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

SCALING OF ROUGH-WALL TURBULENCE IN A TRANSITIONALLYROUGH REGIME

Guo-Zhen Ma
Chun-Xiao Xu
Department of Engineering Mechanics
Department of Engineering Mechanics
Tsinghua University
Tsinghua University
Haidian District, Beijing, 100084, P. R. China Haidian District, Beijing, 100084, P. R. China
mgz17@mails.tsinghua.edu.cn
xucx@tsinghua.edu.cn

Hyung Jin Sung
Wei-Xi Huang
Department of Mechanical Engineering
Department of Engineering Mechanics
KAIST
Tsinghua UniversityDaejeon 34141, Republic of Korea
Haidian District, Beijing, 100084, P. R. China
hjsung@kaist.ac.kr
hwx@tsinghua.edu.cn

ABSTRACT

literature for rough-wall turbulence. In the present study, theReynolds number and the wall roughness parameters are comprehensively considered, to parameterize the turbulent statistics.

Direct numerical simulations (DNSs) were performed fora turbulent channel flow over three-dimensional sinusoidalrough walls to test a roughness scaling method in a transitionally rough regime. Three groups of rough cases were employed to assess the scaling formula by systematically varying
the friction Reynolds number Reτ and the ratio of the roughness height k to the channel half-height δ , and their predictions
were compared with Nikuradse’s experimental data and otherroughness types. A new coupling scale Reτ /(k̄+ S)n is thenproposed by combining Reτ and in a logarithmic form, where
n is the scaling exponent, k̄+ is the viscous-scaled mean roughness height and S is the roughness steepness. All the simulated
data for the roughness function and the peak of the streamwise
turbulent velocity fluctuations collapse into single curves withthis coupling scale. Our investigation of the rough-wall scaling
behavior with DNS data can serve as a basis for supplementing
Moody’s data in the transitionally rough regime.

NUMERICAL SIMULATION
The problem under consideration is the fully developed turbulent channel flow over three-dimensional sinusoidalrough walls. A schematic diagram of the channel is shown infigure 1. A right-handed Cartesian frame fixed in the physical space is employed with x, y and z axes along the streamwise, vertical and spanwise coordinates, respectively. The corresponding velocity components in the three directions are u, vand w. The flow is driven by a mean pressure gradient, which
is dynamically adjusted to keep the constant flow rate in time.
For the roughness parameters, k is the semi-amplitude of thesinusoidal roughness and λ is the wavelength of the roughness elements. The governing equations are the dimensionlessincompressible Navier-Stokes and continuity equations:

INTRODUCTION
Turbulent flow over a rough wall almost always produceshigher drag than that of a smooth wall. The increase in drag isoften quantified by the roughness function ∆U + , which reflects
the downward shift of the mean velocity profile (Hama, 1954).
In practice, the increase of drag usually brings a lot of harmand efficiency loss, and predicting the drag on a rough wall istherefore of crucial importance to most engineering problems.
A central goal in the present study is to predict the roughness
function directly from the given rough surface and flow conditions, and then predict the wall resistance.
The dependences of the roughness function on the roughness height and other geometric features such as the roughness density and shape have been explored extensively for various types of roughness (Chung et al., 2021). Some scales forthe equivalent sand grain roughness have been found (Flack &Schultz, 2010). However, the influence of the Reynolds number is usually not considered in these scaling methods. Furthermore, the Reynolds-number scaling of the second-order statistics has to the best of our knowledge not been reported in the ∂ u + u · ∇u = −∇p + 1 ∇2 u∂tReb∇·u = 0(1)where u = (u, v, w) are the velocity components, p is the pressure and Reb is the bulk Reynolds number. The above equations are discretized by using a pseudo-spectral method in thex- and z-directions and a second-order finite-difference schemeon the staggered grids in the y-direction. A third-order timesplitting method is employed for time advancement. To deal
with the deformation boundary, we use a coordinate transformation to transform the irregular physical domain (x, y, z) intoa rectangular computational domain (ξ1 , ξ2 , ξ3 ) based on theboundary-fitted system. The periodic condition is applied in
the streamwise and spanwise directions, and the no-slip condition is applied to the walls of the channel. More details regarding the numerical methods and analysis procedures can befound in Zhang et al. (2019).

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

Figure 1. Schematic diagram of the turbulent channel flow
over three-dimensional sinusoidal rough walls.
Figure 3. Profiles of the mean streamwise velocity for k/δ =1/18. The circle symbols represents smooth-wall cases, thesquare symbols represent the rough-wall cases, and the colors
represent different Reynolds numbers.

logarithmic law for a smooth wall and a rough wall respectively can be expressed as

Ur+ =Figure 2. Instantaneous vortical structures over rough walls.
Contours of the instantaneous streamwise velocity are plottedin the x − y plane and the y − z plane.

