Dynamic stall model for wind turbine airfoils
Introduction
Wind turbines are designed with increasingly slender blades and blade controls that can react rapidly to changing loading conditions. Thus, it becomes increasingly important to account for nonstationary load effects. The basic nonstationary mechanisms are illustrated in Fig. 1, showing a typical airfoil profile with relative flow velocity V, angle of attack , and lift force L per unit length. The lift force is represented via the nondimensional lift coefficient , defined bywhere c is the chord length. It is well established that under stationary attached flow conditions the lift force L acts approximately in the forward quarter-point, named the aerodynamic centre, and is approximately a linear function of the angle of attack , when is sufficiently small (Bisplinghoff, 1996).
At a limiting value of the angle of attack the flow pattern changes, and the lift force passes through a maximum. This is the phenomenon of static stall. A typical curve of the static lift coefficient is shown as the dashed line in Fig. 2. Stall is associated with separation of the flow at the suction side of the profile as illustrated in Fig. 1(b). The degree of separation can be represented in an approximate way by the distance cf from the leading edge to the separation point, where the nondimensional parameter f is a measure of the degree of separation. For fully attached flow , and for separated flow f decreases towards zero with increasing separation.
Under nonstationary conditions, it takes some time for the flow to approach the stationary flow pattern. For fully attached flow, analytic solutions can be obtained for harmonic motion of the airfoil as well as for a step function change in position (Fung, 1993, pp. 206). In principle, these solutions involve translation as well as rotation of the airfoil, but for the present purpose it is sufficient to combine the effect of the motion into an effective angle of attack. This is described in Section 4. The analytical solutions indicate delayed lift during fully attached conditions with a lower lift at increasing and higher lift at decreasing compared to the quasi-static solution. This effect is also seen on the dynamic lift in Fig. 2 at low angles of attack. When the flow is separated during the motion, the degree of attachment, represented by the parameter f, also exhibits delay with respect to its stationary value. The delay in the flow and the degree of separation for harmonic motion influences the stall phenomenon. A typical dynamic stall curve is plotted as the full line in Fig. 2. It is seen that for a harmonic variation of the angle of attack between and , the occurrence of stall is delayed, and the lift during the phase of decreasing is considerably lower than during the increasing phase. Thus, dynamic stall typically involves an increased range of attached flow and different branches for increasing and decreasing angle of attack.
In addition to trailing edge separation, a separation bubble may develop in the boundary layer at the suction side of the profile close to the leading edge, a phenomenon characteristic for thin profiles. In front of the bubble the boundary layer is attached, whereas behind the bubble two situations can occur. In one case, the boundary layer downstream of the bubble is turbulent, enhancing the tendency of attachment. This situation is illustrated in Fig. 1(b). In the other case, the boundary layer detaches fully, creating separation from the leading to the trailing edge. Under quasi-static conditions, the situation is very unstable, and in fact both situations can occur at the same angle of attack. Hence, two different lift curves can occur (Bak et al., 1999). This phenomenon is called double stall. Under dynamic oscillations, the flow will separate immediately at the separation bubble. Even though the flow is fully separated over the profile, experimental data show an almost linearly increasing lift force (Leishman, 2000). This increase in lift is generated by a rising pressure peak at the leading edge and a large vortex created in the turbulent wake as the trailing edge separation point moves abruptly to the leading edge. At low pitch velocities, leading edge separation is usually no problem, but in recent years a wish for active control of tower and blade vibrations has led to pitch control of large wind turbine wings with pitch velocities of up to . Also, large deformations due to flexible wings may produce high amplitude changes in the angle of attack. Under such conditions, leading edge separation may contribute significantly to the lift. As the angle of attack is increased, the vortex builds up in strength and finally detaches from the leading edge travelling downstream. CFD calculations indicate the creation of a secondary vortex with opposite circulatory contribution at the trailing edge during dynamic stall conditions. The trailing edge vortex builds up during dynamic stall, and at some point it detaches and convects downstream similar to the leading edge vortex. A CFD calculation of a NACA-0015 profile in dynamic stall conditions is illustrated in Fig. 3. The dynamic stall calculation has been performed by Risø National Laboratory, Wind Energy and Atmospheric Physics Department within the EC project VISCWIND (VISCWIND, 1999). The dots indicate particles added to illustrate the flow characteristics. Fig. 3(a) illustrates the onset of leading edge separation with the entire boundary layer starting to detach. Fig. 3(b) shows the build-up of the leading edge vortex, which in Fig. 3(c) detaches and moves downstream, while a trailing edge vortex starts building up. Finally, Fig. 3(d) shows the detachment of the trailing edge vortex and breakdown of the leading edge travelling vortex. Both experimental data and CFD calculations indicate that the flow changes, caused by the leading edge separation vortex, generate an increased suction contribution, leading to an increased lift even after flow separation has occurred. This effect may be seen in Fig. 2 as the dynamic curve that continues to increase above the static stall angle.
