Data-driven prediction of the equivalent sand-grain height in rough-wall turbulent flows

Abstract

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 k s for 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 Re τ = 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 k s with 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 formulas. They also identified the surface porosity and the effective slope of roughness in the spanwise direction as important factors in drag prediction.

Full text

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/Afx,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
¯
θis 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
LxLzV
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
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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
Atx,z
kdA,(2.4)
Ra=1
Atx,z
|k−kavg|dA,(2.5)
krms =1
Atx,z
(k−kavg)2dA,(2.6)
Sk=1
Atx,z
(k−kavg)3dAk3
rms,(2.7)
Ku=1
Atx,z
(k−kavg)4dAk4
rms,(2.8)
Ex=1
Atx,z
∂k
∂x
dA,(2.9)
Ez=1
Atx,z
∂k
∂z
dA,(2.10)
Po=1
Atkckc
0
Afdy,(2.11)
Ix=tan−11
2Sk∂k
∂x,(2.12)
Iz=tan−11
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.
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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.
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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/kc0.017,
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
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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+
s50, in which case ksis equal to ˆ
ks.
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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:
Lxmax 1000δν,3Lz,λr,x,(2.14a)
Lykc/0.15,(2.14b)
Lzmax 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 Lykc/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).
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Equivalent sand-grain height prediction of rough walls
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U+
(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).
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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.
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Equivalent sand-grain height prediction of rough walls
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2.5
–2.5
15
10
5
0
10
0
0
1
0
0.5
1.0
1.0
1.5
1
2.5
–2.5
1.5
1.0 0 1 0 0–2.5 2.510010 200.5 1.01.5
1.0 0 100–2.5 2.5
100
10 20
0.5 1.0
1.5
1.0 0 1 0 0–2.5 2.510010 200.5 1.0
1.5
1.0 0 100–2.5 2.5
100
10 20
0.5 1.0
1.5
1.0 0 1 0 0–2.5 2.510010 200.5 1.0
1.5
1.0 0 1 0 0–2.5 2.510010 200.5 1.0
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
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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.
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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
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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
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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
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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
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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.
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