Rortex A New Vortex Vector Definition and Vorticity Tensor and Vector Decompositions

TL;DR

This paper introduces Rortex, a mathematically rigorous vortex vector that accurately captures local fluid rotation, along with new tensor and vector decompositions, enhancing vortex identification and analysis in turbulence research.

Contribution

It provides a new vortex vector definition, Rortex, with a mathematical proof of its rotational axis, a fast computation algorithm, and novel tensor and vector decompositions for fluid rotation analysis.

Findings

Rortex accurately identifies swirling strength and rotational axes.

The proposed decompositions distinguish rotational and deformation components.

Demonstrations on flow cases validate Rortex's effectiveness in vortex detection.

Abstract

A vortex is intuitively recognized as the rotational/swirling motion of the fluids. However, an unambiguous and universally-accepted definition for vortex is yet to be achieved in the field of fluid mechanics, which is probably one of the major obstacles causing considerable confusions and misunderstandings in turbulence research. In our previous work, a new vector quantity which is called vortex vector was proposed to accurately describe the local fluid rotation and clearly display vortical structures. In this paper, the definition of the vortex vector, named Rortex here, is revisited from the mathematical perspective. The existence of the rotational axis is proved through real Schur decomposition. Based on real Schur decomposition, a fast algorithm for calculating Rortex is also presented. In addition, new vorticity tensor and vector decompositions are introduced: the vorticity tensor is decomposed to a rigidly rotational part and an anti-symmetric deformation part, and the vorticity vector is decomposed to a rigidly rotational vector and a non-rotational vector. Several cases, including 2D Couette flow, 2D rigid rotational flow and 3D boundary layer transition on a flat plate, are studied to demonstrate the justification of the definition of Rortex. It can be observed that Rortex identifies both the precise swirling strength and the rotational axis, and thus it can reasonably represent the local fluid rotation and provide a new powerful tool for vortex dynamics and turbulence research.