12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

ASSESSMENT OF OUTER-LAYER SIMILARITY IN TURBULENT FLOWSOVER CANOPIES

Zishen Chen
Department of Engineering
University of Cambridge
Trumpington Street, Cambridge, CB2 1PZ, UK
zc328@cam.ac.uk

Ricardo Garcı́a-Mayoral
Department of Engineering
University of Cambridge
Trumpington Street, Cambridge, CB2 1PZ, UK
r.gmayoral@eng.cam.ac.uk

ABSTRACTapplications.
The surface topology has a direct impact on the flow
within the ’roughness sublayer,’ which can extend up to 2 − 3roughness heights or spacings above the roughness crests, depending on the density regime (Jiménez, 2004; Brunet, 2020).
Above this height, it is widely accepted that the turbulenceis essentially undisturbed and exhibits outer-layer similarity(Clauser, 1954; Townsend, 1976). The only effect is then aconstant shift, ∆U + , in the mean velocity profile, while boththe Kármán constant, κ ≈ 0.4, and the ‘wake’ region are unaffected. 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, τw , or the friction velocity, uτ = (τw /ρ)1/2 , andthe characteristic length scale provided by the distance to thewall, y. However, whether surface effects extend beyond the’roughness sublayer’ and into the outer layer is still a topic ofdebate as conflicting evidence exists for some surfaces (Wu &Christensen, 2007; Flack & Schultz, 2010). Experimental andnumerical studies for turbulent flows over porous media carried out by Breugem et al. (2006), Suga et al. (2010), Manes
et al. (2011), and Okazaki et al. (2021) have reported the lossof outer-layer similarity and values for κ very different fromκ ≈ 0.4 at Reτ ≈ 400−5800, suggesting a more in-depth modification of the flow by the substrates.
The canonical logarithmic form of the mean velocity profile for a turbulent flow is

Turbulent flows over rough and complex surfaces generally exhibit a similar character to those over smooth walls sufficiently far above the surface (Townsend, 1976). The innerlength and velocity scales for these flows are typically evaluated at the reference height, y∗ = 0, used as the virtual origin to
obtain a smooth-wall-like logarithmic layer. However, recent
studies have reported the loss of this outer-layer similarity inturbulence over highly obstructing layouts, such as canopies
and permeable substrates (Okazaki et al., 2021; Manes et al.,2011). In this study, we probe the existence of outer-layer similarity for a series of direct numerical simulations of canopies
of varying density. By the assessment of the diagnostic function of the mean velocity profile, it is shown that as canopy
density decreases, the overlying turbulent flow perceives a
deeper virtual origin into the canopy. Based on the lengthand velocity scales provided by the virtual origin, the turbulent flow over the densest canopy with λ f ≈ 2.04 is essentiallysmooth-wall-like. However, canopies with λ f ≈ 0.01 − 0.91
result in values of Kármán constant, κ ≈ 0.34 − 0.43, different
from the smooth-wall value, κ ≈ 0.4, while turbulent statistics
are essentially smooth-wall-like in the logarithmic layer andabove. This suggests a modified outer-layer similarity whereκ is not 0.4, but otherwise the canopy flows are outer-layer
similar to a smooth-wall flow.

Introduction
Turbulent boundary layers over rough and complex surfaces are ubiquitous and are of significant environmental andindustrial interest (Jiménez, 2004; Finnigan, 2000). Studies
of wall-bounded turbulence have provide the tools for understanding these rough-wall flows, with engineering models
that treat roughness as a small perturbation to the smoothwall boundary layer (Flack et al., 2007; Wu & Christensen,
2007). However, if the roughness-induced perturbation propagates into the outer layer, the scaling based on smooth-wall
similarity could produce erroneous results. Understanding theextent of roughness effects and whether this smooth-wall similarity holds true is therefore of great importance to various

U+ =1
log(y+ − ∆y+ ) + A − ∆U + ,κ(1)where κ is the Kármán constant and κ ≈ 0.4 if outer-layer
similarity recovers, A is the log-law intercept for a smoothwall flow, ∆U + is the velocity deficit caused by the substrate,and ∆y+ is the zero-plane displacement that recovers outerlayer similarity for the mean velocity profile, U + . However,recent studies suggest that the scaling for wall turbulence isessentially local, and is given by the local mean shear and pro-1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
(a)

(b)

