Comparison Of Momentum And Vortex Methods For The Aerodynamic Analysis Of Wind Turbines

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2 American Institute of Aeronautics and Astronautics
I.
Introduction
Understanding the complicated unsteady aerodynamics of a wind turbine1–3 is very important to predict accurately its blade loads and power output. However, the aerodynamics of a wind turbine are extremely complicated, and as of yet are not well understood2, 4 or easily predictable.5, 6 It is known that the operation of wind turbines involves the creation of three-dimensional, unsteady aerodynamic forces on the rotating blades. Compounding the prediction of these forces are effects such as atmospheric turbulence, the ground boundary layer, directional and spatial variations in wind shear, and the effects of an upstream support structure (tower shadow).2, 4, 7 Over the last few decades, performance methods based on the blade element momentum (BEM) theory have been used predominantly to model the aerodynamics of wind turbines. BEM methods are simple and fast, but are strictly valid only for a limited range of flow conditions and break-down in the turbulent wake state (TWS) and the vortex ring state (VRS). Modified BEM methods have been used based on empirical corrections8–10 to overcome this limitation. However, these corrections are not applicable for all operating conditions and often fail at higher tip-speed ratios. Some of the limitations of the BEM methods have been addressed in Refs. 11 and 12. On the other hand, the FVM has been used extensively in rotorcraft research.13 However, despite the potential flexibility for use in wind turbine applications, the FVM has yet seen only limited use.14–16 Prescribed vortex wake models17, 18 have been used, but these methods are limited in their formulation by the unavailability of experimental data for wind turbine wakes over a full operating range. Sudden changes in the wind direction or turning the wind turbine out of the wind to limit the power captured, causes the wind turbines to operate in a yawed flow condition. This causes unsteady airloads on the blades. The BEM approach is often acceptably accurate for predicting the axisymmetric distribution of inflow in unyawed flow, but additional inflow estimates at the rotor disk are required to apply the BEM methods to yawed flows. This introduces empiricism into the calculations. Various linear inflow models19–21 have been used to approximate the inflow at the rotor disk in yawed flow, but most of these methods are valid only for small yaw angles. Recent developments of inflow models for yawed flow condition has been discussed by Snel.22 The coefficients of these inflow models are derived empirically from experiments19, 20 or numerical simulations21 and their applicability is limited to a restricted range of wind turbine operating conditions. Another shortcoming of the BEM methods is their inability to capture the transient behavior of the power output in yawed flow. However, in the FVM the induced velocities from the skewed wake in yawed flow can account for the time-accurate, asymmetric distribution of inflow over the rotor disk and no additional approximations are needed. This study compares the results from the BEM theory and FVM for a 2-bladed rotor with ideal twist for a range of tip-speed ratios. The various flow states of a wind turbine where BEM methods fail have been identified. A universal curve for the thrust and power coefficients has been derived. A FVM calculation for a wind turbine yawing out of the wind has also been performed. The ability of the FVM to account for the non-linearities involved in the turbulent wake state and yawed flow, shows the flexibility and robustness of the FVM to capture the time-accurate aerodynamic response of a wind turbine.
Abbreviations
BEM FVM NWS TSR TWS VRS WBS Blade-element momentum Free-vortex method Normal working state Tip-speed ratio Turbulent wake state Vortex ring state Windmill brake state
Nomenclature
a a A c Cl Clα CP CPhel CPwt CT CThel CTwt Dψ Dζ F Froot
∗ †
Axial induction factor, vi /V∞ Tangential induction factor, vt /Ωr Area of the rotor disk, m2 Blade chord, m Lift coefficient Sectional lift curve slope, rad−1 3 Power coefficient, P/ 1 2 ρAV∞ Helicopter power coefficient, P/ρA(ΩR)3 3 Wind turbine power coefficient, P/ 1 2 ρAV∞ 2 Thrust coefficient, T / 1 2 ρAV∞ Helicopter thrust coefficient, T /ρA(ΩR)2 2 Wind turbine thrust coefficient. T /0.5ρAV∞ Temporal difference operator Spatial difference operator Prandtl loss factor Prandtl root loss factor