Complex Chemistry ModelingofDiesel Spray CombustionP A Niklas Nordin

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CONTENTSContentsPapers vAbstra t viNomen lature viiA knowledgments ix1 Introdu tion 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Turbulent Diesel Spray Combustion . . . . . . . . . . . . . . . . . . . . . . 11.3 Lagrangian Spray Cal ulations . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 The CFD odes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Stru ture of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Governing Equations 42.1 The Gas (Eulerian) Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.1 The Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . 42.1.2 The Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 The Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.4 The Turbulen e Equations . . . . . . . . . . . . . . . . . . . . . . . 62.1.5 The Chemistry Equations . . . . . . . . . . . . . . . . . . . . . . . 82.2 The Liquid (Lagrangian) Phase . . . . . . . . . . . . . . . . . . . . . . . . 92.2.1 The Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 The Droplet Energy Equation . . . . . . . . . . . . . . . . . . . . . 102.2.3 The Droplet Mass Equation . . . . . . . . . . . . . . . . . . . . . . 112.2.4 The Atomization Model . . . . . . . . . . . . . . . . . . . . . . . . 132.2.5 The Breakup Model . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.6 The Collision Model . . . . . . . . . . . . . . . . . . . . . . . . . . 153 The Turbulen e/Chemistry Intera tion Model 173.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Ba kground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 The Referen e Spe ies Te hnique . . . . . . . . . . . . . . . . . . . . . . . 183.4 The Partially Stirred Rea tor Con ept . . . . . . . . . . . . . . . . . . . . 203.5 The PaSR Model: Observations and Impli ations . . . . . . . . . . . . . . 254 Implementing a Lagrangian Representation of the Spray 294.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Tra king . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29ii

CONTENTS4.3 The Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3.1 Turbulent Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . 324.3.2 Lagrangian-Eulerian Des ription Dis repan y . . . . . . . . . . . . 334.3.3 Lagrangian-Eulerian Coupling . . . . . . . . . . . . . . . . . . . . . 354.4 The Liquid Mass and Heat Transfer . . . . . . . . . . . . . . . . . . . . . . 364.5 The Breakup Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.6 The Lagrangian Solution Pro edure . . . . . . . . . . . . . . . . . . . . . . 375 Results and Dis ussion 395.1 Mesh Dependen e Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 Turbulent Parameters Analysis . . . . . . . . . . . . . . . . . . . . . . . . 415.3 Collision Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.4 Mi ro Mixing Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 Con lusions 467 Future Work 478 Summary of Papers 48Numeri al Evaluation of n-Heptane Spray Combustion at Diesel-like Conditions 48Numeri al Evaluation of Dual Oxygenated Fuel Setup for DI Diesel Appli ation 48Computer Evaluation of DI Diesel Engine Fueled with Neat Dimethyl Ether . . 48Neat Dimethyl Ether: Is It Really Real Diesel Fuel of Promise? . . . . . . . . . 48Evaluation of Ignition Quality of DME at Diesel Engine Conditions . . . . . . . 493-D Diesel Spray Simulations Using a New Detailed Chemistry Turbulent Com-bustion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49A Robust Fa e-To-Fa e Tra king 50B The turbulen e/spray intera tion onstant Cs 51Comment 52Referen es 53

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Make everything as simple as possible, butnot simpler. - Albert Einstein -

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PapersThis thesis is based on the work arried out in the following papers.Paper I. N. Nordin and V. Golovit hev,Numeri al Evaluation of n-Heptane Spray Combustion at Diesel-like Conditions,7th International KIVA Users Meeting at the SAE Congress, Feb 23, 1997, DetroitMI, Book of Abstra ts, pp. 1-5Paper II. V. Golovit hev, N. Nordin and J. Chomiak,Numeri al Evaluation of Dual Oxygenated Fuel Setup for DI Diesel Appli ation,SP-1276 Advan es in SI and Diesel Engine Modeling, SAE Te hni al Paper Series,SAEpaper 971596The 1997 SAE Spring Fuels & Lubri ants Meeting, May 5-7, Dearborn MIPaper III. N. Nordin, V. Golovit hev and J. Chomiak,Computer Evaluation of DI Diesel Engine Fueled with Neat Dimethyl Ether,Pro eedings of the 22nd CIMAC International Congress on Combustion Engines,Copenhagen 18-21 May 1998, Vol IPaper IV. V. Golovit hev, N. Nordin and J. Chomiak,Neat Dimethyl Ether: Is It Really Real Diesel Fuel of Promise?,SAEpaper 982537 presented at theInternational Fall Fuels & Lubri ants Meeting & ExpositionSan Fran is o, CA, O tober 19-22, 1998Paper V. V. Golovit hev, N. Nordin, J. Chomiak, K. Nishida and K. Wakai,Evaluation of Ignition Quality of DME at Diesel Engine Conditions,4th International Conferen e, ICE99Internal Combustion Engines: Experiments and Modeling, pp. 299-306September 12-16, 1999, Capri - NaplesPaper VI. V. Golovit hev, N.Nordin, R. Jarna ki and J. Chomiak,3-D Diesel Spray Simulations Using a New Detailed Chemistry Turbulent Combus-tion Model,Presented 19-22 June 2000, Paris, Fran e at:CEC/SAE International Spring Fuels & Lubri ants Meeting and Expositionv

Abstra tThis thesis illustrates the appli ation of omputational uid dynami s (CFD) toturbulent rea tive two-phase ows in piston engines.The fo us of the thesis lies on numeri al simulations of spray ombustion phe-nomena with an emphasis on the modeling of turbulen e/ hemistry intera tion ef-fe ts using a detailed hemistry approa h. The turbulen e/ hemistry intera tionmodel a ounts for the e�e ts of turbulent mi ro-mixing on the hemi al rea tionrates. The models have been implemented in the KIVA3-V ode [1, 2, 3℄ and su - essfully applied to spray ombustion analysis in a onstant volume and a DI Dieselengine. The limitations and diÆ ulties of representing the spray in a Lagrangianfashion are also adressed.Three di�erent liquid fuels have been used in the simulations: n-heptane, methanoland dimethyl ether (DME). Detailed and redu ed hemi al me hanisms have beendeveloped and validated for all these fuels and reasonable agreement between ex-perimental data and numeri al simulations has been obtained.

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Nomen lature When I use a word it means just whatI hoose it to mean - neither more nor less.- Humpty DumptyLatin Letters Molar on entration mol=m3CD Drag oeÆ ient � p Spe i� heat at onstant pressure J=kgK l;d Liquid spe i� heat at onstant volume J=kgKD Mass di�usion oeÆ ient m2=sD Droplet diameter kge Spe i� internal energy J=kgfm Chemi al sour e term kg=m3shm Spe i� enthalpy J=kgI Identity matrix �J Heat ux ve tor W=m2k Turbulent kineti energy m2=s2md Droplet mass kg_md Droplet evaporation rate kg=smp Par el mass kgNr Number of rea tions �Ns Number of spe ies �p Gas pressure Par Droplet radius mS Rate of strain tensor 1=sT Gas temperature KTd Liquid droplet temperature Ku Gas velo ity m=sud Droplet velo ity m=sup Par el velo ity m=sV Cell volume m3Xv;s Mass fra tion fuel vapor at droplet surfa e -Xv;1 Mass fra tion fuel vapor far away -vii

Greek Letters�t Integation step s" Turbulent kineti energy dissipation rate m2=s3� Gaseous dynami vis osity kg=ms� Gaseous kinemati vis osity m2=s�� Rea tive mass fra tion �� Rea tion rate multiplier �� Thermal ondu tivity W=mK� Gas density kg=m3�l Droplet/par el/liquid density kg=m3�m Partial density of the gaseous spe ies kg=m3_�s Total liquid evaporation rate per unit volume kg=m3s� Vis ous stress tensor N=m2� Integration step s�u Momentum relaxation time s�e Evaporation relaxation time s�h Heat transfer relaxation time s�m Turbulent mixing time s_! Chemi al rea tion rate mol=m3sDimensionless NumbersNu Nusselt total / ondu tive heat transfer ratioOh Ohnesorge vis ous / ( inertia * surfa e tension )(1=2) for e ratioPr Prandtl momentum / thermal di�usion ratioRe Reynold inertia / vis ous for e ratioS S hmidt momentum / mole ular di�usion ratioSh Sherwood mass / mole ular di�usion ratioT Taylor OhpWeWe Weber inertia / surfa e tension for e ratio

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A knowledgments Teamwork is essential - it allows you to blame somebody else- AnonymousThis work was arried out at the Department of Thermo and Fluid Dynami s at ChalmersUniversity of Te hnology and the Thermo uids Se tion of Department of Me hani alEngineering at Imperial College of S ien e Te hnology and Medi ine. I am grateful tomany people, but I wish to express my sin ere gratitude espe ially to the people below,who have ontributed more than others. If someone feels left out I apologize, it is onlybe ause my memory serves me ill.Chalmers University of Te hnologyI am deeply grateful to Ass. Prof. Valeri Golovit hev for his enthusiasm and the manyhelpful dis ussions. Without his help this would have looked very di�erent and a lotworse. I am also grateful to Prof. Jerzy Chomiak for making invaluable omments andsuggestions. Both of them have also been a great sour e of inspiration.Jan Hanson for, to my big surprise, hiring me in the �rst pla e.Monika Orrba ke, Ulla Lindbergh-Thieme, Sandra Arvidsson and Birgitta Hultman fortaking are of the basi 'stu�', thus leaving me with more time to surf the internet.All friends and olleagues at TFD for reating a friendly and enjoyable atmosphere towork in. I am also grateful to B�orje Sennung and the MDC gang, for the omputersupport.Imperial College/Computational Dynami s Ltd./Nabla Ltd.I am also very glad to have met Henry Weller who, besides introdu ing me to FOAM andC++, also made my time at Imperial both fun and edu ational. It has made resear h atrue joy. I am also glad to have met Henrik Rus he and Dr. Marek Duszynski, for themany, always friendly and helpful, dis ussions.I am also grateful to Prof. David Gosman for letting me ome to Imperial and makingme feel wel ome, and toDr. Hrvoje Jasak and Dr. Chris Marooney for being a great help and always answeringmy questions in a friendly way.

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OthersFor their indire t support and always enjoying, but not always fruitful, ontribution I amgrateful to all my friends, and espe ially Martin Asplund for writing a thesis that gaveme good ideas on how to write my own (BIF rulez), and also to Lars R�onneg�ard, H�akanOlsen, Mats Purs he, Eva-Lena Tolstoy.For her love - Camilla Ljus, it would de�nately not be as good without you.Finan ial SupportFinally, I would like to thank the Combustion Engine Resear h Center (CERC) for pro-viding me with the means to do the resear h summarized in this thesis.This do ument has been typeset using LATEX.

