poinsot and veynante - 2005 - theoretical and numerical combustion.pdf

Contents

Preface xi

1 Conservation equations for reacting flows 11.1 General forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Choice of primitive variables . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Conservation of momentum . . . . . . . . . . . . . . . . . . . . . . . . . 131.1.3 Conservation of mass and species . . . . . . . . . . . . . . . . . . . . . . 131.1.4 Diffusion velocities: full equations and approximations . . . . . . . . . . 141.1.5 Conservation of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2 Usual simplified forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.2.1 Constant pressure flames . . . . . . . . . . . . . . . . . . . . . . . . . . 221.2.2 Equal heat capacities for all species . . . . . . . . . . . . . . . . . . . . 231.2.3 Constant heat capacity for the mixture only . . . . . . . . . . . . . . . . 24

1.3 Summary of conservation equations . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Laminar premixed flames 272.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Conservation equations and numerical solutions . . . . . . . . . . . . . . . . . . 282.3 Steady one-dimensional laminar premixed flames . . . . . . . . . . . . . . . . . 30

2.3.1 One-dimensional flame codes . . . . . . . . . . . . . . . . . . . . . . . . 302.3.2 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4 Theoretical solutions for laminar premixed flames . . . . . . . . . . . . . . . . . 352.4.1 Derivation of one-step chemistry conservation equations . . . . . . . . . 352.4.2 Thermochemistry and chemical rates . . . . . . . . . . . . . . . . . . . . 372.4.3 The equivalence of temperature and fuel mass fraction . . . . . . . . . 402.4.4 The reaction rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.4.5 Analytical solutions for flame speed . . . . . . . . . . . . . . . . . . . . 442.4.6 Generalized expression for flame speeds . . . . . . . . . . . . . . . . . . 512.4.7 Single step chemistry limitations and stiffness of reduced

schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.4.8 Variations of flame speed with temperature and pressure . . . . . . . . . 55

iii

iv CONTENTS

2.5 Premixed flame thicknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.5.1 Simple chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.5.2 Complex chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.6 Flame stretch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.6.1 Definition and expressions of stretch . . . . . . . . . . . . . . . . . . . . 592.6.2 Stretch of stationary flames . . . . . . . . . . . . . . . . . . . . . . . . . 622.6.3 Examples of flames with zero stretch . . . . . . . . . . . . . . . . . . . . 622.6.4 Examples of stretched flames . . . . . . . . . . . . . . . . . . . . . . . . 63

2.7 Flame speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662.7.1 Flame speed definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 662.7.2 Flame speeds of laminar planar unstretched flames . . . . . . . . . . . . 682.7.3 Flame speeds of stretched flames . . . . . . . . . . . . . . . . . . . . . . 70

2.8 Instabilities of laminar flame fronts . . . . . . . . . . . . . . . . . . . . . . . . . 79

3 Laminar diffusion flames 813.1 Diffusion flame configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.2 Theoretical tools for diffusion flames . . . . . . . . . . . . . . . . . . . . . . . . 83

3.2.1 Passive scalars and mixture fraction . . . . . . . . . . . . . . . . . . . . 843.2.2 Flame structure in the z-space . . . . . . . . . . . . . . . . . . . . . . . 863.2.3 The steady flamelet assumption . . . . . . . . . . . . . . . . . . . . . . . 883.2.4 Decomposition into mixing and flame structure problems . . . . . . . . 893.2.5 Models for diffusion flame structures . . . . . . . . . . . . . . . . . . . . 89

3.3 Flame structure for irreversible infinitely fast chemistry . . . . . . . . . . . . . 933.3.1 The Burke-Schumann flame structure . . . . . . . . . . . . . . . . . . . 933.3.2 Maximum local flame temperature in a diffusion flame . . . . . . . . . . 953.3.3 Maximum flame temperature in diffusion and premixed flames . . . . . 963.3.4 Maximum and mean temperatures in diffusion burners . . . . . . . . . . 96

3.4 Full solutions for irreversible fast chemistry flames . . . . . . . . . . . . . . . . 993.4.1 Unsteady unstrained one-dimensional diffusion flame with infinitely fast

chemistry and constant density . . . . . . . . . . . . . . . . . . . . . . . 993.4.2 Steady strained one-dimensional diffusion flame with infinitely fast chem-

istry and constant density . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.4.3 Unsteady strained one-dimensional diffusion flame with infinitely fast

chemistry and constant density . . . . . . . . . . . . . . . . . . . . . . . 1063.4.4 Jet flame in an uniform flow field . . . . . . . . . . . . . . . . . . . . . . 1093.4.5 Extensions to variable density . . . . . . . . . . . . . . . . . . . . . . . 111

3.5 Extensions of theory to other flame structures . . . . . . . . . . . . . . . . . . . 1123.5.1 Reversible equilibrium chemistry . . . . . . . . . . . . . . . . . . . . . . 1123.5.2 Finite rate chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.5.3 Summary of flame structures . . . . . . . . . . . . . . . . . . . . . . . . 1163.5.4 Extensions to variable Lewis numbers . . . . . . . . . . . . . . . . . . . 116

3.6 Real laminar diffusion flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

CONTENTS v

3.6.1 One-dimensional codes for laminar diffusion flames . . . . . . . . . . . . 1183.6.2 Mixture fractions in real flames . . . . . . . . . . . . . . . . . . . . . . . 118

4 Introduction to turbulent combustion 1254.1 Interaction between flames and turbulence . . . . . . . . . . . . . . . . . . . . . 1254.2 Elementary descriptions of turbulence . . . . . . . . . . . . . . . . . . . . . . . 1264.3 Influence of turbulence on combustion . . . . . . . . . . . . . . . . . . . . . . . 130

4.3.1 One-dimensional turbulent premixed flame . . . . . . . . . . . . . . . . 1304.3.2 Turbulent jet diffusion flame . . . . . . . . . . . . . . . . . . . . . . . . 131

4.4 Computational approaches for turbulent combustion . . . . . . . . . . . . . . . 1324.5 RANS simulations for turbulent combustion . . . . . . . . . . . . . . . . . . . . 140

4.5.1 Averaging the balance equations . . . . . . . . . . . . . . . . . . . . . . 1404.5.2 Unclosed terms in Favre averaged balance equations . . . . . . . . . . . 1424.5.3 Classical turbulence models for the Reynolds stresses . . . . . . . . . . . 1434.5.4 A first attempt to close mean reaction rates . . . . . . . . . . . . . . . . 1464.5.5 Physical approaches to model turbulent combustion . . . . . . . . . . . 1484.5.6 A challenge for turbulent combustion modeling: flame flapping and in-

termittency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1494.6 Direct numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

4.6.1 The role of DNS in turbulent combustion studies . . . . . . . . . . . . . 1534.6.2 Numerical methods for direct simulation . . . . . . . . . . . . . . . . . . 1534.6.3 Spatial resolution and physical scales . . . . . . . . . . . . . . . . . . . . 159

4.7 Large eddy simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1624.7.1 LES filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1624.7.2 Filtered balance equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1644.7.3 Unresolved fluxes modeling . . . . . . . . . . . . . . . . . . . . . . . . . 1654.7.4 Simple filtered reaction rate closures . . . . . . . . . . . . . . . . . . . . 1694.7.5 Dynamic modeling in turbulent combustion . . . . . . . . . . . . . . . . 1714.7.6 Limits of large eddy simulations . . . . . . . . . . . . . . . . . . . . . . 1724.7.7 Comparing large eddy simulations and experimental data . . . . . . . . 173

4.8 Chemistry for turbulent combustion . . . . . . . . . . . . . . . . . . . . . . . . 1774.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1774.8.2 Global schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1774.8.3 Automatic reduction - Tabulated chemistries . . . . . . . . . . . . . . . 1804.8.4 In situ adaptive tabulation (ISAT) . . . . . . . . . . . . . . . . . . . . . 181

5 Turbulent premixed flames 1835.1 Phenomenological description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

5.1.1 The effect of turbulence on flame fronts: wrinkling . . . . . . . . . . . . 1835.1.2 The effect of flame fronts on turbulence . . . . . . . . . . . . . . . . . . 1865.1.3 The infinitely thin flame front limit . . . . . . . . . . . . . . . . . . . . . 189

5.2 Premixed turbulent combustion regimes . . . . . . . . . . . . . . . . . . . . . . 196

vi CONTENTS

5.2.1 A first difficulty: defining u . . . . . . . . . . . . . . . . . . . . . . . . 1965.2.2 Classical turbulent premixed combustion diagrams . . . . . . . . . . . . 1975.2.3 Modified combustion diagrams . . . . . . . . . . . . . . . . . . . . . . . 200

5.3 RANS of turbulent premixed flames . . . . . . . . . . . . . . . . . . . . . . . . 2145.3.1 Premixed turbulent combustion with single one-step chemistry . . . . . 2145.3.2 The no-model or Arrhenius approach . . . . . . . . . . . . . . . . . . 2165.3.3 The Eddy Break Up (EBU) model . . . . . . . . . . . . . . . . . . . . . 2165.3.4 Models based on turbulent flame speed correlations . . . . . . . . . . . . 2185.3.5 The Bray Moss Libby (BML) model . . . . . . . . . . . . . . . . . . . . 2195.3.6 Flame surface density models . . . . . . . . . . . . . . . . . . . . . . . . 2245.3.7 Probability density function (pdf) models . . . . . . . . . . . . . . . . . 2335.3.8 Modeling of turbulent scalar transport terms ui . . . . . . . . . . . 2395.3.9 Modeling of the characteristic turbulent flame time . . . . . . . . . . . . 2435.3.10 Kolmogorov-Petrovski-Piskunov (KPP) analysis . . . . . . . . . . . . . 2465.3.11 Flame stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

5.4 LES of turbulent premixed flames . . . . . . . . . . . . . . . . . . . . . . . . . . 2525.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2525.4.2 Extension of RANS models: the LES-EBU model . . . . . . . . . . . . . 2535.4.3 Artificially thickened flames . . . . . . . . . . . . . . . . . . . . . . . . . 2535.4.4 G-equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2555.4.5 Flame surface density LES formulations . . . . . . . . . . . . . . . . . . 2575.4.6 Scalar fluxes modeling in LES . . . . . . . . . . . . . . . . . . . . . . . . 258

