[Paraview] SWKey: Definitions of 'Q-Criterion' and the calculator's mag(*) operator? Revised w/ Answers

[Andy Bauer]

Hi Samuel,

The issue with the symmetric tensor magnitude calculation is that ParaView
doesn't know it's a symmetric tensor so it doesn't know that it needs to
sum the off-diagonal values twice. All that is know for these arrays are
how many components there are. A three-component array could be
representing either a 3D vector or a 3D anti-symmetrix, second order tensor
so it's the user's responsibility to compute the magnitude properly where
"shortened" representations of tensors are used.

Regards,
Andy

On Sat, May 23, 2015 at 12:03 PM, Samuel Key <samuelkey at bresnan.net> wrote:

>  Reference: ParaView Version 4.3.1
>
> I have two technical questions:
>
>  (Q1) In the PV filter 'Gradient Of Unstructured Data Set' is the
> "QCriterion" function, the Q-Criterion defined by J. C. R. Hunt, A. Wray,
> and P. Moin (1988) "Eddies, stream, and convergence zones in turbulent
> flows," Center for Turbulence Research Report CTR-S88, and referenced by G.
> Haller in "*An objective definition of a vortex*," J. Fluid Mech. (2005),
> vol. 525, pp.1–26?
>
> (Answer to Q1) Courtesy of Don Simons of Egility Corp., Yes, the formula
> being
> Q=-0.5*(Vx,x*Vx,x + Vy,y*Vy,y + Vz,z*Vz,z) - (Vx,y*Vy,x + Vx,z*Vz,x +
> Vy,z*Vz,y)
>
> (Q2) In the PV filter 'Calculator' what is  the formula used by the "mag"
> function to calculate mag(*S*) where *S* is a symmetric 2nd-order tensor?
> Using matrix notation, the tensor *S* would have the following components:
>
> ---------------
> | Sxx Sxy Sxz |
> | Sxy Syy Syz |
> | Sxz Syz Szz |
> ---------------
>
> (Answer to Q2) The PV formula for mag(*S*) is the square root of the sum
> of the squares of the 6-tuple (Sxx, Syy, Szz, Sxy, Sxz, Syz).
>
> For consistency with a three-component vector, I would have expected it to
> be the 6-tuple (Sxx, Syy, Szz, 2Sxy, 2Sxz, 2Syz)?
>
>
> Samuel W Key
> FMA Development, LLC
> 1005 39th Ave NE
> Great Falls, Montana 59404
> USA
>
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