In the present work, a consistent method for calculating the lift and drag forces from the 2-D airfoil data for the dihedral or coned horizontal-axis wind turbines when using generalized lifting-line methods is described. The generalized lifting-line methods refer to the models that discretize the blade radially into sections and use 2-D airfoil data, for example, lifting-line (LL), actuator line (AL), blade element momentum (BEM) and blade element vortex cylinder (BEVC) methods. A consistent interpretation of classic unsteady 2-D thin airfoil theory results reveals that it is necessary to use both the relative flow information at one point on the chord and the chordwise gradient of the flow direction to correctly determine the 2-D aerodynamic force and moment. Equivalently, the magnitude of the force should be determined by the flow at the three-quarter-chord point, while the force direction should be determined by the flow at the quarter-chord point. However, this aspect is generally overlooked, and most implementations in generalized lifting-line methods use only the flow information at one calculation point per section for simplicity. This simplification will not change the performance prediction of planar rotors but will cause an error when applied to non-planar rotors. In this work this effect is investigated using the special case, where the wind turbine blade has only out-of-plane shapes (blade dihedral) and no in-plane shapes (blade sweep), operating under steady-state conditions with uniform inflow applied perpendicular to the rotor plane. The impact of the effect is investigated by comparing the predictions of the steady-state performance of non-planar rotors from the consistent approach of the LL method with the simplified one-point approaches. The results are verified using blade-geometry-resolving Reynolds-averaged Navier–Stokes (RANS) simulations. The numerical investigations confirmed that the full method complying with the thin airfoil theory is necessary to correctly determine the magnitude and direction of the sectional aerodynamic forces for non-planar rotors. The aerodynamic loads of upwind- and downwind-coned blades that are calculated using the LL method, the BEM method, the BEVC method and the AL method are compared for the simplified and the full method. Results using the full method, including different specific implementation schemes, are shown to agree significantly better with fully resolved RANS than the often used simplified one-point approaches.
With the scientific and engineering advancements in the design optimization and manufacturing of horizontal-axis wind turbines (HAWTs), modern wind turbine blades are generally more flexible and may have more out-of-plane shapes compared to conventional stiff machines. Also, research on downwind turbines proposed for land-based low-rated-wind applications
According to unsteady 2-D thin airfoil aerodynamics, it is necessary to use the flow information at both the quarter-chord point and the three-quarter-chord point to correctly determine the magnitude and direction of the lift and drag forces
The structure of the present work is as follows: the highlights from the unsteady 2-D thin airfoil theory are firstly summarized in Sect.
Unsteady 2-D thin airfoil theory is of vital importance to correctly model the aerodynamics of wind turbine blades with dihedral using generalized lifting-line methods.
The reason is related to the ideas underlying generalized lifting-line methods, which are considered to be the application of perturbation theory
In this section, some of the important conclusions and results from unsteady 2-D thin airfoil theory are briefly described. These conclusions will be used to derive the non-circulatory forces in Sect.
For the present work, the simplest representation of an uncambered airfoil, a flat plate, with a chord length of
Definitions of the coordinate system and positive directions used in the derivations of the unsteady 2-D thin airfoil theory.
Essentially, the fundamental output from the thin airfoil analysis is the local unsteady pressure difference over the airfoil and the leading edge suction force. At the leading edge of an airfoil, there is generally a low pressure because the air is accelerated around the relatively small leading edge radius. In thin airfoil theory the airfoil thickness, and along with it the leading edge radius, tends to zero. This results in the pressure tending to minus infinity at the leading edge This corresponds to the singularity of the resulting bound vorticity distributions at the leading edge of the thin airfoil.
Projection of the normal and tangential forces (
The lift force that is defined according to the effective flow direction that deviates by an angle of
In the equation above,
Using the expression for the normal force In the case where the forcing is arbitrary, the 2-D shed wake effect can be modeled by a time-lag filter on the term
The first term of the lift force is the circulatory lift force,
To have a better understanding of the circulatory lift equation in Eq. (
By setting
Since an angle of attack is defined as the angle between a flow direction and the chord line, in this case using the small-angle approximation,
This means the magnitude of the circulatory lift is correctly determined by using the angle of attack at the three-quarter-chord point.
