1
2
2
2
En vue de l'obtention du
DOCTORAT DE L'UNIVERSITÉ DE TOULOUSE
Délivré par :
Institut National Polytechnique de Toulouse (INP Toulouse)
Discipline ou spécialité :
Energétique et Transferts
Présentée et soutenue par :
M. RAPHAËL MARI
le jeudi 25 juin 2015
Titre :
INFLUENCE DES TRANSFERTS THERMIQUES SUR LA STRUCTURE
ET LA STABILISATION DE FLAMME A HAUTE PRESSION DANS LES
MOTEURS FUSEES CRYOTECHNIQUES
Ecole doctorale :
Mécanique, Energétique, Génie civil, Procédés (MEGeP)
Unité de recherche :
Centre Européen de Recherche et Formation Avancées en Calcul Scientifique (CERFACS)
Directeur(s) de Thèse :
MME BENEDICTE CUENOT
M. LAURENT SELLE
Rapporteurs :
Mme PASCALE DOMINGO, CORIA
M. THIERRY SCHULLER, ECOLE CENTRALE PARIS
Membre(s) du jury :
M. RICHARD SAUREL, UNIVERSITE AIX-MARSEILLE 1, Président
M. LAURENT SELLE, INP TOULOUSE, Membre
Mme BENEDICTE CUENOT, CERFACS, Membre
Mme DANY ESCUDIE, INSA LYON, Membre
i
La fusée descendait à travers l’espace. Elle venait des étoiles et des vertiges noirs,
des scintillantes orbites et des silencieux golfes interstellaires. [...] Elle laissait
derrière elle un sillage ardent, net et silencieux. [...] Maintenant elle perdait de la
vitesse au contact des atmosphères supérieures de Mars.
Il en émanait toujours force et beauté.
Ray Bradbury, Chroniques martiennes, 1955
iii
Remerciements
Je tiens tout d’abord à remercier Thierry de m’avoir accueilli au CERFACS au sein
de l’équipe CFD. Sa disponibilité et ses conseils ont toujours été précieux. Un grand
merci à Bénédicte de m’avoir encadré pendant tout ce travail de thèse, d’avoir su
me faire comprendre quand je faisais fausse route et de m’avoir laisser retrouver le
chemin par moi-même. Un merci tout spécial pour Laurent qui, malgré la distance
CERFACS-IMFT a su me faire parvenir ses conseils que je tacherai de garder en
mémoire. Un grand merci à toute l’équipe des seniors d’avoir répondu à mes nombreuses questions: Florent, Eléonore, Antoine, Olivier, Gabriel, Jérôme et Laurent.
Je n’oublie pas les sorciers de l’équipe CSG qui sont toujours là pour régler nos
problèmes informatiques, nous faire de la place sur des disques et nous guider dans
nos premiers pas de HPC. Mes remerciements vont également à toutes les personnes
essentielles au fonctionnement du CERFACS et de ses équipes: sa directrice Catherine Lambert, Michelle, Chantal (pour le rire communicatif et les fraises tagada) et
également les secrétaires de l’équipe CFD: Marie et Nicole. Merci à tous les doctorants, stagiaires, post-doctorants que j’ai pu croiser au détour d’un couloir ou d’une
bière. Les citer tous serait trop risqué mais une pensée particulière va à Corentin,
Charlie, Adrien, Greg, Geoff, Sandrine, Damien, Antony, Mickael, Dimitrios, Luc,
Lucas, Manqi, Thomas (Leader F), Dodo, Abdullah, Jean-Christophe, Thomas J et
David.
Une pensée toute particulière va à ma famille, mes parents et ma soeur qui m’ont
donné la curiosité et le goût d’apprendre et qui m’ont soutenu durant ses longues
années d’études sans jamais remettre en cause mes envies et mes choix. Cette thèse
leur est dédiée.
Enfin une tendre pensée pour Frédérique, qui a eu la patience de me supporter
durant ces derniers mois de travail.
Contents
1 Introduction
1.1 History of rockets . . . . . . . . .
1.1.1 Antiquity . . . . . . . . .
1.1.2 Modern Era . . . . . . . .
1.2 Industrial Context . . . . . . . .
1.2.1 Satellites Evolution . . . .
1.2.2 New Market Actors . . . .
1.2.3 European Strategy . . . .
1.3 Scientific context . . . . . . . . .
1.4 State of the Art . . . . . . . . . .
1.4.1 Mixing and atomization .
1.4.2 Structure and dynamics of
1.4.3 Flame stabilization . . . .
1.4.4 Modeling and simulations
1.5 Main Goal of the Study . . . . .
I
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supercritical flames
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Real-Gas Equations & Numerical framework
2 Governing Equations
2.1 Introduction . . . . . . . . . . . . . . . . . . .
2.2 Chemical Kinetics . . . . . . . . . . . . . . .
2.3 Navier-Stokes Equations . . . . . . . . . . . .
2.3.1 Species diffusion flux . . . . . . . . . .
2.3.2 Viscous stress tensor . . . . . . . . . .
2.3.3 Heat flux vector . . . . . . . . . . . .
2.4 Filtered Equations for LES . . . . . . . . . .
2.4.1 The filtered viscous terms . . . . . . .
2.4.2 Subgrid-scale turbulent terms for LES
2.5 Models for the subgrid-stress tensor . . . . . .
2.5.1 Smagorinsky model . . . . . . . . . . .
2.5.2 WALE model . . . . . . . . . . . . . .
2.6 Equation of state . . . . . . . . . . . . . . . .
3 Real Fluid Thermodynamics
3.1 Introduction . . . . . . . . . . . . . . . . . . .
3.2 Critical Point . . . . . . . . . . . . . . . . . .
3.3 Equation of State . . . . . . . . . . . . . . . .
3.3.1 The Principle of Corresponding States
3.3.2 The Virial Equation of State . . . . .
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27
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vi
Contents
3.4
3.3.3 The van der Waals Equation of State . . . . .
3.3.4 The Peng-Robinson Equation of State . . . .
3.3.5 The Soave-Redlich-Kwong Equation of State .
3.3.6 The Benedict-Webb-Rubin Equation of State
3.3.7 Thermodynamic Coefficients . . . . . . . . . .
3.3.8 Conclusion for the EoS . . . . . . . . . . . . .
Transport properties in mixture . . . . . . . . . . .
3.4.1 Chung correlation . . . . . . . . . . . . . . .
3.4.2 Mass diffusivity . . . . . . . . . . . . . . . . .
4 Multiphysics and Code Coupling
4.1 Introduction . . . . . . . . . . . . . . . . . . . . .
4.2 Thermal conduction in solid materials . . . . . .
4.2.1 The heat equation . . . . . . . . . . . . .
4.3 Fluid/Thermal coupled problem . . . . . . . . . .
4.3.1 Conjugate Heat Transfer . . . . . . . . . .
4.3.2 Coupled Flame Wall Interaction . . . . .
4.3.3 The AVTP solver . . . . . . . . . . . . . .
4.3.4 Fluid Solver Wall Boundary Condition . .
4.3.5 Evaluation of the wall resistance constant
II
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45
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57
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73
Experimental validation
75
5 Methane Chemistry
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
5.2 Methane Chemistry . . . . . . . . . . . . . . . . . .
5.3 Chemistry for oxycombustion of methane . . . . .
5.3.1 Chemical Equilibrium . . . . . . . . . . . .
5.3.2 Kinetic scheme . . . . . . . . . . . . . . . .
5.3.3 The LU analytical mechanism . . . . . . . .
5.3.4 Implementation of the LU scheme in AVBP
6 Coaxial CH4-O2 burner - NEMO
6.1 Introduction . . . . . . . . . . . .
6.2 Numerical Setup . . . . . . . . .
6.2.1 Mesh . . . . . . . . . . . .
6.2.2 Boundary conditions . . .
6.2.3 Numerical scheme . . . .
6.3 Non-reactive Coaxial Jets . . . .
6.3.1 Characteristic numbers . .
6.3.2 Results . . . . . . . . . .
6.3.3 Hot Wire results . . . . .
6.4 Reacting flow simulations . . . .
6.4.1 Flame shape and location
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121
121
Contents
6.5
III
vii
6.4.2 Flame stabilization and chemical structure . . . . . . . . . . . 133
6.4.3 Heat flux and temperature field in the solid . . . . . . . . . . 145
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Applications to Liquid Rocket Engine Configurations
161
7 Laminar H2-O2 Flames
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Chemistry model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Article submitted to Combustion & Flame . . . . . . . . . . . . . .
163
164
164
167
8 2D Splitter Plate
8.1 Introduction . . . . . . . . . . . . . . . . . . . .
8.2 Configuration . . . . . . . . . . . . . . . . . . .
8.2.1 Mesh, Injection & Boundary conditions
8.2.2 Thermodynamic properties . . . . . . .
8.3 The heat transfer problem . . . . . . . . . . . .
8.3.1 Thermal balance . . . . . . . . . . . . .
8.3.2 Dirichlet boundary conditions . . . . . .
8.3.3 Neumann boundary conditions . . . . .
8.3.4 Convection coefficients . . . . . . . . . .
8.4 Theoretical Results . . . . . . . . . . . . . . . .
8.4.1 Dirichlet boundary conditions . . . . . .
8.4.2 Von Neumann boundary conditions . . .
8.5 Reactive LES Results . . . . . . . . . . . . . . .
8.5.1 Flame structure . . . . . . . . . . . . . .
8.5.2 Flame stabilization . . . . . . . . . . . .
8.5.3 Temperature fields and thermal fluxes .
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211
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226
IV
Conclusions and Perspectives
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231
9 Conclusion and Perspectives
233
Bibliography
235
List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
1.14
1.15
1.16
1.17
1.18
1.19
1.20
1.21
Chinese soldier launches fire-arrow. From http://www.grc.nasa.gov/WWW/k12/TRC/Rockets/history_of_rockets.html. . . . . . . . . . . . . . . .
2
Principle of Newton’s third law applied to a rocket.(from NASA http://exploration.grc.nasa.gov/education) . . . . . . . . . . . . . . .
3
Dr Robert Goddard 1926 rocket. From http://www.grc.nasa.gov/WWW/k12/TRC/Rockets/history_of_rockets.html. . . . . . . . . . . . . . . .
4
German V2 missile. From http://www.grc.nasa.gov/WWW/k-12/TRC/Rockets/history_of_roc
Ariane 5 at take-off uses the cryogenic engine Vulcain 2. SSME cryo8
genic engine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Operating principle of Vulcain2 engine. [Snecma 2011] . . . . . . . .
9
Pressure chamber related to engine thrust for several kinds of engines. [Yang 2004] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Oxygen phase diagram. Extracted from [Masquelet 2013] . . . . . . . 11
Visualizations of central N2 (vN 2 = 5 m.s−1; TN 2 = 97 K) jet surrounded by an annular He (vHe = 5 m.s−1 ; THe = 280 K) jet.
(a) subcritical chamber pressure of 1 MPa. (b) transcritical chamber
pressure of 6 Mpa. [Mayer 2003] . . . . . . . . . . . . . . . . . . . . . 13
Evolution of a cryogenic N2 jet in a quiescent N2 atmosphere with
the pressure ratio P/Pc . Left: P/Pc = 0.91; Center: P/Pc =1.22;
Right: P/Pc = 2.71. [Chehroudi 2002] . . . . . . . . . . . . . . . . . 14
Potential core reduced length lc /dl , where dl is the dense jet injector
diameter, as a function of the momentum flux ratio J. ⋄ sub-critical
cases; △,◦ near or super-critical cases. [Candel 2011] . . . . . . . . . 15
Instantaneous image of OH* emission from a cryogenic H2 /O2 flame.
Pressure is 6.3 MPa. [Singla 2006] . . . . . . . . . . . . . . . . . . . . 15
Time averaged and Abel transformed images of cryogenic H2 /O2
flame. Pressure is 7 Mpa. [Juniper 2001] . . . . . . . . . . . . . . . . 16
Diagrams of the flame structure. (a) below the critical pressure of
oxygen; (b) above the critical pressure of oxygen. [Candel 2006] . . . 16
OH-PLIF of a transcritical LOx/GH2 flame. Pressure is P=6.3 Mpa. [Candel 2006] 17
Abel transformation OH* emissions. Enlarged image of the vicinity
of the injector lip. [Juniper 2003c] . . . . . . . . . . . . . . . . . . . . 18
OH-PLIF image of the anchoring region. H2 (O2 ) stream is above
(resp. below) the splitter plate1 . Lip thickness is lp = 0.3 mm.
Pressure is P=6.3 MPa. [Singla 2006] . . . . . . . . . . . . . . . . . . 18
Transcritical flame general feature and response to strain rate. [Lacaze 2012] 19
Schematic of coaxial jet injector and the near-field mixing layers. [Schumaker 2009] 20
Temperature field in the near injector vicinity. [Oefelein 1997] . . . . 20
Flow configurations at different values of ψ. [Juniper 2003a] . . . . . 21
x
List of Figures
1.22 Coaxial transcritical O2 / supercritical CH4 injection. LES results
from [Schmitt 2010a]. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.23 Visualization of a transcritical LOx/GCH4 flame: (top) direct visualization from experiment [Singla 2005]; (bottom) T = (T max + T
min)/2 isosurface from LES [Schmitt 2010a]. [Ruiz 2012] . . . . . . .
21
22
2.1
Energy activation of forward and reverse reactions. [Pearson-Education 2011] 29
3.1
3.2
3.3
Thermodynamic properties of a pure substance . . . . . . . . . . . . 39
Disappearance of the meniscus at the critical point. [Poling 2001] . . 41
Compressibility factor Z of nitrogen as a function of reduced pressure (Pr = P/Pc ) for various isothermals (reduced temperature Tr =
T /Tc ). [Hirschfelder 1954] . . . . . . . . . . . . . . . . . . . . . . . . 42
Density predictions for pure oxygen compared with experimental data
points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Heat capacity of different pure substances: H2 , O2 and H2 O with the
Peng-Robinson EoS. Pressure is P = 10 MPa. (◦) NIST database [Lemmon 2007];
(–) AVBP PR EoS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Oxygen transport properties. Pressure is P = 10 MPa. (◦) NIST
database [Lemmon 2007]; (–) Chung model. . . . . . . . . . . . . . . 54
Water transport properties. Pressure is P = 10 MPa. (◦) NIST
database [Lemmon 2007]; (–) Chung model. . . . . . . . . . . . . . . 55
Hydrogen transport properties. Pressure is P = 10 MPa. (◦) NIST
database [Lemmon 2007]; (–) Chung model. . . . . . . . . . . . . . . 56
3.4
3.5
3.6
3.7
3.8
4.1
4.2
4.3
4.4
4.5
4.6
5.1
5.2
5.3
5.4
Sequential Coupling Strategy (SCS). Solid arrows represent data exchange, dashed arrows represent computation path. . . . . . . . . . .
Parallel Coupling Strategy (PCS). Solid arrows represent data exchange, dashed arrows represent computation path. . . . . . . . . . .
Basic features of a flame interacting with a wall. . . . . . . . . . . .
Nodes and cells used for the computation of the diffusion term ∇2 T .
Typical cooling device configuration. . . . . . . . . . . . . . . . . . .
Heat fluxes balance in the wall fluid cell. Wall flux (black arrow),
chemical source term (black spot) and heat flux from the flow (striped
arrow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Species mass fractions at equilibrium for a stoichiometric mixture at
1bar : (a) CH4 /Air ; (b) CH4 /O2 . . . . . . . . . . . . . . . . . . .
Equilibrium temperature as a function of the equivalence ratio φ at
1 bar : (a) CH4 /Air ; (b) CH4 /O2 . . . . . . . . . . . . . . . . . . .
Auto Ignition time in a PSR. . . . . . . . . . . . . . . . . . . . . . .
Auto Ignition delays as a function of the initial temperature, for a
perfectly premixed stoichiometric mixture at 1 bar for CH4 - Air. (-) LU ; (•) Gri-3.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
63
65
69
70
73
82
83
84
85
List of Figures
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.19
5.20
5.21
5.22
5.23
5.24
5.25
6.1
Auto Ignition delays as a function of the initial temperature, for a
perfectly premixed stoichiometric mixture at 1 bar for CH4 - O2 . (-) LU ; (•) Gri-3.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Flame speed as a function of equivalence ratio for (a)methane/air and
(b)methane/O2 flames. (- -) Lu scheme ; (•) Gri3.0 . . . . . . . . . .
Relative error on flame speed ∆Sl /Sl : (△) CH4 - Air; () CH4 - O2 .
Burnt gas temperature as a function of equivalence ratio for (a)
methane/air and (b) methane/O2 flames. (- -) Lu scheme ; (•) Gri3.0
Relative error ∆Tf /Tf on burnt gas temperature: (△) CH4 - Air;
() CH4 - O2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Flame thickness as a function of equivalence ratio for (a) methane/air
and (b) methane/O2 flames. (- -) Lu scheme ; (•) Gri3.0 . . . . . . .
Relative error on flame thickness ∆δl /δl : (△) CH4 - Air; () CH4 O2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Profiles for heat release rate, temperature and species mass fractions.
CH4 - Air; φ = 0.6. (- -) LU ; (–) Gri-3.0 . . . . . . . . . . . . . . .
Profiles for heat release rate, temperature and species mass fractions.
CH4 - Air; φ = 1.0. (- -) LU ; (–) Gri-3.0 . . . . . . . . . . . . . . .
Profiles for heat release rate, temperature and species mass fractions.
CH4 - Air; φ = 1.4. (- -) LU ; (–) Gri-3.0 . . . . . . . . . . . . . . .
Profiles for heat release rate, temperature and species mass fractions.
CH4 - O2 ; φ = 0.6. (- -) LU ; (–) Gri-3.0 . . . . . . . . . . . . . . . .
Profiles for heat release rate, temperature and species mass fractions.
CH4 - O2 ; φ = 1.0. (- -) LU ; (–) Gri-3.0 . . . . . . . . . . . . . . . .
Profiles for heat release rate, temperature and species mass fractions.
CH4 - O2 ; φ = 1.4. (- -) LU ; (–) Gri-3.0 . . . . . . . . . . . . . . . .
Counterflow strained diffusion flame. . . . . . . . . . . . . . . . . . .
Maximum temperature as a function of the strain rate at 1 bar and
fresh gas temperature of 300 K for CH4 - Air. (- -) LU ; (•) Gri-3.0.
Maximum temperature as a function of the strain rate at 1 bar and
fresh gas temperature of 300 K for CH4 - O2 . (- -) LU ; (•) Gri-3.0.
Profiles for heat release rate, temperature and species mass fractions.
CH4 - Air; a = 50 s−1 . (- -) LU ; (–) Gri-3.0 . . . . . . . . . . . . . .
Profiles for heat release rate, temperature and species mass fractions.
CH4 - Air; a = 300 s−1 . (- -) LU ; (–) Gri-3.0 . . . . . . . . . . . . .
Profiles for heat release rate, temperature and species mass fractions.
CH4 - Pure O2 ; a = 1000 s−1 . (- -) LU ; (–) Gri-3.0 . . . . . . . . .
Profiles for heat release rate, temperature and species mass fractions.
CH4 - Pure O2 ; a = 10000 s−1 . (- -) LU ; (–) Gri-3.0 . . . . . . . . .
Profiles for heat release rate, temperature and species mass fractions.
CH4 - O2 ; φ = 1.0. (- -) CANTERA ; (–) AVBP . . . . . . . . . . .
xi
85
86
87
88
88
89
90
92
93
94
95
96
97
98
99
99
101
102
103
104
106
The NEMO experiment : (a) Sketch of the injector, (b) Flame direct
visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
xii
List of Figures
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
6.14
6.15
6.16
6.17
6.18
6.19
6.20
Full 3D atmosphere and coaxial burner geometry. . . . . . . . . . . .
2D slices of the mesh used for NEMO. Top: Global view ; Bottom:
Injector lip region. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CH4 mass fraction and axial velocity u: mean (left) and rms(right)
fields. 2D cut in the transverse mid-plane. In figure (a) the white line
represents the iso-contour of stoichiometric methane mass fraction
YCH4 ,s = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q-criterion iso-surface colored by axial vorticity. Case M1A1. . . . .
CH4 mass fraction and axial velocity u: mean (left) and rms(right)
fields. 2D cut in the transverse mid-plane. In figure (a) the white line
represents the iso-contour of stoichiometric methane mass fraction
YCH4 ,s = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q-criterion iso-surface colored by axial vorticity. Case M10A1. . . . .
CH4 mass fraction and axial velocity u: mean (left) and rms(right)
fields. 2D cut in the transverse mid-plane. In figure (a) the white line
represents the iso-contour of stoichiometric methane mass fraction
Ys = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q-criterion iso-surface colored by axial vorticity. Case M1A10. IsoQ =
2e9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mean axial velocity profiles along a transverse line at 1mm above
the injector. Comparison between (◦) experiment (EM2C) and (–)
AVBP. (a) Case M1A1 ; (b) Case M1A10 and M10A1. . . . . . . . .
Instantaneous temperature and heat release rate fields in the median
plane down to x/din = 25. Case M 1A1, Coupled. . . . . . . . . . . .
Abel transform of a visualization with an OH ∗ filter (top) and numerical 2D cut of OH mass fraction (bottom). Case M 1A1 . . . . .
OH-PLIF (top) and numerical 2D cut of OH mass fraction (bottom).
(a) Mean fields; (b) rms fields. Case M 1A1 . . . . . . . . . . . . . .
Spreading angle θs based on the iso-contours of OH. . . . . . . . . .
Axial velocity at four downstream sections: (◦) Experimental PIV;
(–) Coupled simulation; (- -) Adiabatic simulation. Case M 1A1. Top:
mean profiles ; Bottom: rms profiles. . . . . . . . . . . . . . . . . . .
Mean profiles at four downstream sections: (–) Coupled simulation;
(−−) Adiabatic simulation. Case M 1A1. Top: Mean CH4 mass
fraction ; Bottom: Mean temperature. . . . . . . . . . . . . . . . . .
Instantaneous temperature and heat release rate fields in the median
plane down to x/din = 30. Case M 1A1, Coupled. . . . . . . . . . . .
Abel transform of direct visualization with an OH∗ filter (top) and
numerical 2D cut of OH mass fraction (bottom). (a) Mean fields; (b)
rms fields. Case M 1A10. . . . . . . . . . . . . . . . . . . . . . . . .
OH-PLIF (top) and numerical 2D cut of OH mass fraction (bottom).
(a) Mean fields; (b) rms fields. Case M 1A10. . . . . . . . . . . . . .
Mean profiles at four downstream sections: (–) coupled simulation;
(- -) adiabatic simulation. Case M 1A10. . . . . . . . . . . . . . . . .
110
111
115
115
116
117
118
119
120
122
123
124
125
126
127
129
130
131
132
List of Figures
6.21 Mean OH (left) and HO2 (right) mass fractions fields. Close view on
the injector internal lip. Case M 1A1. Coupled and Adiabatic cases. .
6.22 Radical species profiles at four downstream locations: (–) coupled
simulation; (- -) adiabatic simulation. Case M 1A1. . . . . . . . . . .
6.23 Sketch of the boxes used for the scatter plots. Boxes extend over the
whole azimuthal angle. . . . . . . . . . . . . . . . . . . . . . . . . . .
6.24 Temperature, heat release rate and radical mass fractions as functions
of the mixture fraction. (black dots) coupled simulation; (red dots)
adiabatic simulation; (grey line) reference counterflow diffusion flame
at a strain rate a = 500 s−1 computed with CANTERA. Box 1. O2
(CH4 ) is injected above (resp. below) the splitter plate. Case M 1A1.
6.25 Temperature, heat release rate and radical mass fractions as functions
of the mixture fraction. (black dots) coupled simulation; (red dots)
adiabatic simulation; (grey line) reference counterflow diffusion flame
at a strain rate a = 700 s−1 computed with CANTERA. Box 2. O2
(CH4 ) is injected above (resp. below) the splitter plate. Case M 1A1.
6.26 Heat release rate iso-contours superimposed to an instantaneous temperature field. Iso-contours range from 5 108 to 1 1010 W.m−3 and
temperature from 300 to 2960 K. O2 (CH4 ) is injected above (resp.
below) the splitter plate. Case M 1A1. . . . . . . . . . . . . . . . . .
6.27 Radical mean mass fractions fields. Close view on the injector internal
lip. O2 (CH4 ) is injected above (resp. below) the splitter plate. Case
M 1A1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.28 Radical species profiles at four downstream locations: (–) coupled
simulation; (- -) adiabatic simulation. Case M 1A10. . . . . . . . . .
6.29 Temperature, heat release rate and radical mass fractions as functions
of the mixture fraction. (black dots) coupled simulation; (red dots)
adiabatic simulation; (grey line) reference counterflow diffusion flame
at a strain rate a = 2000 s−1 computed with CANTERA. Box 1.
Case M 1A10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.30 Heat release rate iso-contours superimposed to an instantaneous temperature field. Iso-contours range from 8 108 (light grey line) to 5 1010
(blue line) W.m−3 and temperature from 300 to 2930 K. O2 (CH4 ) is
injected above (resp. below) the splitter plate. Case M 1A10. . . . .
6.31 Temperature, heat release rate and radical mass fractions as functions
of the mixture fraction. (black dots) coupled simulation; (red dots)
adiabatic simulation; (grey line) reference counterflow diffusion flame
at a strain rate a = 1000 s−1 computed with CANTERA. Box 2.
Case M 1A10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.32 Profile of the temperature at the center line of the injector wall. (△
and ◦) experimental data; (–) and (- -) coupled simulations. Case
M 1A1 and M 1A10. . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.33 2D cut of the mean fluid and solid temperature field. Injector field is
truncated at 2/3 of its length. Case M 1A1. . . . . . . . . . . . . . .
xiii
133
134
135
136
137
138
139
140
142
143
144
146
147
xiv
List of Figures
6.34 Sketch of the inner tube of the injector. The red crosses represent the
thermocouples positions. Lines used to plot the temperature: (–) O2
External skin; (- -) Half width line; (-.-) Internal skin. . . . . . . . .
6.35 Temperature in the injector. (–) External O2 skin; (- -) Injector center
line; (-.-) Internal CH4 skin. Case M 1A1. . . . . . . . . . . . . . . .
6.36 Heat flux along the faces of the injector. Case M 1A1. . . . . . . . .
6.37 Temperature in the solid and fluid density field. Case M 1A1. . . . .
6.38 Density along the injector walls. (–) External O2 skin; (-.-) Internal
CH4 skin. Case M 1A1. . . . . . . . . . . . . . . . . . . . . . . . . .
6.39 y + along the injector walls. (–) External O2 skin; (-.-) Internal CH4
skin. Case M 1A1. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.40 2D cut of the mean fluid and solid temperature field. Case M 1A10. .
6.41 Temperature in the injector. (–) External O2 skin; (- -) Injector center
line; (-.-) Internal CH4 skin. Case M 1A10. . . . . . . . . . . . . . .
6.42 Heat flux along the faces of the injector. Case M 1A10. . . . . . . . .
6.43 Temperature of the injector and impact on the density field. Case
M 1A10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.44 Density profiles along the injector walls. (–) External O2 skin; (-.-)
Internal CH4 skin. Case M 1A10. . . . . . . . . . . . . . . . . . . . .
6.45 y + along the injector walls. (–) External O2 skin; (-.-) Internal CH4
skin. Case M 1A10. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.46 Axial heat flux along centerline of the injector. (–) Case M 1A1 ; (-) Case M 1A10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
148
148
150
151
151
152
153
154
155
156
157
158
159
7.1
Flame-Wall interaction: Head-On Quenching. . . . . . . . . . . . . . 164
8.1
8.2
8.3
198
201
Computational domain and injection parameters. [Ruiz 2011b] . . . .
Heat fluxes in an elementary surface. φc is for conductive flux. . . . .
Temperature profile along the 1D splitter. Dirichlet boundary condition with Teq = 445K . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4 Penetration length in the splitter as a function of the equivalent temperature Teq at the front face of the splitter. . . . . . . . . . . . . . .
8.5 Temperature profile along the 1D splitter. Neumann boundary condition with φ = 6.89 107 W.m−2 . . . . . . . . . . . . . . . . . . . . .
8.6 Instantaneous field of heat release rate. Zoom on the splitter plate
region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.7 Instantaneous field of heat release rate. The boxes represent two
different zones used for the scatter plots. . . . . . . . . . . . . . . . .
8.8 Scatter plot of the temperature vs. mixture fraction in the splitter
configuration. Box 1. . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.9 Scatter plot of the Heat Release Rate vs. the mixture fraction in the
splitter configuration. Box 1. . . . . . . . . . . . . . . . . . . . . . .
8.10 Scatter plot of the H2 O2 radical vs. the mixture fraction in the
splitter configuration. Box 1. . . . . . . . . . . . . . . . . . . . . . .
205
206
208
210
211
211
212
212
List of Figures
8.11 Scatter plot of the HO2 radical vs. the mixture fraction in the splitter
configuration. Box 1. . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.12 Scatter plot of the OH radical vs. the mixture fraction in the splitter
configuration. Box 1. . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.13 Scatter plot of the Temperature vs. the mixture fraction in the splitter configuration. Box 2. . . . . . . . . . . . . . . . . . . . . . . . . .
8.14 Scatter plot of the Heat Release Rate vs. the mixture fraction in the
splitter configuration. Box 2. . . . . . . . . . . . . . . . . . . . . . .
8.15 Scatter plot of the H2 O2 radical vs. the mixture fraction in the
splitter configuration. Box 2. . . . . . . . . . . . . . . . . . . . . . .
8.16 Scatter plot of the HO2 radical vs. the mixture fraction in the splitter
configuration. Box 2. . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.17 Scatter plot of the OH radical vs. the mixture fraction in the splitter
configuration. Box 2. . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.18 Schematic representation of the flame stabilized in the recirculation
zone behind the injector. Extracted from [Singla 2006] . . . . . . . .
8.19 Instantaneous field of heat release rate in the splitter configuration. .
8.20 Instantaneous field of the OH mass fraction in the splitter configuration. Liquid oxygen and gaseous hydrogen are injected below and
above the step, respectively. . . . . . . . . . . . . . . . . . . . . . . .
8.21 Position of the maximum OH mass fraction in the near vicinity of
the injector at several instants for the splitter configuration. . . . . .
8.22 Instantaneous field of H2 O2 mass fraction. For clarity reasons scaling maximum is 2.778 10−5 in the adiabatic case and 2 10−3 in the
coupled case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.23 Instantaneous field of HO2 mass fraction. For clarity reasons scaling
maximum is 5.976 10−4 in the adiabatic case and 1.2 10−3 in the
coupled case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.24 Instantaneous reaction rate of Reaction 10. For clarity reasons the
scale maximum is 3.936 104 in the adiabatic case and 5.283 105 in
the coupled case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.25 Instantaneous field of temperature. Adiabatic simulation. . . . . . .
8.26 Instantaneous field of temperature. Coupled simulation of the splitter
case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.27 Solid stationary thermal fields. . . . . . . . . . . . . . . . . . . . . .
8.28 Time evolution of the heat flux on a probe located on the front face
of the splitter close to the flame anchoring region. . . . . . . . . . . .
xv
213
213
214
214
215
215
216
217
218
220
221
223
224
225
226
227
228
229
List of Tables
3.1
3.2
Critical point data for some species. . . . . . . . . . . . . . . . . . . 40
Empirical constants and strain-rate employed in the BWR EoS (Eq. 3.18)
for methane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1
Summary of the isothermal methods. Rc is a relaxation coefficient
[J.m−2 .K −1 ], δt the time step [s], Rw the wall resistance [W.m−2 .K −1 ]. 72
5.1
5.2
5.3
5.4
Specific impulse of some ergols. . . . . . . . . . . . . . . . . . . . . . 78
The three set of species. . . . . . . . . . . . . . . . . . . . . . . . . . 81
Schmidt numbers used in AVBP for the LU mechanism. . . . . . . . 105
Flame speed, burnt gas temperature and flame thickness for 1D premixed flame. P = 1.0 bar, T = 300 K and φ = 1.0. . . . . . . . . . . 105
6.1
6.2
Mesh properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Boundary condition parameters. The Relax term refers to a partially
non-reflecting inlet/outlet. . . . . . . . . . . . . . . . . . . . . . . . .
Numerical parameters. . . . . . . . . . . . . . . . . . . . . . . . . . .
Test cases of the NEMO configuration. . . . . . . . . . . . . . . . . .
Injector Steel 316L properties and characteristic numbers. . . . . . .
