Evaluation of Liquid Fuel Spray Models for Hybrid RANS/LES ... · PDF fileRANS/LES and DLES...

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

iv

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


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


7.3.1 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





5.25 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





5.31 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

xiii

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




6.23 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


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


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.


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


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,


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


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].


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


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].


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].


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


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


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].


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


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].


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)


∂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]:


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


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)


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.


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].


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].


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


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)


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)


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)


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


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


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.


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


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.


Figure 5.6: Mean axial gas velocity for

case 1 using RANS (m/s).


case 1 using DLES (m/s).


case 1 using LES (m/s).


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.


Figure 5.10: Topview of the droplets for

case 1 using wave model.

Figure 5.11: Spatial variation of the

droplets using TAB model.


droplets using wave model.


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


Figure 5.14: Spanwise distribution of the

droplets for RANS.


droplets for DES.


droplets for LES.


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


Figure 5.18: Droplet diameter found by Sen et al. [12].

Figure 5.19: Droplet diameter for case 1

using RANS turbulence model.


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










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


Figure 5.25: Droplet axial velocity for case

1 using RANS turbulence model.


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










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)



using wave and RANS models.


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



using wave and RANS models (delayed

breakup).


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



using wave and RANS models.


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


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.



using RANS.


using DES.


using LES.


using DLES.



3 using RANS.


3 using DES.


3 using LES.


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


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].


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


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.


Figure 6.11: Axial velocity for the coarse

mesh (film Thickness=2.5× 10−4m).


mesh (film thickness=1.5× 10−4m).






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.


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


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.


Figure 6.19: Axial velocity (spray an-

gle=45, 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


Figure 6.23: Experimental droplet radial

velocity [13].

Figure 6.24: Experimental droplet tangen-

tial velocity [13].

Figure 6.25: Radial velocity (spray an-


Figure 6.26: Tangential velocity (spray an-


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


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.


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.


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].


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


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.


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.


7.3.1 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


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].


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


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.


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.


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.


Figure 7.15: Droplet breakup using RANS and wave models

Figure 7.16: Spanwise distribution of the droplets using RANS and wave models.


Figure 7.17: Droplet breakup using RANS and KHRT models.

Figure 7.18: Spanwise distribution of the droplets using RANS and KHRT models


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.

References

[1] Lefebvre. Atomization and Sprays. Hemisphere publishing corporation, New York,

USA, 1989.

[2] Anonymous. ANSYS Fluent Theory Guide Release 13.0. ANSYS Inc., Canonsburg,

Pennsylvani, 2010.

[3] P. Walzei. Spraying and Atomizing of Liquids. Essen, Germany, 2005.

[4] C. Bekdemir. Numerical modeling of diesel spray formation and combustion. Mas-

ter’s thesis, Eindhoven University of Technology, Eindhoven, 2008.

[5] B. Magnussen and Hjertager.B. On mathematical models of turbulent combustion

with special emphasis on soot formation and combustion. Symposium International

on Combustion, 16(1):719–729, 1977.

[6] S. Brookes and J. Moss. Prediction of soot and thermal radiation in confined

turbulent jet diffusion flames. Combustion and Flames, 116(4):486–503, 1999.

[7] M. Carvalho, T. Farias, and P. Fontes. Predicting radiative heat transfer in ab-

sorbing, emitting and scattering media using the discrete transfer method. Funda-

mentals of Radiation Heat Transfer, 160(1):17–26, 1991.

[8] E. Chui and G. Raithby. Computation of radiant heat transfer on a non orthogonal

mesh using the finite volume method. Numerical Heat Transfer, 23:269–288, 1993.

[9] H. Barths, N. Peters, A. Brehm, M. Mack, Pfitzner, and V Smiljanowski. Simu-

lation of pollutant formation in a gas turbine combustor using unsteady flamelets.

27th Symp on Combustion, the Combustion Institute, 27(2):1841–1847, 1998.

[10] G. Faeth, L. Hsiang, and P. Wu. Structure and breakup properties of sprays.

International Journal of Multiphase Flow, 21:99–127, 1995.

