Multicomponent Flow Modeling

1. Introduction.- 2. Fundamental Equations.- 2.1. Introduction.- 2.2. Conservation equations.- 2.2.1. Species, momentum, and energy.- 2.2.2. Total mass conservation.- 2.2.3. Species mass fractions.- 2.2.4. Kinetic and internal energy.- 2.2.5. Species independent specific forces.- 2.3. Thermodynamics.- 2.3.1. Density and internal energy.- 2.3.2. Enthalpy.- 2.3.3. Mole fractions and molar concentrations.- 2.3.4. Enthalpy and temperature equations.- 2.3.5. Entropy and Gibbs function.- 2.3.6. Alternative formulations.- 2.3.7. Thermodynamic data.- 2.4. Chemistry.- 2.4.1. Elementary reactions.- 2.4.2. Maxwellian production rates.- 2.4.3. Total mass conservation.- 2.4.4. Notation for three-body reactions.- 2.4.5. Chemistry data.- 2.5. Transport fluxes.- 2.5.1. Viscous tensor, species mass fluxes, and heat flux.- 2.5.2. Diffusion velocities and diffusion matrix.- 2.5.3. Alternative formulations.- 2.5.4. Transport coefficients.- 2.6. Entropy.- 2.6.1. Entropy differential.- 2.6.2. Entropy equation.- 2.6.3. Entropy production.- 2.7. Boundary conditions.- 2.7.1. Dirichlet and Neumann boundary conditions.- 2.7.2. Porous walls.- 2.7.3. Catalytic plates.- 2.8. Notes.- 2.9. References.- 3. Approximate and Simplified Models.- 3.1. Introduction.- 3.2. One-reaction chemistry.- 3.2.1. One-reaction kinetics.- 3.2.2. Approximations for one-reaction kinetics.- 3.2.3. Simplified equations.- 3.2.4. Deficient reactants.- 3.3. Small Mach number flows.- 3.3.1. Orders of magnitude.- 3.3.2. Momentum equation and pressure splitting.- 3.3.3. Energy equation.- 3.3.4. Isobaric equations.- 3.3.5. Vorticity-velocity formulation.- 3.3.6. Strained flows.- 3.3.7. Shvab-Zeldovitch formulation.- 3.4. Coupling.- 3.4.1. Coupling of partial differential equations.- 3.4.2. Dilution approximation.- 3.4.3. Constant density approximation.- 3.5. Notes.- 3.6. References.- 4. Derivation from the Kinetic Theory.- 4.1. Introduction.- 4.2. Kinetic framework.- 4.2.1. Distribution functions.- 4.2.2. Macroscopic properties.- 4.2.3. Boltzmann equations.- 4.2.4. Scattering source terms.- 4.2.5. Reactive source terms.- 4.2.6. Examples.- 4.3. Kinetic entropy.- 4.3.1. Definition of the kinetic entropy.- 4.3.2. Kinetic entropy equation.- 4.3.3. Positivity of entropy production.- 4.4. Enskog expansion.- 4.4.1. Asymptotic orders.- 4.4.2. Collisional invariants of the fast operator.- 4.4.3. Macroscopic equations.- 4.5. Zero-order approximation.- 4.5.1. Maxwellian distributions.- 4.5.2. Zero-order macroscopic equations.- 4.5.3. Zero-order time derivatives.- 4.6. First-order approximation.- 4.6.1. Linearized Boltzmann operator.- 4.6.2. Linearized Boltzmann equations.- 4.6.3. Expansion of perturbed distributions.- 4.6.4. Macroscopic equations and transport fluxes.- 4.6.5. Transport coefficients.- 4.6.6. Chemistry source terms.- 4.6.7. Thermodynamics.- 4.7. Transport linear systems.- 4.7.1. Galerkin method.- 4.7.2. Basis functions.- 4.7.3. Structure of transport linear systems.- 4.7.4. Sparse transport matrix.- 4.7.5. Vanishing mass fractions.- 4.8. Notes.- 4.9. References.- 5. Transport Coefficients.- 5.1. Introduction.- 5.2. Transport algorithms.- 5.2.1. Transport linear systems.- 5.2.2. Mathematical structure.- 5.2.3. Direct inversion.- 5.2.4. Iterative methods.- 5.2.5. Empirical expressions.- 5.2.6. Operational count.