In the present work, a computationally efficient engineering model for the aerodynamic load calculation of non-planar wind turbine rotors is proposed. The method is based on the vortex cylinder model and can be used in two ways: either used as a correction to the currently widely used blade element momentum (BEM) method or used as the main model, replacing the BEM method in the engineering modeling complex. The proposed method needs the same order of computational effort as the ordinary BEM method, which makes it ideal for time-domain aero-servo-elastic simulations. The results from the proposed method are compared with results from two higher-fidelity aerodynamic models: a lifting-line method and a Navier–Stokes solver. For planar rotors, the aerodynamic loads are identical to the current BEM model when the drag force is excluded during the calculation of the induced velocities. For non-planar rotors, the influence of the blade out-of-plane shape, measured by the difference of the load between the non-planar rotor and the planar rotor, is in very good agreement with higher-fidelity models. Meanwhile, the existing BEM methods, even with a correction of radial induction included, show relatively large deviations from the higher-fidelity method results.

The blade element momentum (BEM) method has long been dominant in the low-fidelity aerodynamic modeling of horizontal-axis wind turbines. Until now, it is the main working horse for wind turbine aero-servo-elastic simulations and is widely used in the wind turbine design and optimization framework. There are many explicit and implicit assumptions in the BEM method. The BEM method explicitly assumes that uniform inflow is applied to the rotor that is operating at a high tip-speed ratio and the stream tubes are independent of each other. The model also implicitly assumes a planar rotor with straight blades and using quasi-steady aerodynamics. There has been extensive work on the modifications and corrections to the BEM method, such as dynamic stall model

The results from the BEM method generally show surprisingly good agreement with higher-fidelity models, at least on the integral level. However, due to the progress of wind turbine technology, modern multi-megawatt designs are generally more flexible than the stiff machines of the 1980s. It implies that modern wind turbine blades typically have more prebend, larger cone angle and larger deformations. The influence of blade out-of-plane shapes on the aerodynamics is then more pronounced and can not simply be neglected. Some new developments have even more pronounced out-of-plane shapes. For example, a downwind wind turbine designed for low-wind conditions could have large cone and prebend and possibly dramatic out-of-plane deformations

On the other hand, rotor-resolved CFD and lifting-line method are computationally too expensive for extensive use in current and near-future design optimization processes. Navier–Stokes solvers with fully resolved rotor geometry have no difficulties predicting non-planar rotor effects. But for the lifting-line method, care should be taken on the influence of curved bound vortex on itself

In order to correctly account for the out-of-plane shapes of the wind turbine blades in the low-fidelity model, the physics behind the problem should be analyzed, and then the most important aspects should be proactively modeled while less important features can be neglected. In the present work, the force on the non-planar rotor is firstly analyzed in a physically consistent way using the Kutta–Joukowski theorem. The conclusion from the analysis is that the streamwise-shifted starting position of the trailed vorticity, due to the non-planar bound vortex surface swept by the blades, will influence both axial and radial induction, and both have direct influences on the aerodynamic loads. Therefore, we consider the vortex cylinder model

In the present work, a detailed analysis of the vortex cylinder model for non-planar rotors will be performed. A method based on the vortex cylinder model for the aerodynamic load calculation of such non-planar rotors is then proposed. The description of the implementation of the proposed method is in the framework of the HAWC2 code

For the planar rotor with straight blades, the Kutta–Joukowski analysis was previously used to derive the similarity between the superposition of the vortex cylinders and the BEM method by

The coordinate system is defined as follows and is illustrated in Fig.

The definition of the coordinate system of the wind turbine with only out-of-plane shapes. The

The local dihedral angle is defined to be positive when the blade is tilting upwind and can be calculated using the blade main-axis geometry.

The relative velocity experienced by the blade section is

With the Kutta–Joukowski theorem in three-dimensional vector form, the lift force on the blade is obtained.

