Wind Energ. Sci., 5, 819–838, 2020
https://doi.org/10.5194/wes-5-819-2020
© A uthor(s) 2020. This work is distr ib uted under
the Creativ e Commons Attr ib ution 4.0 License.
Car tographing dynamic stall with mac hine learning
Matthew Lennie 1 , Johannes Steenb uck 1 , Ber nd R. Noack 2 , and Christian Oliver P aschereit 1
1 T echnische Uni versität Berlin, Institut für Strömungsmechanik und T echnische Akustik, Berlin, German y
2 LIMSI, CNRS, Uni versité P aris-Saclay , Bât 507, rue du Belvédère,
Campus Uni versitaire, 91403 Orsay , France
Correspondence: Matthe w Lennie (matthe w [email protected] )
Recei ved: 27 June 2019 – Discussion started: 1 July 2019
Re vised: 22 April 2020 – Accepted: 5 May 2020 – Published: 29 June 2020
Abstract. Once stall has set in, lift collapses, drag increases and then both of these forces will fluctuate strongly .
The result is higher fatigue loads and lo wer ener gy yield. In dynamic stall, separation first dev elops from the trail-
ing edge up the leading edge. Ev entually the shear layer rolls up, and then a coherent vorte x forms and then sheds
do wnstream with its lo w-pressure core causing a lift ov ershoot and moment drop. When 50 + experimental c ycles
of lift or pressure v alues are av eraged, this process appears clear and coherent in flo w visualizations. Unfortu-
nately , stall is not one clean process b ut a broad collection of processes. This means that the analysis of separated
flo ws should be able to detect outliers and analyze cycle-to-c ycle v ariations. Modern data science and machine
learning can be used to treat separated flo ws. In this study , a clustering method based on dynamic time warping
is used to find dif ferent shedding behaviors. This method captures the f act that secondary and tertiary vorticity
v ary strongly , and in static stall with surging flo w the flo w can occasionally reattach. A con v olutional neural
network w as used to extract dynamic stall v orticity con v ection speeds and phases from pressure data. Finally ,
bootstrapping was used to pro vide best practices reg arding the number of experimental repetitions required to
ensure experimental con ver gence.
1 Intr oduction
Beyond small angles of attack, airfoil boundary layers ha v e
to contend with strong adverse pressure gradients. When the
boundary layer does not ha ve enough momentum, a flo w re-
versal occurs and e v entually the flo w separates from the sur -
face of the airfoil (Abbott and Doenhof f, 1959). Once this
occurs, viscous ef fects dominate and any assumption of po-
tential flo w falls apart (Schlichting and Gersten, 2016). This
means that modeling separated flo ws has alw ays been a chal-
lenging part of designing wind turbines or e ven understand-
ing experimental and field data. Even in the age of computa-
tional fluid dynamics (CFD), attempts to simulate stall with
unsteady Reynolds-a v eraged Navier –Stokes (URANS) equa-
tions ha ve not yet yielded good-quality results (Strangfeld
et al., 2015; Rumsey, 2008; Rumse y and Nishino, 2011).
Lar ge-eddy simulations (LESs) show promise b ut are still
too computationally expensi v e to be used as an ordinary de-
sign and analysis tool (Rumsey and Nishino, 2011). In the
wind industry , semiempirical models (Andersen et al., 2007;
W endler et al., 2016; Holierhoek et al., 2013) are still the
main analysis tools for stalled airfoil flo ws. These models
ha ve to mak e simplifications to be viable in terms of av ail-
able computational po wer and input boundary conditions.
The ke y questions are as follo ws. What information is lost?
If we had better understanding and better models, ho w much
could we improv e wind turbine designs?
Fortunately , the recent sur ge in de velopment of machine
learning techniques has provided a ne w set of tools to answer
these types of questions. The foundational idea at the basis
of this paper is that modern machine learning approaches
are accessible to aerodynamics practitioners and can help
us better understand experimental data and better recreate
that physics in simulations. W e will pro vide a number of
demonstrations on ho w machine learning can help dissect
stalled airfoil data. W e will also pro vide a road map for cre-
ating a machine-learned semiempirical dynamic stall model.
It should be obvious by the end of this paper not only that
Pub lished by Copernicus Publications on behalf of the European Academ y of Wind Energy e.V .

820 M. Lennie et al.: Car togr aphing dynamic stall with machine lear ning
these ne w methods are po werful and accessible but also that
they are of vital importance for dealing with airfoil stall.
Stall is the term used to describe a broad range of phe-
nomena that occur during boundary layer separation. There
are two broad characteristics that help us pro vide a loose def-
inition.
1. A flo w re versal in the boundary layer results in the
stream-wise streamline no longer follo wing the surface
of the airfoil (Abbott and Doenhof f, 1959). The region
of flo w re versal will usually ha ve a neutral pressure.
2. Instabilities, such as shear layer instabilities or wak e
mode (v ortex shedding) instabilities (Hudy and Naguib,
2007), are present. These instabilities make the pressure
footprint on the airfoil highly unsteady .
While the follo wing explanations of the cate gories of stall
will di ve deep into details, these tw o features remain the ba-
sic underlying phenomena.
Let us begin by considering a stationary airfoil. As the
angle of attack increases, the airfoil will encounter trailing-
edge (light) stall (McCroske y, 1981). Light stall will de-
velop at moderate angles of attack and is more likely to be
present on airfoils with a well-rounded leading edge (Green-
blatt and W ygnanski, 2002; Leishman, 2006). The adv erse
pressure gradient ov ercomes the momentum of the bound-
ary layer some where do wnstream of the point of minimum
pressure (Abbott and Doenhof f, 1959). The vertical size of
the viscous region will be on the order of the airfoil thick-
ness (McCroske y, 1982). A well-rounded leading edge will
result in a smooth de velopment of trailing-edge stall, whereas
a sharp leading edge may cause trailing-edge stall to be by-
passed rapidly (Leishman, 2006). The separated region will
not contrib ute to the lift, implying a smooth roll of f of the
lift, increase in drag and a nose-up moment. Ev en on a sta-
tionary airfoil, the boundaries of the separated region will be
unsteady (Mulleners and Rütten, 2018) and will v ary along
the span and chord.
At higher angles of attack, deep stall will de velop on the
airfoil (McCroske y, 1982). Deep stall is characterized by
separation occurring at the leading-edge re gion. As the angle
of attack increases, the point of minimum pressure will mo ve
closer to the leading edge as the stagnation point mo ves more
to wards the pressure side of the airfoil (Abbott and Doenhof f,
1959). Here the airfoil leading-edge geometry is critical as a
tight radius will cause a stronger adverse pressure gradient
which can lead to deep stall initiating from the leading edge,
thus bypassing light stall. Ev en though the stall occurs at the
leading edge, the definition of “leading-edge stall” usually
in v olv es a laminar bubble b ursting b ut the mechanism can
more simply be trailing-edge stall that engulfs the entire suc-
tion side of the airfoil (Leishman, 2006). In the steady case,
deep stall will cause a plummet in the lift being produced
and a sharp increase in drag. The v ertical size of the viscous
region will be on the order of the airfoil chord (McCrosk ey,
1982). The viscous region will be home to v arious instabili-
ties such as shear layer mode or wak e mode shedding (Hudy
and Naguib, 2007), essentially dif ferent types of shedding
phenomena leading to fluctuating airfoil forces.
Flo w that detaches from the leading edge can reattach due
to transition of the shear layer (Abbott and Doenhof f, 1959)
or a re-thickening of the airfoil; for e xample, wind turbine
airfoils can ha ve dents due to manufacturing (Madsen et al.,
2019). This phenomenon is called a separation b ubble. Bub-
bles are a sensiti ve phenomenon and small changes to bound-
ary conditions can make them disappear completely (W ard,
1963). Inflo w turbulence, leading-edge surface erosion, foul-
ing or ice will often cause forced transition (Pires et al.,
2018). Earlier transition will tend to reduce or remov e b ub-
bles (W ard, 1963). Even without outside influences, b ubbles
are an unstable phenomena due to shear -layer disturbances
which lead to transition and e ventual reattachment or b urst-
ing (Kirk and Y aruse vych, 2017). F or certain older airfoil
families, i.e., N A CA 63-2nn, the presence or lack of a bubble
may cause an airfoil to switch between leading and trailing-
edge stall; this phenomenon is kno wn as double stall (Bak
et al., 1998). While double stall might no longer be as rel-
e v ant in ne w generations of airfoils on wind turbines with
pitch regulation, b ubbles can af fect stall beha vior and the
e ventual performance of the airfoil.
What happens when the airfoil starts mo ving? When an
airfoil mov es from lo w angles of attack into light stall
regimes, there will be a phase lag between the angle of at-
tack and the separation. This ef fect becomes stronger as the
airfoil pitches faster and can be seen as a resistance to stall
when compared to the stationary case. One can interpret this
ef fect in a fe w ways:
1. The wak e has not yet forgotten the pre vious flo w ar -
rangement, meaning the ef fecti ve angle of attack is still
catching up with the geometric angle of attack, i.e., cir -
culatory lift delay .
