# Examination of outer-layer similarity in wall turbulence over obstructing
surfaces

Zishen Chen1  Ricardo García-Mayoral1<EMAIL_ADDRESS>1Department of
Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ,
UK

###### Abstract

Turbulent flows over canopies of rigid filaments with different densities,
$\lambda_{f}$, are studied using direct numerical simulations at moderate
Reynolds numbers $Re_{\tau}\approx 550-1000$. The canopies have heights
$h^{+}\approx 110-220$, and are used as an instance of obstructing substrate
for the assessment of outer-layer similarity. We show that conventional
methods used to determine the zero-plane displacement, $\Delta y$, can be at
odds with proper outer-layer similarity and may not be applicable for flows at
moderate $Re_{\tau}$. Instead, we determine $\Delta y$ and the length and
velocity scales that recover outer-layer similarity by minimising the
difference between the smooth-wall and canopy diagnostic function everywhere
above the roughness sublayer, not just in the logarithmic layer. In addition,
we explore the possibility of the zero-plane displacement and the friction
velocity being set independently, but find that outer-layer similarity is more
consistently recovered when they are coupled. We observe that although the
Kármán constant, $\kappa$, may not have smooth-wall-like values, the flow
statistics are smooth-wall-like in the logarithmic layer and above if the
surface effect is limited within the near-wall region. This suggests a
modified outer-layer similarity, where $\kappa$ is not 0.39, but turbulence is
otherwise smooth-wall-like. When the canopy is dense, the flow above the tips
is essentially smooth-wall-like, with smooth-wall-like $\kappa\approx 0.39$
and origin essentially at the tip plane. For canopies with intermediate
density, the overlying flow perceives a deeper zero-plane displacement into
the canopy, which is consistent with observations reported by previous
studies, but exhibits a lower Kármán constant, $\kappa\approx 0.34-0.36$. For
sparse canopies, $\kappa$ tends back to its smooth-wall value, and the zero-
plane-displacement height is at the canopy bed. For all canopies studied, the
decrease in $\kappa$ never exceeds $15\%$, which is significantly less than
that obtained in some previous works using conventional methods to assess
outer-layer similarity.

###### keywords:

turbulent boundary layers, canopy flow, roughness

##  1 Introduction and background

Turbulent boundary layers over rough and complex surfaces are ubiquitous and
are of significant environmental and industrial interest. Surface roughness
can induce significant frictional drag or pressure drop for flows in
engineering settings, as summarised in the reviews by Flack & Schultz (2010,
2014), and Chung et al. (2021). Vegetation canopies are of great ecological
importance to terrestrial and aquatic ecosystems, as reviewed by Finnigan
(2000), Belcher et al. (2012), Nepf (2012a, b), and Brunet (2020). Porous
substrates are also present in a variety of settings (Wood et al., 2020), such
as river beds (Vollmer et al., 2002; Breugem et al., 2006), heat exchangers
(Lu et al., 1998; Dixon et al., 2012), and catalytic reactors (Lucci et al.,
2017). In addition, engineered surfaces exposed to turbulent flows generally
degrade and roughen due to erosion, fouling, and cumulative damage (Wu &
Christensen, 2007). For these reasons, understanding the impact of complex
surfaces on turbulence is essential for the modelling and control of practical
flows and to improve environmental and engineering practices.

The surface topology has a direct impact on the flow within the roughness
sublayer, which generally extends up to $2-3$ roughness heights, $h$, or
spacings, $s$, above the roughness crests, depending on the density regime
(Jiménez, 2004; MacDonald et al., 2018; Brunet, 2020). Above this height, it
is widely accepted that the turbulence is essentially undisturbed and exhibits
outer-layer similarity (Hama, 1954; Clauser, 1956; Townsend, 1976). The only
effect is then a constant shift, $\Delta U^{+}$, in the mean velocity profile,
while both the Kármán constant, $\kappa\approx 0.39$, and the wake region
remain unaffected. Experimental evidence of outer-layer similarity was
provided by Perry & Abell (1977) and Andreopoulos & Bradshaw (1981), who
reported smooth-wall-like mean-velocity profiles and turbulent statistics in
the outer layer for flows over rough walls. The recovery of outer-layer
similarity has also been observed in flows over a wide range of surface
topologies, including 2D ribs and grooves (Krogstad et al., 2005; Leonardi et
al., 2007; MacDonald et al., 2018; Zhang et al., 2020), sand grain (Flack et
al., 2005; Connelly et al., 2006; Amir & Castro, 2011; Flack & Schultz, 2023),
prismatic roughness (Castro, 2007; Yang et al., 2016; Sadique et al., 2017;
Placidi & Ganapathisubramani, 2018; Sharma & García-Mayoral, 2020a; Xu et al.,
2021), and practical rough surfaces (Shockling et al., 2006; Allen et al.,
2007; Wu & Christensen, 2007). Jiménez (2004) argued that the recovery of
outer-layer similarity relies on a large scale separation, $h/\delta<1/40$,
where $\delta$ is the boundary layer thickness. Numerical studies of roughness
and riblets have nevertheless observed outer-layer similarity for roughness
with larger blockage ratios, $h/\delta=1/8$ for cubes in an open channel and
$h/\delta=1/7$ for sinusoidal roughness in a pipe (Leonardi & Castro, 2010;
García-Mayoral & Jiménez, 2011; Chan et al., 2015; Abderrahaman-Elena et al.,
2019; Sharma & García-Mayoral, 2020a). As summarised in Chung et al. (2021),
the smooth-wall similarity in the wake region remains robust and holds even
for intrusive roughness with $h/\delta\gtrsim 0.15$. In this obstacle regime
(Jiménez, 2004), the protruding surface effect can completely disrupt
similarity in the logarithmic layer, but the similarity is still recovered in
the outer wake region (Flack & Schultz, 2010, 2014).

Studies of wall-bounded turbulence have provided the tools for analysing and
modelling rough-wall flows, with engineering models that treat roughness as a
small perturbation to the smooth-wall flow (Flack et al., 2005, 2007).
However, if the roughness-induced perturbation propagates into the outer
layer, the scaling based on smooth-wall similarity could result in inaccurate
predictions for turbulent statistics and integral quantities. Understanding
the extent of roughness effects and whether smooth-wall similarity holds true
is therefore of great importance to various applications. Townsend (1976)
proposed the outer-layer similarity hypothesis, articulating that at a
sufficiently high Reynolds number, the turbulent eddies in the outer layer
would be essentially unaffected by the surface topology. The surface affects
the flow only through providing the relevant scales, the wall shear stress,
$\tau_{w}$, or the friction velocity, $u_{\tau}=(\tau_{w}/\rho)^{1/2}$, and
the characteristic length scale provided by the wall-normal distance to the
wall, $y$. Townsend’s hypothesis is essentially a dimensional argument stating
that given $\delta^{+}\gg 1$ and $h/\delta\ll 1$, surface effects are confined
within the roughness sublayer, and thus the only relevant scales for the flow
above are $u_{\tau}$ and $y$, independent of the surface topology. Note that
$u_{\tau}$ and $y$ are well defined for smooth-wall flows but may not be
easily estimated for flows over rough and complex surfaces where the ’wall’ is
not obvious (Schultz & Flack, 2007; Squire et al., 2016).

The canonical logarithmic form of the mean velocity profile is

$U^{+}=\frac{1}{\kappa}\log(y^{+}+\Delta y^{+})+A-\Delta U^{+},$ (1)

where $\kappa$ is the Kármán constant and $\kappa\approx 0.39$ if outer-layer
similarity recovers, $A$ is the log-law intercept for a smooth-wall flow,
$\Delta U^{+}$ is the velocity deficit caused by the drag induced by surface
roughness, $y^{+}$ is the wall-normal distance, and $\Delta y^{+}$ is the
zero-plane displacement that recovers outer-layer similarity for the mean-
velocity profile, $U^{+}$. The displacement $\Delta y^{+}$ is typically
measured from the roughness tip or trough, and the zero-plane-displacement
height, $y^{+}=-\Delta y^{+}$, corresponds to the height of the origin
perceived by the outer-layer flow (Breugem et al., 2006; Manes et al., 2011).

Despite substantial evidence that supports Townsend’s outer-layer similarity
hypothesis in the presence of diverse surface topologies, some experimental
studies cast doubt on its universal validity, reporting that the roughness
effects can extend well into the outer layer (Krogstad et al., 1992; Krogstadt
& Antonia, 1999; Tachie et al., 2003; Bhaganagar et al., 2004). In these
works, it was observed that the presence of roughness significantly alters the
intensities of turbulent fluctuations, especially the wall-normal velocity
fluctuations and Reynolds shear stress, and the mean-velocity profile even in
the wake region. Additionally, recent experimental and numerical studies for
turbulent flows over rough and complex surfaces, as summarised in Table 1,
have reported the existence of a logarithmic layer but with values for
$\kappa$, logarithmic slope, very different from the smooth-wall value,
$\kappa_{s}\approx 0.39$. Moreover, for studies with $\delta^{+}\approx
1000-10000$ and $h/\delta\ll 1$, a decrease in $\kappa$ is still observed with
an increase in Reynolds number for the same roughness, suggesting an in-depth
modification of the flow by the substrates. (Suga et al., 2010; Manes et al.,
2011; Fang et al., 2018; Okazaki et al., 2021, 2022). Some studies have
observed that permeable roughness could lead to approximately $50\%$ drop in
$\kappa$, which is more substantial than that induced by the impermeable
roughness with the same geometry, implying that permeability may enhance the
extent of and the intensity of roughness effects (Okazaki et al., 2021, 2022;
Karra et al., 2022; Esteban et al., 2022). Nevertheless, the prediction of
$u_{\tau}$, which is of great importance for the assessment of outer-layer
similarity, remains a challenge in experiments (Chung et al., 2021).
Generally, $u_{\tau}$ is evaluated at the zero-plane-displacement height,
which is typically between the tip and trough of the obstacles, and $u_{\tau}$
is therefore not necessarily given by the total drag, $\tau_{w}$, exerted on
the surface. Depending on flow conditions and apparatus, uncertainties in
$u_{\tau}$ and turbulent statistics are typically $\sim\pm 1-5\%$ (Schultz &
Flack, 2007, 2013; Squire et al., 2016).

