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J. Fluid Mech. (2021), vol.912, A8, doi:10.1017/jfm.2020.1085

Data-driven prediction of the equivalent

sand-grain height in rough-wall turbulent ﬂows

Mostafa Aghaei Jouybari1,†,JunlinYuan

1,GilesJ.Brereton

1and

Michael S. Murillo2

1Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824, USA

2Department of Computational Mathematics, Science and Engineering, Michigan State University,

East Lansing, MI 48824, USA

(Received 23 June 2020; revised 1 October 2020; accepted 27 November 2020)

This paper investigates a long-standing question about the effect of surface roughness on

turbulent ﬂow: What is the equivalent roughness sand-grain height for a given roughness

topography? Deep neural network (DNN) and Gaussian process regression (GPR) machine

learning approaches are used to develop a high-ﬁdelity prediction approach of the

Nikuradse equivalent sand-grain height ksfor turbulent ﬂows over a wide variety of

different rough surfaces. To this end, 45 surface geometries were generated and the ﬂow

over them simulated at Reτ=1000 using direct numerical simulations. These surface

geometries differed signiﬁcantly in moments of surface height ﬂuctuations, effective slope,

average inclination, porosity and degree of randomness. Thirty of these surfaces were

considered fully rough, and they were supplemented with experimental data for fully rough

ﬂows over 15 more surfaces available from previous studies. The DNN and GPR methods

predicted kswith an average error of less than 10 % and a maximum error of less than 30 %,

which appears to be signiﬁcantly more accurate than existing prediction formulae. They

also identiﬁed the surface porosity and the effective slope of roughness in the spanwise

direction as important factors in drag prediction.

Key words: turbulence modelling

1. Introduction

At sufﬁciently high Reynolds numbers all surfaces are hydrodynamically rough, as

is almost always the case for ﬂows past the surfaces of naval vessels. Reviews of

roughness effects on wall-bounded turbulent ﬂows are provided by Raupach, Antonia &

Rajagopalan (1991) and Jiménez (2004). The most important effect of surface roughness

in engineering applications is an increase in the hydrodynamic drag (Flack 2018), which is

†Email address for correspondence: aghaeijo@msu.edu

© The Author(s), 2021. Published by Cambridge University Press 912 A8-1

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M. Aghaei Jouybari, J. Yuan, G.J. Brereton and M.S. Murillo

due predominantly to the pressure drag generated by the small-scale recirculation regions

associated with individual roughness protuberances.

For the foreseeable future, the most practical approach to making predictive

ﬂow calculations for many realistic applications is to use engineering one-point

closures of turbulence, such as two-equation turbulent eddy-viscosity models to the

Reynolds-averaged Navier–Stokes equations. Existing rough-wall corrections to this type

of closure typically model the increase in hydrodynamic drag on a single length scale –

the equivalent sand-grain height (Nikuradse 1933)ks– without physically resolving the

surface or changing the governing equations. In the fully rough ﬂow regime, where the

wall friction depends on the roughness alone and is independent of the Reynolds number,

kswas observed to quantify the increase in hydrodynamic drag through the empirical

relation with the roughness function (deﬁned as the offset of the log–linear velocity proﬁle

of a rough-wall ﬂow relative to that of a smooth-wall one),

U+=1

κln k+

s−3.5,(1.1)

where κ=0.41 is the von Kármán constant and +represents normalization in wall units.

A universal length scale (e.g. ksin Nikuradse’s relation, or in the Moody diagram

Moody 1944) that can predict accurately the surface drag coefﬁcient is not known a

priori and does not appear to be equivalent to any single geometrical length scale, such

as an average or a root mean square (r.m.s.) of roughness height (Flack 2018). It is also

well-established that kscan depend on many geometrical parameters such as the effective

slope (Napoli, Armenio & De Marchis 2008; Yuan & Piomelli 2014a)andtheskewnessof

the roughness height distribution (Flack & Schultz 2010). Readers are referred to Flack &

Schultz (2010) and Bons (2002) for extensive reviews on this topic. Empirical expressions

for ksbased on a small number of geometrical roughness parameters include, among

others,

ks=c1kavg(α2

rms +c2αrms), ks=c1kavgΛc2

sand ks=c1krms (1+Sk)c2,

(1.2a–c)

proposed by Bons et al. (2001), van Rij, Belnap & Ligrani (2002) and Flack & Schultz

(2010), respectively. Here kavgis the average height, αis the local streamwise slope angle

and Λs=(S/Sf)(Sf/Ss)−1.6(where S,Sf,Ssare, respectively, the platform area, the total

frontal area and the total windward wetted area of the roughness) while krms and Skare the

r.m.s. and skewness of the roughness height ﬂuctuations and c1and c2are constants.

The hydrodynamic length scale ksappears to be correlated with different sets of

geometrical parameters for each type of rough surface and no universal correlation

currently exists for ﬂow over surfaces of arbitrary roughness. For example, for synthetic

roughness comprising closely packed pyramids (Schultz & Flack 2009) and random

sinusoidal waves (Napoli et al. 2008), it has been shown that ksscales on the effective

slope when the surface slope is gentle (i.e. within the ‘waviness’ regime), whereas the

skewness and r.m.s. height, but not slope magnitude, become important when the slope is

steeper (i.e. within the ‘roughness’ regime). The boundary between these two regimes has

been shown to be surface dependent (Yuan & Piomelli 2014a).

Some more recent studies of kscorrelations are summarized below. Thakkar, Busse

& Sandham (2017) carried out direct numerical simulation (DNS) of transitionally

rough turbulent ﬂows for different irregular roughness topographies. They found that

the roughness function is inﬂuenced by solidity, skewness, the streamwise correlation

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Equivalent sand-grain height prediction of rough walls

length scale and the r.m.s. of roughness height. Flack, Schultz & Barros (2020) performed

several experiments to systematically investigate the effects of the skewness and amplitude

of roughness height on the skin friction. They found that the r.m.s. and skewness of

roughness height ﬂuctuations are important scaling parameters for the prediction of

roughness function; however, the surfaces with positive, negative and zero skewness values

needed different correlations. Also, Chan et al. (2015) simulated turbulent pipe ﬂows over

sinusoidal roughness geometries and conﬁrmed strong dependence of roughness function

on the average height and streamwise effective slope.

In previous studies, the small number of roughness parameters used to devise ks

correlations tended to limit their application to a narrow range of surface roughness. Since

it appears that many geometrical parameters, such as porosity, moments of roughness

height (e.g. r.m.s., skewness and kurtusis), effective slope and surface inclination angle

might affect ks, it is useful to employ a data science approach suited to modelling large

multivariate/multioutput systems.

Speciﬁcally, we use machine learning (ML) to explore ks-prediction approaches that

depend on a large set of surface-topographical parameters, with the expectation that

the resulting models may be applied accurately to a wider range of surfaces. Since the

prediction of ksfrom surface topography is essentially a labelled regression problem,

supervised ML operations were performed using deep neural networks (DNN) and

Gaussian process regressions (GPR). Both methods are explained thoroughly in §3.

Readers are referred to the monograph by Rasmussen & Williams (2006) and the review

provided by LeCun, Bengio & Hinton (2015) for detailed descriptions of these methods.

An initial ensemble of 60 sets of data on ksas a function of topographical parameters –

45 DNS results and 15 experimental results – was considered. All experimental data sets

are fully rough, and of the DNS data, 30 are considered fully rough ﬂows; all fully rough

cases were used for ML training and testing. To the best of our knowledge, this ensemble

of roughness geometries is the most extensive used for developing a ks-prediction method.

In this paper, we ﬁrst present the governing equations, solution methodologies,

simulation parameters and different roughness topographies, and then discuss the

post-processed DNS results used to calculate ksfor each surface. Finally, we describe the

ML models, their predictions of ksand their uncertainty.

2. Problem formulation

2.1. Governing equations

The governing equations of incompressible continuity and linear momentum – the

Navier–Stokes equations – for a constant-property Newtonian ﬂuid, were solved by DNS.

These equations are written in indicial notation as

∂ui

∂xi

=0,(2.1a)

∂ui

∂t+∂uiuj

∂xj

=−

∂P

∂xi

+ν∂2ui

∂xj∂xj

+Fi,(2.1b)

where i,j=1,2,3, x1,x2and x3(or x,y,z) are the streamwise, wall-normal and spanwise

coordinates, with corresponding velocity components of u1,u2and u3(or u,v,w)andP

is deﬁned as p/ρ,wherepis the pressure and ρis the ﬂuid density; νis the kinematic

viscosity. An immersed boundary method (Yuan & Piomelli 2014b) was used to enforce

the ﬁne-grained roughness boundary conditions on a non-conformal Cartesian grid.

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M. Aghaei Jouybari, J. Yuan, G.J. Brereton and M.S. Murillo

The corresponding body force Fiis added to the right-hand side of the momentum

equations to impose a no-slip boundary condition at the ﬂuid–roughness interface. To

solve the equations, second-order central differencing was used for spatial discretizations

and second-order Adams–Bashforth semi-implicit time advancement was employed. The

numerical solver was parallelized using a message passing interface (known as MPI)

method (Keating 2004).

A double-averaging decomposition (Raupach & Shaw 1982) was used to resolve

turbulent and dispersive components of ﬂow variables in the presence of roughness. In

this decomposition, any instantaneous ﬂow variable θmay be decomposed into three

components, as

θ(x,t)=¯

θ(y)+θ(x,t)+˜

θ(x), (2.2)

where the time-averaging operator is ¯

θand the intrinsic spatial-averaging operator is θ=

1/Afx,zθdA(and Af(y)is the area occupied by ﬂuid at an elevation y). The Reynolds and

dispersive ﬂuctuating components are then θ=θ−¯

θand ˜

θ=¯

θ−¯

θ,respectively. Here

¯

θis called the double-averaged component.

