Michael Graham - Wind Energy Handbook

which is precisely the same as for the non‐rotating wake case.

The momentum theory of Section 3.1uses the concept of the actuator disc across which a pressure drop develops, constituting the energy extracted by the rotor. In the rotor disc theory of Section 3.3, the actuator disc is depicted as being swept out by a multiplicity of aerofoil blades, each represented by a radial vortex of constant strength ΔΓ that denotes the bound circulation around each blade section (the totality of spanwise vorticity in the blade surface sheets). Each of these vortex lines is usually considered to lie along the quarter‐chord line of the blade but cannot terminate in the flow field at the tip. Therefore, each vortex is shed at the tip of the blade and convects downstream with the local flow velocity, forming a wake vortex in the form of a helix with strength ΔΓ. If the number, B, of blades is assumed to be very large but the solidity of the total is finite and small, then the accumulation of helical tip vortices will form the surface of a tube. As the number of blades approaches infinity, the tube surface will become a continuous tubular vortex sheet; see Figure 3.6.

Figure 3.6 Helical vortex wake shed by rotor with three blades each with uniform circulation ΔΓ.

From the root of each blade, assuming it reaches to the axis of rotation, a line vortex of strength ΔΓ will extend downstream along the axis of rotation, contributing to the total root vortex of strength Γ(= BΔΓ ). The streamtube will expand in radius as the flow of the wake inside the tube slows down. Because the axial convection of the tip vortices is therefore slowing from the rotor to the far wake, their spacing decreases and hence the vorticity density on the tube sheet representing the tip vortices increases. The vorticity is confined to the surface of this tube, the root vortex, and to the bound vortex sheet swept by the multiplicity of blades to form the rotor disc; elsewhere in the wake and everywhere else in the entire flow field the flow is irrotational.

The nature of the tube's expansion cannot be determined by means of the momentum theory but is known from numerical simulations to be usually fairly small. Therefore, as an approximation, the tube is considered to remain cylindrical, as shown in Figure 3.7. The Biot–Savart law is used to determine the induced velocity at any point in the vicinity of the actuator disc. The cylindrical vortex model allows the whole flow field to be determined and is accurate within the limitations of the non‐expanding cylindrical wake.

Figure 3.7 Simplified helical vortex wake ignoring wake expansion.

In the limit of an infinite number of blades and ignoring expansion the tip vortices form a cylinder with surface vorticity that follows a helical path with a helix angle ϕ t ,which is the same as the flow angle at the outer edge of the disc. The strength of the vorticity is , where Δ n is the distance along the tube surface in a direction normal to ΔΓ between two successive tip vortices. g has components g θin the azimuthal direction and g xin the axial direction. Due to g θthe axial (parallel to the axis of rotor rotation) induced velocity u dat the rotor plane is uniform over the rotor disc and can be determined by means of the Biot–Savart law as

(3.27)

In the far wake the axial induced velocity u wis also uniform within the cylindrical wake and is

(3.28)

The ratio of the two induced velocities corresponds to that of the simple momentum theory and justifies the assumption of a cylindrical vortex sheet.

The total circulation on all of the multiplicity of blades is Γ ,which is shed at a uniform rate into the wake in one revolution. So, from Figure 3.8in which the cylinder has been slit longitudinally and opened out flat, we must have for the strength of the axial vorticity that

(3.29)

since irrespective of the vortex convection velocities the whole circulation Γ is distributed over the peripheral length 2 π R.

Figure 3.8 The geometry of the vorticity in the cylinder surface.

To evaluate the strength of the azimuthal vorticity, we require the axial spacing over which it is distributed, i.e. the axial spacing of any tip vortex between one vortex and the next. Vortices and sheets of vorticity must be convected at the velocity of the local flow field if they are to be force‐free. This velocity can be evaluated as the velocity of the whole flow field at the vortex or vorticity element location less its own local (singular) contribution. In the case of a continuous sheet, it is the average of the velocities on the two sides of the sheet. For axial convection in the ‘far’ wake the two axial velocities are:

so that the axial convection velocity is U ∞ (1 − a) . However, the vortex wake also rotates relative to stationary axes at a rate similarly calculated as halfway between the rotation rate of the fluid just inside the downstream wake = 2 a ′Ω R and just outside = 0. Therefore, the helical wake vortices (or vortex tube in the limit) rotate at a ′Ω R . The result is that the pitch of the helical vortex wake (see Figure 3.8) is

(3.30)

Using this value we obtain

(3.31)

where λ = Ω R/U ∞the tip speed ratio and the rotation period = 2π/ Ω.

So, the total circulation is related to the induced velocity factors

(3.32)