On particle dynamics in steady axial rotor flows

We investigate the effect of rotor velocity induction on the distribution of particles impinging on rotor blades and model the delayed response of a particle to the rotor-induced velocity field. We consider as reference a wind turbine rotor and a small-scale propeller in axial flow conditions. We first show that the classical 2D modeling of the multi-phase flow can generate a systematic error with respect to the 3D solution. We consider two limiting cases: particles in equilibrium with the rotor-induced velocity field, where the carrier phase is computed using the section's aerodynamic velocity vector, and induction-independent particles, where the geometric velocity vector is used. The 3D solution differs from the two limiting cases when particles are in partial equilibrium with the induced velocity. We introduce an induction Stokes number $\mathit{Stk}_\text{ind}$ and identify a transition regime between the two limiting solutions for $0.1 \lesssim \mathit{Stk}_\text{ind} \lesssim 10$. Then, we present a simple 1D delay model to evaluate the induced component of the particle velocity at the rotor disk as a function $\mathit{Stk}_\text{ind}$. We validate the model by showing that it allows capturing the transition regime in 2D simulations. The model only requires knowledge of the aerodynamic and geometric velocity vectors, i.e., of the axial and tangential induction factors.