Spectral Descriptor Approach For Solving Hydrodynamic PDE Models Of Swirling Flows With Applications Diana Alina Bistrian 1 , Florica Ioana Dragomirescu 2 , Sebastian Muntean 3 , Romeo Susan-Resiga 4 , George Savii 5 1 Department of Electrical Engineering and Industrial Informatics, Engineering Faculty of Hunedoara, ”Politehnica” University of Timisoara, 331128 Hunedoara, Romania 2 Department of Mathematics, ”Politehnica” University of Timisoara, 300006 Timisoara, Romania 3 Centre of Advanced Research in Engineering Sciences, Romanian Academy - Timisoara Branch, 300222 Timisoara, Romania 4 Department of Hydraulic Machinery, ”Politehnica” University of Timisoara, 300222 Timisoara, Romania 5 Department of Mechatronics, Mechanical Engineering Faculty, ”Politehnica” University of Timisoara, Mihai Viteazu Nr.1, 300222 Timisoara, Romania Keywords: Swirling flow, Spectral collocation, Singularities, Vortex rope 1 Abstract Swirling flow behavior and stability in various technical applications has long been an intensive subject of research in hydropower turbine design. A special challenge in Francis turbine design is the ability to efficiently op- erate at partial loads because, at partial loads, this hydropower turbine often exhibit strong swirl at the runner outlet [8] as the incoming flow decelerates in the diffuser cone. The vortex rope hydrodynamic instability creates high-pressure unsteady fluctuations on the walls of the draft tube that could lead to fatigue damage over time. This phenomenon is especially severe when the frequency of the oscillations of the vortex rope matches the resonant frequency of the turbine or circuit. Our paper presents the mathematical and numerical methodology to investi- gate the stability of the fluid system downstream the Francis runner, in order to simulate the frequency, pressure pulsation amplitude and other parameters un- der various operating conditions. The spatial stability problem is reduced to the study of a difficult nonlinear eigenvalue problem with nonconstant coefficients. The approach presented in this paper is different from the traditional optimiza- tion methods, since the spectral collocation technique that we developed has the peculiar feature that can approximate the perturbation field for all types of boundary conditions, especially when the boundary limits are described by sophisticated expressions [7]. For the case of axisymmetrical mode [1], the study involves a new mathemat- ical model in descriptor formulation and simulation algorithms that translate the