In-plane moment for U = 10 m/s

0 30 60 90 120 150 180 210 240 270 300 330 360

Azimuth (degrees)

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 downwind 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, UM, giving the stream function:

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

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 velocity deficit on the flow axis of symmetry is equal to UM(D/2x)2 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 UM(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 = 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 = 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

Velocity Profile at rotor plane

Figure 5.12 Tower Shadow Parameters where d is the total width of the deficit region. The slight enhancement of velocities beyond the deficit region is usually ignored (see also Section 6.13.2).

The sharp dip in blade loading caused by tower shadow is more prone to excite blade oscillations than the smooth variations in load due to wind shear, shaft tilt and yaw, and this aspect is considered in the section on blade dynamic response.

Wake effects

Within a wind-farm it is common for one turbine to be operating wholly or partly in the wake of another. In the latter case, which is more severe, the downwind turbine is effectively subjected to horizontal wind shear, and the blade load fluctuations can be analysed accordingly.

Figure 5.13 Profile of Velocity Deficit due to Tower Shadow at Different Distances x= D Upwind of Tower Centreline

5.7.3 Gravity loads

Gravity loading on the blade results in a sinusoidally varying edgewise bending moment which reaches a maximum when the blade is horizontal, and which changes sign from one horizontal position to the other. It is thus a major source of fatigue loading. For the blade 'TR' (see Example 5.1), the maximum gravity moment, J0 m(r)r dr is 134 kNm, so the edgewise bending moment range due to gravity is 268 kNm. This dwarfs the variations in edgewise moment due to yaw or wind shear, which are typically one tenth this value or less. The spanwise distribution of gravity bending moment is shown in Figure 5.15 for blade 'TR'.

5.7.4 Deterministic inertia loads

Centrifugal loads

For a rigid blade rotating with its axis perpendicular to the axis of rotation, the centrifugal forces generate a simple tensile load in the blade which at radius r* is given by the expression Q2 Jr* m( r) r dr. As a result, the fluctuating stresses in the blade arising from all loading sources always have a tensile bias during operation. For blade 'TR' rotating at 30 r.p.m., the centrifugal force at the root amounts to 134 kN - approximately seven times its weight.

Thrust loading causes flexible blades to deflect downwind, with the result that the centrifugal forces generate blade out-of-plane moments in opposition to those due to the thrust. This reduction of the moment due to thrust loading is known as

Figure 5.14 Variation of Blade Root Bending Moments with Azimuth due to Tower Shadow, for Typical 40 m Diameter Stall-regulated Upwind Machine Operating in Steady, Uniform Winds of 10 m/s and 20 m/s

Radius (m)

Figure 5.15 Blade 'TR' Gravity Bending Moment Distribution

Radius (m)

Figure 5.15 Blade 'TR' Gravity Bending Moment Distribution centrifugal relief. The phenomenon is non-linear, so iterative techniques are required to arrive at a solution. Greater centrifugal relief can be obtained by coning the rotor so that the blades are inclined downwind in the first place. A balance can be struck so that the maximum forward out-of-plane moment due to centrifugal loads in very low wind is approximately equal to the maximum rearward out-of-plane moment due to the thrust loading in combination with centrifugal loads during operation in rated wind.

Gyroscopic loads

When an operating machine yaws, the blades experience gyroscopic loads perpendicular to the plane of rotation. Consider the point A on a rotor rotating clockwise at a speed of Q rad/s, as illustrated in Figure 5.16. The instantaneous horizontal velocity component of point A due to rotor rotation is Qz, where z is the height of the point above the hub. If the machine is yawing clockwise in plan at a speed of A rad/s, then it can be shown that point A accelerates at 2QAz towards the wind, assuming the rotor is rigid. Integrating the resulting inertial force over the blade length gives the following expression for blade root out-of-plane bending moment a/ n z

Q = Speed of rotor rotation A = Speed of yawing

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