In this work, a computationally efficient engineering model for the aerodynamics of swept wind turbine blades is proposed for the extended blade element momentum (BEM) formulation. The model is modified based on a coupled near- and far-wake model, in which the near wake is assumed to be the first quarter revolution of the non-expanding helical wake of the own blade. For the special case of in-plane trailed vorticity, the original empirical equations determining the steady-state value of the near-wake induction are replaced by the analytical results, which are in the form of incomplete elliptic integrals. For the general condition of helical trailed vorticities, the steady-state near-wake induction is approximated based on the results of the special conditions and a correction factor. The factor is calculated using empirical equations with influence coefficient tensors, to minimize the computational effort. These influence coefficient tensors are pre-calculated and are fitted to the results from the numerical integration of the Biot–Savart law. With the indicial function approach, it is not necessary to explicitly save the information of the vorticities that were trailed in the previous time steps. This engineering approach is a combination of analytical results and numerical approximations, with low and constant computational effort for each time step. The proposed model is practically applicable to time-marching aero-servo-elastic simulations. The results of the swept blades with uniform inflow perpendicular to the rotor calculated from the proposed model are compared with the results from a BEM code, a lifting-line solver and a Navier–Stokes solver. The significantly improved agreement with the higher-fidelity models compared to the BEM method highlights the performance of the proposed method.

With the technological advancements in the design optimization and manufacturing of horizontal-axis wind turbines, the turbine blades are becoming increasingly flexible. Thus, there could be significant in-plane and out-of-plane deformations due to the aeroelastic loads. In addition, there is an increasing interest in the backward swept blades because of the possibility to achieve passive load alleviation with geometric bend–twist coupling

In the spectrum of the lower-fidelity models, the most commonly used blade element momentum (BEM) method implicitly assumes a planar rotor with straight blades. If the actuator disc (AD) is not planar, the induction deviates from what the BEM model predicts as demonstrated by

This led to the formulation of the coupled model (usually referred to as the near-wake model) by

Since the first version of the model in 2004, there have been several improvements. Integration in the multibody aeroelastic HAWC2 code is presented in

There has been previous work by

In the present work, the background of the engineering aerodynamic models for horizontal-axis wind turbines is first briefly described. Then, the details of the near-wake model, including the analytical solutions as well as the engineering approaches for a computationally efficient implementation, are described. Afterwards, the far-wake model and the coupling method are briefly discussed. Finally, the aerodynamic loads of the swept blades under the special condition of uniform inflow perpendicular to the rotor plane predicted by the proposed model are compared with the results from a BEM code, a lifting-line solver and a CFD Reynolds-averaged Navier–Stokes (RANS) solver.

For the application of aeroelastic simulations of wind turbines, there are multiple low- and mid-fidelity engineering aerodynamic models with different assumptions.
An example of a low-fidelity model is the polar grid implementation of the blade element momentum (BEM) method with unsteady aerodynamics

An example of the higher-fidelity model is the lifting-line method, which models each blade of the rotor with a bound vortex line. This is under the assumption that the bound vorticity of a blade is concentrated into a line vortex at the quarter-chord line.
Vortices are trailed from the bound vortex line, with the trailed vorticity strength equal to the spanwise gradient of the bound vorticity.
The trailed vortices are modeled with helical vortex filaments and could possibly include the wake expansion effect.
There is also shed vorticity for the unsteady conditions.
Compared to the BEM method, the lifting-line method models the blade and the wake using vortex line filament and helical vortex filaments instead of using superposition of actuator discs and concentric vortex cylinders. The assumption that the blades are straight and are located in the rotor plane can be relaxed.
In addition, the influence of the non-straight bound vortex on itself should also be explicitly included

The coupled near- and far-wake model is considered as a hybrid of the aforementioned two methods.
For the first quarter revolution of the own wake of every blade, which corresponds to the near wake, the model is similar to the lifting-line method without wake expansion.
In the modified coupled model by

Illustration of the modeling of the blade and the wake in the three different engineering aerodynamic models. In the lifting-line method