Studies | Wall | Roughness | $\delta^{+}$ | $\sqrt{K^{+}}$ | $k_{s}^{+}$ | $\kappa$
---|---|---|---|---|---|---
B06 | IPW | Packed bed | 353-678 | 0.31-9.35 | 0.11-49.6 | 0.23-0.40
S10 | IPW | Foamed ceramics | 150-1086∗ | 0.81-11.05 | 0.3-63.0 | 0.23-0.39
M11 | IPW | Polymer foam | 1856-5840 | 1.9-17.2 | - | 0.31-0.33
R17 | IW | Elastic Wall | 180.8-307.0 | - | - | 0.20-0.36
K17 | APW/IPW | Pore arrays | 120-399 | 0-11.6 | 0-19 | 0.21-0.41
S20 | APW/IPW | Spheres | 395 | 2.62 | - | 0.32-0.33
K21 | APW | Spanwise rods | 150-1137 | 3.5-25.9 | 0.4-60.9 | 0.21-0.40
K22 | APW/IW | Spheres | 270 | 2.56 | 2.65-6.65 | 0.32-0.35
E22 | IW | Riblets | 850-3840 | - | - | 0.30-0.37
F18 | IPW | Spheres | 862-10174 | 0.7-109 | 108-295∗ | 0.32-0.34∗
IW | Hemisphere | 986.8 | - | 10.6 | 0.41
O21 | IW | Solid ribs/grooves | 160-2100 | - | 642-1831 | 0.38-0.41
IW | Porous ribs/grooves | 380-1730 | 1.7-16.0 | 185-1049 | 0.19-0.29
O22 | IW | Solid ribs | 220-2050 | - | 17-1850 | 0.38-0.41
IW | Porous ribs | 240-1780 | 1.0-17.0 | 38-1340 | 0.22-0.33
Table 1: Studies that observe the modification of the logarithmic layer by the
surface topology. The wall types are IPW (isotropic permeable wall), APW
(anisotropic permeable wall) and IW (Impermeable wall). $\delta^{+}$,
$\sqrt{K^{+}}$, $k_{s}^{+}$ and $\kappa$ are the reported friction Reynolds
number ($\delta^{+}=\delta u_{\tau}/\nu$, where $\delta$ is the boundary-layer
thickness or channel half-height, $u_{\tau}$ is the characteristic friction
velocity and $\nu$ is the kinematic viscosity), permeability Reynolds number
($\sqrt{K^{+}}=\sqrt{K}u_{\tau}/\nu$ where $K$ is the permeability), roughness
height ($k_{s}^{+}=k_{s}u_{\tau}/\nu$ where $k_{s}$ is the equivalent sand-
grain roughness height) and Kármán constants, respectively. The abbreviations
for the studies are B06 (Breugem et al., 2006), S10 (Suga et al., 2010), M11
(Manes et al., 2011), R17 (Rosti & Brandt, 2017), K17 (Kuwata & Suga, 2017),
S20 (Shen et al., 2020), K21 (Kazemifar et al., 2021), K22 (Karra et al.,
2022), E22 (Endrikat et al., 2022), F18 (Fang et al., 2018), O21 (Okazaki et
al., 2021), and O22 (Okazaki et al., 2022). Note that $\delta^{+}$ denoted by
* is estimated from $k_{s}^{+}$ and $k_{s}/\delta$ from S10. The values for
$k_{s}^{+}$ and $\kappa$ from F18, denoted by *, were provided only for some
of their cases.

Recent studies carried out by Tuerke & Jiménez (2013) and Lozano-Durán & Bae
(2019) suggest that the scaling for wall turbulence is essentially local, and
is set by the local mean shear and production rate of turbulent kinetic
energy, with no explicit reference to the wall-normal distance, $y$. This
implies that the traditional scaling based on $y$ and $u_{\tau}$ happens to
hold because of the one-to-one correspondence between the latter and the local
production and shear, but this correspondence does not need to hold
necessarily for flows over non-smooth walls. As part of this work, we
investigate, for flows that exhibit an apparent loss of outer-layer
similarity, whether the local scale can still have correspondence to a
friction velocity, $u_{\tau}^{\star}$, and a length scale, $y_{*}$, where
$y_{*}$ is the wall-normal distance to the zero-plane-displacement height,
$y_{*}=0$, but $u_{\tau}^{\star}$ is not necessarily evaluated at $y_{*}=0$.
In this work, superscript $(\cdot)^{\star}$ denotes wall units defined by
$\nu$ and $u_{\tau}^{\star}$ decoupled from $y_{*}=0$, and superscript
$(\cdot)^{+}$ denotes wall units defined by $\nu$ and $u_{\tau}^{*}$ evaluated
at $y_{*}=0$. Subscript $(\cdot)_{*}$ denotes outer units that are normalised
by the bulk velocity, $U_{b}$, and outer length scale, $y_{*}$.

The diagnostic function of the mean-velocity profile in equation (1) is

$\beta=y_{*}^{+}\frac{\partial U^{+}}{\partial y_{*}^{+}},$ (2)

where $y_{*}=(y+\Delta y)/(\delta+\Delta y)$ is the wall-normal distance from
the zero-plane-displacement height, at $y_{*}=0$, which would exhibit a
plateau $\beta\approx 1/\kappa$ in the logarithmic layer, if outer-layer
similarity recovers (Mizuno & Jiménez, 2011; Luchini, 2018). This diagnostic
function is useful because deviations from the log-law profile are typically
more apparent in $\beta$ in equation (2) than in $U^{+}$ in equation (1). Many
previous studies therefore rely on the existence of this plateau in $\beta$ to
determine the extent of the logarithmic layer and the inner scaling for flows
over roughness (Breugem et al., 2006; Suga et al., 2010). Particularly, the
linear relation between $U^{+}$ and $\log(y_{*}^{+})$ in equation (1) is
enforced by choosing a $\Delta y$ that yields a plateau in $\beta(y_{*}^{+})$.
The inner velocity and length scales are then determined based on
$u_{\tau}^{*}$ evaluated at the reference height, $y_{*}=0$, yielding values
for $\kappa$ that are not necessarily smooth-wall like, as listed in Table 1.
However, a logarithmic layer with a plateau in $\beta$ emerges only in flows
at very high $Re_{\tau}$ (Lee & Moser, 2015; Hoyas et al., 2022). More
importantly, outer-layer similarity, by definition, refers to the similarity
in not just the logarithmic layer but also the wake, the whole outer region.
In the present work, we argue that for flows at all but the highest
$Re_{\tau}$, neglecting smooth-wall similarity in the wake region while
enforcing a plateau in the diagnostic function could result in spurious
predictions of parameters, including $\Delta y$, $u_{\tau}^{*}$ and $\kappa$,
and friction-scaled turbulent statistics. In this study, we determine the
zero-plane displacement, $\Delta y$, by minimising the deviation compared to a
smooth-wall flow of the diagnostic function not only in the logarithmic layer
but also above. We assess the validity as a scaling velocity of the friction
velocity, $u_{\tau}^{*}$ and $u_{\tau}^{\star}$, both measured at the height
of zero-plane displacement and set as an independent, free parameter.
Additionally, we examine whether the value of $\kappa$ is modified by the type
of surface or not. We probe the existence of outer-layer similarity in an
extensive dataset of canopy flows. This choice is motivated by canopies being
an instance of porous-like complex surfaces which are particularly obstructing
and intrusive to the flow (Ghisalberti, 2009).

The paper is organised as follows. The numerical method and relevant canopy
parameters are presented in §2. Results, with particular emphasis on scaling
for the outer-layer turbulence, are discussed in §3. Finally, the conclusions
are summarised in §4.

##  2 Direct numerical simulations

We present results for a series of direct-numerical simulations (DNS) of
closed and open channels with canopies of rigid filaments covering the walls
at moderate Reynolds numbers, $Re_{\tau}\approx 500-1000$. We note that these
$Re_{\tau}$ are sufficiently high for convective effects to be dominant, such
that the interpretation of the turbulent statistics in these canopy flows may
be extrapolated to cases with higher $Re_{\tau}$ (Sharma & García-Mayoral,
2020a). The streamwise, spanwise, and wall-normal directions are $x$, $z$, and
$y$, respectively. A schematic of the numerical domain is portrayed in figure
1. The dimensions of the closed channels are $L_{x}\times L_{z}\times
L_{y}=2\pi\delta\times\pi\delta\times 2(\delta+h)$, where $h$ is the canopy
height and $\delta=1$ is the distance between the channel centre and the
canopy-tip planes. The canopy region is below $y=0$ for the bottom wall, and
above $y=2\delta$ for the top wall. This domain size is large enough to
reproduce the one-point statistics for the friction Reynolds numbers
considered in this study, without imposing artificial constraints on the
largest turbulent eddies (Lozano-Durán & Jiménez, 2014).

$(a)$$(b)$

Figure 1: Schematics of the numerical domain of $(a)$ full-channel case
C108550 and $(b)$ open-channel case O400550. An instantaneous realisation of
the streamwise velocity is shown in the orthogonal planes.

We vary the canopy density by changing the spacing between elements, resulting
in frontal densities $\lambda_{f}\approx 0.01-2.04$, defined as the ratio
between the frontal area of the obstacles and the total plan area. This covers
a broad range from sparse to dense canopies based on the notional limit
$\lambda_{f}\approx 0.1$ proposed by Nepf (2012a). All canopies in the closed
channel consist of collocated prismatic posts with thickness
$\ell_{x}^{+}=\ell_{z}^{+}\approx 24$ and height $h^{+}\approx 110$. Relevant
simulation parameters are listed in table 2. For the canopy simulations,
letters C and O denote closed and open channels, and the number that follows
denotes the approximate spacing, $s^{+}=L_{x}^{+}/n_{x}=L_{z}^{+}/n_{z}$,
between the canopy elements, where $n_{x}$ and $n_{z}$ are the numbers of
elements in the streamwise and spanwise directions, respectively. The number
in the subscript is the approximate friction Reynolds number of the flow.
Cases C216900, C288900, and C432900 conducted at $Re_{\tau}\approx 900$ match
the geometry parameters, $s^{+}$, $l^{+}$ and $h^{+}$, of the sparse and
intrusive cases C216550, C288550, and C432550 in inner units, respectively.
These cases at high Reynolds numbers are conducted to verify that outer-layer
similarity can recover, even for flows over intrusive textures, provided a
large enough core flow unperturbed by the roughness. Cases C550, C600, C900,
O550, and O1000 are reference smooth-wall simulations.

