Wind Energy Handbook. Michael Barton Graham

prime Subscript b"/> defined as here will be used in this section where required to distinguish them.

      The high value of the axial flow induction factor a_b at the tip, due to the proximity of the tip vortex, acts to reduce the angle of attack in the tip region and hence the circulation so that the circulation strength Γ(r) cannot be constant right out to the tip but must fall smoothly through the tip region to zero at the tip. Thus, the loading falls smoothly to zero at the tip, as it must for the same reason as on a fixed wing, and this is a manifestation of the effect of tip‐loss on loading. The result of the continuous fall‐off of circulation towards the tip means that the vortex shedding from the tip region that is equal to the radial gradient of the bound circulation is not shed as a single concentrated helical line vortex but as a distributed ribbon of vorticity that then follows a helical path. The effect of the distributed vortex shedding from the tip region is to remove the infinite induction velocity at the tip, and, through the closed loop between shed vorticity, induction velocity and circulation, converge to a finite induction velocity together with a smooth reduction in loading to zero at the tip. The effect on the loading is incorporated into the BEM method, which treats all sections as independent ‘2‐D’ flows, by multiplying a suitably calculated tip‐loss factor f(r) by the axial and rotational induction factors a_b and that have been calculated by the uncorrected BEM method. Because the blade circulation must similarly fall to zero at the root of the blade, a similar ‘tip‐loss’ factor is applied there in the same way.

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 3.30 and 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 = ρU_∞(1 – (r)).2πrδr.

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

      The rate of change of axial momentum = 4πρU_∞²(1 – (r)).(r)δr.

      The blade element forces are and , where W and C_l are 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.

      3.9.3 Prandtl's approximation for the tip‐loss factor

      The function for the tip‐loss factor f(r) is shown in Figure 3.29 for 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