Figure 1. Schematics of (a) the numerical domain and (b) a zoomed cross-section for case D54. An instantaneous realisation of thestreamwise velocity is shown in the orthogonal planes.
Table 1. Simulation parameters: Reτ = uτ δ /ν is the friction
Reynolds number based on δ = 1, the half-channel height, uτ
calculated from the total shear at the canopy tips, and kinematic viscosity, ν; λ f is the frontal density; h and s are thecanopy height and streamwise and spanwise spacing; ∆+
x and
+
+
∆+
z are the streamwise and spanwise resolutions; ∆y, f and ∆y,t
are the wall-normal resolutions at the floor and canopy tips.

duction rate of turbulent kinetic energy, with no explicit reference to the wall-normal distance (Lozano-Durán & Bae, 2019;
Tuerke & Jiménez, 2013). This implies that the traditional
scaling based on y and uτ happens to hold because of the oneto-one correspondence between them and the local production
and shear, but this correspondence does not need to hold necessarily for every flow. As part of this work, we have investigated, for flows that exhibit an apparent loss of outer-layer similarity, whether the local scale can still have correspondence toa friction velocity, u⋆τ , and a length scale, y∗ where y∗ is zeroat the height of zero-plane displacement, but u⋆τ is not necessarily evaluated at y∗ = 0. Here, superscript (·)⋆ denotes wall
units defined by ν and u⋆τ decoupled from y∗ = 0, and superscript (·)∗ denotes wall units defined by ν and u∗τ evaluated aty∗ = 0. Subscript (·)∗ denotes outer units that are normalisedby the bulk velocity and outer length scale, y∗ , measured fromthe virtual origin.
The diagnostic function of the mean velocity profile is(Luchini, 2018)+
β (y+∗ ) = y∗

∂U +
,
∂ y+
∗

(2)

+
where y+∗ = (y−∆y) is the wall-normal distance from the virtual origin at y∗ = 0, which would exhibit a plateau β ≈ 1/κ inthe logarithmic layer, if outer-layer similarity recovered. Many
previous studies rely on the existence of this plateau to determine the extent of the logarithmic layer and the inner scaling inflows over roughness (Suga et al., 2010; Breugem et al., 2006).
Particularly, the logarithmic law in Eq.(1) can be recoveredby choosing a ∆y that yields a plateau in (y − ∆y)dU/dy withvalue u∗τ /κ. The inner velocity and length scales are then determined based on u∗τ evaluated at the reference height, y∗ = 0.
However, a logarithmic layer with a plateau in β emerges
only in a flow at high Reτ . According to numerical evidence,Reτ ≈ 5200 is not yet high enough for the diagnostic function of a wall-bounded turbulent flow to exhibit a completely
flat plateau (Lee & Moser, 2015; Yamamoto & Tsuji, 2018;
Luchini, 2018). Additionally, outer-layer similarity, by definition, refers to the similarity in not just the logarithmic layer
but also the ’wake’ region. For a flow at limited Reτ , neglecting similarity in the ’wake’ region while enforcing a plateau
in the diagnostic function could therefore result in erroneouspredictions of parameters including ∆y, u∗τ , and κ .
In this study, we determine the zero-plane displacement,∆y, by minimising the deviation of the diagnostic function in
the logarithmic layer and above, compared to that of a smoothwall flow at roughly the same Reτ . We assess the validity asa scaling velocity of the friction velocity both measured at the

Case

Reτ

λf

s/h

∆+
x

∆+
z

∆+
y, f

∆+
y,t

S

550.6

-

-

9.01

4.50

0.27

-

D36

550.3

2.04

0.33

2.00

2.00

2.95

0.5

D54

550.1

0.91

0.49

3.00

3.00

2.53

0.50

I72

551.9

0.51

0.65

4.01

4.01

2.01

1.00

I108

548.5

0.23

0.98

5.98

2.99

2.00

1.00

I144

547.0

0.13

1.31

5.97

2.98

1.19

1.19

S216

549.0

0.06

1.96

5.99

2.99

1.00

1.00

S288

547.9

0.03

2.62

5.98

2.99

0.79

0.79

S432

548.9

0.01

3.93

5.99

2.99

0.60

0.60

height of zero-plane displacement and set as an independent,
free parameter. With this, we probe the existence of outer-layer
similarity in turbulence over canopies of varying density.

