Large- and Very Large-scale Structures - Transport and Coherent Structures in Wall Turbulence - page 400

Figure 6.21. Temporal signals of fluctuations in streamwise velocity at
15 y + , according to [MAT 09]. Raw signal of streamwise velocity (top),

large-scale component after high-pass filtering (middle) and small-scale
component after low-pass filtering (bottom).

=

Re

τ =

7, 300

Mathis et al. [MAT 09] use the notion of an analytical
signal [PAP 84] to detect amplitude modulation of the outer
structures on the streamwise fluctuations of the velocity in
the buffer sublayer. The low-pass filtered small-scale signals

are indicated by the subscript S and the large-
scale component

L

Λ< 1

0

is indicated by the subscript L .

L

Λ> 1

0

The envelope of the signal

u is

( ) () () ()

2

ˆ

2

[6.2]

E ut ut ut = +

S

() ()

ˆ ut is the Hilbert transform of

where ut . Remember 7

that this is simply the imposition of phase shifting of ± /2

on each Fourier component with negative and positive
frequency, respectively. The envelope

()

E u with high-pass

S

7 See Chapter 4.