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Evaluation of Liquid Fuel Spray Modelsfor Hybrid RANS/LES and DLES
Prediction of Turbulent Reactive Flows
by
Ali Afshar
A thesis submitted in conformity with the requirementsfor the degree of Masters of Applied Science
Graduate Department of Aerospace EngineeringUniversity of Toronto
Copyright © 2014 by Ali Afshar
Abstract
Evaluation of Liquid Fuel Spray Models for HybridRANS/LES and DLES Prediction of Turbulent
Reactive Flows
Ali Afshar
Masters of Applied Science
Graduate Department of Aerospace Engineering
University of Toronto
2014
An evaluation of Lagrangian-based, discrete-phase models for multi-component liquid
sprays encountered in the combustors of gas turbine engines is considered. In partic-
ular, the spray modeling capabilities of the commercial software, ANSYS Fluent, was
evaluated. Spray modeling was performed for various cold flow validation cases. These
validation cases include a liquid jet in a cross-flow, an airblast atomizer, and a high
shear fuel nozzle. Droplet properties including velocity and diameter were investigated
and compared with previous experimental and numerical results. Different primary and
secondary breakup models were evaluated in this thesis. The secondary breakup mod-
els investigated include the Taylor analogy breakup (TAB) model, the wave model, the
Kelvin-Helmholtz Rayleigh-Taylor model (KHRT), and the Stochastic secondary droplet
(SSD) approach. The modeling of fuel sprays requires a proper treatment for the tur-
bulence. Reynolds-averaged Navier-Stokes (RANS), large eddy simulation (LES), hybrid
RANS/LES, and dynamic LES (DLES) were also considered for the turbulent flows in-
volving sprays. The spray and turbulence models were evaluated using the available
benchmark experimental data.
ii
Acknowledgements
I would like to thank Professor C.P.T Groth for giving me the opportunity to study under
his supervision at the Aerospace department of the University of Toronto. I appreciate
his guidance, support and supervision through the course of my Masters studies.
I would like to thank Professor P.S Sampath for reading my thesis and helping me with
all the questions I had on sprays and atomization.
I would also like to thank Jonathan West for helping me understand the basics of turbu-
lence modeling and assisting me with all the questions I had.
Finally, I would like to thank my family and all my friends for their support and en-
couragement during various stages of my research. This thesis is dedicated to all of
them.
The financial support of my studies was provided by Pratt and Whitney Canada (P&WC).
Computational resources of my research were provided by the SciNet High Performance
Computing Consortium available at the University of Toronto.
Toronto, 2014 Ali Afshar
iii
Contents
Abstract ii
Acknowledgements iii
Contents iii
List of Tables ix
List of Figures x
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Review of Combustion Models in ANSYS Fluent . . . . . . . . . . 2
1.2.2 Assessment of Various Spray Models for Turbulent Flows . . . . . 3
1.2.3 Evaluation of Hybrid RANS/LES and DLES Prediction of Turbu-
lent Flows Involving Liquid Sprays . . . . . . . . . . . . . . . . . 4
1.3 Simulation Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Thesis Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Fundamentals of Liquid Sprays 8
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2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Primary Breakup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Secondary Breakup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Atomizers and Injectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Turbulence Modeling 19
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Reynolds-averaged Navier-Stokes (RANS) . . . . . . . . . . . . . . . . . 20
3.3 Large Eddy Simulation (LES) . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Hybrid RANS/LES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Spray Models in ANSYS Fluent 24
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 Droplet Collision Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3 Secondary Breakup Modeling . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3.1 TAB Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3.2 Wave Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3.3 KHRT Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3.4 SSD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.4 Spray Injection Modeling for Primary Breakup . . . . . . . . . . . . . . . 28
4.4.1 Single Injection Model . . . . . . . . . . . . . . . . . . . . . . . . 29
4.4.2 Group Injection Model . . . . . . . . . . . . . . . . . . . . . . . . 29
4.4.3 Cone Injection Model . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4.4 Solid Cone Injection Model . . . . . . . . . . . . . . . . . . . . . 30
4.4.5 Surface Injection Model . . . . . . . . . . . . . . . . . . . . . . . 30
v
4.4.6 File Injection Model . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4.7 Plain Orifice Atomizer Model . . . . . . . . . . . . . . . . . . . . 30
4.4.8 Pressure Swirl Atomizer Model . . . . . . . . . . . . . . . . . . . 32
4.4.9 Airblast Atomizer Model . . . . . . . . . . . . . . . . . . . . . . . 33
4.4.10 Flat Fan Atomizer Model . . . . . . . . . . . . . . . . . . . . . . 33
4.4.11 Effervescent Atomizer Model . . . . . . . . . . . . . . . . . . . . . 34
4.5 Spray Models Investigated in the Current Thesis . . . . . . . . . . . . . . 34
5 Numerical Results for Liquid Jet in a Cross-flow 35
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2 Experimental Cases and Computational Setup . . . . . . . . . . . . . . . 37
5.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.3.1 Contours of Velocity . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.3.2 Spatial Variation of the Droplets . . . . . . . . . . . . . . . . . . 41
5.3.3 Droplet Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.3.4 Droplet Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3.5 Sensitivity Analysis for Spray Model Constants . . . . . . . . . . 47
5.3.6 Delayed Breakup . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.3.7 Primary Breakup . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.3.8 Analysis of Mesh Sensitivity . . . . . . . . . . . . . . . . . . . . . 50
6 Numerical Results for Airblast Atomizer 54
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.2 Experimental and Computational Setup . . . . . . . . . . . . . . . . . . 55
6.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
vi
6.3.1 Contours of Air Velocity . . . . . . . . . . . . . . . . . . . . . . . 59
6.3.2 Film Thickness Sensitivity Analysis . . . . . . . . . . . . . . . . . 60
6.3.3 Comparison of Breakup Models . . . . . . . . . . . . . . . . . . . 61
6.3.4 Spray Angle Sensitivity Analysis . . . . . . . . . . . . . . . . . . 61
6.3.5 Radial and Tangential Velocities . . . . . . . . . . . . . . . . . . . 62
6.3.6 Comparison of the Turbulence Modeling . . . . . . . . . . . . . . 63
6.3.7 Results for Optimal Parameter Selection . . . . . . . . . . . . . . 64
7 Numerical Results for High Shear Fuel Nozzle 68
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
7.2 Experimental and Computational Setup . . . . . . . . . . . . . . . . . . 70
7.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.3.1 Contours of Gas Velocity . . . . . . . . . . . . . . . . . . . . . . . 71
7.3.2 Spatial Variation of the Droplets . . . . . . . . . . . . . . . . . . 75
8 Conclusions and Future Research 80
8.1 Conclusions I: Liquid Jet in a Cross-flow . . . . . . . . . . . . . . . . . . 80
8.2 Conclusions II: Airblast Atomizer . . . . . . . . . . . . . . . . . . . . . . 81
8.3 Conclusions III: High Shear Fuel Nozzle . . . . . . . . . . . . . . . . . . 81
8.4 Recommendations for Future Research . . . . . . . . . . . . . . . . . . . 82
References 83
A Chemical Kinetics Models in ANSYS Fluent 94
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
A.2 Species Transport and Finite Rate Chemistry . . . . . . . . . . . . . . . 95
vii
A.2.1 Volumetric Reactions . . . . . . . . . . . . . . . . . . . . . . . . . 95
A.2.2 Wall Surface Reactions . . . . . . . . . . . . . . . . . . . . . . . . 98
A.2.3 Particle Surface Reactions . . . . . . . . . . . . . . . . . . . . . . 99
A.3 Non-Premixed Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . 99
A.3.1 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
A.3.2 Steady Flamelet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
A.3.3 Unteady Flamelet . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
A.3.4 Diesel Unsteady Flamelet . . . . . . . . . . . . . . . . . . . . . . 102
A.3.5 Adiabatic versus Non-Adiabatic . . . . . . . . . . . . . . . . . . . 102
A.4 Premixed Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
A.4.1 C Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
A.4.2 G Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
A.4.3 Extended Coherent Flame Model . . . . . . . . . . . . . . . . . . 105
A.4.4 Zimont Flame Speed Model . . . . . . . . . . . . . . . . . . . . . 105
A.4.5 Peters Flame Speed Model . . . . . . . . . . . . . . . . . . . . . . 106
A.5 Partially Premixed Combustion . . . . . . . . . . . . . . . . . . . . . . . 106
A.6 Composition PDF Transport . . . . . . . . . . . . . . . . . . . . . . . . . 107
A.6.1 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
A.6.2 Eulerian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
B Radiation Models in ANSYS Fluent 111
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
B.2 Rosseland Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
B.3 P1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
viii
B.4 Discrete Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
B.5 Surface to Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
B.6 Discrete Ordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
C Emissions and Soot Formation Models in ANSYS Fluent 118
C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
C.2 NOx Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
C.2.1 Thermal NOx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
C.2.2 Prompt NOx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
C.2.3 Fuel NOx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
C.2.4 N2O Intermediate . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
C.2.5 NOx Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
C.3 SOx Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
C.4 Soot Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
C.4.1 One Step Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
C.4.2 Two Step Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
C.4.3 Moss Brookes Model . . . . . . . . . . . . . . . . . . . . . . . . . 126
C.4.4 Moss Brookes Hall Model . . . . . . . . . . . . . . . . . . . . . . 126
C.5 Decoupled Detailed Chemistry Model . . . . . . . . . . . . . . . . . . . . 127
ix
List of Tables
5.1 Operating conditions for the three test cases. . . . . . . . . . . . . . . . . 38
5.2 Meshes used for LJIC simulations (dimensions in microns). . . . . . . . . 39
5.3 Mesh analysis (dimensions in microns). . . . . . . . . . . . . . . . . . . . 51
6.1 Meshes used for the airblast atomizer (dimensions in mm). . . . . . . . . 56
x
List of Figures
2.1 Atomization breakup [10]. . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Schematic of the breakup process [11]. . . . . . . . . . . . . . . . . . . . 10
2.3 Primary breakup regimes [22, 23]. . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Primary breakup regimes as a function of Ohnesorge and Reynolds num-
bers (Zone A=Rayleigh breakup regime, Zone B=first wind-induced breakup
regime, Zone C=second wind-induced breakup regime, Zone D=atomization)
[22, 24]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Primary breakup regimes as a function of Reynolds number and gas density
(Zone A=Rayleigh breakup regime, Zone B=first wind-induced breakup
regime, Zone C=second wind-induced breakup regime, Zone D=atomization)
[22, 23]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.6 Primary breakup regimes as a function of Reynolds number, Ohnesorge
number, and gas density (Zone A=Rayleigh breakup regime, Zone B=first
wind-induced breakup regime, Zone C=second wind-induced breakup regime,
Zone D=atomization) [22, 25]. . . . . . . . . . . . . . . . . . . . . . . . . 14
2.7 Secondary breakup regimes [22, 26]. . . . . . . . . . . . . . . . . . . . . . 15
2.8 Rayleigh-Taylor and Kelvin-Helmholtz instabilities [22, 27]. . . . . . . . . 15
2.9 Atomizers and injectors [1]. . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.10 Pressure swirl atomizer [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.1 Liquid jet in a cross-flow [52]. . . . . . . . . . . . . . . . . . . . . . . . . 36
xi
5.2 Experimental setup used by Leong and Hautman [55]. . . . . . . . . . . . 37
5.3 Mesh 1 (67000 nodes) [15]. . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.4 Mesh 2 (213000 nodes). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.5 Mesh 3 (866000 nodes). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.6 Mean axial gas velocity for case 1 using RANS (m/s). . . . . . . . . . . . 41
5.7 Mean axial gas velocity for case 1 using DLES (m/s). . . . . . . . . . . . 41
5.8 Mean axial gas velocity for case 1 using LES (m/s). . . . . . . . . . . . . 41
5.9 Mean axial gas velocity for case 1 using DES (m/s). . . . . . . . . . . . . 41
5.10 Topview of the droplets for case 1 using wave model. . . . . . . . . . . . 42
5.11 Spatial variation of the droplets using TAB model. . . . . . . . . . . . . 42
5.12 Spatial variation of the droplets using wave model. . . . . . . . . . . . . 42
5.13 Spatial variation of the droplets using KHRT model. . . . . . . . . . . . 42
5.14 Spanwise distribution of the droplets for RANS. . . . . . . . . . . . . . . 43
5.15 Spanwise distribution of the droplets for DES. . . . . . . . . . . . . . . . 43
5.16 Spanwise distribution of the droplets for LES. . . . . . . . . . . . . . . . 43
5.17 Spanwise distribution of the droplets for DLES. . . . . . . . . . . . . . . 43
5.18 Droplet diameter found by Sen et al. [12]. . . . . . . . . . . . . . . . . . 44
5.19 Droplet diameter for case 1 using RANS turbulence model. . . . . . . . . 44
5.20 Droplet diameter for case 1 using wave breakup model. . . . . . . . . . . 44
5.21 Droplet diameter for case 2 using RANS turbulence model. . . . . . . . . 45
5.22 Droplet diameter for case 3 using RANS turbulence model. . . . . . . . . 45
5.23 Droplet diameter for case 2 using wave breakup model. . . . . . . . . . . 45
5.24 Droplet diameter for case 3 using wave breakup model. . . . . . . . . . . 45
5.25 Droplet axial velocity for case 1 using RANS turbulence model. . . . . . 46
xii
5.26 Droplet axial velocity for case 1 using wave breakup model. . . . . . . . . 46
5.27 Droplet axial velocity for case 2 using RANS turbulence model. . . . . . 47
5.28 Droplet axial velocity for case 3 using RANS turbulence model. . . . . . 47
5.29 Droplet axial velocity for case 2 using wave breakup model. . . . . . . . . 47
5.30 Droplet axial velocity for case 3 using wave breakup model. . . . . . . . . 47
5.31 Droplet diameter for case 1 using wave and RANS models. . . . . . . . . 48
5.32 Droplet axial velocity for case 1 using wave and RANS models. . . . . . . 48
5.33 Droplet diameter for case 1 using wave and RANS models (delayed breakup). 49
5.34 Droplet axial velocity for case 1 using wave and RANS models (delayed
breakup). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.35 Droplet diameter for case 3 using wave and RANS models. . . . . . . . . 50
5.36 Droplet axial velocity for case 3 using wave and RANS models. . . . . . . 50
5.37 Droplet diameter for case 3 using RANS. . . . . . . . . . . . . . . . . . . 52
5.38 Droplet diameter for case 3 using DES. . . . . . . . . . . . . . . . . . . . 52
5.39 Droplet diameter for case 3 using LES. . . . . . . . . . . . . . . . . . . . 52
5.40 Droplet diameter for case 3 using DLES. . . . . . . . . . . . . . . . . . . 52
5.41 Droplet axial velocity for case 3 using RANS. . . . . . . . . . . . . . . . 53
5.42 Droplet axial velocity for case 3 using DES. . . . . . . . . . . . . . . . . 53
5.43 Droplet axial velocity for case 3 using LES. . . . . . . . . . . . . . . . . . 53
5.44 Droplet axial velocity for case 3 using DLES. . . . . . . . . . . . . . . . . 53
6.1 Prefilming airblast atomizer [13]. . . . . . . . . . . . . . . . . . . . . . . 55
6.2 Computational domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.3 Sideview of the airblast mesh. . . . . . . . . . . . . . . . . . . . . . . . . 56
6.4 Contour of the air velocity magnitude for the fine mesh. . . . . . . . . . . 57
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6.5 Contour of the air axial velocity for the fine mesh. . . . . . . . . . . . . . 57
6.6 Computational results for sideview of the droplet breakup using the fine
mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.7 Computational results for spanwise distribution of the droplets using the
fine mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.8 Experimental droplet axial velocity [13]. . . . . . . . . . . . . . . . . . . 58
6.9 Axial velocity for the fine mesh (film thickness=2.5× 10−4m). . . . . . . 58
6.10 Axial velocity for the fine mesh (film thickness=5× 10−5m). . . . . . . . 58
6.11 Axial velocity for the coarse mesh (film Thickness=2.5× 10−4m). . . . . 59
6.12 Axial velocity for the coarse mesh (film thickness=1.5× 10−4m). . . . . . 59
6.13 Axial velocity for the coarse mesh (film thickness=2× 10−4m). . . . . . . 59
6.14 Axial velocity for the coarse mesh (film thickness=5× 10−5m). . . . . . . 59
6.15 Droplet axial velocity for wave model. . . . . . . . . . . . . . . . . . . . . 60
6.16 Droplet axial velocity for KHRT model. . . . . . . . . . . . . . . . . . . . 60
6.17 Droplet radial velocity for wave model. . . . . . . . . . . . . . . . . . . . 60
6.18 Droplet radial velocity for KHRT model. . . . . . . . . . . . . . . . . . . 60
6.19 Axial velocity (spray angle=45, atomizer dispersion angle=15). . . . . . . 62
6.20 Axial velocity (spray angle=0, atomizer dispersion angle=45). . . . . . . 62
6.21 Axial velocity (spray angle=30, atomizer dispersion angle=15). . . . . . . 62
6.22 Axial velocity (spray angle=0, atomizer dispersion angle=15). . . . . . . 62
6.23 Experimental droplet radial velocity [13]. . . . . . . . . . . . . . . . . . . 63
6.24 Experimental droplet tangential velocity [13]. . . . . . . . . . . . . . . . 63
6.25 Radial velocity (spray angle=0, atomizer dispersion angle=45). . . . . . . 63
6.26 Tangential velocity (spray angle=0, atomizer dispersion angle=45). . . . 63
xiv
6.27 Radial velocity using UDF (spray angle=0, atomizer dispersion angle=45). 64
6.28 Tangential velocity using UDF (spray angle=0, atomizer dispersion an-
gle=45). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.29 Axial velocity using RANS and wave models. . . . . . . . . . . . . . . . . 65
6.30 Axial velocity using DLES and wave models. . . . . . . . . . . . . . . . . 65
6.31 Axial velocity using LES and wave models. . . . . . . . . . . . . . . . . . 65
6.32 Axial velocity using DES and wave models. . . . . . . . . . . . . . . . . . 65
6.33 Comparison of experimental and numerical results for axial velocity [13]. 66
6.34 Comparison of experimental and numerical results for radial velocity [13]. 66
6.35 Comparison of experimental and numerical results for tangential velocity
[13]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7.1 High shear fuel nozzle [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.2 Computational domain for the high shear fuel nozzle [14]. . . . . . . . . . 69
7.3 Near injector mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7.4 High shear fuel nozzle mesh. . . . . . . . . . . . . . . . . . . . . . . . . . 70
7.5 Contours of mean velocity magnitude. . . . . . . . . . . . . . . . . . . . 71
7.6 Contours of mean axial velocity. . . . . . . . . . . . . . . . . . . . . . . . 72
7.7 Experimental air velocity magnitude [14]. . . . . . . . . . . . . . . . . . . 72
7.8 Numerical air velocity magnitude [14]. . . . . . . . . . . . . . . . . . . . 73
7.9 Air velocity for RANS model. . . . . . . . . . . . . . . . . . . . . . . . . 73
7.10 Air velocity for LES model. . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.11 Spatial variation of the droplets [14]. . . . . . . . . . . . . . . . . . . . . 74
7.12 Near-injector particles for the wave model. . . . . . . . . . . . . . . . . . 74
7.13 Numerical results for droplet breakup [14]. . . . . . . . . . . . . . . . . . 75
xv
7.14 Experimental results for droplet breakup [14]. . . . . . . . . . . . . . . . 75
7.15 Droplet breakup using RANS and wave models . . . . . . . . . . . . . . 77
7.16 Spanwise distribution of the droplets using RANS and wave models. . . . 77
7.17 Droplet breakup using RANS and KHRT models. . . . . . . . . . . . . . 78
7.18 Spanwise distribution of the droplets using RANS and KHRT models . . 78
7.19 Droplet breakup using LES and wave models. . . . . . . . . . . . . . . . 79
7.20 Spanwise distribution of the droplets using LES and wave models . . . . 79
xvi
Chapter 1
Introduction
1.1 Motivation
Nowadays, liquid fuel is used as an energy source for various types of combustors including
internal combustion engines and gas turbines. In many situations, the liquid fuel is
atomized into small droplets, which enhances the rate of vaporization of the fuel by
increasing the fuel surface exposed to the surrounding gas. The atomization process
therefore has direct influence on the combustion performance and level of emissions [1].
It also involves various complex processes including primary breakup, secondary breakup,
droplet collisions, droplet evaporation, and coalescence [2].
Several studies have investigated the spray formation and behavior in combustion devices.
Spray and atomization analysis may contribute to combustion efficiency and safety by
decreasing emissions and increasing combustion performance [3]. Various previous ex-
perimental investigations have been conducted on the atomization and droplet breakup
phenomena. The experimental facilities utilize high-level optical methods as diagnos-
tic approaches to determine atomization characteristics. Even though extremely useful,
these experimental approaches may be costly and may not be able to fully diagnose the
breakup phenomenon especially for the region near the atomizer [4].
Computational fluid dynamics (CFD) models have become popular in spray and com-
bustion modeling due to their potentially lower financial cost. Several CFD models have
been proposed for spray modeling. Despite the progress made in CFD development in
1
Chapter 1. Introduction 2
recent years, an accurate simulation of the atomization process occurring in a combustor
is still a challenging task.
