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Turbulent boundary layer flow over regular multiscale roughness

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This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1
Turbulent boundary layer flow over
regular multiscale roughness
T. Medjnoun1, E. Rodriguez-Lopez1, M. A. Ferreira1, T. Griffiths1
J. Meyers2and B. Ganapathisubramani1
1Aerodynamics and Flight Mechanics Research Group, University of Southampton,
Hampshire, SO17 1BJ, UK
2KU Leuven, Mechanical Engineering, Celestijnenlaan 300, B3001 Leuven, Belgium
(Received xx; revised xx; accepted xx)
In this experimental study, multiscale rough surfaces with regular (cuboid) elements
are used to examine the effects of roughness scale hierarchy on turbulent boundary
layers. Three iterations have been used with a first iteration of large-scale cuboids onto
which subsequent smaller cuboids are uniformly added, with their size decreasing with
a power-law as the number increases. The drag is directly measured through a floating-
element drag balance, while particle image velocimetry allowed the assessment of the flow
field. The drag measurements revealed the smallest roughness iteration can contribute
to nearly 7% of the overall drag of a full surface, while the intermediate iterations are
responsible for over 12% (at the highest Reynolds number tested). It is shown that the
aerodynamic roughness lengthscale between subsequent iterations varies linearly, and can
be described with a geometrical parameter proportional to the frontal solidity. Mean and
turbulent statistics are evaluated using the drag information, and highlighted substantial
changes within the canopy region as well as in the outer flow, with modifications to the
inertial sublayer (ISL) and the wake region. These changes are shown to be caused by
the presence of large-scale secondary motions in the cross-plane, which itself is believed
to be a consequence of the largest multiscale roughness phase (spacing between largest
cuboids), shown to be of the same order of magnitude as the boundary layer thickness.
Implications on the classical similarity laws are additionally discussed.
Key words: Authors should not enter keywords on the manuscript
1. Introduction
Understanding turbulent flows over rough-walls is paramount to a wide range of
practical problems. In engineering applications, they manifest in internal flow systems
such as in pipes and ducts, marine transportation or turbomachinery (Shockling et al.
2006; Monty et al. 2016; Bons 2010). They are also ubiquitous in natural environments
such as in the atmospheric boundary-layer (ABL), urban areas or in fluvial processes
(Garratt 1994; Cheng & Castro 2002; Nezu et al. 1993; Nikora & Roy 2012). In all the
aforementioned examples, roughness plays a fundamental role, influencing the overall
drag, turbulent mixing and transport properties (Jim´enez 2004).
One of the longstanding questions remains the characterisation of the drag by means
of the surface properties, however, there is a plethora of arrangements and shapes under
Email address for correspondence: t.medjnoun.soton.ac.uk
2Boundary layers over multiscale roughness
which roughness can be present. Following Grinvald & Nikora (1988) and Stewart et al.
(2019) definitions, roughness can be classified into two categories. It can be considered
discrete (regular), when presented as an amalgamation of discrete elements defined by a
combination of linear lengthscales such as height, width and pitch (e.g. cube roughness).
Alternatively, it can be interpreted as continuous (irregular), when the roughness is
randomly distributed among a wider range of scales and defined by its statistical moments
such as standard deviation, skewness and flatness (e.g. naturally corroded pipes). The
seminal work of Nikuradse (1933) followed by the comprehensive reviews by Schlichting
(1937) and Colebrook et al. (1939) served as the basis for the early drag predictions
performed by Moody (1944) for pipe flows, using the equivalent sandgrain roughness
height hs. The latter has since become the common currency for predicting drag, with
the aim of correlating surface properties to this aerodynamic roughness lengthscale hs.
Several studies have attempted to provide relations that would predict drag at high
Reynolds numbers based on either laboratory measurements or numerical simulations.
However, the existence of a universal correlation that would generically represent any
rough surface is unlikely, due to its simultaneous dependence on various parameters such
as front and plan solidities (Van Rij et al. 2002), effective slope (Napoli et al. 2008),
roughness skewness (Flack & Schultz 2010), roughness directionality (Nugroho et al.
2013), spanwise heterogeneity (Anderson et al. 2015) among other parameters.
There exists an extensive body of research dedicated to the study of flows over idealised
two- and three-dimensional topographies such as cubes, bars, pyramids or spheres (see
e.g. Cheng & Castro 2002; Volino et al. 2009; Schultz & Flack 2009). However, these
surfaces have typically a limited range of scales through which they can interact with the
turbulent flows. Fractal-like geometries on the other hand, have an inherent capacity to
interact with turbulent flows through a much broader range of lengthscales. These flows
have received special attention in the previous two decades from both the numerical and
experimental communities, such as in flows encountering fractal grids or trees (see e.g.
Meneveau & Katz 2000; Mazzi & Vassilicos 2004; Chester et al. 2007; Hurst & Vassilicos
2007; Graham & Meneveau 2012; Gomes-Fernandes et al. 2012). Particularly, realistic
surfaces such as urban areas and natural landscapes are generally multiscale, exhibiting
a fractal-like behaviour with a height spectral slope ranging between -1 and -3 (Huang
& Turcotte 1989; Passalacqua et al. 2006; Wan & Port´e-Agel 2011). These have only
till the last decade started earning a wider attention from researchers, especially the
modelling community. The readers are encouraged to refer to the recent study by Zhu
& Anderson (2018) which offers an excellent and comprehensive review of the different
contributions in the literature. One of the challenges when examining these flows is the
determination of the smallest spatial resolution needed in order to accurately predict the
bulk flow quantities over surfaces featuring a multitude of scales. In fact, unless direct
numerical simulations (DNS) or well-controlled laboratory measurements are accessible,
computational fluid dynamics (CFD) methods such as large-eddy simulations (LES) or
Reynolds-averaged Naviers-Stokes (RANS) require modelling of the unresolved roughness
features.
Both “discrete” and “continuous” types of rough surfaces have recently been docu-
mented within the multiscale roughness framework. Using LES data obtained from an
ABL flow over various irregular topographies, Wan & Port´e-Agel (2011) examined the
effect of the subgrid-scale roughness on the effective aerodynamic roughness lengthscale,
and found the latter quantity to linearly increase with the root-mean-square (r.m.s.) of
the unresolved roughness topography. In an LES study of a channel flow over multiscale
fractal-like surfaces, Anderson & Meneveau (2011) developed a dynamic roughness model
whereby the unresolved small-scale topography fluctuations are modelled using a local
Turbulent boundary layer over regular multiscale roughness 3
equilibrium wall model. The effective roughness lengthscale is parameterised through
the r.m.s. height and a roughness parameter determined a posteriori and validated with
an invariance resolution test. They showed that, by including more and more subgrid-
scale roughness, the wall shear stress increased, indicating the importance of correctly
modelling the unresolved roughness fluctuations.
In an effort to examine the effect of the intermediate and small topographical length-
scales, Mejia-Alvarez & Christensen (2010, 2013) experimentally studied a turbulent
boundary-layer flow over a highly irregular surface roughness replicated from a turbine-
blade damaged by deposition of foreign materials, from which they created filtered
models. By comparing the original surface to its lower order representations, they
observed the mean and turbulent statistics in the outer region to be self-similar as they are
predominately governed by the large topographical scales. However, they reported that
the filtered models failed to appropriately reproduce the contributions of the most intense
Reynolds shear stress events (sweeps and ejections) within the roughness sublayer (RSL).
In a similar approach to Anderson & Meneveau (2011), Barros et al. (2018) investigated
a channel flow over systematically generated multiscale rough surfaces with varying
spectral slopes, with predefined statistical moments. They showed that the surfaces
with shallower spectral slopes led to higher drag, suggesting that high wavenumbers
(small scales) can noticeably contribute to the overall drag. Stewart et al. (2019) have
examined three different self-affine bed roughness in two different open-channel flows,
and reported similar findings to Barros et al. (2018), such that decreasing the roughness
spectral exponent leads to higher drag. They explained this behaviour by analytically
demonstrating the existence of a relation between the effective slope with the roughness
spectral exponent (also validated with experimental data), which increases as the spectral
exponent decreases. They suggested that this drag increase is due to contributions from
an increase in the flow separation around steeper, scaling-range roughness features.
For the discrete roughness type, Yang & Meneveau (2017) used LES to examine tur-
bulent boundary layers over randomly distributed prism-like roughness elements with a
fractal-like size distribution, and investigated the flow response to the addition of smaller
roughness generations. To account for the drag that rises from the unresolved scales,
they developed a dynamic roughness model which assumes the flow to become scale-
invariant when it interacts with the scale-invariant roughness, which was then validated
by the independence of the mean flow profiles on grid resolution. They also proposed
an analytical model that explicitly accounts for the sheltering effect within the canopy
flow, however, its predictions seem to depend on the number of generations included
in the roughness. In a similar approach to Yang & Meneveau (2017), Zhu & Anderson
(2018) performed a series of LES simulations of a turbulent channel flow over cube-like
fractal topographies, with varying fractal dimensions (spectral slope). They assessed the
drag penalty associated with changing the spectral slope and the number of smaller
generations. They showed that the drag associated with the unresolved generations can
be parametrised with the first few generations, and proposed a logarithmic law-based
model to model their contributions. They further added that the turbulence statistics
are predominantly controlled by the first generation of elements.
