Lambda2 method

The Lambda2 method, or Lambda2 vortex criterion, is a detection algorithm that can adequately identify vortices from a three-dimensional velocity field.^[1] The Lambda2 method is Galilean invariant, which means it produces the same results when a uniform velocity field is added to the existing velocity field or when the field is translated.

Definition

The Lambda2 method consists of several steps. First we define the gradient velocity tensor $\mathbf{J}$ ;

$\mathbf{J} \equiv \nabla \vec{u} = \begin{bmatrix} \partial_x u_x & \partial_y u_x & \partial_z u_x \\ \partial_x u_y & \partial_y u_y & \partial_z u_y \\ \partial_x u_z & \partial_y u_z & \partial_z u_z \end{bmatrix},$

where $\vec{u}$ is the velocity field. The gradient velocity tensor is then decomposed into its symmetric and antisymmetric parts:

$\mathbf{S} = \frac{\mathbf{J} + \mathbf{J}^\text{T}}{2}$ and $\mathbf{\Omega} = \frac{\mathbf{J} - \mathbf{J}^\text{T}}{2},$

where T is the transpose operation. Next the three eigenvalues of $\mathbf{S}^2 + \mathbf{\Omega}^2$ are calculated so that for each point in the velocity field $\vec{u}$ there are three corresponding eigenvalues; $\lambda_1$ , $\lambda_2$ and $\lambda_3$ . The eigenvalues are ordered in such a way that $\lambda_1 \geq \lambda_2 \geq \lambda_3$ . A point in the velocity field is part of a vortex core only if at least two of its eigenvalues are negative i.e. if $\lambda_2 < 0$ . This is what gave the Lambda2 method its name.

Using the Lambda2 method, a vortex can be defined as a connected region where $\lambda_2$ is negative. However, in situations where several vortices exist, it can be difficult for this method to distinguish between individual vortices ^[2] . The Lambda2 method has been used in practice to, for example, identify vortex rings present in the blood flow inside the human heart ^[3]

References

↑ J. Jeong and F. Hussain. On the Identification of a Vortex. J. Fluid Mechanics, 285:69-94, 1995.
↑ Jiang, Ming, Raghu Machiraju, and David Thompson. "Detection and Visualization of Vortices" The Visualization Handbook (2005): 295.
↑ ElBaz, Mohammed SM, et al. "Automatic Extraction of the 3D Left Ventricular Diastolic Transmitral Vortex Ring from 3D Whole-Heart Phase Contrast MRI Using Laplace-Beltrami Signatures." Statistical Atlases and Computational Models of the Heart. Imaging and Modelling Challenges. Springer Berlin Heidelberg, 2014. 204-211.

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