In Eq. (

The force

For a non-planar rotor with upwind direction dihedral (

There are two tracks to modify the BEM method to model the non-planar effects. The first track is based on the previous work on the development of the radial induction correction for the application in the BEM method, derived based on an analytical 2-D actuator disc/strip model combined with an engineering fit of the numerical actuator disc simulations

Apart from the momentum theory, which effectively only applies to a planar rotor disc, it is possible to calculate the induced velocity, including the radial component, from analytical equations at each blade section with the superposition of vortex cylinders. So, the second track is based on the vortex cylinder model where the assumption of the planar rotor in the previous work by

The vortex cylinder model is a simplified representation of the vortex system of a horizontal-axis wind turbine rotor. The model consists of superposition of bound vortex discs, non-expanding vortex cylinders with both tangential and longitudinal vorticity, and root vortices

Illustration of different components of a right vortex cylinder. The bound vortex disc (in blue) with radial vorticity

This model can be considered as the special case of the Joukowski rotor model for the limiting case of the number of blades tending to infinity.
It has been shown by

One advantage of the vortex cylinder model over the momentum theory is that the induced velocity at any arbitrary point in the flow field is known. In contrast, from the momentum theory, only the axial and tangential velocity at the rotor disc and at infinitely far upstream and downstream of the rotor plane are known. This advantage has been used in the application of the vortex cylinder model in the calculation of the induction zone of a wind turbine

When applying the vortex cylinder method to a non-planar rotor, the starting position of the cylindrical vortex sheets follows the curved bound vortex surface and will be displaced upstream or downstream compared to the case of a planar rotor. The induced velocity on the non-planar rotor surface will therefore be different from the induced velocity of a planar rotor. This effect can be modeled by the superposition of the vortex cylinders according to the curved bound vortex surface swept by the blades that have out-of-plane shapes. However, the possibility of using the vortex cylinder model for the non-planar rotor is not well recognized and is thus not widely utilized. The work of

The equations of the inductions of a right vortex cylinder have been derived in detail by

The tangential vorticity contributes to both axial and radial induced velocities. For the vortex cylinder with radius

The other components of the vortex cylinder, which are the bound vortex disc, the longitudinal vorticity and the root vortex line, only have contribution to the tangential velocity. The tangential induced velocity of the entire flow field is derived by

Consider the superposition of the Joukowski rotor model to achieve radially varying bound circulation. There will be helical trailed vorticities emanated along each blade with the strength equal to the derivative of the bound circulation strength with respect to the radius. In Joukowski's rotor model, it is assumed that the radial distribution of the bound circulation of all blades is the same. Then consider the corresponding vortex cylinder model that is the limiting case of Joukowski's rotor model, where the number of blades tends to infinity. It is consisted of a superposition of cylindrical vortex sheets with both tangential and longitudinal vorticity. Details of the superposition of vortex cylinders have been described by

With the superposition of the vortex cylinders, the bound circulation is assumed to be piecewise constant along the blade. The blade is discretized radially into

Sketch of the superposition of the cylindrical vortex system. The blade is extending from

The closure of the system determines the tangential vorticity strength of each vortex cylinder. The closure of the system is determined at the far wake, and the cylindrical vortex sheet is assumed to convect at a constant speed equal to the mean of the two far-wake velocities surrounding the vortex sheet

With the system closure, the axial induction factor of section

The effective thrust coefficient

The wake rotation effect increases toward the rotor rotational axis and decreases when the tip-speed ratio increases. For typical modern wind turbine designs the effect of this term is rather small and may be neglected.

In Eqs. (

Considering Eqs. (

In the vortex cylinder model, the relationship between the axial induction factor and the effective thrust coefficient is in the same form as in the momentum theory:

The system closure of the vortex cylinder model for planar rotors has been described in Sect.

For illustration, we show the superposition of the vortex cylinders for a non-planar rotor and the corresponding planar rotor with the same radial discretization in Fig.

Side view of the vortex system of the non-planar rotor and the corresponding planar rotor. The two vortex systems have the same radial discretization and radial distribution of bound vorticity. The tangential and longitudinal trailed vorticity strengths of the two systems will be identical.

