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Γ Γ + dΓ
Γ Γ Γ+ dΓ
dΓ Γ + dΓ
Γ
Γ
Γ
(a)
Fig. 2.1 Bound and trailing vortices
34 Bramwell’s Helicopter Dynamics
opposite sense and the resultant trailing-vortex strength is dΓ, Fig. 2.1(b). Thus,
when the circulation varies along the span there is an associated distribution of
trailing vortices forming a vortex sheet springing from the blade’s trailing edge. In
principle, once the distribution of vortex lines trailing from the rotor is determined,
the induced velocity at a given point of the flow can be calculated by applying the
Biot–Savart law to an element of the sheet and integrating over the sheet to obtain its
total effect. The velocity distribution induced by the bound vortices and the vortex
sheets of all the blades constitutes the rotor slipstream. Unfortunately, the geometry
of the vortex sheet is extremely difficult to calculate, especially for the important
case of the hovering rotor, since the flow through the rotor is determined largely by
the velocities induced by the sheet itself; i.e. the sheet geometry and the velocity field
it gives rise to are interdependent. In contrast, for a propeller operating under normal
flight conditions, the velocities induced by the trailing vortices are found to be small
compared with the relatively high axial velocity and can usually be regarded merely
as perturbations to be superimposed on the otherwise uniform axial and rotational
flow components.
We shall leave a detailed discussion of the flow pattern induced by the vortex
wake until later in the chapter, since much useful information about the performance
of the rotor can be gained from a simple flow pattern which can be treated by
momentum methods. The method is known as the classical actuator disc theory.
2.2 Actuator disc theory
In the actuator disc analysis, the following assumptions are made.
(i) The thrust is uniformly distributed over the rotor disc across which there is a
sudden jump of pressure Δp. The uniform thrust distribution can be interpreted
as an assumption that the rotor has an infinite number of blades.
(ii) No rotation or ‘swirl’ is imparted to the flow. This is not a necessary restriction,
since the effects of rotation can be included in the analysis2. However, the
problem becomes more complicated than is really justified, particularly as the
swirl velocities in typical helicopter operation, as will be shown later, are
usually negligible.
(iii) The slipstream of the rotor is a clearly defined mass of moving air outside
which the air is practically undisturbed.
A further assumption of the classic actuator disc theory is that the pressure in the
ultimate slipstream is the same as the pressure of the surrounding undisturbed air.
This assumption implies that the slipstream is like a jet whose velocity is unrelated
to that of surrounding air, but the above description of the vortex wake generated by
the rotor requires us to look at the assumption more critically. Since an element of the
vortex sheet moves with the local velocity generated by the rest of the sheet, there is
no normal flow component relative to the sheet itself. But this is also the boundary
Rotor aerodynamics in axial flight 35
condition for a solid surface moving through the fluid, so we deduce that the velocity
field generated by the vortex sheet is the same as if the sheet were a rigid membrane
whose constituent elements move with the local induced velocity. An important
particular case has been considered by Betz3 who showed that for a certain blade
loading, called the ‘ideal loading’, the power loss is a minimum and the trailing
vortices lie on a helical surface of constant pitch which moves axially at constant
velocity. This case is analogous to that for the minimum induced drag of a wing for
which the span loading is known to be elliptic and the induced velocity along the
span is constant. As mentioned above, the velocity field caused by the motion of the
helical surface relative to the surrounding air constitutes the rotor ‘slipstream’. The
formidable hydrodynamical problem of calculating this velocity field for a propeller
having a finite number of blades was solved by Goldstein4 in 1929.
Actually, as will be discussed later, the ‘ideal’ blade loading does not occur in
practice, and the vortex wake does not conform to the simple helical surface referred
to above. Nevertheless, let us suppose that, close to the rotor at least, the wake
consists of well defined sheets moving away from the rotor with constant velocity w;
Fig. 2.2 shows the sheet springing from one blade. Imagine an observer stationed at
　
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