Numerical Investigation of Transonic Resonance with a Convergent-Divergent Nozzle

Abstract

At pressure ratios lower than the design value, convergent ‐divergent nozzles often undergo a e ow resonance accompaniedbytheemissionofacoustictones.Thephenomenon,drivenbytheunsteadyshockwithinthedivergent section of the nozzle, has been studied experimentally previously. The space ‐time conservation element solution element method is employed to numerically investigate the phenomenon. The computations are performed for a given nozzle geometry forseveral different pressure ratios. Sustained limit-cycleoscillations are encountered in all cases. The oscillation frequencies and their variation with pressure ratio, including a stage jump, agree well with the experimental results. The unsteady e ow data cone rm that stage 1 of the resonance (fundamental) involves a one-quarter standing wave, whereas stage 2 (third harmonic ) involves a three-quarter standing wave within the divergent section of the nozzle. Details of the shock motion, the e ow, and the near acoustic e eld are documented for one case each of stages 1 and 2. I. Introduction T HIS paper is concerned with an aeroacoustic resonance often encountered with convergent‐ divergent nozzles when run near transonic conditions. The resonance is usually accompanied by the emission of intense acoustic tones. Whereas a casual observer may easily confuse it with the well-known screech tone, it has been shown to be different in character as well as origin. The frequency of the tone increases with increasing plenum pressure. The frequency variation may involve a staging behavior, that is, an abrupt jump in frequency. Whereas odd harmonic stages take place at lower pressures, the fundamental takes place over a wide rangeof higher pressures. Depending on nozzle geometry, the fundamental has been found to persist to pressure ratios as high as 5. The phenomenon has been identie ed and studied experimentally by Zaman etal. 1 For a discussion of background, technological relevance, and pertinent past work from the literature, see Ref. 1. A discussion of numerical works from the literature apparently capturing the same phenomenon, and the implication of those results, will be deferred to the discussion in Sec. V. During the course of the experimental study a numerical study was initiated to complement the investigation. The space‐ time conservation-element and solution-element (CE/SE) method, to be elaborated shortly, was employed because of its past success with e ows involving shocks and acoustic waves. 2 The calculations were e rst performed for a nozzle geometry and pressure ratio that corresponded to an experimental test condition. The result was encouraging in that the e ow not only exhibited a quasi periodicity, but the frequency was also in agreement with the experimental result. Subsequent calculations at two other pressures captured the right trend in the frequency variation as well as the stage jump. These promising results prompted a further study. Thiswas deemedwell-justie ed