6.3
6.4
6.5
111
112
113
113
145
7.1
Rate coefficients in Arrhenius form k = AT n exp(−E/R0 T ) as in [Boivin 2011]
.
a Units are mol, s, cm3 , kJ and K.
b Chaperon efficiencies H : 2.5, H O : 16.0, 1.0 for all other species.
2
2
Troe falloff with Fc = 0.5.
c Chaperon efficiencies H : 2.5, H O : 12.0, 1.0 for all other species.
2
2
d Chaperon efficiencies H : 2.5, H O : 6.0, 1.0 for all other species.
2
2
Troe falloff with Fc = 0.265 exp(−T /94) + 0.735 exp(−T /1756) +
exp(−T /5182). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
8.1
8.2
8.3
Injection properties. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Injection characteristics. . . . . . . . . . . . . . . . . . . . . . . . . .
Species critical-point properties (temperature Tc , pressure Pc , molar
volume Vc and acentric factor ωac ) and Schmidt numbers Sc. . . . .
Fluid properties & parameters using NIST database [Lemmon 2009]
for the splitter case. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Penetration length as a function of the equivalent front face temperature Teq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4
8.5
199
200
200
204
206
Chapter 1
Introduction
Contents
1.1
1.2
History of rockets . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.1.1
Antiquity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.1.2
Modern Era . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Industrial Context . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2.1
Satellites Evolution . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2.2
New Market Actors
6
1.2.3
. . . . . . . . . . . . . . . . . . . . . . .
European Strategy . . . . . . . . . . . . . . . . . . . . . . . .
7
1.3
Scientific context . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.4
State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.5
1.4.1
Mixing and atomization . . . . . . . . . . . . . . . . . . . . .
12
1.4.2
Structure and dynamics of supercritical flames . . . . . . . .
14
1.4.3
Flame stabilization . . . . . . . . . . . . . . . . . . . . . . . .
17
1.4.4
Modeling and simulations . . . . . . . . . . . . . . . . . . . .
17
Main Goal of the Study . . . . . . . . . . . . . . . . . . . . . .
23
2
1.1
1.1.1
Chapter 1. Introduction
History of rockets
Antiquity
F
irst recorded use of a "rocket" was by the Chinese (Fig. 1.1) during the 13th
century with the invention of the gunpowder used to propel the "fire arrow". This
technology was then known by the Europeans and the Arabs thanks to the invasion
of the Mongol army during the invasions of the Middle Age. Arab literature of the
end of the 13th century describes gunpowder recipes, 22 of which are described for
rockets. They all use saltpeter as ground component.
Figure 1.1: Chinese soldier launches fire-arrow. From http://www.grc.nasa.gov/WWW/k12/TRC/Rockets/history_of_rockets.html.
During the 16th century, Conrad Haas, a hungarian military engineer, wrote a book
describing the technology of multi-stage rockets, liquid fuel, delta-shaped fins and
bell-shaped nozzles. During the 18th century, first iron-cased rockets were created,
increasing the range of a rocket up to 2 km.
1.1.2
Modern Era
The theory of interplanetary rocket was developed by [Tsiolkovsky 1903], inspired by
the writings of Jules Verne. He developed the so-called Tsiolkovsky equation Eq. 1.1.
His work advocated the use of liquid hydrogen and oxygen for propellants, calculating their maximum exhaust velocity:
m0
∆v = ve ln
(1.1)
m1
where m0 is the initial mass of the rocket, m1 the final mass, ve the exhaust velocity
and ∆v is the maximum change of velocity. Acceleration is obtained by the ejection of burnt gas. Through the third Newton’s law, the so-called "action-reaction
principle", the momentum of the burnt gas creates the thrust as shown in Fig. 1.2
“ For every action, there is an equal and opposite reaction force."
1.1. History of rockets
3
Figure 1.2: Principle of Newton’s third law applied to a rocket.(from NASA http://exploration.grc.nasa.gov/education)
Independently, Robert Esnault-Pelterie, a french engineer, re-developed the rocket
equation Eq. 1.1 in [Esnault-Pelterie 1913]. He also calculated the energy required
to make round trips to the Moon and proposed the use of atomic power.
In the United States, Robert Goddard published in [Goddard 1919] three major
improvements that are still in use:
• the fuel should be burned in a small combustion chamber in order to increase
the pressure.
• the rocket should be composed of several stages.
• the exhaust speed should be increased to beyond the sound speed by using a
nozzle.
In 1926, he was the first to launch a liquid-fueled rocket (Fig. 1.3) with an attached
supersonic de Laval nozzle to the combustion chamber. This technology doubled
the thrust and increased the efficiency of the engine from 2% to 64%.
During the 1930s, the Soviet Union built over 100 experimental engines under the
direction of Valentin Glushko. They developed many technologic improvements such
as regenerative cooling, hypergolic propellants, swirling and bi-propellants mixing
injectors.
In 1927, a group of amateur rocket engineers formed the German Rocket Society,
and launched in 1931 their first liquid propellant rocket using oxygen and gasoline.
A young rocket scientist of the German Rocket Society, Wernher von Braun, joined
the military research team founded in Peenemünde. In 1943 they built the V-2
rocket which is similar to most modern rockets with turbopumps and inertial guidance.
After World War II and the capture of the german engineers by the Allies and the
Russians, the V-2 rocket, shown in Fig. 1.4, evolved into the Redstone rockets in
4
Chapter 1. Introduction
Figure 1.3: Dr Robert Goddard 1926 rocket. From http://www.grc.nasa.gov/WWW/k12/TRC/Rockets/history_of_rockets.html.
the USA and into the R-1, R-2 and R-5 missiles in the USSR.
Figure 1.4:
German V2 missile.
12/TRC/Rockets/history_of_rockets.html.
From
http://www.grc.nasa.gov/WWW/k-
This launched the modern era of the rocket industry and space conquest of the
second part of the 20th century. The race for space between the United States and
the USSR really started in 1957 with the first artificial satellite orbiting around the
1.1. History of rockets
5
Earth: Sputnik 1. In 1958, the United States created the National Aeronautics and
Space Agency (NASA) with "the goal of peaceful exploration of space for the benefit
of all humankind."1
Since then, many satellites were launched into space for various uses: weather forecast, earth and universe observation, telecommunications, GPS...
1
quotation from http://www.grc.nasa.gov/WWW/k-12/TRC/Rockets/history_of_rockets.html.
6
1.2
1.2.1
Chapter 1. Introduction
Industrial Context
Satellites Evolution
Commercially, rocketry is the enabler of all space technologies, particularly satellites, which impact people’s everyday lives. Scientifically, the launch of satellites and
space probes, such as the recent Rosetta probe with the landing of the robot Philae
on a comet, opens a window on the universe. Moreover the manned spaceflights
have enabled scientific research in low gravity environment. Astronautical launch
mastery is an important challenge to ensure satellites orbit. A good launcher must
be powerful enough to carry heavy loads, reliable to ensure loads safety and its
functional costs must be as low as possible. For the past decades, the increase of
satellite weights, especially for telecommunication, has implied an increase of rocket
engines size and thrust.
Nevertheless, in the last decade the space industry has been affected by budget reductions. This situation has encouraged the use of "micro-satellites" (satellites with
a mass < 100 kg) either for scientific missions or for commercial use, as an option
to develop space activities with limited budgets ([Assembly 2008]). For example, in
2012 the Defense Advanced Research Project Agency (DARPA) announced a program that is intending to release a constellation of 24 micro-satellites (∼ 20 kg) used
for Earth observation, each with 1-meter imaging resolution ([Lindsey 2012]).
On the other hand, big geostationary satellites are still launched at a rate of 20 per
year, e.g. for telecommunications.
1.2.2
New Market Actors
While micro-satellites has been carried to space as secondary payloads with large
launchers, an increasing number of commercial companies are currently developing micro-satellites dedicated launch vehicles. In 2012, Virgin Galactic announced
LauncherOne, a vehicle designed to launch small loads of 100 kg to low-Earth orbit.
The development of air-launched vehicle is supported by Boeing and propose to
drive launch costs down to 7000 US$/kg. The SpaceX company, has recently built
and tested a re-usable launch vehicle, the Falcon 9 Reusable Development Vehicle,
that is able to take-off and to land vertically. Several tests have been conducted in
the last few years.
China is currently developing a large launcher, Changzeng 5 with a low operating
cost.
A Swiss company, Swiss Space System (S3), has announced the development of a
suborbital spaceplane that would launch small satellites up to 250 kg into low-Earth
orbit.
In New Zealand, the RocketLab company develops its own 18-meter carbon composite rocket with a LOx-Kerosene engine. They announce a launch cost less than
US$5 million.
1.2. Industrial Context
1.2.3
7
European Strategy
In Europe, the main commercial launcher is the Ariane 5 (Fig. 1.5(a)) developed
more than 20 years ago by the European Space Agency (ESA) and the French Center for Space Studies (CNES). The maximum payload is 21 tons for low-Earth orbit
(LEO) and ∼ 10 tons for geostationary transfer orbit (GTO). In 2009, Arianespace,
the company that exploits Ariane 5, thanks to 63 consecutive successful launches
hold more than 60% of the GTO international market. Ariane 5 is built by a consortium of ∼ 1100 companies under the supervision of Airbus Defense & Space.
Another launcher, Vega, with a smaller payload between 300 kg and 2.5 tons has
been launched for the first time in 2012 and has orbited 9 earth observation small
satellites to LEO. Its propulsion is mainly ensured by solid rocket for the three lower
stages and by a liquid ergol engine for the last stage.
A picture of the Ariane 5 launcher and its principal cryogenic engine is provided
in Fig. 1.5(a) and Fig. 1.5(b). Such a launcher uses several kinds of propellants:
solid for the boosters and liquid for the main engines of lower and upper stages.
The launcher is able to orbit 2 satellites per launch. This characteristic is the major
drawback of Ariane 5 because of the difficulty to find two satellites compatible for the
same launch. This capacity is essential to ensure the rentability of a launch. Despite
the success of the Ariane 5, the evolution of the satellites, the rising of new actors
and their ability to lower the launch cost by ∼ 30% has driven the main european
companies, Airbus Defense & Space and Safran, to regroup their space activities in
a joint venture [Gallois 2014]. They also decided to propose the configuration of the
new Ariane 6, so far decided by the ESA and the CNES. This decision was made on
a triple objective: to lower the development cost, to reduce the development delay
and to minimize the operation cost. The european governments have restricted the
budget to 800 million euros.
The main conclusions of a report of the CNES in 2009, confirmed the necessity of
a re-ignitable cryogenic LOx/LH2 upper stage engine: the Vinci. It also concluded
that the first stage should be propelled only by solid propellers. The latest design
of Ariane 6 includes two or four solid rocket boosters P120 for the first stage, the
Vulcain 2 engine (Fig. 1.5(b)) which equips the first stage of Ariane 5 for the second stage and the Vinci re-ignitable engine for the upper stage [CNES 2015]. The
modularity of the first stage allows to launch either one intermediate size satellite
with two solid boosters, or, one big or two intermediate size satellites with the four
boosters configuration.
8
Chapter 1. Introduction
(a) Ariane 5 at take-off.
(b) SSME (Space Shuttle Main Engine) rocket cryogenic
engine test firing. from NASA Stenis space center.
Figure 1.5: Ariane 5 at take-off uses the cryogenic engine Vulcain 2. SSME cryogenic
engine.
1.3. Scientific context
1.3
9
Scientific context
In a liquid rocket engine (LRE), oxygen is injected at very low temperature, typically
around 100 K, and low velocity, whereas the hydrogen is injected at very high
velocity (200 m.s−1 ). Figure 1.6 shows Vulcain 2 operating principle. Some of
the propellants are burned in a gas generator and the resulting hot gas supply the
turbopumps. Liquid reactants are injected in the combustion chamber thanks to the
high velocity turbopumps, where, thanks to turbulence and to molecular diffusion,
they can mix, burn and expand increasing the pressure. Thanks to the nozzle, the
velocity of the burnt gas increases converting the chamber pressure into thrust.
Figure 1.6: Operating principle of Vulcain2 engine. [Snecma 2011]
As shown by Eq. 1.2, thrust is proportional to pressure chamber therefore the need
for high thrust levels leads to high pressure combustion chamber.
F = ṁ ve + (Pe − Patm )Ae
(1.2)
where F is the thrust, ṁ the ejected mass per time unit, ve the ejection velocity, Pe
(Patm ) the ejection (resp. the atmospheric) pressure and Ae the nozzle exit section.
This phenomenon has been highlighted by [Yang 2004] as shown in Fig. 1.7.
10
Chapter 1. Introduction
Figure 1.7: Pressure chamber related to engine thrust for several kinds of engines. [Yang 2004]
1.3. Scientific context
11
High-pressure, low-temperature combustion using liquid fuels is a desirable mode
of operation in state-of-the-art propulsion and power devices. Depending on the
system, this combination of design attributes has the potential to provide both
high-performance and/or minimal emissions. These potential benefits are offset by
a variety fundamental questions. Many of these questions center around the fact that
combustion processes within the chamber are inherently turbulent and in many cases
operating pressures approach the critical point. Indeed operation at elevated pressure significantly increases the system Reynolds number, i.e., integral-scale Reynolds
number are of the order of O(106 ). Moreover at near-critical or supercritical conditions, thermodynamic non-idealities and transport anomalies can become dominant
necessitating a important modification of fluid and transport descriptions. Figure 1.8
exhibits the different states of pure oxygen in a pressure-temperature diagram. As
shown when the pressure is higher than the critical pressure, the liquid-vapor phase
line vanishes and the fluid becomes supercritical. Arrows represent the different
transformations of an injected fluid. These concepts will be developed in Chap. 3.
Figure 1.8: Oxygen phase diagram. Extracted from [Masquelet 2013]
12
Chapter 1. Introduction
1.4
State of the Art
Research over the past decade has provided significant insights into the structure and dynamics of multiphase flows at high pressures as in [Bellan 2000] and
[Oschwald 2006]. Research have been conducted on the trans- and supercritical
thermodynamic regimes phenomena: atomization, vaporization and mixing in non
reactive conditions ; structure, dynamics and stabilization of cryogenic flames. First
experimental observations reported in the literature are described then a review of
the different numerical studies is conducted.
1.4.1
Mixing and atomization
Non-reactive experiments at DLR [Mayer 1998, Mayer 2003, Oschwald 1999] focusing on mixing and vaporization of nitrogen or oxygen, have shown that, depending
on the pressure, injected liquid jets exhibit two different sets of evolutionary processes as shown in Fig. 1.9.
• When the chamber pressure is lower than the critical pressure of the fluid
(subcritical regime), the classical situation is observed where a well defined
interface separates the liquid from the gas due to surface tension. Interactions
between dynamic shear forces and surface tension lead to primary atomization
and the well known cascade of processes associated with the formation of
a heterogeneous spray. This phenomenon has been studied in experiments
by [Mayer 2004].
• The situation becomes quite different, however, as chamber pressures approach
or exceed the critical pressure of the injected fluid. Under these conditions,
liquid jets undergo a transcritical change of state as interfacial fluid temperatures rise above the critical temperature. Surface tension vanishes and the
distinct gas-liquid interface can not be identified anymore. Thus, instead of
two distinct phases, the coaxial jet evolves as a single phase jet with very large
thermophysical gradients.
These types of configurations have also been considered by [Chehroudi 2002] with
a nitrogen jet into a chamber of gaseous nitrogen at rest. Figure 1.10 illustrates
the different jet behaviors as pressure increases. At subcritical pressure (P/Pc =
0.91), the jet exhibits a classical two-phase spray behavior, with breakup structures
and droplets. When the pressure increases, (P/Pc = 1.22 and P/Pc = 2.71) the
dense jet dissolves in the ambient gas with no evidence of droplet generation. At
reduced pressure slightly larger than unity (P/Pc = 1.22), "finger-like" structures
develop at the jet edge. As pressure is increased, the jet evolves similarly to a
variable density turbulent gas flow. The fractal dimension of the jet mixing layer
boundary in the transcritical case was found similar to the one of classical turbulent
jets [Chehroudi 2002, Chehroudi 2004]. Many studies [Oschwald 1999, Mayer 2003,
Oschwald 2006, Branam 2003, Oschwald 2002] confirmed that quantitative measurements of density, density spreading rate and density decay coefficients obey the same
1.4. State of the Art
13
Figure 1.9: Visualizations of central N2 (vN 2 = 5 m.s−1; TN 2 = 97 K) jet surrounded by
an annular He (vHe = 5 m.s−1 ; THe = 280 K) jet. (a) subcritical chamber pressure of 1
MPa. (b) transcritical chamber pressure of 6 Mpa. [Mayer 2003]
14
Chapter 1. Introduction
laws as variable density flows. It was also found that the high density contrast in
the injector near-field leads to a strong anisotropy of the turbulent structures.
Figure 1.10: Evolution of a cryogenic N2 jet in a quiescent N2 atmosphere with
the pressure ratio P/Pc . Left: P/Pc = 0.91; Center: P/Pc =1.22; Right: P/Pc =
2.71. [Chehroudi 2002]
Analytical developments conducted by [Lasheras 1998, Rehab 1997, Villermaux 1998,
Villermaux 2000] have shown that coaxial mixing is essentially governed by the momentum flux ratio J between the dense inner jet and the "gaseous" outer jet :
J = ρg vg2 /ρl vl2 . The length of the central dense core is inversely proportional to
the momentum flux ratio J: lc ∝ 1/J α . As shown in Fig. 1.11, in sub-critical cases
the exponent α is close to 0.2 and in near or super-critical cases α is around 0.5
indicating that pressure has a strong influence on the jet development.
1.4.2
Structure and dynamics of supercritical flames
Many cryogenic flames researches started in 1991 with the Onera Mascotte test
bench [Habiballah 1996]. A similar experiment was set up later at DLR [Mayer 1996,
Mayer 2003]. Analysis is based on backlighting and OH* emission imaging integrated along the line of sight. Examples of OH* visualizations are presented
in Fig. 1.12 and Fig. 1.13.
As for pure mixing cases, experimental investigations at supercritical pressures show
high density gradients. No drops or ligaments are created and highly wrinkled fingerlike structures are dissolved into the surrounding fuel jet as shown in Fig. 1.14. One
can observe big pockets of dense fluid mixing with the burnt gas downstream. The
flame expansion angle is also smaller in supercritical than in subcritical regime. The
droplets spray at subcritical pressure needs to evaporate before burning leading to a
wider flame brush. At trans- or supercritical pressures, the coaxial flame is thinner
and stays closer to the cold dense oxygen jet.
1.4. State of the Art
15
Figure 1.11: Potential core reduced length lc /dl , where dl is the dense jet injector diameter, as a function of the momentum flux ratio J. ⋄ sub-critical cases; △,◦ near or
super-critical cases. [Candel 2011]
Figure 1.12: Instantaneous image of OH* emission from a cryogenic H2 /O2 flame. Pressure is 6.3 MPa. [Singla 2006]
16
Chapter 1. Introduction
Figure 1.13: Time averaged and Abel transformed images of cryogenic H2 /O2 flame.
Pressure is 7 Mpa. [Juniper 2001]
Figure 1.14: Diagrams of the flame structure. (a) below the critical pressure of oxygen;
(b) above the critical pressure of oxygen. [Candel 2006]
1.4. State of the Art
17
In comparison with direct OH* visualization, time-averaged or Abel-transformed,
the Planar Laser Induced Fluorescence (PLIF) technique provides more detailed images of the flame stabilization region. OH PLIF images were obtained by [Singla 2006]
in the near injector field for a LOx/GH2 flame at pressure P=6.3MPa. They showed
in Fig. 1.15 that the flame is well established in the wake of the inner coaxial injector
lip.
Figure 1.15: OH-PLIF of a transcritical LOx/GH2 flame.
Mpa. [Candel 2006]
1.4.3
Pressure is P=6.3
Flame stabilization
Flame stabilization mechanisms influence the engine behavior and can lead to strong
flame oscillations, pressure fluctuations and combustion abnormalities. Experiments [Juniper 2003c, Singla 2006] on coaxial LOx/GH2 or LOx/GCH4 cryogenic
injectors indicate that the flame is anchored in the near vicinity of the LOx injector
lip as shown in Fig. 1.16.
[Singla 2006] showed that the flame is well established in the very near region of the
injector lip. Even though the GH2 velocity can be very important, leading to very
high strain rates, the flame remains anchored at the injector lip. A zoom on the
injector near-field region is shown in Fig. 1.17. These visualizations showed that the
maximum of the OH* emission is located at a short distance of the order of one lip
thickness from the injector.
1.4.4
Modeling and simulations
Mixing layers at high pressure levels have been considered since the beginning of the
2000s with [Bellan 2000, Okong’o 2002a, Miller 2001, Yang 2000, Okong’O 2003]. A
2
Image has been vertically flipped from the original paper in order to keep the consistency with
the rest of the present manuscript.
18
Chapter 1. Introduction
Figure 1.16: Abel transformation OH* emissions. Enlarged image of the vicinity of the
injector lip. [Juniper 2003c]
Figure 1.17: OH-PLIF image of the anchoring region. H2 (O2 ) stream is above (resp. below) the splitter plate2 . Lip thickness is lp = 0.3 mm. Pressure is P=6.3 MPa. [Singla 2006]
1.4. State of the Art
19
review can be found in [Bellan 2006]. As a general conclusion of these studies, one
can retain that due to the presence of high density gradients, anisotropy plays an
important role in mixing processes.
The impact of supercritical thermodynamics was also studied in more realistic configurations such as round jets. LES computations were carried out by [Schmitt 2010b,
Zong 2004, Zong 2006, Petit 2013] reproducing the experimental results of [Chehroudi 2002,
Mayer 2003]. Because of the lack of real gas models, the subgrid scale models used
for these studies is issued from the perfect gas theory and a good agreement was
found with experimental results.
Under transcritical conditions, the local flame layer is very thin and as a consequence
can be described as a strained flame. Then trans- or supercritical strained flames
have been studied recently in [Ribert 2008, Pons 2009a, Lacaze 2012, Giovangigli 2011].
In the flame region, the temperature is very high and it was concluded that real gas
effects have a negligible influence. Moreover as the Damkohler number is large, the
flamelet assumption is valid to describe the flame, and the heat release rate per
flame surface unit is proportional to the square root of strain rate and pressure.
It was also shown that the extinction strain rate increases linearly with respect to
pressure and can reach very high values up to 106 s−1 [Juniper 2003c].
(a) Temperature field and flow topology of a (b) Strain rate effect on the H2/O2 counter
typical counterflow diffusion flame.
flow flame: maximum flame temperature.
(TO2 = 120 K; TH2 = 295 K and P = 7
MPa).
Figure 1.18: Transcritical flame general feature and response to strain rate. [Lacaze 2012]
Many researches have been conducted in a coaxial injector configuration as in Fig. 1.19
where a transcritical oxygen jet is surrounded by an annular supercritical hydrogen or methane jet as in [Kim 2011, Masquelet 2009, Tucker 2008, Schmitt 2010a,
Ruiz 2011c].
[Oefelein 1997, Oefelein 2005, Oefelein 2006, Zong 2007] have chosen a detailed approach and the numerical domain covers only the near vicinity of the coaxial injector.
It was shown that because of the weak diffusion of the transcritical oxygen, a rich
20
Chapter 1. Introduction
Figure 1.19:
Schematic of coaxial jet injector and the near-field mixing layers. [Schumaker 2009]
mixture appears in the shear layer. The simultaneous presence of a strong recirculating zone, hot burnt gas and this rich mixture allow the flame to firmly anchor
at the tip of the injector as shown in Fig. 1.20.a. In comparison in Fig. 1.20.b,
because of the high oxygen temperature, the two reactants mix faster and the flame
tip oscillates closer to the H2 layer. All these studies confirm that combustion take
place in a diffusion flame regime. However, in [Ruiz 2011a] it was shown that partially premixed rich pockets of reactants can detach from the splitter plate, move
downstream and finally burn during an engulfment process in the dense oxygen, in
"finger-like" structures.
(a) O2 transcritical injection temperature.
(b) O2 supercritical injection temperature.
Figure 1.20: Temperature field in the near injector vicinity. [Oefelein 1997]
The stabilization of the flame tip at the injector rim has later been confirmed in other
experimental [Juniper 2000, Singla 2007] and numerical studies [Juniper 2003a]. In [Juniper 2003a]
1.4. State of the Art
21
the authors have identified that the flame anchoring process is essentially determined
by the ratio ψ of the lip height to the flame thickness. When ψ > 1 the flame edge
is stuck behind the lip and the flame is stable. Contrary, when ψ < 1 the flame is
thicker than the lip and, as it becomes sensitive to the high velocity flow, it can be
unstable or blown away. Both situations are represented in Fig. 1.21.
(a) ψ < 1
(b) ψ > 1
Figure 1.21: Flow configurations at different values of ψ. [Juniper 2003a]
Very recent work studying transcritical oxygen and supercritical methane in a 3D
coaxial injection [Schmitt 2010a] uses an infinitely fast flame model assumption.
The flame is found stable in the near region of the injector lip before wrinkling
under the effect of the turbulent structures created by the jet as shown in Fig. 1.22.
The flame global structure is found to be in good agreement with [Singla 2005] as
shown in Fig. 1.23.
(a) Isosurface of temperature (T = 1750 K) (b) Turbulent fluctuations represented by the
colored by axial velocity.
Q-criterion colored by axial velocity.
Figure 1.22: Coaxial transcritical O2 / supercritical CH4 injection.
from [Schmitt 2010a].
LES results
22
Chapter 1. Introduction
Figure 1.23: Visualization of a transcritical LOx/GCH4 flame: (top) direct visualization from experiment [Singla 2005]; (bottom) T = (T max + T min)/2 isosurface from
LES [Schmitt 2010a]. [Ruiz 2012]
1.5. Main Goal of the Study
1.5
23
Main Goal of the Study
In the above studies, either adiabatic or isothermal conditions have been considered.
However, as in subcritical combustion systems, heat transfer plays a crucial role in
flame structure and stabilization. In addition, heat fluxes transmitted to the solid
parts of the injector and chamber are very strong and capable of important damages
The work presented throughout this thesis attempts to evaluate and understand
the influence of the heat transfer in the injector of a liquid rocket engine on the
structure and the stabilization mechanisms of cryogenic flames. It relies on multiphysics calculations by coupling unsteady solvers in the context of high performance
computing (HPC).
The outline of this document is as follows:
Part 1: Tools The solver, namely AVBP and the LES filtered equations are presented. The supercritical thermodynamics and its related equations are then
described. Special attention is paid to the transport and mixture phenomena. The multi-physics and the coupling methodology, along with the heat
conduction solver, AVTP, are presented.
Part 2: Validation This methodology is then evaluated on a methane/oxygen
coaxial burner, namely NEMO, studied at EM2C laboratory by Thomas Schmitt
and Philippe Scoufflaire.
Part 3: Application The coupled methodology is finally applied to H2 /O2 realgas cases. First, we assess the influence of the pressure up to 100 bar in a 1D
flame wall interaction configuration. Finally the 2D configuration of a splitter
plate, representative of a lip of a coaxial injector, is studied to evaluate the
influence of the heat transfer on the structure and the stabilization mechanisms
of the flame.
Part I
Real-Gas Equations & Numerical
framework
Chapter 2
Governing Equations
Contents
2.1
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.2
Chemical Kinetics . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.3
Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . .
30
2.4
2.5
2.6
2.3.1
Species diffusion flux . . . . . . . . . . . . . . . . . . . . . . .
31
2.3.2
Viscous stress tensor . . . . . . . . . . . . . . . . . . . . . . .
32
2.3.3
Heat flux vector . . . . . . . . . . . . . . . . . . . . . . . . .
32
Filtered Equations for LES . . . . . . . . . . . . . . . . . . . .
33
2.4.1
The filtered viscous terms . . . . . . . . . . . . . . . . . . . .
34
2.4.2
Subgrid-scale turbulent terms for LES . . . . . . . . . . . . .
34
Models for the subgrid-stress tensor . . . . . . . . . . . . . .
35
2.5.1
Smagorinsky model . . . . . . . . . . . . . . . . . . . . . . . .
36
2.5.2
WALE model . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
Equation of state . . . . . . . . . . . . . . . . . . . . . . . . . .
36
28
Chapter 2. Governing Equations
2.1
Introduction
This chapter presents the conservation equations of aerothermochemistry. Combustion implies working with a multi-species and multi-reactions system. Each species
k is characterized by several quantities:
• the mass fraction Yk = mk /m defined as the ratio between the mass mk of the
species and the total mass m in a given volume V ,
• the density ρk = ρYk where ρ is the mixture density,
• the atomic weight Wk of species k,
• the specific heat capacity at constant pressure Cp,k ,
• the mass enthalpy hk = hs,k +∆h0f,k . The mass enthalpy hk is composed of the
RT
sensible enthalpy hs,k given by hs,k = T0 Cp,k dT , and the chemical enthalpy
equal to the mass enthalpy of formation ∆h0f,k at temperature T0 .
The mean molecular weight W of a mixture composed of N species is then given
by:
N
X Yk
1
=
W
Wk
(2.1)
k=1
The mole fraction Xk of species k is defined as the ratio between the number of
moles nk of species k and the total number of moles n of the mixture:
Xk =
W
nk
=
Yk
n
Wk
(2.2)
The molar concentration of species k is then defined as the moles of species k per
unit volume:
[Xk ] =
2.2
nk
nk
Xk
=
=ρ
V
nW/ρ
W
(2.3)
Chemical Kinetics
During combustion reactants are transformed into products thanks to a set of reactions which occur if the energy of the system is high enough. Generally N species
react through M reactions as:
N
X
k=1
′
νkj
Mk ⇋
N
X
k=1
′′
νkj
Mk
(2.4)
′ and ν ′′ are the molar stochiometric
where Mk is the symbol for species k, νkj
kj
coefficients of species k for reaction j such as:
2.2. Chemical Kinetics
N
X
k=1
29
′′
′
(νkj
− νkj
)Wk =
N
X
νkj Wk = 0
(2.5)
k=1
This equation ensures the mass conservation.
The mass species reaction rate per unit volume ω̇k describes the rate of production
(resp. destruction if negative) of species k due to chemical reactions. Each reaction
j contributes to the source term of the species k, ω̇k following its progress rate Qj :
ω̇k = Wk
M
X
j=1
(2.6)
νkj Qj
The reaction rate progress Qj is expressed as:
Qj = Kjf
N
Y
k=1
′
[Xk ]nkj − Kjr
N
Y
′′
[Xk ]nkj
(2.7)
k=1
where n′kj and n′′kj are the forward and reverse orders of reaction j for species k. Kjf
and Kjr are the forward and reverse reaction constants for reaction j. In its simplest
formulation, the forward reaction constant Kjf is expressed using an Arrhenius law:
Eja
Kjf = Aj T βj exp −
RT
(2.8)
where Aj is the pre-exponential constant which models the collision frequency, the
geometry and the orientation of the molecules during collisions, βj describes the
thermal excitation of the molecules and Eja is the activation energy, i.e. the minimum
amount of energy necessary to enhance the reaction. Forward and reverse reactions
are characterized by two different activation energies as shown on Fig. 2.1.
Figure 2.1: Energy activation of forward and reverse reactions. [Pearson-Education 2011]
30
Chapter 2. Governing Equations
Equation 2.8 describes the probability that an atom exchange occurs during molecular collisions. The reverse reaction constant is expressed as:
Kjr = Kjf /Kjeq
(2.9)
where Kjeq is the equilibrium constant defined as:
Kjeq
p P N
k=1 νkj
0
=
exp
RT
∆Sj0 ∆Hj0
−
R
RT
!
(2.10)
where p0 = 1 bar. ∆Hj0 and ∆Sj0 are respectively the enthalpy and the entropy
changes of the reaction j.
In a constant pressure system, the heat released by combustion is obtained by summing the heat of creation of each species k and subtracting the heat of creation
ω̇T = −
N
X
∆h0f,k ω̇k
(2.11)
k=1
where ∆h0f,k is the enthalpy of formation of species k at temperature T0 .
So far all the equations have been written with the "perfect gas" assumption. As
the equilibrium constant Kjeq implies thermodynamic considerations, real gas effects
have to be taken into account in its computation and will be develop in Chap. 7.
2.3
Navier-Stokes Equations
This section presents the conservation equations implemented in the numerical solver
AVBP used in later DNS and LES studies.
Throughout this document, it is assumed that the conservation equations for a supercritical fluid flow are similar to a perfect-gas variable-density fluid. The only
difference between the two is that they are coupled to different equations of state
which will be developed in details in Chap. 3. Throughout this part, the index notation (Einstein’s rule of summation) is adopted for the description of the governing
equations. Note however that index k always refers to the k th species and will not
P
follow the summation rule unless specifically mentioned or implied by the
sign.
