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
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