The wall shear stress (including both viscous and pressure drag contributions on a rough

wall) was determined by integrating the time-averaged immersed boundary method body

force in the x-direction F1as

τw=ρ

LxLzV

F1(x,y,z)dxdydz,(2.3)

where Vrepresents the simulation domain volume below the roughness crest and Lxiis the

domain length in the xi-direction. Readers are referred to Yuan & Piomelli (2014b,c)for

details of the implementation and validation of the immersed boundary method and the τw

calculation.

2.2. Surface roughness

In ﬁgure 1, surface plots of the 45 roughness geometries used in these simulations

are displayed; their statistical properties are given in table 1. Each case name in

ﬁgure 1 and table 1 begins with the letter C or E, which denotes whether the data

is computational or experimental, followed by an identifying index for that particular

surface. For computational cases, this index is followed by: a characteristic length scale

(as a percentage of δ) used for roughness synthesis; an identiﬁer of whether the surface

roughness is regular (reg) or random (rnd); and ﬁnally an identiﬁer for one additional

surface feature and its position in a series of surfaces with different sizes of that feature.

These features were: the streamwise inclination angle Ixin surfaces C01 to C12; the

porosity Poin surfaces C13 to C24; and the streamwise effective slope Exin surfaces

C25 to C30. For the experimental data two indices were assigned to each surface. The

ﬁrst denotes the year in which the data were published and the second is the surface

designation in that publication. Thus surfaces with index 16 are from Flack et al. (2016),

those with index 18 are from Barros, Schultz & Flack (2018)and those with index 19

are from Flack et al. (2020). Note that these experimental data were obtained from fully

developed channel ﬂows, where the drag was measured through the pressure drop along

the channel. Thus their results are expected to be more accurate than those of boundary

layer studies where the drag is usually inferred.

Surfaces C01 to C24 were created using ellipsoidal elements (Scotti 2006)of

different size, aspect ratio and inclination. For regular roughness, each element had the

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Equivalent sand-grain height prediction of rough walls

C01,r4,reg,inc1

C07,r4,rnd,inc1

C13,r4,reg,por1

C19,r4,rnd,por1

C25,r4,reg,ES1 C26,r4,reg,ES2 C27,r4,reg,ES3 C28,r6,reg,ES1 C29,r6,reg,ES2 C30,r6,reg,ES3

C31,r4,rnd,SGR C32,r4,rnd,RND1 C33,r4,rnd,RND2 C34,r4,rnd,RND3 C35,r4,rnd,RND4 C36,r4,rnd,RND5

C37,r6,rnd,SGR C38,r6,rnd,RND1 C39,r6,rnd,RND2 C40,r6,rnd,RND3 C41,r6,rnd,RND4 C42,r6,rnd,RND5

C20,r4,rnd,por2 C21,r4,rnd,por3 C22,r6,rnd,por1 C23,r6,rnd,por2 C24,r6,rnd,por3

C14,r4,reg,por2 C15,r4,reg,por3 C16,r6,reg,por1 C17,r6,reg,por2 C18,r6,reg,por3

C08,r4,rnd,inc2 C09,r4,rnd,inc3 C10,r6,rnd,inc1 C11,r6,rnd,inc2 C12,r6,rnd,inc3

C02,r4,reg,inc2 C03,r4,reg,inc3 C04,r6,reg,inc1 C05,r6,reg,inc2 C06,r6,reg,inc3

C43,SG C44,TB C45,CB

Figure 1. Roughness geometries – each plot is a section of size δ×0.5δin the x–zplane. Cases C43 to C45

are from simulations with regular domain sizes (Yuan & Piomelli 2014a; Aghaei Jouybari, Brereton & Yuan

2019).

same orientation and semiaxis lengths, (λ1,λ2,λ3)=(1.0,0.7,0.5)kc,wherekcis the

peak-to-trough height (also called the crest height). For random roughness, the elements

had random orientations and semiaxis lengths (with uniform distributions of the random

variables). The average orientation and semiaxis lengths for random roughness were the

same as the corresponding regular surface. Surfaces C25 to C30 comprised sinusoidal

waves in the x-direction, of the same magnitude but different wavelengths, to generate

different values of effective slope Ex. The wavelengths were 3δ/4, 3δ/8andδ/6. Surfaces

C31 and C37 comprised the random sand-grain roughness of Scotti, which were produced

by randomly oriented ellipsoidal elements with ﬁxed semiaxes of (1.0,0.7,0.5)kc.

Surfaces C32 to C36 and C38 to C42 were generated as the low-order (the ﬁrst 5, 10, 20,

30 and 50) modes of Fourier transforms of white noise in the streamwise and spanwise

directions; they therefore describe random surfaces with large- to small-wavelength

roughness. Cases C43, C44 and C45 are DNS results from full-span channel computations

912 A8-5

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M. Aghaei Jouybari, J. Yuan, G.J. Brereton and M.S. Murillo

of ﬂow over surfaces of: random sand-grain roughness; the roughness found on a turbine

blade (Yuan & Aghaei Jouybari 2018); and arrays of cubes (from the study of Aghaei

Jouybari et al. (2019)), respectively. Case C46 is a full-span DNS of case C21, generated

to validate the minimal-channel approach of the preceding cases. A baseline smooth-wall

ﬂow was also simulated using a full-span channel (Yuan & Aghaei Jouybari 2018).

The geometric parameters reported for each surface in table 1 are: roughness

peak-to-trough height (also termed crest height) kc(i.e. distance between the highest and

the lowest surface points); mean peak-to-trough height kt(i.e. the average of peak-to-trough

heights obtained from surface tiles of size δ×δ, similar to Forooghi et al. 2017); mean

roughness height kavg; ﬁrst-order moment of height ﬂuctuations Ra;root mean square krms,

skewness Skand kurtosis Kuof the roughness height ﬂuctuations; surface porosity Po;

effective slope in the xi-direction Exi; and inclination angle (in radians) in the xi-direction

Ixi, together with the hydrodynamic length scale ksdeduced from the mean velocity ﬁeld

using (1.1).

These geometrical parameters are deﬁned as

kavg=1

Atx,z

kdA,(2.4)

Ra=1

Atx,z

|k−kavg|dA,(2.5)

krms =1

Atx,z

(k−kavg)2dA,(2.6)

Sk=1

Atx,z

(k−kavg)3dAk3

rms,(2.7)

Ku=1

Atx,z

(k−kavg)4dAk4

rms,(2.8)

Ex=1

Atx,z

∂k

∂x

dA,(2.9)

Ez=1

Atx,z

∂k

∂z

dA,(2.10)

Po=1

Atkckc

0

Afdy,(2.11)

Ix=tan−11

2Sk∂k

∂x,(2.12)

Iz=tan−11

2Sk∂k

∂z,(2.13)

where k(x,z)is the roughness height distribution and Af(y)and At(y)are the ﬂuid and

total planar areas at each ylocation. Here Sk(∂k/∂ xi)is the skewness of the ∂k/∂xi

distribution. In table 1,kavg,kc,krms and ksare then normalized by the ﬁrst-order moment

of height ﬂuctuations Raand were incorporated in the ML algorithms in this form. All

surfaces considered were in the ranges kc/δ 0.17 and Ra/δ 0.04.

912 A8-6

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Equivalent sand-grain height prediction of rough walls

Case name kavgkcktkrms RaIxIzPoExEzSkKuks

C01,r4,reg,inc1 0.026 0.043 0.043 0.013 0.011 −0.801 −0.089 0.535 0.584 0.510 −0.544 2.177 —

C02,r4,reg,inc2 0.030 0.059 0.059 0.021 0.019 0.012 0.032 0.609 1.029 0.562 −0.265 1.597 —

C03,r4,reg,inc3 0.025 0.043 0.043 0.013 0.011 0.821 −0.078 0.537 0.600 0.485 −0.459 2.052 —

C04,r6,reg,inc1 0.032 0.064 0.064 0.022 0.019 −0.978 0.016 0.597 0.595 0.590 −0.167 1.601 0.064

C05,r6,reg,inc2 0.038 0.088 0.088 0.033 0.030 0.025 0.064 0.654 0.916 0.643 0.109 1.436 0.124

C06,r6,reg,inc3 0.031 0.064 0.064 0.022 0.019 0.955 0.121 0.599 0.588 0.558 −0.087 1.590 0.059

C07,r4,rnd,inc1 0.025 0.086 0.084 0.022 0.019 −0.860 0.033 0.774 0.511 0.559 0.560 2.244 0.136

C08,r4,rnd,inc2 0.027 0.116 0.115 0.030 0.025 −0.007 0.048 0.819 0.861 0.604 0.870 2.627 0.322

C09,r4,rnd,inc3 0.025 0.083 0.081 0.021 0.018 0.829 0.002 0.753 0.517 0.482 0.514 2.292 0.131

C10,r6,rnd,inc1 0.026 0.125 0.120 0.030 0.025 −0.957 −0.019 0.835 0.498 0.578 0.967 2.874 0.269

C11,r6,rnd,inc2 0.033 0.172 0.169 0.044 0.037 0.076 0.138 0.842 0.758 0.543 1.150 3.176 0.536

C12,r6,rnd,inc3 0.032 0.127 0.121 0.032 0.027 0.923 0.032 0.784 0.508 0.471 0.758 2.642 0.272

