Computational Optimization of InternalCombustion Engines

Yu Shi • Hai-Wen Ge • Rolf D. Reitz

Computational Optimizationof Internal CombustionEngines

123

Dr. Yu ShiDepartment of Chemical EngineeringMassachusetts Institute of TechnologyBldg. 66-26477 Massachusetts AvenueCambridge, MA 02139USAe-mail: yushi@mit.edu

Dr. Hai-Wen GeEngine Research CenterUniversity of Wisconsin-Madison1500 Engineering Dr.Madison, WI 53706USAe-mail: heavengerc@gmail.com

Prof. Rolf D. ReitzEngine Research CenterUniversity of Wisconsin-Madison1500 Engineering Dr.Madison, WI 53706USAe-mail: reitz@engr.wisc.edu

ISBN 978-0-85729-618-4 e-ISBN 978-0-85729-619-1

DOI 10.1007/978-0-85729-619-1

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Preface

Striking progress has been made in internal combustion engine design due to thedevelopment of computer models and optimization techniques. In this book westrive to document the state of the art in predictive IC engine modeling andoptimization. The fact that this is an important topic for research and developmentis emphasized by society’s reliance on IC engines for transportation, commerceand power generation. Indeed, the world as we know it would be a quite differentplace were it not for the remarkable internal combustion engine! It drives allmanner of utility devices (e.g., pumps, mowers, chain-saws, portable generators,etc.), as well as earth-moving equipment, tractors, propeller aircraft, ocean linersand ships, personal watercraft and motorcycles. However, its major application ispowering the 600 million passenger cars and other vehicles on our roads today.250 million vehicles (cars, buses, and trucks) were registered in 2008 in the UnitedStates alone. According to the International Organization of Motor VehicleManufacturers, about 50 million cars were made world-wide in 2009, compared to40 million in 2000. Much of this dramatic increase comes from increased pros-perity in China, which became the world’s second-largest car market in 2010. Athird of all cars are produced in the European Union, and about 50% of those arepowered by diesel engines. Thus, IC engine research spans both gasoline anddiesel powerplants.

The world’s economic expansion has been powered by cheap oil. It has beenargued that the increase in population from 1.9 billion in the 1920s to today’s 6.6billion has been made possible, in part, by fossil fuel combustion and by the Ha-ber–Bosch process to make crop fertilizer. 80% of the roughly 80 billion barrels ofcrude oil consumed annually world-wide is used in IC engines for transportation.In the United States, 10 million barrels of oil are used per day in automobiles andlight-duty trucks, and 4 million barrels per day are used in diesel engines, withtotal oil usage of about 2.5 gallons per day per person. Of this, 62% is imported oil,which at today’s $80/barrel, costs the US economy $1 billion/day. This cost iscertain to increase as more-and-more economic development drives increased de-mand for automotive fuels world-wide.

v

Associated with our massive oil use is the accompanying annual emission of 37billion tons of CO2 (6 tons each for each person in the world) and other pollutantemissions, including nitric oxides (NOx) and particulates (soot). Pollutant emis-sions have serious environmental and health implications, and thus most govern-ments have imposed stringent vehicle emissions regulations that are continuallybeing tightened further. In addition, CO2 emissions contribute to Green HouseGases (GHG), which some fear could lead to climate change with unpredictableconsequences. Drastic reductions in fuel usage will be required to make appre-ciable changes in GHG trends.