In the modified coupled model proposed in the present work, the assumption of straight blades in the original coupled model is partially relaxed. The bound vortex can be curved but is constrained to the rotor plane, which means the blades can be swept forward or backwards.
There are two key features of the modified model, and they correspond to two impacts of the blade sweep on the vortex system.
The first one is the influence of the curved bound vortex on itself, which has been described by

The modified near-wake model is similar to the modified lifting-line model for curved wind turbine blades that is labeled as the “LL-test” in

The trailing function represents the induction due to an elementary trailed vorticity arc, depending on its azimuthal location relative to the blade. In a previous work

The coordinate system used in the present work is consistent with the commonly used conventions for wind turbine aerodynamics.
In this work, we assume the blade has no prebend, which means the out-of-plane component of the geometry is assumed to be zero.
However, if prebend exists, the projection of the blade main axis into the rotor plane should be used to calculate the sweep geometry for the input of the model proposed here.
The origin of the coordinate system is located at the rotational center of the rotor, and it is locally defined for every blade and every section. The

The front view of a backward swept wind turbine blade that is rotating clockwise with the rotational speed

The radius of the trailing point is denoted as

It is assumed that the near-wake part of the trailed vorticity convects downstream with the velocity determined at the blade. This is because the first quarter revolution of the wake is generally very close to the rotor plane where it is emitted.
The in-plane and out-of-plane components of the flow velocity at the trailing points are

The

The relative velocity

In the previous work by

Assuming both

The

The other components of the two vectors of

For the infinitesimally trailed vorticity element

The elementary axial and tangential induced velocity, which are the

In the above equations, the length of the elementary trailed vorticity arc is

Recall that the near-wake part of the trailed vorticity is defined as the first quarter revolution of the wake of the own blade. Thus, the integral of the trailing functions in Eqs. (

The value of

The numerical implementation of the lifting-line method and the coupled method requires the radial discretization of the blade. If the blade is discretized into

For the free-wake lifting-line method that is implemented as a time-marching fashion for numerical computations, the vortex wake system is evolving and its size is growing in time. The information of the vorticities trailed and shed in the previous time steps has to be explicitly stored. For every single vortex element, there will be influence from all other vortex elements on it.
For each time step, the size of the problem is of the order of

In the near-wake model, the trailing functions in Eqs. (

Assuming

When the value of

In Eq. (

One of the important features of the near-wake model is the use of exponential functions to approximate the trailing function that is based on the Biot–Savart law. The approximated trailing function can then be integrated using the indicial function approach instead of using direct numerical integration. With this approach, the information of the individual trailed vortex elements emitted from the previous time steps is implicitly stored.
For every time step, it is only necessary to calculate the decrement of the induction at the previous time step and the increment of the induction at the current time step.

The fast and slow response terms are calculated separately and then summed together to get the complete near-wake induction.

The problem is now of the order of

It could be confusing that the approximated value of the steady-state near-wake induction in Eq. (

For the analytical near-wake induction

For the approximated induction in Eq. (

The difference between the analytical and the approximated near-wake induction is illustrated in Fig.

Illustration of the difference between the analytical and the approximated normalized trailing functions

The different methods of obtaining the normalized steady-state near-wake induction

In the original implementation of the near-wake model by

There are two major limitations when using the empirical functions in Eq. (

There has been previous work by

Secondly, the influence of the helix angle on the near-wake induction is modeled by introducing the convective correction.
The value of

The weight

This approach has a very low computational cost, which is crucial for the efficiency of the coupled near- and far-wake model.
The approximated steady-state axial induction of a straight blade after these corrections has a reasonably good accuracy.
In addition, the near-wake part of the tangential induction is included in the modification. It is argued by

Recall the procedures to approximate the steady-state near-wake induction in the previous modifications by

Firstly, for the special condition of zero helix angle (in-plane trailed vorticity), modification is needed to get the correct steady-state results for the swept blades.
In the original empirical equation of

Secondly, the idea of convective correction for the general case of an arbitrary helix angle is used, but the definition is adjusted.
The convective correction is now defined as the function to obtain the steady-state induction from the special condition of in-plane trailed vorticity (

In the previous modifications by

The relative position

In order to solve this problem, the equivalent relative position

Another procedure to ease the process of obtaining the influence coefficient tensors is to normalize the sweep angle

