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 NUMERICALSIMULATIONS

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

pendent of the Reynolds number for a given roughness geometry and size, k+ (Flack et al., 2007). In turn, the change inC f depends on the Reynolds number through the referencesmooth Uδ+ . The offset increases with k+ , but how it varies
greatly depends on the roughness geometry and is difficult topredict a priori (Jiménez, 2004; Chung et al., 2021). To circumvent this difficulty, an equivalent “sand-grain roughness”,
ks+ , 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 problem from predicting ∆U + to predicting ks+ , as there is a oneto-one relationship between both quantities (Bradshaw, 2000;
Abderrahaman-Elena et al., 2019). Furthermore, the ratio ks /kfor a given surface only becomes constant in the fully roughregime (Jiménez, 2004; Chung et al., 2021), when it becomes+ /k+ and the curve k+ -∆U + becomes universal.
equal to ks,∞s,∞
This implies that ks+ , like ∆U + , does not exclusively depend on
the surface geometry but also on the flow, i.e., it is a hydraulicproperty, and only becomes a geometric property in the fullyrough regime. It is therefore important to understand the physical mechanisms at play in determining ∆U + , or ks+ , up to theroughness size for which the flow becomes fully rough. Beyond this point, once the curve becomes universal, the practical interest in understanding the physical mechanisms is more
limited.

We present results of fully resolved, direct numericalsimulations (DNSs) of turbulent flows over regular arrays ofcylindrical roughness elements. DNSs were conducted fork+ = 5, 10, 15, and 20 at Reτ ≈ 190 and for k+ = 20, 40, and
60 at Reτ ≈ 380, ranging from the onset of the transitionallyrough regime through to the fully rough regime. Data fromthe DNSs are presented and discussed for the roughness function, equivalent sand-gain roughness, mean flow velocity profiles, turbulence statistics, and spectral energy densities. Results suggest there is a progressive departure from smoothwall-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 thebackground turbulence and plan to assess the importance of
this mechanism in future work.

Introduction
Many engineering surfaces are rough and cause additional drag compared to smooth surfaces, and it is of industrial interest to quantify this drag. Sufficiently far above theroughness elements, it is commonly accepted that smoothand 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 offset in the mean velocity profile of a rough wall relative tothat of a smooth wall in the log layer. This is given by
the roughness function, ∆U + = US+ − UR+ , measured in thelog layer, where the superscript ‘+’ indicates wall-unit scaling with kinematic viscosity and the friction velocity, andthe subscripts ‘S’ and ‘R’ indicate smooth and rough walls,
respectively. Defining the skin-friction coefficient as C f =τw /(ρUδ2 /2) = 2/Uδ+2 , the impact of ∆U + on C f through thedecrease in Uδ+ for rough walls becomes immediately apparent(Spalart & McLean, 2011; Garcı́a-Mayoral & Jiménez, 2011;
Garcı́a-Mayoral et al., 2019; Chung et al., 2021). When expressed in wall units, ∆U + is generally believed to be inde-

Toward this aim, we conduct fully resolved, direct numerical simulations (DNSs) of turbulent flows over regular arrays of cylindrical roughness elements. The roughness function varies from the onset of the transitionally rough regime
through to the fully rough regime. When the roughness ismuch smaller than the smallest eddies in the near-wall flow,k+ << 20, the overlying turbulent flow perceives the near-wallflow to be smooth-wall-like (Abderrahaman-Elena et al., 2019;
Ibrahim et al., 2021) and the roughness is perceived as a homogenised boundary condition by the overlying flow (Bottaro,2019). As the roughness size becomes comparable to the sizeof the near-wall turbulent eddies, the overlying flow begins to1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 1. Roughness height, frictional Reynolds number, andresolution near the walls in the streamwise and spanwise directions for the cases investigated. Simulations marked withthe (∗ ) superscript are from Adams (2021).

Figure 1. Schematic plan view of the staggered cylindricalroughness elements investigated, from Vishwanathan et al.
(2021).

perceive the non-homogeneity of the texture. In the context
of alternating slip/no-slip textures, Fairhall et al. (2019) observed that the overlying flow still perceived a homogenised
boundary condition from the surface up to k+ ≈ 50, but notedthat the texture caused additional dissipation in the flow abovefrom k+ ≈ 15. Fairhall et al. (2019) proposed that this additional dissipation was caused by the nonlinear interaction between the texture-coherent flow and the background, textureincoherent turbulence. We aim to assess the importance of thismechanism for roughness, and ultimately to aid in the development of physics-based models that can predict ∆U + a priori
without resorting to costly experiments, simulations, or correlations (if available) to similar surfaces.

