12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
ROUGHNESS FUNCTION AND TURBULENT STATISTICS WITH
INCREASING ROUGHNESS SIZE FROM DIRECT NUMERICAL
SIMULATIONS
Hiten Mulchandani
Department of Engineering
University of Cambridge
Trumpington Street, Cambridge CB2 1PZ
hm650@cam.ac.uk
Ricardo Garcia-Mayoral
Department of Engineering
University of Cambridge
Trumpington Street, Cambridge CB2 1PZ
r.gmayoral@eng.cam.ac.uk
ABSTRACT
We present results of fully resolved, direct numerical
simulations (DNSs) of turbulent flows over regular arrays of
cylindrical roughness elements. DNSs were conducted for
k+ = 5,10,15, and 20 at Reτ ≈ 190 and for k+ = 20,40, and
60 at Reτ ≈ 380, ranging from the onset of the transitionally
rough regime through to the fully rough regime. Data from
the DNSs are presented and discussed for the roughness func-
tion, equivalent sand-gain roughness, mean flow velocity pro-
files, turbulence statistics, and spectral energy densities. Re-
sults suggest there is a progressive departure from smooth-
wall-like turbulence for all cases except the smallest roughness
size investigated. We hypothesise the differences are related to
the nonlinear interaction of the texture-coherent flow with the
background turbulence and plan to assess the importance of
this mechanism in future work.
Introduction
Many engineering surfaces are rough and cause addi-
tional drag compared to smooth surfaces, and it is of indus-
trial interest to quantify this drag. Sufficiently far above the
roughness elements, it is commonly accepted that smooth-
and rough-wall turbulence behave in a similar manner in what
is known as outer-layer similarity (Clauser, 1956; Townsend,
1956). The effect of the roughness reduces, then, to an off-
set in the mean velocity profile of a rough wall relative to
that of a smooth wall in the log layer. This is given by the
roughness function, ∆U+ = U+S −U+R , measured in the log
layer where the superscript ‘+’ indicates wall-unit scaling with
kinematic viscosity and the friction velocity, and the subscripts
‘S’ and ‘R’ indicate smooth and rough walls, respectively.
Defining the skin-friction coefficient as C f = τw/(ρU2δ /2) =
2/U+2δ , the impact of ∆U
+ on C f through the decrease in
U+δ for rough walls becomes immediately apparent (Spalart
& McLean, 2011; Garcı´a-Mayoral & Jime´nez, 2011; Garcı´a-
Mayoral et al., 2019; Chung et al., 2021). When expressed
in wall units, ∆U+ is generally believed to be independent
of the Reynolds number for a given roughness geometry and
size, k+ (Flack et al., 2007). In turn, the change in C f de-
pends on the Reynolds number through the reference smooth
U+δ . The offset increases with k
+, but how it varies greatly
depends on the roughness geometry and is difficult to pre-
dict a priori (Jime´nez, 2004; Chung et al., 2021). To cir-
cumvent this difficulty, an equivalent “sand-grain roughness”,
k+s , is often employed to characterize the effect of the surface,
so that the actual surface is referred to the sand-grain surface
that produced the same ∆U+ in the pioneering experiments of
Nikuradse (1933). This, however, simply transfers the prob-
lem from predicting ∆U+ to predicting k+s , as there is a one-
to-one relationship between both quantities (Bradshaw, 2000;
Abderrahaman-Elena et al., 2019). Furthermore, the ratio ks/k
for a given surface only becomes constant in the fully rough
regime (Jime´nez, 2004; Chung et al., 2021), when it becomes
equal to k+s,∞/k
+ and the curve k+s,∞-∆U+ becomes universal.
This implies that k+s , like ∆U+, does not exclusively depend on
the surface geometry but also on the flow, i.e., it is a hydraulic
property, and only becomes a geometric property in the fully
rough regime. It is therefore important to understand the phys-
ical mechanisms at play in determining ∆U+, or k+s , up to the
roughness size for which the flow becomes fully rough. Be-
yond this point, once the curve becomes universal, the practi-
cal interest in understanding the physical mechanisms is more
limited.