DNSs of sparse canopies, $\lambda_{f}\approx 0.01$, from Sharma & García-
Mayoral (2020a) are included for the assessment of outer-layer similarity in
open-channel flows. These channels are bounded by a bottom no-slip wall and a
top free-slip surface at $y=\delta$, as shown in figure 1$(b)$. Case O400550
consists of prismatic posts with sides $\ell_{x}^{+}=\ell_{z}^{+}\approx 20$,
and height $h^{+}\approx 110$. The canopy of O4001000 matches the dimensions
of O400550 in inner units, with thickness $\ell_{x}^{+}=\ell_{z}^{+}\approx
20$, and height $h^{+}\approx 110$, while the canopy of C800${}_{\text{1000}}$
matches the dimensions of O400550 in outer units, with thickness
$\ell_{x}/\delta=\ell_{z}/\delta\approx 0.04$, and height $h/\delta\approx
0.2$. The reference friction velocity, $u_{\tau}$, in table 2 is calculated
from the total shear stress at the canopy tips for the full-channel cases and
from the net drag for the open-channel cases. This is the reference friction
velocity used in $Re_{\tau}=\delta u_{\tau}/\nu$ and in the other friction-
scaled variables discussed in this section.

| Case | $Re_{\tau}$ | $\lambda_{f}$ | $N_{x}\times N_{z}$ | $s/h$ | $\Delta x^{+}$ | $\Delta z^{+}$ | $\Delta y_{f}^{+}$ | $\Delta y_{t}^{+}$
---|---|---|---|---|---|---|---|---|---
Channel flow | C550 | 550.6 | - | - | - | 9.01 | 4.50 | 0.27 | -
C600 | 603.3 | - | - | - | 8.77 | 4.39 | 0.29 | -
C950 | 950.2 | - | - | - | 7.77 | 3.89 | 0.32 | -
C36550 | 550.3 | 2.04 | $96\times 48$ | 0.33 | 2.00 | 2.00 | 2.95 | 0.50
C54550 | 550.1 | 0.91 | $64\times 32$ | 0.49 | 3.00 | 3.00 | 2.53 | 0.50
C72550 | 551.9 | 0.51 | $48\times 24$ | 0.65 | 4.01 | 4.01 | 2.01 | 1.00
C108550 | 548.5 | 0.23 | $32\times 16$ | 0.98 | 5.98 | 2.99 | 2.00 | 1.00
C144550 | 547.0 | 0.13 | $24\times 12$ | 1.31 | 5.97 | 2.98 | 1.19 | 1.19
C216550 | 549.0 | 0.06 | $16\times 8$ | 1.96 | 5.99 | 2.99 | 1.00 | 1.00
C288550 | 547.9 | 0.03 | $12\times 6$ | 2.62 | 5.98 | 2.99 | 0.79 | 0.79
C432550 | 548.9 | 0.01 | $8\times 4$ | 3.93 | 5.99 | 2.99 | 0.60 | 0.60
C216900 | 895.1 | 0.06 | $32\times 16$ | 1.96 | 8.26 | 4.13 | 1.1 | 1.1
C288900 | 895.7 | 0.03 | $24\times 12$ | 2.62 | 8.26 | 4.13 | 1.1 | 1.1
C432900 | 892.4 | 0.01 | $16\times 8$ | 3.93 | 8.26 | 4.13 | 1.1 | 1.1
Open-channel flow | O550 | 537.1 | - | - | - | 8.79 | 4.39 | 0.21 | -
O1000 | 994.6 | - | - | - | 8.14 | 4.07 | 0.32 | -
O400550 | 529.7 | 0.01 | $8\times 4$ | 3.65 | 4.33 | 4.33 | 0.20 | 1.77
O4001000 | 1062.4 | 0.01 | $16\times 8$ | 3.53 | 4.35 | 4.35 | 0.35 | 2.76
O8001000 | 1001.8 | 0.01 | $8\times 4$ | 3.61 | 4.10 | 4.10 | 0.33 | 3.45
Table 2: Simulation parameters: $Re_{\tau}=\delta u_{\tau}/\nu$ is the
friction Reynolds number based on $\nu$, $\delta$ and $u_{\tau}$ evaluated at
the canopy tips; $\lambda_{f}$ is the frontal density; $N_{x}$ and $N_{z}$ are
the numbers of canopy elements in the streamwise and spanwise directions,
respectively; $h$ and $s$ are the canopy height and the spacing in the
streamwise and spanwise directions; $\Delta x^{+}$ and $\Delta z^{+}$ are the
streamwise and spanwise resolutions; $\Delta y_{f}^{+}$ and $\Delta y_{t}^{+}$
are the wall-normal resolutions at the floor and canopy tips. The wall-normal
resolution is validated in Appendix for reference. O550 and open-channel
canopy cases are from Sharma & García-Mayoral (2020a).

The DNS code implemented in this study is from Sharma & García-Mayoral (2020a,
b) and has been validated in Sharma (2020). It is summarised here for
reference. The numerical method solves the three-dimensional incompressible
Navier-Stokes equations,

$\displaystyle\frac{\partial\mathbf{u}}{\partial{t}}+\mathbf{u}\cdot\nabla\mathbf{u}$
$\displaystyle=-\nabla{p}+\frac{1}{Re}\nabla^{2}\mathbf{u},$ (3)
$\displaystyle\nabla\cdot\mathbf{u}$ $\displaystyle=0,$ (4)

where $\mathbf{u}$ is the velocity vector $\langle u,w,v\rangle$ with
components in the streamwise, spanwise and wall-normal directions,
respectively, $p$ is the kinematic pressure, and $Re$ denotes the bulk
Reynolds number $Re=U_{b}\delta/\nu$ based on $U_{b}$, $\delta$, and the
kinematic viscosity, $\nu$. No-slip and no-penetration boundary conditions are
enforced at both walls. The canopy elements are explicitly resolved using a
direct-forcing, immersed-boundary method (Iaccarino & Verzicco, 2003; García-
Mayoral & Jiménez, 2011). The numerical domain is periodic in the wall-
parallel directions, which are discretised spectrally. A second-order central
difference scheme on a staggered grid is used in the wall-normal direction to
avoid the ’chequerboard’ problem (Ferziger & Perić, 2002). The wall-normal
grid is stretched with $\Delta y_{max}^{+}\approx 4.5$ at the channel centre
for the closed-channel simulations. For the open-channel simulations, $\Delta
y_{max}^{+}\approx 2.2$ when $Re_{\tau}\approx 550$ and $\Delta
y_{max}^{+}\approx 5.3$ when $Re_{\tau}\approx 1000$. $\Delta y_{min}^{+}$
occurs at the floor or tips, wherever the mean shear is the highest; $\Delta
y_{min}^{+}\approx 0.5-1$ is at the tips for the intermediate to dense
canopies ($\lambda_{f}\gtrsim$0.1), and $\Delta y_{min}^{+}\approx 0.3-0.8$ is
at the floor for the sparse canopies ($\lambda_{f}\lesssim$0.1). The wall-
normal grid resolutions are listed in table 2.

The typical wall-parallel resolutions are $\Delta x^{+}\lesssim 8$ and $\Delta
z^{+}\lesssim 4$ for the DNS of smooth-wall turbulent flows (Jiménez & Moin,
1991). However, for the filament canopies considered in this study, the
element-induced eddies are typically of the order of or smaller than the
element thickness (Poggi et al., 2004). Therefore, the wall-parallel grids are
smaller than $\ell_{x}^{+}$ and $\ell_{z}^{+}$ to resolve the eddies induced
by the canopy elements, as presented in table 2. To resolve the turbulence
within and above the roughness sublayer without inducing excess computational
cost, the numerical domain is partitioned into blocks with different wall-
parallel resolutions (García-Mayoral & Jiménez, 2011). The blocks that contain
the roughness sublayer have a more refined resolution than the block
encompassing the channel centre. In the fine blocks, the grid resolution
resolves not just the turbulent scales but also the canopy geometry and
element-induced eddies. The height of these blocks is chosen such that the
small and rapid element-induced eddies naturally diffuse and damp out before
reaching the coarse block at the channel centre, which has a standard $\Delta
x^{+}\approx 8$ and $\Delta z^{+}\approx 4$ resolution. This is verified _a
posteriori_ by examining the spectral densities of turbulent fluctuations near
the interface to ensure that any small-wavelength signal has already vanished.

The time advancement uses a fractional-step method with a three-substep Runge-
Kutta scheme where pressure is corrected to enforce incompressibility (Le &
Moin, 1991; Perot, 1993),

$\displaystyle\left[\text{I}-\Delta
t\frac{\beta_{k}}{Re}\text{L}\right]\mathbf{u}^{n}_{k}$
$\displaystyle=\mathbf{u}^{n}_{k-1}+\Delta
t\left[\frac{\alpha_{k}}{Re}\text{L}\mathbf{u}^{n}_{k-1}-\gamma_{k}\text{N}\mathbf{u}^{n}_{k-1}-\zeta_{k}\text{N}\mathbf{u}^{n}_{k-2}-\left(\alpha_{k}+\beta_{k}\right)\text{G}p^{n}_{k}\right],$
(5) $\displaystyle\text{DG}\phi^{n}_{k}$ $\displaystyle=\frac{1}{\Delta
t\left(\alpha_{k}+\beta_{k}\right)}\text{D}\mathbf{u}^{n}_{k},$ (6)
$\displaystyle\mathbf{u}^{n}_{k+1}$ $\displaystyle=\mathbf{u}^{n}_{k}-\Delta
t\left(\alpha_{k}+\beta_{k}\right)\text{G}\phi^{n}_{k},$ (7) $\displaystyle
p^{n}_{k+1}$ $\displaystyle=p^{n}_{k}+\phi^{n}_{k},$ (8)

where $k=1,2,3$ are the Runge-Kutta substeps (e.g. $u^{0}_{0}=u^{0}$,
$u^{0}_{3}=u^{1}$), $\Delta t$ is the time step, I is the identity matrix, L,
G and D are the discretised Laplacian, gradient and divergence operators, N is
the advective term dealiased with the 2/3-rule (Canuto et al., 2012), and
$\alpha_{k}$, $\beta_{k}$, $\gamma_{k}$ and $\zeta_{k}$ are the integration
coefficients adapted from Le & Moin (1991). The channel is driven by a
constant mean pressure gradient, with the flow rate adjusted to obtain the
targeted friction Reynolds number. Each simulation is run for at least 10
largest-eddy-turnover times, $\delta/u_{\tau}$, to wash out any initial
transients. Once the flow reaches a statistically steady state, statistics are
collected over another $20\delta/u_{\tau}$.