83

REFERENCES 84

[11] M. Battistoni and C. Grimaldi. Numerical analysis of injector flow and spray

characteristics from diesel injectors using fossil and biodiesel fuels. Applied Energy,

97:656–666, 2012.

[12] B. Sen, Y. Guo, R. McKinney, F. Montanari, and F. Bedford. Pratt and whitney

gas turbine combustor design using ansys fluent and user defined functions. In

Proceedings of ASME Turbo Expo 2012, Copenhagen, Denmark, 2012.

[13] R. Gurubaran, R. Sujith, and S. Chakravarthy. Characterization of a pre-filming

airblast atomizer in a strong swirl flow-field. Journal of Propulsion and Power,

24(5):1124–1132, 2008.

[14] X Li, M Soteriou, and J. Kim, W. Cohen. High fidelity simulation of the spray

generated by a realistic swirling flow injector. In Proceedings of ASME Turbo Expo

2013, Texas, USA, 2013.

[15] R. Madabhushi, M. Leong, and D. Hautman. Simulation of the breakup of a liquid

jet in cross flow at atmospheric conditions. In Proceedings of ASME Turbo Expo

2004, Vienna, Austria, 2004.

[16] J. Smagorinsky. General circulation experiments with the primitive equations, 1,

the basic experiment. 91(3):99–164, 1963.

[17] B Launder and D. Spalding. Lectures in mathematical models of turbulence. Aca-

demic Press, London, England, 1972.

[18] Anonymous. ANSYS Fluent User’s Guide Release 13.0. ANSYS Inc., Canonsburg,

Pennsylvani, 2010.

[19] M. Germano, u. Piomelli, P. Moin, and W. Cabot. A dynamic subgrid scale eddy

viscosity model. Physics of Fluid, pages 1760–1765, 1991.

[20] A. Chorin. Numerical solution of navier-stokes equations. Mathematics of Compu-

tation, pages 745–762, 1968.

[21] T. Barth and D. Jespersen. The design and application of upwind schemes on

unstructured meshes. Technical report, AIAA 27th Aerospace Sciences Meeting,

Reno, Nevada, 1989.

REFERENCES 85

[22] A. Kadosca. Modeling of spray formation in diesel engines. PhD thesis, Budapest

University of Technology and Economics, Budapest, 2007.

[23] B. Schneider. Experimentelle Untersuchung zur spraystruktur in transparenten,

verdampfenden und nicht verdampfenden brennstoffstrahlen unter hochdruck. PhD

thesis, Eidgenossische Technische hochschule Zurich, Zurich, 2003.

[24] R. Reitz. Atomization and other breakup regimes of a liquid jet. PhD thesis, Me-

chanical and Aerospace department of Princeton University, Princeton, NJ, USA,

1978.

[25] G. Bower, S. Chang, M. Corradini, M. El-beshbeeshy, J. Martin, and J. Krueger.

Physical mechanisms for atomization of a jet spray a comparison of models and

experiments. Technical report, SAE technical paper series, 1988.

[26] M. Pilch and C. Erdmann. Use of breakup time data and velocity history data to

predict the maximum size of stable fragments. International Journal of Multiphase

Flow, 13:741–757, 1987.

[27] R. Rotondi, G. Bella, C. Grimaldi, and L. Postrioti. Atomization of high pressure

diesel spray experimental validation of a new breakup model. Technical report,

SAE technical paper series, 2001.

[28] R. Reitz. Modeling atomization processes in high-pressure vaporizing sprays. At-

omization and Spray Technology, 3:309–337, 1987.

[29] E. Tan, M. Anwar, R. Adnan, and M. Idris. Biodiesel for gas turbine application,

an atomization characteristics study. 2013.

[30] V. Yakhot and S. Orszag. Renormalization group analysis of turbulence, basic

theory. Journal of Scientific Computing, 1:1–51, 1986.

[31] T. Shih, W. Liou, A. Shabbir, Z. Yang, and J. Zhu. A new k-e eddy viscosity

model for high reynolds number turbulent flows, model development and validation.

Computers Fluid, 24:227–238, 1995.

