Dynamic stall phenomena carry the risk of negative damping and instability in wind turbine blades. It is crucial to model these phenomena accurately to reduce inaccuracies in predicting design driving (fatigue and extreme) loads. Some of the inaccuracies in current dynamic stall models may be due to the fact that they are not properly designed for high angles of attack and that they do not specifically describe vortex shedding behaviour. The Snel second-order dynamic stall model attempts to explicitly model unsteady vortex shedding. This model could therefore be a valuable addition to a turbine design software such as Bladed. In this paper the model has been validated with oscillating aerofoil experiments, and improvements have been proposed for reducing inaccuracies. The proposed changes led to an overall reduction in error between the model and experimental data. Furthermore the vibration frequency prediction improved significantly. The improved model has been implemented in Bladed and tested against small-scale turbine experiments at parked conditions. At high angles of attack the model looks promising for reducing mismatches between predicted and measured (fatigue and extreme) loading, leading to possible lower safety factors for design and more cost-efficient designs for future wind turbines.

Wind turbines operate in highly unsteady aerodynamic environments (Leishman,
2002). For design and certification, design load cases (DLCs) have been set
which describe the conditions that wind turbine designs have to withstand
(DNV GL, 2016). Some of the design driving DLCs are those for parked and
idling conditions where wind turbine blades will experience high angles of
attack (AoAs), leading to (dynamic) stall behaviour (Schreck et al., 2000).
The wind turbine yaw angle is defined as the angle in the horizontal plane
between the free-stream wind direction and the wind turbine rotor shaft. It
can be noted that when the turbine is parked, and the blades are pitched to
90

Dynamic stall is a phenomenon leading to larger variations in lift, drag, and pitching moments on the aerofoil than would be observed during steady operation (Choudry et al., 2014). This then creates larger aerodynamic forces on the blades than expected during steady conditions (Leishman, 2002). Dynamic stall happens with dynamic variation in the inflow and/or the effective angle of attack and can be viewed as a delay in the onset of stall. Recirculation of flow after the static stall angle starts near the trailing edge and rapidly moves towards the leading edge, leading to the formation of a large dynamic stall clockwise vortex at the leading edge at increasing angles of attack. The dynamic stall vortex will travel along the suction side, leading towards the trailing edge before detaching completely. Full separation will occur when the dynamic stall vortex is completely detached. This moment is called the “break” or “dynamic stall onset”. As a result, low lift remains until reattachment of the flow. However, a time delay for reattachment of the flow is present as well. After reattachment the process repeats, creating a hysteresis loop. Dynamic stall phenomena carry the risk of negative damping and instability, especially if the aerofoil is oscillating in and out of stall (McCroskey, 1981). A visual description for dynamic stall is presented in Fig. 1.

Classical visual representation of dynamic stall (Leishman, 2002).

When keeping the aerofoil pitched in (deep) stall for longer periods of time,
periodic vortex shedding will occur. A single large dynamic stall
vortex will no longer be shed, but rather multiple periodic vortices from
both the leading and
trailing edge will be shed. This will induce time-varying loads on the blades
(Riziotis
et al., 2010). The periodic vortex shedding is characterized by the Strouhal
number representing the dimensionless frequency of shedding (Pellegrino and
Meskell, 2013). The Strouhal number is defined following Eq. (1):

This paper will have the following outline.

The Snel model is validated against experimental data.

Proposed changes are presented based on the validation results to improve the model predictions. These include a dimensional analysis, calculation of the slope for the potential lift coefficient, application of the normal force coefficient, an investigation into the downstroke and vortex shedding predictions of the model, and finally an optimization for empirical constants.

Attention will be paid to vibration prediction as this influences turbine fatigue loading, which has a large impact on the design of the wind turbine.

An absolute error analysis is carried out before and after the improvements to quantify the increase in performance.

The improved model will be tested with actual small-scale turbine experimental data to assess performance in combination with Bladed.

This section will validate the Snel model and propose adaptations to the
model to improve the performance. The description of the model is based on
the description in both Snel (1997) and Hollierhoek et al. (2013).
Snel (1997) derived a dynamic stall model based on the work of Truong (1993),
who proposed that the dynamic lift coefficient can be distinguished into two
parts, namely

In the original model of Truong the first part is based on the
Beddoes–Leishman dynamic stall model. The Snel model uses the SIMPLE model
from
Montgomerie (1996) as a departure point for the first-order correction
while Truong (1993) uses the Beddoes–Leishman (B–L) model to
calculate

The Snel model as described in the section above is implemented in a MATLAB
environment. The numerical implementation follows the steps described in
De Vaal (2009). The time step used is 0.001 s to capture higher-frequency
events from the Ohio State University experiments. The Ohio State
University (OSU) experiments (Hoffmann et al., 1996) will be used for
validation of the initial implementation of the Snel model. In the OSU
experiments an extensive set of aerofoils have been tested for both unsteady
and steady data. The measurements recorded responses to forced sinusoidal
pitching oscillations. Different amplitudes, mean angles of attack,
oscillation frequencies, and Reynolds numbers were tested. The focus in this
paper will be on the NACA4415 and S809 aerofoils. The model parameters
obtained from the OSU database, as displayed in Table 1, will be analysed.
The oscillation frequencies of the cases are set such that the forcing angle
of attack matches the OSU experiments. Figures 2 and 3 show the time series
of the lift coefficient for cases with both low and high reduced frequencies
and low and high mean angles of attack. The following observations can been
made for the current model:

The current model overpredicts the loss of lift during the downstroke.

