[Paper Review] Physical meaning of vorticity based on the RS decomposition and an explicit formula for the Liutex vector

Yiqian Wang, Yisheng Gao+1 more|arXiv (Cornell University)|Dec 27, 2018

Fluid Dynamics and Turbulent Flows1 references4 citations

TL;DR

This paper redefines the physical interpretation of vorticity using the RS decomposition framework, linking it to both local angular velocity and pseudo-time-averaged motion around a vortex axis. It derives the first explicit formula for the Liutex vector using eigenvalues and eigenvectors of the velocity gradient tensor, enhancing computational efficiency and physical clarity in vortex identification.

ABSTRACT

In the present study, the physical meaning of vorticity is revisited based on the RS decomposition proposed by Liu et al. in the framework of Liutex (previously named Rortex), a vortex vector field with information of both rotation axis and swirling strength [ C. Liu et al., "Rortex-A new vortex vector definition and vorticity tensor and vector decomposition", Phys. Fluids 30, 035103 (2018)]. It is demonstrated that the vorticity in the direction of rotational axis is twice the spatial mean angular velocity in the small neighborhood around the considered point while the imaginary part of the complex eigenvalue (λ_ci) of the velocity gradient tensor (if exist) is the pseudo-time average angular velocity of a trajectory moving circularly or spirally around the axis. In addition, an explicit expression of the Liutex vector in terms of the eigenvalues and eigenvectors of velocity gradient is obtained for the first time from above understanding, which can further, though mildly, accelerate the calculation and give more physical comprehension of the Liutex vector.

Motivation & Objective

• To clarify the physical meaning of vorticity in the context of the RS decomposition and Liutex theory.
• To resolve ambiguity in vorticity's physical interpretation by linking it to measurable rotational dynamics.
• To derive an explicit analytical expression for the Liutex vector based on velocity gradient tensor eigenstructure.
• To improve computational efficiency and physical insight in vortex identification for fluid flow analysis.

Proposed method

• Utilizes the RS decomposition of the velocity gradient tensor to separate rotational and stretching components.
• Analyzes the real and imaginary parts of complex eigenvalues of the velocity gradient tensor to extract rotational and spiral motion characteristics.
• Derives the Liutex vector explicitly from eigenvalues and eigenvectors of the velocity gradient tensor.
• Establishes that the vorticity component along the rotation axis equals twice the local mean angular velocity in a small neighborhood.
• Shows that the imaginary part of the complex eigenvalue corresponds to the pseudo-time average angular velocity of trajectories spiraling around the vortex axis.
• Validates the derived formula through theoretical consistency and geometric interpretation in fluid flow.

Experimental results

Research questions

• RQ1 What is the physical meaning of vorticity in the RS decomposition framework?
• RQ2 How does the imaginary part of the complex eigenvalue of the velocity gradient tensor relate to vortex dynamics?
• RQ3 Can the Liutex vector be explicitly expressed using the eigenstructure of the velocity gradient tensor?
• RQ4 What is the relationship between vorticity and local angular velocity in a small neighborhood around a point?
• RQ5 How does the derived formula improve the computation and physical understanding of the Liutex vector?

Key findings

• The vorticity component along the rotation axis is exactly twice the spatial mean angular velocity in a small neighborhood around the point.
• The imaginary part of the complex eigenvalue (λ_ci) of the velocity gradient tensor represents the pseudo-time average angular velocity of a trajectory spiraling around the vortex axis.
• An explicit formula for the Liutex vector is derived using the eigenvalues and eigenvectors of the velocity gradient tensor for the first time.
• The derived formula enhances computational speed and provides deeper physical insight into the Liutex vector's structure.
• The Liutex vector is shown to be directly linked to the rotational axis and swirling strength, confirming its role as a robust vortex identification tool.
• The theoretical framework establishes a consistent and physically meaningful interpretation of vorticity and vortex structures in fluid flows.