Today’s gasoline IC engine powered vehicle equipped with its 3-way catalystfor emission control converts only about 16% of the chemical energy in the fuel touseful work—the rest is lost to the environment. The modern automotive dieselengine is 20 to 40% more efficient than its gasoline counterpart. However, mea-sures introduced to meet emissions mandates, such as the use of non-optimal fuelinjection timings, large amounts of Exhaust Gas Recirculation (EGR) or ultra-highinjection pressures reduce diesel engine fuel efficiencies, and also increase engineexpense. Many diesel engine manufacturers have elected to use Selective CatalyticReduction (SCR) exhaust after-treatment for NOx reduction. However, with SCRthere is also a fuel penalty since a reducing agent such as urea (carbamide) must besprayed into the exhaust stream at rates (and cost) of about 1% of the fuel flow ratefor every 1 g/kWh of NOx reduction desired. Soot control is achieved using DieselParticulate Filters (DPF), which generally require periodic regeneration. This isachieved by adjusting the fuel-air mixture strength so as to increase exhausttemperatures to burn off the accumulated soot, which imposes as much as a 3%additional fuel penalty.

From these discussions it is clear that new technologies are urgently needed toimprove the efficiency of both gasoline and diesel engines. For further improve-ments, engines need to be optimized to balance emissions, fuel cost, and marketcompetitiveness. As described in this book, this task can be efficiently attackedusing state-of-the-art computational models and optimization methods. This hasbeen made possible, in part, by dramatic increases in computer speeds that haveincreased 10,000-fold in the past 15 years. Engine development is now greatlyfacilitated using multi-dimensional Computational Fluid Dynamic (CFD) tools andoptimization algorithms, supported by significantly reduced requirements for ex-perimental testing, which is extremely expensive.

An additional enabling factor for engine CFD modeling has been the devel-opment of predictive models for the physical processes occurring in the com-bustion chamber. Many of these models are reviewed in this book, together withdiscussion of strategies to reduce computational cost and numerical inaccuracies.Example applications are presented for the optimization of 2-stroke spark-ignitiongasoline and 4-stroke heavy- and light-duty diesel engines. The effects of designparameters including nozzle design, injection timing and pressure, swirl, EGR,engine size scaling, and piston bowl shape are considered, together with explo-ration of fuel effects for low temperature combustion strategies. It is also dem-onstrated how optimization results can be used in combination with regression

vi Preface

analysis to explore and explain the complex interactions between engine designparameters.

The present example applications also demonstrate that current multi-dimen-sional CFD tools are mature enough to guide the development of more efficientand cleaner internal combustion engines. New low temperature combustion con-cepts, such as Homogeneous Charge Compression Ignition (HCCI), PremixedCharge Compression Ignition (PCCI) and Reactivity Controlled CompressionIgnition (RCCI) offer the promise of dramatically improved engine efficiencies.For example, optimized dual fuel RCCI operation (port injection of gasolinetogether with optimized in-cylinder multiple diesel fuel injections) was discoveredwith computer simulations using the models and tools described in this book(Kokjohn et al. 2009). The computer simulations predicted high-efficiency, low-emissions operation with excellent combustion phasing control at high and lowengine loads without excessive rates of pressure rise. Subsequent engine experi-ments have confirmed the model predictions, and have demonstrated that US EPA2010 NOx and soot emissions mandates can be met in-cylinder without after-treatment, while achieving up to 57% gross indicated thermal efficiency (Kokjohnet al. 2011).

The adoption of RCCI combustion engines could improve fuel efficiencies byup to 20% over standard diesel operation, while also providing dramatic costreductions through the elimination of the need for exhaust after-treatment. RCCI isapplicable with a wide range of fuels, including conventional gasoline and diesel,as well as biofuels such as ethanol and biodiesel and their blends. The implicationsof such improvements in fuel efficiency are very significant. For example, if RCCIwere adopted to replace the relatively inefficient spark-ignition engine it is esti-mated that US transportation oil usage could be reduced by 34%, which equals100% of the current US oil imports from Persian Gulf. If these efficiencyimprovements were combined with electric hybrid technologies in the vehicle,even greater reductions in oil usage would be possible.

The ultimate goal of engine modeling is to guide designers to improve engineperformance and to reduce pollutant emissions. The goal of this book is to providean up-to-date reference to current developments and future directions in the fieldof engine modeling. We hope that you will think that we have achieved this goal.

