# Investigation of unsteady secondary flows and large-scale turbulence in
heterogeneous turbulent boundary layers

D. D. Wangsawijaya1,2<EMAIL_ADDRESS>N. Hutchins1  1Department
of Mechanical Engineering, University of Melbourne, Victoria 3010, Australia
2Aerodynamics and Flight Mechanics Research Group, Faculty of Engineering and
Physical Sciences, University of Southampton SO17 1BJ, United Kingdom

###### Abstract

Following the findings in Wangsawijaya et al. (2020), we re-examine the
turbulent boundary layers developing over surfaces with spanwise heterogeneous
roughness of various roughness wavelengths $0.32\leq S/\overline{\delta}\leq
3.63$, where $S$ is the width of the roughness strips and $\overline{\delta}$
is the spanwise-averaged boundary-layer thickness. The heterogeneous cases
induce counter-rotating secondary flows, and these are compared to the large-
scale turbulent structures that occur naturally over the smooth wall. Both
appear as meandering elongated high- and low-momentum streaks in the
instantaneous flow field. Results here based on the triple decomposed velocity
fluctuations suggest that the secondary flows might be spanwise-locked
turbulent structures, with $S/\overline{\delta}$ governing the strength of the
turbulent structures and possibly the efficacy of the surface in locking the
structures in place (most effective when $S/\overline{\delta}\approx 1$). In
terms of unsteadiness, we find an additional evidence from these conditional
averages showing that the secondary flows exhibit maximum unsteadiness (or
meandering) when $S/\overline{\delta}\approx 1$. Conditional averages of the
fluctuating velocity fields of both spanwise heterogeneous and smooth wall
cases result in structures that are strongly reminiscent of the streak-vortex
instability model. Secondary flows and large-scale structures coexist in the
limits where either $S/\overline{\delta}\gg 1$ or $S/\overline{\delta}\ll 1$,
where the secondary flows scale on $\delta$ or $S$, respectively. When
$S/\overline{\delta}\gg 1$, the secondary flows are locked about the roughness
transition, while relatively unaltered large-scale structures occur further
from the transition. In the case where $S/\overline{\delta}\ll 1$, $S$-scaled
secondary flows are confined close to the surface, coexisting with unaltered
larger scale turbulent structures that penetrate much deeper into the layer.

###### keywords:

##  1 Introduction

We consider a specific case of heterogeneous roughness where the roughness
varies in the spanwise direction. Spanwise heterogeneity is imposed by
alternating rough and smooth strips to form a test surface over which a
turbulent boundary layer is developed (figure 2). This type of roughness
heterogeneity induces secondary flows in the form of counter-rotating
streamwise rollers, which are apparent in the time-averaged velocity field. An
examination of the instantaneous velocity field, however, reveals the
secondary flows as elongated meandering high- and low-speed streaks, not
unlike the large-scale/very-large-scale motions (LSM/VLSM) that occur
naturally in wall-bounded turbulence (Wangsawijaya et al., 2020).

LSMs have commonly been associated with hairpin packets (Adrian et al., 2000;
Tomkins & Adrian, 2003) and $\delta$-scaled meandering high- and low-momentum
streaks (Hutchins & Marusic, 2007a, b) in the outer layer of wall-bounded
turbulent flows, where $\delta$ is either the boundary layer thickness,
channel half-height, or pipe radius. In the logarithmic region, clusters of
hairpin packets agglomerate to form VLSM or ‘superstructures’ (Kim & Adrian,
1999; Guala et al., 2006; Balakumar & Adrian, 2007; Hutchins & Marusic, 2007a;
Dennis & Nickels, 2011b; Lee & Sung, 2011; Hutchins et al., 2012; Wu et al.,
2012). The imprint of symmetrical hairpin vortex packets is evident in certain
statistical analyses of these structures. For example, two-point correlations
of the fluctuating flow field exhibit an elongated low-momentum streak flanked
by high-momentum streaks in the spanwise direction (Tomkins & Adrian, 2003;
Ganapathisubramani et al., 2003, 2005; Hutchins et al., 2005; Hutchins &
Marusic, 2007a; Marusic et al., 2010; Dennis & Nickels, 2011a). However,
Johansson et al. (1991) cautioned against this interpretation of the two-point
correlation contours. Ensemble averaging and assumptions of spanwise
homogeneity, as typically applied in the computation for smooth wall-bounded
turbulence, enforce a plane of symmetry in the resulting conditional average.
Asymmetrical large-scale structures (one-sided roll modes) have, in fact, been
observed instantaneously in turbulent boundary layers formed over smooth walls
(Lozano-Durán et al., 2012; Kevin et al., 2019a), heterogeneous roughness
(Vanderwel et al., 2019; Wangsawijaya et al., 2020), and converging-diverging
(C-D) riblets (Kevin et al., 2017). Elsinga et al. (2010) observed that
hairpin packets are most likely comprised of arch-like or cane-shaped
structures. Similar to the near-wall cycle, an alternative model for the
formation mechanism has been suggested for coherent structures in the outer
layer. Large-scale elongated streaks with a sinuous instability and associated
asymmetric staggered quasi-streamwise vortices have been observed in the
logarithmic region and beyond (Flores & Jiménez, 2010; Cossu & Hwang, 2017; de
Giovanetti et al., 2017), which are similar to, but at a much larger scale
than, structures associated with the streak-vortex instability, which was
initially developed as a model for near-wall streak formation (Jeong et al.,
1997; Waleffe, 2001; Schoppa & Hussain, 2002).

The naturally occurring large-scale structures and the secondary flows induced
by spanwise heterogeneity share some similarities: both are characterised by
roll modes and elongated high- and low-momentum streaks. An unsteadiness or
streamwise periodic behaviour of the secondary flows have also been inferred
from previous studies of spanwise heterogeneous roughness through the 1-D
energy spectrograms (Nugroho et al., 2013; Awasthi & Anderson, 2018; Medjnoun
et al., 2018; Zampiron et al., 2020; Wangsawijaya et al., 2020), two-point
correlation maps (Kevin et al., 2017, 2019b; Wangsawijaya et al., 2020), and
POD (proper orthogonal decomposition) of the turbulent fluctuation fields
(Vanderwel et al., 2019). It was suggested in Wangsawijaya et al. (2020) that
this unsteadiness is a function of the spanwise roughness wavelength
$\Lambda=2S$, where $S$ is the width of the roughness strip. The secondary
flows strongly meander when $S/\delta\approx 1$, with a greater meandering
amplitude as compared to the LSM/VLSM of smooth-wall turbulent flows and the
secondary flows in the limiting cases where $S/\delta\gg 1$ or
$S/\overline{\delta}\ll 1$. These observed similarities and differences pose
questions about the nature of the secondary flows imposed by surface roughness
in comparison to the large-scale structures in wall-bounded turbulence. The
similarities might suggest that secondary flows and large-scale structures
share the same formation mechanism (a notion alluded to by Townsend, 1976, p.
328–331). If that is the case, (i) is it possible that secondary flows are
just phase-locked large-scale structures (and, large-scale structures are just
non-phase-locked secondary flows)? Given the unsteadiness of secondary flows
noted above, (ii) can this be explained as the natural meandering process of
the large-scale structures? In Chung et al. (2018), the isovels of the mean
streamwise velocity showed that in the limiting case scenarios where
$S/\delta\gg 1$ or $S/\overline{\delta}\ll 1$ the secondary flows are confined
within certain parts of turbulent boundary layer while other regions approach
locally homogeneous conditions. This leads to the final question about (iii)
the extent to which the secondary flows and the naturally present turbulent
structures coexist in boundary layers formed over spanwise heterogeneous
roughness. To answer these questions, we conduct an analysis of the
fluctuating velocity components obtained from particle image velocimetry (PIV)
measurements on turbulent boundary layers developing over surfaces composed of
streamwise-aligned, spanwise-alternating sandpaper and cardboard strips
(figure 2). The test surfaces cover a range of spanwise wavelength $0.32\leq
S/\overline{\delta}\leq 3.63$, where $\overline{\delta}$ is the spanwise-
averaged boundary layer thickness at the measurement station.

Figure 1: Triple-decomposition of (a) instantaneous snapshot streamwise
velocity $u$ into (d) $yt$-average $\langle U\rangle_{\Lambda}$, (e) time-
average, spatial fluctuation $\widetilde{U}$, and (f) instantaneous snapshot
of turbulent fluctuation $u^{\prime}$ for case SR50
($S/\overline{\delta}=0.62$). In (b), $\Lambda_{ci}$ is the mean swirl
strength multiplied by the sign of streamwise vorticity
$\Omega_{x}/|\Omega_{x}|$ and normalised by $\overline{\delta}/U_{\infty}$.
Red dashed lines illustrate the extent of non-zero $\Lambda_{ci}$, whose width
is $l_{y}$. (c) shows the variance of streamwise turbulent fluctuation
$\overline{u^{\prime}u^{\prime}}$. (g) contours of time-averaged streamwise
velocity $U$. Vectors indicate $V$ and $W$, downsampled for clarity. Solid
white lines mark the three spanwise locations related to the mean secondary
flows: common flow up ( WAH1gu), centre of a mean secondary flow ( WAH1gc),
and common flow down ( WAH1gd). Black and white patches indicate ‘rough’ and
‘smooth’ strips, respectively. (h) Contours of $\widetilde{u}^{\prime}$, which
is defined as $\widetilde{u}^{\prime}\equiv\widetilde{U}+u^{\prime}$.

The axis system in this study $\boldsymbol{x}=(x,y,z)$ is defined as the
streamwise, spanwise, and wall-normal directions, respectively, which
corresponds to the instantaneous velocity components $\boldsymbol{u}=(u,v,w)$.
As illustrated in figure 1, $\boldsymbol{u}$ is triple decomposed into its
temporal-, spatial-average, and the fluctuations (Raupach & Shaw, 1982;
Finnigan, 2000; Coceal et al., 2006),

$\displaystyle\boldsymbol{u}(y,z,t)$ $\displaystyle=$
$\displaystyle\boldsymbol{U}(y,z)+\boldsymbol{u^{\prime}}(y,z,t)$ (1)
$\displaystyle=$ $\displaystyle\boldsymbol{\langle
U\rangle}_{\Lambda}(z)+\boldsymbol{\widetilde{U}}(y,z)+\boldsymbol{u^{\prime}}(y,z,t)$
(2) $\displaystyle=$ $\displaystyle\boldsymbol{\langle
U\rangle}_{\Lambda}(z)+\boldsymbol{\widetilde{u}^{\prime}}(y,z,t)$ (3)

where $\boldsymbol{U}$ (figure 1a) is the Reynolds (temporal) average, and
$\boldsymbol{u^{\prime}}$ (figure 1f) is the turbulent fluctuations about this
Reynolds average. $\boldsymbol{U}$ is further decomposed into its $yt$-average
$\boldsymbol{\langle U\rangle}_{\Lambda}$ (figure 1d) and the spatial
fluctuations about this mean $\boldsymbol{\widetilde{U}}$ (figure 1e). Since
$\langle V\rangle_{\Lambda}=\langle W\rangle_{\Lambda}=0$,
$\boldsymbol{\widetilde{U}}=(\widetilde{U},V,W)$. With this chosen
decomposition method for spanwise heterogeneous roughness,
$\boldsymbol{\widetilde{U}}$ can be considered as the stationary components of
the secondary flows (or the dispersive components) and
$\boldsymbol{u^{\prime}}$ are both the advecting turbulence _and_ the
unsteadiness of the secondary flows. We also introduce the quantity
$\boldsymbol{\widetilde{u}^{\prime}}$ (see 3), which is defined as
$\boldsymbol{\widetilde{u}^{\prime}}\equiv\boldsymbol{\widetilde{U}}+\boldsymbol{u^{\prime}}$
(figure 1h). It should be noted that for the reference smooth-wall case SW-2,
as a result of spanwise homogeneity, $\boldsymbol{\widetilde{U}}=0$ and
$\boldsymbol{\widetilde{u}^{\prime}}=\boldsymbol{u^{\prime}}$.

