Global modes and superdirective acoustic radiation in low-speed axisymmetric jets

Abstract

Global linear stability theory is used to study the resonances in a slowly diverging axisymmetric jet. The absolute frequency ω 0 is calculated as a function of slow axial position X , and analytic continuation into the complex X -plane allows a saddle point in ω 0 to be identified. A key element in the analysis is the approximation of ω 0( X) by rational functions which identifies a well-defined saddle point and leading-order global frequency. A preferred-mode Strouhal number, S D =0.44 , is calculated which compares well with existing experimental values. The global frequency has negative imaginary part and the jet is interpreted as being marginally globally stable, so that forcing in the vicinity of the resonance frequency produces a large response above background, rather like that of a slightly damped linear oscillator. The axial shape of the global-mode amplitude is Gaussian and yields a superdirective acoustic field.