C13,r4,reg,por1 0.038 0.059 0.059 0.018 0.015 0.024 0.067 0.498 1.0 43 0.523 −0.820 2.508 —

C14,r4,reg,por2 0.018 0.059 0.059 0.022 0.020 0.021 0.038 0.776 0.613 0.456 0.708 1.840 0.141

C15,r4,reg,por3 0.010 0.059 0.059 0.019 0.014 0.022 0.063 0.877 0.334 0.253 1.64 6 4.094 0.157

C16,r6,reg,por1 0.051 0.089 0.089 0.030 0.026 0.041 0.149 0.529 1.137 0.534 −0.538 1.873 0.077

C17,r6,reg,por2 0.022 0.089 0.089 0.031 0.027 0.041 0.080 0.801 0.537 0.403 0.982 2.308 0.260

C18,r6,reg,por3 0.013 0.089 0.089 0.026 0.020 0.057 0.126 0.886 0.307 0.230 1.849 4.839 0.247

C19,r4,rnd,por1 0.027 0.112 0.108 0.021 0.017 0.025 −0.107 0.806 0.487 0.486 0.732 3.422 0.158

C20,r4,rnd,por2 0.013 0.095 0.087 0.017 0.014 0.032 −0.646 0.896 0.311 0.323 1.343 4.126 0.106

C21,r4,rnd,por3 0.009 0.098 0.094 0.016 0.012 0.321 −0.741 0.929 0.219 0.233 2.168 7.72 8 0.103

C22,r6,rnd,por1 0.035 0.139 0.139 0.029 0.024 −0.070 −0.245 0.791 0.456 0.499 0.591 2.830 0.277

C23,r6,rnd,por2 0.017 0.123 0.111 0.025 0.020 −0.672 −0.841 0.885 0.305 0.325 1.467 4.347 0.175

C24,r6,rnd,por3 0.014 0.152 0.145 0.027 0.019 0.189 −0.056 0.926 0.254 0.257 2.371 8.740 0.260

C25,r4,reg,ES1 0.020 0.040 0.040 0.014 0.013 0.046 0.006 0.510 0.106 0.009 −0.032 1.503 —

C26,r4,reg,ES2 0.021 0.040 0.040 0.014 0.013 0.039 −0.001 0.510 0.212 0.020 −0.071 1.505 0.065

C27,r4,reg,ES3 0.023 0.040 0.040 0.014 0.012 0.006 −0.023 0.510 0.609 0.032 −0.214 1.544 —

C28,r6,reg,ES1 0.030 0.059 0.059 0.021 0.019 0.044 0.018 0.504 0.158 0.015 −0.031 1.499 0.071

C29,r6,reg,ES2 0.031 0.059 0.059 0.021 0.019 0.028 −0.069 0.504 0.316 0.022 −0.071 1.503 0.112

C30,r6,reg,ES3 0.034 0.059 0.059 0.020 0.018 0.015 −0.069 0.505 0.917 0.048 −0.203 1.543 0.064

C31,r4,rnd,SGR 0.025 0.059 0.059 0.011 0.009 0.104 −0.039 0.648 0.370 0.398 0.378 2.784 0.049

C32,r4,rnd,RND1 0.040 0.075 0.072 0.013 0.010 0.117 0.108 0.479 0.068 0.169 −0.069 2.991 —

C33,r4,rnd,RND2 0.041 0.088 0.084 0.013 0.011 0.109 0.078 0.553 0 .117 0.308 0.004 2.763 —

C34,r4,rnd,RND3 0.042 0.080 0.071 0.010 0.008 0.070 0.051 0.508 0.175 0.458 −0.002 3.031 —

C35,r4,rnd,RND4 0.043 0.077 0.066 0.008 0.007 0.039 0.042 0.488 0.218 0.558 0.013 2.941 —

C36,r4,rnd,RND5 0.045 0.084 0.067 0.009 0.007 0.035 0.037 0.535 0.378 0.841 0.075 3.018 —

C37,r6,rnd,SGR 0.037 0.088 0.088 0.018 0.015 0.312 0.180 0.640 0.428 0.463 0.323 2.686 0.109

C38,r6,rnd,RND1 0.060 0.106 0.091 0.016 0.012 0.045 0.028 0.444 0.077 0.183 −0.220 3.258 —

C39,r6,rnd,RND2 0.061 0.098 0.095 0.012 0.009 0.111 0.057 0.400 0.108 0.285 −0.020 3.267 —

C40,r6,rnd,RND3 0.064 0.121 0.112 0.016 0.013 0.061 0.022 0.512 0.280 0.760 0.037 2.977 0.050

C41,r6,rnd,RND4 0.065 0.130 0.130 0.015 0.012 0.045 0.037 0.546 0.374 0.989 0.028 3.036 —

C42,r6,rnd,RND5 0.068 0.118 0.116 0.013 0.010 0.037 0.025 0.503 0.547 1.204 0.052 2.933 —

C43,SG 0.036 0.089 0.087 0.017 0.014 0.288 0.156 0.649 0.425 0.441 0.476 2.970 0.093

C44,TB 0.055 0.125 0.088 0.018 0.014 0.007 −0.006 0.569 0.097 0.081 0.200 3.493 0.024

C45,CB 0.010 0.070 0.070 0.023 0.016 0.420 0.508 0.878 0.249 0.247 2.101 5.569 0.150

C46,r4,rnd,por3,FS 0.009 0.098 0.094 0.016 0.012 0.321 −0.715 0.929 0.219 0.234 2.168 7.728 0.104

E01,16,2 0.138 0.261 0.254 0.020 0.016 −0.005 0.011 0.472 0.720 0.835 −0.711 3.843 0.052

E02,16,3 0.143 0.252 0.252 0.021 0.016 −0.021 0.010 0.432 0.740 0.868 −0.338 3.159 0.050

E03,16,7 0.133 0.365 0.254 0.019 0.014 −0.038 0.000 0.638 0.618 0.705 −1.169 5.292 0.058

E04,16,8 0.126 0.298 0.227 0.017 0.013 −0.034 0.009 0.579 0.587 0.682 −1.445 5.421 0.056

E05,16,9 0.112 0.308 0.167 0.018 0.014 −0.031 0.015 0.637 0.636 0.753 −0.738 3.714 0.043

E06,16,15 0.081 0.191 0.191 0.013 0.010 −0.027 0.003 0.578 0.621 0.713 −0.687 3.854 0.035

E07,18,1 0.121 0.241 0.227 0.026 0.021 −0.013 −0.183 0.500 0.181 0.188 0.107 2.941 0.053

E08,18,2 0.143 0.276 0.255 0.032 0.025 −0.019 0.194 0.483 0.162 0.164 0.093 2.967 0.034

E09,19,1 0.204 0.398 0.344 0.046 0.036 0.042 −0.096 0.487 0.227 0.230 −0.080 2.989 0.065

E10,19,2 0.389 0.763 0.689 0.088 0.070 0.046 0.002 0.492 0.447 0.452 −0.065 2.925 0.200

E11,19,3 0.477 0.730 0.679 0.088 0.070 −0.029 −0.245 0.348 0.434 0.432 −0.660 3.274 0.160

E12,19,4 0.459 0.751 0.710 0.089 0.071 −0.052 0.036 0.391 0.455 0.459 −0.351 3.041 0.180

E13,19,5 0.292 0.732 0.650 0.090 0.072 −0.058 −0.004 0.602 0.445 0.452 0.346 3.051 0.245

E14,19,6 0.202 0.711 0.604 0.087 0.069 0.004 −0.010 0.716 0.391 0.400 0.812 3.559 0.435

E15,19,7 0.522 0.967 0.894 0.114 0.092 −0.050 −0.235 0.462 0.557 0.562 −0.066 2.794 0.230

Table 1. Statistical parameters of roughness topography and the equivalent sand-grain height ksfor each

roughness geometry. Here Ra,kavg,kc,kt,krms and ksvalues from DNS are normalized by the channel

half-height δ, while corresponding experimental values are given in mm; ksis not listed for cases thought

to be transitionally rough.

912 A8-7

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M. Aghaei Jouybari, J. Yuan, G.J. Brereton and M.S. Murillo

2.3. Simulation parameters

Direct numerical simulation was used to calculate the velocity and pressure ﬁelds in

turbulent open-channel ﬂows over 45 different rough surfaces and one smooth one, at

a constant frictional Reynolds number Reτ=uτδ/ν =1000, where uτis the friction

velocity and δis the channel half-height. In these simulations, the domain sizes were

(Lx,Ly,Lz)=(3,1,1)δ. The origin of the yaxis was the elevation of the lowest trough

for each rough surface. The number of grid points was (nx,ny,nz)=(400,300,160).A

uniform mesh was used in the x-andz-directions, yielding grid sizes of x+=7.5and

z+=6.3, where +denotes normalization in wall units. For all cases, the mesh was

stretched in the y-direction with a hyperbolic tangent function, with the third grid point

from the origin at y+<1. For the rough-wall cases, at the roughness crest, y/kc0.017,

with this ratio taking its highest value for case C11. The maximum grid size was y+

max =

9.5 at the channel centreline, where the Kolmogorov length scale η+≈6. Moin & Mahesh

(1998) have proposed that one requirement for obtaining reliable ﬁrst- and second-order

ﬂow statistics is that the grid resolution must be ﬁne enough to capture accurately most of

the dissipation, while Moser & Moin (1987) noted that most of the dissipation in curved

channel ﬂow occurs at scales greater than 15η(based on average dissipation). It follows

that for DNS computations of these kinds of ﬂow statistics in channel and boundary-layer

ﬂows, x/η and z/η are typically chosen between 7 to 15 and 4 to 8, respectively (see, for

example Kim, Moin & Moser 1987; Spalart 1988; Yuan & Piomelli 2014c). The grid sizes

in this study were chosen accordingly and were x/η < 7.5, y/η < 4.0and z/η < 6.5.

