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

AN ACTIVE LEARNING APPROACH FOR THE
PREDICTION OF HYDRODYNAMIC ROUGHNESS PROPERTIES

Jiasheng Yang
Alexander Stroh
Institute of Fluid Mechanics
Institute of Fluid Mechanics
Karlsruhe Institute of Technology Karlsruhe Institute of Technology
Karlsruhe, Germany
Karlsruhe, Germany
jiasheng.yang@kit.edu
alexander.stroh@kit.edu

Bettina Frohnapfel
Institute of Fluid Mechanics
Karlsruhe Institute of Technology
Karlsruhe, Germany
bettina.frohnapfel@kit.edu

Pascal FriederichInstitute of Theoretical Informatics
Karlsruhe Institute of Technology
Karlsruhe, Germany
pascal.friederich@kit.edu

Pourya Forooghi
Department of Mechanical & Production Engineering
Aarhus University
Aarhus, Denmark
forooghi@mpe.au.dk

the (Hama) roughness function ∆U + . Prediction of the roughness function is of key importance for the estimation of dragforce and modeling of turbulent flow over a rough surface.
However, ∆U + is not known a priori for an arbitrary roughsurface, and for every new roughness topography at differentReτ , ∆U + needs to be determined using a laboratory experiment or high-fidelity numerical simulation, both of which arecostly and time consuming. As an alternative, predictive empirical ’roughness correlations’ have been extensively studied
by researchers, a review is provided by Chung et al. (2021).
These correlations predict the roughness function based on statistical properties of the roughness height distribution. Dueto the complexity of the roughness geometry, these roughness
correlations proposed in the past research have limits in theirgeneralizability to other types of roughness topographies.
It is understood that a comprehensive database is the fundament of an universally valid roughness predictive model.
Apart from the difficulties in constructing a massive roughness database that contains realistic surfaces in terms of time
consumption and technical difficulties investigating realisticroughness has its disadvantages because of the lack of flexibility for parametric studies. In contrast, regular roughness
that contains the specific shape of roughness elements e.g.
cubes (Yang et al., 2016), cones (Forooghi et al., 2017), ellipsoids (Jouybari et al., 2021), sinusoidal waves (MacDonald
et al., 2016) etc. are frequently used for systematic investigation of roughness effect in respect to single, or a few, roughness parameter(s). However, regular roughness has its shortcomings in representing roughness that are encountered in realistic applications. Recently, mathematical irregular roughness generation methods have received increasing attention inthis field of study. The mathematically generated irregularrough surfaces are able to reflect the stochastic nature of realistic roughness while their geometrical statistics can be manipulated. Therefore, the generation method proposed by PérezRàfols & Almqvist (2019) is employed to systematically produce the rough surfaces that can be regarded as surrogates ofrealistic surfaces.
Under the common goal of predicting roughness-induced
drag of an arbitrary roughness based on its geometrical prop-

ABSTRACT
Realistic surfaces of flow-related equipment are often hydraulically rough due to wear or fouling. Predicting the skinfriction exerted by such rough surfaces is a challenging task
since the topography of these surfaces is inherently irregularand complex. Recent developments in data-driven methods
and increasing affordability of high-fidelity direct numericalsimulations (DNS) have created new possibilities for estimation of drag on irregular rough surfaces. In the present work we
aim to demonstrate a viable approach to efficiently train a predictive model for the estimation of drag for irregular roughness
based on its height probability density function (PDF) and thesurface height power spectrum (PS). An active learning (AL)framework is employed to efficiently navigate the constructionof a training database. Training data is generated by conducting direct numerical simulations of a flow over artificially generated rough surfaces in minimal channels in order to minimize
the computational effort. An ensemble neural network (ENN)
model is trained based on the database. The ENN model shows
promising potential in predicting the skin friction as well as estimating the epistemic (model) uncertainty. Furthermore, themodel – trained on artificial surfaces – is tested on five realistic
surface scans, showing that a maximum error of 8.7% between
the predicted roughness function ∆U + and the DNS results isachieved. Overall, the AL framework shows a great potential
as a basis for future research towards a universal predictive
tool for any arbitrary roughness.

