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1 Introduction
2 Mathematical issues on hydrodynamic stability of swirling flows
2.1 Linearized disturbance equations
2.2 The method of normal modes analysis
2.3 Definition of temporal and spatial instability
2.4 Studies upon stability of swirling flows cited in literature
3 Mathematical model for a swirling system - a Francis turbine runner case
3.1 Discrete operator formulation of the hydrodynamic model
3.2 Axis and wall boundary conditions
4 Orthngonal decomposition method for stability eigenvalue problems
4.1 Motivation of using the spectral methods in hydrodynamic stability problems
4.1.1 The L2 -Projection method
4.1.2 The collocation method
4.2 0rthogonal polynomial decomposition base
4.2.1 Considerations on shifted Chebyshev polynomials
4.2.2 Orthogonality of the shifted Chebyshev polynomials
4.2.3 Evaluation of the shifted Chebyshev derivatives
4.3 Computational domain and grid setup
5 Numerical approach for non-axisymmetrie stability investigation
5.1 Boundary adapted decomposition
5.1.1 Description of the method
5.1.2 Interpolative derivative matrix
5.1.3 Implementation of the boundary adapted decomposition
5.2 Summary of this chapter
6 Numerical approach for axisymmetric and bending modes stability investigation
6.1 A modified L2-Projection method based on orthogonal decomposition
6.1.1 Description of the method
6.1.2 Implementation of the projection method using symbolic and numeric conversions
6.2 Summary of this chapter
7 Spectral descriptor technique for hydrodynamic stability of swirling flows
7.1 The analytical investigation of the eigenvalue problem
7.2 Numerical approach based on collocation technique
7.2.1 Interpolative derivative operator
7.2.2 Parallel implementation of the spectral collocation algorithm
7.3 Summary of this chapter
8 Validation of the numerical procedures on a Q-vortex problem
8.1 The Q-vortex profile
8.2 Radial boundary adapted method validation and results
8.3 L2 -Projection method validation and results
8.4 Spectral descriptor method validation and results
8.5 Comparative results
9 Parallel and distributed investigation of the vortex rope model
9.1 Considerations about parallel computing
9.2 Theoretical model and computational domain
9.3 Influence of discharge coefficient on hydrodynamic stability
9.3.1 Investigation of axisymmetric mode
9.3.2 Investigation of bending modes
9.4 Study of absolute and convective instability of the swirl system with discrete velocity profiles
9.4.1 Computational aspects
9.4.2 Validations with experimental results
9.5 Accuracy and convergence of the algorithm
9.6 Evaluation of the parallel algorithm performance
9.7 Summary of this chapter
10 Conclusions
10.1 Book summary
10.2 Final remarks
Bibliography and references
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