##  3 Results and discussion

In this section, we present and discuss the scaling for the outer-layer flow,
aiming to show that for canopy flows that exhibit an apparent loss of outer-
layer similarity, a modified outer-layer similarity can be recovered when
using the appropriate velocity and length scales.

###  3.1 Depth of roughness layer

Before we set out to investigate outer-layer similarity, it is important to
establish a lower bound for $y$ from which it can be expected to hold. In the
immediate vicinity of a complex surface, the flow cannot be expected to be
universal, but specific to the particular surface topology. Outer-layer
similarity should not be expected within the roughness sublayer, where
turbulence is directly perturbed by the element-induced flow. Thus, we first
need to identify the height above which the direct effect of the texture,
manifesting as a texture-coherent signature in the flow field, vanishes
effectively. The height of the roughness sublayer assumed here to be the
height beyond which the element-induced flow vanishes, is generally a function
of the element spacing or height, depending on the density regime (Jiménez,
2004; Brunet, 2020). On the basis of canopy geometry and configuration, the
frontal density $\lambda_{f}$ gives a notional measure of canopy density
(Wooding et al., 1973; Nepf, 2012a). For conventional sparse canopies
($\lambda_{f}\lesssim 0.1$) with element spacing larger than height, the
roughness sublayer thickness is typically a function of the canopy height
(Poggi et al., 2004; Flack et al., 2007; Abderrahaman-Elena et al., 2019;
Sharma & García-Mayoral, 2020a). However, for dense canopies
($\lambda_{f}\gtrsim 0.5$), the flow within the obstacles is ’sheltered’ from
the turbulent flow as the elements interact with turbulence only in the
vicinity of the tips, and thus the height of the roughness sublayer depends on
the element spacing (MacDonald et al., 2018; Placidi & Ganapathisubramani,
2018; Sharma & García-Mayoral, 2020b). In the very dense limit, where the
element spacings are vanishingly small, the eddies are essentially precluded
from penetrating within the texture, and the overlying flow essentially
perceives a smooth wall at the tips (Sharma & García-Mayoral, 2020b; Brunet,
2020).

To measure the extent of roughness effects, here we quantify the intensity of
the element-induced flow using standard triple decomposition (Reynolds &
Hussain, 1972)

$\displaystyle\mathbf{u}(x,y,z,t)$
$\displaystyle=\mathbf{U}(y)+\mathbf{u^{\prime}}(x,y,z,t),$ (9)
$\displaystyle\mathbf{u^{\prime}}(x,y,z,t)$
$\displaystyle=\tilde{\mathbf{u}}(x,y,z)+\mathbf{u^{\prime\prime}}(x,y,z,t),$
(10)

where $\mathbf{u}$ is the full instantaneous velocity vector field $\langle
u,w,v\rangle$, $\mathbf{U}$ is the mean velocity profile, and
$\mathbf{u^{\prime}}$ is the full temporal and spatial turbulent fluctuation,
decomposed into a time-averaged but spatially varying component,
$\tilde{\mathbf{u}}$, and the remaining time-varying fluctuation,
$\mathbf{u^{\prime\prime}}$. $\mathbf{U}$ is the velocity averaged in time and
in the wall-parallel directions, and $\tilde{\mathbf{u}}$, often termed the
dispersive flow (Castro et al., 2021; Modesti et al., 2021), is obtained from
the average of the flow in time only. Therefore, the r.m.s. of
$\tilde{\mathbf{u}}$ at each height gives a measure of the intensity of the
coherent spatial fluctuation induced by the canopy elements. Abderrahaman-
Elena et al. (2019) argued that $\tilde{\mathbf{u}}$ does not contain the
whole element-coherent signal, but it nevertheless gives a good measure of its
intensity.

$(a)$$(b)$$(c)$$(d)$$(e)$$(f)$$(g)$$(h)$$(i)$$(j)$$(k)$$(l)$$(m)$$(n)$$(o)$$(p)$

Figure 2: R.m.s. velocity and pressure fluctuations of the element-induced,
dispersive flow normalised by $u_{\tau}$ evaluated at the canopy tips.
$(a,e,i,m)$ and $(c,g,k,o)$, full-channel cases; $(b,f,j,n)$ and $(d,h,l,p)$,
open-channel cases. The solid lines from blue to red are cases C36550 to
C432550 in the first and third columns and cases O400550, O4001000, and
O8001000 in the second and fourth columns; the dashed-dotted lines  ,  and
are cases C216900, C288900 and C432900 that have similar geometry parameters
in inner units to those of cases C216550 , C288550 , and C432550 . The square
markers represent height $y=s$ for cases at $Re_{\tau}\approx 550$, and the
triangle markers represent that for those at $Re_{\tau}\approx 900-1000$.

As shown in figure 2, the intensity of the element-induced fluctuations decays
exponentially with $y$ above the tips. A similar decaying pattern has been
observed in flows over superhydrophobic surfaces (Seo et al., 2015), three-
dimensional sinusoidal roughness (Chan et al., 2018), prismatic roughnesses
(Abderrahaman-Elena et al., 2019), and filament canopies (Sharma & García-
Mayoral, 2020b). The cause can be traced to the pressure in this region
satisfying a Laplace equation with two components, one forced by the nonlinear
terms of the overlying flow, which is essentially texture-incoherent, and one
forced by the effective boundary conditions at $y=0$, induced by the texture.
The latter then takes the form $\sim e^{-y/\lambda}$ for each excited
wavelength, for which the first texture harmonic, $\lambda=s$, decays more
slowly and dominates (Kamrin et al., 2010; Seo et al., 2015). The velocities
satisfy in turn their own corresponding Laplace equations, with additional
source terms from this texture-induced pressure, leading to similar
exponential decays. Figure 2 evidences this exponential decay with $y/s$,
which is particularly clear for the pressure, as well as for the wall-normal
velocity, and to a lesser extent for the tangential velocities, which is to be
expected given their more intense source terms in their respective Laplace
equations.

For all canopies considered, the texture-coherent pressure and velocity
fluctuations essentially vanish at approximately one canopy spacing above the
tips, as depicted in figure 2. However, this implies that the sparse
($\lambda_{f}\lesssim 0.1$) and tall ($h\approx 0.2\delta$) canopies, C216550,
C288550, C432550, O400550 and O8001000, are significantly more intrusive to
the overlying flow compared to the other canopies with either intermediate to
high density ($\lambda_{f}\gtrsim 0.1$) or small height ($h\approx
0.1\delta$), as their element-induced flows penetrate into the channel as far
as $y\approx 0.5\delta$, or even beyond. This implies that the roughness
sublayer of these intrusive canopies can extend well into the overlying flow
and reach the channel centre. Sharma & García-Mayoral (2020b) have reported a
similar behaviour for the element-induced flow over dense canopies, for which
the element-induced velocity fluctuations become negligible at one canopy
spacing above the tips regardless of the canopy height. However, their
element-induced flows caused a more profound modification of the background
turbulence, which only became smooth-wall-like at heights $y/s>2-3$ above the
tips. For the present flows, however, it will be demonstrated in §3.3 and §3.4
that, for the cases which exhibit it, outer-layer similarity recovers above a
roughness sublayer that extends only to a height $y/s\approx 1$ above the
tips.

###  3.2 Logarithmic velocity profiles over smooth and rough walls

In this section, we discuss and appraise the conventional methods used to
assess the existence of a logarithmic layer and find the zero-plane-
displacement height, $y_{*}=0$, that sets the velocity and length scales,
$u_{\tau}^{*}$ and $y_{*}$, for turbulent flows over roughness. Their values
are generally determined by using the total drag (Jackson, 1981; Raupach,
1992; Cheng et al., 2007; Leonardi & Castro, 2010; Squire et al., 2016), by
fitting $U^{+}$ to be proportional to $\log(y_{*}^{+})$ in the logarithmic
layer (Clauser, 1956; Flack & Schultz, 2014), or by, equivalently, enforcing a
plateau in $\beta$ (Breugem et al., 2006; Suga et al., 2010; Manes et al.,
2011).

In experiments, friction velocity and zero-plane-displacement height are
generally estimated based on the total drag exerted on the surface, as
summarised in Chung et al. (2021). For small and sparsely distributed
roughness, where the overlying flow penetrates all the way to the floor, this
method generally yields $u_{\tau}^{*}$ and $\Delta y$ that recover outer-layer
similarity (Flack et al., 2005; Wu & Christensen, 2007; Schultz & Flack,
2013). However, for dense and tall roughness, the total drag method could
result in nonphysical prediction for both $u_{\tau}^{*}$ and $\Delta y$. The
element-induced drag is significant for dense roughness, and therefore the
point of action of the total drag, $\Delta y=\int_{h}D(y)ydy/\int_{h}D(y)dy$
(Jackson, 1981), is located at an intermediate point in the roughness
sublayer. However, DNSs of dense filament canopies have illustrated that the
zero-plane-displacement height approaches the tips as the overlying turbulence
interacts only with the upper part of the obstacles and cannot perceive the
floor (Sharma & García-Mayoral, 2020b; Brunet, 2020). In turn, for sparse
canopies, the zero-plane-displacement height approaches the floor, even when
most of the drag is still exerted by the canopy elements (Sharma & García-
Mayoral, 2020a). Consequently, the total drag may not necessarily be directly
relevant for the assessment of outer-layer similarity and the estimates of the
zero-plane displacement, $\Delta y$, and the friction velocity evaluated at
$y_{*}=0$, $u_{\tau}^{*}$.