- 5.2.7. Stability for vanishing mass fractions.- 5.3. Molecular parameters.- 5.3.1. Interaction potentials.- 5.3.2. Collision integrals.- 5.3.3. Viscosity of pure gases and binary diffusion.- 5.3.4. Relaxation and diffusion of internal energy.- 5.3.5. Transport data.- 5.4. Shear viscosity.- 5.5. Volume viscosity.- 5.6. Diffusion matrix.- 5.7. Thermal conductivity.- 5.8. Thermal diffusion ratios.- 5.9. Partial thermal conductivity.- 5.10. Thermal diffusion coefficients.- 5.11. Notes.- 5.12. References.- 6. Mathematics of Thermochemistry.- 6.1. Introduction.- 6.2. Thermodynamics with volume densities.- 6.2.1. State variables (T, ?1, …, ?n).- 6.2.2. Energy and enthalpy per unit volume.- 6.2.3. Entropy and Gibbs function per unit volume.- 6.2.4. Assumptions.- 6.2.5. Differentials and convexity.- 6.3. Thermodynamics with mass densities.- 6.3.1. State variables (T, p, Y1, …, Yn).- 6.3.2. Energy and enthalpy per unit mass.- 6.3.3. Entropy and Gibbs function per unit mass.- 6.3.4. Assumptions.- 6.3.5. Differentials and convexity.- 6.3.6. Miscellaneous.- 6.4. Chemistry sources.- 6.4.1. Chemical reactions.- 6.4.2. Maxwellian production rates.- 6.4.3. Assumptions.- 6.4.4. Mass weights.- 6.4.5. Mass conservation.- 6.4.6. Creation and destruction rates.- 6.4.7. Symmetric formulation for the rates of progress.- 6.5. Positive equilibrium points.- 6.5.1. Definition of equilibrium points.- 6.5.2. Equilibrium points with T and ? fixed.- 6.5.3. Equilibrium points with h and ? fixed.- 6.5.4. Smoothness of equilibrium points.- 6.6. Boundary equilibrium points.- 6.6.1. Definition of boundary equilibrium points.- 6.6.2. Decomposition chain property.- 6.7. Inequalities near equilibrium.- 6.7.1. Production rates and chemical dissipation.- 6.7.2. Entropy difference and chemical dissipation.- 6.8. A global stability inequality.- 6.9. Notes.- 6.10. References.- 7. Mathematics of Transport Coefficients.- 7.1. Introduction.- 7.1.1. Definition of transport fluxes.- 7.1.2. Diffusion velocities.- 7.1.3. Alternative formulations.- 7.2. Assumptions on transport coefficients.- 7.3. Properties of diffusion matrices.- 7.3.1. First properties of the diffusion matrix D.- 7.3.2. First properties of the flux diffusion matrix C.- 7.3.3. Flux splitting.- 7.3.4. Generalized inverses of C and D.- 7.3.5. Modified diffusion coefficients.- 7.4. Properties of other coefficients.- 7.4.1. Alternative coefficients.- 7.4.2. Waldmann coefficients.- 7.5. Diagonal diffusion.- 7.5.1. Irreducibility of C and D.- 7.5.2. Matrix E and mass fraction gradients.- 7.5.3. Irreducibility of CE and DE.- 7.5.4. Diagonal diffusion of C and D over U?.- 7.5.5. Diagonal diffusion of CE and DE over U?.- 7.5.6. Diagonal diffusion of C and D for n — 1 species.- 7.5.7. Diagonal diffusion of CE and DE for n — 1 species.- 7.6. Diffusion inequalities.- 7.6.1. Fundamental diffusion inequality.- 7.6.2. Positivity properties of C.- 7.7. Stefan-Maxwell equations.- 7.7.1. Matrices ? and D.- 7.7.2. Matrices ? and C.- 7.7.3. Diagonal first-order diffusion.- 7.7.4. Asymptotic expansions of D and C.- 7.8. Notes.- 7.9. References.- 8. Symmetrization.- 8.1. Introduction.- 8.2. Vector notation.- 8.2.1. Conservative and natural variables.- 8.2.2. Vector equations.- 8.3. Quasilinear form.- 8.3.1. The map Z ? U.- 8.3.2. Dissipation matrices and quasilinear form.- 8.4. Symmetrization and entropic variables.