Simulations are performed using open-source software , where the incompressible, transient flow solver pimpleDyMFoam is coupled to the in-house non-linear structural finite element beam solver BeamFOAM . The structural beam implementation makes use of the finite element GEBT formulation that was originally proposed by and and is capable of resolving large deformations of wind turbine blades. Within this study, 49 iso-parametric beam elements per blade are used corresponding to the generic NREL design, each consisting of two nodes with 6 degrees of freedom. Each beam accounts for gyroscopic effects and stiffening due to centrifugal forces. The temporal integration made use of the second-order generalized-α scheme . The coupling of fluid and structure is embedded inside the OpenFOAM framework without the necessity of external code communication and operates in a so-called loose coupling manner, which means that information is exchanged once per time step.

The incompressible, transient flow is simulated using the hybrid Spalart–Allmaras delayed detached eddy simulation method . To advance the solution in time, a second-order implicit backward method was used. The spatial discretization makes use of a second-order accurate Gauss linear scheme for the gradient terms and a first-order Gauss upwind scheme for the divergence terms. The rotation of the blades is accounted for by using sliding mesh interfaces between the three blade grids, the rotor disk and the farfield grid.

This work focuses on the blade flexibility; therefore the complexity of the setup is reduced. The inflow is uniform and perpendicular to the rotor plane. The rotor is neither coned nor tilted, which leads to symmetric inflow conditions. The tower and nacelle are not modeled either, and the blades are extruded at the blade roots towards the center of rotation to compose a closed hub region of the three cylindrical structures. Rotor operation is set towards rated conditions. All settings are listed in Table .

The boundary conditions of the rectangular domain are shown in Table . All sides are modeled as slip walls. The values of the turbulent quantities are set for laminar flow at the inlet. Fully turbulent conditions are assumed in the boundary layer on the blade surface.

In order to investigate the influence of flexibility of the blades on the aerodynamic behavior and near-wake effects, three simulations are performed: one with rigid blades (called rigid case in the following), one with flexible blades (called flexible case), and one with flexible blades excluding gravity effects (flexible-noG case).

The numerical grid for all simulations is based on the numerical grid used by . They showed that a reasonable convergence for investigating aerodynamic effects taking into account fluid-structure interactions was reached for a mesh consisting of 36.38 million cells. This mesh was called the medium-sized mesh and is composed of five mesh parts, namely three blade meshes, a farfield and a rotor mesh.

The blade meshes were created using the BladeBlockMesher utility , which composes 2D sectional meshes into a 3D structured blade volume mesh, consisting only of hexahedral cells. The blade meshes are generated with a resolution of 260 cells in spanwise direction. The chordwise component contains 300 cells, and 40 cells are distributed in blade normal direction. The cells normal to the surface follows a ratio of 1.2, and in order to limit the computational costs and to circumvent high aspect ratio cells inside the boundary layer, an adaptive wall function is applied (Fig. c). This wall function is capable of blending automatically between a high-Re and a low-Re approach, depending on the local y+ value. For the majority of the cells inside the first layer, a y+ value between 30 and 70 is applied. Here, the wall function ensures a consistent turbulent viscosity profile for all simulated walls. In situations with low velocities close to the blades and smaller y+ values, the wall function is switched off automatically to increase accuracy for, for example, flow separation.

The domain measures 5 rotor diameters (D) upstream and 15 D downstream of the rotor plane. All sides are located 3.5 D from the rotational center (Fig. a). The overall domain setup was kept, but as this study highlights the behavior of the tip vortex trajectory, additional mesh refinement was introduced in the wake of the blade tips to ensure a proper resolution of the flow in that region. This refinement is accounted for using a ring-shaped zone downstream of the blade tips, as shown in Fig. b. As shown in the literature , the trajectory of the tip vortex is expected to have a characteristic inboard motion on the order of 1 %–2 % of blade length. This movement needs to be captured properly. With the level of refinement used in this study, we ensure a minimum of seven cells within this scale. The highest resolution is located around the blade tip, where the vortex and blade surface are closest. Since the trajectory of the tip vortex is part of both the rotor mesh and the farfield mesh, both grid regions are equipped with an increased number of cells, leading to a total grid size of 73.05 million cells. This already doubles the numerical costs of the medium case mesh. The total number of cells and their distribution after the refinements is shown in Table . In contrast to , the setup used here allows very detailed vortex core tracking in the vicinity of the blade tip. An even higher cell refinement would exceed 100 million cells and would not be manageable by the used infrastructure.

