This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1
Modulation of near-wall turbulence in the
transitionally rough regime
Nabil Abderrahaman-Elena, Chris T. Fairhall and
Ricardo Garc´ıa-Mayoral†
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2
1PZ, UK
(Received xx; revised xx; accepted xx)
Direct numerical simulations of turbulent channels with rough walls are conducted in
the transitionally rough regime. The effect that roughness produces on the overlying
turbulence is studied using a modified triple decomposition of the flow. This decomposi-
tion separates the roughness-induced contribution from the background turbulence, with
the latter essentially free of any texture footprint. For small roughness, the background
turbulence is not significantly altered, but merely displaced closer to the roughness crests,
with the change in drag being proportional to this displacement. As the roughness size
increases, the background turbulence begins to be modified, notably by the increase
of energy for short, wide wavelengths, which is consistent with the appearance of a
shear-flow instability of the mean flow. A laminar model is presented to estimate the
roughness-coherent contribution, as well as the displacement height and the velocity at
the roughness crests. Based on the effects observed in the background turbulence, the
roughness function is decomposed into different terms to analyse different contributions
to the change in drag, laying the foundations for a predictive model.
1. Introduction
Although one of the oldest problems in fluid mechanics, roughness in turbulent flows
remains an active area of research. The seminal works by Nikuradse (1933) and Colebrook
& White (1937), offering a systematic study of rough surfaces, set the foundations for the
field. Based upon Nikuradse’s results on pipes roughened with sand of equal grain size,
Schlichting (1936) introduced the concept of equivalent sand grain size, ks, in order to
classify rough surfaces. Surfaces with different roughness height k that produce the same
friction as a particular sand-grain size would share the same ks. However, there are two
important shortcomings in this strategy. First, as stated by Bradshaw (2000), it “simply
defines a useful common currency for roughness size—like paper money, valueless in itself
but normally acceptable as a medium of exchange”, i.e. ks is merely a classifier, an a
posteriori parameter that is of limited use for prediction purposes. Second, Colebrook
(1939) observed that, despite sharing the same value of ks in the fully rough regime, the
ratio ks/k varies for a given surface along the transitionally rough regime. In other
words, different surfaces depart from the hydraulically smooth regime differently. In
figure 1, these differences are highlighted by using k+s∞(Jime´nez 2004). In this figure,
a roughness lengthscale ks∞ , of fixed proportion to k, is chosen so that in the fully
rough regime ks =ks∞and the roughness function ∆U
+ depends only on k+s∞ . The figure
highlights that, while such lengthscale can collapse the results in the fully rough regime,
differences can still be observed in the transitionally rough regime. It is also worth
† Email address for correspondence: r.gmayoral@eng.cam.ac.uk
2 N. Abderrahaman-Elena, C. T. Fairhall and R. Garc´ıa-Mayoral
∆U+
10
5
0
k+s∞
100 101 102
Figure 1. Roughness function ∆U+ in the transitionally rough regime as a function of
k+s∞ . ◦, uniform sand (Nikuradse 1933); △, uniform packed spheres (Ligrani & Moffat 1986);
, galvanized iron; , tar-coated cast iron; , wrought-iron; , interpolation
(Colebrook 1939). Adapted from Jime´nez (2004).
nothing that ks∞ can only be obtained a posteriori from experimental or numerical
results with varying k+s∞ . Many studies in the recent years have aimed to find a suitable
combination of parameters to describe roughness surfaces and predict their friction. Some
authors have explored the effective slope, the solidity, as well as different moments of the
roughness height, mainly the mean, the standard deviation and the skewness (Flack &
Schultz 2010). On the simulation front, it has recently been proposed that ∆U+ can
be calculated from direct numerical simulations in minimal boxes, which reduces their
cost significantly (Chung et al. 2015; MacDonald et al. 2017). Predictive models are also
beginning to emerge. Yang et al. (2016) and Yang & Meneveau (2016) have recently
proposed a model based on the flow sheltering by the roughness elements, capturing the
behaviour of k+s for a wide variety of roughness elements. In the transitionally rough
regime, however, the gradual changes in the flow that lead to the eventual departure
from the hydraulically smooth friction have not been fully understood, and a quantitative
account of the different ∆U+observed for different surfaces with the same k+s∞would be
needed. The approach taken in this work is to understand the modifications produced on
the flow by the roughness texture. As stated by Marusic et al. (2010), “without further
theoretical advances, there is a risk of needing a catalogue of roughness results”.
In the hydraulically smooth regime, roughness is of small size and has a negligible
effect on the flow, causing no change in terms of wall friction. As a result, the flow is
effectively equivalent to canonical smooth-wall turbulence. In the fully rough regime,
on the other hand, roughness dominates. The friction coefficient becomes independent
of viscosity and the roughness size determines the skin friction drag. Between these
two regimes, in the so-called transitionally rough regime, both the viscous and the
rough effects have comparable importance. Nikuradse (1933) observed that sand grain
roughness was hydraulically smooth for k+s < 5, fully rough for k
+
s > 70, and transitional
between those extremes. However, Flack & Schultz (2010) compile experimental results
for typologies of roughness other than sand grain roughness, finding that the limits
vary considerably. In particular, the literature shows the transitionally rough regime
spanning 1.4–15 < k+s < 18–70. The lower bound of this regime has classically been
Turbulence in the transitionally rough regime 3
seen as a threshold below which the surface completely behaves as hydraulically smooth.
However, Bradshaw (2000) proposes a gradual transition between these regimes, without
a defined, hard boundary. A recent work by Thakkar et al. (2018) also seems to point in
this direction. The interest in transitional roughness resides in understanding the effects
that small surface texture starts having on the flow; effects that, as size increases, will
lead to the departure from its smooth-wall behaviour and eventually to the fully rough
regime. This interest is also growing amongst the high-Reynolds-number community
(Marusic et al. 2010). Apparently, smooth surfaces, with a well-controlled micro-texture,
may eventually enter the transitionally rough regime as viscous scales become sufficiently
small for increasing Reynolds number.
Far enough above conventionally rough walls, the mean velocity profile exhibits the
same logarithmic region found over smooth walls. Effectively, roughness only modifies
the intercept of the logarithmic velocity profile, while the Ka´rma´n constant, κ, and the
wake function are unaffected (Nikuradse 1933; Clauser 1956). In the logarithmic layer,
the mean velocity profile, U , can then be expressed as
U+ = κ−1 ln
(
y+
)
+B +∆U+ = U+0 +∆U
+, (1.1)
where B is the smooth-wall intercept of the logarithmic velocity profile. The roughness
function, ∆U+ (Hama 1954), depends on the roughness texture and its size. The 0
subscript denotes smooth-wall flow quantities. The + superscript indicates scaling in
wall-units, with the friction velocity, uτ , and the kinematic viscosity, ν. The change in
friction coefficient, ∆Cf , can be directly related to ∆U
+. At the centreline of a channel
or at the edge of the boundary layer, denoted by the subscript δ, equation (1.1) becomes
(2/Cf )
1/2
= U+δ = U
+
δ0
+∆U+. (1.2)
From the above equation, the relative change in drag ∆Cf (Spalart & McLean 2011;
Garc´ıa-Mayoral et al. 2018) is
∆Cf
Cf0
= −
1(
1 +∆U+/U+δ0
)2 − 1. (1.3)
The above expression shows the dependence of the drag on the roughness function. Notice
that ∆Cf/Cf0 also depends weakly on the Reynolds number through U
+
δ0
. Equation (1.3)
shows that an increase in friction is accompanied by a downward displacement of the
velocity profile towards the wall, and vice versa. In the context of drag reduction, for
small ∆Cf , linearised forms of equation (1.3) have often been used (Luchini 1996; Garc´ıa-
Mayoral & Jime´nez 2011a). In this work, we characterise the change in drag using ∆U+,
rather than ∆Cf/Cf0 , to avoid the dependence on the Reynolds number mentioned
above. Note also that in periodic, numerical channels the drag force can only be defined
unambiguously if the cross-section area is homogeneous (MacDonald et al. 2018).
In this work we explore the transitionally rough regime. The ultimate goal is to
understand and predict the effect of transitionally rough surfaces on the flow. The range
of roughness sizes covered is therefore small, but of more general relevance as this regime
marks the onset of the rough behaviour. To explore it, we conduct a campaign of direct
numerical simulations (DNSs) of turbulent channels. The channels are symmetric, with
roughness on both walls. Four different geometries are considered, as depicted in figure 2.
To explore the behaviour of each geometry, direct numerical simulations are conducted
for different values of k, while their shape is kept fixed, also resulting in constant solidity
and effective slope. These surfaces exhibit the classic k-roughness behaviour, which is
characteristic of most three-dimensional rough surfaces (Jime´nez 2004).
4 N. Abderrahaman-Elena, C. T. Fairhall and R. Garc´ıa-Mayoral
a)
b)
c)
d)
sx
sx
sx
sx
sz
sz
sz
sz
k
k
1
2
k
k
k
k
k
k
k
k
Figure 2. Schematics of the roughness patterns studied. (a) Collocated; (b) Collocated with two
heights; (c) Spanwise-staggered and (d) Streamwise-staggered. Left panels, three-dimensional
representation of the roughness surface, where the mean flow is from bottom left to top right.