[32] M. Shur, P. Spalart, M. Strelets, and A. Travin. Detached eddy simulation of

an airfoil at high angle of attack. In International Symposium on Engineering

Turbulence Modeling and Experiments, Corsica, France, 1999.

REFERENCES 86

[33] F. Menter, M. Kuntz, and R. Langtry. Ten years of experience with the sst turbu-

lence model. Turbulence, Heat and Mass Transfer, pages 625–632, 2003.

[34] P. O’Rourke. Collective drop effects on vaporizing liquid sprays. PhD thesis, Prince-

ton University, Princeton, NJ, USA, 1981.

[35] G. Taylor. The shape and acceleration of a drop in a high speed air stream.

Scientific papers of Taylor, G. ed., Batchelor, G., 1963.

[36] P. O’Rourke and A. Amsden. The tab method for numerical calculation of spray

droplet breakup. Technical report, SAE technical paper series, 1987.

[37] A. Liu, D. Mather, and R. Reitz. Modeling the effects of drop drag and breakup

on fuel sprays. Technical report, SAE technical paper series, 1993.

[38] R. Reitz. Mechanisms of atomization processes in high pressure vaporizing sprays.

Atomization and Spray Technology, 3:309–337, 1987.

[39] R. Reitz and F. Bracco. Mechanisms of breakup of round liquid jets. The Ency-

clopedia of Fluid Mechanics, 3:223–249, 1986.

[40] M. Patterson and R. Reitz. Modeling spray atomization with the kelvin-

helmholtz/rayleigh-taylor hybrid model. Atomization and Sprays, 9:623–650, 1999.

[41] V. Levich. Physicochemical Hydrodynamics. Prentice Hall, 1962.

[42] S. Apte, M. Gorokhovski, and P. Moin. Les of atomizing spray with stochastic

modeling of secondary breakup. International Journal of Multiphase Flow, 29:1503–

1522, 2003.

[43] C. Soteriou and M. Andrews, R. Smith. Direct injection diesel sprays and the effect

of cavitation and hydraulic flip on atomization. Technical report, SAE technical

paper series, 1995.

[44] W. Nurick. Orifice cavitation and its effects on spray mixing. Journal of Fluids

Engineering, 98(9):681–687, 1976.

[45] D. Schmidt and M. Corradini. Analytical prediction of the exit flow of cavitating

orifices. Atomization and Sprays, 7(6):603–616, 1997.

REFERENCES 87

[46] W. Ranz. Some experiments on orifice sprays. Canadian Journal of Chemical

Engineering, pages 175–181, 1958.

[47] P. Wu, L. Tseng, and G. Faeth. Primary breakup in gas liquid mixing layers for

turbulent liquids. Atomization and Sprays, 2:295–317, 1995.

[48] D. Schmidt, I. Nouar, P. Senecal, C. Rutland, J. Martin, and R. Reitz. Pressure

swirl atomization in the near field. Technical report, SAE technical paper series,

1999.

[49] Z. Han, S. Perrish, P. Farrell, and R. Reitz. Modeling atomization processes of

pressure swirl hollow cone fuel sprays. Atomization and Sprays, 7(6):663–684, 1997.

[50] C. Weber. Zum zerfall eines flussigkeitsstrahles. ZAMM, 11:136–154, 1931.

[51] R. Reitz and F. Bracco. Mechanism of atomization of a liquid jet. Physics of

Fluids, 25(10):1730–1742, 1982.

[52] P. Wu, K. Kirkendall, R. Fuller, and A. Nejad. Breakup processes of liquid jets in

subsonic crossflows. Journal of propulsion and Power, 13:64–73, 1997.

[53] J. Mazallon, Z. Dai, and G. Faeth. Primary breakup of nonturbulent round liquid

jets in gas crossflows. Atomization and Sprays, 9:291–311, 1999.

[54] K. Sallam, C. Aaalburg, and G. Faeth. Breakup of round non-turbulent liquid jets

in gaseous crossflows. AIAA paper, 42:2529–2540, 2004.

[55] M. Leong and D. Hautman. Near-field spray characterization of a liquid fuel jet

injected into a crossflow. In Proceedings of ILASS-Americas 2002, pages 350–354,

Madison, Wisconsin, USA, 2002.

