Coherent Wall Structures: Dynamics and Contribution to Turbulent Activity - Transport and Coherent Structures in Wall Turbulence - page 217

Figure 4.16. Spanwise spacing between the high- and low-velocity streaks
in the viscous sublayer as a function of the Reynolds number, based on the
momentum thickness. This figure is adapted from [SMI 83], and combines
the various measurements obtained by visualizations, hot-wire/hot-film
anemometry and electrochemical methods

Nakagawa and Nezu [NAK 81] and Smith and Metzler
[SMI 83] show that the probability density p

of the
spanwise spacing of the streaks follows a log-normal law 8 for
1

()

y + ≤ 30 , with a negligible dependence on the Reynolds

number when

≤

Re θ < 6, 000 . The probability density function

()

is expressed by

p

⎧

2

⎫

⎛ ⎞

⎪ 11 λ

ψλ

⎪

exp − ln

⎨ ⎬

⎜ ⎟

2

⎝ ⎠

⎪ ⎪

⎩ ⎭

0 0

()

[4.4]

p λ =

()

1/ 2

λψ 2 π

0

where , is the standard variation,

ψ λ = σ λ λ

σ λ

( )

1/ 2

−
=+

1/ 2

( )

2

and ⎡ ⎤

⎦ .

λλ ψ 1 ψ = ln 1 + ψ

2

⎣

0 λ 0 λ

8 The logarithm of spacing of the streaks follows a normal law. A random
variable may be modeled by a log-normal distribution, when it is the
product of a large number of independent positive variables.