5.5 DNS of turbulent premixed flames . . . . . . . . . . . . . . . . . . . . . . . . . 2625.5.1 The role of DNS in turbulent combustion studies . . . . . . . . . . . . . 2625.5.2 DNS database analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2635.5.3 Studies of local flame structures using DNS . . . . . . . . . . . . . . . . 2675.5.4 Complex chemistry simulations . . . . . . . . . . . . . . . . . . . . . . . 2735.5.5 Studying the global structure of turbulent flames with DNS . . . . . . . 2765.5.6 DNS analysis for large eddy simulations . . . . . . . . . . . . . . . . . . 285

6 Turbulent non-premixed flames 2876.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2876.2 Phenomenological description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

6.2.1 Typical flame structure: jet flame . . . . . . . . . . . . . . . . . . . . . . 2886.2.2 Specific features of turbulent non-premixed flames . . . . . . . . . . . . 2886.2.3 Turbulent non-premixed flame stabilization . . . . . . . . . . . . . . . . 2896.2.4 An example of turbulent non-premixed flame stabilization . . . . . . . . 297

6.3 Turbulent non-premixed combustion regimes . . . . . . . . . . . . . . . . . . . 3006.3.1 Flame/vortex interactions in DNS . . . . . . . . . . . . . . . . . . . . . 3016.3.2 Scales in turbulent non-premixed combustion . . . . . . . . . . . . . . . 3066.3.3 Combustion regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

6.4 RANS of turbulent non-premixed flames . . . . . . . . . . . . . . . . . . . . . . 311

CONTENTS vii

6.4.1 Assumptions and averaged equations . . . . . . . . . . . . . . . . . . . . 3116.4.2 Models for primitive variables with infinitely fast chemistry . . . . . . . 3146.4.3 Mixture fraction variance and scalar dissipation rate . . . . . . . . . . . 3176.4.4 Models for mean reaction rate with infinitely fast chemistry . . . . . . . 3206.4.5 Models for primitive variables with finite rate chemistry . . . . . . . . . 3216.4.6 Models for mean reaction rate with finite rate chemistry . . . . . . . . . 327

6.5 LES of turbulent non-premixed flames . . . . . . . . . . . . . . . . . . . . . . . 3336.5.1 Linear Eddy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3336.5.2 Infinitely fast chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . 3346.5.3 Finite rate chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

6.6 DNS of turbulent non-premixed flames . . . . . . . . . . . . . . . . . . . . . . . 3376.6.1 Studies of local flame structure . . . . . . . . . . . . . . . . . . . . . . . 3376.6.2 Autoignition of a turbulent non-premixed flame . . . . . . . . . . . . . . 3416.6.3 Studies of global flame structure . . . . . . . . . . . . . . . . . . . . . . 3436.6.4 Three-dimensional turbulent hydrogen jet lifted flame with complex

chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

7 Flame/wall interactions 3497.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3497.2 Flamewall interaction in laminar flows . . . . . . . . . . . . . . . . . . . . . . 352

7.2.1 Phenomenological description . . . . . . . . . . . . . . . . . . . . . . . . 3527.2.2 Simple chemistry flame/wall interaction . . . . . . . . . . . . . . . . . . 3557.2.3 Computing complex chemistry flame/wall interaction . . . . . . . . . . . 356

7.3 Flame/wall interaction in turbulent flows . . . . . . . . . . . . . . . . . . . . . 3597.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3597.3.2 DNS of turbulent flame/wall interaction . . . . . . . . . . . . . . . . . . 3607.3.3 Flame/wall interaction and turbulent combustion models . . . . . . . . 3647.3.4 Flame/wall interaction and wall heat transfer models . . . . . . . . . . . 365

8 Flame/acoustics interactions 3758.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3758.2 Acoustics for non-reacting flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

8.2.1 Fundamental equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 3768.2.2 Plane waves in one dimension . . . . . . . . . . . . . . . . . . . . . . . . 3788.2.3 Harmonic waves and guided waves . . . . . . . . . . . . . . . . . . . . . 3808.2.4 Longitudinal modes in constant cross section ducts . . . . . . . . . . . . 3828.2.5 Longitudinal modes in variable cross section ducts . . . . . . . . . . . . 3838.2.6 Longitudinal/transverse modes in rectangular ducts . . . . . . . . . . . 3848.2.7 Longitudinal modes in a series of constant cross section ducts . . . . . . 3878.2.8 The double duct and the Helmholtz resonator . . . . . . . . . . . . . . . 3908.2.9 Multidimensional acoustic modes in cavities . . . . . . . . . . . . . . . . 3938.2.10 Acoustic energy density and flux . . . . . . . . . . . . . . . . . . . . . . 395

viii CONTENTS

8.3 Acoustics for reacting flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3988.3.1 An equation for ln(P) in reacting flows . . . . . . . . . . . . . . . . . . 3988.3.2 A wave equation in low Mach-number reacting flows . . . . . . . . . . . 3998.3.3 Acoustic velocity and pressure in low-speed reacting flows . . . . . . . . 4008.3.4 Acoustic jump conditions for thin flames . . . . . . . . . . . . . . . . . . 4018.3.5 Longitudinal modes in a series of ducts with combustion . . . . . . . . . 4038.3.6 Three-dimensional Helmholtz tools . . . . . . . . . . . . . . . . . . . . . 4048.3.7 The acoustic energy balance in reacting flows . . . . . . . . . . . . . . . 4078.3.8 About energies in reacting flows . . . . . . . . . . . . . . . . . . . . . . 410

8.4 Combustion instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4138.4.1 Stable versus unstable combustion . . . . . . . . . . . . . . . . . . . . . 4148.4.2 Interaction of longitudinal waves and thin flames . . . . . . . . . . . . . 4158.4.3 The (n ) formulation for flame transfer function . . . . . . . . . . . . 4168.4.4 Complete solution in a simplified case . . . . . . . . . . . . . . . . . . . 4178.4.5 Vortices in combustion instabilities . . . . . . . . . . . . . . . . . . . . . 421

8.5 Large eddy simulations of combustion instabilities . . . . . . . . . . . . . . . . 4248.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4268.5.2 LES strategies to study combustion instabilities . . . . . . . . . . . . . . 428

9 Boundary conditions 4319.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4319.2 Classification of compressible Navier-Stokes equations formulations . . . . . . . 4339.3 Description of characteristic boundary conditions . . . . . . . . . . . . . . . . 434

9.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4349.3.2 Reacting Navier-Stokes equations near a boundary . . . . . . . . . . . . 4369.3.3 The Local One Dimensional Inviscid (LODI) relations . . . . . . . . . . 4409.3.4 The NSCBC strategy for the Euler equations . . . . . . . . . . . . . . . 4429.3.5 The NSCBC strategy for Navier-Stokes equations . . . . . . . . . . . . . 4449.3.6 Edges and corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

9.4 Examples of implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4479.4.1 A subsonic inflow with fixed velocities and temperature (SI-1) . . . . . . 4489.4.2 A subsonic non-reflecting inflow (SI-4) . . . . . . . . . . . . . . . . . . . 4499.4.3 Subsonic non-reflecting outflows (B2 and B3) . . . . . . . . . . . . . . . 4499.4.4 A subsonic reflecting outflow (B4) . . . . . . . . . . . . . . . . . . . . . 4519.4.5 An isothermal no-slip wall (NSW) . . . . . . . . . . . . . . . . . . . . . 4519.4.6 An adiabatic slip wall (ASW) . . . . . . . . . . . . . . . . . . . . . . . . 452

9.5 Applications to steady non-reacting flows . . . . . . . . . . . . . . . . . . . . . 4529.6 Applications to steady reacting flows . . . . . . . . . . . . . . . . . . . . . . . . 4569.7 Unsteady flows and numerical waves control . . . . . . . . . . . . . . . . . . . . 459

9.7.1 Physical and numerical waves . . . . . . . . . . . . . . . . . . . . . . . . 4599.7.2 Vortex/boundary interactions . . . . . . . . . . . . . . . . . . . . . . . . 463

9.8 Applications to low Reynolds number flows . . . . . . . . . . . . . . . . . . . . 466

CONTENTS ix

10 Examples of LES applications 47310.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47310.2 Case 1: small scale gas turbine burner . . . . . . . . . . . . . . . . . . . . . . . 474

10.2.1 Configuration and boundary conditions . . . . . . . . . . . . . . . . . . 47410.2.2 Non reacting flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47510.2.3 Stable reacting flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

10.3 Case 2: large-scale gas turbine burner . . . . . . . . . . . . . . . . . . . . . . . 48410.3.1 Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48410.3.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48410.3.3 Comparison of cold and hot flow structures . . . . . . . . . . . . . . . . 48510.3.4 A low-frequency forced mode . . . . . . . . . . . . . . . . . . . . . . . . 48610.3.5 A high-frequency self-excited mode . . . . . . . . . . . . . . . . . . . . . 489

10.4 Case 3: self-excited laboratory-scale burner . . . . . . . . . . . . . . . . . . . . 49110.4.1 Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49110.4.2 Stable flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49210.4.3 Controlling oscillations through boundary conditions . . . . . . . . . . . 492

References 497

Index 519

x CONTENTS

Preface

Since its initial publication in 2001, many readers have commented that Theoretical and Nu-merical Combustion is useful in that it provides a coherent summary of research in a fieldwhere advances occur rapidly, especially in the area of numerical simulation. This SecondEdition recognizes the research advances of the past four years. In addition, however, we ac-knowledge that the fast pace of research makes teaching combustion difficult, especially whenthe numerous prerequisites required to grasp the essence of modern combustion science areconsidered. Therefore, our two primary objectives in preparing this Second Edition were:

to capture the most recent advances in combustion studies so that students and re-searchers alike can (hopefully) find the most essential information in this text; and

to simplify, as much as possible, a complex subject so as to provide a reasonable startingpoint for combustion beginners, especially those involved in numerical simulations.

Obviously, numerical techniques for combustion are essential tools for both engineers andresearch specialists. Along with many of our colleagues, we strongly believe that this evolutiontoward more simulations is dangerous if it is not accompanied by a minimum expertise incombustion theory. Obtaining expertise in combustion theory and modeling is much moredifficult than simply being able to run off-the-shelf codes for combustion. Yet, consideringthe enormous stakes associated with the design of combustion devices in terms of both economyand human safety, it is imperative that this deep understanding not be shortcut. We implore allmembers of the combustion community to strive to ensure that the capabilities and limitationsof numerical simulations are well understood. We would like this book to continue to be oneof the tools available to achieve this understanding. To accomplish this we have maintainedour presentation of the basic combustion theories, and have sought to establish necessaryconnections and interrelationships in a logical and comprehensible way.