Preface vii

Acknowledgments

This book expands on recent computational optimization studies of internal com-bustion engines performed at the Engine Research Center of the University ofWisconsin-Madison. The present work would not have been possible without thesolid research foundation that our ERC colleagues have built over the past dec-ades.We would like to express our sincere gratitude to them. During the preparation ofthis book, we also received valuable suggestions from our colleagues, Dr. ShiyouYang, Dr. Yuxin Zhang and Mr. Yue Wang, to whom we are indebted.

The work included in this book was supported financially by several govern-ment and industry research projects. We are grateful to the US Department ofEnergy, Caterpillar Inc., Ford Motor Company, General Motors, and DetroitDiesel Company for their long term support. We also thank Dr. David Wickman ofWisconsin Engine Research Consultants for allowing use of the Kwickgrid soft-ware. ESTECO provided access to optimization software (modeFRONTIER),which facilitated some of the assessment studies in this book.

We thank the Society of Automotive Engineers (SAE), American Society ofMechanical Engineers (ASME), American Chemical Society (ACS), SAGE Pub-lications Ltd., Elsevier, and Taylor & Francis for allowing us to use figures andother materials from previously published articles. We also thank Springer forinviting us to write and helping us to prepare this book.

Finally, we very much appreciate our families for their love, encouragement,support, and their understanding in our lives, in our research work, and in thepreparation of this book.

December 31, 2010 Yu ShiHai-Wen Ge

Rolf D. Reitz

ix

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Roles of Internal Combustion Engines . . . . . . . . . . . . . . . . . . . 11.2 Modeling of Internal Combustion Engines . . . . . . . . . . . . . . . . 21.3 Computational Optimization of Internal Combustion Engines . . . 4

1.3.1 Engine Optimization with Parametric Studies . . . . . . . . . 41.3.2 Engine Optimization with Non-Evolutionary Methods . . . 71.3.3 Engine Optimization with Evolutionary Methods . . . . . . 9

2 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1 Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 Comparison of Different Optimization Algorithms . . . . . 152.1.2 Multi-Objective Genetic Algorithms . . . . . . . . . . . . . . . 222.1.3 Genetic Algorithm Source Code and Software . . . . . . . . 26

2.2 Engine Modeling with Computational Fluid Dynamics. . . . . . . . 272.2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2.2 Physical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2.3 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.2.4 CFD Codes and Software for Engine Simulations . . . . . . 70

2.3 Regression Analysis Methods . . . . . . . . . . . . . . . . . . . . . . . . . 71

3 Acceleration of Multi-Dimensional Engine Simulationwith Detailed Chemistry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.1 Methods for Reducing Mesh- and Timestep-Dependency

in Engine CFD Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.2 Efficient Methods for Reaction Mechanism Reduction . . . . . . . . 79

3.2.1 Overview of Reaction Mechanism Reduction . . . . . . . . . 793.2.2 Automatic Mechanism Reduction of Hydrocarbon Fuels

for HCCI Engines Based on DRGEP and PCA Methodswith Error Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.3 An Adaptive Multi-Grid Chemistry (AMC) Model . . . . . . . . . . 94

xi

3.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.3.2 Model Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 100

3.4 An Extended Dynamic Adaptive Chemistry (EDAC) Scheme . . . 1053.4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1053.4.2 Model Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063.4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 114

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4 Assessment of Optimization and Regression Methodsfor Engine Optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.1 Assessment of Multi-Objective Genetic Algorithms . . . . . . . . . . 1254.2 Assessment of NSGA II: Niching Technique, Convergence

and Diversity Metrics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324.2.1 Design- and Objective-Space Niching of NSGA II . . . . . 1324.2.2 Convergence and Diversity Metrics . . . . . . . . . . . . . . . . 1344.2.3 Assessment of Niching Strategies . . . . . . . . . . . . . . . . . 135

