Delegates are invited to meet and discuss with the poster presenters in this topic directly after the session 'Aerodynamics and rotor design' taking place on Wednesday, 12 March 2014 at 09:00-10:30. The meet-the-authors will take place in the poster area.

**Witold SkrzypiÅ„ski**Technical University of Denmark, Denmark

Co-authors:

Witold SkrzypiÅ„ski (2)

^{F}

^{P}Frederik Zahle (2) Christian Bak (2)

(1) DTU, Roskilde, Denmark (2) Technical University of Denmark, Roskilde, Denmark

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## Abstract

**Parametric approximation of airfoil aerodynamic coefficients at high angles of attack
****Introduction**

Authors dealing with blade stability at standstill [1,2] indicate that the aerodynamic damping of blades at high angles of attack (AOAs) is dependent on their aerodynamic characteristics. Often those characteristics are assumed to be as of a flat plate due to the fact that at high AOAs, airfoils resemble flat plates as the flow already separates at the leading edge. According to [1,2], a more accurate representation of airfoil characteristics at high AOAs may be necessary to tackle the problem of aerodynamic stability at standstill. This paper presents an attempt to approximate relevant airfoil characteristics with an engineering model.

**Approach**

At high AOAs, it is often assumed that the aerodynamic response of the airfoil is identical to that of a flat plate. Alternative approaches are presented by Apostolyuk [3] for the estimation of the aerodynamic coefficients of modern jet fighter planes, and Viterna and Corrigan [4], developed with the scope on stall-regulated turbines.

In the present work, the aforementioned methods were verified against the 360Â° wind tunnel measurements of the DU96-W-180 airfoil carried out in the wind tunnel of the Delft University of Technology by Timer and van Roij [5,6]. The most optimal method was chosen and modified in order to produce a satisfactory approximation of airfoil aerodynamic coefficients in the whole AOA range with the lowest possible number of data points necessary to tune the model. Also, a method for the approximation of the moment coefficient (Cm) similar to those proposed by Apostolyuk was found. Additionally, polynomial approximation was used in those AOA ranges which could not be effectively approximated by the aforementioned methods.

Note that obtaining the aerodynamic characteristics in the AOA range approximately between -30Â° and 30Â° is typically easy as this data may be computed with computationally inexpensive, two-dimensional (2D) Computational Fluid Dynamics (CFD) or is often readily available from wind tunnel measurements. Therefore, this AOA range is outside the scope of present work. On the other hand, accurate prediction of airfoil aerodynamic characteristics at high AOAs by means of 2D CFD or wind tunnel measurements proves very difficult. CFD computations capable of resolving the physics of flows at high AOAs are three-dimensional (3D) and computationally expensive. This increases the need for an engineering model capable of delivering reliable airfoil characteristics at high AOAs by using a relatively low number of reference data points, obtained by 3D CFD or cautious wind tunnel measurements.

**Main body of abstract**

The level of accuracy of the three methods mentioned in the preceding section is visualized in Figure 1 and Figure 2 by comparing the Cl and Cd curves of the DU96-W-180 airfoil approximated in the whole AOA range with the coefficients measured by Timmer and van Rooij [5,6].

Figure 1: Three different methods for predicting Cl in deep stall are compared with the reference measurement data.

Figure 2: Three different methods for predicting Cd in the whole AOA range are compared with the reference measurement data.

The method by Viterna and Corrigan [4] estimated the lift coefficient (Cl) curve well only the in the AOA range between stall and 120Â°. The two other methods worked well as low fidelity approximations. The methods seem incapable of representing different maximum Cl values in the negative and positive AOA regions. Also, they do not seem to capture the peculiarities of the curve around 180Â°.

All three approximations for the drag coefficient (Cd) curve presented in Figure 2 produced effectively the same result, working well as low fidelity approximations but not predicting the different maxima in the positive and negative AOA regions.

The equations proposed by Apostolyuk were chosen for further investigation, with the even sine approximation for Cl, and even cosine approximation for Cd. The most effective approach to capture the different maxima in the positive and negative AOA regions of both Cl and Cd curves was to find independent approximations for the curves in both regions, and blend them with the known characteristics in the AOA region between -30Â° and 30Â°. The results of such approximations for Cl and Cd are presented in Figure 3 and Figure 4, respectively.

Figure 3: Even sine approximation of Cl, blended with the reference data for AOAs between -30Â° and 30Â°; â€˜Ref Dataâ€™ shows the points used for tuning the model

Figure 4: Even cosine approximation of Cd, blended with the reference data for AOAs between -30Â° and 30Â°; â€˜Ref Dataâ€™ shows the points used for tuning the model

An accurate estimation of the curves was produced in AOAs between -160Â° and 160Â° for Cl, and in the whole AOA range for Cd. The reference data was used directly in the AOA region between -30Â° and 30Â°.