The current thesis focuses on an evaluation of liquid fuel spray models for turbulent flows.
The commercial CFD package, ANSYS Fluent, was used to evaluate a range of primary
and secondary breakup models in a Lagrangian based simulation of the atomization
process for a range of problems representative of spray processes found in gas turbine
engines. The spray models were also assessed in conjunction with different turbulence
models including Reynolds-averaged Navier-Stokes (RANS), large eddy simulation (LES)
and Hybrid RANS/LES.
1.2 Objectives
1.2.1 Review of Combustion Models in ANSYS Fluent
In the first phase of this study, the primary combustion models in ANSYS Fluent were
reviewed. The findings of this initial review is now briefly summarized. Appendices A-C
provide more details on the available combustion models.
Firstly, spray modeling in ANSYS Fluent can be done using a Lagrangian-based discrete
phase approach. In this approach, the fluid phase is modeled using Navier Stokes equa-
tions while the modeling of the dispersed phase is done by tracking a number of droplets
or particles. ANSYS Fluent offers various secondary droplet breakup models including
the Taylor analogy breakup (TAB), wave, Kelvin-Helmholtz Rayleigh-Taylor (KHRT),
and Stochastic secondary droplet (SSD) models. ANSYS Fluent also provides different
injection models as primary breakup methods.
ANSYS Fluent enables users to model the emissions and soot formation resulted from
combustion or other chemical processes. Pollutant modeling in ANSYS Fluent normally
consists of NOx formation and SOx formation modelings. The soot formation is also
modeled using different approaches such as the one step, two step [5], and Moss Brookes
models [6].
ANSYS Fluent offers various approaches for radiation modeling. The simplest radiation
model is the Rosseland model which is valid for optically thick mediums [7]. Other
Chapter 1. Introduction 3
radiation models in ANSYS Fluent include the P1 model, the discrete transfer radiation
model, the surface to surface technique, and the discrete ordinates method (DOM)[8].
The chemical kinetics models in ANSYS Fluent were also reviewed. The mixing com-
bustion models reviewed in ANSYS Fluent include non-premixed combustion, premixed
combustion and partially premixed combustion models. ANSYS Fluent also offers dif-
ferent flamelet models including steady, unsteady and diesel unsteady flamelet methods
[9]. The chemical kinetics present in turbulent reactive flows can be also modeled via
probability density function (PDF) transport models.
1.2.2 Assessment of Various Spray Models for Turbulent Flows
The main focus of this thesis was on modeling of liquid fuel sprays and their effects
on the combustion. It was deemed particularly important that the modeling properly
incorporates the key physical processes associated with liquid fuel sprays. The flow arising
from a fuel injector normally includes two different regions of multiphase flow. The initial
multiphase region consists of a liquid core and a dispersed flow region beyond the liquid
surface [10]. The liquid fuel spray experiences primary break as the liquid is injected. The
formation of irregular liquid fuel elements along the liquid core contributes to primary
breakup. The liquid then enters the dilute spray region. Secondary breakup occurs in
the dilute spray region due to further irregularities [11]. Secondary breakup results in
further diameter reduction of the droplets. The atomization process may also involve
other chemical and physical processes such as coalescence, collisions, and evaporation.
As mentioned previously, ANSYS Fluent offers different breakup models. Various breakup
models were evaluated using previous experimental and numerical data. The following
validation cases were used to evaluate the spray modeling capabilities of ANSYS Fluent:
• Liquid jet in a cross-flow [12];
• Airblast atomizer [13]; and
• High shear fuel nozzle [14].
The liquid jet in a cross-flow case presented by Sen et al. was used as one of the valida-
tion cases [12]. Simulations were performed for three experimental test cases originally
Chapter 1. Introduction 4
presented by Madabhushi et al. [15]. Droplet diameter and axial velocity were collected
on a plane one inch downstream the orifice. The results were compared for different tur-
bulence and breakup models. An airblast atomizer was also used to validate the primary
and secondary spray breakup models in ANSYS Fluent in conjunction with different
turbulence models. This airblast atomizer was originally presented by Gurubaran et al.
[13]. Finally, simulations were also performed for a high shear fuel nozzle [14]. The air
velocity was compared for different spray breakup models. The spatial distribution of
the droplets was also compared with previous experimental and numerical data. The
following secondary breakup models were evaluated for all of these cases:
• TAB;
• Wave;
• KHRT; and
• SSD.
Several primary breakup models such as single injection model, plain orifice atomizer,
and airblast atomizer were also evaluated for turbulent flows.
1.2.3 Evaluation of Hybrid RANS/LES and DLES Prediction
of Turbulent Flows Involving Liquid Sprays
A component of this thesis was also concerned with the evaluation of different turbulence
models for prediction of the atomization process occurring in the combustor of gas turbine
engines. The following turbulence models were investigated in this thesis:
• RANS;
• LES;
• Dynamic LES; and
• Hybrid RANS/LES.
Chapter 1. Introduction 5
LES is one of several computational methods currently available for the prediction and
modeling of turbulence. LES methods tend to resolve the large eddies directly, while
requiring modeling of only the smaller eddies [16]. Pure RANS-based methods can also be
used for the numerical treatment of turbulence. RANS methods solve the time-averaged
Navier-Stokes equations for the mean motion of the fluid and the turbulence is unresolved
and must be modeled [17]. Because of this, RANS methods generally require considerable
less CPU time and memory to solve a particular turbulent flow problem; however, in cases
involving complex geometries and complex turbulent flows the RANS model may not
accurately predict the turbulence. While RANS-based methods are still typically used in
most engineering and practical applications, the potential of LES and hybrid RANS/LES
methods is now being recognized. Hybrid approaches, combining both RANS and LES
methods in a single computation, so-called hybrid RANS/LES methods are also possible,
although there have been very few studies of hybrid RANS/LES methods for reactive
flows to date [18].
As noted above, both LES and hybrid RANS/LES methods were considered for the pre-
diction of turbulent flows. The dynamic LES or DLES approach is one LES method
that was also investigated in this research. DLES uses two filters including a grid LES
filter and a test LES filter for resolving the sub-grid turbulent stress tensor [19]. The
turbulence models were evaluated using the validation cases discussed earlier. The com-
patibility of the spray models for LES of flows within combustor was also evaluated as a
part of this thesis.
1.3 Simulation Methodology
The commercial CFD package ANSYS Fluent was used for the simulations. All the
simulations were performed using the pressure-based solver available in ANSYS Fluent
[20]. In the pressure-based solver, the velocity field is calculated by manipulating the
momentum equations. The continuity and momentum equations are also used to obtain
a pressure equation. This pressure equation is further solved to calculate the pressure
field [2]. ANSYS Fluent provides two different algorithms for the pressure-based solver,
which include a coupled approach and a segregated algorithm. A coupled pressure-based
algorithm was used for all the simulations as this is recommended by ANSYS Fluent when
Chapter 1. Introduction 6
using the discrete phase model (DPM) to obtain more accurate results. The coupled
approach provides advantages over the segregated model by solving the pressure and
momentum continuity equations together. Such a coupled scheme is desired when the
mesh is not highly refined, or when large time steps are used for the transient simulation
[2]. Furthermore, a second-order upwind scheme was used for turbulent kinetic energy
and turbulent dissipation rate. The second-order upwind method uses a Taylor series
expansion to improve the accuracy of the results at the cell faces [21].
Spray modeling was achieved using DPM which accounts for an Euler-Lagrange approach
where the continuous phase is solved using time-averaged Navier-Stokes equations, and
the dispersed flow is solved by tracking a number of droplets or parcels through the flow
field of the primary phase [2]. Unsteady particle tracking was used to track the droplets.
A two-way coupling was used for the interactions between the particles and the gas phase.
In the two-way coupled approach, the dispersed flow interacts with the continuous phase
by exchanging energy, mass, and momentum. For this approach, the flow field of the
continuous phase should first be solved. After achieving a converged solution for the
primary phase, the discrete phase can be introduced using the available injection models
in ANSYS Fluent. Consequently, ANSYS Fluent recalculates the primary phase flow field
and the discrete phase trajectories until a converged solution is obtained. In addition,
the droplets resulted from the breakup are assumed to be spherical [18].
1.4 Thesis Summary
The next chapter explains the fundamentals of liquid sprays, introducing the different
types of breakup processes that occur during atomization. Chapter 2 concludes with
a discussion of common injectors and atomizers used in gas turbine engines. Chapter
3 discusses various treatments for the turbulence within gas turbine engines. Chapter
4 explains the spray models available in ANSYS Fluent and evaluated as part of this
study. Chapter 5 presents the simulation results found for the liquid jet in a cross-flow
benchmark case. This chapter is followed by Chapter 6, which describes the numerical
results obtained for the airblast atomizer case. The results found for the high shear fuel
nozzle follow in Chapter 7. Finally, Chapter 8 includes a discussion of conclusions and
future recommendations. The current thesis also includes reviews of chemical kinetics,
Chapter 1. Introduction 7
radiation, and emission models in ANSYS Fluent, which as previously mentioned are
presented in the appendices.
Chapter 2
Fundamentals of Liquid Sprays
2.1 Introduction
The flow arising from a fuel injector consists of two different regions of multi-phase flow
including a dense spray region and a dilute spray region [10]. As is shown in Figure
2.1, the dense spray region consists of an intact liquid core and a multiphase mixing
layer where the liquid core is not completely disintegrated. The liquid core consists
of a completely intact liquid column and it includes non-atomized fluid elements. The
dispersed flow region however includes separate discrete droplets moving in the gaseous
phase. The multiphase mixing layer is further developed into a dilute spray region, which
consists of smaller droplets.
The atomization in a liquid fuel spray experiences a primary breakup as the liquid exits
the atomizer or injector. The primary breakup accounts for the breakup of the intact
liquid core and leads to formation of irregular liquid elements such as ligaments along the
surface of the liquid core. The length of the liquid core is directly related to the primary
breakup process.
Secondary breakup and droplet collisions also occur in the fuel spray atomization process.
Secondary breakup accounts for diameter reduction of the droplets outside the primary
breakup length mostly in the dilute spray region [11]. The secondary breakup is due
to further irregularities and decrease in the surface tension of droplets. High pressure
combustion present in typical atomizers results in conditions where the liquid surface
8
Chapter 2. Fundamentals of Liquid Sprays 9
Figure 2.1: Atomization breakup [10].
approaches the thermodynamic critical point. As a consequence, the droplets tend to
breakup in the dilute spray region. Figure 2.2 shows the primary and secondary breakups
occurring in an atomization process. Furthermore, droplet collisions also occur in both
dense and dilute spray regions.
In the sections to follow further details of primary and secondary breakup are discussed.
This chapter concludes with a discussion of typical atomizers and injectors used in gas
turbine engines.
2.2 Primary Breakup
The external and internal forces present on the surface of the liquid core create per-
turbations and oscillations, which can result in the disintegration and breakup of the
liquid column into small droplets [1]. This breakup mechanism is referred to as primary
breakup. The primary breakup process can be divided into four different categories.
These breakup types can be categorized based on three non-dimensional parameters that
include the Weber number, the Reynolds number, and the Ohnesorge number.
The Weber number in the gas and liquid is defined as the ratio of aerodynamic force to
the force generated by surface tension and given by
Chapter 2. Fundamentals of Liquid Sprays 10
Figure 2.2: Schematic of the breakup process [11].
Weg =u2ρgD
σ(2.1)
Wel =u2ρlD
σ(2.2)
where u is the jet velocity, D is the nozzle diameter, ρ is density, and σ represents the
surface tension. The Reynolds number is introduced as the ratio between inertial and
viscous forces, and is widely used in turbulence calculations. It is given by
Re =uρlD
µl(2.3)
where µ is the dynamic viscosity. Finally, the Ohnesorge number is defined as the ratio
between viscous forces and surface tension forces and has the form
Oh =WelRe
=µl√σDρl
(2.4)
The Ohnesorge number only contains liquid properties and assumes a small and negligible
gas viscosity during the breakup process.
As mentioned, the primary breakup process is normally divided into four different types
based on the breakup regime as defined by the Weber, Reynolds, and Ohnesorge num-
bers. The four primary breakup regimes are shown in Figure 2.3 and include the Rayleigh
regime, the first wind-induced regime, the second wind-induced regime, and the atom-
ization regime [22].
Chapter 2. Fundamentals of Liquid Sprays 11
Figure 2.3: Primary breakup regimes [22, 23].
Rayleigh Breakup Regime: Zone A
The Rayleigh breakup regime accounts for breakup at low jet velocities. The surface
tension plays the most important role in the breakup process of this regime by introduc-
ing small perturbations leading to axisymmetric oscillations. These oscillations finally
lead to breakup of the liquid core. The Rayleigh breakup introduces droplets with larger
diameter than the nozzle. The breakup length is long and it can be increased, by in-
creasing the jet velocity [22].
First Wind-Induced Breakup Regime: Zone B
As the Weber number is increased, aerodynamic forces also play a role in the spray
breakup process. The Weber number for the first wind-induced breakup regime is di-
rectly related to the relative velocity between the liquid and the surrounding gas. The
first wind-induced breakup normally leads to droplets with similar diameters to the orig-
inal nozzle diameter. The breakup length is larger than the nozzle diameter and it can
Chapter 2. Fundamentals of Liquid Sprays 12
be decreased by increasing the jet velocity [22].
Second Wind-Induced Breakup Regime: Zone C
As the Weber number is further increased it leads to turbulence within the nozzle, in-
stabilities, and growth of short surface waves, which finally result in droplet breakup.
The breakup process in this so-called second wind-induced breakup regime is more rapid
than that of the first wind-induced breakup regime, resulting in formation of droplets
with smaller diameters than the nozzle diameter. The breakup length can be decreased
by increasing the jet velocity [22].
Atomization: Zone D
Breakup occurs directly at the nozzle orifice for higher Weber numbers. The intact core
length is either zero or very short to be detected with modern measurements techniques.
The atomization process results in much smaller droplets compared to the original nozzle
diameter.
Regime Diagram for Primary Breakup
As it was mentioned earlier, the preceding regimes for the primary breakup process can
be categorized based on the non-dimensional parameters, We, Re, and Oh. The regime
diagram of Figure 2.4 is based on the work done by Reitz [24] and shows the four breakup
regimes as a function of the Ohnesorge and Reynolds numbers.
The primary breakup process is also influenced by the gas density. As the gas density
is increased, the atomization is enhanced and the division lines of the chart are shifted
to the left. Figure 2.5 shows a chart created by Schneider to categorize the primary
breakup process as a function of Reynolds number, Weber number, and gas density [23].
The primary breakup regimes are illustrated in Figure 2.6 for the full three-dimensional
parameter space based on the three parameters of gas density, Ohnesorge number, and
Reynolds number [25].
Chapter 2. Fundamentals of Liquid Sprays 13
Figure 2.4: Primary breakup regimes as a function of Ohnesorge and Reynolds numbers
(Zone A=Rayleigh breakup regime, Zone B=first wind-induced breakup regime, Zone
C=second wind-induced breakup regime, Zone D=atomization) [22, 24].
Figure 2.5: Primary breakup regimes as a function of Reynolds number and gas density
(Zone A=Rayleigh breakup regime, Zone B=first wind-induced breakup regime, Zone
C=second wind-induced breakup regime, Zone D=atomization) [22, 23].
Chapter 2. Fundamentals of Liquid Sprays 14
Figure 2.6: Primary breakup regimes as a function of Reynolds number, Ohnesorge
number, and gas density (Zone A=Rayleigh breakup regime, Zone B=first wind-induced
breakup regime, Zone C=second wind-induced breakup regime, Zone D=atomization)
[22, 25].
2.3 Secondary Breakup
The fluid particles may experience further breakup and diameter reduction in the dilute
spray region due to further irregularities of the liquid elements. This breakup process
is referred to as the secondary breakup. Secondary breakup mostly occurs due to the
aerodynamic forces and decrease of surface tension [10]. The relative velocity between
the droplets and the surrounding gas is the main breakup contributor. The gas Weber
number is used to identify the breakup process [4].
With increasing values of the relative velocity, the gas Weber number increases and
contributes to increased secondary breakup. This secondary breakup process can be cat-
egorized based on the gas Weber number, as is shown in Figure 2.7 [22]. As the gas
Weber number is increased, breakup becomes more catastrophic and more droplets are
produced.
Vibrational Breakup
Chapter 2. Fundamentals of Liquid Sprays 15
Figure 2.7: Secondary breakup regimes [22, 26].
Figure 2.8: Rayleigh-Taylor and Kelvin-Helmholtz instabilities [22, 27].
For low gas Weber numbers the main secondary breakup process is referred to as vi-
brational breakup. The surrounding flow contributes to the oscillation of the droplets,
which finally leads to breakup. The droplets are slowly decomposed into fragments cre-
ating smaller and smaller droplets [22, 26].
Bag Breakup
Bag breakup refers to the secondary breakup process in which the droplets are deflected
into a disc before decomposing. The center of the disc is then transformed into a bal-
loon moving parallel to the flow direction. As a result, the balloon experiences breakup
creating small droplets. The surrounding ring also decomposes into droplets, which are
Chapter 2. Fundamentals of Liquid Sprays 16
mostly larger than the droplets produced by the balloon. Bag breakup is also sometimes
referred to as parachute breakup [22].
Bag and Stamen Breakup
Bag and stamen breakup is also referred to as umbrella breakup and is similar to bag
breakup. However, this breakup process also creates a liquid column at the center of the
ring parallel to the flow direction. The liquid column then also disintegrates with the
ring [22, 26].
Stripping Breakup
As the relative velocity and gas Weber number are further increased, the droplets expe-
rience sheet stripping breakup. The shear forces at the equatorial region of the droplets
play the most important role in this secondary breakup process by pulling apart the
boundary layer. The main difference between sheet stripping breakup and bag breakup
is the initiation of the disintegration process, which occurs on the peripheries of the disc
for sheet stripping as opposed to the bag breakup where disintegration starts from the
center [22].
For larger Weber numbers greater than 350, the droplets may experience wave crest
stripping. Large amplitude surface waves with small wavelength play the most impor-
tant role in the droplet secondary breakup. This type of secondary breakup is due to the
pressure difference resulted from an initial perturbation on the droplet surface leading
to continuous diameter loss of the droplet. This breakup process is produced by the
Kelvin-Helmholtz instability [28].
Catastrophic Breakup
The droplets may also experience a catastrophic breakup. For larger amplitude surface
waves having short wavelength the perturbations lead to droplet disintegration form-
ing smaller fragments. As a result, these fragments are also further disintegrated. This
breakup process is produced by the Rayleigh-Taylor instability [25]. Figure 2.8 illustrates
the two physical mechanisms leading to catastrophic breakup [22].
Chapter 2. Fundamentals of Liquid Sprays 17
Figure 2.9: Atomizers and injectors [1].
Figure 2.10: Pressure swirl atomizer [2].
2.4 Atomizers and Injectors
Various types of injectors and atomizers are used in gas turbine engines. Examples of
some typical injection types are shown in Figures 2.9 and 2.10. Most of the atomizers
consist of single substance pressure nozzles. The pressure swirl atomizer, which is also a
Chapter 2. Fundamentals of Liquid Sprays 18
single substance injector, produces a hollow cone as the liquid is pushed against the walls,
and it is shown in Figure 2.10 [1]. The swirl atomizer is used in different applications in
gas turbine combustors. Other single substance pressure nozzles include flat fan nozzles
and full cone injectors [3]. In addition, pneumatic atomizers, which use gases to produce
small droplets as a result of high relative velocities, are also becoming popular in gas
turbine industry [1]. Twin fluid nozzles, airblast atomizers, and effervescent injectors
are some examples of pneumatic atomizers which are used in various industries including
biodiesel and gas turbine combustors [29]. Other complex atomizers used in aero-based
gas turbine engines include atomizers with propellants, rotary atomizers and liquid jet
in a cross-flow [3].
Chapter 3
Turbulence Modeling
3.1 Introduction
The combustors of gas turbines for aviation and industrial applications all operate with
the spray and reactive flows lying well within the turbulent regime and are characterized
by high Reynolds numbers. For this reason, the modeling of multi-component fuel spray
requires a treatment for the turbulence within both the liquid and gaseous phases.
Turbulent flows can be simulated using different approaches. Direct numerical simula-
tion (DNS) accounts for the turbulence by numerically solving the full unsteady Navier-
Stokes equations. DNS simulations are computationally very expensive and are therefore
mostly used as a research tool while having few practical industrial applications. RANS
and LES are more common treatments for the turbulence, which decrease the computa-
tional costs relative to those of DNS. Turbulence modeling can be also achieved using
hybrid approaches, combining both RANS and LES methods in a single computation,
so-called hybrid RANS/LES methods. In this study, the spray models of interest will be
assessed for use in conjunction with a range of turbulence models including RANS, LES,
and hybrid RANS/LES methods. Each of the methodologies considered are now briefly
reviewed.
19
Chapter 3. Turbulence Modeling 20
3.2 Reynolds-averaged Navier-Stokes (RANS)
RANS-based methods are one of the computational approaches that can be used for
the numerical treatment of turbulence. RANS methods solve the time-averaged Navier-
Stokes equations for the mean motion of the fluid where the turbulence is unresolved
and must be modeled. Because of this, RANS methods generally require considerable
less time and memory to solve a particular turbulent flow problem; however, in cases
involving complex geometries and complex turbulent flows the RANS models may not
accurately predict the turbulence.
