Near-Wall Coherent Structures: History, Identification and Detection - Transport and Coherent Structures in Wall Turbulence - page 183

, according to Cheng and
Cantwell [CHE 96] and the model put forward by Martin and
Dopazo [MAR 95]. By substituting this expression into
equations [3.55] and [3.56] and using

i H is linked to

Ha ∝− =− β a

ij ij ij

aa =− 2 Q and

ik ki

aa a =− 3 R , we construct the dynamical system

ik kn ni

dQ

Q = − − 2 β Q 3 R

dt
dR

[3.57]

2 2

R == − QR 3 β

dt 3

The Lagrangian transport equation for the discriminant
results directly from its definition [3.32], i.e. Δ

d Δ 27 dR dQ 2

[3.58]

= R + 3 Q

dt 2 dt dt

This relation is simply reduced to

d
dt

Δ

=− 6 β Δ [3.59]

if we use in isotropic homogenous turbulence. It is
interesting to note that the discriminant is preserved
( ) when . This scenario corresponds to the
Eulerian restriction in the absence of viscosity, and was
analyzed by Cantwell [CAN 92]. However, for finite values of
, the trajectories of the fluid particles should tend
toward the curve

H ij =−β a ij

Δ= constant

β = 0

β > 0

( )

2

for t , because

Δ = 27 4 RQ + = 0 →∞

( )

according to equation [3.59].

Δ ∝− exp 6 β t

Readers will note that relation [3.57] is written in the
Lagrangian coordinate system, and that the material
derivatives are replaced by the temporal accelerations
of the fluid particles. The characteristics of this 2D
dynamic system can be determined using the concepts

DDt

ddt