Methods in hydrodynamic stability, systems and control theory are applied to spatially developing flows, where the flow is not required to vary slowly in the streamwise direction. A substantial part of the thesis presents a theoretical framework for the stability analysis, input-output behavior, model reduction and control design for fluid dynamical systems using examples on the linear complex Ginzburg-Landau equation. The framework is then applied to high dimensional systems arising from the discretized Navier–Stokes equations. In particular, global stability analysis of the three-dimensional jet in cross flow and control design of two-dimensional disturbances in the flat-plate boundary layer are performed. Finally, a parametric study of the passive control of two-dimensional disturbances in a flat-plate boundary layer using streamwise streaks is presented.

In this thesis direct numerical simulations are used to investigate two phenomenain shear flows: laminar-turbulent transition over a flat plate and periodicvortex shedding induced by a jet in cross flow. The emphasis is on understanding and controlling the flow dynamics using tools from dynamical systems and control theory. In particular, the global behavior of complex flows is describedand low-dimensional models suitable for control design are developed; this isdone by decomposing the flow into global modes determined from spectral analysisof various linear operators associated with the Navier–Stokes equations.Two distinct self-sustained global oscillations, associated with the sheddingof vortices, are identified from direct numerical simulations of the jet incrossflow. The investigation is split into a linear stability analysis of the steadyflow and a nonlinear analysis of the unsteady flow. The eigenmodes of theNavier–Stokes equations, linearized about an unstable steady solution revealthe presence of elliptic, Kelvin-Helmholtz and von K´arm´an type instabilities.The unsteady nonlinear dynamics is decomposed into a sequence of Koopmanmodes, determined from the spectral analysis of the Koopman operator. Thesemodes represent spatial structures with periodic behavior in time. A shearlayermode and a wall mode are identified, corresponding to high-frequency andlow-frequency self-sustained oscillations in the jet in crossflow, respectively.The knowledge of global modes is also useful for transition control, wherethe objective is to reduce the growth of small-amplitude disturbances to delaythe transition to turbulence. Using a particular basis of global modes, knownas balanced modes, low-dimensional models that capture the behavior betweenactuator and sensor signals in a flat-plate boundary layer are constructed andused to design optimal feedback controllers. It is shown that by using controltheory in combination with sensing/actuation in small, localized, regionsnear the rigid wall, the energy of disturbances may be reduced by an order of magnitude.