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

GLOBAL FRICTION OF
UNIFORM VS. HETEROGENEOUS SANDPAPER ROUGHNESS

Lars H. von Deyn, Alexander Stroh, Davide Gatti and Bettina Frohnapfel
Institute of Fluid Mechanics
Karlsruhe Institute of Technology
76131 Karlsruhe, Germany
von-deyn@kit.edu

Abstract
Skin-friction measurements of streamwise aligned
sandpaper strips are presented in Nikuradse-type diagrams obtained in a dedicated experimental facility.
These novel results shed light onto the global friction
drag of heterogeneous rough surfaces. They indicate thedrag-increasing presence of secondary motions known
to exist over spanwise heterogeneous roughness. The
Reynolds number dependency of the skin-friction coefficient appears to be a superposition of the smooth walland homogeneous rough wall behavior. The results suggest that the present heterogeneous rough surface does
not reach a fully rough behavior in which the friction coefficient becomes independent of Reynolds number.

1

protruding sandpaper strips

Introductionsmooth ridges

Predicting the global friction drag of rough surfaces
poses a challenge which is subject of ongoing research.
Especially heterogeneous roughness effects are not yet
included in existing predictive frameworks (Chung et al.,2021). Nevertheless, inhomogeneous roughness is a key
feature of many environmental and engineering surfaces (Colombini & Parker, 1995; Bons et al., 2001). The
present work aims at providing reliable friction measurements for spanwise heterogeneous roughness overa broad Reynolds number range and thus novel data forpossible future engineering correlations.
Surfaces with inhomogeneity normal to the mean
flow direction are known to produce large-scale secondary motions of Prandtl’s second kind that can alter the drag and heat transfer properties of a turbulent boundary layer (see e.g. Stroh et al., 2019) and arethus important to be considered in predictive frameworks. From literature, it is known that protrudingand δ -spaced ridges (δ is the characteristic outer lengthscale, here mean half-channel height) produce strong
secondary currents (Vanderwel & Ganapathisubramani,
2015; Medjnoun et al., 2020). This simplified configuration is often termed ridge-type roughness. Additionally,
spanwise alternating strips of high and low wall friction (such that the roughness elevation can be neglected)
termed strip-type roughness are often investigated due toits easy accessibility especially in numerical simulations(Chung et al., 2018).
Medjnoun et al. (2020) investigated the friction behavior of smooth protruding ridges of various shapes

submerged sandpaper strips
Figure 1: Experimental setup. bottom:

schematic representation of measured
surface configurations. Top: Photograph of s ≈ 4δavg spaced submergedroughness strips built in the wind tunnel test section (view from tunnel exit)

and reported an upward shift of the smooth reference
curve dependent on the structures shape, i.e. magnitude of the secondary flows. In accordance with thesefindings, we choose two spanwise wavelengths s ≈ 2δavg
s ≈ 4δavg spanwise spacing and add sandpaper roughness on top of the smooth ridges as a step closer to“real world inhomogeneous roughness”. δavg denotes
the geometrically averaged channel half-height throughout the manuscript. In addition to these protruding
roughness strips, we also investigate streamwise stripsof submerged sandpaper roughness which are similar tothe configuration studied by Wangsawijaya et al. (2020).
The results are presented in Nikuradse-type and roughness function diagrams in comparison with smooth andhomogeneous rough references.

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
ID
smoothhomogen rgh
protruding rgh 2δ
protruding rgh 4δ
ridge 2δridge 4δsubmerged rgh 2δ
submerged rgh 4δ

s
[mm]
25
50
25
50
25
50

kavg
[mm]
0.67
0.67
.67
0.67
0.67
0.67
0.67

havg
[mm]
0.67
0.335
0.335
0.335
0.335
0
0

s
δavg

kavg
δavg

havg
δavg

s
w

kavg
ks

2.038
4.076
1.926
3.852
1.976
3.952

0.055
0.054
0.054
0.053
0.053
0.054
0.054

0.055
0.027
0.027
0.026
0.026
0
0

1
1
1
1
1
1

1.169
0.568
0.429
0.505
0.429

s
havg

kavg
w

Table 1: Overview of investigated geometries. δavg denotes the average half-channel height defined

Settling chamber∆powith havg .

