We show that the upscaling of wind turbines from rotor diameters of 15–20 m to presently large rotors of 150–200 m has changed the requirements for the aerodynamic blade element momentum (BEM) models in the aeroelastic codes. This is because the typical scales in the inflow turbulence are now comparable with the rotor diameter of the large turbines. Therefore, the spectrum of the incoming turbulence relative to the rotating blade has increased energy content on

The blade element momentum (BEM) model

When describing an aeroelastic code, it is often just mentioned that BEM is the model for computing the aerodynamic forces and that the model is further extended with submodels for tip loss, yawed conditions, dynamic inflow and dynamic stall. This is an incomplete description, as implementation details such as the way the models are coupled together can influence the computational results considerably. The most important aspect is how the BEM model is implemented to model the induction response due to the unsteady and non-uniform loading over the rotor caused by the atmospheric turbulent inflow, wind shear or control actions like pitch and flap control.

The purpose of the present article is to present in detail a complete unsteady BEM induction model for non-uniform inflow and loading that can be readily implemented.

The non-uniform unsteady loading over the rotor disk due to the atmospheric inflow increases with rotor size. Thus, the requirements to the BEM modeling capability have changed considerably from the 15–20 m diameter rotors in the 1980s to 100–200 m rotors today. This important effect of turbulence scales relative to rotor size was already described by

To illustrate how the upscaling of rotors leads to a more non-uniform inflow and thus non-uniform loading of the rotor when operating in turbulent inflow (no shear), we simulate two turbines with the aeroelastic code HAWC2 (Horizontal Axis Wind turbine simulation Code;

As the turbine blades rotate through the turbulent vortex structures, the spectrum of the free wind speed at the tip of the blades has energy concentrated on multiples of the rotational frequency

Rotational sampling of turbulence for different turbine sizes.

The increasingly non-uniform rotor loading with turbine size is also caused by inflow with shear. The largest modern turbines with the blade tips at top positions around heights of 300 m now span most of atmospheric boundary layer containing the main part of the shear

Other challenging wind situations comprise non-stationary wind conditions containing trends, such as wind shear developing over time. For very large rotors, these situations are important for the extreme load levels during operation. Thus, they need attention in the modeling phase if turbine designers shall be able to counteract such events using either active or passive load alleviation techniques.

Besides the upscaling trend, turbine design has changed in the same time span of years, which results in new requirements for the aerodynamic modeling in the aeroelastic codes. Pitch control is now the common power regulation method; therefore, situations like pitch fault have to be simulated for certification. Such a situation with, e.g., one blade pitch differing from the pitch of the other blades with, e.g., 20

The subject of sheared and turbulent inflow was part of the work in the EU-funded UpWind project (2006–2011) with the main objective to study upscaling of turbines to 8–10 MW. The aerodynamic flow mechanisms at high shear in the inflow were investigated by simulating the sheared inflow on the 5 MW reference wind turbine

Similar work was continued in the EU-funded AVATAR project (2013–2017) with a focus on even bigger turbines (10 MW and higher) than in the UpWind project. A summary of the findings has been presented by

The basic BEM formulation originates from Glauert and was developed for airplane propellers

When the momentum part of the BEM theory is used in aeroelastic simulations,
the actual flow conditions violate most assumptions in the basic theory: (1) turbulent and sheared inflow compared with the assumption of uniform, steady inflow; (2) non-uniform load in contrast to the assumed uniform loading and (3) skewed inflow in contrast to assumed axial inflow, just to mention the most important violations. To compensate for this, a number of submodels are introduced like dynamic inflow and skewed wake models. However, there is no real consensus on how the different phenomena should be modeled and how the submodels should interact. Therefore, we often see considerable deviations for BEM simulations on complex inflow cases

Many researchers have contributed over time to the development of the BEM theory for wind turbines but only a few will be mentioned here.

Later, a comprehensive description of the BEM modeling is presented in the handbook of

In Sect. 3, we present a detailed description of the implementation of the grid BEM approach. However, first, in that section, we give a short introduction to the origin of the CFD simulations of the actuator disk flow used heavily in developing and tuning the submodel for yawed flow, the dynamic induction model and a submodel for radial induction. The mechanism of induction in turbulent and sheared flow is explored in Sect. 4, and we present the load and power impact for two turbines for design load case (DLC) 1.2 load cases. In Sect. 5, a selection of validation cases is presented, followed by conclusions in Sect. 6.

