The effect of specific cavity dimensions of circular concentric Helmholtz resonators is investigated theoretically, computationally, and experimentally. Three analytical models are employed in this study: (1) A two-dimensional model developed to account for the nonplanar wave propagation in both the neck and the cavity; (2) a one-dimensional solution developed for the limit of small cavity length-to-diameter ratio, l/d, representing a radial propagation in the cavity; and (3) a one-dimensional closed-form solution for configurations with large l/d ratios which considers purely axial wave propagation in the neck and the cavity. For low and high l/d, the resonance frequencies determined from the two-dimensional approach are shown to match the one-dimensional predictions. For cavity volumes with l/d>0.1, the resonance frequencies predicted by combining Ingard’s end correction with one-dimensional axial wave propagation are also shown to agree closely with the results of the two-dimensional model. The results from the analytical methods are then compared with the numerical predictions from a three-dimensional boundary element method and with experiments. Finally, these approaches are employed to determine the wave suppression performance of circular Helmholtz resonators in the frequency domain.
The quarter-wave resonator, which produces a narrow band of high acoustic attenuation at regularly spaced frequency intervals, is a common type of silencer used in ducts. The presence of mean flow in the main duct, however, is likely to promote an interaction between these acoustic resonances and the flow. The coupling for some discrete flow conditions leads to the production of both large wave amplitudes in the side branch and high noise levels in the main duct, thereby transforming the quarter-wave silencer into a noise generator. The present approach employs computational fluid dynamics (CFD) to model this complex interaction between the flow and acoustic resonances at low Mach number by solving the unsteady, turbulent, and compressible Navier-Stokes equations. Comparisons between the present computations and the experiments of Ziada [PVP-Vol. 258, ASME, 35-59 (1993)] for a system with two coaxial side branches show that the method is capable of reproducing the physics of the flow-acoustic coupling and predicting the flow conditions when the coupling occurs. The theory of Howe [IMA J. Appl. Math. 32, 187-209 (1984)] is then employed to determine the location and timing of the acoustic power production during a cycle.
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