INTRODUCTION
Roughness is encountered in a variety of engineering applications since many industrial processes inherently include
rough surface conditions. The roughness on solid surfaces inflow-related applications can enhance the near-wall momentum transfer, thus it also increases the skin friction drag. Thiscan translate into a significant deterioration of equipment performance. The increase of skin friction is manifested by adownward shift in the mean velocity profile in respect to asmooth surface profile at identical friction Reynolds numberReτ . This downward shift in the logarithmic layer is known as1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
(NN) with identical architectures but trained with random initial weights. Each NN is constructed with one input layer with
64 neurons, two hidden layers with 64 and 128 non-linear neurons with rectified linear unit (ReLU) activations, respectively
and one linear neuron in the output layer. 10% samples inthe database are used for validation while 90% samples areused for training. Diversity of the NNs is ensured by selecting the distribution of the training-validation database independently. L2-regularization is applied to the loss function.
Adaptive momentum estimation (Adam) is employed to train
the model. A sketch of the NN is shown on the right panelof Figure 1. The input vector I of the NN contains the roughness PDF and PS along with 4 additional characteristic features of the rough surface, i.e. the roughness peak-to-troughheight kt , the 99% confidence interval of the PDF k99 , the normalized largest/ smallest roughness wavelength λ0∗ = λ0 /k99 ,and λ1∗ = λ1 /k99 , respectively. The input vectors that represent the roughness PDF and PS are obtained by discretizing
the roughness PDF and PS into each 30 values equidistantly
within the height range 0 < k < kt and the wavenumber range
2π/λ1 > 2π/λ > 2π/λ0 , respectively. The input elements in I
are normalized with the standard deviation values of each corresponding element calculated from the repository. The final
prediction of the ENN is the averaged prediction over all 50++
NNs, namely µ∆U + = ∑50
i=1 ∆Ûi /50. Where ∆Û represents
the prediction of a single NN, index i indicates the number of
NN. The uncertainty is measured
q by the standard deviation oferties several empirical roughness correlations have been built
based on different statistical properties of the roughness, forinstance the models based on correlation length Lcorr (Thakkar
et al., 2017), skewness Sk (Flack et al., 2020) and effective
slope ES (Forooghi et al., 2017). As a matter of fact, it is
widely reported in the literature that the skin friction of roughness can be partly determined by a wide range of geometrical statistics. In order to incorporate as much aspects ofroughness statistical properties as possible into the prediction,
multi-variant machine learning approaches have drawn muchattention in the field. Jouybari et al. (2021) applied neural network to predict the equivalent sand-grain size ks based on 17
roughness statistics. Lee et al. (2022) employed transfer learning method to enhance the performance of the machine learning model with three empirical roughness correlations, significant enhancement of the model is achieved with relative lower
amount of training data.
Recently, Yang et al. (2021) suggested that the flow properties of an arbitrary roughness can be uniquely determinedby a reduced-order representation of roughness, namely theroughness PDF and PS. In the present work, a machine learning framework is employed for the purpose of predicting theroughness function ∆U + of homogeneous irregular roughness
at Reτ = 500 based on its PDF and PS. Active learning (AL)approach is applied to navigate the construction of the training
database with the aim of strengthening the model iteratively inan efficient manner. In order to quantify the model uncertainty,
ensemble neural network (ENN) is used.
The structure of the paper is as following: in section 2 we
present the methodology. The advantage of AL framework as
well as the performance of ENN are analysed in section 3.1. In
section 3.2 the evaluation of the model is demonstrated with 5
realistic surfaces. Finally in section 3.3 model interpretation iscarried out with sensitivity analysis method.

the predictions, i.e. σ∆U + =+2∑50
i=1 (∆Ûi − µ∆U + ) /50.

Minimal channel simulation
The rough surfaces from the repository are investigatedin closed turbulent channel driven by constant pressure gradient with DNS. The immersed boundary method following
Goldstein’s method (Goldstein et al., 1993) is employed. The
Navier-Stokes equation writes

METHODOLOGY
Active learning (AL)In the present work, we aim at developing an efficient AL
framework to predict ∆U + at Reτ = 500 of given surfaces andquantify the uncertainty for each prediction. For this purpose aroughness topography repository with systematic variation ofroughness properties is constructed, containing 400 surfaces.
It is prohibitively expensive to conduct DNS investigation for
the entire repository. Therefore, it is important to efficiently
select the representative roughness samples in the repository
with the aim of sparing the computational effort for the roughness with similar properties to the already labeled samples. To
this end, AL is proposed to navigate the selection of the most
relevant samples based on the model confidence, or model uncertainty (Aggarwal et al., 2014). The workflow of the AL is
illustrated on the left panel of figure 1. Within the present AL
framework, the model is firstly trained by an initial roughness
data set with 20 randomly selected and DNS-annotated surfaces, and subsequently enhanced by a small amount of roughness samples (∼ 20) guided by AL.

