Study of combustion dynamics has gained significant research attention since low-emission systems are increasingly employed in the industry. In particular, combustion noise and thermo-acoustic instabilities are of great importance in highly critical applications such as power generation, jet propulsion engines, and rocket propulsion systems. Recently, porous inert media (PIM), also referred to as foam insert, has shown promise in mitigating combustion noise and thermo-acoustic instabilities in lean premixed (LPM), swirl-stabilized combustion at atmospheric pressure and elevated pressures. In this study, the flow field without and with PIM is investigated to understand the underlying mechanisms responsible for mitigating thermo-acoustic instabilities. Experiments are conducted for LPM combustion and lean direct injection (LDI) combustion. First, time-resolved PIV technique is utilized to measure the non-reacting flow field without and with PIM. Although the flow field inside the annulus of the foam insert was optically inaccessible, measurements immediately downstream provide insight into the instantaneous flow field and turbulence characteristics. The study highlights the role of the foam insert on vorticity, velocity, shear layer spreading angle, recirculation zone dynamics, and turbulent kinetic energy; which ultimately affects the acoustics behavior of the combustor in a favorable manner. The effect of PIM on the dominant turbulent structures in the flow field is quantified using proper orthogonal decomposition (POD) technique. Next, flow field measurements are acquired for LPM swirl-stabilized combustion without and with PIM. The turbulent structures similar to the non-reacting flow field are also present in the reacting flow field, with notable difference in size and shape. The instantaneous and average flow fields provide insight into the effects of PIM on the velocity and turbulence fields. POD analysis is used to quantify the effect of PIM on the dominant turbulent structures, and PIM is shown to distribute the turbulent energy from the large scale structures to smaller scale structures. By harmonically reconstructing the flow field at the frequency of thermo-acoustic instability, the feedback mechanism is found to be the vortical structures in the corner recirculation zones, and PIM is shown to eliminate the feedback mechanism. The efficacy of PIM in mitigating combustion noise and thermo-acoustic instabilities is demonstrated for liquid fuel combustion utilizing the LDI concept. In this system, the flame stabilizes downstream of the dump plane due to a balance of flow velocity and flame speed of the fuel-air mixture created upstream. The ring shaped PIM is placed at the dump plane of the combustor to alter the flow field in an advantageous manner. Sound pressure levels (SPL) and CO and NOx emissions are measured for combustion without and with PIM inserts. Effect of atomizing air to liquid mass ratio on SPL suggests equivalence ratio oscillations are the driving force for thermo-acoustic instabilities. Results show that the PIM insert reduces broad band combustion noise, mitigates peak instabilities occurring at the first longitudinal mode of the natural frequency of the combustor, and facilitates thermal feedback from the flame to the fuel atomization process. Different insert geometries were examined and they all reduced SPLs, but the converging foam geometry provided the best performance. Finally, flow fields of LDI combustion are experimentally measured using time-resolved PIV technique without and with PIM. The instantaneous flow field highlights the role of PIM on the fluctuating velocity field. The driving mechanism for thermo-acoustic instability is identified by analyzing the fluctuating flow field, and PIM is found to decrease the driving force for thermo-acoustic instability. The average flow field is used to show the effect of PIM on the turbulence and POD analysis is used to quantify the effect of PIM on the turbulent structures. The study identifies spatial and temporal non-homogeneities in equivalence ratio as the feedback mechanisms for exciting thermo-acoustic instabilities in LDI swirl-stabilized combustion. In general, PIM decreases the driving force while increasing the dampening force in both LPM and LDI combustion systems.