The predicted vortex shedding does not always happen at the correct time.

There is currently no unsteady vortex shedding at higher angles of attack.

The shedding frequency is not dependent on the reduced frequency while the experiments do show a dependency.

Selected cases from the OSU experiments.

Lift coefficient time of the initial Snel model (NACA 4415 aerofoil
with mean angle of attack of 14

Lift coefficient time of the initial Snel model (NACA 4415 aerofoil
with mean angle of attack of 18.4

To quantify the accuracy of the Snel model, the total absolute error between
model predictions and experimental data is calculated. This will give an
objective measure to assess proposed improvements. The Snel model is
interpolated along the angle of attack to obtain the dynamic lift coefficient
output at the precise angles of attack of the OSU experiment. The errors are
then evaluated according to

This section will outline proposed modifications to the Snel model. The
following areas will be investigated for modification:

a dimensional analysis of the model,

the calculation method of the slope for the potential lift coefficient,

application of the normal force coefficient instead of the lift force coefficient,

the downstroke prediction of the model,

prediction of vortex shedding,

and lastly an optimization for empirical (aerofoil-specific) constants.

It can be seen in the formulation of the model that Eqs. (8) and (10) are
cast in a dimensional form. For different values for chords, wind speed, and
pitching frequency, the current model will not produce identical results. In
order to make them dimensionless, the time constant used in dynamic stall
(Eq. 11) is added. The new constants will be such that the initial value of
the constant is kept. The equations will now be

Steady polars of the NACA 4415 aerofoil.

The Snel model uses 2

The Snel model uses the lift coefficient of the steady aerofoil data.
However, the lift coefficient tends to zero when angles of attack reach
90

The consistent differences between the implementation of the Snel
second-order model and earlier implementations are lower values in the
downstroke.
Figure 5 shows the first- and second-order part of the model as a function of
time. The second-order part (

Analysis of the dynamic lift coefficient.

Equation (10) uses

Constants used for optimization.

The shedding frequency will depend on the angle of attack. The current model
does not predict a dependency and so it has to be improved in this aspect.
The Strouhal number uses the chord or the projected chord perpendicular to
the incoming flow. Because the projected chord length is driven by the angle
of attack, it is proposed to add the projected chord length to the “spring”
term (cf

Absolute error analysis of the optimization runs for all cases from Table 1.

Lift coefficient over time of the improved Snel model (NACA 4415
aerofoil with mean angle of attack of 14

An unconstrained minimization algorithm in MATLAB is used to optimize
empirical constants. The algorithm searches for the lowest summation of the
absolute errors, from Eq. (13), of all cases considered. The constants
selected for this analysis are shown in the initial row of Table 2. The
constants are selected since they are included in equations affected by the
modifications
from this section and also because they have a high influence on the output
of the Snel model. Three optimization analyses have been carried out: a
global optimization which covers all cases, an optimization which focussed
only on the cases with a mean angle of attack of 14 or 20

C1 from Table 2, which will be called the first-order coefficient and

C2 from Table 2, which will be called the second-order forcing coefficient.

Lift coefficient over time of the improved Snel model (NACA 4415
aerofoil with mean angle of attack of 18.4

Lift coefficient over time of the improved Snel model (S809 aerofoil
with mean angle of attack of 18.8

The modified Snel model is implemented as described in Sect. 2.1 and tested against the same set of experiments as in Table 1. The results of the modified Snel model are shown in Figs. 7 and 8. In comparison to Figs. 2 and 3 it is noted that the shedding prediction at low reduced frequency is improved. For the higher reduced frequency the model also captures the shedding slightly better, even though there is less shedding present. Furthermore, it can be seen that the frequency changes between both cases as desired. It is important to investigate the impact changes on different situations and aerofoils. Figure 9 displays the updated model in combination with the S809 aerofoil. The improved model still predicts slightly lower values than the experiments but the overall trends are followed nicely, and shedding frequency matches the experiment well.

Power spectral density of experimental data and the improved Snel
model (NACA 4415 aerofoil mean angle of attack of 14

Power spectral density of experimental data and the improved Snel
model (S809 aerofoil with mean angle of attack of 18.8

Another goal of the proposed changes to the Snel model was to capture, predict, and match the vortex shedding and the shedding frequency of aerofoils in dynamic stall conditions. In order to check the validity of these changes in a quantitative way, a frequency domain analysis has been performed. The power spectral density (PSD) estimate is calculated using Welch's method. The Hamming window is set equal to the number of data points and the number of overlapped values to 50 % of the window length. The forcing frequency is removed from the plot as the shedding frequencies are higher and the forcing frequency will take up a large proportion of the PSD. The results for both aerofoils are shown in Figs. 10 and 11. From the figures it becomes clear that the Snel model captures, for both aerofoils at different mean angles of attack and different reduced frequencies, the self-induced shedding frequencies fairly correct. However, as shown in Figs. 7–9, there is still room for improvement here. All predicted shedding frequencies match frequencies observed in the measurements, whereas the intensity is not always correct. Care must be taken with the higher frequencies as the OSU measurement has a relatively low sampling frequency and might therefore not fully capture some higher-frequency dynamics.