##  2 Experimental set-up

###  2.1 Test surfaces

Spanwise heterogeneous roughness
---
Case000 | $S$ | $\overline{\delta}$ | $S/\overline{\delta}$ | $x$ | $U_{\infty}$ | $Re_{x}$ | $Re_{\overline{\delta}}$ | $Re_{\overline{\theta}}$ | $\overline{\delta}^{+}$ | $z_{sheet}/\overline{\delta}$ | $z_{sheet}/S$
| [mm] | [mm] |  | [m] | [ms-1] | $[\times 10^{6}]$ | $[\times 10^{4}]$ |  |  |  |
SR250-2 | 250 | 68.8 | 3.63 | 4.00 | 15.4 | 3.91 | 6.72 |  09090 | – | 0.49 | 0.13
SR16000 | 160 | 70.0 | 2.28 | 4.00 | 15.3 | 3.91 | 6.84 |  09200 | – | 0.48 | 0.21
SR10000 | 100 | 73.9 | 1.35 | 4.00 | 15.2 | 3.94 | 7.27 | 10150 | – | 0.46 | 0.34
SR50000 |  050 | 80.7 | 0.62 | 4.00 | 15.3 | 3.92 | 7.92 | 10220 | – | 0.32 | 0.51
SR25000 |  025 | 77.6 | 0.32 | 4.00 | 15.5 | 3.86 | 7.49 |  09990 | – | 0.18 | 0.55
Reference smooth wall
Case000 | $S$ | $\delta_{s}$ | $S/\delta_{s}$ | $x$ | $U_{\infty}$ | $Re_{x}$ | $Re_{\delta_{s}}$ | $Re_{\theta_{s}}$ | $\delta_{s}^{+}$ | $z_{sheet}/\delta_{s}$ | $z_{sheet}/S$
| [mm] | [mm] |  | [m] | [ms-1] | $[\times 10^{6}]$ | $[\times 10^{4}]$ |  |  |  |
SW-2000 | – | 56.4 | – | 4.00 | 15.2 | 3.99 | 5.62 | 7320 | 2030 | 0.46 | –
SW-2000 | – | 56.4 | – | 4.00 | 15.2 | 3.99 | 5.62 | 7320 | 2030 | 0.24 | –
Table 1: Summary of spanwise heterogeneous roughness cases and the reference
smooth-wall cases. $\overline{\delta}$ is the spanwise-averaged 98% boundary-
layer thickness of the surface with spanwise heterogeneity, while $\delta_{s}$
is the 98% boundary-layer thickness of the reference smooth-wall case.
$\overline{\theta}$ is the spanwise-averaged momentum thickness of the
spanwise heterogeneous surfaces, while $\theta_{s}$ is the momentum thickness
of the reference smooth-wall case. Reynolds number definitions are:
$Re_{x}\equiv xU_{\infty}/\nu$, $Re_{\delta}\equiv\delta U_{\infty}/\nu$,
$Re_{\theta}\equiv\theta U_{\infty}/\nu$, and $\delta^{+}\equiv\delta
U_{\tau}/\nu$, where $U_{\tau}$ is the friction velocity. $z_{sheet}$ is the
wall-normal location of the wall-parallel PIV laser sheet, measured from the
wall.

The measurements are performed in an open return boundary layer wind tunnel in
the Walter Basset Aerodynamics Laboratory at the University of Melbourne using
the same experimental set-up and test surfaces as Wangsawijaya et al. (2020).
The spanwise heterogeneous roughness (‘SR’) surfaces are constructed from
spanwise-alternating strips of P-36 grit sandpaper (‘rough’ patch) and
cardboard (‘smooth’ patch) of equal width $S$ (figure 2), with minimal
variations in surface elevation between the two. Throughout this report, the
rough patches are shaded black and the smooth strips are shaded white. This
study considers measurements over heterogeneous surfaces of various $S$ at
$x=4$ m downstream of the sandpaper trip located at the inlet of the wind
tunnel test section, covering a range of $0.32\leq S/\overline{\delta}\leq
3.63$. For comparison to SR cases, measurements are also conducted over a
reference smooth-wall (‘SW’) case at the same $Re_{x}\equiv xU_{\infty}/\nu$
as the SR cases, where $U_{\infty}$ is the freestream velocity. The details of
spanwise heterogeneous and smooth wall reference cases are summarised in table
1. Note that the ‘-2’ suffix in SR250 and SW cases refer to measurements at
$x=4$ m, such that it is consistent with the nomenclature in Wangsawijaya et
al. (2020).

###  2.2 Particle image velocimetry (PIV)

Figure 2: Schematic of PIV experimental set-up: stereoscopic PIV (SPIV) in
$yz$-plane and wall-parallel PIV (WPPIV) in plane $xy$-plane. Measurements in
these planes are not conducted simultaneously.

PIV measurements are conducted over the test surfaces in two planes:
stereoscopic PIV (SPIV) in the cross-stream ($y$–$z$) plane and wall-parallel
PIV (WPPIV) in the streamwise-spanwise ($x$–$y$) plane (figure 2).
Measurements in both planes are non-time-resolved and non-simultaneous. The
SPIV plane is located at $x=4$ m downstream of the trip, coinciding with the
WPPIV plane which spans $3.5\leq x\leq 4.01$ m. SPIV images are captured by
two pco.4000 cameras in forward scatter arrangement, which gives the field of
view (FOV) of $4\delta_{s}\times 3\delta_{s}$ ($y\times z$). For WPPIV, the
FOV is obtained by stitching images from three pco.4000 cameras, resulting in
a FOV of total size $9\delta_{s}\times 5\delta_{s}$ ($x\times y$). The wall-
normal location of the WPPIV laser sheet relative to the wall $z_{sheet}$
varies between SR cases (see table 1), depending on whether the secondary flow
sizes are governed by $\overline{\delta}$ or $S$ (see equation 3.1 in
Wangsawijaya et al., 2020). For the cases where $S/\overline{\delta}\geq 1$
(SR250-2, SR160, and SR100), the sheet is located at
$z_{sheet}/\overline{\delta}\approx 0.5$, while for $S/\overline{\delta}<1$
(SR50 and SR25), the sheet is located at $z_{sheet}/S\approx 0.5$. To
accommodate the variation of $z_{sheet}$, WPPIV measurements for the reference
smooth-wall case SW-2 are conducted in two $xy$-planes, at
$z_{sheet}/\delta_{s}=0.46$ and 0.24. It should be noted that for SR250-2 and
SR160 cases, only half of the spanwise roughness wavelength is captured in the
SPIV and WPPIV images due to the limited FOV width. The complete description
of the SPIV and WPPIV experimental set-ups are given in the appendix A and B
of Wangsawijaya et al. (2020).

##  3 Meandering of secondary flows

In Wangsawijaya et al. (2020), meandering of the low-speed streaks, presumably
the manifestation of the secondary flows in the instantaneous flow field, is
inferred from the $y$–$z$ plane by a spanwise leaning behaviour and asymmetry
of the low-speed features. That is, the structures lean sideways (in $y$)
depending on the sign of $v^{\prime}$, with the intermediate cases
($S/\overline{\delta}\approx 1$) showing the strongest leaning amplitude
(figure 18 in Wangsawijaya et al., 2020). In the wall-parallel ($x$–$y$)
plane, meandering was implied by the two-point correlations of $u^{\prime}$,
showing a strong periodicity in the cases where $S/\overline{\delta}\approx 1$
when computed at certain $y$ locations (figure 19 in Wangsawijaya et al.,
2020). In the current study, analysis of the WPPIV data (for the reference
smooth wall case and spanwise heterogeneous roughness) is extended to permit
further examination of this meandering behaviour. Further analysis of the
similarities (and differences) between the secondary flows and naturally
occurring LSMs/VLSMs in homogeneous canonical flows is also possible.

Figure 3: Detected low-speed structures for (a) reference case SW-2 at
$z_{sheet}/\delta_{s}=0.46$ and SR cases: (b) SR250-2, (c) SR160, (d) SR100
($z/\overline{\delta}=0.46$), (e) SR50, (f) SR25, about the center of the
secondary flows. Grey coloured contours are the low-speed structures,
$\widetilde{u}^{\prime}/U_{\infty}<-0.03$. ‘+’ marks the minima (in terms of y
location) of a low-speed structure. The spines of the detected low-speed
structures are shown in solid lines (from PIV data:  , smoothed:  ). Dashed
lines ( ) are the spanwise locations of the common flow up of the secondary
flows (marked by WAH1gu, see figure 1g for these locations in the $yz$-plane).
The low-speed structures related to the secondary flows due to spanwise
heterogeneity are assumed to occur inside the red-shaded area, spanning
$2l_{y}$ about the location of common flow up, as shown in (b) and figure 4.
In (b–f), white and black patches illustrate the arrangement of smooth and
rough strips, respectively, underneath the WPPIV planes. Figure 4: Contours of
mean swirl strength $\Lambda_{ci}$ in $yz$-plane multiplied by the sign of
vorticity $\Omega_{x}/|\Omega_{x}|$ for case: (a) SR250-2
($S/\overline{\delta}=3.63$), (b) SR100 ($S/\overline{\delta}=1.35$), (a) SR25
($S/\overline{\delta}=0.32$). Contours are normalised by
$\delta_{s}/U_{\infty}$. Vectors indicate $V$ and $W$, downsampled for
clarity. WAH1gu is the location of common flow up. Red ( ) and blue ( ) solid
lines illustrate the WPPIV laser sheets ($xy$-plane in figure 3) of each case.
Secondary flows are assumed to occur along the red lines, LSM/VLSM along the
blue lines. Red arrows indicate $l_{y}$, the spanwise extent of the mean
secondary flows.

We attempt to reconstruct the meandering of large-scale structures and
secondary flows through conditional averaging of the fluctuating velocity
field. Figure 3 shows the instantaneous low-speed structures from a single
representative snapshot taken from the WPPIV measurements for the reference
smooth wall case SW-2 (figure 3a) at $z/\delta_{s}=0.46$ and all SR cases
(figure 3b–f) at a wall-normal location that corresponds to the approximate
centre of the roll-modes associated with the mean secondary flows. Gray
coloured contours show negative fluctuations of
$\widetilde{u}^{\prime}\equiv\widetilde{U}+u^{\prime}$ (total velocity with
the global $yt$-average subtracted),
$\widetilde{u}^{\prime}/U_{\infty}<-0.03$. To limit the analysis to long,
large-scale structures, the velocity field is filtered with a box filter of
size $0.1\delta_{s}\times 0.1\delta_{s}$ for SW-2 and
$0.1\overline{\delta}\times 0.1\overline{\delta}$ for SR cases and only
structures with length $\geq 3\overline{\delta}$ ($\geq 3\delta_{s}$ for SW-2)
are retained for analysis. The ‘spine’ of each detected low-speed region is
constructed from the spanwise midpoint of the structure at all streamwise
locations along the length of the detected feature (gray solid lines,  in
figure 3). This ‘spine’ is further smoothed with a 1-D low-pass filter whose
length is $\overline{\delta}$ ($\delta_{s}$ for SW-2), shown by the black
solid lines ( ) in figure 3. Conditional averaging of the turbulent
fluctuation $u^{\prime}$ (instead of $\widetilde{u}^{\prime}$) is computed at
the minima (in $y$) of the smooth spines fitted to the detected low-speed
structures, as marked by the ‘+’ symbols in figure 3.

Figure 3(d,e) also shows how the meandering of low-speed structures is clearly
‘phase-locked’ about the spanwise location of the common flow up of the mean
secondary flows (marked by dashed lines,  and WAH1gu). This is expected since
$\widetilde{u^{\prime}}$ and $\widetilde{U}$ is phase-locked. Ensemble
averaging of the total velocity field, as shown in the contours of swirl
strength $\Lambda_{ci}$ (figure 4), reveal time-averaged large-scale secondary
flows, even for cases where $S/\overline{\delta}\gg 1$ (figure 4a) and
$S/\overline{\delta}\ll 1$ (figure 4c). Based on the time-averaged secondary
flows depicted in figure 4, it can be assumed that the secondary flows in the
instantaneous velocity fields meander about a certain spanwise location for
all $S/\overline{\delta}$ cases. Here, it is assumed that the secondary flows
due to spanwise heterogeneity cause low-speed streaks to meander about
$y_{u}$, the spanwise location of common flow up ( WAH1gu in figure 3 and 4),
spanning the area shaded by red in figure 3. This area spans $y_{u}\pm l_{y}$
(figure 3b), where $l_{y}$ is the width of the mean secondary flow roll-mode,
as illustrated in figure 1(b) and 4. Under this assumption, all detected low-
speed structures could be classified as either secondary flows due to spanwise
heterogeneity (red shaded area) or LSM/VLSM over homogeneous roughness regions
in the cases where $S/\overline{\delta}>1$ (blue shaded area in figure 3b–d).
In the cases where $S/\overline{\delta}<1$ (figure 3e,f), $l_{y}=S$ and all
detected low speed structures will be categorized as belonging to the
secondary flows; hence this categorisation is somewhat flawed. It should be
noted that although the secondary flows fill the entire spanwise extent of the
turbulent boundary layers in these cases, larger structures whose scale is
$\overline{\delta}$ also coexist with the secondary flows. However, these two
cannot be distinguished in cases where $S/\overline{\delta}\leq 1$ because the
currently available WPPIV snapshots in the $x$–$y$ plane are obtained at a
$z_{sheet}$ height that is centred on the roll modes and also because the
method used to separate secondary flows and LSM/VLSM is based only on the
spanwise location of the secondary flows (and not the spanwise lengthscale or
wall-normal extent of the detected structure). A method to discriminate
secondary flows from large scale structures for the case where
$S/\overline{\delta}\ll 1$ based on POD (proper orthogonal decomposition)
filtering is proposed in §4.2.2. The locations of the wall-parallel slices
relative to the secondary flows and the locations of red- and blue-shaded
regions in $y$–$z$ plane are illustrated in figure 4. For all subsequent
conditional averaging, features are assigned to the red or the blue regions
based on which region the detected y-minima of the low-speed structures (which
is the condition vector) resides.