The first similarity is the calculation of the thrust coefficient of the non-planar rotor and the corresponding planar rotor. For the non-planar rotor, the Kutta–Joukowski thrust coefficient of section

For the planar rotor, the thrust coefficient is also obtained using Eq. (

Since

The second similarity is that the two vortex wake systems have the same tangential and longitudinal vorticity strength distribution. For the two rotors with the same radial distribution of bound circulation, it can be easily shown that the trailed vorticity strengths between each section are the same.

According to the assumption, the closure of the superposition of the vortex cylinders is determined at the far wake (infinitely far downstream). Therefore, there is no influence of the changed starting position of the vortex cylinders. As a result, both the tangential and longitudinal vorticity of the non-planar rotor wake is the same as that of the corresponding planar rotor that has the same bound circulation distribution.

According to the description in Sect.

It is assumed that the radial distribution of the tangential induction factor of the non-planar rotor is the same as the corresponding planar rotor in Sect.

Side view of a part of the axisymmetric vortex system of the planar rotor. The bound vorticity and trailed vorticity are highlighted. The two points of interest are marked with

Consider the point

Recall the definition of circulation:

The relationship between the velocity along the contour line

So, the tangential velocity at point

For the point

There is a jump of the tangential velocity when the flow passes through the bound vorticity disc. The tangential velocity at the disc should be the mean value of the tangential velocities at the two sides of the disc.

The same result was obtained for the planar rotor case by

Side view of a part of the axisymmetric vortex system of the non-planar rotor. The bound vorticity and trailed vorticity are highlighted. The two points of interest are marked with

Similar as for the planar rotor case, consider the point

For the point

The relationship between the vortex cylinder model and the momentum theory will be described separately for the planar rotor and the non-planar rotor.

For a planar rotor at a high tip-speed ratio, when excluding the contribution of drag to the momentum balancing for determining the induced velocities, the converged results from the momentum theory are equal to those from the vortex cylinder model

For a non-planar rotor, the inductions from the momentum theory with wake rotation effect included are equivalent to the inductions from the vortex cylinder model for the corresponding planar rotor and excluding the radial induced velocity. This means the momentum theory equivalently assumes the rotor is planar when calculating inductions. Then, the vortex cylinder model for the non-planar rotor is equivalent to the momentum theory with the following corrections: for the annulus axial induced velocity, the correction is the difference of the results of the non-planar rotor and the corresponding planar rotor from the vortex cylinder model. The tangential induced velocity from both methods are the same, as described in Sect.

These relationships between the momentum theory and the vortex cylinder model for the non-planar rotor are summarized in the equations as follows, where the superscript “np” represents the non-planar rotor.

Some important aspects of the implementation of the low-fidelity models that use blade element theory and rely on the 2-D airfoil data are briefly discussed. They are important for the load calculation and to get good agreement with the higher-fidelity models.

For the blade with out-of-plane shapes, it is necessary to include the unsteady airfoil aerodynamics model (usually referred to as the dynamic stall model), even for the steady-state simulation. Otherwise, the results of the tangential forces will have a visible error. The reason originates from the conclusions of unsteady 2-D aerodynamics: the correct circulatory lift can be obtained if the magnitude of the effective angle of attack is determined at the three-quarter-chord point, but the direction of it should be determined by the velocity at the quarter-chord point

For blades with in-plane or out-of-plane shapes, the curved blade length in an elementary annulus (

The BEM method is the blade element theory coupled with the momentum theory. Similarly, the vortex cylinder model should be coupled with the blade element theory for the aerodynamic load calculation of a rotor with finite number of blades. The link between the blade element theory and the vortex cylinder method is the relationship of the blade bound circulation and the trailed vorticity strength of the vortex cylinder. The trailed vorticity strength is calculated from the total bound circulation of the two neighboring sections using Eq. (

The blade bound circulation is calculated from the circulatory part of the lift coefficient

In this section, the coupling of the vortex cylinder model and the blade element theory will be firstly described in Sect.

According to the relationship between the momentum theory and the vortex cylinder model described in Sect.