Illustration of the variation of the range of the sweep angle

The spread of the realistic points in the 2-D plot of

In addition, when the value of

As a result, it is favorable to normalize the sweep angle to spread the realistic design space more evenly inside the cubic parameter space for data fitting and also to proactively enlarge the spread of the realistic conditions for a small value of

Since the data fitting is practically performed in a cuboid parameter space, it is necessary to determine the range of each variable. For the value of

To obtain the range of the normalized sweep angle, an initial numerical study is performed by calculating the value of

The parameterization of the swept blade with sweep ratio

The purpose of this preliminary study is to determine the range and also the Pareto front of the design variables. So, the range of the geometric variables for this numerical study is chosen to represent the blades with relatively large sweep.
The range of the sweep ratio is from 0.25 to 0.75. The ratio of the sweep magnitude over the sweep ratio is set to vary between 0.2 and 1. So, the swept magnitude

The scatter plot of the realistic conditions of the normalized sweep angle

The range of

Secondly, according to the scatter plot in Fig.

In this section, the modified convective correction is described in detail. The idea is similar to the method of calculating the corrected value of

For the trailing functions of

The steady-state value of the near-wake base induction corresponds to the integral of the base trailing functions in Eqs. (

The normalized base axial and tangential inductions are also introduced; they are defined similar to the normalized axial and tangential induction in Eqs. (

For the special condition of in-plane trailed vorticity

If the shape of the blade does not change (or the change is within a threshold) between two time steps, only the helix angle

The convective correction is an empirical composite function of three independent variables which corresponds to three layers.
These empirical functions are based on polynomial functions and rational functions. The composite functions are designed so that there is only one independent variable for each layer.
Then, an optimum approach will be letting the helix angle

For a given combination of the three design variables

Following the aforementioned description, the function of the convective correction is a triple composite function that has the form as in Eq. (

The influence coefficient tensors for the axial and the tangential induction are different and will be described separately. In addition, the whole design space is divided into several sub-spaces with their own influence coefficients, which is for the ease of data fitting.
The empirical functions for both the axial and tangential normalized base induction and for all the regions are the same and are as follows:

For the approximation of the normalized axial induction

The first region corresponds to the first and fourth quadrant of the design space of

The definition of the three regions for the parameter space of the equivalent relative position

The second region corresponds to the third quadrant of the design space of

The third region corresponds to the second quadrant of the design space of

The influence coefficient tensors of

For the approximation of the normalized tangential induction

The first region corresponds to the first and fourth quadrant of the design space of

The second region corresponds to the second and third quadrant of the design space of

As for the axial induction, the influence coefficient tensors of

The definition of the two regions for the parameter space of the equivalent relative position

The quality of the fitted influence coefficients for the modified convective correction described in Sect.

The numerical integration is calculated using the Runge–Kutta algorithm with Dormand–Prince method implemented in the ode45 function in MATLAB version 2020a

It can be seen that for both the axial and the tangential induction, the results calculated using the convective correction method with the fitted influence coefficient tensors have relatively high accuracy. In addition, for both the base axial and tangential induction, and for all regions, the relative error is always zero when

The user of the coupled model should bear in mind that the model has its limitations with a certain range of validity. The data fitting was performed on a relatively large range, which is intended to cover most of the swept blades. However, it is possible that the input value is outside of the range of validity. As a result, it is necessary to put a limit to the input parameters for the model to avoid catastrophic failure of the model. The range of the input variables and the corresponding physical representation are explained. Then, the limits on the input variables and their effects are described.

For the helix angle

For the normalized relative position

For the normalized sweep

The indicial function described in Sect.

For the axial induction, the modified indicial functions are

For the tangential induction, the modified indicial functions are

The algorithm of computing the axial and tangential near-wake induction using the convective correction is summarized in this section. The algorithm corresponds to the calculation from the dynamic bound vorticity strength

The basis for the far-wake model is the BEM model implemented in the HAWC2 code

The far-wake axial induction is calculated as a function of the scaled thrust coefficient

For the case of straight blades, previous studies have illustrated that the coupling factor is able to be automatically adjusted during the computation. Indeed, the dynamic response of the coupled model shows improved agreement with higher-fidelity models and experiments, when compared to the BEM method

For the application of the steady-state aerodynamic load calculation of swept blades under uniform inflow that is perpendicular to the rotor plane, it is also possible to fix the coupling factor equal to that of the baseline straight blade. As will be described in Sect.