2. The current boundary layer still has the higher momen-
tum from the former more fa v orable flo w state.
3. The surface of the airfoil accelerates the boundary layer
during the motion.
When moving from light stall angles of attack do wn to at-
tached flo w , the flow attachment is delayed for the same rea-
son. This appears in polar diagrams as hysteresis loops b ut
can also be interpreted as a dangerous phase dif ference be-
tween the angle of attack and the lift, moment and drag. In
this context, phase dif ferences mean that the structure will
absorb or dissipate ener gy (Bowles et al., 2014; Lennie et al.,
2016). In short, this phase dif ference can lead to single-
degree-of-freedom pitch flutter also kno wn as stall flutter
(McCroske y, 1982). If the unstable nature of separated flo ws
leads to the extent and phase of light stall being v ariable be-
Wind Energ. Sci., 5, 819–838, 2020 https://doi.org/10.5194/wes-5-819-2020

M. Lennie et al.: Car togr aphing dynamic stall with machine lear ning 821
tween cycles of pitching, then it follo ws that the aeroelastic
damping of the airfoil will also be v ariable between cycles 1 .
When an airfoil mov es rapidly from attached flo w into
deep stall, it creates an ef fect kno wn as dynamic stall. The
separation mov es rapidly from the trailing edge up to the
leading edge; the shear layer becomes unstable and then rolls
up into a v ortex with a strong lo w-pressure core (Mullen-
ers and Raf fel, 2013). The v ortex then tra vels do wnstream,
causing a spike in lo w pressure across the airfoil, which
presents as a strong spike in lift and a strong dump in the
moment. A full description of dynamic stall would be e x-
traneous here b ut excellent re vie ws can be found in McAl-
ister et al. (1978), McCroske y (1982), McCroske y (1981),
Leishman (2002) and Carr (1987). More modern e xperimen-
tal works can be found in Granlund et al. (2014), Mulleners
and Raf fel (2013), Mulleners et al. (2012), Mulleners and
Raf fel (2012), Müller-V ahl et al. (2017), Müller -V ahl et al.
(2015), Strangfeld et al. (2015), Balduzzi et al. (2019), and
Holst et al. (2019). For the discussion here, it is suf ficient
to note that as the strength and phase of the leading v ortex
v aries, so will the aeroelastic stability .
T o revie w the pre vious section,
1. there are dif ferent types of stall that occur dif ferently in
static or dynamic conditions,
2. the spatiotemporal v ariation is in both span and chord,
and
3. dif ferences in stall behavior will also lead to changes in
aeroelastic stability .
So ho w are these v ariations treated? T reating stall as
a stochastic process is a relati vely recent idea. As early
as 1978, one sees ackno wledgment that stall is v ariable in
literature such as McAlister et al. (1978), an e xperimental
report that described taking measurements of 50 cycles of a
pitching airfoil under going dynamic stall to ensure con v er-
gence of the lift. While these researchers did ackno wledge
the v ariability of the data, they still used a simple a v erage to
represent the data. This was a reasonable choice at the time
gi ven that man y of the more adv anced tools no w av ailable
did not exist nor w as the requisite computational po wer av ail-
able. Only more recently ha ve researchers be gun to address
the spatial and temporal v ariability of stall in experimental
work. Mulleners and Raf fel (2013) were able to sho w that
dynamic stall could be described by two stages of a shear
layer instability and that the de velopment of these instabili-
ties v aried across cycles. In light stall, it w as shown that the
trailing-edge separation region had tw o modes, resulting in
either a v on Kármán shedding pattern or a stable dead water
zone (Mulleners and Rütten, 2018). The separation pattern
1 While we may normally consider an operating state to be stable
or unstable on a long range trajectory , we may hav e to consider that
each operating state can display short-term behaviors that appear
unstable.
fluctuates unreliably and when v orticity is present, the vorte x
con v ection speed is also v ariable.
Experimental data from Manolesos serve as a detailed re-
minder that stall happens in three dimensions (Manolesos
et al., 2014; Manolesos, 2014). Even on a simple 2D wind
section, flo w visualization sho wed four dif ferent separation
patterns (Manolesos, 2014). These patterns are referred to as
stall cells, and they create complicated v orte x patterns on and
behind the airfoil. Even more complicated still are the sepa-
ration patterns on wind turbine blades due to the changes of
airfoil shape, twist and chord length (v arious surface visu-
alizations can be found in Manolesos, 2014; Lennie et al.,
2018b; V ey et al., 2014). W ind turbines uniquely e xperience
very high angles of attack, where the spatial patterns create
further complications (Skrzypinski et al., 2014; Skrzypinski,
2012; Gaunaa et al., 2016; Lennie et al., 2018b). The pic-
ture that should no w be clear is that stall is a continuum of
beha viors rather than a small number of defined cases.
So v ariability is rampant in stall. Ho w should we measure
and interpret airfoil stall beha vior? This paper will attempt to
demonstrate that machine learning has provided a ne w set of
tools that can be helpful for these very tasks. This paper will
demonstrate
1. a clustering method to group similar time series to-
gether ,
2. a computer vision method for extracting v orte x con vec-
tion speeds from pressure data, and
3. ho w to detect outliers and inspect the con v ergence of
the dataset.
Furthermore, we will provide a future perspecti v e on the way
that machine learning may help us in modeling airfoil stall in
simulations. While the specific methods used in this paper
should prov e to be useful and while we will point out some
specific aerodynamic ef fects in the examples section, these
are only examples. This paper is trying to communicate that
machine learning more broadly is approachable and useful
for unsteady aerodynamics, wind ener gy and other adjacent
fields.
Before jumping into the ne w methods we should establish
what kind of techniques ha ve been used pre viously . It should
be clear gi ven the discussion so f ar that simple av eraging
or e ven phase a v eraging will remov e important data (Riches
et al., 2018). In dynamic stall, for example, phase-a v eraged
flo w visualizations and pressure data appear v astly cleaner
and more coherent than a single cycle. The c ycle-to-cycle
v ariations and outliers are an important part of the dataset and
should not be smeared out. Manolesos (2014) suggested con-
ditional a veraging to produce better airfoil polar diagrams.
Mulleners and Rütten (2018) also performed a kind of con-
ditional a veraging using the orbits of POD coordinates dis-
played onto recurrence plots. Furthermore, Holst et al. (2019)
also suggest a binning approach, especially when consider -
ing very deep stall. Conditional a v eraging is an interesting
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822 M. Lennie et al.: Car togr aphing dynamic stall with machine lear ning
approach, b ut the important question becomes the follo wing:
what rules should we use to split the data and is it possible to
automate this process to some degree?
Fluid dynamics has alw ays been a natural case for dimen-
sionality reduction. In particular , there is ab undant literature
using singular v alue decomposition (SVD) methods such
as proper orthogonal decomposition– principle component
analysis (POD–PCA) (T aira et al., 2017), dynamic mode de-
composition (DMD) (Schmid, 2010; Kutz et al., 2015; Brun-
ton et al., 2015) and spectral proper orthogonal decomposi-
tion (SPOD) (Sieber et al., 2015). These methods generally
do not perform well in cases with any kind of tra v eling wa ve
beha vior (T aira et al., 2017; Riches et al., 2018; Hosseini
et al., 2016). The reason for this lies in the creation of fix ed
spatial functions and basis functions. If the shedding is con-
sistent, the system will be sparse, a sensible reduced-order
system can be found. Ho we ver , introduce phase jitter and the
small number of basis functions no longer does a good job in
representing the shedding; so more mode shapes are needed.
Even for a simple c ylinder shedding, up to 50 modes were
required to represent the system reasonably well (Loiseau
et al., 2018). Dynamic stall con vection v elocities v ary con-
tinuously (Mulleners and Rütten, 2018); therefore we cannot
expect a sparse set of spatial functions to represent the sys-
tem well.
Fortunately the SVD and simple a veraging-type methods
are not the only forms of dimensionality reduction techniques
a vailable. It turns out the dimensionality reduction is a cor -
nerstone technique of machine learning; an interacti ve sum-
mary can be found on Christopher Olah’ s website (Olah,
2019). In this paper , we will show ho w multidimensional
scaling (MDS) (O’Connell et al., 1999) and clustering (Mai-
mon and Rokach, 2006) can be used as a reliable analysis
technique for airfoil stall. Nair et al. (2019) ha ve demon-
strated one approach to clustering for separated flo ws in the
context of cluster -based feedback control. Cao et al. (2014)
also demonstrated the use of time series clustering in the con-
text of comb ustion. The advantage of cluster -type methods
is that they break the data do wn into similar neighborhoods
rather than assuming that a set of global basis functions can
describe the whole domain. Both Loiseau et al. (2018) and
Ehlert et al. (2019) ha ve demonstrated that local linear em-
bedding (LLE), a neighborhood-type method, can create a
sparse representation of the system. In this paper , we will
focus on clustering and MDS, although other methods also
sho w promise.