[56] R. Madabhushi. A model for numerical simulation of breakup of a liquid jet in

crossflow. Atomization and Sprays, 13:413–424, 2003.

[57] C. Engelbert, Y. Hardalupas, and J. Whitelaw. Breakup phenomena in coaxial

airblast atomizers. In Proceedings of the royal society a mathematical physical and

engineering sciences, pages 189–229, 1995.

REFERENCES 88

[58] H. Liu, X. Gong, W. Li, F. Wang, and Z. Yu. Prediction of droplet size distribution

in sprays of prefilming airblast atomizers. Chemical Engineering Science, pages

1741–1747, 2006.

[59] C. Dumouchel. On the experimental investigation on primary atomization of liquid

streams. Exp Fluids, pages 371–422, 2008.

[60] P. Watanawanyoo, H. Mochida, H. Hirahara, and S. Chaitep. Flow characteris-

tics of fluid droplet obtained from air assisted atomizer. Joint Fluids Engineering

Conference, pages 89–94, 2011.

[61] T. Shedd, M. Corn, J. Cohen, M. Arienti, and M. Soteriou. Liquid film formation

by an impinging jet in a high velocity air stream. In AIAA Aerospace Sciences

Meeting, Orlando, Fl, USA, 2010.

[62] M. Arienti, L. Wang, M. Corn, X. Li, M. Soteriou, T. Shedd, and M. Herrmann.

Modeling wall film formation and breakup using an integrated interface track-

ing/discrete phase approach. Journal of Engineering for Gas Turbines and Power,

133, 2011.

[63] M. Arienti and M Soteriou. Time-resolved proper orthogonal decemposition of

liquid jet dynamics. Physics of Fluids, 21, 2009.

[64] J. Becker and C. Hassa. Breakup and atomization of a kerosene jet in crossflow at

elevated pressure. Atomization and Sprays, pages 49–67, 2002.

[65] X. Li and M Soteriou. Prediction of high density ratio liquid jet atomization

in crossflow using high fidelity simulations on hpc. In AIAA Aerospace Sciences

Meeting, Nashville, TN, 2012.

[66] X. Li, M. Arienti, M. Soteriou, and M. Sussman. Towards an efficient high fidelity

methodology for liquid jet atomization computations. In AIAA Aerospace Sciences

Meeting, Orlando, Fl, USA, 2010.

[67] M. Sussman, K. Smith, M. Hussaini, M. Ohta, and R. Zhi-Wei. A sharp interface

method for incompressible two phase flows. Journal of Computational Physics,

pages 469–505, 2007.

REFERENCES 89

[68] M. Sussman and D. Dommermuth. The numerical simulation of ship waves using

cartesian grid methods. In Proceedings of the Twenty Fifth Symposium on Naval

Hydro, St. John, New Foundland and Labrador, Canada, 2004.

[69] X. Liu, R. Fedkiw, and M. Kang. A boundary condition capturing method for

poisson’s equation on irregular domains. Journal of Computational Physics, pages

157–178, 2000.

[70] D. Spalding. Mixing and chemical reaction in steady confined turbulent flames.

13th Symp on Combustion, the Combustion Institute, 13:649–657, 1971.

[71] B. Magnussen. On the structure of turbulence and a generalized eddy dissipation

concept for chemical reaction in turbulent flow. In Nineteenth AIAA Meeting, St.

Louis, 1981.

[72] I. Gran and B. Magnussen. A numerical study of a bluff-body stabilized diffusion

flame part 2 influence of combustion modeling and finite rate chemistry. Combustion

Sience and Technology, 119:191–217, 1996.

[73] P. O’Rourke and F. Bracco. Two scaling transformations for the numerical com-

putation of multidimensional unsteady laminar flames. Journal of Computational

Physics, 32:185–203, 1979.

[74] I. Smith. The combustion rates of coal chars: a review. Symposium International

on Combustion, 19:1045–1065, 1982.