In order to assess the performance of the proposed coupled near- and far-wake model, the results from two higher-fidelity aerodynamic models are used for the comparison. In particular, a version of the lifting-line method implemented in the MIRAS code

In the lifting-line method used for comparison, the bound vorticity is represented by the concentrated lifting line that is located at the quarter-chord line of the blade. This is where the trailed vortices emanate from and will form the helical vortex wake system. The induced velocity due to the trailed vorticities is evaluated at the quarter-chord line, with a possible contribution from the shed vorticity in the unsteady case.
The influence of the curved bound vortex is modeled by adding the difference of the induced velocity due to the 3-D bound vorticity and an imaginary 2-D bound vorticity (infinitely long line vortex) evaluated at the three-quarter-chord point to the induction of the blade section.
This implementation of the lifting-line method is labeled as the LL-test in the previous work of

Apart from the lifting-line method, the results from a rotor-resolved Navier–Stokes solver were also used for comparison.
The in-house finite-volume code EllipSys3D solves the incompressible Navier–Stokes equation on a structured grid. Several approaches are available in EllipSys3D for dealing with turbulence. In the present study, the RANS formulation in combination with the k-

The modified coupled near- and far-wake model is implemented in a test version of the in-house aero-servo-elastic simulation tool HAWC2 based on the release version 12.8

The BEM method implemented in the HAWC2 code version 12.8 is also used for the comparison

In this section, the aerodynamic loads calculated from different models are compared. The blades are assumed to be stiff, which means the effect of elastic deformation is not included.

In the previous work of

The previous argument is erroneous and will be illustrated using the vortex theory. It has been described in Sect.

Instead, the reason is discovered to be the inconsistent definition of the loads.
Recall the procedures to obtain the aerodynamic loads in the lifting-line-like methods that rely on 2-D airfoil data, such as the BEM method, the lifting-line method and the coupled near- and far-wake model.
For each blade section, the 3-D velocity at the calculation point consists of the induced velocity, the blade motion, and the onset flow and is projected into the 2-D airfoil section. After subtracting the 2-D bound vorticity induction at this section, the angle of attack and the relative velocity are calculated from the velocity triangle. Then, the 2-D lift and drag force can be calculated and are projected with respect to the rotor plane to obtain the in-plane and out-of-plane loads.
The resulting aerodynamic loads should correspond to force per unit length of curved blade length, since they are from the 2-D aerodynamic loads. If we want to have other definitions of the load, we have to multiply the load with the corresponding scaling factor. For example, to get the loads with the definition of force per unit radius, the factor

In this work, the in-plane and out-of-plane loads are defined as force per unit length of

The wind turbine blades that are used for the comparison are modified based on the IEA-10.0–198 10 MW reference wind turbine (RWT)

The operational condition is the same as the in previous work by

The parameters of the planforms of four backward swept blades

The top view of the backward swept blades Blade-1 to Blade-4 together with the baseline straight blade.

A set of rotor-resolved meshes were used for the CFD simulations, each of them corresponding to a different blade geometry. They were generated in two consecutive steps, which were fully scripted in order to ensure a similar resulting grid quality. Firstly, a structured mesh of the blade surface was generated with the openly available Parametric Geometry Library (PGL) tool

For the lifting-line method, each time step corresponds to

For the modified coupled near- and far-wake model and the BEM method implemented in the HAWC2 code, each time step corresponds to 0.01 s and each simulation is calculated for 600 s. Each blade is discretized radially into

Firstly, the loads of the baseline straight blade calculated from the BEM method, the modified coupled model (NW), the lifting-line method (LL) and the Navier–Stokes solver (CFD) are compared in Fig.