The set of conservation equations describing the evolution of a compressible flow
2.3. Navier-Stokes Equations
31
with chemical reactions reads:
∂ρ
∂
+
(ρ uj ) = 0
∂t
∂xj
(2.12)
∂ρ ui
∂
∂
+
(ρ ui uj ) = −
[p δij − τij ]
∂t
∂xj
∂xj
(2.13)
∂
∂
∂ρ E
+
(ρ E uj ) = −
[ui (p δij − τij ) + qj ] + ω̇T
∂t
∂xj
∂xj
(2.14)
∂ρk
∂
∂
+
(ρk uj ) = −
[Jj,k ] + ω̇k .
∂t
∂xj
∂xj
where
ρ=
X
ρk
(2.15)
(2.16)
k
In Eq. 2.12 to 2.15, which respectively correspond to the conservation laws for mass,
momentum, total energy and species, the following symbols (ρ, ui , E, ρk ) denote
the density, the velocity vector, the total energy per unit mass (E = ec + e, with ec
the kinetic energy and e the internal energy). p denotes the pressure, τij the stress
tensor Eq. 2.24, qj the heat flux vector (see Eq. 2.26) and Jj,k the vector of the
diffusive flux of species k (see Eq. 2.23). The species source term ω̇k in the species
transport equation Eq. 2.15) comes from the consumption or production of species
by chemical reactions as defined in Eq. 2.6. The heat source term ω̇T in the total
energy equation in Eq. 2.14) is the heat release rate, which comes from the energy
variation associated to the species source terms Eq. Eq. 2.11
The fluxes of the Navier-Stokes equations can be split into two parts:
• Inviscid fluxes
• Viscous fluxes
2.3.1


ρ ui uj + P δij
 (ρE + P δij ) uj 
ρk u j


−τij
 −(uj τij ) + qj 
Jj,k
(2.17)
(2.18)
Species diffusion flux
In multi-species flows the total mass conservation implies that
N
X
Yk Vik = 0
(2.19)
k=1
where Vik are the components of the diffusion velocity of species k. They are often expressed as a function of the species molar concentration gradients using the
32
Chapter 2. Governing Equations
Hirschfelder-Curtis approximation:
Xk Vik = −Dk
∂Xk
,
∂xi
(2.20)
where Xk is the molar fraction of species k as defined in Eq. Eq. 2.2 Reformulating Eq. 2.20 in terms of mass fraction gives:
Yk Vik = −Dk
Wk ∂Xk
W ∂xi
(2.21)
Summing Eq. 2.21 over all k, shows that the expression Eq. 2.21 does not necessarily comply with Eq. 2.19 that expresses mass conservation. As a consequence, a
~ c is added to the diffusion velocity V k to ensure global
correction diffusion velocity V
i
mass conservation [Poinsot 2005]:
Vic =
N
X
k=1
Dk
Wk ∂Xk
W ∂xi
The diffusive species flux for each species k finally writes:
Wk ∂Xk
c
− Y k Vi
Ji,k = −ρ Dk
W ∂xi
(2.22)
(2.23)
Here, Dk is the diffusion coefficients of species k in the mixture (see Chap. 3).
Using Eq. 2.23 to determine the diffusive species flux implicitly verifies Eq. 2.19.
2.3.2
Viscous stress tensor
The stress tensor τij is given by:
1
τij = 2µ Sij − δij Sll
3
(2.24)
where Sij is the rate of strain tensor and µ is the dynamic viscosity (see Chap. 3).
∂uj
1 ∂ui
Sij =
+
(2.25)
2 ∂xj
∂xi
2.3.3
Heat flux vector
For multi-species flows, an additional heat flux term appears in the diffusive heat
flux. This term is due to heat transport by species diffusion. The total heat flux
vector then takes the form:
N
N X
X
∂T
∂T
Wk ∂Xk
qi =
−λ
− Yk Vic hk = −λ
+
Ji,k hk
−ρ
Dk
∂xi
W ∂xi
∂xi
k=1
| {z }
{z
}
| k=1
Heat conduction Heat flux through species diffusion
(2.26)
2.4. Filtered Equations for LES
33
where λ is the heat conduction coefficient of the mixture (see Chap. 3) and hk the
mass enthalpy of the species k [Meng 2003].
Dufour terms in Eq. 2.26 and Soret terms in Eq. 2.23 have been neglected, because
they are supposed to be second-order terms for the type of flow investigated here. In
similar conditions to the present document, [Oefelein 2006] showed that they were
negligible compared to the other terms.
2.4
Filtered Equations for LES
The basic concept of Large-Eddy Simulation (LES) is to solve the filtered NavierStokes equations: the filtered quantity f is resolved in the numerical simulation
whereas the subgrid-scale quantity f ′ = f − f is modeled. Modeling the small scales
of turbulence, is based on the principle that they have a universal dissipation role
of the turbulent kinetic energy.
For variable density ρ, a mass-weighted Favre filtering of f is introduced such as:
ρfe = ρf
(2.27)
The conservation equations for LES are obtained by Favre-filtering the instantaneous
equations 2.13, 2.14 and 2.15:
∂ρ uei
∂
∂
+
(ρ uei uej ) = −
[P δij − τij − τij t ]
∂t
∂xj
∂xj
e
∂
∂ρ E
e uej ) = − ∂ [uj (P δij − τij ) + qj + qj t ] + ω̇T
+
(ρ E
∂t
∂xj
∂xj
fk
∂
∂ρ Y
fk uej ) = − ∂ [Jj,k + Jj,k t ] + ω̇k
+
(ρ Y
∂t
∂xj
∂xj
(2.28)
(2.29)
(2.30)
In equations 2.28, 2.29 and 2.30, there are now four types of terms to be distinguished: inviscid fluxes, viscous fluxes, source terms and subgrid-scale terms.
In this section, the subgrid-scale models for heat, mass and momentum fluxes are
assumed to bear the same form as in the perfect-gas case.
Inviscid fluxes:
These terms are equivalent to the unfiltered equations except that they now contain
filtered quantities:

ρuei uej + P δij
e uej + P uj δij 
 ρE
ρk uej

(2.31)
Viscous fluxes:
The viscous terms take the form:

−τij
 −(uj τij ) + qj 
Jj,k

(2.32)
34
Chapter 2. Governing Equations
Filtering the balance equations leads to unclosed quantities, the subgrid-scale fluxes,
which need to be modeled, as presented in Sec. 2.4.2.
Subgrid-scale turbulent fluxes:
The subgrid-scale fluxes are:


−τij t
 q t 
(2.33)
 j 
t
Jj,k
2.4.1
The filtered viscous terms
The laminar filtered stress tensor τij is given by the following relations (see [Poinsot 2005]):
τij
and
= 2µ(Sij − 31 δij Sll ),
≈ 2µ(Seij − 31 δij Sell ),
1
Seij =
2
∂e
uj
∂e
ui
+
∂xj
∂xi
The filtered diffusive species flux vector is:
(2.34)
(2.35)
,
c
k ∂Xk
Ji,k = −ρ Dk W
−
Y
V
i
k
W ∂xi
e
Wk ∂ X
fk Vei c ,
≈ −ρ Dk W ∂xik − Y
(2.36)
where higher order correlations between the different variables are assumed negligible.
The filtered heat flux is :
PN
∂T
qi = −λ ∂x
+
k=1 Ji,k hk
i
(2.37)
P
e
f
e ∂T + N J h
≈ −λ
∂xi
k=1
i,k
k
where hk is the enthalpy of species k. These expressions assume that the spatial
variations of molecular diffusion fluxes are negligible and can be modeled through
simple gradient assumptions.
2.4.2
Subgrid-scale turbulent terms for LES
As highlighted above, filtering the transport equations yields a closure problem,
which requires modeling of the Subgrid-Scale (SGS) turbulent fluxes (see Eq. 2.4).
The Reynolds tensor is introduced as:
τij t = −ρ (ug
ei u
ej )
i uj − u
(2.38)
This tensor is usually modeled with the turbulent-viscosity hypothesis (or Boussinesq’s hypothesis):
1 e
t
e
(2.39)
τij = 2 ρ νt Sij − δij Sll ,
3
2.5. Models for the subgrid-stress tensor
35
which relates the SGS stresses to the filtered rate of strain, similar to the exact
stress and rate of strain (see Eq. 2.24). Models for the SGS turbulent viscosity νt
are presented in Sec. 2.5.
The subgrid-scale diffusive species flux vector is:
t
Ji,k = ρ ug
ei Yek ,
(2.40)
i Yk − u
t
Ji,k is modeled with a gradient-diffusion hypothesis:
t
Ji,k = −ρ
ek
∂X
fk Vei c,t
−Y
W ∂xi
Wk
Dkt
with
Dkt =
!
,
νt
t
Sc,k
(2.41)
(2.42)
t = 0.6 is the same for all species. The turbulent
The turbulent Schmidt number Sc,k
correction velocity reads:
Veic,t =
N
X
ek
µ t Wk ∂ X
,
t
ρSc,k W ∂xi
(2.43)
k=1
with νt = µt /ρ.
The subgrid-scale heat flux vector is:
e
qi t = ρ(ug
ei E),
iE − u
(2.44)
where E is the total energy. The SGS turbulent heat flux qet also takes the same
form as its molecular counterpart (see Eq. 2.26):
N
∂ Te X
t f̄
qi = −λt
+
Ji,k h
k,
∂xi
t
(2.45)
k=1
with
λt =
µt Cp
.
Prt
(2.46)
The turbulent Prandtl number Prt is set to a constant value of 0.6.
2.5
Models for the subgrid-stress tensor
Models for the subgrid-scale turbulent viscosity νt are an essential part of a LES.
The SGS turbulence models are derived on the theoretical ground that the LES filter
is spatially and temporally invariant. Variations in the filter size due to non-uniform
meshes are not directly accounted for. Change of cell topology is only accounted for
1/3
through the use of the local cell volume, defined as △ = Vcell .
36
2.5.1
Chapter 2. Governing Equations
Smagorinsky model
In the Smagorinsky model, the SGS viscosity νt is obtained as:
q
2
νt = (CS △) 2 Seij Seij
(2.47)
where CS is the model constant set to 0.18 but can vary between 0.1 and 0.18 depending on the flow configuration. The Smagorinsky model [Smagorinsky 1963] was
developed in the 1960s and heavily tested for multiple flow configurations. This
closure is characterized by its globally correct prediction of kinetic energy dissipation in homogeneous isotropic turbulence. However, it predicts non-zero turbulent
viscosity levels in flow regions of pure shear, which makes it unsuitable for many
wall-bounded flows [Nicoud 1999]. This also means that it is also too dissipative in
transitioning flows [Sagaut 2002], such as turbulent jets.
2.5.2
WALE model
In the WALE model, the expression for νt takes the form:
νt = (Cw △)2
with
(sdij sdij )3/2
(Seij Seij )5/2 +(sdij sdij )5/4
(2.48)
1
1
2
2
(f
gij 2 + gf
gf
(2.49)
ji ) −
kk δij
2
3
Cw = 0.4929 is the model constant and geij denotes the resolved velocity gradient.
The WALE model [Nicoud 1999] was specifically developed for wall bounded flows
and allows to obtain correct scaling laws near the wall in turbulent flows. This
model allows the transitioning of shear flows in LES.
sdij =
2.6
Equation of state
The above conservation equations, both non-filtered and filtered, must be completed
by a relation between P, T and ρ, i.e. the equation of state (EoS ). In the perfect
gas regime, the EoS is simply:
P = ρrT
(2.50)
where r = R/W .
This controls the evolution of the energy and the acoustics in the system and has a
direct impact on the boundary conditions. In the case of real gas, cubic EoS must
be used as described in the next chapter (Chap. 3).
Chapter 3
Real Fluid Thermodynamics
Contents
3.1
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
3.2
Critical Point . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
3.3
Equation of State . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.4
3.3.1
The Principle of Corresponding States . . . . . . . . . . . . .
44
3.3.2
The Virial Equation of State . . . . . . . . . . . . . . . . . .
44
3.3.3
The van der Waals Equation of State . . . . . . . . . . . . . .
45
3.3.4
The Peng-Robinson Equation of State . . . . . . . . . . . . .
45
3.3.5
The Soave-Redlich-Kwong Equation of State . . . . . . . . .
46
3.3.6
The Benedict-Webb-Rubin Equation of State . . . . . . . . .
46
3.3.7
Thermodynamic Coefficients . . . . . . . . . . . . . . . . . .
49
3.3.8
Conclusion for the EoS . . . . . . . . . . . . . . . . . . . . . .
51
Transport properties in mixture
. . . . . . . . . . . . . . . .
52
3.4.1
Chung correlation . . . . . . . . . . . . . . . . . . . . . . . .
52
3.4.2
Mass diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . .
53
38
Chapter 3. Real Fluid Thermodynamics
3.1
Introduction
In thermodynamics, a system is defined as a quantity of matter of given mass and
chemical composition. The system interacts with the external environment through
a set of boundaries. An isolated system is one that is not influenced in any way
by the surroundings. This means that no heat or work cross the boundary of the
system. Each substance can exist in the gas, liquid or solid phase and is homogeneous
throughout this phase. In each phase, the substance may exist at various pressures
and temperatures or, in thermodynamic terms, various states. A state is defined by
several macroscopic properties, which can be temperature, pressure or volume. A
property can be defined as any quantity that depends on the state of the system
and is independent of the path to reach this state. Properties are also called state
functions.
Thermodynamic properties can be divided into two general classes : intensive and
extensive. An intensive property is independent of mass, while an extensive property
varies proportionally with mass. Thus, if a quantity of matter in a given state
is divided into two equal parts, each part will have the same value of intensive
properties and half the original value of the extensive properties. For example, mass
and volume are extensive. Extensive properties expressed per unit mass become
intensive properties such as the specific volume.
An equation of state is a relation between state functions. More specifically, an
equation of state is a thermodynamic equation describing the state of matter under
a given set of physical conditions. It is a constitutive equation which provides a
mathematical relationship between two or more state functions associated with the
matter, such as its temperature, pressure, volume, or internal energy. Equations of
state are useful to describe the properties of fluids, mixtures of fluids, solids, and
even the interior of stars. 1
1
Perrot, Pierre (1998). A to Z of Thermodynamics. Oxford University Press
3.2. Critical Point
3.2
39
Critical Point
As shown in Fig. 3.1(a), the phase diagram of a pure substance in (P, T ) coordinates
shows several unique phase zones (i.e. pure gas, liquid or solid), separated by lines
where two phases coexist. In the liquid phase, density is very high, molecular
interaction is strong and Van der Waals forces stabilize molecules around their
equilibrium position. In the gaseous phase, the temperature is high and molecular
interactions is negligible.
(a) Pure species phase diagram. [Clifford 2002]
(b) Isothermals in the pressure - volume diagram.
Figure 3.1: Thermodynamic properties of a pure substance
Let us consider a system where both phases, liquid and gas, coexist. Inside a volume
of liquid each molecule is attracted by others in all directions in a similar way, so that
the resultant forces are zero. At the interface between liquid and gas, the situation
is completely different, forces toward the inside of the liquid are much stronger than
40
Chapter 3. Real Fluid Thermodynamics
toward the gas, resulting in a sharp interface. This surface disequilibrium generates
a force called surface tension. At a temperature, Fig. 3.1(b) shows that vaporization
vanishes. Instead, the critical point is an inflection point where the saturated-liquid
and the saturated-vapor states are identical. This point is characterized by its temperature Tc , its pressure Pc , and its volume Vc . The critical point data for some
substances are given in Tab. 3.1.
Substance
Formula
Hydrogen
Nitrogen
Oxygen
Water
H2
N2
O2
H2 O
Tc
(K)
Pc
(MPa)
Vc
3
(m /kmol)
33.3
126.2
154.8
647.3
1.30
3.39
5.08
22.09
.0649
.0899
.0780
.0568
Table 3.1: Critical point data for some species.
In Fig. 3.1(a), pressure and temperature increase together, along the gas-liquid coexistence line, while liquid density decreases due to thermal expansion and gas density
increases due to pressure increase. As a consequence, the disequilibrium between
liquid and gas molecular interaction reduces and the surface tension diminishes.
This phenomenon is illustrated in Fig. 3.2, where it can be seen that the separating interface thickens and finally completely disappears as both temperature and
pressure increase beyond the critical point. In supercritical state, liquid and gas
states can not be clearly identified anymore and are simply replaced by a "fluid
state". However, beyond Pc , temperatures below the critical temperature (T < Tc )
will be called compressed liquid, while temperature above the critical temperature
(T > Tc ) will be called superheated gas. Nevertheless, it should be kept in mind
that these two "pseudo-states" do not correspond to separate states but rather to
different thermodynamics conditions.
3.2. Critical Point
41
Figure 3.2: Disappearance of the meniscus at the critical point. [Poling 2001]
Another peculiar characteristic of supercritical fluids is that their thermodynamic
and transport properties are intermediate between those of a gas and a liquid, as
outlined by [Bellan 2000, Oschwald 2006]. Supercritical fluids exhibit important
characteristics such as compressibility, homogeneity, and a continuous change from
“gas-like" to “liquid-like" properties. For example, surface tension vanishes and solubility is close to that of a gas. On the other hand, density and thermal diffusivity can
be comparable to that of a liquid. These properties are characteristic of conditions
inside the hatched area in Fig. 3.1(a).
At the highest temperatures (T ≫ Tc ), isotherms exist only at high molar volumes
(Fig. 3.1(b)) and follow a law of the form P ∝ 1/V , i.e. that of a perfect gas. As
the temperature is lowered, an inflexion in the isotherms appears corresponding to:
∂ 2 p =0
(3.1)
∂V 2 T
As the temperature is decreased further, the slope at the point of inflexion also
decreases until it becomes zero at T = Tc . Hence, at the critical point it can be
written:
∂ 2 p ∂p =
=0
(3.2)
∂V 2 T =Tc
∂V T =Tc
As a consequence the isothermal compressibility, κT defined by:
−1 ∂V κT =
V ∂p T
(3.3)
42
Chapter 3. Real Fluid Thermodynamics
tends to infinity at the critical point.
κT gives the rate of change in volume with pressure at constant temperature. κT
is high in the critical region which is also characteristic of supercritical and transcritical fluids. Another quantity can be used to describe fluid behavior around
the critical point, namely the compression, or compressibility, factor Z = pV /RT ,
where R = 8.314 J mol−1 K−1 is the universal gas constant. For ideal gas Z = 1.
For many substances, values of Z at the critical point are Zc ∼ 0.28 (except for
some highly polar substances like water). Figure 3.3 illustrates the evolution of the
compressibility factor of nitrogen. Important deviation from unity is observed in
the vicinity of the critical point which means that the behavior of the fluid is no
more that of a perfect gas.
Figure 3.3: Compressibility factor Z of nitrogen as a function of reduced pressure (Pr =
P/Pc ) for various isothermals (reduced temperature Tr = T /Tc ). [Hirschfelder 1954]
3.3. Equation of State
3.3
43
Equation of State
The equation of state (EoS in the following) must be adapted to supercritical fluids to ensure accuracy of the thermodynamics. Indeed, the density or the pressure
dependence of the heat capacities are dramatically changed compared to ideal gas
regime and directly driven by the EoS. Another example is the prediction of acoustics, as the speed of sound directly results from the EoS. From a practical point of
view, the EoS is a model resulting from a trade off between accuracy and computational cost. Cubic equations of state are usually good candidates in this respect
for supercritical fluids. So far seven thermodynamics functions of state have been
discussed:
• pressure P
• molar volume V = W/ρ where W is the molar mass and ρ the density
• temperature T
• isothermal compressibility κT
• compressibility factor Z
• internal energy e
• enthalpy h
Now two additional functions of state are introduced:
• heat capacity at constant volume Cv
• heat capacity at constant pressure Cp
The state of a system at equilibrium is defined by any two of these nine functions,
most commonly V and T or P and T . It is indeed possible to write relationships for
any function of state in terms of any two others. Thus if V and T are known for a
particular system, the remaining functions of state can be determined. For example
P ≡ P (V, T )
(3.4)
This equation is the most common form for an EoS. For a perfect gas one can write:
P =
RT
V
(3.5)
Then the EoS allows to write all the thermodynamic functions from two functions
via standard thermodynamic relationships.
44
3.3.1
Chapter 3. Real Fluid Thermodynamics
The Principle of Corresponding States
The principle of corresponding states is based on the reduced pressure Pr , the reduced temperature Tr and the reduced volume Vr :
Pr = P/Pc
;
Tr = T /Tc
;
Vr = V /Vc
(3.6)
The reduced quantities can be related through an equation of state:
Pr ≡ Pr (Tr , Vr )
(3.7)
According to [Hirschfelder 1954], the function Pr (Tr , Vr ) is a universal function for
all substances. Thus in principle the behavior of any substance can be predicted
from that of another substance, using the critical parameters. The principle extends to any thermodynamic variable when reduced by suitable critical constants,
and predicts a constant Zc as previously mentioned.
However, the principle of corresponding states does not make very accurate predictions for non-spherical molecules. This has been improved by introducing the
concept of acentric factor ω, defined by [Pitzer 1955] in Eq. 3.8:
ω = − ln pr (Tr = 0.7) − 1
(3.8)
The extended EoS is modified to integrate the acentric factor ω as follows:
Pr ≡ Pr (Tr , Vr , ω)
3.3.2
(3.9)
The Virial Equation of State
An equation for the pressure of a substance can be written in terms of temperature
and volume in the following form of the virial EoS :
A(T ) B(T ) C(T ) D(T )
P
=
+
+
+
+ ···
RT
V
V2
V3
V4
(3.10)
The infinite series on the right-hand side of Eq. 3.10 is found to converge above the
critical temperature and is perfectly general, as the virial coefficients A(T ), B(T ),...
are arbitrary functions of temperature. However, at very large V , a substance tends
to behave as a perfect gas yielding A(T ) = 1 and B = C = ... = 0. It can be shown
by statistical mechanics that the second and higher order terms on the right-hand
side arise from molecular interactions of the same order. In a situation where bimolecular interaction cause an appreciable deviation from perfect gas behavior, but
ter-molecular and higher interactions are negligible, the substance can be described
as a dilute gas and only the second order term is kept. Thus we have:
1
B(T )
P
≈
+
RT
V
V2
(3.11)
3.3. Equation of State
45
The second virial coefficient B(T ) is equal to the volume occupied by one mole of
the gas less the volume the gas would occupy at the same pressure and temperature
if it were behaving like a perfect gas. Hence B(T ) is negative at low temperature,
where attractive forces are important, and positive at higher temperature, where
the effect of attractive forces is no more significant.
3.3.3
The van der Waals Equation of State
The Van der Waals EoS in its simplest form is not very accurate, but very useful
to show the qualitative behavior of supercritical fluids. For a one-component fluid
it is given by:
P =
a
RT
−
V −b V2
(3.12)
where a and b are constants, known as the Van der Waals parameters. The above
equation is an adaptation of the perfect-gas EoS in which the volume has been
reduced by b, the so-called excluded volume, to account for the physical size of
the molecules, and the pressure has been increased by a/V 2 , to account for the
attraction between the molecules. As an example the critical volume and density
given by the Van der Waals EoS are erroneous by about 50%. Although the Van
der Waals can not be retained for our computations due to its low accuracy, it gives
a qualitative results that can still be useful, for example reproducing the behavior
seen in Fig. 3.1(a).
3.3.4
The Peng-Robinson Equation of State
A more complex and realistic EoS has been proposed in [Peng 1976], and widly
applied in the field of supercritical fluids. The Peng-Robinson (PR) EoS is of the
family of cubic EoSs as the Van der Waals EoS. For the PR EoS, the second term of
the Van der Waals equation is modified by making a a function of the temperature
(a ≡ a(T )) and introducing b in the second term, as shown below
P =
a(T )
RT
− 2
V − b V + 2V b − b2
(3.13)
where a and b are calculated from the critical temperature Tc and pressure Pc
a(Tc ) = 0.45724 R2 Tc2 /Pc
;
b = 0.07780 RTc /Pc
(3.14)
The function a(T ) was then obtained by Peng and Robinson by fitting the EoS on
experimental data for hydrocarbon vapor:
a(T ) = a(Tc ){(1 + 0.37464 + 1.5422ω − 0.26992ω 2 )(1 − Tr0.5 )}2
where ω is the acentric factor and Tr is the reduced temperature.
(3.15)
46
3.3.5
Chapter 3. Real Fluid Thermodynamics
The Soave-Redlich-Kwong Equation of State
The Soave-Redlich-Kwong (SRK) EoS, given by [Reid 1987, Chapter 3], is another
form of cubic EoS with a good compromise between accuracy and numerical cost.
It writes:
P (ρ, T ) =
where
a=
and
ρRT
ρ2 a
−
1 − ρb 1 + ρb
i2
Ωa R2 Tc2 h
1 + f (ω)(1 − Tr1/2 )
Pc
;
(3.16)
b=
Ωb RTc
Pc
f (ω) = 0.48 + 1.57ω − 0.176ω 2
with Ωa and Ωb constants. This equation is an extension of the original RedlichKwong EoS that introduces a temperature dependence into the cohesive energy
terms a and b and incorporates the acentric factor ω. When evaluated in bar,
mol/liter, and K units the constants Ωa and Ωb take the values of 0.42748 and
0.086640, respectively. Introducing the compressibility factor, Eq. 3.16 can be
rewritten as:
P = ρZRT
(3.17)
where Z is the solution of:
with
Z 3 − Z 2 + (A − B − B 2 )Z − AB = 0
A=
3.3.6
aP
R2 T 2
;
B=
bP
RT
The Benedict-Webb-Rubin Equation of State
The 32-terms Benedict-Webb-Rubin (BWR) EoS is given by [Jacobsen 1973]. Contrary to the EoS previously introduced the BWR EoS is not a cubic equation. It
writes:
P (ρ, T ) =
9
X
n=1
an (T )ρn +
15
X
an (T )ρ2n−17 e−γρ
2
(3.18)
n=10
where
a1 (T ) = RT
a2 (T ) = N1 T + N2 T 1/2 + N3 + N4 /T + N5 /T 2
a3 (T ) = N6 T + N7 + N8 /T + N9 /T 2
a4 (T ) = N10 T + N11 + N12 /T
a5 (T ) = N13
a6 (T ) = N14 /T + N15 /T 2
a7 (T ) = N16 /T
a8 (T ) = N17 /T + N18 /T 2
a9 (T ) = N19 /T 2
a10 (T ) = N20 /T 2 + N21 /T 3
a11 (T ) = N22 /T 2 + N23 /T 4
a12 (T ) = N24 /T 2 + N25 /T 3
a13 (T ) = N26 /T 2 + N27 /T 4
a14 (T ) = N28 /T 2 + N29 /T 3
a15 (T ) = N30 /T 2 + N31 /T 3 + N32 /T 4
3.3. Equation of State
47
The quantities Ni , i = 1, . . ., 32, and γ represent semi-empirical constants that
give the best fit through experimentally acquired data. The quantity γ is called the
strain-rate. The relevant constants for methane, as given by [Ely 1992], are listed in
Table 3.2. Eqs. 3.17 and 3.18 are all written in terms of molar quantities to remain
consistent with the constants given in the literature.
Fig. 3.4 illustrates the effectiveness of this EoS in predicting the behavior of oxygen
within the thermodynamic regimes of interest. Here, density versus temperature
is plotted over the interval 40 ≤ T ≤ 1000 K and for pressures of 1, 10, 50, 100,
200, and 400 atmospheres. Predicted values are compared with the experimental
data obtained by [Vargaftik 1975, Chapter 6]. The average deviation between the
experimental data and the calculated densities is less than 1%.
Figure 3.4: Density predictions of pure oxygen compared with experimental data points
— : BWR equation of state ; , N, •, , △, ◦ : data points obtained by [Vargaftik 1975]
48
Chapter 3. Real Fluid Thermodynamics
Table 3.2: Empirical constants and strain-rate employed in the BWR EoS (Eq. 3.18) for
methane.
N1 =
−1.184347314485 × 10−2
N17 =
1.071143181503 × 10−5
N2 =
7.540377272657 × 10−1
N18 =
−9.290851745353 × 10−3
N3 =
−1.225769717554 × 10+1
N19 =
1.610140169312 × 10−4
N4 =
6.260681393432 × 10+2
N20 =
3.469830970789 × 10+4
N5 =
−3.490654409121 × 10+4
N21 =
−1.370878559048 × 10+6
N6 =
5.301046385532 × 10−4
N22 =
1.790105676252 × 10+2
N7 =
−2.875764479978 × 10−1
N23 =
1.615880743238 × 10+6
N8 =
5.011947936427 × 10+1
N24 =
6.265306650288 × 10−1
N9 =
−2.821562800903 × 10+4
N25 =
1.820173769533 × 10+1
N10 =
−2.064957753744 × 10+5
N26 =
1.449888505811 × 10−3
N11 =
1.285951844828 × 10−2
N27 =
−3.159999123798 × 10+1
N12 =
−1.106266656726 × 10+0
N28 =
−5.290335668451 × 10−6
N13 =
3.060813353408 × 10−4
N29 =
1.694350244152 × 10−3
N14 =
−3.174982181302 × 10−3
N30 =
8.612049038886 × 10−9
N15 =
5.191608004779 × 10+0
N31 =
−2.598235689063 × 10−6
N16 =
−3.074944210271 × 10−4
N32 =
3.153374374912 × 10−5
Units for pressure, density, and temperature are given in bar, mol/liter, and K.
The strain-rate parameter γ = 0.04 (mol/liter)−2 .
3.3. Equation of State
3.3.7
49
Thermodynamic Coefficients
At low pressure, the thermodynamic coefficients such as the heat capacity only depend on the temperature: Cp0 ≡ Cp0 (T ). A standard technique consists in tabulating
or using polynomial fits to allow for the temperature dependence. At high pressure,
these coefficients also depend on the pressure and their formulation must be modified to take into account real gas effects. This procedure can be extended to account
for pressure dependence by keeping the tabulation for low pressure reference values
and use departure functions based on the EoS to compute the influence of pressure [Poling 2001]. For example to calculate the constant-pressure heat capacity
Cp , one starts to write the Gibbs function G as:
Z v0
(3.19)
G(P, T ) = G0 + P v − RT +
P (v̄, T )dv̄
v
where v0 and G0 are respectively the molar volume and the Gibbs energy at a
reference low pressure. The enthalpy h is then classically defined as:
∂G
(3.20)
h=G−T
∂T P
as well as the constant-pressure heat capacity:
∂h
Cp =
∂T P
(3.21)
This technique is applied for all the thermodynamics properties such as: the internal
energy e, the enthalpy h, the entropy s and the heat capacities Cp and Cv . Figure 3.5 shows results of heat capacity evaluation at P=10 MPa with the technique
describes above using the PR EoS. The agreement is good except for the hydrogen
where the effect of tabulation method is observable at very low temperature. This
points out that low-pressure data, combined with the PR-EoS, allow to compute all
thermodynamic properties of the fluid at high pressure. More details can be found
in [Schmitt 2009].
50
Chapter 3. Real Fluid Thermodynamics
4
1.6x10
-1
-1
Cp [J.kg .K ]
1.5
1.4
1.3
1.2
150
200
250
300
350
400
450
500
Temperature [K]
(a) H2
3000
-1
-1
Cp [J.kg .K ]
4000
2000
1000
100
200
300
400
500
Temperature [K]
(b) O2
4
1.0x10
-1
-1
Cp [J.kg .K ]
0.8
0.6
0.4
0.2
1000
2000
3000
Temperature [K]
(c) H2 O
Figure 3.5: Heat capacity of different pure substances: H2 , O2 and H2 O with the PengRobinson EoS. Pressure is P = 10 MPa. (◦) NIST database [Lemmon 2007]; (–) AVBP PR
EoS.
