AeroModule AM; is a state-of-the-art wind turbine aerodynamic library that includes both a BEM solver and a FVW model called AWSM (Aerodynamic Wind turbine Simulation Module; ). AM can be either coupled to a structural code to perform aeroelastic load case calculations or used as a standalone tool for simple aerodynamic design iterations featuring a rigid turbine model. Thanks to the source code modularity, BEM and AWSM share multiple routines that guarantee fully consistent inflow modelling and airfoil aerodynamic properties (the same polar lookup tables are used) including rotational effects and dynamic stall models. This allows the user to switch between the two solvers from the same input file and hence facilitates the cross-verification of results. Both models benefit from a long validation record e.g..

While the BEM method has to rely on engineering sub-models to account for dynamic inflow effects, AWSM's detailed wake modelling allows predicting the unsteady wake evolution with high fidelity . In AWSM the wake is modelled by vortex filaments, accounting for both trailed and shed vorticity, and the induced velocity at each point in space is estimated via the Biot–Savart law. Therefore, unlike blade-resolved CFD and experiments, the instantaneous induced velocities at the blade lifting lines are readily available without the uncertainty associated with induction extraction techniques . This makes the local induced velocities obtained with AWSM directly comparable to those of BEM, facilitating the verification of results. In this work, a special AWSM version that neglects the contribution of shed vortices in the induced-velocity calculations (both at the lifting lines and at the wake points) has also been developed, aiming to gain insight into the physical effect of shed vortices on the results presented in Sect. .

This section provides a brief description of the dynamic inflow models for BEM considered in this work. More detailed information on their derivation and parameter tuning can be found in the cited references, while their implementation in AM is addressed in Sect. .

In AM's BEM solver, the BEM equations for each control element in each annulus are solved implicitly by means of the iterative Newton–Raphson algorithm. When the ECN dynamic inflow model is used, a term depending on the first derivative of the annulus induction (an average between the local blade element inductions) is added to the axial momentum equation (Eq. ). This couples the BEM equations of the different blades, making it hard to guarantee the local convergence at each element when non-uniform inflow conditions are considered .

Using the two-constant models listed in Sect.  avoids such a shortcoming. All these models, indeed, correct the quasi-steady axial induction value obtained by solving the steady-state momentum equations, and they are applied outside the iterative solution loop. Thanks to their similarities, a common implementation strategy has been devised following the process depicted in Fig. .

Indicating the generic annulus with subscript j and the current time instant with superscript t, the induced-velocity correction (Δujt) can be defined as 8Δujt=u‾jt-u‾QSjt, with u‾jt being the annulus average axial induced velocity accounting for dynamic inflow effects and u‾QSjt the corresponding value from steady-state momentum theory, i.e. accounting for possible skewed wake effects as described by but not dynamic inflow.

Selecting the Øye, TUD-VR, or DTU dynamic inflow model only changes the way the induced-velocity correction is computed. In fact, the routine that applies the induced-velocity correction is common to all models and it just requires the annulus radius (rj) and the annulus average free-stream axial wind velocity (V‾0jt) to be provided as inputs, along with u‾QSjt. The correction expressions for each model are reported in Appendix .

Note that in non-uniform conditions, the value of V‾0jt depends on the reference velocity chosen for each blade. The local wind speed at each blade control point is used in AM, and the average of these values yields V‾0jt . To include the loading dependency of the model parameters, the quasi-steady annulus average axial induction factor is evaluated a‾QSjt=-u‾QSjtV‾0jt before computing the correction according to the selected model.

Following AM's BEM solver philosophy, an annulus approach has been chosen to compute and apply the dynamic inflow correction. Therefore all the steps highlighted in this section are repeated for each control annulus assuming complete radial independence. This is just one of the various implementation choices possible , and it might not be the optimal one (provided that a general optimum exists). These choices have no impact on axial uniform inflow cases (Sect.  and ), but they do affect generic non-uniform cases. Nevertheless, the consistency of all the implementation choices for the different models guarantees the validity of this comparative study where the relative performance is assessed.

