Tangential induced velocity in the vortex cylinder model for non-planar rotors

This work examines the tangential induced velocity in a general non-planar vortex cylinder framework for non-planar rotors, including coned rotor and rotors with prebent or deflected blades. Compared with the planar rotor case, the planar bound vortex disc is replaced by a curved axisymmetric bound vortex surface, and the longitudinal trailed vortex cylinders start from the corresponding curved surface. The compact expression for the tangential induced velocity was previously obtained by Li et al. (2022a) using the definition of circulation. That compact result already contains the influence of the curved bound vortex surface. The purpose of the present work is to provide an independent verification of the same result directly from the Biot–Savart law by separately evaluating and then combining the contributions from the curved bound vortex surface and the longitudinal trailed vortex system. Numerical checks using direct numerical integration of the Biot–Savart law and elliptic integral formulations support the analytical verification for both planar and curved cases.