3.3. Equation of State
3.3.8
51
Conclusion for the EoS
At supercritical pressures, density shows strong non-linear variations, that have to
be correctly predicted in order to capture the right dynamics of the flow. The BWR
EoS is applicable over a broader range of temperatures and pressures than the cubic
EoS. Even though BWR is more accurate, its numerical cost represents an obstacle
that make LES simulations prohibitively expensive. Thus this EoS will not be used
in this manuscript. Both SRK and PR equations, however, are less complex and
easier to implement numerically. According to [Schmitt 2009], accuracy comparisons
of the EoS must be done in the critical region because of its criticality. In the vicinity
of the critical point, where the density is highly sensitive to temperature, PR is the
most accurate and SRK generates around 12% of error. When pressure decreases,
SRK error increases whereas PR remains highly accurate. However, SRK is more
accurate than PR in the transcritical zone where the temperature is low and the
pressure elevated. Finally for the rest of the work presented in this manuscript the
PR EoS has been chosen. More details can be found in [Schmitt 2009, Pons 2009b,
Petit 2014]
52
Chapter 3. Real Fluid Thermodynamics
3.4
Transport properties in mixture
In real gas conditions, transport properties depend non-linearly on temperature and
pressure, but also molar fractions. Several constraints have to be accounted for:
• the pressure-temperature validity range must be large typically 100-4000 K
and 1-300 bar.
• the model must remain easy to implement and relatively cheap in terms of
computation time.
[Congiunty 2003] exposed several models for the computation of viscosity and thermal conductivity at high pressure. Two models are generally used in real gas simulations:
• the Chung model [Chung 1984, Chung 1988] is based on the Chapman-Enskog
equation issued from kinetic theory of gases at low pressure corrected with an
empiric expression for high pressure.
• the Ely & Hanley model [Ely 1981a, Ely 1981b] is based on the principle of
the corresponding states and uses an empirical relation based on a reference
fluid.
3.4.1
Chung correlation
For the viscosity µ and the thermal conductivity λ Chung model writes:
µ = µ0 F1 (T, ρ, Tc , ρc , ω)
(3.22)
λ = λ0 F2 (T, ρ, Tc , ρc , ω)
(3.23)
where µ0 and λ0 represent the low pressure reference values obtained from the
Chapman-Enskog theory. Functions F1 and F2 are semi-empirical function that
account for high pressure deviation. More details can be found in [Schmitt 2009].
In Fig. 3.6 to Fig. 3.7, the transport coefficients computed with the Chung method
along with the Peng-Robinson EoS for the density evaluation are compared to the
NIST database reference [Lemmon 2007]. As shown in Fig. 3.6 the dynamic viscosity decreases with temperature in the "liquid-like" phase and increases with the
temperature in the "gas-like" phase for O2 . This behavior is well reproduced by
Chung method at 10 MPa for pure oxygen. For water2 (Fig. 3.7) results still exhibit
a reasonable behavior compared to the NIST database. The same comparison is
made for H2 in Fig. 3.8. Unlike the other species, the viscosity predicted by Chung
model differ from the NIST database up to ∼ 15% and the thermal conductivity up
to ∼ 30%. However the coefficient follow the good general behavior and the error
remains acceptable.
2
for water, the NIST database ranges from 273K to 1275K
3.4. Transport properties in mixture
3.4.2
53
Mass diffusivity
The computation of the species diffusion coefficients Dk is a specific issue. These
coefficients should be expressed as a function of the binary coefficients Dij obtained
from kinetic theory [Hirschfelder 1954]. The mixture diffusion coefficient for species
k, Dk , is computed as [Bird 1960]:
1 − Yk
D k = PN
j6=k Xj /Djk
(3.24)
The Dij are functions of collision integrals and thermodynamic variables. In addition
to the fact that computing the full diffusion matrix appears to be prohibitively
expensive in a multidimensional unsteady CFD computation, there is a lack of
experimental data to validate these diffusion coefficients. Therefore, a simplified
approximation is used for Dk . The Schmidt numbers Sc,k of the species are supposed
to be constant so that the binary diffusion coefficient for each species is computed
as:
µ
(3.25)
Dk =
ρ Sc,k
This is a strong simplification of the diffusion coefficients since it is well-known that
the Schmidt numbers are not constant in a transcritical flame and can vary within
several orders of magnitude between a liquid-like and a gas-like fluid [Oefelein 2006].
Models exist to qualitatively take into account this variation [Hirschfelder 1954,
Bird 1960, Takahashi 1974], although no experimental data is available to validate
them. However, the effects of the constant-Schmidt simplification on the flame
structure is actually very small, as shown in [Ruiz 2012], and thus appears to be
reasonable for the present studies of combustion.
54
Chapter 3. Real Fluid Thermodynamics
-4
4x10
µ [Pa.s]
3
2
1
100
200
300
400
500
Temperature [K]
(a) Dynamic viscosity
-7
4x10
2
-1
ν [m .s ]
3
2
1
100
200
300
400
500
Temperature [K]
(b) Kinematic viscosity
0.15
-1
-1
λ [W.m .K ]
0.20
0.10
0.05
100
200
300
400
500
Temperature [K]
(c) Thermal conductivity
Figure 3.6: Oxygen transport properties.
database [Lemmon 2007]; (–) Chung model.
Pressure is P = 10 MPa.
(◦) NIST
3.4. Transport properties in mixture
55
-3
1.5x10
µ [Pa.s]
1.0
0.5
1000
2000
3000
Temperature [K]
(a) Dynamic viscosity
-6
10x10
ν [m .s ]
8
2
-1
6
4
2
1000
2000
3000
Temperature [K]
(b) Kinematic viscosity
-1
-1
λ [W.m .K ]
1.0
0.5
0.0
1000
2000
3000
Temperature [K]
(c) Thermal conductivity
Figure 3.7: Water transport properties.
database [Lemmon 2007]; (–) Chung model.
Pressure is P = 10 MPa.
(◦) NIST
56
Chapter 3. Real Fluid Thermodynamics
-5
µ [Pa.s]
1.5x10
1.0
0.5
150
200
250
300
350
400
450
500
450
500
450
500
Temperature [K]
(a) Dynamic viscosity
-6
3x10
2
-1
ν [m .s ]
2
1
0
150
200
250
300
350
400
Temperature [K]
(b) Kinematic viscosity
0.30
-1
-1
λ [W.m .K ]
0.25
0.20
0.15
0.10
150
200
250
300
350
400
Temperature [K]
(c) Thermal conductivity
Figure 3.8: Hydrogen transport properties.
database [Lemmon 2007]; (–) Chung model.
Pressure is P = 10 MPa.
(◦) NIST
Chapter 4
Multiphysics and Code Coupling
Contents
4.1
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
4.2
Thermal conduction in solid materials . . . . . . . . . . . . .
58
4.2.1
4.3
The heat equation . . . . . . . . . . . . . . . . . . . . . . . .
58
Fluid/Thermal coupled problem . . . . . . . . . . . . . . . .
60
4.3.1
Conjugate Heat Transfer . . . . . . . . . . . . . . . . . . . . .
60
4.3.2
Coupled Flame Wall Interaction . . . . . . . . . . . . . . . .
65
4.3.3
The AVTP solver . . . . . . . . . . . . . . . . . . . . . . . . .
67
4.3.4
Fluid Solver Wall Boundary Condition . . . . . . . . . . . . .
70
4.3.5
Evaluation of the wall resistance constant . . . . . . . . . . .
73
58
4.1
Chapter 4. Multiphysics and Code Coupling
Introduction
Many industrial applications are subject to strong thermal interactions between fluids and solids. For example in an aeronautical combustion chamber, an external
cold flow is injected through the multi-perforated walls establishing a thin cooling
film. This kind of configuration exhibits intense aerothermal phenomena, complex
flows and geometries, which strongly interacts with other combustion phenomena
such as combustion instabilities, flame quenching or ignition.
Many software solutions are now being developed for large-scale, complex, coupled
multiphysics simulations on massively parallel platforms. Applications are aerodynamic design [Nikbay 2009], climate and weather prediction [Valcke 2012], combustion engine [Shankaran 2001], biological and biochemical systems such as cardiac
mechanics [Nordsletten 2011], hypersonic space re-entry [Oppattaiyamath 2011].
At CERFACS, the OpenPALM software has been developed and applied for example to: heat transfer in gas turbines [Amaya 2010b, Poitou 2011, Jauré 2011,
Duchaine 2013], evaluation of coke deposit in petrochemical pipes [Pedot 2012] and
fluid-structure interaction in rocket boosters [Richard 2011].
The main goal of this chapter is to introduce the coupled phenomena involving reactive fluid dynamics and heat conduction in solids.
4.2
Thermal conduction in solid materials
Heat conduction in solids is governed by the heat equation formulated from the first
principle of thermodynamics,ie. the law of conservation of energy. It describes the
heat propagation and the temperature distribution in a domain. The theoretical
formulation of heat propagation was developed during the beginning of the 19th
century by Joseph Fourier.
4.2.1
The heat equation
The heat equation is written as:
S
ρ C
S
∂
∂T
=
∂t
∂xi
∂T
λ
∂xi
S
+ Q̇
(4.1)
where T (x, t) is the temperature in the solid at location x and time t, ρS the density
of the solid material, C S its heat capacity, λS its thermal conductivity and Q̇ a thermal source term. These material properties can be combined in a single parameter:
α the thermal diffusivity.
λS
αS = S S
(4.2)
ρ C
This quantity evaluates the time required by a solid to change its temperature under
the influence of an external or internal source. It measures the ability of a material
to conduct thermal energy relative to its ability to store thermal energy. It also
4.2. Thermal conduction in solid materials
59
characterizes the penetration length into a semi-infinite solid subjected to a heat
flux at its surface.
To properly define the heat problem boundary conditions must be applied. Let be
V a domain and S its frontier, several types of boundary conditions can be defined:
Dirichlet boundary conditions the value of the temperature is fixed in a hard
way by imposing: T (xw , yw , zw , t) = Tw (t), where w sub-scripted quantities
refer to quantities imposed at the limiting surface S.
Neumann boundary conditions the temperature is not fixed, only the heat flux
is imposed at the surface S as
∂T
−λ
= qw
(4.3)
∂n w
60
Chapter 4. Multiphysics and Code Coupling
4.3
Fluid/Thermal coupled problem
The main purpose of this chapter is to expose the coupling methodology used for
the computations of the manuscript. Several aspects of the coupling methodology
have to be investigated:
Exchanged data: The choice of exchanged data and exchange frequency must be
chosen in accordance with stability criteria to ensure computation convergence.
Convergence acceleration: Because the fluid and the solid have, in general, very
different characteristic times, a strategy to accelerate the convergence of the
solid domain is used and needs to be assessed.
Exchange frequency: the coupling frequency is a key parameter of a coupling
methodology.
Interpolation: The interpolation methodology at the interface must ensure conservativity of the variables.
4.3.1
4.3.1.1
Conjugate Heat Transfer
Exchanged data at the interface
Without chemistry effects due to ablation or pyrolysis, the thermal exchange at the
solid surface is limited to the Fourier heat flux. The energy conservation imposes
F = −q S 1 . For the same reasons the temperature of the molecules of fluid and
qw
w
solid in contact at the interface is the same, thus TwF = TwS . In order to close
the mathematical system of the coupled problem equations at the interface can be
expressed as:
Dirichlet problem: the temperature continuity is imposed at the interface
(
TwF ← TwS
TwS
← TwF
Neumann problem: the heat flux continuity is imposed at the interface
(
F
S
qw
← qw
S
qw
F
← qw
(4.4)
(4.5)
Mixed Neumann-Dirichlet problem: the temperature is imposed in one side of
the interface, while the heat flux is imposed in the other
(
F
S
qw
← qw
(4.6)
TwS ← TwF
1
superscript XF/S refers to fluid/solid quantities
4.3. Fluid/Thermal coupled problem
or
(
S
qw
TwF
61
F
← qw
← TwS
(4.7)
[Giles 1997] studied the stability of a basic one dimensional conjugate heat transfer
problem using different coupling schemes. [Giles 1997] demonstrates that the coupled problem is stable as long as at the interface a Dirichlet condition is applied to
the fluid domain and a Neumann condition to the solid (case 4.7). This result was
proved for a simulation with equal time steps in each domain and data exchange
at each time step, which is a very restrictive condition for industrial configurations. This is called a strong coupling. On the contrary loose coupling refers to two
systems which synchronize at a given frequency. [Duchaine 2013, Jauré 2013] have
proven that for very tightly loose coupled systems, the Dirichlet/Neumann coupling
set remains stable.
4.3.1.2
Acceleration strategy
The heat conduction problem is simply solving a Laplace equation that for a set
of fixed boundary conditions has a unique solution. Thus if the the boundary conditions transferred from the fluid to the solid are physical the thermal solver will
converge to the physical solution. Unfortunately the coupling process is an iterative
process with an unsteady fluid solver. Hence the boundary conditions imposed at
the solid surface may vary greatly from one coupling step to another. Loose coupling
refers to two systems which synchronize at a given frequency. Thus there is a linear
relation between the two simulation times tF and tF such as
tS = α S τ S
t
F
F F
= α τ
η =
αS
αF
(4.8)
(4.9)
(4.10)
where αF (αS ) is the number of iteration of the fluid (resp. solid) solver between
two coupling updates, τ F (τ S ) is the time step of the fluid (resp. solid) solver
enforced by stability criteria of the individual solver. As long as the goal of the
coupled simulation is to obtained the average solution in the solid the ratio η has
no physical meaning and can be used to optimize load balancing. [Jauré 2013]
has investigated the influence of fluid unsteady features on a 1D solid temperature
domain. They show that low frequency excitations do have an impact on the solid
temperature. Also due to the acceleration methodology the apparent frequencies
of the fluid unsteady features are lowered, hence the perturbation depths are over
predicted. Therefore convergence acceleration can yield unrealistic instantaneous
solutions. However at each point in space the temperature oscillates in time around
the average solution at the forcing frequency, meaning that averaging over a period
of the forcing frequency should yield the converged time independent solution. Two
62
Chapter 4. Multiphysics and Code Coupling
methods for resource distribution are generally introduced for parallel computing,
either for LES [Duchaine 2013] or RANS [Radenac 2005, Radenac 2014] applications: the Sequential Coupling Strategy (SCS) and the Parallel Coupling Strategy
(PCS). In the first method, illustrated by Fig. 4.1, each code performs a simulation
sequence one after the other, using all the processors at once. Each simulation duration is independent and defined by the number of iteration, αF for the fluid solver
and αS for the solid solver, and the time step τ F (resp. τ S ). The result is a chained
simulation where each code has access to the full computational resources when
running. The main advantage of this method is the simple CPU synchronization
(one code begins when the other ends), but the total restitution time can be high.
In the Parallel Coupling Strategy (PCS) all codes run simultaneously on different
processors, and can be synchronized in physical time and CPU time (Fig. 4.2). In
general, because the heat equation problem is easier to solve and smaller in terms
of number of nodes, less CPUs are allocated to the heat solver and an efficient load
balancing between the fluid and the solids solvers can be achieved. This strategy
allows to perform fast simulations and will be used in the rest of the work.
F (n)
qw
; TwF (n)
(
(
Fluid Solver
W F (n)
S(n)
BC : Tw
Solid Solver
T S(n)
F (n) F (n)
BC : Tw , qw
α S τS
S(n+1)
qw
; TwS(n+1)
(
Fluid Solver
F (n)
W
S(n+1)
BC : Tw
(
Solid Solver
T S(n+1)
F (n) F (n)
BC : Tw , qw
α F τF
(
Fluid Solver
W F (n+1)
S(n+1)
BC : Tw
Figure 4.1: Sequential Coupling Strategy (SCS). Solid arrows represent data exchange,
dashed arrows represent computation path.
4.3. Fluid/Thermal coupled problem
(
Fluid Solver
W F (n)
S(n) S(n)
BC : Tw , qw
F (n)
qw
; TwF (n)
S(n)
; TwS(n)
qw
63
(
Solid Solver
S(n)
T
F (n) F (n)
BC : Tw , qw
α F τF
(
Fluid Solver
α S τS
F (n+1)
qw
; TwF (n+1)
F (n+1)
W
S(n+1) S(n+1)
BC : Tw
, qw
α F τF
Fluid Solver
S( +1)
+1)
S(n+1)
S(n+1)
qw
; TwS(
(
Solid Solver
S(n+1)
T
F (n+1) F (n+1)
BC : Tw
, qw
α S τS
Solid Solver
Figure 4.2: Parallel Coupling Strategy (PCS). Solid arrows represent data exchange,
dashed arrows represent computation path.
4.3.1.3
Coupling frequency
The loose coupling strategy of instantaneous quantities can be seen as a sampling
process of the quantities every αF τ F (αS τ S ) for the fluid (resp. the solid).Therefore
to ensure the convergence of the solid, the Nyquist-Shannon Eq. 4.11 theorem has
to be respected.
fs > 2 fmax
(4.11)
where fs is the sampling frequency and fmax the maximum frequency of the signal. Not respecting this theorem may lead to aliasing and artificially create low
frequency. This is very inconvenient for the heat penetration because the solid is
more sensitive to low than to high frequencies.
4.3.1.4
Interpolation
The interpolation is made using a linear interpolation method. [Jauré 2013] has
demonstrated that this technique avoids aliasing if the data are interpolated from
a coarse to a fine grid. For this reason, the solid mesh will always be finer than
64
Chapter 4. Multiphysics and Code Coupling
the fluid mesh. As the heat equation solver is at least an order of magnitude faster
than the fluid solver it does not penalize the computation rendering time. Moreover
as the heat solver smoothes the temperature gradients, because of the hyperbolic
nature of the heat equation, the interpolation from the fine grid of the solid to the
coarse mesh of the fluid does not exhibit any aliasing problem.
Conclusion In conclusion, an accurate solution can be obtained thanks to:
• a Neumann/Dirichlet coupling strategy
• a tight coupling that ensures that the coupling scheme remains stable and the
aliasing problems are avoided
• a mesh of the solid finer than the solid one for interpolation reasons
4.3. Fluid/Thermal coupled problem
4.3.2
65
Coupled Flame Wall Interaction
(a) Schematic representation of a flame-wall interaction. φw represents the heat flux
exchanged at the wall.
Wall Heat Flux
Quenching Distance
Chemical Strucuture
Solid Temperature Field
(b) Coupling chain between a flame and a wall.
Figure 4.3: Basic features of a flame interacting with a wall.
The final objective of the manuscript is the application of the multi-physics system
to an industrial astronautical case, in order to show the feasibility of unsteady codes
coupling, in particular with the use of real gas thermodynamics, reactive LES and
realistic flame kinetic mechanism.
As shown in Fig. 4.3, when interacting with a non adiabatic wall, the flame looses
energy through heat transfer to the wall. This wall heat flux modifies the temperature field in the solid leading to modifications of the flame structure by enhancing
or disabling chemical reactions, even quenching the flame and on a second order of
magnitude modifying the fluid velocity fields.
In many physical problems, the interface conditions is assumed constant through
time. However in reality, its variations are caused by the interactions with the
surroundings (or complementary subsystems). When the interaction between two
66
Chapter 4. Multiphysics and Code Coupling
subsystems is handled by the exchange of information at their interface, the joint
problem is called a coupling, and the exchanged information is called the interface
variables. To close the system the exchange of interface variables must be subject
to closing rules, called the interface equations which are commonly based on conservation laws.
In this work conjugate heat transfer is performed by running in parallel two solvers
(one for the fluid and one for the solid) which exchange boundary conditions at the
solid surface, using the coupling chain AVBP-AVTP developed at CERFACS with
the code coupler OpenPALM [Duchaine 2009b].
4.3. Fluid/Thermal coupled problem
4.3.3
67
The AVTP solver
This section presents the unsteady thermal solver used in the coupled computations.
The validation of this tools on simple test cases was made in [Amaya 2010a], and
on more complex cases such as industrial refinery furnaces [Pedot 2012] or gas turbines [Jauré 2012]. The solid thermal solver AVTP is a parallel numerical code that
solves the heat equation (4.1) on unstructured hybrid meshes. AVTP has been derived from the fluid solver AVBP, therefore it shares the computational structure of
AVBP ([Amaya 2010b, Chapter 4]). Explicit numerical methods used in AVBP are
not optimal for solving the heat equation thus an implicit scheme has been added
to the solver.
The motivation for AVTP is to have access to a reliable and fast solver of the heat
equation in solid components in a combustion system compatible with the LES
solver AVBP. The main targeted applications are aerothermal unsteady coupled
simulations with both AVTP and AVBP.
4.3.3.1
Boundary Condition
Several boundary conditions are available in AVTP. Most are similar to those used
in AVBP.
Adiabatic: the heat flux is always equal to zero
qw = 0
(4.12)
Isothermal (Dirichlet condition): the temperature is imposed equal to a reference value Tref .
Tw = Tref
(4.13)
Heat Loss (Neumann condition): the heat flux is imposed at the wall equal to
a value given by
qw = qref
(4.14)
Convective Flux: the heat flux is imposed with a convective approach
qw = href (Tw − Tref )
(4.15)
Mixed Condition (Robin Condition): a heat flux is imposed at a value combining a given flux plus a convective flux (through a convective heat transfer
coefficient and a reference temperature)
qw = href (Tw − Tref ) + qw,ref
(4.16)
68
Chapter 4. Multiphysics and Code Coupling
4.3.3.2
Time integration
Two algorithms are available for the time integration:
• an explicit 1st order forward Euler scheme:
∂T
T n+1 − T n
=
+ o(δt)
∂t
∆t
(4.17)
where T n (T n+1 ) represents the temperature at the nth (resp. nth +1) instant.
In order to ensure the scheme stability the time step ∆t must verify the Fourier
condition given by:
α∆t
< 0.5
(4.18)
Fo =
∆x2
where ∆x is the characteristic length of the smallest cell of the mesh: ∆xmin =
√
3
Volmin
• an implicit solver : the implicit system is solved thanks to a matrix free
conjugate gradient method well suited for parallel applications [Fraysse 2008].
4.3.3.3
Diffusion term
The diffusion term can be calculated thanks to two different methods:
• a cell-vertex approach based on a discretized element (Fig. 4.4)
• a method derived from a finite element approach centered on the computation
nodes.
More details on these operators can be found in [Lamarque 2007].
4.3. Fluid/Thermal coupled problem
(a) Nodes and cells used for the computation of the diffusion term at the central node using the 4∆ operator.
Extracted from [Lamarque 2007].
(b) Nodes and cells used for the computation of the diffusion term at the central node using the 2∆ operator.
Extracted from [Lamarque 2007].
Figure 4.4: Nodes and cells used for the computation of the diffusion term ∇2 T .
69
70
Chapter 4. Multiphysics and Code Coupling
4.3.4
Fluid Solver Wall Boundary Condition
In AVBP, several wall boundary thermal conditions are available.
Adiabatic : the heat flux is always equal to zero
(4.19)
qw = 0
Isothermal : the temperature is imposed equal to a reference value Tref .
(4.20)
Tw = Tref
Heat Loss : the heat flux is imposed at the wall equal to a value given by
qw = −
Tw − Tref
Rw
(4.21)
where Tref is a target temperature and Rw the wall thermal resistance.
The particular formulation of the heat loss condition is used to mimic convective
heat transfer in cooling devices. A typical situation is shown in Fig. 4.5, where a
wall separates the hot side and a cold side where the temperature Tref is known (e.g.
by experimental data). By knowing the solid properties such as λS , ρS , CS and its
thickness e, the wall resistance reads:
Rw =
e
λS
(4.22)
Cold Flow
Tref
Solid
e
λS , ρS , CS
Tw
Hot Flow
Figure 4.5: Typical cooling device configuration.
As explained in Subsec. 4.3.1, the temperature of the solid is given to the fluid solver
and used as a boundary condition. To impose the fluid temperature at the domain
frontier, various possibilities exist:
• The wall temperature Tw may be directly imposed, resulting in the correction
of the transported variables : ρ or ρE.
4.3. Fluid/Thermal coupled problem
71
• A heat loss condition with a very small wall resistance may be used in order
to keep the wall temperature very close to the reference temperature.2
Table 4.1 summarizes all the methods with their advantages and their drawbacks.
2
more details available at : http://www.cerfacs.fr/∼avbp/AVBP_V6.X/AVBPHELP/ASCIIB/help_asciiBound.php
Wall Resistance
Relaxation Time
Pressure Correction
Density Correction
(ρ)
Implementation
n+1
qw
n+1
qw
(T n − Tref )
=− w
Rw
(T n − Tref ) · Rc
=− w
δt
Tw = Tref
Tw = Tref
Mass conservation
Mass conservation
T strictly imposed
Mass conservation
T strictly imposed
+
imposition
Weak temperature
relaxation time
Sensitivity to
Pressure oscillations
No strict mass conservation
-
Table 4.1: Summary of the isothermal methods. Rc is a relaxation coefficient [J.m−2 .K −1 ], δt the time step [s], Rw the wall resistance
[W.m−2 .K −1 ].
HeatLoss
Weak condition
IsoThermal
Hard condition
Boundary Condition
72
Chapter 4. Multiphysics and Code Coupling
4.3. Fluid/Thermal coupled problem
73
First two methods in Tab 4.1 are isothermal which means the temperature is strictly
imposed at the wall Tw = Tref with a strong formulation. However this induces either density correction which is not mass conservative and may lead to severe mass
gains / losses especially in real gas cases where the density variations can be very
strong, or to pressure correction, which might create pressure oscillations of a significant importance and eventually leading to code crashing.
The two last methods are based on wall heat flux and are very similar. The relaxation time Rc is a measure of the time needed to reach the target temperature. If the
phenomenon is very unsteady (for example a premixed flame quenching on a wall)
the relaxation time might be too long to reach the target temperature. Similarly
the wall resistance Rw , that mathematically plays the same role as Rc /δt has to be
properly calculated to ensure a wall temperature at the target.
4.3.5
Evaluation of the wall resistance constant
To find the value of Rw , let consider a wall cell and the balance of heat fluxes and
source terms:
Wall
qq̇c
qw
qλ
Figure 4.6: Heat fluxes balance in the wall fluid cell. Wall flux (black arrow), chemical
source term (black spot) and heat flux from the flow (striped arrow).
The energy balance in the wall cell writes:
qw = qλ + q̇c
(4.23)
where qw is the flux evacuated at the wall and q̇c the total (= cell integrated)
chemical source term. As at this location the flow speed is close to zero u ∼ 0, the
transported heat flux reduces to the conductive flux qλ .
74
Chapter 4. Multiphysics and Code Coupling
qw = −
Tw − Tref
Rw
(4.24)
dT
(4.25)
dx
To mimic an isothermal wall boundary condition the wall cell temperature must keep
constant through time, ie. all fluxes entering the cell have to be evacuated to the
wall. Knowing the temperature right surface of the wall cell (ie., in the neighboring
fluid cell) Tright :
qλ = λ
qλ = λ
Tright − Tref
∆x
(4.26)
Rw =
Tw − Tref
(4.27)
Tw − Tref
(4.28)
One obtains, if q̇c = 0:
or if q̇c 6= 0:
Rw =
ql0
λ
Tright −Tref
∆x
+λ
Tright −Tref
∆x
Part II
Experimental validation
Chapter 5
Methane Chemistry
Contents
5.1
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
5.2
Methane Chemistry . . . . . . . . . . . . . . . . . . . . . . . .
78
5.3
Chemistry for oxycombustion of methane . . . . . . . . . . .
80
5.3.1
Chemical Equilibrium . . . . . . . . . . . . . . . . . . . . . .
80
5.3.2
Kinetic scheme . . . . . . . . . . . . . . . . . . . . . . . . . .
84
5.3.3
The LU analytical mechanism . . . . . . . . . . . . . . . . . .
84
5.3.4
Implementation of the LU scheme in AVBP . . . . . . . . . .
105
78
Chapter 5. Methane Chemistry
5.1
Introduction
The main objective of this part is to validate the coupled methodology presented
in Chap. 4 on a CH4-O2 coaxial burner, called NEMO, experimentally studied at
EM2C laboratory in Ecole Centrale, Paris by T. Schmitt and P. Scouflaire. A sketch
of the coaxial burner is shown in Fig. 6.1. Gaseous methane is injected through the
inner injector and is surrounded by pure gaseous oxygen. Both reactants are injected at ambient temperature (300 K) and atmospheric pressure (1 bar) without
combustion chamber.
One particularity of this configuration is that because CH4 burns with pure O2 , the
combustion kinetics must be accurate enough to capture all the flame features and
a special attention is payed to the chemical kinetics in this chapter.
Moreover, in the spatial industry, methane is presented as a good substitute to
hydrogen. Several advantages in comparison with hydrogen are to be cited :
• because of its more important density, methane needs smaller fuel tanks.
• as its boiling temperature is more elevated it can be injected at higher temperature which increases combustion efficiency.
• its production cost is lower
• among all hydrocarbon fuels, its specific impulse (Isp ) is the largest (cf. Tab. 5.1).
Propellant mix
Isp (s)
LOx - GH2
LOx - LCH4
LOx - RP1 Kerosene
NTO - Aerozine 50
450
370
∼ 300
∼ 300
Table 5.1: Specific impulse of some ergols.
In this chapter, a chemical mechanism for oxycombustion of methane is described
and evaluated. This chemical scheme will be used to compute the NEMO configuration, presented in the next chapter.
5.2
Methane Chemistry
The methane/air combustion chemistry is well-known and has been studied for
many years. The Gas Research Institute has developed the well-known GRI-Mech
since the 90s. Several versions are available : GRI-1.2 [Frenklach 1995], GRI2.11 [Bowman 1998] which accounts for nitrogen oxides formation and more recently
GRI-3.0 [Smith 2000]. The last version, GRI-3.0, accounts for 53 species and 325
5.2. Methane Chemistry
79
reactions. It has been validated in many configurations and over a wide range of
thermodynamic conditions1 in shock-tube experiments where ignition delays, species
concentration profiles and laminar premixed flame speed have been measured. It is
considered as a reference for methane combustion.
To reduce the computational cost, reduced mechanisms derived from detailed mechanisms. They are built so as to reproduce the main features of flames such as flame
speed, ignition delays and adiabatic temperature. When increasing the number of
species and reactions predictions get more accurate and more behaviors can be predicted such as the response to stretch or the chemical flame structure.
Reduced mechanisms can be classified in a increasing level of complexity:
Global mechanisms are the most simple ones. They are fitted against reference
data and contain very few reactions (e.g. < 4) which follow the Arrhenius law. The
validity of such fitted mechanism is limited to a very small range of thermodynamic
condition usually only the ones on which properties have been fitted. Several reduced
mechanisms have been derived from the GRI detailed mechanisms. Examples can be
found in [Westbrook 1981, Jones 1988, Najm 1997, Frassoldati 2009, Franzelli 2010].
In [Chen 1997] the authors propose a 12 species and 9 (if NOx chemistry is ignored)
or 12 reactions reduced mechanism.
Analytical mechanisms are more complex but still reasonable. They use the
quasi-steady state (QSS) approximation for some species and partial equilibrium
for some reactions. When the creation rate of a species is slow compared to its
consumption rate, the species can be considered in a quasi steady state and its net
rate may be considered equal to 0. Then it reads for species k:
(5.1)
ω̇k = 0
=
M
X
j=1
=
M
X
j=1
(5.2)
νkj Qj
"
νkj Kjf
N
Y
k=1
[Xk ]
n′kj
− Kjr
N
Y
k=1
[Xk ]
′′
nkj
#
(5.3)
The concentration of the species k is then directly computed from Eq. 5.3 and not
from its conservation equation which can be ignored.
Partial equilibrium hypothesis can be assumed when both the forward and the backward rates of reaction k are fast compared to all other reactions. Partial equilibrium
1
more details are available at : http://combustion.berkeley.edu/gri-mech/version30/text30.html
80
Chapter 5. Methane Chemistry
of reaction j writes:
Qj
(5.4)
= 0
= Kjf
N
Y
k=1
′
[Xk ]nkj − Kjr
N
Y
′′
[Xk ]nkj
(5.5)
k=1
Several methane schemes are based on this technique such in [Peters 1985, Chen 1991,
Lu 2008]. The Peters [Peters 1985] and the Seshadri [Chen 1991] mechanisms account for 8 species (CH4 , O2 , CO2 , CO, H2 O, H2 , H, N2 ). The Peters scheme is a
18 reactions mechanism and describes the basic features of the flame, ie. the flame
structure for both premixed and diffusion flames, at fresh gas temperature lower than
500K. The Seshadri kinetic model is a mechanism of 25 reactions and exhibits similar behaviors than the Peters scheme. More details can be found in [Franzelli 2011,
Chapter 3].
5.3
5.3.1
Chemistry for oxycombustion of methane
Chemical Equilibrium
The burnt gas state is fully determined by the set of species at equilibrium. In other
words, the species mandatory for oxycombustion of methane can be determined
with simple equilibrium computations. Assuming constant enthalpy and pressure,
equilibrium is reached when the Gibbs free energy of the system is minimum. This
means that the derivative of the Gibbs energy vanishes signaling a stationary point.
