Direct numerical simulations are carried out to investigate the flow features responsible for secondary tones arising in trailing-edge noise at moderate Reynolds numbers. Simulations are performed for a NACA 0012 airfoil at freestream Mach numbers 0.1, 0.2 and 0.3 for angle of incidence 0 deg. and for Mach number 0.3 at 3 deg. angle of incidence. The Reynolds number based on the airfoil chord is fixed at Re c = 10 5 . Flow configurations are investigated where noise generation arises from the scattering of boundary layer instabilities at the trailing edge. Results show that noise emission has a main tone with equidistant secondary tones, as discussed in literature. An interesting feature of the present flows at zero incidence is shown; despite the geometric symmetry, the flows become non-symmetric with a separation bubble only on one side of the airfoil. A separation bubble is also observed for the non-zero incidence flow. For both angles of incidence analyzed, it is shown that low-frequency motion of the separation bubbles induce a frequency modulation of the flow instabilities developed along the airfoil boundary layer. When the airfoil is at 0 deg. angle of attack an intense amplitude modulation is also observed in the flow quantities, resulting in a complex vortex interaction mechanism at the trailing edge. Both amplitude and frequency modulations directly affect the velocity and pressure fluctuations that are scattered at the trailing edge, what leads to secondary tones in the acoustic radiation.

A large eddy simulation is performed to study secondary tones generated by a NACA0012 airfoil at angle of attack of $\alpha = 3^{\circ}$ with freestream Mach number of $M_{\infty} = 0.3$ and Reynolds number of $Re = 5 \times 10^4$ . Laminar separation bubbles are observed over the suction side and near the trailing edge, on the pressure side. Vortex shedding occurs aft of the suction side separation bubble, and vortex interaction results in merging or bursting such that coherent structures or turbulent packets are advected towards the trailing edge. This mechanism modulates the amplitude of the incident pressure signal, leading to different levels of noise emission. Despite the intermittent occurrence of laminar–turbulent transition, the noise spectrum depicts a main tone with multiple equidistant secondary tones. To understand the role of flow instabilities on the tones, the linearised Navier–Stokes equations are examined in their operator form through biglobal stability and resolvent analyses, and by time evolution of disturbances using a matrix-free method. These linear global analyses reveal amplification of disturbances over the suction side separation bubble. Spanwise-averaged pressure fluctuations elucidate aspects of the acoustic feedback loop mechanism in the nonlinear solutions. This feedback process is self-sustained by acoustic waves radiated from the trailing edge, which reach the most sensitive flow location near the leading edge, as identified by the resolvent analysis. Flow disturbances arising from secondary diffraction and phase interference among the most unstable frequencies computed in the eigenspectrum are shown to have an important role in the feedback loop.

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Lagrangian simulations of unsteady vortical flows are accelerated by the multi-level fast multipole method, FMM. The combination of the FMM algorithm with a discrete vortex method, DVM, is discussed for free domain and periodic problems with focus on implementation details to reduce numerical dissipation and avoid spurious solutions in unsteady inviscid flows. An assessment of the FMM-DVM accuracy is presented through a comparison with the direct calculation of the Biot-Savart law for the simulation of the temporal evolution of an aircraft wake in the Trefftz plane. The role of several parameters such as time step restriction, truncation of the FMM series expansion, number of particles in the wake discretization and machine precision is investigated and we show how to avoid spurious instabilities. The FMM-DVM is also applied to compute the evolution of a temporal shear layer with periodic boundary conditions. A novel approach is proposed to achieve accurate solutions in the periodic FMM. This approach avoids a spurious precession of the periodic shear layer and solutions are shown to converge to the direct Biot-Savart calculation using a cotangent function.

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