Direct Numerical Simulations
We carry out a series of direct numerical simulations(DNSs) of closed channels with canopies of rigid filaments
covering the walls at moderate Reynolds numbers (Reτ ≈ 550).
The size of the numerical domain is Lx × Lz × Ly = 2πδ ×
πδ × 2(δ + h), where δ is the half-channel height and h is
the canopy height. We vary the canopy density by changing the spacing between elements, resulting in frontal densities, λ f ≈ 0.01−2.04 (Nepf, 2012; Sharma & Garcı́a-Mayoral,
2020a,b). All canopies considered consist of collocated prismatic posts with sides lx+ = lz+ ≈ 24 and height h+ ≈ 110, as
depicted in Figure 1. Relevant simulation parameters are listed
in Table1. Case S is a reference smooth-wall channel. For
the canopy simulations, letters ’D’, ’I’, and ’S’ denote dense,
intermediate, and sparse configurations (Nepf, 2012; Brunet,2020), and the number that follows denotes the approximate
spacing, s+ , between the canopy elements.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
6

0.4

0

0.3

-0.2

0.2

-0.4

0.1

-0.6

[

]

4

[

2
[

0

]

0
10

-1

10

0.1

0

0.2

]

-0.8
0.3

0.4

0.1

0.2

0.3

0.4

Figure 2. (a) Canopy diagnostics function for case I144 scaled with y∗ and u⋆τ that minimise the r.m.s. deviation from the smooth-wall
profile above a critical height.
, smooth-wall statistics at Reτ ≈ 550;,,
, canopy statistics based on, ycrit = 0.098;, ycrit = 0.173;, ycrit = 0.248. The wall-normal coordinate for the shifted profiles is y∗ = (y − ∆y)/(1 − ∆y). (b), r.m.s.
deviation between βs and βc ;, once and twice the baseline error; [ and ], bounds for ycrit . (c), zero-plane displacement, ∆y;, height where u⋆τ is evaluated at, yu⋆τ .

3

2
1.5

2
1
1
0.5
0

0
0

0.5

1

10-1

Figure 3. (a) Streamwise r.m.s. velocity fluctuation for case I144 scaled with
Logarithmic relationship in Eq.(3), κc · y⋆ ∂U ⋆ /∂ y∗ ;, κc · y+ ∂U + /∂ y+∗;reference statistics at Reτ ≈ 550.

100
, u∗τ and, u⋆τ based on ycrit = 0.173. (b), κs βs = 1. In (a) and (b),, smooth-wall∆y and u⋆τ that recover a smooth-wall-like βc , we therefore
minimise the deviation between βs and βc , the smooth-walland canopy diagnostic functions, above ycrit . Additionally, because ycrit can vary depending on the canopy density regime,
ycrit need to be determined separately for each canopy flow,(Jiménez, 2004; Brunet, 2020). By this method, we avoid enforcing a plateau in βc artificially, relying on just the logarithmic layer. Instead, we recover a smooth-wall-like βc in thelogarithmic layer and above.
We first assess the sensitivity of the results to the choice of
ycrit . As an example, Figure 2(a) depicts the diagnostic function of case I144 for different values of ycrit . For ycrit = 0.098,
βc is not smooth-wall-like above this height even if the deviation between βs and βc is minimised. This is because ycrit is sosmall that the flow just above ycrit is still within the ’roughness
sublayer’ and is perturbed by the canopy elements. However,for ycrit ≥ 0.173, βc is smooth-wall-like for all y∗ greater thanycrit . As shown in Figure 2(b), the r.m.s. deviation between
βc and βs remains stable and small when ycrit ≥ 0.173. Nevertheless, the deviation between βc and βs persists even forlarge ycrit . The major source of this persistent deviation may
be attributed to the numerical noise in βs and βc , as depicted
in Figure 2(a).
In the above, we have left the friction velocity as an independent parameter. To determine ∆y and u⋆τ , which are functions of ycrit as depicted in Figure 2(c), we need to set thelower and upper bounds for ycrit . In Figure 2(b), the lower

The simulations solve the three-dimensional incompressible turbulent flow in a symmetric channel with rigid canopy
elements attached to both walls, as portrayed in Figure 1. Noslip and no-penetration boundary conditions are enforced atboth walls. The canopy elements are explicitly resolved using
a direct-forcing, immersed-boundary method (Garcı́a-Mayoral& Jiménez, 2011). The domain is periodic in the wall-parallel
directions (x, z), which are discretised spectrally. A secondorder central difference scheme on a staggered grid is used in
the wall-normal direction (y), for which the grid is stretchedwith ∆+y ≈ 4.5 at the channel centre. The wall-normal grid
resolutions at the floor and the canopy tips, where mean shear
concentrates, are listed in Table 1. The temporal discretisation
uses a semi-implicit Runge-Kutta, fractional-step method (Le& Moin, 1991), and the channel is driven by a constant meanpressure gradient, with viscosity adjusted to obtain Reτ ≈ 550.
Each simulation is run for at least 10 largest eddy-turnovertimes, δ /uτ , to wash out any initial transients. Once the flow
reaches a statistically steady state, statistics are collected over
10-15 δ /uτ . Full details of the numerical method can be foundin Sharma & Garcı́a-Mayoral (2020a,b).