A major evolution of numerical combustion of the past few years is the development ofmethods and tools to study unsteady reacting flows. Two combined influences explain thistrend; namely,

Many of the present challenges in the combustion industry are due to intrinsically un-steady mechanisms: autoignition or cycle-to-cycle variations in piston engines, ignitionof rocket engines, combustion instabilities in industrial furnaces and gas turbines, and

xi

xii PREFACE

flash-back and ignition in aero gas turbines. All these problems can be studied numeri-cally, thus partially explaining the rapid growth of the field.

Even for stationary combustion, research on classical turbulent combustion models(herein called RANS for Reynolds Averaged Navier Stokes) is being replaced by newunsteady approaches (herein called LES for Large Eddy Simulations) because comput-ers allow for the computation of the unsteady motions of the flames. However, thesenew unsteady methods also raise new problems: for example, how are numerical resultscompared with experimental data, and what explains the deviations?

Major changes in the Second Edition actually reflect these evolutionary influences: thechapters on turbulent combustion and modeling (Chapters 4 through 6) incorporate recentadvances in unsteady simulation methods. Chapter 8 has also been extensively modified todescribe combustion/acoustics interaction. Chapter 10, an entirely new chapter, is devoted toimportant examples of LES which illustrate the present state of the art in numerical combus-tion while discussing the specificities of swirled flows.

This book can be read with no previous knowledge of combustion, but it cannot re-place many existing books on combustion (Kuo285, Lewis and Von Elbe304, Williams554,Glassman193, Linan and Williams315, Borghi and Destriau51, Peters398) and numerical com-bustion (Oran and Boris383). We concentrate on what is not in these books: i.e. giving toreaders who know about fluid mechanics all the information necessary to move on to a solidunderstanding of numerical combustion. We also avoid concentrating on numerical methodsfor fluid mechanics. Information on Computational Fluid Dynamics (CFD) may be found inRoache448, Anderson8, Hirsch227, Oran and Boris383, Ferziger and Peric167 or Sengupta477.This text concentrates on which equations to solve and not on how to solve them.

Two important topics are also absent from the present edition:

The presentation is limited to deflagrations, i.e. to flames with low speed. Detonationsconstitute a different numerical challenge which is not considered here (see Oran andBoris383).

The chemistry of combustion is also a topic which requires a book (or many) in itself.This text does not try to address this issue: the construction of chemical schemes andtheir reduction and validation are not discussed here. The impact of chemical schemeson reacting flow computations, however, is discussed; especially in the field of turbulentcombustion. Recent progress in this field, both at the fundamental level and for practicalapplications, has changed the way industrial combustion systems are being designedtoday. The important numerical tools needed to understand this evolution are presented.

The Second Edition is now organized as follows:

Chapter 1 first describes the conservation equations needed for reacting flows and re-views different issues which are specific to the numerical resolution of the Navier Stokesequations for a multi-species reacting flow. Tables summarizing the main conservation

PREFACE xiii

forms used in numerical combustion codes are provided. Specific difficulties associatedwith reacting flows are also discussed: models for diffusion velocities, possible simpli-fications for low-speed flames and simple chemistry approximations. Compared to theprevious edition, the discussion about diffusion velocities was revised, clearly statingexact formulations and usual approximations (1.1.4).

Chapter 2 provides a short description of numerical methods for laminar premixedflames. It also includes a summary of many significant theoretical results which areuseful for numerical combustion. Most of these results come from asymptotic theory.They are given here to not only provide an understanding of the results and limitationsof numerical combustion codes, but also to provide insight into how to initialize them,determine necessary grid resolutions, and to verify their results. Extended definitionsand examples of flame speeds, flame thicknesses and flame stretch are discussed.

Chapter 3 introduces laminar diffusion flames and two specific concepts associated withsuch flames: mixture fraction and scalar dissipation. Asymptotic results and the struc-ture of the ideal diffusion flame are used to provide an accurate picture of the phe-nomenology of these flames before computation.

Chapter 4 introduces the basic concepts used to study turbulent combustion. Elementaryconcepts of turbulence and flame/turbulence interaction are described. Averaging andfiltering procedures are discussed. A classification of the different methods (RANS:Reynolds Averaged Navier Stokes, LES: Large Eddy Simulation, DNS: Direct NumericalSimulation) used in numerical combustion for turbulent flames is given. New sectionshave been added to discuss the incorporation of complex chemistry features in turbulentflame computations (4.8), the classification of physical approaches to model turbulentcombustion phenomena according to recent papers (4.5.5), and the comparison betweenLES and experimental data (4.7.7).

Chapter 5 presents turbulent premixed flames. After a description of the main phenom-ena characterizing these flames, a review of recent results and theories is presented forRANS, LES and DNS approaches. Implications for turbulent combustion computationsare discussed. The close relations between all numerical techniques used in the last tenyears (especially DNS results used to develop RANS or LES models) are emphasized.

Chapter 6 presents turbulent diffusion flames. These flames present even more complex-ities than premixed flames and numerical investigations have recently helped to uncovermany of their specificities. Models are also very diverse. The topology of diffusion flamesis first described and RANS methods used for turbulent non premixed combustion areclassified for CFD users. Recent advances in the field of LES and DNS are also described.

Chapter 7 addresses the problem of flame/wall interaction. This issue is critical in manycombustion codes. Asymptotic results and DNS studies are used to illustrate the maincharacteristics of this interaction. Models including this interaction in RANS codes are

xiv PREFACE

described. Since the presence of a flame strongly modifies the turbulence as well as thedensity and the viscosity near walls, models for wall friction and heat transfer in reactingflows are also discussed.

Chapter 8 describes a series of theoretical and numerical tools used to study the cou-pling phenomena between combustion and acoustics. This coupling is the source ofnot only noise, but also of combustion instabilities which can significantly modify theperformances of combustors and sometimes lead to their destruction. Basic elements ofacoustics in non reacting flows are described before extending acoustic theory to react-ing flows. This chapter then focuses on three numerical tools for combustion instabilitystudies: (1) one-dimensional acoustic models to predict the global behavior of a fullcombustion system submitted to longitudinal waves, (2) three-dimensional Helmholtzcodes to identify possible acoustic modes in a burner and (3) multi-dimensional LargeEddy Simulation codes to investigate the detailed response of the combustion chamberitself which is a critical building block for acoustic models. Examples of applicationswith complex geometries are provided in Chapter 10.

Chapter 9 presents recent techniques to specify boundary conditions for compressibleviscous reacting flows. Modern simulation techniques (LES or DNS) as well as recentapplications of CFD (such as combustion instabilities described in Chapter 8) requireelaborate boundary conditions to handle unsteady combustion and acoustic waves aswell as to adjust for numerical schemes which do not provide large levels of dissipation.This chapter provides an overview of such methods and offers a list of test cases forsteady and unsteady flows which can be used in any code.

The new Chapter 10 describes three very recent applications of large eddy simulationsto complex-geometry swirled combustors both to predict the mean flow structure andto analyze unsteady hydrodynamic and acoustic phenomena. These examples provideinformation on the flow physics of these combustors while also demonstrating how recentnumerical tools can be applied today.

To assist the reader, this book uses two distinctive pedagogical devices throughout. First,fundamental and frequently used formulae are boxed for easy identification. Second, an in-novative citation system has been adopted to provide rapid access to a comprehensive set ofreferences. In addition, unsteady flow animations are available on the web to simplify thedescription of certain unsteady phenomena in Chapter 10.

Thierry POINSOTInstitut de Mecanique des FluidesUMR CNRS/INP/UPS 5502

Institut National Polytechniquede Toulouse/ENSEEIHT

Denis VEYNANTELaboratoire E.M2.C.UPR CNRS 288

Ecole Centrale Paris

Chapter 1

Conservation equations forreacting flows

1.1 General forms

This section presents the conservation equations for reacting flows and highlights the threemain differences between these equations and the usual Navier-Stokes equations for non-reacting cases:

a reacting gas is a non-isothermal mixture of multiple species (hydrocarbons, oxygen,water, carbon dioxyde, etc.) which must be tracked individually. Thermodynamicdata are also more complex than in classical aerodynamics because heat capacities in areacting gas change significantly with temperature and composition,

species react chemically and the rate at which these reactions take place requires specificmodeling,

since the gas is a mixture of gases, transport coefficients (heat diffusivity, species diffu-sion, viscosity, etc.) require specific attention.

The derivation of these equations from mass, species or energy balances may be foundin such standard books as those by Williams554, Kuo285 or Candel78. This chapter concen-trates on the various forms used in combustion codes and on their implications for numericaltechniques.

1.1.1 Choice of primitive variables

Combustion involves multiple species reacting through multiple chemical reactions. TheNavier-Stokes equations apply for such a multi-species multi-reaction gas but they requiresome additional terms.

1

2 CHAPTER 1. CONSERVATION EQUATIONS FOR REACTING FLOWS

First, species are characterized through their mass fractions Yk for k = 1 to N where N isthe number of species in the reacting mixture. The mass fractions Yk are defined by:

Yk = mk/m (1.1)

where mk is the mass of species k present in a given volume V and m is the total mass of gasin this volume. The primitive variables for a three-dimensional compressible reacting flow are:

the density = m/V , the three dimensional velocity field ui, one variable for energy (or pressure or enthalpy or temperature T ), and the mass fractions Yk of the N reacting species.Going from non reacting flow to combustion requires solving for N +5 variables instead of

5. Knowing that most chemical schemes involve a large number of species (N is larger than50 for most simple hydrocarbon fuels), this is the first significant effort needed to computereacting flows: increase the number of conservation equations to solve.