For the out-of-plane loads, the results from all the models have good agreement. At the

The differences between the CFD and LL are assumed to be related to the 2-D airfoil aerodynamic coefficients retrieved from the lookup table involved in the lifting-line approach. This source of disagreement is also to be considered for BEM and for the coupled method. The relative difference of the loads calculated from BEM and the coupled method compared to the loads from LL is relatively small. This means both the BEM and the coupled method can be used in the design optimization of a straight blade with acceptable accuracy.

Comparison of out-of-plane load

The steady-state results of the swept blades are also calculated from the BEM method, the modified coupled model, the lifting-line models and the CFD. In order to clearly show the influence of the backward sweep on the loads, the difference between the loads of the backward swept blade Blade-1 with respect to the baseline straight blade is shown in Fig.

Comparison of the difference between the out-of-plane load

For both the out-of-plane and in-plane load of the backward swept blade, the results from the coupled method of either automatically adjusted or fixed coupling factor are very similar.
For the offset of the out-of-plane load, the result from the coupled method is in good agreement with the lifting-line method. The results are also in harmony with the result from CFD. For the inboard part of the swept blade in which the main axis is still straight, the out-of-plane load of the swept blade is almost identical to that of the baseline straight blade. When moving towards the blade tip, the out-of-plane load of the swept blade is lower compared to the baseline straight blade until approximately halfway until the blade tip. Then, when moving further towards the tip, the load of the swept blade is higher compared to the baseline straight blade until almost all the way until the blade tip. This pattern was also observed in the previous work

The BEM method is not able to correctly predict this pattern of the radial redistribution of the loads. For the out-of-plane load, it predicts a maximum increase of the load near the blade tip of approximately 100

The results of the other backward swept blades are shown in Appendix

The difference between the loads of the forward swept blade Blade-5 with respect to the baseline straight blade is shown in Fig.

Comparison of the difference between the out-of-plane load

For the coupled method with fixed coupling factor, the results of both out-of-plane load and in-plane load are in good agreement with the higher-fidelity lifting-line method and CFD. However, for the coupled method with automatically adjusted coupling factor, the loads have significant offsets compared to the higher-fidelity models. This means the current coupling method is not capable of correctly adjusting the coupling factor automatically.

Similar to the backward swept blade cases, the BEM method is not able to predict the radial redistribution of the loads but predicts an increase of the load compared to the baseline straight blade near the blade tip. For the in-plane load, the BEM method predicts that the in-plane loads of the swept blade and the straight blade are almost identical along the span.

The results of other forward swept blades are shown in Appendix

The integrated aerodynamic rotor 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 on each blade section is

The negative

According to Eq. (

The aerodynamic thrust of the rotor is the total contribution of the out-of-plane force of all

The aerodynamic power and thrust of the rotor with baseline straight blades as well as the rotors with swept blades predicted by the LL method, the BEM method and the coupled method with fixed coupling factor are calculated.
It is difficult to directly draw conclusions from the absolute values of power and thrust.
To better illustrate and compare the integral effects of the blade sweep represented by the aerodynamic power and thrust predicted using different methods, the relative difference of the aerodynamic power and thrust with respect to the baseline rotor from each method are calculated and are summarized in Tables

The aerodynamic power (in kW) of the baseline straight blade and the relative difference in aerodynamic power (in %) of the different swept blades with respect to the baseline blades calculated using different aerodynamic models. The operational condition has a uniform wind speed of 8 m s

The aerodynamic thrust (in kN) of the baseline straight blade and the relative difference in aerodynamic thrust (in %) of the different swept blades with respect to the baseline blades calculated using different aerodynamic models. The operational condition has a uniform wind speed of 8 m s

For the aerodynamic power, the magnitude of the relative difference predicted by the BEM method is underestimated compared to the prediction by LL for all blades. Compared to the BEM method, the relative change of power predicted by the proposed coupled method with fixed coupling factor is in significantly improved agreement with the predictions by LL. For backward swept blades, the maximum error of predicted increment is 0.76 %, which is smaller than the prediction of 1.36 % by the BEM. For forward swept blades, the maximum error of predicted decrement is 1.68 %, which is smaller than the prediction of 2.65 % by the BEM.

