Lambda2 method

<h2 id="description">Description</h2>
The <a href="/facts/Flow_velocity/amIT4Ht8">flow velocity</a> of a fluid is a <a href="/facts/Vector_field/ccFNuPPp">vector field</a> 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 <a href="/facts/Velocity/Q3o1LVDe">velocity</a> of an <a href="/facts/Fluid_parcel/yk70LICD">element of fluid</a> 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).

<h2 id="definition">Definition</h2>
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 <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric</a> and <a href="/facts/Skew-symmetric_matrix/nw5CQeLN">antisymmetric</a> 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 <a href="/facts/Transpose/8wmsagGS">transpose operation</a>. Next the three <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalues</a> 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
<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
. The Lambda2 method has been used in practice to, for example, identify <a href="/facts/Vortex_ring/N5IpqxrV">vortex rings</a> present in the blood flow inside the <a href="/facts/Human_heart/fF4KSGdk">human heart</a>
<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">J. Jeong and F. Hussain. On the Identification of a Vortex. J. Fluid Mechanics, 285:69-94, 1995. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Jiang, Ming, Raghu Machiraju, and David Thompson. "Detection and Visualization of Vortices" The Visualization Handbook (2005): 295. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">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. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
</ol>

Lambda2 method open-in-new

Lambda2 method