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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 U∞are the air kinematic viscosity and free stream

velocity respectively, have been observed to range between 1–5×10−8from 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 ms−1, 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 x≈30δfrom 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 ms−1. 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 ms−1for 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∞(ms−1)δS δh ∆U +hsδ Π Reτ×10−3CF×10−3β(%)

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

a≈−15 and b≈0.058, 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=(P−P∞)q∞, with Pand P∞being the

mean static and free stream pressure, while q∞is 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

(y≥0.2δ). 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 yδ≈0.15, 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 0.2−0.3δ. 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 (y−d)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 U∞were 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 Ξ≈1κbetween 0.2≤yδ≤0.3, 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 0.2−0.3δ, 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 Br≈8. (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=Br−B). 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 (y−d)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 <(y−d)hs<3, whereas Iter1and Iter13 collapse better within 3 <(y−d)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 Br≈8. 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 κ≈0.41. In contrast,

boundary layer ﬂows have a relatively smaller value of κ≈0.39 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,

ρhs≡h(i)

s

h(1)

s−1=aα(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 δS≈1.1 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 yδ≈0.15), 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 yδ≈0.4), 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 yδ≈0.15) 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 (Reτ≫1) and (ii)

a small relative roughness height to the depth of the ﬂow (hδ≪1), 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,

U∞−U

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

κ2−wy

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 y≈2hsfor Iter123 while its occurrence is seen at y≈5hsfor

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˜uUare plotted

against the normalised streamwise velocity UU∞, 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 δS≈O(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 ≤Reτ≤9700 and 0.05 ≤hsδ≤0.1 ), 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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