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

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

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 $\mathbf {u}$ 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 $\mathbf {J}$;

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

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

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

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