Lambda2 method

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

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 gradient velocity tensor J ;

J ≡ ∇ u → = [ ∂ x u x ∂ y u x ∂ z u x ∂ x u y ∂ y u y ∂ z u y ∂ x u z ∂ y u z ∂ z u z ] ,

where u → is the velocity field. The gradient velocity tensor is then decomposed into its symmetric and antisymmetric parts:

S = J + J T 2 and Ω = J − J T 2 ,

where T is the transpose operation. Next the three eigenvalues of S 2 + Ω 2 are calculated so that for each point in the velocity field u → there are three corresponding eigenvalues; λ 1 , λ 2 and λ 3 . The eigenvalues are ordered in such a way that λ 1 ≥ λ 2 ≥ λ 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 λ 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 λ 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