Optimal perturbations in a time-dependent round jet

We analyse the influence of the specific features of a time-dependent round jet undergoing the Kelvin-Helmholtz instability on the development of three-dimensional secondary instabilities. We proceed to a non-modal linear stability analysis based on a direct-adjoint approach in order to determine the fastest growing perturbation over a single period of the time-evolving two-dimensional base flow during a given time interval, [t0; T]. An optimisation loop allows to adapt the control criteria depending on the quantity we want to maximise. In particular, it enables to explore the sensitivity of the optimal perturbation to the azimuthal periodicity of the mode, m, as well as the Reynolds number, Re. We focus on the influence of the azimuthal wavenumber, m, on the optimal energy gain as a function of the horizon time, T. We also consider the impact of the injection time, t0, on the transient dynamics of the optimal perturbation and, in particular, on the contribution of the Orr mechanism. Through a detailed analysis of the kinetic energy budget, we identify the different types of instability (elliptical, hyperbolic) and their coexistence associated with the three-dimensionalisation process of the round jet.