Figure 5: Histogram of the spanwise location of the detected minima of low-
speed structures $y_{ref}$ for (a) reference case SW-2 at $z/\delta_{s}=0.46$
and SR cases: (b) SR250-2, (c) SR160, (d) SR100, (e) SR50, (f) SR25 about the
center of the secondary flows. Red bars ($\blacksquare$) indicate $y_{ref}$
detected inside the red-shaded area in figure 3, while blue bars
($\blacksquare$) are detected inside the blue-shaded area in the same figure.
Dashed lines ( ) are the spanwise locations of the common flow up of the mean
secondary flows (marked by WAH1gu).

Figure 5 shows the histogram of the detected low speed streaks (identified in
the manner depicted in figure 3) for all SR cases, split into those that we
crudely classified as secondary flows (occuring within the red regions of
figure 3) and LSM/VLSM (occurring within the blue regions). The histogram
counts the observed $y_{ref}$ minima (denoted by the black $+$ sign in figure
3) in these two regions. The probability of the occurrence of long, low-speed
structures are distributed equally across the span of the FOV in the reference
smooth wall case SW-2 (figure 5a). For the case SR250-2
($S/\overline{\delta}=3.63$, red bars in figure 5b) and SR160
($S/\overline{\delta}=2.28$, figure 5c), the low-speed structures related to
the secondary flows comprise 41% and 74% of all structures detected across the
FOV, respectively (as a reference the red shaded regions associated with the
secondary flows consist of 38% and 53% respectively of the available total
area in these cases). The secondary flows dominate as $S/\overline{\delta}$
approaches 1 (86% of detections occur within the red regions for case SR100,
which occupy 40% of the total area, see figure 5d), and they are also clearly
‘phase-locked’ to the location of common flow up (case SR50, figure 5e). For
the smallest $S/\overline{\delta}$ case (case SR25, figure 5f), the spines are
more evenly distributed across the span compared to cases SR100 and SR50
(figure 5d,e), with a hint of some residual spanwise locking of the structures
(higher probabilities over common flow up regions).

Figure 6: Contours of filtered turbulent fluctuation (a,c) $u_{f}^{\prime}$
and (b,d) $v_{f}^{\prime}$ conditionally averaged at (a,b) the minima and
(c,d) maxima of the detected low-speed structures illustrated in figure 3 for
the reference case SW-2 at $z/\delta_{s}=0.46$.

Figure 6 shows the conditional average of filtered streamwise $u_{f}^{\prime}$
(figure 6a,c) and spanwise $v_{f}^{\prime}$ (figure 6b,d) turbulent
fluctuations for the reference smooth wall case SW-2 at
$z/\overline{\delta_{s}}=0.46$. The subscript ‘$f$’ denotes the velocity
fields filtered with a box filter of $0.1\delta_{s}\times 0.1\delta_{s}$ size
($0.1\overline{\delta}\times 0.1\overline{\delta}$ for SR cases). Figure
6(a,b) shows $u_{f}^{\prime}$ conditionally averaged at the local $y$ minima
of the spines of the detected low-speed structures as illustrated in figure
3(a), while figure 6(c,d) is conditionally averaged at the maxima of the
detected low-speed structures (not shown in figure 3a). The contours of
$u_{f}^{\prime}$ show a low-speed structure flanked by two high-speed
structures meandering to the left and right depending on the reference
location (minima or maxima) where the conditional average is computed. The
meandering tendency is also shown in $v_{f}^{\prime}$, where the conditional
average at the minima of low-speed structures largely correspond to
$v_{f}^{\prime}>0$ (figure 3b) and the maxima to $v_{f}^{\prime}<0$ (figure
3d). Diagonal alignment of $v^{\prime}$, similar to that observed in Sillero
et al. (2014) and de Silva et al. (2018), is also apparent in these contours.

To obtain a complete picture of how the secondary flow meanders, similar
conditional averaging is also computed in the $yz$-plane from SPIV
measurements. Since the minima and maxima of the spine-fitted low-speed
structure cannot be observed in the $yz$-plane, a different conditional
averaging approach must be taken to detect these same events. Informed by the
conditional averages shown in figure 6, we elect to use the a simultaneous
detection criteria of negative $\widetilde{u}^{\prime}$ and either
$v^{\prime}>0$ or $<0$ to approximately detect the minima and maxima of the
fitted structure spines (cases shown in figure 6a,b and c,d respectively). The
conditional average is computed at the common flow up ( WAH1gu in figure 1g)
in $y$ and at $z/\overline{\delta}=0.1$, as close to the wall as the FOV
permits. For the reference case SW-2, the average is computed at any point in
$y$ along $z/\delta_{s}=0.1$ where the conditions are satisfied. The
conditions $\widetilde{u}^{\prime}<0$, $v^{\prime}>0$ and
$\widetilde{u}^{\prime}<0$, $v^{\prime}<0$ are each satisfied for 25% of the
smooth wall realisations. The velocity fields in the $yz$-plane are filtered
with a box filter of size $0.1\delta_{s}\times 0.1\delta_{s}$
($0.1\overline{\delta}\times 0.1\overline{\delta}$ for SR cases). It should be
highlighted that these condition vectors for the ensemble averaging differ
between the $x$–$y$ and $y$–$z$ plane and are only intended to show the
representation of the high- and low-speed structures to complement the
conditional averages in the $x$–$y$ plane. As a direct comparison, the
conditional average at the exact same condition points in $x$–$y$ and $y$–$z$
plane has also been computed, showing a good agreement between both planes
(not shown here for brevity, but available in Wangsawijaya, 2020, p. 157–158).

Figure 7: Contours of conditionally averaged turbulent fluctuation
$u^{\prime}$ for (a) reference case SW-2 at $z/\delta_{s}=0.46$ and all SR
cases: (b) SR250-2, (c) SR160, (d) SR100, (e) SR50 and (f) SR25. Conditions:
(i) $\widetilde{u}^{\prime}<0$ and $v^{\prime}>0$ at the $y$ location of
common flow up ( WAH1gu in figure 3), (ii) minima of detected low-speed
structures ($\widetilde{u}^{\prime}/U_{\infty}<-0.03$) assumed to be the
secondary flows (red-shaded regions in figure 3). In (a)(ii), solid black line
( ) is the ‘spine’ extracted from the conditionally averaged low-speed
structure and $\hat{y}$ is the amplitude of meandering.

Conditionally averaged $u_{f}^{\prime}$ for the reference smooth wall case
SW-2 and all SR cases computed for features within the red-shaded region in
figure 3 are shown in figure 7. The plots show a low-speed structure flanked
by two high-speed structures in both the $yz$\- (i) and $xy$-planes (ii).
Subplot (i) shows the tendency of the structures in the cross-plane to lean
sideways to the right when $v_{f}^{\prime}>0$, corresponding to the minima of
$y_{ref}$ in the wall-parallel plane (ii). The left-leaning tendency in the
cross-plane for $v_{f}^{\prime}<0$, corresponding to the maxima of $y_{ref}$
in the wall-parallel plane, is not shown for brevity. The leaning is strongly
one-sided for the cases where $S/\overline{\delta}\approx 1$ in figure
7(d,e)(i), where the low-speed structures lean to one side and are flanked by
asymmetric high-speed structures which are highly one-sided. For the reference
smooth wall case SW-2 in figure 7(a) and the heterogeneous cases where
$S/\overline{\delta}\gg 1$ (figure 7b,c), the sideways leaning of the low-
speed event is less prominent with a more symmetric arrangement of flanking
high-speed events. The conditionally averaged velocity fields in the
$xy$-plane (ii) further confirm the differences between the smooth wall and
the limiting cases compared to the $S/\overline{\delta}\approx 1$ cases. In
the wall-parallel plane, the structures exhibit stronger meandering and
asymmetry in figure 7(d,e)(ii) when $S\rightarrow\overline{\delta}$ compared
to the reference smooth wall and the limiting cases where
$S\gg\overline{\delta}$ or $S\ll\overline{\delta}$ in figure 7(a–c,f). The
shape of these structures provide additional evidence for the meandering
inferred from the contours of normalised two-point correlation of $u^{\prime}$
in figure 19(e,f) in Wangsawijaya et al. (2020), where a clear streamwise
anti-phase pattern was exhibited by the $S/\overline{\delta}\approx 1$ cases.
The approximate streamwise wavelength observed in figure 7 is
$\lambda_{x}/\overline{\delta}\approx 4$ (see figure 7(e)ii), similar to that
observed in the contours of two-point correlation coefficients and
spectrograms from hot-wire anemometry measurements for the same surfaces in
Wangsawijaya et al. (2020). Structures resulting from this conditional average
for the limiting cases where $S/\overline{\delta}\gg 1$ and
$S/\overline{\delta}\ll 1$ are further discussed in §4.2.1 and §4.2.2,
respectively.

Figure 8: (a) Illustration of meandering (spanwise leaning) angle $\psi$ in
$y$–$z$ plane, constructed from meandering amplitude $\hat{y}$ and the wall-
normal location of the WPPIV laser sheet $z_{sheet}$. (b) Meandering amplitude
$\hat{y}$ measured from the ‘spines’ of the conditionally averaged structures
(see figure 7(a)ii) and normalised by $z_{sheet}$ as a function of
$S/\overline{\delta}$.  : $\hat{y}$ measured from the structures assumed to be
LSM, when available (blue-shaded regions in figure 3) and  : $\hat{y}$
measured from the structures assumed to be secondary flows (red-shaded regions
in figure 3). Measured $\hat{y}$ of the reference smooth-wall case SW-2,  :
$z_{sheet}/\delta_{s}=0.24$ and  : $z_{sheet}/\delta_{s}=0.46$. $\times$:
$\hat{y}$ of structures assumed to be LSM, $\times$: $\hat{y}$ of structures
assumed to be secondary flows, both extracted from $u^{\prime}$ instead of
$\widetilde{u}^{\prime}$ (see appendix A for details).

We next attempt to quantify the meandering of the conditionally averaged
structures in the $x$–$y$ plane. This will supplement the quantification of
the leaning angle in the $y$–$z$ plane conducted in Wangsawijaya et al.
(2020). ‘Spines’, similar to those fitted to the instantaneous low-speed
structures in figure 3, are now fitted to the conditionally averaged low-speed
structures in figure 7. The meandering amplitude $\hat{y}$ is defined as the
average distance between the $y$ location of the spines at $\Updelta
x/\overline{\delta}=0$ and $\Updelta x/\overline{\delta}\pm 2$ (see figure
7(a)ii). It should be noted that the wall-normal location of the $x$–$y$ plane
(i.e. the WPPIV laser sheet location $z_{sheet}$) differs between cases,
depending on the size of the secondary flows (table 1) – hence the measured
$\hat{y}$ is not directly comparable between SR and SW cases. Alternatively,
meandering can also be quantified with the spanwise leaning angle in the
$y$–$z$ plane, $\psi\equiv 2\tan^{-1}(\hat{y}/2z_{sheet})$ (figure 8a). It
should be noted that the definitions of $\hat{y}$ and $\psi$ are similar to
those extracted from the conditionally averaged structures in the $y$–$z$
plane for Wangsawijaya et al. (2020) (cf. figure 17e and 18).