Despite the burgeoning interest in this area, there is still a need for experimental
studies that relate to flows over such complex surfaces. To our knowledge, the systematic
investigation of the multiscale roughness hierarchy effect on turbulent flows in regular
roughness has only been reported in a limited number of works (e.g. Yang & Meneveau
2017, Zhu & Anderson 2018 and Vanderwel & Ganapathisubramani 2019), and remain
experimentally largely unexplored. It is also worth mentioning that when simulating
turbulent flows over these types of surfaces, the previous LES studies usually rely on
4Boundary layers over multiscale roughness
channel flows conditions, in which case the conclusions may not be fully applicable
for instance to external flow systems such as an ABL flow. Therefore, wind tunnel
experiments in addition to the modelling techniques can prove beneficial with the different
lines of enquiries mentioned above. In fact, understanding the dynamics of turbulent flows
over multiscale rough surfaces will result in appropriate modelling strategies and better
predictions of flows such as in urban environments.
To this end, a comprehensive experimental study is designed aimed at examining the
effect of roughness scale hierarchy on turbulent boundary layers. The design used is
similar to the one employed by Yang & Meneveau (2017) and Zhu & Anderson (2018)
which first builds large-scale cuboids, subsequently adding smaller iterations where their
size decreases with a power-law as the number increases (see e.g. the study of Zhu &
Anderson (2018) for a detailed description of an iterated function system). The main
difference between the current investigation and the former studies lies in the layout of
the roughness in the horizontal plane, currently chosen to be uniform instead of randomly
distributed. To uncover the effects of the multiscale roughness on the turbulent boundary
layer, direct wall-drag as well as flow field measurements are conducted using an in-
house floating element drag balance (Ferreira et al. 2018) and particle image velocimetry
(PIV). The remainder of the manuscript is presented in three sections. The experimental
methodology is described in section §2 depicting the multiscale roughness topography,
drag and flow field measurement techniques. The results and discussion are presented
in section §3, analysing the near-wall and outer-flow regions. Finally, a summary and
concluding remarks are provided in section §4.
2. Experimental methodology
2.1. Facility
Measurements were carried out in an open-circuit suction-type wind-tunnel at the
University of Southampton. The working section follows a 7:1 contraction and extends 4.5
m in the streamwise direction, with a cross-section of 0.6 m ×0.9 m ×4.5 m in wall-normal
and spanwise directions, respectively. The test section was designed with a weak diverging
cross-section to allow a constant free stream along the streamwise direction and a growth
of a turbulent boundary layer with a nominally zero pressure gradient. The acceleration
parameter K=ν
U
dU
dx , where νand Uare the air kinematic viscosity and free stream
velocity respectively, have been observed to range between 1–5×108from various recent
studies performed in this facility (see e.g. Ferreira et al. 2018; Medjnoun et al. 2018).
The turbulent boundary layer grows over a flat surface composed of five equally-sized
wooden boards onto which the roughness was mounted. In order to assess the skin friction,
the boundary-layer plate at the measurement location was cut out to allow the floating
element drag balance to be inserted in. The boundary layer plates were preceded by
a ramp of 0.2 m long inclined by four degrees to the horizontal, ensuring a smooth
transition of the flow from the bottom floor of the test section. The free stream velocity
can reach up to 30 ms1, with a turbulence level less than 0.5%, and was monitored
and acquired using the Micromanometer FCO510 by Furness controls. To account for
air density variations, temperature was also acquired, and its standard deviation for an
average run was less than ±0.5oCelsius. The facility schematic as well as the experimental
procedures employed in this study are shown in figure 1.
Turbulent boundary layer over regular multiscale roughness 5
Figure 1: (a) Schematics of the experimental arrangements including the planar- and
stereo-PIV set-ups along with (b) a close-up on the floating-element drag balance
developed by Ferreira et al. (2018). (c) Example of instantaneous planar and stereoscopic
velocity fields performed at two streamwise and spanwise symmetry planes. Both PIV
and drag balance measurements are performed at a fetch distance x30δfrom the
leading edge, with δbeing the boundary layer thickness at this fetch determined from
the horizontally-average velocity profiles.
2.2. Multiscale roughness
The surface roughness considered in the current investigation is based on a multiscale
distribution pattern as illustrated in figure 2. The design is inspired by a Sierpinski
carpet model, which essentially builds the roughness by superimposing size-decreasing
self-similar cuboid elements. The model offers an easy control of the geometrical charac-
teristics of the roughness, allowing numerical simulations to be replicated in wind tunnels
and vice-versa. To generate the fractal-like roughness, a similar method employed by Zhu
& Anderson (2018) is used, which considers an iterated function system to construct the
successive generations with h(i+1)=rh(i),h(i)and rbeing the cuboid height at the ith
iteration and the scale-reduction factor respectively. The cuboid-side width is defined as
w(i)=Arh(i), with Arbeing the width-to-height aspect ratio. These roughness elements
are uniformly distributed in the horizontal (x, z)-plane with each individual cuboid of a
given iteration being equidistant from its neighbouring ones, that is S(i)
x=S(i)
z=S(i).
For the first iteration, S(1)=2w(1), whereas for the following iterations, S(i)=w(i). It
should be noted that for scaling purposes, S(1)will exclusively be referred to as Sin
6Boundary layers over multiscale roughness
Figure 2: Schematics of the different multiscale roughness combinations tested in the
present investigation labelled Iter1, Iter12 , Iter13 and Iter123.
the remainder of the manuscript. The subsequent self-similar cuboids following the first
iteration are overlaid both at the surrounding as well as on top of the previous parent
cuboid.
Due to both roughness manufacturing constraints and flow considerations, the height
of the first iteration roughness element was set to h(1)=8 mm, so that δh(1)10. This
constrained the number of iterations to three due to the lowest cuboid height that can
be manufactured. The scale-reduction factor rand the width-to-height aspect ratio Ar
were fixed to 1/4 and 4.2 respectively, whereas the number of elements was set to 1, 36
and 576 for the three successive iterations namely; large, intermediate and small cuboids
respectively. The roughness elements were confined in a 100 mm ×100 mm repeating
unit tile, which was 3D printed then replicated using a moulding-casting manufacturing
process to cover the floor of the wind tunnel test section.
In order to investigate the effect of the multiscale roughness on the turbulent boundary
layer flow and assess the contributions of different length scales, four combinations are
considered. (i) Iter1: which considers the presence of a single iteration with a cuboid
height h(1). (ii) Iter12: which looks at the presence of two iterations with cuboid heights
h(1)and h(2). (iii) Iter13: which examines the presence of two iterations with cuboid
heights h(1)and h(3). Finally, (iv) Iter123: which investigates the effect of presence of
all three iterations with cuboid heights h(1),h(2)and h(3). Schematics for the different
cases are illustrated in figure 2 while a summary of their different statistical attributes
are presented in table 1.
2.3. Floating element
The wall shear stress was directly measured by means of a floating element drag
balance, developed and validated by Ferreira et al. (2018) in both smooth- and rough-
wall flow conditions. This method relies on the determination of the streamwise net force
Fdacting on a finite and structurally independent area A, representative of the surface
being investigated. It makes use of the parallel-shift linkage principle, through pairs of
bending beam transducers which allow the isolation of both the streamwise load as well
as the induced pitching moment. The wall shear stress is then deduced via τw=FdA.
Turbulent boundary layer over regular multiscale roughness 7
Case h(mm) ¯
h(mm) hrms (mm) hskw λf(%) λp(%) α
Iter18 0.88 2.51 2.48 2.66 11.09 0.94
Iter12 10 1.38 2.65 2.14 3.66 33.13 1.90
Iter13 8.5 1.01 2.52 2.45 2.91 33.13 1.16
Iter123 10.5 1.50 2.66 2.12 3.91 49.79 2.20
Table 1: Essential geometrical characteristics of the different multiscale rough surfaces
with h,¯
h,hrms,hsk w being the maximum, mean, root-mean-square and skewness of
the roughness height, while λf,λprepresent the frontal and plan solidities within one
repeating unit. The parameter α=¯
h
hrms
λfis a geometrical parameter used to describe
the aerodynamic roughness lengthscale of the surfaces discussed in section §3.1.3.
The floating element used consists of a 200 ×200 mm2surface area, on top of which four
roughness tiles are placed. The balance was flush-mounted with the wind tunnel floor
and positioned around 3.1 m downstream the leading edge. Additional care was taken
when setting up the surrounding roughness owing to the tight tolerance of the air gap
which is only 0.5 mm wide.
To assess whether these surfaces can reach the fully-rough regime, the floating element
was exposed to a series of nine free stream speeds ranging from 9 up to 27 ms1. Each
acquisition lasted 30 s with a sampling rate of 150 Hz (corresponding to 2500 eddy
turnover time at the lowest speed), with a total of five repetitions per speed. Pre- and
post-calibrations were performed for each configuration without notable discrepancies. A
more detailed discussion on the complete design, acquisition procedure and uncertainty
of the measurement technique can be found in Ferreira et al. (2018).
2.4. Particle Image Velocimetry
The turbulent boundary layer flow was diagnosed in both the streamwise-wall-normal
plane (x, y)as well as in the cross-plane (y, z ), using planar (2D2C) and stereoscopic
(2D3C) PIV measurements respectively. Both types of PIV measurements were performed
at approximately 3.2 m downstream of the contraction. As illustrated in figure 1(c), two
2D2C-(x, y)measurements were conducted at the spanwise symmetry planes zS= 0,0.5
(at the roughness valley and ridge), and two additional 2D3C-(y, z)were also acquired
at two successive streamwise symmetry planes, at an x=3.2 m and 3.25 m. The flow was
seeded with vaporised glycerol-water particles generated by a Magnum 1200 fog machine,
then illuminated with a laser light sheet sourced by a two-pulse Litron Nd:YAG laser
operating at 200 mJ. A LaVision optical system for the beam focus/expansion of the
light sheet was used, which comprised of convex and concave lenses in order to focus
the beam, and a cylindrical lens in order to expand the sheet with relatively constant
thickness in the measurement plane (1 mm thickness).