The overall idea with the present BEM implementation is to model the rotor as an actuator disk (AD) that is updated at each time step in stationary grid points covering the rotor disk. In an aeroelastic simulation, the loading will normally be non-uniform and unsteady as discussed above. The input to the computation of the induced velocities is thus the distributed normal and tangential loading on the AD, and it will be shown in Sect.

The general purpose CFD code FIDAP (Fluid Dynamics Analysis Program), based on the finite element method, is used for the AD computations. It was one of the first commercially available CFD codes and has an unstructured mesh capability which reduces the requirements to the total number of nodes.

In the past, the code has been used for several studies of the flow through an actuator disk model. In a first setup from 1996, the computations show good correlation with the momentum theory with one-third induction at the rotor disk and two-thirds in the far field for a prescribed uniform loading corresponding to a thrust coefficient of 0.89

Later, in 1999, the AD model was coupled to the aeroelastic code HAWC

The CFD mesh and model from this setup is used for the present simulations with a prescribed uniform loading on the disk (Fig.

The CFD mesh used for the AD yaw computations. The velocity contours for computation of a 30

The fundamental part of the BEM model

For thrust values causing higher induced velocities than

For different reasons explained below, we use a BEM implementation where the induction in the whole operational range from negative

The approximation of the basic momentum relation between

For

One important reason for using a polynomial fit to Eq. (

A next step in implementing the BEM model is to couple the momentum theory to the blade element theory where the forces on a blade section are derived by means of two-dimensional airfoil characteristics. We apply Eq. (

Illustration of the BEM approach.

Besides the elemental thrust d

Applying the angular momentum equation across the disk, we get

The relation between thrust and induced velocities (Eqs.

Even though the BEM relationship is originally derived for a full rotor, it is generally implemented on an annular element form as proposed by

In order to model azimuthal variations of induction due to azimuthal variations of blade loading as discussed above, we propose to expand the annular BEM approach. Dividing the annular elements into azimuthal sub-elements leads to a polar grid BEM approach; see Fig.

The induced velocity is found in each grid point using the

The computation of the local torque is done in the same manner.
Then, the two resulting thrust and torque coefficients are interpolated based on the azimuth angle

It is evident that skewed inflow to the disk violates the conditions for the basic momentum equation (Eq.

A comprehensive investigation of yaw and dynamic inflow models for wind turbines and dynamic inflow modeling was carried out in the EU-funded project “Joint Investigation of Dynamic Inflow Effects and Implementation of an Engineering Method”

The general equation relating the thrust and induction at a rotor operating in yaw (see Fig.

Based on these results, a reduction factor

The parameters

The values

The wake skew angle

Top view of the velocity vectors and angles used for the skew wake expression. The

The wake skew angle

The wake skew deflection angle

Comparison of axial velocity through a vertical line (

Comparison of axial velocity through a horizontal line through the rotor disk. The rotor loading is prescribed to a constant loading of

As the wake in the yawed conditions is skewed behind the rotor disk expressed by the skew angle

A very general equation for the azimuthal variation of the induction was presented by

Coefficients for different yaw models

This is close to the model of Coleman, as seen in Table

In Figs.

As seen in Fig.

Results for the horizontal plane are depicted in Fig.

The response of the axial velocity to a step change in loading at the actuator disk at different radial position. Panel

In summary, it can be concluded that the present yaw model is in close alignment with some of the models derived and presented in

A time-varying loading of the AD will cause a time delay of the velocities at the disk as the whole wake flow has to adapt to the new loading. This phenomenon, the dynamic inflow effect, was also part of the abovementioned EU-funded project “Joint Investigation of Dynamic Inflow Effects and Implementation of an Engineering Method”

As for the yaw modeling, we use the AD-CFD model results again to develop and tune an engineering submodel for the dynamic inflow. The AD simulations are carried out with a uniform loading and a step change in

Comparing the decay in velocity for the different radial positions in Fig.

Approximating the response with an engineering model led to the conclusion that two time constants are necessary to obtain an accurate fitting to the AD data. We use the following expression for the two first-order filters:

We use a numerical optimization routine to find the set of parameters that minimizes the difference between the AD-CFD step response curves in Fig.

