling axis and the magnitude of it is the rotation strength, but the predictions by vorticity does not match the experiment very well which drove scientists to develop the second-generation methods. All the second-generation methods are scalar-valued and as a result they are unable to locate the swirling. Another problem is even though second-generation methods can predict vortex better than vorticity, they are more or less contaminated by shear. To solve these problems, Liu innovated Liutex recently. Liutex is a vector quantity whose direction represents rotation axis and whose magnitude represents rotation strength. Firstly, the physical essence of rotation axis is provided coming with the mathematical definition of swirling axis which is the direction of Liutex. Secondly, orthogonal transformations are used to find out the rigid rotation speed and that speed is defined as the strength of Liutex. Wang later proposed an explicit formula to calculate Liutex strength without doing orthogonal transformations which makes it much easier to apply Liutex method. Some theories dependent on Liutex concept have been proposed these years. Liutex core lines represent the core of vortex which can uniquely and clearly show vortex structure. Principal coordinate is a special coordinate in which it is easy to decompose the velocity gradient tensor into rotation, shear and stretching parts correctly. And principal decomposition is the decomposition in principal coordinate. The Liutex theory system is gradually formed and they are considered as the third generation by Liu.

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