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Chapter 3 Fluid mechanics conservation principles, PDE notation

  • A. J. Baker and James D. Freels
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Error Freed CFD Mathematics
This chapter is in the book Error Freed CFD Mathematics

Libretto

Compressible, incompressible fluid mechanics continuum conservation principles are expressed by Navier-Stokes (NS) partial differential equation (PDE) systems

NS PDE system alterations to their isentropic (Euler), time-averaged (RaNS), and space-filtered (LES) PDE statements are detailed in vector and tensor notation.

A brief exposure to heuristic “physics-based” closure modeling for RaNS/LES PDE systems documents the significant limitations of current practice formulations.

Libretto

Compressible, incompressible fluid mechanics continuum conservation principles are expressed by Navier-Stokes (NS) partial differential equation (PDE) systems

NS PDE system alterations to their isentropic (Euler), time-averaged (RaNS), and space-filtered (LES) PDE statements are detailed in vector and tensor notation.

A brief exposure to heuristic “physics-based” closure modeling for RaNS/LES PDE systems documents the significant limitations of current practice formulations.

Chapters in this book

  1. Frontmatter I
  2. Dedication V
  3. Contents VII
  4. Prologue 1
  5. Chapter 1 Why this monograph? 3
  6. Chapter 2 A brief pertinent CFD chronology 6
  7. Chapter 3 Fluid mechanics conservation principles, PDE notation 13
  8. Part A: Space-time discretization error annihilation, algebraic instability
  9. Chapter 4 Legacy CFD difference algebra derived stabilizations, O(h2) truncation error annihilation (TEA) theory, NS error freed PDE mathematics, monotonicity, stability 21
  10. Chapter 5 Navier-Stokes error freed CFD mathematics, continuous weak formulation, FE theory, asymptotic convergence, error estimates, validations, monotone continuum shock interpolation 42
  11. Chapter 6 Time-averaged (RaNS) Navier-Stokes error freed CFD weak formulation, theory, asymptotic convergence, validations 73
  12. Chapter 7 Annihilation of NS/RaNS PDE space-time discretization-induced O(m2, m3) phase dispersion and O(Δt3) truncation errors 97
  13. Chapter 8 Hypersonics, aerothermodynamics, radiation CFD issues 109
  14. Part B: Analytical closure of spatially filtered NS PDE systems
  15. Chapter 9 Mathematically rigorous closure of spatially filtered NS PDE systems 125
  16. Chapter 10 FaNS theory O(1;δ2) PDE system rendered bounded domain well-posed 139
  17. Chapter 11 Validation, FaNS, NS PDE systems same code/mesh predictions,100 < Re < 4,000 159
  18. Chapter 12 Validation, ∀ Re pertinent FaNS theory prediction of laminar BL profile O(1) velocity insipient transition, separation, turbulent BL profile reattachment, then relaminarization, Re ≈ 12,000 175
  19. Chapter 13 Epilogue, error freed CFD mathematics continuing impact 201
  20. Nomenclature
  21. Appendix A: The finite element toolbox 207
  22. Appendix B: Theory, weak formulation optimal continuous Galerkin FE p = 1,2,3 basis CFD algorithms 221
  23. Appendix C: Error freed CFD mathematics altered continuous Galerkin FE basis algorithm and Newton jacobian matrix statements 233
  24. Subject index 345
https://doi.org/10.1515/9783112206430-004

Chapters in this book

  1. Frontmatter I
  2. Dedication V
  3. Contents VII
  4. Prologue 1
  5. Chapter 1 Why this monograph? 3
  6. Chapter 2 A brief pertinent CFD chronology 6
  7. Chapter 3 Fluid mechanics conservation principles, PDE notation 13
  8. Part A: Space-time discretization error annihilation, algebraic instability
  9. Chapter 4 Legacy CFD difference algebra derived stabilizations, O(h2) truncation error annihilation (TEA) theory, NS error freed PDE mathematics, monotonicity, stability 21
  10. Chapter 5 Navier-Stokes error freed CFD mathematics, continuous weak formulation, FE theory, asymptotic convergence, error estimates, validations, monotone continuum shock interpolation 42
  11. Chapter 6 Time-averaged (RaNS) Navier-Stokes error freed CFD weak formulation, theory, asymptotic convergence, validations 73
  12. Chapter 7 Annihilation of NS/RaNS PDE space-time discretization-induced O(m2, m3) phase dispersion and O(Δt3) truncation errors 97
  13. Chapter 8 Hypersonics, aerothermodynamics, radiation CFD issues 109
  14. Part B: Analytical closure of spatially filtered NS PDE systems
  15. Chapter 9 Mathematically rigorous closure of spatially filtered NS PDE systems 125
  16. Chapter 10 FaNS theory O(1;δ2) PDE system rendered bounded domain well-posed 139
  17. Chapter 11 Validation, FaNS, NS PDE systems same code/mesh predictions,100 < Re < 4,000 159
  18. Chapter 12 Validation, ∀ Re pertinent FaNS theory prediction of laminar BL profile O(1) velocity insipient transition, separation, turbulent BL profile reattachment, then relaminarization, Re ≈ 12,000 175
  19. Chapter 13 Epilogue, error freed CFD mathematics continuing impact 201
  20. Nomenclature
  21. Appendix A: The finite element toolbox 207
  22. Appendix B: Theory, weak formulation optimal continuous Galerkin FE p = 1,2,3 basis CFD algorithms 221
  23. Appendix C: Error freed CFD mathematics altered continuous Galerkin FE basis algorithm and Newton jacobian matrix statements 233
  24. Subject index 345
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