The particle images were recorded by high-resolution LaVision Imager LX 16 MP CCD
cameras fitted with 200 and 300 mm AF Micro Nikon lenses for the 2D2C- and 2D3C-
PIV measurements, respectively. One camera was used for the planar-PIV set-up, and
positioned at nearly 1 m away from the object plane, whereas two cameras mounted on
Scheimpflug adapters to account for the oblique view angle (±42o) were needed for the
stereo-PIV measurements, and placed at nearly 1.3 m from the object plane at either
sides of the test section, also depicted in figure 1(a). A single and a double-sided dual
plane calibration target aligned with the laser light sheet were used to determine the
mapping function for each set-up, using a third-order polynomial fit. This resulted in a
8Boundary layers over multiscale roughness
field of view of approximately 0.9δ×1.4δin the (x, y)-plane and 1.3δ×δin the (y, z )-
plane for the planar and stereoscopic PIV, respectively. In terms of the largest roughness
lengthscale S, this was equivalent to 1.1S×1.4Sin the (x, y)-plane, whereas in the
(y, z )-plane it spanned 1.5S×1.1S, for the planar and stereoscopic PIV, respectively.
A total of 3000 statistically independent realisations of image pairs were acquired for
each case at 0.6 Hz, with a time delay between pulses of 50 and 20 µs for the 2D2C and
2D3C PIV respectively. In order to allow a comparison between the different surfaces,
the PIV measurements were performed at different speeds to obtain a relatively matched
roughness function ∆U+, and ensure they reached a fully-rough regime (although it will
be seen later that Iter1never reached the fully-rough condition). The free stream speeds
were adjusted from case to case, with U= 18.7, 10.3, 18.5 and 10.2 ms1for the cases
Iter1, Iter12,Iter13 and Iter123 , respectively. The velocity vector fields were then obtained
by interrogating particle images using a decreasing multipass scheme starting from 48
pixels ×48 pixels down to a final pass of 16 pixels ×16 pixels for the 2D2C and 24
pixels ×24 pixels for the 2D3C. Using a 50% interrogation window overlap, the resulting
effective vector spacing ranged between 0.4–0.5 mm for the different cases both in the
planar and stereoscopic PIV measurements.
3. Results and discussion
This section focuses on the analysis of results reported from the floating element drag
balance with the data from both planar- and stereoscopic-PIV measurements. Section
§3.1 discusses the results of the direct drag estimates along with the flow topology in
the canopy region, with the assessment of the aerodynamic parameters associated with
the mean flow profiles. The section §3.2 reveals the multiscale roughness effects on the
flow topology in cross-plane, focusing on the turbulence properties in the outer region as
well as the assessment of the outer-layer similarity hypothesis.
3.1. Inner region
3.1.1. Surface drag
Results from the floating-element drag balance are presented in figure 3. They describe
the response of the wall shear stress to the four multiscale rough surfaces investigated.
Figure 3(a) examines how the skin-friction coefficient Cfvaries with respect to Rex(with
varying free stream speed) in a linear-logarithmic scale. At the lowest tested Reynolds
numbers (Rex<3×106), the roughest surface Iter123 is distinctively shown to experience
the largest frictional drag, and is noticeably higher that Iter12. On the other hand, the
surfaces Iter1and Iter13 exhibit relatively similar values and are the weakest among the
four cases. Beyond a certain Reynolds number (Rex>3×106), the skin-friction coefficients
is seen to weakly vary for the cases Iter12, Iter13 and Iter123 with an average variation
about the mean less than ±2%, which is within the uncertainty of the measurements.
Conversely, a substantial decay of Cfis observed for the Iter1case, perhaps with a steeper
slope in comparison with the smooth-wall curve represented by Schlichting’s power-law
Cf=bRea
x(Schlichting 1979), highlighted with the black solid line.
It is also observed that at the highest Reynolds numbers of the facility, Iter12 expe-
riences a reduction in Cfof about 4% from its mean value across Reynolds numbers.
These two different behaviours demonstrate that Iter1and maybe Iter12 still remain
transitionally rough, whereas the cases Iter13 and Iter123 have reached the fully-rough
regime, i.e. Reynolds number invariance of Cf(Jim´enez 2004; Flack & Schultz 2014).
As demonstrated by other studies (see e.g. Napoli et al. 2008; Yuan & Piomelli 2014),
Turbulent boundary layer over regular multiscale roughness 9
Case Symbol U(ms1)δS δh ∆U +hsδ Π Reτ×103CF×103β(%)
Iter118.68 1.11 13.87 11.93 0.05 0.16 7.95 7.07 82.8
Iter12 10.23 1.15 11.5 12.23 0.09 0.25 4.84 8.15 93.4
Iter13 18.48 1.16 13.65 12.50 0.06 0.17 8.50 7.43 87.8
Iter123 10.14 1.14 10.85 12.74 0.11 0.21 4.97 8.83 100
Table 2: Aerodynamic parameters of the turbulent boundary layer flow over the different
multiscale rough surfaces. The boundary layer thickness δwas identified as the wall-
normal distance at which the horizontally-averaged streamwise velocity reached 99%
of the free-stream speed U, while S=S(1). The quantities ∆U+,hsand Πnamely;
roughness function, aerodynamic roughness lenghscale and wake strength parameter
respectively are discussed in §3.1.3.
the fully-rough regime stems from pressure drag contributions prevailing compared to
the viscous drag. This suggests that the viscous drag contributions still represent a
considerable fraction in the Iter1case (at least within the range of Rextested herein),
as opposed to the other cases, whose pressure drag contributions dominate owing to the
presence of additional roughness scales. Although this result seems counter-intuitive since
the large cuboid is essentially and sharp-edged bluff body, it should be recalled that the
frontal and plan solidities of Iter1are noticeably smaller in comparison with the other
cases. This is further investigated in section §3.1.2.
The magnitude of the skin-friction coefficient is also considerably affected by the
roughness content of the surface, such that the addition of subsequent smaller scales
enhances the shear stress acting at the wall. This is clear when examining the skin-friction
coefficient averaged over the different Reynolds numbers CFas shown in figure 3(b). It
indicates that the magnitude of the skin-friction coefficient CFincreases monotonically
with increasing α(with α=¯
h
hrms
λfbeing a non-dimensional roughness parameter
proportional to the frontal solidity). There are many alternatives and possible ways that
can be used to correlate the drag with roughness statistics. In this instance, the use of
this surface parameter is justified by the fact that its variation approaches a linear trend
with the drag, while other parameters such as λpwould fail to capture a clear trend.
To quantify the difference between these surfaces, a relative drag-increase coefficient β
(where β=100% ×C(i)
FC(4)
Fwith i=1–4 for Iter1, Iter12, Iter13 and Iter123 ) is shown in
figure 3(c) for the different cases. We observe that the largest cuboid alone is responsible
for over 80% of the drag of the full surface. It is also shown that the drag increase is
higher for Iter12 than Iter13 (93% against 87% of drag of the full surface), indicating
that the drag increment that stems from the intermediate iteration is more prominent
than the one caused by the smaller iteration despite Iter12 and Iter13 having a similar
plan solidity λp. Therefore, it can be implied that the contributions to the full multiscale
surface from the smallest cuboid roughness alone amount to β(3)=β(123)β(12)7%.
Similarly, the contributions from the intermediate roughness scale alone amount to β(2)=
β(123)β(13)12%. These results are in agreement with the observations reported in
the previous LES studies, highlighting the importance of small roughness features and
emphasising the attention needed when modelling the unresolved subgrid-scale roughness
in numerical simulations (Yang & Meneveau 2017; Zhu & Anderson 2018).
10 Boundary layers over multiscale roughness
Figure 3: (a) Skin-friction coefficient estimates for one repeating unit at various moderate
Rex, with the black solid line representing Schlichting’s power-law for smooth-wall with
a15 and b0.058, and the blue stars representing the smooth-wall data from
Medjnoun et al. (2018). The red stars highlight the skin-friction estimates corresponding
to the flow conditions where PIV measurements are performed. (b) Variation of the
skin-friction coefficient and (c) relative drag-increase for the different multiscale rough
surfaces with respect to the non-dimensional roughness parameter α. The data points in
both right-hand-side plots represent values averaged over the different Reynolds numbers.
3.1.2. Flow topology in the canopy
The effect of the multiscale rough surfaces on the flow topology is explored by examin-
ing the normalised mean streamwise velocity maps from the planar-PIV measurements,
at the peak symmetry plane (zS= 0.5). Two cases, Iter1and Iter123 are illustrated for
comparison in figure 4. Results show that, away from the wall, the flow is unaffected by
the surface condition, while in the roughness-affected layer it undergoes strong changes
in the streamwise direction (as shown by the streamline contours). Specifically within
the surface canopy, the maps highlight the presence of a separation bubble past the
large cuboid, whose size and intensity appear to depend on the surface condition. This is
illustrated in figure 4(b) which shows that by adding the intermediate and smaller scales,
Iter123 produces a weaker streamwise velocity deficit within the canopy.
The impact of the multiscale roughness on the recirculation region is further assessed
by examining the separation length as shown in brown in figures 4(a) and (b), and
highlighted in an appropriate scaling in figure 4(c). The results show the extent of the
zero contour-level curves to be relatively conserved for the different cases, irrespective of
the surface condition, and is approximately 3h(1). It is however reported that the wall-
Turbulent boundary layer over regular multiscale roughness 11
normal extent subtly weakens with addition of iterations. This behaviour is believed to be
caused by the increased turbulent mixing and wall drag at the top of the large cuboid,
owing to the interaction of a broader range of roughness scales with the separating
shear layer. To explore further the effect of the multiscale roughness on turbulence, the
normalised Reynolds shear stress uv+and its associated turbulence production Pxy δU3
τ
(with Pxy =uvdU dy) fields are examined, as shown in left and right panels of figure
5, respectively. For Iter1, the shear layer formed at the top surface of the large cuboid
separates at its trailing edge, from which strong vortices are shed towards the canopy.
