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.

💡 Research Summary

The paper investigates how the velocity induction generated by a rotating axial‑flow rotor influences the trajectories of dispersed particles that impact the rotor blades. Using both a full three‑dimensional (3D) Reynolds‑averaged Navier‑Stokes (RANS) simulation in a rotating reference frame and two-dimensional (2D) sectional models, the authors compare three approaches: (i) a 2D model that includes the induced velocity (denoted “2D Ind”), (ii) a 2D model that completely neglects induction (denoted “2D Geom”), and (iii) the reference 3D solution.

A key contribution is the introduction of an “induction Stokes number” ( \mathrm{Stk}{\text{ind}} = \tau{p}/\tau_{\text{ind}} ), where (\tau_{p}) is the particle relaxation time and (\tau_{\text{ind}}) characterises the time scale of the rotor‑induced velocity field. This dimensionless number quantifies how quickly a particle can adjust to the induced flow, independently of the classic Stokes number that measures particle inertia relative to the carrier flow.

The authors demonstrate that for ( \mathrm{Stk}{\text{ind}} \ll 0.1 ) (small, highly responsive particles) the particle velocity rapidly aligns with the induced flow, and the 2D Ind model reproduces the 3D results. Conversely, for ( \mathrm{Stk}{\text{ind}} \gg 10 ) (large, inertia‑dominated particles) the induced velocity has negligible effect, and the 2D Geom model becomes accurate. The intermediate range ( 0.1 \lesssim \mathrm{Stk}_{\text{ind}} \lesssim 10 ) constitutes a transition regime where neither limiting 2D model alone can capture the correct impingement locations, impact angles, or subsequent phenomena such as ice accretion or erosion.

To bridge this gap, a simple first‑order delay model is proposed. The model assumes that the induced component of the particle velocity at the rotor disk is multiplied by a response function ( f(\mathrm{Stk}{\text{ind}}) = 1 - \exp(-\mathrm{Stk}{\text{ind}}) ). The particle velocity is then expressed as
\