This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics
1
Simultaneous skin friction and velocity
measurements in high Reynolds number pipe and
boundary layer flows
R. Baidya
1
†, W. J. Baars
1
, S. Zimmerman
1
, M. Samie
1
, R. J. Hearst
2,3
, E. Dogan
2
,
L. Mascotelli
4
, X. Zheng
4
, G. Bellani
4
, A. Talamelli
4
, B. Ganapathisubramani
2
, N.
Hutchins
1
, I. Marusic
1
, J. Klewicki
1
and J. P. Monty
1
1
Department of Mechanical Engineering, University of Melbourne, Melbourne, Victoria 3010, Australia
2
Aerodynamics and Flight Mechanics Research Group, University of Southampton, Hampshire, SO17
1BJ, UK
3
Department of Energy & Process Engineering, Norwegian University of Science & Technology,
Trondheim, NO-7491, Norway
4
Department of Industrial Engineering, CIRI Aerospace, Alma Mater Studiorum, Università di Bologna,
47100 Forlì, Italy
(Received xx; revised xx; accepted xx)
Streamwise velocity and wall-shear stress are acquired simultaneously with a hot-wire and an
array of azimuthal/spanwise-spaced skin friction sensors in large-scale pipe and boundary layer
flow facilities at high Reynolds numbers. These allow for a correlation analysis on a per-scale
basis between the velocity and reference skin friction signals to reveal which velocity-based
turbulent motions are stochastically coherent with turbulent skin friction. In the logarithmic
region, the wall-attached structures in both the pipe and boundary layers show evidence of self-
similarity, and the range of scales over which the self-similarity is observed decreases with
an increasing azimuthal/spanwise offset between the velocity and the reference skin friction
signals. The present empirical observations support the existence of a self-similar range of
wall-attached turbulence, which in turn are used to extend the model of Baars et al. (J. Fluid
Mech., vol. 823, 2017, R2) to include the azimuthal/spanwise trends. Furthermore, the region
where the self-similarity is observed correspond with the wall height where the mean momentum
equation formally admits a self-similar invariant form, and simultaneously where the mean and
variance profiles of the streamwise velocity exhibit logarithmic dependence. The experimental
observations suggest that the self-similar wall-attached structures follow an aspect ratio of
7 : 1 : 1 in the streamwise, spanwise and wall-normal directions, respectively.
Key words:
1. Introduction
Following the discovery of quasi-periodic features within wall-bounded turbulence that are
thought to be associated with the physical mechanism that governsturbulence — from production
at the expense of the mean flow to eventual dissipation due to viscous forces at the fine scales
(Robinson 1991), substantial efforts have been made to better understand these critical processes
(Jiménez 2011). The quasi-periodic features remain coherent across a finite three-dimensional
† Email address for correspondence: baidyar@unimelb.edu.au
2 R. Baidya et al.
domain, and in this study we focus on the coherent structures that reside in the inertial range of
the energetic scales that become increasingly prominent at high Reynolds number, and therefore
account for a large portion of overall turbulence production (Smits et al. 2011). One of the
challenges of examining high Reynolds number flows is to capture the broadbandturbulence over
the extensive range of scales, which by definition of Reynolds number (Re) corresponds to the
separation of scales between the smallest and the largest energetic motions present within a flow.
From an experimental point of view, the range of scales accessible is typically constrained by
the physical size of the sensor at the small scales and the facility for the large scales. Hence, our
approach here is to utilise large-scale facilities, that allow high Re flows while still maintaining
the smallest energetic length scales such that they are accessible using conventional techniques.
Here, we present novel pipe flow experiments where an azimuthal array of skin friction signals
are simultaneously sampled with a velocity sensor. Due to the simplicity of axially symmetric
mean flow, the fully developed pipe flow is a classical configuration to examine wall-bounded
turbulence. The results from the pipe flow are also compared against a turbulent boundary layer
dataset, where simultaneous skin friction measurements with velocity have been acquired.