Using the open source software CANTERA [Goodwin 2009], equilibrium states are
calculated from given initial composition and conditions and a given set of species.
These computations give the adiabatic flame temperature and the composition of
the burnt gas. It uses an element potential method where the chemical potential of
each species in equilibrium is the sum of the contribution from each atom. It reads:
X
µk =
akm λm
(5.6)
m
where µk is the chemical potential of the species k, λm the chemical potential of the
atom m and akm the number of atoms m in species k.
Three different sets of species are tested, summarized in Tab. 5.2. In the following,
each set will be referred to by the name of the associated chemical kinetic scheme
even though no chemical scheme is used for equilibrium computations. The most
complete set of 53 species, corresponds to the GRI-3.0 mechanism. The most simplified set is the one involved in the BFER scheme [Franzelli 2010] which accounts
for CO-CO2 equilibrium and counts 5 species. The BFER scheme is then extended
to 10 species by adding 5 radical species : H2 , OH, O, H, HO2 and is named
BFER_EXT. These radicals have been chosen using GRI-3.0 equilibrium results.
5.3. Chemistry for oxycombustion of methane
81
For combustion in air N2 has to be taken into account and adds one species (N2
oxidation is not considered).
Name
Number of species
Species
GRI3.0
53
cf GRI-Mech website
BFER
5
CH4 , O2 , CO2 , H2 O, CO
BFER_EXT
10
CH4 , O2 , CO2 , H2 O, CO
OH, O, H, H2 , HO2
Table 5.2: The three set of species.
Fresh gas temperature is 300 K and pressure is 1 bar. As coaxial injectors used in
rocket engines, essentially leads to diffusion flames all CANTERA computations are
here conducted at φ = 1.
Results of equilibrium computations with CANTERA are provided in Fig. 5.1 and 5.2
for the three set of species in terms of composition and temperature for methane
oxidation in both Air and pure O2 .
When methane is burnt with air at stoichiometry, the burnt gas composition involves only 5 major species. However when methane is burnt with pure oxygen,
CANTERA is not even able to reach an equilibrium state with BFER and initial
and final compositions are the same. The BFER_EXT set of species on the contrary allows a good prediction of the composition at equilibrium.
In terms of adiabatic temperature, Fig. 5.2 the same conclusions can be made. CANTERA does not converge with only 5 major species except for very lean mixtures.
The chemical kinetic scheme therefore requires to include, at least, the 10 species of
the BFER_EXT set. At temperatures higher than 2500 K, the CO − CO2 equilibrium is in favor of CO and the H2 − H2 O equilibrium leads to a significant amount
of H2 . In addition the dissociation phenomena have a significant impact in limiting
the heat release. Simplified mechanisms such as BFER, do not account for dissociation reactions and do not include radicals. This explain why they do not reach the
good equilibrium.
82
Chapter 5. Methane Chemistry
GRI3.0
BFER_EXT
BFER
0.13
0.12
CH4 / AIR
Φ = 1.0
0.11
0.10
0.09
Y [-]
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
CH4
O2
CO2
H2O
CO
OH
O
H2
H
HO2
(a) CH4 /Air ; φ = 1
0.8
GRI3.0
BFER_EXT
BFER
0.7
CH4 / O2
Φ = 1.0
0.6
Y [-]
0.5
0.4
0.3
0.2
0.1
0.0
CH4
O2
CO2
H2O
CO
OH
O
H2
H
HO2
(b) CH4 /O2 ; φ = 1
Figure 5.1: Species mass fractions at equilibrium for a stoichiometric mixture at 1bar :
(a) CH4 /Air ; (b) CH4 /O2
5.3. Chemistry for oxycombustion of methane
83
2200
2100
T [K]
2000
1900
1800
1700
CH4 / AIR
1600
GRI3.0
BFER_EXT
BFER
1500
0.6
0.8
1.0
φ
1.2
1.4
(a) CH4 /Air
3500
GRI3.0
BFER_EXT
BFER
3400
CH4 / O2
T [K]
3300
3200
3100
3000
2900
0.6
0.8
1.0
φ
1.2
1.4
(b) CH4 /O2
Figure 5.2: Equilibrium temperature as a function of the equivalence ratio φ at 1 bar :
(a) CH4 /Air ; (b) CH4 /O2
84
5.3.2
Chapter 5. Methane Chemistry
Kinetic scheme
Based on the results of chemical equilibrium, several fitted chemical schemes for
methane oxycombustion involving at least the ten necessary species have been
tested: in particular [Najm 1997] and [Frassoldati 2009]. Unfortunately even if these
schemes predict correctly the adiabatic flame temperature and burnt gas composition, they are not usable in LES computations for several reasons: [Najm 1997]
has too many species and would be too expensive in terms of computational cost
and [Frassoldati 2009] was designed for diffusion flames and the flame speed, which
is a critical flame parameter to preserve, is too far from the GRI-3.0 reference.
[Lu 2008] developed a scheme with 17 species, 4 of them being treated as QSS
species (CH2 / CH2 (S) / HCO / CH2 OH), and 73 elementary reactions. Results
are detailed in the next section.
5.3.3
5.3.3.1
The LU analytical mechanism
Ignition delays
In order to characterize the ignition delay of a mixture, the chemical kinetic scheme
is applied in a 0D perfectly stirred reactor (PSR) providing the evolution of the temperature of the mixture as a function of the residence time. As shown in Fig. 5.3
the auto-ignition time is arbitrary defined here as the time when the temperature
gradient reaches its maximum. This definition is similar to what is used in a simulation using AVBP with a 0D domain where the unburnt mixture is at rest waiting
to ignite. It reads:
τai = t | ∇T = ∇Tmax
(5.7)
!"
∇T
τai
#"
Figure 5.3: Auto Ignition time in a PSR.
Auto-ignition times have been computed for a perfectly premixed mixture at atmospheric pressures and stoichiometric equivalence ratio φ = 1.0 for various initial
temperature. As for premixed flame, computations for both oxidizers air and O2
5.3. Chemistry for oxycombustion of methane
85
have been made. Results are shown in Fig. 5.4 for air combustion and Fig. 5.5 for
O2 combustion. The LU mechanism exhibits with the same slope. However autoignition delays with LU are overall one order of magnitude too high, which means
the flame will be more difficult to stabilize. However the correct trend is reproduced
by the LU mechanism.
1
10
0
10
-1
τai [s]
10
-2
10
-3
10
-4
10
-5
10
0.5
0.6
0.7
0.8
0.9
1.0
1000/T [1/K]
Figure 5.4: Auto Ignition delays as a function of the initial temperature, for a perfectly
premixed stoichiometric mixture at 1 bar for CH4 - Air. (- -) LU ; (•) Gri-3.0.
1
10
0
10
-1
τai [s]
10
-2
10
-3
10
-4
10
-5
10
0.5
0.6
0.7
0.8
0.9
1.0
1000/T [1/K]
Figure 5.5: Auto Ignition delays as a function of the initial temperature, for a perfectly
premixed stoichiometric mixture at 1 bar for CH4 - O2 . (- -) LU ; (•) Gri-3.0.
5.3.3.2
Premixed flames
Unstrained premixed flames are computed using CANTERA with thermodynamic
conditions corresponding to the target configuration: the inlet temperature of the
86
Chapter 5. Methane Chemistry
fresh gas is 300 K and the pressure is 1 bar. Both air and oxygen are tested. Results
obtained with the LU mechanism are compared to the GRI-3.0 scheme taken as a
reference.
Flame speed In Fig. 5.6, the flame speed obtained with the reduced scheme LU
is compared to the GRI-3.0 mechanism.
3.2
0.4
-1
Sl [m.s ]
-1
Sl [m.s ]
3.0
0.3
0.2
2.8
2.6
0.1
0.6
0.8
1.0
1.2
1.4
2.4
0.6
0.8
1.0
1.2
Equivalence Ratio [-]
Equivalence Ratio [-]
(a) CH4 - Air
(b) CH4 - O2
1.4
Figure 5.6: Flame speed as a function of equivalence ratio for (a)methane/air and
(b)methane/O2 flames. (- -) Lu scheme ; (•) Gri3.0
When the oxidizer is air even though the flame speed is slightly overpredicted around
stoichiometry (maximum error of 6%), the agreement with the reference GRI-3.0 is
satisfactory for the whole range of equivalence ratio. For rich mixtures the decrease
of the flame speed is well reproduced except for very high equivalence ratios where
the relative error reaches almost 30%. With pure O2 , due to faster chemistry, the
flame speed is largely increased reaching ∼ 3 m.s−1 , ie. about 10 times larger than
the stoichiometric air flame. The agreement with the reference scheme is very good
and the error on flame speed is less than 2% for all equivalence ratios. Relative
errors ∆Sl /Sl are plotted in Fig. 5.7 for both flames.
5.3. Chemistry for oxycombustion of methane
87
Relative error [%]
2
10
8
6
4
2
1
8
6
4
2
0.1
0.6
0.8
1.0
1.2
1.4
Equivalence Ratio [-]
Figure 5.7: Relative error on flame speed ∆Sl /Sl : (△) CH4 - Air; () CH4 - O2 .
88
Chapter 5. Methane Chemistry
Burnt gas temperature Figure 5.8 reports the adiabatic flame temperature evaluated with both GRI-3.0 and LU schemes for air and pure O2 oxidizers.
2200
3100
3050
T [K]
T [K]
2000
3000
1800
2950
1600
0.6
0.8
1.0
1.2
2900
0.6
1.4
0.8
Equivalence Ratio [-]
1.0
1.2
1.4
Equivalence Ratio [-]
(a) CH4 - Air
(b) CH4 - O2
Figure 5.8: Burnt gas temperature as a function of equivalence ratio for (a) methane/air
and (b) methane/O2 flames. (- -) Lu scheme ; (•) Gri3.0
Relative error [%]
2.0
1.5
1.0
0.5
0.0
0.6
0.8
1.0
1.2
1.4
Equivalence Ratio [-]
Figure 5.9: Relative error ∆Tf /Tf on burnt gas temperature: (△) CH4 - Air; () CH4
- O2 .
As expected, the introduction of additional species, allows a very good prediction
of the burnt gas temperature. Very small errors are shown in Fig. 5.9.
5.3. Chemistry for oxycombustion of methane
89
Flame thickness The flame thickness is an important feature of premixed flames
because it strongly affects their interaction with turbulence. Moreover the flame
thickness also determines the grid refinement necessary to resolve the flame in a
simulation. In the following results the flame thickness δl is calculated as
δl =
∆T
(∇T )max
(5.8)
where ∆T is the temperature difference between burnt and fresh gas and (∇T )max
is the maximum of the temperature gradient. As shown in Fig. 5.10, for CH4 - Air
combustion the LU mechanism is able to recover the flame thickness for lean and
around stoichiometry mixtures. The error becomes non negligible for rich mixtures
(∼ 40%). For CH4 - O2 combustion, due to faster chemistry, the flame thickness is
around one order of magnitude lower, reaching values of ∼ 10−4 m. The comparison
between GRI-3.0 and Lu schemes shows a low error for all equivalence ratios, around
3% as shown in Fig. 5.11.
-4
-3
1.15x10
1.5x10
1.10
δl [m]
δl [m]
1.05
1.0
1.00
0.95
0.90
0.5
0.85
0.6
0.8
1.0
1.2
Equivalence Ratio [-]
(a) CH4 - Air
1.4
0.6
0.8
1.0
1.2
1.4
Equivalence Ratio [-]
(b) CH4 - O2
Figure 5.10: Flame thickness as a function of equivalence ratio for (a) methane/air and
(b) methane/O2 flames. (- -) Lu scheme ; (•) Gri3.0
90
Chapter 5. Methane Chemistry
Relative error [%]
4
2
10
8
6
4
2
1
8
6
0.6
0.8
1.0
1.2
1.4
Equivalence Ratio [-]
Figure 5.11: Relative error on flame thickness ∆δl /δl : (△) CH4 - Air; () CH4 - O2 .
5.3. Chemistry for oxycombustion of methane
91
Flame profiles Spatial profiles of heat release rate, temperature, major species
: CH4 , O2 , CO2 , H2 O, CO and some radicals H2 , HO2 , OH are plotted and
compared to the GRI-3.0 mechanism. Three equivalence ratios are tested in order
to evaluate the behavior of the LU scheme in lean, stoichiometric and rich conditions
(φ = 0.6, 1.0 and 1.4). Both oxidizers, air and pure O2 are used.
Air oxidizer Figure 5.12 and 5.13 show the results for a free propagating flame
at lean and stoichiometric conditions φ = 0.6 and φ = 1.0 respectively. Temperature profiles of the LU and the Gri-3.0 schemes are almost superimposed
and heat release rate profiles are very similar. Major species are in very good
agreement and radical species profiles are comparable in shape and value. OH
radical is of great importance because it is usually measured to capture the
flame position. It is well predicted by the LU scheme for these equivalence
ratios.
For rich conditions φ = 1.4, predictions are less accurate as shown in Fig. 5.14.
Note that this condition is extreme and had already been identified as problematic. It is shown here to evaluate the maximum error of LU scheme. From the
temperature profile one can observe that the flame thickness is over-predicted,
as already mentioned in Fig. 5.11. Regarding radical species, H2 has become
the main radical produced by the flame. Its value is under-predicted by the LU
scheme in the post flame zone. However OH levels are still in good agreement
with the reference.
Pure O2 oxidizer Same graphs are plotted in Fig. 5.15 to 5.17. When burning
with pure O2 the flame thickness decreases by one order of magnitude, the
adiabatic temperature reaches 3000 K and the flame speed increases to around
3 m.s−1 . In this configuration, the agreement between GRI-3.0 and the LU
scheme is even better than in Air combustion. For the three equivalence
ratios tested most profiles are super-imposed. Only H2 seems to be slightly
underpredicted. Heat release rates are comparable and OH profiles are very
well predicted by the LU mechanism (Fig. 5.16 and Fig. 5.17).
92
Chapter 5. Methane Chemistry
6x10
8
1800
HRR
4
1300
2
800
T [K]
-3
-1
HRR [J.m .s ]
T
0
1
2
3
4
5
300
7
6
x [mm]
(a) Temperature and HRR
0.25
O2
0.20
Y [-]
0.15
CO2
0.10
H2O
0.05
CH4
CO
0.00
1
2
3
4
5
6
7
x [mm]
(b) Major species
-3
1.5x10
OH
1.0
Y [-]
HO2 (*5)
0.5
H2
0.0
1
2
3
4
5
6
7
x [mm]
(c) Radicals
Figure 5.12: Profiles for heat release rate, temperature and species mass fractions. CH4
- Air; φ = 0.6. (- -) LU ; (–) Gri-3.0
5.3. Chemistry for oxycombustion of methane
5x10
93
9
2100
HRR
T
1500
3
T [K]
-3
-1
HRR [J.m .s ]
4
2
900
1
0
1
2
3
4
300
6
5
x [mm]
(a) Temperature and HRR
0.25
O2
0.20
0.15
Y [-]
CO2
H2O
0.10
0.05
CH4
CO
0.00
1
2
3
4
5
6
7
x [mm]
(b) Major species
5x10
-3
OH
H2 (*3)
4
Y [-]
3
2
1
HO2 (*5)
0
1
2
3
4
5
6
7
x [mm]
(c) Radicals
Figure 5.13: Profiles for heat release rate, temperature and species mass fractions. CH4
- Air; φ = 1.0. (- -) LU ; (–) Gri-3.0
94
Chapter 5. Methane Chemistry
8x10
8
2100
T
6
-1
HRR [J.m .s ]
HRR
T [K]
-3
1500
4
900
2
0
1
2
3
4
5
300
7
6
x [mm]
(a) Temperature and HRR
0.25
O2
0.20
0.15
Y [-]
H2O
0.10
CO
CO2
0.05
CH4
0.00
1
2
3
4
5
6
7
x [mm]
(b) Major species
6x10
-3
H2
Y [-]
4
2
HO2 (*5)
OH
0
0
2
4
6
8
10
x [mm]
(c) Radicals
Figure 5.14: Profiles for heat release rate, temperature and species mass fractions. CH4
- Air; φ = 1.4. (- -) LU ; (–) Gri-3.0
5.3. Chemistry for oxycombustion of methane
95
11
3000
2.0x10
HRR
1.5
-1
HRR [J.m .s ]
T
T [K]
-3
2100
1.0
1200
0.5
0.0
2.9
3.0
3.1
3.2
300
3.4
3.3
x [mm]
(a) Temperature and HRR
1.0
O2
0.8
Y [-]
0.6
0.4
H2O
0.2
CO2
CH4
CO
0.0
2.9
3.0
3.1
3.2
3.3
3.4
x [mm]
(b) Major species
6x10
OH
-2
Y [-]
4
H2 (*10)
2
HO2 (*5)
0
2.9
3.0
3.1
3.2
3.3
3.4
x [mm]
(c) Radicals
Figure 5.15: Profiles for heat release rate, temperature and species mass fractions. CH4
- O2 ; φ = 0.6. (- -) LU ; (–) Gri-3.0
96
Chapter 5. Methane Chemistry
3x10
11
3100
2400
2
-3
-1
T
1700
T [K]
HRR [J.m .s ]
HRR
1
1000
0
2.9
3.0
3.1
300
3.3
3.2
x [mm]
(a) Temperature and HRR
0.8
O2
Y [-]
0.6
0.4
H2O
CO
0.2
CH4
CO2
0.0
2.9
3.0
3.1
3.2
3.3
x [mm]
(b) Major species
8x10
-2
OH
6
Y [-]
H2 (*5)
4
2
HO2 (*5)
0
2.9
3.0
3.1
3.2
3.3
x [mm]
(c) Radicals
Figure 5.16: Profiles for heat release rate, temperature and species mass fractions. CH4
- O2 ; φ = 1.0. (- -) LU ; (–) Gri-3.0
5.3. Chemistry for oxycombustion of methane
2x10
97
11
3100
T
-3
-1
2400
1
1700
T [K]
HRR [J.m .s ]
HRR
1000
0
2.9
3.0
3.1
3.2
300
3.4
3.3
x [mm]
(a) Temperature and HRR
0.8
O2
Y [-]
0.6
H2O
0.4
CO
0.2
CO2
CH4
0.0
2.9
3.0
3.1
3.2
3.3
3.4
x [mm]
(b) Major species
5x10
-2
OH
4
H2 (*2)
Y [-]
3
2
1
HO2 (*5)
0
2.9
3.0
3.1
3.2
3.3
3.4
x [mm]
(c) Radicals
Figure 5.17: Profiles for heat release rate, temperature and species mass fractions. CH4
- O2 ; φ = 1.4. (- -) LU ; (–) Gri-3.0
98
Chapter 5. Methane Chemistry
Diffusion flame To evaluate the scheme in diffusion flame regime, the counterflow
laminar diffusion flame (Fig. 5.18) is computed giving the response to strain rate
and the flame structure.
Figure 5.18: Counterflow strained diffusion flame.
The strain rate is evaluated here as follows:
a=
∆u
H
(5.9)
where ∆u is the velocity difference between oxidizer and fuel inlets and H is the
distance between the two inlets.
Maximum temperature and extinction limit are important features when
studying diffusion flame is the response to strain, evaluated with the maximum
temperature and the extinction limit. In Fig. 5.19 the comparison between LU and
GRI-3.0 is plotted for CH4 - air combustion. The maximum temperature decreases
with increasing the strain rate similarly to the GRI scheme. The extinction strain
rate predicted by LU of 346 s−1 is very close to the value of 356 s−1 obtained with
the GRI-3.0.
For pure O2 oxidizer, the extinction limit is much higher than for Air. The GRI-3.0
and the LU mechanism both predict an extinction strain rate of ∼ 3.15 104 s−1 . The
decrease of temperature is slightly faster with the detailed mechanism after strain
rates of ∼ 5000 s−1 , ie. very high values.
5.3. Chemistry for oxycombustion of methane
99
2000
T [K]
1500
1000
500
3
4
5
6 7 8 9
2
3
4
5
100
-1
StrainRate [s ]
Figure 5.19: Maximum temperature as a function of the strain rate at 1 bar and fresh
gas temperature of 300 K for CH4 - Air. (- -) LU ; (•) Gri-3.0.
3000
T [K]
2100
1200
300
6
8
2
3
10
4
6
8
2
4
4
10
-1
StrainRate [s ]
Figure 5.20: Maximum temperature as a function of the strain rate at 1 bar and fresh
gas temperature of 300 K for CH4 - O2 . (- -) LU ; (•) Gri-3.0.
100
Chapter 5. Methane Chemistry
Flame structure Profiles of flame quantities are compared on steady strained
flames for both air and O2 oxidizers at two values of the strain rate.
Air oxidizer For air oxidizer the two selected strain rates: 50 s−1 and 300 s−1
correspond to a weakly and strongly (resp.) strained flame (see Fig. 5.19). As
shown in Fig. 5.21 for low strain rate the agreement between both schemes
is very good for the temperature, heat release rate and major species. For
radical species the agreement remains good even if the H2 species profiles
exhibit some discrepancies.
In Fig. 5.22, the strain rate has been increased to 300 s−1 . The reactive zone
thickness decreases by ∼ 30% compared to the low strain rate case. The
maximum of the heat release rate increases to ∼ 2 109 J.m−3 .s−1 whereas
the maximum temperature has only slightly decreased. The profiles compare
again favorably between GRI-3.0 and LU mechanism. In the high strain rate
case, the HO2 maximum value is multiplied by more than 2 compared to the
low strain rate case.
Pure O2 oxidizer When burning with pure O2 , the flame is much more resistant
to strain compared to burning with air. Thus strain rates are chosen much
higher: a = 1000 s−1 and a = 10000 s−1 consistently with Fig. 5.20. Because of
this high strain rate, the flame thickness drastically decreases. O2 combustion
leads to higher heat release rate and temperature, leading to higher radical
levels especially CO. In Fig. 5.24, as for the air combustion case, the HO2
species profile is doubled in the high strain rate case. As for combustion with
air, the agreement between the LU and the GRI-3.0 mechanisms is very good.
Conclusion All the comparisons made between the LU mechanism and the Gri3.0 taken as a reference have shown the ability of the Lu mechanism to reproduce
methane combustion in air and with pure O2 in premixed and diffusion flames. Thus
this scheme was implemented in the LES solver AVBP to be applied in the target
configuration.
5.3. Chemistry for oxycombustion of methane
HRR [J.m .s ]
6x10
101
8
2100
T
1200
2
T [K]
-3
-1
4
HRR
0
24
27
300
33
30
x [mm]
(a) Temperature and HRR
Y [-]
1.0
CH4
0.5
H2O (*2)
O2
CO2
0.0
24
27
30
33
x [mm]
(b) Major species
6x10
-2
CO
Y [-]
4
H2(*10)
2
OH(*5)
HO2(*300)
0
24
27
30
33
x [mm]
(c) Radicals
Figure 5.21: Profiles for heat release rate, temperature and species mass fractions. CH4
- Air; a = 50 s−1 . (- -) LU ; (–) Gri-3.0
102
Chapter 5. Methane Chemistry
2100
-1
HRR [J.m .s ]
2x10
9
-3
1200
1
T [K]
T
HRR
0
27
300
31
29
x [mm]
(a) Temperature and HRR
Y [-]
1.0
CH4
0.5
O2
H2O (*2)
CO2
0.0
27
29
31
x [mm]
(b) Major species
6x10
-2
CO
4
Y [-]
HO2(*300)
2
H2(*10)
OH(*5)
0
27
30
x [mm]
(c) Radicals
Figure 5.22: Profiles for heat release rate, temperature and species mass fractions. CH4
- Air; a = 300 s−1 . (- -) LU ; (–) Gri-3.0
5.3. Chemistry for oxycombustion of methane
-1
3000
T
10
1.5x10
HRR [J.m .s ]
103
2100
T [K]
-3
1.0
0.5
1200
HRR
0.0
2.3
300
4.5
3.4
x [mm]
(a) Temperature and HRR
1.0
CH4
O2
Y [-]
H2O (*2)
0.5
CO2
0.0
2.3
3.4
4.5
x [mm]
(b) Major species
CO
0.4
Y [-]
OH(*5)
H2(*10)
0.2
HO2(*300)
0.0
2.3
3.4
4.5
x [mm]
(c) Radicals
Figure 5.23: Profiles for heat release rate, temperature and species mass fractions. CH4
- Pure O2 ; a = 1000 s−1 . (- -) LU ; (–) Gri-3.0
104
Chapter 5. Methane Chemistry
11
3000
1.0x10
-1
HRR [J.m .s ]
T
T [K]
-3
2100
0.5
1200
HRR
0.0
3.2
3.4
3.6
300
4.0
3.8
x [mm]
(a) Temperature and HRR
1.0
CH4
O2
Y [-]
H2O (*2)
0.5
CO2
0.0
3.2
3.4
3.6
3.8
4.0
x [mm]
(b) Major species
CO
0.3
Y [-]
OH(*5)
0.2
H2(*10)
0.1
HO2(*300)
0.0
3.4
x [mm]
(c) Radicals
Figure 5.24: Profiles for heat release rate, temperature and species mass fractions. CH4
- Pure O2 ; a = 10000 s−1 . (- -) LU ; (–) Gri-3.0
5.3. Chemistry for oxycombustion of methane
5.3.4
105
Implementation of the LU scheme in AVBP
Due to the presence of the QSS species, the implementation of the Lu scheme in
AVBP is not straightforward. It implies to write a specific routine where the source
terms relationship are written and used to calculate QSS species. In addition, the
Lu scheme was developed with complex transport, meaning that Schmidt numbers
must be determined for each species, as explained in Chap. 2.
The viscosity is computed as:
n
T
(5.10)
µ = µ0
Tref
with µ0 = 1.8405e−5 Pa.s−1 , Tref = 300 K and n = 0.6759. Constant Prandtl number is used and equal to 0.7. Schmidt numbers are set according to [Franzelli 2011]
and are summarized in Tab. 5.3.
Species
Schmidt number
CH4
0.69
O2
0.75
CO2
0.96
H2 O
0.55
CO
0.76
N2
0.74
Species
Schmidt number
O
0.49
H
0.12
H2
0.21
HO2
0.75
CH3
0.68
CH2 O
0.87
OH
0.5
Table 5.3: Schmidt numbers used in AVBP for the LU mechanism.
Figure 5.25 shows the results for the 1D premixed flame at 1 bar and 300 K. Results
from AVBP are compared to CANTERA ones and exhibit a very good agreement
between the detailed transport model used in CANTERA and the simplified one in
AVBP.
In Tab. 5.4 several global flame quantities are compared. The low relative errors
demonstrate the good behavior of the LU scheme in AVBP.
Code
CANTERA
AVBP
Relative error (%)
Scheme
LU
LU
Sl
[m.s−1 ]
Tadia
[K]
δl
[m]
3.16
3.14
0.6
3052
3158
3.5
8.87e-05
8.49e-05
4.2
Table 5.4: Flame speed, burnt gas temperature and flame thickness for 1D premixed
flame. P = 1.0 bar, T = 300 K and φ = 1.0.
106
Chapter 5. Methane Chemistry
3x10
11
-1
HRR [J.m .s ]
3000
2
T [K]
-3
2100
1200
300
-0.3
0.0
1
0
-0.3
0.3
0.0
x [mm]
x [mm]
(a) Temperature
(b) HRR
0.8
8x10
O2
6
0.4
Y [-]
Y [-]
-2
OH
0.6
0.2
0.3
H2O
CH4
4
2
10*HO2
0.0
-0.3
0.0
x [mm]
(c) Major species
0.3
0
-0.3
0.0
0.3
x [mm]
(d) Radicals
Figure 5.25: Profiles for heat release rate, temperature and species mass fractions. CH4
- O2 ; φ = 1.0. (- -) CANTERA ; (–) AVBP
Chapter 6
Coaxial CH4-O2 burner - NEMO
Contents
6.1
Introduction
6.2
Numerical Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3
6.4
6.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.2.1
Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
110
6.2.2
Boundary conditions . . . . . . . . . . . . . . . . . . . . . . .
111
6.2.3
Numerical scheme . . . . . . . . . . . . . . . . . . . . . . . .
112
Non-reactive Coaxial Jets . . . . . . . . . . . . . . . . . . . . 113
6.3.1
Characteristic numbers . . . . . . . . . . . . . . . . . . . . .
113
6.3.2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
114
6.3.3
Hot Wire results . . . . . . . . . . . . . . . . . . . . . . . . .
120
Reacting flow simulations
. . . . . . . . . . . . . . . . . . . . 121
6.4.1
Flame shape and location . . . . . . . . . . . . . . . . . . . .
121
6.4.2
Flame stabilization and chemical structure
. . . . . . . . . .
133
6.4.3
Heat flux and temperature field in the solid . . . . . . . . . .
145
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
108
6.1
Chapter 6. Coaxial CH4-O2 burner - NEMO
Introduction
As shown in Fig. 6.1, the NEMO burner is instrumented with thermocouples which
are placed inside the internal lip of the injector to measure the wall temperature.
The inner tube can be recessed from 0 to 4mm. In the chosen configuration for this
study, the recess is fixed to 0mm.
Multiple diagnostics have been made to study the flame and its stabilization:
Hot Wire The cold flow has been characterized with hot wire measurements. For
several Reynolds values of methane and oxygen flows, measurements have
been made at 1 mm from the injector exit. However because the hot wire
technique depends on the density of the measured fluid, oxygen and methane
could not be measured simultaneously. As a consequence, each injection has
been measured independently, the other being turned off. The measured flow
is therefore different from the coaxial injection flow, but can still be used for
validation of the simulation methodology, applied in the same conditions.
Direct flame visualization Using a high frequency camera, images of the visible
light were recorded. These images are integrated along the line of sight. Even
though this technique is rather simple and allows to have a general picture of
the flame behavior it is difficult to compare to computed quantities.
OH* emissions With the high frequency camera and by adding a OH* filter, images of the excited OH* radical have been made marking the presence of
reaction. These images are integrated along the line of sight. Note that the
measured excited OH* is different from the radical OH computed by the fluid
solver.
Planar Laser Induced Fluorescence (PLIF) Planar Laser Induced Fluorescence
is a technique that targets minor combustion species, such as OH, CH, and NO
by exciting electrons in those species to an excited energy level, then measuring the fluorescence emitted when the molecules relax back to a lower energy
state. The OH-fluorescence intensity can be used as a direct indicator of OH
mole fraction. The obtained images are 2D cuts in the flow which make them
easily comparable to numerical computations.
Particle Image Velocimetry (PIV) Particle Image Velocimetry is an optical method
of flow visualization used to obtain instantaneous velocity measurements in
fluids. The fluid is seeded with tracer particles which, if sufficiently small,
perfectly follow the flow dynamics. In this experiment, solid particles are
injected in both methane and oxygen flows.
6.1. Introduction
109
(a)
(b)
Figure 6.1: The NEMO experiment : (a) Sketch of the injector, (b) Flame direct visualization
110
6.2
6.2.1
Chapter 6. Coaxial CH4-O2 burner - NEMO
Numerical Setup
Mesh
The Nemo experimental set up coaxial burner in a free atmosphere environment. In
the numerical simulation the atmosphere is represented by a large rectangular box.
Its length is 30 Linj and its width is 10 Linj where Linj = 2 cm is the injector length,
as shown in Fig. 6.2. In the near-field injector region, mixing is a key phenomenon
and must be captured accurately from the largest to the smallest scales, i.e. in a
DNS-like approach. A very refined mesh was then used in this zone, leading to a
mesh size of 0.05 < ∆x < 0.2 mm, and imposed for x/din < 15, where din is the
internal jet diameter. This leads to around 25 points in the lip height. No sub-grid
turbulent combustion model has been used for the computations. However the Wale
sub-grid scale is used for the LES.
2D slices of the mesh are provided in Fig. 6.3. The mesh used for the solid part
contains ∼ 3 millions of nodes and ∼ 17 millions of tetrahedra. For interpolation
reason, at the interface the resolution are equivalent between the fluid and the solid
meshes. A summary of mesh properties is given in Tab. 6.1.
Figure 6.2: Full 3D atmosphere and coaxial burner geometry.
6.2. Numerical Setup
111
Figure 6.3: 2D slices of the mesh used for NEMO. Top: Global view ; Bottom: Injector
lip region.
Mesh
Fluid M4
Solid S3
Elements
Number of Nodes
Number of Cells
Tetrahedra
Tetrahedra
∼ 4.5M
∼ 3M
∼ 26M
∼ 17M
Table 6.1: Mesh properties.