Reynolds averaging basically decomposes the instantaneous Navier-Stokes equations into
fluctuating and mean components. For velocity, pressure, or other scalar quantities the
Reynolds averaging is defined as follows [2]:
~ui = ui + ui (3.1)
φ = φ+ φ (3.2)
where ui is the mean velocity, ui is the fluctuating velocity, and φ represents a scalar
such as energy or pressure. Using the preceding definition and applying Reynolds time
averaging to the Navier-Stokes equations yields the following set of equations for time
averaged equations [2]:
∂ui∂xi
= 0 (3.3)
∂ui∂t
+ uj∂ui∂xj
=1
ρ
∂
∂xj
[−pδij + µ(
∂ui∂xj
+∂uj∂xi
)− τij]
(3.4)
where δij is the Kronecker delta, p is the pressure, µ represents the viscosity, and ρ is the
density. The so-called Reynolds stress term, τij, is given by:
τij = ρujui (3.5)
This term represents the influence of fluctuations on the mean flow. ANSYS Fluent offers
various RANS modeling options. The k-ε model is the main RANS model used in this
thesis. This model is based on the transport equations for the kinetic energy, k, and the
dissipation rate, ε [17]. ANSYS Fluent offers different k-ε models including the standard,
RNG [30], and realizable k-ε models [31].
Chapter 3. Turbulence Modeling 21
The standard k-ε model solves two transport equations to determine a turbulent length
and time scale. The standard k-ε model has various engineering applications and is rea-
sonably accurate for most of the turbulent flows. The equations describing the transport
of the turbulent kinetic energy and its rate of dissipation are [17]
ρ∂k
∂t+ ρ
∂
∂xi(kui) =
∂
∂xj
[(µ+
µtσk
)∂k
∂xj
]+Gk +Gb − ρε− YM + Sk (3.6)
ρ∂ε
∂t+ ρ
∂
∂xi(εui) =
∂
∂xj
[(µ+
µtσε
)∂ε
∂xj
]+ C1ε
ε
k(Gk + C3εGb)− C2ερ
ε2
k+ Sε (3.7)
where Gk is the turbulent kinetic energy generated due to mean velocity gradients and
Gb represents the turbulent kinetic energy generated due to buoyancy. The term YM
represents the dissipation rate due to fluctuating dilatation, and Sk and Sε are user-
defined source terms. σk and σε represent turbulent Prandtl numbers [2]. Finally, C1ε
C2ε C3ε are constants.
The RNG k-ε model includes an additional term in its second transport equation. This
term is used to improve the accuracy of the model. RANS modeling can be also achieved
using the realizable k-ε model. The realizable k-ε model uses a different expression to
predict the turbulent viscosity. Consequently, a modified transport equation is used to
predict the dissipation rate [2]. Note that the standard k-ε model was used for all of the
RANS simulations of this thesis.
3.3 Large Eddy Simulation (LES)
LES is one of several other computational methods currently available in ANSYS Fluent
for the prediction and modeling of turbulence. LES methods tend to resolve the large
eddies directly, while requiring modeling of only the smaller eddies. LES models filter the
Navier-Stokes equations into either configuration space or Fourier space. The filtering
process filters out the smaller eddies to be modeled, while the larger eddies are directly
resolved. The following equations are obtained for filtered solution quantities φ, after
filtering the continuity and momentum equations [2]:
∂ui∂xi
= 0 (3.8)
Chapter 3. Turbulence Modeling 22
∂ui∂t
+ uj∂ui∂xj
=1
ρ
∂
∂xj(−pδij + σij − τij) (3.9)
where σij is the stress tensor resulted by molecular viscosity and τij represents the LES
subgrid-scale stress tensor.
The filtering operation results in the subgrid-scale stresses, τij, which are unknown and
require modeling. The default LES model in ANSYS Fluent uses the Smagorinsky-Lilly
approach to model the eddy viscosity as follows [16]:
µt = ρL2s
√2Sij sij (3.10)
Ls = min(κd, CsV1/3) (3.11)
where Ls is the mixing rate of the subgrid scales, κ is the von Karman constant, Cs is the
Smagorinsky constant, V is the volume of the computational cell, and d represents the
distance to the closest wall. LES modeling can be also done using a dynamic approach
for modeling the eddy viscosity. The dynamic LES or DLES approach is one LES method
that will be investigated in this research. DLES uses two filters including a grid LES
filter and a test LES filter for resolving the sub-grid turbulent stress tensor [19]. Both
the Smagorinsky LES and DLES treatments for the turbulence were investigated in this
thesis.
3.4 Hybrid RANS/LES
Hybrid RANS/LES methods tend to combine the RANS and LES models in a single
computation. In general, hybrid RANS/LES methods can be applied in either a wall-
bounded or an embedded approach. The wall-bounded method applies a RANS method
near walls or in other necessary places and LES methods for other regions. The embedded
methods tend to use LES in an unsteady region and RANS for other sections of the
flow. The wall-bounded hybrid RANS/LES methods available in ANSYS Fluent include
detached-eddy simulation (DES) and delayed DES (DDES) [18].
The RANS region of the DES methods in ANSYS Fluent can be modeled using three dif-
ferent approaches including the Spalart Allmaras model, the realizable k-ε method, and
the shear stress transport (SST) k-ω model. The Spalart-Allamaras based DES model
replaces d in all the equations with a new length scale [32]:
Chapter 3. Turbulence Modeling 23
d = min(d, Cdes∆max) (3.12)
where Cdes is a constant equal to 0.65, and ∆max represents the largest grid spacing. The
realizable k-ε based DES model uses an alternative equation for the dissipation term [31]:
Yk =ρk3/2
min(lrke, lles)(3.13)
lrke =k3/2
ε(3.14)
lles = Cdes∆max (3.15)
where ∆max represents the maximum local grid spacing, and Cdes is a constant equal
to 0.61. Finally the SST based DES model uses a modified equation for the dissipation
term [33]:
Yk = ρβ∗kωmax(Lt
Cdes∆max, 1) (3.16)
Lt =
√k
β∗ω(3.17)
where ω is the specific dissipation rate. ANSYS Fluent also offers the delayed DES
method, which keeps the simulation in RANS mode for boundary layer [18]. In this
work, the SST based DES model in ANSYS Fluent was used for hybrid RANS/LES
simulations.
Chapter 4
Spray Models in ANSYS Fluent
4.1 Introduction
Spray modeling can be done using the discrete phase approach available in ANSYS
Fluent. The discrete phase model uses a combination Eulerian-Lagrangian approach
where the fluid phase is modeled using Navier-Stokes equations while the dispersed phase
is modeled by tracking a number of representative droplets or particles throughout the
computational domain. In contrast, with other multiphase models available in ANSYS
Fluent, the discrete phase approach assumes low volume fraction for the dispersed second
phase.
ANSYS Fluent uses the following equation to predict the trajectory of a discrete phase
droplet [2]:
d~updt
= FD(~u− ~up) +~g(ρp − ρ)
ρp+ ~F (4.1)
where ~u is the fluid velocity, ~up represents the particle velocity, ρ is the fluid density,
and ρp represents the particle density. In equation (4.1), ~F is an additional acceleration
term, FD(~u − ~up) represents the acceleration due to the drag force, and FD is the drag
constant. Droplet collision and secondary breakup models are also available in ANSYS
Fluent. Furthermore, ANSYS Fluent offers various spray injection models which can be
used to prescribe spray droplet size and velocity distributions associated with various
24
Chapter 4. Spray Models in ANSYS Fluent 25
primary breakup models. The spray models in ANSYS Fluent are further discussed in
this chapter, as mentioned in the ANSYS Fluent theory and user manuals [2, 18].
4.2 Droplet Collision Modeling
Droplet collision modeling in ANSYS Fluent can be done by tracking the droplets. For
modeling collision of N droplets, 12N2 total collision partners should be considered. AN-
SYS Fluent reduces the computational costs by using liquid parcels, each representing
several droplets. ANSYS Fluent uses the O’Rourke algorithm where two parcels may
only collide in the case of being in the same cell of the continuous phase [34]. Moreover,
the O’Rourke method is second order accurate and can reduce the computational costs
of predicting droplet collisions. Furthermore, the collision method in ANSYS Fluent also
determines the type of collision to be either bouncing or coalescence. The probability of
each collision is found using the collisional Weber number which is dependent on Urel,
the relative velocity, and D, the mean diameter of two parcels. The collisional Weber
number is given by
Wec =ρU2
relD
σ(4.2)
The droplet collision model in ANSYS Fluent is suitable for low Weber numbers under
about 100 where the effects of droplet shattering can be ignored [2].
4.3 Secondary Breakup Modeling
ANSYS Fluent offers various secondary breakup models for the liquid droplets. The TAB
model is mostly used for low Weber numbers. The wave breakup model is recommended
for Weber numbers greater than 100. Furthermore, ANSYS Fluent also offers the KHRT
and the SSD models. Features of each of these four models are now described.
4.3.1 TAB Model
The Taylor analogy approach models an oscillating and distorting droplet based on a
spring mass system according to the surface tension, droplet drag and droplet viscosity
Chapter 4. Spray Models in ANSYS Fluent 26
forces [35]. The TAB method which is suitable for low Weber numbers can be modeled
using the equation for a damped forced oscillator as follows [36]:
~F − k~x− dd~xdt
= md2~x
dt2(4.3)
Using the Taylor analogy and by assuming the breakup requirement for the distortion to
be x > Cbr, the governing equations for a droplet are then given by
y =x
Cbr(4.4)
d2y
dt2=CFρgu
2
Cbρlr2− Ckσ
ρlr3y − Cdµl
ρlr2
dy
dt(4.5)
where ρl is the discrete phase density, ρg represents the continuous phase density and
u is the relative velocity of the droplet. The variable Cb is a constant equal to 0.5.
Additionally, CF , Ck, and Cd are also dimensionless constants [37]. It should be noted
that breakup occurs when y>1. Furthermore, the size of the child droplet can be also
determined using the energy of the child droplet given by [36]
E = 4πr2σr
r32
+π
6ρlr
5(dy
dt)2 (4.6)
where r32 represents the Sauter mean radius of the droplet [2].
4.3.2 Wave Model
The wave droplet breakup model is suitable for higher Weber numbers where the droplet
breakup is resulted from the relative velocity between the liquid and gaseous phases.
The wave model, which was first proposed by Reitz, accounts for the influence of Kelvin-
Helmholtz instabilities [38]. The wave method uses a jet stability analysis for a viscous
cylindrical jet with a radius of a, in order to predict the desired dispersion relation [39].
The maximum growth rate, Ω, and the wavelength, δ, found from the wave breakup
model analysis are defined by
δ = 9.02a(1 + 0.45Oh0.5)(1 + 0.4Ta0.7)
(1 + 0.87We1.672 )0.6
(4.7)
Ω =σ(0.34 + 0.38We1.5
2 )
ρ1a3(1 + Oh)(1 + 1.4Ta0.6)(4.8)
Chapter 4. Spray Models in ANSYS Fluent 27
where Oh is the Ohnesorge number and Ta represents the Taylor number. Here, ρ1 is the
liquid density and We2 represents the gas Weber number. The radius of the new droplet
is proportional to the wavelength found above and can be written as [2]
r = B0δ (4.9)
where B0 is a constant usually taken to be 0.61 [38]. In addition, the rate of change in
the droplet radius is modeled as
da
dt= −δΩ(a− r)
3.726B1a(4.10)
where B1 is a constant set to a value of 1.73 [37].
4.3.3 KHRT Model
ANSYS Fluent also offers the KHRT model which combines the wave breakup model
with Rayleigh-Taylor instabilities [40]. The KHRT method models droplet breakup by
considering breakup to occur as a result of the fastest growing instability. The KHRT
approach is suitable for high Weber numbers and assumes the presence of liquid core in
the near nozzle region [2]. ANSYS Fluent uses the wave breakup approach within the
liquid core, while both KH and RT methods are considered for regions outside the liquid
core. The length of the liquid core can be predicted using the theory of Levich as follows
[41]:
L = CLd0
√ρlρg
(4.11)
where d0 is the reference nozzle diameter and CL represents the Levich constant. It
should be noted that the KHRT method uses an effective droplet diameter based on Ca,
the contraction coefficient, with the effective diameter defined by
De =√Cad0 (4.12)
In addition, the frequency of the fastest growing wave and its corresponding wavelength
using the RT method are found using [2]
Ω =
√2(−gt(ρp − ρg))1.5
3√
3σ(ρp + ρg)(4.13)
δ =
√−gt(ρp − ρg)
3σ(4.14)
where gt represents the droplet acceleration.
Chapter 4. Spray Models in ANSYS Fluent 28
4.3.4 SSD Model
The SSD breakup model uses the Fokker-Planck equation to find a probability distribu-
tion of the secondary droplet size [42]. Unlike other secondary breakup models, droplet
diameter distribution in the SSD model is a random event and is independent of the di-
ameter of the parent droplet. The SSD model introduces and defines the critical breakup
radius and breakup time as
rcr =Wecrσlρgu2
rel
(4.15)
tbu = B
√ρlρg
r
|urel|(4.16)
where Wecr represents the critical Weber number and is originally set to 6. The parame-
ter B is a user-defined constant having a default value of 1.73. When the droplet radius
is larger than the critical radius, the breakup time increases. As the breakup time in-
creases and becomes larger than the critical breakup time, the parent droplet experiences
breakup.
4.4 Spray Injection Modeling for Primary Breakup
ANSYS Fluent provides 11 different injection types including atomizer and non-atomizer
injection models which can be used to represent various types of primary spray breakup
behaviour. ANSYS Fluent also offers different particle types for each injection. The
available particle types include massless particle, inert, droplet, combusting particle,
and multicomponent particle. ANSYS Fluent provides various laws for each particle
type. The massless particle has no mass or other physical properties, and it follows the
temperature and flow of the continuous phase. The inert particle, which is available for all
injections, represents a particle obeying the force balance and the heating or cooling law.
Furthermore, a droplet particle also obeys the vaporization and boiling laws in addition
to the laws obeyed by an inert particle. Droplet particles are also available where the
heat transfer model is active. The combusting particle is a solid particle which obeys
the surface reaction law and the devolatilization law in addition to the force balance
and heating laws. ANSYS Fluent also offers the wet combustion option which considers
evaporation and boiling of the combusting particle. Finally, the multicomponent particle
includes several droplet particles and is governed by multicomponent droplets law [2].
Chapter 4. Spray Models in ANSYS Fluent 29
ANSYS Fluent provides different approaches for determining the diameter distribution.
The diameter distribution modeling in ANSYS Fluent includes the uniform, Rosin-
Rammler, and Rosin-Rammler logarithmic approaches. The linear or uniform diameter
distribution is the default model in ANSYS Fluent [18].
The Rosin-Rammler method uses the following equation for mass fractions greater than
d:
Y = e−(d/d)n (4.17)
where d represents the size constant and n is the size distribution parameter. Further-
more, the mass fraction for diameters smaller than d are taken to be given by
Y = e−(d/d)n − 1 (4.18)
In addition, ANSYS Fluent also offers stochastic tracking and cloud tracking as turbulent
dispersion models. The stochastic model considers the contributions of turbulent velocity
fluctuations while the particle cloud tracking accounts for tracking the changes of a cloud
consisting of different particles [2]. In what follows, each of these injection models are
briefly reviewed in turn.
4.4.1 Single Injection Model
The single injection is used when a single value is desired for each of the initial boundary
conditions. The position, velocity, diameter, mass flow rate, and duration of injection
are specified as the point properties of single injection.
4.4.2 Group Injection Model
A range of different values are specified for the initial conditions at the particles using
the group injection model. ANSYS Fluent uses the first and last points of the position,
velocity, diameter, temperature, and flow rate to predict the intermediate values as follows
Φi = Φ1 +ΦN − Φ1
N − 1(i− 1) (4.19)
where Φ represents the desired initial condition [18].
Chapter 4. Spray Models in ANSYS Fluent 30
4.4.3 Cone Injection Model
Hollow spray cone injections in 3D cases are modeled using the cone injection method.
The point properties for cone injection include position, diameter, temperature, axis,
velocity, cone half angle, radius, and mass flow rate. Swirl fraction is also one of the
point properties for the hollow cone injection which specifies the fraction of the swirling
velocity of the particles [18].
4.4.4 Solid Cone Injection Model
The solid cone injection has similar properties to cone injection and is used for solid cones
instead of hollow spray cases.
4.4.5 Surface Injection Model
The surface injection available in ANSYS Fluent enables users to model spray injection
from a surface. The point properties used in surface injection is similar to single injection
except the initial position of the streams which is not defined in the surface injection [18].
4.4.6 File Injection Model
The file injection in ANSYS Fluent uses an input file containing the specified position,
diameter, velocity, temperature and mass flow rate for the droplets introduced for the
spray modeling.
4.4.7 Plain Orifice Atomizer Model
ANSYS Fluent provides the modeling of plain orifice atomizer which is the most common
form of atomizers in industrial applications. The flow in a plain orifice atomizer may
experience a single phase, cavitating, or flipped region [43]. The exit velocity and droplet
diameter highly depends on these internal regions. The complex internal regions in a plain
orifice atomizer are computationally expensive to model. Consequently, ANSYS Fluent
uses models found from previous experimental data for the spray modeling. The list of
Chapter 4. Spray Models in ANSYS Fluent 31
governing parameters for the internal regions of the nozzle include nozzle diameter, nozzle
length, radius of curvature of the inlet, upstream and downstream pressures, viscosity,
density, and vapor pressure, pv [2]. The Reynolds number and cavitation parameter are
found using the following expressions:
Re =dρ
µ
√2(p1 − p2)
ρ(4.20)
K =p1 − pvp1 − p2
(4.21)
where p1 is the upstream pressure and p2 represents the downstream pressure. In addition,
the coefficient of contraction is found using Nurick’s equation [44]
Cc =1√
1C2
ct− 11.4r
d
(4.22)
where Cct represents a constant equal to 0.611. Furthermore, ANSYS Fluent uses the
following equation to find the coefficient of discharge based on azimuthal stop angle,
Φstop, and azimuthal start angle, Φstart [2]:
Cd =2πm
A(Φstop − Φstart)√
2ρ(p1 − p2)(4.23)
where m is the mass flow rate. ANSYS Fluent uses different equations for the three
possible internal regions. The following equations show the exit velocity for a single
phase nozzle, cavitating nozzle, and flipped nozzle, respectively [45]:
u =meff
ρA(4.24)
u =2Ccp1 − p2 + (1− 2Cc)pv
Cc√
2ρ(p1 − pv)(4.25)
u =meff
ρCctA(4.26)
where meff represents the effective mass flow rate. The spray angle, θ, for a flipped
nozzle is assumed to be 0.02 [46]. In addition, ANSYS Fluent uses the following equation
for predicting the spray angle for single phase and cavitating nozzles:
θ = 2tan−1[4π
3 + l3.6d
√3ρg36ρl
] (4.27)
Chapter 4. Spray Models in ANSYS Fluent 32
Finally, ANSYS Fluent uses the following equation for estimating the droplet diameter
for single phase nozzle flows [47]:
d32 = 133d
8We−0.74 (4.28)
where We is the Weber number. For the case of cavitating flow, the effective diameter
of the exiting fluid jet, deff , replaces d in the equation above. Furthermore, the droplet
diameter for a flipped nozzle is obtained using [2]
d0 = d√Cct (4.29)
4.4.8 Pressure Swirl Atomizer Model
The pressure swirl atomizer is also a common type of atomizer used in combustion, oil
furnaces and spark ignited automobile engines. The fluid in the pressure swirl atomizer
flows through a central swirl chamber where a hollow air cone is created as a result of
the liquid moving towards the wall of the chamber [2]. ANSYS Fluent uses the linearized
instability sheet atomization (LISA) method first developed by Schmidt as the pressure
swirl atomizer model [48]. The LISA model in ANSYS Fluent consists of a film formation
section, a sheet breakup section and an atomization section.
