This study aims to advance the fundamental flow physics of acoustic source generation, propagation, and mitigation using large-eddy simulations with specific emphasis on flows around an airfoil.First, a wavelet-based recursive denoising algorithm is applied to airfoil flow field. The pressure field around the airfoil is decomposed into coherent contributions corresponding to denoised pressure and incoherent pressure corresponding to background noise. It is found that the denoised pressure represents physical phenomena associated with near-wall hydrodynamic wavy structures and sound propagation generated near the boundary-layer tripping region and trailing edge. On the other hand, the incoherent pressure or background noise exhibits a small and constant amplitude closely adhering to the Gaussian distribution. Second, wavenumber-frequency decomposition and Amiet's theory are used to separate total pressure into two distinct components: hydrodynamic (incident) pressure and acoustic (scattered) pressure. The findings reveal that hydrodynamic pressure consists of turbulent coherent structures characterized by high-energy spectra but acts as non-propagating sources due to destructive interference within the streamwise correlation length. In contrast, acoustic pressure exhibits an in-phase nature that facilitates efficient sound propagation. Amiet's theory shows similarities in magnitude and directivity patterns, but the disparity between numerical and analytical studies exists, which is investigated in detail. Third, this study delves into the detailed sound generation and propagation mechanisms associated with trailing-edge scattering and flow perturbations of boundary-layer tripping. Two distinct boundary-layer tripping techniques, namely a geometrically resolved stair strip and an artificially modeled trip using suction and blowing, are investigated. It is found that boundary-layer tripping notably leads to intricate, scenario-specific noise generation: there is an interaction between the laminar separation bubble and tripping for the stair strip case, whereas laminar boundary layer instability is evident for the suction and blowing scenario. Aerodynamic flow fields involving acoustic noise sources, their propagating natures near the wall, and far-field acoustics are cross-examined in detail. Fourth, a cross-spectrum method is proposed to identify the origin of noise sources and understand sound production mechanisms. This novel method shows its potential by effectively detecting the acoustic source and propagation that are on par with or surpass those of dynamic mode decomposition and spectral proper orthogonal decomposition modes while simultaneously providing the spectral magnitude and phase topologies when applied to different flow transition mechanisms of an airfoil. Fifth, the effect of misaligned flow on trailing-edge noise is studied to examine fundamental mechanisms governing noise mitigation. Flow misalignment observed in the swept airfoil is found to generate destructive interference, playing a crucial role in the noise reduction mechanism. Critical flow physics responsible for the noise source attenuation is discovered in the numerical simulations, which are cross-examined with Amiet's swept trailing-edge noise theory. Sixth, the effects of trailing-edge morphing on aerodynamic and aeroacoustic performances are studied. Concave (M1) and convex (M2) shapes are imposed near the trailing edge of the airfoil. The M1 airfoil demonstrates enhanced performance in both aerodynamics and aeroacoustics, while the M2 airfoil shows deteriorated performances compared to the baseline airfoil. Flow features impacting the airfoil performances are examined in detail.