Central panels, top view of a periodic unit element of the pattern; right panels, side view of an
isolated post; where the mean flow is from left to right.
This paper is organised as follows. In §2 the details of the numerical method are
presented. A novel decomposition of the flow is presented in §3 and it is used in §4 and
§5 to study the two components of the velocity. In §6 the appearance of a shear-flow
instability is discussed, and the relationship between the roughness function and the
virtual origin of turbulence is analysed in §7. The key conclusions are discussed in §8.
Turbulence in the transitionally rough regime 5
2. Methodology
The numerical experiments are conducted in an incompressible, turbulent channel
with roughness on the top and bottom walls. The domain is periodic in the wall-parallel
directions. The channel half-height is δ measured from the roughness crests, and the
length and width are Lx = 2πδ and Lz = πδ, respectively. The streamwise, wall-normal
and spanwise coordinates are x, y and z, with u, v and w the corresponding components
of the velocity u. The origin of the wall normal coordinate, y = 0, is set at the roughness
crests of the bottom wall.
Table 1 includes the geometric parameters used in our simulations. Note that, although
in this work δ only includes the height above the roughness crests, for the largest
roughness cases studied, C24 and C36, the roughness-to-channel-height ratio is δ/k < 10.
Under these conditions, roughness can be expected to perturb the whole logarithmic
region (Jime´nez 2004). Nevertheless, previous works have shown that such low δ/k
produce acceptable results for the dynamics within the first few roughness heights above
the surface and for the resulting ∆U+(Garc´ıa-Mayoral & Jime´nez 2012; Chung et al.
2015; MacDonald et al. 2017). In all the present simulations, we observe that outer-layer
similarity (Townsend 1976) is recovered no later than y+ ≈ 90.
The numerical method is adapted from that of Garc´ıa-Mayoral & Jime´nez (2011b) for
riblet channels to simulate fully three-dimensional roughness and is briefly summarised
here. The temporal integrator is a fractional-step method combined with a three–sub-step
Runge-Kutta and with pressure correction at the final sub-step only (Le & Moin 1991;
Perot 1993). The spatial discretisation is pseudo-spectral. The two periodic directions, x
and z, are discretised using Fourier series, while the wall-normal direction, y, is discretised
using a second order finite difference scheme in a collocated grid. The chequerboard
effect, typical of collocated grids (Ferziger & Peric´ 2002), is addressed using a quasi-
divergence-free formulation (Nordstro¨m et al. 2007; Garc´ıa-Mayoral & Jime´nez 2011b).
Near the walls, the high velocity gradients and the roughness texture require a high
spatial resolution for the flow to be correctly resolved, while for the core of the channel,
a coarser, less costly resolution would suffice. To alleviate the computational cost of the
method while still resolving all scales, two strategies are implemented. Along the wall-
normal direction the finite-difference discretisation allows for the grid to be stretched,
with a higher density of points near the walls. Above the roughness, the wall-normal grid
spacing is ∆y+
min
≈ 0.3 at the roughness crests, and progressively grows to ∆y+max ≈ 3.1
at the centre of the channel. Within the roughness region, i.e. between roughness crests
and troughs, the grid spacing is ∆y+ ≈ 0.3. Additionally, to increase the x- and z-
resolutions near the walls, the domain is vertically divided into three blocks. This multi-
block technique allows a high number of grid points to be set near the walls, with a
coarser resolution in the block resolving the core of the channel. In this latter region
the resolution is set to resolve all turbulent scales. In the blocks containing the walls,
it is also necessary to resolve the flow around the roughness elements, which requires
a higher resolution for small k+. The resolution in the central block is ∆+xc ≈ 6 and
∆+zc ≈ 3 along the x and z directions respectively, while in the two wall blocks a finer
resolution is used to represent each roughness element, with at least 12 points along x
and z. The roughness elements are resolved using a direct-forcing immersed boundary
method (Mohd-Yusof 1997; Fadlun et al. 2000; Iaccarino & Verzicco 2003). The unit
element is a cube of side k, repeated along x and z on both walls. These unit elements
are described using 12 × 12 points for the cases with k+ . 9, 24 × 24 points for the
cases with 12 . k+ . 25, and 48 × 48 points for the case with k+ ≈ 36. This choice
results from the need to solve both the turbulent scales as well as the flow around the
6 N. Abderrahaman-Elena, C. T. Fairhall and R. Garc´ıa-Mayoral
Case sx/k sz/k δ/k k
+ k+s ∆U
+ Reτ U
+
0 ℓ
+
U ℓ
+
u ℓ
+
J ℓ
+
uv
Smooth Channel SC − − − 0.0 0.0 0.0 183.9 0.0 0.0 0.0 0.0 0.0
Collocated
C06 2 2 30.6 6.0 9.6 0.5 183.9 0.5 0.5 0.5 0.6 1.2
C09 2 2 20.4 8.8 14.1 0.7 180.8 0.7 0.7 0.7 0.7 1.5
C12 2 2 15.4 11.7 18.7 1.5 180.0 1.1 1.2 1.2 1.2 3.2
C15 2 2 12.3 14.4 23.1 2.4 178.7 1.3 1.5 1.6 1.6 4.5
C18 2 2 10.3 17.4 27.8 3.5 179.0 1.5 1.9 1.9 2.3 6.3
C24 2 2 7.7 22.5 36.0 4.7 174.5 1.6 2.5 2.6 3.8 8.4
C36 2 2 5.2 35.7 57.2 6.7 186.7 2.1 4.4 4.8 8.8 −
Collocated
two heights
CC06 4 4 30.7 5.8 − 0.8 178.7 1.2 1.2 1.4 1.3 2.1
CC09 4 4 20.6 8.7 − 1.8 179.8 1.6 1.7 2.1 1.7 3.7
CC12 4 4 15.5 11.7 − 3.3 182.6 2.1 2.7 3.0 2.6 6.2
CC15 4 4 11.7 15.4 − 4.5 180.1 2.4 3.6 3.4 3.5 8.3
CC18 4 4 10.4 17.0 − 4.9 176.7 2.4 3.9 3.5 4.0 9.1
Spanwise
staggered
SZ06 2 4 30.6 5.9 − 0.4 183.0 0.5 0.5 0.6 0.6 1.0
SZ09 2 4 20.4 8.9 − 0.9 182.7 0.7 0.7 0.8 0.8 1.7
SZ12 2 4 15.4 11.8 − 1.6 181.8 1.2 1.3 1.3 1.3 3.4
SZ15 2 4 11.6 16.2 − 3.2 188.3 1.4 1.8 1.8 2.1 6.0
Streamwise
staggered
SX06 4 2 30.6 5.9 − 0.6 181.6 0.5 0.5 0.6 0.5 1.1
SX09 4 2 20.4 8.8 − 1.2 179.5 0.6 0.6 0.9 0.7 2.0
SX12 4 2 15.4 11.9 − 2.6 183.1 1.0 1.1 1.4 1.2 4.2
SX15 4 2 11.5 16.0 − 4.1 185.3 1.2 1.4 1.7 1.6 6.0
SX18 4 2 10.3 17.6 − 4.3 180.6 1.2 1.5 1.8 1.8 6.2
Table 1. Main characteristics of the simulations. The roughness-element width and maximum
height is k, and the streamwise and spanwise pitches of the pattern are sx and sz. The half-height
of the channel, measured from the roughness crests, is denoted by δ. The mean velocity at the
roughness crests is U0. ℓ
+ denotes the depth of virtual origins, measured from the tips, as
introduced in §5, where ℓ+U refers to the virtual origin of the mean velocity profile; ℓ
+
u is the
virtual origin of the streamwise rms fluctuations; ℓ+J is Jackson’s displacement height (Jackson
1981); and ℓ+uv refers to the virtual origin of the Reynolds shear stress.
roughness elements while keeping a moderate computational cost. Note that, although
the resolution for the smallest k+ values is marginal, the simulations carried out by
Thakkar et al. (2018) on random roughness, performed using a 12 point-per-element
discretisation, are in good agreement with experimental results.
The simulations are conducted at constant mass flow rate, which is adjusted to achieve
a friction Reynolds number Reτ ≈ 185 for all simulations. Reτ is computed using δ
′ =
δ + ℓU , where ℓU is the virtual origin of the mean velocity profile, i.e. where the mean
velocity profile tends to zero when extrapolated from y > 0. The friction velocity uτ
is obtained by extrapolating the total shear stress to δ′. Statistically converged initial
conditions are obtained by simulating and discarding the flow during 10 δ/uτ . Statistics
are then collected over at least 15 δ/uτ . Relevant parameters and results of the simulations
are given in table 1.
Turbulence in the transitionally rough regime 7
0 0.5 1 1.5 2
α
+
100
103
0 500 1000
x+
0
1
0
1
100
103
0
1
100
103
a.1) a.2)
b.1) b.2)
c.1) c.2)
Figure 3. Instantaneous realisation of the streamwise velocity, for case C06, at y+ = 0.3, in a
section of constant z+ through the middle of a row of posts. (a.1) and (a.2) , full signal;
, background turbulent component; , ensemble average over time and over the relative
position in the periodic unit of texture. (b.1) and (b.2) , full signal minus the ensemble
average; , background turbulence. (c.1) and (c.2) , low-pass filtered full signal; ,
difference between the latter and the full signal. The left panels represent the signals in physical
space, and the right panels in Fourier space, where α is the wavenumber.