ANSYS Fluent uses the following equation to determine the effective mass flow rate based
on the film thickness, t, and the injector exit diameter, dinj:
meff = πρut(dinj − t) (4.30)
It should be noted that the axial velocity, u, can be found using the Han approach based
on the total velocity, U, and the spray angle, θ, as follows [49]:
u = Ucosθ = kvcosθ
√2∆p
ρ(4.31)
where kv, the velocity coefficient, is estimated using Lefebvre’s method [2]:
kv = max[0.7,4meff
d20cosθ
√1
2ρ∆p] (4.32)
Furthermore, ANSYS Fluent uses different approaches for modeling the sheet breakup
in a pressure swirl atomizer. The droplet atomization is also modeled using Weber’s
analysis as follows [50]:
d0 = 1.88dL(1 + 3Oh)1/6 (4.33)
Chapter 4. Spray Models in ANSYS Fluent 33
where dL represents the diameter of ligaments formed at the point of breakup. dL for
short waves is related to Ks, the wave number of the maximum growth, as follows:
dL =2πCLKs
(4.34)
where CL represents the ligament constant. In addition, for longer waves the ligament
diameter is predicted using the sheet thickness, H, and given by [2]
dL =
√4H
Ks
(4.35)
4.4.9 Airblast Atomizer Model
The airblast atomizer introduces an additional air stream to facilitate the droplet breakup
and the atomization process. Air assist in the airblast atomizer contributes to sheet
instability and may also prevent collisions between droplets. The airblast atomizer uses
the same specifications of the pressure swirl atomizer model for short waves with the
exception of the sheet thickness which can be specified by the user. In addition, the
maximum relative velocity resulted from the sheet and air is also specified in the airblast
atomizer [2]. The ligament diameter in the airblast atomizer model is estimated using
the same equation for short waves in the pressure swirl model:
dL =2πCLKs
(4.36)
4.4.10 Flat Fan Atomizer Model
The flat fan atomizer is also similar to the pressure swirl atomizer. The flat fan model
assumes a flat sheet for droplet breakup instead of using swirl. The flat fan atomizer is
only used for three dimensional problems and the origin of the fan should be specified
[2]. The ligament diameter for short waves in a flat fan atomizer is estimated using
dL =
√8H
Ks
(4.37)
Chapter 4. Spray Models in ANSYS Fluent 34
4.4.11 Effervescent Atomizer Model
ANSYS Fluent also offers the effervescent atomizer model where a super heated liquid
is directed to the liquid injected through the nozzle [2]. As a result, the liquid’s phase
changes rapidly resulting in droplet break up [51]. The initial velocity of the droplets is
dependent on the effective mass flow rate and specified using
u =meff
ρCctA(4.38)
In addition, the droplet size depending on the angle between the injection direction and
stochastic trajectory of the droplet, θ, is found using the following equation:
d0 = dmaxe−(θ/Θs)2 (4.39)
dmax = d√Cct (4.40)
where Θs represents the dispersion angle multiplier. The dispersion angle can be esti-
mated using the dispersion constant, Ceff , and the mass flow rates of vapor and liquid,
and written as
Θs =mvapor
Ceff (mvapor + mliquid)(4.41)
4.5 Spray Models Investigated in the Current Thesis
As discussed in Chapter 1 of the thesis, the spray modeling capabilities of ANSYS Fluent
for applications in the combustors of gas turbine engines were evaluated in this work by
considering different benchmark validation cases. Note that not all of the primary spray
modeling techniques described above were considered in this thesis. The single injection
model was the main primary breakup method used in all of the present simulations. In
addition, the airblast atomizer and the plain orifice atomizer models were also evaluated
for the experimental cases of interest. Nevertheless, all of the secondary breakup models
in ANSYS Fluent, including the TAB, wave, KHRT, and SSD models, were compared
and evaluated as part of this thesis.
Chapter 5
Numerical Results for Liquid Jet in
a Cross-flow
5.1 Introduction
The numerical simulation of a liquid jet in a cross-flow (LJIC) was investigated as part
of this thesis. ANSYS Fluent was used to model the breakup of the liquid jet. Different
breakup and treatments for the turbulence were examined. The problem of liquid jet in
a cross-flow has many applications such as afterburners for gas turbines, lean premixed
prevaporized ducts and augmentors. As Figure 5.1 shows, the primary breakup occurs
as the liquid is injected and consists of a liquid column region which is followed by
the formation of large ligaments and droplets. The primary breakup is followed by a
dilute spray region consisting of smaller droplets created during secondary breakup. The
drag forces generated by the cross-flow bends the liquid jet and contributes to column
breakup. The column fracture depends on several parameters including the momentum
flux, cavitation in the injection nozzle, pressure fluctuations, and turbulence [15].
Several previous experimental and numerical studies have investigated the breakup of a
liquid jet in subsonic or supersonic cross-flows. For example, Wu et al. developed corre-
lations for droplet locations, velocities, and diameters by solving momentum equations
for a spherical droplet [52]. Mazaloon et al. and Sallam et al. performed investigations
on the primary breakup of the jet using holograph techniques [53]. They found similar-
35
Chapter 5. Numerical Results for Liquid Jet in a Cross-flow 36
Figure 5.1: Liquid jet in a cross-flow [52].
ities between breakup of a liquid jet in a cross-flow with secondary breakup of droplets
subjected to shock wave disturbances [54]. Leong and Hautman also measured the spray
characteristics close to the orifice using Jet-A as the test liquid [55]. Correlations for
column waves present along the jet was developed by Sallam et al. [54]. Several studies
have also reported satisfactory agreement between the experimental data and simulated
results using various sub models for the jet breakup. Madabhushi calculated the water
jet atomization using complex sub models and reported reasonable agreement with the
experimental data [56]. Madabhushi et al. further enhanced the models developed by
Madabhushi and modified the model based on correlations developed by Sallam et al.
[15]. The results were also compared with the near field experimental spray data re-
ported by Leong and Hautman [55]. A recent study conducted by researchers at Pratt
& Whitney converted the model developed by Madabhushi into a user-defined function
(UDF) and applied together with ANSYS Fluent solver to model the breakup process
[12].
The focus of the current study is to use ANSYS Fluent to model the experimental setup
presented by Leong and Hautman, and compare the results with experimental and nu-
merical data found by Madubhushi and Sen et al.. The experimental setup used by Leong
and Hautman is shown in Figure 5.2 [55]. The test facility consisted of a cross-flow air
injection system, a test section where the liquid was injected, and an exhaust chamber.
The air is injection to the system using a plenum air-feed and bellmouth inlet. The
Chapter 5. Numerical Results for Liquid Jet in a Cross-flow 37
Figure 5.2: Experimental setup used by Leong and Hautman [55].
exhaust chamber includes quartz windows having a height of 101.6 mm and a width of
50.8 mm.
For the cases of interest, the liquid injector is located 114.3 mm away from the entrance
of the test section. Two different liquid orifices with diameters of 0.762 mm and 1.778
mm are used for the experiment. Both of these orifices have a length of 12.7 mm [15].
Droplet velocities and diameters were measured by Leong and Hautman using phase
doppler interferometry (PDI) technique. The measurements were obtained in a plane
25.4 mm downstream the orifice exit.
5.2 Experimental Cases and Computational Setup
The experiments performed by Leong and Hautman were conducted under atmospheric
temperature and pressure conditions. Experiments were performed for four different test
cases. Madubhushi modeled these four cases and compared the results obtained for flow
rate, droplet size, and droplet velocity [15]. Sen et al. also used case 3 as described in
[15] to validate the UDF model created for the spray breakup [12]. The current thesis
focuses on validating both breakup and turbulence models in ANSYS Fluent using three
cases originally used by Leong and Hautman for their experiments. Table 5.1 shows the
Chapter 5. Numerical Results for Liquid Jet in a Cross-flow 38
Table 5.1: Operating conditions for the three test cases.
Case 1 Case 2 Case 3
ml (kg/h) 40.6 15.3 132.27
Diameter(mm) 0.762 0.762 1.778
Vj(m/s) 31 11.65 18.5
Wel 24711 3490 20534
Figure 5.3: Mesh 1 (67000 nodes) [15].
details of the conditions for the cases considered. This table also includes the Reynolds
and Weber numbers for both the liquid and cross-flow gas. The liquid Reynolds and
Weber numbers are found using liquid properties and orifice diameter. The gas Reynolds
number is a function of gas properties and the hydraulic diameter of the test section. The
gas Weber number is also computed using the gas properties and liquid surface tension.
The mesh and computational domain created by Sen et al. was used as the base mesh
when modeling the breakup using different spray breakup and turbulence models in
ANSYS Fluent. A schematic of the side view of this three dimensional computational
domain, referred to here as mesh 1, is shown in Figure 5.3. The test section consists of
364,000 tetrahedral elements and 67,000 nodes. The resolution of the mesh is higher for
the region near the fuel injector to capture the primary and secondary breakup proce-
dures. The mesh resolution is sufficient for RANS simulations. However, higher mesh
resolutions are needed for LES and hybrid RANS/LES simulations in order to capture
the turbulence. Consequently, two finer meshes were created using ICEM CFD. Mesh
2 contains 213,000 nodes while the mesh 3 has the highest resolution with consisting of
approximately 866,000 nodes. All the meshes have higher resolution near the fuel orifice.
Chapter 5. Numerical Results for Liquid Jet in a Cross-flow 39
Figure 5.4: Mesh 2 (213000 nodes). Figure 5.5: Mesh 3 (866000 nodes).
Table 5.2: Meshes used for LJIC simulations (dimensions in microns).
Mesh 1 Mesh 2 Mesh 3
Number of nodes 67000 213000 866000
Minimum grid spacing 736 373 75
Number of points in the orifice diameter for a
diameter of 1.778 mm3 6 20
Number of points across the duct at 1 inch
downstream the injector35 50 75
Upstream portion of fuel orifice Not included Included Included
The details of the three meshes used are shown in Table 5.2. Mesh 1 has a minimum grid
spacing of 736 microns, while including 3 points in the orifice diameter. The minimum
grid spacings for mesh 2 and 3 are 373 and 75 microns respectively. Mesh 2 includes 6
points in the orifice diameter, while 20 points are present in the orifice diameter of mesh
3. In addition, mesh 2 and 3 also include a discretization of the upstream portion of the
Chapter 5. Numerical Results for Liquid Jet in a Cross-flow 40
fuel orifice (the grid is extended 12.7 mm upstream of the orifice).
The simulations were performed using the discrete phase model available in ANSYS Flu-
ent, which, as stated previously, uses a Lagrangian-based treatment for the liquid spray.
ANSYS Fluent offers various injection and atomizer models for the primary breakup.
The breakup models available in ANSYS Fluent were described previously in Chapter
4. The liquid jet in a cross-flow was modeled by injecting several droplets at the liquid
orifice using the single injection model as the primary breakup. The plain orifice atom-
izer model available in ANSYS Fluent was also validated using the previous experimental
and numerical data. The simulations were performed using different secondary breakup
models including the TAB, wave, KHRT, and SSD model. Additionally, the simulations
were performed using RANS, LES, DLES and DES treatments for the turbulence. Fur-
thermore, pressure inlet and pressure outlet were used as boundary conditions of the test
section. All the simulations were performed using the computational mesh 1 with the
single injection model unless stated otherwise.
5.3 Numerical Results
5.3.1 Contours of Velocity
Simulations were performed for the three test cases presented by Madubhushi as shown
in Table 5.1. Figures 5.6 to 5.9 show and compare the mean axial gas velocity for the
first test case on mesh 1 using the various turbulence models. The single injection model
with multiple droplets was used to represent the primary breakup and the wave breakup
model was used to model secondary breakup and obtain these results. The axial gas
velocity reaches a maximum of approximately 155.75 m/s before decreasing near the
liquid orifice. The results obtained from these simulations show similar behavior for the
mean velocity of the gas phase. The figures specifically show similar results for LES,
DES, and DLES simulations. This result could be due to the coarse mesh used for these
simulations. The resolution of this mesh may not be sufficient to capture the turbulence
for LES and Hybrid RANS/LES simulations. Simulations were also performed for finer
meshes and are described in Section 5.3.8 to follow.
Chapter 5. Numerical Results for Liquid Jet in a Cross-flow 41
Figure 5.6: Mean axial gas velocity for
case 1 using RANS (m/s).
Figure 5.7: Mean axial gas velocity for
case 1 using DLES (m/s).
Figure 5.8: Mean axial gas velocity for
case 1 using LES (m/s).
Figure 5.9: Mean axial gas velocity for
case 1 using DES (m/s).
5.3.2 Spatial Variation of the Droplets
Figures 5.10 to 5.13 show the spatial variation of the droplets for different secondary
breakup models available in ANSYS Fluent. The standard k-ε RANS turbulence model
on mesh 1 was used to obtain all of the results. These figures are presented to pro-
vide a better understanding of the breakup process. The result obtained from the TAB
breakup model shows a narrow distribution of the droplets. The results obtained using
the wave and KHRT methods are in good agreement. These figures also show the data
collection plane 1 inch downstream the orifice used to collect the particle information to
be compared with previous experimental and numerical data. The wave breakup model
is mostly used for higher Weber numbers, when the breakup is due the relative velocity
between the gas and liquid phases. Spanwise distribution of the droplets at the data
collection plane is shown for different turbulence models in Figures 5.14 to 5.17, again
for mesh 1. These figures again show similar results for different turbulence models.
Chapter 5. Numerical Results for Liquid Jet in a Cross-flow 42
Figure 5.10: Topview of the droplets for
case 1 using wave model.
Figure 5.11: Spatial variation of the
droplets using TAB model.
Figure 5.12: Spatial variation of the
droplets using wave model.
Figure 5.13: Spatial variation of the
droplets using KHRT model.
5.3.3 Droplet Diameter
The predicted droplet diameter is compared for different secondary breakup and turbu-
lence models. Figure 5.18 shows the previous results found by Sen et al. [12]. Figures
5.19 and 5.20 compare the breakup and turbulence models for the first case. The ex-
perimental data show an initial increase in droplet Sauter mean diameter (SMD) as we
move away for the wall. Note that the SMD of a droplet is defined as the diameter of a
sphere having the same ratio between volume and surface area of the droplet. The overall
droplet size increases as the transverse distance is increased. There is slight droplet di-
ameter decrease present at around 10 and 20 mm away from the wall. The initial increase
in droplet SMD is because of the thickening of the boundary layer, which results in an
Chapter 5. Numerical Results for Liquid Jet in a Cross-flow 43
Figure 5.14: Spanwise distribution of the
droplets for RANS.
Figure 5.15: Spanwise distribution of the
droplets for DES.
Figure 5.16: Spanwise distribution of the
droplets for LES.
Figure 5.17: Spanwise distribution of the
droplets for DLES.
increase of the wavelength of the surface waves. This trend continues up to the point of
column fracture. A wider range of droplet sizes are introduced as the result of column
fracture and secondary breakup. Consequently, the droplet diameter slightly decreases
before increasing again. The larger droplets having lower accelerations also tend to move
to the edge of the spray [15]. The results obtained for the wave and KHRT models show
identical results, which are in good agreement with the experimental data. On the other
hand, the SSD and TAB models show a narrow distribution of the droplets. The RANS
simulation generates better results compared to other turbulence models. The results
obtained for finer meshes are compared in Section 5.3.8. The wave model when used in
conjunction with the RANS turbulence model, produces reasonable results up to around
25 mm away from the wall. These results however fail to predict the presence of larger
Chapter 5. Numerical Results for Liquid Jet in a Cross-flow 44
Figure 5.18: Droplet diameter found by Sen et al. [12].
Figure 5.19: Droplet diameter for case 1
using RANS turbulence model.
Figure 5.20: Droplet diameter for case 1
using wave breakup model.
droplets at the outer edge of the spray.
Figures 5.21 and 5.23 compare the results obtained for the second test case using mesh
1. This case accounts for a low momentum jet case where the column fractures occurs
close to the liquid orifice. The results again show a narrower distribution of the droplets
for TAB and SSD simulations. The numerical results show similar behavior for all the
turbulence models. LES and DES results are in good agreement and correctly predict
the final decrease in droplet SMD due to presence of large droplets at the edge of the
spray. Figures 5.22 and 5.24 illustrate the results obtained for the third case, again with
mesh 1. The third case has a larger orifice diameter resulting in a higher momentum
Chapter 5. Numerical Results for Liquid Jet in a Cross-flow 45
Figure 5.21: Droplet diameter for case 2
using RANS turbulence model.
Figure 5.22: Droplet diameter for case 3
using RANS turbulence model.
Figure 5.23: Droplet diameter for case 2
using wave breakup model.
Figure 5.24: Droplet diameter for case 3
using wave breakup model.
jet. The droplet diameter is mostly underpredicted for different turbulence and breakup
models, especially for the region close to the wall. ANSYS Fluent produces similar results
for LES, DES, and DLES models. All of the models fail to predict the final increase in
droplet SMD.
5.3.4 Droplet Velocity
The predicted distributions of the droplet axial velocity are also compared for different
breakup and turbulence models in ANSYS Fluent using mesh 1. All the data is collected
at a plane 1 inch downstream the orifice. Figures 5.25 and 5.26 compare the results
Chapter 5. Numerical Results for Liquid Jet in a Cross-flow 46
Figure 5.25: Droplet axial velocity for case
1 using RANS turbulence model.
Figure 5.26: Droplet axial velocity for case
1 using wave breakup model.
obtained for the first test case. The droplet velocity initially decreases up to the point
of column fracture. As the air goes around the jet it creates a low velocity wake which
results in an initial decrease in droplet velocity. Column flattening, which is present in
the wake region results in lower velocities for the droplets. The velocity then starts to
increase before a final decrease due the presence of larger droplets at the outer edge of
the spray [15]. The increase in droplet velocity after the column fracture is because of
the introduction of smaller droplets due to secondary breakup. The wave breakup model
is in a better agreement with the experimental data and correctly predicts the initial
decrease in droplet velocity, however it fails to predict droplets closer to the outer edge
of the spray. The RANS turbulence model shows a lower point for the column fracture,
which is in a better agreement with the experimental data as opposed to other turbulence
models. The overall droplet velocity is overpredicted for the region close to the wall.
Figures 5.27 and 5.29 show the droplet axial velocity for the second test case. The
experimental results dont show an initial decrease in droplet velocity. This is due to the
fact that column fracture occurs close the wall before the first experimental point. The
wave and KHRT models however correctly predict the initial decrease in droplet velocity.
Droplet velocity is also compared for the third test case having the highest jet momentum.
The results show a narrow distribution of the droplets for the TAB and SSD models while
the wave and KHRT models are in better agreement with the experimental data. Near
wall predictions of droplet velocity are higher than the experimental measurements. The
wake effect may not be properly modeled in ANSYS Fluent, which results in higher
Chapter 5. Numerical Results for Liquid Jet in a Cross-flow 47
Figure 5.27: Droplet axial velocity for case
2 using RANS turbulence model.
Figure 5.28: Droplet axial velocity for case
3 using RANS turbulence model.
Figure 5.29: Droplet axial velocity for case
2 using wave breakup model.
Figure 5.30: Droplet axial velocity for case
3 using wave breakup model.
velocity values for droplets close to the wall.
5.3.5 Sensitivity Analysis for Spray Model Constants
The wave secondary breakup model was chosen to be the best breakup option based on
previous results. As a result, all the following simulations are performed only for the wave
secondary breakup model. The wave breakup model includes a constant B1 as given by
[2]:
τ =3.726B1a
ΛΩ(5.1)
Chapter 5. Numerical Results for Liquid Jet in a Cross-flow 48
Figure 5.31: Droplet diameter for case 1
using wave and RANS models.
Figure 5.32: Droplet axial velocity for case
1 using wave and RANS models.
where a is the radius of the parent droplet, Λ represents the wavelength of the fastest
growing surface wave of the original droplet, τ is the breakup time, and Ω represents
the maximum growth rate. The recommended value for B1 is 1.73. Figures 5.31 and
5.32 show predicted results for the SMD and droplet velocity for a range of values of
B1 from 1.73 to 30. Simulations were also performed for B1 values smaller than 1.73.
These values did not improve the results and are not presented in this section. As Figure
5.31 shows, ANSYS Fluent under predicts the overall droplet SMD, as the value of B1 is
increased. The results for droplet axial velocity are also in a better agreement with the
experimental data when the default value of 1.73 is used.
5.3.6 Delayed Breakup
The single injection model was used as the primary breakup model for all the previous
simulations. The single injection model injects particles at the liquid orifice without
taking into account the presence of an intact liquid core. All the previous results also
fail to show the presence of larger droplets at the outer edge of the spray. Therefore, a
delayed breakup mechanism was used in which particles are injected at positions further
downstream above the fuel orifice in an attempt to improve the results. Figures 5.33 and
5.34 show the results obtained for this delayed primary breakup for a range of values of
initial injection height from 5×10−5 to 0.5 mm. Despite showing the presence of droplets
Chapter 5. Numerical Results for Liquid Jet in a Cross-flow 49
Figure 5.33: Droplet diameter for case 1
using wave and RANS models (delayed
breakup).
Figure 5.34: Droplet axial velocity for case
1 using wave and RANS models (delayed
breakup).
at the edge of the spray, as the initial height of the breakup point is increased, the results
do not match the experimental data and the overall droplet velocity is over predicted.
ANSYS Fluent also offers a plain orifice atomizer as a primary breakup model. The
results obtained using this model is analyzed in the following section.
5.3.7 Primary Breakup
All the previous simulations were conducted using the single injection model, which sim-
ply injects a single droplet into the domain. The single injection model does not account
for the presence of the intact liquid core. By increasing the number of particle injections
however, the number of droplets increases and we obtain a more realistic model for the
breakup process. The previous simulations were performed by injecting 37 particles at
the orifice at each time step. The simulations were repeated for the third case using the
plain orifice atomizer model available in ANSYS Fluent. The plain orifice atomizer is
one of the most common atomizers in industrial applications. The plain orifice atomizer
is further discussed in Chapter 4. Three different meshes were created as discussed pre-
viously in Section 5.2. Mesh 2 and 3 contain the upstream portion of the liquid nozzle to
be used in conjunction with the plain orifice atomizer. Figures 5.35 and 5.36 compare the
droplet diameter and axial velocity using mesh 2 and 3. The plain orifice model tends to
under predict the initial droplet diameter. The results obtained with the single injection
Chapter 5. Numerical Results for Liquid Jet in a Cross-flow 50
Figure 5.35: Droplet diameter for case 3
using wave and RANS models.
