How should the lift and drag forces be calculated from 2-D airfoil data for dihedral or coned wind turbine blades?

Abstract

In the present work, a consistent method for calculating the lift and drag forces from the 2-D airfoil data for the dihedral or coned horizontal-axis wind turbines when using generalized lifting-line methods is described. The generalized lifting-line methods refer to the models that discretize the blade radially into sections and use 2-D airfoil data, for example, lifting-line (LL), actuator line (AL), blade element momentum (BEM) and blade element vortex cylinder (BEVC) methods. A consistent interpretation of classic unsteady 2-D thin airfoil theory results reveals that it is necessary to use both the relative flow information at one point on the chord and the chordwise gradient of the flow direction to correctly determine the 2-D aerodynamic force and moment. Equivalently, the magnitude of the force should be determined by the flow at the three-quarter-chord point, while the force direction should be determined by the flow at the quarter-chord point. However, this aspect is generally overlooked, and most implementations in generalized lifting-line methods use only the flow information at one calculation point per section for simplicity. This simplification will not change the performance prediction of planar rotors but will cause an error when applied to non-planar rotors. In this work this effect is investigated using the special case, where the wind turbine blade has only out-of-plane shapes (blade dihedral) and no in-plane shapes (blade sweep), operating under steady-state conditions with uniform inflow applied perpendicular to the rotor plane. The impact of the effect is investigated by comparing the predictions of the steady-state performance of non-planar rotors from the consistent approach of the LL method with the simplified one-point approaches. The results are verified using blade-geometry-resolving Reynolds-averaged Navier–Stokes (RANS) simulations. The numerical investigations confirmed that the full method complying with the thin airfoil theory is necessary to correctly determine the magnitude and direction of the sectional aerodynamic forces for non-planar rotors. The aerodynamic loads of upwind- and downwind-coned blades that are calculated using the LL method, the BEM method, the BEVC method and the AL method are compared for the simplified and the full method. Results using the full method, including different specific implementation schemes, are shown to agree significantly better with fully resolved RANS than the often used simplified one-point approaches.