Brief communication: On the local disagreement of two engineering models for the radial induced velocity of non-planar wind turbine rotors

Abstract. Blade element momentum (BEM) codes traditionally neglect the radial induced velocity u_r, although for the strongly deflected or prebent blades of current multi-MW turbines it produces an additional in-plane driving force. Two engineering models proposed by Li et al. (2022) — the smooth correction attributed to Madsen, driven by the area-averaged thrust coefficient, and the superposition of discrete semiinfinite vortex cylinders formed from the converged annulus inductions — were implemented independently in an open-source blade design and analysis code. For the NREL 5 MW reference rotor with an operational tip deflection of 6 m both models change thrust and power by less than 1 %, and their integral effects on the power coefficient agree to within 0.05 %. This agreement conceals a pronounced local disagreement: at mid-span the vortex-cylinder u_r changes sign, following the non-monotonic radial derivative of the axial induction, and at the outermost section it grows logarithmically with the number of blade sections, u_r ≃ 0.78 ln N_sec + 3.05 (m s⁻¹), in close agreement with the analytic slope |γ_t   ^tip|/2π = 0.82, reaching 6.6 m s⁻¹ at N_sec = 70 against 2.3 m s⁻¹ for the Madsen correction. We show that the integral agreement is systematic rather than coincidental, and that local load predictions near the tip of non-planar rotors are model- and grid-dependent and require an explicit, resolution-independent regularisation.