Near-Wall Coherent Structures: History, Identification and Detection - Transport and Coherent Structures in Wall Turbulence - page 163

Figure 3.20. Streamlines around a focal point a) and a critical
point node/saddle/saddle b). The directions of the streamlines
depend on the sign of R. This figure is adapted from [CHA 96]

Δ> 0

Δ< 0

Remember that the vorticity is linked to the Ω

i

antisymmetrical tensor

9 by

A ij

Ω=− ε A

i ijk jk

[3.34]

1
2

A =− ε Ω

ij ijk k

The antisymmetrical tensor is defined such that:

ε ijk

- if two of the indices are arbitrarily identical

0
ε 112 = ε 122 = ε 313 =

ε =

ijk

( );

0

- if the indices are all different and for even
numbers of permutations of the indices in the clockwise
direction (

ε ijk = 1

);

ε 123 = ε 231 = ε 312 = 1

- if the indices are all different and for odd
numbers of permutations of the indices in the anticlockwise

ε =− 1

ijk

9 This antisymmetrical tensor is also called the rotation rate tensor.