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V having the same diameter as the rotor, flowing past and around the rotor and being
deflected downwards so that far downstream it has an induced velocity component
normal to the rotor disc of 2vi, Fig. 3.1. This interpretation, and the fact that eqn 3.1
reduces to the ‘momentum’ formula of hovering flight, may lead one to think that the
formula is obvious from momentum considerations: it should be appreciated, however,
that the flow depicted in Fig. 3.1 is quite fictitious and that eqn 3.1 is far from
obvious. More will be said about this point later in section 3.5.
Equation 3.1 can be expressed as a unique curve if, as in Chapter 2, we define
v0 = √(T/2ρA) as the ‘thrust velocity’. Then, for αD = 0 and if
v i0 = vi0 /v0 and V = V/v0
eqn 3.1 can be written
v v i0
4
i0
+ V 2 2 – 1 = 0 (3.2)
This equation is shown plotted in Fig. 3.2.
The dotted line shows the relations vi0 = 1/V , which for V > 1 lies extremely
close to the true result. This implies that, for forward speeds greater than about
10 m/s, the induced velocity is much smaller than the forward speed and is equivalent
to an approximation to eqn 3.1 in the form
vi0 = T/2ρAV (3.3)
Although eqn 3.1 provides a very simple and useful formula for the estimation of
the mean induced velocity, it was appreciated by Glauert that the induced velocity
over the rotor is far from uniform. From aerofoil and wing theory one would expect
V
V sin(–α D)
vio
V
2vio
V′
V
V cos αD
Fig. 3.1 Flow interpretation of Glauert’s formula (showing disc incidence αD as negative for normal helicopter
flight case)
–αD
Rotor aerodynamics and dynamics in forward flight 79
an upwash at the leading edge of the rotor and an increase of induced velocity
towards the trailing edge. Accordingly, Glauert proposed a second formula:
vi = vi0(1 + Kx cos ψ) (3.4)
where vi is the general induced velocity and vi0 is the induced velocity at the rotor
centre (and also the mean induced velocity), taken as the value given by eqn 3.1,
x = r/R, and K is a factor chosen to be slightly greater than unity. A typical value
taken for K is 1.2, so that on the longitudinal axis (ψ = 0 or π) there is an upwash at
the leading edge and a linear increase of induced velocity towards the trailing edge;
in fact, the value of K denotes the slope of the induced velocity distribution along the
longitudinal axis.
An attempt to calculate the longitudinal induced velocity distribution and to find
a theoretical value of K was made by Coleman, Feingold, and Stempin. Their analysis
was based on the fact that a rotor carrying a uniform load and moving through the air
at an angle to its plane leaves behind a vortex wake in the form of an elliptical
cylindrical shell, Fig. 3.3.
This cylindrical shell can be regarded as a continuous distribution of vortex rings
whose planes are parallel to the rotor plane and whose ‘strength’ is determined by the
magnitude of the load and the forward speed of the rotor. By applying the Biot–
Savart law to the cylindrical wake, the induced velocity could be obtained in the form
of a double integral. The integral could not be evaluated at a general point of the rotor
disc, but an exact expression was obtained for points on the longitudinal axis. However,
even this result was quite complicated and expressible only in the form of elliptical
1.0
0.8
0.6
0.4
0.2
0 1 2 3 4 5 6
V
Fig. 3.2 Non-dimensional induced velocity as a function of forward speed
v i0
80 Bramwell’s Helicopter Dynamics
χ
V
αD
Fig. 3.3 Vortex ring representation of rotor wake in forward flight
3
2
1
–3 –2 –1 0 1 2 3
3
2
1
χ = 26.6°
3
2
1
0
3
2
1
0
–1
–1
x
vi /vi0
χ = 0°
χ = 63.4°
χ = 90°
Fig. 3.4 Longitudinal induced velocity distribution for uniformly loaded rotor
integrals. A number of longitudinal distributions depending on the wake angle χ are
shown in Fig. 3.4. It can be seen that the induced velocity distribution is not exactly
linear but can be taken as approximately so over most of the rotor diameter. The slope
of the ratio vi /vi0 at the rotor centre was found from the analysis to be tan (χ/2), so
that Glauert’s formula, eqn 3.4, could be written as
vi = vi0(1 + x tan (χ/2) cos ψ) (3.5)
Since χ ≤ 90°, this approximation to the slope fails to give the required upwash at
　
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