After furthermore noting that under the thin airfoil approximations, the relative wind speed is equal to
The non-circulatory lift in Eq. (
According to Eq. (
One of the key conclusions that can be drawn from a full unsteady 2-D thin airfoil theory analysis regarding application in generalized lifting-line methods is usually overlooked: a consistent definition of the direction with which lift and drag forces are defined. Even though the details may at first glance seem overwhelmingly focused on unimportant details, the effect of skipping these details can in some cases lead to completely unphysical behaviors. This is shown by an example in Appendix
As stated previously, the local angles of attack observed at different locations on the chord line differ from each other in the general unsteady case due to the pitching or torsional motion of the blade. By use of Eq. (
Using this angle of attack as the reference with which the drag force is defined, which means applying
The tangential force, which in the uncambered airfoil case stems entirely from the leading edge suction force, is given by
Inserting the results from Eqs. (
Each component of the drag force in Eq. (
The term
In Eq. (
The last term
In analogy with Eq. (
For cases with non-negligible pitch rate, the differences in the two different drag values, in Eqs. (
These important details about the drag are generally overlooked but are important for the performance prediction of dihedral blades, which is investigated in Sect.
When applying these results from thin airfoil aerodynamics in an aeroelastic model, it might seem as if it needs to be carefully considered whether components of the relative motion of the airfoil with respect to the surrounding air are due to a change in flow speed (e.g., due to a gust) or due to a motion of the airfoil itself.
To avoid this issue, it is chosen in
The only airfoil motion that remains to be treated individually is the torsion rate of the airfoil, which does not cause a constant change in velocity along the chord (like the gusts with the assumption described above or a translation of the airfoil). Instead, the torsion rate causes a velocity that varies linearly with the position on the chord. The torsion rate is then the total rotation rate of the airfoil section about an axis perpendicular to the cross-section, including influences from rotor rotation, blade torsion and any movement of the substructure that contributes to this rotation.
In this section, the focus is on the special case that the rotor is non-planar, and the blades have no sweep. For this special case, the blade pitch angle is set to be zero since pitching the blade will result in blade in-plane geometries. The blades are operating at steady-state, with uniform inflow applied perpendicular to the rotor plane, and the rotor has zero tilt and no yaw error. The influence of the non-circulatory lift on such a pure dihedral blade is derived analytically in this section.
Following the conventions in the HAWC2 code, the main axis of the blade is chosen to be the half-chord line. The airfoils are aligned perpendicular to this main axis.
Since it is necessary to perform the projection of the 3-D motion of the blade section into the 2-D airfoil section for the analysis, it is convenient to introduce different coordinate systems and the corresponding transformation matrices between them. Three coordinate systems, which are the blade coordinate system, the sectional coordinate system and the local flow coordinate system are used in the present work. The three different coordinate systems and the relationship among them are illustrated in Fig.
Illustration of the blade coordinate system
It is assumed that the turbine has three identical blades. The blade coordinate system is a rotating system following a blade that is chosen for reference.
For the blade coordinate system, the
The connection between the 3-D flow and motion projected into the local flow system and the unsteady 2-D airfoil theory introduced in Sect.
For a blade section, the position vector
The dihedral angle
The transformation matrix from the blade coordinate system
For the airfoil section with the local flow angle of
For a given transformation matrix, the reverse transformation matrix is equal to its transposed matrix. This is because the transformation matrices are orthonormal.
For the pure dihedral blade, the projection of the centrifugal acceleration from the blade coordinate system into the local flow coordinate system is obtained using the transformation matrices in Eqs. (
According to Eq. (
There is also an effective streamwise acceleration that is equal to the negative value of
Since the term This can also be derived directly from Eq. (
For non-planar rotors, the projection of the angular velocity into the 2-D airfoil section will also result in an effective pitching motion of the airfoil if assuming the flow seen by the airfoil is uniform. This is shown by projecting the angular velocity vector from the blade coordinate system into the local flow coordinate system.
The airfoil pitch rate of the effective pitching motion is the
The existence of the effective pitch rate
The resulting non-circulatory lift due to this effective pitching motion is obtained using Eq. (
The total non-circulatory lift is then the sum of the contribution of the mid-chord acceleration in Eq. (
If assuming the flow angle
In addition, if assuming the twist angle
Since the non-circulatory lift is perpendicular to the airfoil, there should be an effective non-circulatory drag
In summary, for a pure dihedral blade operating under steady-state conditions, the total non-circulatory lift and the corresponding effective non-circulatory drag are negligible. This conclusion is tested numerically in Sect.