Figure 5.36: Droplet axial velocity for case
3 using wave and RANS models.
model are in better agreement with the experimental data. The plain orifice atomizer
is unable to properly take into account the wake effect formed behind the jet and under
predicts the initial droplet diameter. Based on the results obtained from this section it
can be concluded that the single injection model still provides better results for liquid
jet in a cross-flow.
5.3.8 Analysis of Mesh Sensitivity
This section investigates and compares the results obtained for mesh 2 and 3. Figures 5.37
to 5.40 show the droplet diameter obtained for the third case using different turbulence
models. As the mesh resolution is increased, it is expected that the turbulence of the
gaseous phase is captured more accurately. However, the figures show better results for
coarser meshes. Mesh 3, which is the finest mesh with containing 866,000 nodes, is unable
to produce accurate results. Further evidence for this is provided by Figures 5.41 to 5.44,
which compare the droplet axial velocity for the third case. The predictions obtained
from mesh 1 and 2 are again in a better agreement with the experimental data. Mesh 1
and 2 correctly predict an increase in droplet velocity after the point of column fracture.
Mesh 3 however, shows a continuous decrease in droplet velocity as we move away form
the wall. ANSYS Fluent predicts similar results for all the turbulence models.
Insight into what may be happening is provided by Table 5.3. The table compares the
Chapter 5. Numerical Results for Liquid Jet in a Cross-flow 51
Table 5.3: Mesh analysis (dimensions in microns).
Mesh 1 Mesh 2 Mesh 3
Number of nodes 67000 213000 866000
Minimum grid spacing 736 373 75
Overall droplet SMD 52 59 76
Ratio of the maximum droplet size to minimum
grid spacing2.4 4.7 23.6
Ratio of the overall droplet SMD to minimum
grid spacing0.07 0.158 1.01
particle diameter with the grid size for the three meshes used for the simulations. Mesh
3, which is the finest mesh, has a minimum grid spacing in the range of the droplet
diameter. It seems that ANSYS Fluent cannot produce accurate predictions for this case
when the particle diameter is on the order of the grid spacing. Therefore, while increased
resolution may be required for the accurate prediction of the turbulence, there appears
to be limitations in the DPM of ANSYS Fluent when the grid spacing approaches the
droplet diameter such that the droplet occupies a significant portion of the computational
cell. This casts doubt on the possibility of using Lagrangian-based spray models with
LES, DES, and DLES for describing the turbulence within ANSYS Fluent.
Chapter 5. Numerical Results for Liquid Jet in a Cross-flow 52
Figure 5.37: Droplet diameter for case 3
using RANS.
Figure 5.38: Droplet diameter for case 3
using DES.
Figure 5.39: Droplet diameter for case 3
using LES.
Figure 5.40: Droplet diameter for case 3
using DLES.
Chapter 5. Numerical Results for Liquid Jet in a Cross-flow 53
Figure 5.41: Droplet axial velocity for case
3 using RANS.
Figure 5.42: Droplet axial velocity for case
3 using DES.
Figure 5.43: Droplet axial velocity for case
3 using LES.
Figure 5.44: Droplet axial velocity for case
3 using DLES.
Chapter 6
Numerical Results for Airblast
Atomizer
6.1 Introduction
Airblast atomizers have an additional air stream compared to single substance injectors,
to facilitate the breakup of the liquid sheet. The liquid sheet formed by the nozzle is
atomized by the additional air stream. The assisting air stream contributes to sheet
instability and produces smaller droplets. Most gas turbine engines utilize a prefilming
type of airblast atomizer, where the liquid is spread out on a surface to produce a thin
sheet. The sheet experiences breakup as it is subjected to the high velocity air. Prefilming
atomizers improve the atomization efficiency and reduce the exhaust smoke and soot
formation.
The airblast atomizer investigated in this thesis consists of two separate airflows to allow
atomization of the liquid stream on both sides of the sheet formed in the nozzle. A strong
swirling flow is also used to create a conical spray by deflecting the droplets outward in
the radial direction [1].
Airblast atomizers have been investigated in previous experimental and numerical stud-
ies. A previous experimental study on an airblast atomizer, was decided to be chosen
to evaluate the capabilities of spray modeling within ANSYS Fluent. Various relevant
experimental cases were reviewed as possible validation cases. Breakup phenomena in a
54
Chapter 6. Numerical Results for Airblast Atomizer 55
Figure 6.1: Prefilming airblast atomizer [13].
coaxial airblast atomizer was reviewed by Engelbert et al. [57]. Liu et al. introduced
an experimental setup for finding the droplet size distribution of an airblast atomizer
using a finite stochastic breakup model (FSBM) as the numerical modeling method for
comparison with the experimental results [58]. Dumouchel et al. presented experimental
investigation on primary atomization of airblast atomizers [59]. The atomizer model pre-
sented by Watanawanyoo et al. also investigated the flow characteristics and droplet size
using distilled water as the test liquid [60]. Gurubaran et al. investigated the atomization
of a prefilming airblast atomizer in a strong swirling flow [13].
The study conducted by Gurubaran et al. was chosen as the validation case for the
airblast atomizer model in ANSYS Fluent. The schematic of the prefilming airblast
atomizer used by Gurubaran et al. is shown in Figure 6.1. The experimental setup
consisted of an air supply system, an injector, a fuel supply system, and a mechanism to
collect particle information. The prefilming airblast atomizer is surrounded by a swirler
having an outer diameter of 40 mm, and an inner diameter of 33 mm [13].
6.2 Experimental and Computational Setup
Gurubaran et al. investigated the atomization process using various experimental flow
conditions. One of the flow conditions from the original paper, the one for which there
Chapter 6. Numerical Results for Airblast Atomizer 56
Figure 6.2: Computational domain. Figure 6.3: Sideview of the airblast mesh.
Table 6.1: Meshes used for the airblast atomizer (dimensions in mm).
Coarse Mesh Fine Mesh
Number of nodes 273000 1280000
Minimum grid spacing 0.44 0.24
Number of points across the inlets of the atomizer 25 40
Number of points across the swirler 12 20
Number of points across the diameter of the duct 150 245
exists the most experimental data, was selected for the present simulations. For this case,
the air is entered the system with a flow rate of 1250 liters per minute. The liquid flow
rate is 270 cc/m leading to a 5.43 air to liquid ratio. The air having an initial velocity
of 10.4 m/s enters the atomizer via two separate inlets to improve atomization. The
atomizer is surrounded by a swirler having a swirl number of 1.09 [13].
Chapter 6. Numerical Results for Airblast Atomizer 57
Figure 6.4: Contour of the air velocity
magnitude for the fine mesh.
Figure 6.5: Contour of the air axial veloc-
ity for the fine mesh.
Figure 6.6: Computational results for side-
view of the droplet breakup using the fine
mesh.
Figure 6.7: Computational results for
spanwise distribution of the droplets using
the fine mesh.
Several important boundary conditions for the liquid phase are missing from the original
paper presented by Gurubaran et al. [13]. Consequently, a sensitivity analysis was
performed for these various boundary conditions of the airblast atomizer. In particular,
the sensitivity of the predicted spray solutions to the flow rate, spray angle, and film
thickness were all investigated. The droplet breakup process and velocity were also
investigated for the different spray breakup and turbulence models.
Two different meshes were generated and used for this case. The details of these meshes
are shown in Table 6.1. The coarse mesh contains 273,000 nodes and has a minimum
grid spacing of 0.44 mm. This mesh includes 25 points across the air inlets and 12 points
Chapter 6. Numerical Results for Airblast Atomizer 58
Figure 6.8: Experimental droplet axial velocity [13].
Figure 6.9: Axial velocity for the fine mesh
(film thickness=2.5× 10−4m).
Figure 6.10: Axial velocity for the fine
mesh (film thickness=5× 10−5m).
across the swirler. The fine mesh consists of 1,280,000 nodes resulting in a minimum
grid spacing of 0.24 mm. For this mesh, the air inlets include 40 points, while the swirler
consists of 20 points.
The simulations were performed using the DPM model available in ANSYS Fluent. Dif-
ferent turbulence models were validated for discrete phase spray modeling within ANSYS
Fluent. The simulations were performed using RANS, LES, DLES and DES turbulence
models. The airblast atomizer model available in ANSYS Fluent was used as the pri-
mary breakup model. The simulations were performed using different secondary breakup
models including wave and KHRT methods.
Chapter 6. Numerical Results for Airblast Atomizer 59
Figure 6.11: Axial velocity for the coarse
mesh (film Thickness=2.5× 10−4m).
Figure 6.12: Axial velocity for the coarse
mesh (film thickness=1.5× 10−4m).
Figure 6.13: Axial velocity for the coarse
mesh (film thickness=2× 10−4m).
Figure 6.14: Axial velocity for the coarse
mesh (film thickness=5× 10−5m).
6.3 Numerical Results
6.3.1 Contours of Air Velocity
Figure 6.4 shows the contour of air velocity magnitude for the fine mesh. The velocity
reaches a maximum of around 51.7 m/s. The recirculation zones created by the swirler
are also shown in Figure 6.5. This figure shows the axial air velocity in the mid plane of
the domain.
Chapter 6. Numerical Results for Airblast Atomizer 60
Figure 6.15: Droplet axial velocity for
wave model.
Figure 6.16: Droplet axial velocity for
KHRT model.
Figure 6.17: Droplet radial velocity for
wave model.
Figure 6.18: Droplet radial velocity for
KHRT model.
6.3.2 Film Thickness Sensitivity Analysis
The liquid atomization and breakup occur as the liquid phase interacts with the sur-
rounding air. The overall breakup process is shown in Figures 6.6 and 6.7. The spanwise
distribution of the droplets shows a maximum velocity of 21.5 m/s for the droplets. The
droplets velocity tends to increase as we move away in the radial direction. Consequently,
droplet velocity reaches a maximum before finally decreasing. Droplet information was
collected at five different planes downstream the atomizer. Figure 6.8 shows droplet axial
velocity on the downstream planes found by Gurubaran et al. [13]. The experimental
data show an initial increase in droplet velocity as we move away from the z axis. The
Chapter 6. Numerical Results for Airblast Atomizer 61
velocity then reaches a maximum of around 20 m/s, before decreasing. A sensitivity
analysis on the film thickness was performed for both meshes. Figures 6.9 and 6.10 show
the results obtained for two different film thicknesses for the fine mesh.
The simulations were repeated for the coarse mesh and are shown in Figures 6.11 and 6.14.
The figures show similar results for both meshes. Simulations were also performed for
two other film thicknesses shown in Figures 6.12 and 6.13. The results correctly predict
the final decrease in droplet axial velocity, however, they cannot predict the presence of
droplets closer to z axis. The overall droplet axial velocity is increased, by decreasing
the film thickness. Furthermore, the maximum velocity of 20 m/s is correctly predicted
in Figure 6.12 when a film thickness of 1.5× 10−4 m is used. The rest of the results are
obtained using the coarse mesh.
6.3.3 Comparison of Breakup Models
The previous simulations were all conducted using the wave secondary breakup model.
Simulations were repeated with a film thickness of 1.5×10−4 m using the KHRT breakup
model. As is shown in Figures 6.15 to 6.18, the KHRT breakup model produces similar
results to those found previously with the wave breakup method. The wave breakup
model is mostly suitable for higher Weber numbers where the breakup is due to the
relative velocity between the liquid and gaseous phases. The rest of the simulations were
performed using the wave breakup model.
6.3.4 Spray Angle Sensitivity Analysis
The experimental data predict the presence of droplets close to the z axis. The airblast
atomizer model in ANSYS Fluent enables users to specify the spray and the atomizer
dispersion angles. The default value for the dispersion angle is 6. A sensitivity analysis
was also performed for the spray angle and the dispersion angle. Figures 6.19 to 6.22 show
the axial droplet velocity found for different spray angles. The results obtained using
a dispersion angle of 45 are in best agreement with the experimental data, correctly
predicting the presence of droplets closer to the central axis, a maximum velocity of
around 20 m/s, and the final decrease in droplet axial velocity.
Chapter 6. Numerical Results for Airblast Atomizer 62
Figure 6.19: Axial velocity (spray an-
gle=45, atomizer dispersion angle=15).
Figure 6.20: Axial velocity (spray an-
gle=0, atomizer dispersion angle=45).
Figure 6.21: Axial velocity (spray an-
gle=30, atomizer dispersion angle=15).
Figure 6.22: Axial velocity (spray an-
gle=0, atomizer dispersion angle=15).
6.3.5 Radial and Tangential Velocities
Gurubaran et al. also investigated the radial and tangential components of droplet veloc-
ity as it is shown in Figures 6.23 and 6.24 [13]. Figures 6.25 and 6.26 show the radial and
tangential components of droplet velocity found for a dispersion angle of 45. ANSYS
Fluent fails to predict accurate results for the radial and tangential components of ve-
locity. The default airblast atomizer in ANSYS Fluent does not include a model for the
swirler. As a result, the tangential component of the initial droplet velocity is neglected.
A UDF file was used to add the tangential component of the initial droplet velocity. The
results found for this UDF are shown in Figures 6.27 and 6.28. The airblast atomizer
model in ANSYS Fluent still needs modifications to properly model the complex physics
Chapter 6. Numerical Results for Airblast Atomizer 63
Figure 6.23: Experimental droplet radial
velocity [13].
Figure 6.24: Experimental droplet tangen-
tial velocity [13].
Figure 6.25: Radial velocity (spray an-
gle=0, atomizer dispersion angle=45).
Figure 6.26: Tangential velocity (spray an-
gle=0, atomizer dispersion angle=45).
behind droplet breakup and atomization in a prefilming airblast atomizer.
6.3.6 Comparison of the Turbulence Modeling
The previous simulations were repeated using the first mesh for other turbulence models
including LES, DES, and DLES. The results found for all the turbulence models are
compared in Figures 6.29 to 6.32. The figures show similar results for all the turbulence
models. The results were obtained using the first mesh which consists of 128000 nodes
and has a minimum grid spacing of 0.24 mm. Finer meshes are recommended to be used
for grid converged RANS, as well as LES, DES, and DLES. However, fine meshes may
Chapter 6. Numerical Results for Airblast Atomizer 64
Figure 6.27: Radial velocity using UDF
(spray angle=0, atomizer dispersion an-
gle=45).
Figure 6.28: Tangential velocity using
UDF (spray angle=0, atomizer dispersion
angle=45).
not produce accurate results when the grid spacing is smaller than the particle diameter.
6.3.7 Results for Optimal Parameter Selection
Figures 6.33 to 6.35 show the most accurate set of results obtained from the previous
sections and compare the findings directly with the experimental data. A film thickness
analysis was performed and mentioned in details in Section 6.3.2. Based on the film
thickness analysis, a film thickness of 1.5× 10−4 m generates the best results and shows
a maximum droplet axial velocity of around 20 m/s in agreement with the experimental
data, as is shown in Figure 6.33. As was mentioned before, most of the models fail to
show the precense of droplets close to z axis. However, the results obtained for a disper-
sion angle of 45, when the atomizer angle is 0, correctly predict the initial increase in
droplet axial velocity as we move away in the radial direction. Nevertheless, the overall
droplet axial velocity is over-predicted by ANSYS Fluent. Figures 6.34 and 6.35 show
the results obtained for the radial and tangential components of the velocity using a UDF
to incorporate the initial swirl factor of droplet velocity. In general, the results predict
the observed trends for the experimental radial and tangential velocities. However, it
would seem that further improvements or modifications to the spray modeling approach
in ANSYS Fluent are needed to obtain more accurate results.
Chapter 6. Numerical Results for Airblast Atomizer 65
Figure 6.29: Axial velocity using RANS
and wave models.
Figure 6.30: Axial velocity using DLES
and wave models.
Figure 6.31: Axial velocity using LES and
wave models.
Figure 6.32: Axial velocity using DES and
wave models.
Chapter 6. Numerical Results for Airblast Atomizer 66
Figure 6.33: Comparison of experimental and numerical results for axial velocity [13].
Figure 6.34: Comparison of experimental and numerical results for radial velocity [13].
Chapter 6. Numerical Results for Airblast Atomizer 67
Figure 6.35: Comparison of experimental and numerical results for tangential velocity
[13].
Chapter 7
Numerical Results for High Shear
Fuel Nozzle
7.1 Introduction
As a last benchmark validation case for the spray modeling study, a high shear fuel nozzle
was examined. The high shear fuel nozzle case studied in the present work is based on
the work done by Li et ll., who investigated the spray atomization and droplet transport
created by a complex nozzle system consisting of different swirlers [14]. Figure 7.1 shows
the injector used by Li et al [14]. The injector is surrounded by two swirlers. The inner
swirler contributes to the breakup process by increasing the hydrodynamic forces between
the air and the liquid, which is injected using six orifices. As a result, the liquid reaches
the wall of the swirler creating a thin film. This thin sheet of liquid is further atomized
by the second swirling air flow [14]. The swirling air streams also increase the tangential
component of the droplets velocity. The two-step atomization produces finer droplets
and increases the efficiency.
Several previous studies have investigated the physics behind the atomization process for
the high shear fuel nozzles. Most of the studies focused on the breakup of the liquid jet
in a cross-flowing air. Shedd et al. investigated the atomization of the liquid film by a
high velocity air stream [61]. Arienti et al. modeled the wall film formation and droplet
breakup using a discrete phase approach [62]. The dynamics of the atomization process
68
Chapter 7. Numerical Results for High Shear Fuel Nozzle 69
Figure 7.1: High shear fuel nozzle [14].
Figure 7.2: Computational domain for the high shear fuel nozzle [14].
and liquid jet decomposition were further investigated by Arienti et al. [63]. Becker et al.
investigated the breakup process of a kerosene jet in a cross-flowing air at high pressures
[64]. Li et al. investigated the physics behind the near-field distribution of the droplets
in a liquid jet in a cross-flow configuration [65]. The far-field atomization details for a
liquid jet in a cross-flow were also investigated [66].
As mentioned above, the high shear fuel nozzle validation case is based on the work
done by Li et al., who investigated the air velocity and spatial variation of the droplets
Chapter 7. Numerical Results for High Shear Fuel Nozzle 70
Figure 7.3: Near injector mesh. Figure 7.4: High shear fuel nozzle mesh.
for a complex swirler/nozzle configuration [14]. A coupled level set and volume of fluid
(CLSVOF) method was used by Li et al. to capture the interactions between the liquid
and gaseous phases [67]. This method directly solves the Navier-Stokes equations for
incompressible flow, without using LES subgrid models. The complex geometry of the
injector was modeled using an embedded boundary approach [68]. The model created by
Li et al. also used an adaptive mesh refinement (AMR) method and a ghost of fluid (GF)
approach to enhance the accuracy of the results [69]. A Lagrangian-based method was
also used to track the droplets. The results obtained by Li et al. were further compared
with the experimental results found at the ambient spray facility of United Technologies
Research Center (UTRC). A phase doppler interferometry (PDI) technique was used to
capture droplet statistics. Furthermore, a mechanical patternator device was used to
calculate fuel fluxes [14].
7.2 Experimental and Computational Setup
Figure 7.2 shows the computational domain used for the simulations. Air enters the
swirler from two different air inlets with a velocity of 70 m/s. Liquid particles are injected
to the system from six orifices using the single injection method available in ANSYS
Fluent. The droplets are injected with an initial velocity of 8.43 m/s which results in
a momentum flux ratio of 9.4 [14]. The liquid droplets are treated and tracked using
a Lagrangian-based method. The liquid droplets can exchange mass, momentum, and
energy with the continuous phase. Simulations were repeated for different turbulence and
secondary breakup models. Figure 7.2 also shows the two measurement planes located
downstream the nozzle.
Chapter 7. Numerical Results for High Shear Fuel Nozzle 71
Figure 7.5: Contours of mean velocity magnitude.
Figures 7.3 and 7.4 show the unstructured mesh used for the simulations. The mesh
consists of 163,000 nodes, and has a minimum grid spacing of 75 microns. This mesh
includes around 3 nodes across each of the fuel orifice diameters and approximately 15
nodes across each of the air inlets. As was mentioned earlier in Chapter 5, ANSYS Fluent
may not be able to predict accurate results for meshes having smaller grid spacing than
the particle diameter. Refining the mesh used for the high shear fuel nozzle, creates
smaller grid spacings, which may not be compatible with the Lagrangian-based spray
modeling in ANSYS Fluent. For these reasons, a finer mesh was not considered for this
case.
7.3 Numerical Results
7.3.1 Contours of Gas Velocity
Contours of mean gas velocity are shown for the high shear fuel nozzle. For the results
shown, the simulations were performed using the wave secondary breakup model and
Chapter 7. Numerical Results for High Shear Fuel Nozzle 72
Figure 7.6: Contours of mean axial velocity.
Figure 7.7: Experimental air velocity magnitude [14].
a RANS turbulence approach. Figure 7.5 shows the contour for the mean gas velocity
magnitude. Air enters the inlets at a velocity of 70 m/s and reaches a maximum of
145 m/s downstream the injector. Figure 7.6 shows the air mean axial velocity clearly
showing the recirculation zones where the axial velocity is negative.
Li et al. collected the air velocity at two planes downstream the injector [14]. These
planes are located 1.1 inch and 1.6 inch away from the injector. Figures 7.7 and 7.8
show the air velocity magnitude found by Li et al. at the data collection planes [14].
