3.1. Coning the Rotors from the Same Position with Different Cone Angles
(1) Six coning with Ttrans = 5/R.
For the rotor starting to cone at
r = 5 m, the spanwise distributions of rotor force and torque (sum of the values from three blades) at a wind speed of 9 m/s are depicted in
Figure 6. The tangential force drives the rotor to rotate along the tangential direction and the normal force is parallel to the rotor axis. The tangential force per unit span length
Ft varies along the radial position and between the different rotors. It is interesting to note that the lines of
Ft intersect at the approximate position of
r = 2
R/3, as shown in
Figure 6a. When
r < 2
R/3, the downwind rotor configurations, with a positive
Ccone, have larger
Ft than the corresponding upwind rotor configurations. The situation reverses in the outer part of the blades where larger forces appear on the upwind configurations and lower forces on the downwind cases. What is more, the downwind coning with
Ccone = 4 and the upwind counterpart with
Ccone = −4 have almost the reverse
Ft distribution relative to the standard rotor (with in-plane straight blades). Other corresponding groups (
Ccone = ±8, ±16) show the same phenomenon as well. The distribution of torque per unit span length
Tt, which equals
Ft multiplied by
r, is shown in
Figure 6b. The differences of
Tt between the coned and the standard rotor are exaggerated (especially in the outer part of the blade) by a radial position
r, compared to the deviations of
Ft in
Figure 6a. The increment of
Ft near the blade tip is more beneficial to power production than that near the blade root.
The distribution of normal force per unit length
Fn is shown in
Figure 6c,d. The downwind rotors have larger
Fn than the straight rotor when
r < 2
R/3, and gradually give lower
Fn lines when
r is further towards the blade tip. However, the
Fn lines of upwind rotors gradually get close to that of the straight rotor when
r > 2
R/3, which is different from the enlargement phenomenon of
Ft lines. Although the largest downwind cone angle (
Ccone = 4) leads to the lowest
Fn near the blade tip, the largest upwind cone angle (
Ccone = −4) gives the overall lowest
Fn. Furthermore,
Ccone = −4 brings the largest
Ft and
Tt values near the blade tip, which is beneficial for wind turbine design.
The integrated thrust
T and torque
Q of these rotors are listed in
Table 3. As a higher torque and a lower thrust are favorable, it comes very naturally that we use the torque to thrust ratio (torque
Q divided by thrust
T) to judge the overall influence of conning. This parameter is defined as
QT and listed in
Table 3. The relative variations of these parameters are denoted as
δT,
δQ, and
δQT, respectively, which are also listed in
Table 3. The parameter
δT is defined as Equation (2), and
δQ,
δQT follow similar definitions.
The integrated thrust of Ccone = −4 is the lowest among these seven cases, which is 4.53% less than that of the straight rotor. Though this largest upwind conning design leads to a 1.07% reduction of torque, it still has the highest torque to thrust ratio QT due to the spectacular dwindling of thrust. In short, Ccone = −4 gives the best result. The downwind counterpart with Ccone = 4 produces the largest thrust, lowest torque, and lowest QT, which makes it the most undesirable design. The upwind configuration surpassing their downwind counterpart was also revealed in the pairs of Ccone = ±8 and ±16. The pair Ccone = ±16 (with the smallest conning slope) gives the slightest variations of T, Q, and QT. The pair Ccone = ±8 shows median variations, which is in line with our intuition.
(2) Coning at Ttrans = 1/3 and 2/3.
For the rotors with
Ttrans = 1/3 and 2/3, the spanwise force distributions at a wind speed of 9 m/s are shown in
Figure 7. It is interesting to see that
Tt also intersects at the approximate position of
r = 2
R/3, which is similar to the phenomenon shown in
Figure 6. It also holds true that at
r < 2
R/3, the downwind and upwind rotors have a larger and a smaller
Tt force than the straight rotor, respectively. And at r > 2
R/3, the opposite distribution appears. For
Fn lines, the situation is also similar to those in
Figure 6. The downwind rotors have a larger
Fn at
r < 2
R/3, and gradually give lower
Fn lines when r > 2
R/3. The upwind rotors get close to the straight rotor when
r > 2
R/3.
The upwind coning configurations have a lower
T and a larger
Q as compared with their downwind counterparts, as listed in
Table 4 and
Table 5, which implies the priority of the upwind design. From the view of
QT, the configuration
Ccone = −4 (upwind rotor with the largest cone angle) also gives the best results. It gives the largest thrust reduction with a minor loss of torque. The downwind counterpart
Ccone = 4 has the worst performance with the largest thrust and the lowest torque.