The MDS and clustering methods rely on a distance metric
to gauge the similarity between the time series of lift of v ar -
ious experimental repetitions. As already discussed, the data
will contain phase jitter which may cause simple distance
metrics such as Euclidean metrics to ov erestimate the dif-
ference between cycles (Ratanamahatana and K eogh, 2004).
The problem is amplified by the strong gradients present
around the time of v ortex con vection. This is a common time
series problem, and dynamic time warping (DTW) w as cre-
ated for this purpose (Morel et al., 2018; Ratanamahatana
and K eogh, 2004). DTW allo ws for the time series to be
stretched and squashed a small amount to allo w for an ef-
fecti ve comparison between e xperimental repetitions. The
approach of using a cycle-to-c ycle distance metric (in this
case DTW) is dif ferent to making time-independent clusters
used in the work of Nair et al. (2019). The dif ference in ap-
proach comes from intended application. In this paper , we
will create clusters and MDS plots by comparing the entire
time series of separate pitch cycles.
Methods such as clustering and MDS belong to a branch of
machine learning called unsupervised learning, i.e., learning
from the data without ha ving the answer ahead of time. Con-
versely , supervised learning uses a labeled dataset to learn a
mapping between input and outputs. Once a model is trained,
we can then map ne w data. This is the nature of our sec-
ond example, e xtracting the vorte x con vections from pres-
sure data. W e manually create a small set of examples by
clicking on the v ortex patterns. W e then use these data to
train a model that can do the same job ov er the whole dataset
ef ficiently . Manually clicking on the patterns is a laborious,
time-wasting and unpleasant task. F or these reasons, we want
to do this only for the bare minimum number of examples.
Fortunately , we can lev erage the concept of transfer learning
to minimize the ef fort.
The concept of transfer learning exploits the f act that once
a model has been trained for one task, it can be easily re-
molded to complete similar tasks (Bro wnlee, 2017). In prac-
tice this means that a neural network can be trained for
a specific computer vision task and then easily be reused;
i.e., a network originally trained for classifying breeds of
dogs within photographs can be easily reused on aerodynam-
ics data (the F AST AI project has a lecture series e xpanding
at length on this theme; Ho ward et al., 2019). This may seem
like an e xotic claim but there is solid reasoning underpinning
the claim. Pictures are displayed in pix els, which is an incred-
ibly high-dimensional space (modern cameras ha ve a 10 MP
range). If we randomly choose pixel v alues, the chances of
getting a sensible picture are almost zero; we would usually
only get noise. This means that sensible pictures with geo-
metric features such as lines and circles e xist in an incredibly
small neighborhood. That is to say , any real picture (of an
elephant, a calculator , a cloud or ev en a plot of our pressure
vs. time) is more similar to any other real picture than it is
to a picture of the kind of random static noise we kno w from
old tele vision sets. Why does this matter? It means that we
can use any general picture dataset to get our neural netw ork
to the right neighborhood, that is, being able to recognize
real geometry . It turns out that as far as the neural netw ork
is concerned, the pressure plots look close enough to real
world pictures that it only needs a small amount of retrain-
ing. Therefore, instead of requiring millions of training data
examples, we only needed roughly 700.
In this paper , we will demonstrate the utility of trans-
fer learning by using a pre-trained con volutional neural net-
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M. Lennie et al.: Car togr aphing dynamic stall with machine lear ning 823
work (CNN) to e xtract v ortex con vection speeds from airfoil
pressure plots. A huge challenge of working with e xperimen-
tal data is that it is e xceptionally difficult to e xtract features
from data in an automated fashion. One e xample of this is ex-
tracting the con vection speed of a v ortex from pressure data.
T o the human eye it is a fairly ob vious stripe in the pres-
sure plot; ho we ver it is challenging to e xtract this feature au-
tomatically based on basic rules. Computer vision machine
learning is perfect for such cases. While the v ortex con vec-
tion speeds are themselves an interesting result, the e xample
should demonstrate to readers the incredible po wer of using
pre-trained neural networks for e xtracting features from data.
Deep neural networks are becoming increasingly used within
the wind industry for applications, e.g., for predicting rotor
icing (Y uan et al., 2019), power -curve estimation (K ulkarni
et al., 2019) or e ven for rotor –blade inspections (Shiha vud-
din et al., 2019). W e hope to demonstrate with this paper that
modern machine learning tools and infrastructure can pro-
vide a useful boost to research in unsteady aerodynamics,
wind ener gy and other adjacent fields.
2 Experimental data
Most machine learning methods are hea vily reliant on an ini-
tial dataset 2 . The analyses sho wn in the rest of this paper
rely on two e xisting datasets. W e use these two datasets to
demonstrate dif ferent approaches as they feature dif ferent in-
teresting ef fects. The wind tunnel dataset is the primary data
source and unless explicitly stated will be used in all figures,
graphs and discussions. The to wing-tank dataset provides a
great example for outlier detection. The following introduc-
tions aim to provide some conte xt but do not e xhaustiv ely
describe the experimental setups or the data they retrie ved.
The original references provide a f ar more detailed vie w into
the setups.
2.1 Wind tunnel
The first dataset was collected by Müller -V ahl (2015). Ex-
tensi ve unsteady aerodynamic e xperiments were conducted
in a blo wdo wn wind tunnel po wered by a 75 kW backward
bladed radial blo wer . The test section is depicted in Fig. 1
and is 610 mm per 1004 mm. The model is mounted on two
circular , rotatable plexiglas windo ws and the wind speed is
measured with two hot-wire probes. The pressure around the
model is captured by 20 pressure sensors on both suction and
pressure sides (40 in total). The N A CA 0018 airfoil model
has two control slots at 5 % and 50 % chord for additional
blo wing. The model has a chord length of 347 mm and a span
of 610 mm. More information about the tunnel can also be
found in Greenblatt (2016), and excerpts of the dataset can be
2 Most b ut not all. For e xample, reinforcement learning can use
self-play as a training mechanism.
Figure 1. V iew of the test section sho wing the pitching mechanism
and the approximate location of the airfoil model. From Müller -
V ahl (2015).
found at https://www .flo wcontrollab .com/data- resource (last
access: 13 September 2019).
The wind tunnel data cov er a comprehensiv e collection of
experiments with v arying boundary conditions. The dataset
has been thoroughly explored in pre vious publications and
appears to be of good quality . It ranges from static baseline
in vestigations o ver oscillating pitching and v ariation in free-
stream velocity (and a combination of both). In order to ma-
nipulate the boundary layer , blo wing was added. One pecu-
liarity of this dataset is that boundary layer tripping can be
induced by the taped-ov er blowing slots on the suction side
of the airfoil. For the purposes of our analysis, this detail w as
not critical.
2.2 T owing tank
The second dataset comes from a lar ge towing-tank f acility
at the T echnische Uni versität Berlin. This dataset is used to
demonstrate outlier detection as the test configuration used
in these data did ha ve some peculiar stall beha vior on some
cycles. The w ater tank dimensions are 250 m in length, 8.1 m
in width, and about 4.8 m in av erage depth. A carriage runs
on rails, to wing a rig (and the model) through the water with a
maximum speed of 12.5 m s − 1 . On it the complete measuring
system is installed. The rig consists of two side plates with a
length of 1.25 m, a height of 1 m and a thickness of 0.035 m
prohibiting lateral flo w around the model. In between the side
plates, the model, with a span size up to 1 m, can be inserted
at arbitrary angles of attack. The model resembles a flat plate
with an elliptical nose and blunt trailing edge. It has a span
of 0.95 m, 0.5 m chord and a thickness of 0.03 m. The surface
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824 M. Lennie et al.: Car togr aphing dynamic stall with machine lear ning
Figure 2. Cross section of the mounted flat plate. Red dots indicate
position of pressure sensors. From Jentzsch et al. (2019).
is cov ered in aluminum, and 12 pressure ports are inserted
at the specified locations in Fig. 2. The airfoil model is an
unusual form b ut only some qualitati ve demonstrations are
made with this dataset. A more detailed description is gi ven
in Jentzsch et al. (2019).
3 Machine learning appr oaches
In this paper , we aim to provide a demonstration of a fe w
machine learning methods and ho w they can be applied to
unsteady aerodynamics data. A brief ov erview of the algo-
rithms is provided to gi ve a sense of what each of the al-
gorithms is doing. The first algorithm demonstrates ho w to
train and use a relati vely simple machine learning algorithm,
clustering, from scratch. The second example demonstrates
the more adv anced deep-learning approach and sho ws a fe w
tricks to make it possible to do so with a modest amount of
data and computational po wer . Usually each task will call for
a dif ferent algorithm and dif ferent approach, but man y of the
principles discussed in the follo wing section should transfer
well onto other problems. This is especially true for the deep-
learning training tricks.
