book

Error Freed CFD Mathematics

Name: Error Freed CFD Mathematics
ISBN: 9783112206461

by A. J. Baker, James D. Freels

November 2025

Intermediate

262 pages

7h 46m

English

De Gruyter

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Frontmatter
Contents
Prologue
Chapter 1 Why this monograph?
Chapter 2 A brief pertinent CFD chronology
2.1 Evolution, legacy CFD theorizations2.2 Legacy CFD NS potential flow shock capturing algorithms2.3 Legacy CFD NS aerodynamics algorithms2.4 Evolution of integral calculus expressed CFD algorithms2.5 Taylor series correlation of legacy NS CFD stabilizations
Chapter 3 Fluid mechanics conservation principles, PDE notation
3.1 Textbook appearance of Navier-Stokes (NS) PDE systems, monograph notation3.2 Textbook time averaged Navier-Stokes (RaNS) PDE systems3.3 Textbook spatially filtered Navier-Stokes (LES) PDE systems
Part A: Space-time discretization error annihilation, algebraic instability
Chapter 4 Legacy CFD difference algebra derived stabilizations, O(h2) truncation error annihilation (TEA) theory, NS error freed PDE mathematics, monotonicity, stability4.1 CFD algorithm O(h2) TE instability, Fourier modal analysis4.2 Functional correlation of legacy CFD derived stabilizations4.3 Algebraic instability, O(h2) numerical diffusion infused legacy CFD4.4 Derivation, NS DP O(h2) truncation error continuum nonlinear functional, n = 14.5 Nonlinear fidelity, NS TEA altered O(h4) DmP GWSh FE algorithm, Newton Jacobian4.6 Validation, Burgers TEA O(h4) DmP GWSh FE DOF monotonicity ∀Re ≤ 1064.7 Validation, compressible NS TEA O(h4) PDE monotone shock interpolation, Re ≤ 1054.8 O(h4) error freed CFD mathematics diagnosed physics of fluids anomaliesChapter 5 Navier-Stokes error freed CFD mathematics, continuous weak formulation, FE theory, asymptotic convergence, error estimates, validations, monotone continuum shock interpolation5.1 Introduction, unstagnation options for current practice NS (and RaNS) codes5.2 CFD fundamentals, verification, validation, uncertainty quantification (VVUQ)5.3 Derivation, weak formulation continuous Galerkin NS CFD algorithm5.4 Compressible NS TEA altered O(h4) error freed CFD mathematics PDE system5.5 Weak formulation continuous Galerkin NS TEA O(h4) PDE FE basis algorithms5.6 Weak formulation finite volume NS TEA altered O(h4) PDE algorithm5.7 Theory synopsis, weak formulation error freed CFD, asymptotic convergence, optimal continuous GWSh FE p = 1,2,3 trial space basis algorithms5.8 Validation, shock continuum interpolation, monotonicity, stability, weak formulation compressible NS TEA O(h4) error freed PDE GWSh FE p = 1 basis algorithm5.9 Validation, weak formulation theory, asymptotic convergence, compressible NS TEA-altered O(h4) error freed PDE GWS FE p = 1,2,3 basis algorithms5.10 Validation, weak formulation theory, error quantification, compressible NS TEA altered O(h4) error freed PDE GWSh FE p = 1,2,3 basis algorithms5.11 Validation, weak formulation theory, optimal mesh identification, compressible NS TEA O(h4) error freed PDE GWSh FE p = 1,2,3 basis algorithms5.12 Incompressible NS (INS) error freed TEA altered O(h4) PDE system5.13 Validation, weak formulation theory, INS TEA altered O(h4) PDE GWSh FE p = 1 basis algorithm, driven cavity monotone DOF5.14 Validation, weak formulation theory, asymptotic convergence, INS TEA O(h4) error freed DmE GWSh FE algorithm, thermal stagnation point 5.15 Incompressible NS DM differential constraint, pressure Poisson PDE, DMh algorithm5.16 Validation, INS pressure, TEA O(h4) error freed DmP:O(h2)DMh + NBC GWSh FE p = 1 basis algorithm, n = 3 step wall diffuser 5.17 Validation, pressure, multiply connected domains, INS O(h4) error freed DmP: O(h2)DMh + NBC:DP + NBC GWSh + θTS FE p = 1 basis algorithm5.18 Conclusion, weak formulation Navier-Stokes continuous GWSh + θTS FE p = 1,2,3 basis algorithms cast on error freed CFD