By comparing the area within the encapsulated contour level in the left panel of figure
5, we observe that the separated shear layer carries a weaker vortical activity as more
roughness lengthscales are imposed. However, a plausible explanation for this behaviour
can be related to the range over which turbulent and roughness lengthscales interact
with each other. In fact, when a turbulent flow encounters a broader range of scale, such
as in the case of Iter123, the h(1)-scale vortical structures as seen for Iter1break down
to smaller eddies. This means that while the overall drag increases when increasing the
roughness content, the shear stress activity is redistributed among scales whereby the
intermediate and smaller scales are responsible for a larger portion of drag. Consequently,
this results in a relatively shorter extent and a weaker separation bubble and more drag
emanating from these additional scales. These observations are further substantiated by
the turbulence production maps which show stronger shear layers to be associated with
substantial turbulence production. In contrast, weaker shear layers are accompanied with
weaker turbulence production.
Using the streamwise-wall-normal PIV measurements, the mean pressure distributions
are additionally estimated at the peak symmetry plane (zS= 0.5), by means of the
two-dimensional Reynolds-averaged Naviers-Stokes equations. For more details on the
methodology as well as the numerical integration schemes employed, the reader should
refer to the study by Ferreira & Ganapathisubramani (2020). The mean pressure is
expressed by its non-dimensional form as Cp=(PP)q, with Pand Pbeing the
mean static and free stream pressure, while qis the freestream dynamic pressure.
An example of a pressure field is shown in figure 6(a) for the Iter1case. The coefficient
of pressure field is shown to be dominated by an alternating high and low pressure region
in the canopy, whereas it quickly recovers to the free stream pressure in the outer region
(y0.2δ). In fact, the high pressure is seen to be associated with accelerating flow regions
while low pressures are accompanied with decelerating flow regions. More specifically, the
highest pressure regions are recorded at the windward side of the cuboid, while the lowest
pressure magnitude is shown to occur right past the leading edge of the upper surface
of the cuboid, reflecting the presence of a strong shear layer shedding off from the blunt
leading edge. This negative pressure is shown to trail downstream, forming a weak patch
of low pressure within the recirculation region till the reattachment point. The pressure
transitions to a positive magnitude as the flow re-accelerates again past the reattachment
point.
The profile of the streamwise pressure difference across the largest cuboid is addi-
tionally examined and shown in figure 6(b). The pressure difference is expressed as
∆P (y)=Pw(y)Pl(y)(with the wand lsubscripts referring to the windward and leeward
sides of the cuboid respectively), normalised by the wall-normal-averaged pressure ∆P ,
and plotted against the wall-normal distance normalised by the cuboid height. Although
the profile is not fully resolved down to the wall (due to the spurious region below 1
mm height), the drag profile shows similarities with the results of urban-like roughness
(Ferreira & Ganapathisubramani 2020), with a maximum at nearly two-thirds of the
cuboid height, and decreasing when getting close to the wall. The profile is subsequently
12 Boundary layers over multiscale roughness
Figure 4: Contour maps of the (x, y)-plane normalised mean streamwise velocity for
(a)Iter1and (b)Iter123 measured at the peak symmetry plane zS= 0.5. The cross-
sections of the roughness geometries are included at the bottom of the figures to scale for
reference. The brown solid line represents the zero-velocity contour level illustrating the
separation length, while the mean in-plane streamlines are superimposed to highlight the
recirculation region. (c) Contours of the separation lengths for the different multiscale
rough surfaces normalised by the first iteration cuboid height h(1), with ˆxbeing the
streamwise distance from the leeward side of the cuboid.
used to get an estimate of the pressure drag produced by the large cuboid. This is done
by integrating the streamwise pressure difference profile over the cross sectional area
h(1)×W(1)(assuming the pressure distribution around the cuboid is uniform in the
spanwise direction). The drag force and the friction velocity over the large cuboid are
expressed as
F(1)=W(1)h(1)
0
∆P dy, U (1)
τ=F(1)
ρh(1)W(1)1/2
,(3.1a,b)
The results reveal that the ratio between the friction velocity estimated from the largest
cuboid to the total friction velocity estimated with the drag balance is approximately
60% for Iter1. Assuming this ratio to remain constant throughout the different cases,
the pressure drag contributions from the addition of intermediate and small roughness
Turbulent boundary layer over regular multiscale roughness 13
Figure 5: Contour maps of the (x, y)-plane normalised (left) Reynolds shear stress and
(right) turbulence production for (a, e)Iter1,(b, f )Iter12,(c, g)Iter13 and (d, h)Iter123
measured at the peak symmetry plane zS= 0.5. The cross-sections of the roughness
geometries are included at the bottom of the figures to scale for reference.
scales ultimately leads to the fully-rough regime. In fact, for the other three surfaces, the
intermediate and small roughness lengthscales proportionally add form drag contribu-
tions to the overall drag, resulting in the Reynolds number invariance of Cfobserved in
figure 3. In contrast, the pressure drag contribution for Iter1that stems from the largest
cuboid alone is insufficient to result in a flow that is fully-rough. Due to the viscous drag
which amounts to nearly 40% of the total drag, the skin-friction coefficient Cfremains
dependant of Rexas seen in figure 3. It should be noted that despite the inherent degree
of uncertainty that rises from the PIV-based pressure and the assumption of the spanwise
uniformity of the profiles, this method gives a reasonable indication of the expected form
drag.
3.1.3. Aerodynamic parameters
In order to estimate the aerodynamic quantities that characterise the rough-wall flow,
both the peak and valley symmetry planes (zS= 0, 0.5) PIV data are used. Figure 7(a)
compares the wall-normal distribution of the mean streamwise velocity at the symmetry
planes for Iter1. The angled brackets  in figure 7(b) refer to the horizontally-averaged
velocity between both planes, comparing profiles of the different cases with the smooth-
wall. Substantial differences between the peak and valley profiles can be observed from
figure 7(a). Below the canopy layer, the velocity is shown to be higher at the valley than
at the peak as expected due to the blockage of the large obstacles. At yδ0.15, it is
observed that UP eak =UV alley . However, beyond this point till almost two thirds of the
boundary layer thickness, the velocity above the peak becomes higher than that at the
valley. This indicates that the outer flow presents a degree of spanwise heterogeneity,
probably caused by the surface condition. It is also observed that at the individual
planes, the velocity profiles do not exhibit a clear wall-normal logarithmic distribution.
However, when both planes are averaged, a velocity range appears to vary logarithmically,
and occurs approximately between 0.20.3δ. It is further shown from the profiles of the
different surfaces plotted in figure 7(b), that all the cases deviate from the smooth-wall
14 Boundary layers over multiscale roughness
Figure 6: (a) Normalised mean pressure field reconstructed from the 2D2C-PIV for Iter1
case measured at the peak symmetry plane zS= 0.5. The cross-sections of the roughness
geometries are included at the bottom of the figures to scale for reference. The brown solid
line represents the zero-velocity contour level illustrating the separation length, while
the mean in-plane streamlines are superimposed to highlight the recirculation region.
(b) Mean streamwise pressure difference profile assessed from a single cuboid element of
height h(1)for Iter1.
Figure 7: (a)Wall-normal distribution of the mean streamwise velocity at both symmetry
planes for Iter1.(b)Comparison of the horizontally-averaged mean streamwise velocity
profiles between the different rough surfaces.
behaviour. More specifically, the richer the roughness content is, the higher the profile
deficit becomes.
The mean streamwise velocity profile over the rough surfaces can also be expressed
using the modified law-of-the-wall,
U+U
Uτ=1
κln (yd)Uτ
ν+B∆U+hsUτ
ν,(3.2)
where κand Brepresent the slope and intercept of the logarithmic region, respectively,
similar to a smooth-wall. In contrast with a flat smooth wall, rough walls will give
rise to the quantities dand ∆U+in the form of wall-normal and velocity shifts in the
Turbulent boundary layer over regular multiscale roughness 15
Figure 8: Inner scaling of the horizontally-averaged streamwise velocity profile for the
different multiscale rough surfaces, compared with the DNS turbulent boundary layer
profile of Sillero et al. (2013). The value of the log-law slope κand the smooth-wall
intercept Bused in the current investigation are 0.39 and 4.5 respectively.
logarithmic region, which are termed the zero-plane displacement and the roughness
function respectively. The former is interpreted as the “virtual” origin representative of
the height at which the mean drag acts (Jackson 1981). On the other hand, the latter
provides a quantification of the momentum loss (if positive) or gain (if negative) due to
surface roughness, and depends on the roughness Reynolds number h+
s. The wall-normal
distribution of the inner-normalised mean streamwise velocity profiles are presented in
figure 8, compared with the DNS turbulent boundary layer profile of Sillero et al. (2013).
As expected, the profiles are shown to be affected by the surface condition as they
exhibit both wall-normal as well as velocity shifts in comparison with the smooth-wall.
The rough-wall profiles appear to have relatively similar vertical shifts, However, it is
worth recalling that the free stream speeds Uwere not kept constant among cases.
The zero-plane displacement dis estimated by making use of the modified law-of-the-
wall. This is achieved by taking the derivative of equation 3.2 with respect to y, to obtain
what is called the indicator function, expressed as:
Ξ=(y+d+)dU+
dy+=1
κ.(3.3)
This equation is classically used to determine the extent of the inertial sublayer as well as
the logarithmic slope κfor smooth-wall flows, once Uτis known a priori (¨
Osterlund et al.