It is important to note that the equations derived previously in this section are only applicable to a pure dihedral blade without sweep under steady-state operating conditions. For unsteady cases, there could be net non-circulatory forces. As a result, for the unsteady conditions, it is important to include all the non-circulatory terms described in Sect.
According to the conclusions from the unsteady 2-D thin airfoil theory described in Sect. It is argued by some researchers that the drag force should be excluded during the convergence calculation but should be included when calculating the aerodynamic loads after the convergence, which can be considered to be the post-processing of the converged results.
The generalized lifting-line methods are usually implemented as the one-point approach that only utilizes the flow information at one chordwise location per section for simplicity.
This simplification will not change the performance prediction of planar rotors because the flow angle is constant along the chord for planar rotors under steady-state conditions. This can be shown using Eq. (
However, with the known effective pitch rate
The difference between the magnitude of the relative velocity at the three-quarter-chord point (
One of the common one-point implementations of the generalized lifting-line method is placing the calculation point at the quarter-chord point for each section. The flow information at the quarter-chord point is used to determine both the magnitude and direction of the lift force. This simplified approach is labeled as QC in the present work. An example of this is the most common implementation of the lifting-line (LL) method
Another commonly used one-point approach of the generalized lifting-line methods is placing the calculation point at the three-quarter-chord point, such as the BEM method implemented in the HAWC2 code. This implementation uses the flow information at the three-quarter-chord point for both the magnitude and direction of the lift force. This simplified approach is labeled as 3QC. Then, the magnitudes of the circulatory lift and drag coefficients are correctly calculated, but the direction of the lift force is erroneous, which will result in an additional effective drag force. Then, the calculated tangential load will have an offset if the blade has dihedral, which will result in an error in the aerodynamic power prediction.
One possible method of applying the correction is using the angle of attack at the three-quarter-chord point and the pitch rate
Alternatively, it is possible to include an additional pitch rate drag force in the flow direction at the three-quarter-chord point by modifying the quasi-steady drag coefficient in Eqs. (
Both implementations should give almost identical results when the difference between
Apart from the two most common choices of the calculation point described previously, other definitions of the calculation point are possible. The general correction for the one-point approach of the generalized lifting-line methods that use an arbitrary chordwise location as the calculation point is given.
When placing the calculation point at the
The approximated angle of attack
Details of different approaches that are investigated in the present work are summarized in Table
Details of different approaches of the generalized lifting-line method that are investigated in the present work. The differences are the chordwise locations where the flow information is used to determine the sectional force magnitude and force direction. The correction methods for different approaches are also listed.
The blades used for the numerical tests in the present work are based on the IEA-10.0-198 10 MW reference wind turbine (RWT)
In the present work, the test cases are mostly using the baseline straight blade with zero coning, 15
Side view of the main axes of the blades used for different cases of the comparison. The blades from left to right are the baseline straight blade with zero cone angle, 15
In this section, the different aerodynamic models with different numerical fidelities used for comparison are described. The highest-fidelity model used for the comparison is based on blade-geometry-resolving RANS simulations. The generalized lifting-line methods with different fidelities are compared, which are the actuator line (AL) method, the lifting-line (LL) method, the BEVC method and the BEM method. The airfoil data used in all generalized lifting-line methods in the present work are the same, and they were generated with 2-D fully turbulent RANS computations
For the numerical simulations that are used as a reference in the present study, the three-dimensional computational fluid dynamics (CFD) solver EllipSys3D
Detail of the surface mesh for the straight variants used in the fully resolved CFD simulations, showing two of the blades. Two different configurations are included, accounting for a cone angle of 0 and 15
The lifting-line module in the aerodynamic solver MIRAS
The two-point approach in the LL method refers to the explicit calculation of the flow information at both the quarter- and the three-quarter-chord point and is labeled as LL-2P. This approach aligns with the conclusions from the unsteady 2-D thin airfoil theory, as previously described in Sect.
The one-point approaches only use the flow information at a single chordwise location for each section.
Two variants of the one-point approach, which are representative of the most common implementations, are used for the comparison.
The first one uses only the quarter-chord point as the calculation point (labeled as LL-QC); the second one uses only the three-quarter-chord point (labeled as LL-3QC).
The corrections previously described in Sect.
