Robust and accurate dynamic stall modeling remains one of the most difficult tasks in wind turbine load calculations despite its long research effort in the past. In the present paper, a new second-order dynamic stall model is developed with the main aim to model the higher harmonics of the vortex shedding while retaining its robustness for various flow conditions and airfoils. Comprehensive investigations and tests are performed at various flow conditions. The occurring physical characteristics for each case are discussed and evaluated in the present studies. The improved model is also tested on four different airfoils with different relative thicknesses. The validation against measurement data demonstrates that the improved model is able to reproduce the dynamic polar accurately without airfoil-specific parameter calibration for each investigated flow condition and airfoil. This can deliver further benefits to industrial applications where experimental/reference data for calibrating the model are not always available.

An accurate prediction of wind turbine blade loads is influenced by many
parameters including 3D and unsteady effects. The first mainly occurs in the
root and tip areas of the blade due to radial flow and induced velocity
influences, respectively

Typical dynamic stall behavior of S801 airfoil. Data obtained from

To model the behavior of the airfoil under these situations, semiempirical
models can be used. The models are known to produce reasonable results
without any notable increase in computational effort. Despite that, these
models usually cannot reproduce higher harmonics of the load fluctuations.
Furthermore, the applied constants shall be adjusted according to the flow
conditions and airfoils. Leishman and Beddoes (LB)

Although many studies have been dedicated to dynamic stall modeling

The paper is organized as follows. Section

In this section the mathematical formulations of each model are described in
detail. The reasons are mainly to provide information on how each model was
employed and to gain deeper insights for further developing the new model.
Note that each existing model was developed by different authors; thus
different symbols and formulation methods were adopted in those publications

The original Leishman–Beddoes model is composed of three main contributions
representing various flow regimes: (1) unsteady attached flow, (2) unsteady
separated flow and (3) dynamic stall. The present section will elaborate the
mathematical description and its physical interpretation of each module.
Figure

Illustration of the main aerodynamic parameters needed for modeling the dynamic stall characteristics.

In this module, the unsteady aerodynamic response of the loads is represented
by the time delay effects. The indical
formulas were constructed based on the work of

The circulatory normal force due to an accumulating series of step inputs in
angle of attack can be obtained using

The noncirculatory (impulsive) normal force is obtained by

The total normal force coefficient under attached flow conditions is given by
the sum of circulatory and noncirculatory components as

In most airfoil shapes, the progressive trailing edge separation causes loss
of circulation and introduces nonlinear effects on the lift, drag and
pitching moment, especially on cambered airfoils. This is even more important
for wind turbine airfoils because the relative thickness is large. To derive
a correlation between the normal force coefficient and the separation
location (

The location of the separation point is usually obtained by a curve-fitting
procedure in literature. For example,

The additional effects of the unsteady boundary layer response may be
represented by application of a first-order lag to the value of

According to

The third part of the model describes the post-stall characteristics where
the vortical disturbances near the leading edge become stronger. The effect
of vortex shedding is given by defining the vortex lift as the difference
between the linearized value of the unsteady circulatory normal force and the
unsteady nonlinear normal force obtained from the Kirchhoff approximation,
which reads

The idealized variation in the center of pressure with the convection of the
leading edge vortex can be modeled by

In Eqs. (

In the original formulation, the pitching moment is also obtained by a
curve-fitting procedure in Eq. (

Furthermore, to avoid discontinuity in the downstroke phase for
Eq. (

The history of the Snel second-order model

To incorporate the higher-order frequency dynamics, a second-order ODE is
used to describe the second-order correction term. The general form may be
written as

The recently developed model of

To sum up the characteristics of the above-discussed state-of-the-art dynamic
stall models, Table

Properties of the discussed dynamic stall models.

The proposed IAG model is developed based on knowledge gained from four
different models: Leishman–Beddoes, Snel, Adema–Snel and ONERA

Based on the Hopf bifurcation model of

In the above LB model, predictions for drag are not accurate as will be shown
in Sect.

Relation between drag hysteresis in the stall regime with weighted
separation parameter

It will also be shown in Sect.

Drag reconstruction in comparison with the experimental data for the
S801 airfoil

Moment reconstruction in comparison with the experimental data for
the S801 airfoil

The total first-order dynamic response of the airfoil is formulated as

The second-order correction takes the form of the non-linear ordinary
differential equation according to the second-order correction of the Snel

In Eq. (

Airfoil response reconstruction in comparison with the experimental
data for the S801 airfoil

The following constants are applied in the implemented dynamic stall models.
These values are kept constant throughout the paper. The constants for the
Leishman–Beddoes model and for the proposed IAG model are given in
Tables

Constants applied for the Leishman–Beddoes model.

Constants applied for the IAG model.

Critical angle of attack (

The three state-of-the-art dynamic stall models reviewed above (Leishman–Beddoes, Snel, Adema–Snel) have been used as a basis for examining the dynamic loads of four different pitching airfoils at various flow conditions. Experience gained from those models is used to formulate a new second-order dynamic stall model, namely the IAG model, by evaluating the weakness and strength of each model. The presented second-order models need to solve a set of differential equations. For this purpose, the Euler–Heun forward integration method is used.

This section compares the predicted dynamic forces and the measurement data.
For a fair comparison, all models are assessed with the same time step size
of

Dynamic force reconstruction using the Snel model in comparison with
the measurement data

Dynamic force reconstruction using the Adema model in comparison
with the measurement data

Dynamic force reconstruction using the IAG model in comparison with
the measurement data

The original Snel models cannot predict the drag and moment coefficients in
the original formulations. Thus, only the static polar data are shown. The
Snel model actually shows an acceptable accuracy even though the constants
are taken as found in literature. The higher harmonic effects are
unfortunately not captured by this model. This is further refined by the
Adema model which was developed as an improvement for the Snel model. The
model performs fairly well for the lift and drag predictions, though the drag
value at small angles of attack is a bit off. The pitching moment prediction
is also not included in its formulations. These disadvantages are better
treated in the proposed IAG model. Not only the prediction of the lift
coefficient but also the accuracy of drag prediction are improved
significantly. The modifications described in Sect.