$(a)$$(b)$$(c)$$(d)$$(e)$$(f)$

Figure 3: Mean velocity and velocity deficit profiles for $(a,c,e)$ cases
C144550 (solid colour lines) and C432550 (dashed colour lines), and $(b,d,f)$
cases O4001000 (solid colour lines) and O8001000 (dashed colour lines). From
blue to red, results are based on $(a,c)$ $\Delta y=0$ to $0.25\delta$ and
$(b,d)$ $\Delta y=0.05\delta$ to $0.15\delta$; the black solid lines are
reference smooth-wall profiles $(a)$ C550 and $(b)$ O1000; the black dashed
lines are the smooth-wall log-law profile, $1/\kappa\log(y^{+})+A$; and the
shaded area marks the logarithmic region for smooth-wall flows, $y^{+}=80$ to
$y=0.3\delta$ for closed channels and $y^{+}=80$ to $y=0.2\delta$ for open
channels. The upper bound of the log region in open channels is lower than
that in a closed channel because the free-slip surface induces a ’cutoff’ to
the wake region, which limits the extent of the log layer.

Hama (1954) and Clauser (1956) noted that roughness generally leads to a
downward but otherwise parallel shift, $\Delta U^{+}$, in the mean-velocity
profile. It stems from this that $\Delta y$ and $u_{\tau}^{*}$ can be obtained
by matching the shape of the mean-velocity profile over roughness, $U^{+}$, to
a $\log(y_{*}^{+})$, assuming the latter accurately represents the
corresponding smooth-wall profile. This matching is usually done iteratively.
Figures 3$(e,f)$ illustrate how, for the flows over our canopies, $U^{+}$ can
be made logarithmic by selecting a suitable value of $\Delta y$ and taking
$u_{\tau}^{*}$ based on the total shear stress at the zero-plane-displacement
height. However, $U^{+}$ is not exactly logarithmic even in the logarithmic
layer of a smooth-wall flow, so matching $U^{+}$ to a $\log(y_{*}^{+})$ as in
figures 3$(e,f)$ is different from matching $U^{+}$ to the smooth-wall profile
in the logarithmic layer as in figures 3$(c,d)$. As an example, $\Delta
y\approx 0.1\delta-0.15\delta=0.5h-0.75h$ enforces a logarithmic mean-velocity
profile for case C144550, but with a non-smooth-wall-like $\kappa_{c}$, as
evidenced in figures 3$(a,e)$. In addition, the mean-velocity profile above
the logarithmic layer is different from a corresponding smooth-wall profile,
suggesting a breakdown of outer-layer similarity, even though $U^{+}$ was made
logarithmic. Alternatively, by imposing $\Delta y\approx 0.1\delta=0.5h$, we
may recover a smooth-wall-like logarithmic layer, as depicted in figure
3$(c)$, where $\kappa_{c}\approx\kappa_{s}\approx 0.39$. Nevertheless, the
outer-wake region is still not smooth-wall-like, which would still break full
outer-layer similarity. Moreover, for the intrusive cases C432550 and
O8001000, where the near-wall turbulence is completely disrupted by the
element-induced flow, the mean-velocity profile could still be enforced to
take a logarithmic or smooth-wall-like shape within the ’logarithmic layer,’
as shown in figures 3$(c-f)$. Nevertheless, the flow above this ’logarithmic
layer’ is never smooth-wall-like. In contrast, outer-layer similarity may
still be achieved without recovering a complete smooth-wall-like logarithmic
region, as illustrated for case O4001000 in figure 3$(b,d,f)$, where the lower
part of the logarithmic region is perturbed by the canopy but the flow above
$y_{*}^{+}\approx 130$ is essentially smooth-wall-like when $\Delta
y=0.1\delta$. The above suggests that outer-layer similarity cannot be
recovered simply by artificially matching $U^{+}$ to a $\log(y_{*}^{+})$, or a
smooth-wall profile, exclusively in the logarithmic layer. The matching should
be for any height above the roughness sublayer.

$(a)$$(b)$

Figure 4: $(a)$ Diagnostic function and $(b)$ defect-law velocity profile,
scaled with $y_{*}$ and $u_{\tau}^{*}$ for case C36550.  , reference smooth-
wall profile C550;  ,  ,  , canopy statistics based on $\Delta y=0$,
$0.05\delta$, and $0.1\delta$, respectively; and  ,  ,  values of $1/\kappa$
for smooth-wall and canopy flow, where $\kappa_{s}\approx 0.39$ and
$\kappa_{c}\approx 0.32$ and 0.28, respectively.

$(a)$$(b)$

Figure 5: Diagnostic function for smooth-wall channel flows at $Re_{\tau}$
increasing from blue to red: $Re_{\tau}\approx 1000,2000,5000$, and $10000$.
The solid lines represent $y^{+}dU^{+}/dy^{+}$, with $\Delta y=0$; the dash-
dotted lines in $(a)$ represent $\beta$ with $\Delta y^{+}\approx 40,27,13,7$,
so that a plateau is enforced within the logarithmic layer as in Manes et al.
(2011) and Okazaki et al. (2021, 2022); the dash-dotted lines in $(b)$
represent $\beta$ with $\Delta y^{+}\approx 80,139,311,1004$, so that the
extent of plateau is maximised as in Breugem et al. (2006), Suga et al.
(2010), Kuwata & Suga (2017), Rosti & Brandt (2017), Fang et al. (2018), Shen
et al. (2020), Kazemifar et al. (2021), Okazaki et al. (2021, 2022), and Karra
et al. (2022). Notice that Okazaki et al. (2021, 2022) use both methods (see
figures 5$(c)$ and 5$(a,b,d)$ in Okazaki et al. (2021) and figures 12 and 21
in Okazaki et al. (2022)). With $Re_{\tau}$ increasing, the horizontal dotted
lines correspond to $\kappa_{c}\approx(0.32,0.34,0.37,0.40)$ in $(a)$ and
$\kappa_{c}\approx(0.29,0.29,0.31,0.30)$ in $(b)$, and the vertical lines
represent the upper bound of the logarithmic layer, $y/\delta\approx 0.3$.
$Re_{\tau}\approx 1000,5000$ data are from Lee & Moser (2015),
$Re_{\tau}\approx 2000$ data is from Hoyas & Jiménez (2006), and
$Re_{\tau}\approx 10000$ data is from Hoyas et al. (2022).

Within the logarithmic layer, enforcing $U^{+}$ to be logarithmic, or smooth-
wall-like, is essentially equivalent to enforcing $\beta$ to have a plateau,
or a smooth-wall-like region. While both the shape of $U^{+}(y_{*}^{+})$ and
$\beta(y_{*}^{+})$ contain the same information, $\beta$ is more sensitive to
deviations from the smooth-wall reference profile, and it directly portrays
the value of $1/\kappa$ if it exhibits a plateau in the logarithmic layer. For
these reasons, some recent studies rely on $\beta$ to predict $\Delta y$ and
$u_{\tau}^{*}$ (Mizuno & Jiménez, 2011; Kuwata & Suga, 2017; Okazaki et al.,
2021, 2022). Breugem et al. (2006) and Suga et al. (2010) argued that the
slope of $U^{+}$ v.s. $\log(y_{*}^{+})$ must be constant within the
logarithmic layer, and the profile of $(y+\Delta y)dU/dy$ should therefore
exhibit a plateau with value $u_{\tau}^{*}/\kappa$. It is worth mentioning
that although the method and argument by Breugem et al. (2006) and Suga et al.
(2010) have been adapted by some recent studies, these studies actually
maximise the extent of the plateau in the entire flow, instead of just within
the logarithmic layer (e.g. figure 5 in Breugem et al. (2006), figure 6 in
Rosti & Brandt (2017), and figure 8 in Shen et al. (2020)). However, let us
illustrate using one of our present cases that there are instances when
enforcing a plateau within the logarithmic layer or maximising the extent of
the plateau may result in a breakdown of outer-layer similarity. Figure 5$(a)$
shows the diagnostic function of case C36550, which consists of closely packed
elements. A plateau in $\beta$ emerges within the logarithmic layer when
picking $\Delta y=0.05\delta=0.25h$, and the extent of this plateau maximises
when picking $\Delta y=0.1\delta=0.5h$. The resulting Kármán constant,
$\kappa_{c}\approx 0.32$ or $0.29$ respectively, is much smaller than the
smooth-wall value, $\kappa_{s}\approx 0.39$, implying a significant
modification of the outer-layer mean-velocity profile by the substrate. For
dense canopies like C36550, however, we would expect the overlying turbulent
flow to ’skim’ over the canopy tips, as the turbulent eddies are precluded
from penetrating within and interacting with the full height of the canopy, to
the point that the flow above the tips resembles a smooth-wall flow (Sharma &
García-Mayoral, 2020b; Brunet, 2020). Figure 5$(b)$ illustrates that the
scaling based on $\Delta y=0$ produces a smooth-wall-like diagnostic function
for case C36550, suggesting that the overlying flow indeed perceives an origin
in the vicinity of the canopy-tip plane. The underlying problem is that the
emergence of a logarithmic layer relies on a large Reynolds number,
$\delta^{+}$, such that the only available length scale in the overlap region,
$\nu/u_{\tau}\ll y\ll\delta$, is the wall-normal distance, $y$ (Townsend,
1976). Within this overlap region, only a dimensionless constant,
$\kappa=u_{\tau}/(y\partial U/\partial y)$, can be constructed (Luchini,
2017). However, if $\delta^{+}$ is not sufficiently large,
$u_{\tau}/(y\partial U/\partial y)$ need not exhibit a flat plateau, as shown
in figure 5. For smooth-wall flows, even $Re_{\tau}\approx 5000$ is not yet
sufficient for the diagnostic function to exhibit a completely flat plateau,
according to numerical evidence (Lee & Moser, 2015; Hoyas et al., 2022). The
local value of $\beta\approx 1/\kappa$ in the logarithmic layer exhibits a
dependence on $y/\delta$, or $y^{+}/Re_{\tau}$, caused by the contamination
from the wake above (Jiménez & Moser, 2007; Mizuno & Jiménez, 2011; Luchini,
2018). As shown in figure 5, even for smooth-wall flows, so long as
$Re_{\tau}\lesssim 5000$, enforcing a plateau in $\beta$ would overlook this
dependence, and result in nonphysical zero-plane displacements and values of
$\kappa$ down to 0.3, as those listed in the figure. The above results for
smooth walls suggest that artificially prescribing a plateau in $\beta$, while
not enforcing the similarity in the wake region, could result in an apparent
but false breakdown of outer-layer similarity and values for $\kappa$ that are
consistently lower than the true smooth-wall value, as those listed in table
1. It is worth mentioning that the logarithmic layer of a smooth-wall channel
is generally understood to span from $y^{+}\approx 80$ to $y/\delta\approx
0.3$. It is therefore not possible to define $\kappa$ meaningfully for flows
at Reynolds numbers $\delta^{+}\lesssim 300$, for which there is no
significant range of $y$ in which a logarithmic layer can manifest. Note that
some of the flows in table 1 fall in this low-$Re$ range.