[75] Y. Sivathanu and G. Faeth. Generalized state relationships for scalar properties in

non premixed hydrocarbon/air flames. Combustion and Flame, 82:211–230, 1990.

[76] W. Jones and J. Whitelaw. Calculation methods for reacting turbulent flows:a

review. Combustion and Flame, 48:1–26, 1982.

[77] K. Bray and N. Peters. Laminar flamelets in turbulent flames. P.A Libby and F.A

Williams, editors of Turbulent Reacting Flows, pages 63–114, 1994.

[78] N. Peters. Laminar diffusion flamelet models in non premixed combustion. Energy

Combustion Science, 10:319–339, 1984.

REFERENCES 90

[79] H. Pitsch, H. Barths, and N. Peters. Three dimensional modeling of nox and soot

formation in di-diesel engines using detailed chemistry based on the interactive

flamelet approach. Technical report, SAE technical paper series, 1996.

[80] H. Barths, C. Antoni, and N. Peters. Three dimensional simulation of pollutant

formation in a di-diesel engine using multiple interactive flamelets. Technical report,

SAE technical paper series, 1998.

[81] V. Zimont, W. Polifke, M. Bettelini, and W. Weisenstein. An efficient computa-

tional model for premixed turbulent combustion at high reynolds number based

on a turbulent flame speed closure. Journal of Gas Turbines Power, 120:526–532,

1998.

[82] N. Peters. Turbulent Combustion. Cambridge University press, Cambridge, Eng-

land, 2000.

[83] T. Poinsot and D. Veynante. Theoretical and Numerical Combustion. McGraw

Hill, New york, USA, 2001.

[84] D. Veynante and L. Vervisch. Turbulent combustion modeling. Progress in Energy

and Combustion Science, 28:193–266, 2002.

[85] O. Colin, C. Angelberger, and A. Benkenida. 3d modeling of mixing, ignition and

combustion phenomena in highly startified gasoline engines. Oil and Gas Science

and Technology, 58:47–62, 2003.

[86] A. Teraji, T. Tsuda, T. Noda, M. Kubo, and T. Itoh. Development of a novel

flame propagation model for si engines and its application to knocking prediction.

Technical report, SAE 2005 world congress and exhibition, 2005.

[87] J. Ewald. A level set based flamelet model for the prediction of combustion in

homogeneous charge and direct injection spark ignition engines. PhD thesis, Aachen

University, Aachen, 2006.

[88] S. Pope. Pdf methods for turbulent reactive flows. Progress Energy Combustion

Science, 11:119–192, 1985.

REFERENCES 91

[89] R. Cao and S. Pope. Numerical integration of stochastic differential equations,

weak second order mid point scheme for application in the composition pdf method.

Journal of Computational Physics, 185:194–212, 2003.

[90] S. Pope. Computationally efficient implementation of combustion chemistry using

in situ adaptive tabulation. Combustion Theory and Modelling, 1:44–63, 1997.

[91] J. Janicka, W. Kolbe, and W Kollmann. Closure of the transport equation for

the pdf of turbulent scalar fields. Journal of Non-equilibrium Thermodynamics, 4,

1979.

[92] C. Dopazo and E. O’Brien. Functional formulation of nonisothermal turbulent

reactive flows. Physics Fluids, pages 1968–1975, 1974.

[93] S. Subramaniamand and S. Pope. A mixing model for turbulent reactive flows

based on euclidean minimum spanning trees. Combustion and Flame, 115:487–514,

1998.

[94] R. Fox. Computational Models for Turbulent Reacting Flows. Cambridge University

press, Cambridge, England, 2003.

[95] S. Pope. Turbulent Flows. Cambridge University press, Cambridge, England, 2000.

[96] R. Siegel and C. Spuckler. Thermal Radiation Heat Transfer. Hemisphere publish-

ing corporation, Washington DC, USA, 1994.

[97] P. Cheng. Two dimensional radiating gas flow by a moment method. AIAA Journal,

pages 1662–1664, 1964.

[98] N. Shah. A new method of computation of radiant heat transfer in combustion

chambers. PhD thesis, Imperial College of Science and Technology, London, Eng-

land, 1979.