Once Δujt is known, the new annulus average axial induced velocity is obtained from Eq. (). Finally, in order to evaluate the new tangential induction (resulting from the coupling of the momentum equations) and update the loads on each element, it is assumed that the same correction that applies to the annulus average value (i.e. the average among the blades) also applies to the annulus axial induced velocity of each individual blade.

Within the project, a 220 m diameter variable-speed rotor design rated at 12 MW has been developed following the current design process for large rotor blades to provide the research community with a realistic reference turbine. The rotor has been designed for a rated wind speed of 10.5 m s-1 with an average induction close to 1/3 at the optimal tip speed ratio. For this rotor, a new variable-speed and individual pitch controller has been designed by TNO. The resulting state-of-the-art offshore wind turbine design has been used to assess the impact of the aerodynamic modelling activities carried out within the project.

The dynamic inflow model verification and benchmarking campaign has been performed in two steps:

Step 1. Standard dynamic inflow tests based on prescribed blade pitch actuations in axial uniform wind conditions have been considered using a simplified rigid turbine model (standalone AM simulations) without the shaft tilt angle to avoid skewed wake effects (Sect.  and ). These simple cases allowed isolating the effect of the dynamic inflow models on BEM results, verifying their predictions by comparing them with AWSM.

The numerical setup for the two steps is described in the following subsections.

Blade pitch steps represent the traditional test case to investigate the performance of dynamic inflow models e.g.. This is because the slow response of the induction to sudden pitch changes leads to load overshoots that may affect the fatigue life of components. These pitch variations are often designed to be representative, at least to some extent, of fast controller actions.

A pitch step case has been considered in this work as well. The test selection has been inspired by a numerical case simulated in the international code comparison round of IEA Task 29 , where a very fast pitch variation was imposed on the AVATAR turbine. The very same pitch step variation has been prescribed in this case, starting from the design blade pitch value, suddenly raising it by 2∘ towards feathering, holding it for 300 s, and finally returning to the initial condition (Fig. b) with a maximum pitch rate of ±10∘ s-1. The fast pitching rate helps highlight differences between the models.

The test has been run near rated conditions (10 m s-1 wind speed) where dynamic inflow effects are expected to be most significant. Indeed, pitch control actions only start occurring once the rated power is reached, but the higher the wind speed, the lower the induction factor, leading to weaker dynamic wake effects. Therefore, the only standard operating region where pitch actions are likely to occur at high rotor induction (typically close to the Betz optimum) is the one near rated power. Rotational speed variations below rated conditions might also play a role, but they are usually much slower due to the great inertia of large variable-speed rotors and thus trigger few dynamic wake effects.

In order to isolate the unsteady behaviour from discrepancies in the steady-state values (always present between BEM and AWSM results, albeit small), the time histories of all quantities have been normalized as follows: 9X*(t)=X(t)-X0∘X0∘-X2∘, with X(t) being the time history of a generic quantity, X0∘ and X2∘ the steady-state values before and after the feathering step, and X*(t) the resulting normalized time history. Despite the normalization helping to better compare the results of different models, the radial dependency of the induction time histories hinders a synthetic visualization.

The axial induction responses, normalized via Eq. (), at four spanwise locations are shown in Fig. . As expected, the ECN model predictions provide the worst match with AWSM, with the single time constant leading to a faster induction reconfiguration in both pitch step directions, which confirms the findings of previous studies e.g.. This is especially true in the tip region where fa approaches zero (Fig. a) and the ECN model response tends to be quasi-steady. Slightly better predictions have been obtained with the Øye model, which gives similar results to the ECN model in the inboard part of the blade but finds a slower decay close to the tip. The TUD-VR and DTU models provide the best match with AWSM for the case considered. Their predictions appear to be almost equivalent and both very similar to the free-vortex results until the blade tip, where a slightly faster decay than AWSM is observed. It is important to notice that, due to the very fast pitch rate, the AWSM results show a sudden spike in the induction response immediately after the step (i.e. around 300 and 600 s) that cannot be replicated by the BEM models. Section  shows this behaviour to be a result of the sudden change in the blade bound circulation induced by the pitch manoeuver, which produces strong shed vorticity in the near wake.

