Page 260 - Wind Energy Handbook

234 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

towers than for lattice towers and, in the case of tubular towers, is larger on the
downwind side because of flow separation. As a consequence, designers of down-
wind machines usually position the rotor plane well clear of the tower to minimize
the interference effect.
The velocity deficits upwind of a tubular tower can be modelled using potential
flow theory. The flow around a cylindrical tower is derived by superposing a
doublet, i.e., a source and sink at very close spacing, on a uniform flow, U 1 , giving
the stream function:

!
(D=2) 2
ł ¼ U 1 y 1 (5:20)
2
x þ y 2

where D is the tower diameter, and x and y are the longitudinal and lateral co-
ordinates with respect to the tower centre (see Figure 5.12). Differentiation of ł with
respect to y yields the following expression for the flow velocity in the x direction:

!
2
2
2
(D=2) (x y )
U ¼ U 1 1 (5:21)
2 2
2
(x þ y )
The second term within the brackets, which is the velocity deficit as a proportion of
the undisturbed wind speed, is plotted out against the lateral co-ordinate, y,
divided by tower diameter, for a range of upwind distances, x, in Figure 5.13. The
2
velocity deficit on the flow axis of symmetry is equal to U 1 (D=2x) and the total
width of the deficit region is twice the upwind distance. Consequently the velocity
gradient encountered by a rotating blade decreases rapidly as the upwind distance,
x, increases.
The effect of tower shadow on blade loading can be estimated by setting the local
velocity component at right angles to the plane of rotation equal to U(1 a) in place
of U 1 (1 a), and applying blade element theory as usual. Results for blade root
bending moments for the example 40 m diameter stall-regulated machine are given
in Figure 5.14, assuming a tower diameter of 2 m and ignoring dynamic effects. The
plots show the variation of in-plane and out-of-plane root moments with azimuth
during operation in wind speeds of 10 m=s and 15 m=s, for a blade-tower clearance
equal to the tower radius i.e., for x=D ¼ 1. Note that the dip in out-of-plane bending
moment is more severe at the lower wind speed. Also shown are 10 m=s plots for
x=D ¼ 1:5, which exhibit a much less severe disturbance.
In the case of downwind turbines, the flow separation and generation of eddies
which take place are less amenable to analysis, so empirical methods are used to
estimate the mean velocity deficit. Commonly the profile of the velocity deficit is
assumed to be of cosine form, so that

ðy
2
U ¼ U 1 1 k cos (5:22)
ä