mathematicsChapter 6 Time-averaged (RaNS) Navier-Stokes error freed CFD weak formulation, theory, asymptotic convergence, validations6.1 Derivation, compressible RaNS:SA TEA O(h4) DmM:DmP:DmE:DmSA PDE system6.2 Validation, weak formulation continuous RaNS:SA TEA O(h4) error freed PDE GWSh FE p = 1 basis algorithm, shock monotone continuum interpolation, stability6.3 Validation, first-order effects predictions, same code/same mesh RaNS:SA O(h4) PDE GWSh and O(h2) PDE non-GWSh numerically stabilized CFD algorithms6.4 Validation/failure, GWSh/non-GWSh CFD first-, higher-order effects predictions6.5 Validation, shock-turbulent BL interaction CFD diagnostics, error freed PDE prediction of the dominant physics of fluids process6.6 Derivation, incompressible RaNS TEA O(h4) DmP:DmTKE:O(h2)DMh PDE system continuous GWSh FE basis algorithm, Newton Jacobian6.7 Validation, RaNS:TKE O(h4) PDE system GWSh + θTS FE p = 1 basis algorithm6.8 Spectral distribution of TEA annihilated O(h2) error, RaNS:TKE O(h4) PDE system6.9 Conclusion, RaNS error freed CFD mathematics O(h4) PDE systemsChapter 7 Annihilation of NS/RaNS PDE space-time discretization-induced O(m2, m3) phase dispersion and O(Δt3) truncation errors7.1 Characterization, NS scalar convective transport dispersion error mechanisms7.2 Derivation, NS DM scalar transport CFD algorithm amplification factor, n = 17.3 Derivation, NS mass transport DmY O(mp) TE annihilation, p = 2, 3, n = 17.4 Derivation, NS mass transport DmY PDE O(mp) TE annihilation, p = 2, 3, n-D7.5 Validation, space-time discretization aliasing error annihilation, NS O(mp) DmY, p = 2, 37.6 Derivation, trapezoidal rule O(Δt3) TE-annihilated NS/RaNS GWSh + θTS algebraic statementChapter 8 Hypersonics, aerothermodynamics, radiation CFD issues8.1 Issues uniquely challenging to hypersonic aerodynamics CFD8.2 Aerothermodynamics impact on hypersonic BL CFD prediction8.3 Radiation, weak formulation Stefan-Boltzmann RBC, Newton Jacobian8.4 Radiation, differential radiosity theory weak formulation NBC, Newton Jacobian8.5 Differential radiosity weak formulation, GWSh algorithm8.6 Validation, asymptotic convergence, differential radiosity weak formulation GWSh FE p = 1 basis algorithm8.7 Validation, long time stability, weak formulation gray body unsteady NS DE + NBC GWSh + θTS FE p = 1 basis trapezoidal rule algorithm8.8 Conclusion, NS/RaNS PDE system error-freed CFD mathematics
Part B: Analytical closure of spatially filtered NS PDE systems
Chapter 9 Mathematically rigorous closure of spatially filtered NS PDE systems9.1 Legacy “physics-based” closure models for spatially filtered NS (LES) PDE systems9.2 Rigorous analytical closure of spatially filtered NS PDE systems9.3 Resolution of the auxiliary problem, analytical FaNS theory PDE systems9.4 Compilation, RaNS, LES, SA, FaNS theory tensor closures9.5 Derivation, FaNS theory O(δ3) unresolved scale tensor and vectors9.6 Conclusion, analytical FaNS theory O(h4,m4,Δt4,δ4) PDE systemsChapter 10 FaNS theory O(1;δ2) PDE system rendered bounded domain well-posed10.1 Incompressible FaNS theory O(1) DmP‾ completion for pressure10.2 Resolution, FaNS uniform measure δ filter penetration of NS no-slip BC ∂Ω10.3 Derivation, weak formulation BCE integral convolution GWSh algorithm10.4 Characterization, incompressible FaNS theory DmE‾:AD-BCEh GWSh(t) algorithm10.5 Characterization, FaNS theory mesh resolution requirement, turbulent/laminar BL profile O(1) velocity efflux NBC10.6 Validation, FaNS theory O(h4) DmP:AD-BCEh-NBCh GWSh algorithm10.7 Derivation, FaNS theory O(δ2) state-variable nonhomogeneous Dirichlet BC10.8 Commentary, a priori identified FaNS theory O(δ2) state-variable DBCChapter 11 Validation, FaNS, NS PDE systems same code/mesh predictions,100 << Re << 4,00011.1 Design of FaNS theory validation, the n = 3 virtual fluid dynamics laboratory (VFDL)11.2 Validation, NS and FaNS theory same code/mesh