3. Flow decomposition
In the proximity of the wall, roughness induces fluctuations that alter, and even destroy,
the near-wall cycle (Jime´nez 2004). The intensity of these fluctuations decreases expo-
nentially away from the surface, and they are generally considered to be confined within
a region near the wall of height ∼ 3k, known as the roughness-sublayer (Raupach et al.
1991; Flack et al. 2007). In this region the fluctuations induced by roughness cannot be
neglected and their effect is noticeable in the flow. These roughness-coherent fluctuations
are of the same order of magnitude as the background turbulence fluctuations, leaving
their footprint on the velocity field and the energy spectrum.
In the roughness-sublayer the flow can be thought of as formed of a roughness-
coherent component, which is coherent in time and space with the roughness surface,
and a background-turbulent component of a chaotic nature. This is analogous to the
decomposition from Reynolds & Hussain (1974) into a coherent and a turbulent com-
ponent. To illustrate this concept, figure 3(a.1) displays an instantaneous realisation of
the streamwise velocity close to the roughness crests. We observe two contributions. The
first, with longer wavelengths, has a characteristic lengthscale of the order of turbulent
eddies. The second, whose dominant wavelength is that of the roughness texture, smaller
than the overlying eddies, can be obtained by ensemble averaging over time and over
the relative position in the periodic unit of texture. If the roughness-coherent component
is subtracted from the full velocity, as in figure 3(b.1), we observe that the footprint of
the roughness elements does not completely disappear. The reason can be explained by
means of their corresponding Fourier transform, shown in figures 3(a.2) and 3(b.2). The
8 N. Abderrahaman-Elena, C. T. Fairhall and R. Garc´ıa-Mayoral
ensemble average is composed of the wavelength of the texture and its subharmonics.
The full velocity exhibits these same wavelengths, induced by the texture; however,
they are modulated in amplitude and thereby surrounded by energy in the neighbouring
wavelengths. When the ensemble average is subtracted from the full velocity, the most
energetic part, which corresponds to the wavelength of the texture and its subharmonics,
is removed, but all the energy neighbouring these wavelengths is not, failing to remove
the footprint as observed in figure 3(b.1). In figures 3(c.1) and 3(c.2) we set a threshold
to identify what wavelengths correspond to background turbulence and thus what the
remainder represents in physical space. The wavelengths are then divided into long and
short ones. In figure 3(c.1), we observe that the contribution from long wavelengths closely
resembles smooth-wall turbulence. Wavelengths shorter than those of turbulence are
similar to the ensemble average, shown in figure 3(a.1), but they appear to be modulated
in amplitude by the overlying, long-wavelength signal of the background turbulence.
Following the above analysis, in Abderrahaman-Elena & Garc´ıa-Mayoral (2016) we
proposed that the flow over a roughness texture of small size is not simply the sum of
the ensemble average plus a turbulent component. Instead, the roughness-coherent flow
is modulated by the overlying turbulence. For a particular roughness size, inducing a
coherent flow uRC,u, and with a background turbulence uBT , the instantaneous velocity
can intuitively be modelled as
u ≈ U + uBT +
U + uBT
U
uRC,u, (3.1)
where U is the temporal and x-z-spatial averaged streamwise velocity.
Equation (3.1) can also be inferred by considering small roughness in the viscous limit.
In the spirit of Luchini et al. (1991), the vanishingly small roughness limit reduces the
problem to a shear driven, purely viscous flow, which results in a self-similar solution
that scales with k and with the overlying shear, proportional to U + uBT .
The triple decomposition proposed by Reynolds & Hussain (1974) is similar to equa-
tion (3.1) except for the amplitude modulation by U+uBT . The triple decomposition has
been widely used to characterise the texture-coherent flow in turbulence over a variety
of complex surfaces (Choi et al. 1993; Jime´nez et al. 2001; Garc´ıa-Mayoral & Jime´nez
2011b; Jelly et al. 2014; Seo et al. 2015). However, if we are to study the modifications in
the background turbulence, the modulation of the roughness-coherent component needs
to be accounted for. Figure 4(a) shows the streamwise velocity above the roughness crests
within the roughness layer, where the coherent signal of the roughness elements is strong.
If the ensemble average, the roughness-coherent contribution, is subtracted we observe in
figure 4(b) that, although reduced, the footprint of the surface texture remains. However,
when we account for the modulation of the coherent flow, using equation (3.1), the
signature of the roughness-coherent flow on the background turbulence is substantially
weaker, as shown in figure 4(c) where it is negligible.
The instantaneous overlying flow induces around roughness elements a velocity field
with components in all three directions. At the same time, the background turbu-
lent flow has components in all three directions, so for instance the spanwise back-
ground shear induces a texture-coherent cross-flow. Hence, the three velocity components
have a roughness contribution induced by all components of the overlying flow. Let
us take the streamwise velocity signal as an example. It would have contributions
from the mean streamwise velocity, U , the background-turbulent streamwise component,
uBT , a roughness-coherent component, uRC,u, driven and modulated by U + uBT , and
a roughness-coherent component, uRC,w, driven and modulated by the background-
turbulent spanwise component, wBT . The simplified model presented in equation (3.1)
Turbulence in the transitionally rough regime 9
a)
b)
c)
L+z
L+z
L+z
L+x
Figure 4. Instantaneous streamwise velocity at a distance y+ ≈ 0.3 above the roughness crests,
for case C09.(a) Full signal. (b) Turbulent contribution obtained using triple decomposition,
u− U − uRC,u. (c) Turbulent contribution obtained using equation (3.1).
can then be extended to all components of the full velocity,
u ≈ U+uBT +
U + uBT
U
uRC,u +
wBT
w˜
uRC,w +
vBT
v˜
uRC,v, (3.2a)
w ≈ wBT +
U + uBT
U
wRC,u +
wBT
w˜
wRC,w +
vBT
v˜
wRC,v, (3.2b)
v ≈ vBT +
U + uBT
U
vRC,u +
wBT
w˜
vRC,w +
vBT
v˜
vRC,v, (3.2c)
where uRC,w is the roughness-coherent streamwise velocity induced by the overlying,
background spanwise shear of wBT , etcetera. Note that, while the modulating signal
U + uBT is always positive, wBT and vBT can either be positive or negative, and due to
symmetry, their mean values cancel out. To obtain a measure of the intensity of these
components, w˜ and v˜ denote the conditional averages of w and v, which account for their
mean direction over individual roughness elements, such that they are only included
into the average when they are positive (Garc´ıa-Mayoral & Jime´nez 2011b). In the cube
roughness geometries studied in this work we observe no significant flow induced by v
in the streamwise and spanwise directions, although the corresponding terms have been
included in equations (3.2) for completeness. In textures with inclined planes, such as
pyramids or cones, for instance, we would expect these contributions, uRC,v and wRC,v,
to be relevant. Also, the principal directions of the surfaces studied in the present work
are aligned with x and z. In a more general case, the tensorial relationship between the
10 N. Abderrahaman-Elena, C. T. Fairhall and R. Garc´ıa-Mayoral
L+x L
+
x
L+z
L+z
a.1)
b.1)
a.2)
b.2)
Figure 5. Instantaneous realisation for case C09. (a) spanwise velocity, at y+ ≈ 0.3, above the
roughness crests. (b) wall-normal velocity at y+ ≈ 1. (a.1) and (b.1), full signals; (a.2) and (b.2),
turbulent contributions obtained using equations (3.2).
driving mean pressure gradient and the resulting mean flow (Kamrin et al. 2010; Luchini
2013) would cause the latter to have a component in z. This would introduce additional
terms in the decomposition of (3.2), but would not alter its spirit. The effect would be
small in irregular surfaces with no clear preferential directions, but could be important
in patterned surfaces.
Figures 4 and 5 show instances of all three velocity components and their corresponding
turbulent contributions, obtained by removing the roughness-coherent terms following
equations (3.2). The present decomposition, accounting for the modulation of the over-
lying turbulence, reduces significantly the signature of the roughness-coherent flow. The
remaining signature is overall much weaker than in the full signal, and is mostly prevalent
in vBT . For uBT , our results show essentially no footprint of the texture, even for the
larger k+ studied. In any event, the decomposition is based on a fundamentally linearised
approach, assuming that the roughness lengthscales are much smaller than those over
which the background overlying shear varies (Zhang & Chernyshenko 2016), and can be
expected to fail eventually as k+ increases.
The modulation of the roughness-coherent flow by the background turbulence is
analogous to the modulation of near-wall turbulence by the outer-layer dynamics (Mathis
et al. 2009). Zhang & Chernyshenko (2016) have recently revisited this problem, and
propose a formulation similar to equations (3.2). Also recently, Anderson (2016) observed
that the modulation of near-wall turbulence is present in flows over rough surfaces. In
our direct numerical simulations, Reτ is not sufficiently high to reproduce large-scale
dynamics, so the background turbulence is essentially that of near-wall, small-scale
dynamics. The larger scales would be significant at higher Reτ and if the simulation
domain was sufficiently large (Flores & Jime´nez 2010). In such case, we could expect
a cascading modulation, where large scales modulate small scales in the background
turbulence, and the full background turbulence modulates the texture-coherent flow.