One of the conceptual models for wall-bounded flows which has received considerable atten-
tion is the attached eddy hypothesis (AEH) proposed by Townsend (1976). The AEH idealises
wall-bounded flow as a collection of inertia-driven coherent structures that are self-similar and
are randomly distributed in the plane of the wall. The AEH prescribes these coherent structures,
or eddies, to scale with the distance from the wall with the height of the eddies following a
geometric progression (Perry & Chong 1982). To assess the self-similarity of coherent structures
in a wall-bounded flow, del Álamo et al. (2006) examined their size based on dimensions of
a vortex core identified by thresholding the discriminant of the velocity gradient tensor, using
direct numerical simulation (DNS) datasets. They find that the tall vortex clusters which extend
from the near wall (below 20 viscous units) to the logarithmic region, scale with wall height.
Furthermore, work by Hwang (2015) suggests that these self-similar structures can self-sustain
and therefore play a key role in driving the wall-bounded turbulence. Experimentally, fully
resolving the velocity gradient tensor is challenging, and typically the streamwise velocity is
used as a surrogate. For example, Hellström et al. (2016) uses Proper Orthogonal Decomposition
(POD) on instantaneous snapshots of the streamwise velocity from a radial–azimuthal plane in a
pipe flow. They find that the POD mode shapes of the radial–azimuthal structure within the pipe
flow follow a self-similar progression that obeys wall scaling. That is, the various POD mode
shapes show a one-to-one relationship between azimuthal mode number and their characteristic
wall-normal extent. An excellent overview of key assumptions and limitations associated with
AEH is provided by Marusic & Monty (2019).
When examining flow data in the context of the AEH, a common objective is to search
for scaling laws in energy spectra (Baidya et al. 2017; Nickels et al. 2005) and wall-normal
profiles of the turbulent stresses (Marusic et al. 2013). In energy spectra of the streamwise
velocity component,a k
−1
x
behaviour in the inertial range (where k
x
corresponds to the streamwise
wavenumber) would reflect a self-similar wall-attached structure of the turbulence envisioned
in Townsend’s AEH. Likewise, a semi-logarithmic wall-normal decay in the variance of the
streamwise velocity also reflects such a structure. For several decades it has been challenging to
observe these scaling laws in raw velocity data and conclusive empirical evidence has remained
elusive (Marusic et al. 2010). Davidson & Krogstad (2008) propose that the one-dimensional
spectrum is not an ideal tool to investigate the self-similar behaviour due to an aliasing effect,
whereby a large wavenumber,whose wavenumber vector is inclined with respect to the direction
of measurement, is interpreted as a contribution to the one-dimensional u-spectra at a lower
wavenumber (Tennekes & Lumley 1972), presumably contaminating the k
−1
x
behaviour. Indeed,
a clear trend towards self-similarity is evident in two-dimensional u-spectra (here the spectral
energetic content can be examined as a function of streamwise and spanwise wavenumbers) with
Skin friction and velocity measurements in pipe and boundary layer flows 3
increasing Re, although this behaviour does not necessarily translate to a more prominent k
−1
x
behaviour once an integration is performed along the spanwise wavenumbers to obtain the one-
dimensional spectra (Chandran et al. 2017). Instead of k
−1
x
behaviour in the one-dimensional
u-spectra, Davidson & Krogstad (2008) advocate
(∆u)
2
∼ log(r
x
) scaling for the structure
function as an indicator of self-similarity, where ∆u = u(x + r
x
) − u(x) is the difference in the
streamwise fluctuating velocity, u, separated in the streamwise direction, x, with a displacement
r
x
. Subsequent assessment performed by de Silva et al. (2015) showed a decade of log(r
x
)
behaviour in the structure function for high Re boundary layer flow. Noticeable deviations from
the logarithmic behaviour occur however for the pipe flow at a comparable Re (Chung et al.
2015). Chung et al. (2015) suggest that these differences in the (∆u)
2
behaviour between the
pipe and boundary layer flows are due to a crowding effect in the pipe, whereby a restriction
is imposed on the width of the structures by its geometry. Furthermore, at an even higher Re
(Re
τ
∼ 10
6
) universality of
(∆u)
2
∼ log(r
x
) behaviour is retained between the pipe and boundary
layer flows. More recently, Yang et al. (2017) generalise the scaling of structure functions in
arbitrary directions in the three-dimensional space based on the AEH.
