Figure 6 shows the impact of changing t/c on the turbine CP for the studied range of xt/c, I and λ. Figure 7 shows the instantaneous moment coefficient Cm versus θ for selected t/c and xt/c. The following can be observed:

regarding the lowest value of I=4.5 (see Figs. 6a–e and 7), generally speaking, the trend of CP–t/c for different λ is similar, except for some noticeable differences. That is, by increasing λ, the CP shows higher sensitivity to t/c. This is reflected as higher |ΔCP| and can be explained by the following: by changing t/c, the pressure gradient changes over the airfoil; therefore, the transition point, the separation and stall characteristics, and eventually the resultant aerodynamic loads also change. However, when the flow is fully separated in the post-stall regime, changing t/c no longer has a significant impact on CP. By increasing λ, and thus a more limited variation of α, the blade passes over a range of fewer azimuth angles in the post-stall regime (see Fig. 5). Due to this, changing the t/c is influential within a wider range of effective θ. This can be recognised by the improved CP for higher λ. At λ=2.5, the CP follows a non-monotonic trend for xt/c≤30 % and a monotonic upward trend for xt/c≥35 %. Nevertheless, with the exception of xt/c≤22.5 % at λ=5.5 where the CP monotonically decreases by increasing t/c, the trend remains non-monotonic for different values of xt/c at the studied range of λ. That is, by changing the t/c to higher values, the CP experiences an initial growth to its maximum value at topt/c, followed by a reduction for t/c>topt/c. This can be recognised from the Cm plots, where by changing t/c to its optimal value at topt/c, the sudden drop in Cm,max, which indicates the instant of moment stall, is observed at higher θ; the consequent fluctuation is alleviated, and the mean value of Cm increases, thus making consistency with the highest CP at topt/c for a fixed xt/c (see Fig. 7a–i for selected xt/c). This can be attributed to the following observations from the skin-friction, lift and drag coefficients (Cf, Cl and Cd): when the turbine is operating at low values of λ≤3.5, increasing t/c from 10 % to topt/c changes the stall type from mixed stall for t/c=10 % to trailing-edge stall for thicker airfoils; an earlier formation of laminar separation bubble (LSB) and trailing-edge separation (TES) is observed; TES–LSB merging (i.e. full-flow separation) is discovered to occur at a higher azimuth, indicating a more extended favourable pressure gradient for topt/c(see, for example, Fig. 8 for xt/c=27.5 % at λ=2.5); and a lighter dynamic stall is observed, that is, lift and drag jump, which indicate the onset of dynamic stall, reduce and shift to a higher azimuth, and the consequent post-stall load fluctuations are alleviated (see, for example, Fig. 9 for xt/c=27.5 % at λ=2.5). However, an earlier stall is found to occur for t/c>topt/c due to more pronounced earlier merging of TES–LSB. This is reflected by lower CP and Cm,max for t/c>topt/c (see Figs. 6a–c and 7a–i). Note that the monotonic growth in CP-t/c for xt/c≥35 % at λ=2.5 can also be explained with the aforementioned reasoning, yielding the CP,max at the highest thickness of topt/c=24 % (see Fig. 6a).

Table 3 shows the topt/c corresponding to each xt/c (i.e. topt,xt/c) at different λ. The topt/c corresponding to each λ is indicated by a star sign. It can be seen that by increasing xt/c, which means a longer favourable pressure gradient on the blade, a higher thickness is needed for the airfoil to be optimal. Note that increasing λ influences the shape of the optimal airfoil by decreasing its thickness. In other words, the higher λ is, the thinner the optimal airfoil is. This is consistent with the findings documented in Healy (1978) and Subramanian et al. (2017), where it shows the superior performance of thick airfoils at low λ. This may be attributed to the turbine operational regime as follows: when the turbine goes into regimes with higher λ and more pronounced reduction in the variation of α, higher values of Cl at lower α have the most impact on the turbine CP. Therefore, thinner airfoils with a higher lift curve slope outperform the thicker ones with a lower slope of the Cl-α. Eventually, this results in less pronounced sensitivity of the topt/c to xt/c and shifting the peak in CP-t/c (i.e. topt/c) towards the lowest t/c=10 % and 12 % in the non-dynamic stall regime with λ≥4.5 (see Table 3). The analysis also shows a drag increment for thicker airfoils at λ=4.5 and 5.5, which is a result of the earlier formation of LSB and TES. This is consistent with the reduction in CP and Cm,max for t/c>topt/c (see Figs. 6d, e and 7j–o). Note that for xt/c≤22.5 % at λ=5.5 the same reasoning results in a monotonic decrease of CP, yielding the CP,max at the lowest thickness of t/c=10 %.

