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

EFFECT OF THE SURFACE MORPHOLOGY ON ROUGH WALLTURBULENCE

Yoshiki Yamamoto, Shinya Tabata, Yusuke Kuwata*, Kazuhiko Suga
Department of Mechanical Engineering,Osaka Prefecture University
Sakai, Osaka 599-8531, Japan.
kuwata@me.osakafu-u.ac.jp

ABSTRACT

rough surface. In addition, a further difﬁculty arises in predicting the effects of wall roughness on ∆U + in the transitionallyrough regime, because ∆U + in this regime depends not only
on ks but also on the other parameters (Flack & Schultz, 2010;
Flack et al., 2012). The effects of the surface morphology
on the roughness function or the equivalent sand-grain roughness have been extensively studied. One of the well-known
important parameters that have a large impact on the roughness effect is the skewness factor Sk, which is deﬁned as thestatistical moment of the surface elevation. Flack & Schultz(2010); Forooghi et al. (2017); Kuwata & Kawaguchi (2019);
Kuwata & Nagura (2020) showed that the surfaces with thepositive skewness yield larger frictional resistance. Anotherwell-studied parameter is the effect slope, which quantiﬁes thesteepness of the surface undulations (Napoli et al., 2008). The
recent direct numerical simulation (DNS) study by (Ma et al.,2020) showed that ∆U + can be expressed as a function of a
new coupling scale ESk+ , which is a product of the effective
slope ES and the roughness height k:

To clarify the effects of the surface morphology on roughwall turbulence, we conducted experiments on turbulent ﬂows
over three-dimensional irregular rough surfaces in which theskewness factor and effective slope were systematically varied.
For the rough surface, the root-mean-square roughness was remained ﬁxed while the skewness factor was Sk = ±0.4, and theeffective slope, ES, was varied from 0.09 to 0.72. It is foundthat the roughness function ∆U + for the positively-skewed surfaces (Sk = +0.4) is larger than that for the negatively-skewed
surfaces (Sk = −0.4), and ∆U + increases with increasing theeffective slope. The transitional behavior to the fully roughregime strongly depends on ES but not on Sk: the steep surfaces tend to lead to a sudden transition to the fully roughregime, whereas the transitional behavior for the wavy surfaces with the small ES value is close to the Colebrook-typetransition. The equivalent sand-grain roughness linearly increases with increasing the ES value from 0.09 to 0.36, while
the increasing trend slows down when the ES value further increases. It is also found that the effects of the skewness factor
on the equivalent sand-grain roughness do not largely dependon the ES values.

∆U + = 2.66[log(ESk+ ] +C,(2)

where C = 1.46 is adopted for the three-dimensional sinusoidalrough walls. However, it is still unclear on the combined effects of those parameters, and this motives us to examine theeffects of ES and Sk on the rough wall turbulence from thetransitionally rough to fully rough regimes. In this study, we
focus on the two typical morphological parameters of roughsurfaces, namely the skewness factor Sk and effective slope
ES, and we aim to clear the effect of Sk and ES from the transitionally rough regime to the fully rough regime.

Background
Surfaces in engineering systems typically have roughness,and cannot be regarded as hydraulically smooth, particularly
for high Reynolds number ﬂows. Familiar examples include asurface of a turbine blade in harsh operating conditions and asurface with deposition of incombustible ash in internal combustion engines. It is well established that in the fully roughregime where wall roughness protrudes into the logarithmic
region, a downward shift value in the inner-scaled mean velocity ∆U + follows a universally-accepted correlation (Flack& Schultz, 2010):