Results and Discussion
Because the surface morphology has a direct impact on
the flow within the ’roughness sublayer’, of thickness ycrit , we
expect no outer-layer similarity below y∗ = ycrit . To determine

3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
2
3
1.5
2
1
1

0.5

0

0
10-1

100

10-1

100

+
+
Figure 4. (a) Modified defect-law velocity profiles and (b) modified diagnostic functions, κ · y+∗ ∂U /∂ y∗ , scaled with the friction∗
velocity, uτ , evaluated at the virtual origins, as shown in Figure 7.
, smooth-wall reference statistics;
to, dense to sparse
canopy statistics.

3

1.2

2

0.8

1

0.4

0
1.5

0
1
0.75

1
0.5
0.5
0.25
0
-0.2 00.5

0-0.2 01

0.5

1Figure 5. Turbulent r.m.s velocity fluctuations of smooth-wall and canopy flows scaled with the friction velocity, u∗τ , evaluated at thevirtual origins, as shown in Figure 7. The colour scheme is as in Figure 4.
bulent velocity fluctuations are better scaled by u∗τ than byu⋆τ , as illustrated in Figure 3(a). Alternatively, the diagnostic
function can be shown to exhibit the same smooth-wall-likebehaviour if, rather than setting the friction velocity independently of the zero-plane displacement height, κ is allowed tobe different from its smooth-wall value, as shown in Figure3(b). In the logarithmic layer, Eq.(2) can be expressed asbound for ycrit is given by the intersection between the r.m.s.
error curve and twice the baseline error—the r.m.s. βc − βswhen ycrit is large. Above this lower bound, the deviation between βc and βs is small, and a smooth-wall-like βc can berecovered based on ∆y and u⋆τ obtained from all ycrit above
the lower bound. We set the upper bound 0.1 above the lower
bound, as the r.m.s. βc − βs varies little beyond this.
As shown in Figure 2, the critical height where outer-layer
similarity recovers for I144 is ycrit = 0.173 ± 0.05, which issmaller than the typical roughness layer thickness (1-2h above
the canopy tips). Additionally, because ycrit /δ is smaller than
the typical upper bound for the logarithmic region (y ≈ 0.3δ ),we expect the recovery of a smooth-wall-like logarithmic layer
for the canopy flow (Luchini, 2018; Yamamoto & Tsuji, 2018).
Based on the confidence interval of ycrit , we obtain the zeroplane displacement, ∆y = −0.113 ± 0.005, and the height
where u⋆τ is evaluated at, yu⋆τ = −0.574 ± 0.063—note thatyu⋆τ ̸= ∆y.

κc y+
∗

∂U +
= 1, κc = (u∗τ /u⋆τ )κs ,
∂ y+
∗

(3)

where κs ≈ 0.4 is the smooth-wall Kármán constant, and κc is
the new canopy-flow Kármán constant.
Both the diagnostic function, κc βc , and the defect law
for κU + exhibit a good collapse with smooth-wall data, as
shown in Figure 4. This implies the recovery of a logarithmic
layer, but with a Kármán constant different from κs ≈ 0.4. Figures 5 and 6 show that the turbulent velocity fluctuations andReynolds shear stress are significantly perturbed by the presence of the canopy in the near-wall region, y∗ ≲ 0.2, while in

Despite y∗ and u⋆τ yielding a smooth-wall-like βc , the tur-4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

103

102

103

102

102

103

102

103

Figure 6. Pre-multiplied spectral energy densities at y∗ = 0.5δ for the smooth-wall (filled contours) and canopy flows (line contours):(a) kx kz Euu ; (b) kx kz Evv ; (c) kx kz Eww ; (d) kx kz Euv . The contours in (a − c) are at 0.1, 0.3 and 0.5 times the maximum level. The
contours in (d) are at 0.05, 0.2 and 0.5 times the maximum level. The colour scheme is as in Figure 4.

