In this investigation, pull–release tests of an approximately 14 m wind turbine blade were chosen, as experimental results were available for validation. The blade is designed for use on upwind horizontal axis turbines and equips a combination of FFA-W3 airfoils of varying thicknesses

Due to confidentiality constraints, the exact geometry of the blade cannot be given in this document.

The blade model for HAWC2 was discretized into 19 Timoshenko beam elements grouped in 10 bodies in the multi-body formulation. The blade was clamped at the root and was oriented horizontally with an initial angle of 3.5∘ at the root, like the experimental setup; see Fig. . The simulations, as well as the experimental tests, were conducted pressure side up. Structural properties were calibrated using the known cross sections throughout the blade, and structural damping was tuned to 1 % logarithmic decrement through Rayleigh damping for the first and second modes of the blade. The impact of structural damping on the overall result is assessed in Appendix . A concentrated mass of 4.8 kg was added at 13 m to resemble the saddle, which was attached to the blade during the experiment. For FSI-BET calculations, the blade was described through seven aerodynamic profiles. Each profile had a look-up table of force coefficients for calculations of aerodynamic forces along the blade.

To study the flow around the oscillating blade, the so-called Q criterion found using the FSI-CFD results is depicted as isobars in Fig. . The Q criterion defines vortices as the spatial region where the vorticity dominates the strain rate, i.e., showing the whirling structures.

As seen in Fig. b, vortices are created and shed from the blade as the blade is released, and at the turning point (Fig. c) many vortices are present in the wake. The blade travels through the vortices (Fig. d) and creates a new wake of vortices.

From the experimental test, the acceleration at 12.8 m was measured with an accelerometer. This measured acceleration will in the following be used for comparison between simulations and the experiment.

As seen in Fig. , the measurements and the FSI-CFD results compare well. For all cases, a decay factor ζ was estimated for the first part of the free oscillation. This decay factor does not consist of the aerodynamic damping alone but is rather a way of comparing the effective total damping between experiments and simulations. The aerodynamic damping will change during the oscillations, as it is dependent on the squared velocity and the drag coefficient, which might not be constant either. The decay factors are seen to be very comparable with a difference of 7 %. Note that these decays are approximated for a limited number of cycles only, using the logarithmic decrement. The same number of cycles is used from both the experimental and simulation results.

Looking at the power spectral density (PSD) of the test and simulation acceleration signals, it is also evident that the three mode frequencies are found at around 2.15, 4.6 and 7.0 Hz for both tests and simulations.

From the beam moments, found through simulations, the strains were found for the locations of the strain gauges used in the experiment, using Eq. (), omitting any axial forces and any effects related to the stiffness matrix cross-talk term EIxy. 2ϵ=MxEIxy+MyEIyx Strain gauge calibration was carried out beforehand by means of a static blade pull. As seen in Fig. , the calculated and measured strain agree well but have calculations overshooting the strain, especially at the suction side of the blade. This overshoot happens before release as well, where CFD has no influence, thus the reason must be a mismatch between the structural model and the tested setup or the coordinates of the strain gauges. A cumulative uncertainty is present, as Young's modulus (E), moments of inertia (Ix and Iy), strain gauge positions (x and y) and moments (Mx and My) will all have discrepancies between the model and the real blade. As a quantitative example, the relative difference in strain amplitude between experiment and simulations of the peaks at ≈4.68 s yields overshoots of 30.3 % at suction side and 6.0 % at the pressure side.

As mentioned earlier, the standard practice for using the BET quasi-steady aerodynamics for simulations of tests is to increase the drag coefficient CD to consider the additional drag resistance in test cases compared to normal flow conditions. In the following, FSI-BET results are shown for multiple different drag coefficients and compared with the FSI-CFD. The case of 600 mm initial tip displacement is used throughout. The calibration of drag in the present case was done by simply shifting the entire drag curve by constants, such that CD at 90∘ angle of attack reaches a chosen value. In this case, with an almost pure translatory motion, the angle of attack will be close to ±90∘ at all times. For other tests with motion in different directions, such as a bi-axial fatigue test, the pure shift of CD might not be an ideal method, as one could imagine that the added resistance due to vortices will be quite limited for small angles of attack. The original drag coefficient of the profile at 90∘, CD,90∘, found for operational conditions through 3D CFD is 1.3. This CD,90∘ is lower than the range stated by , due to 3D effects included in the simulations, thereby reducing the drag. Drag coefficients known to be used by industry and research groups of 1.8, 2.2, 2.7, 4.5 and 5.30 were applied to the FSI-BET simulations. As seen in Fig. , showing the tip displacement of the blade, the drag coefficients reach 4.5–5.30 before achieving similar damping as the FSI-CFD simulation. This value fits well with the drag coefficients stated by for the calibration with 2D CFD simulations of 5.30 and 4.45 for 1 and 2 m amplitudes, albeit on another profile.

Taking a close look at the first few oscillations of the FSI-BET simulations shown in Fig. , the best fitting drag coefficient is around 2.7, while after a few seconds the simulations using a CD of 5.30 fit much better, indicating a varying drag coefficient through the vibration. Peaks at 1.15 s yield a 1.2 % overshoot for CD=2.7 and -2.2 % for CD=5.30. At 7.2 s, however, CD=2.70 results in a 8.0 % overshoot, while CD=5.30 overshoots with only 0.6 %. In this way, CD=5.30 represents the total simulation better when using a constant drag coefficient, which is also visible in Fig. .