Experimental methods
1B − ∆U + + ln(ks+ ) = 8.5,
κExperiments were conducted in the rectangular duct witha bottom rough surface. A sketch depicting the experimental
facility is provided in Figure 1. Tap water from a header tankgoes into a ﬂow straightener where the ﬂow is conditioned byhoneycomb-bundled, and the ﬂuid temperature was recorded
by a digital thermometer (FD-T1, Keyence). The conditioned
ﬂow develops in the upstream portion of the rectangular ductwhose cross-section is 50mm in height (H), and 400mm inwidth (W ). This gives the aspect ratio of W /H = 8 which is
sufﬁcient to assume that the ﬂow is two-dimensional in the
middle of the duct. The duct consisted of smooth acrylic(1)where κ is the von Kármán constant, B is the log-law intercept for smooth wall turbulence, and ks+ is the inner-scaled
equivalent sand-grain roughness. The correlation suggests thatin the fully rough regime, the effect of wall roughness on themean velocity can be predicted once we obtain ks for the roughsurface of interest. However, there is still no universal correlation that estimates ks from morphological information of the1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Nd-YAG lasersmooth wallflow
water tank

90mm
(10 plane)

50mm

manometer

400mm

3400mmvalve

yrough wall

digital thermometer

2820mm

CCD camera

2000mm

z

flow

x

valve

y

PIV system

4500mmstraightenerx
digital flow meterpump

Figure 1.

Experimental apparatus.

Sk=+0.4
ES=0.09

Sk=+0.4
ES=0.18

Sk=+0.4
ES=0.36

Sk=+0.4
ES=0.72

Sk=-0.4
ES=0.09

Sk=-0.4
ES=0.18

Sk=-0.4
ES=0.36

Sk=-0.4
ES=0.72

Figure 2.

Three-dimensional irregular rough surfaces with the Sk and ES values.

Rough surface

walls, and a rough surface was considered for the bottom of
the duct. The fully developed turbulent ﬂow was measured at
4.5 m (90H) from the duct entrance. A pressure difference wasmeasured by a U-tube manometer with pressure taps along thecenterline of the top wall. The pressure taps were spaced 2mapart, and the tap in the upstream position was located at 2.8m
(56H) from the duct entrance. The ﬂow rate was controlled
by the pump with the power converter and adjusted such thatthe bulk Reynolds number was close to the target value in therange of Re = 2, 000 to 45, 000. Here, the Reynolds number isdeﬁned as Re = Ub (H − km )/ν where km is the mean surface
height, and Ub is the bulk mean velocity.

In this study, we considered 8 rough surfaces in which the
Sk and ES values were systematically varied while the rootmean-square roughness remained ﬁxed: krms = 0.85mm. The
root-mean-square roughness krms represents the variance fromthe mean surface height km , which is deﬁned as follows:
2
krms
=

1
Lx Lz

Z Z
x z

(h(x, z) − km )2 dxdz,(3)where Lx and Lz respectively denote the streamwise and spanwise lengths of the rough surface, and h(x, z) is the surface
height. Here, the mean surface height km is given as follows:

The ﬂow properties were measured by a two-component
PIV system, which consists of a dual pulsed Nd-YAG laserdelivering 532 nm beams with pulse energy of 70 MJ (EverGreen, Quantel), CCD camera of 20.4 fps (FlowSense EO
4M, Dantec Dynamics), the camera lens of Micro-Nikkor
55 mm f/2.8 (Nikon) and computer for data acquisition. A
single recorded frame covers the zone of 50 × 50mm2 with
2080 × 2080 pixel. The nylon powder (Kanomax) was used
for the tracer particle, and the seeding density was adjusted
to obtain more than 30 particle-image pairs in each interrogation window whose size was set to 32 × 32 pixel. Each image
was processed to produce 127 × 127 vectors from the interrogation windows. The image sampling rate is 5Hz (the statistical time is almost 200s) and 1000 image pairs are processedin this study. As the rough surfaces used in this study arethree-dimensional irregular rough surfaces, the mean proﬁles
above the rough surfaces are inhomogeneous in the streamwise
and spanwise directions. Therefore, to obtain the spatial andReynolds averaged proﬁles above the rough surfaces, we measured 10 x − y planes with an offset of 10mm intervals from acenter plane, and an ensemble-averaged value is also averaged
over 10 x − y planes.