Figure 8(b) shows $\hat{y}$ normalised by $z_{sheet}$ (left hand side
abscissa) and $\psi$ (right hand side abscissa) of the secondary flows ( ) as
a function of $S/\overline{\delta}$ ($z_{sheet}$ is roughly the radius of the
secondary flows, see figure 4). The red symbols on this figure show the
meandering amplitude of the secondary flows as a function of
$S/\overline{\delta}$, and clearly this amplitude is maximum at
$S/\overline{\delta}=0.62$ (SR50, $S/\overline{\delta}\approx 1$), similar to
the trend previously reported for the $y$–$z$ plane (figure 18 in Wangsawijaya
et al., 2020). The amplitude of the meandering normalised by the wall-normal
sheet location $\hat{y}/z_{sheet}$ is approximately equal to or less than 1
for the smooth wall case SW-2 and for the limiting cases
($S/\overline{\delta}\gg 1$ and $\ll 1$) and larger than 1 for the
intermediate cases ($S/\overline{\delta}\approx 1$). For the limiting cases
$S/\overline{\delta}\gg 1$ ($S/\overline{\delta}=2.28$ and 3.63), the
magnitude of $\hat{y}$ measured at $z_{sheet}/\overline{\delta}\approx 0.5$ is
closer to that of the reference smooth-wall case SW-2 at
$z_{sheet}/\delta_{s}=0.46$ (dashed line in figure 8b), while in the limit
$S/\overline{\delta}\ll 1$, $\hat{y}$ measured at
$z_{sheet}/\overline{\delta}=0.18$ is approximately equal to that of SW-2 at
$z_{sheet}/\delta_{s}=0.24$ (dash-dot line in figure 8b). This suggests that
in the limiting cases where $S\gg\overline{\delta}$ or
$S\ll\overline{\delta}$, the meandering of the secondary flows reverts to
smooth-like behaviour.

It is also important to note that while the contours in figure 7 show the
conditionally averaged (and filtered) fluctuating velocity field $u^{\prime}$,
the condition points (‘+’ in figure 3) are extracted from
$\widetilde{u}^{\prime}$ for SR cases (which contains the stationary component
of the secondary flows $\widetilde{U}$) and from $u^{\prime}$ for the
reference smooth-wall case SW-2 ($\widetilde{U}=0$ for this case). As a direct
comparison, we compute the same conditional average of the maxima and minima
of the low-speed structures with the spines extracted from $u^{\prime}$ (the
non-stationary component of secondary flow) instead of
$\widetilde{u}^{\prime}$ for all SR cases. An example of this analysis is
given in appendix A for case SR50 ($S/\overline{\delta}=0.62$). The magnitude
of $\hat{y}$ extracted from the $u^{\prime}$ field (‘$\times$’ symbol in
figure 8b) is approximately equal to that extracted from
$\widetilde{u}^{\prime}$ for the limiting cases $S/\overline{\delta}\gg 1$ and
$\ll 1$, and slightly smaller for the intermediate cases
$S/\overline{\delta}\approx 1$. However, in general this illustrates that the
choice of $u^{\prime}$ or $\widetilde{u}^{\prime}$ for the condition vector
makes very little difference in terms of the salient trends exhibited in
figure 8(b). Regardless of the condition vector, the meandering amplitude of
the secondary flows exhibit a clear peak when $S/\overline{\delta}\approx 1$.

##  4 Secondary flows and large-scale structures

Figure 9: Illustrations of meandering low-speed structures flanked by two
high-speed structures constructed by conditionally averaged $u^{\prime}$ from
SPIV and WPPIV plane for (a) the reference case SW-2 and (b) case SR100
($S/\overline{\delta}=1.35$, structures extracted from the red region in
figure 3d), conditioned at the minima of the detected low-speed structures
illustrated in figure 3. Solid red lines ( ) show
$u_{f}^{\prime}U_{\infty}=0.05,0.1,0.2,...,0.5$, blue solid lines ( ):
$-0.5,-0.4,...,-0.1,-0.05$. In (b), gray-shaded contours indicate vorticity of
the conditionally averaged velocity field
$\omega^{\prime}_{x}\overline{\delta}/U_{\infty}=0.2$ and
$\omega^{\prime}_{z}\overline{\delta}/U_{\infty}=0.8$ for case SR100 in
$y$–$z$ and $x$–$y$ plane, respectively.

The similarities and differences between secondary flows (intermediate cases,
figure 7d,e) and the large-scale motions (SW case, figure 7a) raise many
important questions. Firstly, is it possible that secondary flows due to
spanwise heterogeneity and large-scale motions share similar formation
mechanisms? Some parallels regarding the formation mechanism have been
suggested. For example, Lee et al. (2019), when studying VLSMs in canonical
smooth wall pipe flow, have associated their formation with instantaneous
lateral variations of the wall shear stress of homogeneous turbulent pipe
flows. Secondary flows, on the other hand are also associated with stationary
or persistent lateral variations in the wall shear stress (Townsend, 1976). In
this sense, both naturally occurring large-scale structures and secondary
flows are both triggered by lateral stress variations (the former convecting
and transient, and the latter stationary). Figure 9(a,b) show the 3-D
reconstruction of the large-scale streaks from $xy$\- and $yz$-plane for case
SW-2 and SR100 ($S/\overline{\delta}=1.35$), respectively111It should be noted
that the condition vectors for the ensemble averaging differ between the
$xy$\- and $yz$-plane and these figures are only intended to show the
representation of the high- and low-speed structures above the reference
smooth wall case and an intermediate case.. The yawed and inclined asymmetric
roll modes can be inferred from the in-plane vorticity (grey-shaded contours
in figure 9b) calculated from the conditionally averaged velocity fields:
$\omega_{x}^{\prime}\equiv\partial w_{f}^{\prime}/\partial y-\partial
v_{f}^{\prime}/\partial z$ in $y$–$z$ plane and
$\omega_{z}^{\prime}\equiv\partial v_{f}^{\prime}/\partial x-\partial
u_{f}^{\prime}/\partial y$ in $x$–$y$ plane. A cursory look at figure 9
reveals that the meandering is more prominent in the intermediate cases where
$S/\overline{\delta}\approx 1$ (secondary flows, figure 9b) than that of the
reference smooth-wall case (LSM/VLSM, figure 9a). However, both structures
show similarities to the streak-vortex models for the naturally occurring
large-scale structures proposed in Jeong et al. (1997); Waleffe (2001);
Schoppa & Hussain (2002); Flores & Jiménez (2010); Cossu & Hwang (2017); de
Giovanetti et al. (2017). These asymmetric streak-vortex structures have also
been elucidated in the log region from both PIV of converging-diverging (C-D)
riblets, which also induce secondary flows, and in direct numerical
simulations (DNS) of smooth-wall channel flow (Kevin et al., 2019a). This
leads to other questions: if it is possible that both secondary flows and
large-scale structures share a similar formation mechanism, is it also
possible that the secondary flows are just phase-locked large-scale
structures? If that is the case, why do the secondary flows strongly meander
only when $S/\overline{\delta}\approx 1$, as shown in figure 8(b)? Finally, we
ask if the secondary flows and the large-scale structures coexist with each
other in certain heterogeneous wavelengths. In the next section of this study,
we attempt to address these questions.

###  4.1 Phase locking

The 1-D spanwise energy spectra, related to spanwise wavenumber $k_{y}$ and
wavelength $\lambda_{y}$, are computed from the fluctuating velocity field
obtained from SPIV measurements. At any wall-normal location $z$ in the FOV,
the non-normalised two-point correlation of the fluctuating velocity component
$u^{\prime}$ is given by

$\hat{R}_{u^{\prime}u^{\prime}}(\Updelta
y,z)=\overline{u^{\prime}(y,z)u^{\prime}(y+\Updelta y,z)}$ (4)

where the overbar denotes ensemble averaging at all $y$ and for all SPIV
realizations and $\Updelta y$ is the spanwise shift. The Fourier
transformation of $\hat{R}_{u^{\prime}u^{\prime}}$ yields the energy spectra
of the fluctuating velocity component $u^{\prime}$

$\Upphi_{u^{\prime}u^{\prime}}=\int_{-\infty}^{\infty}\hat{R}_{u^{\prime}u^{\prime}}\;\mathrm{e}^{-j2\pi
k_{y}\Updelta y}\;\mathrm{d}(\Updelta y)$ (5)

where $k_{y}$ is the spanwise wavenumber and $\lambda_{y}=2\pi/k_{y}$ is the
spanwise wavelength. The relationship between the Reynolds stress component
$\overline{u^{\prime}u^{\prime}}$ and $\Upphi_{u^{\prime}u^{\prime}}$ is given
by

$\overline{u^{\prime}u^{\prime}}=\int_{0}^{\infty}\Upphi_{u^{\prime}u^{\prime}}\;\mathrm{d}(k_{y})$
(6)

Due to the limited spanwise extent of the FOV, the velocity field is padded
with zeros in $y$ to increase the length of the signal to three times the
total width of the FOV (from $4\delta_{s}$ to about $12\delta_{s}$) and thus
increase the resolution in $k_{y}$ and $\lambda_{y}$. This method for 1-D
spanwise energy spectra computation, however, must be applied with caution for
the spanwise heterogeneous roughness cases. Figure 1(f) shows the
instantaneous turbulent fluctuation component $u^{\prime}$ for case SR50
($S/\overline{\delta}=0.62$), whose mean (for all snapshots) across the
repeating period of the spanwise heterogeneous roughness is zero. Its variance
$\overline{u^{\prime}u^{\prime}}$, however, is heterogeneous in $y$ (see
figure 1c). Using this method, integration of the energy spectra across
$k_{y}$ in (6) results in the _average_ of $\overline{u^{\prime}u^{\prime}}$
in $y$, and yields no information regarding the spanwise heterogeneity of the
flow.

Figure 10: Contours of premultiplied energy spectra (i)
$k_{y}\Upphi_{u^{\prime}u^{\prime}}$ and (ii)
$k_{y}\Upphi_{\widetilde{U}\widetilde{U}}$ as functions of $\lambda_{y}$ and
$z$ for (a) reference case SW-2 and SR cases: (b) SR250-2
($S/\overline{\delta}=3.63$), (c) SR50 ($S/\overline{\delta}=0.62$), (d) SR25
($S/\overline{\delta}=0.32$). Data are obtained from SPIV measurements. In
(i), ‘+’ marks the peak of energy spectra in the outer layer at
$z/\overline{\delta}=0.2$ and $\lambda_{y}/\overline{\delta}=1$ (b–d),
$z/\delta_{s}=0.2$ and $\lambda_{y}/\delta_{s}=1$ in (a). Black boxes are the
area of integration in (7) In (ii), ‘+’ marks the spanwise wavelengths related
to the mean secondary flows. White boxes are the area of integration in (8)
and (9).

Figure 10(i) shows the premultiplied energy spectra
$k_{y}\Upphi_{u^{\prime}u^{\prime}}$ for the reference smooth wall case SW-2
(figure 10a(i)) and SR cases: SR250-2, SR50, and SR25 (figure 10b(i)–d(i),
respectively). In general, all SR cases have higher energy than the reference
smooth wall case SW-2, which is expected since the presence of surface
roughness results in higher magnitude of energy (when normalised by
$U_{\infty}$). Further, all cases (SR and SW-2) also show an outer peak (‘+’)
at $z/\overline{\delta}\approx 0.2$ and $\lambda_{y}/\overline{\delta}\approx
1$ ($z/\delta_{s}\approx 0.2$ and $\lambda_{y}/\delta_{s}\approx 1$ for SW-2)
which is associated with the width and spacing between large-scale low- and
high-speed streaks for the large-scale and very large-scale motions
(LSM/VLSM). It is important to note that $k_{y}\Upphi_{u^{\prime}u^{\prime}}$
does not contain information regarding the stationary (time-averaged)
secondary flows. Instead, this is contained in $\widetilde{U}$ (figure
10(ii)). Figure 10(i) contains information about non-stationary turbulent
fluctuations, and these seem largely similar between the smooth and
heterogeneously rough surfaces. The contours of spanwise energy spectra for SR
cases calculated from the SPIV data also do not contain a clear imprint of the
time-dependant (meandering) secondary flows, unlike the 1-D streamwise energy
spectra $k_{x}\Upphi_{u^{\prime}u^{\prime}}$ (see Wangsawijaya et al., 2020,
figure 13 and 14), which is expected since this assumed meandering behaviour
is related primarily to the phase of the spanwise Fourier modes which are not
analysed in the amplitude spectra (although increased meandering might affect
the magnitude of the resolved $k_{y}\Upphi_{u^{\prime}u^{\prime}}$ energy).