To address the above issues, we propose to obtain $u_{\tau}^{*}$ and $\Delta
y$ by minimising the deviation between the smooth-wall and canopy diagnostic
function everywhere above the roughness sublayer, not just in the logarithmic
layer. It will be demonstrated in §3.3 and §3.4 that this method consistently
recovers outer-layer similarity.

###  3.3 Sensitivity of canopy diagnostic function

Outer-layer similarity can only be expected to appear above the roughness
sublayer, of height $y_{r}$, above which the canopy diagnostic function,
$\beta_{c}$, should be smooth-wall-like. Because the extent of roughness
effects can vary depending on the canopy density, as shown in figure 2,
$y_{r}$ needs to be determined separately for each canopy (Jiménez, 2004;
Brunet, 2020). Care must be taken, because if part of the roughness sublayer
is included in the region where outer-layer similarity is sought, the values
of $\Delta y$ and $u_{\tau}$ may be distorted. Therefore, the latter region
needs to be sufficiently far away from the wall, $y>y_{r}$, such that all
surface effects have vanished. We determine the values of $\Delta y$ and
$u_{\tau}$ that recover a smooth-wall-like diagnostic function, or mean
velocity profile, as those that minimise the deviation between $\beta_{s}$ and
$\beta_{c}$, the smooth-wall and canopy diagnostic functions, above $y_{r}$.
By this method, we recover a smooth-wall-like $\beta_{c}$ in the outer layer,
including both the logarithmic layer and the ’wake’ region.

As an example, for case C144550, figure 7 portrays $\beta_{c}$ scaled with
$y_{*}$ and $u_{\tau}^{\star}$, the friction velocity decoupled from
$y_{*}=0$, based on three tentative values of $y_{r}$. For
$y_{r}=0.074\delta=0.37h=0.28s$, $\beta_{c}$ is not smooth-wall-like, even if
the deviation between $\beta_{c}$ and $\beta_{s}$ above $y_{r}$ is minimised.
This is because the flow at this $y_{r}$ is still directly perturbed by the
canopy elements, as $y_{r}$ is too small compared to $s$, which provides an
estimate of the roughness sublayer thickness, as evidenced in figure 2. In
turn, adopting $y_{r}=0.274\delta=1.37h=1.04s$ overestimates the roughness
sublayer thickness, as $\beta_{c}$ is already smooth-wall-like based on
$y_{r}\gtrsim 0.174\delta=0.87h=0.66s$, implying that the height affected by
the canopy in this case is lower than $y_{r}/s\approx 2-3$ reported in
Abderrahaman-Elena et al. (2019) and Sharma & García-Mayoral (2020b). For this
canopy, we can conclude that outer-layer similarity is recovered when
$y_{r}\approx 0.66s$, and beyond this limit $\beta_{c}$ and $\Delta y$ are
insensitive to the lower bound from which outer-layer similarity is enforced,
as illustrated in figure 7 and figure 7$(b)$. The r.m.s. deviation between
$\beta_{c}$ and $\beta_{s}$ above $y_{r}$ is minimised if $y_{r}$ is
sufficiently far away from the canopy, as shown in figure 7$(a)$.

$(a)$$(b)$

Figure 6: Diagnostic function of case C144550 scaled with $\Delta y$ and
$u_{\tau}^{\star}$, the friction velocity decoupled from $y_{*}=0$, that
minimise the r.m.s. deviation from the smooth-wall profile above $y_{r}$.  ,
reference smooth-wall profile at $Re_{\tau}\approx 550$;  ,  ,  , canopy
statistics based on  $y_{r}=0.074\delta$,  $y_{r}=0.174\delta$, and
$y_{r}=0.274\delta$, respectively.

$(a)$$(b)$

Figure 7: $(a)$ R.m.s. deviation between $\beta_{s}$ and $\beta_{c}$ above
$y_{r}$ and $(b)$ zero-plane displacement, $\Delta y$, versus $y_{r}$ for case
C144550. The horizontal dashed lines mark once and twice the baseline error;
and the thick black vertical markers denote the lower and upper bounds for
$y_{r}$ above which outer-layer similarity recovers. The vertical dashed lines
are as in figure 7.

In the above, we have let $\kappa_{c}=\kappa_{s}$ and $u_{\tau}^{\star}$ be
independent of $\Delta y$, and we obtain the values of $\Delta y$ and
$u_{\tau}^{\star}$ for each $y_{r}$ by minimising the difference between
$\beta_{c}$ and $\beta_{s}$ for any $y$ above $y_{r}$. However, if $y_{r}$ is
not carefully chosen, the corresponding $\Delta y$ and $u_{\tau}^{\star}$ may
not result in a smooth-wall-like $\beta_{c}$ above $y_{r}$, as illustrated in
figure 7. Therefore, we need to identify an appropriate lower bound for
$y_{r}$ to correctly assess outer-layer similarity, sufficiently far away from
the wall for all surface effects to vanish. Starting at too low values, as
$y_{r}$ increases, the r.m.s. deviation of $\beta$ in figure 7$(a)$ decreases
and eventually stabilises at a baseline-error level, which may be used to
guide the selection of the lower bound of $y_{r}$. Here, we propose a lower
bound of $y_{r}$ such that the r.m.s. error is twice the baseline error. Above
this lower bound, the deviation between $\beta_{c}$ and $\beta_{s}$ is small,
suggesting that $\beta_{c}$ scaled with $y_{*}$ and $u_{\tau}^{\star}$ is
essentially smooth-wall-like. On the other hand, $y_{r}$ should not be adopted
excessively far above the surface, because outer-layer similarity cannot be
properly examined on too small a portion of the flow. We propose an upper
bound for $y_{r}$ that is $0.1\delta$ above the lower bound, as the r.m.s. of
$\beta_{c}-\beta_{s}$ varies little beyond this.

$(a)$$(b)$$(c)$$(d)$

Figure 8: Mean-velocity profiles of the canopy and smooth-wall flows. $(a,c)$,
full-channel cases with less intrusive canopies, C36550 to C144550 and C216900
to C432900; $(b,d)$, open-channel cases. The coloured solid and dash-dotted
lines represent cases as in in figure 2. The black solid lines represent
smooth-wall reference profiles at $Re_{\tau}\approx 900-1000$, and the black
dashed lines represent smooth profiles at $Re_{\tau}\approx 550$. The square
markers represent the lower bound of $y_{r}$ for cases at $Re_{\tau}\approx
550$, and the triangle markers represent that for those at $Re_{\tau}\approx
900-1000$.

Using the method we propose, $y_{r}$ results in $\Delta y$ and
$u_{\tau}^{\star}$ that consistently recover outer-layer similarity, with the
resulting $\Delta y$ being insensitive to the particular choice of $y_{r}$
within the range proposed above, as evidenced by the flat region in Figure
7$(b)$. As shown in figure 7$(a)$, the height above which outer-layer
similarity recovers for case C144550 is $y_{r}=(0.174\pm 0.05)\delta=(0.87\pm
0.25)h=(0.66\pm 0.19)s$. This is smaller than the typical roughness sublayer
thickness. which extends up to $2-3h$ above the tips (Jiménez, 2004; Brunet,
2020). Nevertheless, figure 2 illustrates that the dominant length scale for
the intensity of the texture-coherent flow is the element spacing, and these
texture-coherent flows essentially vanish at one spacing above the tips. For
intermediate to dense cases with $\lambda\gtrsim 0.1$, where $s/h\lesssim 1$,
we observe the recovery of outer-layer similarity for $y_{r}\approx s\lesssim
h$. Additionally, because $y_{r}/\delta$ is smaller than the typical upper
bound of a logarithmic layer, $y\approx 0.3\delta$, we expect the recovery of
a smooth-wall-like logarithmic layer. Based on the confidence interval for
$y_{r}$, we obtain the zero-plane displacement, $\Delta y=(0.113\pm
0.005)\delta=(0.57\pm 0.03)h$, as shown in figure 7$(b)$, indicating that the
outer-layer flow perceives an origin $y_{*}=0$ at a depth roughly half-way
between the floor and the canopy tips. However, the height of
$u_{\tau}^{\star}$, corresponding to a smooth-wall-like $\kappa$, is
$y_{u_{\tau}^{\star}}=(0.553\pm 0.036)\delta=(2.77\pm 0.18)h$, which is below
the canopy floor (note that $y_{u_{\tau}^{\star}}$ is decoupled from
$y_{*}=0$).

The fact that outer-layer similarity can only be assessed above a minimum
height $y_{r}$ implies that such similarity cannot be assessed for certain
flows/substrates. In figures 8$(c,d)$ and 9$(c)$, we observe a breakdown of
outer-layer similarity for the open-channel cases O400550 ($y_{r}\geq
0.636\delta$) and O8001000 ($y_{r}\geq 0.274\delta$). This is because these
intrusive canopies perturb the overlying flow extensively, well into the
region where a logarithmic layer would otherwise have been found. In contrast,
for intermediate to dense cases, C36550 to C144550, C216900 to C432900, and
O4001000, which have less intrusive canopies, the canopy flow becomes smooth-
wall-like no higher than $y_{r}=0.224\delta$, allowing for the recovery of a
smooth-wall-like outer layer. The canopy diagnostic functions in figure
9$(a,b)$ also show that whether outer-layer similarity recovers depends on the
extent of the roughness layer, which is associated with $y_{r}$. In contrast,
prescribing a constant slope for $U^{+}$ v.s. $\log(y_{*})$, or enforcing a
plateau in $\beta_{c}$, can consistently result in a breakdown of outer-layer
similarity and a Kármán constant different from the smooth-wall one, as
illustrated in figures 9$(c,d)$.

