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
Appendix F - Rigorous Adaptation to Compressible-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
This model upgrade is intended to embrace a variety of high-speed noncontact seals and bearings traditionally used in gas turbine applications. In this case, the heatenergy exchange between the hardware and the working medium may be far from being negligible. This very fact calls for including a separate energy equation in the set of flow-governing equations cited in Appendix E. The boundary conditions on insertion of this equation involve such variables as the local convection heat transfer coefficient and the local “wall” and flow-stream temperatures.
Given the fact that the rotor-to-housing clearance width is extremely small, the problem of friction choking (in a Fanno-process type of mechanism) is indeed part of this compressible flow problem. Unfortunately, the occurence of this choking type requires external intervention during the flow solution process. During the iterative procedure, where the momentum-equations convection terms are continually linearized, and once the nodal magnitudes of velocity vector are attained, the intervention process begins by computing the corresponding nodal magnitudes of Mach number. These are then examined to see if the Mach number is in excess of unity anywhere in the computational domain (the seal exit station in particular), which is impossible in a subsonic nozzle-like passage. Referring to the simple annular seal in Figure 16.20, the term nozzle here is applicable in the sense that the blockage effect of the boundary layer growth over the solid walls causes, in effect, a streamwise reduction in the cross-flow area, turning what is physically a constant-area passage into a subsonic nozzle in the sense of rising displacement thickness (a fraction of the boundary layer thickness that depends on the boundary-layer velocity profile).
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- Publisher: Cambridge University PressPrint publication year: 2013