- Source: 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
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
. The Lambda2 method has been used in practice to, for example, identify vortex rings present in the blood flow inside the human heart