3.1 Dynamic time warping, cluster ing and
multidimensional scaling
In this section, we will describe a method of grouping simi-
lar data together called clustering. For clustering to w ork we
need two parts, a distance metric/measurement and a cluster -
ing algorithm. The distance metric gi ves us a measurement of
similarity between our data 3 . The clustering algorithm takes
the distances and groups the data into clusters.
Dynamic time warping (DTW) is a distance measurement
that allo ws for squashing and stretching of the time series
in order to reach a best fit. In practice, it is comparable to
taking a winding path through a grid where each box corre-
sponds to a time step from the two paths being compared (see
Fig. 5). The dynamic time warping algorithm is particularly
useful in this case because it will still indicate that time series
are similar e ven if there is a slight phase dif ference in vorte x
shedding or other stall phenomena (this ef fect is sho wn in the
wa ve form in Fig. 5). The general rule of thumb is that a small
amount of warping is a good thing; a lot can end up distorting
3 In this example, we are comparing a time series of a single
experimental repetition ag ainst another . Clustering can also work
with much more simple distance metrics.
Figure 3. Soft-DTW centroid for clustered time series with strong
phase jitter . (Example data from pressure sensor reading from to w-
ing tank.)
reality . Therefore, DTW algorithms are usually implemented
with either global or local constraints, and these constraints
ha ve a bonus of increasing the computational ef ficiency .
A useful extension to the DTW algorithm creates a com-
posite of multiple time series called a centroid (see Fig. 3).
Normally the problem with dynamic stall time series is that
the v ortex shedding is smeared out when simple means are
taken. The onset of static stall can also appear to be a smooth
process rather than a sudden separation that occurs at v ari-
able phases across dif ferent cycles of the e xperiments (see
Fig. 3). The barycenter extension to DTW creates an a verage
that preserves these features. This means that the resulting
centroid will be far more representati ve of a real stall process.
In short, it is just a pseudo-a verage using dif ferent mathemat-
ics in the background, b ut it provides a better answer to the
follo wing question: for these boundary conditions, what does
the stall process typically look like?
For this research, the soft-DTW algorithm w as used to
compute the barycenter and was tak en from the Python mod-
ule tslearn by T av enard (2008). The algorithm was first pro-
posed by Cuturi and Blondel (2017). T o create the clusters,
it is necessary to compare e very time series within a group
to each other . This means the complexity of the algorithm
is O ( N 2 ). T wo steps were taken to scale the process; firstly
the data were do wnsampled, thus reducing “ N ”, and sec-
ondly the code was scaled using D ASK (Dask De velopment
T eam, 2016). D ASK is a Python library designed to paral-
lelize standard Python functions onto cluster architecture.
The second step may at first appearance seem extreme. In
practice the po wer required was more than a standard desk-
top b ut one or two compute nodes were more than suf ficient.
For the e xamples computed in this paper , one to two w orkers
(nodes with eight cores each) would process a single e xperi-
ment within a fe w minutes. A combination of parallelization
and do wnsampling was used in this study 4 .
4 Combining the soft-DTW algorithm and the D ASK module did
require some programming ef fort, but as both tools were well de vel-
oped, the ef fort was smaller than it perhaps first appears. In partic-
ular , tools like D ASK allow people with v ery modest programming
skill to run cluster -scale code.
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M. Lennie et al.: Car togr aphing dynamic stall with machine lear ning 825
Figure 4. T ime series clustering algorithm.
Reducing the number of samples gi ves a significant speed
boost as the complexity of the distance measurement is based
on the number of time steps. While reducing the sample size,
the spectral resolution is reduced about the same factor . The
frequency of the e xpected phenomena limits the amount of
do wnsampling. In order to improv e the cluster results the
data are, in addition to do wnsampling, filtered. Dynamic time
warping is noise sensiti ve as the algorithm shifts and bends
the time series in order to match similar v alues. Fortunately ,
tuning these steps is not dif ficult as a visual inspection of the
Figure 5. Euclidean distance vs. DTW distance between two time
series.
resulting data will indicate whether the algorithm is making
sensible groups or not. This topic is explored in greater detail
in the related work from Steenb uck (2019).
Clustering is a method of dimensionality reduction based
on the principle that the dataset can be ef ficiently described
by a set of subgroups. These subgroups are formed on the as-
sumption that the description of the cluster is a useful enough
generalization for each member of the cluster . This means
that the groups are formed on the basis of similarity . Cluster -
ing is an unsupervised method in the sense that there is no
correct answer defined ahead of time. Usually unsupervised
methods will re veal underlying data structures. This is not to
say that we can just passi vely use these algorithms and use-
ful results will ensue. Each clustering algorithm will perform
well for some datasets and will deli ver nonsense for others;
care is required. T o ensure good results, users will usually
ha ve to tune hyper -parameters for the dataset, and the sim-
plest of these parameters is the number of clusters. A first
estimation about a reasonable number can be made from in-
specting the dendrogram or the MDS plot as described belo w .
Another approach is to calculate the mean silhouette score of
all elements for a range of cluster numbers (Fig. 6). The sil-
houette score is calculated by comparing the distance from a
data element to its o wn cluster center to the distance to the
center of the closest neighboring cluster (Raschka, 2015). By
calculating the mean silhouette score for a number of dif fer-
ent clusters, we can see that once we get to four clusters, we
only mar ginally change the quality of the clusters (sho wn in
Fig. 6). This means that breaking the dataset up into further
smaller pieces is not going to improv e our analysis.
For this application hierarchical clustering turned out to
produce groups that were physically meaningful and shared
features. Hierarchical clustering creates links between data
points (in our case a single cycle of a dynamic stall test)
to form a dendrogram as seen in Fig. 4. This process es-
sentially takes the distances that we pre viously calculated
and starts collecting similar data together recursi vely which
is what is sho wn in the dendrogram. The dendrogram is
then cut at a height which results in a gi ven number of
clusters. As longer branches indicate bigger dif ferences,
the height of cutting should be chosen so that the longest
branches are cut. The clustering was implemented using
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826 M. Lennie et al.: Car togr aphing dynamic stall with machine lear ning
Figure 6. Mean of silhouette scores per cluster number .
SciPy’ s (Jones et al., 2019) hierarchical clustering algorithm
(scipy .cluster .hierarchy) with the ward method as a measure
for distances between ne wly formed clusters. Hierarchical
clustering was chosen after e xploratory analysis sho wed that
other basic algorithms such as k -means tended to perform
poorly for these data. Fortunately , well-dev eloped machine
learning libraries such as Sci-Kit Learn make it v ery simple
to trial dif ferent algorithms.
Another way of presenting the data is to use multidimen-
sional scaling (MDS) (O’Connell et al., 1999). MDS essen-
tially takes a cloud of data points with high dimensionality
and squashes the points onto a lo w-dimension plane while at-
tempting to maintain the distance between the points. In our
case, each time step of a single series represents a dimension
or feature which results in dimensionality that is incredibly
dif ficult to interpret. No w take each series as a single data
point and then squash it onto a 2D plane, and the data re veal
an underlying structure. W e can then color each point and
use a k nearest-neighbor classifier to color the background
as seen in Fig. 4. The resulting point cloud (hopefully) in-
herits distinct clusters. The number of clusters encountered
here gi ves a good first estimation about a reasonable cluster
number for further analysis. So instead of creating a chaos of
ov erlapping time series, the data appear as a lo w-dimensional
representation image with each color representing time se-
ries with similar beha vior . In some circumstances, the coor-
dinates of the image will e ven ha ve a clear physical meaning;
i.e., dimension 1 could correlate with the Reynolds number .
A broad ov erview of the algorithm used in this paper can be
found in Fig. 4.
An example of the cluste r analysis is depicted in Fig. 4. In
this figure, we summarize all of the time series of a single
cluster by displaying only its centroid. W e can see that each
of the centroids represents a slightly dif ferent beha vior , par-
ticularly during the secondary v ortex shedding. Each cluster
has a small uncertainty band sho wn by the standard de vi-
ation. As the dataset can be represented by three centroids
instead of trying to compress the entire data into a single a v-
erage, the representation is concise b ut still provides a more
accurate vie w of the process.
3.2 Con volutional neur al networks
In the pre vious section, we looked at ho w we can cluster to-
gether similar experimental samples. This section aims to see
if we can extract some interesting features from our data us-
ing machine learning. For this e xample, we will attempt to
use computer vision (machine learning applied to pictures)
to extract information about the dynamic stall v ortex.
