Dynamics and Optimal Control of Self-Sustained Instabilities in Laminar and Turbulent Swirling Flows: Application to the Part Load Vortex Rope in Francis Turbines

In 2050, the European commission plans to achieve actual energetic transition, which contributes in meeting the climate change challenges by moving away from the fossil fuels and by developing a competitive low-carbon economy. As a consequence, a foreseeable massive introduction of intermittent renewable energy sources in the transmission and distribution systems is going to occur, which will impact randomly the balance of the electric consumption and production, and could jeopardize the electric grid stability. In this context, the present research work aims at increasing the operation flexibility of Francis turbines at part load regime, which is thought to be one of the main solutions to mitigate the large power fluctuations of the electric grid arising from these intermittent power productions. An intense cavitation vortex rope is however known to appear in these operating conditions and to prohibit power generation due to large pressure fluctuations at a well-defined frequency, which causes risks of operating instability, and fatigue of the mechanical structures. The control of the part load vortex rope is therefore addressed in this research work using optimal flow control techniques. Minimization algorithms need a physically-based target characterizing this single helical structure, which is obtained by investigating the hydrodynamic instability properties of this turbulent open swirling flow. This approach is first developed on a benchmark open flow, the spiral vortex breakdown in laminar swirling flow. Global stability analysis about the mean flow is computed and reveals up to two unstable eigenmodes, which well predict the frequencies of the vortical structures of the nonlinear dynamics. Moreover, the computations of the nonlinear interactions of these purely hydrodynamic self-sustained instabilities help identifying a Ruelle-Takens-Newhouse route to chaos as the Reynolds number increases. The optimal control of spiral vortex breakdown, implemented through a minimization algorithm targeting the largest eigenvalue growth rate of the mean flow, results in a distributed force which succeeds in stabilizing the self-sustained instability. This promising approach next requires to transpose the uncluttered theoretical framework of laminar swirling flow to the complexity inherent to industrial design, including in particular turbulence modeling to reach high Reynolds number flows, Re = O(10^6). The present work bridges this gap and presents an optimal control technique suitable for stabilizing the flow in hydraulic turbines. The part load vortex rope is identified as a global unstable eigenmode which originates from a single helical disturbance of the mean flow. This first result provides a linear framework to investigate the flow in hydraulic turbines and enables us to obtain the generation mechanism of the synchronous pressure fluctuations as a fluid-solid interaction using an asymptotic expansion. Based on the hydrodynamic instability propert