Regeneration and Self-Maintaining of Coherent Structures - Transport and Coherent Structures in Wall Turbulence - page 328

The eigenvalues of the matrix on the right-hand side of
equation [5.63]

{ }

are clearly negative, which,
a priori , indicates that the solutions will decrease
exponentially over time. However, a detailed analysis of the
behavior of

− 1, Re − 2, e

τ τ

+ which is governed by

ω

y

( ) ( ) ( )

[5.65]

+ + + + + +

ω = vRe exp − t Re + ω exp − 2 t Re − exp − 2 t Re

y 0 τ τ y 0 τ τ

indicates that caution must be exercised. The expression in a
Taylor series for short times of the forcing term on the right-
hand side of the latest equation, and only of the last term,
gives us

+

3 v

( ) ( )

[5.66]

+ + + + + + 2

vRe exp − t Re − exp − 2 t Re = vt −

0

t + ...

0 τ τ τ 0

Re

τ

This relation clearly shows that

+ grows linearly over

ω

y

( )

time when , because, quite precisely, the

eigenvectors of the matrix given in equation [5.63] - i.e.

+

t ≤ O e

τ

Φ

⎛ ⎝ ⎞

⎠

Φ 2 ≡

⎛
⎝

⎞
⎠ ⎟

1
Re

0
1

1

[5.67]

≡

⎜ ⎟ ⎜ 1

2

1 + Re

τ

τ

- are not orthogonal. On the contrary, the angle between
the eigenvectors

( )

− 1 2

tends toward zero

θ = cos Re 1 + Re

τ τ

within the boundary Re τ →∞ . Consequently, although the

individual norms decrease exponentially

because of the stability of the eigenvalues, it is by no means
impossible for the resultant of

and

Φ Φ

1 2

and to increase

Φ Φ

1 2

( )

transiently over time when (Figure 5.27). Thus,

transient growth may trigger nonlinear mechanisms over
time

+

t ≤ O e

τ

( )

if the basic flow is subcritical . If not, the
resultant mimics the temporal behavior of the larger of the

+

t ≤ O e

τ