The jet edge-tone feedback cycle; linear theory for the operating stages

Abstract

The paper presents a linear analytical model to predict the frequency characteristics of the discrete oscillations of the jet-edge feedback cycle. The jet is idealized as having top-hat profile with vortex-sheet shear layers, and the nozzle from which it issues is represented by a parallel plate duct. At a stand-off distance h, a flat plate is inserted along the centreline of the jet, and a sinuous instability wave with real frequency w is assumed to be created in the vicinity of the nozzle and to propagate towards the splitter plate. Its interaction with the splitter plate produces an irrotational feedback field which, near the nozzle exit, is a periodic transverse flow producing singularities at the nozzle lips. Vortex shedding is assumed to occur, alleviating the singularities and allowing a trailing-edge Kutta condition to be satisfied; this Kutta condition is claimed to be the phase-locking criterion. The shed vorticity develops into a sinuous spatial instability, and the cycle of events is repeated periodically.Problems corresponding to the various physical processes described are analysed, for in viscid flow with vortex-sheet shear layers and aligned flat-plate boundaries, and solved in an appropriate asymptotic sense by Wiener-Hopf methods. Calculation of the phase changes occurring in the constituents of the cycle gives an equation for the frequency w in the Nth ‘stage’ as a function of jet width 2b, jet velocity U0, standoff distance h, and stage label N: \[ \omega b/U_0 = (b/h)^{\frac{3}{2}}[4\pi(N-{\textstyle\frac{3}{8}})]^{\frac{3}{2}}. \] The variations with b, U0, h and N are in excellent agreement with edge-tone experiments; the principal disagreement lies in the overall numerical factor and explanations are given for this. Possible effects associated with the inclusion of displacement thickness fluctuations in the splitter-plate boundary layers, and the enforcement of a leading-edge Kutta condition, are also considered and shown not to affect the frequencies of the operating stages.