Sound Radiation from a Co-Planar and Buried Nozzle with Jet and Bypass Flow and Arbitrary Edge Condition

In this report, the radiation of sound from an annular duct exit with buried inner nozzle system, with jet and bypass flow, is studied. This problem represents a simplified model of the radiation of fan and turbine noise from engine exhausts. The model consist of two semi-infinite ducts. The smaller duct (inner nozzle), whose exit plane coincides with the larger one or extends beyond it, is situated inside the larger semi-infinite duct. The inner nozzle issues the core flow inside the bypass jet flow. The bypass nozzle issues the bypass jet flow inside the ambient co-flow. Two vortex sheets, attached to the duct exit trailing edges, separate the different flows from each other. These vortex sheets are unstable due to this mean velocity discontinuity. The application of the Kutta condition at the respective trailing edges guarantees shedding of vorticity which excites these instabilities. The system is set up to respond to an incident annular duct mode, but the analysis is very similar for an inner duct mode. Formulation of the boundary value problem, by following the classical approach, leads to a couple of simultaneous Wiener-Hopf equations. These equations produce a matrix equation system, which is formally decoupled by the introduction of an infinite sum of poles with coefficients to be determined. This method is known as Idemen’s method of “weak factorisation” or “the pole removal technique”. The uncoupled scalar equations are solved independently by a standard application of analytical continuation. The final solution includes unknown coefficients which are determined by solving an infinite linear algebraic system numerically. The contribution of the instability waves are separated from the rest of the solution. The asymptotic far field is found by a standard application of the steepest descent method. Finally a series of practical examples are given.