This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1
A comparative study of the velocity and
vorticity structure in pipes and boundary
layers at friction Reynolds numbers up to 10
4
S. Zimmerman
1
†, J. Philip
1
, J. Monty
1
, A. Talamelli
2
, I. Marusic
1
,
B. Ganapathisubramani
3
, R. J. Hearst
3,4
, G. Bellani
2
, R. Baidya
1
,
M. Samie
1
, X. Zheng
2
, E. Dogan
3
, L. Mascotelli
2
, and J. Klewicki
1
1
Department of Mechanical Engineering, University of Melbourne, Melbourne, VIC 3010,
Australia
2
DIN, Alma Mater Stu diorum – Universit`a di Bologna, I-47100 Forli, Italy
3
Aerodynamics and Flight Mechanics Research Group, University of Southampton,
Southapton SO17 1BJ, UK
4
Department of Energy & Process Engineering, Norwegian University of Science &
Technology, Trondheim N O-7491, Norway
(Received xx; revised xx; accepted xx)
This study presents findings from a first-of-its-kind measurement campaign that includes
simultaneous measurements of the full velocity and vorticity vectors in both pipe and
boundary layer flows under matched spatial resolution and Reynolds number conditio ns.
Compariso n of canonical turbulent flows offers insight into the role(s ) played by features
that are unique to o ne or the other. Pipe and zero pressure gradient bo undary layer
flows are often compared with the g oal of elucidating the roles of geometry and a free
boundary co nditio n on turbulent wall-flows. Prior experimental efforts towards this end
have focused primarily o n the streamwise compo nent of velocity, while direct numerical
simulations are at relatively low Reynolds numbers.
In contrast, this study presents
experimental measurements of all thre e components of both velocity and vorticity
from
5000 . Re
τ
. 10000. Differences in the two transverse Reynolds normal stresses are
shown to exist throughout the log-layer and wake layer at Reynolds numbers that excee d
those of existing numerical data sets. The turbulence enstrophy profiles ar e also shown
to exhibit differences spanning from the outer edge of the log-layer to the outer flow
boundary.
Skewness and kurtosis profiles of the velocity and vorticity components imply
the existence o f a ‘quiesce nt core’ in pipe flow, as described by Kwon et al. (J. Fluid
Mech., vol. 751, 2014, pp. 228–254) for channel flow at lower Re
τ
, and characterise the
extent of its influence in the pipe. Observed differences between statistical profiles of
velocity and vorticity ar e then discussed in the context of
a structural difference between
free-strea m intermittency in the boundary layer and ‘quiescent core’ intermittency in the
pipe
that is detectable to wall-distances as small as 5% of the layer thickness.
Key words:
† Email address for correspondence: zimmermans@unimelb.edu.au
2 S. Zimmerman and others
1. Introduction
The degree to which turbulent zero pre ssure gradient (ZPG) boundary layer and
pipe flows can be treated as similar has been a subject of debate for much o f the
last decade (e.g. see Monty et al. (2009), Jim´enez & Hoyas (2008)). While the no-slip
condition forces similarity between boundary layers and pipes when scaled with friction
velocity (U
τ
≡
p
τ
w
/ρ) and length (l
v
≡ ν/U
τ
) s c ales sufficiently close to the wall,
the wall-distance at which this similarity breaks down (and which flow feature s begin
to deviate) remains an open question. Possible sour ces o f dissimilarity include differing
outer boundary conditions (turbulent pipe centreline versus non-turbulent free stream
in boundary layers), geometry (outer flow boundary exists along 1D line in pipes versus
2D plane in boundary layers), and differences in contributions to the mean momentum
balance (mean pressure gra die nt in pipe s versus mean advection in boundary layer s).
Both physical experiments and numerical simulations have been c onducted towards
clarifying the onset and causes of discrepancies. Experimental results, however, are
primarily limited to those pertaining to the streamwise component of velocity—largely
owing to the relative difficulty of measuring the other two components. Monty et al.