Periodic boundary conditions were imposed in the streamwise and spanwise directions,

with no-slip and symmetry boundary conditions at the bottom and top boundaries,

respectively. After each simulation had reached statistical stationarity, data were collected

for ensemble averaging over 10 large-eddy turn-over times (δ/uτ). In these simulations,

the time step τ+0.04 and so was signiﬁcantly smaller than the largest acceptable one

of τ+≈0.2 recommended by Choi & Moin (1994) for DNS.

The surface Taylor microscales λT,xand λT,z,inthex-andz-directions, were used to

evaluate the adequacy of the grid resolution for resolving details of ﬂow in the roughness

sublayer, following Yuan & Piomelli (2014b). These geometric microscales were obtained

by ﬁtting a parabola to the two-point autocorrelation of the surface height ﬂuctuation in

the respective direction. They represent the size of an equivalent ‘roughness element’ in

the context of random multiscale roughness. The streamwise and spanwise values of λT,

rescaled by uτ/ν as λ+

T, and the respective grid sizes are given in table 2 (part I). For

each case, λ+

T,xiis of order 10 to 102, indicating that the average size of the roughness

element is large in viscous units. On average, roughness elements were well resolved by

the grid, with typically 4 to 12 grid points per λT,ximicroscale in each direction. For

reference purposes, Yuan & Piomelli (2014a) reported a resolution of λT,x/x≈4in

their large-eddy simulations of channel ﬂow over surfaces with sand-grain roughness. The

cases in table 2 for which λTwas not well resolved in at least one direction (λT,x/x<3

or λT,z/z<3) may also not have been fully rough ﬂows (as discussed in the following

section), and so were not included in the ensemble of ﬂows for ML training and testing.

In rough-wall ﬂows, the pressure drag is caused primarily by the local ﬂow

structures and separation in the vicinity of individual roughness protuberances, which

are predominately near-wall phenomena. To carry out the 46 separate DNS simulations

for determining ksefﬁciently, with sufﬁcient near-wall resolution, a small-span channel

simulation approach was employed. The concept of minimal-span simulation was

introduced by Jimenez & Moin (1991). Chung et al. (2015) and MacDonald et al. (2017)

carried out analyses of the performance of DNS over small spanwise domains for full and

912 A8-8

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Equivalent sand-grain height prediction of rough walls

Part I Part II

Case name λ+

T,xλT,x/xλ+

T,zλT,z/zd/δ ˆ

k+

s

C01,r4,reg,inc1 19.72.621.13.4 0.032 19.4

C02,r4,reg,inc2 20.42.733.15.3 0.046 49.7

C03,r4,reg,inc3 19.82.622.93.7 0.033 31.0

C04,r6,reg,inc1 27.73.728.44.5 0.038 64.4

C05,r6,reg,inc2 31.64.239.16.20.057124.4

C06,r6,reg,inc3 29.94.030.04.8 0.045 58.9

C07,r4,rnd,inc1 33.84.526.74.3 0.036 136.2

C08,r4,rnd,inc2 26.13.532.75.2 0.052 322.3

C09,r4,rnd,inc3 35.54.730.14.8 0.039 131.1

C10,r6,rnd,inc1 38.25.129.74.8 0.042 268.9

C11,r6,rnd,inc2 38.15.147.07.5 0.070 536.4

C12,r6,rnd,inc3 47.96.440.26.4 0.053 271.7

C13,r4,reg,por1 17.82.432.75.20.04741.4

C14,r4,reg,por2 27.53.734.25.5 0.032 140.6

C15,r4,reg,por3 31.54.239.46.3 0.028 157.1

C16,r6,reg,por1 25.63.446.17.4 0.066 76.7

C17,r6,reg,por2 40.15.347.87.6 0.044 259.8

C18,r6,reg,por3 44.45.954.88.8 0.039 246.5

C19,r4,rnd,por1 32.74.431.15.00.042158.2

C20,r4,rnd,por2 35.64.731.35.0 0.026 105.7

C21,r4,rnd,por3 37.45.034.25.50.027102.7

C22,r6,rnd,por1 44.65.935.35.6 0.053 276.8

C23,r6,rnd,por2 47.16.339.76.4 0.038 175.1

C24,r6,rnd,por3 47.16.344.47.1 0.045 260.3

C25,r4,reg,ES1 89.011.9 — — 0.024 25.6

C26,r4,reg,ES2 66.58.9 — — 0.026 65.3

C27,r4,reg,ES3 27.13.6 — — 0.035 45.5

C28,r6,reg,ES1 90.612.1 — — 0.033 71.2

C29,r6,reg,ES2 66.88.9— —0.040112.0

C30,r6,reg,ES3 27.23.6— —0.05464.0

C31,r4,rnd,SGR 27.83.725.04.0 0.032 48.7

C32,r4,rnd,RND1 131.217.554.18.70.041 8.4

C33,r4,rnd,RND2 96.312.842.16.70.04317.6

C34,r4,rnd,RND3 56.47.522.43.6 0.045 22.5

C35,r4,rnd,RND4 39.55.315.82.50.04618.3

C36,r4,rnd,RND5 25.13.311.41.80.05123.4

C37,r6,rnd,SGR 36.54.931.95.10.046108.8

C38,r6,rnd,RND1 88.511.872.611.6 0.060 12.0

C39,r6,rnd,RND2 93.812.535.75.7 0.062 17.1

C40,r6,rnd,RND3 57.07.622.83.6 0.070 50.4

C41,r6,rnd,RND4 40.55.415.62.50.07348.7

C42,r6,rnd,RND5 24.53.311.31.80.07643.8

C43,SG 35.26.033.55.70.04493.0

C44,TB 132.110.4168.513.2 0.058 24.1

C45,CB 25.74.525.54.4 0.039 149.9

C46,r4,rnd,por3,FS 37.65.034.65.50.027104.2

Table 2. Part I: streamwise and spanwise values of the surface Taylor microscale λT. Part II: ﬂow-related

parameters obtained from DNS. The ﬂow is assumed fully rough if ˆ

k+

s50, in which case ksis equal to ˆ

ks.

912 A8-9

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M. Aghaei Jouybari, J. Yuan, G.J. Brereton and M.S. Murillo

open channel ﬂows on rough and smooth walls and showed that minimal-span simulations

captured the essential near-wall dynamics and yielded accurate computations of wall

friction, and of mean velocities and Reynolds stresses as far from the wall as y≈0.3δ,

when the following constraints were met:

Lxmax 1000δν,3Lz,λr,x,(2.14a)

Lykc/0.15,(2.14b)

Lzmax 100δν,kc/0.4,λr,z,(2.14c)

where δν=ν/uτand λr,xiis the characteristic roughness wavelength in the xi-direction.

Alternatively, the surface Taylor microscale may be used as the length scale in this

constraint. Conditions (2.14a,c) were satisﬁed by choosing domain sizes L+

xand L+

zof

3000 and 1000, respectively, while condition (2.14b) was met for all cases except C11,

which fell below the Lykc/0.15 constraint by approximately 10 % – C11 is a case with

random geometry; protuberances beyond 0.15δexist but are rare.

Thecriteriaof(2.14) were developed originally for simulations of ﬂow over surfaces

with uniformly distributed roughness elements. In this study, the random roughness

geometries used require an additional criterion on the sufﬁciency of the domain size: the

area LxLzshould be large enough to achieve statistical convergence of surface parameters,

such as krms and Exi, and of the ﬂow parameter ks. To check the adequacy of the chosen

domain size, an additional simulation was carried out of case C21, the surface comprising

the largest dominant spatial wavelength (and consequently the most limited sampling of

random geometrical components with this wavelength) and a long-tailed height-ﬂuctuation

probability density function (p.d.f.) with a kurtosis of around 8. In this validation

simulation, denoted case C46, the domain sizes were doubled in xand z, by duplicating

C21 in these directions. The double-averaged velocity proﬁles U+=¯u+(y+)for cases

C21 and C46 are in a very good agreement over the log–linear region, as shown in ﬁgure 2.

Each surface statistic differs by no more than 3 %, with the greatest discrepancy found in

Iz, while the equivalent sand-grain roughness height ksis almost equal in the two cases.

The chosen domain size was therefore considered sufﬁcient for accuracy and convergence

of statistics describing ﬂow over the random roughness geometries of this study.

3. Results

3.1. Post-processed results

In ﬁgure 2, the streamwise double-averaged velocity proﬁles computed in these

simulations are shown. The proﬁles in the logarithmic region are described for the

smooth-wall case and the fully rough rough-wall cases as

¯u+=1

κln(y+)+5.0and (3.1a)

¯u+=1

κln y−d

ks+8.5,(3.1b)

respectively, where dis the zero-plane displacement, obtained as the location of the

centroid of the wall-normal proﬁle of the averaged drag force (Jackson 1981). The shift

in the ycoordinate by daccounts for the ﬂow blockage by surface roughness elements,

and the values of dare given in table 2 (part II).

