*3.1. Influence of Eccentricity of Deformed Blades on the Turbine Performance*

Numerical simulations of the air flow in the Savonius rotor with deformable blades were carried out for different magnitudes of eccentricity and different angular positions of the eccentricity line with respect to the direction of the incoming wind. The eccentricity magnitude *E* was defined as a distance between the axis of turbine rotation (marked with a black X in Figure 6) and the center of the blade tip trajectory (marked with a red X), whereas the eccentricity line connects these centers. Selected angular positions of the eccentricity line marked by purple arrows are shown in Figure 6. The rotor blades are presented for their highest deformation. Three magnitudes of eccentricity *E* = 50 mm (the ratio of eccentricity to the guide ring diameter *E*/*D* = 5%), 100 mm (10%) and 150 mm (15%) were investigated for the whole 360◦ range of angular positions of the eccentricity line with respect to the direction of the incoming wind.

**Figure 6.** Angular positions of the eccentricity line.

The polar plot of the power coefficient distribution is shown in Figure 7, where data markers indicate all the simulations performed. The black line in the figure indicates the case with a non-deformable rotor. The aim of the investigations was to find the optimal eccentricity position. Therefore, the initial screening calculations were performed with eccentricity changes every 45◦. Then, the additional calculations were conducted for the range of 75–180◦. In every case 10 revolutions were simulated and average values of the power coefficient out of the last three revolutions were calculated and shown in the figure. Finally, for the position of 105◦, an additional 5 revolutions were simulated to determine more precisely the average values for further analysis. However, no significant difference (less than 2%) of the averaged pressure coefficient was obtained with respect to the previously obtained value.

**Figure 7.** Polar plot of the power coefficient *Cp* for deformable blade rotors with different magnitudes and positions of eccentricity.

As one can see in Figure 7, some gain of the Savonius rotor aerodynamic performance due to blade deformation is obtained for the eccentricity angular position in the range 45–180◦ (with respect to the direction of the incoming wind—0◦). This range diminishes with a growth in the eccentricity magnitude. The optimal position of eccentricity is around 105◦ for all eccentricity values, however, differences in performance in the range 90–120◦ are low. In these cases, the advancing blade is expanded and the returning blade is contracted. For the eccentricity position of 105◦, the power coefficient values are 0.284, 0.344 and 0.393 for 5%, 10% and 15% of the eccentricity magnitude, respectively. It provides a 37%, 66% and 90% increase with respect to the power coefficient of the non-deformable rotor (*Cp* = 0.207).

A proper angular position of the eccentricity is crucial. It can be easily noticed that for misplaced eccentricity, negative effects are very strong. For eccentricity magnitudes higher than 5%, the power coefficient is negative in a wide range of angular positions, which means that the turbine cannot convert the wind energy. Therefore, the mechanism to position the rotor with respect to the incoming wind is of a key importance in the case of this design.

The results presented in the next figures and their detailed analysis refer to the rotor with deformable blades with the eccentricity of 100 mm (10% of the rotor diameter) in the position of 105◦. In this angular position (or in its close vicinity), the investigated turbine reached the maximum of its performance at any eccentricity. The eccentricity value selected for this analysis is high enough to reach significant improvement in the aerodynamic performance of the rotor and to see clearly deformation of the blades in the pressure field plots. On the other hand, deformation is not excessively high as far as the fatigue of the blade and the form of the blade deformation due to mechanical constraints are concerned. The results for this rotor configuration are compared with the results for the rotor with non-deformable blades (eccentricity 0).

## *3.2. Changes in the Power Coe*ffi*cient during Rotor Revolution*

Changes in the power coefficient during one revolution of the rotors for deformable blades and non-deformable blades are presented in Figure 8. They are sums of contributions of the individual blades shown in Figure 9. The hatched areas between deformable and non-deformable blade lines in both figures indicate positive (+) or negative (−) effects of deformable blade geometries onto the power coefficient readings. Both blades were analyzed separately for better understanding of the occurring phenomena. Blade A is marked blue and Blade B is indicated by red lines in Figure 9. Small changes can be distinguished between characteristics for each blade of the particular rotor type. They resulted from not fully repeatable loads of each blade due to random structures influenced by numerical errors (mesh deformation and remeshing). Still, 5 rotor revolutions selected for data averaging seem to be sufficient to reveal differences between the deformable and non-deformable rotors.

**Figure 8.** Comparison of the averaged values of the power coefficient *Cp* for one revolution of the rotors with non-deformable and deformable blades. Subsequent positions of the turbine illustrated in the schemes above the graph correspond to the turbine revolution angle.

**Figure 9.** Comparison of the averaged values of the power coefficient *Cp* for each rotor blade for one revolution of the rotors with non-deformable and deformable blades. Subsequent positions of the turbine illustrated in the schemes above the graph correspond to the turbine revolution angle.

An increase in the power coefficient of the deformable blade rotor with respect to the one with fixed shape can be observed in Figure 8 during all their revolutions. The most significant increase is observed for 30–70◦ and 210–250◦ parts of the cycle. However, one can see that for the part of the period when *Cp* reaches its minimum (90–135◦ and 270–330◦), a significant increase is also observed. This positive effect results from an increase in the performance of both blades for almost all the revolutions, as one can see in Figure 9. A slight negative effect can be spotted for the advancing blade in the ranges 85–130◦ and 265–325◦ and a very minor one for the returning blade at 160–220◦ and 340–15◦.