The axial velocity contour in plane
x = 0 m (of the coordinate system shown in
Figure 3) of three rotors are shown in
Figure 8. Two rotors with
Ccone = ±4 and
Ttrans = 1/3 are compared along with the straight rotor. It is found that the downwind rotor has the strongest wake deficit. According to the momentum theory, this will lead to the largest thrust force, which is consistent with the results shown in
Table 4. The upwind rotor has the weakest deficit and the minimum thrust force. What is more, the upwind rotor has the lowest spanwise wake expansion. More wind turbines can be installed on a wind farm if upwind rotors are utilized instead of the downwind and straight rotors. So, the priority of upwind rotors lies not only in the force performance but also in the wind farm construction.
3.2. Coning the Rotors from Different Positions with the Same Cone Angle
From
Section 3.1, it is found that the rotors with
Ccone = −4 have better performanceinspite of different conning locations (
Ttrans = 5/
R, 1/3, 2/3). And the rotors with
Ccone = 4 have worse performance. So, in this section, the rotors with the same cone angle are compared together.
(1) Rotors with Ttrans = 5/R, 1/3.
Firstly, four upwind rotors configurations with
Ttrans = 5/
R, 1/3 are compared, whose conning configurations are shown in
Figure 9a. In the range of
R/3 <
r <
R, the two upwind rotors have the same conning angle. In addition, as has been described in
Section 2.1, the two rotors have the same chord, twist, and airfoil at the same radial location. In other words, the running condition on rotor
Ccone = −4 and
Ttrans = 5/
R are the same as that on rotor
Ccone = −4 and
Ttrans = 1/3, as long as
r >
R/3. Based on the above knowledge, we can guess intuitively that the two upwind rotors will perform similarly in the range
R/3 <
r <
R. And the simulation results, as shown in
Figure 9b–d, have revealed our initial guess. In the range of
R/3 <
r <
R, the respective
Ft and
Fn distributions of the two upwind rotors are very close.
In the range of 5 m <
r <
R/3, the discrepancy between lines in
Figure 9b,c are obvious. This discrepancy is caused by the different conning configurations where
Ttrans =
5/R has a cone angle and
Ttrans = 1/3 is without cone. This phenomenon indicates that the conning slope is a fundamental parameter when considering the effects of the cone. The two downwind rotors also show a similar phenomenon. Although rotor with
Ttrans = 1/3 has zero
Zcone in this range, it does not produce the same force distributions as the straight one. This implies that different cone configurations at
r >
R/3 influence the blade sections at 5 m <
r <
R/3, even if they all have zero cone angle there.
(2) Rotors with Ttrans = 1/3, 2/3.
Secondly, the up/downwind counterparts of
Ttrans = 1/3 are compared with counterparts of
Ttrans = 2/3. The conning configurations are shown in
Figure 10a. In the range of 2
R/3 <
r <
R, the two upwind rotors have the same cone angle. The two downwind rotors cone with the same slope as well. As is shown in
Figure 10b–d, the
Ft and
Fn lines of the two upwind rotors almost coincide and as well as the downwind rotors. This phenomenon agrees well with
Section 3.2(1).
In the range of
R/3 <
r < 2
R/3, the two upwind rotors have different cone configuration so that the
Ft and
Fn lines do no coincide. The same holds true for the downwind rotors. Although
Ttrans = 2/3 coincides with the straight configuration as is shown in
Figure 10a, they produce different force distributions. In the range of
r <
R/3, all the rotors are not conning. However, their force distributions deviate from each other. Similar to
Section 3.2(1), conning on the blade tip will influence the spanwise region extending to the blade root. However, the different cone designs on the spanwise region away from the blade tip (such as
R/3 <
r < 2
R/3) do not influence the blade tip region (
r > 2
R/3). Cone on regions with larger
r has priority over smaller
r, which shows the importance of blade tip design.
3.3. Coning the Rotors from Different Positions with the Same Blade Tip Position
It has been found in
Section 3.2 that the different cone designs on the middle span region (such as
R/3 <
r < 2
R/3) do not influence the blade tip region (
r > 2
R/3). However, the cone on the blade tip region could influence the rest of the blade. This implies the importance of the blade tip. In this section, rotors with the same tip locations are selected for comparison.
As is shown in
Figure 11, the rotor of
Ccone = 8,
Ttrans = 1/3 and the rotor of
Ccone = 4,
Ttrans = 2/3 have the same tip displacement
Zcone =
R/12. For
Ccone = −8, −4, the tip displacement is
Zcone = −
R/12. In the range of
r <
R/3, all the rotors have zero
Zcone. It is interesting that the
Ft and
Fn lines of the two upwind rotors almost coincide at
r <
R/3, which never appears in figures of
Section 3.1 and
Section 3.2. The downwind rotors also show the same rule. It is validated that, when considering the effects of conning on the non-cone part, the blade tip position is a critical parameter. If the partially coned configurations have the same tip position, the part without cone will perform the same.