The normalized aerodynamic rotor thrust and power are plotted in Fig. . Their behaviour is consistent with the observations made on the induction field, with Øye and ECN models predicting a faster decay after the overshoot than TUD-VR, DTU, and AWSM. All BEM models slightly overestimate the overshoot peak, especially in terms of power where peaks are larger, and have been highlighted with small zoomed-in windows in Fig. b. This is a direct consequence of the induction spike caused by the shed vorticity that sharply increases the instantaneous induction right after the step and hence reduces and delays the overshoot.

One of the main drawbacks of considering pitch steps to verify the performance of different dynamic inflow models is the difficulty of visualizing results in a compact yet comprehensive way. Prescribing sinusoidal collective blade pitch variations helps a lot in that sense. Observing that the spectrum of the induction response to a mono-harmonic blade pitch variation is dominated by the component at the pitch actuation frequency, the resulting unsteady axial induction can be fully characterized by the amplitude and phase of that harmonic. This allows characterizing all radial locations in one pair of plots.

Sinusoidal variations were also considered by , but there the use of a uniformly loaded actuator disc allowed a direct prescription of the thrust coefficient variations. Here more detailed lifting-line rotor models are considered instead; therefore the rotor loading is only varied indirectly through the blade pitch angle. Due to the non-linear dependency of CT on the blade pitch angle (especially at high loading) and the non-uniform spanwise loading of a real rotor, the present results are hardly comparable to those of though some conclusions are alike.

Both near- and above-rated conditions have been considered in this case, with wind speeds of 10 and 15 m s-1 and average angles of attack of ∼7 and ∼4∘, respectively. A sinusoidal blade pitch variation with respect to the design pitch setting has been prescribed, always with an amplitude of Ap=2∘, which triggers significant dynamic wake effects while guaranteeing to avoid stall effects in the last 75 % of the blade span. Two different frequencies have been considered: fp=10 and fp=50 mHz. These correspond to a reduced rotor frequency (defined as k=fpRV0, with fp being the pitch frequency, R the rotor radius, and V0 the free-stream wind speed) of k≃0.11 and k≃0.55 near rated conditions and k≃0.07 and k≃0.36 above rated conditions. The frequencies have been selected as multiples of the simulation time step to prevent leakage effects in the spectra that have been obtained via fast Fourier transform (FFT).

The near-rated results for the two pitch frequencies are shown in Figs.  and , respectively. The first plot in the two figures (Figs. a and a) shows the spanwise evolution of the reduced airfoil frequency (defined as kairfoil=πfpcW, with fp being the pitch frequency, c the airfoil chord, and W the airfoil effective wind speed). The mean effective wind speed found by AWSM during a pitch cycle has been used to compute kairfoil. Even in the higher-frequency case, the values remain below the conventional 5% threshold from 25 % span onwards, with maximum values at the root never exceeding 10 %. This guarantees that unsteady airfoil effects are limited and standard polars can be relied upon in all the cases considered.

Looking at the amplitude-versus-span plots for the two frequencies (Figs. b and b), it can be noticed how dynamic inflow effects reduce the axial induced-velocity oscillations with respect to the quasi-steady momentum theory prediction (the black line in the figures), especially for the higher pitch frequency. Compared to AWSM, ECN and Øye models tend to slightly overestimate the oscillation amplitudes at the lower frequency (Fig. b). A better agreement is found for the faster pitch variation (Fig. b), except for the ECN model prediction in the outboard sections, which rises to reach the quasi-steady value at the tip. As in the pitch step case, the TUD-VR and DTU models provide very similar results that match the AWSM predictions well.