Thermochemistry

For a mixture of N perfect gases, total pressure is the sum of partial pressures:

p =Nk=1

pk where pk = kR

WkT (1.2)

where T is the temperature, R = 8.314 J/(moleK) is the perfect gas constant, k = YK andWk are respectively the density and the atomic weight of species k. Since the density of themulti-species gas is:

=Nk=1

k (1.3)

the state equation is:

p = R

WT (1.4)

where W is the mean molecular weight of the the mixture given by:

1W

=Nk=1

YkWk

(1.5)

Mass fractions are used in most combustion codes, but other quantities are also commonlyintroduced to measure concentrations of species (see Table 1.1):

1.1. GENERAL FORMS 3

the mole fraction Xk is the ratio of the number of moles of species k in a volume V tothe total number of moles in the same volume.

the molar concentration [Xk] is the number of moles of species k per unit volume. It isthe quantity used to evaluate kinetic rates of chemical reactions (see Eq. (1.24)).

Quantity Definition Useful relations

Mass fraction Yk Mass of species k / Total Mass Yk

Mole fraction Xk Moles of species k / Total moles Xk =WWk

Yk

Molar concentration [Xk] Moles of species k / Unit volume [Xk] = YkWk

= XkW

Mean molecular weight W 1W

=N

k=1

YkWk

and W =N

k=1XkWk

Table 1.1: Definitions of mass fractions, mole fractions, molar concentrations and useful relations.

For a reacting flow, there are multiple possible variables to represent energy or enthalpy:Table 1.2 gives definitions for energy (ek), enthalpy (hk), sensible energy (esk) and sensibleenthalpy (hsk) for one species.

Form Energy Enthalpy

Sensible esk = TT0CvkdT RT0/Wk hsk =

TT0CpkdT

Sensible+Chemical ek = esk +hof,k hk = hsk +h

of,k

Table 1.2: Enthalpy and energy forms for species k. Enthalpies and energies are related by esk =hsk pk/k and ek = hk pk/k.

The mass enthalpy of formation of species k at temperature T0 is written hof,k. Inprinciple, any value could be assigned to the reference temperature T0. T0 = 0 would havebeen a logical choice (Reynolds and Perkins444) but gathering experimental information onformation enthalpies at 0 K is difficult so that the standard reference state used to tabulateformation enthalpies is usually set to T0 = 298.15K.

In addition to the reference temperature T0, a reference enthalpy (or energy) value mustalso be chosen. This level is set up by assuming that the enthalpy hk is such that:

hk = TT0

CpkdT sensible

+ hof,k chemical

(1.6)


so that the sensible enthalpy hsk is zero at T = T0 for all substances (even though theenthalpy at T = T0 (hk = hof,k) is not). Furthermore, sensible energies are defined to satisfyhsk = esk + pk/k. This choice requires the introduction of the RT0/Wk term in esk: thesensible enthalpy hsk is zero at T = T0 but the sensible energy esk is not: esk(T0) = RT0/Wk.i

In all these forms, energies and enthalpies are mass quantities: for example, the formationenthalpies hof,k are the enthalpies needed to form 1 kg of species k at the reference tem-perature T0 = 298.15K. These values are linked to molar values h

o,mf,k (which are often the

values given in textbooks) by:hof,k = h

o,mf,k /Wk (1.7)

For example, the mass formation enthalpy of CH4 is 4675 kJ/kg while its molar formationenthalpy ho,mf,k is 74.8 kJ/mole. Table 1.3 gives reference values of hof,k and ho,mf,k forsome typical fuels and products used for examples in this book.

Molecular Mass formation Molar formationSubstance weight Wk enthalpy hof,k enthalpy h

o,mf,k

(kg/mole) (kJ/kg) (kJ/mole)

CH4 0.016 4675 74.8C3H8 0.044 2360 103.8C8H18 0.114 1829 208.5CO2 0.044 8943 393.5H2O 0.018 13435 241.8O2 0.032 0 0H2 0.002 0 0N2 0.028 0 0

Table 1.3: Formation enthalpies (gaseous substances) at T0 = 298.15K.

The heat capacities at constant pressure of species k (Cpk) are mass heat capacities relatedto molar capacities Cmpk by Cpk = C

mpk/Wk. For a perfect diatomic gas:

Cmpk = 3.5R and Cpk = 3.5R/Wk (1.8)

In practice, the changes of Cmpk with temperature are large in combusting flows and Cmpk

values are usually tabulated as temperature functions using polynomials (Stull and Prophet496,Heywood217). Fig. 1.1 shows heat capacities Cmpk (divided by the perfect gas constant R) asa function of temperature. While N2 and H2 heat capacities are of the order of 3.5R at lowtemperatures, they deviate rapidly from this value at high temperature.

iAnother solution would be to define esk = TT0

CvkdT and ek = esk + hof,k RT0/Wk where eof,k =

hof,k RT0/Wk would be a formation energy at constant volume. In this case esk and hsk pk/k wouldbe different. The final equations are equivalent.


The mass heat capacities Cpk of usual gases are displayed in Fig. 1.2. CO2, CO and N2have very similar values of Cpk. Water has a higher capacity (2000 to 3000 J/(kgK)) and H2mass heat capacities are much higher (of the order of 16000 J/(kgK)).

The mass heat capacities Cvk at constant volume are related to the Cpk by:

Cpk Cvk = R/Wk (1.9)

For a mixture of N species, Table 1.4 summarizes the different forms used for energy andenthalpy (eight because of the possible combinations between sensible, kinetic and chemicalparts). The enthalpy h is defined by:

h =Nk=1

hkYk =Nk=1

( TT0

CpkdT +hof,k

)Yk =

TT0

CpdT +Nk=1

hof,kYk (1.10)

while the energy e = h p/ is given by, using Eq. (1.4), (1.5) and (1.9):

e =Nk=1

( TT0

CpkdT RT/Wk +hof,k)Yk =

Nk=1

( TT0

CvkdT RT0/Wk +hof,k)Yk

= TT0

CvdT RT0/W +Nk=1

hof,kYk =Nk=1

ekYk (1.11)

Most compressible codes use the total non chemical energy (E) and enthalpy (H) forms fornon reacting flows.

Form Energy Enthalpy

Sensible es = TT0CvdT RT0/W hs =

TT0CpdT

Sensible+Chemical e = es +N

k=1hof,kYk h = hs +

Nk=1

hof,kYk

Total Chemical et = e+12uiui ht = h+

12uiui

Total non Chemical E = es +12uiui H = hs +

12uiui

Table 1.4: Enthalpy and energy forms used in conservation equations.

The heat capacity at constant pressure of the mixture, Cp, is:

Cp =Nk=1

CpkYk =Nk=1

CmpkYkWk

(1.12)


7

6

5

4Sca

led

mol

ar h

eat c

apac

ity

2500200015001000500Temperature (K)

CO2

C O

H2O

H2N2

Ideal diatomic gas

Figure 1.1: Scaled molar heat capacities at constant pressure Cmpk/R of CO2, CO, H2O, H2 and N2.

3000

2500

2000

1500

1000

Mas

s he

at c

apac

ity

2500200015001000500Temperature (K)

CO2

C O

H2O

N2

Figure 1.2: Mass heat capacities (J/(kgK)) at constant pressure Cpk of CO2, CO, H2O and N2.


Eq. (1.12) shows that the mixture heat capacity Cp is a function both of temperature (T)and composition (Yk). It may change significantly from one point to another. However, inmost hydrocarbon/air flames, the properties of nitrogen dominate and the mass heat capacityof the mixture is very close to that of nitrogen. Furthermore, this value changes only from1000 to 1300 J/(kg K) when temperature goes from 300 K to 3000 K so that Cp is oftenassumed to be constant in many theoretical approaches ( 2 and 3) and some combustioncodes.

The heat capacity of the mixture at constant volume, Cv, is defined as:

Cv =Nk=1

CvkYk =Nk=1

CmvkYkWk

(1.13)

where the heat capacities Cvk of individual species are obtained by: Cvk = Cpk R/Wk formass values or Cmvk = C

mpk R for molar values.

Viscous tensor

The velocity components are called ui for i = 1 to 3. The viscous tensor ij is defined by:

ij = 23ukxk

ij + (uixj

+ujxi

)(1.14)

where p is the static pressure and is the dynamic viscosity. The kinematic viscosity is = /. The bulk viscosity is supposed to be zero (Kuo285). ij is the Kronecker symbol:ij = 1 if i = j, 0 otherwise.

Viscous and pressure tensors are often combined into the ij tensor defined by:

ij = ij pij = pij 23ukxk

ij + (uixj

+ujxi

)(1.15)

Molecular transport of species and heat

The heat diffusion coefficient is called . The diffusion coefficient of species k in the rest ofthe mixture (used in Ficks law, see below) is called Dk. Diffusion processes involve binarydiffusion coefficients (Dkj) and require the resolution of a system giving diffusion velocities.This is not done in most combustion codes (Ern and Giovangigli161): solving the diffusionproblem in a multi-species gas is a problem in itself. Simplified diffusion laws (usually Fickslaw) are used in a majority of combustion codes and this text is restricted to this case. TheDk coefficients are often characterized in terms of Lewis number defined by:

Lek =

CpDk=DthDk

(1.16)


where Dth = /(Cp) is the heat diffusivity coefficient. The Lewis number Lek compares thediffusion speeds of heat and species k. Section 2.4 shows that this parameter is importantin laminar flames. Lek is a local quantity but, in most gases, it changes very little from onepoint to another. The kinetic theory of gases (Hirschfelder et al.228) shows that changesroughly like T 0.7, like 1/T and Dk like T 1.7 so that Lek is changing only by a few percentsin a flame. This property is used in 1.1.3.

The Prandtl number, Pr, compares momentum and heat transport:

Pr =

/(Cp)=Cp

=Cp

(1.17)

The Schmidt number, Sck, compares momentum and species k molecular diffusion:

Sck =

Dk= Pr Lek (1.18)

Soret (molecular species diffusion due to temperature gradients) and Dufour (heat fluxdue to species mass fraction gradients) effects are neglected for this presentation (Ern andGiovangigli161, Giovangigli185).

Chemical kinetics

Consider a chemical system of N species reacting through M reactions:

Nk=1

kjMk Nk=1

kjMk for j = 1,M (1.19)

where Mk is a symbol for species k, kj and kj are the molar stoichiometric coefficients of

species k in reaction j. Mass conservation enforces:

Nk=1

kjWk =Nk=1

kjWk or

Nk=1

kjWk = 0 for j = 1,M (1.20)

wherekj =

kj kj (1.21)

For simplicity, only mass reaction rates are used. For species k, this rate k is the sum ofrates kj produced by all M reactions:

k =Mj=1

kj =WkMj=1

kjQj with kjWkkj

= Qj (1.22)

where Qj is the rate of progress of reaction j.