Prandtl's tip-loss factor is commonly applied to the BEM method to account for the difference between a finite number of blades and the assumption of an infinite number of blades in the momentum theory

The tip-loss correction is applied by scaling the thrust coefficient with the inverse of the tip-loss factor when calculating the blade axial induction.

Care should be taken when applying the tip-loss correction to the non-planar rotor. The first aspect is the angle to use when calculating the tip-loss factor in Eq. (

The second aspect is that the tip-loss factor is only to model the amplified axial induction at the blade compared to the annulus-averaged axial induction, and it should not directly change the trailed (tangential or longitudinal) vorticity strength of the vortex cylinders. Recall the similarity of the vortex cylinder model for the non-planar rotor and the corresponding planar rotor with the same circulation distribution described in Sect.

In the first step, the axial induction factor at the blade of the corresponding planar rotor with the tip-loss correction is calculated using Eq. (

The tangential vorticity of the vortex cylinder is calculated from the annulus axial induction of the planar rotor

Then, the annulus axial induction factor of the non-planar rotor

The tip-loss factor is only applied to the axial induction but not applied to the tangential or radial induction, which is following the application of the tip-loss correction in the BEM module in the HAWC2 code.

If the model is used without clearly distinguishing between the axial induction on the blade and the annulus-averaged axial induction, the resulting system closure could be wrong. If using the blade axial induction factor

For the planar rotor with straight blades, the calculated aerodynamic loads on the blade using the erroneous method will still be correct. The tangential vorticity from the erroneous method is wrong and the radial induction calculated using Eq. (

The tip-loss correction used in the present work is scaling the thrust coefficient when calculating the blade axial induction and is actually only applied to the planar part of the axial induction. There are other definitions of the tip-loss factor, such as the ratio of the blade axial induction and the annulus-averaged axial induction

However, consider that the tip-loss factor is originally developed for planar rotors. As in Prandtl's simple model of system of material sheets, the flow will go around the vortex disc edges. Also for the modern definition of the tip-loss factors

As has been described previously in this section, the proposed vortex cylinder model can be used in two ways: either used as a correction to the BEM method (BEM-VC) or solely used and coupled with the blade element theory (BEVC). Details of the implementation of both methods have been described previously in this work and are summarized in Algorithm 1.

The higher-fidelity models for the comparison are the Navier–Stokes solver EllipSys3D

The pressure-based incompressible three-dimensional solver EllipSys3D was used to solve the Reynolds-averaged Navier–Stokes equations, using a finite-volume discretization. An inlet/outlet strategy was followed for the boundary conditions of the outer limit of the CFD domain. The flow was assumed to be fully turbulent, and the

Several rotor-resolved meshes were built. They were generated in two consecutive steps, which were fully scripted in order to ensure a similar resulting grid quality. First, a structured mesh of the blade surface was generated with the openly available Parametric Geometry Library (PGL) tool

Visualization of straight blade CFD mesh.

While a steady solver was used, unsteady separation is expected near the root of the wind turbine blade in operation. To mitigate the effects that this can have on the conclusions of the present work, all the CFD quantities were averaged for the last 350 iterations.

The lifting-line module in the aerodynamic solver MIRAS

Three low-fidelity aerodynamic models are used for the comparison. The first one is the BEM method implemented in the HAWC2 code version 12.8

In this section, the distributed aerodynamic load in the axial and tangential direction, as well as the integrated loads of aerodynamic thrust and power from different low-fidelity models, is compared with results from higher-fidelity models. The axial and tangential loads are defined to be positive when aligned with

There are five different wind turbine blades used for the comparison; all of them are based on the IEA-10.0-198 10 MW reference wind turbine (RWT)

The parameters of these upwind dihedral blades are summarized in Table

The parameterization of the dihedral blade with dihedral ratio

Side view of the main axes of the four different upwind dihedral blades used for the comparison. The dihedral blades from left to right are W-1 to W-4.

The parameters of the planforms of the four upwind dihedral blades used for the comparison.