6.2.2
Boundary conditions
Boundary conditions are set accordingly to Tab. 6.2. For the inlets and the outlets,
the characteristic NSCBC procedure [Poinsot 1992] has been followed. Fuel (CH4)
and oxydizer (O2) inlets have been set with a strong relax coefficient to impose
strictly the velocity and the temperature. On the contrary, the outlet relax coefficient is quite small in order to simulate an open boundary to the atmosphere. With
such coefficients pressure waves will not be reflected into the domain and no acoustic mode will develop. For numerical stability reasons, AVBP computations use a
112
Chapter 6. Coaxial CH4-O2 burner - NEMO
coflow inlet at the bottom of the injector. The axial velocity of the coflow has been
arbitrary set to 1 m.s−1 . Walls are treated with the mass conservative boundary
condition HeatLoss as described in Chap. 4.
For the solid injector boundaries are treated using a Neumann condition for the
coupled wall: the flux is imposed from the fluid computation. At the bottom of the
injector, the temperature is assumed to be equal to the injection temperature 300
K and is imposed using a Dirichlet boundary condition. This hypothesis will be
verified a posteriori.
Boundary
Fluid Domain
Internal Injector
Inlets CH4 / O2
Inlets CoFlow
Outlets
External Injector
Internal Injector
Bottom Face
Lateral Walls
Front Lip
Condition type
Relax_U_T_Y
Relax_U_T_Y
Relax_P_3D
No Slip / Adiabatic
No Slip / Adiabatic or HeatLoss
IsoThermal
Flux
Flux
Table 6.2: Boundary condition parameters. The Relax term refers to a partially nonreflecting inlet/outlet.
6.2.3
Numerical scheme
The Navier-Stokes equations are solved using AVBP presented in Chap. 2. The
TTGC scheme is used which is third order accurate in space and time. The subgridscale turbulence model is the WALE model described in Sec. 2.5.2. This model sets
the turbulent viscosity to zero in pure-shear regions, which allows the development
of coherent structures by shear instability and tends to zero near walls making it
compatible with the physics of wall-bounded flows. This feature is very important
to capture the laminar to turbulent transition of jets. This model has been used in
LES of transcritical and supercritical non reacting jets by [Schmitt 2010b] as well as
transcritical LOx/GCH4 reacting flows [Schmitt 2010a]. For numerical stability reasons some artificial viscosity is locally added using the Colin’s sensor [Colin 2000].
Numerical parameters are described in Tab. 6.3.
The heat conduction equation is solved using AVTP presented in Chap. 4. The
implicit time integration is used in order to reduce computation cost. The maximum number of iterations for the inversion is set to 1000 and the tolerance for the
convergence of the inversion is 10−6 . As the scheme used is implicit the Fourier
number can exceed 1 without causing numerical instabilities, and is set here to 15.
6.3. Non-reactive Coaxial Jets
Code
AVBP
AVTP
113
Parameter
Numerical Scheme
SubGrid-Scale Model
Artificial viscosity
Velocity injection profile
Numerical Scheme
Number of iterations / Tolerance
Fourier number
3rd order TTGC
WALE
Colin sensor
Poiseuille / 71 power
Implicit CG
1000 / 10−6
15
Table 6.3: Numerical parameters.
6.3
Non-reactive Coaxial Jets
Table. 6.4 summarizes relevant quantities for the computed cases. Both methane
and oxygen Reynolds number can be independently varied from a laminar (Re =
1000) to a strong turbulent regime (Re = 10000).
Case
M1A1
M10A1
M1A10
ReCH4
[-]
ReO2
[-]
uCH4
[m.s−1 ]
uO2
[m.s−1 ]
ṁCH4
[g.s−1 ]
ṁO2
[g.s−1 ]
Rρ
[-]
J
[-]
Re∗
[-]
1K
10K
1K
1K
1K
10K
4.42
44.2
4.42
4.60
4.60
46.0
0.0379
0.379
0.0379
0.291
0.291
2.91
2
2
2
2.2
0.02
217
2850
9094
28496
Table 6.4: Test cases of the NEMO configuration.
In this section results for cases M1A1, M1A10 and M10A1 are presented. The
main features of the jets are analyzed in order to show and understand the major
differences between the various cases.
6.3.1
Characteristic numbers
Experimental studies [Snyder 1997, Lasheras 1998, Favre-Marinet 2001] have identified the density ratio Rρ (Eq. 6.11 ) and the momentum flux ratio J (Eq. 6.2) as
two influencing numbers for mixing efficiency in coaxial configurations:
Rρ =
J
=
ρout
ρin
(6.1)
ρout u2out
ρin u2in
(6.2)
For coaxial jets the effective Reynolds number Re* is defined as the Reynolds number
1
Xin (resp. Xout ) refers to a quantity evaluated in the inner (resp. outer) jet.
114
Chapter 6. Coaxial CH4-O2 burner - NEMO
of an equivalent jet having the same total momentum flux. It writes:
"
2 #1/2
ρ
d
u
1
−
J
d
out out out
in
Re∗ =
1+
µout
J
dout
(6.3)
where d is the jet diameter, µ the viscosity and J the momentum flux ratio.
One useful characteristic parameter of the overall mixing process is the stoichiometric mixing length Ls . The mixture fraction z is defined as the mass fraction of the
inner jet fluid since both jets contain only one species. The stoichiometric contour
is located where the mixture fraction equals the stoichiometric value zs , which for
methane is equal to zs = 0.2. The mixing length Ls is defined as the location along
the centerline where z reaches zs .
6.3.2
Results
Case M1A1 First the laminar case M 1A1 is presented, where both methane and
oxygen Reynolds numbers are equal to 1000 leading to an equivalent Reynolds number of Re∗ = 2850. This case serves as a first validation as the flows in the tubes
remain laminar. Figure 6.4a shows the methane mean mass fraction with an isoline of zs = 0.2 colored in white. This allows to evaluate the stoichiometric mixing
length non-dimensionalized by the internal diameter Ls /din = 9.3. The mean velocity (Fig. 6.4c shows a recirculation zone (in blue) which is an important feature
for the flame stabilization as it will be demonstrated in Sec. 6.4.
For these moderate values of momentum flux ratio and effective Reynolds number,
the CH4 core develops smoothly. As seen in the rms field, mixing occurs along the
potential core and follows a V-shape. As shown in Fig. 6.4b and c, the mixing of
CH4 with pure O2 jet occurs in the wake of the internal lip of the injector and in the
recirculation zone. Figure 6.5 shows an iso-surface of Q-criterion defined by Eq. 6.4:
1
1
Q=
Sij Sij − k~
ωk
(6.4)
2
2
~ is the vorticity. This
where Sij is the rate of strain tensor defined in Eq. 2.25 and ω
quantity allows to exhibit rotating structures in jets [Hunt 1988, Hussain 1995]. This
iso-surface is colored by the axial component of the vorticity ωx .
Two zones can be identified :
• in the near vicinity of the injector, one can identify toroidal vortices where the
axial vorticity is close to zero. These structures results from a primary instability derived from Kelvin-Helmholtz mechanism at the exit of the injector.
• further downstream, the vorticity structures tend to align with the x-axis.
At x/din ∼ 7.2, the coherent vortices disappear and transition to a more perturbated flow. The axial vorticity increases, leading to the tri-dimensionalization
of the flow and the development of turbulence.
6.3. Non-reactive Coaxial Jets
115
Figure 6.4: CH4 mass fraction and axial velocity u: mean (left) and rms(right) fields.
2D cut in the transverse mid-plane. In figure (a) the white line represents the iso-contour
of stoichiometric methane mass fraction YCH4 ,s = 0.2.
Figure 6.5: Q-criterion iso-surface colored by axial vorticity. Case M1A1.
116
Chapter 6. Coaxial CH4-O2 burner - NEMO
Case M10A1 In the M10A1 case the methane Reynolds number has been increased to 10 000. The momentum flux ratio J = 0.02 is divided by 100 compared to previous case M1A1 whereas the effective Reynolds number is increased to
Re∗ = 9094. As shown in Fig. 6.6 the structure of the coaxial jet is very different
from the M1A1 case. As the momentum flux is very small and the effective Reynolds
number is large the jet evolves almost like a simple free jet in the atmosphere. As
illustrated by Fig. 6.6b and Fig. 6.6d, rms fields show that for values of x/din < 2,
mixing occurs in the high shear zone in the wake of the injector internal lip. For
greater values of x/din turbulence develops fully and exhibits the structure of a
turbulent jet.
Figure 6.6: CH4 mass fraction and axial velocity u: mean (left) and rms(right) fields.
2D cut in the transverse mid-plane. In figure (a) the white line represents the iso-contour
of stoichiometric methane mass fraction YCH4 ,s = 0.2.
Figure 6.7 shows an iso-surface of Q criterion Q = 1e8 s−2 . The general feature is
largely different from the previous case. Axial tubes of vorticity appear very close
to the exit of the injector and vortex rings due to the Kelvin-Helmholtz instability
generate very quickly 3D turbulence structures. Longitudinal filaments starts at
x ∼ 1.5 din and the turbulent activity is more important than in case M1A1. This
indicates a better mixing at the tip of the methane potential core.
6.3. Non-reactive Coaxial Jets
Figure 6.7: Q-criterion iso-surface colored by axial vorticity. Case M10A1.
117
118
Chapter 6. Coaxial CH4-O2 burner - NEMO
Case M1A10 In the high oxygen Reynolds number case M1A10, the shape of the
methane potential core is very different compared to the low Reynolds case M1A1
and M10A1 (Fig. 6.8a. Figure 6.8c exhibits a strong recirculation zone in the center
of the methane jet. This negative axial velocity zone changes drastically the shape
of the methane jet. As shown in Fig. 6.8b the methane rms mass fraction field
exhibits a D-shape due to the oxygen jet which encloses the central jet leading to a
very small length of the methane core. Even though the methane core is different
from the case M1A1, rms velocity fields (Figure 6.8d) are quite similar.
Figure 6.8: CH4 mass fraction and axial velocity u: mean (left) and rms(right) fields.
2D cut in the transverse mid-plane. In figure (a) the white line represents the iso-contour
of stoichiometric methane mass fraction Ys = 0.2.
Figure 6.9 shows an iso-surface of Q criterion Q = 2e9 s−2 . Axial tubes of vorticity
appear earlier than in case M1A1 and the turbulent activity is more important.
This indicates again a better mixing at the tip of the methane potential core.
6.3. Non-reactive Coaxial Jets
119
Figure 6.9: Q-criterion iso-surface colored by axial vorticity. Case M1A10. IsoQ = 2e9
120
Chapter 6. Coaxial CH4-O2 burner - NEMO
6.3.3
Hot Wire results
Figure 6.10 compares mean axial velocity of hot wire results from experiment performed at EM2C with the present simulations.
10
70
Exp.
AVBP
60
8
50
U [m.s ]
40
-1
-1
U [m.s ]
6
4
2
30
20
10
0
0
-10
-2
-20
-3
-6x10
-4
-2
0
2
4
6
r [m]
(a) ReCH4 = 1000 and ReO2 = 1000
-3
-6x10
-4
-2
0
2
4
6
r [m]
(b) ReCH4 = 10000 and ReO2 = 10000
Figure 6.10: Mean axial velocity profiles along a transverse line at 1mm above the injector.
Comparison between (◦) experiment (EM2C) and (–) AVBP. (a) Case M1A1 ; (b) Case
M1A10 and M10A1.
In the M1A1 case, the low Reynolds numbers lead to standard Poiseuille profiles,
indicating laminar flow. When Reynolds numbers are increased to 10K, the flow
inside the pipe becomes turbulent and the velocity profile takes a flat shape. The
recirculation zones in the wake of the injector lip is predicted by the simulation
but do not appear in the hot wire measurements. This can be explained by some
limitations of the hot wire technique in very low velocity and separation regions
due to the difficulty to distinguish the flow direction. Even though a qualitative
agreement can be found, important discrepancies appear between the experimental
data and AVBP. Two main reasons may explain these differences :
• The hot wire technique needs to be recalibrated when fluid properties change.
Thus when O2 (resp. CH4 ) velocity is measured, no CH4 (resp. O2 ) is
injected. This modifies largely the recirculation zone in front of the injector
internal lip.
• As previously mentioned, AVBP computations use a coflow inlet at the outside
of the injector with a velocity of 1 m.s−1 . This coflow is responsible for some
outer jet entrainment, and may contribute to the over-prediction of velocity
in the oxygen jet. However the coaxial jet flow interactions with the ambient
air have a very small influence on the inner methane and the external oxygen
jets.
6.4. Reacting flow simulations
6.4
121
Reacting flow simulations
Based on the cold flow feature presented previously only two configurations have
been selected for the reacting flow simulations. Case M1A1 with relatively low
effective Reynolds number and Case M1A10 with a large velocity ratio as in a
rocket injector, are both representative of the phenomena but still present different
features. For each case, two types of computations have been made:
Adiabatic In this case the flux at the injector walls is set to zero, the flame does not
loose any heat at the lip. This case will be used as a reference for comparison
with non-adiabatic simulations, in terms of flame anchoring and structure in
the vicinity of the injector and in the far field.
Coupled To investigate heat transfer, the coupled framework described in Chap. 4
is applied. The solid temperature field predicted by the numerical framework
will be compared with the measured values, in an attempt to validate the
coupling methodology.
6.4.1
Flame shape and location
As adiabatic and coupled simulations are very close in terms of global flame shape,
only coupled results are presented here.
Case M1A1 A 2D cut of the instantaneous temperature and heat release rate
fields is shown in Fig. 6.11 (up to x/din = 25). As shown in Sec. 6.3 for non reacting
jets, annular vortices appear at the injector exit and propagate slowly downstream
along the outer part of the hot gas region. The central CH4 jet core is shorter than
non reacting case, while under the effect of hot gas expansion , the outer jet gets
much larger.
Looking at the heat release rate field, one can distinguish two zones:
• In the injector wake, a very thin and very strong heat release rate stands in the
recirculation zone described in the previous section. This zone corresponds to
the flame anchoring region where the methane of the inner jet reacts with the
oxygen of the surrounding jet. The oxy-combustion leads to very high levels
of heat release.
• Further downstream the methane reacts with the ambient air and the heat
release rate levels are much lower.
Figure 6.12 shows a comparison of the Abel transform of mean flame visualization
using an OH ∗ filter with the mean OH field of the coupled computation. This image
demonstrates the presence of OH ∗ in the near vicinity of the injector tip as predicted
by the LES. The flame lengths are comparable in the LES and the experiment and
extend to ∼ 5.2 din .
122
Chapter 6. Coaxial CH4-O2 burner - NEMO
(a)
(b)
Figure 6.11: Instantaneous temperature and heat release rate fields in the median plane
down to x/din = 25. Case M 1A1, Coupled.
6.4. Reacting flow simulations
123
Figure 6.12: Abel transform of a visualization with an OH ∗ filter (top) and numerical
2D cut of OH mass fraction (bottom). Case M 1A1
Figure 6.13 shows a comparison of OH-PLIF with the LES result both mean and
rms. In Fig. 6.13a it is seen that the overall shape is well reproduced and the maximum of OH is reached at the same distance from the injector exit at ∼ 2.3 din . Even
though the mean OH mass fraction compares favorably to the experimental field,
the OH-PLIF image shows no signal in the vicinity of the injector tip, whereas OH is
produced in this region in the computation. This phenomenon can be explained by
an important OH production rate at a few injector diameters downstream, which
may have led to a saturation of the sensor, so that a weaker production rate in the
vicinity of the lip could not be recorded by the camera.
124
Chapter 6. Coaxial CH4-O2 burner - NEMO
(a) Mean fields
(b) rms fields
Figure 6.13: OH-PLIF (top) and numerical 2D cut of OH mass fraction (bottom). (a)
Mean fields; (b) rms fields. Case M 1A1
6.4. Reacting flow simulations
125
Figure 6.14: Spreading angle θs based on the iso-contours of OH.
Moreover the spreading angles based on the iso-contours of OH (Fig. 6.14) are
comparable: experimental value is θsexp = 18.0◦ whereas the coupled simulation
predicts θsCP LD = 16.1◦ and the adiabatic computation (not shown here) predicts
θsADIA = 15.8◦ .
As shown in Fig. 6.13(b) the rms fields of OH are also in good agreement. One
can observe a double peak located at either side of the main axis at a distance of
x/din ∼ 2.7. These zones correspond to high shear where eddies are generated,
which modify the flame stretch, leading to variations of the chemical source terms
as the OH production.
The mean and fluctuating axial velocity profiles are now compared to PIV measurements in Fig. 6.15 at four sections downstream the injector exit: x/din = 0.03,
0.3, 1.3 and 2.6 (resp. 0.1, 1, 5 and 10mm). Both adiabatic and coupled simulation results are reported. The overall agreement is very good for both simulations
and the opening angle is well recovered. However the mean axial velocity tends to
be overpredicted at the downstream locations. The rms levels are also fairly well
reproduced in the central jet at all locations. However in the external O2 jet the
fluctuations are smaller in the numerical simulations at small axial distances, and
increase as the jet evolves. This can be due to the fact that no turbulence was
injected in the injection tubes. Turbulence is mostly generated at the exit of the
injector and takes a short distance to develop.
Figure 6.16 compares profiles of the mean CH4 mass fraction and temperature at
the same four locations for the adiabatic and coupled simulations. In the adiabatic
computation, the maximum temperature is slightly lower at x = 0.1 mm than at
x = 1 mm. This is due to a higher level of strain rate in the near vicinity of
the injector lowering the maximum temperature reached. One can observe that in
both simulations the profiles of the main species are almost superimposed, while
temperature profiles show some differences in particular close to the injector tip
where the main impact of the heat losses is expected. The last position differences
can be explained by the fact that the solution used are not fully converged and should
be taken with caution for positions far from the injector tip where the turbulence
levels are higher.
126
Chapter 6. Coaxial CH4-O2 burner - NEMO
x = 1mm
x = 0.1mm
x = 5mm
x = 10mm
6
4
2
0
-2
-4
-6
Expe
Coupled
Adiab
0
4
8
12
(a) Mean axial velocity.
x = 1mm
x = 0.1mm
x = 5mm
x = 10mm
6
Expe
Coupled
Adiab
4
2
0
-2
-4
-6
1
2
3
(b) rms axial velocity
Figure 6.15: Axial velocity at four downstream sections: (◦) Experimental PIV; (–)
Coupled simulation; (- -) Adiabatic simulation. Case M 1A1. Top: mean profiles ; Bottom:
rms profiles.
6.4. Reacting flow simulations
x = 1mm
x = 0.1mm
127
x = 5mm
x = 10mm
6
Coupled
Adiab
4
2
0
-2
-4
-6
0.0
0.5
1.0
(a) Mean CH4 mass fraction.
x = 1mm
x = 0.1mm
x = 5mm
x = 10mm
6
4
2
Coupled
Adiab
0
-2
-4
-6
300
1600
2900
(b) Mean temperature.
Figure 6.16: Mean profiles at four downstream sections: (–) Coupled simulation; (−−)
Adiabatic simulation. Case M 1A1. Top: Mean CH4 mass fraction ; Bottom: Mean temperature.
128
Chapter 6. Coaxial CH4-O2 burner - NEMO
Case M1A10 Figure 6.17 shows a 2D cut of the instantaneous temperature and
heat release rate fields for Case M 1A10. As previously described in Sec. 6.3 for
non reacting jets, the high level of turbulence developing at a few injector diameters downstream the exit surrounds and closes the flame leading to very strong
mixing levels. On the contrary of Fig. 6.11(b), the maximum heat release rate is
not produced in the close vicinity of the injector but at a few injector diameters
downstream where the turbulence develops. This is due to the high levels of shear,
stretching the flame which increases the flame power.
As for the previous case, the overall shape of the flame is compared with the Abel
transform of the direct flame visualization with a OH ∗ filter in Fig. 6.18 and with
OH-PLIF measurements in Fig. 6.19. Two main aspects of the flame can be discussed:
Flame shape Unlike the low Reynolds M 1A1 case where the flame exhibits a
rather simple "V-shape", here the flame first bends towards the center jet
of the coaxial flow, then re-expands towards the external oxygen jet. This
feature observed in the experiment is also predicted in the computation. The
attachment point is located on the outer side of the lip, close to the high velocity oxygen stream. This is observed in both experimental and computational
field.
Flame length As seen in Fig. 6.18 the flame length predicted by computation
differs from experiment, with a shorter and thicker flame in the simulation.
The location of maximum OH is also different and the computation predicts a
much shorter flame. This could be explained by the high levels of turbulence,
as shown in Fig. 6.9 conjugated with the response to the stain rate of the LU
chemistry used for the computation.
Despite these differences on the OH mean field, the rms field is well reproduced by
the computation (Fig. 6.19(b)), exhibiting two branches corresponding to the areas
of high shear and vortices creation.
Figure 6.20 compares profiles of the mean CH4 mass fraction and temperature at
the same four locations for the adiabatic and coupled simulations. Similarly to Case
M 1A1, in both simulations the profiles of the main species are almost superimposed
except at the last location. The temperature profiles differ only at the first location,
close to the injector lip, ie. where the influence of the heat loss is maximum.
6.4. Reacting flow simulations
129
(a)
(b)
Figure 6.17: Instantaneous temperature and heat release rate fields in the median plane
down to x/din = 30. Case M 1A1, Coupled.
130
Chapter 6. Coaxial CH4-O2 burner - NEMO
Figure 6.18: Abel transform of direct visualization with an OH∗ filter (top) and numerical
2D cut of OH mass fraction (bottom). (a) Mean fields; (b) rms fields. Case M 1A10.
6.4. Reacting flow simulations
131
(a) Mean fields
(b) rms fields
Figure 6.19: OH-PLIF (top) and numerical 2D cut of OH mass fraction (bottom). (a)
Mean fields; (b) rms fields. Case M 1A10.
132
Chapter 6. Coaxial CH4-O2 burner - NEMO
x = 1mm
x = 0.1mm
x = 5mm
x = 10mm
6
Coupled
Adiab
4
2
0
-2
-4
-6
0.0
0.5
1.0
(a) Mean CH4 mass fraction.
x = 1mm
x = 0.1mm
x = 5mm
x = 10mm
6
Coupled
Adiab
4
2
0
-2
-4
-6
300
1600
2900
(b) Mean temperature.
Figure 6.20: Mean profiles at four downstream sections: (–) coupled simulation; (- -)
adiabatic simulation. Case M 1A10.
6.4. Reacting flow simulations
6.4.2
133
Flame stabilization and chemical structure
Case M1A1 Focus is now made on the region close to the injector to compare
in detail the adiabatic and coupled cases. These observations are limited to the
kinetics of the radicals. In the coupled simulation, as the temperature at the wall is
modified by the heat flux, the chemical equilibrium is also modified compared to an
adiabatic flame. Figure 6.21 shows 2D cuts of mean OH and HO2 mass fractions
for both adiabatic and coupled simulations.
In the coupled case the production of OH is lifted at a short distance from the lip
(Fig. 6.21a), whereas it appears right on the adiabatic wall. On the contrary HO2
is produced closer to the internal jet in the coupled case (Fig. 6.21b) and a strong
HO2 production peaks at the wall. OH production is due to chemical reactions
with a large energy of activation thus it decreases when the temperature decreases.
On the contrary, HO2 is produced by reactions with low or zero energy of activation
and becomes dominant close to cooler walls.
(a) OH mass fraction.
6.5 10−2 .
Maximum value is (b) HO2 mass fraction.
4.4 10−4 .
Maximum value is
Figure 6.21: Mean OH (left) and HO2 (right) mass fractions fields. Close view on the
injector internal lip. Case M 1A1. Coupled and Adiabatic cases.
This trend is also observable on profiles at very close locations to the injector
in Fig. 6.22. OH radical is more important in the adiabatic case than in the coupled
simulation. The opposite behavior occurs for HO2 , which is larger in the coupled
simulation.
In order to compare the flame chemical structures, scatterplots of several quantities
as functions of the mixture fraction z are used. The mixture fraction z is defined
by [Bilger 1990], based on the atomic mass fractions:
z=
O−z )
2 zC + 1/2zH + (zO
O
F
F
O
2 zC + 1/2zH + zO
(6.5)
134
Chapter 6. Coaxial CH4-O2 burner - NEMO
x = 1mm
x = 0.1mm
x = 5mm
x = 10mm
6
4
2
Coupled
Adiab
0
-2
-4
-6
0
x10
-3
25
50
(a) Mean OH mass fraction
x = 1mm
x = 0.1mm
x = 5mm
x = 10mm
6
4
2
Coupled
Adiab
0
-2
-4
-6
0
x10
1
-4
2
(b) Mean HO2 mass fraction.
Figure 6.22: Radical species profiles at four downstream locations: (–) coupled simulation;
(- -) adiabatic simulation. Case M 1A1.
The mixture fraction allows to identify the local fuel/oxidizer ratio going from pure
fuel (z = 1) to pure oxidizer (z = 0). The superscripts F (O) refers to a quantity
evaluated in the fuel (resp. the oxidizer) stream. The atomic mass fraction zi is
6.4. Reacting flow simulations
135
Figure 6.23: Sketch of the boxes used for the scatter plots. Boxes extend over the whole
azimuthal angle.
defined as
zi =
Ns
X
nk,i Yk
k=1
Wk
(6.6)
where Ns is the number of species, nk,i the number of atom i in species k, Yk the
mass fraction and Wk the molar mass of species k.
Figures 6.24 and 6.25 display instantaneous scatter plots of temperature, heat release
rate and radical mass fractions versus mixture fraction for the coupled and adiabatic
simulations, obtained in two boxes defined in Fig. 6.23. Reference results from
Cantera calculation are also reported.
Each box extends over the whole azimuthal angle.
Box 1 This box extends from the lip to 0.3 mm downstream and its thickness
is equal to the thickness of the lip. This corresponds to the flame anchoring region. The adiabatic temperature scatter plot (red dots) in Fig. 6.24(a)
shows that there are no points on the mixing line, indicating that the flame
is ignited everywhere and no premixing occurs. Points are very close to the
laminar flame (grey line) in a counterflow with a very low strain rate a = 500
s−1 . On the contrary, for coupled wall boundary (black dots), the temperature
does not reach the maximum temperature of the 1D counterflow flame. This
can be explained by the heat flux loss at the wall. The coupled simulation
temperature follows three distinct lines from pure oxidizer (z = 0) to pure fuel
(z = 1). These lines result from a strong modification of the chemical path due
to the heat loss at the tip of the injector. Even though the three reaction zones
observable in Fig. 6.26 could look like a triple flame structure, as shown in the
scatterplots there is no premixing process. Thus the three branches (called
lean side, main and rich side) observable in Fig. 6.26 correspond to a diffusion
flame modified by the heat loss at the wall: lean and rich side correspond to
the traces, in the vicinity of the injector, of the radicals produced at the wall.
Note that when z tends to zero or one, ie. pure O2 or pure CH4 , the reactants
temperatures does not tend to 300 K. This exhibits the pre-heating of the
reactants by the coupled injector lip. In Fig. 6.26, in the adiabatic case, this
flame structure is not present.
136
Chapter 6. Coaxial CH4-O2 burner - NEMO
These phenomena lead to an important modification of the heat release rate
as shown in Fig. 6.24b. In the adiabatic case, the flame follows the pattern
of a counterflow flame with a unique peak located at z = 0.32, similar to
the results obtained with CANTERA (grey line in Fig. 6.24). In the coupled
simulation, a more powerful peak is observable at z = 0.2. This peak is larger
by ∼ 44%.
As the chemical path is modified by the wall heat flux, the production of
radicals is different between both boundary conditions: it stays very close to
the 1D counterflow flame in the adiabatic case but has a lower OH and a much
higher production of HO2 for the coupled case (Fig. 6.24c and Fig. 6.24d).
This high production of HO2 close to the wall is responsible for the large heat
release at the wall. This phenomenon is explained in details in Chap. 7.
(a) Temperature
(b) Heat release rate
(c) OH
(d) HO2
Figure 6.24: Temperature, heat release rate and radical mass fractions as functions of the
mixture fraction. (black dots) coupled simulation; (red dots) adiabatic simulation; (grey
line) reference counterflow diffusion flame at a strain rate a = 500 s−1 computed with
CANTERA. Box 1. O2 (CH4 ) is injected above (resp. below) the splitter plate. Case
M 1A1.
6.4. Reacting flow simulations
137
Box 2 This box is located further downstream, but still in the reaction zone.
Boundaries are xmin = 3 mm, xmax = 6 mm and rmin = 1 mm, rmax = 4
mm. The combustion for both adiabatic and coupled cases is very similar
to the 1D counterflow diffusion flame as shown in Fig. 6.25. Indeed the wall
heat flux is no more influencing the flame in this zone. The heat release rate
is slightly lower than the adiabatic case in Box 1, while OH and HO2 levels
have increased.
(a) Temperature
(b) Heat release rate
(c) OH
(d) HO2
Figure 6.25: Temperature, heat release rate and radical mass fractions as functions of the
mixture fraction. (black dots) coupled simulation; (red dots) adiabatic simulation; (grey
line) reference counterflow diffusion flame at a strain rate a = 700 s−1 computed with
CANTERA. Box 2. O2 (CH4 ) is injected above (resp. below) the splitter plate. Case
M 1A1.
138
Chapter 6. Coaxial CH4-O2 burner - NEMO
Figure 6.26: Heat release rate iso-contours superimposed to an instantaneous temperature
field. Iso-contours range from 5 108 to 1 1010 W.m−3 and temperature from 300 to 2960 K.
O2 (CH4 ) is injected above (resp. below) the splitter plate. Case M 1A1.
6.4. Reacting flow simulations
139
Case M1A10 Figure 6.27 compares the radical mass fraction fields in the close
wake of the injector lip for Case M 1A10. Unlike Case M 1A1 where the Reynolds
numbers of the injection jets are low, the fields are here very similar in the adiabatic
and coupled cases. In Fig. 6.27a the OH mass fraction shows a lifted reaction zone
above the lip for both wall boundary conditions, i.e. even in the adiabatic case because of the high velocity of the oxygen stream. Still, in this case, OH production is
slightly more intense and starts closer to the lip as in M 1A1. The same conclusions
held for HO2 radical fields, which exhibited large discrepancies in case M 1A1, but
are very similar here. The maximum production is located at the external part of
the lip and only the intensity is slightly higher in the coupled case, as in M 1A1 case.
(a) OH mass fraction. Maximum value is (b) HO2 mass fraction. Maximum value
6.2 10−2 for both cases.
is 6.7 10−4 .
Figure 6.27: Radical mean mass fractions fields. Close view on the injector internal lip.
O2 (CH4 ) is injected above (resp. below) the splitter plate. Case M 1A1.
These similitudes between the coupled and the adiabatic cases are also observable in
1D profiles of the radicals plotted in Fig. 6.28. For all distances profiles are almost
superimposed at all axial distances, except at the last location.
140
Chapter 6. Coaxial CH4-O2 burner - NEMO
x = 1mm
x = 0.1mm
x = 5mm
x = 10mm
6
4
2
Coupled
Adiab
0
-2
-4
-6
0
x10
-4
1
2
3
(a) Mean HO2 mass fraction.
x = 1mm
x = 0.1mm
x = 5mm
x = 10mm
6
4
2
Coupled
Adiab
0
-2
-4
-6
0.0
x10
2.5
-2
5.0
(b) Mean OH mass fraction
Figure 6.28: Radical species profiles at four downstream locations: (–) coupled simulation;
(- -) adiabatic simulation. Case M 1A10.
6.4. Reacting flow simulations
141
As for M 1A1 case and using the same boxes, scatter plots are used to investigate
the flame structure.
Box 1 As for the low Reynolds case, this region corresponds to the anchoring region.
Scatter plots for coupled and adiabatic conditions are compared to the 1D
counterflow diffusion flame with a strain rate of a = 2000 s−1 , chosen by
comparing the maximum heat release rate with that of the LES computations.
In Fig. 6.29(a), contrary to case M 1A1, the maximum temperature reached in
the adiabatic case is lower than the one predicted by the 1D flame (grey curve).
Note that the scatter plot is more dispersed than in case M 1A1 indicating a
larger range of strain. The maximum temperature of the coupled simulation
is much lower than the adiabatic case and exhibits again three distinct lines.
The heat release in the adiabatic simulation (Fig. 6.29b) follows a diffusion
flame structure. Note that the peak heat release for the adiabatic case is
lower than the reference diffusion flame (grey line), consistent with the lower
maximum temperature. In the coupled case as shown in Fig. 6.30 a small "lean
side" reaction zone stands at the wall. This is responsible for the heat released
in lean condition (z ∼ 0.15) in Fig. 6.29b but the main peak is located at the
same mixture fraction than the adiabatic condition (z = 0.25). The main peak
of the coupled heat release rate is larger than the adiabatic computation by
∼ 38%.