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1 INTRODUCTION1 Introdu tion \The time has ome", the Walrus said,\to talk of many things"- L. Carroll1.1 MotivationThe Diesel engine is fa ing a hallenging task. Either redu e emissions or be legislated outof existen e. Future environmental legislation will require both soot and NOx emissions tobe redu ed drasti ally. It is highly doubtful that the Diesel engine an meet these demandswithout post/exhaust treatment, sin e developing and testing a new Diesel engine hasbeen, and still is, done by experiments, whi h is a time onsuming and expensive task.Even though experimental resear h is reliable it o�ers little feedba k (in terms of what an be improved and how to improve it) as the engine essentially is a bla k box. What ame out and what went in is known, but the important pro ess in-between is not wellknown. By using omputational uid dynami s (CFD) in onju tion with experiments itis possible to drasti ally redu e the time and ost of the engine development pro edure.Feedba k and ideas of how to improve details an also be obtained, sin e CFD allows theengineer to see what is a tually happening. Even though CFD is still not reliable enoughto be trusted ex lusively, its value annot be argued. Reliable CFD models are, therefore,the key to better and more predi tive omputations.1.2 Turbulent Diesel Spray CombustionDiesel spray ombustion is one of the most diÆ ult problems of applied ma ros opi physi s as it involves the most diÆ ult problems of turbulen e, hemistry and two phase ows. These are problems that are tightly oupled and highly non-linear, with time andlength s ales that are so small that it's questionable if they will ever be possible to resolve.The turbulen e- hemistry intera tion is very strong and it is therefore essential to havea reliable intera tion model for this pro ess if a urate predi tions of emissions are to beperformed. To a ount for the in uen e of the turbulent u tuations on the rea tion ratethe Partially Stirred Rea tor (PaSR) approa h has been used. This model is des ribed indetail in Se tion 3.A urate pollutant predi tions require a number of spe ies and rea tions, in whi h thenumber is limited by the omputational resour es. A realisti approximation typi allystates that a omplex hydro arbon ombustion system ontains 1000's of spe ies with10000's of rea tions. Using this many spe ies and rea tions is not possible today in aCFD appli ation. Still, as always, the more spe ies and rea tions that are in luded, thebetter the a ura y. It is therefore attra tive to be able to in lude and handle a ompre-hensive hemi al me hanism.1

1 INTRODUCTIONRedu ing both soot and NOx is a well investigated problem, known as 'The Dieseldilemma.' It has haunted generations of engineers and will probably ontinue to do so.Its onsequen e is that by redu ing the soot, NOx is in reased, and vi e versa. However,by redu ing the inje tor ori� e and in reasing the fuel inje tion pressure it is possible toredu e them both [17℄. Another re ent approa h is to use oxygenated fuel, in whi h theoxygen atoms are readily available in the fuel itself. Among these fuels, methanol anddimethyl ether, CH3OH and (CH3)2O, respe tively, have been analyzed, and their qualityas a possible substitute for hydro arbon fuels has been evaluated.1.3 Lagrangian Spray Cal ulationsThe Diesel spray reates a highly turbulent �eld with very strong gradients. The diameterof the liquid spray is on the order of 0.1 mm and the liquid velo ity around 200-400m/s. The subsequent ignition and ombustion involve length s ales that are even smallerand resolving this would require an enormous amount of memory and omputationaltime. Hen e, using a oarse omputational mesh is today a ne essity. But, together withmodels for the sub-grid pro esses, it is possible to obtain results. Treating the spray ina Lagrangian way is today also a ne essity sin e it does not require the nozzle region tobe fully resolved by the omputational mesh. In the Lagrangian des ription, the spray isrepresented by points, often referred to as par els or droplets. The more points, the morea urate representation of the spray. These points are then assigned properties whi h maybe as many as desired, some basi ones are lo ation, velo ity, diameter, mass, temperatureand fuel omposition. Sin e these points are of zero dimension and do not o upy anyspa e in the domain, they only serve one purpose: To a t as a marker. The Eulerianequations, for the gas, need to know in whi h ell the liquid/gas ex hange takes pla e,so that the intera tion terms an be distributed in the orre t position. These points arethen tra ked through the domain, in whi h they move from ell to ell, a ording to theimplemented physi s and distribute mass, momentum and energy.The other approa h, where one treats the spray in an Eulerian way, requires the nozzleto be resolved and be ause of the large di�eren e in s ale between the inje tor ori� e andthe bore of the ylinder, and limitations in omputer power, the omputational time usingthis approa h is mu h longer and often too long for pra ti al purposes. Even though om-puter speed is doubled every 18-24 months, a ording to Moore's law, a pra ti al Euleriantreatment of the spray, with all the physi al pro esses in luded, will not be realizable forsome time. However, with the available omputer power it is now possible to have a omputational mesh in whi h the ells are of the same order as the nozzle diameter. Thisis already a reality when performing Large Eddy Simulations with sprays and it leads tosome intri ate problems. A simple question like- In whi h ell(s) is this par el/droplet lo ated?is suddenly a very omputationally expensive question to ask, espe ially sin e the par- el/droplet may o upy multiple ells. Thus, in order to treat the spray in a truly orre t2

1 INTRODUCTIONway and obtain a orre t sour e term distribution of the Lagrangian intera tion termsthe \point-representation" has to be repla ed by a \sphere-representation", or rather anon-zero dimensional representation. Another problem related to this situation, is thefa t that a omputational ell may be o upied by only liquid and no gas. Numeri allythis leads to singularities, sin e the gaseous volume tends towards zero, and it has stillnot been established whether or not this phenomenon is important enough to in lude.While some laim that it an be ignored, it is the authors feeling that ignoring physi s an never be the orre t path to follow. This is, however, a question for future resear h.1.4 The CFD odesTwo di�erent CFD odes have been used in the work arried out in this thesis. All ofthe omputations have been arried out using the KIVA3-V ode [2℄ developed at LosAlamos National Laboratories, mainly by A. Amsden. All of the work arried out onimplementing the Lagrangian des ription of the spray has been done using the FOAM(Field Operation And Manipulation) pa kage [36℄, developed at Imperial College, mainlyby H. Weller. The transition from KIVA to FOAM was based on the limitations in thefortran77 programming language, whi h KIVA is written in. Sin e FOAM is written inC++, it is a more attra tive environment, as the omplexity of the CFD ode is in reasedsubstantially when a Lagrangian des ription is added to represent the liquid phase.1.5 Stru ture of the ThesisThe basi equations for both the gaseous and liquid phase, together with the sub modelsfor the spray, are presented in Se tion 2. This is followed by a detailed des ription anddis ussion of the turbulen e/ hemistry intera tion model in Se tion 3. In Se tion 4 theauthors experien e from implementing a Lagrangian des ription of the spray is shared.The results in Se tion 5 are �ndings that have not been in luded in the papers. The on lusions are drawn in Se tion 6, after whi h future work is dis ussed in Se tion 7.

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2 GOVERNING EQUATIONS2 Governing Equations Now is a good time to put your workon a �rm theoreti al foundation.- Sam MorganThis se tion presents the governing equations for both the Eulerian and Lagrangian rep-resentation of the gas and liquid phase. Here, unless stated otherwise, the governingequations have been averaged.Due to the Eulerian/Lagrangian oupling the Eulerian equations will have extra sour eterms to a ount for the phase intera tion. The numeri al treatment and implementationof these sour e terms will be dis ussed in Se tion 4.2.1 The Gas (Eulerian) PhasePerforming a al ulation involving ombustion requires a minimum of 3 omponents (fuel,oxidizer and produ ts) and the governing equations for a multi- omponent mixture musttherefore be used.2.1.1 The Continuity EquationThe ontinuity equation, or rather spe ies transport equation, for one omponent in amulti omponent mixture reads��m�t +r � (�mu) = r � "�Dr �m� !# + fm + _�smÆm1 (1)where �m is the mass density of spe ies m, � the total gaseous mass density, u the gasvelo ity, fm the sour e/sink term due to hemistry (25) and _�sm the sour e due to theevaporation of the liquid1. (By onvention the fuel is ommonly set to spe ies/ omponent1.) D is the mass di�usion oeÆ ient, where we assume Fi k's law of binary di�usion witha single di�usion oeÆ ient as de�ned in Equation (23). D in ludes the turbulent di�u-sivity repla ing the orrelations �0mu0, whi h are modeled using a Boussinesq hypothesis.Note that all spe ies di�use equally, whi h is only true where the di�usion is dominatedby turbulent di�usion, as it is here. The Souret and Dufour e�e ts together with thermalradiation have been negle ted. By forming the sum, � =Xm �m, for (1) over all spe ies,the global ontinuity equation is obtained���t +r � (u�) = _�s (2)1This thesis only deals with single omponent fuels. Thus, the Dira fun tion is limited to just j = 1in the general ase with Æmj 4

2 GOVERNING EQUATIONSThe equation for the liquid evaporation rate, _�s, is given by the relationXV Np _md = � ZV _�sdV = �V _�s (3)where the l.h.s sum is the sum over all the spray par els in the ell volume, V is the volumeof the ell, Np the statisti al number of droplets in the par el and _md is the evaporationrate for a single droplet, given by Equation (42)2.1.2 The Momentum EquationThe averaged momentum equation for the gas reads�(�u)�t +r � (�uu) = �rp+r � � + Fs + �g � 23r(�k) (4)where (uu)ij = uiuj, p is the gas pressure, � the vis ous stress tensor, de�ned below, Fsthe rate of momentum gain/loss per unit volume due to the spray [3, p.16℄ and g thespe i� body for e, whi h is assumed to be onstant.� = 2�S+ �r � uI (5)where S = 12 hru + (ru)T i (6)where � and � are the �rst and se ond oeÆ ients of vis osity (in luding turbulent vis- osity), see Equation (22).2.1.3 The Energy EquationThe internal energy equation reads�(�e)�t +r � (�ue) = �pr � u�r � J+ �"+ _Q + _Qs (7)where e is the spe i� internal energy, ex lusive of hemi al energy, and J is the heat uxve tor (8) whi h is the sum of heat ondu tion and enthalpy di�usion:J = ��rT � �DXm hmr �m� ! (8)where T is the gas temperature and hm is the spe i� enthalpy of spe ies m. The sour eterms _Q and _Qs are due to the hemi al heat release, Equation (9), and spray intera tion[3, p.19℄, respe tively. _Q = NrXr=1 qr _!r (9)5

2 GOVERNING EQUATIONSwhere _!r is de�ned in Equation (26) and qr is the heat of rea tion r at the referen etemperature qr = NsXs=1(#0sr � #00sr)(hs)reff (10)where (hs)reff is the heat of formation of spe ies s at the referen e temperature.2.1.4 The Turbulen e EquationsAlthough there are numerous turbulen e models available in the literature today, the k-"model and the RNG version of it, are the only models that have been used throughout thework arried out in this thesis. For sake of ompleteness they will be presented here as well.Sin e the k-"-models originally was developed for in ompressible stationary ows,they have been modi�ed to a ount for the e�e ts of ompressibility (� : ru) and sprayintera tion ( _W s), see [3, p.19℄.The turbulent modulation due to the presen e of a Diesel spray is a omplex and notwell known pro ess. In the standard versions of the KIVA odes the spray sour e in thek-equation an only a t as a sink, i.e., the turbulent energy is redu ed in the presen e of aliquid par el, see [3℄. This is, however, in omplian e with the assumption that the liquidvolume fra tion is negligible and the droplets are small in omparison to the turbulentlength s ales. For a more detailed dis ussion on this topi see [22℄.The k-"-model reads�(�k)�t +r � (�uk) = �23�kr � u+ � : ru+r � �� �Prk�rk�� �"+ _W s (11)and �(�")�t +r � (�u") = ��23C"1 � C"3� �"r � u +r � �� �Pr"�r"�+ "k hC"1� : ru� C"2�"+ Cs _W si (12)where � : ru = �ij�ui=�xj and the di�erent onstants are given in Table 1.The value of Cs = 1.5 has been suggested by O'Rourke in [27℄, based on the postulateof length s ale onservation in spray/turbulen e intera tion. This an easily be shown.Assume onstant density and an absen e of gradients. Equations (11) and (12) are thenredu ed to2 �k�t = _W s; �"�t = Cs "k _W s (13)2If a on lusion, whether the onstant Cs should be 1.5, or larger/smaller, one must also negle t ".Futher dis ussion of this topi an be found in Appendix B6

