Chapter 9 - Liutex and proper orthogonal decomposition for hairpin vortex generation

Abstract

Mechanism of hairpin vortex formation is important in turbulence research. In this chapter, the new Liutex core line method and the proper orthogonal decomposition (POD) are applied to support the hypothesis that the ring-like vortex is formed by the Kelvin–Helmholtz (K-H) instability. The new modified Liutex-Omega method is efficiently used to visualize the shapes of vortex structures. The POD analysis shows that the eigenvalues and eigenvectors of POD modes are shown in pairs, which is typical in the K-H instability. The same characteristics such as mode shapes and fluctuations are also established in pairs. This evidence is a strong proof that K-H instability plays a key role in the hairpin vortex formation from the Λ-vortex, which transforms the non-rotational vorticity or shear to the rotational vorticity or Liutex. This further confirms that the theory of Lambda vortex self-deformation to hairpin vortex has no solid scientific foundation. And then, the Liutex core line method and the snapshot POD are utilized in a supersonic wake flow of MVG at Ma = 2.5 and Reθ = 5670. The K-H instability is confirmed again as the driven force of vortex generation. This study clearly shows that shock wave and boundary layer interaction (SWBLI) is really shock-Liutex interaction and the frequency of shock oscillations is determined by the Liutex spectrum, which may solve the problem with low frequency noise generated by the supersonic vehicles. The Liutex vectors are used as the direct input for POD analysis. The result shows the same conclusions as the velocity input.