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

Results and Analysis
The measured roughness function against equivalent
sand-grain roughness is shown in Figure 3 for the cases studied, overlaid with values for other rough surfaces from Jiménez(2004). The cases investigated range from the onset of the transitionally rough regime, for k+ ≈ 5, through to the fully roughregime beyond k+ = 15. The roughness function measured
from the DNSs, ∆Um+ , is obtained by subtracting the roughwall 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 predicted values. The predicted roughness function, ∆U p+ , isobtained as follows. We define y+ = 0 at the plane of theroughness tips. For small roughness, k+ ≈ 5, turbulence issmooth-wall-like, except for an offset, ℓ+
T , such that it perceives an apparent ‘virtual’ origin at y+ = −ℓ+
T (Luchini, 1996;
Ibrahim et al., 2021). Turbulence then remains essentiallyunchanged compared to that over a smooth wall, except for
an offset given by the virtual origin. We find the value ofthe offset by fitting the curve representing the Reynolds shear
stress to smooth-wall data, as shown in Figure 5(f ). As longas turbulence remains smooth-wall-like, the smooth-wall andrough-wall Reynolds stresses collapse. From a mean momentum balance, this implies that the mean velocity profiles forrough and smooth walls will curve in the same way above thetips, but will be offset by a constant value which can, for instance, be measured at the plane of the tips (Gómez-de Segura& Garcı́a-Mayoral, 2019). At this plane, the mean velocity
for the rough case is Ut+ , while at the corresponding plane thesmooth-wall mean velocity is US+ (y+ = ℓ+
T ). The predictedroughness function assuming smooth-wall-like turbulence is+thus ∆U + = US+ (y+ = ℓ+
T ) − Ut , as portrayed in Figure 4.
Discrepancies in the results between the measured and predicted ∆U + for k+ > 5 suggest that a virtual origin framework
is insufficient to accurately estimate the roughness function beyond the smallest roughness size.
Mean velocity profiles, turbulent velocity fluctuations,and Reynolds stresses are shown in Figure 5 for k+ = 5 tok+ = 20 at Reτ ≈ 190. For small roughness, k+ = 5, there isgood collapse of the mean streamwise velocity and the wallnormal and spanwise root mean square (r.m.s.) velocity fluctuations shifted by the virtual origin onto the smooth-wall profile. The streamwise r.m.s. velocity fluctuations, however,Numerical Method
The DNSs solve the incompressible flow in a periodic
channel driven by a constant mean pressure gradient withroughness on the top and bottom walls, imposed using immersed boundaries, using a code adapted from Sharma &Garcı́a-Mayoral (2020a,b). The channel is of size 2πδ x2(δ + k) x πδ in the streamwise, wall-normal, and spanwise
directions, respectively, where δ is the channel half-heightfrom 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 asecond-order central difference scheme on a staggered grid isemployed in the wall-normal direction. The grid is stretched+such that ∆y+min ≈ 0.4 near the walls and ∆ymax ≈ 4 in the center. The code uses a ‘multiblock’ grid which allows finer resolution near the walls compared to the channel center to properly resolve the flow between the roughness elements. At thecenter of the channel, the grid resolution is standard for DNSs,
with ∆x+ ≤ 8 and ∆z+ ≤ 4. The resolution near the walls inthe streamwise and spanwise directions is given in Table 1 forthe list of the cases investigated at their respective frictional
Reynolds numbers. Time integration is carried out using athree-step Runge–Kutta scheme with a fractional step, pressure 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 cylindrical 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 realizationsin Figure 2 show simulation results over the texture geometry
for k+ = 10, 15, and 20 at Reτ ≈ 190.

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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.

ture the effect on the outer layers of the flow, i.e., the roughness function (Abderrahaman-Elena et al., 2019). To analyze
the effect of frictional Reynolds number on the mean velocity profile and turbulence statistics, we compare the results atReτ ≈ 187 with Reτ ≈ 376 for k+ = 20, in contrast with theresults 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 frictional Reynolds number. For rough-wall data, the trends inthe mean velocity profiles are very similar at the two frictional 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 corresponding frictional Reynolds numbers, which is indicative of
the recovery of outer-layer similarity. Minor differences in thecollapse are caused by small differences in the corresponding
frictional Reynolds numbers. These arguments support the useof DNSs at a low frictional Reynolds number.

k+ = 60

10k+ = 40∆U +k+ = 20k+ = 155k+ = 10k+ = 5

0
100

101

102

ks+∞

Figure 3. Roughness function against equivalent sand-grainroughness for the present regular staggered cylinders (squares
for Reτ ≈ 190 and diamonds for Reτ ≈ 380), compared withuniform sand (circles), uniform packed spheres (white triangles), galvanised iron (dotted line), tar-coated cast iron (dashedline), wrought iron (dotted dashed line), interpolation (solidline) and riblets (black triangles). Adapted from Jiménez(2004).

