Lambda2 method

The Lambda2 method is a detection algorithm that can adequately identify vortices from a three-dimensional 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.

Definition

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