Michael Graham - Wind Energy Handbook

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.

When flow reversal and breakdown into turbulence in the wake of a porous plate occurs, typically starting when the resistance coefficient K (= Δp/(½ ρ U 2)) exceeds 4, experimental measurements show that the axial force on the body departs from the well‐known theory of Taylor (1944) for ordered flow through a porous plate. Similarly, experimental measurements of the thrust force coefficient for a rotor – for example, reported by Glauert (1926) and plotted in Figure 3.16– show a departure from the actuator disc momentum theory C T= 4 a (1 − a ). In both cases the measured forces are larger than the predictions of theory, and in both cases the point of break‐away is near the maximum predicted by the momentum theory.

Figure 3.16 Comparison of theoretical and measured values of C T.

The thrust (or drag) coefficient for a simple, flat circular plate is given by Hoerner (1965) as 1.17 but, as demonstrated in Figure 3.16, the thrust on the rotor reaches a higher value. A major difference between the wake of the circular plate and of the rotor is that the latter contains a strong rotating component even after flow reversal in the wake has started.

It would follow from the above arguments that for high values of the axial induction factor a large part of the pressure drop across the disc is not simply associated with blade circulation, just as it is absent in the case of the circular plate. Circulation would cause a pressure drop similar to that given by the momentum theory determined by the very low axial velocity of the flow that actually permeates the disc.

A suitable straight line through the experimental points would appear to be possible, although Glauert proposed a parabolic curve, and provides an empirical solution to the problem of the thrust on a heavily loaded turbine (a rotor operating at a high value of the axial flow induction factor).

Most authors assume that the entire thrust on the rotor disc is associated with axial momentum change. Therefore, for the empirical line to be useful it must be assumed that it applies not only to the whole rotor but also to each separate streamtube. Let C T1be the empirical value of C Twhen a = 1. Then, as the straight line must be a tangent to the momentum theory parabola at the transition point, the equation for the line is

(3.58)

and the value of a at the transition point is

By inspection, C T1must lie between 1.6 and 2: C T1= 1.816 would appear to be the best fit to the experimental data of Figure 3.16, whereas Wilson et al. (1974) favour the lower value of C T1= 1.6. Glauert fits a parabolic curve to the data [replacing a in the mass flow expression by 4a(1 − a)/(0.6 + 0.61a + 0.79a 2 ) when a > 1/3 ] giving much higher values of C T1at high values of a but he was considering the case of an airscrew in the windmill brake state where the angles of attack are negative. De Vaal et al. (2014) suggest a be replaced by 0.25a(5 − 3a), similarly giving a somewhat lower windmill brake state result.

The flow field through the turbine under heavily loaded conditions cannot be modelled easily, and the results of this empirical analysis must be regarded as being only approximate at best. They are, nevertheless, better than those predicted by the momentum theory. For most practical designs the value of the axial flow induction factor rarely exceeds 0.6 and for a well‐designed rotor will be in the vicinity of 0.33 for much of its operational range.

For values of a greater than a T ,it is common to replace the momentum theory thrust in Eq. (3.9)with Eq. (3.58), in which case Eq. (3.54a)is replaced by

(3.59)

However, as the additional pressure drop is caused by breakdown of the streamline wake, this course of action is questionable, and it may be more appropriate to retain Eq. (3.54).