The optimization gave the following polynomials for the time constants:

The

A further result of the optimization is the weighting constants of the two filters which gave the following result:

Finally, the functions for the local flow speed to adjust the time constants were determined as

This result shows that the highest time constant (

As a test case of the implementation of the above-described dynamic inflow model implemented in the HAWC2 model, we run the same prescribed variation of

In a time-marching formulation with non-dimensional time step

Comparing the present dynamic inflow model with the models derived and presented in

For non-straight blades with sweep/prebend or in-plane and out-of-plane deflection, the radial distance between adjacent grid points is not equal to the distance along the curved blade. Therefore, both

The calculation of d

Sketch of a non-straight blade with in-plane and out-of-plane deflections.

The curved length

The standard BEM theory does not give information about the radial induction component, and for plane rotors this induction component will only have minor influence on the loading. However, for rotors with out-of-plane bending blades or rotors with coning, the radial induction component will have an impact on the angle of attack (AOA) and thus also on the loading. An analytical expression for the lateral induction for a 2-D actuator disk is presented in

We test the radial induction model by a comparison with the AD-CFD solution for a constant loading of

The radial induction computed with an engineering submodel in comparison with the AD-CFD result for a constant loading with a thrust coefficient of 0.89.

An overview of the complete aerodynamic model is shown in Algorithm 1. The algorithm includes references to the relevant equations in this article and can be used as a manual for implementation of the grid BEM algorithm. It is crucial that the dynamic inflow filter is applied at the very end of the algorithm to prevent nonphysical rapid induction changes due to any of the submodels. Otherwise, for example, a change in yaw angle at one time instant at the rotor disk in turbulent inflow would lead to an immediate change of the induced velocities, even though the wake did not have time to deflect.

The aerodynamic model as described here is the aerodynamic model in HAWC2. However, it is also found in a stand-alone version HAWC2_Aero which can run the same type of simulations with turbulent inflow, pitch actions and rpm variations as HAWC2 but for a stiff structure. In this version, the simulation speed with all input/output operations is on the order of 7–10 times real time on a 2016 workstation laptop. This means that the computational time for the aerodynamic part is still small (10 %–20 %) relative to the total computational time for the aeroelastic simulations although we, in this BEM implementation, update the induction over the whole disk at each time step. One reason for this is that no sub-iterations in the induction modeling are necessary.

At very low rotor speeds below 0.1 rad s

Unsteady airfoil aerodynamics effects (dynamic stall and Theodorsen effects in attached flow) are not included in the computation of the induced velocities. This is possible because unsteady airfoil aerodynamics occur at much faster timescales with time constants that depend on the half chord divided by the relative speed. For comparison, the dynamic inflow time constants scale with the rotor diameter divided by the free wind speed. After the induced velocities are computed, the unsteady airfoil aerodynamics are determined using the Beddoes–Leishman-type model described by

Illustration of the dynamic induction mechanism in turbulent inflow showing the blade scanning through the field of slow-varying induction velocities but transferring to higher frequencies due to the rotational sampling of the turbulence.

In this section, we demonstrate the impact of the present grid BEM implementation on the induction and load characteristics based on simulations of the AVATAR 10 MW reference wind turbine (RWT)

Data for the DTU and AVATAR 10 MW reference wind turbines

The impact is evaluated by comparing with an “annular mean BEM” version computing the mean induced velocities in an annular element. This annular mean BEM version was incorporated in a test version of HAWC2 for the present investigation. Because the version is only a test version, the mean annular approach was implemented in a crude way by executing the loop two times. During the first loop, the local three wind speed components were summed in new variables for each grid point. At the end of the first loop, the mean of the velocity components for a constants radius (a ring element) was derived and then used in the second loop instead of the local wind speed components.

The induction mechanism simulated with the grid BEM implementation for turbulent inflow is illustrated in Fig.

An important mechanism of the induction of the presented BEM implementation on a polar grid is that each grid point has a memory effect incorporated. Thus, past loading changes at a grid point (e.g., due to a pitch action in this region, a local gust, an instantaneous shear, a blade passing with another pitch angle offset) will influence the induction of the blade passing that grid point. The weighting of the impact of these past events is controlled by the dynamic inflow filter.

To illustrate further the characteristics of the induced velocities from the AVATAR rotor case mentioned above, the time trace of the induced velocity at a radius of 43 m is shown in Fig.

As expected, the PSD of the induced velocity computed with the annular mean method has no peaks and has some resemblance with the PSD of the hub wind speed.

We will now illustrate the mechanism behind the load impact of using a mean annular BEM approach and a grid BEM model, respectively. Again, it is a simulation example for the AVATAR rotor.

