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The hydrodynamic instability arising under the form of a precession flow, named the vortex rope, creates high-pressure unsteady fluctuations on the walls of the draft tube of a Francis hydropower turbine, leading to a poor performance of the turbine including fatigue damage. Modeling of the phenomena and the numerical investigation of the hydrodynamic instability of this swirling system were tasks that have been assessed in this book. As an alternative to classical methods, the present book proposes a new approach of the analysis of the swirling flows based on a recently mathematical method of spectral collocation with orthogonal decomposition. The mathematical model presented in this book can recover the information in the prediction of the turbine characteristics without computing the full three-dimensional unsteady flow. As a result, this book provides valuable tools for assessing the turbine behavior at off-design operating regimes in the early stages of runner design. Mathematical instruments offered in this book give a transferable knowledge in approaching interdisciplinary problems that occur in modern science, which can be applied to a number of different disciplines.