1 +lny +C − ∆U + ,κ(2)where κ is the von Kármán constant and C is the offset constant, here κ = 0.40 and C = 5.3. The velocity profiles decrease significantly with the increase of Reynolds number, thatis, and the main factor affecting the roughness function is theroughness height k+ . In our previous study (Ma et al., 2020),
we found that the roughness function is strongly correlatedwith the proportion of wall pressure drag and total drag, while
the roughness height has a significant impact on the pressuredrag. In addition, the mean velocity profiles of three groups ofrough cases all satisfy the log law from a certain position above
the crest of the roughness elements. Therefore, from the perspective of first-order statistics, the hypothesis of outer-layer
similarity is still satisfied for rough-wall cases (Townsend,1976). For the other two groups of cases, the variation trend issimilar to that of k+ = 1/18.
The presence of roughness elements has a significant effect on the velocity fluctuations. According to the phase average and triple decomposition, the second-order velocity correlation can be decomposed into three parts, i.e.

In this work, three groups of numerical cases were simulated with reference to Nikuradse’s pipe experiment: (1)k/δ = 1/18; (2) k/δ = 1/54; (3) k/δ = 1/108. For eachgroup, the friction Reynolds number was systematically variedin the range 180 to 1080, while the physical geometrical sizeof the roughness element relative to the half-channel height
was kept constant. The size of the computational domain is2πδ × 2δ × πδ for the all cases, where δ is the half channelheight. The mesh is uniformly spaced in the streamwise andspanwise directions, and stretched in the wall-normal direction
according to a cosine distribution. A detailed list of the roughness parameters and computational parameters can be found inTable 1.

RESULTS
Figure 2 shows the instantaneous three-dimensional vortical structures, to visually describe the differences in turbulent flow field caused by roughness elements. Here, we takethe rough case k+ = 30, λ + = 212 at Reτ = 540 of as an example. The overlying vortical structures are visualised via theisosurface of the local swirling strength λci . The vortical structures of the rough cases are dense and finely fragmented vortical structures appear near the roughness elements as comparedwith the smooth-wall case. The streamwise size, inclination
angle and distribution pattern of these vortices are modulatedby the rough form and rough parameters (De Marchis et al.,2015). Note that the low-speed zones are lifted up in the y-z
plane, corresponding to the formation of the streaky structures.
To study more quantitatively the effects of roughness, turbulent statistics are presented and discussed in the following.
The increase in wall drag caused by the surface roughness is manifested in the streamwise mean velocity profile as adownward shift in the logarithmic region, known as the roughness function ∆U + , as shown in figure 3. The roughness function is itself a function of the roughness Reynolds number. The

ui u j == ui u j + ũi ũ j + u′i u′j ,(3)where the second and third terms on the right-hand side denotethe dispersive and Reynolds stresses, which correspond to thewave-induced and turbulent components, respectively. Figure
4 shows the streamwise Reynolds stress profiles with respect
to the wall-normal height for k/δ = 1/18. Compared with thesmooth-wall case, the roughness elevates the wall-normal location about a roughness height, and the peak intensity of theReynolds stresses decreases with increasing k+ , which indicates that the typical coherent structures near the wall are disrupted, and the turbulent fluctuations are weakened. The larger
roughness viscous scale causes more significant suppression of
the near-wall turbulence. In the outer region, the profiles collapse with the smooth-wall case at higher vertical positions,
supporting the Townsend’s outer-layer similarity hypothesis.
Further, figure 5 shows the streamwise dispersive stressprofiles with respect to the wall-normal height for k/δ = 1/18.
+For the smooth-wall case, ũũ should be zero. The profiles

212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 1. Reynolds number, roughness parameters and computational setup. ∆x+ and ∆z+ characterize the grid resolution in the+streamwise and spanwise directions, respectively; ∆y+min and ∆yc are the minimum grid resolution near the bottom boundary and themaximum grid resolution near the channel centerline in the vertical direction, respectively.

Casek/δ = 1/18k/δ = 1/54k/δ = 1/108

Reτ

k+λ+Nx , Ny , Nz++∆x+ , ∆y+min , ∆yc , ∆z

180

10

71

144, 144, 144

7.85, 0.04, 4.0, 3.9

360

20

141

192, 144, 192

11.78, 0.08, 7.9, 5.9

540

30

212

288, 192, 288

11.78, 0.07, 7.9, 5.9

720

40

283

384, 288, 384

11.78, 0.04, 7.9, 5.9

1080

60

424

576, 384, 576

11.78, 0.04, 8.9, 5.9

180

3.33

24

384, 288, 384

2.94, 0.01, 1.9, 1.47

360

6.67

47

384, 288, 384

5.89, 0.02, 3.9, 2.9

540

10

71

384, 288, 384

8.83, 0.03, 5.9, 4.4

720

13.33

94

384, 288, 384

11.78, 0.04, 7.9, 5.9

1080

20

141

576, 384, 576

11.78, 0.06, 8.9, 5.9

180

1.67

12

576, 384, 576

1.96, 0.01, 1.5, 1.0

360

3.33

24

576, 384, 576

3.93, 0.01, 2.9, 2.0

540

5

35

576, 384, 576

5.89, 0.02, 4.4, 2.9

720

6.67

47

576, 384, 576

7.85, 0.02, 5.9, 3.9

1080

10

71

576, 384, 576

11.78, 0.04, 8.9, 5.9

Figure 4. Profiles of the streamwise Reynolds stress fork/δ = 1/18. The definitions of the lines and symbols are thesame as those in Figure 2.