To account for blade deformations inside the CFD domain, a mesh update algorithm developed by was used. This method aims at providing a cost-efficient and robust algorithm specifically developed for wind turbine simulations in OpenFOAM. Here, new mesh point coordinates are calculated in each time step without the need to solve a set of equations, where the deformation calculation costs are similar to those of a rigid mesh rotation. Within this method, all mesh volume cells surrounding the blade, as well as the blade surface cells, are projected onto the moving blade beam using a Newton algorithm. Each cell is assigned to a specific beam element to account for the respective blade deformation magnitude. All mesh cells are assigned into one of three mesh regions that are differently affected by the blade deformation. The closest region surrounding the blade surface moves rigidly with the blade, making sure no boundary layer cells are deformed during the process. Between the rigidly moved mesh region and the outer fixed zone, a polynomial of selectable order smooths out the blade mesh motion. Within this study, a first-order polynomial is used. All cells outside of this transition region are not affected by any deformation.

The wake induction is extracted from the CFD simulation using the in-house “3-point” method first introduced by . This method gives the opportunity to account for unsteady blade deformations and non-symmetric inflows for all azimuthal positions. Other aerodynamic quantities, i.e., angle of attack (AoA), lift and drag coefficients, on the rotor blades are derived from the obtained induction factors and the calculated pressure distribution, as well as the viscous forces. This method uses six points, three points on the pressure and three points on the suction side, which are distributed parallel to the chord of each analyzed blade section. The distance of the points towards the local chord is kept constant, to account for blade deformation during the simulation. These points are used to interpolate a representative AoA and local induced velocity while excluding effects of bound circulation as well as reducing the up- and downwash of the blade. This method gives the possibility to account for dynamic effects like yaw misalignment, different azimuth positions and deforming blades also close to the blade root and tip, which is considered important for the investigation of blade tip vortex trajectories. Since the aerodynamic quantities are based on probing locations inside the fluid domain, the results are influenced by vortices trailing from the blade surface. Therefore, the resulting quantities at the tip are to be considered a qualitative estimate to assess the aerodynamic behavior for a relative comparison between different blade setups.

Simulations are performed on the EDDY high-performance computing cluster of the University of Oldenburg . A total of 480 computing cores are used in each simulation run, computing a total of 220 s, i.e., 44 full rotor rotations. One simulation run took about 25 d, with less than 1 % more computational effort for the flexible case setup in comparison to the rigid calculation. Simulation results were extracted from the last full rotor rotation if not stated differently.

Firstly, the overall global thrust T and power P quantities are computed by 〈T(Θ)〉rtn and 〈P(Θ)〉rtn, where Θ describes the azimuth angle and 〈〉rtn denotes the average over the last rotor revolution 0∘≤Θ<360∘, which corresponds to a time span of Δt=4.959 s; 0∘ and 360∘ define a blade pointing in upwards direction. The averaged quantities for the three simulation cases are in good agreement with the literature. Table  shows a comparison to various sources for rigid and flexible simulations of the NREL5 MW rotor, taking into account slightly different configurations with respect to rotor tilt, pre-cone or tower inclusion. In the current investigation, the total thrust shows slightly higher loads when including blade flexibility compared to the rigid case, while the total aerodynamic power slightly decreases. This is also confirmed in the study by , which uses a similar setup.

