Nonlinear development of acoustical instabilities in supersonic jets

In contrast with the roll-up of fluid interfaces through Kelvin-Helmholtz instability, recent numerical simulations with small amplitude perturbations of supersonic jets reveal another very different coherent mode of nonlinear acoustical instability of jets through the appearance of regular zig-zag shock patterns which traverse the interior of the jet and amplify as time evolves. In this paper, through a combination of appropriate ideas from linear and nonlinear high frequency geometric optics, the authors develop a quantitative theory which predicts the nonlinear development of zig-zag modes with a structure like those observed in the numerical simulations. The perturbation analysis is developed via a systematic application of nonlinear small amplitude high frequency geometric optics to the complex free surface problem defined by the perturbed jet; this procedure automatically yields simplified asymptotic equations which are analyzed explicitly and lead to the development of regular amplifying "zig-zag" shock structures in the jet. For a given streamwise period, Mach number, and jet width, the asymptotic theory gives explicit criteria for the number and structure of different regular zig-zag shock patterns which amplify with time. For Mach numbers M < 1, there are no amplifying acoustic zig-zig modes while for M > 1, there are a finite number of such modes depending on Mach number, jet width, and streamwise period. Explicit criteria to select the most destabilizing of these nonlinear eigenmodes are developed as well as several new quantitative predictions regarding the nonlinear development of acoustical instabilities in supersonic jets including the phenomenon of "super-resonance" for special values of the streamwise period.