(2009) compared stre amwise velocity spectr a and the first four statistical moments
of the streamwise velocity collected in pipe, channel, and boundary layer flows at a
friction Reynolds number of approximately 3000, where Re
τ
≡ U
τ
δ/ν and δ refers to the
boundary layer height and/or the pipe ra dius
/channel half-height, where applicable. They
found that the statistical structure of the streamwise velocity fluctuations was virtually
the same in all three flows from the wall to at least 0.5δ. Despite this statistica l invariance,
the authors also found that eddies with streamwise wavelength & 10δ contribute more to
the streamwise variance in the log-layer for internal (pipe/channel) flows than they do for
external (boundary layer) flows. That the strea mwise statistical invariance is apparently
maintained despite the difference in spatial organization motivates an investiga tion into
the be haviours of other flow variables such as the cross-stre am velocities and the vorticity.
While experimentally determined profile statistics of the wall-normal and span-
wise/azimuthal components of velocity are available independently for both pipes and
boundary layers, no single experimental study has presented data for both flows ac quired
with the same probe and da ta-reduction scheme under matched probe resolutio n and
Reynolds number conditions. Cons equently, it is difficult to differentiate between
flow-dependent features and experimental scatter based on a collection of existing
experimental results alone. This is illus trated in Jim´enez & Hoyas (2008), where a
selection of existing experimental da ta from both internal and external flows is presented
alongside the results of a set of direct numerical simulations (DNS) of channel flow.
One way to approach the issue o f experimental scatter is to compare DNS results
of internal and external flows directly, as in Jim´enez et al. (2010) and Chin et al.
(2014). Such comparisons, however, have thus far been limited to friction Reynolds
numbers of Re
τ
≈ 1000 or less. Since it is unclear whether wall-flows of Re
τ
. 1000
contain a well-develo ped inertial layer (Morrill-Winter et al. 2017), it remains to be
seen whether features observed in the transverse velocity variance profiles persist at
higher Re
τ
. Furthermore, to the authors’ knowledge, third and fourth order statistics o f
the tra nsverse velocity comp onents have not yet been reported in a comparative study
of internal and external flows. Such statistics contain valuable information about the
probability distribution functions of turbulence quantities, as they clarify the relative
dominance of positive versus negative, or large versus small fluctuatio ns, and the
depe ndence of these measures on wall-distance. Additionally, the normalised third and
fourth order moments and, in pa rticular, how these compa re to those a ssociated with
A comparative study of pipes and boundary layers 3
Gaussian processes, may be used to evalua te existing models of wall-bounded flow, and
inform new models/modifications to existing models.
Differences in wake structure between internal and external flows have been discussed in
the context of turbulent/non-turbulent intermittency since the early s tudies by Schubauer
(1954) and Klebanoff (1955). Both authors asserted that the distribution of velocity
fluctuations was most likely the same in the ‘turbulent’ patches of the boundary layer
as they are in the pip e. External boundary layers are bounded by irrotational potential
flow, the entrainment of which is commensurate with flow development in the streamwise
direction. Fully developed internal flows, however, have no such source of ir rotational flow
and do not develop in the streamwise direction. Despite this fact, Kwon et al. (2014)
identified a large-scale region, or ‘quiescent core’, in channel flows at Re
τ
≈ 1 000–4000
having characteris tics reminiscent of thos e of the boundary layer free- stream.
Example
snapshots of the turbulent/non-turbulent interface (TNTI) in a boundary layer from
Chauhan et al. (2014b) a nd the quiesc ent core bounda ry in a channel from Kwon et al.
(2014) are shown in figures 1(a) a nd (b) respectively. Although the boundary of the
quiescent core is qualitatively similar to the TNTI, its influence (if any) on turbulence
statistics at Reynolds numbers hig he r than Re
τ
≈ 4000 is presently unknown. In this
study, we show that normalised thir d- and fourth-order sta tis tical moments of pipe flow
are indicative of intermittency associated with a quiescent core, and that differences in
the intermittency between pipe and boundary layer flow can explain many of the observed
differences between the two flows.
