Wind Energy Handbook. Michael Barton Graham

Wind Energy Handbook - Michael Barton Graham


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of multiple wakes interacting with each other and with the incident atmospheric boundary layer (ABL) and impinging on downstream rotors. Wake interactions have a very significant effect on power generated by turbines downstream of others (see, e.g. Argyle et al. 2018) and on the buffeting of downstream rotors. The usual method of carrying out these computations is to embed actuator models of the turbines within much larger numerical grid representations of the flow through and surrounding the whole wind farm. This outer large‐scale flow is solved numerically on the grid by conventional, and now well‐established, CFD Reynolds averaged Navier–Stokes (RANS) or higher fidelity but much more computationally expensive large eddy simulation (LES) computer codes. The actuator model embedded in the grid to represent the action of each turbine may be at the simplest level of an actuator disc model, in which the thrust force on the rotor disc is inserted as a momentum sink, i.e. a step change in momentum in the streamwise direction across grid cells that are intersected by the rotor disc. However, it is usually found desirable to go to a higher level of representation including swirl and embed an actuator line model for each turbine blade in the grid. The rotating actuator lines are now the momentum sinks of both axial and azimuthal forces including the radial variations, which are projected at each timestep onto the adjacent grid points (see, for example, Soerensen and Shen 2002).

      3.4.11 Conclusions

      Despite the exclusion of wake expansion, the vortex theory produces results in agreement with the momentum theory and enlightens understanding of the flow through an energy extracting actuator disc. However, the infinite radial velocity predicted at the outer edge of the disc is further evidence that the actuator disc is physically unrealisable.

      3.5.1 Introduction

      The aerodynamic lift (and drag) forces on the spanwise elements of radius r and length δr of the several blades of a wind turbine rotor are responsible for the rate of change of axial and angular momentum of all of the air that passes through the annulus swept by the blade elements. In addition, the force on the blade elements caused by the drop in pressure associated with the rotational velocity in the wake must also be provided by the aerodynamic lift and drag. As there is no rotation of the flow approaching the rotor, the reduced pressure on the downwind side of the rotor caused by wake rotation appears as a step pressure drop just as is that which causes the change in axial momentum. Because the wake is still rotating in the far wake, the pressure reduction associated with the rotation is still present and so does not contribute to the axial momentum change.

      It is assumed that the forces on a blade element can be calculated by means of two‐dimensional (2‐D) aerofoil characteristics using an angle of attack determined from the incident resultant velocity in the cross‐sectional plane of the element. Applying the independence principle (see Appendix A3.1), the velocity component in the spanwise direction is ignored. Three‐dimensional (3‐D) effects are also ignored.

      The velocity components at a radial position on the blade expressed in terms of the wind speed, the flow factors, and the rotational speed of the rotor together with the blade pitch angle will determine the angle of attack. Having information about how the aerofoil characteristic coefficients Cl and Cd vary with the angle of attack, the forces on the blades for given values of a and a can be determined.

      From Figure 3.14, the resultant relative velocity at the blade is

      (3.43)upper W equals StartRoot upper U Subscript infinity Baseline Superscript 2 Baseline left-parenthesis 1 minus a right-parenthesis squared plus r squared normal upper Omega squared left-parenthesis 1 plus a prime right-parenthesis squared EndRoot

Schematic illustration of a blade element sweeps out an annular ring. Schematic illustration of the blade element velocities and forces: (a) velocities, and (b) forces.

      The angle of attack α is then given by

      The basic assumption of the blade element theory is that the aerodynamic lift and drag forces acting upon an element are the same as those acting on an isolated, identical element at the same angle of attack in 2‐D flow.

      The lift force on a spanwise length δr of each blade, normal to the direction of W, is therefore

delta upper L equals one half rho upper W squared italic c upper C Subscript l Baseline delta r

      and the drag force parallel to W is

delta upper D equals one half rho upper W squared italic c upper C Subscript d Baseline delta r

      The axial thrust on an annular ring of the actuator disc is