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

REYNOLDS NUMBER EFFECTS IN 2D SQUARE-BAR ROUGHNESSWALL-BOUNDED TURBULENT FLOWSJiahao Kong
School of Mechanical Engineering
University of Adelaide
Adelaide, 5005 South Australia, Australia.
jiahao.kong@adelaide.edu.au

Luke G. Bennetts
School of Mathematical Sciences
University of Adelaide
Adelaide, 5005 South Australia, Australia.

Bagus Nugroho
Department of Mechanical Engineering
University of Melbourne
Melbourne, 3010 Victoria, Australia

Rey ChinSchool of Mechanical Engineering
University of Adelaide
Adelaide, 5005 South Australia, Australia.

pfriction velocity is defined as Uτ = τw /ρ , where τw is thewall shear stress and ρ is the fluid density.
Although many roughness indeed obey Townsend’s outerlayer similarity hypothesis, many 2D roughness studies haveshown a contradictory view on the similarity hypothesis, particularly for developing boundary layers (Krogstad & Antonia,
1999; Lee & Sung, 2007; Volino et al., 2011). For 2D squarebar roughness, direct numerical simulation (DNS) results (Lee& Sung, 2007) showed that the 2D roughness effect extends
to the outer region (y/δ99 ≥ 0.5) for TBLs with δ99 /k = 22
at, Reτ ≈ 500 (a relatively low Reynolds numbers). Intriguedby these opposing views, Efros & Krogstad (2011) performed
a 2D square-bar roughness boundary layer experiment over awide range of Reynolds numbers (Reτ = 4, 200 – 13, 300) toinvestigate the roughness effects in the outer region at highReτ , and high boundary layer height and roughness height ratio δ99 /k ≈ 130. Their results indicate that the outer-layer similarity hypothesis is preserved, stipulating that an increase inscale separation ratio and Reynolds number provide a different conclusion for 2D roughness flow. From the discussion
above, it is clear that Reynolds numbers Reτ and the ratio of
TBL thickness to the roughness height δ99 /k play significant
roles for the validity of the wall-similarity hypothesis in 2D
roughness TBLs. Moreover, these two parameters will also
affect the drag coefficient C f
For 2D square-bar roughness, they are known to experience increases in drag with p/k, where it reaches the maximum drag at p/k = 8. Beyond this value, the drag starts dodecrease. For p/k = 8, the DNS results of (Lee & Sung, 2007)shows C f ≈ 0.012 for a low Reτ TBL flow, while the high Reτ
experiment (Efros & Krogstad, 2011) resulting in C f = 0.0068
for TBLs at Reτ ≥ 9900. In addition, Djenidi et al. (2018)
conducted a TBL study over 2D circular-rod roughness at a
range of Reynolds numbers, 620 ≤ Reτ ≤ 7000, and reported
C f ≈ 0.007 at the highest Reτ , which is close to the resultof Efros & Krogstad (2011). However, to the best of the authors’ knowledge, there are not many studies that investigate
2D square-bar roughness TBLs at moderate Reynolds numbers, 600 ≤ Reτ ≤ 9000. To reduce the uncertainty causedby different cross-sectional shapes, there is a need to exam-

ABSTRACT
Turbulent boundary layers (TBLs) above twodimensional (2D) square-bar roughness are studied experimentally to investigate Reynolds number effects on the
turbulent flow properties at moderate friction Reynolds numbers (Reτ ). Previous studies have examined 2D square-bar
roughness at low Reτ numerically and high Reτ experimentally. Inspired by their results, a set of experiments wereconducted for a classical k-type 2D roughness with the pitch p
to height k ratio p/k = 8, at Reτ = 1840, 3900 and 4950. The
roughness effect increases with increasing Reτ and extends
to the wake region in terms of the streamwise turbulence
intensity. The results shows that the drag coefficient C fconverges for the fully-rough, indicating that it behaves like aclassical k-type roughness at a fully-developed state when it
reaches Reτ ≥ 3900 with δ99 /k ≥ 53.

