Lambda2 method

Template:Short description 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 $𝐮$ of a fluid is a vector field

$𝐮=𝐮(x,y,z,t),$

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 $𝐉$;

$𝐉\equiv \mathrm{\nabla }\overrightarrow{u}=\left[\begin{array}{ccc}{\partial }_x{u}_x& {\partial }_y{u}_x& {\partial }_z{u}_x\\ {\partial }_x{u}_y& {\partial }_y{u}_y& {\partial }_z{u}_y\\ {\partial }_x{u}_z& {\partial }_y{u}_z& {\partial }_z{u}_z\end{array}\right],$

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

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

where T is the transpose operation. Next the three eigenvalues of ${𝐒}^2+{\mathrm{\Omega }}^2$ are calculated so that for each point in the velocity field $\overrightarrow{u}$ there are three corresponding eigenvalues; ${\lambda }_1$, ${\lambda }_2$ and ${\lambda }_3$. The eigenvalues are ordered in such a way that ${\lambda }_1\ge {\lambda }_2\ge {\lambda }_3$. 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