The effect of flow curvature on aerodynamic loading is another important physical phenomenon to take into account in predicting the performance of VAWTs. Because of the angular velocity of the turbine rotor blades, the relative flow direction continuously varies along the airfoil chord, and thus the blades experience curved streamlines. As a result of this, a symmetrical airfoil with zero pitch angle in the circular path of a VAWT rotor behaves as if it is a cambered airfoil with a non-zero pitch angle in a straight flow (Migliore et al., 1980; Rainbird et al., 2015). The flow curvature effects become less pronounced on a curved airfoil (Coiro et al., 2005). In addition, a blade hinge located at 50 % chord length significantly alleviates the flow curvature effects. However, among all the parameters, the ratio of blade chord to turbine rotor radius (c/R) has the greatest impact on flow curvature effects (Migliore et al., 1980). For low values of c/R (i.e. low solidity), the blade surface pressure distribution shows negligible differences with respect to that of the no-lift condition (Coiro et al., 2005), indicating less pronounced effects of flow curvature on the performance of low-solidity turbines (Rainbird et al., 2015). In this study, due to the low value of c/R=0.12 (i.e. low σ), the contribution of flow curvature effects is considered small.

Regarding the moderate and highest values of I=6.0 and 7.5 (see Fig. 6f–j and k–o), the overall trend for CP is very similar to that of the lowest I=4.5; however, it shows comparatively lower values of |ΔCP|), especially for λ≤3.5. The impact of changing the rLE on the turbine CP is separately discussed in detail in Sect. 4.1.3; therefore, it is not included here.

Figure 10 shows the variation of the turbine CP versus xt/c at the studied range of t/c, I and λ. Figure 11 shows the instantaneous moment coefficient Cm versus azimuth for selected xt/c and t/c. The following can be seen:

regarding the lowest value of I=4.5 (see Fig. 10a–e), the overall trend of CP-xt/c for different λ is very similar, except for the following differences. By increasing λ, the turbine CP shows higher |ΔCP|. This is due to the similar reasoning discussed earlier in Sect. 4.1.1 and summarised as follows: changing the xt/c results in changing the boundary layer and stall characteristics. On the other hand, increasing λ is associated with a lower variation of α, i.e. a more limited azimuthal range of the post-stall regime. As a result, the impact of changing the xt/c becomes significant over a wider range of θ, resulting in improved CP.

For t/c≤12 % in the dynamic stall regime with λ≤3.5, the CP monotonically decreases by increasing the xt/c, yielding the CP,max with the lowest xt/c of 20 % (see Fig. 10a–c). However, apart from t/c=10 % at λ=4.5, where CP monotonically decreases, the trend for thin airfoils changes to non-monotonic at λ≥4.5 (see Fig. 10d and e). In other words, by increasing the xt/c from 20 % to 40 %, the CP grows to its maximum value at xtopt/c, before decreasing for xt/c>xtopt/c. The monotonic behaviour of CP for thin airfoils at low λ can be explained based on the observations of the skin-friction coefficient Cf as follows: the dynamic stall for t/c≤12 % is preceded by either (i) gradual extension of the LSB towards the trailing edge (thin-airfoil stall) or (ii) a sudden upstream propagation of the TES (leading-edge stall). Changing the xt/c to higher values results in either an earlier downstream extension of the LSB or an earlier formation and abrupt upstream propagation of the TES and, consequently, an advanced stall on the blade. This is evident from the Cm plots for t/c=12 % (see Fig. 11a–c), where the abrupt drop in Cm,max occurs at a lower θ, indicating an earlier moment stall due to increasing the xt/c. The overall lower values of Cm for higher xt/c justify the monotonic reduction in CP. For brevity, the Cf plots are not presented here.

