Regeneration and Self-Maintaining of Coherent Structures - Transport and Coherent Structures in Wall Turbulence - page 318

ˆ

( )

from streamwise vorticity Uyz

therefore depends only on y and z . The authors assume
that the field

. The basic flow

ω

,

x

ˆ

( ) () ( ) ( ) ( )

Uyz , = Uy + Δ u 2 cos β zgy

s

[5.55]

ˆ ˆ

VW == 0

accurately models the flow engendered by a streak extending
outside of the buffer sublayer , rather than only in the near-
wall sublayer at , where the quasi-streamwise

structures are mainly concentrated. The distribution

y + ≤ 30

()

Uy
in [5.55] is the turbulent profile of the mean lo ca l velocity of
the channel flow, the Reynolds number is Re = U h ν = 2, 000 ,

c

()

g y is an amplitude function which satisfies the non-slip
conditions at the wall and is of the form

( )

()

.
The analysis is limited solely to the bottom wall of the
channel. The adjustment parameter

g yy ∝ exp − ς y

2

can be used to locate
the maximum wall-normal vorticity surrounding the streak

ς

ˆ

( )

[5.56] ω∂ ˆ = Uz ∂ = β Δ u 2

y max s

in the buffer sublayer 20 , in accordance with what is

observed in fully developed near-wall turbulence. As the
streaks are typically spaced around 100 inner variables
apart, the wavenumber

y ≤≤ 30

πβ + = . The flow

is periodic in the spanwise direction. The periodicity imposed
means that the disturbances must be in accordance with
Floquet theory 20 [DRA 81], i.e. it must be of the form

β is such that 2 100

s

s

20 Numerous references are to be found in the existing literature on the
stability of periodic flows in general and time -periodic flows in particular.
For example, readers could consult [AKH 91] for a specific application of
the Floquet theory to oscillating Stokes flows.