1
Lx Lz

km =

Z Z

(h(x, z)) dxdz.

(4)

x z

The skewness factor Sk quantiﬁes the asymmetry of the probability density function of the roughness height elevation, which
is deﬁned as follows:
Sk =

1
3 L L
krms
x z

Z Z
x z

(h(x, z) − km )3 dxdz.

(5)

The effective slope ES represents the steepness of the roughsurface undulation, which is deﬁned as follows:

ES =

1
Lx Lz

Z Z
x z

∂ h(x, z)
dxdz.
∂x

(6)

We numerically generated the original surface with ES = 0.09
and Sk = +0.4, by superimposing differently sized hyperbolic2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
shape roughness elements (Kuwata & Nagura, 2020). Basedon the original rough surface, we changed the sign of Sk by
inverting the surface height, whereas the ES value was increased by reducing the surface width in the streamwise andspanwise directions while preserving the surface height. The
numerically generated 8 rough surfaces were duplicated by a
3D printer as shown in Figure 2, in which the ES value ranges
from 0.09 to 0.72, and the Sk value takes either of the values−0.4 or +0.4.

Results and discussions
Figure 3 shows the inner-scaled mean velocity proﬁlesfor the surfaces with ES = 0.36 at Re ≈ 2.0 × 103 , 4.0 × 103 ,and 1.5 × 104 together with the smooth wall proﬁle at Re ≈4.1 × 104 (Kawamura, 2008). Here, the normal distance fromthe mean surface location km is used as the distance from therough surface. The ﬁgure conﬁrms that the mean velocity proﬁles over the rough surface are considerably lower than thesmooth wall proﬁle, while the proﬁles seem to maintain thelogarithmic proﬁle away from the rough walls. The downward
shift value from the smooth wall proﬁle, which is referred to as
the roughness function ∆U + , is 11.9 for the positively-skewed
surface (Sk = +0.4) at Re ≈ 1.5 × 104 in Fig.3(a), while it is
9.9 for the negatively-skewed surface (Sk = −0.4) at the same
Reynolds number in Fig.3(b). That is, ∆U + for the positivelyskewed surfaces is larger than that for the negatively-skewed
surfaces. We also observe that the ∆U + progressively increases with increasing Re. Given that the roughness function
measures an increase in the skin friction coefﬁcient, the ﬁgure conﬁrms that the positively-skewed surfaces yield larger
friction resistance compared with the negatively-skewed surfaces. This is consistent with the previous experimental andDNS studies (Flack & Schultz, 2010; Forooghi et al., 2017;
Kuwata & Kawaguchi, 2019; Kuwata & Nagura, 2020).
Figure 4 shows the mean velocity proﬁle in velocitydefect form for the surfaces with ES = 0.36 together with thesmooth wall proﬁles at Re ≈ 3.2 × 103 (Iwamoto et al., 2002)
and Re ≈ 4.1 × 104 (Kawamura, 2008). Here, Uδ+ denotes the
maximum peak value of U + , and δ is the boundary layer thickness for the rough wall side, which is computed as the distancefrom the mean surface height to the maximum velocity location. The ﬁgure conﬁrms that for the cases with Sk = +0.4
and Sk = −0.4, the mean velocity proﬁle in the velocity-defectform shows similarity to some extent for the cases. However,the Reynolds number dependence on the mean velocity defectis not consistent with the smooth wall cases. The possible explanation is the effects of the top smooth wall. The ﬂows forsmooth wall cases are bounded by two parallel smooth walls,
whereas the ﬂows in the present experiments are bounded by
the top smooth and bottom rough walls. Hence, the top smoothwall may affect the mean velocity proﬁles on the rough wallside.
To better understand the Reynolds number dependenceof the roughness function, we focus on the behavior of ∆U +against ks+ in Figure 5. The solid line shows the experimental result for the sand-grain surface by Nikuradse (Nikuradse,
1933), and the broken line shows the experimental result for
the wavy surface by Colebrook (Colebrook, 1939). Here, theequivalent sand-grain roughness ks is determined from Eq.(1).
Fig.5(a) clearly conﬁrms that ∆U + in the fully rough regime
depends solely on ks+ , whereas the transitional behavior to
the fully rough regime signiﬁcantly differs depending on thesurface geometry. The same trend can be observed for thenegatively-skewed surfaces in Fig.5(b). In Fig.5 (a,b), it is

