During the last decades, many efforts have been invested by the scientific community in minimising exhaust emissions from aeronautical gas turbine engines. In this context, many advanced ultra-low NOx combustion concepts, such as the Lean Direct Injection treated in the present study, are being developed to abide by future regulations. Numerical simulations of these devices are usually computationally expensive since they imply a multi-scale problem. In this work, a non-reactive Large Eddy Simulation of a gaseous-fuelled, radial-swirled Lean-Direct Injection (LDI) combustor has been carried out through the OpenFOAM Computational Fluid Dynamics (CFD) code by solving the complete inlet flow path through the swirl vanes and the combustor. The geometry considered is the gaseous configuration of the CORIA LDI combustor, for which detailed measurements are available. Macroscopical analysis of the main turbulent features related to the swirling flow and the generated Central Recirculation Zone (CRZ) are well established in the literature. Nevertheless, a more in-depth characterization is still required in this area of active research since theory and experimental data are not yet able to predict which unstable mode dominates the flow. This work aims at using Large Eddy Simulation for a complete characterisation of the unsteady flow structures generated within the combustion chamber of a gaseous methane injection immersed in a strong non-reactive swirling flow field. To do so, a spectral analysis of the flow field is performed to identify the frequency, intensity and instabilities associated to the phenomena occurring at the swirler outlet region. A coherent structure known as Precessing Vortex Core (PVC) is identified both at the inner and the outer shear layers, resulting in a periodic disturbance of the pressure and velocity fields. The pressure and velocity fluctuations predicted by the CFD code are used to compute the spectral signatures through the Sound Pressure Level (SPL) amplitude at multiple locations. This allows investigating both the complex behaviour of the PVC and its associated acoustic phenomena. The acoustic characteristics computed by the numerical model are first validated qualitatively by comparing the spectrum with available experimental data. In this way, the use of dimensionless numbers to characterise the most energetic structures is coherent with the experimental observations and the characteristics of the PVC. Then, the numerical identification of the main acoustic modes in the chamber through Dynamic Mode Decomposition (DMD) allows overcoming the Fast Fourier Transform (FFT) shortcomings and better understanding the propagation of the hydrodynamic instability perturbations. This investigation on the main non-reacting swirling flow structures inside the combustor provides a suitable background for further studies on combustion instability mechanisms.