912 A8-10

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Equivalent sand-grain height prediction of rough walls

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(y – d)+(y – d)+(y – d )+(y – d)+(y – d)+

(y – d)+

C01,r4,reg,inc1

C07,r4,rnd,inc1

C13,r4,reg,por1

C19,r4,rnd,por1

C25,r4,reg,ES1 C26,r4,reg,ES2 C27,r4,reg,ES3 C28,r6,reg,ES1 C29,r6,reg,ES2 C30,r6,reg,ES3

C31,r4,rnd,SGR C32,r4,rnd,RND1 C33,r4,rnd,RND2 C34,r4,rnd,RND3 C35,r4,rnd,RND4 C36,r4,rnd,RND5

C37,r6,rnd,SGR

C43,SG C44,TB C45,CB C46,SM

C46,r4,rnd,por3,FS

C38,r6,rnd,RND1 C39,r6,rnd,RND2 C40,r6,rnd,RND3 C41,r6,rnd,RND4 C42,r6,rnd,RND5

C20,r4,rnd,por2 C21,r4,rnd,por3 C22,r6,rnd,por1 C23,r6,rnd,por2 C24,r6,rnd,por3

C14,r4,reg,por2 C15,r4,reg,por3 C16,r6,reg,por1 C17,r6,reg,por2 C18,r6,reg,por3

C08,r4,rnd,inc2 C09,r4,rnd,inc3 C10,r6,rnd,inc1 C11,r6,rnd,inc2 C12,r6,rnd,inc3

C02,r4,reg,inc2 C03,r4,reg,inc3 C04,r6,reg,inc1 C05,r6,reg,inc2 C06,r6,reg,inc3

Figure 2. Proﬁles of streamwise double-averaged velocity plotted against a zero-plane-displacement shifted

logarithmic yabscissa. The dashed lines are u+=y+and u+=2.5ln(y−d)++5.0. The red dot-dash line

in plot C46 is that of C21.

To determine whether a particular ﬂow was within the fully rough regime, (3.1b)was

applied to the computed logarithmic velocity proﬁle to yield a test value of ks, denoted

as ˆ

ksin table 2 (part II). With ˆ

ksdetermined for all cases, those with ˆ

k+

sgreater than a

threshold value of 50 were deemed to be in the fully rough regime (30 surfaces), in which

case kswas set to equal ˆ

ks. Those below the threshold were possibly transitionally rough

(15 surfaces) and so were not included in ML predictions in this study. The threshold value

of k+

s– the lower end of the fully rough regime – has been observed to vary signiﬁcantly for

different types of roughness and is typically between 20 and 80. For example, the threshold

values for surfaces C43 and C44 are roughly 80 and 20 (Yuan & Piomelli 2014a)and 50

for surface C45 (Bandyopadhyay 1987).

912 A8-11

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M. Aghaei Jouybari, J. Yuan, G.J. Brereton and M.S. Murillo

The threshold value of k+

swhich signiﬁes the beginning of the fully rough regime

was not determined more precisely because of the cost of carrying out, for each surface,

simulations at successively higher values of k+

suntil ks/Rabecame invariant with the

Reynolds number. In the GPR prediction, potential uncertainties in kswhich might arise

through treating all ﬂows with k+

s>50 as fully rough, and other sources of possible error,

were compensated for by incorporating an assumed 10% noise level in the learning stage

of the prediction of ks, as discussed in § 3.2. The values of k+

s=50 as the threshold

for fully rough ﬂows and the assumed noise level were chosen as part of a trade-off to

maximize the number of usable data, to avoid overﬁtting, while acknowledging possible

uncertainties in the modelling data.

In ﬁgure 3, pair plots of the different topographic roughness parameters are shown

as scatter plots (lower left), joint p.d.f.s (upper right) and distribution p.d.f.s (diagonal).

Pair scatter plots for the true (DNS and experimental) value of ksand other roughness

parameters are along the bottom row of this ﬁgure. It can be seen that, for the roughness

cases chosen, there is some correlation between kurtosis and r.m.s. roughness (column 1,

row 6), kurtosis and skewness (column 5, row 6) and skewness and porosity (column 2,

row 5). The relationship between others appears to be more random. From the graphs in

the bottom row, it can be seen that ks/Rascales on porosity to some power, albeit with

some scatter (column 2, row 7). It also appears that ks/Ramight decrease with skewness

for surfaces with Sk<0 and increase with skewness in cases with Sk>0 (column 5,

row 7). Surfaces with positive skewness yielded higher values of kscompared with those

with negative skewness, consistent with the observation of Flack et al. (2020). Beyond

these observations, there does not appear to be a clear linear correlation between ksand

any individual roughness parameter, which makes the search for a functional dependence

of kson these parameters a problem well suited to ML. The measures of inclination, Ixand

Iz, showed no clear correlation with other variables or with ks/Ra.

3.2. ML predictions of the equivalent sand-grain height

The ML techniques of DNN and GPR were employed to predict ksfrom the data

sets described in the previous section. The objectives of this exercise were to generate

and collect data, and make qualitative comparisons between ML predictions and those

from conventional correlations, rather than evaluating and comparing the performance of

various ML procedures per se.The DNN and GPR approaches were used because our

experience was that they predicted kswith high accuracy, notwithstanding their simplicity.

Other approaches such as the support vector machine technique were considered initially,

but their preliminary predictions were not as accurate as those found using DNN and GPR

approaches.

The main characteristics of DNN and GPR methods are described below.

(i) The inputs for both techniques were 17 roughness geometrical parameters, eight of

which were the primary variables krms/Ra,Ix,|Iz|,Po,Ex,Ez,Skand Ku(deﬁned in

(2.4)to(2.13)). The other nine were products of the primary variables, which were

added to improve the efﬁciency of each learning stage. They were p1=ExEz,p2=

ExSk,p3=ExKu,p4=EzSk,p5=EzKu,p6=SkKu,p7=E2

x,p8=E2

zand p9=

S2

k. These particular products were chosen because of their perceived importance for

certain types of roughness.

(ii) The database consisted of 45 different sets: 30 DNS of turbulent channel ﬂows over

different surfaces at Reτ=1000, and 15 experimental data sets at higher Reynolds

numbers, with all data sets in the fully rough turbulent-ﬂow regime.

912 A8-12

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Equivalent sand-grain height prediction of rough walls

k

rms

/R

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1.5

Figure 3. Pair plots of geometrical parameters and ks,withksplots in the bottom row and the ﬁrst column,

DNS data (blue), experimental data (red).

(iii) The DNN architecture was a multilayer perceptron, with three hidden layers (with

18, 7 and 7 neurons, respectively). The activation functions at all nodes were of the

rectiﬁed linear unit kind, and kernel regularization was used to avoid overﬁtting.

The network had 521 trainable weights in total. The preset parameters to the

algorithm were optimized based on available data, through a hyperparameter tuning

process. Speciﬁcally, 270 conﬁgurations were ﬁrst generated with different lengths

(representing the number of layers) and widths (representing the number of neurons).

For each conﬁguration, the DNN compiler was performed 1000 times with random

selections of training (70 % of total) and testing (30 % of total) datasets to identify

the best performance of the conﬁguration. The conﬁguration that yielded the best

results was considered as the optimal one, the results of which are presented here.

The cost of data ﬁtting for one iteration (out of 1000) for each DNN conﬁguration

was approximately one second. In total, it took approximately 75 hours to obtain

the optimal DNN network. This architecture was found to provide suitable accuracy

in modelling without overﬁtting, for this particular multivariate labelled regression

problem.

(iv) The GPR procedure used rational quadratic kernels to represent ksas a superposition

of scaled Gaussian functions of the independent variables of the modelling problem.

Similar to the DNN method, the training and testing data were chosen randomly, with

respective ratios of 70% and 30 % of the total data points. The preset parameters

912 A8-13

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M. Aghaei Jouybari, J. Yuan, G.J. Brereton and M.S. Murillo

0

0

0

0

0.05

0.10

25

–25

–25

10

10

ks/Ra

0025

Error (%)

Error (%)

10

ks/Ra

ksp /Ra

ksp /Ra

p.d.f.

0

0

0

0

0.05

0.10

25

–25

–25

10

10 0025

Error (%)

10

p.d.f.

(a)(b)(c)

(d)(e)(f)

Figure 4. (a,d) Scatter plot of true ksand predicted ks,(b,e)scatterplotoftrueksand relative error, (c,f)

p.d.f.s of relative error, for (a–c) DNN and (d–f) GPR predictions, with DNS data (blue), experimental data

(red).

(e.g. kernel type, number of iterations, etc.) were also tuned with the available data

by running the GPR compiler approximately 8000 times. It took approximately

35 hours to obtain the optimal ﬁt. The GPR method has the capability of

incorporating uncertainty or noise in the determination of model parameters in the

learning stages. Such noise might arise through: numerical and discretization errors;

uncertainty in the form and model coefﬁcients of equation (1.1); the applicability

and ﬁtting range of equation (1.1) (which was deduced from high Reynolds number

experiments) to simulations at much lower Reynolds numbers; and the possibility

that some of the training data may have been from simulations in which the ﬂow

was not quite fully rough. A noise level of 10% in ks/Ravalues was chosen as an

upper estimate of the likely uncertainty from these sources. Noise levels of 5% and

15% were also tested, but little sensitivity of the ksprediction was found to the

assumed noise level within the tested range.

The values of kspredicted from the surface topography parameters, henceforth called

ksp, are compared with the actual ksvalues in ﬁgure 4, for the DNN and GPR methods,

respectively. Scatter plots of ksp and the true value of ksin ﬁgures 4(a)and4(d) reveal

a tight clustering of data along the y=xdiagonal, with only a few outlying points. This

very high degree of correlation between ksp and ksimplies that both techniques have been

applied with equal success to this prediction problem. The error range, ﬁgures 4(b)and

4(e), is less than ±30% (L∞norm) and the average error (L1norm) is less than 8 %, for

both techniques.

