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.[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 ${\displaystyle \mathbf {u} }$ of a fluid is a vector field

${\displaystyle \mathbf {u} =\mathbf {u} (x,y,z,t),}$

which gives the velocity of an element of fluid at a position ${\displaystyle (x,y,z)\,}$ and time ${\displaystyle t.\,}$ The Lambda2 method determines for any point ${\displaystyle \mathbf {u} }$ 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 ${\displaystyle \mathbf {J} }$;

${\displaystyle \mathbf {J} \equiv \nabla {\vec {u}}={\begin{bmatrix}\partial _{x}u_{x}&\partial _{y}u_{x}&\partial _{z}u_{x}\\\partial _{x}u_{y}&\partial _{y}u_{y}&\partial _{z}u_{y}\\\partial _{x}u_{z}&\partial _{y}u_{z}&\partial _{z}u_{z}\end{bmatrix}},}$

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

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

where T is the transpose operation. Next the three eigenvalues of ${\displaystyle \mathbf {S} ^{2}+\mathbf {\Omega } ^{2}}$ are calculated so that for each point in the velocity field ${\displaystyle {\vec {u}}}$ there are three corresponding eigenvalues; ${\displaystyle \lambda _{1}}$, ${\displaystyle \lambda _{2}}$ and ${\displaystyle \lambda _{3}}$. The eigenvalues are ordered in such a way that ${\displaystyle \lambda _{1}\geq \lambda _{2}\geq \lambda _{3}}$. 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 ${\displaystyle \lambda _{2}<0}$. This is what gave the Lambda2 method its name.

Using the Lambda2 method, a vortex can be defined as a connected region where ${\displaystyle \lambda _{2}}$ 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

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.