Figures c and c show the radial distribution of the phase shift between the axial induced-velocity oscillation (at the pitching frequency) and the blade pitch actuation signal. This phase shift was evaluated by considering the difference between the pitch frequency harmonic in the phase spectrum of the local induced-velocity signal (evaluated via FFT) and the phase of the sinusoidal blade pitch variation prescribed. Greater differences can be observed in these plots, with an average phase difference with respect to AWSM of about 30∘ throughout the span for the fast-frequency case. Smaller discrepancies are also noticeable for the slower pitch variation. Besides the ECN line that always goes to zero at the tip (i.e. quasi-steady behaviour), the discrepancies between AWSM and the other BEM models are not fully understood. Searching for the critical parameters affecting such a phase shift, it was found that a reduction in the far-wake correction in the TUD-VR and DTU models generally improves the phase match with AWSM, although with conflicting effects on the amplitude . Shed vorticity also contributes to the observed phase differences, tending to reduce the phase delay in AWSM (as discussed in Sect. ) and thus increasing the mismatch with BEM results.

Very similar conclusions can be drawn from the above-rated simulations. The slower blade pitch results have not been reported here, as dynamic inflow effects were barely visible due to the low rotor loading. The usual plots for the higher-frequency case of the above-rated simulations are shown in Fig. . As expected, the quasi-steady induction variation amplitudes are smaller than the corresponding near-rated case, leading to milder dynamic wake effects (Fig. b). In terms of model performance, the comments made for the rated case apply to these results as well, with all two-constant models behaving similarly and overestimating the phase shift compared to AWSM.

To assess the impact of the different dynamic inflow models for BEM on a practical design load case, the procedure outlined in Sect.  has been followed. With DLC 1.2 being a fatigue load case, the main load time histories have been transformed into 1 Hz equivalent load cycles following the guidelines of IEC 61400-1. Figure  shows a comparison of the equivalent out-of-plane bending, in-plane bending, and torsion moments at the root of blade 1, along with the tower base fore–aft moment for several wind speeds along the turbine operating curve. The values have been normalized by the corresponding equivalent moments obtained by running steady-state BEM simulations (MT), i.e. without dynamic inflow models. This helps highlight the relative differences in various operating conditions.

In general, the use of a dynamic inflow model leads to higher equivalent loads of up to ∼10 % for the out-of-plane blade root bending moment (Fig. a). The tower base moment (Fig. d), which is driven by the rotor thrust and thus includes the contributions of all the blades, shows up to ∼20 % greater loads. Lower increments are observed in the equivalent torsion moment at the root of blade 1 as well (Fig. c). In contrast, the in-plane root bending moment (Fig. b) is almost insensitive to the dynamic inflow models as it is largely dominated by gravitational load cycles.

Provided that similar wind speed, rotor speed, and blade pitch time histories have been prescribed, the general equivalent load increase can be attributed to the damping of the induced-velocity fluctuations that results in greater aerodynamic load variations when dynamic inflow models are considered. The maximum equivalent load increase is always located around rated conditions (at 10 m s-1 in this case) because this is where the blade pitch regulation starts and the rotor loading is still high. As the wind speed increases above rated conditions, the average induction lowers, leading to milder dynamic wake effects despite the relevant blade pitch regulation. The opposite is true below rated conditions, where the high tip speed ratio makes the rotor more sensitive to dynamic wake effects but the constant blade pitch reduces the chances of sharp loading variations occurring. In fact, unless considerable wind gusts occur, the large inertia of a more than 200 m diameter rotor makes rotational speed variations less prone to triggering strong dynamic inflow effects.

Provided that a dynamic inflow model is used in the BEM simulations, the choice of the specific model seems to matter significantly less in terms of equivalent load predictions. The differences found among the models always fall within 5 %, with larger discrepancies found at the wind speeds where the induction is higher. The TUD-VR and DTU models yield similar predictions to in the uniflow inflow cases (Sect.  and ), with the Øye model typically falling between those and the ECN model results.

Once more, these results have been obtained by prescribing the time histories of the rotor operating conditions to aid their interpretation. The actual differences for a fully realistic load case including the original turbine controller and considering six seeds per wind speed have been found to be higher, especially below rated conditions (as shown in Fig. ). This is because different dynamic inflow models lead to different controller actions and thus different operating conditions.