Regeneration and Self-Maintaining of Coherent Structures - Transport and Coherent Structures in Wall Turbulence - page 336

streamwise and spanwise directions. The form of the
disturbance analyzed here is strictly equivalent to that
introduced by Henningson et al. [HEN 93], and the function
()

y defined by f

( )

q

f () yy =

p

2 − y

, exactly like in [HEN 93]. The quantities in
these relations are rendered dimensionless in relation to the
half-height of the channel and the velocity at its center.
Thus,

where pq ==

2

correspond, respectively, to the lower
and upper walls. The initial velocity field engendered by the
disturbance is

and y =

0

y =

2

( )

( )

uvw ,, = 0, ψψ , −

z y

with the subscripts indicating the partial derivatives in the
corresponding directions, and . Note

ψ = ∂ψ ∂ z

ψ = ∂ψ ∂ y

z y

that the initial streamwise component is and that the

growth of the streamwise disturbance in time and space
constitutes a direct proof of the mechanism of transitory
amplification, which has been abundantly discussed in the
existing literature. The initial disturbance is superimposed
on the basic flow, and the flow field is resolved by DNS,
tracked both in time and space. The low-intensity
disturbances essentially develop as a wave packet (WP),
extending in the spanwise and wall-normal directions before
gradually decreasing over time under the influence of the
viscous effects. If, however, the parameter

u = 0

in equation
[5.85] is sufficiently large, a turbulent “spot” develops, with a
rapid transition being the key. Figure 5.28 shows the
temporal evolution of the total energy determined
throughout the volume V of the calculation

ε