On the other hand, the non-monotonic trend of CP for thin airfoils at λ≥4.5 (i.e. non-dynamic stall regime) can be recognised from the Cm plots. For example, by changing the xt/c from 20 % to xtopt/c=25 % for t/c=12 % λ=5.5, the Cm,max slightly increases before decreasing for xt/c≥27.5 % (see Fig. 10e). This can be explained by the skin-friction coefficient Cf, where it shows an earlier formation and upstream propagation of the TES and, thus, a promoted TES for xt/c>xtopt/c(see Fig. 12). Note that, when the adverse effects of dynamic stall are suppressed at λ≥4.5, increasing xt/c shows a marginal positive impact on the CP for thin airfoils, reflecting a non-monotonic trend of CP versus xt/c. However, the value of t/c for thin airfoils plays a more crucial role in this regime. This can be observed from the sharp downward trend of CP-xt/c for t/c=10 % at λ=4.5, while it changes to a non-monotonic trend for t/c=12 %. This may be attributed to the more pronounced formation and propagation of TES and, thus, an earlier stall due to increasing xt/c for t/c=10 %. However, the trend of CP-xt/c for t/c of 10 % remains non-monotonic at λ=5.5, showing less sensitivity to TES at higher λ.

For the medium- and high-thickness airfoils (i.e. t/c≥15 %), the turbine CP follows a trend with a defined maxima at xtopt/c (see Fig. 10a–e). As previously discussed in Sect. 4.1.1, this non-monotonic trend is a consequence of thicker-airfoil stall type, which is triggered by the formation of a flow reversal near the trailing edge (McCroskey, 1981; Sharma and Visbal, 2019; Frolov, 2016; Meseguer et al., 2007). Therefore, when xt/c changes to its optimal value, the adverse pressure gradient becomes less severe, resulting in improved stall characteristics. This can be recognised by either dynamic stall alleviation at low values of λ≤3.5 or a postponed stall at non-dynamic stall regimes with λ≥4.5. Table 4 gives the xtopt,t/c (i.e. the xtopt/c at each t/c) in terms of CP,max for each λ. The corresponding xtopt/c for different λ is indicated by a star sign. For λ≤3.5, by increasing t/c, the xtopt,t/c also increases. However, by increasing λ from 2.5 to 3.5 and thus encountering a comparatively lighter dynamic stall and more limited variation of α, the xtopt/c and its corresponding t/c decrease (see also Fig. 10). The reason for the outperformance of thin airfoils at higher λ is explained earlier in Sect. 4.1.1. Nevertheless, in the dynamic stall regime, the outperformance of moderate to high values of xt/c for thicker airfoils at a fixed λ is readily apparent from the turbine Cm for selected t/c=18 % and 24 % (see Fig. 11f–h and k–m). It can be seen that increasing xt/c to its optimal value results in an increase in the Cm curve peak, a delay in the sudden drop of Cm,max, less pronounced subsequent fluctuations and higher values of Cm in the turbine downwind quartile. This is due to alleviated dynamic stall and is more pronounced for t/c=24 % (see Fig. 11k–m). A further increase in xt/c>xtopt/c is found to have a negative effect on Cm and finally leads to an earlier stall. This is because increasing the xt/c higher than xtopt/c promotes the formation of LSB and TES and results in an earlier full-flow separation and drop in Cl,max. Please note that for better illustration the Cm plots are not presented for all the studied values of xt/c. For λ≥4.5, by increasing xt/c for t/c≥15 %, the CP shows less sensitivity to xt/c, and the corresponding xtopt/c changes marginally (see Fig. 10d, e and Table 4). This is consistent with the turbine Cm plots for selected t/c=18 % and 24 %, where the Cm,max and the azimuth of moment stall are almost invariant to xt/c (see Fig. 11i–j and n–o).

Regarding the moderate and highest values of I=6.0 and 7.5 (see Fig. 10f–j and k–o): the CP-xt/c shows a similar trend to that of I=4.5. However, in dynamic stall regime (i.e. λ≤3.5), the turbine CP shows a considerably smaller |ΔCP|, especially for higher xt/c. On the other hand, in non-dynamic stall regime with λ≥4.5, a marginal reduction in |ΔCP| is observed. However, the CP-xt/c shows more pronounced sensitivity to changing xt/c for the moderate and thick airfoils.

