LITMIR.BIZ

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.