$(a)$$(b)$$(c)$$(d)$

Figure 9: Diagnostic functions of the canopy and smooth-wall flows. $(a,c)$,
full-channel cases; $(b,d)$, open-channel cases. $(a,b)$ enforce smooth-wall-
like $\beta_{c}$ above $y_{r}$; $(b,d)$ enforce plateaus in $\beta_{c}$. The
lines and markers are as in figure 8.

###  3.4 Universality or non-universality of the Kármán constant

In the above, the friction velocity, $u_{\tau}^{\star}$, was not computed from
the stress at the zero-plane-displacement height, $y_{*}=0$, and was instead
set as an independent variable, such that $\beta_{c}$ was smooth-wall-like and
$\kappa$ was universal, $\kappa_{c}=\kappa_{s}$. Alternatively, the friction
velocity, $u_{\tau}^{*}$, could be computed from the stress at $y_{*}=0$, and
$\kappa_{c}$ found by enforcing a match of the diagnostic function to that of
a smooth-wall profile. In the logarithmic layer, equation (2) can be expressed
as

$(a)$$(b)$

Figure 10: $(a)$ Modified diagnostic function, $\kappa\beta$, and $(b)$
Reynolds shear stress of case C144550.  , reference smooth-wall profiles at
$Re_{\tau}\approx 550$;  , canopy profile scaled with $u_{\tau}^{\star}$ and
$y_{*}$;  , canopy profile scaled with $u_{\tau}^{*}$ and $y_{*}$.

$(a)$$(b)$$(c)$$(d)$

Figure 11: Pre-multiplied spectral energy densities for case C144550 (line
contours) and reference smooth-wall case C550 (filled contour) normalised by
their respectively r.m.s. level at $y_{*}=0.3\delta$.  , canopy statistics
scaled with $u_{\tau}^{\star}$ and $\nu$;  , canopy statistics scaled with
$u_{\tau}^{*}$ and $\nu$. The contours are at 0.01, 0.05 and 0.1 times the
r.m.s. level.

$\displaystyle
y_{*}\frac{dU}{dy_{*}}\frac{\kappa_{s}}{u_{\tau}^{\star}}=\kappa_{s}\beta(y_{*},u_{\tau}^{\star})$
$\displaystyle\approx 1,$ (11) $\displaystyle
y_{*}\frac{dU}{dy_{*}}\frac{\kappa_{c}}{u_{\tau}^{*}}=\kappa_{c}\beta(y_{*},u_{\tau}^{*})$
$\displaystyle\approx 1,$ (12)

from which we can observe that the group $\kappa/u_{\tau}$ needs to have a
certain value, but the mean-velocity profile does not provide information on
whether this should be achieved by fixing $\kappa$ and finding $u_{\tau}$, or
vice versa, or else. As long as
$\kappa_{c}/u_{\tau}^{*}=\kappa_{s}/u_{\tau}^{\star}$, the modified
logarithmic profile in equation (12) is recovered. Figure 10$(a)$ shows that
whether $u_{\tau}^{\star}$ or $\kappa_{c}\neq\kappa_{s}\approx 0.4$ is set as
an independent parameter, the resulting mean-velocity profile or diagnostic
function is still smooth-wall-like. In this section, we discuss the
implications of fixing either $u_{\tau}$ or $\kappa$. While it is not possible
to identify which option provides a complete outer-layer similarity by
inspecting just the mean-velocity profile, other turbulent statistics collapse
with the smooth-wall data when the friction velocity is evaluated at the zero-
plane-displacement height. As an example, the Reynolds shear stress profile in
figure 10$(b)$ collapses to smooth-wall data when scaled with $u_{\tau}^{*}$,
compared to $u_{\tau}^{\star}$. If, alternatively, a universal $\kappa_{c}$ is
enforced, the Reynolds shear stress would differ from smooth-wall values by a
factor $(u_{\tau}^{*}/u_{\tau}^{\star})^{2}$. The spectral density maps in
figure 11 also show that outer-layer similarity is achieved with
$u_{\tau}^{*}$ obtained from the shear at the zero-plane-displacement height,
$y_{*}=0$, and $\kappa_{c}$ having a non-smooth-wall value.

$(a)$$(b)$$(c)$$(d)$$(e)$$(f)$

Figure 12: R.m.s. velocity fluctuations scaled with $y_{*}$ and
$u_{\tau}^{*}$. $(a,c,e)$, full-channel cases; $(b,d,f)$, open-channel cases.
Lines are as in figure 8.

$(a)$$(b)$

Figure 13: Reynolds shear stress scaled with $y_{*}$ and $u_{\tau}^{*}$.
$(a)$, full-channel cases; $(b)$, open-channel cases. The lines are as in
figure 8.

$(a)$$(b)$$(c)$$(d)$$(e)$$(f)$$(g)$$(h)$$(i)$$(j)$$(k)$$(l)$

Figure 14: Pre-multiplied spectral energy densities for canopy-flow (line
contours) and reference smooth-wall cases (filled contours) normalised by
their respective r.m.s. level at $y_{*}=0.3\delta$. $(a-d)$ represent the
full-channel cases; and $(e-h)$ and $(i-l)$ represent the open-channel cases.
The line colour scheme is as in figure 8. The contours are at 0.01, 0.05 and
0.1 times the mean-square fluctuation level.

$(a)$$(b)$

Figure 15:  $(a)$ Ratio of zero-plane-displacement depth to canopy height,
$-\Delta y/h$, and $(b)$ Kármán constant, $\kappa_{c}$, versus canopy density,
$\lambda_{f}$, for all cases considered. Markers $\bigcirc$, $\ast$, $\times$,
and $\Delta$ represent full-channel cases at $Re_{\tau}\approx 550$ and
$Re_{\tau}\approx 900$ and open-channel cases at $Re_{\tau}\approx 550$ and
$Re_{\tau}\approx 1000$, respectively. The colour scheme is as in figure 2.
Transparent markers represent cases with substantial texture-induced coherent
flows in the outer layer, and coloured markers represent cases with a smooth-
wall-like outer layer. In $(a)$,  represents $\Delta y$ corresponding to zero-
plane-displacement height at the canopy tip ($-\Delta y/h=0$) and at the floor
($-\Delta y/h=-1$). In $(b)$,  represents $\kappa_{s}\approx 0.39$.

For the cases with less intrusive canopies, C36550 to C144550, C216900 to
C432900, and O4001000, the r.m.s. velocity fluctuations, Reynolds shear stress
and pre-multiplied spectral energy densities scaled with $u_{\tau}^{*}$
essentially collapse with their respective smooth-wall values above a height
$y_{*}\approx 0.3\delta$, as shown in figure 12, 14 and 14. This suggests that
$u_{\tau}^{*}$, and not $u_{\tau}^{\star}$, is the velocity scale for the
outer-layer turbulence. However, for the more intrusive cases, for example
O8001000, the roughness sublayer extends well into the overlying flow, which
no longer exhibits a smooth-wall-like logarithmic layer, as evidenced by the
footprint of the element-induced flow in figure 14$(i-l)$. These intrusive
textures cause a more in-depth modification of the outer-layer flow, and can
lead to the breakdown of outer-layer similarity. Similar observations have
been reported on large riblets and porous walls, where outer-layer similarity
is limited if surface effects penetrate into a significant portion of $\delta$
(Breugem et al., 2006; Manes et al., 2011; Endrikat et al., 2022).

By minimising the r.m.s. deviation of $\beta_{c}$ from $\beta_{s}$, as
discussed in §3.3 & 3.4, we obtain the values of $\Delta y$ and $u_{\tau}^{*}$
that provide the scales for the outer-layer turbulent flow. The trend of the
zero-plane displacement in figure 15$(a)$ suggests that the flows over dense
canopies perceive an origin close to the tips, and that this origin becomes
deeper as the canopy density decreases. In figure 15$(b)$, the densest case
has a smooth-wall-like Kármán constant, $\kappa_{c}\approx 0.4$, and the value
of $\kappa_{c}$ decreases as the canopy density decreases. However, for the
sparsest cases, $\kappa_{c}$ appears to tend back to its smooth-wall value. We
note that the decreases in $\kappa_{c}$ for all canopies in this study never
exceeds $15\%$ of its smooth-wall value, which is significantly smaller than
obtained by fitting $U^{+}$ to a $\log(y_{*}^{+})$ profile as in Breugem et
al. (2006), Suga et al. (2010), Rosti & Brandt (2017), Kuwata & Suga (2017),
Kazemifar et al. (2021), and Okazaki et al. (2021, 2022), where the values of
$\kappa_{c}$ reported were as low as $\kappa_{c}\approx 0.2$.

The present results for dense and sparse canopies are consistent with the
results in Nepf (2012a), Brunet (2020) and Chung et al. (2021), who summarised
that for roughness in the dense or sparse regime, the zero-plane-displacement
height approaches the roughness crest or trough, respectively. For instance,
for the dense canopy C36550, $\lambda_{f}\approx 2.04$, the smooth-wall-like
overlying flow perceives an origin at the tips, and the Kármán constant has a
smooth-wall value, $\kappa_{c}\approx\kappa_{s}\approx 0.39$, as illustrated
in figure 15. MacDonald et al. (2016) and Sharma & García-Mayoral (2020b) have
also observed such skimming flow for closely-pack sinusoidal roughness and
filament canopies, where $s^{+}\lesssim 3$. Particularly, in the dense regime,
where the spacing between elements is comparable to the viscous length scale,
$\nu/u_{\tau}$, the overlying turbulence is essentially precluded from
penetrating the roughness, and the unmodified overlying flow is smooth-wall-
like with a zero-plane-displacement height at the tips.

