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If, despite the views of Wilson et al. (1974) – see Section 3.5.3, the effects of drag are included in the determination of the flow induction factors, we must return to Eq. (3.48) and follow the same procedure as described for the drag‐free case.

In the current context, the effects of drag are dependent upon the magnitude of the lift/drag ratio, which, in turn, depends on the aerofoil profile but largely on Reynolds number and on the surface roughness of the blade. A high value of lift/drag ratio would be about 150, whereas a low value would be about 40.

Unfortunately, with the inclusion of drag, the algebra of the analysis is complex. Polynomial equations have to be solved for both a and a^′. The details of the analysis are left for the reader to discover.

In the presence of drag, the axial flow induction factor for optimal operation is not uniform over the disc because it is in the hypothetical drag‐free situation. However, the departure of the axial flow distribution from uniformity is not great, even when the lift/drag ratio is low, provided the flow around a blade remains attached.

The radial variation of the axial and tangential flow induction factors is shown in Figure 3.23 for zero drag and for a lift/drag ratio of 40. The tangential flow induction factor is lower in the presence of drag than without because the blade drags the fluid around in the direction of rotation, opposing the general rotational reaction to the shaft torque.

From the torque/angular momentum Eq. (3.52), the blade geometry parameter becomes

(3.76)

Figure 3.24 compares the blade geometry parameter distributions for zero drag and a lift/drag ratio of 40, and, as is evident, drag has very little effect on blade optimal design.

Figure 3.23 Radial variation of the flow induction factors with and without drag.

Figure 3.24 Spanwise variation of the blade geometry parameter with and without drag.

Figure 3.25 Variation of inflow angle with local speed ratio with and without drag.

Figure 3.26 The variation of maximum C_P with design λ for various lift/drag ratios.

A similar result is apparent for the inflow angle distribution (Figure 3.25), in which drag is also seen to have little influence.

As far as blade design for optimal operation is concerned, drag can be ignored, greatly simplifying the process.

The results of Eq. (3.57) show that the maximum power coefficients for a range of design tip speed ratios and several lift/drag ratios are as shown in Figure 3.26. The flow induction factors have been determined without drag using Eqs. (3.54a) and (3.55), but the torque has been calculated using Eq. (3.57), which includes drag. The losses caused by drag are significant and increase with increasing design tip speed ratio. As will be shown later, when tip‐losses are also taken into account, the losses at low tip speed ratios are even greater.