Figure 3. Inner-scaled streamwise mean velocity proﬁles atdifferent bulk mean Reynolds numbers for ES = 0.36: (a)Sk =+0.4 and (b)Sk = −0.4. The DNS data for smooth wall caseby (Kawamura, 2008) is included for comparison.
found that ∆U + for the steep rough surfaces with ES = 0.36
and 0.72 exhibits a sudden transition to the fully rough regime,
and the onset of the fully rough regime is delayed in comparison with the asymptote for the sand-grain by Nikuradse (Nikuradse, 1933). For the case with ES = 0.18, the asymptotic behavior of ∆U + is close to the Nikuradse-type transition. For
the wavy surface with ES = 0.09, the asymptotic behavior of∆U + is the most moderate and close to the Colebrook-typetransition (Colebrook, 1939). Therefore, it is conceivable thatthe asymptotic behavior to the fully rough regime depends on
the steepness of the rough surfaces but not on the skewness
factor.
To see the Reynolds number dependence of the turbulence intensities, Figure 6 shows the proﬁles of the streamwise and wall-normal root-mean-square velocity ﬂuctuations
for the case with ES = 0.36 and Sk = +0.4. For comparison,
the DNS data from Kawamura (2008); Iwamoto et al. (2002)
are included. The ﬁgure conﬁrms that the proﬁles of u+
rms andv+
rms near the rough surfaces deviate largely from the smoothwall proﬁles, whereas those away from the rough surfaces
are rather close to the DNS data irrespective of the Reynolds
number. For the streamwise component, the maximum peakvalue of u+
rms is signiﬁcantly reduced by the wall roughness,and the maximum peak value decreases with increasing the
Reynolds number. In contrast, for the wall-normal component,
the maximum peak value of v+
rms increases with increasing the
Reynolds number. This trend is consistent with the experimental studies (Flack et al., 2007; Flack & Schultz, 2014) and DNS
studies (Forooghi et al., 2018; Kuwata & Kawaguchi, 2019). It

3

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

Figure 5. Roughness function ∆U + against the inner-scaled
equivalent sand-grain roughness ks+ together with the correlations from (Nikuradse, 1933; Colebrook, 1939): (a)Sk = +0.4
and (b)Sk = −0.4.

Figure 4. Mean velocity proﬁle in velocity-defect form at
different bulk mean Reynolds numbers for ES = 0.36: (a)Sk =+0.4 and (b)Sk = −0.4. For comparison, the DNS data from
Kawamura (2008); Iwamoto et al. (2002) are included.

