VR helicity density and its application in turbomachinery tip leakage flows

Abstract

Helicity is an important quantity that represents the topological interpretation of vortices; however, helicity is not a Galilean invariant. In this study, VR helicity density (HVR) is derived via taking the dot product of vorticity with the unit real eigen vector of the velocity gradient tensor when the complex eigenvalues exist. The analytical solution of HVR is derived to resolve it in a local pointwise manner, and the Galilean invariance of HVR is proved. Tip leakage flow structures in a direct numerical simulation of a tip leakage flow model and a delayed detached eddy simulation of a low-speed large-scale axial compressor rotor are extracted using helicity, eigen helicity density and HVR methods. Results show that the utilization of HVR permits the identification and accentuation of concentrated vortices. Vortices identified by HVR appear in more connective states. As in the case of helicity, the sign of HVR distinguishes between primary and secondary vortices, while eigen helicity density fails. The normalized HVR is superior to the normalized helicity density in locating the vortex axis, especially for the induced vortex structures. Hence, HVR is a strong candidate to replace the helicity density, especially when Galilean invariance is required.