The airfoils are aligned perpendicular to the curved main axis, which is the half-chord line. The chord and twist distribution of the dihedral blades remain unchanged compared to the baseline straight blade. The radius of the dihedral blades is identical to that of the baseline straight blade, but the curved blade length is increased due to the dihedral. For the simulations in this section, the uniform inflow of 8 m s

For the test cases described in Sect.

Firstly, the steady-state results of the baseline straight blade without cone calculated from different models are compared and plotted in Fig.

Comparison of axial load

The steady-state results of the different upwind dihedral blades with zero cone angle are calculated with different aerodynamic models. The axial load and tangential load of the dihedral blade W-1 is shown in Fig.

Comparison of axial load

Comparison of the difference of the axial load

Comparing the distributed load of the baseline straight blade and the dihedral blade W-1, it is difficult to draw conclusions for the axial load or the tangential load, because no clear trends can be seen. In order to clearly show the influence of the blade dihedral on the loads predicted by different aerodynamic models, the difference of the loads of the dihedral blade W-1 with respect to the baseline straight blade is shown in Fig.

It can be seen that for the difference of both the axial and tangential load, both higher-fidelity models (CFD and LL) predict a fairly similar pattern of spanwise load redistribution. For the spanwise location that is further inboard compared to where the blade starts to become dihedral, the axial and tangential loads of the upwind dihedral blade are lower compared to the baseline straight blade. When moving from the spanwise location where the blade starts to become dihedral towards halfway until the blade tip, both axial and tangential loads are also lower compared to the baseline. When moving further towards the tip, both axial and tangential loads are then increased compared to the baseline until the blade tip.

A similar pattern of spanwise load redistribution was also observed for swept blades in the previous works

For both the ordinary BEM method and the BEM method with radial induction correction (BEM-

In comparison, the proposed BEM-VC method correctly predicts the pattern and the magnitude of the load redistribution for both axial and tangential loads. The decrease in the loads further inboard compared to where the blade starts to become dihedral is also well predicted. It should be highlighted that the spanwise location of the crossing of the zero load difference is also in good agreement with the results from higher-fidelity models (CFD and LL). The largest error with the BEM-VC method is mainly for the tip-most part: the increase in the load is overpredicted for both axial and tangential loads. This could be due to the use of Prandtl's tip-loss correction in the model. This model is based on a wake shape corresponding to that from a planar rotor with straight blades. The authors believe that a more advanced aerodynamic model that can replace the current tip-loss correction, if coupled with the proposed vortex cylinder model, could have better agreement with higher-fidelity models. An example is the near-wake model

It should be mentioned that the difference between the higher-fidelity models (CFD and LL) and the BEM-VC method for the baseline straight blade is of similar magnitude as the influence due to blade dihedral. However, since the model predicts the sensitivity of changes in dihedral relatively well, it is favorable to be used for parameter studies and to be eventually integrated in a multi-fidelity aerodynamic optimization framework. For example, in order to design a rotor with dihedral blades, higher-fidelity models could be used for the initial design of a straight blade. Then, the proposed vortex cylinder model could be used to explore the sensitivity of different dihedral parameters on the aerodynamic loads, with a relatively low computational effort.

The results of the other three upwind dihedral blades are shown in Appendix

To exploit the range of validity of the proposed method, a large upwind cone of 15

For the baseline straight blade with 15

Comparison of the difference of the axial load

Comparison of the difference of the axial load

For the straight blade with large cone in Fig.

To further exploit the range of validity of the proposed method, a large downwind cone angle of 15

Comparison of the difference of the axial load

For the upwind dihedral blades with 15

Comparison of the difference of the axial load

It can be seen that for both the straight blade and the upwind dihedral blade with large downwind cone, the results from the proposed method (BEM-VC) are in good agreement with the results from the higher-fidelity models (CFD and LL). On the other hand, the BEM method and the BEM method with radial induction correction (BEM-

The integrated aerodynamic loads, which are the aerodynamic power and thrust from different models, are compared in this section. Please note that when comparing the integrated aerodynamic loads, errors in the distributed loads may cancel out. So, it is important to bear in mind that the performance of the different aerodynamic models is not fully represented by their abilities to predict the total aerodynamic power or thrust of the rotor. The aerodynamic force (per unit length of radius)

