Probability density function and ‘‘plus’’ and ‘‘minus’’ structure functions
in a turbulent channel flow
Miguel Onorato
Dipartimento di Fisica Generale, Universita ` di Torino, Via Pietro Giuria 1, 10125 Torino, Italy
Gaetano Iuso
Dipartimento di Ingegneria Aerospaziale, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
Received 19 May 2000; published 18 January 2001
We consider the statistical properties of the longitudinal velocity increments in a turbulent channel flow at
different distances from the wall. The probability density function PDF of the velocity difference of the
streamwise component near the wall are found to be, especially at small scales, strongly skewed, showing a
very long left tail. We consider ‘‘plus’’ and ‘‘minus’’ structure functions and compute separately the statistics
for the right and left part of the PDF. It is found that the relative scaling exponents for the right tail are less
affected by the presence of the wall and their values are consistent with the ones found in experiments in
homogeneous and isotropic turbulence. A simple phenomenological model that explains the results obtained is
also given.
DOI: 10.1103/PhysRevE.63.025302 PACS numbers: 47.27.Nz, 01.50.Pa
Recently, a number of papers on the intermittency prop-
erties of the streamwise velocity component in turbulent
channel flow have been published 1–5. A major finding,
consistent with all the work published, is that a dependence
of the scaling exponent,
p
, on the distance from the wall is
observed through the extended self-similarity ESS6.A
similar result has been obtained from experimental data in a
magnetically confined turbulent fusion plasma 7. Moreover
in 1, using an eduction method based on the wavelet trans-
form, it was found that strong velocity gradients are respon-
sible of the increase of intermittency close to the wall. In
none of the cited papers the asymmetry of the probability
density functions PDF’s of the longitudinal velocity incre-
ments has been investigated. A different approach that takes
into account the contribution of anisotropy has been intro-
duced by Arad et al. 8. They decompose the structure func-
tions into their irreducible representation of the SO3 sym-
metry group and their finding is that the isotropic
contribution has a wide scaling region and is universal. Their
analysis has been performed at the center and at one fourth
of the channel height. Nevertheless, it has to be pointed out
that close to the wall, besides effects of anisotropy, a differ-
ent dynamics should occur due to the presence of the wall.
One of its major effects is the creation of coherent structures
such as ‘‘low’’ and ‘‘high’’ speed streaks and streamwise
vortices. Their dynamics, as anticipated in 1, must, in some
way, influence the shape of the PDFs and the values of the
relative scaling exponents.
In this Rapid Communication we focus our attention on
the asymmetry of the PDFs of the longitudinal velocity in-
crements. The analysis is carried out by means of the ‘‘plus’’
and ‘‘minus’’ structure functions at different distances from
the wall. We provide tentative evidence that the statistics of
the positive velocity increments is less affected by the pres-
ence of the wall. Moreover, we give a possible explanation
of this behavior in terms of the dynamic of coherent struc-
tures concentrated in the near wall region. The ESS is used
as a statistical method for characterizing and comparing data
at different distances from the wall.
The experiment has been carried out in a rectangular cross
section channel, 7 m long, 70 mm high, and 300 mm wide,
arranged in five plexiglas modules. The Reynolds number
based on the mean velocity at the center of the channel and
on the channel half height was 10 800 which corresponds to
Re
=510 (Re
is the Reynolds number based on the friction
velocity. Validation of the data set for basic statistical quan-
tities such as mean, rms, skewness and flatness are shown in
1. Data have been taken at different positions in the y
+
coordinate that is the coordinate normal to the wall super-
script + indicates that quantities are in wall units. We con-
centrate our analysis in the buffer layer ( y
+
=7, 10, 15, 28,
and 35 and almost at the center of the channel ( y
+
=218
and 310. In Fig. 1 we show the PDFs of the streamwise
velocity difference at y
+
=218 for different values of the
scale r
+
variation in time are converted into space variation
with the Taylor hypothesis. The PDF continually changes its
character as the separation r
+
varies: from a quasi-Gaussian
curve, when r
+
is comparable to large scales, to some expo-
FIG. 1. PDFs of the streamwise velocity differences centerd at
mean and renormalized by rms at y
+
=218.
RAPID COMMUNICATIONS
PHYSICAL REVIEW E, VOLUME 63, 025302R
1063-651X/2001/632/0253024/$15.00 ©2001 The American Physical Society 63 025302-1