Strouhal scaling of brush–serration crossover in trailing-edge noise

Trailing-edge brushes and sawtooth serrations can mitigate turbulent boundary-layer trailing-edge (TBL-TE) noise, yet their relative performance depends on frequency and flow speed. Within the Lighthill–Curle–Amiet framework, we develop a source–scatterer filtering interpretation: brushes act as finite-extent permeable source modifiers whose characteristic frequency scales with velocity, whereas serrations act as coherence-weighted harmonic scattering filters with intrinsically band-limited effectiveness. This framework yields testable predictions, including a velocity-dependent crossover boundary in the frequency–velocity plane that should be organised, for a fixed serration geometry, by a constant serration Strouhal number St_λ^⋆=O(1). Experiments on a NACA 0012 airfoil equipped with baseline, brush, and sawtooth trailing edges at 40–70 m/s provide evidence consistent with these predictions. A planar microphone array with frequency-domain beamforming reveals that serrations provide the strongest reduction around 2 kHz, brushes dominate around 8 kHz, and the intermediate 3.15 kHz band exhibits a crossover that shifts with velocity. Region-of-interest analysis across frequencies extracts a crossover boundary corresponding to a configuration-specific effective value near St_λ^⋆≈1.3. Bandwidth and finite-record checks give similar crossover trends. Nonlinear time-series diagnostics, including multiscale permutation-entropy statistical complexity, multifractal detrended fluctuation analysis, and ordinal-pattern networks, provide temporal signatures consistent with this crossover and show how the treatments reorganise the residual trailing-edge signal in time. The proposed framework and signatures offer a compact basis for selecting or hybridising passive trailing-edge treatments across operating conditions.