Toward this aim, we conduct fully resolved, direct nu-
merical simulations (DNSs) of turbulent flows over regular ar-
rays of cylindrical roughness elements. The roughness func-
tion varies from the onset of the transitionally rough regime
through to the fully rough regime. When the roughness is
much smaller than the smallest eddies in the near-wall flow,
k+ ≤ 20, the overlying turbulent flow perceives the near-wall
flow to be smooth-wall-like (Abderrahaman-Elena et al., 2019;
Ibrahim et al., 2021) and the roughness is perceived as a ho-
mogenised boundary condition by the overlying flow (Bottaro,
2019). As the roughness size becomes comparable to the size
of the near-wall turbulent eddies, the overlying flow begins to
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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Figure 1. Schematic plan view of the staggered cylindrical
roughness elements investigated (Vishwanathan et al., 2021).
perceive the non-homogeneity of the texture. In the context
of alternating slip/no-slip textures, Fairhall et al. (2019) ob-
served that the overlying flow still perceived a homogenised
boundary condition from the surface up to k+ ≈ 50, but noted
that the texture caused additional dissipation in the flow above
from k+ ≈ 15. Fairhall et al. (2019) proposed that this addi-
tional dissipation was caused by the nonlinear interaction be-
tween the texture-coherent flow and the background, texture-
incoherent turbulence. We aim to assess the importance of this
mechanism for roughness, and ultimately to aid in the devel-
opment of physics-based models that can predict ∆U+ a priori
without resorting to costly experiments, simulations, or corre-
lations (if available) to similar surfaces.
Numerical Method
The DNSs solve the incompressible flow in a periodic
channel driven by a constant mean pressure gradient with
roughness on the top and bottom walls, imposed using im-
mersed boundaries, using a code adapted from Sharma &
Garcı´a-Mayoral (2020a,b). The channel is of size 2piδ x
2(δ + k) x piδ in the streamwise, wall-normal, and spanwise
directions, respectively, where δ is the channel half-height
from the tips of the roughness to the center of the channel
and k is the roughness height. A spectral discretisation is
employed in the streamwise and spanwise directions and a
second-order central difference scheme on a staggered grid is
employed in the wall-normal direction. The grid is stretched
such that ∆y+min ≈ 0.4 near the walls and ∆y+max ≈ 4 in the cen-
ter. The code uses a ‘multiblock’ grid which allows finer res-
olution near the walls compared to the channel center to prop-
erly resolve the flow between the roughness elements. At the
center of the channel, the grid resolution is standard for DNSs,
with ∆x+ ≤ 8 and ∆z+ ≤ 4. The resolution near the walls in
the streamwise and spanwise directions is given in Table 1 for
the list of the cases investigated at their respective frictional
Reynolds numbers. Time integration is carried out using a
three-step Runge–Kutta scheme with a fractional step, pres-
sure correction method that enforces continuity (Le & Moin,
1991), for which the time-step is set by a fixed advective CFL
number of 0.7.
A schematic illustration of the staggered pattern of cylin-
drical roughness elements used in the simulations is shown in
Figure 1. The ratio of element spacing to its height was fixed
at s/k = 3.46 and the ratio of element diameter to its spacing
was fixed at d/s = 0.45. The instantaneous flow realizations
in Figure 2 show simulation results over the texture geometry
for k+ = 10,15, and 20 at Reτ ≈ 190.
Results and Analysis
The measured roughness function against equivalent
sand-grain roughness is shown in Figure 3 for the cases stud-
Table 1. Roughness height, frictional Reynolds number, and
resolution near the walls in the streamwise and spanwise di-
rections for the cases investigated. Simulations marked with
the (∗) superscript are from Adams (2021).
Name k+ Reτ ∆x+ ∆z+
K05∗ 5.25 187 0.96 0.96
K10∗ 10.36 188 1.44 1.44
K15∗ 15.61 190 2.16 2.16
K20a∗ 20.89 191 2.89 2.89
K20b∗ 20.73 379 2.89 2.89
K40 42.51 389 2.89 2.89
K60 64.67 399 2.89 2.89
ied, overlaid with values for other rough surfaces from Jime´nez
(2004). The cases investigated range from the onset of the tran-
sitionally rough regime, for k+ ≈ 5, through to the fully rough
regime beyond k+ = 15. The roughness function measured
from the DNSs, ∆U+m , is obtained by subtracting the rough-
wall mean velocity profile from the smooth-wall mean velocity
profile and averaging over the log layer.