INTRODUCTION
Turbulent boundary over a rough surface is an importantcomponent in many engineering applications because it can
influence heat, mass and momentum transfer, and drag coefficient C f (Chung et al., 2021). There are various types ofroughness pattern in the literatures, such as sand grain typeroughness, riblets, etc. One type of roughness that has attracted plenty of attention in the last 60 years is 2D regularly distributed roughness (or simply 2D roughness), which
is known to have spanwise-aligned roughness elements withconstant streamwise spacing p. Review studies by Raupach
et al. (1991) and Jiménez (2004) indicate that many surface roughness patterns, particularly the sand grain type (or“k-type”) obeys Townsend’s outer-layer similarity hypothesis(Townsend, 1974). The hypothesis states that above the roughness sublayer and at a sufficiently large Reynolds number, turbulent flows are independent of the configuration of surface
roughness. In order to obey the similarity hypotheses however, Jiménez (2004) suggested caveats that the roughness pattern need to have boundary layer height and roughness heightratio of δ99 /k ≥ 40 and Reτ ≥ 4000. Here Reτ = δ99Uτ /ν ,where ν is kinematic viscosity and Uτ is friction velocity. Thei

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
ine TBLs over a similar roughness as Lee & Sung (2007) andEfros & Krogstad (2011) at moderate Reτ .
In the present report, we investigate the influence ofReynolds numbers on 2D roughness on the turbulence statistics at moderate Reτ . Hot-wire anemometry (HWA) TBL
experiments are performed at Reynolds numbers Reτ =1840, 3900 and 4950 over 2D square-bar roughness p/k = 8,
which is similar to the studies of Lee & Sung (2007) andEfros & Krogstad (2011). The present study reports the postprocessing approach, including the determination and validation of the friction velocity Uτ . Finally, the mean statistics anddrag coefficient for the rough wall data are analyzed.

For the rough-wall measurements, the rough surface was madeby affixing ABS plastic square bars spanwisely on the aluminum plate downstream of the tripping device. Figure 1
shows a schematic view of the arrangement of the square bars.
The bar height was k=1.5 mm, and the streamwise spacing was
p = 8k, similar to the numerical and experimental studies (Lee& Sung, 2007; Efros & Krogstad, 2011). The spacing value of
p = 8k is specifically chosen because the drag increases withp/k and it reaches the maximum drag at p/k = 8. The roughwall measurements were taken over a range of free-stream velocities, 6 ≤ U∞ ≤ 16 m s−1 at a fixed streamwise location. The
position is identical to the smooth wall measurement, which isalso in the middle of two adjacent roughness bars, as shownin Figure 1. The experimental conditions for the smooth- andrough-wall measurements are summarized in Table 1.
The TBL profiles were measured with hot-wire anemometry (HWA) sensors and taken from the near-wall position to
the free-stream flow at y = 1.5δ99 by using a 1D traverse system. Note that this study defines y as the wall-normal distancefrom the bottom wall for rough wall measurements, as shownin Figure 1. The traverse system comprised a Mitutoyo heightgauge, driven by a stepper motor and controlled by an opticallinear encoder, which has a resolution of 30 µ m. The HWA
sensors were single-wire boundary-type probes, operated byan in-house built hot-wire anemometry. The probes had a tip
spacing of 2 mm. Wollaston wires were soldered to the prong
tips etched to expose a sensor filament of d = 2.5 µ m diameterand l = 0.51 mm length. The length-to-diameter ratio of thesensor filament was l/d ≥ 200, which satisfies the suggestion
by Ligrani & Bradshaw (1987) to minimize attenuation due toend conduction. The velocity signal from HWA was sampledwith frequency fs = 51200 Hz and a duration T = 120 s usinga National Instrument data acquisition board (USB-NI6211).
As shown in Table 1, the viscous sample time interval, defined
as t + = 1/ fs (Uτ2 /ν ), is less than 3 to capture all the relatively
high energy content frequency (Hutchins et al., 2009). Notethat the largest scales of turbulent structures are observed to
exceed 20δ (Adrian et al., 2000), hence, the sampling durationis required to encompass several hundreds of these large-scale
events for the converged statistics (TU∞ /δ99 > 20000).
The hot-wire sensors were statistically calibrated against
a Pitot tube located above the hot-wire probes, approximately 10 mm into the free-stream flow. Calibrations weremade before and after each traverse. The free-stream velocity was determined by the difference between the total andstatic pressures from the Pitot tube and monitored by an electronic barometer (220DD Baratron, MKS). The temperature
condition was also monitored by a RTD-type thermocouple(PT1000) to compensate for the drift of the HWA signal causedby temperature. Fourth-order polynomial curves were used
to fit the pressure data and hot-wire voltage signals. Linear
interpolation was made to correct the temperature drift between pre- and post-calibration. In addition, the free-stream
velocity was recorded throughout each traverse station andused to compare with the hot-wire signals in the free-stream
flow. The whole data set was discarded if the difference of U∞
from the pressure data and hot-wire signals is larger than ±1%(Hutchins et al., 2009).