The blade element momentum (BEM) method implemented in the HAWC2 code is the one-point approach that only uses the three-quarter-chord point as the calculation point
The blade element vortex cylinder (BEVC) method is the combination of the blade element theory and the vortex cylinder model. It has been shown in a previous work
Consistent with the fully resolved CFD computations, the actuator line (AL) simulations also used the EllipSys3D flow solver Even if the described simulations are based on a CFD solver, they are simply referred to as AL in this document – keeping the CFD abbreviation for the fully resolved computations.
The numerical domain discretization follows a verified approach
Actuator line numerical box domain with a structured mesh and uniform spacing around the rotor at its center. Only every eighth grid point is shown.
In the AL method, the velocity is only calculated on the actuator line itself. Considering the equivalence between the AL and the LL method for straight blades without coning, the AL method before the one-point correction is similar to the LL method that only uses the quarter-chord locations in the load computations (LL-QC) and is labeled as AL-QC. Similarly, the AL method following the one-point correction should be equivalent to LL-QC-corr, which is labeled as AL-QC-corr.
In this section, various numerical tests are performed to investigate the different assumptions outlined in the previous sections and also to evaluate the performance of the one-point lifting-line correction for different aerodynamic models.
Firstly, the impact of non-circulatory forces under steady-state conditions is tested using the LL method in Sect.
For all test cases, the pitch angle is zero, and the rotor is operating under uniform inflow of 8
The initial assessment of the performance of the different numerical methods relies on the study of the sectional aerodynamic load distributions. In order to ensure the quality of the comparison among different blade geometries involved in the present study, it is important to define the loads in a consistent manner.
The loads are defined as force per unit radius, which is the same definition used in the previous work
It is shown analytically in Sect.
Comparison of axial load
It can be seen from the figure that the loads from the LL method with or without the non-circulatory force are almost identical and are in good agreement with the results from the fully resolved CFD solver.
To clearly show the magnitude of the non-circulatory forces, the difference in the loads calculated from LL-2P-NC1 and LL-2P-NC2 with respect to the results from LL-2P is calculated. The difference generally increases when moving from the blade tip to the blade root but is negligible. For the spanwise position between a radius of 40 m (of the unconed blade) to the blade tip, the difference compared to the LL-2P method is within 2
For the generalized lifting-line methods, there are two procedures that involve the magnitude of the relative velocity for the calculation.
The first one is the calculation of the quasi-steady bound circulation strength, which is related to the convergence calculation.
The second one is during the calculation of the lift and drag forces, which is to compute the aerodynamic loads on the blades. This procedure is performed after the convergence calculation and can be considered to be the post-processing of the converged results.
There is no clear indication from unsteady thin airfoil theory at which chordwise location to extract the relative velocities for any of these two procedures. For the one-point approach, it is natural to use the relative velocity at the calculation point for both procedures.
For the two-point approach, it is possible to choose the relative velocity at either the quarter-chord point or at the three-quarter-chord point for both procedures. In total, there are four possible combinations, which are summarized in Table
The four different choices of the chordwise location for the magnitude of the relative velocity to use for the calculation of the bound circulation strength and the magnitude of the lift and drag force in the two-point approach of the lifting-line method.
From an intuitive point of view, Case 3 appears as the most correct one because the angle of attack at the three-quarter-chord point is used to determine the lift coefficient, and the flow at the quarter-chord point is used to determine the lift and drag direction.
The difference between these four combinations is tested numerically by comparing the loads calculated using different implementations of LL-2P. For this numerical test, the extreme case with the upwind dihedral blade W-1 with 15
Comparison of the difference in axial load
It can be seen that the difference in the loads calculated using different methods is extremely small compared to the full loads as shown in Fig.
The correction to the generalized one-point lifting-line method described in Sect.
The two-point approach of the LL method (LL-2P) The LL-2P method in the following sections corresponds to Case 3 in Table
Firstly, the axial and tangential loads of the straight blade without coning calculated from different implementations of the LL method are calculated and are plotted together with the fully resolved CFD results in Fig.
Comparison of axial load
It can be seen from the figure that the loads from all LL methods give very similar results for both axial and tangential loads. The results in the figures are almost on top of each other. The loads predicted by the LL methods are also in good agreement with the CFD results. At the near-root region (i.e., up to an approximate radius of 20 m), clear differences between the CFD solution and the rest of the methods were observed. These discrepancies are related to the separation in the near-root region predicted by the CFD solver. This effect has a relatively low influence on the integrated loads and is not the subject of the present investigation.