For the aerodynamic thrust, the predictions by the BEM method still have acceptable agreement with the predictions by LL. The offset in the predicted the aerodynamic thrust by BEM is smaller compared to the offset in the predicted aerodynamic power. For all blades except for Blade-2, the predictions by the coupled method with fixed coupling factor is in improved agreement with LL compared to the BEM method. The maximum difference of the change of thrust predicted by the coupled method and the LL is 0.11 % for all blades except Blade-2, which has an offset of 0.29 %. In comparison, the maximum difference predicted by the BEM method is 0.67 %.

In summary, the proposed coupled method with fixed coupling factor is in better agreement with higher-fidelity models compared to the ordinary BEM method, for both backwards and forward swept blades.

The computational effort to obtain the steady-state results that are used in the present work, measured in CPU time, are summarized in this section. The CFD computations using EllipSys3D were performed on the Jess high-performance computing (HPC) cluster, 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. The computational time for the LL method in the MIRAS code in this study is high, because the settings were chosen to achieve the highest possible fidelity irrespective of the computational cost. Therefore, the MIRAS computational effort should not be directly compared to the CFD simulation that uses EllipSys3D. Settings that increased the computational effort are small time steps, not using far-wake cut-off, etc. The computational time is expected to be largely decreased if efforts are dedicated to improving the simulation setup. When using large time steps, the LL method in the MIRAS code with the same cluster setup can be converged with a wall clock time of approximately 10 min. And for an aeroelastic simulation of 600 s, the computation using the same cluster requires a wall clock time of approximately 12 h. However, this is beyond the scope of the present work. The computations in 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 for 600 s to reach steady state. The simulations require a wall clock time of approximately 520 and 750 s for the BEM method and the proposed coupled method, respectively. The computational effort of the coupled method is similar to the BEM method because the stiff structural properties are used so that the blade geometry does not need to be updated during the calculation. In addition, since the operational condition between two time steps is very similar, it is not necessary to perform sub-iterations. However, for an aeroelastic simulation with the flexibility of the system enabled and assessing highly dynamic load cases (e.g., turbulent inflow), preliminary assessments by the authors indicate that the additional computational cost due to the coupled method remains below 100 % compared to the aeroelastic simulation using the BEM method. For a stand-alone version of the BEM method, one steady-state computation takes much less than 1 s on a single CPU core. For the coupled method, one steady-state computation takes less than 1 s on a single CPU core if using an efficient algorithm to calculate the incomplete elliptic integrals as in the present work. However, the computational time can be extended to approximately 10 s if using an inefficient algorithm, such as direct numerical integration.

A computationally efficient modified coupled near- and far-wake engineering aerodynamic model for the swept wind turbine blades is proposed. The core of the modifications in this work is to obtain the steady-state induction of the near wake, which is defined as the first quarter revolution of the helical trailed vorticity of the own blade. To achieve this, an engineering approach that combines analytical solutions and approximations based on pre-calculated influence coefficient tensors is proposed. The far-wake model is currently based on a far-wake BEM method. The near- and far-wake model are coupled with a coupling factor that is to scale the far-wake induction, so that the thrust of the whole rotor is similar to that calculated from the BEM method. For the calculation of the steady-state condition with the uniform inflow applied perpendicular to the rotor plane, a fixed coupling factor that is determined according to the baseline straight blade can be applied.

The modified model is used to calculate the steady-state loads of the baseline straight blade, four backward swept blades and four forward swept blades that are modified based on the IEA-10.0-198 10 MW reference wind turbine. The influence of the blade sweep on the loads predicted by the proposed method is shown to have good agreement with the prediction from higher-fidelity models, which are a version of the lifting-line solver and a Navier–Stokes solver. The numerical comparison shows that the BEM method is not able to correctly model the influence of blade sweep and has large discrepancies with the results from the two higher-fidelity models. The improvement of the proposed coupled method over the BEM method is significant, and the results from the proposed method have similar performance to the lifting-line method. The proposed method is computationally efficient and favorable for the application of wind turbine aero-servo-elastic simulations and design optimization. The method shows improved agreement with higher-fidelity models compared to the conventional BEM method when the model is carefully used. However, the current coupling method is not suitable for aeroelastic calculation of forward swept blades. Further work on the far-wake model and the coupling method is needed for the method to be confidently used in the aeroelastic simulations for general swept blades.