A simulation was run with a ramp in wind speed from 4 to 20 m s

Now, a simulation is performed for sheared inflow with an exponent of 0.5 and at a wind speed of 8 and 14.5 m s

For the mean annular BEM, the constant induced velocity as function of the local wind speed on the blade is obvious. The mean value might be slightly different from the value at the same wind speed for the turbine operating in uniform inflow due to non-linear effects from computation of the mean loading.

The picture is quite different for the grid BEM method, as shown in Fig.

At 8 m s

At 14.5 m s

The important impact on the loads is that changes in the local wind speed will always be counteracted to some extent by the induced wind speed and thus reduce the variations in AOA and likewise variations of the aerodynamic loads. This will be further explored below for turbulent inflow and quantified for a few test cases.

The characteristics of the induced velocities for turbulent inflow are basically determined by the same mechanism as described above for sheared inflow. As discussed above, the turbulent inflow with dimension of structures less than one rotor diameter cause a non-uniform inflow over the rotor disk. It means as for sheared inflow that a point on the rotating blade will see a local wind speed different from the mean wind speed corresponding to the mean operational conditions of the turbine. In Fig.

The impact of the grid BEM model on fatigue loads and power production according to DLC 1.2

A comparison for the DTU 10 MW RWT of the difference in blade root flapwise fatigue loads

For brevity, this section focuses only on the 1 Hz equivalent load of the flapwise blade-root-bending moment and the mean power. All results are presented as percent relative difference compared to an annular BEM model that includes the yaw correction presented in Sect.

The results for the DTU 10 MW RWT are shown in Fig.

Overall, the grid BEM results in significant lower fatigue loads, up to 8 %, except in a narrow wind speed interval between 7 and 10 m s

The influence of the grid-based BEM for the power production of the DTU 10 MW is very small at roughly

The results for the AVATAR turbine (Fig.

A comparison for the AVATAR turbine of the difference in blade root flapwise fatigue loads

Comparing the two cases, we can conclude that the impact of the grid BEM approach depends on the actual turbine design with an increasing reduction of fatigue loads for lower loaded (low-induction) rotors. For both turbine designs, the load reduction is considerable (8 % to 10 %) for wind speeds above rated power.

We present in this section a selection of validation results in order to illustrate the performance of the grid BEM implementation for different challenging inflow cases. As mentioned above, the grid BEM method is the aerodynamic model in HAWC2 and the cases are simulated with this model. It also means that several validation cases can be found in different articles published in the past and only two of them are explicitly summarized here. The first referenced validation paper contains not only a validation of the aerodynamic model of HAWC2 but of the full aeroelastic model. However, in the second validation reference, the aerodynamic model in HAWC2 is alternated between the grid BEM and full 3-D CFD, which enables a detailed validation of the grid BEM results.

In

In the other validation publication by

The first validation case is to demonstrate the model response to a considerable shear in the inflow for the NREL 5 MW turbine

Side view and front view of the CFD mesh around the NREL 5 MW reference turbine generated with a hub height of 90 m.

The graph shows the contour plot for the velocity field for the sheared inflow case.

The CFD simulations were carried out with the 3-D incompressible Navier–Stokes solver EllipSys3D by

To minimize the computational time, both grid sequencing and time step sequencing were used. To settle the overall induction field, the flow was simulated with a coarse time step of

Grid/time step convergence of the ElliPSys3D simulation, showing mean integral forces computed for the velocity step case at each of the three grid/time step levels.

The normal force

A comparison of HAWC2 simulations on the NM80 turbine and experimental results from the DanAero project. Power spectra of the chordwise aerodynamic force component parallel to the chord in comparison with measured results. Wind speed is 6.1 m s

A user-defined shear flow can be input to a HAWC2 simulation so the case could be simulated by a default setup. When comparing the normal and tangential loading on the blade at azimuth positions of 90 and 270

The case is further analyzed by comparing the integrated normal and tangential blade forces as function of azimuth as shown in Fig.

Detailed aerodynamic measurements on full-scale turbines are very limited. However, in the DanAero project, such measurements were carried out in 2009 on a NM80 turbine with an 80 m diameter

The comparison of PSD spectra of the measured and simulated aerodynamic forces perpendicular to the chord is shown in Fig.

Comparison of the differences in the azimuthal distribution of normal forces

There is a clear tendency for the simulated spectra to fall below the measured one at higher frequencies, in particular for the outboard stations, which might be due to the resolution in the turbulence box which is 1.28 m in the vertical and horizontal directions. Finally, it can be seen that in this case the difference between the two BEM implementations is quite small. This can be explained by the above considerations in Sect. 4: if the local thrust coefficient is high, the slope of the

In the New Mexico experiments

Comparison of differences in out-of-plane

It can be seen that there is a phase shift in the azimuthal force variation between the normal forces at the inboard section (Fig.

