Axial Momentum Theory
across the disc caused by the rate of change of axial momentum as developed in Section 3.2.1 (Equation (3.9)) is additional to the pressure drop associated with the rotation of the wake and is uniform over the whole disc.
If the wake did not expand as it slows down the rotational wake structure together with the rotational pressure gradient would not change as the wake develops whereas the pressure loss caused by the change of axial momentum will gradually reduce to zero in the fullydeveloped wake, as shown in Figure 3.2. The pressure in the fully developed wake would therefore be atmospheric superimposed on which would be the pressure loss given by Equation (3.26). Consequently, the axial force on the fluid in the wake causing it to slow down would be only that caused by the uniform pressure drop across the disc given by Equation (3.9), as is assumed in the simple theory of Section 3.2.1. The rotational pressure drop does not contribute to the change of axial momentum.
In fact, the wake does expand and the full details of the analysis are given by Glauert (1935a). Glauert's analysis is applied to propellers where the flow is accelerated by the rotor but this is only a matter of reversing the signs of the flow induction factors. The inclusion of flow expansion and wake rotation in a fully integrated momentum theory shows that the axial induced velocity in the developed wake is greater than 2a but the effect is only significant at tip speed ratios less than about 1.5, which is probably outside of the operating range for most modern wind turbines. The analysis does, however, demonstrate that the kinetic energy of wake rotation is accounted for by reduced static pressure in the wake. Glauert's conclusion about wake expansion and its interaction with wake rotation is that its inclusion makes little difference to the results obtained from the simple axial momentum theory and so can be ignored. Where, in the same reference, Glauert deals with 'Windmills and Fans' (1935b) he adopts the simple momentum theory but then has to account for kinetic energy of wake rotation, which he does by assuming that it is drawn from the kinetic energy of the flow. The rotational kinetic energy of the wake is therefore regarded as a loss and reduces the level of the energy that can be extracted. Consequently, at low local speed ratios, the inboard sections of a rotor, the local aerodynamic efficiency falls below the Betz limit. Most authors since Glauert have assumed the same conclusion but, in fact, Glauert himself has demonstrated that the conclusion is wrong. The error makes very little difference to the final results for most modern wind turbines designed for the generation of electricity. For wind pumps, where a high starting torque and high solidity are required, the error would probably be very significant because they operate at very low tip speed ratios.
3.4 Vortex Cylinder Model of the Actuator Disc 3.4.1 Introduction
The momentum theory of Section 3.1 uses 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 with radially uniform bound circulation Ar. From the tip of each blade a helical vortex of strength Ar convects downstream with the local flow velocity (Figure 3.6). If the number 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.
From the root of each blade, assuming it reaches to the axis of rotation, a line vortex of strength Ar will extend downstream along the axis of rotation contributing to the total root vortex of strength r. The vortex tube will expand in radius as the flow of the wake inside the tube slows down. Vorticity is confined to the surface of the 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 and so, as an approximation, the tube is allowed to remain cylindrical Figure 3.7. The BiotSavart 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 nonexpanding cylindrical wake.
3.4.2 Vortex cylinder theory
The vortex cylinder has surface vorticity which follows a helical path with a helix angle 0 or, as it has been termed previously, the flow angle at the blade tip. The strength of the vorticity is g = dr/d n, where n is a direction in the tube surface normal to the direction of Ar, and has a component gg = g cos 0t parallel to the rotor disc. Due to gg the axial (parallel to the axis of rotor rotation) induced velocity at the rotor plane is uniform over the rotor disc and can be determined by means of the BiotSavart law as
Figure 3.6 Helical Vortex Wake Shed by Rotor with Three Blades Each with Uniform Circulation Ar x
Figure 3.6 Helical Vortex Wake Shed by Rotor with Three Blades Each with Uniform Circulation Ar
In the far wake the axial induced velocity is also uniform within the cylindrical wake and is
The ratio of the two induced velocities corresponds to that of the simple momentum theory and justifies the assumption of a cylindrical vortex sheet.
3.4.3 Relationship between bound circulation and the induced velocity
The total circulation on all of the multiplicity of blades is r which is shed at a uniform rate into the wake in one revolution. So, from Figure 3.8 in which the cylinder has been slit longitudinally and opened out flat,
therefore
So, the total circulation is related to the induced velocity r _ 4nU1 a(1  a) (332)
3.4.4 Root vortex
Just as a vortex is shed from each blade tip a vortex is also shed from each blade root. If it is assumed that the blades extend to the axis of rotation, obviously not a practical option, then the root vortices will each be a line vortex running axially downstream from the centre of the disc. The direction of rotation of the all the root vortices will be the same forming a core, or root, vortex, of total strength r. The root vortex is primarily responsible for inducing the tangential velocity in the wake flow and in particular the tangential velocity on the rotor disc.
On the rotor disc surface the tangential velocity induced by the root vortex, given by the BiotSavart law, is r
4n r2Q
This relationship can also be derived from the momentum theory: the rate of change of angular momentum of the air which passes through an annulus of the disc of radius r and radial width d r is equal to the torque increment imposed upon the annulus dQ = pU1(1  a)2nrdr2a'Qr2 (3.34)
By the KuttaJoukowski theorem the lift per unit radial width is l _ p(w x r)
Equating the two expressions gives
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