Chapter 7. Numerical Results for High Shear Fuel Nozzle 73
Figure 7.8: Numerical air velocity magnitude [14].
Figure 7.9: Air velocity for RANS model.
Figure 7.10: Air velocity for LES model.
The figures on the left show the velocity for the plane 1.1 inch downstream the injector,
while the figures on the right show the air velocity 1.6 inch downstream the injector. The
results found by Li et al. show an increase in air velocity as we move away from the z
axis. The velocity reaches a maximum of around 80 m/s for both planes before finally
decreasing. Figure 7.9 shows the results found for the RANS turbulence model. ANSYS
Chapter 7. Numerical Results for High Shear Fuel Nozzle 74
Figure 7.11: Spatial variation of the droplets [14].
Figure 7.12: Near-injector particles for the wave model.
Fluent slightly over-predicts the air velocity at the downstream planes, however, the re-
sults are in a reasonable agreement with previous experimental and numerical results.
The results obtained for LES modeling is shown in Figure 7.10. ANSYS Fluent predicts
similar results for both RANS and LES models of the current mesh.
Chapter 7. Numerical Results for High Shear Fuel Nozzle 75
Figure 7.13: Numerical results for droplet breakup [14].
Figure 7.14: Experimental results for droplet breakup [14].
7.3.2 Spatial Variation of the Droplets
Li et al. also investigated the spatial variation of the droplets. Figures 7.11 shows the
breakup process presented by Li et al. using experimental and numerical approaches
[14]. The near injector particles found using ANSYS Fluent show similar regime for the
breakup.
Chapter 7. Numerical Results for High Shear Fuel Nozzle 76
Figures 7.13 and 7.14 show the breakup process and the spray angle found by Li et
al. The results obtained using RANS and wave models are in rather good agreement
with previous experimental and numerical results showing a spray angle of 65. Figure
7.16 shows the spanwise distribution of the droplets and the six orifices used for liquid
injection. The swirling air increases the tangential component of the droplet velocity and
leads to a counter clockwise rotation of the droplets.
The results obtained using the KHRT method show similar predictions for the spray
angle and the breakup process. The simulation was also repeated for the LES model.
Figure 7.19 show a spray angle of 70 found for this case. The overall breakup pro-
cess is similar to the results found by RANS modeling. Figure 7.20 shows the breakup
process to be more random and/or chaotic compared to the previous results shown above.
Chapter 7. Numerical Results for High Shear Fuel Nozzle 77
Figure 7.15: Droplet breakup using RANS and wave models
Figure 7.16: Spanwise distribution of the droplets using RANS and wave models.
Chapter 7. Numerical Results for High Shear Fuel Nozzle 78
Figure 7.17: Droplet breakup using RANS and KHRT models.
Figure 7.18: Spanwise distribution of the droplets using RANS and KHRT models
Chapter 7. Numerical Results for High Shear Fuel Nozzle 79
Figure 7.19: Droplet breakup using LES and wave models.
Figure 7.20: Spanwise distribution of the droplets using LES and wave models
Chapter 8
Conclusions and Future Research
8.1 Conclusions I: Liquid Jet in a Cross-flow
The liquid jet in a cross-flow as studied by Sen et al. was used as one of the validation
cases to evaluate the spray breakup models in ANSYS Fluent [12]. Liquid jet in a cross-
flow has various applications in gas turbines. The liquid experiences primary breakup as
it is injected from the orifice. The primary breakup consists of a liquid column, which
is further developed into ligaments and droplets. The cross-flowing air facilitates the
breakup process by creating drag forces on the liquid column. The discrete phase model
in ANSYS Fluent was used to model the breakup process described by Sen et al. [12].
The results were also compared with previous experimental and numerical data.
The wave secondary breakup option was found to produce the best results for this case.
The wave breakup model is recommended to be used for higher Weber numbers. Most of
the results are in good agreement with previous experimental data. The initial decrease
in droplet velocity due to thickening of the boundary layer is correctly modeled. ANSYS
Fluent however, overpredicts the near wall velocity where the jet wake effect is not
appropriately modeled. While similar results were obtained for LES, DES, and DLES
models on the default mesh, issues with grid convergence of the spray solutions were
identified. For meshes consisting of grid elements of about the same size as the droplet
size, ANSYS Fluent was unable to produce accurate results for droplet diameter and
droplet axial velocity. It would seem that the DPM modeling in ANSYS Fluent cannot
provide accurate predictions when the grid spacing is on the order of the particle diameter.
80
Chapter 8. Conclusions and Future Research 81
This casts doubt on the possibility of using the Lagrangian-based spray models with grid
converged RANS, as well as LES, DES and DLES for describing the turbulence in ANSYS
Fluent.
8.2 Conclusions II: Airblast Atomizer
The spray modeling capabilities of ANSYS Fluent were also evaluated using the airblast
atomizer presented by Gurubaran et al. [13]. In the experiments, droplet information
was collected for a prefilming airblast atomizer surrounded by a strong swirling flow.
This type of atomizer includes a pre-filming surface. The pre-filming surface is used to
produce a thin sheet of liquid. The high velocity air interacts with the sheet and results
in breakup. Sensitivity analysis was performed on various initial boundary conditions of
the case originally investigated by Gurubaran et al.. Particle information was collected
on five measurements planes downstream the atomizer.
The experimental data shows an increase in particle velocity as the radial distance is
increased, and the particle velocity reaches a peak before a final decrease. The numerical
results found using ANSYS Fluent correctly predict the final decrease of particle velocity.
However, most of the models fail to show the presence of droplets closer to the central
z axis. Similar results were found for the wave and KHRT methods. Finally, all the
turbulence models produced quite similar results.
8.3 Conclusions III: High Shear Fuel Nozzle
Finally, a high shear fuel nozzle was used as one of the validation cases [14]. The high
shear fuel nozzle originally presented by Li et al. consists of a nozzle/swirler system.
Two swirlers were used to facilitate the breakup process. The inner swirler creates the
initial breakup and produces a thin film on the wall of the swirler. The thin film is then
atomized by the outer swirler. Simulations were performed in ANSYS Fluent using the
discrete phase model and a Lagrangian-based method for tracking the droplets. The
single injection model was used to inject the particles into the computational domain.
The results obtained for the spray angle and spatial variation of the droplets were inves-
Chapter 8. Conclusions and Future Research 82
tigated and compared with previous experimental and numerical results. The numerical
predictions show reasonable agreement with previous data found by Li et al. [14]. The
wave and KHRT secondary breakup models again predict similar results. ANSYS Fluent
does not include models for liquid jet in a cross-flow and swirling injectors. The single
injection model does not account for the liquid column present in atomization. However,
increasing the number of initial particles streams produces more realistic results.
8.4 Recommendations for Future Research
In this thesis, the spray modeling capabilities of ANSYS Fluent were evaluated. In gen-
eral, ANSYS Fluent is able to successfully predict the overall trends for three spray cases
considered herein. However, future research is required to obtain more accurate results.
One recommendation is to enhance the accuracy of the primary and secondary breakup
models in ANSYS Fluent by introducing UDFs to model different breakup phenomena
including the intact liquid core, swirler injector, and liquid jet in a cross-flow. The dis-
crete phase model in ANSYS Fluent and the Lagrangian-based tracking method should
be also modified to obtain better predictions for finer meshes with LES, DES, and DLES.
The spray modeling capabilities of ANSYS Fluent should also be further investigated by
considering other reacting and non-reacting benchmark spray validation cases.
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Appendix A
Chemical Kinetics Models in
ANSYS Fluent
A.1 Introduction
ANSYS Fluent offers various models for representing the chemical kinetics and transport
of species in reactive flows. Different conservation equations for diffusion, convection
and species transport are solved in chemical kinetics modeling. Mixing and transport of
species can be modeled with considering reactions which could be volumetric reactions,
wall reactions or reactions with particle surfaces. Furthermore, ANSYS Fluent provides
the users with modeling turbulent reacting flames using different mixing approaches.
The mixing combustion models in ANSYS Fluent include non-premixed combustion,
premixed combustion and partially premixed combustion models. For the non-premixed
approach where the fuel and oxidizer enter the system from distinct streams, ANSYS
Fluent offers different flamelet modelings including steady, unsteady and diesel unsteady
flamelet modelings [2]. In the case of premixed combustion where the fuel and oxidizer
are mixed prior to ignition, the combustion modeling in ANSYS Fluent can be done using
either the C equation, the G equation or the extended coherent flame model. In addition,
the flame speed can be modeled using either Zimont or Peters approaches. The finite
rate chemical kinetics present in turbulent reacting flames can be also modeled with the
composition probability density function (PDF) transport model. The composition PDF
transport approach in ANSYS Fluent includes Lagrangian and Eulerian models [2]. The
94
Appendix A. Chemical Kinetics Models in ANSYS Fluent 95
following sections summarize the chemical kinetics models available in ANSYS Fluent as
is mentioned in ANSYS Fluent theory and user manuals [2, 18].
A.2 Species Transport and Finite Rate Chemistry
Mixing and transport of species can be modeled with the species transport and finite rate
chemistry models. Users are able to choose their desired mixture either from the default
mixtures present in ANSYS Fluent’s database or by manually adding the species. The
diffusion flux of species can be considered using various approaches. ANSYS Fluent uses
the Fick’s law approximation to model mass diffusion in dilute laminar flows [2]. The
Fick’s law approach approximates the diffusion flux of the species as follows:
~Ji = −ρDi,m~∇Yi −DT,i
~∇TT
(A.1)
where Di,m represents the mass diffusion coefficient and DT,i is the Soret diffusion coeffi-
cient. The dilute turbulent modeling for the mass diffusion is as follows:
~Ji = −(ρDi,m +µtSct
)~∇Yi −DT,i
~∇TT
(A.2)
where Sct is called the Schmidt number and is equal to uρD. For non-dilute mixtures
the diffusion modeling can be done using the full multicomponent diffusion model where
Fluent uses Maxwell Stefan equations to find the diffusion flux [18]:
~Ji = −N−1∑j=1
ρDi,m~∇Yi −DT,i
~∇TT
(A.3)
where Yi is the mass fraction of species. Furthermore, ANSYS Fluent also offers thermal
and inlet diffusion models. The species transport modeling in ANSYS Fluent can be
done both with or without considering reactions. The reactions models in ANSYS Fluent
include volumetric, wall surface and particle surface reactions [2].
A.2.1 Volumetric Reactions
ANSYS Fluent takes volumetric reactions into account by solving the following conser-
vation equation for convection and diffusion of chemical species:
∂
∂t(ρYi) + ~∇.(ρ~vYi) = −~∇.~Ji + Ri + Si (A.4)
Appendix A. Chemical Kinetics Models in ANSYS Fluent 96
where Yi is the mass fraction of species, Ri represents the net rate of production of species
by chemical reactions and Ji is the diffusive flux. It should be noted that Si represents the
rate of production of species from user functioned sources or dispersed phases. ANSYS
Fluent predicts the reaction rate of species using three approaches including laminar
finite rate, eddy dissipation and eddy dissipation concept [2].
Laminar Finite Rate Chemistry
In the laminar finite rate model, the effects of turbulence are ignored and the reaction
rate is found using Arrhenius expressions. This model is highly recommended for laminar
flames. However, the laminar finite rate model can be also used in conjunction with
small interactions between the turbulence and the chemistry. The net reaction rate of
the species is computed based on the number of reactions, NR, that occur [2]:
Ri = Mw,i
NR∑r=1
Ri,r (A.5)
where Ri,r is the Arrhenius molar rate of either production or destruction of species, and
Mw,i represents the molecular weight of species. ANSYS Fluent is capable of modeling
both reversible and non-reversible reactions [2]. The Arrhenius molar rate of production
or destruction of species of a non-reversible reaction in reaction r can be computed as
follows:
Ri,r = Γv′′i,r − v′i,r)(kf,rN∏j=1
Cn′′j,r+n′
j,r
j,r ) (A.6)
where v′i,r and n′j,r are the reactants’ stoichiometric coefficient of species i and rate ex-
ponent of species j respectively. The stoichiometric coefficient and rate exponent of of
products are represented as v′′i,r and n′′j,r respectively. Furthermore, Cj,r is the molar
concentration of species j, Γ is the net effect of third bodies and kf,r represents the for-
ward rate constant for reaction r. The reaction rate for a reversible reaction can be also
predicted using the following equation [2]:
Ri,r = Γ(v′′i,r − v′i,r)(kf,rN∏j=1
Cn′j,r
j,r − kb,rN∏j=1
Cv′′j,rj,r ) (A.7)
where kb,r is a constant representing the backward rate of reaction for reaction r.
Appendix A. Chemical Kinetics Models in ANSYS Fluent 97
Eddy Dissipation
Eddy dissipation model is a turbulence chemistry interaction model based on equations
derived by Magnussen and Hjertager [5]. The net reaction rate is found using two ex-
pressions and the smaller of those is considered to be the reaction rate:
˙Ri,r = v′i,rMw,iAρε
kmin<(
Y<v′<,rMw,<
) (A.8)
˙Ri,r = v′i,rMw,iABρε
k(
∑pYp∑
v′′j,rMw,j
) (A.9)
where Yp is the mass fraction of products and Y< represents the mass fraction of a
particular reactant [70]. It should be noted that for the LES turbulence model the large
eddy mixing time scale ε/k is replaced by the subgrid scale mixing rate and the reaction
rate equation is changed as follows [2]:
˙Ri,r = vi,rMw,iAρ√
2SijSij min<(Y<
v<,rMw,<) (A.10)
where Sij is the strain rate tensor.
Finite Rate/ Eddy Dissipation
ANSYS Fluent also offers the finite rate/ eddy dissipation model which calculates both
the Arrhenius rate and eddy dissipation mixing rates and uses the smaller of these two
rates [2].
Eddy Dissipation Concept
The Eddy dissipation concept model accounts for turbulence chemistry interactions us-
ing detailed chemistry mechanisms [71]. The eddy dissipation concept model predicts
the reaction rate using small turbulent structures. The length fraction and the volume
fraction of these turbulent structures are as follows [72]:
ξ∗ = Cξ(vε
k2)1/4 (A.11)
τ ∗ = Cτ (v
ε)1/2 (A.12)
Appendix A. Chemical Kinetics Models in ANSYS Fluent 98
where C is the volume fraction constant and Cτ is the time scale constant. Using the
length and volume fraction of the fine scales, the reaction rate can be found:
Ri =ρ(ξ∗)2
τ ∗[1− (ξ∗)3](Y ∗i − Yi) (A.13)
where Y∗ is the fine scale species mass fraction and τ ∗ represents the time [2].
Thickened Flame Model
The thickened flame approach is a laminar flame model available in ANSYS Fluent. Users
are able to model a laminar flame using the thickened model which decreases the reaction
rate [73]. As a result, ANSYS Fluent is able to approximate the laminar flame speed
using the thickening factor. When using the thickened model, all the species thermal
conductivity and diffusion coefficients are multiplied by the thickening factor:
F =N∆
δ(A.14)
where ∆ is the grid size, N represents the number of grid points in the flame and δ is the
laminar flame thickness. It should be noted that the thickened flame model is normally
used with a single step chemical mechanism [2].
A.2.2 Wall Surface Reactions
Fluent also offers the wall surface reactions model where diffusion and chemical kinetics
govern the rate of absorption and desorption of the chemical species. The net rate of the
rth reaction is:
<r = kf,r(
Ng∏i=1
[Ci])(Ns∏j=1
[Sj]) (A.15)
where Kf,r is the forward reaction constant while Ng and Ns represent the total number of
gas species and site species respectively. Using the net reaction rate the rate of production
of gaseous, solid and site species can be computed as follows [2]:
Ri,gas =N∑r=1
(g′′i,r − g′i,r)<r (A.16)
Appendix A. Chemical Kinetics Models in ANSYS Fluent 99
Ri,bulk =N∑r=1
(b′′i,r − b′i,r)<r (A.17)
Ri,site =N∑r=1
(s′′i,r − s′i,r)<r (A.18)
where g′i,r, b′i,r and s′i,r represent the stoichiometric coefficients of reactants and g′′i,r, b
′′i,r
and s′′i,r are the stoichiometric coefficients of the products. Furthermore, modeling of
the heat release resulted from surface reactions can be accomplished using the heat of
surface reactions model. The effects of surface mass transfer can be also included in the
continuity equation using the mass deposition source option. It should be noted that
ANSYS Fluent is only able to model reversible wall surface reactions [2].
A.2.3 Particle Surface Reactions
Particle surface reactions can be considered in the species transport. ANSYS Fluent uses
the following equation to predict the rate of particle surface species depletion [74]:
<j,r = ApnrYjpn<kin,rD0,r
D0,r + <kin,r(A.19)
where Ap is the particle surface area, nr is the effectiveness factor, Yj represents the
mass fraction of species j in the particle, N is the apparent order of the reaction and pn
represents the bulk partial pressure of the gaseous species. The diffusion rate coefficient
of the reaction is shown with D0,r while <kin,r represents the kinetic rate of the reaction
and depends on the number of gas phase reactants [2].
A.3 Non-Premixed Combustion
Non-premixed combustion accounts for cases where the oxidizer and the fuel enter the
system in distinct streams. Some examples of non-premixed combustion include pool
fires and coal furnaces. In this model the combustion process is simplified to a mixing
case where transport equations for one or two mixture fractions are solved. In the non-
premixed approach in ANSYS Fluent the thermo chemistry calculations accounting for
the mixture fraction fields are processed and analyzed using PDFs [2]. The assumed
Appendix A. Chemical Kinetics Models in ANSYS Fluent 100
shape PDFs in ANSYS Fluent can be calculated using either the double delta function
or the beta function. The double delta function is
p(f) =
0.5 for f = f −√f ′2
0.5 for f = f +√f ′2
0 for elsewhere.
(A.20)
In addition the beta function probability density function is given by [2]
p(f) =fα−1(1− f)β−1∫fα−1(1− f)β−1df
(A.21)
α = f [f(1− f)
f ′2− 1] (A.22)
β = (1− f)f [f(1− f)
f ′2− 1] (A.23)
Non-premixed combustion model is not compatible with Spalart Allmaras turbulence
model. ANSYS Fluent also offers the modeling of a secondary stream, empirical sec-
ondary stream and empirical fuel stream in the non-premixed combustion model. When
the species in the non-premixed model are mixed in the reaction zone, the chemistry can
be modeled using different approaches including chemical equilibrium, steady laminar
flamelet and unsteady laminar flamelet[2].
A.3.1 Equilibrium
When the chemistry model is in equilibrium the mixture fraction of the species can be
derived using the mass fractions of oxidizer and fuel [75]:
f =Zi − Zi,ox
Zi,fuel − Zi,ox
(A.24)
The sum of the mixture fractions for the oxidizer, fuel and secondary stream should be
equal to one. It should be noted that the mixture fraction is dependent on the air to fuel
Appendix A. Chemical Kinetics Models in ANSYS Fluent 101
mass ratio, r, and the equivalence ratio as follows
f = φφ+ r (A.25)
The non-premixed model may be used when the flow is turbulent and the Lewis num-
ber is one. The non-premixed combustion model is also compatible with liquid fuel,
coal combustion, flue gas recycle and inert mode. ANSYS Fluent solves the following
conservation equations for the mixture fraction and mixture fraction variance f ′2 [76]
∂
∂t(ρf) + ~∇.(ρ~vf) = ~∇.(µt
σt~∇f) + Sm + Suser (A.26)
∂
∂t(ρf ′2) + ~∇.(ρ~vf ′2) = ~∇.(µt
σt~∇f ′2) + Cgµt(~∇f)2 − Cdρ
ε
kf ′2 + Suser (A.27)
where Sm represents the source term for the mass transfer of reacting particles or liquid
fuel droplets into the gas phase. In addition, Suser is a source term defined by the user,
and Cg, σt and Cd are constants [2]. It should be noted that ANSYS Fluent models the
mixture fraction variance for LES using another approach [2]
f ′2 = CvarL2s(~∇f)2 (A.28)
where Cvar is a constant and Ls is the sub grid length scale.
A.3.2 Steady Flamelet
ANSYS Fluent is able to model flamelets using a single mixture fraction and the beta
probability density function. It should be noted that modeling of secondary and empirical
streams can not be done using the flamelet models in Fluent. A turbulent flame in
ANSYS Fluent is viewed as a combination of laminar flamelet structures [77]. The
laminar flamelets for non-premixed combustion are calculated and then using statistical
probability density function approaches embedded in a turbulent flame. The steady
laminar flamelet model accounts for fast chemistry combustion where the model is near
chemical equilibrium [78]. The relative chemical non-equilibrium is resulted from the
aerodynamic straining caused by turbulence. Aerodynamic straining tends to relax to
zero as the chemistry reacts quickly. In addition, the scalar dissipation, Xst, is used to
analyze the departure from equilibrium [2]:
Xst =asexp(−2[erfc−1(2fst)]
2)
π(A.29)
Appendix A. Chemical Kinetics Models in ANSYS Fluent 102
where as is the characteristic strain rate, fst is the stoichiometric mixture fraction and
erfc−1 represents the inverse complementary error function. ANSYS Fluent solves an
equation for the species mass fraction Yi to find the rate of flamelet generation [79]
ρ∂Yi∂t
= 0.5ρX∂2Yi∂f 2
+ Si (A.30)
Furthermore, steady flamelet model is not compatible with slow chemistry models such
as NOx formation [2].
