Lambda2 method

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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[edit]

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 of a fluid is a vector field

which gives the velocity of an element of fluid at a position and time The Lambda2 method determines for any point 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[edit]

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

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

and

where T is the transpose operation. Next the three eigenvalues of are calculated so that for each point in the velocity field there are three corresponding eigenvalues; , and . The eigenvalues are ordered in such a way that . 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 . This is what gave the Lambda2 method its name.

Using the Lambda2 method, a vortex can be defined as a connected region where 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[edit]

  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.