is also suggested that the turbulence near the porous wall tends
to be more isotropic as the Reynolds number increases.
The effects of the steepness on the turbulence intensities are shown in Figure 7 where the proﬁles of the streamwise and wall-normal turbulence intensities for the cases with
Sk = +0.4 at the Reynolds number of Re ≃ 4000 are shown.
+We observe that the proﬁles of u+
rms and vrms away from therough surfaces are close to each other, whereas the effects ofthe ES values are clearly visible near the rough surfaces. As
the ES value increases, the maximum peak value of u+
rms tends
to decrease but v+
rms tends to increase. Although the results are
not shown here, the effect of the steepness is consistent for theother Reynolds number cases and the negatively-skewed roughsurface cases.
To better understand the effect of Sk and ES on the equivalent sand-grain roughness, Figure8 displays ks /krms for all
rough surface cases. The ﬁrst notable observation is that theks /krms value increases with the ES value, and the ks value
for the positively-skewed surface is consistently larger thanthat for the negatively-skewed surface. The effects of the ES
and Sk values are in line with the earlier ﬁndings by Napoli
et al. (2008); Chan et al. (2015); Kuwata & Kawaguchi (2019);
Flack et al. (2020); Kuwata & Nagura (2020). Interestingly,
the ks value for Sk = +0.4 is approximately 1.7 times larger
than that for Sk = −0.4 irrespective of the ES value. This increase ratio roughly corresponds to the value of 2.4 obtainedby the correlations of ks /krms = 2.48(1 + Sk)2.24 (Sk > 0) andks /krms = 2.73(2 + Sk)−0.45 (Sk < 0) proposed by Flack et al.
(2020), and 1.7 from the correlation of ks /krms = 4.0(1 +Figure 6. Comparison of the streamwise and wall-normal
turbulence intensities at different bulk mean Reynolds numbers together with the DNS data from Kawamura (2008);
Iwamoto et al. (2002). The position of the maximum roughness crest is denoted by the broken line in the ﬁgure.

0.17Sk)4 proposed by Kuwata & Kawaguchi (2019). The otherimportant ﬁndings from Fig.8 is that the ks /krms value linearly
increases with the ES value when ES ≤ 0.36, while the increase ratio shows down when the ES value further increases.
The similar observation was found by (Ma et al., 2020) who
showed that the roughness length scale multiplied by ES wellcorrelates with the roughness function. This suggests that4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022+
between ∆U + against v+
rms does not hold when vrms > 0.9. The
reason is not clear; however, the discrepancy with the correlation of Eq.(7) may be attributed to differences in the ﬂow
conditions and rough surface geometry with those by Orlandi
et al. (2006); Orland & Leonardi (2008).

Figure 7. Effects of the effective slope on the streamwise andwall-normal turbulence intensities together with the DNS datafrom Kawamura (2008). The position of the maximum roughness crest is denoted by the broken line in the ﬁgure

when the ES value is moderate (ES < 0.4), the roughness
length scale multiplied by the ES value, e.g., ESkrms , may be
one of the candidates for predicting the equivalent sand-grainroughness.

Figure 9.
crest.

Variation of ∆U + against v+
rms at the roughness

CONCLUSION
We performed the PIV measurements for threedimensional irregular rough surfaces in which the skewness
factor and effective slope were systematically varied to clarify
the effects of the surface morphology on the frictional resistance. For the present rough surface, the root-mean-squareroughness height was ﬁxed, while the skewness factor was
Sk = ±0.4, and the effective slope, ES, was varied from 0.09
to 0.72. We discuss the effects of ES and Sk on the transitional behavior to the fully rough regime. It is revealed thatthe transition to the fully rough regime becomes steeper as ES
increases, and the transitional behavior does not depend on theskewness factor Sk. Discussion on the effects of ES and Sk on
the equivalent sand-grain roughness shows that the equivalent
sand-grain roughness for the surfaces with Sk = +0.4 is consistently larger than that for Sk = −0.4, and the increase ratio
does not strongly depend on the ES value. It is also found thatthe equivalent sand-grain roughness linearly increases with the
ES value, while the increase ratio slows down for the steep
rough surfaces with ES > 0.36, suggesting that the product of
the roughness length scale krms and ES can be used for predicting the equivalent sand-grain roughness when the ES value ismoderate (ES ≤ 0.36).