Orifice flow meterTin
Air filter

Grids
Radial Fan
Test section

Tout
Flow straightner∆ppamb aq

Scanivalve
multiplexerexc
str hangplateucture ables

Pressure Tap

Figure 2: Schematic of the utilized wind tunnel including measurement instrumentation.

2

Investigated Geometries

spanwise alternating smooth and protruding sandpaper
strips. Thus, the domain-averaged roughness height ishavg > 0 (see figure below table 1 for definitions).

Measurements are carried out in the three different
configurations depicted in figure 1. We consider streamwise aligned spanwise alternating strips of smooth aluminium and P-60 grit sandpaper at two different spanwise wavelengths s/δavg ≈ 2 and s/δavg ≈ 4. The protruding sandpaper roughness configuration denoted protruding rgh xδ , where x represents the approximatedwave length s/δavg , was manufactured by gluing sandpaper strips on a smooth aluminium plate, resulting in

Additionally, notches of the averaged sandpaper
height’s depth were milled in aluminium plates witha high precision milling machine. This configuration
was measured in the wind tunnel in order to quantify the effect of smooth protruding ridges (termedridge xδ in table 1) on friction drag without sandpaperroughness. Afterwards, sandpaper strips were glued

2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
×10−2
smoothDean 19782homogen rgh
Nikuradse 1931

D/ks = 15 ↓1.5

← 2 δavg /ks = 21.3 D/ks = 30.6 ↓Cf

1D/ks = 60 ↑

0.75
0.5

104

Reb

105

106

Figure 3: C f vs. Reb of smooth and homogeneous rough reference vs. experimental data of Nikuradse (1931) and the correlation proposed by Dean (1978) . ks denotes the sand grain size in Nikuradse’s experiments or the equivalent sand grain roughness in case of the current measurements.
The ratio kavg /ks is obtained from figure 5.

terchangeable orifice plates, the skin-friction coefficientC f is deduced. The orifice’s pressure drop is measured
with one of two Setra 239D (125 Pa and 625 Pa full-scale)unidirectional differential pressure transducers with anaccuracy of 0.07 % of the full-scale, switching automatically depending on Reb . Changes in ambient conditions
are accounted for by tracking the systems inlet and outlet temperature via PT100 thermocouples and the ambient pressure and humidity using Adafruit BMP 388 and
BME 280 sensors, respectively.

in the notches, creating submerged roughness strips
with havg = 0. Thus, this configuration is named submerged rgh xδ . The experimental matrix is completedwith measurements of a smooth reference aluminium
plate and homogeneous sandpaper.

3

Facility & Experimental Procedure

The utilized experimental facility (see figure 2) consists of an open-circuit blower tunnel developed byGüttler (2015). It allows the measurement of smallchanges in skin-friction drag by evaluating the staticpressure at 21 pressure taps located along both side wallsof a 314 δ long channel test section with an aspect rationof 12. The test section is divided into three segments of76 δ , 119 δ and 119 δ streamwise extent, where the last
segment is equipped with the surface structures specified in table 1. The pressure-drop measurements in conjunction with an orifice flow meter are used to obtain2 τw
the skin-friction coefficient C f = ρU
2 , which is presented

