Читать книгу Wind Energy Handbook - Michael Barton Graham - Страница 85

Where the axial flow induction factor a becomes large at the blade position, then, by Eqs. (3.44), the inflow angle ϕ will reduce, and for a given pitch angle the angle of attack α and hence the lift force will become small. At the tip the lift force must decrease to zero because the blade surface pressures must be continuous around the tip. The component of the lift force in the tangential direction in the tip region will therefore be small and so will be its contribution to the torque. A reduced torque means reduced power, and this reduction is known as tip‐loss because the effect occurs at the outermost parts of the blades. A similar effect occurs for the same reason at the blade root but being at small radius has much less effect on torque and power.

Figure 3.27 Helical trailing tip vortices of a horizontal axis turbine wake.

To account for tip‐losses, the manner in which the axial flow induction factor varies azimuthally needs to be known, but, unfortunately, this requirement is beyond the abilities of the BEM theory.

Just as a vortex trails from the tip of an aircraft wing so does a vortex trail from the tip of a wind turbine blade. Because the blade tip follows a circular path, it leaves a trailing vortex as a helical structure that convects downstream with the wake velocity. For example, on a two blade rotor, unlike an aircraft wing, the bound circulations on the two blades shown in Figure 3.27 are opposite in sign and so combine in the idealised case of the blade root being at the rotational axis to shed a straight line vortex along the axis with strength equal to the blade circulation times the number of blades. If as is usual in practice the blade root is somewhat outboard of the axis, the two blade root vortices form independent helices similar to the blade tips but of small radius, close together, and the combined straight line axis vortex is not a bad approximation of their effect.

For a single vortex to be shed from the blade at its tip, only the circulation strength along the blade span must be uniform right out to the tip with an abrupt drop to zero at the tip. As has been shown, such a uniform circulation provides optimum power coefficient. However, the uniform circulation requirement assumes that the axial flow induction factor is uniform across the disc. With an infinite number of blades, the tip vortices form a continuous cylindrical sheet of vorticity directed at a constant angle around the surface. Such a sheet is consistent with a uniform value of the axial induction factor over the disc. But, as has been argued above, with a finite number of blades rather than a uniform disc, the flow factor is not uniform. Sustaining uniform circulation until very close to the two ends (tip and root) of a blade results in a very large gradient of the blade circulation at the tips, which in turn induces large radial variations in the induced velocity factors a and a^′ in those regions, with both tending to infinity in the limit of constant circulation up to the tip and root.

As in Figure 3.27, close to a blade tip a single concentrated tip vortex would on its own cause very high values of the flow factor a with an infinite value at the tip such that, locally, the net flow past the blade is in the upstream direction. This effect is similar to what occurs for the simple ‘horseshoe vortex’ model for a fixed wing aircraft showing that this model is not applicable at a blade or wing tip where a more detailed induced flow analysis is required. The azimuthal average of the axial induction a is uniform radially. Higher values of a tend to be induced close to the blades towards root and tip, becoming higher the closer to the tips. Therefore, low values relative to the average must occur in the regions between the blades. The azimuthal variation of a for a number of radial positions is shown in Figure 3.28 for a three blade rotor operating at a tip speed ratio of 6. The calculation for Figure 3.28 assumes a discrete vortex for each blade with a constant pitch and constant radius helix and is calculated from the effect of the shed wake vortices only.

At a particular radial position the ratio of the azimuthal average of a (which from here on will be written as ) to the value a_b(r) at the blade quarter chord is shown in Figure 3.29, being unity for most of the blade span, and only near the tip does it begin to fall to zero. This ratio is called the tip‐loss factor.

Figure 3.28 Azimuthal variation of a for various radial positions for a three blade rotor with uniform blade circulation operating at a tip speed ratio of 6. The blades are at 120°, 240°, and 360°.

Figure 3.29 Spanwise variation of the tip‐loss factor for a blade with uniform circulation.

From Eq. (3.20) and in the absence of tip‐loss and drag the contribution of each blade element to the overall power coefficient is

(3.77)

Substituting for a^′ from Eq. (3.25) gives

(3.78)

From the Kutta–Joukowski theorem, the circulation Γ on the blade, which is uniform, provides a torque per unit span of

where the angle ϕ_r is determined by the flow velocity local to the blade.

If the strength of the total circulation for all three blades is still given by Eq. (3.69), in the presence of tip‐loss, the increment of power coefficient from a blade element is

(3.79)

in agreement with Eq. (3.78), except that the factor a(1 − a), which relates Γ to the angular momentum loss in the wake, must be expressed as in terms of the azimuthally averaged axial flow induction factor , which = 1/3 for optimum operation. However, the final induction term (1 − a_b) relates to the flow angle at the blade and must therefore be in terms of a_b, the axial induction factor at the blade, with a_b = /f, and therefore a_b ≈ except near the tips. The notation 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.