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The sound generated by an acoustic source near a semi-infinite edge with uniform parameters is studied theoretically. The acoustic emission of a vortex ring passing near a semi-infinite porous or elastic edge with uniform properties is formulated as a vortex sound problem and is solved using a Green’s function approach. The time-dependent pressure signal and its directivity in the acoustic far field are determined analytically for rigid porous edges as a function of a single dimensionless porosity parameter. At large values of this dimensionless parameter, the radiated acoustic power scales on the vortex ring speed U and the nearest distance between the edge and the vortex ring L as U^6L^−5, in contrast to the U^5L^−4 scaling recovered in the impermeable edge limit for small porosity values. These analytical findings agree well with the results of a companion experimental campaign conducted at the Applied Research Laboratories (ARL) at Penn State University. A related theoretical analysis of the sound scattered by uniform, impermeable elastic edges admits analytical results in a specific asymptotic limit, in which the acoustic power scales on U^7L^−6. In complement to the analysis of vortex ring sound from edges, the acoustic scattering of a turbulent eddy near a finite edge with a graded porosity distribution is determined numerically and is validated against analytical acoustic directivity predictions from the vortex-edge model problem for a semi-infinite edge in the appropriate high frequency limit. The cardioid and dipolar acoustic directivity obtained in the vortex ring configuration for low and high dimensionless porosity parameter values, respectively, are recovered by the numerical approach. An imposed linear porosity distribution demonstrates no substantial difference in the acoustic directivity relative to the uniformly porous cases at high porosity parameter values, where the local porosity parameter value at the edge determines the scattered acoustic field. However, more modulated behavior of the acoustic directivity is found at a relatively low frequency for the case of a finite edge with small graded porosity distribution. 

The sound of a vortex ring passing near a semi-infinite porous edge is investigated analytically. A Green's function approach solves the associated vortex sound problem and determines the time-dependent pressure signal and its directivity in the acoustic far field as a function of a single dimensionless porosity parameter. At large values of this parameter, the radiated acoustic power scales on the vortex ring speed $$U$$ and the nearest distance between the edge and the vortex ring $$L$$ as $$U^6 L^{-5}$$ , in contrast to the $$U^5 L^{-4}$$ scaling recovered in the impermeable edge limit. Results for the vortex ring configuration in a quiescent fluid furnish an analogue to scaling results from standard turbulence noise generation analyses, and permit a direct comparison to experiments described in Part 2 that circumvent contamination of the weak sound from porous edges by background noise sources that exist as a result of a mean flow. 

The coupled interaction between an unsteady vortical flow and dynamics of an aerodynamic structure is a canonical problem for which analytical studies have been typically restricted to either static or prescribed structural motions. The present effort extends beyond these restrictions to include a Joukowski airfoil on elastic supports and its aeroelastic influence on the incident vortex, where it is assumed that all vorticity in the flow field can be represented by a collection of line vortices. An analytical model for the vortex motion and the unsteady fluid forces on the airfoil is derived from inviscid potential flow, and the evolution of the unsteady airfoil wake is governed by the Brown and Michael equation. The aerodynamic sound generated by the aeroelastic interaction of an incident vortex, shed Brown-Michael vortices, and the moving airfoil is estimated for low-Mach-number flows using the Powell-Howe acoustic analogy. 

The dynamic interactions between an incident line vortex and a symmetric Joukowski airfoil on elastic translational support are formulated analytically and evaluated numerically, where the unsteady shedding of vorticity from the airfoil trailing edge is modeled by the emended Brown and Michael equation. This mathematical framework explores the effects of initial vortex placement, vortex strength, and the system aeroelastic parameters on the selection of the vortex trajectory to pass either above or below the airfoil, where special attention is paid to the conditions where direct impingement occurs.