4.3 Assessment of Regression Methods for Replacing CFDEvaluations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5 Scaling Laws for Diesel Combustion Systems . . . . . . . . . . . . . . . . 1475.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1475.2 Scaling Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.2.1 Combustion Chamber Geometry . . . . . . . . . . . . . . . . . . 1495.2.2 Power Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1495.2.3 Spray Tip Penetration . . . . . . . . . . . . . . . . . . . . . . . . . 1495.2.4 Flame Lift-Off Length . . . . . . . . . . . . . . . . . . . . . . . . . 1505.2.5 Swirl Ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1515.2.6 Summary of Scaling Laws . . . . . . . . . . . . . . . . . . . . . . 152

5.3 Validation of Scaling Laws on a Light-Duty and aHeavy-Duty Diesel Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . 1535.3.1 Engine Specifications . . . . . . . . . . . . . . . . . . . . . . . . . 1535.3.2 Numerical Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545.3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 155

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1776.1 Engine Optimization with Simple Combustion Models. . . . . . . . 177

6.1.1 Optimization of a 2-stroke Direct-InjectionSpark-Ignited Engine . . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.1.2 Optimization of a Caterpillar Heavy-DutyDiesel Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

6.1.3 Optimization of a DDC Heavy-Duty Diesel Engine. . . . . 210

xii Contents

6.1.4 Optimization of a High-Speed Direct-InjectionDiesel Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

6.2 Engine Optimization with Advanced Combustion Models . . . . . 2336.2.1 Optimization of a Heavy-Duty Compression-Ignition

Engine Fueled with Diesel and Gasoline-Like Fuels . . . . 2346.3 Strategies for Simultaneous Optimization of Multiple Engine

Operating Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2596.3.1 A Two-Step Method for Simultaneous Optimization

of Multiple Operating Conditions . . . . . . . . . . . . . . . . . 2596.3.2 A Consistent Method for Simultaneous Optimization

of Multiple Operating Conditions . . . . . . . . . . . . . . . . . 2706.4 Coupling of Scaling Laws with Computational Optimization . . . 271

6.4.1 Downsizing of a HSDI Diesel Engine . . . . . . . . . . . . . . 2726.4.2 Optimization of Downsized Engine . . . . . . . . . . . . . . . . 274

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

7 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Contents xiii

Abbreviations, Nomenclature

AbbreviationsAFR Air-fuel ratioALE Arbitrary Lagrangian–EulerianAMC Adaptive multi-grid chemistryARMOGA Adaptive range multi-objective genetic algorithmATDC After top dead centerBFGS Broyden–Fletcher–Goldfarb–ShannoBML Bray-Moss-LibbyBTDC Before top dead centerCA Crank angleCFL Courant–Friedrichs–LewyCHA ChalmersCI Compression ignitionCDM Continuous droplet modelCFD Computational fluid dynamicsCFM Continuous formulation modelCMC Conditional moment closureCOSSO Component selection and smoothing operatorCSP Computational singular perturbationCTC Characteristic time combustionDAC Dynamic adaptive chemistryDDB Droplet deformation and breakupDDF Droplet distribution functionDDM Discrete-droplet modelDFS Depth first searchDI Direct injectionDICI Direct injection compression ignitionDISC Direct injection stratified chargeDMZ Dynamic multi-zoneDNS Direct numerical simulation