Con volutional layers are the special trick that ha ve turned
neural networks into a wildly ef fectiv e computer vision tool
(Krizhe vsky et al., 2012). Con volutional layers allo w pictures
to maintain their structure as a grid of pixels. Con v olution
operations are applied ov er the picture as a kind of mo ving
windo w shape filter . The shape filters are learned and often
end up resembling recognizable patterns. In the first layer of
the network, the filters will be detecting edges, slo w gradi-
ents and color changes (Zeiler and Fer gus, 2013). As we pro-
ceed deeper into the neural network, the filters be gin to look
like natural features such as a birds e ye, a bicycle wheel or
a doorframe. Each of these filters is created during the train-
ing process where lar ge datasets are fed through the network,
and the error is propagated backwards through the netw ork to
allo w for incremental improv ement. It is helpful to note that
as pictures are just made up of a grid of pixels, a 2D matrix
structure (for single channel), in a great deal of cases, data
can be represented in this form. This means that computer
vision tools can be used on data that can be structured like
a picture. Con volutional neural netw orks are most ef fectiv e
when features are local.
W e hav e discussed neural networks here with a high num-
ber of layers. This is referred to as deep learning. Deep learn-
ing is a field that has seen rapid innov ation due to the abun-
dance of graphical processor units (GPUs) and more recently
tensor processing units (TPUs). Platforms such as PyT orch or
T ensorFlo w provide high-le vel front ends in Python. These
front ends abstract aw ay much of the complexity , meaning
that users a void much of the lo w-le vel matrix algebra and
optimization. Furthermore, it is common practice to pub-
lish well-performing neural network architectures that are
already pre-trained (transfer learning). Many of the once
dif ficult decisions, such as choosing a learning rate, ha ve
no w been made simpler with tools such as learning rate
finder (Ho ward et al., 2019). Cheap computational po wer ,
easy high-le vel coding and the adv ent of transfer learning
means that these incredibly po werful tools are no w av ailable
for aerodynamic applications like detecting boundary layer
transition from microphone data (see Fig. 7). These inno-
v ations mean that non-machine learning specialists can use
deep learning with a lo w barrier to entry .
In this paper , we will provide an e xample of turning aero-
dynamic data into a picture and then using a con volutional
neural network to e xtract useful information. Dynamic stall
v ortices hav e a strong lo w-pressure core which causes a lift
ov ershoot and moment dump. When dynamic stall v ortex
data are a veraged o ver 50 + c ycles, it tends to sho w dynamic
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M. Lennie et al.: Car togr aphing dynamic stall with machine lear ning 827
Figure 7. Identification of a boundary layer state using a recur -
rent neural network (data from Bak et al., 2010) (see code e xam-
ple https://github .com/MatthewLennie/Aerodynamics, last access:
13 September 2019).
stall v orticity as far more clean and coherent than is the case
for a single cycle. The strength of each v ortex, its con vec-
tion speed and onset of con vection v ary between cycles. This
lea ves the follo wing questions. How much do dynamic stall
v ortices con vect dif ferently? Do boundary conditions like the
reduced frequency af fect the v ariability?
The dynamic stall v ortex feature of a pressure vs. time plot
is easily distinguished by the human eye; ho wev er , pulling
this feature from the data is rather dif ficult. The authors at-
tempted the task with a number of more simple approaches
such as simply finding the peak at each chord-wise position, a
Hough transform or e ven Bayesian linear re gression with the
pressure plot interpreted as a probability distrib ution. They
all work ed for a few cases b ut failed to generalize and in the
end did not perform well enough to be usable. Each v ortex
is dif ferent and therefore manually creating a rule to auto-
matically pull the dynamic stall v ortex feature from the data
was not tri vial. Howe ver , this is a standard computer vision
task very similar to a dri verless car identifying a c yclist in
a picture. Fortunately , heavy de velopment in the computer
vision field has resulted in some incredibly po werful pre-
trained models such as the RESNET family of models (He
and Sun, 2016) 5 . The model is a con volution neural netw ork
that has been pre-trained on a massi ve dataset of real w orld
images. This means that the con volutional layers of the net-
work already ha ve a set of shape filters that are broadly appli-
cable to all natural pictures. This means that with a relati vely
small amount of training data and computational ef fort, we
are able to simply remold the con volutional layers to identify
dynamic stall v ortices’ s and gi ve the con v ection speed and
phase.
Pre-trained neural networks can be b uilt and retrained us-
ing any of the typical frame works such as PyT orch, K eras or
5 Note that while we were able to get acceptable results from
the RESNET models, a higher le vel of accurac y may be obtained
by network architectures that were b uilt specifically for this kind of
localization task.
T ensorFlo w . In this case, we used a RESNET50 model within
the F AST AI architecture which is a high-le vel interface b uilt
on top of PyT orch (How ard et al., 2019). The F AST AI archi-
tecture implements se veral current best practices as def aults
such as cyclical learning rates, drop-out, training data aug-
mentation and data normalization. W e can think of the pre-
trained neural network as a template: most of the training has
already been done, and we only need to retrain the network
to react correctly to our dataset. This approach is cheap in
terms of data v olume and computational power .
The final layer of the neural network w as replaced with
two outputs to represent a linear fit of the v ortex con v ection
(slope, of fset). For this analysis, acceleration of the v ortex
was ignored, though the code could be easily e xtended. The
pressure data were represented as a picture where the hori-
zontal dimension represents phase, and the v ertical dimen-
sion represents the suction side of the airfoil with the bottom
of the picture being the leading edge (an example of an al-
ready processed picture is in Fig. 8, where the training data
do not ha ve the blue line identifying the v ortex b ut are oth-
erwise the same). T raining data were created by manually
clicking (and storing) the positions of the v ortex on 733 im-
ages (an attempt with only 300 pictures tended to ov er -fit on
RESNET50 or ha ve high bias on smaller models). The man-
ual clicking does introduce some measurement error , but a
fe w practice runs sho wed that the error was much smaller
than the ef fect of the physical phenomena. The images were
selected from a wide range of cases with randomized test
training splitting within each case to ensure good general-
ization of the fitted model. Ho we ver , data were limited to
examples with a strong w ake mode shedding, meaning that
the v orticity is easily visible on the pressure footprint. The
training was done in tw o stages, first with the internal lay-
ers of the RESNET model frozen. This means we train only
the very last layers that output the slope and onset. Once the
training error reached stopped improving, the internal layers
were unfrozen to mold the internal layers for a small number
of epochs (training repetitions).
Initially 80 % of the data were taken as the training set and
the training was completed with 20 epochs with the con v o-
lutional layers frozen so that the ne wly added layers could
quickly con ver ge. The training was stopped at 20 epochs
once the v alidation error began to increase. The con volu-
tional layers were then unfrozen and the training was con-
tinued for a further 20 epochs. During training no geometri-
cal augmentations on the training were undertaken, b ut the
brightness of the images was augmented 6 . The error statis-
tics were still unsatisfactory and additional training did not
improv e the performance further . Howe ver , the current set-
tings of the hyper -parameter settings and training procedure
had seemed to extract the best model gi ven the a v ailable data.
The training procedure was repeated e xactly the same a sec-
6 Geometric augmentations would ha ve been the ne xt method to
improv e the model if the process had not worked well enough.
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828 M. Lennie et al.: Car togr aphing dynamic stall with machine lear ning
Figure 8. Example CNN output. Color intensity refers to suction
pressure, and the blue line is regressed fit. Pressure is standardized;
therefore the colors represent Z scores. No units or color bar are
provided for this reason.
ond time, with the same hyper -parameters and the same num-
ber of epochs; ho we ver this time the dataset was not split
into test and training sets, thus neglecting v alidation error (a
practice described by Goodfello w et al., 2016). This may be
percei ved as opening up the risk of o ver -fitting; howe ver the
training procedure and hyper -parameters were already tested
and the neural network did not o ver -fit. Furthermore, usu-
ally additional data will help reduce ov er -fitting. W e there-
fore ha ve confidence that with this procedure the v alidation
error will not increase and that the training error is repre-
sentati ve across the dataset. Thankfully the additional data
did reduce the training error enough to make the model us-
able (see an example results in Fig. 8). The residuals of both
the slope and constant were distrib uted roughly as Gaus-
sians with standard de viations of 0.15. In total, the train-
ing took on the order of 30 min of computational time on
a GPU. Readers are encouraged to vie w the source code at
https://github .com/Matthe wLennie/V ortexCNN (last access:
13 September 2019). The repository contains training sets
and final data used to produce the follo wing analysis.
The resulting model incurs a small measurement error so
the resulting distrib utions hav e be adjusted. Fortunately , the
measurement error could be quantified. Both the error and
the resulting v ortex con vection v alues can be approximated
as Gaussian. The real distrib ution is sought by guessing a
distrib ution, running a Gaussian con volution filter o ver the
distrib ution and then measuring the difference between the
resultant distrib ution and the data. Essentially , we knew the
measurement error distrib ution roughly , we knew the output
distrib ution, and we can work backwards on a statistical ba-
sis. This error term is fed into a optimizer , thus giving an
estimation of the real data distrib ution without the error in-
curred by the neural network inference. In practice, this re-
duces the standard de viations of both the slope and intercept
by roughly 30 %. W e should note that we can not “repair”
the measurement data and locate the true con vection speed
of each measurement, b ut on a statistical basis, we can get
closer to a true estimation. It is also worth mentioning that
this neural net will find the speed that the v ortex footprint
tra vels across the airfoil, and the v ortex will usually ha ve an
additional component normal to the airfoil.
