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 u {\displaystyle \mathbf {u} } of a fluid is a vector field
u = u ( x , y , z , t ) , {\displaystyle \mathbf {u} =\mathbf {u} (x,y,z,t),}which gives the velocity of an element of fluid at a position ( x , y , z ) {\displaystyle (x,y,z)\,} and time t . {\displaystyle t.\,} The Lambda2 method determines for any point u {\displaystyle \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 J {\displaystyle \mathbf {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 ] , {\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}},}
where u → {\displaystyle {\vec {u}}} is the velocity field. The velocity gradient tensor is then decomposed into its symmetric and antisymmetric parts:
S = J + J T 2 {\displaystyle \mathbf {S} ={\frac {\mathbf {J} +\mathbf {J} ^{\text{T}}}{2}}} and Ω = J − J T 2 , {\displaystyle \mathbf {\Omega } ={\frac {\mathbf {J} -\mathbf {J} ^{\text{T}}}{2}},}
where T is the transpose operation. Next the three eigenvalues of S 2 + Ω 2 {\displaystyle \mathbf {S} ^{2}+\mathbf {\Omega } ^{2}} are calculated so that for each point in the velocity field u → {\displaystyle {\vec {u}}} there are three corresponding eigenvalues; λ 1 {\displaystyle \lambda _{1}} , λ 2 {\displaystyle \lambda _{2}} and λ 3 {\displaystyle \lambda _{3}} . The eigenvalues are ordered in such a way that λ 1 ≥ λ 2 ≥ λ 3 {\displaystyle \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 λ 2 < 0 {\displaystyle \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 λ 2 {\displaystyle \lambda _{2}} is negative. However, in situations where several vortices exist, it can be difficult for this method to distinguish between individual vortices 2 . The Lambda2 method has been used in practice to, for example, identify vortex rings present in the blood flow inside the human heart 3
References
J. Jeong and F. Hussain. On the Identification of a Vortex. J. Fluid Mechanics, 285:69-94, 1995. ↩
Jiang, Ming, Raghu Machiraju, and David Thompson. "Detection and Visualization of Vortices" The Visualization Handbook (2005): 295. ↩
ElBaz, Mohammed SM, et al. "Automatic Extraction of the 3D Left Ventricular Diastolic Transmitral Vortex Ring from 3D Whole-Heart Phase Contrast MRI Using Laplace-Beltrami Signatures." Statistical Atlases and Computational Models of the Heart. Imaging and Modelling Challenges. Springer Berlin Heidelberg, 2014. 204-211. ↩