The OH production is more similar in adiabatic and coupled computations
than in case M 1A1 and widely smaller than the reference 1D flame (Fig. 6.29c).
The HO2 radical in the coupled simulation is mainly produced at very low
mixture fraction whereas in the adiabatic case a non negligible part is produced
at larger mixture fraction.
142
Chapter 6. Coaxial CH4-O2 burner - NEMO
(a) Temperature
(b) Heat release rate
(c) OH
(d) HO2
Figure 6.29: Temperature, heat release rate and radical mass fractions as functions of the
mixture fraction. (black dots) coupled simulation; (red dots) adiabatic simulation; (grey
line) reference counterflow diffusion flame at a strain rate a = 2000 s−1 computed with
CANTERA. Box 1. Case M 1A10.
6.4. Reacting flow simulations
143
Figure 6.30: Heat release rate iso-contours superimposed to an instantaneous temperature field. Iso-contours range from 8 108 (light grey line) to 5 1010 (blue line) W.m−3 and
temperature from 300 to 2930 K. O2 (CH4 ) is injected above (resp. below) the splitter
plate. Case M 1A10.
144
Chapter 6. Coaxial CH4-O2 burner - NEMO
Box 2 In the box located downstream, the adiabatic and the coupled simulations
are very close. They compared favorably to a reference 1D counterflow diffusion flame with a strain rate of a = 1000 s−1 . Conclusions made for M 1A1,
Box 2 remain valid in this case.
(a) Temperature
(b) Heat release rate
(c) OH
(d) HO2
Figure 6.31: Temperature, heat release rate and radical mass fractions as functions of the
mixture fraction. (black dots) coupled simulation; (red dots) adiabatic simulation; (grey
line) reference counterflow diffusion flame at a strain rate a = 1000 s−1 computed with
CANTERA. Box 2. Case M 1A10.
6.4. Reacting flow simulations
6.4.3
145
Heat flux and temperature field in the solid
This last section discusses heat transfer phenomena, which was the main motivation for the study of NEMO. This is indeed one of the first direct validation of the
methodology for conjugate heat transfer simulations in an academic but still representative configuration. In the experimental set up, thermocouples have been used
at four locations in the internal injector: x = −0.5, −2.5, −4.5 and −6.5 mm. The
metal of the injector is steel 316L. Its properties are summarized in Tab. 6.5.
Variable
Density
Heat capacity
Heat conductivity
Heat diffusivity
Lip thickness
Characteristic time
[Dimension]
ρS
CS
λS
DS
el
θS
kg.m−3
J.kg−1 .K−1
W.m−1 .K−1
m2 .s−1
m
s
8000
500
16.3
4.1 10−6
1.2 10−3
0.35
Table 6.5: Injector Steel 316L properties and characteristic numbers.
In order to reach convergence for the solid temperature and as the solid thermal
characteristic time is very large compare to the fluid one, the hypothesis of the
quasi steady state approximation is made. This allows asynchronous simulations,
where the time steps τS and τF for the solid and the fluid respectively, are different and the number of iterations between two coupling events αS and αF differ,
leading to αS τS 6= αF τF . Following the recommendations discussed in Chap. 4,
the coupling frequency is set in order to ensure computation stability. To avoid
interpolation error, the spatial resolution in the solid is identical to that in the fluid
at the interface. This leads to cell size smaller than necessary, but has no impact as
the computational time for the resolution of the solid thermal problem remains small.
6.4.3.1
Validation
Results from the coupled computations for cases M 1A1 and M 1A10 are compared with the experimental temperatures obtained with the four thermocouples
in Fig. 6.32. The lines represent the computations profiles taken along the half
width line of the solid (Fig. 6.34). This choice will be discussed in the next section.
For both cases the computation compares favorably to the data measured by the
thermocouples. In case M 1A1, the simulations slightly over-predicts the temperature whereas in Case M 1A10 the opposite trend is observable. The shape of the
profiles is remarkably well reproduced. As shown in Fig. 6.32 (and later in Fig. 6.46),
the heat flux at the back face of the injector, for Case M 1A1 is not equal to zero.
This means that the boundary condition of imposed temperature is not correct,
and that the simulated injector length should be increased. This will be discussed
in Chap. 8.
146
Chapter 6. Coaxial CH4-O2 burner - NEMO
900
Temperature [K]
,
,
M1A1
M1A10
700
500
300
-20
-15
-10
-5
0
x [mm]
Figure 6.32: Profile of the temperature at the center line of the injector wall. (△ and ◦)
experimental data; (–) and (- -) coupled simulations. Case M 1A1 and M 1A10.
6.4. Reacting flow simulations
6.4.3.2
147
Case M1A1
(a) Global view of the mean fluid temperature field.
(b) Zoom on the lip of the injector.
Figure 6.33: 2D cut of the mean fluid and solid temperature field. Injector field is
truncated at 2/3 of its length. Case M 1A1.
Figure 6.33 displays the mean fluid and solid temperature fields. The temperature
in the solid reaches ∼ 865 K at the tip and decreases upstream. The evolution of the
solid temperature across the injector wall is plotted in Fig. 6.35 at three different
locations as shown in Fig. 6.34: at the inside skin of the CH4 injector, along the
half width line of the solid and on the external skin along the O2 jet. Depending on
the location an important deviation of the temperature up to ∼ 30K can be found
at the tip of the injector. This is the trace of the flame, located at the external side
of the inner injector tube. After one injector thickness upstream, the three profiles
148
Chapter 6. Coaxial CH4-O2 burner - NEMO
Figure 6.34: Sketch of the inner tube of the injector. The red crosses represent the
thermocouples positions. Lines used to plot the temperature: (–) O2 External skin; (- -)
Half width line; (-.-) Internal skin.
merge and follow the same decreasing law.
880
880
860
735
840
TS [K]
820
590
800
-1.5
-1.0
-0.5
0.0
445
300
-20
External
Internal
Middle
-15
-10
-5
0
x [mm]
Figure 6.35: Temperature in the injector. (–) External O2 skin; (- -) Injector center line;
(-.-) Internal CH4 skin. Case M 1A1.
As the solid is heated at the tip by the flame, the heat is transferred to the back of the
injector. As shown in Fig. 6.36(a) the flux at the injector tip reaches a very high value
of φw,x = −1.7 106 W.m2 at the center of the face and then gradually goes to zero
6.4. Reacting flow simulations
149
towards the sides. Then, heat is transferred back from the solid to the surrounding
fluid along the sides of the tube, as shown in Fig. 6.36(b). As the temperature at
the back of the injector is imposed equal to the injected fluid temperature (300 K),
the transverse heat flux is equal to zero at the location (x = −2.0 cm) as shown
in Fig. 6.46. The transverse heat flux sign changes between the internal and the
external side, which confirms that the solid heats the fluid on both sides. Both fluxes
are of the same order of magnitude as the fluid velocities are very close uCH4 = 4.41
m.s−1 and uO2 = 4.6 m.s−1 , leading to a similar convective exchange coefficient in
both reactants.
As the fluid is then pre-heated along the annular tube, the fluid density changes and
creates large thermal boundary layers as shown in Fig. 6.37. As the temperature
evolves linearly in the solid the density along the walls follows also a linear law as
shown in Fig. 6.38.
The wall distance y + is introduced, defined by:
y+ =
ρw u τ y
µw
(6.7)
r
(6.8)
with the friction velocity:
uτ =
τw
ρw
where τw is the wall shear stress defined by Eq. 2.24. As demonstrated in [Cabrit 2009],
the WALE ([Nicoud 1999]) subgrid scale model can be used without a wall law as
long as the y + values at the wall remain small: y + ∼ 5. In the present case, the
values of y + along the injector face are shown in Fig. 6.39. As the mesh used to
resolve the flow in the pipe is very fine, the y + values evolve from 15 at the bottom
of the injector in the O2 jet to less than 1 close to the tip of the injector. For most
part of the injector, the y + value remains below 5. This means that in the close region of the injector tip, where the temperature and heat transfer are maximum, the
boundary layer is well resolved and the heat exchange coefficient is well evaluated.
150
Chapter 6. Coaxial CH4-O2 burner - NEMO
-2
Heat Flux [W.m ]
0.0
-0.5
-1.0
-1.5
6
-2.0x10
1.95
2.35
2.75
3.15
y [mm]
(a) Axial heat flux along the front face of the inner tube (x = 0 mm).
-2
Heat Flux [W.m ]
2
1
0
-1
External
Internal
5
-2x10
-20
-15
-10
-5
0
x [mm]
(b) Transverse heat flux along lateral walls of the inner tube. (–) External O2 skin; (-.-) Internal CH4 skin.
Figure 6.36: Heat flux along the faces of the injector. Case M 1A1.
6.4. Reacting flow simulations
151
Figure 6.37: Temperature in the solid and fluid density field. Case M 1A1.
1.4
External
Internal
-3
ρ [kg.m ]
1.2
1.0
0.8
0.6
0.4
0.2
-20
-15
-10
-5
0
x [mm]
Figure 6.38: Density along the injector walls. (–) External O2 skin; (-.-) Internal CH4
skin. Case M 1A1.
152
Chapter 6. Coaxial CH4-O2 burner - NEMO
15
External
Internal
+
y [-]
10
5
0
-20
-15
-10
-5
0
x [mm]
Figure 6.39: y + along the injector walls. (–) External O2 skin; (-.-) Internal CH4 skin.
Case M 1A1.
6.4. Reacting flow simulations
6.4.3.3
153
Case M1A10
In the high Reynolds case M 1A10, the solid temperature increases under the effect
of the flame, but in a much lower extent than in case M 1A1. The maximum temperature at the tip of the injector does not exceed ∼ 580 K. This is due to the higher
velocity in the external O2 jet, which modifies the flame structure and location in
comparison with case M1A. Moreover the convective exchange coefficient increases
on this side and more heat is extracted from the solid.
(a) Global view of the fluid and solid temperature fields.
(b) Zoom on the lip of the injector.
Figure 6.40: 2D cut of the mean fluid and solid temperature field. Case M 1A10.
As for Case M 1A1 the temperature depends on the position in the solid. As exhibited by Fig. 6.41 the difference between the internal, half width and external
positions vanishes very fast at less than one injector lip thickness.
154
Chapter 6. Coaxial CH4-O2 burner - NEMO
600
TS [K]
500
400
300
-20
600
550
500
-1.5
-1.0
-0.5
0.0
External
Internal
Middle
-15
-10
-5
0
x [mm]
Figure 6.41: Temperature in the injector. (–) External O2 skin; (- -) Injector center line;
(-.-) Internal CH4 skin. Case M 1A10.
Because of the high velocity ratio of case M 1A10, the transverse heat flux along
the injector walls is higher on the external side than on the internal one as shown
in Fig. 6.42(b).
The maximum axial flux (qx = 1.3 106 W.m−2 ) transmitted by the flame to the
injector is smaller than in the laminar case M 1A1 (qx = 1.7 106 W.m−2 ), by ∼ 23%.
This explains the lower temperature in the solid and the low resulting flux on the
side.
Several conjugate effects have to be retained to explain the changes compared to
laminar case M 1A1:
• the increase of the convective coefficient along the oxygen side of the tube,
• the incoming flux from the flame at the tip is lower because the flame is lifted
further,
• the flame shape which appears to be thinner because of the high shear of the
O2 jet,
• the flame structure is modified has shown previously leading to smaller temperature.
6.4. Reacting flow simulations
155
-2
Heat Flux [W.m ]
0.0
-0.5
-1.0
6
-1.5x10
1.95
2.35
2.75
3.15
y [mm]
(a) Axial heat flux along the front face of the injector (x = 0 mm).
3
-2
Heat Flux [W.m ]
2
External
Internal
1
0
-1
-2
5
-3x10
-20
-15
-10
-5
0
x [mm]
(b) Tranverse heat flux along lateral wall of the injector. (–) External
O2 skin; (-.-) Internal CH4 skin.
Figure 6.42: Heat flux along the faces of the injector. Case M 1A10.
156
Chapter 6. Coaxial CH4-O2 burner - NEMO
The lower solid temperature and reactant jet heating have a direct on the density
field compared to case M 1A1. As shown in Fig. 6.43, the thermal boundary layers
developing along the injector walls are much thinner especially in the internal CH4
jet. The density profiles (Fig. 6.44) along the injector walls are no more linear
but remain almost constant for half of the injector length and then drop rapidly.
As for case M 1A1, the y + values remain small in the zone of the injector where
the temperature is high (Fig. 6.45 and the heat convective exchange is thus well
captured.
Figure 6.43: Temperature of the injector and impact on the density field. Case M 1A10.
6.4. Reacting flow simulations
157
1.4
-3
ρ [kg.m ]
1.2
1.0
0.8
0.6
0.4
0.2
-20
External
Internal
-15
-10
-5
0
x [mm]
Figure 6.44: Density profiles along the injector walls. (–) External O2 skin; (-.-) Internal
CH4 skin. Case M 1A10.
158
Chapter 6. Coaxial CH4-O2 burner - NEMO
20
External
Internal
+
y [-]
15
10
5
-20
-15
-10
-5
0
x [mm]
Figure 6.45: y + along the injector walls. (–) External O2 skin; (-.-) Internal CH4 skin.
Case M 1A10.
6.4. Reacting flow simulations
159
The maximum value of the axial heat flux at the tip of the injector is lower than
in the M 1A1 case as shown in Fig. 6.46. In the high Reynolds case M 1A10 the
axial flux transmitted (qx = 1.15 106 W.m−2 ) by the flame to the injector is smaller
than in the laminar M 1A1 case (qx = 1.50 106 W.m−2 ). This represents a decay
of ∼ 30%. Thus the temperature in the solid is lower and the flux to be evacuated
is also smaller. This could be explained by the flame shape which appears to be
thinner because of the high shear of the O2 jet in Case M 1A10.
-2
Heat Flux [W.m ]
0.0
-0.5
-1.0
-1.5
M1A1
M1A10
6
-2.0x10
-20
-15
-10
-5
0
x [mm]
Figure 6.46: Axial heat flux along centerline of the injector. (–) Case M 1A1 ; (- -) Case
M 1A10.
160
6.5
Chapter 6. Coaxial CH4-O2 burner - NEMO
Conclusions
The oxy-combustion of methane in a coaxial burner (NEMO) has been studied in
this chapter. As the reactants are methane and pure oxygen, the equilibrium state is
different from a methane/air flame and usual reduced kinetic schemes fail to predict
correctly burnt gas temperature and composition. Therefore an analytical chemical
scheme [Lu 2008] has been used to compute the heat released by the flame. Test on
a variety of simple 1D cases show good agreement between the analytical Lu scheme
and the detailed GRI-3.0 mechanism.
As temperature measurements are available in the injector walls, 3D simulations
were performed using a conjugate heat transfer approach in a fully coupled framework taking into account the heat transfer within the solid, as described in Chap. 4.
Results show good agreement between the computed and measured temperatures in
the solid, which validates the conjugate heat transfer methodology.
In order to study stabilization mechanisms of the flame behind the tip of the injector
these fully coupled simulations have been compared to adiabatic simulations.
For low Reynolds numbers (Case M 1A1), the overall shape and position of the flame
are well captured by both the adiabatic and the coupled simulations. Far from the
injector, the chemical structure of the flame is close to a 1D laminar counterflow
flame. In the near vicinity of the injector, the chemistry is severely modified by the
boundary condition: when heat flux is withdrawn from the flame by the solid, the
chemical kinetic is strongly modified leading to large amounts of HO2 radical at the
wall and a strong heat release rate.
When the Reynolds number is increased (Case M 1A10), the differences on the flame
stabilization between adiabatic and coupled simulations diminish. The difference
with Case M 1A1 is due to the high shear induced by the high velocity oxygen jet
which increases the strain in the wake of the lip. The flame stabilizes closer to the
oxygen stream and is cooled down by the high heat exchange.
Thanks to these computations, we can identify two mechanisms for flame stabilization:
• in the adiabatic case the flame is anchored in the wake of the lip and behaves
almost as a 1D counterflow diffusion flame.
• in the coupled case the chemical equilibrium is modified and a large amount
of heat release and HO2 radical are produced at the wall leading to a different
flame.
Part III
Applications to Liquid Rocket
Engine Configurations
Chapter 7
Laminar H2-O2 Flames
164
7.1
Chapter 7. Laminar H2-O2 Flames
Introduction
This chapter presents results of H2/O2 laminar premixed 1D unsteady flames propagating towards an inert wall. This chapter is based on an article submitted to Combustion & Flame. Such configuration, shown in Fig 7.1 is called head-on quenching
(HOQ). Even though this situation is a simplification of real conditions, it provides
useful information about the wall heat flux and quenching distance.
Figure 7.1: Flame-Wall interaction: Head-On Quenching.
7.2
Chemistry model
Previous studies [Ezekoye 1992] have shown that a single-step chemistry formalism
is unable to predict correctly quenching phenomena. Moreover such a mechanism
tends to over-predict wall heat flux due to the absence of low activation energy
recombination reactions which play a key role close to the wall. In the present work,
the combustion of hydrogen and oxygen is modeled using a skeletal mechanism
accounting for 8 species and 12 reactions shown in Tab. 7.1 [Boivin 2011].
Fall-Off reaction This mechanism accounts for pressure dependence in some of
the pre-exponential coefficients as in three-body reaction R4 through a fall-off mechanism.
A three-body reaction is a gas-phase reaction of the form
A + B + M ⇋ AB + M
(7.1)
Here M is an unspecified collision partner that carries away excess energy to stabilize
the AB molecule (forward direction) or supplies energy to break the AB bond (reverse direction). Different species may be more or less effective in acting as collision
partner. A species that is much lighter than A and B may not be able to transfer
much of its kinetic energy, and so would be inefficient as a collision partner. On the
other hand, a species with a transition from its ground state that is nearly resonant
7.2. Chemistry model
165
with one in the AB* activated complex may be much more effective at exchanging
energy.
A fall-off reaction is one that has a rate that is first-order in [M] at low pressure, like
a three-body reaction, but becomes zero-order in [M] as [M] increases. Dissociation
/ association reactions of polyatomic molecules often exhibit this behavior. The
reaction rate then writes:
Pr
kf (T, Pr ) = k∞
· F (T, Pr )
(7.2)
1 + Pr
where F (T, Pr ) is the fall-off function and Pr is the reduced pressure given by
Pr =
k0 [M]
k∞
(7.3)
where k0 is the low pressure rate coefficient, k∞ is the high pressure rate coefficient
and [M ] is:
[M ] =
X
ǫk [Ck ]
(7.4)
k
where [Ck ] is the concentration of species k and ǫk its collision efficiency.
Following Troe fall-off model [Gilbert 1983] in its general form, the fall-off function
is:
log10 Fc (T )
(7.5)
log10 F (T, Pr ) =
1 + f12
where
Fc (T ) = (1 − A) exp(−T /T3 ) + A exp(−T /T1 ) + exp(−T2 /T )
(7.6)
f1 = (log10 Pr + C)/(N − 0.14(log10 Pr + C))
(7.7)
C = −0.4 − 0.67 log10 Fc (T )
(7.8)
= 0.75 − 1.27 log10 Fc (T )
(7.9)
N
where A, T1 , T2 and T3 are constants to be specified for each fall-off reaction.
This mechanism is fully implemented in CANTERA but it had to be implemented in
AVBP to be able to compute accurately H2 /O2 flames in the whole pressure range.
166
Chapter 7. Laminar H2-O2 Flames
Aa
Reaction
1
H + O2 ⇌ OH + O
2
H2 + O ⇌ OH + H
3
H2 + OH ⇌ H2 O + H
4
H + O2 + M → HO2 + Mb
5
6
HO2 + H → 2 OH
HO2 + OH ⇌ H2 + O2
7
8
HO2 + OH → H2 O +O2
H + OH + M ⇌ H2 O + Mc
9
2 H + M ⇌ H2 + Mc
10
11
12
2 HO2 → H2 O2 + O2
HO2 + H2 → H2 O2 + H
H2 O2 + M → 2 OH + Md
kf
kb
kf
kb
kf
kb
k0
k∞
kf
kb
kf
kb
kf
kb
k0
k∞
3.52
7.04
5.06
3.03
1.17
1.28
5.75
4.65
7.08
1.66
2.69
2.89
4.00
1.03
1.30
3.04
3.02
1.62
8.15
2.62
n
1016
1013
104
104
109
1010
1019
1012
1013
1013
1012
1013
1022
1023
1018
1017
1012
1011
1023
1019
-0.7
-0.26
2.67
2.63
1.3
1.19
-1.4
0.44
0.0
0.0
0.36
0.0
-2.0
-1.75
-1.0
-0.65
0.0
0.61
-1.9
-1.39
Ea
71.42
0.60
26.32
20.23
15.21
78.25
0.0
0.0
1.23
3.44
231.86
-2.08
0.0
496.14
0.0
433.09
5.8
100.14
207.62
214.74
Table 7.1: Rate coefficients in Arrhenius form k = AT n exp(−E/R0 T ) as in [Boivin 2011]
.
a
Units are mol, s, cm3 , kJ and K.
b
Chaperon efficiencies H2 : 2.5, H2 O : 16.0, 1.0 for all other species.
Troe falloff with Fc = 0.5.
c
Chaperon efficiencies H2 : 2.5, H2 O : 12.0, 1.0 for all other species.
d
Chaperon efficiencies H2 : 2.5, H2 O : 6.0, 1.0 for all other species.
Troe falloff with Fc = 0.265 exp(−T /94) + 0.735 exp(−T /1756) + exp(−T /5182).
7.3. Article submitted to Combustion & Flame
7.3
Article submitted to Combustion & Flame
167
Effect of pressure on Hydrogen/Oxygen coupled
flame-wall interaction
R. Maria , J.P. Rocchia , L. Selleb,c , F. Duchainea , B. Cuenota
a
b
CERFACS, 42 av G. Coriolis 31057 Toulouse, France
Université de Toulouse; INPT, UPS; IMFT (Institut de Mécanique des Fluides de
Toulouse);
Allée Camille Soula, F-31400 Toulouse, France
c
CNRS; IMFT; F-31400 Toulouse, France
Abstract
The design and optimization of liquid-fuel rocket engines is a major scientific
and technological challenge. One particularly critical issue is the heating of
solid parts that are subjected to extremely high heat fluxes when exposed
to the flame. This in turn changes the injector lip temperature, leading to
possibly different flame behaviors and a fully coupled system. As the chamber pressure is usually much larger than the critical pressure of the mixture,
supercritical flow behaviors add even more complexity to the thermal problem. When simulating such phenomena, these thermodynamic conditions
raise both modeling and numerical specific issues. In this paper, the interactions of Hydrogen/Oxygen supercritical flames with solid walls are studied
by use of conjugate heat transfer simulations, to evaluate the resulting heat
flux to the wall, the wall temperature, and the impact on the flame as well as
their sensitivity to high pressure and real gas thermodynamics up to 100 bar
where real gas effects are important. At low pressure, results are found in
good agreement with previous studies in terms of wall heat flux and quenching distance, and the wall stays close to isothermal. On the contrary, due to
important changes of the fluid transport properties and the flame characteristics, the wall experiences significant heating at high pressure condition and
the flame behavior is modified.
Keywords: Real-gas thermodynamics, flame-wall interaction, conjugate
heat transfer
Preprint submitted to Elsevier
March 9, 2015
1. Introduction
Most of high performance propulsion devices such as turbines, rocket engines or scramjets operate in wall-bounded flows. The interaction between
flame and walls has a direct and strong impact on combustion, pollutant
emissions and combustion chamber lifetime. Understanding the mechanisms
at play in flame-wall interaction (FWI) is therefore necessary to further gain
in performance, safety, fuel consumption and unburnt gas emission. As shown
in [1, 2], local FWI may be described in simple laminar flows where generic
flame configurations may be introduced. During the flame-wall interaction
process, the flame speed and thickness decrease, before full quenching at a
few microns away from the wall. When the flame approaches the wall, the
temperature decreases from burnt gases (approximately 3000 K for Hydrogen
(H2 )/Oxygen(O2 ) flames at 1 bar) to wall levels that are maintained in the
300-800 K range to avoid damaging. This high temperature variation occurs
in a very thin layer, less than 1mm, leading to very strong temperature gradients and making experimental observation of FWI quite difficult.
Ezekoye et al. [3] experimentally studied the impact of wall temperature
and the equivalence ratio on the wall heat flux for propane and methane
flames. It was shown that the maximum wall heat flux decreases when the
wall temperature increases. Lu et al. [4] investigated FWI in the side wall
quenching configuration where the flame propagates along the wall and found
that the ratio of the wall heat flux to the heat release in the flame is roughly
constant and equal to 0.3 - 0.4. Based on experimental correlations, Boust et
al. [5] proposed a theoretical relation between the normalized wall heat flux
and the quenching Peclet number, defined as the flame position normalized
by the flame thickness, for methane-air flames where they observe that the
wall heat flux is inversely proportional to the flame quenching distance.
Many numerical studies have been conducted on laminar flame-wall interactions [6, 7, 8, 9, 10]. It has been shown by Popp et al. [9] that in the
low wall temperature regime (300 K < Tw < 400 K) the wall can be assumed chemically inert. Kim et al. [11] experimentally confirmed this result
using several surface materials and wall temperatures. Dabireau et al. [7],
Gruber et al. [12] and Owston et al. [10], exhibited a strongly different behavior for hydrogen flames compared to hydrocarbon flames. Hydrogen flame
quenching occurs much closer to the wall relatively to the flame thickness.
2
Normalized wall heat flux is also largely different from hydrocarbon flames
and equal to ∼ 0.12.
In all these studies, results have been provided for wall boundary conditions either adiabatic or isothermal. However in reality heat transfer occurring between the solid wall and the fluid results in a possible increase of wall
temperature and a non-zero heat flux, ie. neither isothermal nor adiabatic
wall behavior. Moreover, FWI being a transient phenomenon, eventually
leading to flame quenching, the solution can not be searched for as a steady
state solution and simulations describing the unsteady coupling of heat conduction in the wall with fluid dynamics and heat transfer are required [13].
This approach allows the wall temperature to temporally evolve and adapt
to the varying fluid temperature, consequently significantly modifying the
wall heat flux.
To address this issue, the present study considers the unsteady behavior of a
stoichiometric laminar premixed hydrogen-oxygen flame impinging on a cold
wall including conjugate heat transfer. The context is Liquid-fuel Rocket Engines (LRE), which operate at very high pressure where the thermodynamic
properties depart from ideal gas laws. Indeed, beyond the critical point, defined by (Pc , Tc ) values specific to each species, surface tension disappears
and the distinction between gaseous and liquid phases vanishes. This state of
matter is called supercritical, where phase change is replaced by a steep but
continuous variation of the density and thermodynamic properties. Therefore the objective of the study is twofold : first, the role of conjugate heat
transfer in FWI is studied; second, the impact on FWI of supercritical thermodynamic and transport properties is evaluated.
As shown in Fig. 1, the chosen configuration corresponds to Head-On Quenching (HOQ), where the flame propagates towards the wall with the characteristics of a free flame before interacting with the wall. In this simplified
configuration in-depth analysis can be made and a good understanding of
basic phenomena can be achieved. The HOQ configuration appears as a necessary first step to study both effects of high pressure and conjugate heat
transfer prior to the thermal study of realistic configurations.
2. Numerical Setup and Methodology
Simulations were performed by running simultaneously a fluid (AVBP)
and a thermal (AVTP) solver in a coupled framework. For comparison pur3
Figure 1: Flame-Wall Interaction (FWI) : Head-On Quenching (HOQ) configuration. Initial wall temperature Twi is set equal to fresh gas temperature T u .
poses, uncoupled simulations with an isothermal wall were also computed.
In AVBP, the compressible reactive Navier-Stokes equations are solved with
a third order in space and fourth order in time, two-steps Taylor-Galerkin
scheme [14, 15] along with a second order Galerkin scheme for diffusion terms.
The parallel conduction solver AVTP is based on the same data structure
than AVBP and also uses a second order Galerkin diffusion scheme. Time
integration is done with an implicit first order forward Euler scheme. The
resolution of the implicit system is done with a parallel matrix free conjugate
gradient method.
The coupling methodology consists in an exchange of variables at the wall
surface between both codes : the fluid solver sends a heat flux and the heat
conduction code sends back a temperature. Data are exchanged through a
supervisor using OpenPalm libraries [16]. Between two coupling events, the
flow and wall thermal conduction are advanced in time by a quantity αf τf
and αw τw respectively, where τf and τw are the flow and heat conduction
characteristic times. To respect simultaneity, the physical time computed by
the codes must be the same between two data exchanges: αf τf = αw τw .
This ensures continuity of the heat flux and temperature at the wall surface.
More details and validation of the coupling methodology can be found in [17].
Computations were initialized with a free stationary premixed flame previously calculated under the same thermodynamic conditions (pressure and
temperature), and located far enough from the wall to assume no interaction
at the start of the simulation. For the same reason the initial wall temperature Twi was taken equal to the fresh gas temperature T u . The fluid
boundary condition at the open end is a pressure-imposed outlet, using the
4
characteristic formulation for compressible flow [18].
3. Real-gas equations
For high pressure computations, real-gas thermodynamics are accounted
for through the Peng-Robinson equation of state [19] (PR - EOS). The general
form of a cubic equation of state is given by:
a(T )
RT
−
(1)
v − b (v + δ1 b)(v + δ2 b)
where P is the pressure, T the temperature, v the molar volume and R the
perfect-gas constant. The coefficients a and b account respectively for longrange and short-range interactions between√molecules.
√ In the Peng-Robinson
equation the parameters (δ1 , δ2 ) are (1 + 2, 1 − 2). All thermodynamic
coefficients must be modified to take into account real gas effects. At low
pressure, a standard technique consists in tabulating or using polynomial fits
to allow for the temperature dependence. This procedure can be extended to
account for pressure dependence by keeping the tabulation for low pressure
reference values and use departure functions based on the EOS to compute
the influence of pressure [20]. For example to calculate the constant-pressure
heat capacity Cp , one starts to write the Gibbs function G as:
Z v0
G(P, T ) = G0 + P v − RT +
P (v̄, T )dv̄
(2)
P (v, T ) =
v
where v0 and G0 are respectively the molar volume and the Gibbs energy at
a reference low pressure. The enthalpy h is then classically defined as:
∂G
(3)
h=G−T
∂T P
as well as the constant-pressure heat capacity:
∂h
Cp =
∂T P
(4)
This points out that low-pressure data, combined with the PR-EOS, allow
to compute all thermodynamic properties of the fluid at high pressure.
The critical point coordinates of the intermediate species OH, O, H,
H2 O2 , HO2 (for which no experimental values are available) are estimated
5
as in [21], using the Lennard-Jones potential-well depth, and the molecular diameter, taken from the transport database of the San Diego mechanism [22]. Transport coefficients are modeled following the theory of corresponding states for the dynamic viscosity and the thermal conductivity [23],
together with constant Schmidt numbers Sc (cf. Tab. 1). It was indeed verified that the Schmidt numbers of most species do not strongly vary through
the flame in the cases considered in the present study.
The ability of the AVBP solver to accurately reproduce supercritical and
transcritical flows and flames has been demonstrated in various configurations corresponding to LRE conditions [24, 25]. Note that real-gas thermodynamics have an impact on the formulation of boundary conditions and
Jacobian matrices of the numerical schemes.
Parameters
H2
Tc [K]
33
Pc [MPa]
1.28
Sc
0.28
O2
154.6
5.04
0.99
H2 O
647.1
22.06
0.77
O
105.3
7.09
0.64
H
190.8
31.01
0.17
OH H2 O2
105.3 141.3
7.09
4.79
0.65
0.65
HO2
141.3
4.79
0.65
Table 1: Species critical-point temperature Tc and pressure Pc , and Schmidt numbers.
4. Chemical kinetics
Computations were carried out with a pure hydrogen (H2 ) and pure oxygen (O2 ) mixture at stoichiometry. The combustion of hydrogen and oxygen
is modeled using a skeletal mechanism accounting for 8 species and 12 reactions from Boivin et al. [26], reported in Tab.2. It is derived from the
21-step San Diego detailed mechanism [22], used in many hydrogen combustion applications. The so-called San Diego scheme has demonstrated
its ability to predict premixed flame speed, autoignition delay, burnt gases
temperature and extinction limits under many conditions of pressure, temperature and composition [27] and is considered as a reference. In order to
validate Boivin’s scheme in the thermodynamic conditions of interest premixed flames, corresponding to Cases 2a and 2c of Tab. 3, have been computed using CANTERA [28] and compared with the detailed mechanism.
