{{Short description|Mathematical algorithm}} 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.<ref>J. Jeong and F. Hussain. On the Identification of a Vortex. ''J. Fluid Mechanics'', 285:69-94, 1995.</ref> 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 <math>\mathbf{u}</math> of a fluid is a vector field
:<math> \mathbf{u}=\mathbf{u}(x, y, z, t),</math>
which gives the velocity of an ''element of fluid'' at a position <math>(x, y, z)\,</math> and time <math> t.\,</math> The Lambda2 method determines for any point <math>\mathbf{u}</math> 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 <math>\mathbf{J}</math>;
<math> \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}, </math>
where <math>\vec{u}</math> is the velocity field. The velocity gradient tensor is then decomposed into its symmetric and antisymmetric parts:
<math>\mathbf{S} = \frac{\mathbf{J} + \mathbf{J}^\text{T}}{2}</math> and <math>\mathbf{\Omega} = \frac{\mathbf{J} - \mathbf{J}^\text{T}}{2},</math>
where T is the transpose operation. Next the three eigenvalues of <math>\mathbf{S}^2 + \mathbf{\Omega}^2</math> are calculated so that for each point in the velocity field <math>\vec{u}</math> there are three corresponding eigenvalues; <math>\lambda_1</math>, <math>\lambda_2</math> and <math>\lambda_3</math>. The eigenvalues are ordered in such a way that <math>\lambda_1 \geq \lambda_2 \geq \lambda_3</math>. 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 <math>\lambda_2 < 0</math>. This is what gave the Lambda2 method its name.
Using the Lambda2 method, a vortex can be defined as a connected region where <math>\lambda_2</math> is negative. However, in situations where several vortices exist, it can be difficult for this method to distinguish between individual vortices <ref>Jiang, Ming, Raghu Machiraju, and David Thompson. "Detection and Visualization of Vortices" ''The Visualization Handbook'' (2005): 295.</ref> . The Lambda2 method has been used in practice to, for example, identify vortex rings present in the blood flow inside the human heart <ref>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.</ref>
== References == {{reflist}}
Category:Vortices Category:Computational fluid dynamics