2 GOVERNING EQUATIONSand thus Cs "k �k�t = �"�t (14)The turbulent length s ale is de�ned as lt = k�=", where � = 1:5. Di�erentiation yields,�lt�t = ��t k�" ! = k�"2 �"k �k�t � �"�t! [27℄= 0 (15)and for (15) to be zero, Equation (14) gives that Cs = � for lt to be preserved.It an, however, be argued that the length s ale really is onserved, sin e the dispersedphase, as long as the droplets are small, introdu es additional dissipation, without a�e t-ing u0. Consequently it should lead to a redu tion of lt. Thus looking at Equation (15)again, using (13), yields�lt�t = k�"2 �"k �k�t � �"�t! = k��1" _W s(�� Cs) < 0 (16)whi h states that Cs must be less than � = 1:5, sin e _W s < 0 here, for Equation (16) tobe valid. The value of " is onstrained to satisfy the inequality" � C�Pr"(C"2 � C"1)! 12 k3=2Lsgs (17)where Lsgs is an input sub grid length s ale whose value is typi ally taken to be 4Æx,where Æx is a representative omputational ell dimension. This prohibits the turbulentlength-s ale, lt, to grow beyond Lsgs.In the RNG k-" model, as des ribed in [15℄, the "-equation is modi�ed to read�(�")�t +r � (�u") = � 23C"1 � C"3 + 23C�C� k"r � u! �"r � u+r � �� �Pr"�r"�+ "k h(C"1 � C�)� : ru� C"2�"+ Cs _W si (18)where C� = �(1� �=�0)1 + ��3 ; � = Smk" ; Sm = (2SijSij)1=2; (19)and �0 = 4:38; � = 0:012For ideal gases the mole ular vis osity, �air � Tm, where T is the temperature andm=0.5,see Equation (22). C"3 is also dependent on the polytropi exponent, n, in the expression,p=�n = onst, whi h des ribes the polytropi relation in a losed thermodynami system.7

2 GOVERNING EQUATIONSTypi ally, n, ranges from 1.3 to 1.4 and the value 1.4 has been used throughout the omputations. C"3 = �1 + 2C"1 � 3m(n� 1) + (�1)Æp6C�C��3 (20)Æ = ( 1; r � u � 00; r � u > 0 (21)The model onstants for the k-"-models are summarized in Table 1Model C� C"1 C"2 C"3 Prk Pr" CsStandard 0.09 1.44 1.92 -1.0 1.0 1.3 1.5RNG 0.0845 1.42 1.68 Equation (20) 0.7194 0.7194 1.5Table 1: Model onstants for the standard and RNG k-" turbulen e model.and the transport oeÆ ients� = �air "1 +s C��air kp"#2 ; � = �� ; �air = A1T 3=2T + A2 ; � = �23� (22)� = � pPr ; D = �S (23)where the Prandtl and S hmidt numbers, Pr and S , are input onstants andA1 = 1:457 � 10�5; A2 = 110:02.1.5 The Chemistry EquationsAny multi omponent mixture involving hemi al rea tions of the form as in Table 21. C7H16 + O2 ! CH2O + H2O... ...Nr. C7H16 + O *) C7H15 + OHTable 2: Ex erpt of a hemi al kineti me hanismmay be written in the formNsXj=1#0jk j *) NsXj=1#00jk j; k = 1; Nr (24)8

2 GOVERNING EQUATIONSwhere #0 is the matrix of the stoi hiometri oeÆ ients for the forward rea tions, #00 isthe stoi hiometri oeÆ ients for the ba kward rea tions and j the molar number for thej:th spe ies, Nr is the number of rea tions and Ns the number of spe ies.The hemi al sour e term in the spe ies transport equation (1) then readsfm =Mm NrXr=1 (#00mr � #0mr) _!r (25)where _!r = krf NsYs=1 #0srs � krb NsYs=1 #00srs = krf�f � krb�b (26)and kr = ArT nr exp�� EraRT � (27)is the generalized Arrhenius rea tion rate for the forward and ba kward rea tion rate oeÆ ients and nr, Ar and Ea are the orresponding steri fa tor, ollision frequen yexponent and a tivation energy, respe tively.2.2 The Liquid (Lagrangian) PhaseThis se tion presents the equation of motion, evaporation and energy for the dis retephase. How to solve them numeri ally, is dis ussed in Se tion 4. The models for atom-ization, se ondary breakup and ollisions are also onsidered.2.2.1 The Equation of MotionAlthough the momentum equation (Newton's II law) for a dis rete parti le is very simplemdduddt = F (28)where md, ud and F are the droplet mass, droplet velo ity and the for e a ting on thedroplet, respe tively, it presents, as will be shown, some intri ate problems. From thepioneering work of Basset (1888), Boussinesq (1903) and Oseen (1927) the most generalform of F is often referred to as the BBO equation and although it is still a subje tof ontroversy, see [13℄, this is of minor importan e in systems where the density ratiobetween the dis rete and ontinuous phase is of the order 102 or higher. Under these on-ditions the added mass, Basset, Magnus (rotating droplets are not onsidered), Sa�man,pressure and buoyan y for e are often negle ted and the BBO equation is redu ed to justthe drag and gravitational for e (Although the gravitational for e in Diesel sprays analso be negle ted, it is in luded due to its simpli ity). The a ting for e on the droplet isthus, F = ��D28 �CDjud � uj(ud � u) +mdg (29)9

2 GOVERNING EQUATIONSwhere CD is the drag oeÆ ient. There are numerous proposals for the expression of thedrag oeÆ ient CD [37, 32, 8, 22℄.In this work the relation below has been used, ignoring all e�e ts, su h as evaporationrate, deformation and proximity (high liquid volume fra tion) e�e ts.CD = 8><>: 24Red �1 + 16Re2=3d � Re < 10000:424 Re > 1000 (30)where Red = �jud � ujD� (31)When solving the equation of motion in pra ti e, Equations (28) and (29) are ombinedand written in the form duddt = �ud � u�u + g (32)where �u is the momentum relaxation time de�ned below�u = 8md��CDD2jud � uj = 43 �dD�CDjud � uj (33)2.2.2 The Droplet Energy EquationThe liquid droplet re eives its energy from the gas, whi h is used to in rease the liquidtemperature and over ome the latent heat of evaporation in order to evaporate the fuel.Unless the gas is saturated with fuel vapor, evaporation is always present, i.e., _md < 0,the evaporation pro ess will re eive its energy from the droplet. Thus, if the transferredheat, from the gas, is insuÆ ient, the droplet temperature will de rease. The equationfor the heat transfer to the liquid droplet is given bymddhddt = _mdhv(Td) + �D�Nu(T � Td)f (34)where f = zez � 1 ; z = � p;v _md�D�Nu (35)f is a fa tor [35℄ whi h orre ts the rate of heat ex hange due to the presen e of masstransfer. In the review by Gosman [11℄, various orrelations for the Nusselt number aredis ussed in more detail. The orrelation used here isNu = 2:0 + 0:6Re1=2Pr1=3 (36)where the Prandtl number is de�ned asPr = � p� (37)10

2 GOVERNING EQUATIONSand all properties are evaluated using the �lm temperature, i.e., the 1=3-ruleTf = 2Td + T3 : (38)Solving Equation (34) in pra ti e is performed by introdu ing a hara teristi heat transferrelaxation time, �h, de�ned as �h = md l;d�D�Nu: (39)where l;d is the spe i� heat for the liquid. Rearranging Equation (34), wherehd = l;d(Td � Tref)and using Equation (42), for the mass transfer, yieldsdTddt = T � Td�h f � 1 l;d hv(Td)�e (40)where �e is de�ned later in Equation (45).2.2.3 The Droplet Mass EquationIn Diesel engines, the temperature and pressure are very high and the riti al propertiesfor hydro arbons are very low, for instan e the riti al temperature and pressure for n-Heptane are 540.2 K and 2.74 MPa, thus, evaporation in lassi al terms does not reallyo ur. These e�e ts have not been addressed yet and are a subje t for future resear h.They will therefore be ignored here.The ondensation pro ess is not present in Diesel engines, thus, the only transfer ofmass is from the liquid phase to the gas phase. Evaporation from a spheri al liquid droplethas experimentally been established to follow the D2-law, i.e.,dD2dt = Cevap (41)see, for instan e, [16℄. However, the evaporation rate is more ommonly expressed interms of mass, or diameter, and the rate of evaporation for a single droplet is given bythe expressiondmddt = _md = ��DD�vSh ln p� pv;1p� pv;s = ��DD�vSh ln 1 + Xv;s �Xv;11�Xv;s ! (42)where �v is the density of the fuel vapor lose to the surfa e. This is estimated using theideal gas law �v = pRvTm (43)11

2 GOVERNING EQUATIONSwhere p is the gas pressure and assumed to be equal to the fuel vapor pressure lose tothe surfa e. The evaporation rate is in reased by the relative velo ity, whi h is a ountedfor by the Ranz-Marshall orrelation3 Sh, where Sh is the Sherwood number and herede�ned as Sh = 2:0 + 0:6Re1=2S 1=3 (44)In pra ti e Equation (42) is solved by using an evaporation relaxation time, �e, asde�ned below: �e = md�DDSh�v ln(1 +B) (45)where B = Xv;s �Xv;11�Xv;s (46)The evaporation rate is then given by the expressionsdmddt = �md�e ; dDdt = � D3�e (47)Sin e, the mass or diameter is always redu ed, the equation above should be solvedimpli itly as it prevents the diameter, or mass, from be oming negative if too large timesteps are used.Although the pressure in a Diesel engine is typi ally above the riti al onditions forthe fuel, the following observation is important. If the liquid starts to boil, the vaporpressure rises above the ambient pressure, whi h yields B ! 1, and the evaporationrelaxation time tends to zero, whi h in turn leads to the evaporation rate _md !1. In fa tEquation (42) is no longer valid. Unless the liquid is lose to the riti al point, this in�niteevaporation rate is not physi al. The evaporation rate at boiling point therefore has tobe oupled with the energy equation (34), sin e the evaporation rate is then governed byhow fast heat an be supplied to the liquid.Under boiling onditions the evaporation rate equation is dedu ed from (34), wherethe l.h.s is equal to zero, sin e the temperature is onstant under boiling onditions. Theboiling mass transfer rate is then given bydmddt = ��D�Nu p;v ln� p;vhv (T � Td) + 1� (48)Or using a hara teristi boiling relaxation timedDdt = � D�boil (49)where �boil = D2�d p;v2�Nu ln� p;vhv (T � Td) + 1� (50)3Originally proposed by Fr�ossling (1938) [10℄. 12