The consistency between both the kspredictions and error bands for two quite different

ML techniques suggests that they are both well-suited to this kind of problem, and possibly

close to an optimum for this class of ML approach.

The error values as percentages, for the DNN and GPR methods, are given in table 3,

together with the error in the empirical relation

ks=2.91krms (2+Sk)−0.284,(3.2)

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Equivalent sand-grain height prediction of rough walls

Case name errDNN errGPR errB1errB2errB3errB4

C04,r6,reg,inc1 4.0 4.1 −16.7−40.98.6−63.9

C05,r6,reg,inc2 0.7 10.3 −38.3−49.52.0−71.7

C06,r6,reg,inc3 4.2 7.5 −10.4−33.6 24.5 −59.8

C07,r4,rnd,inc1 10.5 −4.7−63.5−63.610.0−73.5

C08,r4,rnd,inc2 −0.6−4.8−80.1−77.6−4.1−81.7

C09,r4,rnd,inc3 6.0 −1.5−63.4−64.28.3−73.4

C10,r6,rnd,inc1 0.2 −2.7−76.3−72.511.8−77.8

C11,r6,rnd,inc2 2.6 −6.1−82.9−78.94.1 −82.2

C12,r6,rnd,inc3 −1.0−18.7−74.7−72.7−2.3−78.7

C14,r4,reg,por2 5.3 0.0 −66.2−64.2−8.7−80.3

C15,r4,reg,por3 4.2 −3.1−76.7−66.5 29.0 −78.8

C16,r6,reg,por1 1.8 5.4 3.5 −41.721.5−59.4

C17,r6,reg,por2 −0.6−1.3−74.5−70.3−10.9−82.7

C18,r6,reg,por3 −5.1−5.1−79.1−68.4 35.5 −78.8

C19,r4,rnd,por1 1.8 −2.3−71.4−69.444.9−67.8

C20,r4,rnd,por2 1.1 17.2 −67.0−56.6 82.1 −66.3

C21,r4,rnd,por3 0.0 −1.8−69.6−50.0254.1−46.2

C22,r6,rnd,por1 −7.1−7.9−77.0−76.7−10.6−78.4

C23,r6,rnd,por2 0.2 3.4 −70.9−60.4 80.8 −67.9

C24,r6,rnd,por3 −0.1−6.7−80.5−66.3 136.7 −66.5

C26,r4,reg,ES2 −5.4−12.7−48.6−61.6−57.6−83.8

C28,r6,reg,ES1 9.6 9.8 −29.2−45.9−51.9−81.2

C29,r6,reg,ES2 −2.6−9.8−54.7−66.2−53.2−83.2

C30,r6,reg,ES3 −1.53.4−21.8−45.78.1−65.7

C31,r4,rnd,SGR −0.63.3−46.7−50.7 65.1 −53.8

C37,r6,rnd,SGR −1.5−7.9−61.3−65.011.9−68.6

C40,r6,rnd,RND3 −3.19.1−23.6−39.6 98.3 −30.8

C43,SG 5.5 2.1 −58.6−60.1 46.3 −62.0

C44,TB −3.322.7 77.6 51.9 31.5 −51.6

C45,CB 1.8 −16.5−70.4−52.0 79.3 −72.8

E01,16,2 −2.13.56.2−47.5 370.2 63.0

E02,16,3 2.3 5.2 3.3 −33.7429.4 79.5

E03,16,7 −2.31.2−2.2−69.1 368.1 38.6

E04,16,8 −3.9−5.71.3−78.8412.4 27.6

E05,16,9 −3.3 12.4 10.9 −46.3 262.1 27.3

E06,16,15 −16.0−2.5−3.0−51.1405.4 79.9

E07,18,1 −29.8−25.817.3 −4.0 208.3 11.2

E08,18,2 28.1 26.1 120.7 79.4 388.8 80.0

E09,19,1 6.2 9.4 69.2 25.9 312.5 56.9

E10,19,2 −8.90.65.8−20.7 258.9 20.6

E11,19,3 8.9 7.4 47.4 −24.1 247.4 32.2

E12,19,4 −6.6 2.1 24.1 −21.0 258.4 32.2

E13,19,5 6.7 19.4 −16.6−23.8287.2 6.6

E14,19,6 5.3 8.9 −56.8−52.5177.2−38.2

E15,19,7 22.3 9.4 19.8 −10.2 342.6 43.0

L15.4 7.8 47.6 52.8 133.8 60.6

L∞29.8 26.1 120.7 79.4 429.4 83.8

Table 3. Errors in ksprediction by DNN and GPR compared with errors of the empirical correlations: errB1

(3.2), errB2(3.4), errB3(3.3)anderrB4(3.5). The four largest errors (in magnitude) for each column are

coloured in red. The errors are percentages.

proposed by Flack et al. (2016)and

ks=1.07kt(1−e−3.5Ex)(0.67S2

k+0.93Sk+1.3), (3.3)

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M. Aghaei Jouybari, J. Yuan, G.J. Brereton and M.S. Murillo

given by Forooghi et al. (2017), as well as their respective recalibrated correlations

ks=1.11krms(2+Sk)0.74,(3.4)

ks=0.04kt(1−e−5.50Ex)(S2

k+2.57Sk+9.82), (3.5)

when extended to all cases in the current database. It is interesting to note that the form

of equation (3.2) was chosen for surfaces generated by grit blasting – closely packed,

random, three-dimensional (3-D) roughness with a wide range of scales (E01–E06), while

many of the simulated surfaces are two-dimensional (2-D), some are characterized by

discrete elements of similar sizes, while others are sparse or wavy (characterized by low

slopes). Equation (3.3), on the other hand, includes a slope parameter and was calibrated

for numerically generated surfaces consisting elements of random sizes and a prescribed

shape.

For most cases, the errors from the DNN and GPR methods were of the same order of

magnitude and much smaller than the error in using (3.2)or(3.3). In the DNN and GPR

predictions of simulation cases, the greatest errors (approximately 25 %–30 %) arose in

cases E07 and E08. The surfaces associated with these cases are characterized by fractal

features (with spectral slopes of −0.5and−1.0, respectively (Barros et al. 2018)). The

size of the errors for these cases might be attributed to the small number of surfaces

with this feature used in the training set (as opposed to the many surfaces that are mostly

characterized by single-scale elements). A close examination of the prediction errors for

the DNS cases showed a subtle trend between relatively high errors and low roughness

solidity (or low Esand insigniﬁcant wake sheltering), in, for example, cases C28 and

C44. Both these cases are characterized by large-wavelength, wavy features, suggesting

an under-representation of sparse roughness in the dataset. Beyond this observation, no

clear correlation was found between the error and other primary roughness parameters

included herein or surface categorizations (2-D/3-D, random/regular).

The errors associated with using (3.2) are small for surfaces E01 to E06, which were

used to calibrate this relation. The errors in using (3.2)and(3.3) over all surfaces in the

database are 120 % and 430 %, respectively. However, when recalibrated against the full

database, (3.4)and(3.5) have a signiﬁcantly smaller error band with maximum values of

79 % and 84 %. The high error values of the empirical correlations, compared with DNN

or GPR prediction, are attributed to the small number of geometrical variables used in

their calibrations and the restricted range of the models’ parameters.

3.3. Uncertainty estimation

In addition to predictions of equivalent sand-grain height, the GPR method provides

conﬁdence margins as functions of each input parameter. These margins can be useful

for indicating the kinds of surfaces for which additional training data could improve

conﬁdence in predictions. This feature of the GPR approach makes it very attractive

for studies of this kind, since DNS and experimental generation of data can be

expensive.

The conﬁdence intervals determined by the GPR technique are shown as functions of

the normalized surface r.m.s. roughness height, effective slope, porosity and skewness in

ﬁgure 5. Wider intervals indicate higher estimated values of predictive error, such as at

roughness porosity of 0.68, and skewnesses of −1.5 and 2.0. Surfaces of roughness with

similar values of porosity and skewness would then be priorities for additional simulations

or experiments.

912 A8-16

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Equivalent sand-grain height prediction of rough walls

0 0.5

0.5 0.6 0.7

P

o

0.8 0.9

0.4

0.6 0.8 1.0

Ex

0.40.2

1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45

Sk

–0.5–1.0–1.5

5

ks/Ra

ks/Ra

ksp /Ra

ks/Ra

10

15

5

10

15

ks/Ra

krms /Ra

5

10

15

ks/Ra

5

10

15

1.0 1.5 2.0 2.5

90% CI

(a)

(b)

(c)

(d)

Figure 5. Conﬁdence interval (CI) of predictions with the GPR method, with predicted values of ks/Rain blue

lines (called ksp) and true values of ks/Rain red dots. The GPR predictions for both training and testing data

sets are shown – ksand ksp are very close to each other for the training data points, while they deviate (less than

30 % of error) for some test data points. Line jaggedness is associated with projection of a high-dimensional

space to one-dimensional ones.

3.4. Sensitivity analysis

The dependence of DNN predictions of kson individual roughness parameters is explored

by determining the change in the error norms when each of the primary surface parameters

is removed from the data from which the DNN prediction was made. In table 4, the actual

error for each surface, and the values of the L1and L∞norms of errors in the prediction

of ksover the 45 surfaces, are reported when the parameter(s) in the ﬁrst row is (are)

the excluded one(s). The errors of the base prediction (which includes all eight primary

parameters) are listed in the second column. In the following discussion, we focus on the

L1norm for ease of comparison over all 45 cases.

