*4.2. GF E*ff*ects in Cycloidal Motion: Impact on Torque Profiles*

If the final expected outcome of the application of GFs to Darrieus turbines is the power enhancement (that will be discussed in detail in the next section), it is worth analyzing from a physical point of view their impact on the functioning of the airfoils during a revolution. The balance of the energy extraction between the upwind and the downwind halves of the revolution is in fact very important not only for the overall efficiency, but also for the possible creation of stresses and vibrations of the turbine.

Figure 13 shows the influence of the Gurney flap length and configuration (i.e., the single facing outward or inward, and the fish tail GF, respectively) on the torque coefficient as a function of the turbine position. Displayed data do refer to the first test case only (solidity equal to 0.057 and NACA0018), even though the physical behavior was of general validity in the case of blades in cycloidal motion. The values indicated as "No GF" show the torque distributions with the smooth airfoil.

Upon examination of the figure, it is apparent that the outward pointing Gurney flap tends to increase the unbalance of the torque distribution. The increment of torque in the upwind part is significant and it is connected with the lift-to-drag ratio increment induced by the GF, which is particularly relevant for the higher AoAs reached upwind. On the other hand, the torque reduction downwind is related to the increased drag at those low AoAs that derive from the low wind speed going through the rotor. The inward pointing Gurney flap instead leads to an increment of the torque coefficient along the downwind part of turbine, leading to a more balanced torque profile, even though the extracted work (i.e., the area below the curve) is pretty much the same. The fish tail configuration finally confirmed the hypothesized change in performance, providing a relative increase of the torque coefficient for both the upwind and the downwind parts of the revolution.

**Figure 13.** (**A**) Outward pointing one-sided, (**B**) inward pointing one-sided, and (**C**) fish tail Gurney flap length influence on the torque distribution in function of turbine position.

The relative impact to the produced power of the two halves of the machine is even more apparent from Figure 14. It is very interesting to notice that the fish tail not only provides an increase of the performance on both halves of the machine, but also a very balanced power between the two.

**Figure 14.** Gurney flap length and configuration influence on the power extraction distribution.

#### *4.3. GF E*ff*ects in Cycloidal Motion: Sensitivity Analysis on GF Characteristics*

As discussed, the scope of the present analysis was to study the prospects of different GF configuration in terms of power augmentation of Darrieus VAWTs using symmetric airfoils. To this end, a large number of simulations were carried out. Figure 15A–C and Figure 16A–C show the results of all the studied cases in a way which allows the reader to have an overall outlook on the main effects. In further detail, the two figures show the influence of the one-sided Gurney flap (Figure 15) and the fish tail configuration (Figure 16), respectively, in terms of power coefficient variation for different airfoils thickness. Graphs A–C refer to the three different equivalent turbine solidities (increasing from A to C). The white color in the color palette indicates the reference value of the power coefficient with the smooth airfoil, the red color indicates an incremented one, and the blue color a decreased one.

**Figure 15.** Numerical model results for one-sided Gurney flap for (**A**) 0.057, (**B**) 0.125, and (**C**) 0.25 equivalent solidity turbines.

**Figure 16.** Numerical model results for the fish tail Gurney flap for (**A**) 0.057, (**B**) 0.125, and (**C**) 0.25 equivalent solidity turbines.

#### *Energies* **2020**, *13*, 1877

Finally, the positive and negative values of Gurney flap length in Figure 15 indicate its outward and inward pointing directions, respectively, with respect to the turbine axis.

Upon examination of the graphs, some interesting observations can be noted:


Due to the complexity of analyzing so much data at a glance, some relevant trends have been extracted and reported in Figures 17 and 18.