∇·u = 0 ,

(1)

∂u
1
1
+ ∇ · (uu) = − ∇p + ν∇2 u − Px ex + fIBM .
∂t
ρ
ρ

(2)

where u = (u, v, w)⊺ is the velocity vector and Px is the meanpressure gradient in the flow direction added as a constant anduniform source term to the momentum equation to drive theflow in the channel. Here p, ex , ρ, ν and fIBM are pressure fluctuation, streamwise basis vector, density, kinematic
viscosity and external body force term due to IBM, respectively. Periodic boundary conditions are applied in the streamwise and spanwise directions. No-slip boundary condition isapplied on the rough walls. The friction Reynolds
p numberis defined as Reτ = uτ (H − kmd )/ν, where uτ = τw /ρ andτw = −Px · (H − kmd ) are the friction velocity and wall shear
stress, respectively. Melt-down height denoted by kmd is themean roughness height measured from the deepest trough. In
the present work, simulations are performed at Reτ = 500.
Due to the unfavorable computational costs for DNS, theconcept of minimal channel simulations (MacDonald et al.,2016) is applied. The minimal channel size Lx × Ly × Lz =3H × 2H × 1H is selected to accommodate both near-wall turbulence and roughness structures. Where H is the channel halfheight. In order to resolve the finest roughness structures, aresolution of 576 × 401 × 192 voxels is selected, which translates to the wall-parallel grid size ∆+ = 2.6. With the present

Ensemble neural network (ENN)
As mentioned in the previous section, the selection of surfaces from the repository for DNS simulations is based on theuncertainty estimation of a ML model. We employ a ENN
to predict ∆U + values of surfaces and - at the same time to estimate the uncertainty of the prediction. The ENN is an
ensemble of 50 individual densely-connected neural networks

212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

PS
PDF
∗ k99
λ1∗ λ0

kt

...

...

Input
64 Hidden
layer
64 Hidden
layer
128

Output∆Uˆ +Figure 1: Left: schematic of the AL-framework. Right: illustration of the NN architecture, where input vector contains

normalized and discretized roughness PS, PDF and 4 roughness properties. λ ∗ = λ /k99 indicates normalized wavelength.
Note, that layers are fully connected.

mesh size, the smallest roughness structures with wavelength
λ1 are resolved by more than 7 grid points in each wall parallel
direction. In wall-normal directions, consine stretching meshis adopted for the Chebychev discretization. A vertical cellnumber of 401 is selected. The mesh independence is tested.
An exemplary instantaneous velocity field at an arbitrary zlocation is shown in Figure 2.

PS a repository with 400 rough surfaces is generated. The
overview of the randomly prescribed PSs and PDFs in therepository along with the realistic PSs and PDFs are shown in
Figure 3(a,b). Exemplary surface realizations in the repository
are shown in Figure 4.

Sensitivity analysis
In recent years, growing number of attempts have been
conducted with the aim to interpret the predictions of neuralnetworks. One of the most frequently used model explanation
methods is the sensitivity analysis (SA). The SA approach attempts to quantify the importance of each input feature. The
SA is the Jacobian matrix of the NN output ∆Û + in respect tothe input elements, which writes:

Roughness generation
Artificial irregular rough surfaces are generated througha mathematical roughness generation method where the PS
and PDF of the roughness can be prescribed (Pérez-Ràfols &Almqvist, 2019). PDF and PS in the repository are set in a random but controlled form to imitate the realistic distribution of
PDF and PS. Weibull, bimodal as well as Gaussian distribution
PDF are selected randomly. The Weibull distribution writes:Kf (k) = Kβ K k(K−1) e−(β k) .

1 50
Sj =
∑
50 i=1(3)where the shape parameters K and β can be randomly prescribed to adjust the shape of the PDF profile. k represents the
local roughness height. The Bimodal distribution is obtainedby generating two Gaussian sequences with random expectation µ and standard deviation σ and take the point-wise minimum. The Bimodal roughness height distribution as a functionof wall-parallel coordinates (x, z) writes:
k(x, z) = min{Φ0,1 (x, z), Φµ,σ (x, z)} .

*∂ ∆Ûi+
∂ Ij

+
.

(5)

Where S j is the sensitivity of the input element I j , the subscript
j represents the j-th component of the input vector I . The
symbol · is the averaging operation over all samples in the
training data set. The derivative ∂ ∆Û + /∂ I j is calculated using
an automatic differentiation method.

