Journal ArticleDOI

# A comparative study of the velocity and vorticity structure in pipes and boundary layers at friction Reynolds numbers up to 10(4)

25 Jun 2019-Journal of Fluid Mechanics (Cambridge University Press)-Vol. 869, pp 182-213

AbstractThis 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 conditions. Comparison of canonical turbulent flows offers insight into the role(s) played by features that are unique to one or the other. Pipe and zero pressure gradient boundary layer flows are often compared with the goal of elucidating the roles of geometry and a free boundary condition on turbulent wall flows. Prior experimental efforts towards this end have focused primarily on the streamwise component of velocity, while direct numerical simulations are at relatively low Reynolds numbers. In contrast, this study presents experimental measurements of all three components of both velocity and vorticity for friction Reynolds numbers 휏 ranging from 5000 to 10 000. Differences in the two transverse Reynolds normal stresses are shown to exist throughout the log layer and wake layer at Reynolds numbers that exceed those of existing numerical data sets. The turbulence enstrophy profiles are 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 of a ‘quiescent core’ in pipe flow, as described by Kwon et al. (J. Fluid Mech., vol. 751, 2014, pp. 228–254) for channel flow at lower 휏 , and characterize the extent of its influence in the pipe. Observed differences between statistical profiles of velocity and vorticity are then discussed in the context of a structural difference between free-stream 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.

Topics: Boundary layer (67%), Reynolds number (62%), Turbulence (61%), Vorticity (60%), Pipe flow (60%)

### 1. Introduction

• The degree to which turbulent zero pressure gradient (ZPG) boundary layer and pipe flows can be treated as similar has been a subject of debate for much of the last decade (e.g. see Monty et al. (2009), Jiménez & Hoyas (2008)).
• One way to approach the issue of experimental scatter is to compare DNS results of internal and external flows directly, as in Jiménez et al. (2010) and Chin et al. (2014).
• 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 pipe.
• Forgoing measurement of two normal gradients (∂u2/∂x2 and ∂u3/∂x3) eliminates the practical requirement for each sub-array to estimate all three components of velocity simultaneously, the merits of which are evidenced by the velocity component variances reported in Zimmerman et al. (2017).
• 3. Calibration Data collected from a two-step in situ calibration procedure are combined to characterize the response of each sensor to a range of flow angles and speeds expected to be encountered in the profile scans.

### 3. Velocity statistics

• This section presents profiles of the statistical moments (up to kurtosis) of the three velocity components and the Reynolds shear stress.
• Again, this is consistent with the differences between the intermittency in the pipe and boundary layer.
• The same conclusion is reached via inspection of the HRNBLWT data, the difference in profiles is the most pronounced in the outer region x2/δ ≈ 0.2.
• The pipe and boundary layer cases both exhibit a positive peak in the u2 skewness in the wake region, although the magnitude of the boundary layer peak far exceeds that of the pipe.
• As is the case with u2 fluctuations, the point at which the boundary layer and pipe u3 variance profiles intersect is approximately coincident with the point at which the kurtosis profiles rapidly diverge, with both features occurring near x2/δ ≈ 0.6.

### 4. Vorticity

• This section presents statistics of all three components of vorticity, as well as the mean enstrophy 1 2 ωiωi.
• This synthetic case corresponds to the least-resolved physical experimental cases (see Table 1), and so all the experimental data is expected to approximately lie between the DNS computations and the synthetic experimental curve in the absence of effects not captured by the synthetic experimental model.
• As with the two zero-mean vorticity components (i.e. ω1 and ω2), the pipe and boundary layer ω3 vorticity RMS profiles closely resemble each other with the exception of the change of concavity observed in the boundary layer wake.
• With the exception of the highest-Reτ boundary layer case, the pipe and boundary layer kurtosis profiles track each other closely.
• The outer peaks observed in the enstrophy ratio profiles coincide with the change-of-concavity discussed above in the context of figures 8 through 10 as well as the outer ‘bumps’ in the boundary layer u2 and u3 variance profiles relative to those of the pipe.

### 5. Intermittency

• Two overarching features of the RMS, skewness, and kurtosis profiles are shown in §3 and §4 to consistently differentiate between pipe and boundary layer flow: outer magnitude peaks in the boundary layer skewness and kurtosis cases that emerge at x2/δ ≈ 0.5; and higher RMS/lower skewness and kurtosis magnitude of boundary layer quantities over a domain roughly spanning 0.1 .
• Early studies of this phenomenon even used the departure of the u1 kurtosis from the Gaussian value as a measure of intermittency (Klebanoff 1955).
• Instead, a range of thresholds for instantaneous enstrophy, 1 2 ω̃iω̃i, are used to identify ‘irrotational’ or ‘quasi-irrotational’ flow.
• Magnitude differences between the two Reτ cases, particularly for the lowest thresholds, are most likely influenced by spatial resolution, and thus conclusions drawn from figure 13 should be limited to those based on the relative magnitudes of the pipe and boundary layer, and the dependence of these relative magnitudes on wall-distance.
• The location of the discrepancy between ‘non-turbulent’ time fractions in the two flows roughly corresponds to the outer ‘bump’ in the boundary layer enstrophy relative to pipe enstrophy, the largest differences in Reynolds stresses between the two flows, and the lower magnitudes of boundary layer skewness and kurtosis profiles for a number of quantities as discussed throughout §3 and §4.

### 6. Conclusions

• A multi-sensor hotwire probe capable of measuring both the velocity and vorticity vectors has been deployed in a set of three turbulent pipe flows and three zero-pressure- gradient boundary layers with nominally matched inner and outer scales.
• Basic statistical results of these measurements are presented and highlight differences between the two flows and identify the subdomain over which they occur.
• A number of the observed differences in the present study match the observations of several lower-Reτ DNS and experimental studies, including those of Jiménez & Hoyas (2008), El Khoury et al. (2013), and Monty et al. (2009).
• The pipe and boundary layer velocity variances also intersect at approximately x2/δ ≈ 0.6, beyond which the boundary layer variances decay to zero while those in the pipe do not.
• The kurtosis coefficients of the vorticity fluctuations of all three components are super-Gaussian across the flow domain, with the kurtosis of the zero-mean components (ω1 and ω2) tending to increase with distance from the wall.

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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:

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
) 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 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
x
1
(a)
x
2
x
1
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
2
/∂x
2
and u
3
/∂x
3
) eliminates the practical requirement for each

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3,258 citations

Journal ArticleDOI
Abstract: Numerical simulations of fully developed turbulent channel flow at three Reynolds numbers up to Reτ=590 are reported. It is noted that the higher Reynolds number simulations exhibit fewer low Reynolds number effects than previous simulations at Reτ=180. A comprehensive set of statistics gathered from the simulations is available on the web at http://www.tam.uiuc.edu/Faculty/Moser/channel.

2,471 citations