EXPERIMENTAL METHOD
In this study, the Cartesian coordinates x and y refer tostreamwise and wall-normal directions, respectively. The corresponding velocity components are u and v, and δ99 denotes
the wall-normal location where the mean streamwise velocity is 99% of the free-stream velocity, U∞ . The superscript+ indicates viscous scale normalization so that, for instance,
U + = U/Uτ and y+ = y × Uτ /ν , where Uτ is the friction velocity.

Flow
k
y
x
p = 8kFigure 1. Schematic views of square-roughened surfaces andthe measurement location

EXPERIMENT CONDITION
Experiments were performed in a closed-loop wind tunnel at the University Adelaide. The test section was 2 m longand had a rectangular cross-sectional area of 0.5 × 0.3 m2 .
The sidewalls were adjusted to maintain zero pressure gradient (ZPG) along the test section with a range of free-stream
velocities, 6 ≤ U∞ ≤ 20 m s−1 , for the smooth and rough wallmeasurements. For various free-stream speed flows, the acceleration parameter, defined as K = Uν2 ∂∂Ux∞ , was less than∞

5 × 10−9 (Volino et al., 2009). The bottom wall of the test
section was fully covered by a flat aluminum plate to serveas the smooth wall for the baseline flow measurement. The
boundary layer was tripped near the leading edge of the aluminum plate with tripping devices, consisting of a 3 mm diameter threaded rod and a 36 grit sandpaper of 100 mm in length
to reach a fully-developed state. A smooth wall measurement
was conducted with the tripping devices at Reτ = 1900 to act
as a baseline case. The smooth wall TBL has the shape factor
H = δ∗ /θ = 1.357, where δ∗ is the displacement thickness andθ is the momentum thickness, which shows a good agreementwith the criterion value of H introduced by Chauhan et al.
(2009) at corresponding Reτ , and indicates the smooth wall
TBL reaches a fully-developed state.
The smooth wall TBL measurement was made at 1.5 m
downstream of the tripping device with a free-stream speed of
20 m s−1 to achieve a friction Reynolds number, Reτ = 1900.

Estimating Uτ
Two methods were used to find Uτ indirectly for thesmooth- and rough-wall data for the present streamwise velocity measurements. The composite profile method of Chauhan
et al. (2007) was used to determine Uτ and Π simultaneously
for the smooth-wall data, where Π is known as the wake pa-ii12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 1.