Beyond instantaneous realisations, the spectral densities of the different flow variables
Turbulence in the transitionally rough regime 11
101
102
λ
+ z
101 102 103
λ+x
101
102
λ
+ z
101 102 103
λ+x
101 102 103
λ+x
101 102 103
λ+x
kxkzE
+
uu kxkzE
+
vv kxkzE
+
ww −kxkzE
+
uv
fu
ll
tu
rb
u
le
n
t
Figure 6. Premultiplied energy spectra of the full velocity signal and its background turbulent
component, kxkzE
+
uu, kxkzE
+
vv and kxkzE
+
ww, and cospectrum of the Reynolds stress, kxkzE
+
uv,
for case C06, at y+ = 2 for v and y+ = 0.5 for the rest.
101
102
λ
+ z
101 102 103
λ+x
101
102
λ
+ z
101 102 103
λ+x
101 102 103
λ+x
101 102 103
λ+x
kxkzE
+
uu kxkzE
+
vv kxkzE
+
ww −kxkzE
+
uv
fu
ll
tu
rb
u
le
n
t
Figure 7. Premultiplied energy spectra of the full velocity signal and its background turbulent
component, kxkzE
+
uu, kxkzE
+
vv and kxkzE
+
ww, and cospectrum of the Reynolds stress, kxkzE
+
uv,
for case C18, at y+ = 2 for v and y+ = 0.5 for the rest.
provide quantitative, statistical evidence to evaluate the proposed decomposition. Fig-
ures 6 and 7 show premultiplied spectra and cospectra of the full velocity, as well as those
of the background turbulence variables, for a roughness of k+ ≈ 6 and one of k+ ≈ 18,
with s+x = s
+
z ≈ 12 and 36, respectively. The full flow signals exhibit a large spike at
the wavelengths of the texture, sx and sz, as well as its corresponding subharmonics.
However, the surrounding wavelengths also contain a substantial amount of energy, in a
similar fashion to the effect observed in figure 3 in one dimension. These regions are the
signature of the amplitude modulation of the roughness-coherent flow. Using the classical
triple decomposition, without the modulation of the roughness-coherent component,
removes the wavelength of the texture only, as shown in figure 3(b.2). In addition to the
wavelengths of the roughness-coherent flow and the background turbulence, we observe
very elongated regions with the wavelength of the roughness in z and the range of
12 N. Abderrahaman-Elena, C. T. Fairhall and R. Garc´ıa-Mayoral
wavelengths of the background turbulence in x, or vice versa. These are the signature
of cross-terms BT–RC in equation (3.2). Let us take for instance the term uBTuRC,u,
which essentially arises from the interaction of overlying streaks with the texture. In
figure 4(a), it can be observed that the region of high u corresponding to a streak is
broken down by the canyons formed by the roughness, so that within the footprint of the
streak there are streamwise-aligned, alternating stripes of high and low u. The signature
of this in spectral space will have the streamwise wavelength of streaks, but the spanwise
wavelength of the texture.
For small roughness textures, as in figure 6, the scale separation between the turbulent
and roughness contributions is large enough that they can essentially be extracted by
Fourier filtering. We exploited this in the example shown in figures 3(c.1) and 3(c.2).
However, for larger k+, as shown in figure 7, the elongated regions mentioned above
overlap with the turbulent contribution, which makes a pass-band filter ill-suited. Nev-
ertheless, equations (3.2), interpreted as a filter, provide a tool to extract the turbulent
contribution when Fourier filtering is not possible due to an overlap of wavelengths. In §5
this decomposition is used to extract the background turbulence component and analyse
the effects of roughness on it. Before that, in §4, the roughness-coherent contribution is
characterised, and a model to predict it is proposed.
4. The roughness-coherent component
In smooth walls, the mean velocity is a function of the wall-normal coordinate alone.
However, in rough surfaces, we can define a time-averaged mean velocity that not
only depends on y, but is also a function of the x and z coordinates. This space-
dependent mean flow minus the conventional mean velocity, U(y), is the roughness-
coherent contribution introduced in the previous section, which is steady and coherent in
space with the roughness texture. If we focus on individual roughness elements, the
coherent flow can be thought of as the flow around the obstacles of the roughness
geometry driven by the overlying shear. As the flow is deflected from and surrounds
the obstacles, it generates a three-dimensional velocity field. The roughness-coherent
component can be obtained by averaging the flow in time. Since all geometries in the
present work are made up of a periodic pattern, the roughness-coherent contribution will
also be periodic, with the same wavelength of the roughness texture. This contribution is
therefore obtained by ensemble averaging over time and over the periodic texture units
as in Seo et al. (2015).
The coherent flow thus obtained decays above the roughness crests with height. In
all our cases, the roughness-coherent components of the velocity have heavily decayed
at a distance of ∼ k, shorter than the commonly accepted height of the roughness-
sublayer,∼ 3k (Raupach et al. 1991; Flack et al. 2007). The rms of the roughness-coherent
components of the velocity, whose squares are commonly referred to as dispersive stresses
(Raupach & Shaw 1982), are depicted in figure 8 for several roughness textures and sizes.
The decay is of the form ∼ exp(−y/k) and is observed to scale with the roughness height,
k, rather than the spacing, s. The rate of decay is observed to be different for the three
velocities, and is fastest for u and slowest for v. For all direct numerical simulations, the
magnitude of the fluctuations, and in particular that of the wall-normal component, is
below 2 percent of uτ at one roughness height above the roughness crests. These results
are consistent with experiments, where dispersive stresses are found to vanish at y ≈ k
(Cheng & Castro 2002; Florens et al. 2013).
For vanishingly small k, the roughness-coherent flow can be thought of as the flow
induced around the roughness elements driven by a steady, homogeneous overlying shear.
Turbulence in the transitionally rough regime 13
u′+Ru
0 0.5 1
y/k
10−3
10−2
10−1
100
v′+Ru
0 0.5 1
y/k
w′+Ru
0 0.5 1
y/k
Figure 8. Rms of the roughness-coherent velocity components above the roughness crests.
, collocated posts; , spanwise-staggered posts; , streamwise-staggered posts; ,
collocated posts of two heights. The reference lines, , have slopes of 1.5π, 1π and 1.25π,
respectively.
k+ = 9
-1 0 1
x/k
-1
0
1
y
/k
k+ = 18
-1 0 1
x/k
k+ = 24
-1 0 1
x/k
Figure 9. Wall-normal velocity for cases (left) C09, (centre) C18 and (right) C24. Shaded,
roughness-coherent contribution from direct numerical simulations; contours, steady laminar
model. Red and solid indicate positive values, blue and dashed indicate negative values.
This concept was already used by Luchini et al. (1991), where they considered riblets
of a vanishingly small size, so that they perceive the overlying turbulence as steady
and homogeneous. Notice that, for small roughness, in the transitionally rough regime,
the time-scales and lengthscales of the overlying background-turbulent fluctuations are
much larger than those of roughness. These latter fluctuations are quasi-steady and quasi-
homogeneous with respect to the characteristic scales of the surface, i.e. they vary slowly
and over long distances. The concept of a quasi-steady, quasi-homogeneous limit was
formalised by Zhang & Chernyshenko (2016) for the interaction between the overlying
large-scales of the outer region and the smaller and faster near-wall fluctuations, and a
similar framework would apply here.
The roughness-coherent flow for small k+, as suggested above, can be modelled as the
flow induced by a steady and homogeneous overlying background flow. Therefore, to ap-
proximate the roughness-coherent contribution, we conduct steady, laminar simulations
using the numerical methodology described in §2, but where the periodic domain only
contains one texture element. The mean shear and viscosity are adjusted to match the
14 N. Abderrahaman-Elena, C. T. Fairhall and R. Garc´ıa-Mayoral
u′+R,u
k+
0
0.5
1
1.5
v′+R,u
k+
0
0.1
0.2
w′+R,u
k+
0
0.1
k+
-1 0 1
y/k
0
0.5
1
1.5
k+
-1 0 1
y/k
0
0.1
0.2
0.3
k+
-1 0 1
y/k
0
0.1
0.2
0.3
Figure 10. Rms of the roughness-coherent contribution for the collocated and
streamwise-staggered roughness. Solid, results from direct numerical simulations; dashed, results
from the laminar model simulations. Blue to red, cases C06 to C18 (top row) and SX06 to SX18
(bottom row).
0 10 20
k+
0
1
2
3
U
+ 0
0 1 2
U+0
0
1
2
u
′+ R
u
m
a
x
a) b)
Figure 11. (a) Mean streamwise velocity at the roughness crests. (b) Maximum streamwise
roughness-coherent velocity as a function of the mean streamwise velocity at the roughness tips.