The procedure described abov e represents a first iteration
of such an approach, a feasibility demonstration. W ith some
more ef fort, a better neural network architecture could be
chosen and the clicking procedure could be replaced with
comparison to flo w visualization. W ith these improv ements,
we could potentially a void the final step where we attempt to
repair the distrib utions. W e would prefer to remo ve this final
step which forces us to assume that the distrib ution is Gaus-
sian. Nonetheless, the current model is workable enough for
our purposes.
4 Examples
So far we ha ve e xplored the idea that stall is v ariable as well
as a fe w machine learning methodologies that could help in-
terpret the data. W e will now pro vide demonstrations of both
of the algorithms. While the specific results are interesting
and we will briefly discuss the physical ef fects observed by
the algorithms, the aim of this section is to pro vide illustra-
ti ve e xamples of the approaches in use. The description of
the physical ef fects is provided merely to moti v ate that the
methods appear to be finding sensible phenomena.
4.1 Extracting v or tex con vection with a con v olutional
neural netw or k
In this section, we provide a demonstration of the neural
network e xtracting dynamic stall vortices’ s from the surface
pressure of the airfoil–time series data. The time series data
come from the wind tunnel dataset which sees an airfoil in a
wind tunnel. The airfoil pitches sinusoidally , the free-stream
velocity can be changed and the leading-edge blo wing is in-
stalled. A number of test configurations with dynamic stall
were chosen and pushed through the neural network. The
first case is relati vely complicated, as it features an oscillat-
ing inflo w velocity (sinusoidal with a v ariation of 50 % in
the mean inflo w), pitching into the dynamic stall range (up
to 25 ◦ ) with leading-edge blo wing acti ve. F our example tests
were compared with dif ferent phase dif ferences between the
angle-of-attack motion and the inflo w velocity . The pitch and
blo wing phases for each case are sho wn in Fig. 9. Medina
et al. (2018) made a v ery similar analysis and found that de-
celerating flo w tended to destabilize the boundary layer and
encourage earlier separation. W ith the con vection speed and
onset data retrie ved by the neural netw ork, it is possible to
sho w that this is true in the specific detail of the dynamic stall
v ortex. Figure 10 sho ws that for cases where the inflow speed
is in phase with the angle of attack, the shedding occurs later .
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M. Lennie et al.: Car togr aphing dynamic stall with machine lear ning 829
Figure 9. Inflow and angle of attack for Figs. 10 and 11.
Figure 10. Probability distrib utions of the con vection speed of
dynamic stall with airfoil blo wing different phases of harmonic
inflo w ( τ ). U amp
U = 0 . 5, k = 0 . 08, R e = 2 . 5 × 10 5 and α 0 = 15 ◦ ,
α amp = 10 ◦ .
Ho we ver , when it does finally occur , the vorte x will shed at
a higher velocity (see Fig. 11). Interestingly the results seem
to indicate a much higher v ariability in the cases where the
flo w is decelerating during the vorte x con vection. Figure 14
also sho ws the relationship between the onset of the v ortex
shedding and the con vection speed. There is a weak correla-
tion ( ∼ 0 . 3 Pearson metric) b ut not strong enough with the
existing data to mak e conclusions about the relationship be-
tween the two. This first e xample sho ws us that we can use
a machine learning tool to better understand ho w our bound-
ary conditions such as inflo w velocity af fect the physical pro-
cess. W e were able to take a large set of test repetitions and
summarize them in a compact yet descripti ve manner without
ha ving to resort to av eraging.
A second example sho ws the ef fect of varying only the
Reynolds number with constant inflo w velocity (see Figs. 13
and 12). W e can see that the mean vorte x con vection v eloc-
ity scales with Reynolds number as we should e xpect. The
v ortex con vection onset has a constant v ariance across both
examples (see Fig. 13). Ho we ver , interestingly the v ariance
of the con vection v elocity gro ws with Reynolds number (see
Fig. 12). This example sho ws us to be very careful about ho w
Figure 11. Probability distributions of the onset of dynamic stall
with airfoil blo wing and different phases of harmonic inflo w ( τ ).
U amp
U = 0 . 5, k = 0 . 08, R e = 2 . 5 × 10 5 and α 0 = 15 ◦ , α amp = 10 ◦ .
Figure 12. Probability distributions of the con vection speed of dy-
namic stall with dif ferent Reynolds numbers, k = 0 . 09, and α 0 =
18 ◦ , α 1 = 7 ◦ .
Figure 13. Probability distributions of the onset of dynamic stall
with dif ferent Reynolds numbers, k = 0 . 09, and α 0 = 18 ◦ , α 1 = 7 ◦ .
we think about v ariability and ho w it applies to each part of
the physical process. While these results and the first e xam-
ple’ s results are interesting and can be expanded upon, the
important lesson is that a small data, lo w computational cost
machine learning method was able to help e xtract a richer set
of information from the dataset.
4.2 Dynamic stall clustering
In the follo wing section, we ha ve chosen a fe w examples
purely for the purpose of demonstrating the clustering ap-
proach and their usefulness in analyzing dynamic stall. In
particular , we would like to see if there are distinct beha viors
possibly stemming from stall cells or other comple x phenom-
ena. By using clustering, we hope to split our dataset into
clusters of dif ferent airfoil behaviors as f ar as they e xist. This
provides us with a w ay of inspecting the data without ha ving
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830 M. Lennie et al.: Car togr aphing dynamic stall with machine lear ning
Figure 14. Relationship between the onset of shedding and con vec-
tion speed for a range of blo wing cases.
to laboriously compare each time series or to simply inspect
a veraged data that will hide these ef fects.
At high angles of attack ( α 0 = 21 . 25 ◦ and α amp = 8 . 25 ◦ ),
we can observe the dif ferent kinds of stall behaviors that can
occur . Figures 15 and 16 show contrasting beha viors for the
same angles of attack. In Fig. 15, a quasi-periodic shedding
appears. W ithout flo w visualization it is hard to determine
the shedding type, b ut the pressure footprint sho ws the vorte x
as weak and smeared. This kind of footprint would indicate
that the v orticity is not close to the surface of the airfoil or
is lar ge and not very coherent. This probably indicates that
we are seeing a shear layer instability rather than very clear
wak e mode examples seen in the pre vious section. The clus-
ters seem to indicate that the shedding beha vior is not reli-
able, with cluster 3 (green) and cluster 4 (red) sho wing am-
plitudes of oscillation dissipating rapidly . Ho we ver , the other
two clusters sho w a more sustained shedding pattern.
No w let us consider a second case with a dif ferent
Reynolds number and reduced frequenc y but with the same
angle of attack range (Fig. 15). The airfoil mov es into stall,
releases one (cluster 2 – orange) or two coherent v ortices
(cluster 1 – blue cluster) and then resolves into weak er small-
scale shedding. That is, we are seeing two dif ferent shedding
patterns for the same boundary conditions.
In Figs. 17 and 18 we can observe the ef fect of changing
the reduced frequency while holding the Reynolds number
and angle of attack constant. The first most obvious dif fer-
ence is that the period between the primary and secondary
v orticities remains constant. The data do otherwise follow
the general wisdom that the lift ov ershoot will increase with
reduced frequency , but it does not happen uniformly . Fur -
thermore, the lo wer reduced frequency seems to create a
much wider v ariance in the primary stall vorte x compared
to the higher reduced frequency where both clusters display
a strong primary v ortex with only a barely visible change in
primary stall. Using the clustering method we are also able to
re veal that, in both cases, one cluster has a strong secondary
v orticity and the other has a nearly nonexistent secondary
v orticity (read carefully , colors do not match). Interestingly
the higher reduced frequency in Fig. 18 seems to suppress the
secondary v ortex as Fig. 17 sho ws strong secondary v orticity
in 55.8 % of the cycles and a some what weaker secondary
v ortex for the other c ycles.
W e hav e observed with these four e xample cases that dif-
ferences in reduced frequency and Re ynolds number will re-
solve into a quite dif ferent type of vorte x shedding. Further-
more, e ven within the same case we can see a strong v ari-
ation in the strength of the shedding mechanism. The insta-
bility mechanism dri ving this shedding is very sensiti ve to
the small v ariations in input conditions. The shedding mech-
anisms sho wn in these four examples are just one of the v a-
riety of shedding beha viors.