A.3.3 Unteady Flamelet
The unsteady laminar flamelet model is suitable for slow chemistry processes such as
pollutant formation [9]. The unsteady flamelet model is computationally less expensive
than Eddy dissipation concept and PDF transport model. ANSYS Fluent solves the
following transport equation for unsteady flamelet probability I:
∂
∂t(ρI) + ~∇.(ρ~vI) = ~∇.(µt
σt~∇I) (A.31)
ANSYS Fluent solves the transport equation for unsteady flamelet probability along with
the flamelet species equation to predict the unsteady laminar flamelet [2]:
ρ∂Yi∂t
= 0.5ρX∂2Yi∂f 2
+ Si (A.32)
A.3.4 Diesel Unsteady Flamelet
ANSYS Fluent also offers the diesel unsteady flamelet modeling which can be only used
in conjunction with the transient solver, in- cylinder dynamic mesh and interaction with
continuous phase mode. The diesel unsteady flamelet model is capable of modeling
ignition, pollutants and formation of products [2]. ANSYS Fluent solves the energy
equations and flamelet species equations simultaneously to predict the unsteady flamelet
[80].
A.3.5 Adiabatic versus Non-Adiabatic
In adiabatic non-premixed combustion model the mass fraction, density and temperature
are solely dependent on the mixture fraction. It should be noted that in the presence of
Appendix A. Chemical Kinetics Models in ANSYS Fluent 103
a secondary stream, the instantaneous values also depend on the secondary partial frac-
tion. However, in the non-adiabatic case, the instantaneous values are also dependent on
instantaneous enthalpy. The non-adiabatic model solves the following transport equation
for mean enthalpy [2]:
∂
∂t(ρH) + ~∇.(ρ~vH) = ~∇.(kt
cp~∇H) + Sh (A.33)
where Sh is a source term resulted from radiation or heat transfer. In an adiabatic
non-premixed model, ANSYS Fluent solves an equation for species mass fractions and
temperature Φ
Φ =
∫ ∫Φ(f,Xst)p(f,Xst)dfdXst (A.34)
On the other hand, the non-adiabatic model assumes negligible influences of heat transfer
on species mass fractions and uses adiabatic mass fractions to model the combustion
process [2].
A.4 Premixed Combustion
Premixed combustion accounts for combustion cases where fuel and oxidizer are mixed
at molecular levels before ignition occurs. Gas leak explosions and internal combustion
engines are examples of premixed combustion process. Unlike non-premixed combustion
process where the model can be simplified to a mixing problem, in premixed combustion
model the net rate of flame propagation is controlled by both the turbulent eddies and
the speed of the laminar flame [2]. The premixed combustion model in ANSYS Fluent is
available when the flow is turbulent and subsonic. Furthermore, the premixed combustion
model can not be used in conjunction with reacting discrete phase particles and pollutant
models. It should be noted that the premixed combustion model is only compatible with
the pressure based solver. Premixed combustion modeling is available for both adiabatic
and non-adiabatic models. For the adiabatic model, ANSYS Fluent uses the following
equation to find the temperature [2]:
T = (1− c)Tu + cTad (A.35)
where Tu is the lowest temperature of unburnt mixture and Tad represents the highest
adiabatic burnt temperature. For non-adiabatic combustion ANSYS Fluent solves the
Appendix A. Chemical Kinetics Models in ANSYS Fluent 104
an energy transport equation to account for heat transfer [2]:
∂
∂t(ρh) + ~∇.(ρ~vh) = ~∇.(kt + k
cp~∇h) + S
h,chem + Sh,rad (A.36)
where h is the sensible enthalpy, Sh,rad represents the heat loss resulted from radiation
and Sh,chem is the heat gain resulted from chemical reactions. The two main premixed
combustion models include the C equation and the G equation models. ANSYS Fluent
also offers Zimont and Peters models for predicting the flame speed [2].
A.4.1 C Equation
ANSYS Fluent uses a scalar variable c to represent the progress of premixed combustion
reaction from unburnt to burnt mode. Mean reaction progress variable varies between
zero and one. Behind the flame in the burnt section C has a value of 1 while ahead of
flame in the unburnt reactants C is equal to zero. In the C equation model the burnt
section and the unburnt section are separated by the flame sheet, and the process does
not consider an intermediate reaction state occurring between the burnt and unburnt
species [2]. The mean progress variable used in this model can be found using the mass
fraction of product species Y
c =
∑ni=1 Yi∑ni=1 Yi,eq
(A.37)
ANSYS Fluent solves the following transport equation for mean reaction progress vari-
able, c, to predict the flame front propagation:
∂
∂t(ρc) + ~∇.(ρ~vc) = ~∇.( µt
Sct~∇c) + ρSc (A.38)
where Sct is the turbulent Schmidt number and Sc represents the reaction progress source
term. The mean reaction rate source term can be approximated using the turbulent flame
speed Ut and the density of unburnt mixture ρu [81]
ρSc = ρuUt|~∇c| (A.39)
A.4.2 G Equation
ANSYS Fluent also offers a premixed flame front tracking model namely the G equation
model. The G equation approach solves the following transport equation [82]:
p∂G
∂t+ pv.~∇G = pUt|~∇G| − pDk|~∇G| (A.40)
Appendix A. Chemical Kinetics Models in ANSYS Fluent 105
where Dk is the diffusivity, Ut is the flame speed and k represents the flame curvature.
The mean progress variable can be found using the mean flame front position G and the
flame position variance G2. The flame distance variance in the G equation model can be
approximated using both the transport equation or the algebraic option. The transport
equation is normally used for RANS modeling while the algebraic approach is suitable
for LES turbulent modeling [2].
A.4.3 Extended Coherent Flame Model
The extended coherent flame model in ANSYS Fluent solves a transport equation for
the area density of the flame represented by Σ, in addition to the transport equation for
the progress variable. The extended coherent flame model is more accurate than other
methods while being computationally expensive [83]. ANSYS Fluent solves the following
equation for the transport of flame area density [2]:
∂Σ
∂t+ ~∇.(~uΣ) = ~∇.( µt
Sct~∇(Σ/ρ)) + PΣ + P4−D (A.41)
where SCt is the turbulent Schmidt number, P represents the sources due to turbulence
interaction, dilatation in the flame and expansion of unburned gas. D represents dissipa-
tion of flame area while P4 is the source due to normal propagation. ANSYS Fluent offers
four different extended coherent flame model variants including the Veynante scheme [84],
the Meneveau scheme [85], Poinsot [85] and the Teraji scheme [86]. These four variants
provide different values for P, P4 and D. It should be noted that when LES is used in
conjunction with extended coherent flame approach the effects of subgrid terms should
be also taken into account [2].
A.4.4 Zimont Flame Speed Model
ANSYS Fluent offers two different turbulent flame speed models. The Zimont model
predicts the flame speed closure using a wrinkled flame front as follows [81]:
Ut = A(u′)3/4U1/2l α−1/4l
1/4t (A.42)
and using a thickened flame front model using the equation below [2]:
Ut = Au′(rtrc
)1/4 (A.43)
Appendix A. Chemical Kinetics Models in ANSYS Fluent 106
where A is the model constant, u′ represents the root mean square velocity, α is the
thermal diffusivity, lt represents the turbulence length scale, rt is the turbulent time
scale and rc represents the chemical time scale. In addition, The Zimont flame speed
model uses the an equation when LES model:
Ut = A(ltr−1sgs)
3/4U1/2l α−1/4l
1/4t (A.44)
where rsgs is the subgrid scale mixing rate. Furthermore, ANSYS Fluent is capable of
modeling wall damping and flame stretching in conjunction with the Zimont mode [2].
A.4.5 Peters Flame Speed Model
ANSYS Fluent solves the following equation in order to estimate the turbulent flame
speed [82]:
Ut = Ul(1 + σt) (A.45)
where σt is a variable dependent on the Ewald’s corrector, turbulent velocity scale, flame
brush thickness and laminar flame thickness [87]. The effects of wall damping can also
be taken into account [2].
A.5 Partially Premixed Combustion
ANSYS Fluent also offers a partially premixed combustion model which is a combination
of non-premixed and premixed methods available in ANSYS Fluent and models premixed
flames with non-uniform equivalence ratios [2]. Species fractions and temperature are
estimated using a probability density function
Φ =
∫ 1
0
∫ 1
0
Φ(f, c)p(f, c)dfdc (A.46)
It should be noted that all the models and limitations of the non-premixed and premixed
combustion models in ANSYS Fluent also apply to the partially premixed model.
Appendix A. Chemical Kinetics Models in ANSYS Fluent 107
A.6 Composition PDF Transport
The composition PDF transport model is similar to the laminar finite rate and eddy
dissipation concept and should be used to model chemical kinetics effects in turbulence.
It should be noted that the composition PDF transport is only available with the pressure
based solver, and is computationally expensive and it may not be easy to use when using
mixing models. The composition PDF transport derives a transport equation for the
probability density function instead of using Reynolds averaging methods and equations
used in finite rate chemistry approaches. The probability density function demonstrates
the time fraction of fluid that spends at each state depending on the temperature, species
and pressure [2]. The probability density function used in the composition PDF transport
includes the transient rate of change of the PDF, PDF change due to convection and PDF
change due to chemical reactions [88]. The < X|Y > notation shows the conditional
probability of event X when event Y occurs:
∂
∂t(ρP ) +
∂
∂xi(ρuiP ) +
∂ψk(ρSkP ) = − ∂
∂xi[ρ < u′′i |ψ > P ] +
∂
∂ψk[ρ <
∂Ji,kρ∂xi
|ψ > P ]
(A.47)
where P is the Favre joint PDF of composition, ui is the Favre mean fluid velocity, ψ
represents the composition space vector and Sk is the reaction rate. Furthermore, the
fluid velocity fluctuation vector and the molecular diffusion flux vector are represented
by u′′i and Ji,k respectively. The composition PDF modeling can be done using either the
Lagrangian or Eulerian methods [2].
A.6.1 Lagrangian
The Lagrangian method is capable of solving N+1 dimensional probability density func-
tion transport equations. The Lagrangian method is normally used with high dimensional
equations and often introduces statistical errors. The Lagrangian method consists of ran-
domly oriented particles moving due to particle convection, mixing and reaction. The
number of particles are controlled and fixed using the values for particles’ mass [89]. The
particle convection process involves two convection steps illustrated below:
x1/2i = x0
i + 0.5u0i∆t (A.48)
x1i = x
1/2i + ∆t(u
1/2i − 0.5u0
i +∂µt
ρSct∂xi+ ξ
√2µt
ρ∆tSct) (A.49)
Appendix A. Chemical Kinetics Models in ANSYS Fluent 108
where xi represents the particle position vector, ui is the Favre fluid velocity vector,
Sct is the turbulent Schmidt number and ξ represents the standardized normal random
vector. The particle time step, shown by ∆t is the minimum of the convection, diffusion
and mixing time steps [2]. The particle reaction modeling in ANSYS Fluent can be
done using either direct integration or the In Situ adaptive Tabulation model (ISAT).
In addition, the particle mixing modeling can be done using either the modified curl
model, the interaction by exchange (IEM) model or the Euclidean minimum spanning
tree (EMST) model [2].
Direct Integration
The direct integration method simply uses the following equation in terms of S, the
chemical source term, to find the particle composition vector, Φ:
Φ1 = Φ0 +
∫ ∆t
0
Sdt (A.50)
The direct integration method can be computationally costly when modeling fast or slow
chemical reactions [2].
ISAT
The ISAT model is an alternative to direct integration and builds a table during the
simulation. The ISAT approach avoids integrations by the usage of look up tables [18].
Users are able to modify the maximum storage, error tolerance and verbosity for the
ISAT model [90].
Modified Curl
The modified curl model randomly selects a few particle pairs and moves their composi-
tion toward a mean value. The composition vectors of particles i and j having masses of
mi and mj respectively, are estimated as follows [91]:
Φ1i = (1− ξ)φ0
i + ξΦ0imi + Φ0
jmj
mi +mj
(A.51)
Φ1j = (1− ξ)φ0
j + ξΦ0imi + Φ0
jmj
mi +mj
(A.52)
Appendix A. Chemical Kinetics Models in ANSYS Fluent 109
IEM
The IEM approach moves the composition of all the particles toward a mean value:
Φ1 = Φ0 − (1− e−0.5C/τt)(Φ0 − Φ) (A.53)
where Φ is the Favre mean composition vector [92].
EMST
The Euclidean minimum spanning tree model which is computationally more expensive
than previous methods, takes into account the localness of the particles and pairs particles
that are close to each other [2, 93].
A.6.2 Eulerian
ANSYS Fluent also offers a less computationally expensive composition PDF transport
method called the Eulerian approach. The Eulerian model selects a shape for the proba-
bility density function and uses eulerian transport equations to model chemical kinetics
while reducing stochastic errors. ANSYS Fluent uses the following probability density
function which is dependent on the product of delta functions and number of species N
[2]:
P (ψ; ~x, t) =Ne∑n=1
pn
Ns∏k=1
δ[ψk− < Φk >n] (A.54)
where pn is the probability, < Φk >n represents a conditional value for the mean compo-
sition of species k and ψk is the composition space variable. Furthermore, two transport
equations for the probability and probability weighted conditional mean composition of
eulerian method can be derived [94]:
∂ρpn∂t
+∂
∂xi(ρuipn) = ~∇(ρΓ~∇pn) (A.55)
∂ρsn∂t
+∂
∂xi(ρuisn) = ~∇(ρΓ~∇sn) + ρ(Mk,n + Sk.n + Ck,n) (A.56)
where Sn is the conditional value for the probability weighted mean composition, Γ
represents the effective turbulent diffusivity, M is the mixing term, S is the reaction term
Appendix A. Chemical Kinetics Models in ANSYS Fluent 110
and C represents the correction term. In addition, the reaction source term Sk,n can be
computed using the net reaction rate of species as follows [2]:
Sk,n = pnS(< Φk >n)k (A.57)
and the correction term can be also found using the equation below
Ne∑n=1
< Φk >mk−1n Ck,n =
Ne∑n=1
(mk − 1) < Φk >mk−1n pnck,n (A.58)
The mixing modeling in the eulerian approach is based on the IEM model and uses the
following equation to compute the mixing term Mk,n [95]:
Mk,n =CΦ
τ(< Φk >n −ψk) (A.59)
where CΦ is the mixing constant and τ represents the turbulence time scale [2].
Appendix B
Radiation Models in ANSYS Fluent
B.1 Introduction
ANSYS Fluent offers various approaches for radiation modeling. The main radiation
models present in Fluent include Rosseland model, P1 model, discrete transfer radiation
model, surface to surface, and discrete ordinates model. These radiation models can
also be simulated using a radiation medium. Furthermore, ANSYS Fluent provides the
users with two different solar radiation modelings including solar ray tracing and discrete
ordinates irradiation. The latter are not of interest here and will not be discussed. In
addition, the discrete transfer radiation model and the discrete ordinates model are the
only two models appropriate for optically thin problems. It should be noted that the
radiation models should be properly used when the radiant heat flux is noticeably large
enough compared to the heat transfer resulted from conduction or convection:
Qrad = σ(T 4max − T 4
min) (B.1)
All the radiation models in Fluent solve the radiative transfer equation for a scattering,
absorbing and emitting medium with a refractive index of n and local temperature of T
[2]:dI
ds+ I(~r, ~s)(a+ σs) = an
2σT 4
π+σs4π
∫ 4π
0
I(~r, ~s′)ΦdΩ′ (B.2)
where ~r is the position vector, ~s is the direction vector, s is the path length and ~s′ is
the scattering direction vector. Furthermore, the absorption coefficient is shown as a,
σs is the scattering coefficient and I is the radiation intensity. The phase function and
111
Appendix B. Radiation Models in ANSYS Fluent 112
solid angle are represented as Φ and Ω′ respectively. In addition, σ which represents
the Stephan Boltzman constant is equal to 5.669 ∗ 10−8W/m2K4. The following sections
discuss the radiation models in ANSYS Fluent based on ANSYS Fluent theory and user
manuals [2, 18].
B.2 Rosseland Model
Rosseland model is the simplest radiation model in ANSYS Fluent which is valid for
optically thick mediums where L(a+ σs) >> 1 [2]. Furthermore, the Rosseland method
is mostly used when the optical thickness is at least 3. The main advantages of the
Rosseland approach are being fast and requiring less memory than other radiation models.
However, the Rosseland model is only compatible with the density based solver [96].
The radiative heat flux of the Rosseland model in a gray medium can be approximated
using the assumption of intensity to be same as the intensity of black body at the gas
temperature [2]
~qr = −Γ~∇G (B.3)
where
Γ =1
(3(a+ σs)− Cσs)(B.4)
~qr = −16σΓn2T 3~∇T (B.5)
It should be noted that the total heat flux can be written as:
~q = −(k + kr)~∇T (B.6)
where k is the thermal conductivity. The heat flux equation can be used to find the bound-
ary conditions at flow inlets and outlets. In addition, ANSYS Fluent offers anisotropic
scattering modeling for the Rosseland approach. Anisotropic scattering is approximated
using a phase function as follows [2]:
Φ(~s′.~s) = 1 + (C~s′.~s) (B.7)
where ~s is the unit vector of scattering, ~s′ is the unit vector of the radiation and C is the
anisotropic phase function coefficient which differs among fluids.
Appendix B. Radiation Models in ANSYS Fluent 113
B.3 P1 Model
The P1 model is an extended approach of the Rosseland model and can be applied to
various geometries and optical thicknesses. However, The P1 model may be less accurate
if the optical thickness is small. The reflection of the incident radiation at the surface
is considered to be isotropic which results in diffusion of all surfaces [96]. The radiative
heat flux of the P1 model in a gray medium can be calculated using
~qr = − 1
(3(a+ σs)− Cσs)~∇G (B.8)
ANSYS Fluent solves the following transport equation for G in order to predict the
radiative heat flux:
~∇.(Γ~∇G)− aG+ 4an2σT 4 = SG (B.9)
where σ is the Stephan Boltzmann constant and SG is a user defined radiation source.
The P1 model also offers the non-gray radiation model with the assumption of constant
absorption coefficient and constant spectral emissivity at walls for each wavelength band
[97]. For the non-gray radiation model, ANSYS Fluent solves the following transport
equation for G, the spectral incident radiation [2]:
~∇.(Γλ~∇Gλ)− aλGλ + 4aλn2σT 4 = SGλ (B.10)
Γλ =1
(3(aλ + σsλ)− Cσsλ)(B.11)
where aλ is the spectral absorption coefficient and SGλ is a user defined radiation source
term. The spectral black body emission is approximated using start and end wavelengths.
It should be noted that the spectral radiative flux can be determined as
~qλ = −Γλ~∇Gλ (B.12)
P1 model in ANSYS Fluent also offers anisotropic scattering modeling same as the Rosse-
land approach. The effects of dispersed second phase particles can also be included in
the P1 model using a transport equation [2]:
~∇.(Γ~∇G) + 4π(an2σT4
π+ Ep)− (a+ ap)G = 0 (B.13)
where Ep is the equivalent emission of particles and ap is the absorption coefficient.
Furthermore, ANSYS Fluent assumes the emissivity of inlets and outlets to be 1 and
same as the black body absorption.
Appendix B. Radiation Models in ANSYS Fluent 114
B.4 Discrete Transfer
The discrete transfer radiation model replaces the radiation leaving the surface with a
single ray. Furthermore, the accuracy of the approximation can be enhanced by increasing
the number of rays. The discrete transfer model does not account for scattering and is
performed assuming gray radiation and considering diffusion for all surfaces [7]. It should
be noted that the Discrete Transfer model is not compatible with parallel processing and
sliding meshes. The change of radiant intensity at the gas local temperature of T along
a certain path can be approximated as
dI
ds+ aI =
aσT 4
π(B.14)
Integration of this equation with the assumption of constant gas absorption coefficient a,
results in the following equation in terms of I0, the intensity at the start of the desired
path [2]:
I =σT 4
π(1− e−as) + I0e
−as (B.15)
The radiant intensity found in this method may be used to determine the radiative heat
transfer for the discrete transfer radiation model [98]. In addition, the rays are traced
using hemispherical solid angle at a point on the surface. Each ray is associated with two
angles, θ and φ, which vary from 0 to π/2 and from 0 to 2π respectively. Furthermore, the
computational costs can be decreased by clustering surfaces and cells [7]. The number
of faces per surface cluster defines the number of faces which is produced by adding the
neighbor faces to the surface cluster. The number of cells per volume cluster option
represents the quality of clustering which is formed by adding a specific cell’s neighbors
to the volume cluster. Furthermore, the surface and volume cluster temperatures are
approximated using area and volume averaging approaches [2].
B.5 Surface to Surface
The surface to surface radiation model neglects the effects of scattering, absorption and
emissions, and only considers the energy exchange between two surfaces. The surface to
surface model is less computational expensive compared to the discrete transfer radiation
model. When the number of surface faces increases the CPU requires more time per
Appendix B. Radiation Models in ANSYS Fluent 115
iteration. The computational costs can be minimized using clustering option available
in ANSYS Fluent [96]. The surface to surface model is not compatible with periodic
boundary conditions or mesh adaption. ANSYS Fluent assumes diffuse and gray surfaces
for modeling surface to surface radiation. It should be noted that in the gray diffuse model
the emissivity is equal to absorptivity [99]:
ε = α (B.16)
The surface to surface model is governed by the energy flux equation leaving the surface.