, collocated posts; , spanwise staggered posts; , streamwise-staggered posts; , collocated
posts of two heights. Empty symbols correspond to estimations from the laminar model. Linear
regression with slopes , 0.82; and , 0.47.
same k+ of the corresponding direct numerical simulations. For small but finite values of
k+, these numerical domains are too small to sustain turbulence (Jime´nez & Moin 1991),
and result in laminar, steady flows, which provide estimates of the roughness-coherent
contribution. For instance, figure 9 compares the wall-normal roughness-coherent velocity
obtained from the direct numerical simulations by ensemble averaging to the laminar
model. Laminar simulations begin to deviate for the intermediate case, C18, but they
still exhibit good qualitative agreement for the larger case, C24. Notice that, even for
the smallest k+, the flow is not symmetric and does not behave as purely viscous. It
is therefore necessary to consider the advective terms, in contrast with the Stokes-
flow analysis of Luchini et al. (1991). This model also allows us to predict the rms
fluctuations of the roughness-coherent flow. Figure 10 depicts data from our direct
numerical simulations compared against those from the laminar model, and shows good
Turbulence in the transitionally rough regime 15
u′+, v′+, w′+
0 0.5 1
y/δ
0
1
2
3
uv′+
0 0.5 1
y/δ
0
0.5
1
ω′+x
0 0.5 1
y/δ
0
0.1
0.2
0.3
u′+
v′+
w′+
a) b) c)
Figure 12. Full rms fluctuations. Blue to red, cases C06 to C36; black, smooth channel. The
arrows indicate increasing roughness size. (a) Streamwise, wall-normal and spanwise velocity
fluctuations; (b) Reynolds stress; (c) streamwise vorticity fluctuations.
agreement for k+ . 15 for all our roughness surfaces. Notice that this method can also
be used to estimate properties of the mean velocity profile close to the rough surface,
such as the mean streamwise velocity at the roughness crests, by averaging along the x
and z directions, as shown in figure 11.
5. The background turbulence component
Roughness does not only excite the wavelengths of the surface texture, but also modifies
the background turbulence. This can be observed in the rms fluctuations that incorporate
effects from both the roughness contribution, studied in the previous section, and the
background turbulence. The streamwise rms fluctuations, shown in figure 12, decrease in
the near-wall region, with the peak of intensity lowering as the roughness size increases,
while a significant growth occurs at the roughness crests in a similar fashion to that in
figure 10. The decrease in the peak of intensity can only be caused by a decrease, and
therefore modification, of the rms fluctuations of the background-turbulent component.
This decrease is consistent with the destruction of the near-wall peak of intensity of the
streamwise rms fluctuations observed in the fully rough regime (Ligrani & Moffat 1986).
The wall-normal and spanwise rms fluctuations present an increase of intensity across
the entire roughness-sublayer, from the roughness crests to their near-wall intensity peak.
The near-wall intensity peak of the streamwise vorticity fluctuation also increases, save
for the largest case, which flattens in magnitude and almost disappears, this being a
symptom of roughness altering the near-wall cycle. Notice that these modifications of
the rms fluctuations extend to a distance of ∼ 2k, larger than the decay distance of
the roughness-coherent signal observed in the previous section. However, beyond that
distance, the rms fluctuations resemble those in smooth-wall turbulence, in agreement
with the outer-layer similarity hypothesis (Townsend 1976).
In this section, we analyse the effect of roughness on the background turbulence and,
in particular, on the rms fluctuations. The rms fluctuations of the full signal, as shown
in figure 12, contain contributions from both the turbulent-background and roughness-
coherent components. However, the decomposition of equations (3.2) can be used to
derive approximate expressions for the rms velocities from which the rms fluctuations of
the background turbulence can be extracted. For the geometries studied in this paper,
16 N. Abderrahaman-Elena, C. T. Fairhall and R. Garc´ıa-Mayoral
the leading terms are
〈u′2〉 =
〈
u2BT
〉
+
〈
u2RC,u
〉
(5.1a)
+
〈
u2RC,u
〉〈u2BT
U2
〉
+
〈
u2RC,v
〉〈v2BT
v˜2
〉
+
〈
u2RC,w
〉〈w2BT
w˜2
〉
,
〈v′2〉 =
〈
v2BT
〉
+
〈
v2RC,u
〉
(5.1b)
+
〈
v2RC,u
〉〈u2BT
U2
〉
+
〈
v2RC,v
〉〈v2BT
v˜2
〉
+
〈
v2RC,w
〉〈w2BT
w˜2
〉
,
〈w′2〉 =
〈
w2BT
〉
+
〈
w2RC,u
〉
(5.1c)
+
〈
w2RC,u
〉〈u2BT
U2
〉
+
〈
w2RC,v
〉〈v2BT
v˜2
〉
+
〈
w2RC,w
〉〈w2BT
w˜2
〉
,
〈u′v′〉 = 〈uBT vBT 〉+ 〈uRC,uvRC,u〉 (5.1d)
+ 〈uRC,uvRC,u〉
〈
u2BT
U2
〉
+ 〈uRC,vvRC,v〉
〈
v2BT
v˜2
〉
+ 〈uRC,wvRC,w〉
〈
w2BT
w˜2
〉
+ 〈uRC,uvRC,v〉
〈uBT vBT
Uv˜
〉
,
where the angled brackets indicate temporal and x-z-spatial averaging, and the prime
refers to fluctuations with respect to the mean. The full expressions including the terms
that are negligible for our geometries can be found in appendix A. Equations 5.1 are
adequate approximations only for small roughness, as long as equations (3.2) hold.
The coherent–background cross terms, which are zero in the conventional triple de-
composition, can be of the same order of magnitude as the coherent–coherent terms.
Equations (5.1) can be used to extract the rms fluctuations of the background-turbulent
contribution from the full and the roughness-coherent signals.
Compared to the flow over a smooth wall that was located at the plane of the crests,
y+ = 0, the rms fluctuations of the background turbulence are shifted towards the surface,
as if they perceived a smooth wall, or a virtual origin, below the roughness crests. In the
first column of panels in figure 13, the rms fluctuations of the background turbulence are
compared with those in smooth-wall turbulence. The decomposition of equations (5.1)
removes from the background turbulence rms’s the peaks in the immediate vicinity of the
surface observed for the full signal in figure 12. For the uv Reynolds stress, those peaks
are already negligible for the full signal, suggesting that dispersive stresses do not play
an essential role in the onset of roughness effects. The fluctuations are shifted towards
the roughness surface, but otherwise display a similar shape to those of smooth-wall
turbulence close to the wall. We refer to these displacements as virtual origins, since the
rms fluctuations behave as if they had an origin below the roughness crests. This can be
interpreted as the height below the roughness crests at which they would to go to zero if
extended as smooth-wall rms fluctuations.
The virtual origin of the streamwise rms fluctuations, ℓ+u , is obtained as the depth below
the roughness crests at which u′+BT would zero out, when extrapolated from its profile
above the crests. When u′+BT is portrayed versus the height measured from that origin, as
in figure 13(b.2), a good collapse with smooth-wall data is observed in the first few wall
units of height. However, this collapse does not extend outside the roughness-sublayer,
where all curves should converge to the smooth-wall case. According to Townsend’s outer-
layer similarity hypothesis (Townsend 1976), sufficiently far from the surface, effectively
outside the roughness sublayer, all turbulent fluctuations are independent of the surface
condition when normalised in wall units. In the second column of panels in figures 13, the
Turbulence in the transitionally rough regime 17
0
0.5
1
−
〈u
T
v T
〉+
0
1
2
3
〈u
2
+
T
〉1
/
2
0
0.5
1
〈v
2
+
T
〉1
/
2
0 50 100
y+
0
0.5
1
1.5
〈w
2
+
T
〉1
/
2
0 50 100
y+ + ℓ+u
0 50 100
y+ + ℓ+uv
a.1) a.2) a.3)
b.1) b.2) b.3)
c.1) c.2) c.3)
d.1) d.2) d.3)
Figure 13. Rms fluctuations of the background turbulent flow. (a) Reynolds stress, uBT vBT ;
(b) streamwise velocity, uBT ; (c) wall-normal velocity, vBT ; (d) spanwise velocity, wBT . Blue to
red, cases C06 to C24; black, smooth channel; the arrow indicates increasing k+.
18 N. Abderrahaman-Elena, C. T. Fairhall and R. Garc´ıa-Mayoral
0 15 30
k+
0
2.5
5
ℓ+ U
,
ℓ+ u
,
ℓ+ J
Figure 14. Virtual origins for different geometries and k+. ×, ℓ+U ; ◦, ℓ+u ; and +, ℓ+J . Blue
collocated posts; green, spanwise-staggered posts; purple, streamwise-staggered posts; red,
collocated posts of two heights.
origin of the rms fluctuations is set at ℓ+u for all variables. We observe that this virtual
origin does not adequately collapse any rms fluctuation other than that of the streamwise
velocity very near the surface.
We observe that the streamwise virtual origin, ℓu, is related to the apparent origin of
the mean velocity profile. Determining the correct zero-plane for the law of the wall is a
question that has always been present in roughness studies. This reference level for the
mean velocity profile is usually referred to as displacement height, and it is in essence
equivalent to what we have defined as virtual origins, as it is a shift in the y-coordinate.