Summing all reaction rates k and using Eq. (1.20), one obtains:

Nk=1

k =Mj=1

(Qj

Nk=1

Wkkj

)= 0 (1.23)

showing that total mass is conserved.The progress rate Qj of reaction j is written:

Qj = KfjNk=1

[Xk]kj Krj

Nk=1

[Xk]kj (1.24)

where Kfj and Krj are the forward and reverse rates of reaction j. Note that kinetic ratesare expressed using molar concentrations [Xk] = Yk/Wk = k/Wk (see Table 1.1).

The rate constants Kfj and Krj constitute a central problem of combustion modeling.They are usually modeled using the empirical Arrhenius law:

Kfj = AfjT j exp( EjRT

)= AfjT j exp

(TajT

)(1.25)

Expressing the individual progress rates Qj for each reaction means providing data for thepreexponential constant Afj , the temperature exponent j and the activation temperatureTaj (or equivalently the activation energy Ej = RTaj). Before giving these constants, evenidentifying which species and which reactions should or should not be kept in a given scheme isthe challenge of chemical kinetics. Describing the construction of chemical schemes and theirvalidation is beyond the scope of this text. In most numerical approaches for reacting flows,the chemical scheme is one of the data elements which must be available for the computation.Certain properties of the kinetics data have a crucial effect on the success of the computation(as discussed in Section 2) so numerical combustion users cannot avoid considering the char-acteristics of these schemes. An example of a kinetic scheme for H2 - O2 combustion proposedby Miller et al.356 is given in Table 1.5 in standard CHEMKIN format (Kee et al.259). First,elements and species which have been retained for the scheme are listed. For each reaction,the table then gives Afj in cgs units, j and Ej in cal/mole. The backwards rates Krj arecomputed from the forward rates through the equilibrium constants:

Krj =Kfj( pa

RT

)Nk=1

kjexp

(S0jR

H0j

RT

) (1.26)where pa = 1 bar. The symbols refer to changes occurring when passing from reactants toproducts in the jth reaction: H0j and S

0j are respectively enthalpy and entropy changes

for reaction j. These quantities are obtained from tabulations.


ELEMENTSH O NENDSPECIESH2 O2 OH O H H2O HO2 H2O2 N2ENDREACTIONSH2+O2=OH+OH 1.700E13 0.0 47780.H2+OH=H2O+H 1.170E09 1.30 3626.H+O2=OH+O 5.130E16 -0.816 16507.O+H2=OH+H 1.800E10 1.0 8826.H+O2+M=HO2+M 2.100E18 -1.0 0.

H2/3.3/ O2/0./ N2/0./ H2O/21.0/H+O2+O2=HO2+O2 6.700E19 -1.42 0.H+O2+N2=HO2+N2 6.700E19 -1.42 0.OH+HO2=H2O+O2 5.000E13 0.0 1000.H+HO2=OH+OH 2.500E14 0.0 1900.O+HO2=O2+OH 4.800E13 0.0 1000.OH+OH=O+H2O 6.000E08 1.3 0.H2+M=H+H+M 2.230E12 0.5 92600.

H2/3./ H/2./ H2O/6.0/O2+M=O+O+M 1.850E11 0.5 95560.H+OH+M=H2O+M 7.500E23 -2.6 0.

H2O/20.0/HO2+H=H2+O2 2.500E13 0.0 700.HO2+HO2=H2O2+O2 2.000E12 0.0 0.H2O2+M=OH+OH+M 1.300E17 0.0 45500.H2O2+H=H2+HO2 1.600E12 0.0 3800.H2O2+OH=H2O+HO2 1.000E13 0.0 1800.END

Table 1.5: 9 species / 19 reactions chemical scheme for H2 - O2 combustion (Miller et al.356). For

each reaction, the table provides respectively Afj (cgs units), j and Ej (cal/mole).

At this point, for numerical combustion users, it is important to mention that data on Qjcorrespond to models: except for certain reactions, these data are obtained experimentallyand values of kinetic parameters are often disputed in the kinetics community (Just248). Formany flames, very different schemes are found in the literature with very comparable accuracyand the choice of a scheme is a difficult and controversial task. Unfortunately, the values ofthe parameters used to compute Qj and the stiffness associated with the determination of Qjcreate a central difficulty for numerical combustion: the space and time scales correspondingto the Qj terms are usually very small and require meshes and time steps which can beorders of magnitude smaller than in non reacting flows. Particular attention must be paidto the activation energy Ej (usually measured in kcal/mole in the combustion community):the exponential dependence of rates to Ej leads to considerable difficulties when Ej takeslarge values (typically more than 60 kcal/mole). A given combustion code for laminar flamesmay work perfectly with a given mesh when Ej is small but will require many more points toresolve the flame structure when Ej is increased even by a small amount (see 2.4.4).


Stoichiometry in premixed flames

Even though multiple radicals may be involved in a flame, certain species are more importantto characterize the combustion regime. The fuel and the oxidizer mass fractions are obviouslysignificant quantities and their ratio is commonly used to characterize the flame. There aredifferent ways to define the equivalence ratio (Fig. 1.3) depending on the configuration of theburner (premixed or non premixed).

Fuel+ Air

Stream 1: Fuel

Stream 2: oxidizer

(a) Premixed flame (b) Diffusion flame

Figure 1.3: Premixed and diffusion flames configurations.

In a premixed combustor (Fig. 1.3a), fuel and oxidizer are mixed before they enter thecombustion chamber. If F and

O are the coefficients corresponding to fuel and oxidizer

when considering an overall unique reaction of the type

FF + OO Products (1.27)

(for example CH4 +2O2 CO2 +2H2O), the mass fractions of fuel and oxidizer correspondto stoichiometric conditions when: (

YOYF

)st

=OWOFWF

= s (1.28)

This ratio s is called the mass stoichiometric ratio. The equivalence ratio of a given mixtureis then:

= sYFYO

=(YFYO

)/

(YFYO

)st

(1.29)

It can also be recast as: = s

mFmO

(1.30)

where mF and mO are respectively the mass flow rates of fuel and oxidizer.


The equivalence ratio is a central parameter for premixed gases: rich combustion is ob-tained for > 1 (the fuel is in excess) while lean regimes are achieved when < 1 (the oxidizeris in excess).

Most practical burners operate at or below stoichiometry. In hydrocarbon/air flames, thefresh gases contain Fuel, O2 and N2 with typically 3.76 moles of nitrogen for 1 mole of oxygen.Since the sum of mass fractions must be unity, the fuel mass fraction is:

YF =1

1 +s

(1 + 3.76

WN2WO2

) (1.31)Table 1.6 shows typical values of the stoichiometric ratio s and of corresponding fuel massfractions for stoichiometric mixtures ( = 1) with air.

Global reaction s Y stF

CH4 + 2(O2 + 3.76N2) CO2 + 2H2O + 7.52N2 4.00 0.055C3H8 + 5(O2 + 3.76N2) 3CO2 + 4H2O + 18.8N2 3.63 0.060

2C8H18 + 25(O2 + 3.76N2) 16CO2 + 18H2O + 94N2 3.51 0.0622H2 + (O2 + 3.76N2) 2H2O + 3.76N2 8.00 0.028

Table 1.6: Stoichiometric ratio s and fuel mass fraction for stoichiometric combustion ( = 1) in airgiven by Eq. (1.31).

Values of the fuel mass fraction at stoichiometry are very small (Table 1.6) so that thepremixed gas entering the chamber contains mostly air: adding fuel to air to obtain a react-ing mixture does not significantly modify the properties of air in terms of molecular weights,transport properties and heat capacities compared to pure air. This is used in many the-ories of premixed flames and in certain codes to simplify the evaluation of transport andthermodynamic properties.

Stoichiometry in diffusion flames

For a diffusion flame, fuel and oxidizer are introduced separately into the combustion chamberthrough two (or more) inlets where flow rates and mass fractions are controlled separately.For the combustor of Fig. 1.3b with only two inlets (one for the fuel stream with a fuel massfraction Y 1F and the other for the oxidizer stream with an oxidizer mass fraction Y

2O), a first

definition of equivalence ratio is: = s(Y 1F /Y

2O) (1.32)

This ratio characterizes the local structure of the flames formed when the two streams (fueland oxidizer) interact. However, it does not represent the overall behavior of the combustor


for which a global equivalence ratio g must be introduced:

g = sm1F /m2O (1.33)

where m1F and m2O are the flow rates of fuel in the first inlet and of oxidizer in the second

respectively. The global (g) and the local () equivalence ratios are linked by:

g = m1/m2 (1.34)

where m1 and m2 are the total flow rates entering inlet 1 and 2 respectively. For premixedcombustors, fuel and oxidizer are carried by the same stream and m1= m2 so that g = .For non premixed systems, these two quantities may differ significantly.ii

1.1.2 Conservation of momentum

The equation of momentum is the same in reacting and non reacting flows:

tuj +

xiuiuj = p

xj+ijxi

+ Nk=1

Ykfk,j =ijxi

+ Nk=1

Ykfk,j (1.35)

where fk,j is the volume force acting on species k in direction j. Even though this equationdoes not include explicit reaction terms, the flow is modified by combustion: the dynamicviscosity strongly changes because temperature varies in a ratio from 1:8 or 1:10. Densityalso changes in the same ratio and dilatation through the flame front increases all speedsby the same ratio. As a consequence, the local Reynolds number varies much more than ina non reacting flow: even though the momentum equations are the same with and withoutcombustion, the flow behavior is very different. A typical example is found in jets: turbulentnon reacting jets may become laminar once they are ignited (Lewis and Von Elbe304).