The axial and tangential loads of the straight blade with 15
Comparison of axial load
It can be seen from the figure that for the axial load, all LL methods except LL-QC give very similar results and are in good agreement with the fully resolved CFD. The axial load is overestimated by LL-QC.
For the tangential load, the LL-3QC method predicts a somewhat lower value compared to the other LL methods, while the other methods show only small differences and are in good agreement with the fully resolved CFD.
To better illustrate the effect of blade coning predicted by different LL methods, the difference in the loads of the coned straight blade with respect to the baseline straight blade without coning is plotted in Fig.
Comparison of the difference in the axial load
For the LL-QC method, the decrement of the axial load is significantly underestimated compared to the predictions by the LL-2P. This is expected since the magnitude of the lift is not correctly calculated using the LL-QC method. After applying the correction, the axial load from LL-QC-corr agrees significantly better with LL-2P. For the tangential load, the result from LL-QC is in reasonably good agreement with the other methods as despite the magnitude of the lift force having an offset, the lift force is applied in the correct direction.
For the LL-3QC method, the calculated axial load is in good agreement with the result from LL-2P because the magnitude of the lift force is correctly calculated. The tangential load calculated from LL-3QC is underestimated compared to LL-2P. This is because the lift force is not applied to the correct direction in LL-3QC. After applying the correction, the tangential load predicted by LL-3QC-corr is in good agreement with LL-2P.
For the straight blade with 15
Comparison of axial load
It can be seen from the figure that similar to the case of upwind coning, all LL methods except LL-QC predict almost identical axial loads and are similar to the prediction by the fully resolved CFD.
The LL-QC method underestimates the axial load compared to other LL methods.
For the tangential load, LL-3QC predicts a significantly higher load compared to the predictions by other LL methods, which only show a small difference between each other and are similar to the fully resolved CFD result.
Similar to the upwind-coning case, the difference in the loads of the downwind-coning straight blade with respect to the straight blade without coning calculated from different LL methods and the fully resolved CFD is plotted in Fig.
Comparison of the difference in the axial load
For the LL-QC method, the axial load is underestimated and shows a relatively large difference compared to the load calculated using LL-2P. As has been explained for the upwind-coned case, the reason is that the magnitude of the lift is not correctly calculated using LL-QC. On the other hand, the tangential load calculated from LL-QC is slightly overestimated compared to the LL-2P method, but they are still in reasonably good agreement. The reason is, as has been explained for the upwind-coning case, that the lift force is applied in the correct direction despite the magnitude of the calculated lift force having some offsets. After applying the correction, the axial load from LL-QC-corr is now in good agreement with LL-2P. The tangential load predicted by LL-QC-corr is also in improved agreement with LL-2P.
For the LL-3QC method, the axial load is in good agreement with LL-2P since the magnitude of the lift is correctly modeled. However, LL-3QC especially overestimates the tangential load compared to LL-2P. As has been explained for the upwind-coning case, this is because the lift force is not applied to the correct direction and results in an additional effective drag force. After applying the correction, the tangential load calculated from LL-3QC-corr is in significantly improved agreement with LL-2P.
In summary, all of the corrected methods have consistently good performance for either upwind or downwind dihedral cases. The performances of LL-QC-corr and LL-3QC-corr are almost identical to LL-2P, and all of them are categorized as the full model since they align with the conclusions from unsteady 2-D airfoil theory. The important aspect is to use a correction such that the modeling effectively mimics the behaviors according to the thin airfoil theory.
The importance of consistently using the 2-D airfoil data when using the BEM method to calculate the loads of the dihedral blades is tested in this section.
Again, the focus is on the special case of a pure dihedral blade without sweep under steady-state operating conditions.
The blades for the test are the straight blade with 15
Comparison of axial load
The axial loads of the upwind-coned blade and the downwind-coned blade have small differences and are lower compared to the blade without coning. For either an upwind- or downwind-coned blade, the axial loads show only a negligible difference with and without the correction.
For the tangential load of the coned blades, when including the one-point correction, the results show a relatively large difference compared to using only the quasi-steady aerodynamics.
This means that when using the BEM method that only uses three-quarter-chord information and without the correction for a dihedral blade operating under steady-state conditions under uniform inflow, if directly using the quasi-steady aerodynamics, the thrust force is correctly calculated, but there will be a visible error for the predicted aerodynamic power. This conclusion can be generalized to unsteady cases as well. As a result, we recommend the use of the BEM module with the unsteady 2-D airfoil model Such as the Beddoes–Leishman-type dynamic stall model in HAWC2.