Sparse canopies, where the ratio between element spacing and height is large,
are generally more intrusive than the intermediate to dense canopies because
the roughness sublayer can extend to $y/s\approx 1$ into the channel, as
discussed in §3.1. For the sparse and tall canopies, C216550, C288550,
C432550, O400550 and O8001000, the surface effects penetrate deep into the
overlying flow and can reach the channel centre, impeding the assessment of
outer-layer similarity. For these substrates, outer-layer similarity could
only be assessed at higher $Re_{\tau}$. This motivated us to conduct
simulations C216900, C288900, C432900, O4001000, for which the canopies have
similar dimensions to C216550, C288550, C432550, O400550, in inner units, but
such that the unperturbed core flow is a larger portion of the channel. As
illustrated in figure 15, the larger-$Re_{\tau}$ flows allow for a clearer
assessment of outer-layer similarity, with zero-plane-displacement height at
the floor and $\kappa_{c}\approx\kappa_{s}\approx 0.39$. Our observations are
consistent with those in Poggi et al. (2004), Nepf et al. (2007) and Sharma &
García-Mayoral (2020a), where the flows over sparse canopies exhibit
characteristics of smooth-wall flows, with reference wall at the canopy floor.

Flows over canopies with an intermediate density ($\lambda_{f}\gtrsim 0.1$)
perceive an origin between the tips and floor, as shown in figure 15$(a)$. For
these intermediate canopies, with $s/h\lesssim 1$, the overlying flow
interacts mainly with the upper part of the obstacles and turbulence does not
penetrate all the way to the floor (Grimmond & Oke, 1999; Luhar et al., 2008;
MacDonald et al., 2018; Sharma & García-Mayoral, 2020b). These intermediate-
canopy flows exhibit values of the Kármán constant $\kappa_{c}\approx
0.34-0.36$, different from the smooth-wall value $\kappa_{s}\approx 0.39$,
implying that the intermediate canopies disrupt the overlying flow more
profoundly than both dense and sparse substrates. Nevertheless, other than for
the change in $\kappa$, the turbulent statistics remain essentially smooth-
wall-like in the logarithmic layer and above, as depicted in figures 8, 9, and
12-14. This suggests a modified outer-layer similarity, where $\kappa_{c}\neq
0.39$, but where otherwise the turbulence is outer-layer-similar to a smooth-
wall flow. For $h\ll\delta$, we could expect that the flow far from the wall
is not influenced by the details of the surface topology, as is the classical
view of wall turbulence (Clauser, 1956). Further work is required to study if
the same canopies would also exhibit $\kappa_{c}\neq 0.39$ at larger
$\delta/h$ ratios, say $\delta/h\approx 40$ as proposed by Jiménez (2004).

##  4 Conclusions

In the present work, we have assessed outer-layer similarity in flows over
canopies ranging from sparse to dense ($\lambda_{f}\approx 0.01-2.03$) at
$Re_{\tau}\approx 550-1000$. We have discussed and appraised the conventional
methods for the assessment of outer-layer similarity, showing that these
methods could be inaccurate for flows over dense roughness and for flows at
moderate $Re_{\tau}$.

To investigate outer-layer turbulence, we first determined the depth of the
roughness sublayer, within which the turbulence is significantly disrupted by
the element-induced flow. It was shown that the roughness sublayer for the
present flows extends to a height equal to the element spacing. For the cases
with tall ($h\approx 0.2\delta$) and sparse ($\lambda_{f}\ll 0.1$) elements,
C216550, C288550, C432550, O400550, and O8001000, the element-induced flows
penetrate effectively into the channel as far as $y\approx 0.5\delta$, or even
above. As a result, these intrusive canopies leave only a small portion of
core flow unperturbed, making the assessment of outer-layer similarity
difficult. To verify whether outer-layer similarity can recover for these
intrusive cases, simulations at higher $Re_{\tau}$, producing a larger
unperturbed region, where required.

Conventionally, outer-layer similarity for a flow over roughness is recovered
by imposing a zero-plane displacement and the corresponding friction velocity
on the mean-velocity profile, such that the logarithmic layer is smooth-wall-
like. We have discussed some caveats of conventional drag-based and mean-
velocity-based methods used to determine these constants. For drag-based
methods, and for densely packed roughness elements, the point of action of the
total drag is located within the elements, while the overlying flow actually
perceives an origin in the vicinity of the tips. For mean-velocity-based
methods, we have shown that at moderate $Re_{\tau}$ even smooth-wall flows do
not exhibit a constant slope for $U^{+}$ v.s. $\log(y_{*}^{+})$, or a plateau
in $\beta$. Therefore, matching the shape of $U^{+}$ to a $\log(y_{*}^{+})$
or, equivalently, enforcing a plateau in the diagnostic function, $\beta$, may
result in an artificial breakdown of outer-layer similarity and spurious
predictions for the zero-plane displacement and the friction velocity at any
but the highest $Re_{\tau}$ available in DNS literature.

To obtain the zero-plane displacement and friction-velocity scale that
recovers a smooth-wall-like diagnostic function, we minimise the deviation
between the canopy and smooth-wall diagnostic function everywhere above the
roughness sublayer, instead of in the logarithmic layer alone. By this method,
we obtain a smooth-wall-like $\beta$ not only in the logarithmic layer, but
also in the wake region, enforcing outer-layer similarity in its full sense.
We also explore the possibility of the zero-plane displacement and the
friction velocity being set independently, but find that outer-layer
similarity is more consistently recovered when they are coupled. For the dense
canopies like C36550, the unmodified overlying flow is smooth-wall-like with a
zero-plane-displacement height at the tips, and the Kármán constant has a
smooth-wall value $\kappa_{c}\approx\kappa_{s}\approx 0.39$. This is the case
because the skimming turbulence is precluded from penetrating into the
obstacles, and perceives an origin at the tips. For sparse canopies
($\lambda_{f}\ll 0.1$), the higher-$Re_{\tau}$ cases, C216900, C288900,
C432900, and O4001000, allow for the assessment of outer-layer similarity,
with zero-plane-displacement height at the floor, and
$\kappa_{c}\approx\kappa_{s}\approx 0.39$. For intermediate canopies
($\lambda_{f}\gtrsim 0.1$), with $s/h\lesssim 1$, the overlying flow interacts
mainly with the upper part of the obstacles, and turbulence does not penetrate
all the way to the floor. These intermediate-canopy flows perceive a zero-
plane-displacement height between the tips and floor, and exhibit values of
the Kármán constant $\kappa_{c}\approx 0.34-0.36$ different from the smooth-
wall value $\kappa_{s}\approx 0.39$, implying that the intermediate canopies
disrupt the overlying flow more profoundly than both dense and sparse
substrates. Nevertheless, other than for the change in $\kappa$, the turbulent
statistics remain essentially smooth-wall like in the logarithmic layer and
above.

## Acknowledgements

This work was partially supported by EPSRC grant EP/S013083/1. For the purpose
of open access, the authors have applied a Creative Commons Attribution (CC
BY) licence to any Author Accepted Manuscript version arising from this
submission. Computational resources were provided by the ‘Cambridge Service
for Data Driven Discovery’ operated by the University of Cambridge Research
Computing Service and funded by EPSRC Tier-2 grant EP/P020259/1 (projects:
cs066 and cs155), and by the ’ARCHER2’ system in the UK, funded by PRACE
(project: pr1u1702) and EPSRC (project: e776). All research data supporting
this publication is directly available within this publication.

## Declaration of interests

The authors report no conflict of interest.

## Appendix. Grid independence and validation

$(a)$$(b)$$(c)$

Figure 16: Instantaneous realisations of the (a) streamwise, (b) wall-normal
and (c) spanwise velocities in a plane intersecting the canopy elements for
case C54550, scaled with $u_{\tau}$ evaluated at the canopy tips. The darkest
red and blue contours represent intensities of $\pm 1.0$, respectively.

In this study, the no-slip condition within the canopy elements is enforced
using the direct-forcing, immersed-boundary method (Iaccarino & Verzicco,
2003) as implemented in García-Mayoral & Jiménez (2011) and modified by Sharma
& García-Mayoral (2020a, b). This method applies a body force term to the
right-hand side of equation (6) and drives the velocity at the immersed-
boundary points to zero. The reader is referred to the works of Sharma &
García-Mayoral (2020b) and Sharma (2020) for a detailed discussion on the
accuracy of the immersed-boundary method implemented and wall-parallel
resolution used. In the present study, the velocity within the rigid canopy
elements is observed to be less than $0.1u_{\tau}$, with $u_{\tau}$ evaluated
at the canopy tips, for all the DNSs conducted. As portrayed in figure 16, the
velocity at the ’solid’ points within the obstacles is significantly smaller
than that at the surrounding ’fluid’ points, illustrating that the immersed-
boundary method resolves the canopy topology.

To analyse grid convergence, we have carried out three DNSs for case C108550
with different wall-normal resolutions, as portrayed in figure 17. Compared to
smooth-wall flows, where the maximum mean shear occurs at the wall,
$dU^{+}/dy^{+}=1$, in our flows it occurs at the canopy-tip plane and is not
as high, in the present case $dU^{+}/dy^{+}\approx 0.16$, peaking again at the
floor at $dU^{+}/dy^{+}=0.1$. Since the local shear is the driver of
turbulence, the wall-normal resolution required for the direct simulation of
these flows is thus lower than for smooth-wall flows in terms of $\Delta
y^{+}$, and can be adjusted following the local shear, as shown in figures 17
and 18$(b)$. Consistent with this discussion, let us also note that the
resolution at the floor in all of our simulations, for the case portrayed
(C108550) $\Delta y^{+}\approx 2$, is actually $\Delta y^{+}\approx 0.3$ when
scaled with the local, dynamically relevant friction velocity proposed in
Sharma & García-Mayoral (2018, 2020b). Figure 18 portrays the results obtained
for the resolution used in this paper plus one finer and one coarser. All
results collapse, evidencing grid convergence.

Figure 17: Wall-normal resolution, scaled with $\nu$ and $u_{\tau}$ evaluated
at the canopy tips, for case C108550 for a fine resolution  , a standard
resolution  , and a coarse resolution  .

$(a)$$(b)$$(c)$$(d)$$(e)$$(f)$

Figure 18: Turbulent statistics based on $\nu$ and $u_{\tau}$ evaluated at the
canopy tips for case C108550. The colour scheme is as in figure 17.

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