1.1.3 Conservation of mass and species

The total mass conservation equation is unchanged compared to non reacting flows (combus-tion does not generate mass):

t+uixi

= 0 (1.36)

The mass conservation equation for species k is written:

Ykt

+

xi((ui + Vk,i)Yk) = k for k = 1, N (1.37)

iiThe excess air, defined by e = 100(1 g)/g , is also used to characterize the overall mixture. It givesthe percentage of air in excess of the flow rate required to consume all the fuel injected into the burner.


where Vk,i is the i- component of the diffusion velocity Vk of species k and k is the reactionrate of species k. By definition:

Nk=1

YkVk,i = 0 andNk=1

k = 0 (1.38)

1.1.4 Diffusion velocities: full equations and approximations

The diffusion velocities Vk of the N species are obtained by solving the system (Williams554):

Xp =Nk=1

XpXkDpk (Vk Vp) + (Yp Xp)

PP

+

p

Nk=1

YpYk(fp fk) for p = 1, N (1.39)

where Dpk = Dkp is the binary mass diffusion coefficient of species p into species k and Xk isthe mole fraction of species k: Xk = YkW/Wk. The Soret effect (the diffusion of mass due totemperature gradients) is neglected.

The system (1.39) is a linear system of size N2 which must be solved in each direction ateach point and at each instant for unsteady flows. Mathematically, this task is difficult andcostly (Ern and Giovangigli161) and most codes use simplified approaches discussed below.

Binary diffusion

If the mixture contains only two species and if pressure gradients are small and volume forcesare neglected, the system (1.39) reduces to a scalar equation where the unknown are the twodiffusion velocities V1 and V2:

X1 = X1X2D12 (V2 V1) (1.40)

Knowing that Y1 + Y2 = 1 and that Y1V1 + Y2V2 = 0 from Eq. (1.38), Eq. (1.40) leads to:

V1Y1 = D12Y1 (1.41)

which is Ficks law (Kuo285). This expression is exact for binary diffusion but must be replacedby a more complex expression for multispecies gases.

Multispecies diffusion

The rigorous inversion of system (1.39) in a multispecies gas is often replaced by the Hirschfelderand Curtiss approximation (Hirschfelder et al.228) which is the best first-order approximationto the exact resolution of system (1.39) (Ern and Giovangigli161, Giovangigli185):

VkXk = DkXk with Dk = 1 Ykj 6=kXj/Djk (1.42)


The coefficient Dk is not a binary diffusion but an equivalent diffusion coefficient of speciesk into the rest of the mixture. If the mixture contains only two species labelled 1 and 2,Eq. (1.42) reduces to V1X1 = (Y2/X2)D12X1 which is exactly Ficks law (1.41): for abinary mixture with no pressure gradient and no volume forces, both Ficks law and theHirschfelder and Curtiss approximation are exact. In all other cases, however, Eq. (1.42)differs from Ficks law and leads to the following species equationiii:

Ykt

+uiYkxi

=

xi

(Dk

WkW

Xkxi

)+ k (1.43)

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

Lew

is n

umbe

r

543210-1-2x [cm]

CH4

O2

H2

CO2

H2O

OH

Figure 1.4: Variations of Lewis numbers (defined by Eq. (1.16)) of the main species in a stoichio-metric laminar methane air flame (B. Bedat, private communication, 1999).

The Hirschfelder and Curtiss approximation for diffusion velocities is a convenient approx-imation because the diffusion coefficients Dk can be simply linked to the heat diffusivity Dthin many flames: the Lewis numbers of individual species Lek = Dth/Dk are usually varying bysmall amounts in flame fronts. Fig. 1.4 shows a computation of Lewis numbers of main speciesfor a premixed stoichiometric methane/air flame plotted versus spatial coordinate through theflame front. Lewis numbers change through the flame front (located here at x = 0) but thesechanges are small.

iiiNote the Wk/W factor in the RHS diffusion term of Eq. (1.43). This term is missing in formulations usingFicks law (1.41). Ficks law is strictly valid for binary diffusion only and should not be used in flames.


Global mass conservation and correction velocity

Mass conservation is a specific issue when dealing with reacting flows. Obviously, the sum ofmass fractions must be unity:

Nk=1 Yk = 1. This equation adds to the N equations (1.37)

while there are only N unknowns (the Yks): the system is over determined. It is easy tocheck that if all species equations (1.37) are added and the identities

Nk=1 YkVk,i = 0 andN

k=1 k = 0 are used, the mass conservation equation is recovered:

t+uixi

= xi

(

Nk=1

YkVk,i

)+

Nk=1

k = 0 (1.44)

This equation shows that there are only N independent equations and that any one of theN species equations (1.37) or the mass conservation equation (1.36) may be eliminated: thisapparent problem is not a difficulty when exact expressions for diffusion velocities are used.However, this is no longer the case when Hirschfelders law is used. In this case, the RHS termof Eq. (1.44) becomes xi (

Nk=1Dk

WkW

Xkxi

) which is not zero: global mass is not conserved.Despite this drawback, most codes in the combustion community use Hirschfelders law (oreven Ficks law) because solving for the diffusion velocities is too complex. Two methods canbe used to implement such laws and still maintain global mass conservation:

The first and simplest method is to solve the global mass conservation equation (1.36)and only N 1 species equations. The last species mass fraction (usually a diluentsuch as N2) is obtained by writing YN = 1

N1k=1 Yk and absorbs all inconsistencies

introduced by Eq. (1.42). This simplification is dangerous and should be used only whenflames are strongly diluted (for example in air so that YN2 is large).

In the second method, a correction velocity V c is added to the convection velocity inthe species equations (1.43):

tYk +

xi(ui + V ci )Yk =

xi

(Dk

WkW

Xkxi

)+ k (1.45)

The correction velocity V ci is evaluated to ensure global mass conservation. If all speciesequations are summed, the mass conservation equation must be recovered:

t+uixi

=

xi

(

Nk=1

DkWkW

Xkxi

V ci)= 0 so : V ci =

Nk=1

DkWkW

Xkxi

(1.46)At each time step, the correction velocity is computed and added to the convecting fieldui to ensure the compatibility of species and mass conservation equations. In this case,it is still possible to solve for (N-1) species and for the total mass but, unlike for thefirst method, the result for YN is correct. This solution is often chosen in laminar flamecodes where diffusion coefficients can be very different.


1.1.5 Conservation of energy

The energy conservation equation requires the greatest attention because multiple forms exist.Note first that because of continuity (Eq. 1.36), the following relation (which may be used inall left hand sides of enthalpy, energy or temperature equations) holds for any quantity f :

Df

Dt=

(f

t+ ui

f

xi

)=f

t+uif

xi(1.47)

where the last form is called conservative. Starting from the conservation equation for totalenergy et (Kuo285):

DetDt

=ett

+

xi(uiet) = qi

xi+

xj(ijui) + Q+

Nk=1

Ykfk,i(ui + Vk,i) (1.48)

where Q is the heat source term (due for example to an electric spark, a laser or a radiativeflux), not to be confused with the heat released by combustion.

Nk=1 Ykfk,i(ui + Vk,i) is

the power produced by volume forces fk on species k. The energy flux qi is:

qi = Txi

+ Nk=1

hkYkVk,i (1.49)

This flux includes a heat diffusion term expressed by Fouriers Law (T/xi) and a secondterm associated with the diffusion of species with different enthalpies which is specific ofmulti-species gas.

Using the relation between energy and enthalpy: ht = et+p/ and the continuity equation(1.36) yields:

DetDt

= DhtDt

DpDt

puixi

and De

Dt=

Dh

Dt DpDt

puixi

(1.50)

Using Eq. (1.50) to eliminate et in Eq. (1.48) gives the conservation equation for ht:

DhtDt

=htt

+

xi(uiht) =

p

t qixi

+

xj(ijui) + Q+

Nk=1


The equation for the sum of sensible and chemical energy (e) is obtained by writing firstthe kinetic energy equation ujuj/2. Multiplying the momentum equation (1.35) by uj :

t

(12ujuj

)+

xi

(12uiujuj

)= uj

ijxi

+ Nk=1

Ykfk,juj (1.52)


Subtracting this equation from Eq. (1.48) gives a balance equation for e:

De

Dt=e

t+

xi(uie) = qi

xi+ ij

uixj

+ Q+ Nk=1

Ykfk,iVk,i (1.53)

The conservation equation for the enthalpy h is then deduced from Eq. (1.53) and (1.50):

Dh

Dt=h

t+

xi(uih) =

Dp

Dt qixi

+ ijuixj

+ Q+ Nk=1

Ykfk,iVk,i (1.54)

where = ijui/xj is the viscous heating source term.The above expressions are not always easy to implement in classical CFD codes because

they use expressions for energy and enthalpy including chemical terms (N

k=1hof,kYk) in

addition to sensible energy or enthalpy and because the heat flux q also includes new transportterms (

Nk=1 hkYkVk,i). Sensible energies or enthalpies are sometimes preferred. From the

definition of hs (hs = h N

k=1hof,kYk), substituting hs for h in Eq. (1.54) and using the

species equation (1.37) leads to:

DhsDt

= T +Dp

Dt+

xi

(T

xi

) xi

(

Nk=1

hs,kYkVk,i

)+ ij

uixj

+ Q+ Nk=1

Ykfk,iVk,i

(1.55)

where T is the heat release due to combustion:

T = Nk=1

hof,kk (1.56)

The sensible enthalpies of species k, hs,k = TT0Cp,kdT appear on the RHS of Eq. (1.55). The

corresponding term xi (N

k=1 hs,kYkVk,i) is zero:

if the mixture contains only one species or if all species have the same sensible enthalpy: Nk=1 hs,kYkVk,i = hsNk=1 YkVk,i = 0.

In all other cases, this term does not vanish even though it is sometimes set to zero becauseit is usually negligible compared to T .


The equation for the sensible energy es may be deduced from Eq. (1.55) and (1.50):iv

DesDt

=est

+

xi(uies) = T +

xi

(T

xi

) xi

(

Nk=1

hs,kYkVk,i

)+ ij

uixj

+ Q+ Nk=1

Ykfk,iVk,i

(1.57)

Another way is to work with the sum of sensible and kinetic energies (the total non-chemical energy in Table 1.4). Adding Eq. (1.57) and (1.52) leads to the equation for E =es + 12uiui:

DE

Dt=E

t+

xi(uiE) = T +

xi

(T

xi

) xi

(

Nk=1

hs,kYkVk,i

)+

xj(ijui) + Q+

Nk=1

Ykfk,i(ui + Vk,i)(1.58)

In the same way, the equation for H = hs+uiui/2 is obtained by adding Eq. (1.55) and (1.52):

DH

Dt=H

t+

xi(uiH) = T +

xi

(T

xi

)+p

t xi

(

Nk=1

hs,kYkVk,i

)+

xj(ijui) + Q+

Nk=1

Ykfk,i(ui + Vk,i)(1.59)

In some codes (low-Mach number or incompressible formulations), an equation for temper-ature is used. Starting from hs =

Nk=1 hs,kYk where hs,k is the sensible enthalpy of species

k (Giovangigli184, Giovangigli and Smooke186), the derivative of hs is:

DhsDt

=Nk=1

hskDYkDt

+ CpDT

Dt(1.60)

ivNote that the sensible energy is defined here by es = TT0

CvdT RT0/W (Table 1.4). A different equationwould be found if the sensible energy is defined by es =

TT0

CvdT :

DesDt

= Nk=1

eof,kk+

xi

(T

xi

) xi

(

Nk=1

(hs,k +RT0/Wk)YkVk,i

)+ij

ui

xj+Q+

Nk=1

Ykfk,iVk,i

where eof,k is the formation energy at TO: eof,k = h

of,k RT0/Wk.