In order to extend the accurately estimated region of the Cl curve from -180Â° to 180Â° AOA, polynomial approximation was applied at the outer parts of the curve and blended with the harmonic approximations at AOAs equal to -160Â° and 160Â° which is presented in Figure 5.

Figure 5: Even sine approximation of Cl, blended with the reference data in the AOA region between -30Â° and 30Â°, and with polynomial approximation above 160Â° AOA and below -160Â° AOA; â€˜Ref Dataâ€™ shows the points used for tuning the models

The combination of the harmonic and polynomial approximations resulted in the Cl curve being accurately approximated in the whole AOA range. As future work, ability of 2D CFD to compute the coefficients in the vicinity of 180Â°, as an alternative to using the polynomial approximation, will be analyzed.

The estimation of the Cl and Cd curves differed from the estimation of the Cm curve as the Cm curve was estimated by a total of four even cosine harmonic functions as well as by the polynomial approximation â€“ two even cosine functions in the positive AOA region and two in the negative. The following AOA regions were approximated with independent even cosine functions: [-163Â° : -135Â°], [-135Â° : -30Â°], and symmetrically in the positive AOA region. Polynomial approximations were used below -163Â° AOA and above 163Â° AOA. The results are presented in Figure 6.

Note that Cm is not approximated in the original work of Apostolyuk [3] while its approximation is relevant in the context of wind turbine aerodynamic stability. The present method produced satisfactory results for the Cl, Cd and Cm curves in the whole AOA range by using nine independent data points to find the modelsâ€™ coefficients, that would need to be computed using 3D CFD or cautiously measured.

Figure 6: Approximation of Cm, blended with the reference data in the AOA region between -30Â° and 30Â°, and with the polynomial approximation above 163Â° AOA and below -163Â° AOA; â€˜Ref Dataâ€™ shows the points used for tuning the models

**Conclusion**

Three methods for estimating the lift and drag coefficient curves in the whole angle of attack range by using harmonic equations were analyzed in the present paper with one assuming aerodynamic response of a flat plate, another utilizing even sine and even cosine functions, proposed by Apostolyuk [3] for the estimation of the aerodynamic coefficients of modern jet fighter planes, and the third method by Viterna and Corrigan [4] developed with the scope on stall-regulated turbines. The method of Apostolyuk was further developed by using two independent harmonic approximations in the positive and negative AOA regions for the estimation of Cl and Cd, by using four independent harmonic approximations for the estimation of Cm, and by additionally using two independent polynomial approximations around 180Â° AOA for the estimation of Cl and Cm. Original data was used directly in the AOA region between -30Â° and 30Â°. The method produced satisfactory results for the Cl, Cd and Cm curves in the whole AOA range by using nine independent data points to find the modelsâ€™ coefficients, that would need to be computed using 3D CFD or measured. Note that the present work relies solely on a single set of data. It is therefore a first step in the creation of a generic and reliable engineering model where further validation, tuning and development should be carried out on additional data sets. Further, ability of 2D CFD to compute the aerodynamic coefficients in the vicinity of 180Â°, as an alternative to using the polynomial approximation, should be analyzed in future.

**Learning objectives**

The paper focuses on more efficient and accurate estimation of airfoil aerodynamic coefficients at high angles of attack which may result in more reliable aeroelastic computations, especially in the context of wind turbine blade vibrations at standstill conditions.

**References**

1) Gaunaa, M.; Larsen, T.J.: Stilstandslaster; chapter in Forskning i Aeroelasticitet; ed. Bak, C.; RisÃ¸-R-1434(DA) in Danish, RisÃ¸ DTU National Laboratory for Sustainable Energy; Roskilde; 2002

2) SkrzypiÅ„ski, W.; Gaunaa, M.: Wind turbine blade vibration at standstill conditions â€“ the effect of imposing lag on the aerodynamic response of an elastically mounted airfoil; submitted to Wind Energy in July 2011

3) Apostolyuk V.: Harmonic Representation of Aerodynamic Lift and Drag Coefficients; AIAA Journal of Aircraft, Vol. 44, No. 4, pp. 1402-1404; July-August 2007

4) Viterna, L. A.; Corrigan, R.: Fixed pitch rotor performance of large Horizontal Axis Wind Turbines; Brook Park, Ohio: NASA Lewis Research Center; 1981

5) Timmer, W.A.; van Rooij, R.P.J.O.M: Summary of the Delft University wind turbine dedicated airfoils; AIAA-2003-0352; 2003

6) Timmer, W.A.: Aerodynamic characteristics of wind turbine blade airfoils at high angles-of-attack; Proceedings of TORQUE 2010 The science of making torque from wind; Heraklion, Crete, Greece; June 28-30, 2010

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