Figure 5. Profiles of the streamwise dispersive stress fork/δ = 1/18. The definitions of the lines and symbols are thesame as those in Figure 2.

between the roughness function and the Reynolds number andthe roughness height. The roughness function ∆Ū + is calcu++
lated by the mean bulk velocity Ub,s−Ub,r. Then ∆Ū + is plot+
ted against k̄ S in figure 6. All the data collapse onto a single
line, i.e.

show an outward shift as the roughness Reynolds number k+increases, but do not show a monotonic variation different
from the Reynolds stress profiles. This may be related to theξ2 -plane average method (Zhang et al., 2019). Even so, thedispersive stresses for all the cases are dominant within theroughness sublayer but drop rapidly to zero above the crest ofroughness elements.
According to the above analysis, it’s known that the mean
velocity profiles and Reynolds stress profiles in the roughwall turbulence are affected by both the Reynolds number andthe roughness height. While our previous study (Ma et al.,2020) showed that the coupling scale k+ S produces a morereliable prediction than the equivalent sand grain roughness
height ks+ for various types of roughness, where S is the roughness steepness, i.e. the absolute streamwise gradient of thesurface. Therefore, we first consider the scaling relationship∆Ū + =

1.2
1ln k̄+ S+ 5.8,
κ(4)where k̄+ is the arithmetic average of the absolute values ofthe profile height (Chan et al., 2015). The goodness-of-fit
R2 is close to 0.97 for the above fitting function equation.
Eq.(4) shows that the roughness function is only affected by
the viscous-scaled roughness height k+ and the rough shapeparameter, i.e. the roughness steepness S, independent ofthe Reynolds number. In addition, some published data are3

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

Figure 6. Variations of the roughness function ∆Ū + with thecoupling roughness scale k̄+ S.

Figure 7. Variations of the roughness function u′p
coupling roughness scale Reτ /(k̄+ S)2 .

added for comparison. Correspondingly, several roughness
forms were chosen: the irregular random two-dimensional sinusoidal roughness (Napoli et al., 2008), close-packed rightangle pyramids (Schultz & Flack, 2009), randomly rotatedellipsoid (Yuan & Piomelli, 2014), three-dimensional ‘eggcarton’ rough pipe (Chan et al., 2015) and graphite and gritblasted roughness. Good agreement with Eq.(4) is evident inall cases, which indicates that the relationship for wall resistance prediction is also valid for other roughness forms. However, different roughness forms bring forth different intercepts
in the scaling formula Eq.(4) (Ma et al., 2020), especially for
random roughness patterns, which can be found in the literature (Ma et al., 2022).
In smooth-wall turbulence, the peak intensity of thestreamwise velocity fluctuations increases with the increase ofReynolds numbers (Marusic et al., 2010). In order to comprehensively consider the effect of Reynolds number and wallroughness, we refer to the above scaling method applied to the+roughness function and make a log-linear fit for u′p as a function of k̄+ S and Reτ as follows,

considered by systematically varying the relative roughness
height k/δ and the friction Reynolds number Reτ . Our current simulations, which are mostly in the transitionally roughregime in term of the Moody’s chart, where the turbulence
statistics are both dependent on the Reynolds number and theroughness height. To this end, we proposed the combinationof Reτ and k̄+ , i.e. Reτ /(k̄+ S)n , consistent with the theoretical derivation through a linear–log fit. Good data collapse
is obtained. While compared with the equivalent sand grainroughness height ks+ , k̄+ S appears to be the more suitable parameter, and indeed achieves a good scaling behaviour in theroughness function and the peak of the streamwise turbulent
velocity fluctuations for various types of roughness forms. The
present work can serve as a basis for substituting the widelyused Moody chart in engineering.

+u′p = 0.13ln"#Reτ2 + 1.5.
k̄+ Swith theACKNOWLEDGEMENTS
The authors acknowledge funding support from the National Natural Science Foundation of China under grant numbers 11772172 and 91752205 and the National Research Foundation of Korea (No. 2020R1A2C2008106).

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grid were performed for turbulent channel flow over threedimensional sinusoidal rough walls. With reference to Nikuradse’s pipe experimental data, three groups of rough cases are4

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