Jackson (1981) proposed a displacement height, ℓJ , based on the centroid of the friction
and pressure forces acting on the wall, that is, the height where the mean drag appears
to act for the overlying flow (Leonardi & Castro 2010). This can also be calculated from
the average displacement of the total shear stress (Jackson 1981),
ℓ+J =
∫
k+
(
dU+
dy+
−〈uv〉
+
)
dy+. (5.2)
Similarly, since this displacement is essentially a virtual origin, we can also define another
virtual origin for the mean velocity profile, ℓ+U , as the distance from the roughness crests
at which the mean velocity would go to zero. This definition is equivalent to that used
above for ℓ+u , where ℓ
+
U can be obtained by linearly extrapolating the mean velocity profile
at the wall. In figure 14, we observe that ℓ+U , ℓ
+
u and ℓ
+
J are generally similar for different
roughness configurations and sizes.
Further away from the wall, where the influence of the roughness-coherent flow is
negligible, the rms fluctuations suggest that turbulence behaves as smooth-wall, canonical
turbulence with a virtual origin ℓ+uv. The virtual origin of the Reynolds shear stress, ℓuv
appears to be the virtual origin of turbulence outside the roughness sublayer (Go´mez-de-
Segura et al. 2018a; Fairhall et al. 2018). Unlike the rms velocity fluctuations, the near-
wall Reynolds shear stress does not change significantly with roughness. In figure 13(a.3),
we observe that using the virtual origin of its rms fluctuations, ℓuv, not only collapses the
region near the surface, but the entire curve. In the third column of panels in figure 13, the
origin of the rms fluctuations is set at ℓuv for all variables. The rms fluctuations, including
the streamwise ones, converge to smooth-wall turbulence rms fluctuations outside the
roughness-sublayer. Very near the rough surface, different variables would extrapolate to
zero at different heights, as is particularly evident for u′+BT and w
′+
BT , which experience
Turbulence in the transitionally rough regime 19
102λ
+ z
102λ
+ z
102λ
+ z
102 103
λ
+
x
102λ
+ z
102 103
λ
+
x
102 103
λ
+
x
kxkzE
+
uu kxkzE
+
vv −kxkzE
+
uv
Figure 15. Two-dimensional premultiplied spectra of the streamwise and wall-normal velocity
components, and cospectrum of the Reynolds stress at y+ = 8, from top to bottom cases C12,
C15, C18 and C24. Shaded, background-turbulent component; , smooth-wall data at the
equivalent height above the corresponding virtual origin; and , positive and negative
difference between the rough and smooth-wall results. Vertical dashed lines indicate λ+x = 100
and 250. Contours indicate the same value in wall units.
virtual origins shallower than ℓ+uv. Beyond this, up to at least k
+ ≈ 15, the curves show an
excellent collapse with smooth-wall data. This suggests that, up to that k+, turbulence
remains essentially canonical, i.e. smooth-wall-like.
To further explore the modifications of the background turbulence by roughness within
the roughness-sublayer, we analyse its two-dimensional energy spectra and cospectra. As
introduced in §3, the decomposition can be used to obtain the two-dimensional energy
spectra of the background turbulence, without the footprint of roughness. Figure 15
shows the turbulent premultiplied spectra of uBT , vBT and premultiplied cospectra of
uBT vBT , together with smooth-wall results at the equivalent height accounting for the
corresponding virtual origin. For this roughness, the spectra present only small changes
20 N. Abderrahaman-Elena, C. T. Fairhall and R. Garc´ıa-Mayoral
for k+ . 12. As size increases the spectra and cospectra start to be modified: the energy
at large λ+x decreases, while at smaller λ
+
x there is an increase of energy. The decrease of
energy at large wavelengths is particularly clear for the streamwise velocity component.
There is also a noticeable change in the spanwise direction. The spectra and cospectra,
especially that of v, display an increase of energy at small λ+x and a wider range of λ
+
z .
This increase of energy is centred approximately at λ+x ≈ 150, being particularly clear
for Evv.
6. Shear-flow instability
The spectral energy densities in figure 15 indicate the regions where energy increases or
decreases with respect to smooth-wall turbulence. Energy increases at short streamwise
wavelengths, λx, for a broad range of λz , both larger and smaller than the characteristic
λz of smooth-wall turbulence. The values of λx at which energy increases, λ
+
x ≈ 100–200,
appear to be independent of k+. Although significantly more intense, similar modifica-
tions of the energy spectra were observed on riblets, a particular case of roughness. This
increase in energy was found to be due to the formation of a shear flow instability (Garc´ıa-
Mayoral & Jime´nez 2011b). These instabilities are a common feature in obstructed flows
(Ghisalberti 2009), and have been observed on flows over plant canopies (Finnigan 2000)
and permeable substrates (Breugem et al. 2006). Although we do not observed them
directly in our numerical simulations, the concentration of energy for a narrow range of
λ+x suggests a receptivity to this wavelengths that could be connected to a similar shear-
flow instability of the mean velocity profile, for which we analyse the stability properties
in this section.
We focus on Kelvin-Helmholtz-like instabilities, which are essentially spanwise coher-
ent, linear and inviscid. Previous studies have shown that the instability is essentially a
property of the mean profile (Beneddine et al. 2016), although it is modulated by the
effect of the complex substrate on the fluctuating flow (Raupach et al. 1991; White
& Nepf 2007; Zampogna et al. 2016), as well as by viscous effects (Jime´nez et al.
2001; Luminari et al. 2016; Go´mez-de-Segura et al. 2018b). Since we aim to assess this
phenomenon only qualitatively, we conduct a simplified analysis, two-dimensional in x
and y, inviscid and linear, where the flow is allow to fluctuate freely around the mean
profile, neglecting the presence of the solid obstacles. We seek wavelike solutions for
the velocity and pressure perturbations, of the form f = fˆ exp[i(αxx+ αzz − ωt)], which
allows for a modal analysis. The linearised problem becomes then Rayleigh’s equation
(Rayleigh 1879) (
(U − c)(∂yy − k
2)− ∂yyU
)
vˆ = 0, (6.1)
where αx and αz are the wave numbers in the streamwise and spanwise directions
respectively, vˆ is the corresponding perturbation mode of the wall-normal velocity, with
vˆ = 0 at the troughs, k2 = α2x + α
2
z , ω is the complex frequency, and c is the complex
phase velocity defined as ω = αxc. Note that, according to Squire’s theorem, for any
streamwise wavelength αx the most amplified mode is two-dimensional, αz = 0. The
mean velocity profiles, U , are directly extracted from our DNSs, and include the region
below the roughness crests. Notice that the effect of roughness is exclusively introduced
through the mean velocity profile. More precise studies include, for instance, a drag force
model below the roughness or canopy tips (Py et al. 2006; Luminari et al. 2016; Sharma
et al. 2017), at the cost of an increased cost and complexity.
The results of the stability analysis, portrayed in figure 16(a), show an instability
for the mean flow predominantly for wavelengths λ+x ≈ 100–150. This result is in
Turbulence in the transitionally rough regime 21
100 101 102 103
λ+x
0
0.05
0.1
σ
+
0 10 20 30 40
k+
0
0.05
0.1
σ
+ m
a
x
a) b)
Figure 16. (a) Growth rate σ+ = Im
(
ω+
)
of the most amplified mode as a function of the
longitudinal wavelength λ+x . The arrow indicates increasing k
+. (b) Maximum growth rate σ+
as a function of the roughness height. Blue to red, results for the mean velocity profiles for cases
C06 to C36.
rough agreement with the modifications observed in the energy spectra and cospectra
in figure 15, where energy concentrates at λ+x ∼ 150, and does not significantly change
with k+. This could be expected, as previous studies have shown that the lengthscales
of the instability are set by the curvature of the mean velocity profile, independently
of the lenghtscales in the substrate geometry (White & Nepf 2007; Garc´ıa-Mayoral &
Jime´nez 2011b; Go´mez-de-Segura et al. 2018b). However, our simplified model exhibits
a maximum for the instability at k+ ≈ 15, as depicted in figure 16(b), while the changes
in the energy spectra increase monotonically with k+. Nevertheless, the results suggest
that the wavelengths in which energy concentrates in the DNSs are the most receptive
from the point of view of the stability of the mean flow.
7. Skin friction: towards a predictive model
In §1 we have discussed the main drawbacks of the equivalent sand roughness, k+s ,
which for historical reasons has been widely used to characterise rough surfaces. However,
predictive models for k+s are only beginning to appear (Yang et al. 2016; Yang &Meneveau
2016), and the ratio ks/k, which becomes constant once fully rough conditions are
reached, varies through the transitionally rough regime. This is explicitly highlighted by
Jime´nez (2004), who shows that a collapse of the fully rough regime does not guarantee
such a collapse in the transitionally rough regime, as shown in figure 1. Therefore, k+s
is not a suitable parameter to predict ∆U+. Likewise, k+ presents similar problems, as
it is roughly proportional to k+s (Schlichting 1936). Figure 17 shows ∆U
+ as a function
of the roughness element height, k+, and of Jackson’s displacement height, ℓ+J . Both
capture the trend of increasing ∆U+ for increasing roughness size, but display a strong
dependence with the type of roughness surface. Orlandi & Leonardi (2006) find a strong
linear correlation between ∆U+−U+0 and the rms fluctuations of the wall-normal velocity
at the roughness crests, v′+t , as shown in figure 17(c) for our simulations. Similarly,
a linear relationship between ∆U+ − U+0 and ℓ
+
uv is also observed in figure 17(d). As
introduced earlier, ℓ+uv is the shift of the Reynolds stress below the roughness crests, and
can be interpreted as the apparent position of the origin for turbulence. Both v′+t and ℓ
+
uv
22 N. Abderrahaman-Elena, C. T. Fairhall and R. Garc´ıa-Mayoral
0 15 30
k+
0
3
6
−
∆
U
+
0 2 4
ℓ+J
0 0.2 0.4
v′+t
0
3
6
−
∆
U
+
+
U
+ 0
0 4 8
ℓ+uv
a) b)
c) d)
Figure 17. Roughness function represented versus (a) roughness height, (b) Jackson’s
displacement height, (c) wall-normal velocity fluctuations at the roughness crests, and (d) virtual
origin of turbulence. , collocated posts; , spanwise-staggered posts; , streamwise staggered
posts; , collocated posts of two heights; , linear regressions 16.6 v′+t and 0.78 ℓ
+
uv
establish a connection between the roughness function and the effect of the roughness
surface on the flow.