2 GOVERNING EQUATIONSSin e Equation (42) be omes singular lose to the boiling point, the mass transfershould be swit hed from (42) to (48) when �boil > �e and the saturation pressure is lose enough to the surrounding pressure, where lose enough is a matter of numeri alrobustness.2.2.4 The Atomization ModelSin e the purpose of the Lagrangian approa h is to remove the ne essity for resolving thenozzle, the initial onditions for the spray par els must be spe i�ed. This an be doneby using either an atomization model or by spe ifying the initial size and spray angle as onstants, whi h an be seen as a very simple atomization model.As both approa hes need to be tuned for optimal performan e, the latter approa h hasbeen used in the urrent work, sin e it is both simpler, faster and more straight-forward.Using an atomization model is, however, physi ally more orre t, but any advantage ofusing an atomization model is louded by the un ertainties in urrent breakup models.2.2.5 The Breakup ModelWhen the liquid par els have been inje ted they start to deform and break up. If theWeber number is high enough that is. The purpose of the breakup model is to redu ethe hara teristi size of the liquid par els, due to aerodynami for es, and, if the modelin ludes it, to introdu e new par els, whi h are being stripped o� the larger sized (parent)par els. Today, there exist a number of breakup models and although the TAB model isthe one that has been used the most (whi h is standard in the KIVA ode), the Kelvin-Helmholtz Rayleigh-Taylor (KHRT) breakup model by Reitz, see [31, 34, 33℄ will bepresented here, for referen e, as it has been implemented in the KIVA3-V [2℄ ode as wellas the FOAM pa kage [36℄. The details of implementation an be found in Se tion 4.The KHRT breakup model has two modes for breakup. The KH mode works as follows.New, hild, par els with the size r = B0� (51)are stripped from the parent par el and the radius of the parent par el is redu ed a ordingto the rate expression drdt = �r � r �kh ; �kh = 3:788B1D� (52)where B0 = 0:61, B1 = 40 are model onstants and = 0:34 + 0:38We1:5(1 +Oh)(1 + T 0:6)s ��dr3 (53)� = 9:02r (1 + 0:45pOh)(1 + 0:4T 0:7)(1 + 0:865We1:67)0:6 (54)13

2 GOVERNING EQUATIONS

Figure 1: S hemati pi ture of the KH instability droplet formation pro ess. (Taken from[31℄)where � is the wavelength of the fastest growing wave , see Figure 1.The Weber number for the gas is de�ned asWe = �jurelj2r� ; (55)the Ohnesorge number is de�ned as Oh = pWel=Rel, and the Taylor number as T =OhpWe. The liquid Weber number Wel is de�ned similar to We, but with the gasdensity repla ed by the liquid density. The liquid Reynolds number is de�ned asRel = �ljureljr�l :The RT mode of breakup works in a slightly di�erent way. The wavelength �t = �=K,where K = s jgt(�l � �)j3� ; gt = (g + duddt ) � udjudj (56)is ompared to the droplet size and if �t < r, it is assumed that RT waves havestarted to grow on the surfa e of the droplets. (The droplet a eleration is obtained usingEquation (32)). The life time of the growing RT waves is then tra ked from then on, andwhen the life time ex eeds the hara teristi RT time �t = 1=t, wheret = vuut 2p27� jgt(�l � �)j3=2�l + � (57) atastrophi breakup o urs, whi h immediately reates mu h smaller droplets. TheRT breakup splits the par el so that the new statisti al number is in reased by a fa torof r=�t and the new droplet size is orre ted to ensure onservation of mass.14

2 GOVERNING EQUATIONS2.2.6 The Collision ModelAmong the spray sub-models, the weakest model is the ollision model. This is due tothe built-in grid dependen y, whi h will be explained below. First, ollisions betweenpar els are only signi� ant when the liquid volume fra tion is high. Although all Dieselsprays are dense at the beginning, this is, however, a question of mesh resolution, sin ea omputational mesh with small ells will yield a higher liquid volume fra tion than amesh with larger ells. The ollision model by O'Rourke [26℄ states that:Collision between two par els o urs when they o upy the same omputational ell and the probability for ollision is higher than athreshold value based on the ollision frequen y,where the ollision frequen y is de�ned asv = �4V Nd;small(Dsmall +Dlarge)2jUd;small � Ud;largejE12 (58)where E12 is the ollision eÆ ien y4 [26℄. Redu ing the ell size will have two e�e ts,in reasing the han e for ollision, a ording to Equation (58). But, it will also redu ethe domain in whi h the par els an meet. And sin e two par els not in the same ellhave zero probability of olliding, the probability in rease in Equation (58), by redu ingV , is of no importan e. In the extreme, if the ell size is of the same order as the dropletsize, only one droplet will o upy the ell and no ollisions will o ur.Another weakness of this formulation is that there is an equal han e for ollisionregardless of whether the par els are moving towards or away from ea h other, see Figure2, and the han e for ollision is mu h higher if the pla e of ollision is inside a ell than��������������������������������

��������������������������������

��������������������������������

��������������������������������

��������������������������������

��������������������������������

��������������������������������

��������������������������������

Figure 2: Two di�erent ollision possibilities, in whi h the top pair of par els have equal han e of olliding as the bottom pair.if it is between two ells, where the latter is diÆ ult to ontrol in a omplex geometry.These e�e ts are investigated and dis ussed in more detail in Se tion 5.Although the ollision models developed by Berlemont [5℄ are more a urate, they are omputationally very demanding.Therefore, the following mesh independent formulation is hereby suggested4In the KIVA odes E12 = 1:0: 15

2 GOVERNING EQUATIONSCollision between two par els o urs if their traje tories interse tand the interse tion point is rea hed at the same time, and withinthe integration step.These onditions are, however, also quite omputationally demanding if they are ap-plied to all pairs of par els. Therefore, to sort out impossible ollisions, two prerequisiteshave to be met. First, from Figure 3 it an be dedu ed that in order for two par els to2Ur1

r2

2X

1

1

X

U

Figure 3: Two par els traveling towards ea h other. ollide they have to travel towards ea h other, or U12 > 0, whereU12 = (U1 �U2) � x2 � x1jx2 � x1j : (59)The se ond ondition is that the par els' relative displa ement must be larger than thedistan e between them, or U12�t > jx2 � x1j � (r1 + r2).If these two requirements are met, the par els have a han e of olliding and thepostulate above an be expressed in mathemati al terms. If, �0 2 [0;�t℄ and �0 2 [0;�t℄,where �0 and �0 is the solution to the systemp1 = x1 + �U1; p2 = x2 + �U2; �jp2 � p1j��; � (�0; �0) = 0: (60) ollision will o ur if a random number, � 2 [0; 1℄, is less than the ollision probability P,where P = r1 + r2max(r1 + r2;�12)!C1 e�C2j�0��0j=�t: (61)where �12 = jp2(�0) � p1(�0)j is the minimum distan e between the two traje toriesand C1 and C2 are model onstants. Where C1 is the spatial probability de ay outsidethe swept ylinders by the droplets and C2 is the temporal probability de ay. C1 allowsfor traje tories that are ' lose' to also have a han e of olliding. Sin e both �0 and �0 ontain the information of when the ollision o urs in time, the term e�C2j�0��0j=�t takesinto a ount that the par els must be at the same pla e at the same time.This new ondition for ollision has been implemented in the KIVA ode and theout ome of the ollision is treated in the same way as the original model.16

3 THE TURBULENCE/CHEMISTRY INTERACTION MODEL3 The Turbulen e/Chemistry Intera tion ModelWe all agree that your theory is razy,but is it razy enough?- Niels BohrThis se tion deals with how the hemi al sour e terms in the spe ies transport equation(1) have been modeled. The modeling approa h uses a spe ial te hnique, in whi h areferen e spe ies is introdu ed to ompute both the rea tion rate and the hara teristi hemi al time. As this pro edure is losely onne ted with the modeling, it is explainedin detail.3.1 Introdu tionIt is not possible, with urrent omputer te hnology, to resolve the ame stru ture forpra ti al purposes. Be ause of the thinness and omplex stru ture of the ame, the omputational ell size has to be several orders of magnitude larger than required toresolve the ame stru ture.And sin e it is only possible to resolve variables, e.g., spe ies on entrations, on a s alewhi h is of the same order as the ell size, the onditions in the ombustion zone are thus,in prin iple, unknown. And sin e the sour e term, fm, in the spe ies transport equation(1) is a fun tion of the ombustion zone parameters, it is dependent on variables, not ona grid level, but on a sub-grid level. Hen e, using only grid level information 'straight on'is, in most ases, not appropriate as the ombustion region stru ture unresolved. Be auseof this, orrelating the sub-grid onditions with grid level onditions is a ne essity, sin ethe grid level onditions are the only information available.The remainder of this se tion deals with one approa h to how to treat this problem.3.2 Ba kgroundWith the introdu tion of the Eddy Break-Up model (EBU) by Magnussen and Hjertager[24℄ in 1976, it be ame possible to treat turbulent di�usion ombustion in a su esfulmanner. Sin e then, the EBU model has be ome widespread and widely used in manyCFD odes. The su ess of the model an mainly be attributed to two things, the sim-pli ity of the model, plus, at the time, la k of other di�usion ame models. Although itwas �rst and simple, it was not too simple. On the ontrary. In a re ent study by Gran[14℄, in whi h di�erent alternatives for a ounting for �nite rate hemistry in turbulent ombustion are investigated, it is on luded that the EBU on ept is the most attra tiveapproa h for pra ti al purposes5. Sin e the introdu tion of the EBU model, other models5Although this study was not dire tly related with spray ombustion.17

3 THE TURBULENCE/CHEMISTRY INTERACTION MODELhave been derived, based on other prin iples, su h as the amelet on ept [29℄, the prob-ability density fun tion (PDF) approa h [30℄ or the Lagrangian approa h [6℄. There havealso been extensions to the EBU on ept. However, all of these approa hes, ex ept forthe Lagrangian, have diÆ ulties handling omplex hemistry. The standard EBU onlyuses one global rea tion. Although it is possible to extend it for omplex hemistry, it isnot lear how to do this. For instan e, one way, di�erent from the one used here, an befound in [7℄.The amelet on ept is based on the assumption that the hemi al time s ale is mu hfaster than the turbulen e time s ale. The ame is, hen e, treated using a laminar ap-proa h, where the turbulen e and hemistry have been de oupled. This is not appropriatefor Diesel spray ombustion as this on ept annot treat the important slow hemistry onditions, sin e this assumption is not valid. And for predi ting the, for emissions, im-portant ame lift-o� [9℄, this is ru ial. Although, the PDF, theoreti ally, is the most or-re t approa h, the number and omplexity of the various (unknown) orrelations betweenspe ies and temperature, make the system untreatable. Using a Lagrangian approa h, theprobabilisti Eulerian Lagrangian (PEUL) model an treat omplex hemistry, althoughslow hemistry is treated separately using a redu ed me hanism. In addition, ful�llingthe balan e equations lo ally is a problem in this approa h. Thus, it leaves some roomfor improvement.The Partially Stirred Rea tor (PaSR) on ept by Karlsson [17℄ is an extension of theEBU approa h. The derivation is, however, somewhat ad ho and based on several pos-tulates. It has, however, proven to work very well for turbulent Diesel spray ombustionand provides the �rst omplex hemistry treatment of the problem.This work is a ontinuation of the work arried out by Karlsson [17℄.3.3 The Referen e Spe ies Te hniqueSolving a hemi al system in a fully oupled fashion, when the number of rea tions isof the order of about 100, is far too expensive. Hen e, to treat a detailed me hanism,another approa h has been developed in whi h the rea tion set is solved sequentially, i.e.,the rea tions are a ounted for one after another and the spe ies are updated after ea hrea tion and fed into the next rea tion. Although the orre tness of this is yet to beproven, it is based on the same reasoning as the widely used time splitting te hnique,whi h is used for instan e in the KIVA odes, and thus, it appears to be a eptable.An alternative pro edure is yet to be developed. Still, due to the sti�ness of the systema spe ial te hnique, using a referen e spe ies for ea h rea tion, has been applied. Thisapproa h is appropriate when the integration step is larger than the smallest hemi altime s ale, sin e the same algorithm an be applied to all rea tions without he kingwhether they are slow or fast (equilibrium).Normally, by using this approa h, the spe ies in danger of being driven to negative on entration restri ts the integration step.18