A peculiar phenomenon that can be seen in Fig.
The importance of the one-point lifting-line correction when using the BEVC method to calculate the loads of the dihedral blades is shown in this section.
The blades for the test are also the straight blade with 15
Comparison of the difference in axial load
For the axial load, the BEVC results with and without the correction are almost identical. This conclusion is the same as for the LL-3QC and the BEM-3QC. The difference in axial loads predicted from BEVC-3QC and BEVC-3QC-corr is in good agreement with the LL-2P. For the tangential load, BEVC-3QC-corr predicts very similar results as the LL-2P. However, if the one-point correction is not included, BEVC-3QC underestimates the tangential loads compared to the predictions by LL-2P. In comparison, BEM-3QC-corr predicts a relatively large difference compared to LL-2P for both axial and tangential loads. This is as expected because the BEM method, even with the one-point correction, is not able to correctly predict the influence of blade out-of-plane geometry on the loads.
For the 15
Comparison of the difference in axial load
For the axial load, as for the upwind-coned case, BEVC-3QC and BEVC-3QC-corr have almost identical results and are in good agreement with LL-2P. For the tangential load, BEVC-3QC-corr predicts similar results as the LL-2P. When directly using the quasi-steady airfoil data without the one-point correction, BEVC-3QC overestimates the tangential load. In comparison, BEM-3QC-corr is not able to correctly predict the influence of blade coning on the axial or tangential loads as expected.
The actuator line (AL) used in this study is a straight line. This is equivalent to having a straight bound vortex. So, the blades used for the comparison in this section are aligned to a straight quarter-chord line instead of aligned to a straight half-chord line as in the previous sections.
The axial load and tangential load of the straight blade with zero cone angle and with 15
Comparison of the difference in axial load
It can be seen from the figure that the axial load predicted by the AL method without the correction is overestimated compared to LL-2P. This behavior is similar to LL-QC. The tangential load from the AL method is slightly overestimated. After the correction, the axial load from AL-QC-corr is in significantly improved agreement with the result from LL-2P. For the tangential load, the shape of the result from AL-QC-corr is in improved agreement with the result from LL-2P. However, for both axial and tangential loads, the results predicted by the AL-QC-corr method are slightly overestimated compared to LL-2P. This could be related to the smearing correction in the AL method that assumes the calculation point and the trailing point are both in the rotor plane
For the case of 15
Comparison of the difference in axial load
The AL method without the correction (AL-QC) underestimates the axial load, and the behavior is similar to LL-QC. The tangential load predicted by AL is slightly overestimated compared to LL-2P. After including the one-point correction, the results from AL-QC-corr are in improved agreement with results from LL-2P, for both axial and tangential loads.
In addition to the distributed loads that are compared in Sect.
The thrust and power of the rotor with an unconed straight blade as well as the upwind- and downwind-coned straight blades calculated from different aerodynamic models are compared. To better show the influence, the relative difference in thrust and power of the coned rotor with respect to the planar rotor with straight blades is defined as follows:
The results from different models are summarized in Fig.
The relative difference in thrust and power of the rotor with straight blade with 15
The relative difference in thrust and power of the rotor with straight blade with 15
For all the one-point approaches of the generalized lifting-line method in this comparison, if the one-point correction is applied, the predicted thrust and power of non-planar rotors have reasonably good agreement with the two-point LL method (LL-2P) and the fully resolved CFD. This conclusion also applies to the BEM method. However, if the one-point correction is excluded, and the quasi-steady polars are used directly, the results will have significant errors in either the predicted thrust or power or both, depending on the choice of the calculation point.
It can also be concluded from the results that the BEVC method does not result in significant improvement compared to the BEM method when predicting the integral thrust and power of the coned rotor with straight blades. This is because the influence of the dihedral on the distributed loads is partially canceled out when calculating the power and thrust of the whole rotor.
In contrast, the improvement of BEVC over BEM when predicting thrust and power is significant when computing the rotors with curved dihedral blades as shown in the previous work
For the comparison of the AL method with the LL-2P method, the blades are aligned with the quarter-chord line instead of aligned with the half-chord line. The rotor thrust and power predicted by the AL method with and without the correction and also the LL-2P are summarized in Fig.
