Perpendicular rod wake/aerofoil interaction: microphone array and TR-PIV insights via SPOD and beamforming analysis

This paper investigates the acoustic and velocity fields due to a circular rod and an aerofoil placed in the wake of, and perpendicular to, a rod. Simultaneous measurements were conducted using a microphone array and time-resolved particle image velocimetry (TR-PIV). The interaction was characterized through acoustic spectra and the coherence between microphone signals and the three velocity components. Coherent structures were identified with Spectral Proper Orthogonal Decomposition (SPOD) using a norm based either on turbulence kinetic energy (SPOD-u) or on pressure (SPOD-p). An advantage of SPOD-p is that it identifies velocity modes associated with a large acoustic energy. Peaks of energy were observed at $\mathit{St} \approx 0.2$ and $0.4$--Strouhal numbers based on rod diameter and free-stream velocity. At $\mathit{St} \approx 0.2$, the dominant feature is von K\'arm\'an vortex shedding from the rod. At $\mathit{St} \approx 0.4$, a wave-train structure in the rod wake impinging on the aerofoil leading edge is captured by the rank-1 SPOD-p mode, with coherence levels reaching 60\% for the $u_2$ component (upwash/downwash relative to the aerofoil). This structure also appears at $\mathit{St} \approx 0.2$, but as the rank-2 SPOD-p mode. A mode-switching occurs around $\mathit{St} \approx 0.3$: below this value, the rank-1 mode corresponds to von K\'arm\'an shedding (cylinder branch), while above it, the rank-1 mode tracks the interaction of the aerofoil with the rod wake (aerofoil branch). Both branches were also identified via beamforming using low-rank cross-spectral matrices derived from SPOD-p modes.