3 THE TURBULENCE/CHEMISTRY INTERACTION MODELThe referen e spe ies te hnique is used in the KIVA odes, [1, 2, 3℄, but is explainednowhere. It will therefore be explained here.Consider the following elementary rea tion#1 1 + #2 2 kf ;kb*) #3 3 + #4 4 (62)where #i and i are the rea tion stoi hiometri oeÆ ients and molar on entrations,respe tively. For simpli ity, it is assumed that all #i = 1, sin e it is more spe i� fordetailed hemistry, where the maximum value of #i is two for the radi al re ombinationrea tions.The rate equation for the above rea tion isd 1dt = d 2dt = � _!; d 3dt = d 4dt = _! (63)where _! is de�ned similar to (26), but without the subs ript.When solving Equation (63), it is ommonly a epted that one use an impli it s heme,sin e the rea tion rate _! is dependent on both temperature and on entration and andi�er in value by several orders of magnitude when the rea tions pro eed. The problemis illustrated by dis retizing (63) for the forward rea tion and assuming _! > 0 n+11 � n1� = � _!x ) n+11 = n1 � � _!x (64)where _!x is either evaluated at x = n, x = n+ 1 or a ombination of both.If _!x is evaluated at x = n, it is lear from (64) that if the time step, � , is suÆ ientlylarge, there is a danger of n+11 be oming negative. Obviously, this is in orre t, sin e the on entrations should tend towards the equilibrium solution _! = 0, whi h for i = 1 yields eq1 = kb 3 4kf 2 !eq (65)To over ome this de� ien y, _!x is evaluated at x = n+1 semi-impli itly in the followingway. Di�erentiating _! with respe t to time yields (kf and kb are held onstant)d _!dt = kf d 1dt 2 + 1d 2dt !� kb d 3dt 4 + 3d 4dt ! (66)whi h together with (63) yieldsd _!dt = � (kf( 2 + 1) + kb( 4 + 3)) _! = �� _! (67)where � = kf( 2 + 1) + kb( 4 + 3) (68)19

3 THE TURBULENCE/CHEMISTRY INTERACTION MODELDis retizing Equation (67) with a semi-impli it s heme yields_!n+1 � _!n� = ��n _!n+1 ) _!n+1 = _!n1 + �n� (69)To update the spe ies using _!n+1, with the above form of _!, would not be appropriateeither sin e the expression n+11 = n1 � � _!n1 + �n� �!1! n1 � _!n�n = n1 � " kf 1 2 � kb 3 4kf( 1 + 2) + kb( 3 + 4)#n (70)does not tend toward the equilibrium on entration eqi , see Equation (65). In order toover ome this de� ien y, the referen e spe ies on ept is introdu ed. The referen e spe iesis de�ned as 'the spe ies most in danger of being driven negative', thus, it is the spe ieswhi h is being onsumed by the rea tion and has the lowest on entration. For the sakeof argument let's assume the referen e spe ies to be r = 1, i.e., spe ies one. The partnerto spe ies one is, thus, 2. Note, thus, that 2 > 1 and by assuming that kf 2 is largerthan the other terms in (68) we have �n � (kf 2)n (71)With the above expression for �, _!n+1 an now be evaluated. For the referen e spe iesthis yields n+11 = n1 � � _!n1 + �n� �!1! n1 � _!n�n = n1 � kf 1 2 � kb 3 4kf 2 !n = kb 3 4kf 2 !n (72)This way, the equilibrium ondition for ea h rea tion is automati ally ensured andthere is no need to treat equilibrium rea tions separately from the slow, kineti ally on-trolled rea tions6.3.4 The Partially Stirred Rea tor Con eptIn the Partially Stirred Rea tor (PaSR) approa h, a omputational ell is split into twodi�erent zones, one zone, in whi h all rea tions o ur, and another, in whi h no rea tionso ur. Thus, the omposition hanges due to mass ex hange with the rea ting zone.Furthermore, the rea ting zone is treated as a perfe tly stirred rea tor (PSR), in whi hthe omposition is homogeneous (every spe ies is assumed to be perfe tly mixed with theother ones). This allows us to disregard any u tuations when al ulating the hemi alsour e terms.6This statement is not really true, sin e eq1 requires the equilibrium values of all the spe ies. However,obtaining the true equilibrium value requires an iterative pro edure whi h is omputationally demandingand therefore avoided. 20

3 THE TURBULENCE/CHEMISTRY INTERACTION MODELBe ause the omputational ell is divided into two zones, the question now arise of howto split the ell into the rea ting and non-rea ting part. How large is the mass fra tionof the mixture taking part in the ombustion, and what governs the omposition of it?As it turns out, the key issue in the PaSR approa h lies in answering these ques-tions, as they des ribe the onne tion between the sub-grid information and the grid levelinformation.The rea tive mass fra tion will be de�ned below as the al ulation is advan ed onetime step, from 0 to 1, see Figure 4.������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������κ *

CC0 1C

Figure 4: Con eptual pi ture of a PaSRFirst, the model distinguishes between three molar on entrations: 0 is the averaged on entration in the feed of the ell and may be onsidered as theinitial averaged on entration in the ell. is the unknown on entration in the rea tion zone on a sub-grid level in the unknownrea tive fra tion of the ell material. 1 is the sought for, time averaged rea tor-exit on entration. This is also the averaged on entration in the ell.Having established this, the rea tor exit on entration an be obtained, from mass onservation onsiderations, looking at Figure 4, as 1 = �� + (1� ��) 0 (73)where �� is the mass fra tion of the mixture that rea ts. Investigating Equation (73)further, it is obvious that 1 an be obtained using a linear interpolation between 0 and . Thus, plotting 0, and 1 in the diagram of Figure 5, the following observations anbe made.The whole pro ess an be divided into two sub-steps, pro eeding in parallel. In Figure5, they are marked as I and II: 21

3 THE TURBULENCE/CHEMISTRY INTERACTION MODELc

c0

1cII

τ τ mix

conc

entr

atio

n

time

I

f(c)

Figure 5: The rea tion/mixing step pro edureI The initial on entration in the rea tion zone hanges from 0 to as it rea ts,II The rea tive mixture, , is mixed with the unrea tive mixture, 0, by turbulen e,resulting in the averaged on entration 1Sin e 1 is the initial value for the next time step, the time between 0 and 1 must be theintegration step, � . And sin e the turbulen e mixes with 0, the time di�eren e between and 1 must be the hara teristi time for turbulen e, �mix. The total rea tion time forpath I in Figure 5, an, thus, be onsidered as being in reased by the mixing time, from� to � + �mix.From Figure 5 the following relations an be obtained, assuming that the slope of the urve is equal to the rea tion rate in the rea tion zone, . 1 � 0� = � 1�mix = fm( ) (74)The last assumption, � 1�mix = fm( ) (75)is also in agreement with the main assumption of the stationary amelet approa h in [20,p.630℄. Thus, 1 � 0� = � 1�mix (76)whi h an be rearranged as 1 = �� + (1� ��) 0; �� = �� + �mix (77)22

3 THE TURBULENCE/CHEMISTRY INTERACTION MODELwhi h learly states that if the integration step is larger than the turbulent mixing time,i.e., �� � 1, then most of what omes out of the rea tor will be from the rea tionzone. Having established the rea tive fra tion, ��, it remains to eliminate the unknownparameter in order to obtain 1.Let us re apitulate the hemi al sour e term in the spe ies transport equation:fm( ) = NrXr=1 f rm = Mm NrXr=1 (#00mr � #0mr) krf NsYs=1 #0srs � krb NsYs=1 #00srs != Mm NrXr=1 #00mrkrf NsYs=1 #0srs + #0mrkrb NsYs=1 #00srs !| {z }�1� Mm NrXr=1 #00mrkrb NsYs=1 #00srs + #0mrkrf NsYs=1 #0srs !| {z }�2= NrXr=1 (�r1 � �r2) = �1 � �2(78)

where �1 and �2 are both positive. Above, the rea tion rate fm( ), is evaluated at therea tion zone onditions . However, sin e is sub-grid information, it must be repla edby something known on a grid level.Using Taylor expansion yieldsfm( ) = fm( 1) + �fm� ( � 1) + 12 �2fm� 2 ( � 1)2 (79)where the diagonal elements in �2fm=� 2 are zero and the o�-diagonal elements are ne-gle ted. It should be noted that the Taylor expansion does not ontain any derivatives ofa higher order than two. Further, it is assumed that the dominating term in the Ja obianis with respe t to the referen e spe ies, i.e., �fm=� r. The hemi al time an then bede�ned from the Ja obian element 1� = ��fm� r : (80)(Obtaining the hemi al time involves the same assumptions as when obtaining (71),whi h is dis ussed in Se tion 3.3). Equation (79) now readsfm( ) = fm( 1)� � 1� (81)By substituting the expression for in (81) with (77), we obtain 1 � 0� = fm( ) = fm( 1)� 1� �� 1�� � 1� ���� 0�� 1� (82)23

3 THE TURBULENCE/CHEMISTRY INTERACTION MODELwhi h leads to 1 � 0� = fm( 1)� 1� �� ( 1 � �� 1 � (1� ��) 0) = fm( 1)� 1� ��� �� ( 1 � 0) (83)The terms ontaining 1 � 0 an, now, be grouped together as�1� + 1� ��� �� � ( 1 � 0) = fm( 1) (84)and after manipulating the lhs, it takes the form�1� + �mix� � � ( 1 � 0) = 1 � 0� � + �mix� = 1 � 0� 1� = fm( 1) (85)whi h �nally yields 1 � 0� = �fm( 1) (86)where � = � � + �mix (87)and the �nal result an be summarized as:fm( ) = �fm( 1) (88)This is the orrelation for the hemi al sour e term on a sub-grid level, based on theresolved information. Note that, fm( 1) is treated impli itly, by using information fromthe next time step, as des ribed in detail in Se tion 3.3.The ex hange pro ess between the fresh, unrea ting mixture and the burned gases isa ounted for by the hara teristi time �mix.There is a wide range of s ales in turbulent ows, from the largest eddies down to themole ular level, and to a ount for all these di�erent s ales using only one hara teristi value is of ourse a great simpli� ation. The question is what to base the representation�mix on. In a study by Kj�aldman [19℄ three di�erent options have been investigated. Theywere related to the Kolmogorov time s ale,�k � (�=")1=2 ;the Taylor time s ale, �t � k=";and a geometri al mean of the Kolmogorov and Taylor time s ales,�k � p�t�kwhi h was proposed by Karlsson and Chomiak [18℄. Kj�aldman reported that the Karlssonand Chomiak approa h was better than using only the Kolmogorov time, but the best24