Inspecting the effective drag coefficients of the FSI-CFD simulations with varying initial deflections, Fig.  displays the node displacement of the 12.8 m node for the first 15 s after release, along with the corresponding force coefficient in the flapwise direction.

It has been shown that the force coefficient increases during the vibration as the amplitude decreases. This relationship is also seen from the initial force coefficients of 5.3, 5.9, 7.0 and 8.4, respectively, for 600, 500, 400 and 300 mm initial tip displacements. It is not possible to conclude anything about the relationship between amplitude and force coefficients with the chosen number of cases, but it is clear that future investigations are relevant. As has been shown, the initial force coefficient fits well with the aforementioned calibrated coefficient of 5.3 using the BET simulations of the 600 mm pull (Fig. ). The reason for the calibrated drag coefficient not needing to increase during the simulation to fit the FSI-CFD results is due to the fact that the aerodynamic force is dependent on the squared velocity. This means that for large amplitudes, i.e., large velocities, the drag force will dominate, while decreasing in impact as the amplitude decreases.

Based on this investigation, drag coefficient calibration of BET-based fatigue test simulations should follow the actual amplitude of the blade during the test. Taking into account the different amplitudes along the blade by having multiple look-up tables for force coefficients depending on amplitude could resolve this. This would, however, require some development on the existing code. The chord length of the blade might also affect this dependency, but this has not been studied further here.

To assess the impact of the increased drag on the fatigue test load simulations, a preliminary study was made. Here, BET-based simulations were conducted to obtain moment distributions along the blade for different drag cases. The simulation results were compared to the target moment distributions of the specific fatigue test. As a case study, a flapwise fatigue test for N=2×106 cycles was considered. The blade was modeled as described in Sect. . A set of tuning masses was installed on the blade, whereas the force excitation was applied by means of a translating mass exciter. Tuning masses were modeled as concentrated masses and located at specific beam nodes. The exciter was modeled as a predefined mass translating on a linear path at the flapwise resonance eigenfrequency, which was evaluated for the blade system including tuning masses and excitation mass. The location and magnitude of the tuning masses and the excitation parameters were determined beforehand. The test target loads Mtgt were also provided as an input based on the blade lifetime fatigue calculations carried out in . A simulation time Tsim=100 s was adopted for all the test cases, excluding the initial transient loading which was disregarded. The rainflow counting algorithm developed in was implemented in a MATLAB script used to evaluate the achieved test bending moment. Equation () was used to determine the so-called 1Hz equivalent load, where i indicates the ith bin, ni the number of counted cycles for the ith bin, Mi the bending moment amplitude for ith bin, m the Stress-Cycle (S-N) curve slope parameter and N1Hz the number of cycles of a 1 Hz signal in a time interval equal to Tsim. The S-N curve slope parameter was chosen equal to 12, which is typical for blades manufactured in glass-fiber-reinforced composite material, as the one used in this study. The test load was subsequently calculated by Eq. (), which accounts for the flapwise resonance testing frequency fflap. 3M1Hz=∑iniMimN1Hz1m4Mtest=M1Hzmfflap1m

Table  describes the test setup adopted during the presented set of simulations, in which the only varied parameter was the drag coefficient. The excitation parameters, i.e., the excitation mass longitudinal position (ze), the excitation mass magnitude (me) and the excitation mass oscillation amplitude (se), were chosen to achieve a relative test bending moment of approx. 100 % in the blade portion and between 20 % and 80 % of the blade span for the test setup, with a maximum value of CD equal to 1.8 for an angle of attack of 90∘. The excitation parameters were then kept constant for varying CD curves with maximum value in the range between 1.3 and 5.0, as done in Sect. , where 1.8 was taken as baseline since 1.3 is expected to underestimate the blade loading.

Figure  shows the variation in the achieved test bending moments along the blade span for a varying CD in the defined range. In Fig. a it can be noticed how the achieved test load decreases with increasing CD. The load reduction is highlighted in Fig. b by representing the variation in the achieved bending moment Mnorm, normalized with respect to the baseline case (CD=1.8), as a function of the drag coefficient. It can be concluded that the decrease in achieved test load follows the same trend for different locations along the blade span. Moreover, it was observed that the achieved load variation is nonlinear and presents a slope which appears to be decreasing for increasing drag coefficients.

As described in Sect. , different configurations were treated to study the possible influence of the confined space around the tested blade. The same four pull distances were tested with and without confinement, and the resulting tip displacements and aerodynamic forces were compared. As shown in Fig. , the confined and unconfined setup yields very similar results for aerodynamic forces and results in essentially identical displacements, indicating that the confinement has no effect in the present setup.

Figure  depicts, respectively, the pressure and velocity field around the blade at the time of the blade passing its equilibrium position, i.e., approximately at maximum velocity. It is evident that the gradients of pressure and velocity near the floor and walls are small, which supports that the effect of the confinement is negligible. As the blade velocity will decrease when moving closer to the floor, the blockage effect will decrease as well. However, it is also shown that a large zone of high pressure, is created beneath the blade, which in cases of longer blades with larger chords, might grow sufficiently to interact with the floor. In large test facilities, blades that are 5–7 times longer than the design studied in the present work are tested. The ratios between blade length and wall and floor distance for these cases will be different than the one studied here. In the minds of the authors, the possible effect should be tested using one of these blades with a similar FSI setup before drawing general conclusions about the effect of confinement.