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