Replacing this derivative in Eq. (1.55) gives:

CpDT

Dt= T +

Dp

Dt+

xi

(T

xi

)(

Nk=1

Cp,kYkVk,i

)T

xi+ ij

uixj

+ Q+ Nk=1 Ykfk,iVk,i(1.61)

Note that the reaction term T is not equal to T , the reaction term found in the equationfor es or hs:

T = Nk=1

hof,kk ; T =

Nk=1

hkk = Nk=1

hskk Nk=1

hof,kk (1.62)

These two terms are both called heat release so that different authors use the same termfor different quantities. They differ by a small amount due to the contribution of sensibleenthalpy terms hsk. Section 1.2.2 shows that they are equal when the heat capacities Cp,ksare supposed equal for all species.

An equivalent form may be written with Cv, starting from the definition of es to have:

DesDt

=Nk=1

eskDYkDt

+ CvDT

Dt(1.63)

so that replacing the time derivative of es in Eq. (1.57) leads to:

CvDT

Dt= T +

xi

(T

xi

)RT

xi

(

Nk=1

YkVk,i/Wk

)

Txi

Nk=1

(YkVk,iCp,k) + ijuixj

+ Q+ Nk=1

Ykfk,iVk,i

(1.64)

where T = N

k=1 ekk is another reaction rate (equal to T when all heat capacities Cpkare equal).

Table 1.7 summarizes the different forms of energy equations.


Form

Energy

Enthalpy

Sensible

e s=hsp/

= T T 0C

vdTRT0/W

hs= T T 0C

pdT

Sensible+Chemical

e=hp/

=e s+ N k=

1ho f,kYk

h=hs+ N k=

1ho f,kYk

TotalChemical

e t=htp/

=e s+ N k=

1ho f,kYk+

1 2uiui

ht=hs+ N k=

1ho f,kYk+

1 2uiui

TotalnonChemical

E=Hp/

=e s+

1 2uiui

H=hs+

1 2uiui

e tDet

Dt=

qi

xi+

xj(

ijui)+Q+ N k=

1Ykf k,i(u

i+Vk,i)

ht

Dht

Dt=

pt

qi

xi+

xj(ijui)+Q+ N k=

1Ykf k,i(u

i+Vk,i)

eDe

Dt=

qi

xi+ijui

xj+Q+ N k=

1Ykf k,iVk,i

hDh

Dt=

Dp

Dt

qi

xi+ ijui

xj+Q+ N k=

1Ykf k,iVk,i

e sDes

Dt=T+

xi(

T

xi)

xi( N k=

1hs,kYkVk,i)+ijui

xj+Q+ N k=

1Ykf k,iVk,i

hs

Dhs

Dt=T+

Dp

Dt+

xi(

T

xi)

xi( N k=

1hs,kYkVk,i)+ ijui

xj+Q+ N k=

1Ykf k,iVk,i

EDE

Dt=T+

xi(

T

xi)

xi( N k=

1hs,kYkVk,i)+

xj(

ijui)+Q+ N k=

1Ykf k,i(u

i+Vk,i)

HDH

Dt=T+

pt+

xi(

T

xi)

xi( N k=

1hs,kYkVk,i)+

xj(ijui)+Q+ N k=

1Ykf k,i(u

i+Vk,i)

Table

1.7:Enthalpyandenergyform

sandcorrespondingbalance

equations.

TheVk,iare

thediffusionvelocities.The

f k,isare

volumeforces

actingonspecieskin

directioni.Q

isthevolumesourceterm

.q i

istheenthalpyfluxdefined

by

q i=

T

xi+ N k=

1hkYkVk,i.Theviscoustensors

are

defined

by ij=2/3uk

xk ij+(ui

xj+

uj

xi)andij= ijp ij.

TheheatreleaseTis N k=

1ho f,kk.Foranyenergyorenthalpyf:Df

Dt=(ft+uif

xi)=

f

t+

xi(uif).


1.2 Usual simplified forms

The complete form of the energy equation is not always needed and simplified forms are oftenutilized in combustion codes. The following sections present such forms.

1.2.1 Constant pressure flames

In deflagrations (Williams554), flame speeds are small compared to the sound speed (typicalvalues for sL range from 0.1 to 5 m/s while the sound speed in the fresh gases c1 varies between300 and 600 m/s in most combustion chambers), allowing some interesting simplifications:

The Mach number Ma in the flow is of the order of sL/c1 and is small: this is sufficientto show that pressure in the state equation may be assumed to be constant. This maybe checked by starting from the conservation of momentum (1.35), written here forsimplicity in one dimension only with a constant viscosity :

u

t+ u

u

x= p

x+

x

u

x(1.65)

This equation may be scaled with:

u+ = u/c1 x+ = x/L + = /1 t+ = c1t/L p+ = p/(1c

21

)(1.66)

where L is a reference distance (for example the size of the burner) and c1 is the soundspeed. Index 1 refers to fresh gas quantities. Introducing the acoustic Reynolds numberRe = 1c1L/1, Eq. (1.65) becomes:

p+

x+= + u

+

t+ o(Ma)

+u+ u+

x+ o(M2a )

+1Re

x+u+

x+ o(Ma/Re)

, (1.67)

Eq. (1.67) shows that, in high Reynolds number steady flows, the changes in meanpressure are of the order of M2a . They are of the order of Ma in unsteady flows. Forsubsonic combustion with low Mach numbers, these variations are negligible and pressurecan be assumed to be constant in the state equation p = R/WT which is replaced byR/WT = p0 = cte. Therefore, the density change through the flame front is directlyrelated to the temperature change through the flame front:

2/1 = T1/T2. (1.68)

In the energy equation, the pressure term Dp/Dt may be set to zero.

In the same way, the viscous heating term = ij(ui/xj) in the temperature equationis of high order in Ma and may be neglected.

1.2. USUAL SIMPLIFIED FORMS 23

The temperature equation (1.61) reduces to:

CpDT

Dt= T +

xi

(T

xi

) T

xi

(Nk=1

Cp,kYkVk,i

)+ Q+

Nk=1

Ykfk,iVk,i (1.69)

1.2.2 Equal heat capacities for all species

Equations for temperature may be simplified by using assumptions on the species heat ca-pacities Cpk. First, assuming that all heat capacities are equal Cp,k = Cp and hs,k = hs(an assumption which is not often true in flames but is often used!), the

Nk=1 Cp,kYkVk,i

term in the temperature equation (1.61) is CpN

k=1 YkVk,i = 0 and the temperature equationbecomes:v

for variable pressure, equal Cp,k flames:

CpDT

Dt= T +

Dp

Dt+

xi

(T

xi

)+ ij

uixj

+ Q+ Nk=1

Ykfk,iVk,i (1.70)

for constant pressure, low speed, equal Cp,k flames:

CpDT

Dt= T +

xi

(T

xi

)+ Q+

Nk=1

Ykfk,iVk,i (1.71)

Here the two reaction rates T and T (Eq. 1.62) are equal:

T = Nk=1

hkk = Nk=1

hs,kk Nk=1

hof,kk = hsNk=1

k Nk=1

hof,kk = T (1.72)

becauseN

k=1 k = 0.The energy and enthalpy equations also have simplified forms if all heat capacities are

equal:

DE

Dt=E

t+

xi(uiE) = T +

xi

(T

xi

)

+

xj(ijui) + Q+

Nk=1


vThe heat capacity Cp may still be a function of temperature.


DH

Dt=H

t+

xi(uiH) = T +

p

t+

xi

(T

xi

)

+

xj(ijui) + Q+

Nk=1


Note that, if all heat capacities are equal but also independent of temperature, the equationfor es is an equation for pressure because:

es =

( TT0

CvdT RT0/W)= (CvT CpT0) = p/( 1) CpT0

where = Cp/Cv. Replacing es in Eq. (1.57) gives and using the continuity equation:

1 1

Dp

Dt= p

1uixi

+ T +

xi

(T

xi

)+ ij

uixj

+ Q+ Nk=1

Ykfk,iVk,i (1.75)

For most deflagrations, p is almost constant: in Eq. (1.75), the first term on the RHS (dueto dilatation) compensates the other terms. When a compressible code is used for reactingflows, this equation controls pressure and explains how such codes react to perturbations. Ifthe heat release term T , for example, is too high, pressure will locally increase. If is notestimated correctly, the term p1 uixi will also induce pressure modifications. Eq. (1.75) isa useful diagnostic tool to identify problems in codes for compressible reacting flows. It mayalso be used to construct a wave equation in reacting flows (see 8.3.2).

1.2.3 Constant heat capacity for the mixture only

It is also possible to assume that the mixture heat capacity Cp is constant because one species(N2 for example) is dominant in the mixture but that the Cp,ks are not equal: this approxi-mation is slightly inconsistent because Cp =

Nk=1 Cp,kYk but it is used in Kuo

285 for exampleto derive the following temperature equation by setting Dhs/Dt = CpDT/Dt in the equationfor hs (Eq. 1.55):

CpDT

Dt= T +

Dp

Dt+

xi

(T

xi

) xi

(T

Nk=1

Cp,kYkVk,i

)

+ijuixj

+ Q+ Nk=1

Ykfk,iVk,i (1.76)

where hs,k was replaced by Cp,kT .

1.3. SUMMARY OF CONSERVATION EQUATIONS 25

1.3 Summary of conservation equations

Table 1.8 summarizes the equations to solve for reacting flows.