The leaving energy flux q is dependent on the energy flux of the surroundings on the
surface qin [2]
q = εkσT4k + ρkqin (B.17)
A geometric function called view factor F is used to account for the surfaces’ geometry,
orientation and distance, and can be calculated using the ray tracing method. ANSYS
Fluent also provides the hemicube method for calculating the view factor. The hemicube
approach is recommended to be used with complex models and geometries. In addition,
blocking and obstruction between the surfaces can also be considered while calculating
the view factor. The energy flux on the surface is related to the energy flux leaving the
surface based on the view factor:
qin = ΣFkjqj (B.18)
where Fkj is the view factor between surface k and surface j. Therefore, the energy flux
leaving the surface k would be approximated as [2]:
q = εkσT4k + ρkΣFkjqj (B.19)
ANSYS Fluent offers face to face and cluster to cluster basis approaches for the view
factor. In the face to face approach the boundary surfaces are used in the view factor
calculations. Cluster to cluster method which is available in three dimensional cases
reduces the CPU usage and computational costs. This method produces polygon faces
for computing the view factor. The clustering process can be done either manually or
automatic. Faces per surface cluster for flow boundary zones may be determined manually
and can be applied to all walls. The automatic clustering method specifies the maximum
faces per surface cluster.
Appendix B. Radiation Models in ANSYS Fluent 116
B.6 Discrete Ordinates
Radiative transfer equations for a number of solid angles are solved in the discrete or-
dinates radiation model. The discrete ordinates radiation model is considered to have
moderate memory requirements and computational costs for simple and typical angular
discretizations. However, the computational costs of the discrete ordinates model may
increase for finer angular discretizations. The radiation modeling in this approach is
available in both gray radiation and non-gray radiation models. As a result, various ra-
diation characteristics including scattering, semi transparent media and anisotropy can
be included in the modeling [8]. The discrete ordinates radiation model solves transport
equations for each solid angle associated with a direction vector ~s. The radiative transfer
equation for direction vector ~s using a gray radiation approach [2]
~∇.(I~s) + (a+ σs)I = an2 2σT 4
π+σs4π
∫ 4π
0
IΦdΩ′ (B.20)
Furthermore, ANSYS Fluent solves the radiative transfer equation for a non-gray model
with wavelength of Iλ using spectral intensity I [2]:
~∇.(Iλ~s) + (aλ + σs)Iλ = aλn2Ihλ +
σs4π
∫ 4π
0
IλΦdΩ′ (B.21)
where aλ is the spectral absorption coefficient and Ihλ is the black body intensity. The dis-
crete ordinates radiation model can be used with either coupled or uncoupled approaches.
In the coupled case, the intensity and energy equations are solved simultaneously at each
cell [100]. The coupled method decreases the computational costs when using high opti-
cal thicknesses or scattering coefficients. The coupled radiation model is not compatible
with shell conduction model and cases involving weak coupling between energy and ra-
diation intensity. It should be noted that the energy coupling method is not available in
some combustion models where enthalpy equation is solved. Furthermore, the coupling
approach is typically used for optical thickness of greater than 10 and in glass melting
applications [100]. On the other hand, the uncoupled approach solves the intensity and
energy equation at each cell one by one [101]. The uncoupled method uses finite volume
scheme to solve the energy and intensity equations.
ANSYS Fluent uses control angles for angular discretization and pixelation in the discrete
ordinates method. The control angles used in the discrete ordinates approach include θ
which is the polar angle and φ which shows the azimuthal angle measured in accordance
Appendix B. Radiation Models in ANSYS Fluent 117
to Cartesian system. The discrete ordinates approach also provides scattering model-
ings. The scattering modeling can be done using a linear phase function described in the
following equation:
Φ(~s′.~s) = 1 + (C~s′.~s) (B.22)
The scattering modeling can also be done using a delta eddington phase function illus-
trated bellow:
Φ(~s′.~s) = (1− f)(1 + (C~s′.~s)) + 2fδ(~s′.~s) (B.23)
where δ is the Dirac delta function and f defines the forward scattering factor. Finally,
ANSYS Fluent also offers a user defined phase function modeling which is [2]
Φ(~s′.~s) = (1− f)Φ∗(~s′.~s) + 2fδ(~s′.~s) (B.24)
Appendix C
Emissions and Soot Formation
Models in ANSYS Fluent
C.1 Introduction
Most of the chemical processes involve production of harmful and unwanted pollutants.
Furthermore, an incomplete combustion of a hydrocarbon may result in soot formation.
ANSYS Fluent enables the users to model the emissions and soot formation resulted
from combustion or other chemical processes. Pollutant modeling in ANSYS Fluent nor-
mally consists of NOx formation and SOx formation modelings. In addition, Decoupled
Detailed Chemistry Model which provides the user with the ability of emission modeling
using any chemical kinetic mechanism is also available in ANSYS Fluent. The soot for-
mation is also modeled using different approaches such as One Step, Two Step or Moss
Brookes model. The details of these models are presented in this chapter as mentioned
in ANSYS Fluent theory and user manuals [2, 18].
ANSYS Fluent uses probability density function approaches for modeling the mean tur-
bulent chemistry interactions. The mean reaction rate form used in the PDF approach
illustrates the instantaneous rate of emissions production integrated over an appropriate
range. Prediction of the mean reaction rate in ANSYS Fluent can be done by using
either one or two variables. ANSYS Fluent uses different equations for determining the
mean reaction rates for the pollutants [2]
118
Appendix C. Emissions and Soot Formation Models in ANSYS Fluent 119
SNO =
∫ρωNO(V1)P1(V1)dV (C.1)
SSO2=
∫ρωSO2
(V1)P1(V1)dV (C.2)
Ssoot =
∫ρωsoot(V1)P1(V1)dV (C.3)
and for two variables as follows:
SNO =
∫ ∫ρωNO(V1, V2)P (V1, V2)dV1dV2 (C.4)
SSO2=
∫ ∫ρωSO2
(V1, V2)P (V1, V2)dV1dV2 (C.5)
Ssoot =
∫ ∫ρωsoot(V1, V2)P (V1, V2)dV1dV2 (C.6)
where V1 and V2 are progress variables and could be temperature or species concentra-
tion, P is the assumed or presumed probability density function (PDF), ω is described as
the turbulated reaction rate, and S is the mean turbulent production rate of species. In
addition, PDF modeling in ANSYS Fluent can be done by using the Gaussian approach
[102]or Beta PDF approach [103] which are beyond the scope of this project.
C.2 NOx Formation
NOx refers to mono nitrogen oxides including mostly nitric oxide (NO) and nitrogen
dioxide (NO2). NOx gases are produced as a result of the reaction between nitrogen and
oxygen gases during combustion. NOx contributes to ozone depletion and formation of
photochemical smog and acid rain [104]. As a result, NOx gases are considered to be
pollutants and should be treated properly. ANSYS Fluent provides various approaches
to model the primary sources of NOx formation which include thermal, prompt, fuel and
intermediate N2O. Furthermore, NOx consumption resulted from re burning in combus-
tion systems can also be modeled in ANSYS Fluent. It should be noted that the NOx
model is not compatible with the premixed combustion model [2].
Appendix C. Emissions and Soot Formation Models in ANSYS Fluent 120
C.2.1 Thermal NOx
The oxidation of atmospheric nitrogen which is present in the combustion air forms
thermal NOx. ANSYS Fluent solves a transport equation for NO species concentration
in order to predict and model thermal NOx resulted from combustion
∂
∂t(ρYNO + ~∇.(ρ~vYNO) = ~∇.(ρD~∇YNO) + SNO (C.7)
where Y is the mass fraction of the species and D defines the effective diffusion coeffi-
cient. The source terms of the species are shown as S. The formation of thermal NOx in
combustion systems is governed by the three following reactions [2]:
O + N2 N+NO (C.8)
N+O2 O+NO (C.9)
N+OH H+NO (C.10)
Studies have revealed the existence of radical concentrations of O and OH in turbulent
diffusion flames. ANSYS Fluent provides different approaches for determining the O rad-
ical concentration present in thermal NOx. These approaches include the equilibrium,
partial equilibrium and predicted O approach. The first method known as the equilib-
rium approach assumes most of the thermal NOx to be formed after the combustion
is completed. In this approach, the hyrdocarbon oxidation rate is notably faster than
the thermal NOx formation rate. This method simplifies the prediction of the thermal
NOx formation by assuming the combustion reactions to be at equilibrium. The partial
equilibrium approach considers the dissociation recombination process of oxygen. This
method which results in a higher partial concentration for an Oxygen atom uses the
following reaction in addition to the thermal NOx formation reactions [2]:
O2 + M 2O+M (C.11)
the Predicated O approach simply predicts the concentration of O atom from the mass
fraction of local O species. Furthermore, the OH radical concentration present in turbu-
lent diffusion flames can be predicted using three models. The third reaction of thermal
NOx formation can be neglected using the exclusion of OH approach or can be consid-
ered using the partial equilibrium approach. The third OH prediction method known as
the predicted OH approach simply determines the OH concentration by using the mass
fraction of the local OH species.
Appendix C. Emissions and Soot Formation Models in ANSYS Fluent 121
C.2.2 Prompt NOx
High speed reactions present at the turbulent flame may result in production of prompt
NOx. The governing equation for prompt NOx transport is same as the transport equa-
tion used for thermal NOx
∂
∂t(ρYNO + ~∇.(ρ~vYNO) = ~∇.(ρD~∇YNO) + SNO (C.12)
Prompt NOx which is mostly present in rich flames can be produced in combustion
environments such as gas turbines and surface burners. Furthermore, short residence
times and low temperature contribute to the prompt NOx formation. The basic reactions
governing the mechanism for prompt NOx production are illustrated as follows [2]:
CH+N2 HCN+N (C.13)
N+O2 O+NO (C.14)
CN+O2 CO+NO (C.15)
HCN+OH CN+H2O (C.16)
Equivalence ratio specifies the quantity of HCN production and has direct effect on the
NOx production rate. Prompt NOx production increases and reaches a peak as the
equivalence ratio is increased. Furthermore, increasing the equivalence ratio after the
peak results in decrease of NOx production as a result of oxygen deficiency [2]. ANSYS
Fluent provides the prompt NOx production model based on a kinetic parameter derived
by De Soete [105]. According to De Soete, the overall prompt NOx production can be
expressed asd(NO)
dt=d(Prompt NOx)
dt− d(Prompt N2)
dt(C.17)
C.2.3 Fuel NOx
Oxidation of the nitrogen present in the fuel results in fuel NOx formation. ANSYS
Fluent solves four transport equation for NO, HCN, NH3 and N2O species in order to
model the fuel NOx formation
∂
∂t(ρYNO + ~∇.(ρ~vYNO) = ~∇.(ρD~∇YNO) + SNO (C.18)
Appendix C. Emissions and Soot Formation Models in ANSYS Fluent 122
∂
∂t(ρYN2O + ~∇.(ρ~vYN2O) = ~∇.(ρD~∇YN2O) + SN2O (C.19)
∂
∂t(ρYNH3
+ ~∇.(ρ~vYNH3) = ~∇.(ρD~∇YNH3
) + SNH3(C.20)
∂
∂t(ρYHCN + ~∇.(ρ~vYHCN) = ~∇.(ρD~∇YHCN) + SHCN (C.21)
It should be noted that ANSYS Fluent uses different approaches for fuel NOx prediction
of gaseous, liquid or coal fuels. Fuel NOx can be formed by the presence of either HCN,
NH3 or coal. The HCN or NH3 formation is governed by the rate of combustion of the
fuel in the case of gaseous fuels. Furthermore, the HCN or NH3 production in liquid fuels
is equal to the rate of the fuel released into the gas as a result of droplet evaporation
[2]. If HCN acts as the intermediate species, the source terms in the species transport
equations are presented as [2]
SHCN = Spl,HCN + SHCN−1+ SHCN−2
(C.22)
SNO = SNO−1+ SNO−2
(C.23)
The source terms in the case of intermediate ammonia are illustrated as follows:
SNH3= Spl,NH3
+ SNH3−1+ SNH3−2
(C.24)
SNO = SNO−1+ SNO−2
(C.25)
ANSYS Fluent is also capable of modeling fuel NOx formed from coal by assuming the
fuel nitrogen to be distributed between the char and the volatiles. The first approach
assumes partially conversion of intermediate species which could be either HCN or NH3
resulted from char N, to NO [2]. The source terms for HCN, NH3 and NO are as follows:
SChar,NO = 0 (C.26)
SChar,NH3=ScYN,CharMw,NH3
Mw,NV
(C.27)
SChar,HCN =ScYN,CharMw,HCN
Mw,NV
(C.28)
where Sc is the char burnout rate and V is the cell volume. The source terms for the
second approach which assumes all the N char to be converted to NO directly are [2]
SChar,NO =ScYN,CharMw,NO
Mw,NV
(C.29)
SChar,NH3= 0 (C.30)
SChar,HCN = 0 (C.31)
Appendix C. Emissions and Soot Formation Models in ANSYS Fluent 123
C.2.4 N2O Intermediate
ANSYS Fluent also offers the modeling of NOx formation using intermediate N2O. The
primary mechanism for intermediate N2O modeling was developed by Melte and Pratt
[106]. The amount of contribution of intermediate N2O in NOx production is approx-
imately 30 percent on average and it could be as much as 90 percent in some certain
combustion mechanisms such as flame less combustion mode [2]. Intermediate N2O is
produced at high pressures and in presence of high amount of oxygen via nitrogen, which
is present during the combustion process
N2 + O+M N2O+M (C.32)
N2O+O 2NO (C.33)
The N2O model can be approached as simple transported model or as Quasi steady model
[18]. The Quasi steady method implies the rate of N2O concentration change to be zero.
C.2.5 NOx Reduction
NOx reduction is also possible by either re burning or selective non-catalytic reduction
(SNCR) [107]. The reduction is probable to occur in the re burn zone of a combustion
system in an either instantaneous or partial equilibrium mechanism. The reaction of NO
with hydrocarbons results in instantaneous reduction of NOx. ANSYS Fluent models the
instantaneous re-burn mechanism using three reactions for temperatures between 1600
to 2100 Kelvin [2]
CH+NO→ O+HCN (C.34)
CH2 + NO→ OH+HCN (C.35)
CH3 + NO→ H2O+HCN (C.36)
The partial equilibrium method was first modeled by Kandamby [108]. The partial
equilibrium approach considers the NOx destruction by CH radicals present in fuel rich
re burn zones. The equivalent fuel types used in the partial equilibrium approach can be
either CH4, CH3, CH2 or CH. The NOx reduction due to reacting with CH radicals may
occur according to the following three reactions:
C+NO→ O+CN (C.37)
Appendix C. Emissions and Soot Formation Models in ANSYS Fluent 124
CH+NO→ O+HCN (C.38)
CH2 + NO→ OH+HCN (C.39)
It should be noted that the destruction of NOx can also be accomplished due to SNCR
mechanism first used by Lyon [109]. The SNCR approach uses an ammonia or urea
injection into the combustion furnace which results in reaction of NO with the reductant.
In addition, the SNCR method may also leads to formation of NOx due to oxidation of
the reductants in some certain temperatures. ANSYS Fluent uses the Ostberg and Dam
Johansen [110] approach for modeling reduction caused by ammonia injection. Simplified
reactions for the selective non-catalytic reduction approach using ammonia injection are
NH3 + NO + 0.25O2 → N2 + 1.5H2O (C.40)
NH3 + 1.25O2 → NO + 1.5H2O (C.41)
Furthermore, the NOx reduction using urea injection is based on the seven step reaction
mechanisms first proposed by Brouwer [111]. The urea injection approach requires AN-
SYS Fluent to solve three additional transport equations for the urea, HNCO and NCO
species [2].
C.3 SOx Formation
The oxidation of fuel sulfur during the combustion process contributes to the formation
of SO2 and SO3. SOx emissions are the main reasons for acid rain resulted from the con-
densation of SOx and formation of sulfuric acid. Furthermore, SO3 may cause corrosion
of combustion equipment and should be considered as a harmful pollutant. It should be
noted that over 50 percent of SO2 emissions is only caused by coal fired boilers [2]. The
formation of SOx commences with sulfur release from the fuel mainly in the form of H2S.
As a result, the released sulfur reacts in the gas phase. Finally, the sorbent particles may
absorb the sulfur emissions. The fuel used in predicting the SOx production may be in a
form of either solid, gas or liquid. ANSYS Fluent predicts the SOx formation by solving
transport equations for SO2 and H2S
∂
∂t(ρYSO2
+ ~∇.(ρ~vYSO2) = ~∇.(ρD~∇YSO2
) + SSO2(C.42)
∂
∂t(ρYH2S + ~∇.(ρ~vYH2S) = ~∇.(ρD~∇YH2S) + SH2S (C.43)
Appendix C. Emissions and Soot Formation Models in ANSYS Fluent 125
Furthermore, the presence of intermediate SO3, SH and SO species may also be taken
into account:∂
∂t(ρYSO3
+ ~∇.(ρ~vYSO3) = ~∇.(ρD~∇YSO3
) + SSO3(C.44)
∂
∂t(ρYSH + ~∇.(ρ~vYSH) = ~∇.(ρD~∇YSH) + SSH (C.45)
∂
∂t(ρYSO + ~∇.(ρ~vYSO) = ~∇.(ρD~∇YSO) + SSO (C.46)
A reduced reaction mechanism first proposed by Kramlich [112] using eight step reactions
with ten species is used to predict the production of SOx species. This method involves
the presence of O and H radicals which can be modeled using equilibrium or partial
equilibrium approaches previously described in the thermal NOx formation process [2].
C.4 Soot Formation
ANSYS Fluent offers a detailed modeling for soot formation during a combustion pro-
cess. The soot formation modeling in ANSYS Fluent is compatible with pressure based
solver and all combustion models except the premixed combustion model. Four different
approaches are used to predict the soot formation in ANSYS Fluent. ANSYS Fluent
approaches the first two methods called the One step and Two step methods according
to the Magnussen combustion rate model [5]. These two methods are only compatible
with turbulent flows. The Moss Brookes [6] and Moss Brookes Hall [113] are other soot
models in ANSYS Fluent which can be used in conjunction with both turbulent and
laminar flows, and are more accurate but computationally more expensive than the first
two methods [2].
C.4.1 One Step Model
The first method called the one step approach was first proposed by Khan and Greeves
[114]. The following transport equation is solved in order to predict the amount of soot
formation:∂
∂t(ρYsoot) + ~∇.(ρ~vYsoot) = ~∇.( µt
σsoot~∇Ysoot) + <soot (C.47)
where < is the net rate of soot generation and σ is the turbulent Prandtl number [2].
Appendix C. Emissions and Soot Formation Models in ANSYS Fluent 126
C.4.2 Two Step Model
The two step method is based on Tesner model [115] which solves transport equations for
both the radical nuclei and soot concentrations. The transport equation used to predict
the soot formation using a two step model is
∂
∂t(ρb∗nuc) + ~∇.(ρ~vb∗nuc) = ~∇.( µt
σnuc~∇b∗nuc) + <∗nuc (C.48)
where bnuc is the radical nuclei concentration. It should be noted that the * superscrip-
tion shows that the variables are normalized [2].
C.4.3 Moss Brookes Model
The Moss Brookes model which is considered to be more accurate than the first two
methods, is based on two transport equations for soot mass fraction and radical nuclei
concentration
∂
∂t(ρYsoot) + ~∇.(ρ~vYsoot) = ~∇.( µt
σsoot~∇Ysoot) +
dM
dt(C.49)
∂
∂t(ρb∗nuc) + ~∇.(ρ~vb∗nuc) = ~∇.( µt
σnuc~∇b∗nuc) +
dN
Ndt(C.50)
where N is the soot particle density and Nnorm defines 1015 particles. Furthermore,
the soot oxidation rate caused by the OH radicals can be modeled according to either
Fenimore and Jones [116] or Lee [117] approaches [2].
C.4.4 Moss Brookes Hall Model
The Moss Brookes Hall approach is an extended Moss Brookes model, which can be used
in conjunction with two ringed or three ringed aromatics based on the following reactions
[2]:
2C2H2 + C6H5 C10H7 + H2 (C.51)
C2H2 + C6H5 + C6H6 C14H10 + H2 + H (C.52)
It should be noted that the Moss Brookes Hall model is only available when C2H2, C6H6,
C6H5 and H2 gases are present in our species model [2].
Appendix C. Emissions and Soot Formation Models in ANSYS Fluent 127
C.5 Decoupled Detailed Chemistry Model
ANSYS Fluent offers the decoupled detailed chemistry approach as an addition to the
current pollutant models. Other pollutant models in ANSYS Fluent consist of fixed
chemical mechanisms and reactions. On the other hand, pollutant modeling and solving
the pollutant transport equations can be done by using any other desired chemical kinetic
equation with the decoupled detailed chemistry model approach. It should be noted that
the decoupled detailed chemistry approach is only compatible with the steady state solver
[2].