4

The obtained C f values of the smooth and rough reference cases are shown in figure 3. It can be seen that thesmooth reference case agrees very well with the correlation proposed by Dean (1978).
The measurements of the homogeneous rough reference consisting of p60 sandpaper are shown as the green
markers in figure 3. For small Reynolds numbers, a
gradual increase of C f is evident indicating a transitionalrough behaviour. Above Reb > 3 × 104 , C f converges
to a constant C f ≈ 0.0135, indicating fully-rough behaviour. For reference, Nikuradse’s original data (Nikuradse, 1931) of his famous rough pipe experiments is included in figure 3. Note that Nikuradse used uniform
sized sand grains of size ks to fabricate the rough pipes
featured in his experiments. The presently investigatedsandpaper is also made of sand grains, but of varying
size. Thus, the averaged height kavg = havg ̸= ks . However, the ratio kavg /ks can be obtained by rescaling theroughness length scale to match the roughness function∆U in fully-rough regime with Nikuradse’s data, as done
in figure 5. In this framework, ks is interpreted as theequivalent sand grain roughness size. The relative roughness size normalized with the respective characteristic
length (half-channel height δavg or pipe radius r) is denoted in figure 3 for the individual data sets.
Overall, the newly presented data set agrees verybin a Nikuradse-type diagram as a function of the bulk2δSkin-Friction Measurement ResultsUb
Reynolds number Reb = avg. τw denotes the wallν
shear stress, ρ and ν the fluid’s density and kinematic
viscosity, respectively, and Ub the bulk velocity. Note τw ,can be interpreted as an effective wall-shear stress that isdefined upon the streamwise pressure gradient and theaveraged half-channel height δavg τw = − ∂∂xp δavg .
1
The facility allows to study C f (Reb ) in the range of
4.5 × 103 < Reb < 8.5 × 104 for fully developed turbulent
channel flow. The flow in the channel can be considered
as fully developed turbulent channel flow for Reb > 4500
downstream of 160 δ after the test section inlet, where themeasurement campaign is carried out (Güttler, 2015).
For the pressure-drop measurement, a MKS Baratron 698A unidirectional differential pressure transducer
with 1333 Pa maximum range and an accuracy of 0.13 %of the reading is employed. In combination with theflow rate V̇ , measured with an orifice flow meter with in-3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
×10−2
smoothprotruding rgh
submerged rgh

1.5

cases. The naming convention introduced in figure 1
attached with 2δavg or 4δavg representing the spanwise
wave length are used in the following. Lighter colors
represent a greater spanwise spacing.
The presence of smooth ridges only (blue datasets)of kavg height produce a drag increase of 3-4 % for s ≈2δavg and 2-3 % for s ≈ 4δavg .
The drag increase of the sandpaper strips exhibits
a mixture of the C f (Reb ) trends of the homogeneousroughness and the smooth ridges. With increasing Reb ,C f (Reb ) decreases for all strip configurations, making
it seem unlikely that a fully rough state is reached for
Reb → ∞. The sandpaper strips reveal a C f (Reb ) similar (but with 40-50% more drag) to smooth ridge results(Medjnoun et al., 2020). This trend can be observed infigure 4b.
Considering the created amount of drag increase of
the roughness strips, the drag increase is dominated by
the presence of the sandpaper, significantly surpassing
the effect of smooth ridges (by approx. one order ofmagnitude). Comparing the protruding (red dataset)and submerged (yellow) sandpaper configurations, theyellow submerged roughness datapoints lay below its
respective protruding counterparts. Interestingly, thedrag increase of the red protruding roughness datapoints exceeds the sum of the smooth (blue) ridges andsubmerged roughness (yellow). These observed differences in drag increase are probably related to varying
strengths of the large-scale secondary motions which are
expected to be strongest for the largest roughness protrusion.
From figure 4 it can be observed that all s ≈ 2δ
spaced configurations produce more friction drag compared to the respective 4 δ configurations. From literature, it is known that the strongest secondary currents
are expected for s ≈ δ spanwise spacing (Vanderwel &Ganapathisubramani, 2015), recently confirmed e.g. by
sPIV measurements of strip-type roughness (Wangsawijaya et al., 2020). Due to the fact that solely the wavelength s varies between cases and the surface parameters
remain constant, the observed increase in friction (between s ≈ 2δ and 2s ≈ 4δ spacing) might be an indicationof stronger secondary currents in case of s ≈ 2δ .

homogen rgh
ridge
Dean 1978

Cf

1.25

1

0.75

0.5
2

4

6
Reb

8
×104

(a)

200
175

∆C f /C f 0

150
125
100
75
50
25
10
0

2

2

4

4

Reb
(b)

6

8

8

6

×104

Reb

×104

Figure 4: a) Skin-friction coefficient C f

and b) relative drag increase ∆C f /C f 0 as
a function of Reb for investigated cases
as listed in table 1. Legend of panel a)also valid for panel b). Dark color: s ≈2δavg , light color: s ≈ 4δavg

