• adjoint methods
• flow control
• Stability analysis

Numerical tools for global linear stability and sensitivity analyses in fluid dynamics are developed in the present work. These tools are applied to the investigation and control of two- and three-dimensional internal flow configurations.

The first application is to the flow in a two-dimensional symmetric channel with a sudden expansion. The laminar flow in such diffusers may produce either symmetric or
nonsymmetric steady solutions, depending on the value of the Reynolds number as compared with
some critical value. The stability properties of the flow are studied in the context of linear theory. In this context, a sensitivity analysis of the flow instability is carried out with respect to perturbations that may be produced by a realistic passive control, thus providing qualitative hints and quantitative information for the control design. Following the so-obtained information, a passive control is built by introducing a small cylinder in the flow with the aim of stabilizing the unstable symmetric flow configuration in the diffuser. The effectiveness of this control is finally assessed by direct numerical simulation. It is shown that the introduction of the cylinder, placed following the indications of the linear sensitivity analysis in the stable asymmetric flow configuration, allows a steady completely symmetric or less asymmetric flow to be recovered. The flow transient between the uncontrolled asymmetric solution and the symmetric controlled one is analyzed in terms of streamlines and
vorticity evolution; the effects of the cylinder introduction on flow dissipation are also assessed.

A further application to the study of the fully three-dimensional instabilities occurring in the flow in micro T-mixers is presented. First, the instability leading to the steady engulfment regime is characterized by the computation of the critical Reynolds number and of the global unstable mode for different T-mixer geometries with fully-developed inlet velocity conditions. Direct numerical simulations (DNS) are also carried out to asses the results of stability analysis. Moreover, the information obtained in DNS, combined with that of stability and sensitivity analyses, is used to investigate the main flow phenomena leading to the onset of the engulfment regime. The sensitivity of this first instability to a generic modification of the base flow is then investigated, thanks to the computation of the mode adjoint to the direct unstable one. This kind of analysis is furthermore specialized to the effect of a perturbation of the velocity distribution at the inlet of the T-mixer. As a particular application, the
sensitivity analysis shows that non-fully developed inlet velocity conditions lead to an increase of the critical Reynolds number, in agreement with the findings of direct numerical simulations and with indications that can be found in the literature. More generally, the sensitivity maps can be used for the design of control strategies aimed at promoting or inhibiting the engulfment. An example is given for the control through suction/blowing from the mixer solid walls. Linear instability and sensitivity analyses and direct numerical simulation are finally used to investigate the secondary instability, which occurs at larger Reynolds numbers and leads to a periodic unsteady flow regime.