It is also worth noting that compared to other SR cases, the magnitude of
$k_{y}\Upphi_{u^{\prime}u^{\prime}}$ is the lowest for case SR50 (figure
10c(i)) – the case which shows the strongest outer peak in
$k_{x}\Upphi_{u^{\prime}u^{\prime}}$ and the most prominent meandering
(Wangsawijaya et al., 2020). To understand this phenomenon, another attempt is
made to examine the characteristics of the secondary flows that appear in the
spanwise energy spectra. Instead of the turbulent fluctuations, the energy
spectrograms are computed for the time-averaged spatial velocity fluctuations
$\widetilde{U}$ (figure 1e), which carries the footprint of the stationary
(time-averaged) secondary flows. The method for computation is similar to that
of turbulent fluctuations in (4)–(6). To ensure zero mean across $y$, the FOV
is reflected for the largest $S$ cases (SR250-2 and SR160) so that it captures
one spanwise roughness wavelength $\Lambda=2S$. For the cases where
$S/\overline{\delta}\lesssim 1$: SR100, SR50, and SR25, the FOV is clipped so
that it captures an integer number of spanwise wavelengths, $\Lambda$,
2$\Lambda$, and 4$\Lambda$, respectively.

Figure 10(ii) shows the premultiplied energy spectra of $\widetilde{U}$. As
expected, the magnitude of $k_{y}\Upphi_{\widetilde{U}\widetilde{U}}$ is zero
for the reference smooth wall case SW-2 in figure 10(a)(ii). Clear, prominent
modes in the energy spectra are observed in all SR cases in figure
10(b)(ii)–(d)(ii). These peaks are related to the mean secondary flows and are
expected to occur at $\lambda_{y}=2S$, i.e. the spanwise roughness wavelength,
except for the case where $S/\overline{\delta}>1$. Here, for case SR250-2
($S/\overline{\delta}=3.63$, figure 10b(ii)), the peak is observed at
$\lambda_{y}/S\approx 0.6$ ($\lambda_{y}/\overline{\delta}\approx 2$). This
observation is in line with the finding in Wangsawijaya et al. (2020); that
the secondary flow size is capped by $\overline{\delta}$ instead of $S$ for
this case (see figure 12a). The prominent mode is locked at $\lambda_{y}=2S$
for cases where $S/\overline{\delta}\leq 1$. Contours of
$k_{y}\Upphi_{\widetilde{U}\widetilde{U}}$ show a very strong mode at
$\lambda_{y}/S=2$ for case SR50 ($S/\overline{\delta}=0.62$, figure 10c(ii))
and a much weaker one for case SR25 ($S/\overline{\delta}=0.32$, figure
10d(ii)). Similar to previous observations regarding the strength of mean
secondary flows (Wangsawijaya et al., 2020), the strongest magnitude of energy
in $\widetilde{U}$ occurs at SR50 ($S/\overline{\delta}=0.62$) and it
decreases as $S$ approaches $S/\overline{\delta}\ll 1$.

The premultiplied energy spectra of $u^{\prime}$ and $\widetilde{U}$ seem to
show opposite tendency for the SR50 case: when compared to the rest of the SR
cases, the magnitude of $k_{y}\Upphi_{u^{\prime}u^{\prime}}$ exhibits a
minimum while the magnitude of $k_{y}\Upphi_{\widetilde{U}\widetilde{U}}$ is
maximum (see figure 10c(i) and 10c(ii)). To further examine this,
$k_{y}\Upphi_{u^{\prime}u^{\prime}}$ is integrated across the area enclosing
the outer peak (black boxes in figure 10(i)):
$0.6\leq\lambda_{y}/\overline{\delta}\leq 1.4$ and $0.1\leq
z/\overline{\delta}\leq 0.4$,

$I_{\Phi_{u^{\prime}u^{\prime}}}=\frac{1}{0.3\overline{\delta}}\int\limits_{0.1\overline{\delta}}^{0.4\overline{\delta}}\int\limits_{10\pi/3\overline{\delta}}^{10\pi/7\overline{\delta}}\frac{\Upphi_{u^{\prime}u^{\prime}}}{U_{\infty}^{2}}\mathrm{d}k_{y}\;\mathrm{d}z$
(7)

For the reference smooth wall cases SW-2, $\delta_{s}$ is used instead of
$\overline{\delta}$. $k_{y}\Upphi_{\widetilde{U}\widetilde{U}}$ for SR cases
is integrated across the area enclosing the prominent $\widetilde{U}$ modes
(white boxes in figure 10b(ii)-d(ii)): $0.5\leq\lambda_{y}/S\leq 3$ and
$0.1\leq z/\overline{\delta}\leq 1$,

$I_{\Phi_{\widetilde{U}\widetilde{U}}}=\frac{1}{0.9\overline{\delta}}\int\limits_{0.1\overline{\delta}}^{\overline{\delta}}\int\limits_{4\pi/S}^{2\pi/3S}\frac{\Upphi_{\widetilde{U}\widetilde{U}}}{U_{\infty}^{2}}\mathrm{d}k_{y}\;\mathrm{d}z$
(8)

For the reference smooth wall case in figure 10(a)(ii), the integration area
encompasses $0.5\leq\lambda_{y}/\delta_{s}\leq 3$ and $0.1\leq
z/\delta_{s}\leq 1$,

$I_{\Phi_{\widetilde{U}\widetilde{U}}}=\frac{1}{0.9\delta_{s}}\int\limits_{0.1\delta_{s}}^{\delta_{s}}\int\limits_{4\pi/\delta_{s}}^{2\pi/3\delta_{s}}\frac{\Upphi_{\widetilde{U}\widetilde{U}}}{U_{\infty}^{2}}\mathrm{d}k_{y}\;\mathrm{d}z$
(9)

Figure 11: Integrated energy spectra of $u^{\prime}$
($I_{\Phi_{u^{\prime}u^{\prime}}}$,  ) and $\widetilde{U}$
($I_{\Phi_{\widetilde{U}\widetilde{U}}}$,  ) as a function of
$S/\overline{\delta}$ for SR cases. Red and blue dashed lines ( and  ) are
$I_{\Phi_{u^{\prime}u^{\prime}}}$ and $I_{\Phi_{\widetilde{U}\widetilde{U}}}$
for the reference smooth wall SW-2, respectively.

Integrated energy spectra of $u^{\prime}$ and $\widetilde{U}$ are shown in
figure 11 as functions of $S/\overline{\delta}$ (dashed lines show the
reference smooth wall case SW-2). The lines in figure 11 suggest a
relationship between $u^{\prime}$ and $\widetilde{U}$: as
$S/\overline{\delta}$ approaches 1, $\widetilde{U}$ and $u^{\prime}$ reach a
maxima and minima, respectively. As $S/\overline{\delta}\gg 1$, the rise of
$u^{\prime}$ is accompanied by a decrease in $\widetilde{U}$. In some sense
this fits with the conceptual picture that the secondary flows are spanwise-
locked large-scale structures. Under this assumption, SR50
($S/\overline{\delta}=0.62$) presumably is the case where the locking is most
complete, since the secondary flows are the strongest and fill the entire
wall-normal and spanwise extent of the boundary layer (Wangsawijaya et al.,
2020). If the secondary flows can really be viewed as locked turbulent
structures, it seems reasonable to assume that for heterogeneous surfaces that
generate secondary flows some of the large-scale structure energy will be
transferred from turbulent fluctuations $u^{\prime}$ to the dispersive
component $\widetilde{U}$. This interplay between turbulent fluctuations and
the dispersive components (as one goes up, the other one goes down and vice
versa) is strongly reminiscent of similar observations by Nikora et al. (2019)
of secondary flows in rough-bed open channel flows and by Modesti et al.
(2018) of secondary flows in a square duct. In the latter case, Modesti et al.
(2018) also observed that when the secondary flows are artificially
suppressed, the turbulent fluctuations rise to compensate the loss of
dispersive component. Taken together, a possible interpretation of these
results would be that the $\widetilde{U}$ component may be an indicator of the
efficacy of the surface at locking turbulent structures in place. For example,
in the smooth surface, $\widetilde{U}$ is zero (blue dashed line in figure 11
and figure 10a(ii)), and the turbulent structures are randomly occurring
(completely unlocked), so all energy ends up in the $u^{\prime}$ component. It
is noted that the cases where $S/\overline{\delta}\ll 1$ and
$S/\overline{\delta}\gg 1$ in figure 11 (SR25, $S/\overline{\delta}=0.32$ and
SR250-2, $S/\overline{\delta}=3.63$) approach this condition, with the
integrated energy in $\widetilde{U}$ approaching the smooth wall reference
case. A possible conclusion from the above analysis is that for
$S/\overline{\delta}\approx 1$, the turbulent structures are optimally locked
or trapped by the surface, and hence energy from $u^{\prime}$ is surrendered
into $\widetilde{U}$ (which respectively reach a minima and a maxima at these
$S/\overline{\delta}$). However, this suggested behaviour is complicated
somewhat by the observed meandering behaviour in §3. It is noted in this
study, and in that of Wangsawijaya et al. (2020), that the meandering of the
secondary flows increases above that of the naturally occurring turbulent
structures when $S/\overline{\delta}\rightarrow 1$. Such behaviour is not
easily reconcilable with the notion of secondary flows as locked turbulent
structures and is perhaps indicative of some additional instability or forcing
when secondary flows are locked at particular spanwise wavelengths.

###  4.2 Coexistence between secondary flows and the large-scale structures

Figure 12: Illustration of the limiting cases in spanwise heterogeneous
roughness: (a) $S/\overline{\delta}\gg 1$ and (b) $S/\overline{\delta}\ll 1$.
Areas bounded by  show regions approaching spanwise homogeneity.

Coexistence between the VLSM and secondary flows was hinted at in Zampiron et
al. (2020) at larger spanwise wavelengths, where these two features exist at a
very different scale in the energy spectrograms. At both limits, when
$S/\overline{\delta}\gg 1$ and when $S/\overline{\delta}\ll 1$, the secondary
flows are confined in certain areas of the surface such that we might expect
the LSM/VLSM to exist in a relatively unaltered form for large areas of the
flow, especially when we are far from the secondary flows, e.g. far from the
roughness transition for the case with $S\gg\overline{\delta}$, or $z>S$ for
the case where $S\ll\overline{\delta}$ (figure 12). What follows is an attempt
to separate the secondary flows due to spanwise heterogeneity and (presumably)
unaltered, naturally occurring LSM/VLSM in the limiting cases:
$S/\overline{\delta}\gg 1$ (§4.2.1) and $S/\overline{\delta}\ll 1$ (§4.2.2).
Conditional averaging and POD analysis are conducted for the two largest
wavelength cases: SR250-2 ($S/\overline{\delta}=3.63$) and SR160
($S/\overline{\delta}=2.28$), and the smallest wavelength, SR25
($S/\overline{\delta}=0.32$).

####  4.2.1 Limiting case: $S/\overline{\delta}\gg 1$

Figure 3(b,c) show the detected low-speed structures and the minima of the
spines of these structures at an instantaneous velocity field of limiting
cases SR250-2 ($S/\overline{\delta}=3.63$) and SR160
($S/\overline{\delta}=2.28$), respectively, taken from WPPIV measurements
approximately at the center of the mean secondary flows
($z/\overline{\delta}\approx 0.5$). Instantaneously, the large-scale
structures induced by spanwise heterogeneity meander about a certain spanwise
location, which results in $\delta$-scaled secondary flows in the time-
averaged velocity field near the interface between rough and smooth strips
(see Wangsawijaya et al., 2020, figure 11). Here, it is assumed that the low-
speed structures related to the secondary flows (upwelling motions) meander
about the spanwise location of common flow up $y_{u}\pm l_{y}$, where $l_{y}$
is the spanwise extent of the mean secondary flows. This area is shaded red in
figure 3. Far removed from the secondary flows, the flow becomes locally
homogeneous (either homogeneously smooth or homogeneously rough). In figure
3(b), areas shaded in blue (closer to the centerline of smooth and rough
strips) represent this condition. It is assumed that secondary flows occur in
the red-shaded area, while LSM/VLSM occur in the blue-shaded area (these two
terms are used here to distinguish between structures that are imposed or
locked by the roughness heterogeneity and naturally occurring turbulent
structures, respectively). Histograms in figure 5(b,c) show that the secondary
flows (red bars) comprise 41% and 74% of all low-speed structures detected
across the FOV for case SR250-2 and SR160, respectively (compared to
respective areas occupied by the red regions of 37% and 54% of the FOV), and
thus LSM/VLSM (blue bars) comprise the rest (59% and 26%).

