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
11 - Use of Nodeless Degrees of Freedom
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
Overview
Of the different problem categories in the remainder of this text, this is the simplest and, appropriately, a good starting point. A potential flow field is one where a single field variable suffices and a single flow-governing equation applies. This variable has typically been chosen as either the stream function ψ or the velocity potential ϕ. This apparent simplicity, nevertheless, may (in the larger picture) underestimate the critical role a potential-flow code often plays in a typical cascade-design setting, as well as the inherent analytical difficulty in securing a single-valued flow solution in a multiply-connected domain, with the latter being the focus of this chapter.
Beginning as early as the 1930s, several methods were devised for the problem of potential flow past a cascade of lifting bodies. Some of these methods were based on the use of conformal transformation [1–5], where one or more transformation step(s) are used in mapping the computational domain into a set of ovals or a flatplate cascade [4, 5]. A separate category of analytical solutions [6] is based on the so-called singularity method, whereby sequences of sources and sinks and/or vortices are used to replace the airfoil itself. Next, the streamline-curvature method was established [7] as a viable approach to the airfoil-cascade-flow problem. With advent of the computer revolution came several numerical models of the problem based on the finite-difference method [9], finite element [9–12], and finite-volume [13] computational techniques.
From an analytical viewpoint a flow passage will conceptually suffer one level of multiconnectivity at any point where two streams with two different histories are allowed to mix together.
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- Publisher: Cambridge University PressPrint publication year: 2013