Investigation of correlation between vorticity, Q, [formula omitted], [formula omitted], [formula omitted] and Liutex

Abstract

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, λci, λ2, Δ 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, λci and Δ 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, λci, λ2 and Δ are better than vorticity, they are still small in strong shear regions. The expression of relation between vorticity, Q, λci, Δ and Liutex reveals how these methods are contaminated by shear or stretching or both.