A quick visual inspection of the time series data would be
unlikely to uproot the v ariable shedding behaviors seen in
these two e xamples. Ho we ver , the cluster centroids or e ven
simply the MDS plots (i.e., Fig. 20) make the dif ferences
clear and easy to interpret. In an y of the example cases,
phase-a veraged results w ould hav e been a poor representa-
tion of the dataset because we would not ha ve an y way of see-
ing the v ariable nature of the results. Better information leads
to better decisions. For e xample, when we calibrate sim-
ulation tools, particularly empirical unsteady aerodynamic
models, we should be aw are of where our models will per-
form poorly because the underlying flo w physics is highly
stochastic, e ven sho wing distinct behaviors. Our model de-
sign choices can be more well informed, i.e., choosing to
fit a model on the most commonly occurring cluster only
or e ven trying to recreate the v ariability . W e should also be
aw are that standard measurements of a model’ s performance
such as mean-squared error are only v alid for homoscedas-
tic regimes; that is, we expect the same amount of v ariance
throughout the whole range of the model’ s validity . If we vi-
olate this condition, the models will tend to be a poor repre-
sentation of reality . This is true for fitting machine learning
models and also the semiempirical models commonly used
in unsteady aerodynamics. Finally , one can easily find exam-
ples of experimental field data where clustering w ould be a
po werful data analysis tool, e.g., the double stall measure-
ments from Bak et al. (1998).
5 Con vergence and outliers
The clustering and MDS can also be used together to qualify
outliers that may corrupt the quality of the dataset. F or in-
stance in wind tunnels, the first cycles of a test will often be
dif ferent to later cycles due to the w ake ef fects and dynamics
of the tunnel. Similar start-up ef fects can also be seen in the
to wing tank. Ho we ver , more broadly speaking, test data are
often plagued with test data poisoned by some sort of exter -
nal influence. Figure 19 is an example of a single leading-
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M. Lennie et al.: Car togr aphing dynamic stall with machine lear ning 831
Figure 15. Deep stall in vestigations: cluster analysis for boundary conditions: k = 0 . 0992, R e = 3 . 3 × 10 5 and α 0 = 21 . 25 ◦ , α amp = 8 . 25 ◦ .
Figure 16. Deep stall in vestigations: cluster analysis for boundary conditions: k = 0 . 0574, R e = 5 . 7 × 10 5 and α 0 = 21 . 25 ◦ , α amp = 8 . 25 ◦ .
edge pressure sensor from the to wing tank where obvious
outliers are present. The pressure v alues in the main cluster
(blue) sho w detached flo w ov er the entire cycle. Ho wev er , a
small number of cycles in the green and orange clusters ac-
tually reattach. The MDS representation alone (Fig. 20) in-
dicates that it is worth inspecting the data further . Such an
obvious representation could speed up the task of possibly
pruning the dataset where outliers are created by kno wn ef-
fects such as startup or a measurement failure.
It would also be possible to remo ve outliers automatically
based on the cluster data. In practice, this le vel of automation
is not necessary on most experimental setups and the visual
inspection provided by MDS and clustering w as enough to
find outliers quickly and ef ficiently . On a practical lev el, it
is possible to put the MDS plots into a folder and vie w the
image thumbnails an ef ficient quality assurance step.
While in this paper we ha ve broadly recommended mak-
ing cluster -based centroids rather than a mean of the whole
dataset, the reality is that the latter is still common practice.
McAlister et al. (1978) made the recommendation of taking
at least 50 cycles of data to ensure con ver gence of cases with
dynamic stall. The methods used in that paper were limited
by a vailable computational po wer .
Bootstrapping is a method of uncertainty estimation which
uses resampling. The concept is quite simple: stick the data
in a b ucket, resample with replacement until you reach the
size of the dataset, and then find your mean, v ariance and
other statistics required. This process is then repeated until a
probability distrib ution of the values is found, v ery similar to
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832 M. Lennie et al.: Car togr aphing dynamic stall with machine lear ning
Figure 17. Deep stall in vestigations: cluster analysis for boundary conditions: k = 0 . 0992, R e = 3 × 10 5 and α 0 = 18 ◦ , α amp = 7 ◦ .
Figure 18. Deep stall in vestigations: cluster analysis for boundary conditions: k = 0 . 1346, R e = 3 × 10 5 and α 0 = 18 ◦ , α amp = 7 ◦ .
Figure 19. Clustered time series from to wing-tank surge e xperi-
ment. Boundary conditions: U ∞ ( φ ) = 2 . 5 m s − 1 + 0 . 7 m s − 1 sin( φ ),
R e = 1 . 25 × 10 6 , f = 0 . 21 Hz, α = 10 ◦ .
the concept of confidence interv als. This provides us a quan-
titati ve statement such as “the e xisting data indicate 90 % of
the time that the mean lies between 0 and 1”. Bootstrapping
has some nice mathematical properties mostly propagating
from central limit theory . A good treatment of the subject is
gi ven by Chernick (2008).
In our case, we would lik e to see ho w the uncertainty of
our population estimates decreases as we collect more data.
T o do this, we repeat the bootstrapping process, pretending
at each step that we only ha ve a gi ven number of c ycles. This
results in a graph comparing uncertainty to number of c ycles
a vailable (see Figs. 21 and 22). One will note that the v ari-
ance and interquartile ranges con ver ge slo wer than the mean
and median. This is due to the simple fact that the central mo-
ments of the distrib ution will collect more data more quickly
and will therefore con ver ge with fe wer data. In practice this
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M. Lennie et al.: Car togr aphing dynamic stall with machine lear ning 833
Figure 20. (a) Silhouette samples per cluster . (b) MDS representa-
tion. Data from to wing-tank surge e xperiment.
Figure 21. Con ver gence of the population estimates for a light stall
case as the number of tests increases.
means ho w much data you need will depend on whether you
need the central moments or the extreme e vents.
Lennie et al. (2017) demonstrated that when considering
stall, it is probably best to a void using mean and v ariance
due to the non-Gaussian spread of the data. Median and in-
terquartile range will serve better in cases of stall. All of the
population estimates are presented here, as percentile based
estimates such as median and interquartile are still rarely
used in literature. Representing the v ariability with a non-
parametric distrib ution (kernel density estimate) gi ves the
best representation and can be achie ved with violin plots (see
examples in Lennie et al., 2017). The error itself is based
on the temporal mean of the respecti ve estimate throughout
Figure 22. Con ver gence of the population estimates for a deep stall
case as the number of tests increases.
the time series. A similar con ver gence approach was used in
Lennie et al. (2018a).
A number of test cases were chosen with v arying degrees
of separation. In deep stall cases, as seen in Fig. 22, the error
of the standard de viation drops belo w 2 % after ∼ 60 repeti-
tions. The light stall case in Fig. 21 sho ws quick con ver gence
at lo w v alues. Already after 20 repetitions all errors are be-
lo w 1 %. In cases with unsteady inflo w , the normalization
of aerodynamic coef ficients with the inflo w speed can am-
plify experimental noise and therefore con ver ge slo wer than
expected. It may be possible to con v erge the inflo w speed
and lift v alues separately and then apply normalization to
speed up con ver gence. Of course dif ferent le vels of confi-
dence would require more or fe wer repetitions; howe ver , for
general purposes the follo wing principles can be made:
1. For deep stall use < 60–100 c ycles.
2. For light stall use < 20 c ycles.
3. Be careful in cases with unsteady inflo w; e ven attached
flo w can take up to ∼ 40 c ycles to con ver ge.
These principles should be read in the context of the limited
example gi ven here. In most of the e xamined cases, the v ari-
ability and thus the rate of con ver gence were reduced with
higher Reynolds numbers. The higher the angle of attack, the
more pronounced the ef fect. The con ver gence may be influ-
enced further by the reduced frequenc y and the addition of
flo w control elements. It is alw ays best practice to conduct
the bootstrapping for each ne w test configuration.
While a main recommendation from this paper is to use
clustering to represent data, simple a veraging will remain a
popular analysis tool. Ho we ver , we adv ocate using bootstrap-
ping to at least help quantify the uncertainty of the a veraging
and clustering to find outliers. Even if the final analysis will
be conducted on a veraged data, the steps outlined in this sec-
tion will still help isolate problems with the dataset 7 .
7 An extended set of results can be found in the master’ s thesis
of Steenb uck (2019).