When the values of L1are considered, the relative importance of these surface

parameters for predicting ksis Ex,Ix,|Iz|,Ez,Po,krms/Ra,Skand, of least importance, Ku.

The L1-norm error is small when all parameters are included (7.4%). Excluding any single

912 A8-17

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M. Aghaei Jouybari, J. Yuan, G.J. Brereton and M.S. Murillo

Excluded

feature(s) None ExEzEx,Ezkrms Kukrms,KuSkPoSk,PoIxIzIx,Iz

C04 2 −23−1−1−215−13 13−12 0 −3

C05 5 −811 6 3−22 −408−4−6−2−11

C06 0 10 −1 1010 251 51868

C07 1 3 −1210−23 01−6−11319

C08 −15 −14 −1−4−19 −24 −19 −2−23 −36 −4−7−9

C09 1846 303 61−265118

C10 0 1 −16 1 −14 0 −1−12 2 −13 15 0 11

C11 −12 −3−3−23 −2−2−5−12 −1−29 1 −2−2

C12 0 −4−40−18 −3−4−1−7−2−30−2

C14 045 515 26 38−6360

C15 1650 029 00−11 −24−54

C16 1 −2−124 −2−2−33−266−114

C17 −4 8 17 17 1 4 8 15 13 −4353

C18 −1−6−10 −11 −2−3−11 −3−21 −17 −10 −25 −16

C19 −10 −15 −11 −12 −34 56−4−11 −1−2−11

C20 133 433 243 023 25 13

C21 921 311 2−10 00814

C22 −3−3−8−9−2−6−8−3−8−9−9−20 −12

C23 0 −2−100−5−17 −10−12−32

C24 0 −21 −1−1−11 000 40−4−7

C26 −6−17 −12 −9−8−5−19 −15 −13 −5−13 −14 −10

C28 18 19 21 26 17 18 −31616 3221 14 20

C29 −9−19 −8−22 −6−5−13 −25 −11 −22 −18 −17 −19

C30 −4611 25 −10 0 6 24 0−8265

C31 22 20 81924 0−218−1−14 9 −19

C37 −2−8−7−310−4−5−1−5−1−9−8−12

C40 −3−6−27 −21 −6−5−70−12−10 −8−18

C43 3 −4−46161 20723−15 −1−12

C44 −615 1 1713 1 420−6−12 −2−16 −21

C45 1 2 1 −4−65 −1−111 1529

E01 12 4 4 −92−3−11 5 11 −10 1 −3−3

E02 −13 6 −6−7−212 1−210 −913 7−2

E03 15 −60−54−6−437−32212

E04 0 −15 −9−9−2−6−6−3−52−240

E05 5175 1749 975 288513

E06 −5−3−6−3−10 −9−10 −6−7−9−10 −10 −5

E07 −21 −21 −24 −18 −16 −21 −18 −17 −23 −41 −25 −25 −24

E08 22 22 25 22 19 18 25 24 72421 22 24

E09 5 −315 27 −122 26 21 −2−21 −322

E10 −18 −19 −5−8−25 −4−51−14 38 −14 8 −2

E11 −1−15 −23 −19 −716 12

−29 290−50

E12 −9−36 0−10 2 −2−15 −10 28 −15 −22 −4

E13 11817 6172 8721 −15 14 25 15

E14 22 61 064 2125 33 95−5

E15 0 18 18 −4 11 9 15 11 19 32 19 23 16

L17.4 8.9 8.2 9.7 7.6 7.1 7.9 7.3 8.0 14.2 8.8 8.6 9.1

L∞22 22 27 27 25 24 26 25 25 41 25 25 24

Table 4. Errors in ksprediction by excluding one or two features. The base prediction includes all primary

variables. The four largest errors (in magnitude) for each column are coloured in red. The errors are percentages.

one of these parameters increases the L1-norm error up to around 9 %. On the other

hand, the exclusion of Kufrom the input parameters does not worsen predictions of ks

signiﬁcantly. Instead, this observation appears to be a consequence of correlation between

912 A8-18

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Equivalent sand-grain height prediction of rough walls

Kuand other surface parameters like krms/Ra(see ﬁgure 3). When such correlations exist

and one correlating parameter is excluded, the DNN process redistributes the weightings

given to other correlated parameters, with little loss in predictive accuracy.

To reduce the correlation between the excluded parameters and the remaining ones, one

may exclude groups of parameters that are thought to characterize the same type of surface

feature. For this reason, a sensitivity analysis was carried out on the effect of groups of

variables on predictions of ks. The characteristics of surface slope, element inclination

angle, porosity and intensity of height ﬂuctuations, are contained in pairs of (Ex,Ez),

(Ix,Iz), (Po,Sk)and(krms,Ku), respectively. Parameters within each pair have been shown

to be correlated to some degree in ﬁgure 3.Table 4 shows how the accuracy of ksprediction

is affected, if any one of these pairs is excluded. According to the table, the prediction of ks

is sensitive to all four pairs, but with greater sensitivities to the surface porosity (described

by Po,Sk) and the surface slope (described by Exand Ez). As expected, the elimination

of both parameters of a pair worsens the prediction more than removing either single

parameter (from around 7–9 % errors to up to 14%).

According to the sensitivity analysis, all parameters considered are of some importance

in the prediction of ks. The effective x-slope Exand roughness height skewness Skhave

been suggested as especially signiﬁcant in earlier studies (Napoli et al. 2008;Flack&

Schultz 2010; Yuan & Piomelli 2014a). The inclination angle in the streamwise direction

Ixmakes a signiﬁcant contribution to the ksprediction because, physically, Ixcharacterizes

the average aerodynamic shape of the roughness elements. Surfaces with Ix>0are

aerodynamically bluff bodies when compared with surfaces of the same size but with

Ix=0, and surfaces with Ix<0 tend to be more streamlined and hence produce less drag.

An important ﬁnding from this study is that the effective z-slope Ezis of similar

importance to accurate ksprediction as Skor Ex. The exclusion of Ezadversely affects

the prediction for a large number of rough surfaces. Physically, Ezdescribes whether the

surface is close to a 2-D roughness with Ez=0 (such as a transverse bar roughness) or a

3-D roughness with ﬁnite Ez. It is known that a k-type 2-D roughness produces a higher

drag than a 3-D roughness with the same height due to the larger spanwise length scale

that the 2-D roughness imparts to the ﬂow (Volino, Schultz & Flack 2009).

3.5. Comparison between ML algorithms and polynomial models

Explicit algebraic data representations, such as polynomial functions, can also be

determined for the data sets of this study, using ﬁtting or minimization procedures. In

such methods, a set of basis functions is proposed for a model, the unknown coefﬁcients

of which are then optimized according to speciﬁed constraints. They are a generalization

of the models of equation (1.2a–c), which were based on experimental observations of

the dependence of kson a small number of surface parameters. A 30-degree-freedom

polynomial basis was proposed as a ‘white-box’ model for ks, analogous to a low-order

Taylor series expansion for ks,

ks/Ra=α0+α1(krms/Ra)α2+α3Ix+α4|Ix|α5+α6|Iz|+α7|Iz|α8

+α9Pα10

o+α11Eα12

x+α13Eα14

z+α15Sk+α16|Sk|α17

+α18(Ku−3)+α19|Ku−3|α20 +α21(krms/Ra)α22 Pα23

o

+α24(krms/Ra)α25 Eα26

z+α27Pα28

oEα29

z,(3.6)

where ai(i=0,1,...,29) are the model coefﬁcients. To keep this model as simple as

possible and to bring the effects of all contributing factors into account, we used terms

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M. Aghaei Jouybari, J. Yuan, G.J. Brereton and M.S. Murillo

0

0

0

0

0.05

0.10

25

–25

–50

10

10

ks/Ra

0050

Error (%)

Error (%)

10

ks/Ra

ksp /Ra

p.d.f.

(a)(b)(c)

Figure 6. (a) Scatter plot of true ksand predicted ks(denoted as ksp), (b) scatter plot of true ksand relative

error and (c)p.d.f. of relative error distribution for prediction using polynomial function deﬁned in (3.6), with

DNS data (blue) and experimental data (red).

as αiθαjfor a test variable θthat take only positive values (e.g. krms), and terms as

αiθ+αj|θ|αkfor those variables that take both positive and negative values (e.g. Sk). For

the latter, the power of θin the ﬁrst term is ﬁxed (at one) instead of ﬁtted, to eliminate

the possibility of an imaginary number. Combinations of six parameters (Ex,Ez,Po,Sk,

krms/Ra and Ku), taken in pairs, were also included. Since, for the present collection of

surfaces, strong correlations were observed between individual variables within the three

pairs of (Ex,Ez),(Po,Sk)and (krms/Ra,Ku), shown in ﬁgure 3, only one variable from

each pair was used for the combination terms in (3.6). Using the other variable from any

of these pairs instead would not lead to a signiﬁcant change in the prediction using (3.6).

The high-dimensional space of aiis poorly suited to curve-ﬁtting and minimization

procedures which use stochastic gradient descent algorithms. However, it is well suited

to robust minimization methods like the differential evolution algorithm (Storn & Price

1997), with which global minima can often be found efﬁciently in spaces of high

dimension. In this case, it is used to determine the values of the coefﬁcients aiwhich

minimize the L1norm.

In ﬁgure 6, the prediction quality of this white-box model with optimized coefﬁcient

values is shown. This method yields an average prediction error of 12 % and a maximum

one of 51% when using all 45 fully rough data sets (to give the best possible prediction

accuracy) for the model training.