0.45

0
-0.2

0.4

-0.4
-0.6

0.35

-0.8
-1

0.3
10-2

10-1

100

10-2

10-1

100

Figure 7. (a) Ratio of zero-plane displacement to canopy height, ∆y/h, and (b) canopy-flow Kármán constant, κc . In (a),,canopy-tip plane, y/h = 0, and floor, y/h = −1. In (b),, smooth-wall Kármán constant, κs ≈ 0.4, obtained from the logarithmic
layer, y+ = 80 to y = 0.25δ . The colour scheme is as in Figure 4.

the outer layer, y∗ ≳0.3, they are essentially smooth-wall-like.
The ratio of zero-plane displacement to canopy height,∆y/h, and the Kármán constant, κc , for each canopy flow are
portrayed in Figure 7. For the densest canopy, with λ f ≈ 2.04,∆y is essentially zero, and κc ≈ 0.4, suggesting that the overlying turbulence is essentially smooth-wall-like and perceives a
virtual origin at the canopy-tip plane. Scaling with the friction
velocity evaluated at the canopy tips has also been reported
in the study of turbulence over dense filament canopies by
Sharma & Garcı́a-Mayoral (2020b). In the densest canopies
investigated, the turbulent flow ’skimmed’ over the canopy
tips, as the turbulent eddies were precluded from penetrating within and interacting with the canopy elements, to thepoint that the flow above the canopy-tip plane resembles asmooth-wall flow (Brunet, 2020). However, canopies withλ f ≈ 0.01 − 0.91 have κc values very different from that ofa smooth-wall flow, as depicted in Figure 7(b), implying a
deeper modification of the flow by the presence of the canopy.
The zero-plane displacement, ∆y, is associated with thedepth that shear can penetrate within a porous media (Jackson,
1981; Clifton et al., 2008), and this shear penetration depth increases with permeability (Okazaki et al., 2021). This trend
is depicted in Figure 7(a). As canopy density decreases, theouter-layer flow perceives a deeper virtual origin, and for thesparse cases with λ f ≈ 0.01 − 0.06, the virtual origins are very
close to the floor. However, for the intermediate to dense caseswith λ f ≈ 0.51 − 2.04, ∆y/h is small because the closelypacked canopy elements obstruct the shear penetration. Figure 7(b) indicates that cases with λ f ≈ 0.03 − 0.91 exhibit
κc smaller than κs ≈ 0.4. We note that κc < 0.4 is also observed by studies that enforce a plateau in (y − ∆y)+ dU + /dy+
(Breugem et al., 2006; Suga et al., 2010). However, the spars-5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
est canopy, with λ f ≈ 0.01, results in κc > κs ≈ 0.4, which tothe best of the authors’ knowledge has not been reported byany previous studies.

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Conclusions
In the present work, we have assessed outer-layer similarity in flows over canopies with various λ f . To obtain the zeroplane displacement that recovers a smooth-wall-like diagnostic
function, we minimise its deviation from smooth-wall results
above the ’roughness sublayer,’ instead of merely enforcing aplateau in the logarithmic layer. We do so because a plateau in
β equal to 1/κ emerges only at high Reτ , and also because we
obtain a smooth-wall-like β not only in the logarithmic layer
but also in the ’wake’ region. However, an independently set
friction velocity results in poor scaling for the turbulent fluctuations, so we measure the friction velocity at the height ofzero-plane displacement, which leads to a Kármán constantdifferent from κs ≈ 0.4. When scaled with u∗τ and y∗ , the turbulent fluctuations, velocity profile and diagnostic function are
similar to those of a smooth-wall flow, indicating the recoveryof outer-layer similarity, but with a Kármán constant different
from κs ≈ 0.4. As the canopy density decreases, the outerlayer flow perceives a deeper virtual origin. For the sparsest
cases, the virtual origins are close to the floor, indicating asignificant flow penetration into the canopy. For intermediate
densities, the Kármán constant is smaller than κs ≈ 0.4, whichis consistent with results from previous studies on outer-layer
similarity (Breugem et al., 2006; Suga et al., 2010). For oursparsest canopy, κc is larger than κs ≈ 0.4.

Acknowledgements
Computational resources were provided by the ‘Cambridge Service for Data Driven Discovery’ operated by
the University of Cambridge Research Computing Service through EPSRC Tier-2 grant EP/P020259/1 and by
the ’ARCHER2’ system through PRACE project pr1u1702ESTONT.

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