Wind Energy Handbook. Michael Barton Graham
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The power developed by the rotor is P = QΩ
The power coefficient is, therefore,
Solving the blade element − momentum Eqs. (3.54a or b) and (3.55) for a given, suitable blade geometrical and aerodynamic design yields a series of values for the power and torque coefficients that are functions of the tip speed ratio. A typical performance curve for a modern, high‐speed wind turbine is shown in Figure 3.15.
The maximum power coefficient occurs at a tip speed ratio for which the axial flow induction factor a, which in general varies with radius, approximates most closely to the Betz limit value of
Figure 3.15 Power coefficient – tip speed ratio performance curve.
3.6 Actuator line theory, including radial variation
Actuator line theory combines the 2‐D blade sectional characteristics used in BEM theory with, usually, a CFD grid calculation of the whole flow field external to the rotor blades including the wake. It is particularly useful for calculating the aerodynamic loads and flow field quantities where wind turbine rotors operate within a larger complex flow field such as a wind farm or a non‐simple ABL topography.
In this method, the outer flow is computed on a finite volume or element grid by some method of numerical simulation (usually viscous and turbulent) of the unsteady flow equations, such as unsteady Reynolds averaged Navier–Stokes (URANS) or LES (see discussion in Chapter 4). A number of open‐source or commercial codes are available to do this with varying degrees of fidelity and cost. Because the computations are carried out over a sequence of timesteps and are spatially 3‐D, this always requires significant computing resource. The discretisation scale should be appropriate to resolve the major structures of the ABL and its turbulence and the rotor (diameters) and the turbulent structures in their wakes. But it does not resolve the flows on the length scales of the blade chords and their boundary layers and hence is orders of magnitude faster than a complete simulation of all scales in the flow field.
Instead of resolving the sectional blade flows, these are replaced, as in BEM theory, by aerofoil characteristics from look‐up tables (or possibly a fast panel method such as XFOIL; Drela 1989). The rotor blades are tracked through the outer flow grid and the velocity field, which has been computed on that grid, is interpolated onto the designated rotor blade sections. The resulting sectional blade forces obtained by interpolating from the blade characteristic look‐up tables are projected back onto the outer grid as a series of momentum sinks for the components of force in the three coordinate directions. These sinks then form part of the grid flow field calculation at the next timestep. This coupling between the inner and outer flow calculations may be either loose going from timestep to timestep as indicated or may be a strong coupling in which the flow is converged within each timestep by iteration or by solving the whole in a single very large matrix. Transfer of force and large‐scale velocities between the inner and outer flow fields is well established, but methods of determining the effective turbulence input from the smaller‐scale structures in the rotor blade flows as sources for the larger‐scale outer flow are not, and further work is required here. Good references for this method are Mikkelsen (2003) and Troldborg et al. (2006).
3.7 Breakdown of the momentum theory
3.7.1 Free‐stream/wake mixing
For heavily loaded turbines, when a is high, the momentum theory predicts a reversal of the flow in the wake. Such a situation cannot actually apply uniformly throughout the far wake as predicted. What happens is that the wake becomes unstable with local flow reversal and breakdown into turbulence. This increases the mixing process, which entrains air from outside the wake, re‐energising the slow moving air that has passed through the rotor.
A rotor operating at increasingly high tip speed ratios presents a decreasingly permeable disc to the flow. Eventually, when λ is high enough for the axial flow factor to be equal to one, the flow field of the disc would appear to have reached a condition like that of a normal solid disc, including the flow in the wake.
As this condition is approached, the flow through a rotor has many of the features of flow through a porous disc of low and decreasing permeability and hence a large increasing resistance to through‐flow. The air that does pass through the rotor emerges into a low‐pressure region and is moving slowly. There is insufficient kinetic energy to provide the rise in static pressure necessary to achieve the ambient atmospheric pressure that exists outside the wake and must exist in the wake far downstream. The air can only achieve this ambient pressure by gaining energy from mixing with the flow that has bypassed the rotor disc and is outside the wake. Castro (1971) has studied in detail the wake of a porous plate as the plate is made increasingly impermeable to flow. At a certain level of resistance, a counter‐rotating vortex pair (in planar 2‐D flow) or a ring vortex (in axisymmetric flow) forms downstream in the wake as a result of the instability of the wake shear. This vortex structure generates a growing region of reversed flow near the plane of symmetry or axis of the wake. As the resistance is increased further, the vortex structure and region of reversed flow moves upstream until it reaches the downstream face of the plate. Depending on the Reynolds number, but increasingly so for a high Reynolds number, the vortex structure develops further instability and the wake becomes turbulent, greatly increasing mixing with the external flow and recovery of kinetic energy. The wake of a rotor has some significant differences from that of a porous disc: in particular that the latter does not have the strong helical vortex structure present in the wake of a rotor. Nevertheless, the behaviour of the rotor wake as its resistance is increased is qualitatively very similar, although the point at which the ordered axial flow through a rotor reverses and breaks down into turbulence is not exactly the same as for a porous disc.
3.7.2 Modification of rotor thrust caused by wake breakdown
When