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      and

(3.53b)

Solving Eqs. (3.51) and (3.52) to obtain values for the flow induction factors a and a' using 2‐D aerofoil characteristics requires an iterative process for which the following equations, derived from (3.51), (3.52), and (3.53a and b), are convenient. The right hand sides are evaluated using existing values of the flow induction factors, yielding simple equations for the next iteration of the flow induction factors:

(3.54a)

(3.55)

If the additional pressure‐drop term Δp_d2 at the rotor due to wake rotation is included in the analysis, following from Eq. (3.48), Eq. (3.54a) becomes

(3.54b)

      Blade solidity σ is defined as total blade area divided by the rotor disc area and is a primary parameter in determining rotor performance. Chord solidity σ_r is defined as the total blade chord length at a given radius divided by the circumferential length around the annulus at that radius:

      (3.56)

It is argued by Wilson et al. (1974) that the drag coefficient should not be included in Eqs. (3.54a or b) and (3.55) because the velocity deficit caused by drag is confined to the narrow wake that flows from the trailing edge of the aerofoil. Furthermore, Wilson and Lissaman reason, the drag based velocity deficit is only a feature of the wake and does not contribute to the velocity deficit upstream of the rotor disc. The basis of the argument for excluding drag in the determination of the flow induction factors is that, for attached flow, drag is caused only by skin friction and does not affect the pressure drop across the rotor. Clearly, in stalled flow the drag is overwhelmingly caused by pressure. In attached flow – see, e.g. Young and Squire (1938) – the modification to the inviscid pressure distribution around an aerofoil caused by the boundary layer has a small effect both on lift and drag. The ratio of pressure drag to total drag at zero angle of attack is approximately the same as the thickness to chord ratio of the aerofoil and increases as the angle of attack increases.

      One last point about the BEM theory: the theory neglects the axial components of the pressure forces at curved boundaries between streamtubes. It is more accurate if the blades have uniform circulation, i.e. if a is uniform. For non‐uniform circulation there is increased radial interaction and exchange of momentum as a result of normal pressure and viscous shear forces between flows through adjacent elemental annular streamtubes. However, in practice, it appears that the error involved is small for tip speed ratios greater than three.

      3.5.4 Determination of rotor torque and power

      The calculation of torque and power developed by a rotor requires a knowledge of the flow induction factors, which are obtained by solving Eqs. (3.54a or b) and (3.55). The solution is usually carried out iteratively because the 2‐D aerofoil characteristics are non‐linear functions of the angle of attack.

To determine the complete performance characteristic of a rotor, that is, the manner in which the power coefficient varies over a wide range of tip speed ratio, requires the iterative solution.

      The iterative procedure is to assume a and a^′ to be zero initially, determining ϕ, C_l, and C_d on that basis, and then to calculate new values of the flow factors using Eqs. (3.54a or b) and (3.55). The iteration is repeated until convergence is achieved.

      From Eq. (3.49), the torque developed by the blade elements of spanwise length δr is

      If drag, or part of the drag, has been excluded from the determination of the flow induction factors, then its effect must be introduced when the torque is calculated [see Eq. (3.49)]:

      The complete rotor, therefore, develops a total torque Q:

(3.57) Скачать книгу