Figure 4 presents a comparison between the measured
roughness function from the DNSs and the corresponding pre-
dicted values. The predicted roughness function, ∆U+p , is ob-
tained as follows. We define y+ = 0 at the plane of the rough-
ness tips. For small roughness, k+ ≈ 5, turbulence is smooth-
wall-like, except for an offset, `+T , such that it perceives an
apparent ‘virtual’ origin at y+ =−`+T (Luchini, 1996; Ibrahim
et al., 2021). Turbulence then remains essentially unchanged
compared to that over a smooth wall, except for an offset given
by the virtual origin. We find the value of the offset by fitting
the curve representing the Reynolds shear stress to smooth-
wall data, as shown in Figure 5f. As long as turbulence remains
smooth-wall-like, the smooth-wall and rough-wall Reynolds
stresses are the same at each height measured from the vir-
tual origin of the turbulence, y+ = −`+T , i.e., they collapse.
From a mean momentum balance, this implies that the mean
velocity profiles for rough and smooth walls will curve in the
same way above the tips, but will be offset by a constant value
which can, for instance, be measured at the plane of the tips
(Go´mez-de Segura & Garcı´a-Mayoral, 2019). At this plane,
the mean velocity for the rough case is U+t , while at the cor-
responding plane the smooth-wall mean velocity is U+S (`
+
T ).
The predicted roughness function assuming smooth-wall-like
turbulence is thus ∆U+ =U+S (`
+
T )−U+t , as portrayed in Fig-
ure 4. Discrepancies in the results between the measured and
predicted ∆U+ for k+ > 5 suggest that a virtual origin frame-
work is insufficient to accurately estimate the roughness func-
tion beyond the smallest roughness size.
Mean velocity profiles, turbulent velocity fluctuations,
and Reynolds stresses are shown in Figure 5 for k+ = 5 to
k+ = 20 at Reτ ≈ 190. For small roughness, k+ = 5, there is
good collapse of the mean streamwise velocity and the wall-
normal and spanwise root mean square (r.m.s.) velocity fluc-
tuations shifted by the virtual origin onto the smooth-wall pro-
file. The streamwise r.m.s. velocity fluctuations, however,
do not collapse onto the smooth-wall profile near the wall.
Ibrahim et al. (2021) argued that this does not essentially af-
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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Figure 2. Instantaneous realizations of the fluctuating streamwise velocity from the DNSs of a regular array of cylindrical roughness
elements at k+ = 10,15, and 20 at Reτ ≈ 190.
k+ = 5
k+ = 10
k+ = 15
k+ = 20
k+ = 40
k+ = 60
102101100
k+s∞
0
5
10
∆U
+
Figure 3. Roughness function against equivalent sand-grain
roughness for the present regular staggered cylinders (squares
for Reτ ≈ 190 and diamonds for Reτ ≈ 380), compared with
uniform sand (circles), uniform packed spheres (white trian-
gles), galvanised iron (dotted line), tar-coated cast iron (dashed
line), wrought iron (dotted dashed line), interpolation (solid
line) and riblets (black triangles). Adapted from Jime´nez
(2004).
fect the structure of near-wall turbulence as the latter’s vir-
tual origin is essentially determined by the origin perceived by
the quasi-streamwise vortices, which mainly induce only wall-
normal and spanwise velocities near the surface. For larger
roughness sizes, the near-wall cycle is more severely disrupted
and turbulence is no longer smooth-wall-like, as evidenced by
a lack of collapse of the data for k+ ≥ 10.
Previous studies suggest that for roughness sizes k+ ≤ 20,
it is sufficient to conduct simulations at Reτ ≈ 200 to cap-
ture the effect on the outer layers of the flow, i.e., the rough-
ness function (Abderrahaman-Elena et al., 2019). To analyze
the effect of frictional Reynolds number on the mean veloc-
ity profile and turbulence statistics, we compare the results at
Reτ ≈ 187 with Reτ ≈ 376 for k+ = 20, in contrast with the
results at Reτ ≈ 180 with Reτ ≈ 360 for smooth-wall data,
as shown in Figure 6. For smooth-wall data, the observed
differences are consistent with changes observed in the fric-
tional Reynolds number. For rough-wall data, the trends in
the mean velocity profiles are very similar at the two fric-
tional Reynolds numbers. Away from the roughness elements,
y+ > 50, the Reynolds stresses from rough-wall simulations
coincide with those from smooth-wall simulations at their cor-
responding frictional Reynolds numbers, which is indicative of
the recovery of outer-layer similarity. Minor differences in the
collapse are caused by small differences in the corresponding
frictional Reynolds numbers. These arguments support the use
of DNSs at a low frictional Reynolds number.