2000). This consequently means that if κis assumed to be universal between smooth-
and rough-walls, dshould be only a function of the velocity gradient and the friction
velocity. Therefore, to avoid solving a two-point fit equation in the present study, the
value of κis assumed constant.
Since the friction velocity Uτwas directly measured through the drag balance, the
zero-plane displacement is found as the value that minimises the difference between
the left-hand-side (Ξ) and the right-hand-side (1κ) of equation 3.3. The results are
illustrated in figure 9(a) and show the appropriate values that minimise the difference
yield a good collapse of Ξfor the different cases in the outer region (yδ>0.2). In the
inner region, Ξis shown to depend on the surface condition with the occurrence of a
peak in intensity coinciding with y=h(1). Unexpectedly, the current profiles are observed
to result in negative values of dfor the cases Iter1, Iter12 and Iter123, with the exception
of Iter13 which reported a positive value. These are shown to range between -0.2 to
0.3h(1). Although the values observed appear to conflict with the interpretation provided
16 Boundary layers over multiscale roughness
Figure 9: (a)Variation of the indicator function Ξand (b)the modified log-law function
Ψfor different multiscale surface roughness cases, compared with the smooth-wall profile
in blue curve. A second-order central-difference scheme was used in order to obtain Ξ.
by Jackson (1981), namely the height where the mean momentum sink occurs (thus d
cannot be negative), these values are merely a solution of the logarithmic distribution
fitting. In fact, given that both the velocity and wall shear stress were directly measured,
a potential reason for this behaviour can arise from the assumed value of the logarithmic
slope 1κ, hypothesised herein to be universal between smooth- and rough-walls. However,
the discussion on the universality of κand Bbetween smooth- and rough-walls is beyond
the scope of the current study. It is further reported that the indicator function exhibits
a plateau Ξ1κbetween 0.2yδ0.3, suggesting the inertial sublayer to have
shifted farther away from the wall, consistently for all the cases. This is in contrast with
the smooth-wall boundary layers which are shown to not exceed an ISL upper limit of
0.15δ(equivalent to the 0.15Reτwith Reτ=δUτνreported by Marusic et al. 2013).
This reveals that the roughness topography may have severely altered the mean and
turbulence flow structure in the outer region.
Once the zero-plane displacement is determined, the difference between the logarithmic
velocity distribution and the measured inner-normalised velocity profiles is examined to
obtain
Ψ=U+1
κln (y+d+)B+∆U+.(3.4)
In the highlighted grey region of figure 9(b), the profiles of Ψreach a plateau at zero
for the appropriate values of ∆U+. Similarly to the indicator function, the plateau of Ψis
clearly identified between 0.20.3δ, indicating that the inertial sublayer of the multiscale
rough surfaces to occur farther than that of a smooth-wall. The values of Ψare reported
to be relatively similar between cases, however, it is recalled that the free stream speeds
were not kept constant. Their values are collated in table 2. It should be mentioned
that despite the unexpected negative results of the zero-plane displacement, the values
of dhave only a marginal influence on the roughness function. In fact, by forcing d=0,
the values of ∆U+change by less than 2%. Futhermore, the profiles exhibit a similar
logarithmic slope as the smooth-wall, albeit with a noticeably less intense wake in the
outer region. This observation suggests that the multiscale rough surfaces might have
substantially affected the wake strength parameter Π, thereby, the outer flow dynamics.
Once ∆U+is determined, the equivalent sandgrain roughness height can be deduced
using the fully-rough asymptote relation of Nikuradse (1933),
h+
s=eκ(∆U++C).(3.5)
Turbulent boundary layer over regular multiscale roughness 17
Figure 10: (a)Variation of the inner-normalised velocity profiles with respect to the wall-
normal distance normalised by the roughness length scale hsfor the different multiscale
rough surfaces. The fully-rough intercept that best fits the logarithmic region is found
to be Br8. (b)Variation of the roughness length scale increment ρhswith respect to
the multiscale geometrical parameter α.
The constant Ccan be empirically determined by exposing the flow to fully-rough
conditions, and is defined as the difference between the fully-rough intercept Brand
the smooth-wall intercept B(C=BrB). In the current study, velocity measurements
are only available at one free stream speed (i.e. one Reynolds number). Hence, the value of
Cis assumed to be 3.5 which means the flow should have met the fully-rough conditions
(Schlichting 1979). Following the Cfresults reported in figure 3, the fully-rough condition
has been met for the three rough surfaces Iter12, Iter12 and Iter123 . At the same time, it is
evident from the same plots that Iter1is yet to reach an asymptotic value, hence should
not be ascribed a sandgrain roughness value. However, for the purpose of characterising
the multiscale rough surface, an hsvalue will also be assumed for Iter1in order to allow
a comparison across all cases.
Equation 3.5 is shown to yield h+
svalues that range between 400 and 560, which are
considerably beyond the h+
s=70 suggested to be the lower limit for a flow to reach the
fully-rough regime (Jim´enez 2004). Their values are tabulated in table 2 as a fraction of δ,
and range between 5–10% of the boundary layer thickness. Figure 10(a) shows the mean
velocity profiles U+plotted against the normalised wall-normal distance (yd)hs, and
highlights a good degree of collapse with the blue-dashed line which follows a logarithmic
distribution. More specifically, the cases Iter12 and Iter123 are seen to adequately collapse
within 2 <(yd)hs<3, whereas Iter1and Iter13 collapse better within 3 <(yd)hs<6.
Two reasons are believed to cause this behaviour and are: (i) The friction Reynolds
numbers for Iter1and Iter13 are nearly a factor of two higher than those of Iter12 and
Iter123, leading to a larger scale separation manifested in a slightly wider log region; (ii)
The equivalent sandgrain roughness height for Iter1and Iter13 is almost twice smaller
than that of Iter12 and Iter123, causing the profiles to shift away from the wall. The profiles
are further shown to have a weaker departure from the logarithmic distribution in the
outer region, confirming the influence of the multiscale rough surface on the outer region.
Interestingly, all profiles are shown to have a fully-rough intercept Br8. It should be
noted that the latter is classically reported to be 8.5 (Schlichting 1979). However, this
difference is believed to stem from the fact that the 8.5 value was originally derived from
pipe flow data, for which a logarithmic slope is usually taken as κ0.41. In contrast,
boundary layer flows have a relatively smaller value of κ0.39 which subsequently results
in a smaller intercept (Nagib et al. 2007).
18 Boundary layers over multiscale roughness
The roughness function (hence the equivalent sandgrain roughness height) is known
to depend on various parameters such as the surface texture, shape as well as the plan
distribution. There have been numerous proposed correlations between the roughness
function and the surface properties (see e.g. Flack & Schultz 2010, 2014). In the current
study, we also attempt to find a relation between the aerodynamic quantity hswith
an appropriate geometrical parameter of the roughness, such that a link between the
different multiscale rough surfaces can be established. Similarly to the observations made
from figure 3(c), the variation of the relative roughness length scale increment ρhs, with
respect to the geometrical parameter αis presented in figure 10(b). In line with the CF
results, ρhsalso shows a reasonable linear behaviour observed for the range of surfaces
tested herein. This is expressed in a non-dimensional form as,
ρhsh(i)
s
h(1)
s1=(i)+b, (3.6)
where h(i)
sand α(i)being the equivalent sandgrain roughness height and the geometrical
parameter for the ith surface (with i=1,4 for Iter1to Iter123) respectively. This means
that for these regular multiscale rough surfaces, the aerodynamic roughness lengthscale
of the successive multiscale surfaces is linearly related to that of the previous iterations,
therefore to the first iteration. This suggests that knowledge of the geometrical properties
along with the aerodynamic properties of the parent rough surface are sufficient to predict
the drag of a given multiscale rough surface. However, it should be noted that in the
general context of multiscale roughness, this linear relation may allow the prediction
of hsonly for a limited range of scales/iterations such as from Iter1to Iter1234 (with a
fourth generation of smaller cuboids). In fact, if we keep adding small scale iterations, the
expected drag (hence its associated roughness lengthscale) that can be generated by the
surface will only weakly increase (as if it reaches a certain asymptotic value of hs), and
can perhaps be more appropriately described by a power-law. In the current study, the
limitation in roughness manufacturing as well as the sensitivity of the drag balance to such
small magnitudes prevented the assessment of the limits of the observed linear behaviour.
It is important to stress that despite the apparent suitability of this relationship for the
discrete-like roughness, it currently lacks generalisation over the broader range of rough
surfaces (e.g. the highly irregular continuous-type roughness). Nonetheless, while more
testing is required to assess its applicability across a wider range, this relationship remains
a potential candidate for a subset of topographies such as the multiscale random/regular
urban-like rough surfaces.
3.2. Outer region
3.2.1. Secondary motions
Figure 11(a) and (b) show the cross-stream fields of the normalised mean streamwise
velocity for the cases Iter1and Iter123, respectively. These maps have been obtained
by averaging two cross-planes at the streamwise locations x=3.2 and x=3.25 m
downstream of the contraction. Unlike the streamwise-wall-normal maps shown in figure
4, the mean cross-plane flow exhibits significant spanwise heterogeneity above the canopy
forming alternating high- and low-momentum pathways (HMPs and LMPs) between the
peaks and valley, respectively. This spanwise undulation in the mean flow is shown to
be accompanied by two streamwise rotating cells flanking the sides of the large cuboids
emphasised with the in-plane velocity vector plot superimposed on top of the maps. These
time-averaged structures were identified using the vorticity-signed swirling strength λci
Turbulent boundary layer over regular multiscale roughness 19
Figure 11: Contour maps of the (y, z)-plane of the normalised mean streamwise velocity
averaged over the two planes at x=3.2 and 3.25 m for (a) Iter1and (b) Iter123, respectively.