The pressure coefficient
Cp at different blade sections is depicted in
Figure 12, whose horizontal axis is normalized by the local chord length. At
r = 15.97 m and 50.20 m, upwind coned configurations have smaller integration areas closed by the
Cp curves. At
r = 79.87 m, downwind rotors have a notably smaller integration area. These are consistent with the smaller
Fn of upwind rotors in
Figure 11. At
r = 61.61 m, the two rotors with
Ttrans = 1/3 have similar
Cp curves as the straight rotor. Among the two rotors with
Ttrans = 2/3, the upwind one has a smaller
Cp integration area, and the downwind has a larger area. They are all compatible with the
Fn lines. However, from
Figure 12c, it is hard to find why the
Ft lines of the two rotors with
Ttrans = 2/3 intersect in
Figure 11b.
To find the relationship with
Ft, the
Cp curves are not drawn along the chordwise direction. Instead, as shown in
Figure 13, they are depicted along the rotor axial direction so that the integration area closed by
Cp curves can reflect the
Ft changes. The horizontal axis is normalized by the maximum thickness of the respected airfoil section. In
Figure 13c, the special
Cp distributions in the axial position range (0.5, 1), which represent the suction side, leads to an equalized
Ft of the two rotors with
Ttrans = 1/3. If the
Cp lines in this range are observed in the clockwise direction (an axial position from 0.5 to 1, then from 1 to 0.5), the observed positions are moving from the leading edge (LE) to the maximum thickness point (MTP) and then to the trailing edge (TE). From LE to the MTP, the upwind rotor with
Ttrans = 1/3 has a lower
Cp than the straight rotor. From MTP to TE, the upwind one also has lower
Cp. The integration areas closed by the
Cp curves of the two rotors with
Ttrans = 1/3 are nearly equal. In
Figure 13a,b,d, the integration areas closed by
Cp curves are different which leads to different
Ft. Downwind coned configurations have larger integration area at
r = 15.97, 50.20 m and smaller integration area at
r = 79.87 m, which is consistent with the
Ft lines in
Figure 11b.
As discussed above, the secrets of equal
Ft lies in the
Cp distributions on the suction side. The lines from MTP to TE in
Figure 13c are different from those in
Figure 13a,b,d. Back to the normally used
Cp curve style in
Figure 12, the suction side lines of
Figure 12c are obviously different from
Figure 12a,b,d. In
Figure 12c, the
Cp difference between up/downwind counterparts appears on the whole suction side. In
Figure 12a,b,d, the
Cp difference mainly show near the leading-edge suction peaks. In short, the
Fn and
Ft force distributions can be explained by the corresponding
Cp distributions.
As is shown in
Figure 12d, the
Cp suction peaks of the two upwind rotors are only slightly higher than that of the straight rotor which explains the nearly coincide
Fn forces near the tip. In
Section 3.1 and
Section 3.2, all the
Fn lines of upwind rotors gradually coincide with that of a straight rotor near the tip. The reason why the
Fn lines of upwind rotors do not obviously surpass that of the straight rotor at
r > 2
R/3 needs to be further studied. Coincidently, Prandtl’s tip loss factor
F starts to be none zero at about 2
R/3 which is shown in
Figure 14. It is preliminarily supposed that the
Fn force near the blade tip is strongly controlled by the tip loss effects, which is caused by the radial flow across the blade tip from the pressure to the suction side.
Fn is directly connected with pressure on both sides. Thrust decreases dramatically near the blade tip because of tip loss effects. Therefore, it is rather hard to produce higher thrust near the blade tip regardless of upwind or downwind configurations. Furthermore, the upwind cone will decrease the spanwise tip loss velocity because of the projection of axial velocity and then decrease the tip loss effects. This is consistent with the slightly higher suction peak of upwind rotors in
Figure 12d. Downwind cone will increase the spanwise velocity and the tip loss, which agrees with the lower suction peaks in
Figure 12d and lower thrust in
Figure 11c.
The vorticity contour on plane
x = 0 m of different rotors are compared in
Figure 15. At approximately
r > 2R/3, the blade sections are surrounded by tip vortex so that the tip loss effect is of vital importance. Consistent with the analysis above, downwind rotors have stronger tip vorticity than the upwind counterparts. Therefore, downwind rotors have larger power loss and lower
Fn force in this range. At approximately
r < 2R/3, the tip loss effect is weak. However, the shed vortex still influences the blade force through induction. According to the Biot–Savart law, the wind speed induction decrease with the square of distance away from the vortex center. So, the distance from the blade section to the tip vortex is of vital importance. And the distances vary with the cone configurations. The downwind rotors push the tip vortex further downstream, which leads to less tip vortex induction and higher axial inflow velocity. As a result, the
Fn force of downwind rotors are larger than the upwind counterparts and the straight rotor in this range, which is consistent with the thrust distributions in
Section 3. In short, the different cone configuration leads to different cone angle, different tip vortex strength and different position of tip vortex, which then determines the final force distributions along the rotor span.