Near-Wall Coherent Structures: History, Identification and Detection - Transport and Coherent Structures in Wall Turbulence - page 161

contains three eigenvalues, two of which are identical. The
sign of determines the stability of the local topology. In
the half-plane , the real parts of the complex conjugate
eigenvalues, or two of the three real eigenvalues, are
negative, and the half-plane is considered to be stable.
Conversely, these quantities are positive in the half-plane
, which is, therefore, linked to an unstable topology.

R

R < 0

R > 0

Figure 3.19. Local topology seen by an observer moving with the flow at
every point. The plane , corresponds to the stable focal points
with stretching (SF/S). The structure is stable node/saddle/saddle
(SN/S/S) at and . In the half-plane shown on the right,
the topology is an unstable focal point contracting (UF/C) at

Δ> 0 R < 0

Δ< 0 R < 0 R > 0

or

Δ> 0

unstable node/saddle/saddle (USN/S/S) at Δ< 0 . This figure is adapted

from [CHO 98]