The cross-sections of the roughness geometries are included at the bottom of the figures to
scale for reference. The black dashed-line represents the velocity contour level U=0.8U.
The mean in-plane velocity vector plot Vand Ware superimposed to highlight the
vortical structures.
as shown in figure 12 and indicate the presence of considerable streamwise circulation in
nearly half the boundary layer thickness irrespective of the multiscale rough surface.
Figure 12 highlights the rotational sense of the vortical structures, with upwash motions
occurring at the valley, while a downwash motion is observed above the peaks. These
cross-plane motions are known to be a manifestation of Prandtl’s secondary flows of
the second kind since they arise from turbulence anisotropy, as opposed to the first kind
which stems from mean flow curvature (Prandtl 1952). These features have recently come
to scrutiny and are now well-documented for instance in boundary layers (Mejia-Alvarez
& Christensen 2013; Nugroho et al. 2013; Willingham et al. 2014; Anderson et al. 2015),
channels and open-channel flows (Wang & Cheng 2006; Yang & Anderson 2017; Chung
et al. 2018; Zampiron et al. 2020) as well as pipe flows (Chan et al. 2018).
A common feature in all these studies is that when the surface spanwise characteristic
length scale is comparable to the dominant length scale of the flow (boundary layer thick-
ness, channel half-height or pipe radius), large-scale secondary motions manifest with
generally upwash and downwash motions occurring above “low-” and “high-roughness”
respectively. In the present investigation, the largest surface lengthscale corresponds to
the largest phase in the multiscale roughness which is Sin both the streamwise and
spanwise direction. This incidentally leads to a δS1.1 for all different multiscale
rough surfaces, which appears to line up with the previous studies, hence, the observed
large-scale secondary motions.
The location of the HMPs and LMPs has recently been questioned since there has
been apparent disagreement between some studies that reported opposite trends, where
upwash and downwash motions can occur above low- and high-roughness, respectively.
In a recent study by Medjnoun et al. (2020), it has been highlighted that these con-
20 Boundary layers over multiscale roughness
Figure 12: Effect of the multiscale rough surfaces on the normalised vorticity-signed
swirling strength distributions for (a)Iter1and (b)Iter123. The black dashed-line
represents the velocity contour level U=0.8U. The mean in-plane velocity vector plot
Vand Ware superimposed to highlight the vortical structures on top of the swirling
motions.
flicting trends are likely to arise from two different driving mechanisms representing
non-equivalent types of surface heterogeneity conditions, which can result in opposite
observations. It is argued that turbulent secondary flows of the second kind can be
generated over two main types of heterogeneous surfaces namely; topography-driven and
skin-friction-driven heterogeneity.
For surfaces purely-driven by skin friction heterogeneity (spanwise-uniform height
distribution), Willingham et al. (2014) and Chung et al. (2018) have demonstrated that
HMPs and LMPs (hence downwash and upwash) systematically occur above regions of
high- and low-skin-friction respectively. For topographically-driven heterogeneity (alter-
nating peaks and valleys), both scenarios have been observed. Several numerical and
experimental studies showed upwash and downwash to occur above low- and high-
roughness (Mejia-Alvarez & Christensen 2013; Barros & Christensen 2014; Yang &
Anderson 2017; Awasthi & Anderson 2018), which is in line with the current observations.
Other studies on the other hand highlighted the opposite behaviour (Nezu & Nakagawa
1984; Wang & Cheng 2006; Vanderwel & Ganapathisubramani 2015; Hwang & Lee
2018). However, one of the key differences between these two sets of studies is the
presence/absence of streamwise heterogeneity in the topography. For surfaces where
HMPs and LMPs have been reported above peaks and valleys respectively, the roughness
clearly exhibited wake producing protrusions which lead to form-drag contributions,
similar to the present study. In contrast, studies that showed upwash and downwash
occurring above the elevated and recessed regions, mainly used streamwise homogeneous
surfaces. It is worth recalling that the location of the HMPs (downwash) and LMPs
(upwash) is a consequence of the averaging procedure. The instantaneous structures on
the other hand, have been shown to be able to reverse this behaviour as both scenarios
can occur in a non-periodic (chaotic) fashion, as recently highlighted by Anderson (2019).
Turbulent boundary layer over regular multiscale roughness 21
Hence, despite the current surfaces being topographically spanwise-heterogeneous, the
presence of the streamwise heterogeneity leads the surfaces to act more like the skin-
friction type heterogeneous surfaces due to the additional pressure drag, unlike the other
set of studies which used surfaces that are predominantly viscous drag producing surfaces.
Therefore, care should be taken when designing numerical simulations or laboratory
experiments, by accounting for both the roughness height spectral content as well as
the phase information, which under certain conditions can dynamically interact with the
δ-scale flow structures, leading large modifications in the boundary layer characteristics.
3.2.2. Turbulent and dispersive stresses
The effect of the multiscale rough surfaces on the turbulence organisation is examined
using the triple decomposition performed on the flow field in the cross-plane. This method
is usually employed in flows where spatial heterogeneity is prevalent, in order to quantify
the magnitude of the stresses originating from the mean flow inhomogeneity. In this
scenario, the dispersive stresses can have substantial contributions to the total stresses
and play a major role in the transport of momentum fluxes (Meyers et al. 2019; Nikora
et al. 2019). In this scheme, the streamwise velocity field can be written as
ui(y, z , t)

Instantaneous
velocity
=
Time-averaged velocity

U(y)S

Spatial-time-
averaged
velocity
+˜u(y, z)

Dispersive
velocity
component
+u(y, z , t)

Random
velocity
component
,(3.7)
where uiis the instantaneous velocity field measured at a fixed streamwise location.
U(y)Sis the time- and horizontally-averaged velocity profile over the spanwise wave-
length S. ˜u(y, z)is the time-invariant spatial deviation field and u(y, z, t)is the time and
space dependant fluctuating part from the Reynolds double decomposition. Using this
method, the different turbulent and dispersive stress tensor terms can be evaluated and
compared. More specifically, the total shear stress term can then be computed as
τxy

total
shear
stress
=ν∂U
∂y

viscous
shear
stress
uv

turbulent
shear
stress
˜u˜v

dispersive
shear
stress
,(3.8)
which typically accounts for viscous, turbulent and dispersive shear stress contributions.
The viscous contribution is only present very near the wall, and very negligible away
from the wall. On the other hand, Nikora et al. (2019) have shown that the dispersive
stresses can be decomposed into roughness-induced and large-scale secondary motion-
induced contributions. They have highlighted that the former can have a notable part
in the total dispersive stress within the canopy region, whereas the secondary motion-
induced part is predominant above the canopy layer. In the present study, we have noticed
that these roughness-induced stresses are very negligible throughout the entirety of the
resolved flow, as the heterogeneity is mainly driven by the secondary motion induced
stresses.
Contour maps of the turbulent, dispersive and total normal stress terms of Iter1are
presented in columns of figure 13, while the shear stress terms are shown in figure 14
respectively. The different normal turbulent stress maps illustrated in figure 13 clearly
exhibit spanwise heterogeneity due to the surface condition. In fact, two behaviours can
be distinguished, in the near canopy region (at yδ0.15), the turbulence intensities
22 Boundary layers over multiscale roughness
Figure 13: Contours maps of the normalised normal components of the (a)streamwise,
(b)wall-normal and (c)spanwise stresses respectively for the Iter1case. The left, middle
and right panels represent the turbulent, dispersive and the total stresses respectively.
The colour scales have been adjusted to highlight the features for the dispersive stresses.
are shown to be much higher above the roughness elements while weaker magnitudes are
seen between the ridges. In the outer region (at yδ0.4), the opposite behaviour occurs
with higher intensities occurring between the ridges while weaker magnitudes are located
above the roughness elements. These heterogeneous cross-plane turbulence intensity
distributions are seen to be related to the rotational sense of the secondary motion, which
itself is imposed by the surface condition. The high-turbulence fluctuations observed near
Turbulent boundary layer over regular multiscale roughness 23
Figure 14: Contours maps of the normalised shear components of the (a)streamwise-
wall-normal, (b)streamwise-spanwise and (c)spanwise-wall-normal stresses respectively
for the Iter1case. The left, middle and right panels represent the turbulent, dispersive
and the total stresses respectively. The colour scales have been adjusted to highlight the
features for the dispersive stresses.
the ridges (at yδ0.15) are advected laterally towards the mid canopy, and upwards
resulting in the LMPs. This is further accompanied by the low-turbulence fluctuations
being advected from the free stream towards the ridges, inducing the observed HMP
above the ridges (clearly emphasised when examining the activity of vv distribution).
This spanwise turbulence inhomogeneity has previously been shown to be associated
24 Boundary layers over multiscale roughness
with an imbalance between turbulence production and viscous dissipation, causing these
amplified lateral motions and sustaining the streamwise vorticity observed in the form
of secondary flows (Hinze 1973; Anderson et al. 2015; Hwang & Lee 2018).
Furthermore, the normal dispersive stresses shown in the mid panel of figure 13 are
shown to be relatively smaller with respect to the turbulent ones, however not negligible.
They are shown to extend for nearly half of the boundary layer thickness although with a
varying intensity along the spanwise direction. Their magnitude is shown to be relatively
weaker in comparison with the turbulent fluctuations (shown with a smaller colormap
scale), while their spatial distribution also exhibiting spanwise variation different to that
observed for the turbulent components. They are in fact shown to be more localised with
upwash regions accompanied with intense activity whereas HMPs are associated with
smaller magnitudes.