Comparison of the total absolute error of both the initial and improved Snel models.

Normal force distribution of the blade at 0

Power spectral density at 82 % span of the blade at 0

Normal force distribution of the blade at 120

To quantify the improvements, the effects of all previous changes on the overall absolute error of the new model are shown in Fig. 12 together with the overall error of the initial model of Sect. 3. It is seen that the improved model outperforms the initial model in almost every case with a single exception. The initial model already gave very accurate results for that case, and the increase in error is very small compared to the reduction achieved in all other cases. It is also noted that the overall prediction of the shedding phenomena has been improved. Hence it can be concluded that the model changes developed and presented in this paper improve the performance of the Snel second-order model.

Model Experiments In Controlled Conditions (MEXICO) was a project
in which an instrumented, three-bladed turbine of 4.5 m rotor diameter was
tested (Schepers et al., 2012). MEXICO was carried out in the Large Low-Speed
Facility (LLF) of the German-Dutch Wind Tunnels (DNW) (Schepers et al.,
2012). The blades were fitted with pressure sensors at 25 %, 35 %,
60 %, 82 %, and 92 % radial positions. Tests were performed with
30 m s

The updated Snel model shows significantly higher-frequency vibrations in the
normal force of the New MEXICO blade. Figure 15 shows the normal force
distribution for the blade at azimuth 120

Investigated cases from the New MEXICO experiments.

A single reason for the higher-frequency prediction has not been found.
Several possibilities are suggested. First, the modal damping is not
specified in the New MEXICO turbine data and was therefore assumed to be
0.5 %. Second, the impact of different aerofoils from the New MEXICO blade
is unknown. The first-order or second-order forcing coefficients might
require other values. Furthermore, the Strouhal number for the New MEXICO
blade is not 0.2 Hz at high angles of attack (Khan, 2018). Research by
Skrypinski et al. (2014) and Zou et al. (2015) shows that for aerofoils with
an effective angle of attack of around 90

The Snel model has been validated with OSU experimental data, and following this validation propositions for improvements have been made. The improvements to the model have been tested and led to a reduction in the overall error between the Snel model and the OSU experimental data. Furthermore, an improvement in the prediction of both the amplitude and frequency of vibrations in the measurements has been accomplished. The improved model is implemented in the Bladed turbine design software and tested against the New MEXICO experimental data. Prediction of normal force distributions along the blade seems to match earlier implementations in other turbine design codes while the mean value of the normal force is not correct. This is a major area for further research and improvement. The Snel model predicts the amplitude of the normal force vibrations well while the predicted frequency is higher than in the experiment. A single reason for this has not been found, and therefore further research into the Snel model is advised. Truong (2017) proposed a similar modification of his original model as Snel (1997) and therefore also similar to the model adaptations presented in this paper. The main difference lies in the incorporation of the steady lift data as described in Sect. 2. The authors highly recommend a study comparing both the model described in Truong (2017) and the model presented in this paper. The model as proposed is formulated on the basis of variations in angles of attack. It is recommended for further research to delve into the possibility to adept this model to dynamic variation in inflow velocity for the case of rotating and vibrating blades.

The proposed Snel second-order dynamic stall model might become a valuable addition to the modelling of dynamic behaviour in stall conditions. As the conditions tested in this paper are often design driving, the Snel model looks promising for more accurate prediction of design-driving (fatigue and extreme) loads and more cost-efficient wind turbine designs.

The paper uses the publicly available OSU oscillating aerofoil experiment data. The implementation of the Snel model in this paper (and the final improved model) in the MATLAB environment is available and can be requested from the corresponding author.

GS has been the thesis supervisor within the MSc programme. He has been involved in writing (reviewing and editing) of the final thesis and the corresponding paper. MK was thesis supervisor from DNV GL and contributed to the implementation of the model in the MATLAB environment. Furthermore, MK has been a part of the investigation, validation, and improvement of the model as well as implementation of the model in the Bladed turbine design software. Finally, he has reviewed and edited the final thesis and corresponding paper.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Wind Energy Science Conference 2019”. It is a result of the Wind Energy Science Conference 2019, Cork, Ireland, 17–20 June 2019.

The author of this paper would like to express his appreciation to Menno Kloosterman and Gerard Schepers for their valuable comments, suggestions, and discussions during the research. Without them this paper would not be possible. The author would also like to thank DNV GL for providing the opportunity to write a MSc thesis (from which this paper is a result) within the company. Lastly, the Association of European Renewable Energy Research Centres, the Hanze University of Applied Sciences, and the National Technical University of Athens are thanked for providing the education leading to this paper.

This paper was edited by Gerard J. W. van Bussel and reviewed by Vasilis A. Riziotis and Xabier Munduate.