The smooth and rough wall experiment condition

Case

p/k

x

U∞

m

(ms−1 )

smooth

–

1.5

Ref 1

–

Ref 2

l+

t+

TU∞ /δ99

2.8

25

0.71

62

2.49

24.5

0.51

17

2.34

25.3

0.48

15

49

7.9

12.76

0.185

9.9

53

6.9

24.7

0.72

19.2

53.3

6.9

31.6

1.13

24

Reτ

δ99 /k

20

1900

–

3.75

20

3470

–

–

6.3

20

4780

–

pk8A

8

1.5

6

1840

pk8B

8

1.5

12.8

3900

pk8C

8

1.5

16

4950

Cf

(a) 30

×10−3

U/uτ

Sym

20
10

Ref 1 and Ref 2 refer to Marusic et al. (2015).

0
x

(b) 8

U+ =

1
ln(y+ ) + B,κu′2 /u2τ

rameter. The method uses all measurement points to fit withthe reference profile. The reference profile is based on a composite function comprising the Musker function for the inner
region and an exponential function for the wake region. Those
equations can be found in Sections II – III of Chauhan et al.
(2007). The composite profile method was compared with theClauser method (Clauser, 1954), which involves forcing themean velocity profile to fit onto a pre-defined log law by adjusting Uτ as

4
2
0

100

101

102
y′ uτ /ν103

104

Figure 2. Comparison of the present smooth wall profiles
scaled by Uτ from two methods and the DNS smooth wall results at Reτ ≈ 1900 (Chan et al., 2021). (a) inner-scaled mean
velocity profile; (b) inner-scaled turbulence intensity profile.
Dash line, the DNS smooth wall result; circle, the present result from the composite method; triangle, the present result
from the Clauser method.

(1)

where κ and B are known as the von Karman constant and theintercept constant, which are suggested as 0.39 and 4.3 for thesmooth wall TBL Chauhan et al. (2009); Marusic et al. (2015);
Squire et al. (2016). The fitting process also requires properly defined
the log region, which are suggested at√ bounds of
+
y+
=
3
Re
and
y
= 0.15Reτ for the smooth-wall TBL
τ
outer
inner
following Marusic et al. (2013). The resulting Uτ between thetwo methods is very similar, with a difference of less than 2%.
In Figure 2, the smooth-wall velocity and turbulence intensity
profiles, scaled by two Uτ from the above methods, are compared with the result of the DNS study (Chan et al., 2021) ata similar Reynolds number Reτ ≈ 1900. Two velocity profiles
show a good agreement with the DNS result at all positions.
The turbulence intensity profiles also show an excellent collapse with the DNS profiles above the log region (y+ ≥ 100).
In the following discussions, the smooth-wall TBL case is normalized by the Uτ that is obtained from the composite method.

& Li, 1990). By adding ε to the experimental data to adjust
the virtual origin from the bottom wall (ε = 2/k) to the crestof roughness (ε = −2/k), several iterations were made by applying the first-order polynomial fit to evaluate a constant line.
The constant line providing the least square error fitting to theexperimental data within the log region is considered the√best
fit line, where the log region is defined from y+ = 3.4 Reτto y/δ = 0.19 for rough-wall TBLs (Squire et al., 2016). The
initially estimated Uτ can be determined from the slope of theconstant line, and the optimized offset value ε was applied todetermine the wall-normal position y′ .
The accuracy of Uτ estimated via the modified Clauser
method (for the rough wall cases) has been investigated by
Flack et al. (2007), and they found that it has uncertainty
around 3 – 5%. To minimise this uncertainty, a novel techniquethat relies on the outer-layer similarity hypothesis (Townsend,1974) was introduced by Monty et al. (2011) (from here on
it is referred to as the ‘defect’ method) and it is used to ‘improve’ the Uτ estimated from the modified Clauser method.
The Uτ estimation was made by fitting the velocity defect profile (U∞+ − U + vs y′ /δ99 ) onto the smooth-wall data from thelower bound of the outer region, y′ /δ99 ≈ 0.1. The mean
velocity and defect profiles for the 2D rough wall TBLs at
Reτ = 1840 and 4950 normalized by the Uτ from the modified
Clauser method and defect method were plotted in Figure 3.
The comparison shows that the differences in terms of Uτ and∆U + between the two methods are small, which are about 4%
and 2%, respectively. These differences are well within the
expected error as indicated by Flack et al. (2007) and Schultz& Myers (2003). Moreover, velocity defect profiles for therough wall cases collapse well with the corresponding smoothwall case, indicating agreement with the outer-layer similarity
hypothesis. Due to this, the improved estimation technique,which is a combination of the modified Clauser method used
to obtain ε and the defect method used to determine Uτ and

