Rortex and comparison with eigenvalue-based vortex identification criteria (1805.03984v2)

Abstract: Most of the currently popular Eulerian vortex identification criteria, including the Q criterion, the Delta criterion and the Lambda_ci criterion, are based on the analysis of the velocity gradient tensor. More specifically, these criteria are exclusively determined by the eigenvalues of the velocity gradient tensor or the related invariants and thereby can be regarded as eigenvalue-based criteria. However, these criteria have been found to be plagued with two shortcomings: (1) these criteria fail to identify the swirl axis or orientation; (2) these criteria are prone to contamination by shearing. In this paper, an alternative eigenvector-based definition of Rortex is introduced. The real eigenvector of the velocity gradient tensor is used to define the direction of Rortex as the possible axis of the local fluid rotation, and the rotational strength obtained in the plane perpendicular to the possible local axis is defined as the magnitude of Rortex. This alternative definition is mathematically equivalent to our previous one but allows a much more efficient implementation. Furthermore, a complete and systematic interpretation of scalar, vector and tensor versions of Rortex is presented to provide a unified and clear characterization of the instantaneous local rigidly rotation. By relying on the tensor interpretation of Rortex, a new decomposition of the velocity gradient tensor is proposed to shed light on the analytical relations between Rortex and eigenvalue-based criteria. It can be observed that shearing always manifests its effect on the imaginary part of the complex eigenvalues and consequently contaminates eigenvalue-based criteria, while Rortex can exclude the shearing contamination and accurately quantify the local rotational strength.

• The paper presents Rortex as a novel method overcoming limitations in eigenvalue-based vortex identification by accurately pinpointing vortex cores under shear flows.
• It demonstrates computational efficiency gains by reducing calculation time by an order of magnitude compared to traditional iterative and decomposition methods.
• The study applies Rortex to DNS data, revealing improved detection of complex vortical structures like hairpin vortices in turbulent boundary layers.

Rortex and Eigenvalue-Based Vortex Identification Criteria: An Expert Overview

The paper "Rortex and comparison with eigenvalue-based vortex identification criteria" by Yisheng Gao and Chaoqun Liu provides an in-depth exploration of a novel vectorial approach termed "Rortex" for identifying vortical structures in fluid dynamics, specifically addressing critical drawbacks inherent in conventional eigenvalue-based criteria. This review synthesizes the core components, comparative analysis, and implications of this paper for the field of fluid mechanics.

Eulerian vortex identification techniques, which evaluate vortex motions based on the eigenvalues of the velocity gradient tensor, are significantly hindered by two predominant issues. Traditional eigenvalue-based metrics, such as the Q and λ2 criteria, fail at pinpointing the orientation of vortex cores and are susceptible to distortions due to shear effects. The researchers introduce Rortex as a decisive remedy, leveraging the real eigenvector of the velocity gradient tensor to determine the axis of local fluid rotation and quantifying rotational strength perpendicularly to this axis. This novel definition optimizes computational efficiency while preserving accuracy.

Key findings of this paper present Rortex as a reliable metric resistant to shearing distortions, thus offering a more precise representation of local rotation than eigenvalue-based criteria. Numerically, this method demonstrates computational efficiency gains, with computation times drastically reduced compared to previous methodologies. For instance, the calculation using present methods slashed time by an order of magnitude relative to the Newton-iterative and real Schur decomposition approaches.

The comparison section reveals that traditional measures like the Q and λ2 criteria exhibit significant deviation in cases of shear superposition, adversely affecting the local swirl strength measurement. Conversely, Rortex remains robust against shear contamination, validating its efficacy as a precise indicator of rotational strength and axis measurement at the local fluid level.

The scalar, vector, and tensor levels of Rortex provide a systematic overview of local rotation, further refining the utility of vortical analyses. In particular, a unique tensor decomposition of the velocity gradient tensor differs from classical symmetric-antisymmetric decompositions, isolating rigid rotational behaviors from shear, stretching, and compression components.

Practical analysis includes applying Rortex to DNS data from boundary layer transitions, which affirms its practical superiority and potential for unveiling complex three-dimensional vortical structures often missed or misrepresented by classical criteria. Rortex's ability to trace the orientation and directionality of vortex lines provides crucial insights into coherent structures like hairpin vortices—a significant advancement over vorticity vectors conventionally used in turbulent flow paper.

The implications of these findings suggest the Rortex approach is not merely a supplemental tool but a fundamental improvement in the mathematical definition of fluid kinematics in vortex identification. Its introduction could influence future research directions, particularly in computational fluid dynamics (CFD) and turbulence modeling, encouraging further exploration of vector-based vortex identification criteria and challenging the predominance of eigenvalue-centric analyses.

In closing, Gao and Liu's presentation of Rortex advocates a shift towards more efficient and accurate vortex identification technologies, promising impactful contributions to fluid dynamic research and applications in engineering, atmospheric science, and beyond.