Figure 13: Contours of conditionally averaged turbulent fluctuation
$u^{\prime}$ for (a) reference case SW-2 at $z/\delta_{s}=0.46$ and (b,c)
SR250-2: conditioned at low-speed structures associated with (b) secondary
flows (SF) and (c) LSMs/VLSMs. Conditions: (i) $\widetilde{u}^{\prime}<0$ and
$v^{\prime}>0$, (iii) $\widetilde{u}^{\prime}<0$ and $v^{\prime}<0$, (ii)
$y$-minima and (iv) $y$-maxima of detected spine-fitted low-speed structures
($\widetilde{u}^{\prime}/U_{\infty}<-0.03$) illustrated in figure 3(b).

Similar to the analysis conducted in §3, the conditional average is also
calculated at the $y$-minima and $y$-maxima of the spines fitted to detected
low-speed structures and in the $yz$-plane from SPIV measurements, using a
different but related condition vector. For the $yz$-plane, the average is
computed at $z_{ref}/\overline{\delta}=0.1$, based on the condition of
$\widetilde{u}^{\prime}$ and $v^{\prime}$ (see §3). Figure 13 shows the
conditional average of case SR250-2 (figure 13b,c) compared to the reference
smooth wall case SW-2 (figure 13a). The conditional average for the secondary
flows (‘SF’) due to spanwise heterogeneity given in figure 13(b) is computed
at the spanwise location of common flow up ( WAH1gu) in subplot (i) and (iii).
It is computed at the minima and maxima of the detected low-speed structures
inside the red-shaded area (figure 3b) in subplot (ii) and (iv). The
conditional average for the LSM/VLSM is computed at $y_{ref}/S=-0.72$, close
to the centerline of a rough strip ($y/S=-1$). A one-sided leaning tendency is
observed in the conditionally averaged secondary flows (figure 13b). However,
both secondary flows and LSM/VLSM (figure 13c) have similar shape and size
(when scaled by $\overline{\delta}$) to the LSM/VLSM in the reference smooth
wall case.

A cursory inspection of figures 13(b) and (c) suggest that the LSM/VLSM are
stronger (higher in magnitude) than the secondary flows. This is expected
since the secondary flows are assumed to occur above the interface between
rough and smooth strips, while the detected LSMs/VLSMs are mostly obtained
above rough strips (figure 4a), where the turbulent energy is higher (note
that the conditionally averaged $u_{f}^{\prime}$ in figure 13 is normalised by
$U_{\infty}$). This observation, although not shown for brevity, also holds
for SR160 ($S/\overline{\delta}=2.28$). Comparing figures 13(a) and (b), we
observe that the LSM/VLSM of the reference smooth-wall case SW-2 is slightly
stronger than the secondary flows of SR250-2 case. However, despite these
minor differences, the overarching impression from figure 13 is the striking
similarity between the smooth wall LSMs/VLSMs and both the secondary flows and
LSMs/VLSMs that occur in the large scale limiting spanwise heterogeneity case
($S\gg\overline{\delta}$).

A comparison of the magnitude of meandering between the secondary flows and
LSM/VLSM is shown by the blue and red symbols in figure 8(b). Recall that this
meandering amplitude is measured from the ‘spines’ extracted from the
conditionally averaged low-speed structures (figure 7a). This comparison is
only possible for the range of spanwise heterogeneous wavelengths ($1.35\leq
S/\overline{\delta}\leq 3.63$), where the red and blue regions of figure 3 can
both be defined. At smaller wavelengths, the secondary flow regions (defined
as $y_{u}\pm l_{y}$) occupy the entire spanwise domain and so the LSM/VLSM
region cannot be identified. The trend shows that $\hat{y}$ (normalised by the
$z_{sheet}$/size of the roll modes) of secondary flows increases as
$S/\overline{\delta}$ approaches 1, but $\hat{y}$ of the LSM/VLSM, whether it
is extracted from $\widetilde{u}^{\prime}$ or $u^{\prime}$ fields, remains
approximately constant. More importantly, as $S/\overline{\delta}$ becomes
large in case SR250-2, we note from figure 8(b) that the meandering amplitude
of both the secondary flows and the LSMs/VLSMs seem to converge to the same
value, which is close to that recorded for the smooth wall. Again this
reconfirms the general underlying similarity between the conditional features
presented for the smooth wall and case SR250-2 in figure 13.

####  4.2.2 Limiting case: $S/\overline{\delta}\ll 1$

For the smallest $S/\overline{\delta}$ case, SR25
($S/\overline{\delta}=0.32$), the WPPIV measurement plane is set as close as
possible to the centre of the secondary flows at $z/\overline{\delta}=0.18$,
while the diameter of the secondary flows is approximately
$0.32\overline{\delta}$. As shown in figure 4(c), the secondary flows fill the
entire spanwise extent at this wall height (see also red shaded area in figure
3f). A similar conditional average at the minima and maxima of the detected
low-speed structures in the $x$–$y$ plane and at the common flow up in the
$y$–$z$ plane as that in §3 and §4.2.1 is also computed for SR25 case. Figure
7(f) shows that the conditionally averaged low-speed structures for this case
do not seem to show a marked difference to the LSM/VLSM in the reference
smooth wall case. The meandering amplitude for this case (figure 8b) is also
approximately equal to that of the reference smooth wall case at
$z/\delta_{s}=0.24$.

These observations for the $S/\overline{\delta}\ll 1$ case reveal the problem
of using spanwise locations to separate secondary flows and LSM/VLSM. This
strategy works for $S/\overline{\delta}\gg 1$ cases because both secondary
flows and LSM/VLSM are both $\delta$-scaled and the secondary flows are
confined about the roughness interface, while the rest of the flow approaches
local homogeneity (smooth wall or homogeneous rough wall). However, for the
other limit where $S/\overline{\delta}\ll 1$, the secondary flows and LSM/VLSM
coexist in different scales. The LSM/VLSM are $\delta$-scaled, while the
secondary flows are scaled by $S$ ($S\ll\delta$) and aligned according to the
roughness strips. For case SR25, specifically, the spanwise scale of the
secondary flows is approximately $0.32\overline{\delta}\approx S$. Hence, to
study the coexistence between LSM/VLSM and secondary flows in this limit where
$S/\overline{\delta}\ll 1$, a different method is necessary to separate the
two.

We propose ‘snapshot’ proper orthogonal decomposition (POD) as a possible
method to separate LSM/VLSM and secondary flows in the SR25 case. POD for
fluid mechanics applications was first introduced by Lumley (1967) and the
algorithm used in the present study is the so-called ‘snapshot’ POD (Sirovich,
1987, see also Meyer et al., 2007 for details). In the context of PIV data,
each instantaneous velocity field is considered as a ‘snapshot’ and the
fluctuating velocity components of snapshot $n$,
$\boldsymbol{u^{\prime}}^{n}=(u^{\prime n},v^{\prime n},w^{\prime n})$, can be
written as a linear expansion

$\boldsymbol{u^{\prime}}^{n}=\sum_{i=1}^{N}\boldsymbol{a}_{i}^{n}\boldsymbol{\phi}^{i}$
(10)

where $\boldsymbol{\phi}^{i}$ is the $i$th POD mode, $\boldsymbol{a}_{i}^{n}$
is the POD coefficient of mode $i$ of snapshot $n$, and $N$ is the total
number of snapshots. In snapshot POD, the number of resolved POD modes is
equal to the number of snapshots, which corresponds to 4800 for the spanwise
heterogeneous cases in the $yz$-plane (1200 for the reference smooth wall case
SW-2) and 1200 snapshots in the $xy$-plane (600 for SW-2). It is noted that
the POD modes $\boldsymbol{\phi}^{i}$ are ordered according to their
contribution to the total turbulent kinetic energy such that the first mode
$\phi^{1}$ has the largest fraction of total energy, followed by the second
mode $\phi^{2}$, and so forth.

Figure 14: POD modes of $u^{\prime}$ for SR25 case in $yz$-plane: (a,d) mode
1, (b,e) mode 5, and (c,f) mode 18. Brackets on top left of the figures show
the fraction of energy of the POD modes. (d,e,f) are 1-D premultiplied energy
spectra of the modes $k_{y}\Upphi_{\phi\phi}$ as a function of $\lambda_{y}$,
computed at $z/\overline{\delta}=0.1$ (solid black lines in a,b,c). The
dominant modes are: (a) $\lambda_{y}/\overline{\delta}\approx 1.5$, (b)
$\lambda_{y}/\overline{\delta}\approx 1$, and (c)
$\lambda_{y}/\overline{\delta}\approx 0.64$ ($\lambda_{y}/S\approx 2$).

Figure 15: Turbulent velocity fluctuations $u^{\prime}$ of case SR25 in
$yz$-plane: (a) all resolved scales, (b) large scales (constructed from
$u^{\prime}$ POD modes with dominant $\lambda_{y}/\overline{\delta}>0.64$),
(c) small scales (constructed from $u^{\prime}$ POD modes with dominant
$\lambda_{y}/\overline{\delta}\leq 0.64$).

Figure 15 shows POD modes of streamwise velocity fluctuation
$\phi_{u^{\prime}}^{i}$ for case SR25 in the $y$–$z$ plane. Three modes are
selected: mode $i=1$, 5, and 18, which contribute to 3.3%, 2.2%, and 0.9% of
the total kinetic energy, respectively. All three selected modes show a
spanwise periodic pattern of low- and high-speed streaks (blue and red filled
contours, respectively). Such streaks could be due to either LSM/VLSM or
heterogeneity-induced secondary flows. To measure the ‘scale’ of turbulence in
each mode, the Fast Fourier Transform (FFT) is computed for each
$\phi_{u^{\prime}}$ at a reference wall-normal location
$z/\overline{\delta}=0.1$ (as close to the wall as the FOV permits, see solid
black lines in figure 15a–c). The 1-D power spectral density of
$\phi_{u^{\prime}}$ at $z/\overline{\delta}=0.1$ is defined as

$\Upphi_{\phi\phi}=\frac{2c_{n}c_{n}^{\ast}}{\Updelta k_{y}}$ (11)

where $c_{n}$ is the $n$th Fourier coefficient, $c_{n}^{\ast}$ is its complex
conjugate, and $k_{y}$ is the spanwise wavenumber. Figures 15(d–f) show the
1-D energy spectra premultiplied by $k_{y}$ of mode 1, 5, and 18,
respectively. These figures show that the spanwise periodic pattern observed
in figure 15(a–c) corresponds to a dominant spanwise wavelength $\lambda_{y}$,
i.e. maximum $k_{y}\Upphi_{\phi\phi}$. Mode 1, for example, has the dominant
$\lambda_{y}/\overline{\delta}\approx 1.5$, while the dominant wavelength for
mode 5 and 18 are $\lambda_{y}/\overline{\delta}\approx 1$ and $\approx 0.64$,
respectively. Dominant spanwise wavelengths are then computed for all 4800
resolved POD modes and separated into two groups: ‘large’ scales (dominant
$\lambda_{y}/\overline{\delta}>0.64$, presumably related to LSM/VLSM) and
‘small’ scales (dominant $\lambda_{y}/\overline{\delta}\leq 0.64=2S$ for case
SR25 which will contain a mix of secondary flows due to spanwise heterogeneity
as well as smaller scale non-stationary turbulent features). With this
threshold for ‘large’/‘small’ scale separation, the POD modes associated with
‘large’ and ‘small’ scales make up 0.6% and 99.4% of all resolved modes, but
contribute to 33.7% and 66.3% of the total kinetic energy, respectively. The
fluctuating velocity fields are then reconstructed from both ‘large’ and
‘small’ scale POD modes in equation 10. Figure 15 shows an instantaneous
fluctuating velocity field $u^{\prime}$ of case SR25
($S/\overline{\delta}=0.32$) in the $yz$-plane, with figure 15(a) showing the
broadband PIV measurement result (all resolved scales), figure 15(b) is the
instantaneous $u^{\prime}$ reconstructed only from POD modes whose dominant
$\lambda_{y}/\overline{\delta}>0.64$ (‘large’ scales), and figure 15(c) is
reconstructed only from ‘small scales’ ($\lambda_{y}/\overline{\delta}\leq
0.64$).