The mean momentum equation is used below to explore this relationship between the
roughness function, ∆U+, and the virtual origin of the Reynolds shear stress, ℓ+uv. The
goal is to obtain an expression for the roughness function. The procedure followed here is
similar to that in Garc´ıa-Mayoral & Jime´nez (2011b) to study the contributions to ∆U+.
In a turbulent channel the mean momentum equation along the streamwise direction is
−〈uv〉+ ν
dU
dy
= u2τ
δ′ − yr
δ′
, (7.1)
where yr is the wall-normal coordinate measured from the virtual origin of the mean
velocity profile, yr = y − ℓU , and the apparent half-height of the channel has previously
been defined as δ′ = δ + ℓU . This allows us to define a common origin yr = 0 for the
mean velocity profiles under different setup configurations. Note that ℓU is positive and
δ′ > δ. Scaling equation (7.1) in viscous units gives
−〈uv〉
+
+
dU+
dy+
=
δ′+ − y+r
δ′+
. (7.2)
Turbulence in the transitionally rough regime 23
This expression is valid above the roughness crests, y+ > 0 or y+r > ℓ
+
U . Integrating
equation (7.2) allows us to obtain U+, and thus an expression for the roughness function,
∆U+, as the difference in U+ between rough and smooth-wall cases. At a distance H+
sufficiently far above the surface, where outer-layer similarity holds and the mean velocity
profile is logarithmic, the roughness function is ∆U+ = U+r (H
+) + U+s (H
+). Let us
denote by the subscripts ‘r’ and ‘s’ the variables in a rough and smooth-wall channel,
respectively. Equation (7.2) can then be integrated between two heights∫ H+
h+s
−〈uv〉
+
s dy
+ + U+s
(
H+
)
− U+s
(
h+s
)
= (H+ − h+s )−
1
2
H+
2
− h+s
2
δ′+s
, (7.3a)∫ H+
h+r
−〈uv〉
+
r dy
+ + U+r
(
H+
)
− U+r
(
h+r
)
= (H+ − h+r )−
1
2
H+
2
− h+r
2
δ′+r
, (7.3b)
where the lower bounds of integration, h+r and h
+
s , can in principle be different for
the rough and the reference smooth case, but for equation (7.1) to hold, h+r > ℓ
+
U is
required. An expression for the roughness function, ∆U+, can be obtained by subtracting
equation (7.3a) from (7.3b). Taking h+ = h+s = h
+
r = ℓ
+
U , we then have
∆U+ = U+r
(
H+
)
− U+s
(
H+
)
= T1 + T2 + T3, (7.4)
where
T1 =−
(∫ H+
h+
−〈uv〉+r dy
+ −
∫ H+
h+
−〈uv〉+s dy
+
)
, (7.5a)
T2 =U
+
r
(
h+
)
− U+s
(
h+
)
, (7.5b)
T3 =−
1
2
(
H+
2
− h+
2
δ′+r
−
H+
2
− h+
2
δ′+s
)
. (7.5c)
The first term, T1, encapsulates the increase in Reynolds stress that roughness produces
compared to that over a smooth wall. Over conventional roughness, the Reynolds stress
increases near the wall and hence the negative contribution of this term towards ∆U+.
Drag-reducing surfaces produce large slip velocities that overcome the effect of T1,
resulting in a net reduction of drag. Similarly, roughness also displays a mean velocity
at the roughness crests, which is captured by the first term in T2. However, when the
change in the virtual origin of the mean velocity profile is accounted for, by the second
term in T2, the net result is generally negative. The first contribution is simply the mean
velocity at the roughness crests, U0. The other contribution to T2 is the mean velocity
of the smooth-wall channel at the roughness crests equivalent height, that is what the
mean velocity would be at the tips if the rough wall had no effect on the mean velocity
profile. The term T2 then represents the difference between the actual mean velocity
at the roughness crests and the ideal velocity that would have been achieved at such
height without roughness. Notice that for small roughness, T2 ≈ 0. As size increases, the
advective terms gain relevance at the roughness crests and the magnitude of T2 increases.
Finally, T3 accounts for Reynolds-number discrepancies between the different simulations,
being zero if δ′+r = δ
′+
s . In essence, the term T3 is obtained by integrating the total stress,
i.e. the linear, right-hand-side of equation (7.2). The intersect of the linear total stress
at different δ+ causes relative variations in T3 of order δ
′+
r /δ
′+
s .
In our simulations, T3 contributes significantly to ∆U
+. The simulations are at slightly
different Reτ , which generates setup-dependent contributions to this term. To isolate
24 N. Abderrahaman-Elena, C. T. Fairhall and R. Garc´ıa-Mayoral
0 10 20 30
0
2
4
6
0 5 10 15
0 5 10 15
k+
0
2
4
0 5 10 15
k+
a) b)
c) d)
Figure 18. Contributions to ∆U+ for (a) collocated posts, (b) collocated posts of two heights,
(c) spanwise-staggered posts, and (d) streamwise-staggered posts. N, T1; , T2; and ⋄, T3; •,
∆U+ = T1 + T2 + T3.
these effects, we propose an alternative breakdown, ∆U+ = T 1 + T 2 + T 3, where
T 1 =−
(
δ′+s
δ′+r
∫ H+
h+
−〈uv〉
+
r dy
+ −
∫ H+
h+
−〈uv〉
+
s dy
+
)
, (7.6a)
T 2 =U+r
(
h+
)
− U+s
(
h+
)
, (7.6b)
T 3 =
(
δ′+s
δ′+r
− 1
)∫ H+
h+
−〈uv〉
+
r dy
+−
1
2
(
H+
2
− h+
2
δ′+r
−
H+
2
− h+
2
δ′+s
)
︸ ︷︷ ︸
T3
.
(7.6c)
These expressions are obtained by adding and subtracting δ′+s /δ
′+
r
∫
〈uv〉+r to equa-
tion (7.5), and rearranging. In this form, the term |T 3| ≪ |T 1|, |T 2|, and the expression
∆U+ = T1 + T2 is still recovered for δ
′+
r = δ
′+
s . By rescaling −〈uv〉
+
r by the ratio
of channel heights, δ+s /δ
+
r , T 1 is less dependent on small variations of the frictional
Reynolds number, and thus cases with slightly different δ+ can be more fairly compared.
Results of equations (7.6) used on our rough geometries are portrayed in figure 18. The
term T 1 contributes the most towards ∆U+. In our geometries, the term T 2 is observed
to always be negative. However, some particular cases with two-dimensional roughness,
such as riblets, have also proven to induce a positive, and therefore drag reducing, T 2
term (Garc´ıa-Mayoral & Jime´nez 2011b).
Turbulence in the transitionally rough regime 25
0 5 10 15
k+
0
2
4
0 5 10 15
k+
0 5 10 15
k+
0
0 5 10 15
k+
0
2
4
a.1) b.1) c.1) d.1)
a.2) b.2) c.2) d.2)
Figure 19. Results from DNSs and estimates based on equations (7.7). N, T1; , T2; •, ∆U+.
(a) Collocated posts; (b) collocated posts of two heights; (c) spanwise-staggered posts; and (d)
streamwise-staggered posts. Solid symbols, estimates; empty symbols, DNSs.
For δ+r = δ
+
s , equations (7.5) and (7.6) simplify to
T 1 = T 1 =
∫ δ′+
s
h+
(
〈uv〉
+
r − 〈uv〉
+
s
)
dy+, (7.7a)
T 2 = T 2 =U+r
(
h+
)
− U+s
(
h+
)
. (7.7b)
Equations (7.7) present in a clearer manner the components of ∆U+, as it is caused
by the change in Reynolds stress as well as the difference between the mean velocity
at the roughness crests and that over a reference smooth wall at the same height from
the virtual origin. Figure 18 indicates that the main contribution to the departure from
the hydraulically-smooth regime is T 1, the change in Reynolds stress. In figure 13, we
observe that up to ∆U+ ≈ 2 the difference embodied in T 1 is simply caused by the offset
of the Reynolds stress, rather than by a change in curve shape.