The

The aerodynamic thrust of the rotor is the total contribution of the axial force of all

The aerodynamic power and thrust of the rotor with baseline straight blades without cone are summarized in Table

The aerodynamic power (in kW) and thrust (in kN) of the rotor with baseline straight blades calculated using different aerodynamic models. The operational condition is with a uniform wind speed of 8 m s

The relative difference of power of the dihedral blades without cone compared to the baseline straight blade. The operational condition is with a uniform wind speed of 8 m s

The relative difference of thrust of the dihedral blades without cone compared to the baseline straight blade. The operational condition is with a uniform wind speed of 8 m s

For the aerodynamic power, the relative change predicted by LL is underestimated compared to the prediction by CFD, but the results are showing similar trends. One of the reasons could be the use of the 2-D airfoil data in the lifting-line method. The ordinary BEM method predicts almost no influence of blade dihedral on power, except for W-1, which predicts the correct direction in which the power increases but underestimates the magnitude. The BEM-

Comparison of the difference of the axial load

For the aerodynamic thrust, the magnitude of the relative decrement is overestimated by approximately 20 % by LL compared to CFD. The BEM method predicts the correct trend that the thrust decreases but the magnitude is underestimated compared to LL and CFD. The BEM-

In summary, the proposed BEM-VC model is in better agreement with higher-fidelity models when predicting the integrated aerodynamic power and thrust of the dihedral rotor, compared to the ordinary BEM method. The BEM method with radial induction correction (BEM-

The results shown up to this point in this work all correspond to fairly high thrust coefficients around approximately 0.9. In this section, the results for operational conditions corresponding to lower thrust coefficients are shown. For simplicity, three of the operational conditions defined in the IEA Wind TCP Task 37 report

The different operational conditions of the lower loading cases used for the comparison.

For the results shown in the previous sections, the lifting-line (LL) results were in good agreement with the CFD results. Therefore, only the LL method is used to generate the higher-fidelity results for the comparison in this section.

First, the differences of the axial load and tangential load of the dihedral blade W-1-U12 and the pitched straight blade are shown in Fig.

Comparison of the difference of the axial load

Comparison of the difference of the axial load

The results for the dihedral blades W-1-U15 and W-1-U20, at wind speeds of 15 and 20 m s

The computational efforts to obtain the steady-state results used in the present work, measured in CPU time, are summarized in this section. The CFD computations using EllipSys3D were performed on DTU's high-performance computing (HPC) cluster Jess, in which each node has 20 cores running at 2.8 GHz. All the CFD simulations of the present work required a wall clock time of approximately 3.5 h when using 216 cores. The lifting-line (LL) computations using the MIRAS code were performed on the Sophia HPC cluster, in which each node has 32 cores running at 2.9 GHz. Each of the LL simulations in the present work required a wall clock time of approximately 100 h when using 32 cores. Note that the computational time for the LL method in the MIRAS code in this study is relatively high, because the settings were chosen to achieve the highest possible fidelity irrespective of the computational cost. Therefore the computational effort for the MIRAS calculations in this work is not indicative of the performance for normal use of the tool. Settings that increased the computational effort in this work are small time steps, not using far-wake cutoff, etc. The computational time is expected to be largely decreased if efforts are dedicated to improving the simulation setup. However, this is beyond the scope of the present work.

The computations using the HAWC2 code were performed on a single core of a 2018 workstation at 4.8 GHz. The simulations were performed with structural properties included and with large stiffness to approximate stiff structures. The simulations were run for 600 s in the simulation time to reach steady state. The simulations required a wall clock time of approximately 600 and 650 s for the BEM method and the BEM-VC method, respectively. For a stand-alone version of the BEM method or the BEM-VC/BEVC method, one steady-state computation can be done in much less than 1 s using a single CPU core.

