在旋进流形式下产生的流体动力不稳定性．称为涡流，在Francis水轮机尾水管的壁面上产生高压不稳定波动，导致了包括疲劳损坏在内的比较差的涡轮性能。该现象的建模和这个涡流系统的流体动力学不稳定性的数值研究是本书调研的任务。作为经典方法的替代方法，本书呈现了一种新的涡流流体解析法，该方法基于最近的一种具有正交分解的谱配置的数学方法。本书呈现的数学模型可以恢复水轮机特性预测中的信息，而无须计算三维的不稳定流。本书提供了有价值的工具，可以在流道设计的早期阶段评估非设计运行状态下的水轮机性能。本书提供的数学工具为解决现代科学中出现的跨学科问题提供了可转移的知识，这些知识可以应用于许多不同的学科之中。

戴安娜·爱丽娜·比斯蒂安，罗马尼亚人，在罗马尼亚的蒂米什瓦拉西部大学学习数学和计算机科学。她是理学学士，后获得了蒂米什瓦拉西部大学的计算机和信息技术博士学位。她的研究方向为计算数学和涡流流体动力的数值方法。

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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