In the present experiments, we simultaneously measure all components of velocity and
vorticity in bounda ry layer and pipe flows for 5000 .Re
τ
. 10000. Thus, the present
data set allows for differentiation between ‘turbulent’ and ‘non-turbulent’ patches by
their instantaneous enstrophy rather than an analogue measure based, for example, on
the streamwise velocity. As such, another aim of this study is to compare the prevalence
and structure of quasi-‘non-turbulent’ flow in pip es and bounda ry layers as well as the
vortical properties of the ‘turbulent’ patches.
Throughout the rest of this text, subscripts 1, 2, and 3 refer to the streamwise,
wall-normal, and spanwise/azimuthal directions, respectively. Superscript ‘+’ indicates
normalisation by viscous sca les. The position x
2
= 0 refers to the wall in both the pipe
and boundary layer cases. Overbar
(·) or capitalisation denotes a time-averaged quantity,
sup erscript prime (·)
′
or lower-case denotes a fluctua ting quantity, and a tilde
˜
(·) denotes
a total quantity. The following are examples of the notation used throughout: the total
streamwise velocity can be decompos e d as ˜u
1
= U
1
+ u
1
; the mean Reynolds shear stress
can be expressed as u
1
u
2
; and the fluctuating component of the instantaneous Reynolds
shear stress can be expres sed as (u
1
u
2
)
′
.
2. Experiments
2.1. Facilities
The present data were collected as part of a collaborative effort between the authors at
the Center for International Collaboration in Long Pipe Experiments (CICLoPE) a nd the
Flow Physics Facility (FPF)—re spectively the largest-scale turbulent pipe flow and zero
pressure gradient boundary layer fac ilities in existence. The former is a closed-loop system
that generates a fully developed turbulent pipe flow in a 90 cm diameter test section over
a development length of 110.9 m (i.e. a length-to-diameter ratio of 123.2). The loop
includes a heat exchanger which keeps the flow temper ature constant to within ±0.2
◦
C,
even for measurement durations in exce ss of 9 hours. A deta ile d design of CICLoPE c an
4 S. Zimmerman and others
U
1
/U
o
1.10.7
1.0
0.5
(b)
x
2
/δ
x
1
/δ
(a)
x
2
/δ
x
1
/δ
0 1 2
0 1 2
2
1
0
1
0
Figure 1. (a) Snapshot adapted from Chauh an et al. (2014b) showing turbulent/non-turbulent
interface in a ZPG boundary layer at Re
τ
≈ 12300. Interface location in (a) based on threshold
of local turbulence kinetic energy (see Chauhan et al. (2014b)). (b) Snapshot adapt ed from Kwon
et al. (2014) showing boundaries of th e quiescent core in channel flow at Re
τ
≈ 1000. Qu iescent
core boundary based on U
1
/U
o
= 0.95 contour, where U
1
is the mean streamwise velocity and
U
o
is the centreline velocity for the channel, and the free-stream velocity for the boundary layer.
Coordinates x
1
and x
2
refer to the streamwise and wall-normal directions, respectively. Ellipse
in (b) highlights instance where the quiescent core boundary nearly reaches the wall.
be found in Talamelli et al. (2009), and initial velocity measurements are reported in
¨
Orl¨u
et al. (2017 ). The FPF, first characterized in Vincenti et al. (2013), is an open circuit
zero pressure gradient wind tunnel in which the boundary layer grows continuously over
a streamwise development length of 72 m, ultimately achieving boundary laye r heights of
up to 75 cm. The spatial development of the boundary layer over this long fetch permits
the outer flow scale to be set to any value up to the maximum by establishing a fixed
measurement station at the corresponding streamwise location. The friction veloc ity at
any streamwise location is constant to within within 0.5% for the central 5 m of the
total 6 m test section span, while the sloped ceiling maintains the free-stream velocity
as constant to within ±1% over the range used herein (Vincenti et al. 2013).