At low-frequency changes of the angle of attack, flow situations including trailing edge separation occur at increasing angle of attack, where the flow separation point moves from the trailing to the leading edge. At decreasing angle of attack, the separation point moves back from the leading to the trailing edge. Both situations are described by the movement of the trailing edge separation point. As the frequency increases, leading edge separation conditions are likely to happen as illustrated in Fig. 3, especially for thin profiles. Under these conditions, no distinct position of the trailing edge separation point can be followed and separation over the entire profile occurs more or less instantaneously at increasing angle of attack. Flow reattachment at decreasing angle of attack mainly happens with the separation point moving from the leading to the trailing edge, as was the case for low-frequency oscillations.
In summary, delay of lift during fully attached conditions and the motion of the separation point as well as leading edge separation and the dynamic interaction between leading and trailing edge vortices are constituent properties of a load model for which dynamic stall phenomena occur. A good model of dynamic stall should describe these phenomena for different frequencies and different amplitude ranges.
There exists a great variety of dynamic stall models in the literature. These models may be categorised into three main groups:
- (1)
The effects of the different flow conditions described above are modelled, e.g. lift reduction due to separation, time delay effects from leading edge separation etc.
- (2)
The characteristics of the lift curve are modelled without resort to the generating physical mechanisms, e.g. a linear growing curve at low angles of attack, a drop in lift at a given stall angle etc.
- (3)
A modification of the angle of attack is made introducing a so-called dynamic angle of attack.
In a series of papers (Leishman and Beddoes, 1986a, Leishman and Beddoes, 1986b; Leishman, 1988), Beddoes and Leishman have developed a model for dynamic stall combining the flow delay effects of attached flow with an approximate representation of the development and effect of separation. This model was developed with helicopter rotor dynamics in mind and therefore includes a fairly elaborate representation of the nonstationary attached flow depending on the Mach number and a rather complex structure of the equations representing the time delays. In contrast, a model proposed by Øye (1991) omits the transient effects of the attached flow, and represents the dynamic stall by introducing a first-order filter on a static equivalent degree of attachment, obtained by a simple interpolation relation. Hansen et al. (2004) developed a reduced version of the Beddoes–Leishman model at Risø National Laboratories, Denmark, omitting the effects of compressible flow and leading edge separation. They introduced an interpolation relation similar to the one used by Øye to make the model valid in the entire range of the angle of attack. The Beddoes–Leishman model, Øye and Risø models may be categorised into the first group of models aiming to model the effects of the flow conditions. The so-called ONERA model by Tran and Petot (1981) is an example of the second category of modelling. In this case, the load coefficients are described by a third-order differential equation. The differential equation is split into a linear domain at low angles of attack determined by a first-order differential equation, and a stall domain determined by a second-order differential equation. Tarzanin (1972) developed a model also named the Boeing–Vertol model, based on a relation between the dynamic stall angle and static stall angle determined by Gross and Harris (1969). From this relationship a dynamic angle of attack is determined and the load coefficients are interpolated from the static data. Obviously, this final model falls into the third category of modelling approaches.
In recent years, Navier–Stokes solvers have also been used to determine airfoil loads in dynamic stall situations. Due to the extensive cost of these calculations, practical applications do not seem possible in the near future, but solving the Navier–Stokes equations gives insight into the flow and pressure changes occurring during a dynamic stall cycle. Srinivasan et al. (1995) used a Navier–Stokes solver to evaluate a variety of turbulence models. Du and Selig, 1998, Du and Selig, 2000 studied 3-D effects on the boundary layer flow of a rotating wind turbine blade by solving the steady boundary layer equations. They found that the separation is slightly postponed due to rotation of the wing, which induces an increase in lift. They suggested a modification of the 2-D static data to incorporate the rotational 3-D effects. Akbari and Price (2003) studied the effects of several parameters including reduced frequency, mean angle of attack, location of the pitch axis and the Reynolds number. They found that the Reynolds number and position of the pitch axis have little effect on the characteristics of the lift cycle, however the position of pitch axis has a major effect on the pitching moment. Wernert et al. (1996) used Particle Image Velocimetry (PIV) and Laser Sheet Visualisation to validate a numerical code based on Navier–Stokes equations. They found that the numerical and experimental results agreed, but some discrepancies were observed. A completely different approach is suggested by Suresh et al. (2003) using neural network for identifying nonlinear unsteady lift. Comparing with experimental data, they show a good agreement with their numerical model. They also argue that the proposed method is easily implemented in available codes and should be less computationally expensive than the ONERA model.