For the rough-wall flow cases, there are three additional
unknowns in the normalized mean velocity profile, namely the
von Karman constant κ , the roughness offset ε and the Hamaroughness function ∆U + (Perry & Joubert, 1963). Based on
the knowledge of 2D roughness TBL velocity profiles from
Leonardi et al. (2003) and Lee & Sung (2007), we assumed
κ = 0.41 for rough-wall measurements. The offset ε accounts
for the effect that the roughness displaces the entire flow awayfrom the bottom wall. In this study, the wall-normal position
from the virtual origin of the mean velocity profile is definedas y′ = y − ε . The classic method to obtain Uτ , ε , and ∆U +was via the modified Clauser method (Perry & Joubert, 1963),
which requires the knowledge of the bounds of the log region.
Here we follow Perry & Li (1990) to optimize ε via iterationon the modified Clauser method. In this method, U/U∞ is plotted as a function of (y − ε )/δ ∗ , where δ ∗ is the displacementboundary layer thickness. In this profile, the equation for thelog region is


U
1 Uτ
y−ε
+C,
=
ln
U∞
κ U∞
δ∗

6

(2)

Uτ
where C = f ( U, Π) is the intercept constant of the line (Perry∞iii

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
∆U + , is used for all 2D roughness data.

0.25
30

0.2

U/Uτ

u′ /U ′

25

0.15

20

0.1

15

0.05

10

0

5
0

100

101

18

102
′
y Uτ /ν

103

0.4

0.5

0.6 0.7
U ′ /U∞′

0.8

0.9

1

Figure 4. The modified diagnostic plot for all rough-wallmeasurements at Reτ = 1840 – 4950 (light to dark square) andthe smooth-wall result (circle). Dashed line: the empirical line
determined from the linear region of the smooth-wall data ofCastro et al. (2013).

104

15
U∞+ −U +0.3

12
at Reynolds numbers Reτ = 1840, 3900 and 4950 (light todark grey). As the baseline flows, the smooth wall TBLs of
Reτ = 1900, 3470 and 4780 are also plotted with the roughwall cases (where the two high Reτ cases are taken from Marusic et al. (2015)). Figure 5(a) shows inner-scaled mean velocity profiles, and clearly indicates the downward shift ∆U + forthe rough-walled case relative to the smooth-walled case. The
effect of the surface roughness on the mean velocity distribution can be expressed in the mean velocity equation for the
logarithmic layer, as

9
6
3
0
0

0.2

0.4
0.6
y′ /δ99

0.8

1Figure 3. Comparison in estimating Uτ of 2D roughness at
Reτ = 1840 and 4950 using the modified Clauser method (Diamond) and the defect method (Square). The result of roughwall case at Reτ = 3900 is not shown here for brevity. As thereference profile for the defect method, the smooth wall TBL
at Reτ = 1900 (open circle) is also plotted. Mean velocity profiles (a) and velocity defect profiles (b).