From the observations of Fairhall et al. (2019) in the con-
text of the alternating slip/no-slip textures mentioned earlier,
the differences that arise from extra Reynolds stresses are hy-
pothesised to be due to the nonlinear interaction of the back-
ground turbulence with the dispersive flow. The spectral en-
ergy densities in Figure 7, which portray the distribution of
energy across different length scales in the flow at y+ = 4
above the roughness tips, demonstrate these effects. There is
a progressive departure from smooth-wall energy densities to-
wards shorter and wider wavelengths as the roughness height
increases from k+ = 5 to k+ = 60. The nonlinear interaction
between the texture-coherent flow and background turbulence
is evident from the distortion in energy contours in the vicinity
of the concentration of energy representing the texture-induced
flow from k+ = 10 onward. We are interested in characteriz-
ing deviations from smooth-wall turbulence and the dispersive
flow from this data.
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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Figure 4. Results for (a) the mean velocity at y+ = 0 for the rough walls, U+t , at Reτ ≈ 190 ( ) and Reτ ≈ 380 (- - - -) and a
smooth wall,U+S (`
+
T ), at Reτ ≈ 190 ( ) and Reτ ≈ 380 (- - - -), and (b) measured values of the roughness function from the DNSs,
∆U+m , at Reτ ≈ 190 ( ) and Reτ ≈ 380 (- - - -) and corresponding predicted values from the virtual origin framework, ∆U+p , at
Reτ ≈ 190 ( ) and Reτ ≈ 380 (- - - -).
Figure 5. Results from DNSs for k+ ≈ 5 ( ), k+ ≈ 10 ( ), k+ ≈ 15 ( ), and k+ ≈ 20 ( ) with Reτ ≈ 190, and
a smooth wall with Reτ ≈ 180 ( ) for (a) the mean streamwise velocity profile with the velocity at the tips subtracted, (b) r.m.s.
velocity fluctuations, (c) Reynolds shear stresses, (d) mean streamwise velocity profile shifted by the turbulent virtual origin with the
predicted velocity at the virtual origin subtracted, (e) r.m.s. velocity fluctuations shifted by the virtual origin, and ( f ) Reynolds stresses
shifted by the virtual origin.
Summary
In this paper, results from DNSs of turbulent flows over
regular arrays of cylindrical roughness elements are presented
and discussed. DNSs were conducted for k+ = 5,10,15, and
20 at Reτ ≈ 190 and for k+ = 20,40, and 60 at Reτ ≈ 380,
ranging from the onset of the transitionally rough regime
through to the fully rough regime. Mean streamwise veloc-
ity profiles, turbulence statistics, and spectral energy densities
from the DNSs suggest there is a progressive departure from
smooth-wall-like turbulence for all cases except the smallest
roughness size investigated. We hypothesise the differences
are related to the nonlinear interaction of the texture-coherent
flow with the background turbulence and plan to assess the im-
portance of this mechanism in future work.
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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Figure 6. Results from the DNSs for k+ ≈ 20a at Reτ ≈ 187 ( ) and Reτ ≈ 376 (- - - -), and a smooth wall at Reτ ≈ 180
( ) and Reτ ≈ 360 (- - - -) for (a) the mean streamwise velocity profile with the velocity at the tips subtracted, (b) r.m.s. velocity
fluctuations, and (c) Reynolds shear stresses.
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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Figure 7. Pre-multiplied two-dimensional spectral energy densities at y+ ≈ 4 above the roughness tips for (a1)− (g1) kxkzE+uu,
(a2)− (g2) kxkzE+vv, (a3)− (g3) kxkzE+ww, and (a4)− (g4) −kxkzE+uv. Results for k+ ≈ 5 ( ), k+ ≈ 10 ( ), k+ ≈ 15
( ), and k+ ≈ 20 ( ) at Reτ ≈ 190, k+ ≈ 20 ( ), k+ ≈ 40 ( ), k+ ≈ 60 ( ) at Reτ ≈ 380, and a smooth wall
at Reτ ≈ 180 and Reτ ≈ 360 (filled contours).
6