It should be noted that larger discrepancies between numerical and experimental results seem to be accepted within the so-called semi-empirical models described above than for the Navier–Stokes solvers. The goals of the semi-empirical models are not to capture every variation in the load, but to model the main characteristics in a fast and efficient way. The main drawback of the semi-empirical models is that all of them are dependent on available static data and use interpolation into tabulated values or curve fitting techniques to determine quasi-static lift values. A semi-empirical model should be able to reproduce these static values for quasi-static rates of the angle of attack, i.e. .
In this paper, a semi-empirical dynamic stall model for the lift is formulated. The model is developed mainly with concern for realistic wind turbine flows, hence compressibility effects are omitted. An essential part is that the model should fit into the first category of semi-empirical models, hence the lift contributions introduced in the model should be explained by certain changes in flow and resulting changes in pressure. The model should be applicable both at high pitch rates during e.g. active control or large gusts, and during normal operation conditions with low pitch rates. The proposed model introduces an analytical solution to the static lift curve in terms of lift at separated flow conditions for known position of the trailing edge separation point, (Krenk, 2006, Thwaites, 1960). This relation serves as a backbone curve for the model, when the position of the trailing edge separation point is assumed to be known for all angles of attack. These curves may be found from static experimental data. The various dynamic effects included in the model are introduced as delay terms in the parameters of the backbone function. The model combines a simple two-term memory kernel for the transient behaviour of the attached flow with a lift reduction due to separation, represented via a first-order filter, giving two state parameters to describe the flow delay under full attachment and one-state parameter to describe the delayed placement of the separation point. The reason for using two timescales under attached flow conditions is to model with sufficient accuracy both high pitch frequencies occurring e.g. during active control or normal operating conditions. Furthermore, an additional first-order filter is used to model the increased lift under leading edge separation. The advantage of this compromise is that within a fairly simple model the transients of the flow are included in a manner that is sufficiently accurate for wind turbines, and the dynamic effect of trailing edge separation in essence only introduces one additional timescale.
The numerical algorithms for the Beddoes–Leishman, Øye, Risø, ONERA and Boeing–Vertol models are described briefly in the Appendix. These models also use the approach of introducing the static lift curve as a backbone curve. The Beddoes–Leishman and the Risø models use the unsteady thin plate approximation adopted in the proposed model. Numerical simulations are made and the performance of the various models are studied and compared with existing experimental data. In the above-mentioned analyses, the profile is assumed to be a thin plate. The effect of camber and thickness under separated quasi-static flow conditions has recently been derived in closed form (Krenk, 2006) and can be included directly in the format of the present model.
Section snippets
Stationary lift and separation
An essential part of a nonlinear load model is the lift reduction due to separation. The lift coefficient of the profile under fully attached flow is denoted and is often linearised for small in the following manner:where is the angle of attack at zero lift. For a thin plate the coefficient , whereas the coefficient is somewhat different for a real profile. The lift coefficient under separated flow can be found from Kirchhoff flow theory using complex
Comparison of models
The main input to all the models are the quasi-static lift data. The dynamic stall models should be able to reproduce these in the entire range of . At low reduced frequencies and at angles of attack below the point of full separation, all the models generate results close to the static lift curve. At angles of attack above this point, the Beddoes–Leishman model is no longer capable of reproducing static data, whereas the proposed model follow the static curve nicely due to the modification
Aeroelastic modelling
All the indicated dynamic stall models are one-degree-of-freedom models in . Assume that the profile has the translation velocities and in the blade and edgewise direction, respectively, the pitch velocity around the aerodynamic centre A, and exposed to turbulence components and . The flow conditions are illustrated in Fig. 12. Assuming the wavelength of the turbulence is significantly larger than the chord-length of the profile, the structural deformation
Concluding remarks
In the present work, a model for determining the dynamic lift coefficient of a wind turbine wing profile has been developed, based on the effects of various flow conditions. In the model three basic features have been included. Firstly, a time delay is introduced under fully attached flow situations using two filter equations. Secondly, the time delay in the motion of the separation point is described by one filter equation. And last, a contribution from leading edge separation vortex and
Acknowledgement
The present work has been supported by the Danish Technical Research Council within the project “Damping Mechanisms in Dynamics of Structures and Materials”.
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