Figure 16: Turbulent velocity fluctuations $u^{\prime}$ of case SR25 in
$xy$-plane ($z/\overline{\delta}=0.18$): (a) all resolved scales, (b) large
scales (constructed from $u^{\prime}$ POD modes with dominant
$\lambda_{y}/\overline{\delta}>0.64$), (c) small scales (constructed from
$u^{\prime}$ POD modes with dominant $\lambda_{y}/\overline{\delta}\leq
0.64$).

Figure 17: Detected low-speed structures $\widetilde{u}^{\prime}$ for case
SR25 ($z/\overline{\delta}=0.18$): (a) large scales (constructed from
$u^{\prime}$ POD modes with dominant $\lambda_{y}/\overline{\delta}>0.64$),
(b) small scales (constructed from $u^{\prime}$ POD modes with dominant
$\lambda_{y}/\overline{\delta}\leq 0.64$). Grey coloured contours are the low-
speed structures, $\widetilde{u}^{\prime}/U_{\infty}<-0.03$. ‘+’ marks the
minima of a low-speed structure. The spines of the detected low-speed
structures with length $\geq 2\overline{\delta}$ are shown in solid lines
(from PIV data:  , smoothed:  ). Dashed lines ( ) are the spanwise locations
of the common flow up of the secondary flows (marked by WAH1gu).

Figure 18: Histogram of the minima detected low-speed structures $y_{ref}$ for
case SR25 ($z/\overline{\delta}=0.18$): (a) large scales (figure 18a) and (b)
small scales (figure 18b). Dashed lines ( ) are the spanwise locations of the
common flow up of the secondary flows (marked by WAH1gu).

A similar strategy to separate turbulent scales related to LSM/VLSM and the
secondary flows is also applied for the fluctuating velocity fields in the
$x$–$y$ plane (WPPIV data) at $z/\overline{\delta}=0.18$. This wall-normal
location of the $xy$-plane is close to the centre of the secondary flows,
which is estimated at $z/\overline{\delta}\approx 0.16$, $z/S\approx 0.5$. To
obtain the ‘scales’ of turbulent flows, FFT is once again computed for each
$\phi_{u^{\prime}}$ in both the $x$\- and $y$-directions. The 2-D power
spectral density of $\phi_{u^{\prime}}$ in the $xy$-plane is defined as

$\Upphi_{\phi\phi}=\frac{4c_{n}c_{n}^{\ast}}{\Updelta k_{x}\Updelta k_{y}}$
(12)

where $k_{x}$ is the streamwise wavenumber. The maxima of the premultiplied
2-D energy spectra of each POD mode $k_{x}k_{y}\Upphi_{\phi\phi}$ corresponds
to the dominant streamwise and spanwise wavelength ($\lambda_{x}$ and
$\lambda_{y}$) of each mode. It should be noted that the accuracy of this
method to detect dominant $\lambda_{x}$ and $\lambda_{y}$ is limited by the
streamwise and spanwise extent of the FOV. Similar to the POD modes of
$u^{\prime}$ in the $y$–$z$ plane, $\phi_{u^{\prime}}$ in the $x$–$y$ plane
are also separated into two groups based on the dominant $\lambda_{y}$:
‘large’ scales (dominant $\lambda_{y}/\overline{\delta}>0.64$) and ‘small’
scales (dominant $\lambda_{y}/\overline{\delta}\leq 0.64=2S$). No such
threshold for scale separation is applied for the dominant $\lambda_{x}$. With
this threshold for ‘large’/‘small’ scale separation, the POD modes in the
$x$–$y$ planes associated with ‘large’ and ‘small’ scales make up 6.3% and
93.7% of all resolved 1200 modes, but contribute to 32.8% and 67.2% of the
total kinetic energy, respectively. Figure 16 illustrates the fluctuating
velocity fields $u^{\prime}$ reconstructed from both ‘large’ and ‘small’ scale
POD modes in equation 10. The conditional average of turbulent fluctuation
$u^{\prime}$ is computed at the minima and maxima of the low-speed structures
in the $x$–$y$ plane. The same algorithm to detect ‘spines’ of the structures
and the minima/maxima in §4.2.1 is also applied for SR25. However, for
previously analysed cases where $S/\overline{\delta}\gg 1$ in §4.2.1 the low-
speed structures were separated into LSM/VLSM and secondary flows based on
their spanwise locations. Here, where $S/\overline{\delta}\ll 1$, each
instantaneous turbulent fluctuation field is separated into the ‘large’ scales
(presumably LSM/VLSM) and ‘small’ scales (presumably containing the secondary
flows) based on the dominant $\lambda_{y}$ of the POD modes. An example of
detected low-speed structures for case SR25 ($z/\overline{\delta}=0.18$) are
shown in figure 18(a) for the ‘large’ scales and in figure 18(b) for the
‘small’ scales (gray filled contours are
$\widetilde{u}^{\prime}/U_{\infty}<-0.03$). Due to the decreasing length of
the structures detected as the ‘small’ scales (figure 16c and figure 18b),
only structures whose length is $\geq 2\overline{\delta}$ are included in the
computation (this criterion for low-speed structure detection is lower than
that in §4.2.1). Figure 18 shows the histogram of the $y$-minima of the
detected low-speed structures constructed from large (figure 18a) and small
scales (figure 18b). The smaller scales exhibit a phase-locking behaviour of
the structures, with higher possibility of the spines detected over the common
flow up. This phase-locking behaviour is still apparent (to lesser extent) in
the larger scales since $\widetilde{u}^{\prime}$ is used for detection (cf.
figure 3f for spine detection without scale separation). With this criterion
for low-speed structure detection, 40% of the detected streaks belong to the
small scales, and 60% belong to the large scales.

Figure 19: Contours of conditionally averaged large-scale turbulent
fluctuation $u^{\prime}$ (constructed from $u^{\prime}$ POD modes with
dominant $\lambda_{y}/\overline{\delta}>0.64$ or
$\lambda_{y}/\delta_{s}>0.64$) for (a) reference case SW-2 at
$z/\delta_{s}=0.24$ and (b) SR25 at $z/\overline{\delta}=0.18$. Conditions:
(i) $\widetilde{u}^{\prime}<0$ and $v^{\prime}>0$, (ii)
$\widetilde{u}^{\prime}<0$ and $v^{\prime}<0$ at ‘+’, (iii) minima and (iv)
maxima of detected low-speed structures (at ‘+’,
$\widetilde{u}^{\prime}/U_{\infty}<-0.03$) illustrated in figure 18(a) (refer
to §3 for the method used to extract the ‘spines’ of low-speed structures). In
(b)(iii) and (iv), dashed lines ( ) show the width of the roughness strips
$S$.

Conditional averages of $u^{\prime}$ are computed in the $y$–$z$ and $x$–$y$
planes for both ‘large’ and ‘small’ scales. In the $y$–$z$ plane, the
condition point of $u^{\prime}$ (‘+’ in figure 19i and iii) is at
$z/\overline{\delta}=0.1$ (as close to the wall as the FOV permits) and at the
spanwise locations of common flow up (marked by WAH1gu, centre of the smooth
strips for this case) for consistency with the previous analysis in §3 and
§4.2.1 (for ‘larger’ scales, the resulting conditionally averaged structures
are insensitive to the spanwise location of the condition vector). The
conditions for the $y$–$z$ plane are as follows: $\widetilde{u}^{\prime}<0$
(low-speed streaks) and $v^{\prime}>0$ or $\widetilde{u}^{\prime}<0$ (low-
speed streaks) and $v^{\prime}<0$. In the $x$–$y$ plane, the condition point
is at the $y$-minima and $y$-maxima of the spines fitted to the detected low-
speed streaks, which corresponds to $v^{\prime}>0$ and $v^{\prime}<0$ (see
figure 6 for an example). To provide a valid comparison, the same POD and
conditions are also applied to the reference smooth wall case SW-2 with
appropriate scaling ($\delta_{s}$ instead of $\overline{\delta}$). For the
smooth case, the velocity field is also separated into ‘large’ (POD modes with
dominant $\lambda_{y}/\delta_{s}>0.64$) and ‘small’ scales
($\lambda_{y}/\delta_{s}\leq 0.64$). In the smooth $y$–$z$ plane, the
condition point of $u^{\prime}$ is at $z/\delta_{s}=0.1$ and spanwise
homogeneity is assumed. For the smooth $x$–$y$ plane, only structures with
length $\geq 2\delta_{s}$ are included and spanwise homogeneity is also
assumed. With this criterion for low-speed structure detection, 13% of the
detected streaks belong to the small scales, and 87% belong to the large
scales.

Figure 19(a,b) show the contours of conditionally averaged ‘large’-scale
$u^{\prime}$ for the reference smooth wall case SW-2 and case SR25
($S/\overline{\delta}=0.32$), respectively. Conditionally averaged
$u^{\prime}$ in the $y$–$z$ plane is shown in subplot (i) and (iii). The
$x$–$y$ plane (subplot ii and iv) is located at $z/\delta_{s}=0.24$ for SW-2
and $z/\overline{\delta}=0.18$ for SR25 (close to the centre of the secondary
flows). The contours show the structures leaning to the left and right,
similar to those shown in §3 and §4.2.1. It is interesting to note, however,
that when only ‘large’ scales (i.e. scales that are larger than the spanwise
heterogeneity) are considered, the structures of SW-2 and SR25 are identical,
further suggesting that in the case of spanwise heterogeneity where the
roughness $S/\overline{\delta}\ll 1$, scales that are $\gtrsim S$ can be
considered as the naturally occurring turbulent structures and they exist in a
relatively unchanged form compared to those observed in the reference smooth
wall. This is also in line with Townsend’s outer layer similarity for rough
walls. As $S/\overline{\delta}$ becomes very small, the wall condition
approaches large scale homogeneity, and we might expect that beyond the
roughness sublayer ($z>S$, see Chan et al., 2018), outer layer similarity
should be preserved. Locking of large-scales now would be minimal, and it
would only be scales that were close to $\lambda_{y}=2S$ that we would perhaps
expect to be locked by the spanwise heterogeneity.

Figure 20: Contours of conditionally averaged small-scale turbulent
fluctuation $u^{\prime}$ (constructed from $u^{\prime}$ POD modes with
dominant $\lambda_{y}/\overline{\delta}\leq 0.64$ or
$\lambda_{y}/\delta_{s}\leq 0.64$) for (a) reference case SW-2 at
$z/\delta_{s}=0.24$, (b) SR25 at $z/\overline{\delta}=0.18$, and (c)
conditionally averaged $\widetilde{u}^{\prime}$ (small-scale $u^{\prime}$
supplemented with $\widetilde{U}$) for case SR25 at
$z/\overline{\delta}=0.18$. Conditions: (i) $\widetilde{u}^{\prime}<0$ and
$v^{\prime}>0$, (ii) $\widetilde{u}^{\prime}<0$ and $v^{\prime}<0$ at ‘+’,
(iii) minima and (iv) maxima of detected low-speed structures (at ‘+’,
$\widetilde{u}^{\prime}/U_{\infty}<-0.03$) illustrated in figure 18(b) (refer
to §3 for the method used to extract the ‘spines’ of low-speed structures). In
(b)(iii) and (iv), dashed lines ( ) show the width of the roughness strips
$S$.