Now we explore the potential of equations (7.7) for predicting ∆U+. Based on the
discussion in §4 and §5, we suggest a model for the contributions T1 and T2 to estimate
∆U+. Let us assume that the statistics for a turbulent flow over a smooth wall at
the desired Reτ are available. Therefore, only the terms from roughness, U
+
r (h
+) and
〈uv〉
+
r (y
+), need to be modelled. For small roughness size, the velocity at the roughness
crests, U+r (h
+), can be estimated from the laminar model for the coherent flow presented
in §4. In §5 we discuss how for small k+ the main effect of roughness on the Reynolds
stress, 〈uv〉
+
r , is a shift ℓ
+
uv towards the surface, but otherwise the Reynolds stress closely
resembles that of smooth-wall turbulence. As a result the effective displacement of the
Reynolds stress is ℓ+uv− ℓ
+
U , since the origin is at a depth ℓ
+
U below the roughness tips. We
can express this relation as if the Reynolds stress of a rough wall was that of smooth-wall
turbulence shifted to the corresponding virtual origin, i.e. 〈uv〉
+
r (y
+) ≈ 〈uv〉
+
s (y
+
⋆ ), where
y+⋆ is an auxiliary wall-normal coordinate that displaces 〈uv〉
+
s . This auxiliary coordinate
y+⋆ is defined such that near the surface, at y
+
r = h
+, 〈uv〉
+
r (y
+
r = h
+) ≈ 〈uv〉
+
s (y
+
r =
h+ + ℓ+uv − ℓ
+
U ). Notice that a mere shift of 〈uv〉
+
s leads to a new δ
′+. Instead, to keep
δ′+ constant, y+⋆ is linearly transformed, with 〈uv〉
+
r (y
+
r = δ
′+) ≈ 〈uv〉
+
s (y
+
r = δ
′+). The
26 N. Abderrahaman-Elena, C. T. Fairhall and R. Garc´ıa-Mayoral
change in Reynolds stress, accounted for in the term T1, can therefore be expressed as
T1 ≈
∫ δ+
h+
(
〈uv〉
+
s (y
+
⋆ )− 〈uv〉
+
s (y
+)
)
dy+, (7.8)
where
y+⋆ =
(
δ′+ − ℓ+uv
δ′+ − ℓ+U
)(
y+r − ℓ
+
U
)
+ ℓ+uv. (7.9)
Results of this model are portrayed in figure 19. In addition, the values of h+r and U0,
used to estimate T2, are estimated from the laminar model for the roughness-coherent
contribution, presented in §4. The model appears to estimate the initial trend of ∆U+
with relatively good agreement up to values of −∆U+ . 2, i.e. capturing the region
of initial increase of drag with ∆Cf/Cf . 25%. For larger roughness, the results begin
to deviate more significantly, as the roughness-coherent flow becomes intense enough to
disrupt the background turbulence, causing a modification of the background Reynolds
stress more profound than a mere displacement (Fairhall et al. 2018). These modifications
would become even more significant in the fully rough regime, for which the model can
be expected to fail.
In any event, the above results show the potential of the model described by equa-
tions (7.7) and (7.8) to estimate ∆U+ as it departs from the smooth, zero value. Further
work needs to be undertaken in order to extend the model to the entire transitionally
rough regime, as well as to obtain an estimate for ℓ+uv. Additionally, its applicability to
random roughness must be analysed, especially for the purpose of industrial applications.
8. Conclusions
In the present work we have investigated the interaction between roughness and near-
wall turbulence. We have focused on the transitionally rough regime, to capture the effects
that trigger the departure from the hydraulically smooth regime. We propose a triple
decomposition where the roughness-coherent contribution is modulated in amplitude
by the overlying background-turbulent flow. Using this decomposition, a background
turbulence component, essentially free of any footprint from the roughness texture, can
be extracted.
The roughness-coherent component, resulting from the ensemble average of the flow
field, is systematically studied as roughness size increases. This component presents an
exponential decay with y/k. The rate of decay depends on the velocity component,
with the wall-normal velocity experiencing the slowest decay. All components seem to
essentially vanish for y & k. However, the region where background turbulence is affected
by the roughness surface, the roughness-sublayer, extends to a height of y/k ≈ 2–3. A
laminar model is proposed to estimate the roughness-coherent component. The model
results agree well with DNS results for roughness sizes that produce offsets of ∆U+ . 2.
It also provides estimates for the mean velocity at the roughness crests, U+0 , and the
virtual origin of the mean velocity profile, ℓ+U .
Using the proposed triple decomposition, the roughness-coherent contribution is ex-
tracted and the changes produced in the background turbulence can be isolated and
analysed. For a large extent of the transitionally rough regime, ∆U+ . 4, the main effect
of roughness is a displacement of the rms fluctuations and the Reynolds stress in the
negative-y direction, towards the surface. This can be interpreted as turbulence perceiving
the wall at a certain virtual origin below the roughness crests. We observe a good
agreement between ℓ+U , ℓ
+
u and ℓ
+
J , the origins perceived by the mean flow, the streamwise
Turbulence in the transitionally rough regime 27
velocity fluctuations and the displacement height, respectively. This supports Jackson’s
methodology for defining the displacement height for the law of the wall. However, the
near-wall turbulence appears to experience a different virtual origin, essentially that of
the Reynolds stress, ℓ+uv. This origin presents a strong correlation with ∆U
+, similar to
that observed for the wall-normal velocity fluctuations by Orlandi & Leonardi (2006).
As the roughness size increases and the effect on the background turbulence ceases to
be a mere shift, the spectral energy density increases at short streamwise wavelengths,
especially for the wall-normal velocity. This increase tends to concentrate at λ+x ≈ 150.
Results from a simplified linear stability model suggest that this concentration is related
to an increased receptivity for λ+x ≈ 100–150, produced by the inflexionality of the mean
velocity profile.
The mean momentum equation has been integrated to identify different contributions
to the roughness function, ∆U+, and in particular to analyse the role of the virtual
origin ℓ+uv. The main contribution is the change in Reynolds stress. For small roughness,
this is essentially due to the shift ℓ+uv, as the Reynolds stress remains otherwise smooth-
wall-like. The second contribution to ∆U+ is proportional to the defect of the mean
velocity at the height of the roughness crests, compared to a smooth-wall flow. This
defect is generally positive, in which case it increases drag.
N.A-E. and C.T.F. were supported by the Engineering and Physical Sciences Re-
search Council under a Doctoral Training Account, grant EP/M506485/1. This work
was partly supported by the European Research Council through the II Multiflow
Summer Workshop. Computational resources were provided under EPSRC-UK Tier-2
grant EP/P020259/1 by CSD3, Cambridge. We also express our gratitude to A. Sharma
for helpful conversations about shear-flow instabilities and their analysis.
Appendix A
Approximate decomposition of the rms fluctuations using equations 3.2
〈u′2〉 =
〈
u2BT
〉
+
〈
u2RC,u
〉
+
〈
u2RC,u
〉〈u2BT
U2
〉
+
〈
u2RC,v
〉〈v2BT
v˜2
〉
+
〈
u2RC,w
〉〈w2BT
w˜2
〉
+2
(
〈uRC,uuRC,v〉
〈uBT vBT
Uv˜
〉
+ 〈uRC,uuRC,w〉
〈uBTwBT
Uw˜
〉
+ 〈uRC,vuRC,w〉
〈vBTwBT
v˜w˜
〉)
(A 1a)
〈v′2〉 =
〈
v2BT
〉
+
〈
v2RC,u
〉
+
〈
v2RC,u
〉〈u2BT
U2
〉
+
〈
v2RC,v
〉〈v2BT
v˜2
〉
+
〈
v2RC,w
〉〈w2BT
w˜2
〉
+2
(
〈vRC,uvRC,v〉
〈uBTvBT
Uv˜
〉
+ 〈vRC,uvRC,w〉
〈uBTwBT
Uw˜
〉
+ 〈vRC,vvRC,w〉
〈vBTwBT
v˜w˜
〉)
(A 1b)
〈w′2〉 =
〈
w2BT
〉
+
〈
w2RC,u
〉
+
〈
w2RC,u
〉〈u2BT
U2
〉
+
〈
w2RC,v
〉〈v2BT
v˜2
〉
+
〈
w2RC,w
〉〈w2BT
w˜2
〉
+2
(
〈wRC,uwRC,v〉
〈uBT vBT
Uv˜
〉
+ 〈wRC,uwRC,w〉
〈uBTwBT
Uw˜
〉
+ 〈wRC,vwRC,w〉
〈vBTwBT
v˜w˜
〉)
(A 1c)
28 N. Abderrahaman-Elena, C. T. Fairhall and R. Garc´ıa-Mayoral
〈u′v′〉 = 〈uBT vBT 〉+ 〈uRC,uvRC,u〉
+ 〈uRC,uvRC,u〉
〈
u2BT
U2
〉
+ 〈uRC,vvRC,v〉
〈
v2BT
v˜2
〉
+ 〈uRC,wvRC,w〉
〈
w2BT
w˜2
〉
+(〈uRC,uvRC,v〉+ 〈uRC,vvRC,u〉)
〈uBTvBT
Uv˜
〉
+(〈uRC,uvRC,w〉+ 〈uRC,wvRC,u〉)
〈uBTwBT
Uw˜
〉
+(〈uRC,vvRC,w〉+ 〈uRC,wvRC,v〉)
〈vBTwBT
v˜w˜
〉
(A 1d)
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