3 THE TURBULENCE/CHEMISTRY INTERACTION MODELresults were obtained using the Taylor s ale. This result is in agreement with the �ndingshere, where �mix has been obtained from the k-" equation,�mix = Cmixk" ; (89)and the model onstant Cmix was set to 0.005. This onstant an, however, vary between0.001-0.3, depending on the ow. It should be noted that using �mix � k=" might notalways yield the best results, depending on the ow situation, as well as the hemi alme hanism. Moreover, the small value of Cmix might be an indi ation that the Karlssonand Chomiak approa h will yield better results using a larger onstant in front of p�t�k.3.5 The PaSR Model: Observations and Impli ationsTreating the rea tive fra tion in a Lagrangian fashion, the governing equation for therea ting spe ies on a sub-grid s ale isd dt = Dsgsr2 + f( ) (90)where Dsgs is the sub-grid di�usion. Solving (90) presents some fundamental problems.First, the problem is not well posed, sin e al ulating (tn+1) requires (tn), whi h isunknown. Se ond, the spatial resolution is too oarse to resolve the dis retized operatorr. Sin e the r operator is unresolved, the operator Dsgsr2 is modeled by the turbulent hara teristi time as Dsgsr2 � � 1�mix (91)where �mix is obtained from the turbulen e model. Note that must be repla ed by � 1in order to satisfy the equilibrium onstraint ! 1 as d =dt ! 0, f( ) ! 0. Equation(90) now reads d dt = � � 1�mix + f( ); (92)whi h is the same equation as used in the probabilisti Eulerian Lagrangian (PEUL)model [6℄; if �mix = �IEM , where �IEM is de�ned as the hara teristi time for Intera tionEx hange with the Mean.However, using relation (74) we obtaind dt = �f( ) + f( ) = 0 (93)implying that is in equilibrium/steady-state.This implies that the rea tions in the rea tive zone pro eed at a rate determined byhow strong the mixing, or intera tion with the mean, is.25

3 THE TURBULENCE/CHEMISTRY INTERACTION MODELFurther, let us onsider the rea tion rate for rea tion if im( ) = �i1 � �i2; (94)and re all the de�nition of the referen e spe ies, r, whi h is used to al ulate the rea tionrate. Linearizing Equation (94) around r, by using the information at the urrent timestep, we �nd that the linear form of (94) is given by (sin e r is only present in �i2)f im( ) = (�i1)0 � (�i2)0 r 0r (95)Di�erentiating (95) yields,df im( )dt = �(�i2)0 0r d rdt = �(�i2)0 0r f im( ); (96)sin e (�i1)0 and (�i2)0 are held onstant. To obtain the rea tion rate at the next time step,a semi-impli it s heme is applied to (96), (supers ript zero is the value at this time stepand supers ript one denotes the value at the next time step)df im( )dt = f 1 � f 0� = �(�i2)0 0r f 1 (97)whi h yields, f 1 = f 0 0r + (�i2)0� 0r = � 1� 0r; � = � 0r + (�i2)0�f 0 (98)Equation (98) yields an expression for both the rea tion rate at the next time step andthe hemi al time. By inserting (98) in (88) we �ndf im( ) = �f im( 1) = � � + �mix f im( 1)(98)= � 0r+(�i2)0�f0� 0r+(�i2)0�f0 + �mix f 0 0r + (�i2)0� 0r= f 0 0r 0r � f 0�mix + (�i2)0� (99)With the above expression for the rea tion rate (99) two limiting extremes an beidenti�ed, the homogeneous rate, �mix ! 0, and, the mixing ontrolled rate, where theterm f 0�mix is dominant.Investigating the �rst ase, �mix ! 0, yieldsf im( ) = f 0 0r 0r � f 0�mix + (�i2)� ! f 01 + (�i2)�= 0r (100)

26

3 THE TURBULENCE/CHEMISTRY INTERACTION MODELwhere the rea tion rate is redu ed to a quasi-laminar approa h, where = 1, a ording toEquation (77), in the rea tion rate terms.Investigating the se ond ase, where f 0�mix dominates, yieldsf im( ) = f 0 0r 0r � f 0�mix + (�i2)� ! � r�mix (101)whi h is equivalent to the eddy break-up rate, in parti ular the Magnussen and Hjertager[24℄ rea tion rate. To illustrate this, let us start with the expression for the rea tion rategiven in [24℄ f( ) = �A"k min� f ; os ; B p1 + s� = �A 1�mix min� f ; os ; B p1 + s� (102)where A and B are model onstants and f , o and p are the fuel-, oxidizer- and produ t on entration, respe tively. It should be noted that the Magnussen model is valid onlyfor one global rea tion, in the form, f + s o ! (1 + s) p (103)If a more detailed me hanism is used with, for instan e, two and/or more than onerea tion, it is not lear how to use the Magnussen model. Moreover, the un ertainties indetermining the model onstants in rease dramati ally. The produ t term, B p=(1+ s) isnot important in our analysis be ause it is introdu ed to des ribe the temperature level orresponding to the moment when mixture ignition is a hieved.Supposing, for simpli ity s = 1, in the rea tion of the form (103), the quasi-laminarrea tion rate is given by the expressionjf( )j = kf f o (104)whi h, essentially reads jf( )PaSRj � f o = f (1� f) (105)whi h redu es to the Magnussen and Hjertager rea tion ratejf( )Magnussenj � min( f ; 1� f) (106)The plot of the di�erent expressions for the rates, jf( )j, an be seen in Figure 6. Thus,to des ribe the Magnussen limiting spe ies, the minimum operator is introdu ed and fromthis, it be omes lear that the Magnussen limiting spe ies and the referen e spe ies areequivalent, as the minimum operator an be seen as approximating the fun tion (1� ).Another interesting observation is that the PaSR rea tion rate (99) an be dedu ed ina ompletely di�erent way.The total ele tri al resistan e, R, of two parallel onne ted resistan es R1 and R2 isobtained in the following way 1R = 1R1 + 1R2 (107)27

3 THE TURBULENCE/CHEMISTRY INTERACTION MODEL

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

fuel concentration

reac

tion

rate

Magnussen

PaSR

PaSR

Figure 6: The rea tion rates as a fun tion of fuel on entration. The two PaSR graphsdi�er by a onstant two.and by viewing the two limiting rea tion rates, (100) and (101) as parallel pro esses, theoverall rea tion rate an be written as1f im( ) = 1f(�mix ! 0) + 1f(�mix !1)= 1 + (�i2)0�= 0rf 0 � �mix 0r= 0r + (�i2)0� � f 0�mixf 0 0r (108)whi h obviously is equivalent to (99).In light of this, an analogy between ele tri al ir uits and ombustion an be drawn:As the resistan e of two parallel onne ted resistan es is determined by their harmoni average7, likewise the overall rea tion rate in the PaSR model is determined by the har-moni average of the quasi-homogeneous rate and the turbulent mixing ontrolled rate. Itis important to note that both rates must be de�ned using the same limiting, or referen e,spe ies.Even though the rea tor is split into two parts, there is only one ommon temperaturefor both zones. It has not been established whether or not this simpli� ation is important.The produ tion of NOx is sensitive to the temperature and may require the onditionsin the rea tion zone to be restored and this may require keeping tra k of two di�erenttemperatures. This is a question for future resear h and has not been investigated here.

7The harmoni average only di�ers by a fa tor of two sin e it is de�ned as 2R = 1R1 + 1R228

4 IMPLEMENTING A LAGRANGIAN REPRESENTATION OF THE SPRAY4 Implementing a Lagrangian Representation of theSpray This one is tri ky - it involves imaginary numbers, like eleventeen.- Bill Watterson (Calvin and Hobbes)4.1 Introdu tionThis se tion deals with how to implement a Lagrangian representation of the spray in anEulerian framework. At �rst glan e, this might seem trivial, but as will be shown below,there are some intri ate details whi h an lead to erroneous behavior if are is not taken.It is the author's experien e that the way the spray models have been implemented isof importan e sin e this an, and must, in uen e the behavior of the spray sub-models.Hen e, omparing spray sub-models for di�erent odes an be misleading, sin e a model'sbehavior, good or bad, might depend on the implementation and not on the model itself.By revealing these details it is the author's wish that by doing so it will help someoneand any mistakes made by the author will be dis overed.The implementation below des ribes how the models have been implemented in FOAM[36℄. This is something that might belong in a manual, but as this is a universal problemfor all CFD odes, this thesis might be the proper pla e. Moreover, this knowledge, orinformation, has not been found anywhere.4.2 Tra kingTra king the par els is the most fundamental operation. This in ludes the 'in-whi h- ell-am-I' fun tionality. Of ourse, the par els must have a position, but they must alsobelong to a ell. This is fundamental if the par els are to intera t with the surroundingEulerian �eld. Currently, there are two main approa hes for tra king a par el, they willhere be referred to, for la k of a better word, the Lose-Find (LF) algorithm and theFa e-To-Fa e (F2F) algorithm. The LF algorithm moves the par el along its path, ud�t,disregarding the mesh, and then �nds out whi h ell it belongs to by performing a sear h.This approa h is very easy to implement and suitable for expli it odes in whi h theintegration step must be small and therefore the par el does not move mu h. The sear halgorithm thus starts by looking for the par el in the same ell (whi h is where it mostoften is). If the par el does not belong to the same ell, but has moved into a new ell,the algorithm sear hes the neighboring ells, until the par el is either found or the sear his expanded further to over the entire mesh. However, sear hing the entire domain is atime- onsuming task and should be avoided; it is most probably an indi ation that thereis a problem in the tra king, or sear h, algorithm.For an impli it ode, in whi h the integration step is mu h larger, this pro edure isinappropriate and there is no guarantee, sin e there is no he k, that the par el passes29

4 IMPLEMENTING A LAGRANGIAN REPRESENTATION OF THE SPRAYevery ell along its path. See Figure 7, in whi h a par el moves from position I to IIduring one time step, thereby missing several ells in-between.��������

��������

��������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

��������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

I

IIFigure 7: Possible s enario using the LF tra king when the time step is too large.This situation is una eptable for both a ura y and stability reasons, as it will reatea sour e �eld whi h is very 'spiky' and has a high intensity, sin e the LF algorithmdistributes the sour e from the par el to only one ell during the full time step. Thislimits the maximum time step and the advantages of using an impli it ode are therebydiminished.The F2F tra king algorithm only requires one sear h, whi h is performed when thepar el is �rst introdu ed into the domain. From then on there is no need to look for itagain (if the tra king algorithm is robust enough). Consider Figure 8, in whi h the par eln̂

n̂

n̂n̂

x

up

2

4

3

1

Figure 8: Par el in a ell.is inside a ell and its position and velo ity are x and up, respe tively. A ell is de�nedby its fa es and the ell fa es are de�ned by their outward pointing fa e normals, n̂i, andsome arbitrary point on the fa e, most ommonly the fa e enter, i. A par el does notleave its ell during the integration step if �i > 1 for all i, where�i = ( i � x) � n̂i(up�t) � n̂i : (109)If the par el does not hange ell, all the Lagrangian sour e terms in the Eulerian equations an be al ulated and the par el moved to the end of its traje tory.30