From the observations of Fairhall et al. (2019) in the context of the alternating slip/no-slip textures mentioned earlier,the differences that arise from extra Reynolds stresses are hypothesised to be due to the nonlinear interaction of the background turbulence with the dispersive flow. The spectral energy densities in Figure 7, which portray the distribution ofenergy across different length scales in the flow at y+ = 4
above the roughness tips, demonstrate these effects. There isa progressive departure from smooth-wall energy densities towards shorter and wider wavelengths as the roughness heightincreases from k+ = 5 to k+ = 60. The nonlinear interaction
between the texture-coherent flow and background turbulenceis evident from the distortion in energy contours in the vicinityof the concentration of energy representing the texture-induced
flow from k+ = 10 onward. We are interested in characterizing deviations from smooth-wall turbulence and the dispersive
flow from this data.

do not collapse onto the smooth-wall profile near the wall.
Ibrahim et al. (2021) argued that this does not essentially affect the structure of near-wall turbulence, as the latter’s virtual origin is essentially determined by the origin perceived by
the quasi-streamwise vortices, which mainly induce only wallnormal 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 bya 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-3

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, Ut+ , at Reτ ≈ 190 () and Reτ ≈ 380 (- - - -) and a) and Reτ ≈ 380 (- - - -), and (b) measured values of the roughness function from the DNSs,
smooth wall, US+ (ℓ+T ), at Reτ ≈ 190 (∆Um+ , at Reτ ≈ 190 () and Reτ ≈ 380 (- - - -) and corresponding predicted values from the virtual origin framework, ∆U p+ , at
) and Reτ ≈ 380 (- - - -).
Reτ ≈ 190 (Figure 5. Results from DNSs for k+ ≈ 5 (), k+ ≈ 10 (), k+ ≈ 15 (), and k+ ≈ 20 () with Reτ ≈ 190, anda 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 thepredicted 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.

Acknowledgments

SummaryThis work has been partially supported by EPSRC-UK
under grant EP/S013083/1. H.M. is supported by the Cambridge Commonwealth, European and International Trust.
Computational resources were provided by the Cambridge
Service for Data Driven Discovery under EPSRC-UK Tier2 grant EP/P020259/1 and by the ARCHER2 system under
PRACE grant pr1u1702-ESTONT.

In this paper, results from DNSs of turbulent flows overregular arrays of cylindrical roughness elements are presented
and discussed. DNSs were conducted for k+ = 5, 10, 15, and20 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 velocity profiles, turbulence statistics, and spectral energy densities
from the DNSs suggest there is a progressive departure from4

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+ ≈ 20 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.

texturing. Fluid Dynamics Research 51 (1), 011410.
Ibrahim, Joseph I., Gómez-de Segura, Garazi, Chung, Daniel& Garcı́a-Mayoral, Ricardo 2021 The smooth-wall-like behaviour of turbulence over drag-altering surfaces: a unifying virtual-origin framework. J. Fluid Mech. 915, A56.
Jiménez, J. 2004 Turbulent flows over rough walls. Annu. Rev.
Fluid Mech 36, 173–196.
Le, H. & Moin, P. 1991 An improvement of fractional step
methods for the incompressible Navier-Stokes equations. J.
Comput. Phys. 92 (2), 369–379.
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John Wiley & Sons Ltd.
Nikuradse, J. 1933 Strömungsgesetze in rauhen rohren. VDIForschungsheft 361.
Gómez-de Segura, G. & Garcı́a-Mayoral, R. 2019 Turbulent
drag reduction by anisotropic permeable substrates – analysis and direct numerical simulations. J. Fluid Mech. 875,
124–172.
Sharma, A. & Garcı́a-Mayoral, R. 2020a Scaling and dynamics of turbulence over sparse canopies. J. Fluid Mech. 888,
A1.
Sharma, A. & Garcı́a-Mayoral, R. 2020b Turbulent flows overdense filament canopies. J. Fluid Mech. 888, A2.
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R. Soc. 369, 1556–1569.
Townsend, A. A. 1956 The Structure of Turbulent Shear Flow.
Cambridge University Press.
Vishwanathan, Vidya, Fritsch, Danny, Lowe, Todd K & Devenport, William J 2021 Analysis of coherent structures
over a smooth wall turbulent boundary layer in pressure
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AIAA Aviation 2021 Forum, p. 2893.

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.

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2019 The control of near-wall turbulence through surface

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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) kx kz Euu
+
+
+
+
+
+
(a2) − (g2) kx kz Evv , (a3) − (g3) kx kz Eww , and (a4) − (g4) −kx kz Euv . 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 wallat Reτ ≈ 180 and Reτ ≈ 360 (filled contours).

6