Unsteady airfoil aerodynamics in attached flow: From unsteady thin-airfoil theory to wind turbine implementation

Abstract. Attached-flow unsteady aerodynamics form the foundation of the dynamic stall models used in wind turbine aerodynamic and aeroelastic codes. These conditions are particularly important over the outboard blade region, which dominates power production and aerodynamic loading. In this region, the angle of attack and relative velocity are unsteady while the flow remains predominantly attached. A complete engineering model for unsteady airfoil aerodynamic loads combines several distinct elements, including airfoil polar lookup, attached-flow shed wake memory, non-circulatory loads, consistent definitions of force magnitude and direction, and, subsequently, separated-flow dynamics. Although the underlying theory is classical, to the authors' knowledge, existing descriptions do not provide a complete and internally consistent route for implementing these attached-flow contributions in wind turbine aerodynamic solvers. This work addresses this gap by focusing on the attached-flow component and presenting all attached-flow contributions in the form of lift, drag, and moment coefficients for implementation in generalized lifting-line methods, including blade-element momentum (BEM), lifting-line (LL), and actuator-line (AL) methods. The formulation is designed for combination with 2-D airfoil polars obtained from measurements or computational fluid dynamics (CFD). Starting from the classical dimensional loads of unsteady thin-airfoil theory, the circulatory and non-circulatory contributions are derived in coefficient form, and the implications of the associated modeling choices are clarified. In particular, the shed wake memory is formulated using the downwash velocity rather than the angle of attack as the aerodynamic state variable. Because the 2-D theory is classical, verification focuses on its rotor-level implementation. The coned straight blade case provides a cross-method benchmark and quantifies the errors in rotor-integrated thrust and power caused by omitting three required contributions that are often omitted. Among these, the lift direction projection contribution has the largest influence on power. Omitting the mid-chord heaving acceleration contribution eliminates the non-circulatory normal force cancellation and produces a thrust error. The zero-onset flow vertical-axis wind turbine (VAWT) case verifies that, in the ideal thin-airfoil limit, all circulatory and non-circulatory contributions cancel, resulting in zero total rotor torque.