In this paper, we will follow the approach of using a correlation-based metric to examine
the wall-attached structure of the wall-bounded turbulence. As opposed to an assessment of
(∆u)
2
∼ log(r
x
) behaviour, which is based on correlation statistics at a single wall height
practically computed by assuming Taylor’s hypothesis, we here employ synchronised measures
of turbulence at two wall-normal positions. A reference turbulent skin friction at the wall is
acquired using hot-film sensors glued to the wall. These measurements are complemented by
a sequence of velocity measurements performed at various wall-height (z) locations. Using the
two-point measurements, it can be revealed to what degree the turbulent scales in the flow remain
coherent with the wall-reference signal. The aim of this paper is to characterise the coherent
part of the velocity signal, associated with the wall-attached structures (Baars et al. 2016), as a
function of its wavelength (λ
x
= 2π/k
x
), transverse offset (∆s) and wall-normal offset (z); and to
extend the observations by Baars et al. (2017) in the λ
x
–z plane to incorporate the ∆s trends.
2. Experimental set-up
Tables 1 and 2 list the experimental conditions and sensors utilised for the pipe and boundary
layer datasets considered in this paper, while details of each experiment are given in subsequent
paragraphs. Here, we report the friction Reynolds number, Re
τ
= L
O
U
τ
/ν, where L
O
, U
τ
and
ν correspond to the outer length scale (pipe radius and boundary layer thickness), mean friction
velocity and kinematic viscosity, respectively. While the pipe radius, R, can be directly measured;
the boundary layer thickness, δ, is determined here by fitting the mean velocity profile to a
modified Coles law of the wall/wake formulation (see Perry et al. 2002). In addition, x, y and
z denote the streamwise, spanwise and wall-normal directions, respectively; and superscript ‘+’
indicates normalisation by viscous units (e.g. U
+
= U/U
τ
, z
+
= zU
τ
/ν and ∆t
+
= ∆tU
2
τ
/ν).
Capitalisation and overbar denote time-averaged quantities, while lower cases correspond to
fluctuations from the time-averaged mean values.
2.1. Pipe flow
The pipe flow experiments are conducted at the Centre for International Cooperation in
Long Pipe Experiments (CICLoPE) facility belonging to the University of Bologna, located
in Predappio, Italy. The inner diameter of this unique large-scale facility is 0.9 m, with the
measurement station located at the downstream end of a 111 m long pipe, where a fully developed
turbulent pipe flow, for the first- and second-order statistics, is established (Örlü et al. 2017).
The large dimension for the outer length scale, R, corresponding in this case to the pipe radius,
4 R. Baidya et al.
Flow type Re
τ
Symbol
U
CL
, U
∞
R, δ U
τ
ν
U
τ
T
s
T
s
U
CL
R
,
T
s
U
∞
δ
(ms
−1
) (m) (ms
−1
) (µm) (s)
Pipe
10000
N 9.8
0.45
0.34 46 115 2500
21500
22.9 0.74 21 100 5000
39500
43.5 1.35 11 60 5800
Boundary
14000 ♦ 20.3 0.33 0.67 23 300 19000
layer
Table 1: Summary of experimental conditions. U
CL
and U
∞
denote the centreline and free-stream
velocities in the pipe and boundary layer flows, while T
s
correspond to the total sampling time
at each wall-normal location, z.
Flow type Re
τ
Hot-wire details Hot-film details
d l l
+
OHR 1/∆t ∆t
+
l l
+
OHR 1/∆t ∆t
+
(µm) (mm) (Hz) (mm) (Hz)
Pipe
10000
2.5 0.5
11
1.8 65000
0.12
1.5
33
1.05 6500
1.2
21500 24 0.59 72 5.9
39500 44 1.97 131 19.7
Boundary
14000 2.5 0.5 21 1.8 60000 0.47 0.9 38 1.05 60000 0.47
layer
Table 2: Summary of velocity and skin friction sensors utilised. OHR denote the overheat ratio
used for each sensor, while 1/∆t corresponds to the sampling frequency.