In the next step, the rotationally averaged force distributions of axial and tangential forces are investigated, i.e., 〈Fax(r,Θ)〉rtn and 〈Ftan(r,Θ)〉rtn. Under flexible conditions and a uniform wind coming from the front, the blades tend to bend towards lower aerodynamic forces in the outboard region of the rotor, where the largest deformations are apparent. This holds for spanwise positions outboard of ≈ 47 m (≈ 75 % relative span) as shown in Fig. , where the rotationally averaged tangential force in the upper graph and the rotationally averaged axial force in the lower graph are smaller for flexible blades, respectively. This is important, since most of the torque, and therefore power of wind turbines, is extracted from the wind in the outer third of the rotor blades due to the corresponding lever arms and the larger swept area. It is also shown that neglecting the gravitational loads barely influences the mean aerodynamic loads, as the corresponding tangential and axial forces are only slightly lower than the loads of the flexible blades including gravity. For spanwise positions between 50 % and 75 % radius, the blade forces including flexibility are slightly larger than for the rigid case, which aligns with the study by . This is due to the fact that the deformation of the blade leads to slightly higher AoAs for the flexible cases throughout the whole span. In the blade region outboard of 75 % r/R the forces are lower for the flexible cases. This can be explained by two counter acting phenomena. On the one hand the higher angle of attack leads to higher loads on the blade. On the other hand, the large flapwise deformation leads to a reduction of the rotor radius and thus in lower a circumferential speed and a change in inflow angle closer to the tip. The contribution of the angle of attack on the loads can be considered approximately linear, whereas the influence of the speed on the blade forces is quadratic and therefore prevails. The resulting forces outboard of 75 % span are therefore smaller for the flexible blades, although a larger AoA is present. Nevertheless, the radial part between 50 % and 75 % of the blade contributes less to the overall power output than the outer part, due to the smaller lever arm. Also a smaller total force difference between the rigid and flexible case than in the most outboard 25 % of blade span makes this region less attractive for this study, which aims at highlighting the differences between these setups. The blade region inboard of 50 % blade span is not shown since the total deformation in all investigated regions is much smaller and the resulting differences in the force distributions are negligible.

It is important to note that even though the curves of flexible and flexible-noG in Fig.  lie almost on top of each other, a detailed look into the distribution resolved over one rotation shows clear differences. For that purpose, the force development of the 0.95 r/R location from Fig.  is plotted against the respective azimuth angle in Fig. . As visible in Fig. a, the time series of the sectional forces at a spanwise position of 95 % span shows almost constant forces for the rigid blade and the flexible-noG case. While both flexible setups, consistent with the averaged forces of Fig. , show lower force values than the rigid case, a sinusoidal behavior within one rotation is clearly dominating the force development of the flexible blade including gravity. The reduction of forces due to blade deformation aligns with the findings of . The lowest loads are given at an azimuth angle of around 90∘, while the maximal loads occur around 270∘ azimuth. A frequency analysis of the forces acting on the flexible blades (with gravity, i.e., flexible case, and without gravity, i.e., flexible-noG case) is shown in Fig. b. The dominant frequency in the flexible case corresponds to the rotationally periodic 1P frequency (vertical dotted magenta line), which does not exist in the flexible-noG case. However, both cases show a clear impact of the second lowest eigenfrequency, which corresponds to the first edgewise mode (second vertical dashed black line). The larger impact of the edgewise component of blade vibrations, in contrast to the flapwise mode, is based on the fact that for attached flows, the aerodynamic damping in flapwise direction is significantly larger than in edgewise direction and is therefore also more pronounced in the force frequencies. In Fig. b the 1P frequency and the other eigenfrequencies (Eigenf.) are clearly distinguishable.

While this frequency only has the second highest power spectral density in the flexible blade case, it dominates the force time series in the flexible-noG simulation. This clearly shows that the external 1P excitation only arises from the gravitation and therefore is not present in the flexible-noG results.

In Fig. , the blade tip deformation is shown, which also corresponds to the maximal deformation in edgewise, flapwise and torsional direction of the whole blade. Although being slightly further outboard than the section at 95 % span from Fig. , it describes an equivalent blade deformation behavior.

The dominating dynamic sinusoidal edgewise deflection component in Fig. a can be directly linked to the 1P gravitational load in Fig. , since the lever arm of gravitation for a blade in horizontal positions at 90 and 270∘ azimuth is the largest. The main cause of the reduction of power of the flexible blades in comparison to the rigid can on the one hand be traced back to the flapwise deflection (Fig. b), which results in a smaller area swept by the turbine blades and therefore less energy that is extracted from the wind. On the other hand, the torsional degree of freedom impacts the loading, since a lower angle of attack is directly linked to a reduction of the local lift and drag components in the “linear region” and therefore impacts the lift-to-drag ratio . As shown in Fig. c, the torsional component is in phase with the edgewise deformation and has its largest influence at 90 and 270∘ azimuth.