Figure 20(a,b) show the contours of conditionally averaged ‘small’-scale
$u^{\prime}$ for the reference smooth wall case SW-2 and case SR25
($S/\overline{\delta}=0.32$), respectively. In the $y$–$z$ plane, the
structures for both SW-2 and SR25 are capped in the wall-normal direction at
$z/\overline{\delta}\lesssim 0.32$ ($z/\delta_{s}\lesssim 0.32$), as suggested
by Chan et al. (2018), who observed that the roughness sublayer height is
approximately half of the spanwise wavelength of the roughness. However, a
clear spanwise periodic pattern is observed in the the SR25 case (figure 20b,
subplot i and iii), which is less apparent in the smooth wall reference
(figure 20a, subplot i and iii). The wavelength of this spanwise periodic
pattern is $2S$ (which is also the spanwise roughness wavelength $\Lambda$),
as shown by the dashed lines in figure 20(b)(iii). A slight misalignment is
attributed to the condition of $v^{\prime}$, such that structures lean
slightly to the left and right. In the $xy$-plane, a similar pattern is also
observed. The pronounced spanwise periodicity for the SR25 case is largely
absent for the smooth wall reference. Furthermore, the streaks for case SR25
appear to be elongated in $x$ (approximately 3–$4\overline{\delta}$ long,
figure 20(b)ii and iv) to a greater extent than those observed for SW-2 ($\sim
2\delta_{s}$). This length is similar to the streamwise wavelength associated
with the additional energy spectra peak (see Wangsawijaya et al., 2020, figure
14f), which has been previously associated with meandering of the secondary
flows. It is important to note that the conditional averages shown in figure
20 are for $u^{\prime}$, the convecting non-stationary turbulent component. A
locked fluctuation such as this is distinct from the stationary secondary
flows (which would appear in $\widetilde{u}^{\prime}$, as shown in figure
20c). A possible explanation for these elongated streaks would be a meandering
behaviour of the secondary flows occurring at the interface between the rough
and smooth strips. The pronounced spanwise periodicity exhibited for case SR25
in figure 20 is more difficult to explain, but implies cross-talk between
adjacent secondary flows such their meandering is in-phase. The results
presented in figures 19 and 20 strongly suggest that the secondary flows and
naturally occurring $\overline{\delta}$-scaled LSM/VLSM co-exist in the case
where $S/\overline{\delta}\ll 1$. With $S$-scaled features residing close to
the wall ($z<S$) that are locked to the phase of the spanwise heterogeneity,
and with superimposed large-scale features that are virtually unchanged from
those occurring over smooth surfaces, exhibiting no spanwise locking or
alteration due to the heterogeneity.

##  5 Conclusions

We conduct an analysis of secondary flows and perturbed turbulent boundary
layers resulting from spanwise varying surface conditions within a range of
spanwise half-wavelength $0.32\leq S/\overline{\delta}\leq 3.63$. In
Wangsawijaya et al. (2020), it was suggested that $S/\overline{\delta}$
governs the size and strength of the secondary flows. The latter is maximum
for intermediate cases ($S/\overline{\delta}\approx 1$), which suggests two
possible scenarios: either (i) that these cases induce the strongest secondary
streamwise vortices or (ii) if secondary flows may be considered as spatially
locked turbulent structures, that these are more effectively locked in place
for these heterogeneous wavelengths. Present results suggest that (ii) is
viable. Following this scenario, we might consider the secondary flows
(instantaneously) as spanwise-locked large-scale turbulent structures. The 1-D
spanwise energy spectra (figure 10 and 11) suggests that there is an interplay
between the stationary secondary flows and the turbulent fluctuations, where
energy of the first might leech into the latter, and this is also governed by
$S/\overline{\delta}$. In the intermediate cases ($S/\overline{\delta}\approx
1$) where the secondary flows are space filling (hence, effectively locked),
the energy deposited into the stationary component and turbulent fluctuations
reach maximum and minimum, respectively. This might further suggest that
$S/\overline{\delta}$ determines the efficacy of roughness wavelengths in
locking the secondary flows/turbulent structures in place.

In addition to being spanwise locked, it was also suggested in Wangsawijaya et
al. (2020) that the secondary flows exhibit a quantifiable streamwise
unsteadiness, which was interpreted as a statistical evidence for meandering
with a preferred streamwise wavelength when $S/\overline{\delta}\approx 1$
(Wangsawijaya et al., 2020, see figure 15, 18, and 19). In the current study
we reveal the strongly meandering structures through conditional averaging of
$u^{\prime}$ for both SR cases and the reference smooth-wall case. The
resulting structures for both of these cases are strongly reminiscent of the
streak-vortex instability model proposed in Jeong et al. (1997); Waleffe
(2001); Flores & Jiménez (2010); Cossu & Hwang (2017) and de Giovanetti et al.
(2017) (figure 9).

In the limits where $S/\overline{\delta}\gg 1$ and $S/\overline{\delta}\ll 1$,
it is possible to examine the coexistence of secondary flows and the naturally
occuring large-scale structures. In the cases where $S/\overline{\delta}\gg 1$
both the secondary flows and large-scale structures are $\delta$-scaled. The
first are locked about the roughness transition and still carry the imprint of
meandering secondary flows, although not as strong as the intermediate cases,
while the latter occur farther from the transition and are identical to those
observed in the reference smooth wall case (figure 13). In the other limit
where $S/\overline{\delta}\ll 1$, secondary flows and large-scale structures
scale on $S$ and $\delta$, respectively. Using a POD-based filter, each
velocity field snapshot can be separated into scales $>S$ or $\leq S$. The
conditional average of $u^{\prime}$ of the larger scales show that the
structures of turbulence in the limiting case ($S/\overline{\delta}\ll 1$) and
the reference smooth wall are identical (figure 19), suggesting that
structures with spanwise scales $>2S$ are unaffected by the heterogeneity. The
smaller scales for the spanwise heterogeneous cases, however, exhibit the
imprint of secondary flows leeching into $u^{\prime}$, characterised as long
meandering spanwise alternating streaks that are locked and aligned with the
roughness strips. One outstanding question that remains unanswered in the
present study is the cause of the prominent meandering of the turbulent
structures, which is only observed when $S/\overline{\delta}\approx 1$. This
will be an interesting question to address in the future.

[Supplementary data]Not available.

[Funding]This research is supported by the Australian Research Council
Discovery Project (DP160102279) and the Office of Naval Research (BRC
N00014-17-1-2307).

[Declaration of interests]The authors report no conflict of interest.

[Data availability statement]The data that support the findings of this study
are available from the corresponding author (D.D.W.).

[Author ORCID]D. D. Wangsawijaya, https://orcid.org/ 0000-0002-7072-4245; N.
Hutchins, https://orcid.org/ 0000-0003-1599-002X

[Author contributions]D.D.W. designed, performed all experiments, and analysed
the data with the supervision of N.H. D.D.W and N.H. wrote the manuscript.

##  Appendix A Stationary vs. non-stationary component of the secondary flows

Figure 21 compares the extracted spines from an instantaneous velocity field
$\widetilde{u}^{\prime}\equiv\widetilde{U}+u^{\prime}$ (figure 21a) and
$u^{\prime}$ (figure 21b) of case SR50 ($S/\overline{\delta}=0.62$). The spine
extraction method for both $\widetilde{u}^{\prime}$ and $u^{\prime}$ velocity
fields is the same: both are filtered with a box filter of
$0.1\overline{\delta}\times 0.1\overline{\delta}$ size, the gray-coloured
contours in figure 21 are $\widetilde{u}^{\prime}/U_{\infty}<-0.03$ and
$u^{\prime}/U_{\infty}<-0.03$, respectively, and only structures longer than
or equal to $3\overline{\delta}$ are kept for the analysis. For comparison
between $\widetilde{u}^{\prime}$ and $u^{\prime}$, we introduce three
parameters related to the extracted spines: the length of the spine
$l_{spine}$ relative to the streamwise extent of the FOV $l_{FOV}$, the
location of the minima/maxima $x_{ref}$ relative to the spine, and the
location of the spine relative to the common flow up ( WAH1gu in figure 21).
Figure 22(a,e) shows the histogram of the length of the extracted spines
$l_{spine}$ from $\widetilde{u}^{\prime}$ and $u^{\prime}$ velocity fields,
respectively. When the $\widetilde{u}^{\prime}$ field is used for extraction,
the spines are likely to extend across the streamwise extent of the FOV
($l_{spine}/l_{FOV}=1$). The spines are mostly shorter
($l_{spine}/l_{FOV}=0.5$) when $u^{\prime}$ is used instead of
$\widetilde{u}^{\prime}$. This is expected since the stationary component of
the secondary flows $\widetilde{U}$ has an infinite mode in $x$, which is
present for $\widetilde{u}^{\prime}$ but removed for $u^{\prime}$. The
histograms of the streamwise location $x_{ref}$ of the minima of the extracted
structures relative to the start of the detected structure (see ‘+’ and the
definition of $x_{ref}$ in figure 21) from $\widetilde{u}^{\prime}$ and
$u^{\prime}$ fields are shown in figure 22(b,f), respectively, while the
histograms for the maxima are shown in figure 22(c,g). The histograms
illustrate the location (of the minima and maxima) relative to the extracted
spines $x_{ref}/l_{spine}$. Here, $x_{ref}/l_{spine}=0$ means that the
minima/maxima is located at upstream end of the spine, while
$x_{ref}/l_{spine}=1$ means that it is located at the downstream end. Figure
22(b,c,f,g) shows that $x_{ref}$ is roughly evenly distributed across 10–90%
length of the spines when either $\widetilde{u}^{\prime}$ or $u^{\prime}$ is
used for extraction. Lastly, we also examine the spanwise locking of the low-
speed structures. We compute the distance between the $y$ location of the
spines $y_{mean}$ and the location of common flow up $y_{up}$ (dashed black
lines in figure 21). Here, $y_{mean}$ is the $y$-average of all extracted
spines (solid black lines in figure 21). Figure 22(d,h) shows the histograms
of the calculated distance between the spines and common flow up normalised by
$S$ when $\widetilde{u}^{\prime}$ and $u^{\prime}$ field are used for
extraction, respectively. Here, $(y_{mean}-y_{u})/S=0$ is the center of a
smooth strip (which coincides with the common flow up) and
$(y_{mean}-y_{u})/S\pm 0.5$ is the interface between a smooth strip and
adjacent rough strips (figure 1). The histograms show that the extracted
spines from $\widetilde{u}^{\prime}$ are more likely to be locked on the
common flow up ($y_{u}$) compared to those extracted from $u^{\prime}$, which
is expected since $\widetilde{U}$ is a minima at $y_{u}$, and hence we are
much more likely to detect a minima in
$\widetilde{u}^{\prime}\equiv\widetilde{U}+u^{\prime}$ at this location.

Figure 21: Detected low-speed structures for case SR50
($S/\overline{\delta}=0.62$) from a WPPIV snapshot. Gray-coloured contours
are: (a) $\widetilde{u}^{\prime}/U_{\infty}<-0.03$ and (b)
$u^{\prime}/U_{\infty}<-0.03$. ‘+’ marks the minima of a low-speed structure.
The spines of the detected low-speed structures are shown in solid lines (from
PIV data:  , smoothed:  ). Dashed lines ( ) are the spanwise locations of the
common flow up of the secondary flows (marked by WAH1gu, see figure 1(g) for
these locations in the $yz$-plane). Figure 22: Histograms of the low-speed
structures for case SR50 ($S/\overline{\delta}=0.62$), detected in the
fluctuating velocity field: (a–d) $\widetilde{u}^{\prime}$ and (e–h)
$u^{\prime}$. (a,e) length of spines of the detected structures relative to
the FOV, (b,f) streamwise position of the detected minima and (c,g) maxima
relative to the length of the spines, (d,h) spanwise position of the spines
relative to the common flow up. Figure 23: Contours of filtered turbulent
fluctuation $u_{f}^{\prime}$ conditionally averaged at (a,b) the minima and
(c,d) maxima of the detected low-speed structures for case SR50
($S/\overline{\delta}=0.62$). Structures are detected in the fluctuating
velocity field: (a,c) $\widetilde{u}^{\prime}$ and (b,d) $u^{\prime}$, as
illustrated in figure 21.

We repeat the computation of the conditionally averaged fluctuating velocity
field previously introduced in §3 at the minima and maxima of the spines
extracted from $u^{\prime}$ field (instead of $\widetilde{u^{\prime}}$). It
should be noted that the conditional average is computed for filtered
turbulent fluctuation $u_{f}^{\prime}$, but the condition points differ
between the spines extracted from $\widetilde{u}^{\prime}$ and $u^{\prime}$
field (see ‘+’ in figure 21). Contours of conditionally averaged
$u_{f}^{\prime}$ of case SR50 ($S/\overline{\delta}=0.62$) are shown in figure
23(a,c) for condition points extracted from $\widetilde{u}^{\prime}$ and
figure 23(b,d) from $u^{\prime}$. Both cases show blue-filled contours (low-
speed structures) flanked by red-filled contours (high-speed structures), but
the strength of flanking high speed events seems stronger and more
symmetrically arranged about the low speed streak for $u^{\prime}$ (figure
23b,d) than $\widetilde{u}^{\prime}$ (figure 23a,c). Further, the amplitude of
meandering for $u^{\prime}$ is only slightly lower than that of
$\widetilde{u}^{\prime}$. This suggests that the choice of
$\widetilde{u}^{\prime}$ or $u^{\prime}$ makes very little difference in the
resulting conditionally averaged structures, and explains the very marginal
influence of choice of condition vector evident in figure 8(b).

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