A resonator system for a turbine engine can improve acoustic performance and cooling effectiveness. During engine operation, a combustor liner exhibits alternating hot and cold regions in the circumferential direction corresponding to the non-uniform temperature distribution of the combustion flame. Accordingly, high flow resonators are formed with the liner in substantial alignment with the hot regions of the fluid flow within the liner, and low flow resonators are formed with the liner in substantial alignment with cold regions of the fluid flow within the liner. As a result, appropriate amounts of cooling can be provided to the liner so that cooling air usage is optimized. Alternatively or in addition, the liner can include two or more rows of resonators, which can provide an enhanced acoustic damping response. The resonators in the first row can be aligned with or offset from the resonators in the second row.