Physics:Lambda2 method

Short description: Mathematical algorithm

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 [math]\displaystyle{ \mathbf{u} }[/math] of a fluid is a vector field

[math]\displaystyle{ \mathbf{u}=\mathbf{u}(x, y, z, t), }[/math]

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

[math]\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}, }[/math]

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

[math]\displaystyle{ \mathbf{S} = \frac{\mathbf{J} + \mathbf{J}^\text{T}}{2} }[/math] and [math]\displaystyle{ \mathbf{\Omega} = \frac{\mathbf{J} - \mathbf{J}^\text{T}}{2}, }[/math]

where T is the transpose operation. Next the three eigenvalues of [math]\displaystyle{ \mathbf{S}^2 + \mathbf{\Omega}^2 }[/math] are calculated so that for each point in the velocity field [math]\displaystyle{ \vec{u} }[/math] there are three corresponding eigenvalues; [math]\displaystyle{ \lambda_1 }[/math], [math]\displaystyle{ \lambda_2 }[/math] and [math]\displaystyle{ \lambda_3 }[/math]. The eigenvalues are ordered in such a way that [math]\displaystyle{ \lambda_1 \geq \lambda_2 \geq \lambda_3 }[/math]. 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 [math]\displaystyle{ \lambda_2 \lt 0 }[/math]. This is what gave the Lambda2 method its name.

Using the Lambda2 method, a vortex can be defined as a connected region where [math]\displaystyle{ \lambda_2 }[/math] 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