The stoichiometric laminar flame speed obtained with the Boivin scheme
(10.76m.s−1 and 9.03m.s−1 , respectively) are very close to the values computed with the reference scheme of San Diego (10.61m.s−1 and 9.46m.s−1 ,
6
respectively). The flame structures shown on Figs. 2 and 3 demonstrate that
both mechanisms are in very good agreement in terms of temperature, heat
release rate, and species (including radicals) mass fraction profiles, at low
and high pressure. In particular, the good prediction of species like HO2 is
critical for FWI, as will be seen later.
R1
Reaction
H + O2 ⇌ OH + O
R2
H2 + O ⇌ OH + H
R3
H2 + OH ⇌ H2 O + H
R4
H + O2 + M → HO2 + Mb
R5
R6
HO2 + H → 2 OH
HO2 + OH ⇌ H2 + O2
R7
R8
HO2 + OH → H2 O +O2
H + OH + M ⇌ H2 O + Mc
R9
2 H + M ⇌ H2 + M c
R10
R11
R12
2 HO2 → H2 O2 + O2
HO2 + H2 → H2 O2 + H
H2 O2 + M → 2 OH + Md
kf
kb
kf
kb
kf
kb
k0
k∞
kf
kb
kf
kb
kf
kb
k0
k∞
Aa
3.52
7.04
5.06
3.03
1.17
1.28
5.75
4.65
7.08
1.66
2.69
2.89
4.00
1.03
1.30
3.04
3.02
1.62
8.15
2.62
1016
1013
104
104
109
1010
1019
1012
1013
1013
1012
1013
1022
1023
1018
1017
1012
1011
1023
1019
n
Ea
-0.7
71.42
-0.26
0.60
2.67
26.32
2.63
20.23
1.3
15.21
1.19
78.25
-1.4
0.0
0.44
0.0
0.0
1.23
0.0
3.44
0.36 231.86
0.0
-2.08
-2.0
0.0
-1.75 496.14
-1.0
0.0
-0.65 433.09
0.0
5.8
0.61 100.14
-1.9 207.62
-1.39 214.74
Table 2: Rate coefficients in Arrhenius form k = AT n exp(−E/R0 T ) as in [26] .
a
Units are mol, s, cm3 , kJ and K.
b
Chaperon efficiencies H2 : 2.5, H2 O : 16.0, 1.0 for all other species.
Troe falloff with Fc = 0.5.
c
Chaperon efficiencies H2 : 2.5, H2 O : 12.0, 1.0 for all other species.
d
Chaperon efficiencies H2 : 2.5, H2 O : 6.0, 1.0 for all other species.
Troe falloff with Fc = 0.265 exp(−T /94) + 0.735 exp(−T /1756) + exp(−T /5182).
7
3000
1.5
-2
-1
J.m .s ]
2500
T [K]
2000
1.0
Q [x10
11
1500
1000
0.5
500
0
2
3
4
0.0
2.5
-3
5x10
x [m]
3.5
-3
4.0x10
x [m]
(a)
(b)
1.5
4
3
-2
Y_H [x10 ]
-3
Y_HO2 [x10 ]
3.0
2
1.0
0.5
1
0
2.5
3.0
0.0
0
-3
3.5x10
x [m]
5
10x10
-3
x [m]
(c)
(d)
Figure 2: 1D flame profiles of (a) temperature T , (b) heat release rate Q̇, (c) HO2 and
(d) H mass fractions for (–) San Diego [22] and (- -) Boivin [26] mechanisms. Case 2a:
pressure is 1 bar and fresh gas temperature 300 K.
5. Flame Wall Interaction (FWI)
Several FWI cases for laminar stoichiometric premixed flames were performed and are summarized in Tab. 3. For all cases, the initial wall temperature Twi and the fresh gas temperature T u are taken the same and noncoupled, isothermal simulations (denoted U ) are compared to fluid-thermal
solid coupled simulations (denoted C ). Case 1a is presented for validation
purposes and will be compared to previous studies [7, 10, 12]. Cases 2a, 2b
and 2c allow to evaluate the influence of the pressure on FWI and extend the
results to very high pressure. Finally Case 3c corresponds to cryogenic flames
8
5
-1
J.m .s ]
4000
4
T [K]
-2
3000
Q [x10
15
2000
1000
0
0
2
4
6
8
10x10
3
2
1
0
0
-6
x [m]
(a)
-6
3
Y_H [x10 ]
2
-3
-2
Y_HO2 [x10 ]
10x10
(b)
3
1
0
1
5
x [m]
2
3
4
5
6x10
2
1
0
0
-6
x [m]
5
10x10
-6
x [m]
(c)
(d)
Figure 3: 1D flame profiles of (a) temperature T , (b) heat release rate Q̇, (c) HO2 and
(d) H mass fractions for (–) San Diego [22] and (- -) Boivin [26] mechanisms. Case 2c:
pressure is 100 bar and fresh gas temperature 300 K.
typical of LREs operating conditions, with very low fresh gas temperature.
The first effect of pressure increase is the reduction of the flame thickness,
which may be approximated by a power law:
α
P
(5)
δl (P ) = δl (P0 )
P0
where P0 is a reference pressure and α depends on the temperature and
the fuel. In the case of stoichiometric hydrogen/oxygen mixture at 300K
α ∼ −1.21 was found. This is expected to have a strong impact on FWI.
9
As shown in Tab. 3, the fresh gas density ρu increases drastically with
increasing pressure and decreasing temperature, up to 200 times (Case 3c)
higher than the reference Case 2a at ambient conditions. Looking at the
compressibility factor, given by:
Z=
P
ρrT
(6)
where r is the specific gas constant, the deviation from the ideal gas law stays
close to 1 as long as the temperature remains relatively high. For these cases
no strong real gas effects are expected. With the decrease of the fresh gas
temperature, Case 3c leads to a compressibility factor of 1.13, i.e. presenting
significant real gas effects.
Case
1a
2a
2b
2c
3c
Tu
Pressure
[K] [bar]
750 1
1
300 10
100
150 100
ρu
[kg.m−3 ]
0.1931
0.4824
4.8476
48.342
108.75
Compressibility
factor
[-]
1.000
1.000
1.004
1.048
1.130
Table 3: Summary of test cases : fresh gases properties at stoichiometry and compressibility factor calculated using NIST software REFPROP [29].
Flame properties, computed with the Boivin scheme, are shown in Tab. 4
for the various cases, together with the mesh resolution. In this table, the
flame thickness δl is the thermal flame thickness calculated from the temperature gradient:
Tb − Tu
(7)
δl =
(∇T )max
where (∇T )max is the maximum of the temperature gradient. This flame
thickness is non-dimensionalized as δl∗ = δl /δ where δ is the diffusive flame
thickness [7] given by:
λu
(8)
δ= u u 0
ρ C p Sl
10
where Sl0 is the laminar flame speed. The laminar flame power Q0l is defined
as:
Q0l = ρu YFu Sl0 ∆H
(9)
where YFu is the fuel mass fraction in unburnt gases and ∆H [J.kg −1 ] the
heat produced per kilogram of fuel consumed. The temperature difference
T b − T u and flame thickness change largely when the pressure increases, from
2770K and 223µm for Case 2a to 3430K and 1.18µm for Case 2c. The flame
thickness has a direct consequence on the mesh cell size that is changed
accordingly to resolve the flame front.
The flame speed first increases with pressure, until ∼ 15bar, where it
reaches ∼ 12.5m.s−1 , before decreasing for higher pressure, to reach ∼ 9.0m.s−1 .
This non-monotonic behavior was already shown in [30] and is due to the
change of chain-branching to straight-chain kinetics. The flame speed also
increases with the fresh gas temperature T u , which has a direct effect on the
u
= λu /ρu Cpu . Increaschemistry but also modifies the thermal diffusivity Dth
ing the fresh gas temperature at ambient pressure leads to a strong increase
of the flame velocity and moderate change of burnt gas temperature and
flame thickness (Case 1a). Finally the cryogenic condition (Case 3c) gives a
hot but slow flame. Its structure is detailed below.
Case
1a
2a
2b
2c
3c
Tu
P
[K] [bar]
750 1
1
300 10
100
150 100
Tb − Tu
S0l
[K]
2380
2770
3090
3430
3544
[m.s−1 ]
34.27
10.76
12.49
9.03
3.96
δl
δ
Q0l
[m]
[m]
[W.m−2 ]
2.59e-4 1.07e-5 8.66e7
2.23e-4 6.96e-6 6.87e7
1.21e-5 5.85e-7 8.22e8
1.18e-6 9.46e-8 6.25e9
1.23e-6 5.93e-8 5.47e9
Mesh
cell size
[m]
2.0e-6
2.0e-6
2.0e-7
1.0e-8
1.0e-8
Table 4: Stoichiometric premixed flame properties and cell size, obtained with the Boivin
scheme.
Cryogenic premixed flame
The cryogenic, supercritical flame (Case 3c) exhibits a particular structure. When compared to Case 2c, the first impact of real gas thermodynamics
is to significantly increase the density in the fresh gas, from 48kg.m−3 (Case
11
2c) to 108kg.m−3 (Case 3c), while the burnt gas temperature is only slightly
lower. The important decrease of the laminar flame speed is mostly related
u
to the decrease of the thermal diffusivity Dth
, from 9.26 10−7 m2 .s−1 in Case
−7 2 −1
2c to 2.23 10 m .s in Case 3c, associated to supercritical transport properties. The most remarkable feature of Case 3c is the change of the chemical
structure in the induction zone ahead of the flame. Figure 4 shows HO2
and H2 O2 mass fraction profiles for Cases 2c and 3c. As already observed in
many studies [6, 7, 12], premixed flames are characterized by chemical reactions occurring in the induction zone between reactants and radical species
that diffuse from the main reaction zone. In the case of H2 /O2 flames, these
reactions lead to the formation of HO2 and H2 O2 in the induction zone. In
Case 3c, the low temperature has a freezing effect on most reactions. In addition, the very high density of the fresh gas strongly limits radical diffusion,
so that even zero-activation, recombination reactions such as R4, R8 or R9 of
Tab. 2 can not occur. As a consequence, radical species do not appear in the
induction zone in Case 3c, as clearly visible on Fig. 4. Cryogenic, supercritical flames therefore have no reactive induction zone and all reactions start
simultaneously when the temperature reaches a sufficiently high value. This
will have direct consequences on the flame-wall interaction for these flames.
-4
-3
8x10
4x10
Case 3c
Case 2c
3
Y_HO2 [-]
Y_H2O2 [-]
6
Case 3c
Case 2c
4
2
2
1
0
0
5
0
-6
0
10x10
x [m]
5
-6
10x10
x [m]
Figure 4: Comparison of H2 O2 (left) and HO2 (right) profiles in free propagating flames
between Case 2c (solid line) and Case 3c (dashed line).
12
Flame Wall Interaction characteristics
Flame-wall interaction is first characterized with the wall heat flux, defined as the conductive flux evaluated at the wall:
∂T (10)
Φw = λw
∂x w
where λw is the thermal conductivity of the fluid evaluated at the wall 1 . The
wall heat flux is non-dimensionalized by the flame power as Φ∗w = Φw /Q0l ,
whereas the non-dimensional flame heat release is Q∗ = Qδ/Q0l . In addition,
the flame characteristic time τ = δ/Sl0 is used to non-dimensionalize the
time as t∗ = t/τ , while space dimensions are non-dimensionalized by the
flame thickness as x∗ = x/δ.
Because of complex chemistry, the definition of the flame position is not
unique. It can be either located at the maximum of heat release rate Q̇max
(xQ̇max ) or at the maximum of fuel consumption rate ω̇F,max (xω̇F,max ). Both
locations are different and may be used to define Peclet numbers which characterize the ratio between diffusion and convective characteristic times :
• the heat release Peclet number is
Pe =
xQ̇max
δ
(11)
P eF =
xω̇F,max
δ
(12)
• the fuel Peclet number is
Assuming that no reaction occurs at the wall, the temperature difference
T −T u divided by the flame quenching distance gives an estimate of the wall
temperature gradient. As shown in [5], this leads to a simple relationship
between the non-dimensional wall heat flux and the Peclet number (either
from the heat release or fuel consumption) Φ∗w ∼ 1/P e or, taking into account
the wall heat loss:
Φ∗w ∼ 1/(1 + P e)
(13)
b
superscript X u/b refers to unburnt / burnt quantities; subscript Xw refers to a quantity evaluated at the wall
1
13
Theoretical model : the Infinitely Fast Flame model
The role and importance of the coupling between the solid and the fluid
thermal problems may be understood from the limit case of Infinitely Fast
Flame [17] (IFF), in which the characteristic flame time scale is negligible
compared to the solid conduction time. In this case the configuration reduces
to the simpler problem of two semi-infinite domains having different temperatures and a common contact surface. Solving this classical heat transfer
problem leads to the following expression for the interface temperature :
TwIF F =
ew Tw + ef Tf
ew + ef
(14)
where Tw (Tf ) is the solid (resp. fluid) temperature, and ew (ef ) the solid
(resp. fluid) effusivity defined by
p
(15)
e = λρCp
where λ is the heat conductivity, ρ the density and Cp the heat capacity of
the solid (w) or the fluid (f ).
Introducing the effusivity ratio parameter κ = ew /ef , Eq. 14 can be written
TwIF F =
κ Tw + T f
κ+1
(16)
Equation 16 shows that the final wall temperature depends on the parameter
κ: for large values of this ratio, the temperature at the solid/fluid interface
stays close to the wall temperature and the wall may be considered isothermal; on the contrary, low values of κ allow significant heating of the wall
which is then neither isothermal nor adiabatic. In this last case the resolution of the unsteady coupled problem is necessary to obtain the correct
wall heat flux. Table 5 summarizes the fluid and wall effusivities for all
test cases. Both quantities increase with temperature, but ef increases even
more strongly with pressure. The resulting interface temperatures predicted
by the IFF model, where the wall temperature has been taken to the initial
wall temperature Twi = T u and the fluid temperature has been taken to the
burnt gas temperature T b , stays close to the initial interface temperature for
high values of κ, in Cases 1a and 2a. As the fluid thermal effusivity increases
from Cases 2a to 2c, the ratio κ decreases and the final wall temperature
moves away from the initial temperature. Finally Cases 2c and 3c, with low
κ, show a significant wall temperature increase.
14
Case
1a
2a
2b
2c
3c
Twi
Pressure
[K] [bar]
750 1
1
300 10
100
150 100
Fluid
effusivity
ef
[SI]
6.09
4.47
13.92
96.61
93.62
Wall
effusivity
ew
[SI]
9280
6186
6186
6186
4914
κ
F
TIF
w
[ - ] [K]
1524 751.6
1383 302
444 307
64 352.8
52.5 216.2
Table 5: Fluid and wall thermal effusivity, effusivity ratio κ and interface temperature
predicted by the IFF model TwIF F . Thermal effusivity unit is [W.m−2 .K−1 .s−1/2 ]
6. Results and Discussion
6.1. Validation Case 1a
Case 1a is first presented and is used for validation purposes, as it has
been studied in previous publications [7, 12, 10]. In Case 1a, the high effusivity ratio κ = 1524 leads to a theoretical wall temperature TwIF F = 751.6 K,
very close to the initial wall temperature. This implies that the isothermal
wall solution will remain valid in this case, with no strong differences with the
coupled solution. Figure 5(left) shows the temperature profiles at several instants, illustrating the time-dependency of FWI and the quenching process.
To allow comparison between cases, time is set to 0 at the start of FWI,
i.e., when the wall heat flux starts to increase. As a consequence the flame
first propagates freely towards the wall, keeping a free flame structure until
t∗ ∼ 0. Then the flame starts to interact with the wall, and becomes thinner
while approaching the wall until t∗ ∼ 20. At this time, there is no sufficient
remaining fuel in the cold gas and the flame quenches. In the same time, a
transient process occurs from the start of FWI, where a very large increase
of the heat release rate at the wall is observed (Fig. 5(right)). This is linked
to a change of the chemical behavior of the induction zone when approaching
the wall. In freely propagating flames, preliminary decomposition of the fuel
occurs in the induction zone through high-energy-activation reactions with
radicals such as R2 and R3 (Tab. 2). During FWI, the temperature in the
induction zone decreases down to the wall temperature and these reactions
15
get frozen, leading to a longer persistence of O2 than H2 near the wall. At
the same time, and for the same reason, zero-activation-energy, exothermic,
radical recombination reactions such as R4 and R8 become dominant, and
lead to the observed peak of heat release rate and production rate of HO2
(Fig. 6(left)). Hence, through the low-activation-energy, propagation reaction R10 hydrogen peroxide (H2 O2 ) is also produced (Fig. 6(right)). This
phenomenon is illustrated in Fig. 7, where net, reaction rates of R4, R8 and
R10 are compared at the time of FWI and in the free propagating regime.
All these chemical mechanisms were already observed in isothermal FWI
[7, 12, 10]. By increasing the wall temperature gradient, the strong peak of
heat release at the wall has a direct impact on the wall heat flux. In addition,
the quenching distance based on the heat release becomes rapidly zero and
can not be used to evaluate the heat flux.
-1
3500
1.0x10
t*
t*
t*
t*
2000
t*
t*
t*
t*
500
0
50
100
=
=
=
=
-2.2
8.7
21.1
52.4
Q* [ - ]
T [K]
0.8
0.6
=
=
=
=
-2.2
8.7
21.1
52.4
0.4
0.2
0.0
0
150
x*
10
20
30
40
50
x*
Figure 5: Profiles of temperature (left) and dimensionless heat release rate (right) at
various instants of FWI. Maximum non-dimensional heat release rate is 0.352. Case 1a,
coupled.
Figure 8(left) shows the time evolution of the wall heat flux and wall
temperature during FWI. The wall temperature progressively increases to a
value of 755.5K, ie., slightly higher than the IFF model value of 751.6K. The
maximum wall heat flux is obtained when the flame quenches at t∗ ∼ 18, and
reaches Φw,Q = 18.9 M W.m−2 (Φ∗w,Q = 0.218). After flame quenching, the
wall heat
√ flux experiences first a fast decrease, then a much slower decrease
(∝ 1/ t∗ ) corresponding to the heat diffusion in the fluid and in the solid.
During FWI, the flame propagates toward the wall until the remaining fuel
16
0.7
Y_HO2 [ x10
=
=
=
=
-2.2
8.7
21.1
52.4
1.2
0.0
0
10
20
30
40
t*
t*
t*
t*
-4
-3
]
t*
t*
t*
t*
Y_H2O2 [ x10 ]
2.5
-2.2
8.7
21.1
52.4
0.3
0.0
0
50
=
=
=
=
10
20
30
40
50
x*
x*
Figure 6: Profiles of HO2 (left) and H2 O2 (right) mass fractions at various instants of
FWI. Case 1a, coupled.
5
6
1.0x10
R4
R8
R10*1000
1.0
0.0
0
50
100
Reaction rates [SI]
Reaction rates [SI]
2.0x10
R4
R8
R10*1000
0.5
0.0
0
150
x*
10
20
30
40
50
x*
Figure 7: Profiles of net reaction rates R4, R8 and R10 in the free propagating flame (left)
and during FWI at t∗ = 14.9 (right). Maximum R4 and R8 are 1.9 106 mol.L−1 .s−1 and
9.9 105 mol.L−1 .s−1 respectively. Case 1a, coupled.
is too low to sustain the flame and compensate for the wall heat loss. The
fuel quenching distance is therefore mainly controlled by the flame power
and the wall temperature. In the present case, the fuel Peclet number at
quenching is found P eFQ = 1.4, as illustrated on Fig. 8(right) showing the time
evolution of the fuel Peclet number during FWI. Both the non-dimensional
flux and the quenching fuel Peclet number are smaller than usual values
obtained in FWI (∼ 0.3 and ∼ 3.0 respectively) and may be explained by
the high wall temperature. The decrease of the maximum wall heat flux with
17
increasing wall temperature was also described in [3]. This may be enhanced
by the high diffusivity of H2 and the high heat release at the wall due to
radical recombination as already mentioned. This trend and the values of
the wall heat flux and quenching distance obtained in Case 1a are in good
agreement with the results of [7, 12] or [10] where a maximum wall heat flux
∼ 18 M W.m−2 was found for the same case.
Finally, Fig. 9 shows the temporal evolution of the temperature in the
solid wall. One can observe that the coupling methodology is able to transfer
the heat flux to the wall, which then diffuses in the solid. Note that the heat
penetration is much slower in the solid than in the fluid, which is consistent
with the higher solid effusivity.
0.25
30
756
25
755
20
0.10
Tw
Φw*
0.00
-10
0
10
20
30
40
50
15
10
752
0.05
F
753
Tw [K]
Φw* [-]
754
0.15
Pe [-]
0.20
Pe
5
751
F
Q
= 1.4
0
750
60
0
20
40
60
t* [-]
t*
Figure 8: Temporal evolution of (left) wall heat flux and wall temperature and (right) fuel
Peclet number. Case 1a, coupled.
6.2. Effect of pressure
In this section the effect of pressure on FWI is investigated with Cases 2a
(1bar) to 2c (100 bar). Although Case 2c is at high pressure, the relatively
high temperature leads to a compressibility factor close to 1 and no real gas
effects are expected here. From the above IFF analysis, results are expected
to be comparable to those obtained in FWI with an isothermal wall for
Cases 2a and 2b. Indeed, the IFF interface temperature does not exceed
the initial wall temperature by more than 2K and 7K respectively. In Case
2c however, the burnt gas effusivity ef = 96.6 W.m−2 .K −1 .s−1/2 being much
higher, the predicted interface temperature increases up to TwIF F = 352 K
18
756
TS [K]
754
t*
t*
t*
t*
=
=
=
=
5.6
40.0
83.8
146.2
752
750
-5
-4
-3
-2
-1
0
x* [-]
Figure 9: Time evolution of temperature profiles in the solid wall. Case 1a, coupled.
and the coupled simulation is expected to give significantly different results
from the corresponding isothermal FWI.
Overall, similar trends as in the validation case are observed, with a heat
release peak and production of H2 O2 and HO2 radicals occurring at the wall
during the FWI. However, as the wall temperature is smaller, the effect is
significantly amplified in comparison to Case 1a. Indeed the non-dimensional
maximum heat release, shown in Fig. 10 is about 2 orders of magnitude larger
during FWI than in the free flame in Case 2a, whereas it was only one order of
magnitude larger in Case 1a (Fig. 5(right)). The effect is however decreasing
with pressure, coming back in Case 2c to the same order of magnitude than
in Case 1a.
Figure 11 shows the temporal evolution of the non-dimensional heat flux and
the temperature at the wall for the three cases. Due to faster chemistry and
smaller flame thickness, FWI is faster at high pressure. The maximum wall
heat flux is obtained when flame quenches at t∗ ∼ 11, t∗ ∼ 8 and t∗ ∼ 5 for
Cases 2a, 2b and 2c respectively, and slightly decreases with pressure, from
Φ∗w,Q = 0.388 for Case 2a to Φ∗w,Q = 0.333 for Case 2c, consistently with the
lower wall heat release effect at high pressure. Overall, the maximum wall
heat flux is little sensitive to pressure and stays in the range 0.3 − 0.4, i.e..,
similar to hydrocarbon flames with low wall temperatures [31, 32, 4]. Note
however that the dimensional wall heat flux increases with pressure, from
Φw,Q = 26.4 M W.m−2 for Case 2a to Φw,Q = 2.09 GW.m−2 for Case 2c, ie.,
19
Q*max [-]
0.15
Case 2a
Case 2b
Case 2c
0.10
0.05
0.00
0
5
10
15
20
25
30
t* [-]
Figure 10: Temporal evolution of the non-dimensional maximum heat release at the wall
for Cases 2a, 2b and 2c, coupled. For all cases, time is set to 0 at the start of FWI.
reaching extremely high values.
As expected from the IFF model, the interface temperature increases
only slightly at low pressure (Cases 2a and 2b), but reaches a much higher
value of 545K for the high pressure Case 2c. Note that the increase is always
stronger than predicted by the IFF model. This difference is due to the strong
heat release, both in the flame and at the wall, during FWI in the coupled
simulations and which is not taken into account in the IFF model. This
makes the heat flux stronger and increases the interface temperature. This
justifies a posteriori the use of fully coupled simulations for the prediction of
heat transfer.
The interface temperature increase also explains the wall heat flux decrease with pressure. Figure 12 shows the evolution of the difference between
the wall heat flux obtained in the uncoupled (calculated with an isothermal wall condition at Tw = 300 K) ΦUw and the coupled computation ΦC
w of
Case 2c. The maximum difference is observed just before quenching, where
the isothermal wall assumption leads to an overestimation of the maximal
wall heat flux by 110M W.m−2 , which is significant for the thermal fatigue
of solid materials. This corresponds to a non-dimensional wall heat flux of
∗
ΦUw,Q
= 0.352, ie., closer to the low pressure cases than the coupled case.
Figure 13(left) shows the fuel Peclet number obtained at quenching for
the three cases. The quenching distance of P eFQ = 4.1 for Case 2a is larger
than for Case 1a due to the lower wall temperature. It is slightly larger than
20
0.4
550
Case 2a
Case 2b
Case 2c
500
Tw [K]
Φw* [-]
0.3
0.2
Case 2a
Case 2b
Case 2c
0.1
0.0
0
5
10
15
20
25
450
400
350
300
0
30
5
10
t* [-]
15
20
25
30
t* [-]
Figure 11: Temporal evolution of non-dimensional wall heat flux (left) and wall temperature (right). Cases 2a, 2b, 2c, coupled.
10
8
-2
∆Φw [W.m ]
10
9
10
10
10
7
6
5
0
1
2
3
4
5
t* [-]
C
Figure 12: Time evolution of wall heat flux difference ∆Φw = ΦU
w −Φw between isothermal
wall condition and coupled computation. Case 2c.
the value of ∼ 3 typically observed in previous numerical and experimental
studies for hydrocarbons fuels [31, 32, 4], which may be due to the high
diffusivity of H2 . When pressure increases, the quenching distance slightly
decreases, down to P eFQ = 3.2 for Case 2c, still staying in the range 3−4. The
slight decrease of P eFQ with pressure may be again attributed to the increase
of the interface temperature which allows fuel oxidation reactions to occur
21
closer to the wall. As already mentioned, the non-dimensional maximum
wall heat flux, also reported on Fig. 13(right), decreases with pressure. This
behavior was already observed in other studies [33, 5] for lower pressure
ranges (0.5 to 3.5 bar) and is confirmed here for higher pressure levels and
conjugate heat transfer. This also demonstrates that, although the simple
expression Eq. 13 still holds in terms of order of magnitude, it is not able to
describe detailed behaviors such as the simultaneous decrease of both Φ∗w,Q
and P eFQ with increasing pressure.
4.5
0.40
0.38
Pe
F
Q
Φw* [-]
[-]
4.0
0.36
3.5
0.34
3.0
0.32
2
1
3
4 5 6
2
10
3
4 5 6
2
100
1
P [bar]
3
4 5 6
2
10
3
4 5 6
100
P [bar]
Figure 13: Effect of pressure on the quenching fuel Peclet number P eF
Q (◦) (left) and
on the dimensionless maximum wall heat flux Φ∗w,Q (◦) (right) for Cases 2a,b,c, coupled
simulations.
6.3. Supercritical case
This section presents the results obtained for the supercritical case (Case
3c) where the fresh gas temperature has been lowered down to T u = 150K.
The compressibility factor in that case is 1.13 meaning that real gas effects
have to be taken into account. As shown in Tab. 5, the effusivity of the burnt
gas is large in such thermodynamic conditions, thus requiring the fluid/solid
thermal coupling to simulate the transient FWI and predict the final wall
temperature. FWI with an isothermal wall at 150K leads to strong water
condensation when the combustion products reach the wall, so that direct
comparison of coupled or uncoupled cases is not possible in this case.
Figure 14(left) reports the temperature profiles during FWI. The overall
process is similar to all previous cases and is comparable to Case 2c, also at
high pressure. As in Case 2c, the interaction is quite fast, with quenching
22
4000
2.2
3000
1.7
Q*max [-]
T [K]
occurring at t∗ ∼ 8, and heat release peak on the wall is still observed
(Fig. 14(right)). However as was observed in the cryogenic free flame, the
induction zone is frozen due to the low temperature and does not interact
with the wall. Neither H2 O2 or HO2 are present outside the flame zone
and they start to build on the wall only when the flame reaches the wall.
Compared to Case 2c, the increase of heat release at the wall is delayed
and starts shortly before quenching. As a result, although the increase is
comparable to Case 2c, its impact on the wall heat flux is reduced.
2000
t*
t*
t*
t*
1000
0
20
40
=
=
=
=
60
0.0
6.3
9.5
18.9
1.1
0.6
0.0
0
80
2
4
6
8
10
t* [-]
x*
Figure 14: Profiles of temperature at various instants of FWI (left) and time evolution of
the maximum heat release at the wall (right). Case 3c, coupled.
In supercritical conditions, the fluid properties differ largely from the
perfect gas, with a thermal diffusivity divided by 4 when compared to Case
2c. (Case 2c : λu /ρu Cpu = 9.26 10−7 m2 .s−1 and Case 3c : λu /ρu Cpu =
2.23 10−7 m2 .s−1 ). This, combined with the low fresh gas temperature,
leads to a large quenching distance corresponding to P eFQ = 6.0. As a
consequence, the wall temperature increases slowly, remaining low during
the quenching process and still increasing after the flame has extinguished
(Fig. 15). As the heat release at the wall stays zero for a long time and
starts to increase just before quenching, it does not contribute much to the
wall temperature increase which stays close to the predicted IFF temperature
(TwIF F = 216.2K). The non-dimensional maximum wall heat flux reaches a
value of 0.36 (Φw,Q = 1.97 GW.m−2 ), ie., stays in the range 0.3 − 0.4, mainly
thanks to the large temperature difference T b − T u . In this case, Eq. 13 does
not hold anymore. This again demonstrates that the quenching distance and
23
the maximum wall heat flux are not directly linked but strongly depend on
the interface temperature, requiring the use of coupled simulations.
0.36
230
Tw
Φw*
210
Tw [K]
Φw* [-]
0.24
190
0.12
170
0.00
0
10
20
30
40
150
50
t* [-]
Figure 15: Temporal evolution of wall heat flux and wall temperature. Case 3c, coupled.
7. Conclusions
The interaction between premixed flames and non-adiabatic walls has
been investigated in a conjugate heat transfer approach, where the fluid and
the solid wall are thermally coupled. To be representative of liquid rocket engines, stoichiometric H2 -O2 mixtures in ambient and cryogenic (low temperature, high pressure) conditions have been considered. A unique framework,
coupling both fluid and heat transfer solvers, was used in order to take into
account the wall heating transient phenomena. It was demonstrated that if
the effusivity of the burnt gas becomes non-negligible compared to that of the
solid, the isothermal assumption does not hold anymore. It was found that
this situation mainly occurs at high pressure, requiring the use of fluid-solid
thermal coupling. When pressure increases, the more powerful and much
thinner flame leads to important quenching distance decrease and maximum
wall heat flux increase by two orders of magnitude compared to atmospheric
conditions. However, when non-dimensionalized with the flame thickness and
flame power, both quantities become almost insensitive to pressure and take
typical values already observed in hydrocarbon flames. Still, the increase of
wall temperature due to conjugate heat transfer, and the heat release at the
24
wall due to radical recombination, are responsible for a slight decrease of the
quenching distance and maximum wall heat flux when pressure increases.
Finally, low-temperature, high-pressure cryogenic conditions which lead to
supercritical fluid properties and the vanishing of the induction zone, give a
very large quenching distance. However the non-dimensional maximum wall
heat flux stays comparable to the previous cases. In this case also, significant impact of the conjugate heat transfer is observed and requires fluid solid thermal coupling to describe accurately the wall temperature and the
flame behavior. These findings may have important implications for flame
stabilization and thermal fatigue in practical systems such as liquid rocket
engine injectors. It is shown that coupled FWI plays a key role and must be
accurately predicted to design optimum geometries.
Acknowledgments
The funding for this research is provided by Snecma, which is the prime
contractor for the European launcher Ariane 5 cryogenic propulsion systems
and CNES (Centre National d’Etudes Spatiales), which is the government
agency responsible for shaping and implementing France’s space policy in
Europe. Their support is gratefully acknowledged.
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