xv

DOI Duration of injectionDPF Diesel particulate filterDPIK Discrete particle ignition kernelDRG Directed relation graphDRGEP Directed relation graph with error propagationEDAC Extended dynamic adaptive chemistryEGR Exhaust gas recirculationEOI End of injectionEPA Environmental Protection AgencyEPFM Eulerian particle flamelet modelEPO Exhaust port openERC Engine Research CenterEVO Exhaust valve openingFTP Federal test procedureGDI Gasoline direct injectionGISFC Gross indicated specific fuel consumptionHCCI Homogeneous charge compression ignitionHSDI High speed direct injectionHTC High throughput computingHRR Heat release rateILDM Intrinsic low-dimensional manifoldsIMEP Indicated mean effective pressureISFC Indicated specific fuel consumptionIVC Intake valve closureKH Kelvin-HelmholtzKN K-nearest neighborsKR KrigingLDEF Lagrangian-Drop Eulerian-FluidLES Large-eddy simulationLHF Locally homogeneous flowLISA Linearized instability sheet atomizationLLNL Lawrence Livermore National LaboratoryMD Methyl decanoateMDDNPS Mean deviation of the distance between neighbor Pareto solutionsMDEPF Mean distance between extreme Pareto solutionsMDPF Mean distance to the Pareto frontMMF Maximum merit functionMOC Method of characteristicsMOEA Multi-objective evolutionary algorithmsMOGA Multi-objective genetic algorithmMOP Multi-objective optimization problemsNMHC Non-methane hydrocarbonNN Neural networksNPR Non-parametric regressionNPS Number of Pareto solutions

xvi Abbreviations, Nomenclature

NSGA Non-dominated sorting genetic algorithmNVO Negative valve overlapODE Ordinary differential equationPAH Polycyclic aromatic hydrocarbonPCA Principal component analysisPDF Probability density functionPFA Path flux analysisPM Particulate matterPPC Partially premixed combustionPPRR Peak pressure rise ratePRF Primary reference fuelPSO Particle swarm optimizationPSR Perfectly stirred reactorQSOU Quasi-second-order upwindQSS Quasi-steady-stateRANS Reynolds-averaged Navier-StokesRBF Radial basis functionsRBFS R-value-based breadth-first searchRIF Representative interaction flameletRNG Renormalization groupROI Radius-of-influenceRSM Reynolds stress modelRT Rayleigh-TaylorSCRE Single-cylinder research engineSF Separated flowSGS Subgrid-scaleSI Spark ignitionSIMPLE Semi-implicit method for pressure-linked equationsSMD Sauter mean diameterSMR Sauter mean radiusSOC Start of combustionSOGA Single-objective genetic algorithmSOI Start of injectionSR Swirl ratioSS-ANOVA Smoothing spline analysis of varianceTAB Taylor analogy breakupTDC Top dead centerUHC Unburnt hydrocarbonWHEAT Wall heat transferWSR Well stirred reactor

NomenclatureA Pre-exponential constant in Arrhenius equation; areaa Speed of sound

Abbreviations, Nomenclature xvii

ad Drop accelerationB0, B1 Model constants in KH modelBm Spalding mass transfer numberbcr Critical impact parameterC Consumption rate of species in chemical reactionCl, Ce,1, Ce,2 Model constants in k-e modelCs, C3 Discharge coefficientCd Drag coefficientCd, sphere Drag coefficient of the spherical dropCs, CRT Model constants in RT modelCl Liquid specific heatCp Constant pressure heat capacityc Progress variablecps Model constant in dispersion modelD Diffusion coefficient; internal diameter of nozzle; distance

between two dropsDd Drag functiond Diameter; nozzle diameterE Activation energye Specific internal energyF Fitness valueF Forcef Drop distribution function; delay coefficient; friction factor;

response functionf* Discrete drop distribution functionfE Fraction of energy dissipationg Gravity forceH Thickness; lift-off lengthH0 Enthalpy of formationh Specific enthalpyI0 Stretch factor; modified Bessel function of the first kindI1 Modified Bessel function of the first kindJ Roughness of the response function; heat fluxK Heat conductivity coefficient; entrainment constantK0, K1 Modified Bessel function of the second kindKf Rate of forward reactionKr Rate of reverse reactionKc Equilibrium constantk Turbulence kinetic energy; wave number; thermal conductivityL Nozzle length; latent heat; lengthlF Laminar flame thicknesslt Turbulence length scaleM Massm massN Number; engine speed