Mass

t +

uixi

= 0

Species: for k = 1 to N 1 (or N if total mass is not used)

With diffusion velocities:

Ykt +

xi

((ui + Vk,i)Yk) = k

With Hirschfelder and Curtiss approximation:

Ykt +

xi

((ui + V ci )Yk) =xi

(Dk WkWXkxi

) + k and V ci =N

k=1DkWkW

Xkxi

Momentum

tuj +

xi

uiuj = pxj +ijxi

+ N

k=1 Ykfk,j

Energy (sum of sensible and kinetic)

Et +

xi

(uiE) = T qixi + xj (ijui) + Q+ N

k=1 Ykfk,i(ui + Vk,i)

with T = N

k=1hof,kk and qi = Txi +

Nk=1 hkYkVk,i

Table 1.8: Conservation equations for reacting flows: the energy equation may be replaced by anyof the equations given in Table 1.7. Q is the external heat source term and fk measures the volumeforces applied on species k.


For most deflagrations, pressure is constant, body forces are zero (fk,j = 0), viscous heatingis negligible so that these equations may be simplified as shown in Table 1.9.

Mass

t +

uixi

= 0

Species: For k = 1 to N 1 (or N if total mass is not used)

With diffusion velocities:

Ykt +

xi

((ui + Vk,i)Yk) = k

With Hirschfelder and Curtiss approximation:

Ykt +

xi

((ui + V ci )Yk) =xi

(Dk WkWXkxi

) + k and V ci =N

k=1DkWkW

Xkxi

Momentum

tuj +

xi

uiuj = pxj +ijxi

Energy (sum of sensible and kinetic)

Et +

xi

(uiE) = T qixi with T = N

k=1hof,kk

Or temperature

CpDTDt =

T +

xi

( Txi ) Txi(N

k=1 Cp,kYkVk,i

)with T =

Nk=1 hkk and qi = Txi +

Nk=1 hkYkVk,i

Table 1.9: Conservation equations for constant pressure, low Mach number flames.

Chapter 2

Laminar premixed flames

2.1 Introduction

The elementary case of a one-dimensional laminar flame propagating into a premixed gas(fuel mixed with air for example) is a basic problem in combustion, both for theory and fornumerical techniques. Numerically solving for laminar premixed flames is of interest because:

It is one of the few configurations where detailed comparisons between experiments,theory and computations can be performed.

It may be used to validate chemical models as discussed in Section 1.1.1. Many theoretical approaches may be used for laminar flames, not only to study theirone dimensional structure but also the various instabilities which can develop on suchfronts (Williams554).

Laminar flames are viewed in many turbulent combustion models as the elementarybuilding blocks of turbulent flames (flamelet theory: see Chapt. 5).

From the point of view of numerical techniques, computing laminar premixed flames is afirst step toward more complex configurations. However, this step is not necessary to computeturbulent flames using models as shown in Chapter 5 because most turbulent combustionmodels essentially use completely different approaches: codes devoted to turbulent combustioncan usually not be used for laminar flames.

There are many ways to compute laminar flame structure and speed depending on the com-plexity of the chemistry and transport descriptions ( 2.2). With complex chemical schemes,there is no analytical solution to the problem and numerical techniques are needed ( 2.3).However, when chemistry and transport are suitably simplified, analytical or semi-analyticalsolutions may be developed and these shed an essential light on the behavior of flames andon the numerical challenges to face when one tries to compute them in simple or complex

27

28 CHAPTER 2. LAMINAR PREMIXED FLAMES

situations ( 2.4). These solutions are useful to check the precision of combustion codes andto provide initial solutions for these codes. They are also needed to determine resolutionrequirements for flames.

2.2 Conservation equations and numerical solutions

For laminar one-dimensional premixed flames, the conservation equations derived in Chapter 1can be simplified starting from Table 1.9 (the index 1 corresponding to the x direction isomitted):

Mass conservation

t+u

x= 0 (2.1)

Species conservation. For k = 1 to N 1:Ykt

+

x((u+ Vk)Yk) = k (2.2)

EnergyCp

(T

t+ u

T

x

)= T +

x

(T

x

) T

x

(Nk=1

Cp,kYkVk

)(2.3)

with T = N

k=1 hkk

These equations describe a wave propagating from the burnt to the fresh gas at a speedwhich reaches a constant value sL when transients are ignored. Solving for the structureof this flame is still a research task for complex chemistry descriptions even when the flamehas reached a constant velocity. When the flame is steady, writing Eq. (2.1) to (2.3) in thereference frame of the flame (moving at speed sL) leads to:i

u = constant = 1sL (2.6)

iNote that the momentum equation is not needed anymore. It can be used to compute the pressure fieldafter all fields have been computed by integrating (viscous terms are neglected):

p

x= uu

xor p(x) = p1 1sL(u(x) u1) (2.4)

The pressure jump through a steady flame front is obtained by integrating this equation between to :p2 p1 = 1u21(1 u2/u1) = 1s2L(1 T2/T1) (2.5)

using the notations of Fig. 2.1. These pressure jumps through flame fronts are small: for a stoichiometricmethane-air flame, with a flame speed of 0.4 ms/s and a temperature ratio T2/T1 close to 7, p2 p1 is of theorder of 1 Pa. Checking in compressible combustion codes whether they capture the correct pressure jump inpremixed flames is a good way to verify that boundary conditions and numerical techniques are adequate.

2.2. CONSERVATION EQUATIONS AND NUMERICAL SOLUTIONS 29

x((u+ Vk)Yk) = k (2.7)

CpuT

x= T +

x

(T

x

) Tx

(

Nk=1

Cp,kYkVk

)(2.8)

This set of equations is closed if a model is given for the reaction rate k (Arrhenius law)and for the diffusion velocities Vk (Ficks law with a correction velocity, for example) and ifproper boundary conditions are provided. For premixed flames, these boundary conditionsmay raise some difficulties. Typical inlet conditions (at location x = 0) correspond to a coldpremixed gas flow (Fig. 2.1): u(x = 0) = u1, Yk imposed for reactants (in proportions imposedby the code user), T (x = 0) = T1 imposed.

7

6

5

4

3

2

1

0

x=l0.0 Abscissa (x)

Normalized temperature T / T 1

Reaction rate

Normalized fuel mass fraction Y

/ YF1

STATE 1: fresh gas STATE 2: burnt gas

Figure 2.1: Basic configuration for computations of one-dimensional premixed flames. State 1: freshgas quantities; State 2: burnt gas quantities.

These conditions are not enough to determine whether the flame actually exists in thecomputational domain:

The temperature T1 is usually low and most reaction terms at this temperature arealmost zero. If this flow is computed in a parabolic way, i.e. by starting from x = 0 andupdating all variables for increasing x, the mass fractions (and other quantities as well)remain constant. Reaction rates rise but so slowly that the ignition point is rejected toinfinity. This is the cold boundary problem (Williams554). From a numerical pointof view, this paradox is easily solved by recognizing that an additional condition mustbe imposed, at least in the initial conditions: the flame must have been ignited beforeit reaches the right side of the domain and the temperature must reach the adiabaticflame temperature at the outlet of the domain (x = l). Another way to express the samepoint is that deflagrations are elliptic phenomena: what happens downstream controls

30 CHAPTER 2. LAMINAR PREMIXED FLAMES

the flame propagation upstream. This feedback mechanism is produced by thermal andspecies diffusion as shown in Section 2.4.ii

Even if the flame is ignited, steady solution can exist only if the inlet velocity u1 is equalto the flame speed sL (the flame remains on a fixed position). This means that theproblem to solve is an eigenvalue problem: the unknowns are the profiles of temperature,species and velocity but also the flame speed sL which is the eigenvalue of the problem.

2.3 Steady one-dimensional laminar premixed flames

The simplest flame configuration is a planar laminar premixed flame propagating in one di-rection. It is a one-dimensional problem. It is also a steady problem if equations are writtenin the reference frame of the flame.

2.3.1 One-dimensional flame codes

Solving Eqs. (2.6) to (2.8) is a numerical problem for which many mathematical tools havebeen developed in the last twenty years. Typical chemical schemes involve two to hundredsof species with one to thousands of reactions. When proper boundary conditions are set upand the problem is discretized on a finite difference grid, the resulting system is a stronglynon linear boundary value problem which can be written:

L(Ui) = 0 (2.9)where Ui = (T, Y1, Y2, ....YN , U)i is the vector of unknowns at point xi.

This system is usually solved with Newton-type methods. There is no need to develop suchcodes today because reliable tools already exist such as PREMIX (Kee et al.260, Kee et al.258,Kee et al.259, see also http://www.reactiondesign.com/webdocs). Fig. 2.2 shows the outputof such codes for a H2 - O2 flame at atmospheric pressure. The flame speed is 32 m/s (suchhigh deflagration speeds are obtained only for pure hydrogen - oxygen combustion and aretypical of cryogenic rocket combustion). The chemical scheme for this computation is givenin Table 1.5. For this computation, PREMIX uses an adaptive grid and the smallest grid sizeis 0.001 mm while the largest one is 0.19 mm.

PREMIX and similar one-dimensional flame codes have certain specificities. Two verydifferent tasks have to be performed with one-dimensional flame codes:

for a given situation, compute a first flame or having this first flame solution, compute other solutions for which physical parameters(equivalence ratio, temperature, pressure, etc) are varied.

iiDetonations are different: they are essentially parabolic phenomena which can be computed by time orspace marching techniques because diffusion phenomena are negligible.

2.3. STEADY ONE-DIMENSIONAL LAMINAR PREMIXED FLAMES 31

0.2 0.4 0.6 0.8 1.0x [cm]

0.0

0.2

0.4

0.6

0.8

1.0

Y H2O2OHH2OOH

0.2 0.4 0.6 0.8 1.0x [cm]

0.0000

0.0005

0.0010

0.0015

Y

HO2H2O2

0.2 0.4 0.6 0.8 1.0x [cm]

0.0

1000.0

2000.0

3000.0

4000.0

Tem

pera

ture

[K]

0.2 0.4 0.6 0.8 1.0x [cm]

5e+11

0

5e+11

1e+12

2e+12

Hea

t rel

ease

poinsot and veynante - 2005 - theoretical and numerical combustion.pdf

Documents

diffusion flame structures

flame codes

diffusion flame configurations

flame speeds512

jet flame

maximum flame temperature

premixed flame thicknesses

flame structure problems