Both fa cilities are ideal for high- fidelity measurements of high Reynolds numb er flows,
as their physical size allows for the generation of a wide range of energy-containing
scales without the smallest of those being unresolvable via conventional measurement
techniques. The two facilities are also particularly well-suited for direct flow comparisons
with one another, as the operational flow speeds and physical dimensions make it possible
to simultaneously match both inner and outer flow sca les at considerable Reynolds
numbers.
It is worth no ting that the open-circuit design of the FPF presents additional exper-
imental challenges relative to smalle r, indoor (o r closed-loop) fa c ilities. As the inflow is
drawn from the atmosphere, compensation is needed for the calibration drift ass ociated
with changes in atmospheric temperature over the course of each measurement. The FPF
data also show slight departures from canonical behaviour in the wake of the generated
boundary layer (e.g. see Vincenti et al. (2013)). Although we do not believe that these
factors impact the conc lusions of this study, additional boundary layer measurements
A comparative study of pipes and boundary layers 5
⊕
x
3
x
2
x
1
u
1
,u
2
u
1
,u
3
u
1
,u
3
u
1
,u
2
∆x
2
= l
w
p
∆x
3
= 2.5 · l
w
p
l
w
p
(d) (c)
(b)
(a)
(b)
(a)
0.5 mm
Figure 2. (a) Probe schematic with relative dimensions. (b) Front-on picture of actual probe.
Lab els (a)-(d) in (a) refer to ×-array ‘sub-arrays’ as referenced throughout text. Probe centroid
is indicated by ⊕. Reference length l
w
p
is the sensor length l
w
projected into the x
2
-x
3
plane,
which for this study is fixed at 0.8mm.
collected in the High Reynolds Number Boundary L ayer Wind Tunnel (HRNBLWT) at
the University of Melbourne (e.g. see Kulandaivelu (2012)) are included in App endix
A for comparison. The HRNBLWT is an indoor open-circuit ZPG boundary layer wind
tunnel with a streamwise development length of 27 m, which allows for generation of a
boundary layer up to 35 cm thick. As such, to achieve matched spatial resolution with
the FPF and CICLoPE measurements, the HRNBLWT measurements are collected at
Reynolds numb ers about 2/3 as large as those obtained at the FPF and the CICLoPE.
2.2. Measurement Probe
All of the data presented herein were acquired via a multi-element hot-wire anemom-
etry probe consisting of 8 independent sensing elements. T he design of this probe and
its capacity to capture key aspects of the velocity and vorticity time-se ries in turbulent
boundary layers are discussed in detail in Zimmerma n et al. (20 17). The arrangement
of the sensing elements, shown in figure 2, is similar to the arrangement deployed by
Antonia et al. (1998) in a grid-generated turbulent flow. Several modifications were made
to this design to reduce the overall measurement volume and better-suit operation in wall-
bounded flows. These include a reductio n of the relative spa c ing betwee n sub-a rrays (a)
and (b) to prioritize resolution of the x
2
gradients, and the use of gold-plated tungsten
wire in place of platinum-core Wollaston wire.
For illustrative purposes, it is useful to des c ribe the present probe as being comp osed
of four individual × -wire sub-arrays. The probe schematic shown in figure 2 is consistent
with this description and demo nstrates one way in which both the velo city and vorticity
vectors may be obtained about the centroid of the measurement volume. In co ntrast
to some other multi-element hot-wire probes deployed in wall-bounded flows (e.g. see
the review of Wallace & Vukoslavˇcevi´c (2010)), the individual sub-array centroids of
the present probe are symmetric about the overall measurement volume centroid. The
advantage of this symmetry is that all gradient estimates (and thus vorticity component
estimates) can be obtained via central finite differences about a single common point.
Another advantage of the present design is the focus on resolving the vorticity vector
specifically rather than the entire veloc ity gra die nt tensor. Forgoing measurement of two
normal gradients (∂u
2
/∂x
2
and ∂u
3
/∂x
3
) eliminates the practical requirement for each