RESULTS
Evaluation of the AL framework(4)

According to the AL workflow, 20 randomly selectedrough surfaces are represented by the red lines in figure 3. The
20 least confident samples are marked with green lines in the
figure. After one round of AL, 40 samples in total are included
into the database for training the ENN. The relative similarityof the samples in terms of PS and PDF can be observed in the
figure 3. It can be seen that the model tends to explore the existing sample parameter space as these newly included samples
exhibit significant dissimilarity in terms of the PS and PDF.

The roughness height is selected to be at the level of kt ≈ 0.1H.
Random values of the roll-off length Lr (Jacobs et al.,2017) as well as the power-law decline rate θPS (Lyashenko
et al., 2013) are prescribed for the modelling of roughness
PS. Random perturbations are added to the PS. The upper aswell as lower bound of the roughness wavelength are set toλ0 = 0.8H and λ1 = 0.04H, respectively. Eventually, by randomly tuning the shape parameters of the roughness PDF and

3

y/H

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
0.2
0.15
0.1
0.05
0

1.1

0
0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.8

1.6

2

2.2

2.4

2.6

2.8

3

u

x/H

Figure 2: Instantaneous streamwise velocity over simulated rough surface. Contour indicates the mean velocity.
roughness repository

initial roughness database

AL selected roughness

realistic roughness

100
15

(a)

(b)

10−1
10

PS

PDF

10−2

10−3
5
10−4

10−5
10−2

10−1

100

0

101

0

0.16

k/H

Wavenumber

Figure 3: PS (a) and PDF (b) of roughness samples in the roughness pool. Realistic surface scans are colored blue.

x/H

0

0.15

samples with
high

{

mid

{

σ̄∆U + =0.55
σ̄∆U + =0.47
σ̄∆U + =0.37
σ̄∆U + =0.29
σ̄∆U + =0.17
initial modelσ̄∆U + =0.12
single AL loop

1

{

low
uncertainty
0
2

0

1

2

z/H

10

20
average error %

30

0

3

Figure 5: Reduction of average prediction error and un-k/Hcertainty for three groups of samples after one AL iteration. Here overbar represents averaging process over the
samples in each uncertainty level.

Figure 4: Exemplary artificial rough surface patches

In order to evaluate the performance of the ENN, multiple samples from the repository at different uncertainty levels are grouped (see Figure 5). The high uncertainty groupcontains the 20 samples with highest uncertainty, the low
uncertainty group contains samples whose σ∆U + < 0.2, themid uncertainty group contains 20 randomly selected samples
in the remaining repository. The averaged prediction error++
Err = (∆UDNS− µ∆U + )/∆UDNS
as well as the averaged uncertainty σ∆U + are compared with the initial model in the figure. A clear correlation between the error and its estimateduncertainty can be observed, i.e. high uncertainty corresponds
to high prediction error. For the most confident sample groupin the testing set with σ∆U + < 0.2, a mean error of 4.3% isachieved. Furthermore, the efficiency of the AL is illustrated

when comparing both uncertainty drop as well as the error
drop from the initial model. We observe a significant reduction in error of 9 percentage points for the high uncertainty
group. It can be seen that at higher uncertainty level more significant improvement of prediction accuracy and reduction in
uncertainty is achieved. A possible explanation is that a cluster of least confident roughness properties is DNS-annotatedand included to the model via AL. This leads to a significant
decrease of the uncertainty for the samples that are similar tothese newly incorporated roughness. Thus, it is expected thatwithin a few AL iterations, the prediction error level converges
to an acceptable level for all considered topographies in therepository.

4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
peak-dominated roughness is mapped to a greater ∆U + value,
while a negatively skewed roughness or pit-dominated roughness translates into a slightly lower value of the velocity retardation.
In figure 6 (c), S PS is plotted as a function of λ + with logarithmic scaling. The coherence function between the localized surface force and the roughness structure is investigatedby Yang et al. (2021). It is reported that a certain range ofthe roughness wavelengths show considerable correlation to
the local streamwise drag force due to the roughness sheltering effect. Beyond this certain wavelength the coherence starts
dropping. This critical roughness wavelength, according to theinvestigation on 12 different types of roughness at Reτ = 500,
locates in the range of 150 < λ + < 300. To qualitatively illustrate the link of this specific length scale with the present research on global drag, the range is added with vertical dashedlines to the figure. It can be seen from the plot that the sensitivity of the model reaches the highest value within the area
between the dashed lines. This observation may suggest thatthe model interprets the PS values at this wavelength range
as being dominant for the global drag of the surface. At thesame time a region with lower sensitivity is observed at smaller
wavelengths. This is supported by the decreasing coherence of
local surface force towards smaller wavelengths (Yang et al.,2021).

