Numerical investigations of double-Helmholtz resonators and duct-end effects

Abstract

The acoustic fields and eigenfrequencies of double Helmholtz resonators (Greenspan viscometers) have been investigated using a boundary-integral-equation technique. The resonator geometry consists of a circular duct connected at both ends to concentric cylindrical cavities. Comparisons of the numerically determined fields with analytic expressions based on a Green’s function expansion establish the high accuracy of the method. The acoustic flow near the duct-cavity orifices has the expected singular behavior at the edge of the duct orifice. The associated duct-end inertance corresponds to a length correction of approximately 0.82 times the duct radii, slightly smaller than the classical formulas of Rayleigh and Ingard [ 1037–1061 (1953)], which are based on an assumed uniform (piston) flow within the orifice. Numerical integrations of the square of the tangential particle velocity over the walls of the duct and the cavity wall surrounding the orifice have been used to estimate the orifice resistance. This quantity was found to correspond to a resistance effective length of approximately 0.79 times the viscous penetration length, a result of importance in the modeling of Greenspan viscometers. [Work supported by the Office of Naval Research.]