The relative difference in thrust and power of the rotor with 15
For the AL method without the one-point correction (AL-QC), the thrust is overestimated for the upwind-coned case and is underestimated for the downwind-coned case. The thrust predicted by the AL method with the correction (AL-QC-corr) is in better agreement with LL-2P for both upwind- and downwind-coned cases.
The power predicted by AL-QC is overestimated compared to LL-2P for both upwind- and downwind-coned cases. After the correction, the power of the downwind-coned case is in improved agreement with LL-2P but is not improved for the upwind-coned case. As is described in Sect.
It is worth mentioning that the power and thrust of the cases with coned straight blades that are aligned to a straight half-chord line in Figs.
The numerical tests in this section are performed on the straight blades with 15
The present work describes a method where the use of key results from the 2-D unsteady thin airfoil theory results in improved performance of generalized lifting-line methods for non-planar rotors. The conclusions from the unsteady 2-D thin airfoil theory, which are that the magnitude of the quasi-steady lift should be determined by the flow at the three-quarter-chord point, and the direction of the quasi-steady lift should be determined by the flow at the quarter-chord point, are highlighted. The impact of the simplification that using only one calculation point and directly using quasi-steady 2-D airfoil data as usually implemented in generalized lifting-line models is investigated. The generalized correction for such a one-point approach is given. The numerical results from the lifting-line (LL) method, the actuator line (AL) method, the blade element momentum (BEM) method and the blade element vortex cylinder (BEVC) method are compared with and without the correction. The results from fully resolved RANS are also included for reference. The results show a large offset on the prediction of the aerodynamic performance of non-planar rotors when only using the quasi-steady aerodynamics and excluding the one-point corrections. The one-point approaches with the correction are in significantly better agreement with high-fidelity CFD results than without the corrections, for both the distributed loads and the rotor thrust and power. It is noted that the effect of the corrections is modeled correctly by a consistently implemented 2-D airfoil aerodynamic model, such as the Beddoes–Leishman-type model in HAWC2. For this reason, it is suggested to keep such models active also for the simulation of steady-state HAWT rotors with dihedral blades.
There are several future works that are of great interest. Firstly, the comparison of the prediction of the unsteady loads using the two-point approach and the one-point approach of the lifting-line method is interesting. Secondly, the impact of correctly determining the magnitude and direction of the lift force for the curved blade with both sweep and dihedral should be investigated. Thirdly, future work on the actuator line model is necessary for the model to correctly predict the loads of blades with complex shapes. An updated smearing correction for actuator lines with curved shapes is an example.
In this section, in order to shed light on the aspect that is described in Sect.
In non-dimensional coefficients this corresponds to a 2-D drag coefficient of
Illustration of two correct methods of defining the direction of the lift and drag forces for a VAWT.
As the last point in this section it should be mentioned that applying the erroneous definition of the direction of the forces in the VAWT case will lead to an effective drag error in the magnitude. For example, for the case of using the three-quarter-chord reference direction but using
The 2-D airfoil data used in this article are generated with 2-D fully turbulent RANS computations
AL conducted the study as part of his PhD research. The present work is based on the previous work on the VAWT by MG and GRP. The unsteady 2-D thin airfoil aerodynamics is described by MG with contribution from AL. The non-circulatory force for the pure dihedral case is described by AL with contribution from MG and GRP. The one-point lifting-line correction is described by AL with contribution from MG and GRP. The implementation of the one-point correction in the lifting-line code is performed by AL. The fully resolved CFD method is introduced by SGH, and its results are computed by SGH. The post-processing of the fully resolved CFD results is performed by SGH with contribution from AL. The lifting-line results, BEM results and BEVC results are computed by AL, and the post-processing of the results are performed by AL. The actuator line method is described by AFM, and the actuator line results are computed by AFM. All authors jointly drew the conclusions of the work and contributed to writing this work.
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The authors would like to thank our colleague Néstor Ramos García for the help and suggestions in the lifting-line simulation using the MIRAS code that was mainly developed by him. The authors would like to thank our colleague Fanzhong Meng for his contribution to the discovery of the disagreement between the aerodynamic load calculated from HAWC2 code version 12.8 and HAWCStab2 code version 2.15 for a wind turbine with large cone angle.
This research has been supported by the Smart Tip project, funded by Innovationsfonden (grant no. 7046-00023B).
This paper was edited by Alessandro Bianchini and reviewed by Vasilis A. Riziotis and one anonymous referee.