The streamwise and wall-normal dispersive stresses are shown to vanish at spanwise
locations between HMPs and LMPs which coincide with weak upwash/downwash mo-
tions. On the other hand, the spanwise dispersive stress is shown to be negligible at
the LMPs and HMPs except in-between. This region is seen to coincide with enhanced
spanwise fluctuations, albeit with a relatively weaker magnitude in comparison with the
normal and streamwise dispersive stresses. The total normal stresses expressed as the
sum of the turbulent and dispersive stress (with the viscous stress contributions above
the canopy being negligible) is illustrated in the right panel of figure 13. The results show
that globally, the total stresses remain relatively similar to the turbulent ones except with
an increased activity between the large roughness elements corresponding to the upwash
regions, which are amplified because of the dispersive stress contributions.
For the shear stress terms presented in figure 14, their spatial distributions are also
seen to undergo a spanwise modulation due to the wall condition. The spatial distribution
of uv presents enhanced shear stress activity above the ridges (in the near canopy) while
a weaker activity is shown in the outer region. Between the ridges, the opposite trend
happens, with weak activity near the canopy and increased shear stress in the outer layer.
At the same time, the dispersive shear stress distribution shows a similar behaviour to
that reported for the normal dispersive stress term ˜u˜u, with localised patches above the
ridges and in valleys, while its magnitude weakens in between. Interestingly, a positive
dispersive shear stress activity is reported near the canopy, more specifically above the
ridges and slightly less at the valley.
The reason for this behaviour comes from the sign of the product of both quantities
˜uand ˜v. ˜uand ˜vare shown to have opposite signs at the HMPs and LMPs, with ˜u>0
and ˜v<0 at the HMPs while with ˜u<0 and ˜v>0 at the LMPs. However, the latter
is true except near the canopy where ˜uexhibits a sign reversal such that ˜u<0 right
above the ridge (flow experiences increased local friction) and, at a similar height at the
valley ˜u>0 (flow experiences weaker local friction), resulting in the observed ˜u˜v>0.
This leads to a relative increase of the total shear stress τxy in the outer region and a
reduction near the canopy. The turbulent, dispersive and total shear stress distributions
of the streamwise-spanwise and the wall-normal-spanwise shear components are shown
in figure 14(b) and (c), respectively. Similarly to the previous stress terms, their spatial
distributions are shown to experience spanwise heterogeneity with varying intensity along
the span. The sign of the stress maps is shown to be switching at the symmetry planes
(zS=0 and ±0.5) which coincide with locations of zero spanwise fluctuations. The extent
of the dispersive shear stress terms is seen to reach at least half of the boundary layer
thickness but with a weaker contribution to the total stresses.
Turbulent boundary layer over regular multiscale roughness 25
3.2.3. Outer-layer similarity
Following the observations made above, outer-layer similarity is also examined. The
similarity between the rough- and smooth-wall structural paradigms stems from an
established concept of flow universality known as Townsend’s wall-similarity, which states
that the influence of viscosity is limited to the near wall region (Townsend 1976). In this
analogy, outer-layer similarity extends this concept to include surface roughness with its
validity relying on two main conditions. (i) A large scale separation (Reτ1) and (ii)
a small relative roughness height to the depth of the flow (hδ1), as emphasised by
Jim´enez (2004). This is generally expressed in the form of the streamwise velocity deficit
and the turbulence quantities which are hypothesised to have a universal form of,
UU
Uτ=fy
δ,uiuj
U2
τ=gij y
δ.(3.9a,b)
This implies that apart from affecting the near-wall region, the main role of roughness is
to set the wall drag (Uτ) and adjust the boundary layer thickness (δ), while the mean and
turbulence structure remain unaffected by the surface condition. To examine the impact
of the multiscale rough surfaces on the outer-layer similarity hypothesis, the horizontally-
averaged mean streamwise velocity profiles are plotted in defect form and shown in figure
15(a), while the streamwise turbulence intensity profiles are depicted in figure 15(b). The
mean velocity profiles indicate a clear lack of collapse in the outer region, showing a
weaker deficit than that of the smooth-wall. However, it is interesting to point out that
between the cases, a reasonable degree of self-similarity is maintained. The absence of
similarity is also shown in the inner-normalised turbulence intensity profiles, both with
respect to the smooth-wall profile as well as among cases. The profiles are shown to
recover similarity only near the edge of the boundary layer thickness.
In order to quantify the changes in the outer flow due to the multiscale rough surfaces,
the wake strength parameter Πis examined. To this end, it is common to assume the
flow in the outer region to be fully described by the composite log-wake profile. This is
expressed as:
U+
composite=U+
log-law+Π
κ2wy
δ,(3.10)
where U+
log-law represents equation 3.2 and wan analytical expression known as Cole’s
wake function (Coles 1956). There are numerous expressions that have been proposed
in the literature, and each one yields a value specific to that function (see e.g. Jones
et al. 2001; Junke et al. 2005; Nagib et al. 2007). The wake strength parameter is then
determined through fitting the measured velocity profiles to the assumed ‘universal’
expression. In order to circumvent the use of a particular expression for the outer
region, the wake strength parameter is instead determined using equation 3.4, and varies
proportionally with the maximum of Ψ. This is expressed as
Π=κ
2max(Ψ)κ
2max U+1
κln(y+d+)+B∆U +.(3.11)
Their values are summarised in table 2 and are shown to be considerably weaker than
that of a smooth-wall (0.57), with Π= 0.18, 0.27, 0.20 and 0.23 for the cases Iter1,
Iter12, Iter13 and Iter123 respectively. These values of Πseem to agree with the defect
velocity profiles which were shown to have a relatively similar deficit across cases, while
being smaller than that of a smooth-wall. The results suggest that universal outer-layer
similarity cannot hold for flows developing over this type of surfaces. In other words,
flows that harbour large-scale secondary motions, yielding a highly three-dimensional
26 Boundary layers over multiscale roughness
Figure 15: Wall-normal distribution of (a)the streamwise velocity deficit and (b)the
streamwise variance profiles for the different multiscale rough surfaces, compared with
the DNS turbulent boundary layer profile.
flow are unlikely to satisfy similarity, as they could have substantially altered the wake
intermittency and the uniform momentum zones which together result in the wake
strength parameter (Krug et al. 2017).
Using the dispersive stress components, the inner-normalised profiles of the total stress
terms τxx+and τzz+are examined in figure 16(a), while τy y +and τxy+are shown
in figure 16(b). The spanwise-averaged shear stresses τxz +and τyz +are null because of
the spanwise anti-symmetry at the symmetry plane zS=0, as illustrated in the figures
14(b) and (c). Despite the contributions of the dispersive stresses, overall, the normal and
the shear stress profiles show a smaller magnitude when compared to their equivalent
smooth-wall. This indicates that the increase in the stresses due to the turbulent and
dispersive fluctuations caused by the roughness is not necessarily accompanied with a
proportional increase in skin-friction, since the profiles do not collapse with the smooth-
wall (i.e. lack of outer-layer similarity). A degree of collapse seems to be recovered only
beyond yδ>0.6, which coincides with the wall-normal distance at which the dispersive
stresses are mitigated as reported in the figures 13 and 14.
Figure 16(c) illustrates the inner-normalised dispersive shear stress profiles with respect
to the wall-normal distance normalised by the roughness lengthscale hs. It shows that
the maximum magnitude of the dispersive shear stresses rises up to nearly 0.16% of
the friction velocity, whereas its location changes across cases. It is observed that the
wall-normal location of this maximum reduces as the roughness content (hs) increases,
with ˜u˜v+
max occurring at y2hsfor Iter123 while its occurrence is seen at y5hsfor
Iter1. The reason for the latter behaviour rises from the increase in hswhen increasing
the roughness content. This leads to a lack of similarity between cases suggesting that
the increment in the aerodynamic roughness lengthscale does not necessarily induce a
proportional change in the secondary motions. However, it is worth noting that ˜u˜v+
max
remains nearly constant across cases. Beyond their maximum value, the dispersive shear
stresses appear to have reasonably similar decay rates in the outer region, despite the
offset in the wall-normal direction.
The quantification of flow heterogeneity for the different surfaces is also examined
in outer-scaling as presented in figure 16(d), illustrating the relative contribution of the
dispersive to the total shear stress. The profiles are shown to have a weak magnitude near
the canopy exhibiting even negative values right above the large cuboid for Iter1, in line
with the observations reported in figure 14. Farther away, the dispersive shear stresses
reach their maximum between 0.2–0.25δ, corresponding to the wall-normal location at
Turbulent boundary layer over regular multiscale roughness 27
Figure 16: Wall-normal distribution of (a, b)the inner-normalised total normal and shear
stresses for the different multiscale rough surfaces compared with the smooth-wall in blue-
solid line. Wall-normal distribution of the (c)inner- and (d)outer-normalised dispersive
shear stress, highlighting the extent of the roughness sublayer.
which the secondary motions are most significant. In this scaling, the dispersive stresses
are shown to amount to a substantial fraction of the total shear stresses with roughly
22% at their maximum, while they seem to have a relatively better collapse in the outer
region as opposed to the scaling presented in figure 16(c). Their contributions rise up
to approximately 0.6δ, corresponding to the turbulent stresses collapsing with the total
shear stress profiles. This means that the inertial sublayer is completely submerged within
the roughness sublayer (indicated in the figure as the RSL). This behaviour is in contrast
with homogeneous rough-wall flows, which generally exhibit a RSL as a small portion
of the flow depth, and does not exceed the upper edge of the ISL. The current results
also indicate that the dispersive stresses are mostly a function of the largest lengthscale
features of the surface (S), while they only weakly vary with addition of small scale
roughness. The multiscale rough surfaces have therefore impacted not only the wall-
normal location of the inertial sublayer as seen in figure 9, but also the overall structure
organisation in the outer region.
