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wake being maintained thereafter. Blade flapping bending modes were included in
the programme, and the calculated loading, blade response, and corresponding induced
velocity were interdependent and were calculated iteratively until all quantities were
self-consistent.
Like Miller, Piziali gave great attention to the unsteady aerodynamic characteristics
due to the shed wakes. To test the validity of the finite element representation of the
shed vorticity, the calculations were made for the two-dimensional aerofoil and compared
Trailing-vortex
filaments
Shed-vortex
filaments
Root and tip
vortices
Fig. 6.23 Piziali’s wake model
kN/m
Rotor aerodynamics in forward flight 217
with the classical exact theory discussed in the previous section. It was found that,
unless the interval between successive vortex lines was much smaller than was really
practical for the computational capabilities that were then current, agreement was
rather poor, but by advancing the shed wake towards the aerofoil by about 70 per cent
of a complete interval the agreement was greatly improved. In a later paper by
Piziali, a smoothing routine was employed to simulate a continuous wake from a
number of finite elements. Comparisons of measured loadings with those from Piziali’s
method are shown in Figs 6.24 and 6.25.
In the free wake model of Landgrebe12, the basic rotor wake is similar to that
described above, but the vortex elements are allowed to take up positions determined
by the free stream velocity and the induced velocity of all the other elements. Only
the bound and trailing vortices were considered in the calculations; the shed vorticity
was accounted for by using two-dimensional unsteady aerofoil data.
One of the difficulties of representing the wake by discrete vortex elements is that
infinite velocities occur at the vortex itself. To overcome this, Landgrebe supposed
that the vortex had a core within which the induced velocity can be neglected. The
core size assumed by Landgrebe was 1 per cent of the rotor radius. Other investigators
have assumed that the velocity within the core varies linearly with radial distance
4
2
0
–2
–4
0 80 160 240 320
Blade loading
ψ
4
2
0
–2
–4
0 80 160 249 240
Blade loading
r/R = 0.90
HU – 1A, μ = 0.26
r/R = 0.75
HU – 1A, μ = 0.26
ψ
Fig. 6.25 Blade loading variation with azimuth angle
0 80 160 240 320
r/R = 0.9
H – 34, μ = 0.18
ψ
0 80 160 240 320
r/R = 0.75
H – 34, μ = 0.18
ψ
Fig. 6.24 Blade loading variation with azimuth angle
2
1
0
–1
–2
Blade loading
kN/m
2
1
0
–1
–2
Blade loading
kN/m
kN/m
kN/m
218 Bramwell’s Helicopter Dynamics
(i.e. as if the core were solid). Measurements conducted to investigate the structure
of the tip vortex have been described by Cook13.
In order to reduce the computational time and expense, not all the vortex elements
were assumed to be free to convect in accordance with the local velocities. Computation
was greatly reduced by assuming that the positions of the vortex elements beyond a
certain distance from the point of interest remained the same as in the original
prescribed wake which formed the starting point of the calculations. An example of
Landgrebe’s calculations is given in Fig. 6.26, which shows the axial displacement of
the tip vortex filament compared with the ‘classical’ rigid wake uniform displacement.
Another example is the theoretical wake boundary at low μ compared with that from
a smoke visualisation study, Fig. 6.27. It can be seen that the vortex filaments at the
front of the disc lie very close to the disc. This might have been expected from the
Glauert or Mangler distributions, since they predict a slight upwash at the leading
edge of the disc. Landgrebe’s calculations also showed that the wake boundary rolls
up, eventually forming two large vortices very similar to those observed by Heyson
and Katzoff, as described in Chapter 3.
Classical wake
Complete vortex interaction
Approx. vortex interaction
0.1
0.2
0.3
0.4
0.5
0 1 2 3
Z/R
Fig. 6.26 Displacement of tip vortex in forward flight
V V/ ΩR = 0.05 Rotor plane
(Like symbols indicate positions of
vortex elements deposited in wake by
blades at same instant of time. The time
interval between spaces corresponds
　
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