4 IMPLEMENTING A LAGRANGIAN REPRESENTATION OF THE SPRAYHowever, if some �i < 1, the par el hits fa e i and � = min(�i) > 0 is the fa e it hits�rst. If the par el hits a fa e, the par el is then moved for ��t time to its new positionon fa e i, whi h is xnew = xold + �up�t. The sour e terms are al ulated for ell i andthe par el now belongs to the neighboring ell of fa e i. This pro edure is then repeateduntil the par el has moved the whole integration step �t.This ensures the par el going through every ell it passes without any error in thesour e term al ulation as it hanges ells and there are no time step restri tions due toits velo ity.Using the F2F tra king te hnique has, however, some quite intri ate loop-holes thatmay result in the par els being lost, if are is not taken. These loop-holes and how toavoid them are dis ussed in Appendix A.As for the 'in-whi h- ell-am-I' fun tionality, the easiest way to he k whether or nota par el belongs to a ell is by he king that �i <= 0 for every fa e, where�i = (x� i) � n̂i (110)x is the position of the par el, i a referen e point on fa e i and n̂i the fa e normal. Thisis a fool-proof method when the ell is a tet, but on a hex mesh, this method will fail ifthe situation looks as in Figure 9. However, using the F2F tra king algorithm on a hex

Figure 9: Par el in a on ave ell, where the lookup will fail using the method suggestedhere.mesh, the tet de omposition has already been performed and thus the tet's are alreadyavailable.4.3 The Equation of MotionDis retizing and integrating the equation of motion for a par el presents some problemsthat an be ategorized under the following ategories:� Turbulent dispersion (Unresolved random u tuations in the Eulerian phase) .� Lagrangian / Eulerian des ription dis repan y.31

4 IMPLEMENTING A LAGRANGIAN REPRESENTATION OF THE SPRAY� Lagrangian / Eulerian oupling.The meaning of ea h one of these will be overed in detail below.All of the three problems mentioned above are present not only in the momentumequation, but in the mass and energy equations as well.4.3.1 Turbulent DispersionBe ause the Eulerian momentum equations have been averaged, in some way, the instan-taneous velo ity u, a ting on the par el, is thus unknown. Writing Equation (32) in termsof the known averaged and unknown u tuating velo ity then readsduddt = �ud � (�u + u0)�u + g (111)where u0 is the velo ity u tuation, whi h has to be modeled, see for instan e [12, 28℄(u0 is also in luded in the evaluation of �u and CD.). In order to obtain the velo ityof the par el at the next time level, Equation (111) has to be integrated, and sin e theanalyti al expression of the r.h.s of (111) is unknown, it is assumed to be onstant duringthe integration step. Hen e, integrating (111) yields:ud(t0 +�t)� ud(t0) = � (ud(t0 +�t)� (�u(t0) + u0)) �t�u + g�t (112)where the par el velo ity on the r.h.s has been evaluated impli itly and the gaseousvelo ity expli itly. Thus, the velo ity for the next time step is obtained fromud(t0 +�t) = ud(t0)�u +�t(�u(t0) + g�u)�t + �u + �t�t + �uu0 (113)The dispersion e�e t will also be present when al ulating the par el position,dxddt = ud; ) xd(t0 +�t)� xd(t0) = Z t0+�tt0 ud(�)d� (114)whi h is evaluated expli itly, and the random u tuation a ounting for the turbulen e,x0 is added xd(t0 +�t) = xd(t0) + �tud(t0) + x0 (115)The purpose of in luding x0 and u0 is to a ount for the in uen e of the randomturbulent u tuations on the par el. The turbulent dispersion is important when therelative velo ity is relatively low, like in parti les transported by air, see for instan e [23℄.In diesel spray appli ations, it is, however, of minor importan e, sin e the relative velo ityis very high and the liquid evaporates qui kly.The implementation of turbulent dispersion will not be des ribed here. It is modeledusing the 'random walk pro edure' as des ribed in [28, 4℄32

4 IMPLEMENTING A LAGRANGIAN REPRESENTATION OF THE SPRAY4.3.2 Lagrangian-Eulerian Des ription Dis repan yBe ause the par el moves ontinously through the omputational mesh and the gaseousvelo ity is stored in �xed positions, either ell enters or verti es, how to evaluate thegaseous velo ity, �u in Equation (113), at the par el position is an open question. Thedi�erent strategies are either to use the velo ity at the position losest to the par el, orto interpolate the velo ity, using a set of the losest positions. In Figure 10, taken from[4℄, the 2-D spray patterns from four di�erent CFD odes are shown. From left to right,they are SPICE, KIVA, CHAMPION and FLUENT. They represent the same ase, at thesame time, and although the turbulent dispersion makes the sprays look more 'random'and di�erent breakup models have been used, one important and obvious, observation an be made. They all are very di�erent. The most signi� ant reason for the di�eren e in

Figure 10: Spray patterns for di�erent CFD odes.the spray pattern is due to the di�eren e in how the evaluation of the relative velo ity isperformed. The SPICE and CHAMPION ode use interpolation, while the others; KIVAand FLUENT, use the value at the losest position. The par els at the tip of the sprayare traveling in an adverse pressure gradient, hen e the radial velo ity omponent, atthe tip, is in the outward dire tion. The e�e t this has on the par els is most visible inthe non-interpolating odes, KIVA and FLUENT, where the par els lose to the axis ofsymmetry tend to go outwards, thus reating a hollow-like shape. For the odes usinginterpolation the tip of the spray is sharper. This behavior depends on the dis ontinuityin the gaseous velo ity �eld, as seen by the par el. Thus, if this behavior is to be avoided,the type of interpolation used is important, sin e not all methods of interpolation are ontinuous. It also matters if the velo ities are stored in a staggered arrangement or not.33

4 IMPLEMENTING A LAGRANGIAN REPRESENTATION OF THE SPRAYThe most ommon and easiest method of interpolation is inverse distan e weighting,where the velo ity, �u, is evaluated by using the losest verti es, or fa es, as�u(x) = 1P�i Xi (�i�ui); �i = 1max(Æ; jx� ij) : (116)where x is the par el position and i the position at whi h the velo ity �ui is stored. Æis some small number to prevent division by zero. As the par el moves from one ell toanother it hanges the set from whi h the velo ity is interpolated and for the methodabove, this swit h makes �u appear to be non- ontinuous. Therefore, another method willbe presented below, in whi h the par el experien es a ontinuous ow �eld.This, however, assumes that the par el is in a tetrahedron. In Figure 11, the distan ei

iα

x

vertex i

β

base iFigure 11: Interpolating inside a tetrahedron.from vertex i, where the gaseous velo ity, ui, is stored, to base i is �i+�i and the distan efrom base i to the par el is �i. The interpolated velo ity at the par el position, x, is thenu(x) = 4Xi=1 �i�i + �iui: (117)Cal ulating all the �i and �i is, however, quite expensive.Using an interpolated velo ity also requires some onsideration when al ulating themomentum ontribution to the gas. If the par el only intera ts with the ell it is in,problems an easily arise. Consider the situation depi ted in Figure 12, in whi h thepar el is in ell 2. The velo ity of the par el is Ud and U2 < Ud < U1.The interpolated Eulerian velo ity at the par el position an, thus, be larger than thevelo ity of the par el. This means that the par el experien es an a eleration and willin rease its velo ity. Thus, it will gain momentum, whi h it must take from the Eulerianphase. And, sin e the par el only intera ts with the ell it is in, this will redu e themomentum in that ell and, hen e, lower the Eulerian velo ity.But, how an a par el gain momentum in a ell whi h has a lower velo ity than thepar el itself? This problem has not been addressed, sin e it is not as serious for the34

4 IMPLEMENTING A LAGRANGIAN REPRESENTATION OF THE SPRAY21

U U1 2

Ud

Figure 12: Par el in a velo ity �eld with a gradient.momentum as it is for the energy, whi h will be addressed in Se tion 4.4. In fa t, itappears to redu e the grid dependen y, whi h will be shown in Se tion 5. It is also aquestion of interpretation. Is the the velo ity in the ell a representative average or justthe value at the position it is stored?4.3.3 Lagrangian-Eulerian CouplingThe last item, regarding the equation of motion, is how to handle the Lagrangian-Eulerian oupling. One of the purposes with the FOAM implementation is portability and mod-ularity. The spray pa kage should fun tion like an add-on module and it should be easyand fast to add to an already existing ode, with a minimum of modi� ations. The La-grangian and Eulerian al ulations have therefore been fully de oupled. Advan ing the al ulation one time step is performed in the following manner:The Eulerian solution is frozen at time level n and the par els are advan ed one byone, to time level n+1. All the sour e terms in the mass, momentum and energy equationare thus evaluated using the information from time level n. When the Lagrangian phaseis done, the Eulerian al ulation is performed using the Lagrangian sour e terms fromtime level n + 1.Performing a al ulation in a fully oupled manner would not only be very expensive,but most likely also unstable, at least for the momentum. This an easily be realized.Imagine a par el traveling along a path, thus modifying the gas �eld. It is not likely thatit will travel along the same path through the same ells in every iteration if the gaseousvelo ity �eld hanges. Thus, if a par el is lose to a fa e, and it passes a di�erent ellevery iteration, onvergen e will never be obtained. And sin e Murphy's Law8 alwaysapplies, oupled momentum al ulations should be avoided. This problem is also linkedwith the par el only intera ting with the ell it is urrently in. Another onsequen e of thede oupling should also be mentioned. Imagine a onstant volume vessel, �lled with zerovelo ity air. The �rst par els inje ted into the vessel will experien e zero gas velo ity,8Edward A. Murphy, Jr. - If there are two or more ways to do something, and one of those ways an result in a atastrophe, then someone will do it (1949). Murphy's Law is, however, more ommonlyknown as - Anything that an go wrong, will, whi h is Finagle's Law of Dynami Negatives35