Figure 13 shows the impact of changing the index of rLE on the CP for selected airfoils at different λ. Figure 14 shows a comparison of the CP-xt/c for different I and selected values of t/c. The analysis is grouped based on the maximum thickness as follows:

regarding the thin airfoils (t/c=10 % and 12 %) (see Figs. 13 and 14a–e), regardless of xt/c, the turbine CP is marginally influenced by the I. This can be attributed to the low dependency of thin airfoils and the relevant aerodynamic loads on rLE, which is due to the geometrical constraints imposed by the airfoil thickness. It can be observed that by increasing the index of rLE for different xt/c at λ≤3.5, CP slightly changes; this minimal difference is in line with the corresponding Cm plots for t/c≤12 %. This can also be recognised from the skin-friction, lift and drag coefficients by the negligible changes in the characteristics of boundary layer events, including LSB and TES, and consequently the onset of dynamic stall and Cd,max. Due to the large volume of the results, the Cm, Cl, Cd and Cf plots are not presented here. For λ≥4.5, except for the NACA0010-I/3.5, where increasing rLE has the most influence on CP, the aerodynamic loads and the turbine Cm show even less sensitivity to I. Note that this is the regime at which the dynamic stall is no longer encountered and thin airfoils outperform the rest of the airfoils. The impact of the index of rLE on the turbine CP for the optimal thin airfoils at λ≥4.5 is shown in Fig. 13f. Figure 15d and e shows the corresponding Cm plots.

Regarding the moderately thick airfoils (t/c=15 % and 18 %) (see Figs. 13 and 14f–j), overall, the turbine CP shows higher dependency and sensitivity to I. The higher dependency is due to the less severe geometrical constraints imposed by the moderately thick airfoils. Thus, changing the I noticeably modifies the airfoil shape and thereby influences the aerodynamic loads. The higher sensitivity is reflected by the noticeable monotonic reduction of CP for most of the xt/c values. This significant decrease can be recognised from the Cm plots, where the curve peak drops by increasing the leading-edge radius index. This may be due to the promoted LSB and TES, which results in higher Cd,max for larger I. For λ≤3.0, the more prominent sensitivity is observed within the range of 22.5 % ≤ xt/c ≤ 35 %; however, the CP shows less sensitivity to rLE for λ=3.5, corresponding to a lighter dynamic stall regime (see Fig. 14f–h). Note that the moderately thick airfoils show superior performance over the thin and thick airfoils at λ=3.0 and 3.5 (i.e. the NACA0018-4.5/2.75 and NACA0015-4.5/2.5, respectively). Figures 13f and 15b and c show the impact of changing rLE on the turbine CP and Cm for the optimal airfoils at λ=3.0 and 3.5. When the turbine goes into the non-dynamic stall regime with λ≥4.5, the range of xt/c within which the index of leading-edge radius is the most influential shifts downstream to 30 % ≤ xt/c ≤ 40 % (see Fig. 14i and j).

Regarding the thick airfoils (t/c=21 % and 24 %) (see Figs. 13 and 14k–o), the analysis shows that at λ=2.5, thick airfoils significantly surpass other airfoils in terms of power performance (see Fig. 13). Aside from the following differences, the overall trend of CP-xt/c is quite similar to that of moderately thick airfoils: CP values are more sensitive to rLE at λ=2.5, 4.5 and 5.5 but less so at λ=3.0 and 3.5 (see Fig. 14k–o). By increasing the rLE, the CP values experience a monotonic reduction, especially for thick airfoils with xt/c≥30 % at λ=2.5, where the variation of I is the most influential on CP. For example, the overall reduction of CP for the NACA0024-I/3.50 at λ=2.5, 3.0, 3.5, 4.5 and 5.5 is 77 %, 21 %, 17 %, 19 % and 23 %, respectively. This can be recognised from the Cm plots, where the Cm values decrease dramatically in both upwind and downwind quartiles, the Cm curve peak drops and the post-stall Cm fluctuations get more significant (see Fig. 15a). This is due to earlier formations of the LSB and TES and thus a higher Cd,max. Thick airfoils with low xt/c show marginal sensitivity to rLE at different λ. The corresponding Cm plots show approximately the same azimuth of moment stall for different I. For brevity, the Cm plots are only presented for the NACA0024-I/3.50, which is the optimal airfoil at λ=2.5 (see Fig. 15a).

