The contribution of this Ph.D consists in a formalism and a methodology to perform linear stability analysis of turbulent flows. The flow dynamics is modelled using the RANS equations closed with a turbulence model, and we focus on the instabilities associated with the large scale structures of turbulence. A global formulation is adopted so as to allow complex geometries analysis. A discrete framework is considered, where the equations are first discretized and then linearized. In particular, the linearization is performed using finite differences. This procedure ensures the generic character of the method regarding the system of equations such as the turbulence model for example, and avoids tedious analytical linearization. Furthermore, it allows to use a numerical code in a black-box manner in order to perform linear stability analysis. Finally, we demonstrate that the sensitivity gradients can be computed within this framework for both laminar and turbulent flows. Sensitivity analysis carries valuable information regarding the location where steady control means can affect the flow unsteadiness, enabling the design of robust strategies for open loop control. The method is first tested on two laminar cases, reproducing former studies concerned with the oscillators dynamics of the wake behind a two dimensional cylinder, and the characterization of a laminar boundary layer as a noise amplifier. The robustness and validity of our procedure is then extensively studied on a compressible turbulent flow over a deep cavity. Numerical validations are performed, ensuring the correctness of our sensitivity gradients up to 3\%, and the flow physics, including unstable mode analysis, acoustics, impact of turbulence modeling, is analysed. In order to enhance the portability and the valuable information carried out by our method, we present several preliminary studies that were performed using our formalism. First, we revisit the transonic buffet over an airfoil, the noise amplifier dynamics of a turbulent shock-boundary layer interaction is then characterized and we conclude with an analysis of the screech phenomenon in under-expanded jets. Finally, we conclude this work by studying the turbulent wake behind a D-shaped cylinder, and show the potential of our method for industrial applications.