The optimized values of aiare

α0=5.312, α1=−1.172, α2=4.264, α3=0.050, α4=−1.283, α5=8.393,

α6=−0.347, α7=−5.771, α8=1.785, α9=7.919, α10 =4.058, α11 =−0.979,

α12 =3.414, α13 =6.380, α14 =1.354, α15 =1.023, α16 =2.969, α17 =1.273,

α18 =−0.946, α19 =−0.762, α20 =0.056, α21 =1.647, α22 =−8.176, α23 =3.523,

α24 =−9.472, α25 =−5.656, α26 =0.580, α27 =−5.425, α28 =0.283, α29 =7.177.

The predictive accuracy of this optimized explicit model equation is considerably lower

than that of the DNN and GPR methods. One reason for this reduced accuracy is that

low-order functions of geometrical parameters do not faithfully represent the dependence

of kson surface parameters because each coefﬁcient in the model is required to take the

same value over the entire surface-parameter space. In ML approaches, such restrictions

need not apply as they are not constrained to low-order polynomial functions but instead

adopt a methodical search for the best representation of ksas a function of the surface

912 A8-20

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Equivalent sand-grain height prediction of rough walls

parameters. This search is carried out through ‘feature selection’ in the ﬁrst layers of DNN

and the properties of the basis functions adopted in GPR, each of which are designed to

yield the same mean and standard deviation of ks/Raas in the original dataset (Rasmussen

& Williams 2006).

4. Concluding remarks

The construction of a predictive model from a large ensemble of datasets for the equivalent

sand-grain height ksof a surface of arbitrary roughness, as a function of many different

measures of surface topography, is a labelled regression problem that is well-suited to

ML techniques. In this paper, data from 45 different rough surfaces (in fully rough ﬂows)

were used to devise DNN and GPR predictions for ksas functions of eight different

surface-roughness parameters.

Both models were able to predict ksfor the 45 surfaces with an average error below

10%, with the largest error for any one surface less than 30%. These predictions were

signiﬁcantly better than those of existing formulae, and of a 30 degree-of-freedom

polynomial model ﬁtted to the same data, where the greatest error for any surface was

approximately 50 %.

Sensitivity analyses revealed that inclusion of nearly all the surface roughness

descriptive parameters was necessary to minimize the average prediction error, but that

exclusion of either measures of porosity or measures of the surface slope increased the

maximum prediction error more signiﬁcantly than omitting other parameters.

Machine learning techniques are well suited to this modelling problem because:

(i) it is complex in so far as different kinds of surface roughness yield different ﬂow

phenomena which are modelled most accurately in different ways, making the prospect

of a general physical model very remote; and (ii) the dependent surface-roughness

variables upon which ksis modelled are a large non-orthogonal set for which robust

multivariable regression techniques are required. As ML methods, they take no account

of physical modelling concepts or observed phenomena within roughness sublayers,

such as recirculation regions, enhanced turbulence production in the wake of roughness

elements, assumed scalings for drag, etc., each of which is applicable to ﬂows over

some rough surfaces but not others. Nor are they hindered by the lack of orthogonality

of the surface roughness parameters as the dependent variables of ks. The techniques

used can be conﬁgured readily to mimic models with very many degrees of freedom

and, when compared with polynomial models, their feature selection properties provide

the equivalent of different values for polynomial coefﬁcients in different regions of the

surface-parameter space. In this application, both approaches of DNN and GPR yielded

models with very similar predictive accuracy, even though the techniques themselves

were very different. We therefore conclude that they yield high-ﬁdelity predictions of the

equivalent sand-grain roughness height for turbulent ﬂows over a wide range of rough

surfaces, as a signiﬁcant improvement over other methods. Improved prediction might be

achieved by enlarging the database to include rough-wall ﬂows with surface parameters

which correspond to the relatively low prediction conﬁdence in the GPR method, and

by including additional roughness parameters as inputs which might describe sparseness

and two-dimensionality, such as the solidity, correlation length scales and other two-point

surface statistics.

In addition to the ksprediction described here, the DNS database and the ML techniques

in general can also be used to uncover relations between roughness geometry and

physics-related quantities, such as the ﬂow pattern around roughness protuberances, ﬂow

912 A8-21

https://doi.org/10.1017/jfm.2020.1085

M. Aghaei Jouybari, J. Yuan, G.J. Brereton and M.S. Murillo

separation locations, characteristics of the shear layers associated with the separation

bubbles, the wake sheltering volume, etc. Speciﬁcally, an ML network trained to correlate

these ﬂow characteristics (as outputs) to the roughness geometry (as inputs) may be

an efﬁcient tool for determining the sets of roughness geometrical features which are

important for characterizing these effects. Knowledge of such a set of signiﬁcant roughness

parameters may also guide the construction of rough-surface databases that yield more

efﬁcient and more widely applicable predictions of ksor other quantities.

Supplementary materials. The rough-wall ﬂow database (including ks, surface height map and surface

parameters) and the trained DNN and GPR networks, called prediction of the roughness equivalent

sand-grain height (PRESH), can be accessed online in the ﬁrst author’s GitHub repository at https://github.

com/MostafaAghaei/Prediction-of-the-roughness-equivalent-sandgrain-height. With this package of data and

programs, interested researchers can: (i) use the ML networks described in this paper to make predictions

of ksfor surfaces of their own roughness topography; (ii) download the code and train new DNN and GPR

networks to predict ksfor a different set of surfaces of arbitrary topography; and (iii) use the database of 45

rough-wall ﬂows for other applications. It is recommended to use the ML conﬁgurations described in this

paper for surfaces with parameters inside the ranges speciﬁed in ﬁgure 3. Extrapolations (using inputs which

are beyond the speciﬁed range) will lead to additional uncertainty.

The PRESH and the database will be actively updated by the authors to improve the prediction accuracy and

universality. We welcome interested researchers to share their datasets with us.

Acknowledgements. The authors gratefully thank Professor K.A. Flack of the US Naval Academy for

providing the experimental data sets.

Funding. The authors gratefully acknowledge the ﬁnancial support of the Ofﬁce of Naval Research (award

no. N00014-17-1-2102). Computational support was provided by Michigan State University’s Institute for

Cyber-Enabled Research.

Declaration of interests. The authors report no conﬂict of interest.

Author ORCIDs.

Mostafa Aghaei Jouybari https://orcid.org/0000-0001-9934-6615;

Junlin Yuan https://orcid.org/0000-0002-4711-6452;

Giles J. Brereton https://orcid.org/0000-0002-7939-0691;

Michael S. Murillo https://orcid.org/0000-0002-4365-929X.

REFERENCES

AGHAEI JOUYBARI,M.,BRERETON,G.J.&YUA N, J. 2019 Turbulence structures over realistic and synthetic

wall roughness in open channel ﬂow at Reτ=1000. J. Turbul. 20, 723–749.

BANDYOPADHYAY, P.R. 1987 Rough-wall turbulent boundary layers in the transition regime. J. Fluid Mech.

180, 231–266.

BARROS, J.M., SCHULTZ,M.P.&FLACK, K.A. 2018 Measurements of skin-friction of systematically

generated surface roughness. Intl J. Heat Fluid Flow 72, 1–7.

BONS, J.P. 2002 St and cfaugmentation for real turbine roughness with elevated freestream turbulence. Trans.

ASME: J. Turbomach. 124, 632–644.

BONS, J.P., TAYLO R, R.P., MCCLAIN,S.T.&RIVIR, R.B. 2001 The many faces of turbine surface

roughness. Trans. ASME: J. Turbomach. 123, 739–748.

CHAN, L., MACDONALD,M.,CHUNG,D.,HUTCHINS,N.&OOI, A. 2015 A systematic investigation of

roughness height and wavelength in turbulent pipe ﬂow in the transitionally rough regime. J. Fluid Mech.

771, 743–777.

CHOI,H.&MOIN, P. 1994 Effects of the computational time step on numerical solutions of turbulent ﬂow.

J. Comput. Phys. 113, 1–4.

CHUNG,D.,CHAN, L., MACDONALD,M.,HUTCHINS,N.&OOI, A. 2015 A fast direct numerical

simulation method for characterising hydraulic roughness. J. Fluid Mech. 773, 418–431.

FLACK, K.A. 2018 Moving beyond Moody. J. Fluid Mech. 842, 1–4.

FLACK,K.A.&SCHULTZ, M.P. 2010 Review of hydraulic roughness scales in the fully rough regime. Trans.

ASME: J. Fluids Engng. 132, 041203.

912 A8-22

https://doi.org/10.1017/jfm.2020.1085

Equivalent sand-grain height prediction of rough walls

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 Turbul. Combust. 104, 317–329.

FLACK, K.A., SCHULTZ, M.P., BARROS,J.M.&KIM, Y.C. 2016 Skin-friction behavior in the

transitionally-rough regime. Intl J. Heat Fluid Flow 61, 21–30.

FOROOGHI,P.,STROH,A.,MAGAG NATO ,F.,JAKIRLIC,S.&FROHNAPFEL,B.2017Towardauniversal

roughness correlation. Trans. ASME: J. Fluids Engng. 139, 121201.

JACKSON, P.S. 1981 On the displacement height in the logarithmic velocity proﬁle. J. Fluid Mech. 111, 15–25.

JIMÉNEZ, J. 2004 Turbulent ﬂows over rough walls. Annu. Rev. Fluid Mech. 36, 173–196.

JIMENEZ,J.&MOIN, P. 1991 The minimal ﬂow unit in near-wall turbulence. J. Fluid Mech. 225, 213–240.

KEATING, A. 2004 Large-eddy simulation of heat transfer in turbulent channel ﬂow and in the turbulent ﬂow