×1.7
×1.4

×1.7
×2.1

Figure 8. Effects of the ES and Sk values on the equivalent
sand-grain roughness. The ratio of the ks /krms for Sk = +0.4
to that for Sk = −0.4 is also shown.
Finally, we discuss the other parametrization for ∆U +with the wall-normal velocity ﬂuctuations. Orlandi et al.
(2006); Orland & Leonardi (2008) reported that the effectsof the wall roughness are strongly connected with the wallnormal velocity disturbances, and the roughness function canbe reasonably predicted by the wall-normal turbulence intensity:

REFERENCES∆U+= (B/κ )v+
rms ,Chan, L., MacDonald, M., Chung, D., Hutchins, N. & Ooi,
A. 2015 A systematic investigation of roughness height andwavelength in turbulent pipe ﬂow in the transitionally roughregime. J. Fluid Mech. 771, 743–777.
Colebrook, C.F. 1939 Turbulent ﬂow in pipes with particular reference to the transitional region between smooth andrough pipe laws. Journal of the Institution of Civil Engineers 12, 393–422.
Flack, K.A & Schultz, M.P 2010 Review of hydraulic roughness scales in the fully rough regime. J. Fluids Eng. 132 (4).

(7)

where B is a log-law intercept for smooth wall, and κ is the vonKármán constant. To investigate the correlation between ∆U +++against v+
rms , Figure 9 presents a variation of ∆U against vrms
at the roughness crest. We can see the correlation between∆U + against v+
rms . However, the experimental data are signiﬁcantly scattered and do not show the qualitative agreementwith Eq.(7). Moreover, we can observe that the linear relation

5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Flack, K.A & Schultz, M.P. 2014 Roughness effects on wallbounded turbulent ﬂows. Phys.Fluids 26, 101305.
Flack, K.A., Schultz, M.P. & Barros, J.M. 2020 Skin friction
measurements of systematically-varied roughness: probing
the role of roughness amplitude and skewness. Flow Turb.
Combust. 104 (2), 317–329.
Flack, K.A., Schultz, M.P. & Connelly, J. 2007 Examination ofa critical roughness height for outer layer similarity. Phys.
Fluids 19, 095104.
Flack, K.A, Schultz, M.P. & Rose, W.B. 2012 The onset ofroughness effects in the transitionally rough regime. Int. J.
Heat Fluid Flow 35, 160–167.
Forooghi, P., Stroh, A., Magagnato, F., Jakirlić, S. &Frohnapfel, B. 2017 Toward a universal roughness correlation. J. Fluids Eng. 139 (12).
Forooghi, P., Stroh, A., Schlatter, P. & Frohnapfel, B. 2018
Direct numerical simulation of ﬂow over dissimilar, randomly distributed roughness elements: A systematic study
on the effect of surface morphology on turbulence. Phys.
Rev. Fluid 3, 044605.
Iwamoto, K., Suzuki, Y. & Kasagi, N. 2002 Database of
fully developed channel ﬂow. http://www.thtlab.
t.u-tokyo.ac.jp/index-j.html.
Kawamura, H. 2008 Direct numerical simulation database forturbulent channel ﬂow with heat transfer. https://www.
rs.tus.ac.jp/t2lab/db/.
Kuwata, Y. & Kawaguchi, Y. 2019 Direct numerical simulation of turbulence over systematically varied irregular roughsurfaces. J.Fluid Mech. 862, 781–815.
Kuwata, Y. & Nagura, R. 2020 Direct numerical simulationon the effects of surface slope and skewness on rough-wall
turbulence. Phys. Fluids 32 (10), 105113.
Ma, G.-Z., Xu, C.-X., Sung, H.J. & Huang, W.-X. 2020 Scaling of rough-wall turbulence by the roughness height andsteepness. J. Fluid Mech. 900.
Napoli, E., Armenio, V. & De Marchis, M. 2008 The effectof the slope of irregularly distributed roughness elements
on turbulent wall-bounded ﬂows. J. Fluid Mech. 613, 385–
394.
Nikuradse, J. 1933 Laws of ﬂow in rough pipes. National Advisory Committee for Aeronautics .
Orland, P. & Leonardi, S. 2008 Direct numerical simulation of
three-dimensional turbulent rough channels: parameterization and ﬂow physics. J. Fluid Mech. 606.
Orlandi, P., Leonardi, S. & Antonia, R.A. 2006 Turbulent
channel ﬂow with either transverse or longitudinal roughness elements on one wall. J. Fluid Mech. 561, 279–305.

6