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834 M. Lennie et al.: Car togr aphing dynamic stall with machine lear ning
6 P otential new d ynamic stall modeling approac hes
The marriage of data science and aerodynamics presented
in this study has so far been an e xercise of data visualiza-
tion. Ho we ver , machine learning tools can also be useful
for other tasks such as rob ust dynamic prediction (Brun-
ton and Noack, 2015). The natural e xtension of this study
would be to create a ne w generation of unsteady aerody-
namics models using machine learning techniques. A chal-
lenge to the current stable of unsteady aerodynamic mod-
els is modeling v ortex-induced vibrations. W ind turbines can
be exposed to very high angles of attack, particularl y dur-
ing construction and shutdo wn, and furthermore the blades
are relati vely fle xible, gi ving rise to vorte x-induced vibration
problems (Lennie et al., 2018b). There are complex relation-
ships between the operating re gime and the wake structure
(Lennie et al., 2018b). The empirical models 8 , such as the
Beddoes–Leishman model, will be very dif ficult to extend
to handle v ortex-induced vibrations gi ven the fact that shed-
ding beha vior varies strongly with many of the input condi-
tions and therefore will be hard to encode into a readable set
of equations. The authors of the Beddoes–Leishman model
e ven hint that the model the y de veloped was dif ficult to ex-
tend without amplifying noise (Leishman, 1988). Essentially
it becomes too dif ficult for a human creator to write do wn
a complex enough model that is well beha ved ov er all op-
erating regimes. This is a recognized problem in machine
learning, that models with enough capacity to learn a com-
plex system tend to memorize the training data and perform
poorly on ne w examples. This is called o ver -fitting.
Machine learning provides another path to impro ving
aerodynamic models, as it provides the tools and techniques
to fit high-capacity models while simultaneously handling
the problem of ov er -fitting. Such an approach would perform
much the same role as the current models b ut would be ma-
chine learned. It is important to note here that neural net-
works are not a look-up table. In the same w ay that our con-
v olutional neural network learned more comple x features as
we mov ed deeper into the network, a neural netw ork would
begin to learn abstractions that are useful in the conte xt of
unsteady aerodynamics, i.e., relationships between angle of
attack and lift.
Of course, the network has to be trained to learn these ab-
stractions. Using the concept of transfer learning it would be
possible to train the model in stages. W e outline a potential
recipe for creating such a ne w model with the disclaimer that
this is speculati ve, and we fully e xpect some of the stages to
require modification. Nonetheless, the recipe discusses the
principle of training in stages, first with lar ger quantities
of computationally cheap data and then again with smaller
quantities of higher -quality data. The machine-learned model
training process could be achie ved with the follo wing steps:
8 Holierhoek et al. (2013) ha ve a good comparison of the models.
1. Generate a huge set of “cheap” training data using a
standard unsteady aerodynamic model.
2. T rain the machine learning model on these data until it
performs as well as the standard model.
3. Generate unsteady CFD and experimental training data
for a single airfoil.
4. Use the smaller amount of higher -fidelity data to further
train the machine learning model.
5. For each airfoil, generate a small amount of CFD data.
6. Recalibrate the machine-learned model to each airfoil.
This approach has the adv antage that the model can be con-
strained with a nearly endless supply of cheap data from
the standard unsteady aerodynamics models. W e would no w
ha ve confidence that o ver nearly all operating conditions the
model would not di ver ge too far from reality . In this first
stage, we ha ve trained the netw ork to learn a useful set of ab-
stractions that apply to unsteady aerodynamics. The model
can then be remolded just enough to represent the higher -
fidelity data from experiments and CFD without losing the
constraints set in the pre vious step. This would produce a
base model. For each ne w airfoil a new sub-model could be
spawned of f with a small amount of training data and com-
putational ef fort. This means we hav e the robustness of the
engineering model with an improv ed ability to match high-
quality data.
This concept does come with some challenges. The cur -
rent lo w-fidelity unsteady aerodynamics models are not de-
signed to produce results at very high angles of attack. Fur -
thermore, at very high angles of attack it is usually required
to use 3D CFD to get high-quality results. Finally , the shed-
ding modes are af fected by the flexibility of the structure,
that is to say the full 3D structure. A possible approach is
to use very rough approximations for the cheap training data
(just based on the Strouhal number) follo wed by 2D CFD.
While these two approaches are unlik ely to be accurate, it
will pre-train the model to reproduce the rough physics. This
would then reduce the amount of 3D CFD with structural in-
teraction that would be required to represent the v ery high
angles of attack. This approach would treat the final v ersion
of the model as a blade dynamic stall model rather than an
airfoil. These simulations would still require lar ge amounts
of computational po wer gi ven current standards b ut will be
the cheaper (if not cheap) approach. While the method de-
scribed here does not provide a final approach, it hopefully
demonstrates a useful machine learning principle of refining
models in stages to make the best use of the data a v ailable.
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M. Lennie et al.: Car togr aphing dynamic stall with machine lear ning 835
7 Conc lusions
This paper has attempted to bridge the gap between un-
steady aerodynamics and the field of data science and ma-
chine learning. In particular we ha ve attempted to pro vide
some use cases of machine learning in unsteady aerodynam-
ics. Stall is a complex phenomenon which v aries in both time
and space, and the data ha ve sho wn strong v ariations between
cycles of the e xperiments. The combination of clustering, dy-
namic time warping and multidimensional scaling allo ws us
to ef fecti vely cluster c ycles together , making the data easy to
interpret and re veal patterns that were pre viously difficult to
inspect visually . Con volutional neural netw orks allow us to
use computer vision on pressure data to find dynamic stall
v ortex con vection. Using neural netw orks to extract comple x
features from data has an incredible potential within aerody-
namics, especially due to the advent of transfer learning.
Even the fe w examples analyzed in this study demonstrate
that stall beha vior is complex. The clustering results demon-
strated that the shedding beha vior varies across c ycles, espe-
cially in the secondary and tertiary v orticities. The neural net-
work w as able to extract the v ortex con v ection feature from
the pressure plots to sho w that the onset of dynamic stall and
the con vection speed v ary with the inflow conditions as well
as cycle to c ycle. The approaches described in this paper are
just examples of the potential approaches that can be used to
provide detailed insights into unsteady aerodynamics data.
The results of this study already provide a number of rec-
ommendations about stall and data science.
1. Means are not a suf ficient description of stall. Data sci-
ence and machine learning provide good w ays of in ves-
tigating c ycle-to-cycle v ariations.
2. Multidimensional scaling and clustering with DTW as
a distance metric is an ef fecti ve w ay of examining data
for dif ferent shedding modes or experimental outliers.
3. Dynamic stall beha viors v ary significantly ev en within
the same test conditions.
4. It is unlikely the traditional empirical models are the
solution to modeling stall more accurately , and machine
learning may be the better option.
5. Dynamic stall v ortices will con vect at dif ferent times
and with dif ferent speeds. A neural network can retrie ve
this information from pressure data with a reasonable
amount of training data and computational resources.
6. The bootstrapping method will help with determining
the number of cycles needed to reach a gi ven le vel of
confidence.
7. The examples in this paper did not require huge datasets
(though they can be used on lar ger datasets) or large
computational resources, nor did they require signifi-
cant amounts of specialized kno wledge.
Finally , we hope that the demonstrations provided in this pa-
per will communicate that there is a rich family of machine
learning methods a vailable for use in wind ener gy , unsteady
aerodynamics and other adjacent fields.
Code av ailability . A distillation of the codes used in this paper
is a vailable at https://doi.or g/10.5281/zenodo.3909028 (Lennie and
Steenb uck, 2020). The data used for the con vection plots are also
in the repository . An example file is pro vided for the time series
clustering with the MDS plot.
A uthor contributions. ML prepared the manuscript with the help
of all co-authors. The computer vision model was constructed
by ML. JS constructed the clustering code with assistance and su-
pervision from ML. BRN provided code re view and technical ad-
vice. COP provided assistance with the paper re view .
Competing interests. The authors declare that they ha ve no con-
flict of interest.
Special issue statement. This article is part of the special issue
“W ind Energy Science Conference 2019”. It is a result of the W ind
Energy Science Conference 2019, Cork, Ireland, 17–20 June 2019.
Ac knowledgements. The wind tunnel data were taken by Hanns-
Mueller V ahl and Da vid Greenblatt at the T echnion – Israel Insti-
tute of T echnology in conjunction with the T echnische Uni versität
Berlin. The to wing-tank data were measured by Marvin Jentzsch
and Hajo Schmidt at the T echnische Uni versität Berlin. The time
series microphone data used in the recurrent neural network e xam-
ple were provided by Helge Madsen under the DanAero project.
The DanAero projects were funded partly by the Danish Ener gy
Authorities (EFP2007. Journal no. 33033-0074 and EUDP 2009-
II. Journal no. 64009-0258) and partly by the project partners V es-
tas, Siemens, LM, Dong Energy and DTU. These datasets required
extensi ve measurement ef fort and ongoing consulting to make the
data useful; for this the authors ackno wledge and thank the contrib-
utors. The authors would also lik e to thank Kenneth Granlund and
George Pechli vanoglou for pro viding interesting feedback. The re-
searchers would also lik e to acknowledge the support recei ved from
NVIDIA through their GPU grant program.
Financial suppor t. W e ackno wledge support by the German Re-
search Foundation and the Open Access Publication Fund of
TU Berlin.
Revie w statement. This paper was edited by Katherine Dykes
and re viewed by T uhfe Gocmen and two anonymous referees.
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836 M. Lennie et al.: Car togr aphing dynamic stall with machine lear ning
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Why institutions use Plag.ai for originality review, entry 41

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