This shows that a wind turbine rotor operating at rated conditions in a uniform inflow faces its largest influence of a dynamic excitation of structural deformation due to gravitational loads. Here, between 0 and 180∘ azimuth, the blade bends forward towards the leading edge, and during the upward rotation between 180∘ and 360∘ azimuth, it bends backwards in the direction of the trailing edge. Due to the structural nature of this blade, the center of mass (CM) is located downstream of the elastic axis (EA) on the chord line. This holds true for all sections along the blade, exemplarily shown in Fig. . In combination with the local twist angles, the gravity force leads to a torsion of the blade. Thus, for a downward-moving blade, the angle of attack is reduced with increasing twist angle, while the AoA is increased for an upward-moving blade, where the twist angle is reduced. This can be explained using the simplified velocity triangle, where the inflow speed Uinf=U∞⋅(1-a) and the circumferential speed Urot=ωr⋅(1+a′) form the relative velocity Urel. Here, a reflects the local axial induction and a′ the local tangential induction factors. The constant ω gives the rotational speed.

The dynamic torsion of the blade (Fig. c) clearly shows the deformation due to the gravitation, forming a sinusoidal shape with a maximal deformation amplitude of |Δϕ̃|=0.5∘. Additionally, higher-frequency components can be found in the time series and superimpose the sinusoidal torsion behavior with oscillations due to the eigenfrequencies of the blade with a much smaller magnitude. Although barely visible in the three investigated deformation components (Fig. ), it can be observed in the resulting forces acting on the blade especially in the frequency domain (Fig. b). In contrast, the flexible-noG case displays nearly constant aerodynamic forces and deformation due to the fact that the gravitational induced loads are completely missing (Fig. ).

In the subsequent analysis of blade flexibility on different aerodynamic quantities, the focus lies therefore on the comparison of the rigid blades with the flexible blades including gravitational loads.

To investigate the effect of a sinusoidal aerodynamic behavior within one rotation of the blade, a characterization of the whole wake cross section at different downstream locations is made. Here, slices at 1/4 D and 1/2 D in the wake of the turbine normal to the flow direction are investigated with respect to the mean speed magnitude for speed in x,y,z direction, i.e., 〈ux2+uy2+uz2〉rot. Here, 〈〉rot denotes the average over the last whole rotation. In Fig. , the mean velocity distribution of the wakes of both simulations are compared, with the rigid blade case in Fig. a and the results for the flexible blades in Fig. b. The velocity magnitude ranges are chosen such that the region of maximal blade deformation becomes apparent. The resulting shape of the visualized speed reduction is forming a uniform ring. The rigid blade case shows a uniform wake velocity deficit over the whole azimuth with minimal wake speed at ≈ 80 % span. This radial location of minimum speed can also be observed in the case of the flexible blades but with a clear shift between left and right, corresponding to 270 and 90∘ azimuth. Here, a velocity difference of ≈ 5 % is visible between the two sides. Also the expansion of the visualized wake speed is larger at 270 than at 90∘ azimuth. This results in a non-uniform thickness of the ring shape, indicating that the wake stream tube expansion itself is deformed in a way that the wake radius is larger at azimuthal positions of lower wake speed and smaller at azimuthal positions with higher wake speed.

Although less significant, similar results are shown for a 1/2 D downstream location as shown in Fig. . The total influenced area seems to expand and smear out. Since mixture of fluid between the outer free stream and the inner wake is not expected within 1 to 2 diameters downstream of the turbine , this can be explained by increasing turbulent mixture inside the stream tube, that is created by the rotor itself. This effect is more dominant at the 1/2 D downstream location compared to the 1/4 D downstream location. The minimal wake speed in the flexible case is shifted slightly towards the ground (dark blue region), which is attributed to the counterclockwise rotation of the wake. Similar rotation is also present in the rigid blade case, but it is not visible here due to the uniform character of the rigid rotor wake. This rotation opposing the clockwise rotation of the turbine is based on the conservation of momentum and has been reported in many studies before . Contrary to the uniform ring shape of the rigid wake, again the shape of the wake of the flexible simulation is deformed, trending towards an oval contour.

