Book contents
- Frontmatter
- Brief Contents
- Contents
- Preface
- 1 The Finite Element Method: Introductory Remarks
- 2 Some Methods for Solving Continuum Problems
- 3 Variational Approach
- 4 Requirements for the Interpolation Functions
- 5 Heat Transfer Applications
- 6 One-Dimensional Steady-State Problems
- 7 The Two-Dimensional Heat-Conduction Problem
- 8 Three-Dimensional Heat-Conduction Applications with Convection and Internal Heat Absorption
- 9 One-Dimensional Transient Problems
- 10 Fluid Mechanics Finite Element Applications
- 11 Use of Nodeless Degrees of Freedom
- 12 Finite Element Analysis in Curvilinear Coordinate
- 13 Finite Element Modeling of Flow in Annular Axisymmetric Passages
- 14 Extracting the Finite Element Domain from a Larger Flow System
- 15 Finite Element Application to Unsteady Flow Problems
- 16 Finite Element-Based Perturbation Approach to Unsteady Flow Problems
- Appendix A Natural Coordinates for Three-Dimensional Surface Elements
- Appendix B Classification and Finite Element Formulation of Viscous Flow Problems
- Appendix C Numerical Integration
- Appendix D Finite Element-Based Perturbation Analysis: Formulation of the Zeroth-Order Flow Field
- Appendix E Displaced-Rotor Operation: Perturbation Analysis
- Appendix F Rigorous Adaptation to Compressible-Flow Problems
- Index
- References
15 - Finite Element Application to Unsteady Flow Problems
Published online by Cambridge University Press: 05 June 2014
- Frontmatter
- Brief Contents
- Contents
- Preface
- 1 The Finite Element Method: Introductory Remarks
- 2 Some Methods for Solving Continuum Problems
- 3 Variational Approach
- 4 Requirements for the Interpolation Functions
- 5 Heat Transfer Applications
- 6 One-Dimensional Steady-State Problems
- 7 The Two-Dimensional Heat-Conduction Problem
- 8 Three-Dimensional Heat-Conduction Applications with Convection and Internal Heat Absorption
- 9 One-Dimensional Transient Problems
- 10 Fluid Mechanics Finite Element Applications
- 11 Use of Nodeless Degrees of Freedom
- 12 Finite Element Analysis in Curvilinear Coordinate
- 13 Finite Element Modeling of Flow in Annular Axisymmetric Passages
- 14 Extracting the Finite Element Domain from a Larger Flow System
- 15 Finite Element Application to Unsteady Flow Problems
- 16 Finite Element-Based Perturbation Approach to Unsteady Flow Problems
- Appendix A Natural Coordinates for Three-Dimensional Surface Elements
- Appendix B Classification and Finite Element Formulation of Viscous Flow Problems
- Appendix C Numerical Integration
- Appendix D Finite Element-Based Perturbation Analysis: Formulation of the Zeroth-Order Flow Field
- Appendix E Displaced-Rotor Operation: Perturbation Analysis
- Appendix F Rigorous Adaptation to Compressible-Flow Problems
- Index
- References
Summary
Introduction
In this chapter we combine the flow-field time dependency with a fully three dimensional solution domain. The result is a large-size computational model requiring a great deal of computer resources. In view of how involved the problem is, several CPU time-saving techniques are devised and implemented.
Example
The relative motion between the stator and rotor subdomains within an axial (Figure 15.1) or centrifugal turbomachinery stage creates an unsteady-flow field that is periodic in time. Limiting the discussion to the axial turbine stage case, the statorcascade wake pattern around the circumference (see Figure 15.1) not only will shape the rotor flow behavior but also will expose its blades to a pattern of oscillating pressure that may very well lead to premature fatigue failure. In fact, the close proximity of the two (stator and rotor) cascades (Figure 15.2) in a predominantly subsonic flow field places the stator vanes in the same fluctuating-stress environment, but with lesser amplitude by comparison.
The small stator/rotor axial-gap length within a typical turbomachinery stage is a double-edge sword. On the one hand, the smaller the gap the less is the total pressure loss within it. This loss is a natural outcome of the boundary layer growth over the endwalls, which, together with the profile losses, constitutes a significant part of the stage losses. However, a small gap length magnifies the cyclic fluctuations within the rotor subdomain as a result of wake cutting, upstream vortex shedding, and potential flow interaction between the stationary and rotating blade rows.
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- Publisher: Cambridge University PressPrint publication year: 2013