Lambda2 method explained

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.^[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.

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

u of a fluid is a vector fieldu = u (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 u 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

J ;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 velocity gradient tensor is then decomposed into its symmetric and antisymmetric parts:S = (J + J ^{T})/(2) and \Omega = (J - J ^{T})/(2) ,

where T is the transpose operation. Next the three eigenvalues of

S ^{2} + \Omega ^{2} are calculated so that for eachpoint in the velocity field \vec{u} there are three corresponding eigenvalues; λ _{1} , λ _{2} and λ _{3} . The eigenvalues are ordered in such a way that λ _{1} \geq λ _{2} \geq λ _{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 λ _{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

λ _{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]

Notes and References

1. J. Jeong and F. Hussain. On the Identification of a Vortex. J. Fluid Mechanics, 285:69-94, 1995.
2. Jiang, Ming, Raghu Machiraju, and David Thompson. "Detection and Visualization of Vortices" The Visualization Handbook (2005): 295.
3. 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.