Lambda2 Method

The Lambda2 method, or Lambda2 vortex criterion, is a vortex core line detection algorithm that can adequately identify vortices from a three-dimensional fluid velocity field. 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.

Description

The flow velocity of a fluid is a vector field which is used to mathematically describe the motion of a continuum.
The length of the flow velocity vector is the flow speed and is a scalar. The flow velocity $\mathbf$ of a fluid is a vector field

:$\mathbf=\mathbf(x, y, z, t),$

which gives the velocity of an '' element of fluid'' at a position $(x, y, z)\,$ and time $t.\,$
The Lambda2 method determines for any point $\mathbf$ in the fluid whether this point is part of a vortex core. A vortex is now defined as a connected region for which every point inside this region is part of a vortex core.

Usually one will also obtain a large number of small vortices when using the above definition. In order to detect only ''real'' vortices, a threshold can be used to discard any vortices below a certain size (e.g. volume or number of points contained in the vortex).

Definition

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

$\mathbf \equiv \nabla \vec = \begin \partial_x u_x & \partial_y u_x & \partial_z u_x \\ \partial_x u_y & \partial_y u_y & \partial_z u_y \\ \partial_x u_z & \partial_y u_z & \partial_z u_z \end,$

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

$\mathbf = \frac$ and $\mathbf = \frac,$

where T is the transpose operation. Next the three eigenvalues of
$\mathbf^2 + \mathbf^2$ are calculated so that for each
point in the velocity field $\vec$ 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

. The Lambda2 method has been used in practice to, for example, identify vortex rings present in the blood flow inside the human heart
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.

References

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Vortices
Computational fluid dynamics