Figure 17 first reports the extracted power for the four tested airfoils (i.e., as a function of the thickness-to-chord ratio of the airfoils) as a function of the GF length in the case of the outward (A), inward (B), mounting, and fish tail configurations (C). The high-solidity test case was selected, even if the same considerations can be repeated for the other three cases. Upon examination of the figure, one can notice that for the inward mounting (i.e., the one privileging the downwind side), the thinner the airfoil, the higher the performance that can be achieved. Also, the optimal GF length decreases monotonically with the airfoil thickness. On the other hand, the thinner NACA0012 airfoil is more sensitive to the GF length, with steeper variation curves. This is due to the larger impact of the GF additional drag on the thin airfoil. Overall, the thicker NACA0021 airfoil shows a quite different behavior than the other ones, with flatter response curves and much larger optimal GF lengths.

On the other hand, in case of the outward mounting, the best performance is achieved for medium-solidity airfoils, where the application of a GF to very thin or very thick ones does not provide benefits. The optimal GF length keeps shifting to lower values as soon as the airfoil thickness decreases. The same analysis is repeated in case of the fish tail configuration (see Figure 17C). One can notice that the fish tail configuration provides consistent benefits for almost all the airfoils, with only the exception of the very thin one, where the draft increase is probably not compensated for by the additional lift.

Figures 18 and 19 compare the optimal configurations found among the tested turbines with different values of solidity.

Upon examination of the figure, some of the relevant trends discussed before are still clearly noticeable. In addition, it is worth noticing that:


**Figure 17.** Turbine power coefficient as a function of the GF length and airfoil thickness for equivalent turbine solidity 0.25: (**A**) inward GF, (**B**) outward GF, (**C**) fish tail.

**Figure 18.** Increment of the torque value for optimal length of GFs for all investigated turbine configurations: (**A**) inward GF, (**B**) outward GF, (**C**) fish tail.

**Figure 19.** Optimal value of the GFs length for all investigated turbine configurations: (**A**) inward GF, (**B**) outward GF, (**C**) fish tail.

#### *4.4. Response Surfaces*

Even though scattered data coming from the simulations already provided some interesting indications about the relevant trends, more detailed results were needed to find optima with a sufficient accuracy. To this end, a radial basic interpolation was carried out on the data. This analysis provided a three dimensional solution of the surface response. As a result, the power coefficient as a function of the airfoil thickness and the Gurney flap length for different configurations are shown in Figures 20 and 21 for the single-side GF and the fish tail GF, respectively. These plots were obtained for interpolation with use of the Inverse Multiquadric (IMQ) basis functions.

**Figure 20.** Power coefficient for one-sided Gurney flap using IMQ functions for (**A**) 0.057, (**B**) 0.125, and (**C**) 0.25 equivalent solidity turbines.

**Figure 21.** Power coefficient for the fish tail configuration using IMQ functions for (**A**) 0.057, (**B**) 0.125, and (**C**) 0.25 equivalent solidity turbines.

Optima were then found on the surfaces. Table 4 reports the values of the power coefficient for the configurations equipped with the most efficient blade and the reference blade, respectively. The power coefficient increment was defined as a ratio of the power coefficient of the blade with changed thickness and of the Gurney flap to the power coefficient of the reference blade.


**Table 4.** Power coefficient and geometrical parameters of the most efficient turbine and the reference one using IMQ functions.

It can be observed that the potential increment of the power coefficient due to the introduction of the Gurney flap can be really significant (up to +89.5%), especially when higher-solidity turbines are considered.

Overall, the increment provided by the fish tail configuration is lower than that of a one-sided GF. The fish tail shape is more suitable for turbines with lower solidity, while the one-sided Gurney flap configuration significantly influences the performance of turbines with higher solidity values. It has to be pointed out, however, that the quantitative results reported in Table 3 only refer to the selected study cases (some of them quite theoretical) and operating conditions. Performance increases to be expected for real rotors are probably lower than the reported values. However, the tendencies are thought to be of general application and they clearly highlight the potential of GFs for use in Darrieus VAWTs. Finally, it is worth noticing that the introduction of the Gurney flap in case of medium-high solidity turbines would suggest that the airfoil thickness should be slightly changed, going toward notably thinner airfoils in the case of the one-sided GF and to slightly thinner ones for the fish tail configuration. In case of low-solidity machines, on the other hand, a medium-thickness airfoil is still the best choice.