This is consistent with the results from the aerodynamic investigation in Sect. , where lower speeds in the wake of azimuthal positions of larger induction or higher angles of attack were observed. Thus, the impact of flexible rotor blades on the aerodynamic behavior of the wind turbine is as well transported into the wake and still visible at least 1/2 D downstream of the turbine.

From the findings of the aerodynamic behavior in Sect.  and the mean wind speeds analyzed in Sect. , a clear difference between azimuth positions of 90 and 270∘ is evident for flexible blades in contrast to the rigid blade representation. Apart from the difference in mean wind speeds, also the shape of the wake stream tube deviates. To quantify these variations, a deeper investigation of the blade tip vortex trajectories is made in the following.

In general, the stream tube, being formed by all particles of the fluid moving through the rotor plane, has a smaller diameter upstream of the turbine, where the speed is higher, than downstream of the rotor disk where the speed is smaller (Fig. a). The larger the reduction of wind velocity, the wider the stream tube is to be expected further downstream as a consequence of the mass conservation. A representative structure, that describes the behavior of the stream tube at the blade tip, is the trajectory of the tip vortex of the blades, since it reflects the most outboard part of the fluid moving through the rotor plane. Based on the comparison of the rigid and flexible blades in Sect. , different behavior for the wake expansion has to be visible at different azimuth positions. For that purpose, the tip vortex trajectories are extracted from the flow field in horizontal planes parallel to the inflow direction making use of the λ2 criterion described in . Figure b shows a representative λ2 field extracted from the rigid blade CFD simulation at different time instants orthogonal to the rotor plane. Here, r/R defines the radial and y/R the axial position normalized by the blade length. The wake age is defined such that at 0∘ the blade tip is located at r/R=1.0 and x/R=0.0. During rotation, the blade passes through the slice, forming a tip vortex that is transported downstream in time, following a characteristic path. For the uniform conditions of the rigid setup, this path is independent of the blade position and can be extracted in any plane perpendicular to the rotor plane. The largest deviations of the aerodynamic quantities (Sect. ) and the wake velocity reductions (Sect.) are present for the azimuth angles, where the blade is positioned horizontally, i.e., 90 and 270∘. Therefore, it is also expected that the largest differences in the tip vortex trajectories are visible under these conditions. To quantify these differences and the influence of blade flexibility on the path, the vortex trajectory is analyzed for tip vortices trailed at 90 and 270∘ for the rigid and flexible simulation. This radial location of the vortex trajectory is defined by the position of maximal values of λ2 inside the vortex core.

Figure  shows the non-dimensionalized radial location of the tip vortex trajectory averaged over the latest five times the blade is passing through the plane of interest. The wake age is limited to 120∘, since this corresponds to the instant the next blade is slicing through the plane, forming a new vortex trajectory heading downstream normal to the rotor plane. The radial position r of the vortex core is normalized by the blade radius R. Vortex core locations are tracked at each 2∘ of the wake age, making sure that the near-wake expansion is properly resolved. A characteristic inboard tip vortex motion is visible for all cases for the first 20∘ wake age before expanding the stream tube and exceeding the actual radius of the rotor blades between 35 and 50∘ wake age. This characteristic trajectory aligns with the findings of and for their investigation for rigid blades, but a comparison between rigid and flexible blades is displayed for the first time in this work.

It is visible that the radial location of the tip vortex of the rigid blade case exceeds the trajectory of the flexible blades at 90∘ azimuth in the whole investigated region (Fig. a), whereas opposing behavior is shown at 270∘ azimuth (Fig. b). A larger stream tube expansion corresponds to a higher velocity reduction and vice versa. This is supported by the findings made in Sect. , where the investigation of axial induction factors in Fig.  showed that the flexible blades induce a higher velocity reduction at 270∘ and induce a lower velocity reduction in the axial component at 90∘ azimuth compared to the rigid blade case. Apart from the impact on the wake flow, this affects the loading of the rotor as well. Non-uniform blade root bending moments are the consequence of each of the three blades, leading to an additional 3P main shaft and tower top excitation in the case of a full wind turbine setup. Consequently, this needs to be considered in the design process of large flexible rotors.