allows access to high Reynolds numberswhile the smallest energetic scales still remain O(10 µm)
and therefore can be resolved using conventional techniques (Talamelli et al. 2009). For further
technical and flow characterisation details on the facility, the readers are referred to Bellani
& Talamelli (2016), Willert et al. (2017) and Örlü et al. (2017). The current experimental
set-up consists of an array of 51 skin friction sensors located at various azimuthal positions
spanning from 0–2π along the pipe circumference simultaneously sampled with a hot-wire probe,
nominally located at the same x location as the hot-films. In addition, the hot-wire probe can be
traversed from near the wall to the pipe centreline as shown in figure 1(a). It should be noted
that the large pipe also allows the skin friction to be better resolved in the azimuthal direction
by virtue of being able to accommodate a larger number of sensors along the circumference (see
figure 1), compared to an alternate approach to high Reynolds numbers that relies on small ν/U
τ
,
typically achieved through reduction of the kinematic viscosity (e.g. Zagarola & Smits 1998).
The velocity sensor is a hot-wire consisting of a Wollaston wire mounted onto a modified
55P15-type Dantec probe and etched to expose a 2.5 µm diameter platinum core of 0.5mm
length. The platinum wire is heated and maintained at a constant temperature with an overheat
ratio of 1.8 using a Dantec StreamLine Pro anemometer system. Furthermore, here we maintain
a hot-wire length (l) to diameter (d) ratio of 200 to avoid contamination from end conduction
(Ligrani & Bradshaw 1987). Before and after each experiment, the hot-wire is traversed close to
the centreline (∼ 0.93R due to the limited range of the traverse) and the mean voltage is calibrated
in situ against the centreline velocity, which is measured using a Pitot-static tube. This allows
construction of a one-to-one relationship between the flow velocity and the anemometer output
Skin friction and velocity measurements in pipe and boundary layer flows 5
(a)
Hot-wire
traverse
S
p
1–19
Log array
S
p
19–36
Linear array I
S
p
36–51
Linear array II
∆S
p
19 ≈ 0.22R
∆S
p
36 ≈ 0.12R
R
∆s
(b)
R = 0.45m
δ ≈ 0.33m
x
y
z
(c)
Hot-wire
traverse
S
b
1–10
Linear array
∆S
b
1 ≈ 0.08δ
δ
∆s
Figure 1: Schematic of experimental set-ups. (a,c) Show locations of hot-film () and hot-wire
sensors (not to scale) for the pipe and boundary layer flows, while (b) shows comparison of the
outer length-scales R and δ respectively, in physical units.
voltage. Although this means that calibration is not performed in a near-potential flow, the ratio
p
u
2
/U at the centreline is sufficiently small to make insignificant differences to the potential flow
calibration (Monty 2005; Örlü et al. 2017). An intermediate calibration relationship between the
hot-wire voltages and velocities is generated for each wall-normal point during the measurement,
based on an assumption that deviations between pre- and post-calibration curves are the result
of a linear drift in the hot-wire voltage with time. Figure 2(a,b) shows comparisons of the
streamwise velocity mean profiles from the pipe experiments (denoted by symbols) against
the dataset of Ahn et al. (2015). Despite the departure from ideal calibration conditions, the
U profiles from the current datasets show good agreement with the DNS of Ahn et al. (2015)
(
) in the inner region when scaled in viscous units. In addition, a good agreement with the
DNS is also observed for the deficit profiles in the outer region. Unlike the mean, the measured
variance profiles are dependent on the sensor spatial resolution (Hutchins et al. 2009; Ligrani &
Bradshaw 1987). For the variance profiles shown in figures 2(c) and (d), the spatial resolution in
the azimuthal direction varies from 6 viscous units between the adjacent grids (at the wall) for
the DNS to 40 viscous units for the Re
τ
≈ 40 000 dataset from the current experiment. However,
since the influence of the spatial resolution diminishes with an increasing z, a good collapse of
the
u
2
profiles is observed for the region z/R > 0.1 in the outer scaling. Our U and u
2
results
and the conclusions drawn agree with the findings of Örlü et al. (2017) from the same facility.
Note, the present analysis is particularly insensitive to slight calibration differences (including the