5

Roughness Function∆C

As an alternative representation to C f 0f discussed
before, the induced drag change of surface roughness
can be expressed in terms of a roughness function ∆U + .
Based on the ∆U + as a function of a viscous scaledroughness length scale k+ , different types of surface
roughness are distinguished in literature (Jiménez, 2004).
Thus, the drag-change measurements expressed as ∆U +are discussed in the following, as presented in figure 5.
∆U + is retrieved from C f and C f 0 measured at thesame Reb via the following relationshipwell with Nikuradse’s measurements. For increasingr/ks , i.e. smaller relative roughness size, the onset of the
fully rough regime is shifted towards higher Reynolds
numbers. The homogeneous sandpaper data set with
δavg /ks = 21.3 satisfies this trend very well as it fits nicely
in between the data sets with r/ks = 15.2 and r/ks = 30.5.
The measurement results for the inhomogeneous
cases are presented in figure 4. As depicted in figure
1, three different lateral inhomogenous configurations
are investigated with its dimensions specified in table
1. Individually, the two different wavelengths s/δavg ≈ 2
and s/δavg ≈ 4 were measured amounting to a total ofsix measurement campaigns not counting the reference

1 Cf∆U = ln
+
κ Cf0
+

s

2
−
Cf0

s

2
,
Cf

(1)

which is obtained from the Prandtl–von Kármán friction
relation as described in Gatti et al. (2020), i.e. by approx+
imating the change ∆UClof centreline velocity with the412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 202212
10

homogen rgh
protruding rgh∆U + (lg+ ) =1
ln k+ + B.
κ

(2)

∆U +

8
with k as a roughness length scale. For choosing k = h+avg
as in figure 5a and a von Kármán constant of κ = 0.39, an
additive constant of B = −3.1 is obtained when matchingequation (2) with the homogeneous rough reference casehomogen rgh. In this scaling, a fully-rough behaviour isevident beyond havg ≈ 70 which matches well with thereported threshold Reynolds number Reb = 3 × 104 as
discussed before.

6
4
2
0
101

h+
avg

Comparing the case protruding rgh 2δ to homogen rgh and the fully-rough reference in figure 5a, a
surprisingly good collapse with the fully rough regime+
between 20 ⪅ h+avg ⪅ 40 is found. Beyond havg ⪆ 50 ansignificantly lower ∆U + compared to the fully-rough reference is found. A similar departure from the fully roughregime was discussed by Gatti et al. (2020) in the context
of purely two dimensional surface structuring aligned instreamwise direction. Bearing in mind that merely thelateral inhomogeneity is a common feature of the present
study and Gatti et al. (2020), it is straightforward to hypothesize the absence of a fully-rough behaviour beyonda certain threshold of k+ is caused by the surface lateral
inhomogeneity.

102

(a)

12
10

homogen rgh
protruding rgh
submerged rgh

∆U +

8
6

The wider-spaced configuration protruding rgh 4δ
reveals a qualitatively very similar ∆U + (h+avg ) behaviour
compared to protruding rgh 2δ in figure 5, although protruding rgh 4δ produces less friction drag resulting insmaller ∆U + for matched h+avg . In order to enable a direct comparison of the sandpaper strip cases including
the cases submerged rgh 2δ with h+avg = 0, the equivalent sand grain roughness concept is employed. For
each data set, the roughness length scale k is adjusted tomatch the relation for the Nikuradse sand grain roughness ∆U + (ks+ ) = κ1 ln ks+ − 3.5 (Chung et al., 2021). The
respective ratio of the averaged roughness height kavg
and equivalent sand grain roughness ks are included intable 1. Note that the homogeneous reference case homogen rgh deviates only by 16% from the equivalentsand grain size.

4
2
0
101

ks+

102

(b)
Figure 5: a) Roughness function ∆U +vs. havg . ∆U + = κ1 ln h+
avg + C with C =−3.1 included for reference. b) Roughness function vs.
equivalent sandgrain roughness ks utilized as lengthscale. The fully rough reference ∆U + =1+κ ln ks − 3.5 is included. κ is set to κ =0.39. dark color: s ≈ 2δavg , light color:s ≈ 4δavg .