U+ =1 (y + ε )Uτln+ B − ∆U + .
κ
ν

(3)

The inner-scaled turbulence intensity plots are shown in
Figure 5(b). As baseline flows, the smooth wall cases have anear-wall peak at y+ = 15, indicating the highly energetic nearwall cycle of streaks and quasi-streamwise vortices (Kline
et al., 1967). For the rough wall cases, the magnitude of theinner peak diminishes from Reτ = 1840 to 3900 and the reduction reaches a maximum at Reτ ≥ 3900. Note that the
hot-wire spatial resolution for those two lower Reτ cases has
l + ≤ 25, which is higher than the smooth wall TBLs and indicates that the spatial attenuation effect is negligible (Hutchins
et al., 2009). Hence, the absence of the inner peak is generally associated with the disturbance of the near-wall cycle ofstreaks and quasi-streamwise vortices near the surface roughness (Schultz & Flack, 2007; Monty et al., 2011; Djenidi et al.,2018). The reduction reaching its maximum indicates the disturbing effect of roughness becomes independent of Reynolds
numbers when Reτ ≥ 3900. Farther from the wall, turbulence
intensity increases with Reτ , particularly in the log region (asshown by the arrow), which is typical for both smooth andrough walls. Interestingly however, the mean velocity profile
and inner scaled turbulence intensity shows that the two higherReτ cases seems to behave like a sand grain roughness (or ktype roughness).
To examine the roughness effect in the outer region, thevelocity defect profiles and the turbulence intensity of smoothand rough-wall TBLs were normalized by the outer length,δ99 , and plotted in Figures 5(c) and (d). The velocity defectprofile shows a good collapse between the rough wall cases

VALIDATION
Alfredsson & Örlü (2010) introduced a diagnostic plot asu′ /U∞ vs U/U∞ to prove the data quality without estimating Uτ
for the smooth-wall measurement, where u′ is the local (rootmean square) velocity fluctuation. The success of the diagnostic tool extended to the wall-bounded flows over roughness
by considering ∆U + (Castro et al., 2013). The modified diagnostic plot is u′ /U ′ vs U ′ /U∞′ , where U ′ = U + ∆U, U∞′ =U∞ + ∆U and ∆U = ∆U + × Uτ . In this plot, the rough-wall
TBL profiles collapse onto the smooth-wall result (∆U = 0) in
the outer layer. Figure 4 plotted all rough-wall profiles with thesmooth-wall result. The smooth-and rough-wall profiles show
a good collapse on the dashed line for 0.6 ≤ U ′ /U∞′ ≤ 0.9.
This observation indicates the smooth- and rough-wall TBLs
in this study have a good quality in terms of the local mean
velocity and velocity fluctuating, and shows agreement of thevalue of ∆U + estimated from the improved technique.

RESULT AND DISCUSSION
Figure 5 shows the distribution of mean velocity and turbulence intensity profiles on the 2D rough wall with p/k = 8iv

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(a) 30

(c)
8
6
2

U/Uτ

u′ /Uτ2

20

4

10
2
0

100

101

(b)

102
y′Uτ /ν

103

0

104

100

101

102
y′Uτ /ν

(d)