[99] M. Modest. Radiative Heat Transfer Series in Mechanical Engineering. Mcgraw-

Hill, 1993.

[100] S. Mathur and J Murthy. Coupled ordinates method for multigrid acceleration of

radiation calculations. Journal of Thermophysics and Heat Transfer, 13:467–473,

1999.

REFERENCES 92

[101] G. Raithby and E. Chui. A finite volume method for predicting a radiant heat

transfer in enclosures with participating media. Journal of Heat Transfer, 112:415–

423, 1990.

[102] HJ. Cao, HQ. Zhang, and WY. Lin. Evaluation of presumed probability density

function models in non premixed flames by using large eddy simulation. Chinese

Physics Letters, 29, 2012.

[103] V. Ramanujachari, S. Balakrishna, and S. Ganesan. Probability density function

approach to non premixed flames. Combustion Science and Technology, 180:2046–

2091, 2000.

[104] TJ. Butler, FM. Vermeylen, M. Rury, GE. Likens, B. Lee, GE. Nowker, and L. Mc-

Cluney. Response of ozone and nitrate to stationary source nox emission reductions

in the eastern usa. Atmospheric Environment, 45:1084–1094, 2011.

[105] G. Desoete. Overall reaction rates of no and n2 formation from fuel nitrogen.

Symposium (International) on Combustion, 15(1):1093–1102, 1975.

[106] P. Melte and D. Pratt. Measurement of atomic oxygen and nitrogen oxides in jet

stirred combustion. Symposium (International) on Combustion, 15(1):1061–1070,

1974.

[107] LJ. Muzio, GC. Quartucy, and JE. Cichanowicz. Overview and status of post

combustion nox control: Sncr, scr and hybrid technologies. International Journal

of Environment and Pollution, 17:4–30, 2002.

[108] N. Kandamby, F. Lazopoulos, F. Lockwood, A. Perera, and L. Vigevano. Math-

ematical modeling of nox emission reduction by the use of reburn technology in

utility boilers. In ASME international Joint Power Generation Conference and

Exhibition, Houston, Texas, USA, 1996.

[109] R. Lyon. The nh3-no-o-o2 reaction. International Journal of Chemical Kinetics,

8:315–318, 1976.

[110] M. Ostberg and K. Dam-Johansen. Emperical modeling of the selective non-

catalytic reduction of no comparison with large scale experiments and detailed

kinetic modeling. Chemical Engineering Science, 49:1897–1904, 1994.

REFERENCES 93

[111] J. Brouwer, M Heap, D. Pershing, and P. Smith. A model for prediction of selective

non catalytic reduction of nitrogen oxides by ammonia, urea and cyanuric acid with

mixing limitations in the presence of co. Symposium International on Combustion,

26:2117–2124, 1996.

[112] J. Kramlich. The fate and behavior of fuel sulfur in combustion systems. PhD

thesis, Washington State University, Washington, USA, 1980.

[113] R. Hall, M. Smooke, and M. Colket. Physical and Chemical Aspects of Combustion.

Gordon and Breach, 1997.

[114] I. Khan and G. Greeves. A method for calculating the formation and combustion

of soot in diesel engines. Heat Transfer in Flames, 1974.

[115] P. Tesner, T. Snegiriova, and V. Knorre. Kinetics of dispersed carbon formation.

Combustion and Flame, 17:253–260, 1971.

[116] C. Fenimore and G. Jones. Oxidation of soot by hydroxyl radicals. Journal for

Physical and Chemistry, 71:593–597, 1967.

[117] K. Lee, M. Thring, and J. Beer. Combustion and flame. Combustion Science and

Technology, 180:137–145, 1962.

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)


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.


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)


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)


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


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


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)


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


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


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)


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)


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.


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)


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)


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


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.


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.


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


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.


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


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].


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.


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

∂


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(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)


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)


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)


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].


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].


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].

Evaluation of Liquid Fuel Spray Models for Hybrid RANS/LES ... · PDF fileRANS/LES and DLES...

Documents

Transcript of Evaluation of Liquid Fuel Spray Models for Hybrid RANS/LES ... · PDF fileRANS/LES and DLES...