Michael Graham - Wind Energy Handbook

It is important to note that tip‐loss factors should only be applied in methods that assume disc‐type actuators (i.e. azimuthally uniform), such as the BEM method, and not, for example, to the line actuator method because methods such as this that compute individual blades and the velocities induced at them already incorporate the tip effect.

Figure 3.30 Spanwise variation of power extraction in the presence of tip‐loss for a blade with uniform circulation on a three blade turbine operating at a tip speed ratio of 6.

The results from Eq. (3.79)with and without this tip‐loss factor are plotted in Figure 3.30and clearly show the effect of tip‐loss on power. Equation (3.78)) has assumed that uniformly over the whole disc, but applying the tip‐loss factor means recognising that cannot be uniform radially. The tip‐loss results from the tip vortices, which generate the induction factor a (effectively the induced drag). It is important to note that there is no additional effective drag associated with tip‐loss.

If the circulation varies along the blade span, vorticity is shed into the wake in a continuous fashion from the trailing edge of all sections where the spanwise (radial) gradient of circulation is non‐zero.

Therefore, each blade sheds a helicoidal sheet of vorticity, as shown in Figure 3.31, rather than a single helical vortex, as shown in Figure 3.27. The helicoidal sheets convect with the wake velocity and so there can be no flow across the sheets, which can therefore be regarded as impermeable. The intensity of the vortex sheets is equal to the rate of change of bound circulation along the blade span and so usually increases rapidly towards the blade tips. There is flow around the blade tips because of the pressure difference between the blade surfaces, which means that on the upwind surface of the blades the flow moves towards the tips and on the downwind surface the flow moves towards the root. The flows from either surface leaving the trailing edge of a blade will not be parallel to one another and will form a surface of discontinuity of velocity in a radial sense within the wake; the axial velocity components will be equal. The surface of discontinuity is called a vortex sheet . A similar phenomenon occurs with aircraft wings, and a textbook of aircraft aerodynamics will explain it in greater detail.

The azimuthally averaged value of can be expressed as a b(r). f (r), where f ( r ) is known as the tip‐loss factor , has a value of unity inboard, and falls to zero at the edge of the rotor disc.

In the application of the BEM theory, it is argued that the rate of change of axial momentum is determined by the azimuthally averaged value of the axial flow induction factor, whereas the blade forces are determined by the value of the flow factor that the blade element ‘senses’. This needs careful interpretation, as discussed in Section 3.9.2.

Figure 3.31 A (discretised) helicoidal vortex sheet wake for a two bladed rotor whose blades have radially varying circulation.

The mass flow rate through an annulus = ρ U ∞(1 – (r)).2 π rδr.

The azimuthally averaged overall change of axial velocity = 2 (r).U ∞.

The rate of change of axial momentum = 4 πρ U ∞ 2(1 – (r)). (r) δ r.

The blade element forces are and , where W and C lare determined using a b(r).

The torque caused by the rotation of the wake is also calculated using an azimuthally averaged value of the tangential flow induction factor 2 (r) with tip‐loss similarly applied for the value at a blade because both induction velocities are induced by the same distribution of shed vorticity.

The function for the tip‐loss factor f ( r ) is shown in Figure 3.29for a blade with uniform circulation operating at a tip speed ratio of 6 and is not readily obtained by analytical means for any desired tip speed ratio. Sidney Goldstein (1929) did analyse the tip‐loss problem for application to propellers and achieved a solution in terms of Bessel functions, but neither that nor the vortex method with the Biot–Savart solution used above is suitable for inclusion in the BEM theory. Fortunately, in 1919, Ludwig Prandtl, reported by Betz (1919), had already developed an ingenious approximate solution that does yield a relatively simple analytical formula for the tip‐loss function.

Prandtl's approximation was inspired by considering that the vortex sheets could be replaced by material sheets, which, provided they move with the velocity dictated by the wake, would have no effect upon the wake flow. The theory applies only to the developed wake. To simplify his analysis Prandtl replaced the helicoidal sheets with a succession of discs, moving with the uniform, central wake velocity U ∞(1 − a ) and separated by the same distance as the normal distance between the vortex sheets. Conceptually, the discs, travelling axially with velocity U ∞(1 − a ), would encounter the unattenuated free‐stream velocity U ∞at their outer edges. The fast flowing free‐stream air would tend to weave in and out between successive discs. The wider apart successive discs the deeper, radially, the free‐stream air would penetrate. Taking any line parallel to the rotor axis at a radius r , somewhat smaller than the wake radius R w(∼ rotor radius R), the average axial velocity along that line would be greater than U ∞(1 − a ) and less than U ∞. Let the average velocity be U ∞(1 − af ( r )), where f ( r ) is the tip‐loss function, has a value less than unity and falls to zero at the wake boundary. At a distance from the wake edge the free stream fails to penetrate, and there is little or no difference between the wake‐induced velocity and the velocity of the discs, i.e. f ( r ) = 1.