A new computationally efficient method for the aerodynamic load calculation of non-planar rotors is described. The method is based on the vortex cylinder model and can be used in two ways: either as a correction to the currently widely used blade element momentum (BEM) method or as the main model, replacing the BEM method in the engineering modeling complex. For uniform inflow that is perpendicular to the rotor plane, the influence of the blade out-of-plane shapes on the distributed aerodynamic loads, measured by the difference of the loads between the non-planar rotor and the planar rotor, is shown to be in good agreement with higher-fidelity models. The predicted distributed and integrated aerodynamic loads are in better agreement with higher-fidelity models than the baseline BEM method and also a BEM method with a radial induction correction. While the present work focused on stiff geometries, the developed framework would be able to handle out-of-plane deflections during aeroelastic simulations accounting for blade elasticity, without any loss of generality. The new model is approximately as numerically efficient as ordinary BEM-based models, which makes it favorable for aero-servo-elastic simulation as well as design optimization of horizontal-axis wind turbines whose blades have out-of-plane shapes. Therefore, the authors recommend the use of the proposed model as a correction to the existing BEM codes.

For the future work on the model applications, it would be interesting to use both the standard BEM method and the proposed method for the aerodynamic or aeroelastic design of a non-planar rotor under the same constraints. Higher-fidelity models, such as CFD or lifting-line method, could be used for the benchmark of the different designs, as done in the present work. The method is also favorable for integration in a multi-fidelity aerodynamic design framework. There are also several ways in which future work could improve the model. Firstly, it would be favorable to have modifications to the existing Prandtl tip-loss correction. For example, it is possible to use the distance between the tip vortex and the calculation point when calculating the correction for a non-planar rotor, instead of using the radial distance as currently implemented in the model. Secondly, it would be beneficial to further develop the model for the application of blades with both in-plane and out-of-plane shapes. One possible track of the development is to couple the vortex cylinder model and the near-wake model, which approximately models the near wake as helical trailed vorticities and is currently coupled with a far-wake BEM method. Thirdly, it would be interesting to investigate the unsteady effects of the non-planar rotor, such as aerodynamic damping and dynamic inflow effect. Fourthly, it would be beneficial to further develop the vortex cylinder model for the application of non-planar rotors in yawed flow. Finally, further development of the model focusing on analytical gradients would be favorable for application in a gradient-based wind turbine design optimization framework.

The difference of the loads of the dihedral blades (W-2 to W-4) with zero cone compared to the baseline straight blade without cone.

Comparison of the difference of the axial load

Comparison of the difference of the axial load

Comparison of the difference of the axial load

The difference of the loads of the dihedral blades (W-2 to W-4) with 15

Comparison of the difference of the axial load

Comparison of the difference of the axial load

Comparison of the difference of the axial load

The difference of the loads of the dihedral blades (W-2 to W-4) with 15

Comparison of the difference of the axial load

Comparison of the difference of the axial load

Comparison of the difference of the axial load

The code is not provided because it is part of the in-house tool HAWC2. However, the reader is able to recover the code using the algorithm provided in this paper.

The airfoil data are from 2-D fully turbulent CFD results (

AL conducted the study as part of his PhD research. The idea of the proposed model originated from MG. The proposed model was jointly developed by MG, AL and GRP. The similarities of the superposition of the vortex cylinder model of non-planar rotor and planar rotor were described by AL and MG. The application of the tip-loss correction as well as high-thrust correction for the non-planar rotor was described by AL and MG. The implementation of the proposed model in HAWC2 code and the computations using the HAWC2 code were performed by AL with contribution from GRP. The CFD method was introduced by SGH, and the CFD results were computed by SGH. The post-processing of the CFD results was performed by SGH with contribution from AL. The lifting-line results were computed by AL, and the post-processing was performed by AL. All authors jointly drew the conclusions of the work and contributed to writing this paper.

DTU Wind Energy develops and distributes HAWC2 on commercial and academic terms.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors would like to thank our colleague Néstor Ramos García in DTU Wind Energy for the suggestions in the lifting-line simulation using MIRAS, a vortex code mainly developed by him.

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 two anonymous referees.