Numerical Investigation of 'Transonic Resonance' with a Convergent-Divergent Nozzle

At pressure ratios lower than the design value, convergent-divergent (C-D) nozzles often undergo a flow resonance accompanied by the emission of acoustic tones. The phenomenon, driven by the unsteady shock within the divergent section of the nozzle, has been studied experimentally by Zaman et al. In this paper, the space-time conservation element solution element (CE/SE) method is employed to numerically investigate the phenomenon. The computations are performed for a given nozzle geometry for several different pressure ratios. Sustained 'limit cycle' oscillations are encountered in all cases. The oscillation frequencies, their variation with pressure ratio including a 'stage jump', agree well with the experimental results. The unsteady flow data confirm that stage 1 of the resonance (fundamental) involves a one-quarter standing wave while stage 2 (third harmonic) involves a three-quarter standing wave within the divergent section of the nozzle. Details of the shock motion, and the flow and near acoustic field, are documented for one case each of stages 1 and 2.