#### *3.3. Blade Loading and Torque Generation*

In order to better understand the contribution of the particular blade loading to the torque generation, torque distributions along the blades (from the axis towards the blade tips) are shown in Figure 10 and pressure fields in Figure 11, correspondingly. The *dT*/*dL* derivative per 1 m of the blade span (actually it is Δ*T*/Δ*L* as it is based on the finite volume simulations) is shown in Figure 10, thus, in order to obtain the torque value, this parameter needs to be integrated along the blade. Three particular instants of the rotor revolution were selected (15◦, 55◦ and 105◦), which is sufficient to display the main differences between the rotors with deformable and non-deformable blades. The data were presented for a half of the cycle, because no significant differences can be distinguished for its second part. The data in Figures 10 and 11 are attained for the 10th rotor revolution, thus there is no perfect agreement with the results in Figure 9 obtained from averaging over 5 revolutions. Animations of changes in parameters for one rotor revolution are presented in Supplementary Materials (Video S1 presents the non-deformable rotor, whereas Video S2 shows the deformable one).

In the 15◦ (also 195◦) instant of the revolution cycle, the non-deformable rotor reaches the maximum of its power output (Figure 8). In this case, one can notice in Figure 11 that a high pressure at the concave side of the advancing blade (A) and a low pressure at its convex side (due to flow acceleration) result in the highest positive torque generated by this blade. It is diminished by the retarding contribution of the returning blade (B), mostly due to a pressure difference for its part near the axis of the turbine. In this instant of the revolution cycle, the blades of the deformable rotor have a semi-circular shape like the blades of the non-deformable rotor. No significant increase in the power coefficient is observed at this moment for the rotor in reference to the non-deformable one (Figure 8). The data presented in Figure 9 reveal slight positive contributions both of the advancing (A) and returning (B) blades. The positive contribution of the advancing blade (blue lines) can be distinguished also in Figure 10. An increase of the torque can be seen for a substantial portion of the blade starting from the rotor axis (internal part), whereas the tip (external) part has slightly lower loading. In the case of the returning blade (red lines), positive and negative contributions of the blade deformation can be observed locally for the internal part of the blade, whereas a positive effect is clearly visible for the external part.

The highest positive effect of the blade deformation with respect to the non-deformable turbine is observed for the 55◦ (also 235◦) instant of the revolution cycle (Figure 8). In this position the deformable rotor reaches the highest power output, whereas it is already significantly diminished for the non-deformable one. In Figure 9 it can be seen that both the blades contribute to the power gain. It is due to the blade deformation, i.e., expansion of the advancing blade (A) and contraction of the returning one (B), as shown in Figure 11. The torque of the advancing blade is significantly higher at its external part (Figure 10). It results from higher velocity at the convex side in a very limited way. The main reason of the higher torque is just a higher arm (radius) due to the blade expansion. Despite the fact that the pressure build-up (stagnation zone) at the convex side of the returning blade is higher than for the non-deformable blade, its contraction and higher pressure at the concave side do not reduce further the torque at its central part. On the other hand, much higher fluid acceleration at the convex side of the deformed blade for its external part results in lower pressure and locally changes its torque to the positive one. Thus, the retarding effect of the contracted returning blade is reduced significantly.

A significantly positive effect of the blade deformation can be observed also for the 105◦ (and 285◦ as well) instant of the revolution cycle (Figure 8). In this position both the deformable and non-deformable rotors reach the lowest power output. The blade deformation in this case is the highest as presented in Figure 11. In Figure 9 one can see that the deformation of the advancing blade (A) decreases the pressure coefficient, nevertheless, it remains positive. A beneficial effect of the higher arm in the torque definition due to blade expansion can be noticed for the external part of this blade (Figure 10). It is lowered by higher pressure at its convex side due to lower intensity of the vortex structure at the blade tip in comparison to the non-deformable rotor (Figure 11). This positive effect does not compensate for a decrease in the torque in the internal part of the blade. This decrease is mainly due to much lower pressure at the concave side of the blade, which is an effect of the flow acceleration in this region. Deformation of the returning blade (B) improves considerably its performance almost along its whole length, excluding the tip only (Figure 10). It is partially due to a much lower torque arm of the contracted blade. Additionally, the higher curvature of the deformed blade yields higher flow acceleration and a significant reduction in the high pressure stagnation zone at its convex side in comparison to the non-deformable rotor (Figure 11). The blade contraction is also followed by a pressure increase at the convex side of the returning blade. Thus, the pressure difference between the convex and concave sides of the blade is significantly diminished. All these aspects contribute to a reduction in the retarding torque of the deformable blade almost to zero (Figure 9).

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**Figure 10.** Torque derivative (*dT*/*dL*) variation along the blade for selected instants (15◦, 55◦ and 105◦) of the rotor revolution with non-deformable and deformable blades.

**Figure 11.** Pressure fields for selected instants (15◦, 55◦ and 105◦) of one revolution of the rotors with non-deformable and deformable blades.