25

103

104

8

(U∞ −U)/Uτ

u′ /Uτ2

20
2

15

6
4

10
2

5
0

10−3

10−2
10−1
′
y /δ99

0

100

10−3

10−2

y′ /δ

10−1

100

99

Figure 5. Comparison of the 2D roughness with p/k = 8 ranging from Reτ = 1840 – 4950 (refer colour codes to Table 1) and thesmooth wall TBL at Reτ = 1900 (open circle) and two higher Reynolds number cases from the literature, Reτ = 3500 (grey dash line)
and Reτ = 4800 (black dash line) from Marusic et al. (2015). Mean velocity profiles (a); inner-scaled turbulence intensity profiles (b);velocity defect profiles (c); outer-scaled turbulence intensity profiles (d).
and the smooth wall results from y′ /δ99 = 0.03. The outerscaled turbulence intensity shows a good collapse between the
low Reτ smooth- and rough-wall results (Reτ ≈ 1900) fromthe center of the log region, y/δ99 ≥ 0.2, all the way to theouter region. For the higher Reτ cases (Reτ ≈ 3470−4950) the
amplitude of the rough walled turbulence intensity is slightly
higher than that of the smooth wall cases, spanning from thelog region to half of the wake region, 0.1 ≤ y/δ99 ≤ 0.6. The
maximum deviation of the turbulence intensity for rough wallcases and the smooth wall cases at high Reτ is about 10% at
y/δ99 ≈ 0.3. Such deviation (despite our relatively high Reτ )may be caused by the low ratio of δ99 /k ≈ 53, which is muchsmaller than that of Efros & Krogstad (2011) (δ99 /k ≈ 130).
Hence although the velocity defect profile indicates a good
agreement with the similarity hypothesis, the outer scaled turbulence intensity is slightly deviate.
The drag coefficient C f can be calculated as
C f = 2(Uτ /U∞ )2 ,ble 1. From the table we can see that the C f seems to converge at s 0.0069, indicating that rough-wall TBLs are fullydeveloped, and will show a similar shape in the outer region of
the mean velocity and turbulence intensity profiles, scaled by
the outer length, δ99 (Djenidi et al., 2018). In addition, δ99 /kincreases with Reτ due to the boundary layer thickness growth,
but the increase of δ99 /k is small when Reτ ≥ 3900. The
slow-growing rate of δ99 /k also indicates that fully-developed
rough wall TBLs have sufficiently large δ99 /k to yield thesame C f . Considering the value of C f = 0.007 from 2D roughness square-bar roughness with p/k = 8 at Reτ ≥ 9900 andδ99 /k ≥ 96 (Efros & Krogstad, 2011), C f can reach its constancy at C f ≈ 0.0069, and the rough-wall TBL achieves its
fully-developed state when Reτ ≥ 3900 and δ99 /k ≥ 53.

By looking at the converges of C f and the plots in figures 5(a) and (b) where the highest Reτ cases behave like asand-grain roughness flow (or k-type roughness), one would
suspect that the flow may indeed a k-type roughness. However we are unable to reach this conclusion, because it requires
more data at higher Reτ which is outside the range of our windtunnel. Despite this, the data indicate all present rough-wallresults are at a fully-rough regime with ∆U + > 10. Hence, thecritical Reτ and δ99 /k at which the rough wall TBL over 2D
square-bar roughness with p/k = 8 achieves a fully-developed
and fully-rough regimes are 3900 and 53, respectively.

(4)

which states that the drag coefficient is related to the ratio of inner and outer velocity scales, namely Uτ and U∞ .
Hence, the amplitude of the free-stream edge of the normalized mean velocity profiles can reflect thepdrag coefficient byconsidering the quantity U∞+ = U∞ /Uτ = 2/C f . The valueof C f for all rough wall cases has been summarized in Ta-v

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

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The experimental study of rough wall TBLs consisting of
spanwise square bars was performed to investigate the drag coefficient and the effects of Reynolds numbers for a 2D roughness with p/k = 8 at moderate Reynolds numbers. For roughwall measurements, the friction velocity Uτ is estimated by an
improved technique, and the modified diagnostic plot shows
agreement with the value of ∆U + obtained from this technique. The rough wall results with p/k = 8 ranging fromReτ = 1840 – 4950 show a slight discrepancy for the turbulence intensity in the outer region between the smooth andrough wall cases, which extends to the middle of the wakeregion at y′ /δ99 = 0.6, despite the fact that the velocity defect data of the rough wall and the smooth wall collapses well.
Such results prevent us to reach a conclusion whether the outerlayer similarity hypothesis is preserved or not. The deviationin the outer scaled turbulence intensity is, despite our relatively high Reτ may be caused by the relatively low ratio ofδ99 /k ≈ 53. The variation of C f against Reτ and δ99 /k showsthat the drag coefficient for the 2D square bars roughness withp/k = 8 converges at 0.0069. The TBL over this 2D roughness may achieve fully-developed and fully-rough regimes at
Reτ = 3900 and δ99 /k = 53, respectively.

ACKNOWLEDGEMENTS
The authors wish to thank their Australian ResearchCouncil (ARC) for the financial support for this research.

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