For the sections further outboard, the influence of the tip vortex becomes more important and the phases of the azimuthal force variation agree well. There is a slight overprediction of the mean loading, especially in the tangential direction. Comparing the integrated out-of-plane and in-plane blade-root-bending moments in Fig.

Comparison of HAWC2 results against measurements of the dynamic inflow case Q0500000 of the NREL/NASA Ames phase VI experiment. The plots show scaled normal

The NREL/NASA Ames phase VI experiments

The measured data have been analyzed by

A comparison of measurements with the dynamic inflow model described in Sect.

The comparison shows good agreement; however, some disagreement is to be expected due to inherent limitations of the actuator-disk-based model. Specifically, the root vortex dynamics are missing and the disk model also assumes an infinite number of blades. Therefore, differences are expected close to the root and the tip of the blade, where the induction from a helical wake deviates most from the induction due to a cylindrical wake. An option to address these limitations is to couple a vortex-based near-wake model to the BEM code

We have presented an implementation of the BEM method on a polar grid in order to simulate more accurately the considerable inflow and load variations over the rotor disk found for large turbines. The model can also be characterized as an engineering actuator disk model where the induced velocities on the stationary polar grid are updated at each time step in an aeroelastic simulation. Further, the detailed integration of submodels for tip correction, yaw and dynamic inflow has been described. Also, a submodel for radial induction important for computations with out-of-plane blades due to elastic effects or coning has been presented.

The load impact mechanism on the flapwise blade root moment from this unsteady induction by the grid BEM is analyzed. It is found that the load impact strongly depends on the turbine design and operating conditions. For operation at low to medium thrust coefficients (conventional turbines at above rated wind speed or low-induction turbines in the whole operating range), it is found that the grid BEM gives typically 8 %–10 % lower 1 Hz blade root flapwise fatigue loads than the classical annular mean BEM approach. At high thrust coefficients, the grid BEM can give slightly increased fatigue loads, in particular for pure shear cases.

Different validation cases have been presented by comparing with experimental data and data from the high-fidelity EllipSys3D code. A challenging half wake in the vertical plane with the double inflow velocity on one side of the rotor relative to the other side is simulated. A good correlation is found with EllipSys3D results for blade loads as function of azimuth.

Results on yawed inflow for the Mexico rotor and dynamic inflow results from the NREL/NASA Ames experiment confirm a satisfactory performance of the submodels for yawed flow conditions and dynamic inflow. Finally, comparing PSD spectra of the simulated local aerodynamic forces at four radial positions on the full-scale NM80 turbine shows excellent agreement with spectra of measured forces originating from the DanAero experiment.

The data for most figures are openly available at:

HAM and TJL developed and implemented the overall grid BEM modeling approach. TJL tested the grid BEM model and increased the robustness of the implementation with contributions from GRP. HAM investigated the load mechanism of the grid BEM method. HAM performed the actuator disk simulations and extracted the data for tuning the yaw and dynamic inflow model. HAM, TJL and GRP wrote the article with contributions from FZ and AL. AL determined the time constants of the dynamic inflow model by means of numerical optimization. TJL and AL derived and implemented the correction for blade in-plane and out-of-plane bending. GRP executed and discussed the validation cases with major contributions from HAM. FZ derived the EllipSys3D setup for the half-wake simulations, conducted the simulations and extracted the data for the validation. All authors jointly finalized the paper.

HAWC2 is developed by DTU Wind Energy. The software can be licensed for research and commercial use.

We thank our colleagues in the AER and LAC section that in one way or another have contributed to this work and the modeling presented. In particular, we acknowledge the setup for automatic preprocessing and post-processing the DLC1.2 simulations presented in Sect. 4.5 developed by David Robert Verelst and Mads M. Pedersen. Also, the valuable contribution from Anders Melchior Hansen, one of the main developers of HAWC2, is acknowledged. We thank the two anonymous reviewers for the feedback. We also thank Frédéric Blondel from Ifpen and David Marten from TU Berlin for pointing out errors in the discussion article.

We also acknowledge the access to the New Mexico NREL/NASA Ames data in the IEA Task 29 database.

This paper was edited by Gerard J. W. van Bussel and reviewed by two anonymous referees.