Finite Element Solutions of a Third-Order Wave Equation for Sound Propagation in Sheared Flows

Abstract

A weak discretization in frequency-domain of the Lilley-Goldstein acoustic analogy equation is presented. This equation is cast into a linear system of two equations that are successively arranged in a convenient form for a Galerkin projection. A conventional nite element discretization is used and the resulting implementation is veri ed against analytical solutions. The method is then applied to compute the sound generated by Dirac ring sources of di erent azimuthal order and radius and propagated through an axi-symmetric parallel jet. The e ects due to the jet temperature and velocity pro le combined with the e ects due to the source frequency, radius and azimuthal order are illustrated through extensive parametric results. It is shown that three-dimensional Green’s functions can be reconstructed by superposition of few axi-symmetric solutions of low azimuthal order when the sources are located inside the jet. Conversely, the three-dimensional reconstruction converges very slowly when the sources are located in the outer part of the jet. Finally, tones from annular sources located in the potential cone of a mixing jet are propagated through the jet shear layer and compared with the tones computed in uniform ow conditions. It is shown that, for conditions that are representative of the thickness and loading noise of a propeller rig located in the potential cone of a wind tunnel jet, the shear layer refraction e ects can be corrected by using a thin-layer approximation. However, the interaction between the uctuating vortex sheet from the nozzle edge and the direct acoustic eld from the source enhances the noise levels in the forward radiation arc.