Wind Energy Handbook. Michael Barton Graham
explanation for tip‐loss
The flow approaching the rotor is expanding because it is slowing down and so is not axial, that is, it is not parallel to the rotation axis or the undisturbed flow direction. Consequently, there is a radial flow velocity component at the upwind side of the rotor that arises because there is a radial pressure gradient with lower pressure in the tip region than in the inner region. The change of radial momentum at a point on the rotor disc is approximately balanced by the equal and opposite radial momentum at the diametrically opposite point. The magnitude of the radial velocity increases with radius, and so its effects will be greatest at the tip region. The kinetic energy associated with the radial flow does not directly affect the energy capture because it does not influence the aerodynamic force on the blade.
At the blade tip the blade chord length becomes zero (usually but not always in a gradual fashion) and so must also the axial force exerted on the air flow beyond the blade tip that bypasses the rotor. The idealised actuator disc theory predicts a logarithmically singular radial velocity at the tip. This is not possible, and the pressure difference across the disc must fall continuously radially over a small tip region to zero at the tip.
Both a and ψ, which is the angle of the resultant flow to the axial direction at the rotor plane, will vary radially and will change according to how the circulation on the disc varies radially. Disc circulation, or the bound vorticity on the disc, must also rise and fall from blade root to blade tip, as shown in Figure 3.41.
Figure 3.41 The variation of circulation along the length of a blade.
Using just the momentum theory, it is not possible to determine the manner of the variation of a and ψ, but it is clear that the integration with respect to radius r of Eq. (3.93) with (3.89) would result in a value for the optimised power coefficient that would be less than the Betz limit.
Throughout the BEM analysis, it is assumed implicitly that the swirl component generated in the wake of the rotor is sufficiently small that its influence on the pressure field may be ignored and specifically that the pressure far downstream in the wake where the momentum balance is calculated is uniform and ambient. However, as discussed earlier at the end of Section 3.3.2, under lower tip speed ratio conditions, typically within streamtubes that pass close to the blade roots so that the local speed ratio λ = Ωr/U∞ < 2, this is increasingly untrue. However, there is not as yet any fully agreed analysis for this effect except that it may offer the possibility of achieving local power coefficients in excess of the Betz limit. In practical terms the possible increase in total rotor power is unlikely to be very significant.
3.10 Stall delay
A phenomenon first noticed on propellers by Himmelskamp (1945) is that of lift coefficients being attained at the inboard section of a rotating blade that are significantly in excess of the maximum value possible in 2‐D static tests. In other words, the angle of attack at which stall occurs is greater for a rotating blade than for the same blade tested statically. The power output of a rotor is measurably increased by the stall delay phenomenon and, if included, improves the comparison of theoretical prediction with measured output. It is noticed that the effect is greater near the blade root and decreases with radius.
The reason for stall delay has been much discussed, but as yet there is no fully agreed explanation. Partly this may be because stall regulation of fixed‐pitch rotors has been largely phased out for modern turbines that use pitch control. Stall occurs on an aerofoil section when the adverse pressure gradient on the surface following the suction peak is sufficiently strong to reduce the momentum in the lower layers of the boundary layer to zero faster than viscous or turbulent diffusion can re‐energise them. At this point flow reversal occurs, and the boundary layer separates from the surface, causing the aerofoil to stall, decreasing or even changing the sign of the lift curve slope and rapidly increasing the drag. However, on a turbine blade, particularly near the blade root, there is a strong outward radial component to the flow, and the pressure gradient following the streamlines in the boundary layer is less adverse than the section and local incidence would suggest. This may explain at least part of the phenomenon.
Aerodynamic analyses (Wood 1991; Snel et al. 1993) of rotating blades using computational fluid dynamic techniques, which include the effects of viscosity, also do show a decreased adverse pressure gradient. It is agreed that the parameter that influences stall delay predominantly is the local blade solidity c(r)/r.
The evidence that does exist shows that for attached flow conditions, below what would otherwise be the static (non‐rotating) stall angle of attack, there is little difference between 2‐D flow conditions and rotating conditions. Due to the rotation, the air that is moving slowly with respect to the blade close to its surface in the boundary layer is subject to strong centrifugal forces. The centrifugal force manifests as a radial pressure gradient, causing a component of velocity radially outwards. Prior to stalling taking place, this effect tends to reduce the adverse pressure gradient along the surface streamlines and hence the growth of boundary layer displacement thickness, thus decreasing the tendency to separate. When stall does occur, the region of slow moving air becomes much thicker throughout a growing separated region, and comparatively large volumes of air flow radially outwards, changing the flow patterns, reducing spanwise pressure gradients in the separated flow regions and hence changing the chordwise surface pressure distributions significantly.
Figure 3.42 Pressure measurements on the surface of a wind turbine blade while rotating and while static by Ronsten (1991).
Blade surface pressures have been measured by Ronsten (1991) on a blade while static and while rotating. Figure 3.42 shows the comparison of surface pressure coefficients for similar angles of attack in the static and rotating conditions (tip speed ratio of 4.32) for three spanwise locations. At the 30% span location, the estimated angle of attack at 30.41° is well above the static stall level, which is demonstrated by the static pressure coefficient distribution. The rotating pressure coefficient distribution at 30% span shows a high leading edge suction pressure peak with a uniform pressure recovery slope over the rear section of the upper surface of the chord. The gradual slope of the pressure recovery indicates a reduced adverse pressure gradient with the effect on the boundary layer that it is less likely to separate. The level of the leading edge suction peak, however, is much less than it would be if, in the non‐rotating situation, it were possible for flow still to be attached at 30.41°.
The situation at the 55% spanwise location is similar to that at 30%; the static pressures indicate that the section has stalled, but the rotating pressures show a leading edge suction peak that is small but significant. At the 75% span location there is almost no difference between static and rotating blade pressure coefficient distributions at an angle of attack of 12.94°, which is below the static stall level: the leading edge suction pressure peak is little higher than that at 30% span, much higher than that at 55%, but the pressure recovery slope is much steeper. The measured pressure distributions are very different from those corresponding to stall, suggesting that the flow may still be attached at the 30% and 55% span locations on the rotating blade. However, the suction pressure peaks are much too low for the corresponding fully