- 8.4.1. Symmetric conservative forms.- 8.4.2. Entropic variables.- 8.4.3. The equivalence theorem.- 8.5. Normal forms.- 8.5.1. Definition of normal forms.- 8.5.2. Nullspace invariance property.- 8.5.3. Description of normal variables.- 8.6. Symmetrization for multicomponent flows.- 8.6.1. Entropy and symmetric conservative form.- 8.7. Normal forms for multicomponent flows.- 8.7.1. Nullspace of dissipation matrices.- 8.7.2. First normal form.- 8.7.3. Natural normal form.- 8.7.4. Intermediate normal form.- 8.8. Notes.- 8.9. References.- 9. Asymptotic Stability.- 9.1. Introduction.- 9.2. Governing equations.- 9.2.1. Abstract system.- 9.2.2. Equilibrium points.- 9.2.3. Entropy equation.- 9.2.4. Functional spaces.- 9.3. Local dissipative structure.- 9.3.1. Linearized equations.- 9.3.2. Locally stable source terms.- 9.3.2. Global dissipative structure.- 9.4. Global existence theorem.- 9.4.1. Main result.- 9.4.2. Local existence.- 9.4.3. A priori estimates.- 9.4.4. More a priori estimates.- 9.4.5. Global existence proof.- 9.5. Decay estimates.- 9.6. Local dissipativity for multicomponent flows.- 9.6.1. Chemical sources.- 9.6.2. Local dissipative structure.- 9.6.3. Linearized source term.- 9.7. Global existence for multicomponent flows.- 9.7.1. Linearized normal form.- 9.7.2. Global existence and asymptotic stability.- 9.8. Notes.- 9.9. References.- 10. Chemical Equilibrium Flows.- 10.1. Introduction.- 10.2. Governing equations.- 10.2.1. Notation associated with equilibrium.- 10.2.2. Atomic species and formation reactions.- 10.2.3. Equations at chemical equilibrium.- 10.2.4. Conservative and natural variables.- 10.2.5. Fluxes at chemical equilibrium.- 10.2.6. Quasilinear form at chemical equilibrium.- 10.3. Entropy and symmetrization.- 10.3.1. Entropy at chemical equilibrium.- 10.3.2. Symmetrized equations.- 10.4. Normal forms.- 10.4.1. Nullspace invariance property.- 10.4.2. Intermediate normal form.- 10.5. Global existence.- 10.5.1. Local dissipativity.- 10.5.2. Global existence.- 10.6. Notes.- 10.7. References.- 11. Anchored Waves.- 11.1. Introduction.- 11.2. Governing equations.- 11.2.1. Conservation equations.- 11.2.2. Thermodynamic properties.- 11.2.3. Maxwellian chemistry.- 11.2.4. Transport fluxes.- 11.2.5. The temperature equation.- 11.2.6. Boundary conditions.- 11.2.7. Equilibrium limit.- 11.2.8. The matrix L.- 11.2.9. Entropy conservation equation.- 11.3. First properties.- 11.3.1. Preliminaries.- 11.3.2. Reduction to a problem on [0, ?).- 11.3.3. Extension to (—?, 0).- 11.4. Existence on a bounded domain.- 11.4.1. Preliminaries.- 11.4.2. Fixed point formulation.- 11.4.3. Existence of the degree.- 11.4.4. Calculation of the degree.- 11.5. Existence of solutions.- 11.5.1. Uniform estimates for c.- 11.5.2. Convergence towards equilibrium.- 11.5.3. Passage to the limit a ? ?.- 11.6. Notes.- 11.7. References.- 12. Numerical Simulations.- 12.1. Introduction.- 12.2. Laminar flame model.- 12.2.1. Governing equations.- 12.2.2. Boundary conditions.- 12.2.3. Chemical mechanism.- 12.3. Computational considerations.- 12.3.1. Discretized equations.- 12.3.2. Multiple time scales.- 12.3.3. Multiple space scales.- 12.3.4. Nonlinear solvers.- 12.3.5. Pseudo-unsteady iterations.- 12.3.6. Thermochemistry and transport software.- 12.4. Hydrogen-Air Bunsen flame.- 12.4.1. Burner geometry.- 12.4.2. Numerical results.- 12.5. References.