predictions, 100 < Re < 400Validation, NS and FaNS theory same code/mesh predictions, 100 < Re < 40011.3 Validation, NS stability theory-predicted VFDL critical Raleigh number RaCR11.4 FaNS filtering annihilation of NS PDE-generated O(≤4 h) multi-scale content, 2,000 < Re < 4,000) multi-scale content, 2,000 < Re < 4,00011.5 Validation, NS/FaNS state variable non-mirror-symmetric VFDL distributions11.6 Legacy CFD unstagnation, FaNS theory O(h4,h2,δ4) PDE system ∀Re < 4,000) PDE system ∀Re < 4,000Chapter 12 Validation, ∀ Re pertinent FaNS theory prediction of laminar BL profile O(1) velocity insipient transition, separation, turbulent BL profile reattachment, then relaminarization, Re ≈ 12,00012.1 Introduction12.2 Global synopsis, FaNS theory prediction of laminar-turbulent transition, Re ≈ 12,00012.3 Validation, ∀ Re FaNS theory O(1; δ²) PDE:AD-BCE-NBC-DBC:O(δ3) SGS algorithm prediction of laminar BL profile O(1) velocity transition to turbulent12.4 Validation, O(δ3) SFS tensor/vector O(h2) TEA functionality12.5 Validation, FaNS state variable O(1;δ2;δ3) significance scaling, TEA non-positive definite SFS tensor/vector, VVUQ pertinence12.6 Validation, ∀ Re FaNS theory O(δ2) resolved-unresolved scale interaction tensor is both realizable and anisotropic12.7 Validation, AD algorithm non-homogeneous DBC for FaNS O(δ2) state variable resolution PDE system12.8 Validation, ∀ Re FaNS theory O(1;δ2;δ3) scaled state variable, VVUQ pertinence12.9 Extraction, NS velocity vector from FaNS spatially filtered O(1) velocity vector12.10 Summary, ∀ Re FaNS theory optimal O(h4,m4,δ4,Δt4) CFD algorithm/codeChapter 13 Epilogue, error freed CFD mathematics continuing impact
Nomenclature
Appendix A: The finite element toolboxA.1 Finite element trial space basisA.2 GWSh elementlevel matrix assemblyA.3 Finite element p = 1, 2, 3 completeness degree trial space basesA.4 Finite element NC basis coordinate transformationA.5 Finite element TP basis coordinate transformationA.6 Nonlinearity, FE p-basis hypermatrix identificationsAppendix B: Theory, weak formulation optimal continuous Galerkin FE p = 1,2,3 basis CFD algorithmsB.1 Derivation, mathematically rigorous weak formulation theoryB.2 Validation, NS PDE GWSh asymptotic convergence, solution optimalityB.3 Error quantification, non-smooth data, asymptotic convergence failureB.4 Theory alterations for error freed CFD mathematics, NS/RaNS O(h4) PDE GWSh FE p = 1,2,3 algorithms, asymptotic convergence, error quantificationReferencesAppendix C: Error freed CFD mathematics altered continuous Galerkin FE basis algorithm and Newton jacobian matrix statementsC.1 NS/RaNS O(h4,m4,Δt4) PDE GWSh + θTS FE p basis algorithm error freed CFD mathematics altered global matrix statementsC.2 Continuous GWShenabled Newton Jacobian nonlinear identificationsC.3 Fidelity, GWSh calculus statements translation to algebraic, Newton JacobianReferences
Subject index

Index

Content preview from Error Freed CFD Mathematics

Part A: Space-time discretization error annihilation, algebraic instability

Chapter 4 Legacy CFD difference algebra derived stabilizations, O(h²) truncation error annihilation (TEA) theory, NS error freed PDE mathematics, monotonicity, stability

Libretto

CFD algorithms cast on the textbook forms of Navier-Stokes (NS) partial differential equation (PDE) systems are invariably (!) algebraically unstable.

The legacy CFD remedy is insertion of a variety of finite difference derived numerical diffusion/stabilization schemes determined optimal in some sense.

Fourier modal analysis quantifies the root cause of algebraic instability is the spatial discretization generated dispersive O(h²) truncation error (TE), h the measure of the mesh.

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Error Freed CFD Mathematics

by A. J. Baker, James D. Freels

Part A: Space-time discretization error annihilation, algebraic instability