Overall, at λ≤3.5, the xtopt/c belongs to the range of xt/c, which corresponds to the highest sensitivity of CP to rLE. For example, the optimal airfoil at λ=2.5 (i.e. the NACA0024-4.5/3.5) has xt/c=35 % that fits in the range of 30 % ≤ xt/c ≤ 40 %, within which the impact of rLE is the most significant. This is while the xtopt/c for λ≥4.5 (i.e. xt/c=22.5 %) does not belong to such a range of xt/c (i.e. xt/c≥30 %). In addition, the most noticeable improvement in CP due to changing the rLE occurs at λ=2.5, where the dynamic stall deeply affects the aerodynamic and power performance of the blade. By increasing λ and thus alleviating or avoiding the dynamic stall, the aerodynamic loads are less affected by the rLE.

The airfoil shape-defining parameters have a coupled impact on turbine performance. Thus, it is of high importance to study the impact of their combined modification on the turbine CP and CT. Figure 16 shows the variation of CP in t/c-xt/c space for different I and λ. Except for λ=5.5, where the combination of topt/c and xtopt/c is achieved by the moderate I=6.0, the Cp,max corresponds to the smallest I=4.5 for λ≤4.5.

For λ=2.5 and I=4.5, the global optimum occurs by a set of high t/c and xt/c (i.e. NACA0024-4.5/3.50). The combination of topt/c and xtopt/c values remains invariant for I=6.0; however, the region of maximum Cp shows lower values of Cp. For I=7.5, the optimal airfoil changes to a thin airfoil with low xt/c, while experiencing lower CP compared to those of I=4.5 and 6.0. The variation of optimal airfoil shape-defining parameters for different λ and the resultant airfoils at each λ are illustrated in Fig. 17.

At λ=3.0, the region of CP,max shows less sensitivity to I, shifting between moderate and high values of t/c and xt/c (see Fig. 16d–f). However, similar to that of λ=2.5, the overall range of Cp values narrows down with increasing I. For λ=3.5, the optimum region of CP remains nearly the same at moderate values of t/c and xt/c for different I (see Fig. 16g–i), while for higher values of λ≥4.5, it stays approximately independent of I, shifting marginally between low values of t/c and xt/c (see Fig. 16j–o). This implies that, by increasing λ, the optimum region of turbine CP is less sensitive to I. Overall, by increasing λ, the local region of optimal airfoil shape-defining parameters changes from the combination of high values of t/c and xt/c for λ=2.5 to moderate t/c and xt/c for λ=3.0 and 3.5 and low values of t/c and xt/c for λ≥4.5.

The results highlight that, in designing morphing blades, single-parameter studies will not provide the overall picture and could lead to unreliable results. The contour plots give a conceptual view of the optimal regions in terms of the airfoil shape-defining parameters, with which the resultant airfoils have their most efficient performance, and also the inefficient regions of the turbine CP, which must be avoided.

Figure 18 shows the turbine CT in t/c–xt/c space. It is interesting to observe that for low λ≤3.5 there is no coincidence between the optimal regions of CT and CP contours, while for λ≥4.5 these two regions overlap. By increasing λ, the optimal region extends marginally towards higher t/c and xt/c, while also experiencing higher values of CT. The non-congruent region of Cp,max and CT,max at low values of λ is different from what is observed in the case of HAWTs. That is, the maximum power output of a HAWT occurs where the highest thrust load is exerted by the turbine blade on the flow. This led to a correlation between the regions of maximum CP and CT. In contrast, the results of the present study show that for VAWTs the same phenomenon only occurs at high values of λ≥4.5, where the turbine goes into non-dynamic stall regimes with more limited variations of α. Therefore, when designing morphing blades for VAWTs, the CT values corresponding to high values of λ are of more importance compared to those of lower λ, where dynamic stall is expected to occur.