Too big to grow : self-consistent model for nonlinear saturation in open shear flows

Open flows, such as wakes, jets, separation bubbles, mixing layers, boundary layers, etc., develop in domains where fluid particles are continuously advected downstream. They are encountered in a wide variety of situations, ranging from nature to technology. Such configurations are characterised by the development of strong instabilities resulting in observable unsteady dynamics. They can be categorised as oscillators which present intrinsic dynamics through self-sustained oscillations, or as amplifiers, which exhibit a strong sensitivity to external disturbances through extrinsic dynamics. Over the years, different linear and nonlinear approaches have been adopted to describe the dynamics of oscillators and amplifiers. However, a simplified physical description that accurately accounts for the nonlinear saturation of instabilities in oscillators as well as that of the response to disturbances in stable amplifier flows is still missing. In this thesis, this question is addressed by introducing a self-consistent semi-linear model. The model is formally constructed by a set of equations where the mean flow is coupled to a linear perturbation equation through the Reynolds stress. The full nonlinear fluctuating motion is thus approximated by a linear equation. The nonlinear dynamics of oscillators is studied in the cylinder wake, where the most unstable eigenmode of finite amplitude is coupled to the instantaneous mean flow for different oscillation amplitudes. This family of solutions provides an instantaneous mean flow evolution as a function of an equivalent slow time. A transient physical picture is formalised, wherein a harmonic perturbation grows and changes the amplitude, frequency, growth-rate and structure due to the modification of the instantaneous mean flow by the Reynolds stress forcing. Eventually this perturbation saturates when the flow is marginally stable. In contrast to standard linear stability analysis around the mean flow, the iterative solution of the model provides a priori an accurate prediction of the instantaneous amplitude, frequency and growth rate, as well as the flow fields, without resorting to any input from numerical or experimental data. Regarding noise amplifiers, the nonlinear saturation of the large linear amplification to external disturbances is studied in the framework of the receptivity analysis of the backward facing step flow. The self-consistent model is first introduced for harmonic forcing and later generalised to stochastic forcing by reformulating it conveniently in frequency domain. The results show an accurate prediction of the response energy as well as the flow fields. Hence, a similar picture is revealed, wherein the Reynolds stress dominates the saturation process. Despite the difference in the dynamics of the described flows, they share the same nonlinear saturation mechanism: the mean flow distortion.