Near-Wall Coherent Structures: History, Identification and Detection - Transport and Coherent Structures in Wall Turbulence - page 184

introduced in section 3.6.1. The system [3.57] has two critical
points

QR
Q
R

== 0

⎧

2

=− 3 β [3.60]

⎪ ⎨

= 2 β

3

⎪ ⎩

The Jacobian [3.21] at the critical point is Q = R = 0

⎛ − 2 β − 3 ⎞

− 2 β − 3

⎛ ⎞

⎜ ⎟

[3.61] J = =

4

⎜ ⎟

⎜ ⎟

Q − 3 β 0 − 3 β

⎜ ⎟ ⎝ ⎠

3

⎝ ⎠ QR == 0

The parameters of the characteristic equation [3.22] are,
respectively, and . As the
discriminant is , it follows that and
. The critical point at the origin of the phase plane
is, therefore, an improper and asymptotically stable node as
shown by Figure 3.19. The nature of the second critical point
is linked to the invariants of the Jacobian

() =−5β () = 6β

2

p = tr J

r = det J

p 2

2

δ = − 4 r = β > 0 > 0

δ r

()

< 0

Q , R

p

⎛ − 2 β − 3 ⎞

− 2 β

β

− 3

⎛ ⎞

⎜ ⎟

J = = [3.62]

4

⎜ ⎟

⎜ ⎟

2

Q − 3 β − 4 − 3 β

⎜ ⎟ ⎝ ⎠

⎝ 3 ⎠

2

Q
R

=−
=

3 β

β

3

2

given by

() =−5β < 0

,

() =−6β

2

and

p 2

2

p = tr J

r = det J < 0

δ = − 4 r = 99 β > 0

( )

is an
unstable saddle point. There is a second critical point in this
system and it is further away from the first one, the higher
the value of . We can easily see that the transport
dynamics of the invariants is dominated by the critical point

Consequently, the critical point

2 3

(,) QR =− 3 ,2 ββ

β