Castro et al. (2013) surveyed a wide range of rough-wall data published in the literature
to examine the outer flow behaviour using the diagnostic-plot method. This method has
first been introduced by Alfredsson et al. (2011) for smooth-wall flows, and consists of
examining the dependence of the streamwise turbulence intensity profile with respect
to the mean flow. Their assessment revealed the existence of a region spanning from
the inertial sublayer up to almost the entire wake which varies linearly, whose extent
increases with increasing scale separation. Accordingly, Castro et al. (2013) examined
the applicability of this method over a comprehensive range of rough-wall data to reveal
whether smooth- and rough-walls also exhibit outer-layer similarity in this form.
28 Boundary layers over multiscale roughness
Figure 17: Effect of the different multiscale rough surfaces on (a)the total normalised
streamwise rms profiles in the diagnostic-plot form as proposed by Alfredsson et al.
(2011). (b)The same data with the modified scaling U=U+∆U +and U
=U+∆U+
proposed by Castro et al. (2013).
Their results showed that just like the smooth-wall flow, rough-wall flows also presented
a linear variation except with a higher slope, shown to be independent of the roughness
morphology. This rise in the turbulence intensity level was shown to be primarily due
to the roughness function ∆U+, with marginal effect from the wake strength parameter.
Using a similar approach, we try to assess whether a self-similar region still exists for the
current multiscale rough surfaces using this scaling.
The total normalised streamwise r.m.s. profiles τxxU=uu +˜u˜uUare plotted
against the normalised streamwise velocity UU, as shown in figure 16(a). The
dispersive stress component has been included to account for the presence of spanwise
mean flow heterogeneity, which would have been non-existent in the case of a two-
dimensional boundary layer flow. The profiles of the different cases have been plotted
along with a smooth-wall profile and two asymptotes; the smooth-wall asymptote of
Alfredsson et al. (2011) and the fully-rough asymptote of Castro et al. (2013). The
results show a good collapse both between the different cases as well as with the fully-
rough line, indicating that all the cases exhibit similar attributes as a fully-rough flow,
including Iter1. This suggests that despite the lack of collapse observed in the profiles
in figure 15, the surface condition influences both the turbulence intensity and the
mean flow proportionally, regardless of the presence of smaller or intermediate roughness
lengthscales. As demonstrated by Castro et al. (2013), the increase in the streamwise
stress level (slope) for rough-wall flows stems from the roughness function ∆U+. As
such, they showed that by rescaling the turbulence intensity profiles with the modified
quantities U=U+∆U+and U
=U+∆U+, all profiles collapsed on to the smooth-wall
asymptote.
After applying the modified scaling approach as seen in figure 16(b), we notice in fact
that the slopes of the different profiles line up relatively well with the smooth-wall asymp-
tote, as reported by Castro et al. (2013). It should be noted that if the dispersive stress
component (˜u˜u) is omitted, the profiles will result in shallower slopes. Figure 16(b)
also suggests that the roughness function embodies both stress contributions; namely
the turbulence which stems from the temporal fluctuations and dispersive stresses which
arise from spatial heterogeneities. This means that despite the large ramifications they
cause to the base flow, the secondary motions are capable of reorganising the flow such
that the mean and turbulence structures maintain a degree of local similarity between
smooth- and rough-walls. However, it is worth recalling that despite this apparent collapse
Turbulent boundary layer over regular multiscale roughness 29
in the diagnostic-plot, Townsend’s similarity hypothesis remains clearly violated as a
consequence of the large-scale secondary motions. Hence, the validity and application
of outer-layer similarity hypothesis should be carefully considered when examining flows
developing over such surfaces.
4. Conclusions
In this experimental study, the characteristics of a turbulent boundary-layer flow
developing over regular multiscale rough surfaces have been examined. Four surfaces
representing a multiscale distribution of roughness elements, based on a Sierpinski fractal-
like pattern, made of superimposing size-decreasing self-similar cuboids are considered.
A floating element drag balance along with PIV measurements allowed us to assess the
impact of these multiscale rough surfaces on the skin friction as well as the mean and
turbulent flow properties.
Drag balance measurements revealed that the skin-friction coefficients reached their
asymptote with Reynolds number (i.e. fully-rough regime) for surfaces containing at least
the intermediate or smaller scales (Iter12 or Iter13), in contrast with the surfaces contain-
ing only the largest roughness element (Iter1) that was shown to be Reynolds number
dependent (despite the high value of ∆U+). The results showed that the magnitude of the
skin friction is dependant of the scale hierarchy of the surface roughness, and increases
with frontal solidity of the added roughness (α).
The impact of the multiscale roughness on the turbulent boundary layer is also
examined using velocity field measurements. In the canopy region, a strong shear layer
formed above the largest cuboid separates at its trailing edge leading to the formation of a
separation bubble past the obstacle. The streamwise extent of this bubble (reattachment
length) was seen to be insensitive to the roughness content while its intensity decreases
with the presence of smaller roughness lengthscales. This behaviour is caused by the
increased turbulent mixing and wall drag at the top of the largest cuboid, owing to
the increased interactions of the shear layer with a broader range of roughness scales.
Additionally, the PIV-based pressure fields showed the canopy to be dominated by alter-
nating high and low pressure zones associated with the flow accelerating and decelerating
upstream and downstream the first roughness iteration respectively. This allowed the
assessment of the cross-sectional drag, and revealed that these large-scale roughness
elements are responsible for nearly 60% of the surface drag.
The aerodynamic parameters are examined using the horizontally-averaged mean ve-
locity profiles. The virtual origin as well as the roughness function have been determined
by means of the indicator function. The latter highlighted an inertial sublayer occurring
between 0.2–0.3δ, which is farther in comparison with the classical smooth- and rough-
walls. The roughness function estimates were then used to determine the equivalent
sandgrain roughness height hswhose values indicated that all the cases can be considered
in the fully-rough regime, including Iter1(h+
s>400). A ratio of the aerodynamic
roughness lengthscale between successive iterations (ρhs) was defined and shown to vary
linearly with a suitable geometrical parameter of the rough surface (α). This relationship
together with the aerodynamic properties of the first iteration can be sufficient to describe
the roughness length scale of the subsequent multiscale surfaces (and hence the surface
drag).
The cross-plane velocity fields revealed a significant spanwise heterogeneity in the outer
layer for all surfaces examined. This spanwise flow modulation is shown to be caused by
the presence of Prandtl’s secondary flows of the second kind, which created high- and
low-momentum pathways alternating above the ridges and the valleys respectively. As
30 Boundary layers over multiscale roughness
shown from previous studies, this feature occurs as consequence of the dominant length
scale of the flow interacting with δ-scale roughness features. In this study, the largest
characteristic length scale of the roughness is the spacing Sbetween the largest cuboid
which results in δSO(1).
Triple decomposition of the velocity fields allowed the qualitative and quantitative
assessment of the spanwise heterogeneity in the form of dispersive stresses. The results
showed that, at their maximum, these form-induced stresses amount to more than
20% of the total stresses and can extend up to 0.6δ. Additionally, the validity of
Townsend’s similarity hypothesis was assessed by examining the velocity deficit as well
as the turbulence intensity profiles. Lack of outer-layer similarity in both the mean
and turbulence statistics is reported despite the required conditions being satisfied
(4600 Reτ9700 and 0.05 hsδ0.1 ), while a good degree of collapse between
cases is observed in the mean flow.
Acknowledgements
The authors gratefully acknowledge the financial support of the Engineering and
Physical Sciences Research Council of the United Kingdom (EPSRC grant Ref No:
EP/P021476/1). J.M. acknowledges the support from the Research Foundation –
Flanders (FWO grant no V4.255.17N) for his sabbatical research stay at the University
of Southampton. The authors would like to also express their acknowledgement
for the insightful comments and pertinent observations of the referees during
the review process. All data presented herein as well as the supporting data of
this study are openly available on the University of Southampton repository at
https://doi.org/10.5258/SOTON/D1765.
Declaration of interests
The authors report no conflict of interest.
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Wall-bounded turbulence, where it occurs in engineering or nature, is commonly subjected to spatial variations in wall shear stress. A prime example is spatially varying roughness. Here, we investigate the configuration where the wall shear stress varies only in the lateral direction. The investigation is idealised in order to focus on one aspect, namely, the similarity and structure of turbulent inertial motion over an imposed scale of stress variation. To this end, we analyse data from direct numerical simulation (DNS) of pressure-driven turbulent flow through a channel bounded by walls of laterally alternating patches of high and low wall shear stress. The wall shear stress is imposed as a Neumann boundary condition such that the wall shear stress ratio is fixed at 3 while the lateral spacing $s$ of the uniform-stress patches is varied from 0.39 to 6.28 of the half-channel height $\unicode[STIX]{x1D6FF}$ . We find that global outer-layer similarity is maintained when $s$ is less than approximately $0.39\unicode[STIX]{x1D6FF}$ while local outer-layer similarity is recovered when $s$ is greater than approximately $6.28\unicode[STIX]{x1D6FF}$ . However, the transition between the two regimes through $s\approx \unicode[STIX]{x1D6FF}$ is not monotonic owing to the presence of secondary roll motions that extend across the whole cross-section of the flow. Importantly, these secondary roll motions are associated with an amplified skin-friction coefficient relative to both the small- and large- $s/\unicode[STIX]{x1D6FF}$ limits. It is found that the relationship between the secondary roll motions and the mean isovels is reversed through this transition from low longitudinal velocity over low stress at small $s/\unicode[STIX]{x1D6FF}$ to high longitudinal velocity over low stress at large $s/\unicode[STIX]{x1D6FF}$ .