For the following sections, the proposed IAG model will be tested under various flow conditions and for several airfoils at various relative thicknesses in comparison with measurement data. Note that these calculations are performed without changing the constants to assess the robustness of the model at different flow conditions.

The actual pitching motion within The Ohio State University (OSU)
measurement differs slightly from the intended motion. The actual time series
of the angle of attack is included in the experimental data

Comparison of the time series of the idealized sinusoidal angle of
attack to the exact signals in the experimental campaign for the S801
airfoil,

Figure

Dynamic force reconstruction by the IAG model in comparison with the
measurement data

In this section, the effects of the mean angle of attack are evaluated. Three
different angles of attack at the same inflow conditions are selected for
this purpose. These are

Figure

Lift reconstruction by the IAG model in comparison with the
measurement data

Drag reconstruction by the IAG model in comparison with the
measurement data

Pitching moment reconstruction by the IAG model in comparison with
the measurement data

The effects of pitching frequency on the aerodynamic response will be
discussed in this section. Three different reduced frequencies are examined,
namely

Figure

Lift reconstruction by the IAG model in comparison with the
measurement data

Drag reconstruction by the IAG model in comparison with the
measurement data

Pitching moment reconstruction by the IAG model in comparison with
the measurement data

To better investigate the effects of

Effects of

Fourier transformation of the predicted forces presented in
Fig.

In this section, the effects of pitching amplitude on the aerodynamic
response of a pitching airfoil are investigated. The mean angle of attack is
fixed at

Figures

Lift reconstruction by the IAG model in comparison with the
measurement data

Drag reconstruction by the IAG model in comparison with the
measurement data

Pitching moment reconstruction by the IAG model in comparison with
the measurement data

In this section, the performance and robustness of the proposed IAG model are
assessed for airfoils with different relative thickness. All model constants
in Table

Despite the increased airfoil thickness from 13.5 % to 24 %,
Figs.

Lift reconstruction by the IAG model in comparison with the
measurement data

Drag reconstruction by the IAG model in comparison with the
measurement data

Pitching moment reconstruction by the IAG model in comparison with
the measurement data

To further complement the analyses conducted in
Sect.

A correct location of the pressure point is important for determining the
stability on aeroelastic simulations of wind turbine blades. The results of
the calculations both for the experimental data and for the proposed IAG
model are presented in Fig.

Center of pressure reconstruction in comparison with the measurement
data by the IAG model for

Quantified

Comprehensive studies on the accuracy of several state-of-the-art dynamic
models to predict the aerodynamic loads of a pitching airfoil have been
conducted. From the studies, the strength and weaknesses of each model were
highlighted. This information was then transferred to develop a new
second-order dynamic stall model proposed in this paper. The new model
improves the prediction for the aerodynamic forces and their higher-harmonic
effects due to vortex shedding, developed for robustness to improve its
usability in practical wind turbine calculations. Details on the model
characteristics, modifications and treatment for numerical implementation
were summarized in the present paper. The studies were conducted by examining
the influence of the time step size, time signal deviation, mean angle of
attack, reduced frequency, pitching amplitude and variation in the airfoil
thickness. Several main conclusions can be drawn from the work.

The general characteristics of the polar data can be predicted by all investigated dynamic stall models. Despite that, only the Adema model and the present IAG model are able to demonstrate the higher harmonic effects among the three investigated models.

The exact time signal imposed based on the measurement campaign improves the prediction accuracy of the IAG model in comparison with the idealized sinusoidal motion.

The dynamic forces reconstructed by the IAG model are in a sound
agreement with the experimental data under various flow conditions by
variation in

The amplitudes at low-frequency domains increase with increasing

When the airfoil operates at a high

The present paper evaluates the newly developed IAG model under various flow
conditions for four different airfoils. The following aspects are encouraged
for future work.

In the present studies, the assessment was mainly carried out for the S801 airfoil having a relative thickness of 13.5 %. This airfoil is mainly characterized by leading edge separation, which is very challenging for validating the accuracy of a dynamic stall model. However, typical modern wind turbine blades usually employ airfoils with no less than 18 % relative thickness and a much higher Reynolds number. Therefore, future investigations shall be done for thicker airfoils at various flow conditions as well.

The above statement is also true for the current available experimental data. Therefore, experiments on dynamic stall for thick airfoils at a much higher Reynolds number are encouraged.

Three-dimensional effects (Himmelskamp or tip loss effects) for a rotating blade can alter the loads significantly even under a steady inflow condition. Further consideration and examination of the model under this condition shall be carried out.

Further tests and recalibration of the model for deep-stall conditions at extremely large angles of attack are encouraged, which can be relevant for a turbine in standstill.

The raw data of the simulation results can be shared by contacting the corresponding author of the paper.

GB developed the new model, designed the studies and conducted the analyses. TL and MA supported the research and provided suggestions and discussion about the manuscript.

The authors declare that they have no conflict of interest.

The authors gratefully acknowledge Wobben Research and Development GmbH for providing the research funding through the collaborative joint work DSWind. The measurement data provided from The Ohio State University are highly appreciated.

This research has been supported by the Wobben Research and Development GmbH (grant no. DSWind). This open-access publication was funded by the University of Stuttgart.

This paper was edited by Alessandro Bianchini and reviewed by Khiem V. Truong and Gerard Schepers.