There are several future works needed to further improve the model. Firstly, it is favorable to also have the parameters representing the dynamics of the indicial functions fitted to numerical results. This can improve the dynamic response of the coupled model. The dynamic response of swept blades from the coupled model should also be compared with results from higher-fidelity models. Secondly, using the method of fixing the coupling factor for forward swept blades reflects the limitation of the current far-wake BEM model. It may be favorable to use the vortex cylinder model as the far-wake model instead. If so, a new method to couple the near-wake model and the far-wake model with a new definition of the coupling factor is needed. Thirdly, it could be useful to have the model further modified for the application of blades with both in-plane and out-of-plane shapes. This will also require the use of the vortex cylinder model as the far-wake model, which has the potential to model the aerodynamic effects of the blade out-of-plane shapes. Finally, it is beneficial to investigate further possible improvements to the lifting-line method for the application of curved wind turbine blades. Then, the coupled near- and far-wake model can be improved according to it. One example is the modeling of the radial viscous drag force, especially for the swept blades.

The analytical solutions for the two special conditions of in-plane trailed vorticity and straight trailed vorticity are derived. They correspond to the lower and upper limit of the helix pitch angle

For the special condition of in-plane trailed vorticity

The integrals of the base induction functions in Eqs. (

For the simplicity of the notation, the steady-state base inductions are normalized and are as follows.

The indefinite integrals corresponding to the definite integral of

In Eqs. (

The advantage of the derived analytical equations in the form of elliptic integrals over the original form is because of the existence of fast approximation methods, such as the work by

The analytical steady-state results for the special condition of in-plane trailed vorticity can then be calculated with low computational efforts. The normalized steady-state value of the base near-wake induction is

To be noted, for this special condition of in-plane trailed vorticity, the near wake, which is the first quarter revolution of the wake of the own blade, is equivalent to one-quarter of a vortex ring.

The reason of defining the first quarter revolution as near wake possibly originates from the introduction of the near-wake model by

Comparing the steady-state value of the axial and tangential near-wake base induction in Eqs. (

It has been proposed by

According to Eq. (

For the special condition of straight trailed vorticity (

Integrating the base trailing function in Eqs. (

The definite integrals are derived as follows. They correspond to the base induction of a semi-infinite line vortex.

So, the normalized based axial and tangential inductions for this special condition of straight trailed vorticity are

The derived analytical equations are further analyzed. Firstly, for the condition of

Secondly, the relationship between the normalized base induction of

For the special condition that the blade is straight without sweep, which means

The difference of the loads of the backward swept blades (Blade-2 to Blade-4) compared to the baseline straight blade.

Comparison of the difference between the out-of-plane load

Comparison of the difference between the out-of-plane load

Comparison of the difference between the out-of-plane load

The difference of the loads of the forward swept blades (Blade-6 to Blade-8) compared to the baseline straight blade.

Comparison of the difference between the out-of-plane load

Comparison of the difference between the out-of-plane load

Comparison of the difference between the out-of-plane load

The influence coefficient tensor in double-precision floating-point format with full digits can be found in the online supplement

The relative error of the convective correction using the influence coefficients with full digits or reduced digits.

A repository

AL conducted the study as part of his PhD research. AL, GRP, MG and HAaM jointly developed the modified coupled near- and far-wake model. AL, GRP and MG contributed to the modification to the near-wake model. HAaM, GRP and MG contributed to the far-wake model and the coupling method. AL derived the analytical equations of steady-state near-wake induction in the form of elliptic integrals. AL proposed the method of generalized relative position and normalization of the sweep angle, with contributions from GRP and MG. The data fitting to calculate the influence coefficient tensors was performed by AL, with contributions from GRP. The results of the coupled method and the BEM method were computed by AL. The lifting-line results were computed by AL. The CFD method was introduced by SGH and the CFD results were computed by SGH.

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 for the help and suggestions in the lifting-line simulation using the MIRAS code. The authors would also like to thank our colleague Alexander Meyer Forsting for discussions on the relevant topics.

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.