On the emergence of secondary tones in airfoil noise

Abstract

We present the results of direct numerical simulations of a NACA 0012 airfoil, with Mach number 0.3 and angle of attack of $3^\circ$ , examining the dynamics of the flow with increasing Reynolds numbers. Two-dimensional simulation results are obtained with chord-based Reynolds numbers in the range $3.2 \times 10^3 \leq Re \leq 2.70 \times 10^4$ , where each simulation uses the last time step of the previous one as a starting point, to capture the evolution of dynamics as a function of $Re$ . The development of the pressure fluctuations with time shows a transition from periodic to quasi-periodic attractor for $2.38 \times 10^4 \leq Re \leq 2.42 \times 10^4$ , leading to the emergence of secondary tones in the wall and acoustic field pressure spectra, different from peaks related to the fundamental frequency $f_1$ and the respective harmonics; a second, incommensurate frequency $f_2$ appears, leading to several secondary tones with frequency $af_1 + bf_2$ , with $a$ and $b$ integers. Further increase of the Reynolds number leads to the emergence of a tertiary frequency, $f_3$ , indicating a route to chaos of the Ruelle–Takens–Newhouse type. Such a mechanism is related to the ladder-type characteristic structure of the tones, indicating that dynamic systems theory is an important tool for understanding airfoil tonal noise.