Page 77 - Wind Energy Handbook

VORTEX CYLINDER MODEL OF THE ACTUATOR DISC 51

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 fully-developed wake, as shown in Figure 3.2. The
pressure in the fully developed wake would therefore be atmospheric super-
imposed on which would be the pressure loss given by Equation (3.26). Conse-
quently, 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 devel-
oped 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