Realistic surface prediction
The applicability of the model is evaluated with realistic rough surfaces. The realistic surface scans are labeled as
Sandpaper, ICE 1∼3 (Velandia & Bansmer, 2019) and IC surface (Forooghi et al., 2018), which originate from sandpaper,ice accretion on simplified aero-engine nacelle and deposit ininternal combustion engines respectively. The PS is obtainedby applying radial averaging about the origin of the frequency
space (Jacobs et al., 2017) i.e. (qx , qz ) = (0, 0), where qx andqz are the wavenumber in wall-parallel x and z directions, respectively. The prediction error as well as the uncertainty regarding each realistic surface are listed in the table 1. Overall,a mean error of 3.2% among all realistic surfaces is achieved.
While the prediction error as well as the uncertainty for allother roughness are relatively low, the model seems to be lessconfident with the surface IC. As observed from Figure 3, thisuncertainty might be caused by the noncompliant shape of the
PS for the IC surface. Therefore, given the satisfactory performance of the AL framework, it is expected that an extendedroughness repository will yield an improved predictive model.

Table 1: prediction error and uncertainty for five realistic

surfaces

Sandpaper
ICE1
ICE2
ICE3
IC
Average

Error [%]
1.9
4.2
3.9
4.0
2.0
3.2

σ∆U + [-]
0.51
0.42
0.26
0.49
0.60
0.46

CONCLUSION
A predictive model based on combination of ENN and AL
approach for characterizing the hydrodynamic properties ofhomogeneous irregular roughness is presented in the present
work. The uncertainty of the prediction, which is measured
by the standard deviation of the ENN outputs, is shown to bea proper quantity for estimation of prediction error level ofthe model. The results show that with 20 selected roughness
samples by the AL framework, a significant decrease of error of 9 percentage points is observed for the group of roughness samples with the highest uncertainty. The performanceof the model is examined on artificial roughness and realistic surfaces. In both cases, very good agreement between theprediction of the ENN and DNS results is observed. The maximum prediction error of the model for the 5 realistic surfaces
is 4.2%, while the mean error is 3.2%. The result of presented
study confirm the potential of the AL approach for predictionof the hydrodynamic roughness properties. The effects of additional AL-loops on the model performance will be investigated in future work and reported in the conference presentation.

Sensitivity analysis
The sensitivity vector of the roughness characteristic
statistics S st , PDF S PDF and PS S PS are separately shown in
Figure 6 (a) to (c), respectively. From figure 6(a) it can be seenthat the roughness height statistics, namely k99 and kt showhigh sensitivity for the prediction. This is due to the fact thatthese values serve as the indications of the physical scale, reminding that the height range for I PDF is scaled with the inputs
kt . A larger roughness height translates to higher skin friction due to the increased effective slope and therefore a higher∆U + (Chan et al., 2015). Lower sensitivity is observed for the
wavelength inputs λ0∗ and λ1∗ . This may indicate that extreme
large and small roughness wavelengths will not dramaticallyalter the roughness induced drag (Yang et al., 2021).
As can be seen from the figure 6, asymmetric distribution of S PDF indicates that the PDF input elements that locate
around and below the median height show higher sensitivity
for the model output. In contrast, the height distribution near
the roughness peaks demonstrates lower sensitivity. This indicates that the neural networks consider the PDF values close
to the roughness troughs to be more important relative to theones at the peaks for the prediction of surface drag. The degree of asymmetry of the PDF curve is usually represented
by the third
central moment of the PDF, namely the skewR
3 . The sensitivity analysis of theness Sk = S (k − kmd )3 dS/krms
model in figure 6(b) may support the previous observations regarding roughness skewness, e.g. Jelly & Busse (2018); Flack(2018), that a positively skewed roughness with Sk > 0 or

ACKNOWLEDGEMENT
Jiasheng Yang and Pourya Forooghi gratefully acknowledge financial support from Friedrich und Elisabeth BoysenFoundation (BOY-151). This work was performed on the supercomputer HoReKa and the storage facility LSDF fundedby the Ministry of Science, Research and the Arts BadenWürttemberg and by the Federal Ministry of Education andResearch.

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512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
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(a)

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k99

λ0∗

kt

λ1∗

Sensitivity of roughness PDF
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0.02
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0.5
k/kt

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Sensitivity of roughness PS

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Figure 6: Sensitivity scores of the input vector I . (a): S st ,(b): S PDF , (c): S PS .

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