xviii Abbreviations, Nomenclature

Nu Nusselt numbern Number densityn Unit normal vector_P Momentum source term in wall film modelPe Peclet numberPr Prandtl numberPk Production term for turbulence kinetic energyp Pressure; production rate of species in chemical reactionpatm Atmosphere pressurepv Equilibrium fuel vapor pressure_Q Source terms in energy equation rate of heat conductionQi Energy flux from inside the drop to the surfaceQd Energy flux at the drop surfaceq Rate of progress of the elementary reactionqw Wall heat fluxR Universal gas constant; response function in gas-jet modelRs Swirl ratioRe Reynolds numberr Radius; mass fraction ratio of products to reactantsr32 Sauter mean radiusS0 Entropy of formationSc Schmidt numberSh Sherwood numberSt Stokes numbers propagation flame speed; spray tip penetrationsL

0 Laminar flame speedT TemperatureT Taylor numbert Timetc Time scale in CTC modeltsc Turbulence time scale in CTC model

tper Turbulence persistence timetturb Turbulence correlation timeU Velocityu Fluctuating velocityu* Shear speed in heat transfer modelV Velocity on the sample space (particle velocity)V VolumeVcell Volume of computational cellVcol Collision volumeW Molecular weight_W Source term in turbulence kinetic energy equation

We Weber numberw Width

Abbreviations, Nomenclature xix

X Molar fractionx Spatial locationY Mass fractionYY Random numbery Distortion from sphericityZ Ohnesorge numberZst Stoichiometric mixture fractionz0 Proportion of fuel oxygen to fuel carbona Liquid surface tension coefficientaT Thermal conductivityv Symbol of speciesDt Time stepd Tensorial Kronecker symbolde Unsteady equilibrium thickness of thermal boundary layere Dissipation rate of turbulence kinetic energy/ Progress equivalence ratiog Wave amplitude; compressibility factor for isentropic flowj Model constant in heat transfer modelK Wavelength of the fastest growing wavek Heat conductivity; wave length; smoothness parameterl Viscosityv Stoichiometric coefficientv0 Forward molar stoichiometric coefficientv00 Reverse molar stoichiometric coefficientvl Liquid kinematic viscosityh Liquid volume fractionq Densityql0

Liquid macroscopic denistyR Flame surface densityrw Wall stress tensorr Surface tensionrk, re Model constants in k-e models Viscous stress tensorsv Response time scale in gas-jet modelX Frequency of the fastest growing wavex Chemical reaction rate; complex growth rate of disturbance; rate

of progress of the reaction

Superscript. Time rate of change~ Favre averaged- Time averaged+ Non-dimensional parameters in heat transfer model0 Fluctuating term in time averaging

xx Abbreviations, Nomenclature

00 Fluctuating term in favre averagingb Body forcen Time steps Spray

Subscript0 Standard conditiona Axialacc Accelerationair Airax Axialb Burnt; backward; breakupbu Breakupc Child; convectionch Chemistrycoll Collisioncrit Criticald Droplet; downstreameff Efficienteq Eqivalentexp Expansionf Forward; film; frictiongroup Group in AMC modelh Thickness of liquid sheeti Inertiaimp Impingementinj InjectionKH KH modelk SpeciesL Ligamentl Liquid; laminarlp Less populousmp More populousn Normal direction to the surfacenoz Nozzlep Pressure; piston; parcelplasma Plasmaprec PrecursorRT RT modelr Reaction; reverse; piston ringrel Relativerst Rate-of-strain tensors Species; soot; surface; oil resistancesf Soot formation

Abbreviations, Nomenclature xxi

so Soot oxidationsp Sprayspk SparkT Turbulencet Turbulence; tangent direction to the surfacetot Totalu Unburned; upstreamvap Vaporizationvena Vena contractaw Walls Turbulence? Outer boundary

xxii Abbreviations, Nomenclature