The ∆U + results rescaled to the equivalent sandgrain size ks+ are shown in figure 5b. In this representation, all sandpaper strip cases collapse with the fullyrough reference as intended and show remarkable agreement with the fully rough regime for 15 ⪅ ks ⪅ 30. Interestingly, all lateral inhomogenous cases experience a
clear departure from the fully rough regime supporting
the hypothesis that the observed behaviour is an effect oflateral surface inhomogeneity. Moreover, the respective
protruding and submerged configuration agree remarkably in the rescaled ks representation, while the widerspaced configurations with s ≈ 4δavg lead to an earlier
departure from the fully rough regime.

change ∆Ub+ of bulk velocity. This methodology is also
employed in other studies dealing with pressure-drop
measurements of flows over irregular roughness (Barros
et al., 2018; Flack et al., 2020).

The presented results highlight the importance to include lateral inhomogenous surface effects in predictive
frameworks for ∆U + . Significant lateral inhomogeneity
leads to the absences of a fully rough regime for highReynolds numbers. Therefore, these effects need to beaccounted for based on reliable data as presented in thecurrent study.

In figure 5a, ∆U + as a function of the domainaveraged height h+avg is shown. Consequently, the cases
submerged rgh xδ are not visible in this representation
as h+avg = 0 by design. The black solid line corresponds tothe fully rough behaviour (Perry et al., 1969)5

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

Conclusions

Dimensional Rectangular Duct Flow. Journal of Fluids
Engineering 100 (2), 215–223.
Flack, K. A., Schultz, M. P. & Barros, J. M. 2020 Skin friction measurements of systematically-varied roughness: Probing the role of roughness amplitude andskewness. Flow, Turbulence and Combustion 104, 317–
329.
Gatti, D., von Deyn, L. H., Forooghi, P. & Frohnapfel, B.
2020 Do riblets exhibit fully rough behaviour? Experiments in Fluids 61 (3), 81.
Güttler, A. 2015 High accuracy determinationof skin friction differences in an air channel
flow based on pressure drop measurements.
https://publikationen.bibliothek.kit.edu/1000048039.
Jiménez, Javier 2004 Turbulent Flows Over Rough Walls.
Annual Review of Fluid Mechanics 36 (1), 173–196.
Medjnoun, T., Vanderwel, C. & Ganapathisubramani,
B. 2020 Effects of heterogeneous surface geometry on
secondary flows in turbulent boundary layers. Journalof Fluid Mechanics 886, A31.
Nikuradse, J. 1931 Strömungswiderstand in rauhen
Rohren. ZAMM - Journal of Applied Mathematics andMechanics / Zeitschrift für Angewandte Mathematik und
Mechanik 11 (6), 409–411.
Perry, A. E., Schofield, W. H. & Joubert, P. N. 1969 Roughwall turbulent boundary layers. Journal of Fluid Mechanics 37, 383–413.
Stroh, A., Schafer, K., Forooghi, P. & Frohnapfel, B. 2019
Secondary flow and heat transfer in turbulent flow
over streamwise ridges p. 31.
Vanderwel, C. & Ganapathisubramani, B. 2015 Effectsof spanwise spacing on large-scale secondary flows inrough-wall turbulent boundary layers. Journal of Fluid
Mechanics 774.
Wangsawijaya, D. D., Baidya, R., Chung, D., Marusic, I. & Hutchins, N. 2020 The effect of spanwise
wavelength of surface heterogeneity on turbulent secondary flows. Journal of Fluid Mechanics 894.

Overall, the presented global friction measurements
add to the ongoing endeavor of characterizing & predicting flow properties of inhomogenous surfaces. Althoughnumerous studies are available in the literature investigating inhomogenous roughness numerically and experimentally, only very few global friction measurements
are available, in particular covering broad Reynolds
number range. The present results provide an indication
on how inhomogenous roughness effects can be embedded in engineering correlations: The amount of drag increase is dominated by the presence of roughness, whilethe “shape” of the C f (Reb ) bears more similarity to thethe trends of smooth ridges. The presented measurements suggest that a fully rough regime is not present
for roughness strips; at least not within the investigatedReynolds number range.

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