4 IMPLEMENTING A LAGRANGIAN REPRESENTATION OF THE SPRAYthe relative velo ity will therefore be higher than it should and the liquid penetrationwill most likely be under-predi ted. Liquid penetration therefore not only depends onbreakup, or the evaporation model, but also on the integration step.4.4 The Liquid Mass and Heat TransferThe de oupling of the Lagrangian and Eulerian al ulation is most important for theenergy equation. Integration step restri tions in impli it CFD odes are mainly due tothe energy/evaporation and too large time steps may result in negative gas temperatures.This is due to the fa t that every par el passing a ell experien es the same temperature.Thus, if many par els pass through the same ell, the energy required to evaporate thepar els may result, when the total energy drainage is summed up, in that there is moreenergy taken out of the ell than is a tually available, hen e, the negative temperature,whi h most ertainly will rash the al ulation. One way to over ome this de� ien y is topass a opy of the enthalpy and spe ies �eld to the Lagrangian phase. When the par elevaporates these opies are modi�ed and subsequent par els experien e a more orre tenvironment. So, when the 100:th par el passes through a ell, the par el will experien ea older environment and the temperature is less likely to be driven to a negative value.As in the previous se tion, the question now arises - How to evaluate the temperature atthe par el position. As mentioned earlier, the gaseous temperature at the par el positionis not obtained by interpolation. This is due to the following reason.Imagine Figure 12 again, but now with the temperature instead of velo ity [25℄. Thedroplet is in ell 2 and T2 < Td < T1.Interpolating the gaseous temperature may thus result in the gaseous temperaturebeing higher than the droplet temperature. Hen e, heat is transferred from the ell, witha lower temperature than the droplet, to the droplet and the 2:nd law of thermodynami shas been violated.And the on lusion is that interpolation is not always better.When solving the mass and energy equation numeri ally, a semi-impli it method ispreferable, espe ially for the evaporation, as this prevents the mass, or diameter, frombeing driven to negative values. Hen e the new diameter of the droplet is obtained asDn+1 �Dn�t = �Dn+13�ne ) Dn+1 = Dn1 + �t=(3�ne ) (118)As the mass of the par el be omes lower than some small value it is removed from the omputations and the mass, momentum and energy added to the gas for onsisten y. Thenew par el temperature is also obtained semi-impli itly from (40) as:T n+1d � T nd�t = (T n � T n+1d ) f�h!n � hv l;d�e!n (119)36

4 IMPLEMENTING A LAGRANGIAN REPRESENTATION OF THE SPRAYwhi h yields T n+1d = "1 + f�h�t#�1 "Td +�t f�hT � hv l;d�e!#n (120)4.5 The Breakup ModelThe purpose of the se ondary breakup model is to redu e the hara teristi size of thepar els due to aerodynami for es. Among the spray sub-models, it is the se ondarybreakup model that has attra ted the most attention. Modifying the breakup model isrelatively straightforward, sin e the mass, heat and momentum transfer depend on thesize of the droplets. It is, thus, by ontrolling the size of the par els possible to in uen eboth evaporation rate history and liquid penetration, with only one model.The rate expression used to al ulate the size of the parent par el in Equation (52) isdis retized using an impli it method:rn+1 � rn�t = �rn+1 � r �kh ; ) rn+1 = �1 + �t�kh��1 �rn + �t�khr � (121)The size of the hild par el, r , stripped from the parent par el, is used to al ulatehow mu h mass that has been stripped from the parent par el. But, sin e adding a par elevery time step would lead to a dependen e of time step and an abundan e of smalldroplets, a limit of how mu h mass must be stripped before a new par el is introdu edinto the al ulation, is used. This is typi ally 5-10 % of the parent par el mass.If a new par el is introdu ed, the hild par el inherits the statisti al number of dropletsfrom the parent par el and its radius is set to r . The new parent radius is dedu ed, asnormal, from the rate equation and the number of droplets in the parent par el is set sothat the liquid mass is onserved.The RT mode of the KHRT breakup model does not introdu e any new par els. WhenRT breakup o urs, the par els new statisti al number of droplets is set toNp;new = max(1; r�t )Np;old; (122)and the new radius is set to onserve mass.4.6 The Lagrangian Solution Pro edureBe ause of the de oupling of the liquid and gas phase, the liquid phase is treated �rst.This is done in the following way:1. Existing par els are moved up to the urrent time, while at the same ex hangingheat, mass and momentum. During this phase the sour e terms for the phaseintera tion are al ulated. 37

4 IMPLEMENTING A LAGRANGIAN REPRESENTATION OF THE SPRAY2. The mass and number of par els to be inje ted during the urrent time step are al ulated and the new par els are inje ted, either sequentially or all at the sametime, and moved up to the urrent time.3. Clean up the spray. Par els with very small mass are removed from the al ulation.4. The various sub-models are applied sequentially: breakup, ollision and turbulentdispersion.The par els have now been moved from time level n to time level n + 1.

38

5 RESULTS AND DISCUSSION5 Results and Dis ussion \ Oh bother", said Pooh- A.A. MilneThe results summarized in this se tion are �ndings that have not been in luded in thepapers. The reader is, therefore, referred to the papers for results regarding experimental omparisons and engine appli ations.In this se tion, the various parametri studies behind the above mentioned resultsare presented, sin e these studies often have to be left out of the papers due to volumerestri tions in the material presented for publi ation.All the studies below have been done in a onstant volume, sin e this allows for more ontrol over the simulation.By in luding many spe ies and rea tions in the simulations, the omputational time isnaturally in reased. However, the in rease is not dramati . Typi al run times are in theorder of one hour to one day for a 2D ase and one to two weeks for a 3D ase, depending,of ourse, on how long the simulation time is.The omputers used to perform the simulations were either LINUX based PCs or anSGI Origin2000.5.1 Mesh Dependen e AnalysisThe liquid penetration is sensitive to the ell size, espe ially in the radial dire tion. As an0.5x0.5mm

0.5x1.0mm

1.0x0.5mm

1.0x1.0mm

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10−3

0

1

2

3

4

5

6

time (s)

2−D, 50 bar, 800 K, 6.0 mg N−Heptane

Figure 13: Leading droplet position for di�erent grids with resolution �r x �z.be seen in Figure 13, by redu ing the ell size in the radial dire tion, the liquid penetrationis in reased and in general, using a �ner mesh will in rease the liquid penetration. Thebreakup model used for obtaining the results in Figure 13 was the TAB model. The TABmodel is known to produ e very small droplets lose to the inje tor. To investigate the39

5 RESULTS AND DISCUSSIONreasons behind the sensitivity to ell size, two di�erent 2-D axi-symmetri grids were usedwith ell sizes �r = 1:0 mm and �r = 0:5 mm, while the ell size along the axis ofsymmetry was 1.0 mm. The breakup model was turned o� and two di�erent sizes of theinitial droplet size were used on ea h grid. They were 1.0 �m and 0.01 �m. The inje tonvelo ity was set to a onstant 223 m/s. Sin e the drag, heat and mass transfer depend onthe relative velo ity, investigating the relative velo ity lose to the inje tor was performedby averaging the relative velo ity for all droplets loser than 3.0 m from the inje tor.

0 0.0005 0.001

Time (s)

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2000

4000

6000

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Rel

ativ

e ve

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ty (

cm/s

)

r0=1.0mu, dr=1.0mur0=0.01mu, dr=1.0mmr0=1.0mu, dr=0.5mmr0=0.01mu, dr=0.5mm

No interpolation

0 0.0005 0.001

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)

r0=1.0mu, dr=1.0mmr0=0.01mu, dr=1.0mmr0=1.0mu, dr=0.5mmr0=0.01mu, dr=0.5mm

No interpolation

Figure 14: Averaged relative velo ity for the par els within 3.0 m from the inje tor andthe orresponding leading droplet position, with no interpolation

0 0.0005 0.001

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r0=1.0mu, dr=1.0mmr0=0.01mu, dr=1.0mmr0=1.0mu, dr=0.5mmr0=0.01mu, dr=0.5mm

Interpolation

0 0.0005 0.001

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r0=1.0mu, dr=1.0mmr0=0.01mu, dr=1.0mmr0=1.0mu, dr=0.5mmr0=0.01mu, dr=0.5mm

Interpolation

Figure 15: Averaged relative velo ity for the par els within 3.0 m from the inje tor withinterpolationIn Figure 14, the di�eren e in relative velo ity for the 1.0 �m droplets is very small,40

5 RESULTS AND DISCUSSIONwhile the di�eren e for the 0.01 �m droplets is very large. The momentum ex hange ismu h larger for the smaller droplets. Hen e, if the ells are small enough to apture thevelo ity gradient lose to the inje tor, this will result in a gaseous jet with a velo ity loseto the liquid par els and, thus, a low relative velo ity. When the grid is too oarse, thenumeri al di�usion, together with the fa t that the momentum in rease yields a lowerin rease in velo ity, results in a mu h higher relative velo ity.In Figure 14, no interpolation was used to al ulate the relative velo ity and to inves-tigate the e�e t of using interpolation, the same ases are shown in Figures 15.As an be seen, by omparing Figure 15 with 14, when using interpolation, the di�er-en e in both relative velo ity and liquid penetration is redu ed. This will, thus, redu egrid sensitivity sin e the phase ex hange depends on, espe ially, the relative velo ity.5.2 Turbulent Parameters AnalysisThe in uen e of the turbulen e parameters on vapor penetration was investigated inFigure 16. The two parameters varied were the turbulent energy, k, and the turbulentlength s ale lt. As an be seen, the largest in uen e on the penetration is obtained by

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

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or P

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800 K, 50 bar, N−Heptane 6.0 mg, dr*dz = 0.5 mm * 1.0 mm

tkei= 10, scli= 0.01tkei= 10, scli= 0.1 tkei= 10, scli=1.0 tkei= 1000, scli= 0.01 tkei= 1000, scli= 0.1 tkei= 1000, scli=1.0 tkei=100000, scli= 0.01 tkei=100000, scli= 0.1 tkei=100000, scli=1.0 Figure 16: Vapor penetration for di�erent ombinations of turbulent parameters.varying the turbulent length s ale, where the general trend is that in reasing the turbulentlength s ale redu es the vapor penetration. This an be explained by the hange inturbulent vis osity �t. �t = C�k2" = C�k1=2lt (123)Hen e, the turbulent vis osity is linearly dependent on the length s ale, while the variationin k is only to the power of 1=2. Thus, in reasing the initial turbulent length s ale willin rease the gaseous e�e tive vis osity, whi h redu es the penetration length. In reasing41

5 RESULTS AND DISCUSSIONthe turbulent energy should, thus, also de rease the penetration length, but the totale�e t is un lear as the longest penetration, for a given lt, is for the medium value of k.The redu tion in penetration is due to the di�usion of the fuel vapor.5.3 Collision ModelingAs pointed out in Se tion 2.2.6, the standard ollision model in the KIVA ode is very griddependent. To show this, two impinging jets are set up in a onstant volume as in Figure17. To minimize the in en e of the gas, the breakup model was turned o� and the size ofthe droplets were set to the size of the nozzle, with a diameter of 0.1 mm. This allows thedroplets to move relatively undisturbed the gaseous �eld, preventing them from followingany eventual gas motion. The turbulent dispersion was also turned o�. To investigatethe in uen e of the size of the grid ells and where the ollision o urs in the mesh, fourdi�erent grids were used, where the number of grid ells in the horisontal dire tion wasvaried between 10, 11, 20 and 21. Sin e the expe ted pla e of ollision is in the enter ofthe domain, the ollisons will o ur on a grid line, for an even number of horisontal ells,and in the enter of a ell, for an odd number of horisontal ells. In Figure 17, the top rowis the 20- ell ase and the bottom row is the 21- ell ase. No ollision model was used (forreferen e) in the left, in the enter the original ollision model is used and to the right themesh independent ollision formulation is used. Looking at Figures 17(b) and 17(e), inwhi h the original ollision model has been used, it is lear that the han e for ollision ispra ti ally redu ed to zero when the par els meet at a grid line. By doubling the ell size,the probability for ollision is redu ed by half. However, by doubling the size of the ell,a larger number of par els o upies the same ell and the e�e t an be seen by omparingFigure 17(e) with 18(e). Many more par els ollide in the larger sized ells ase 18(e),than in 17(e), where the ells are half the size. This feature is very unattra tive, sin e thestandard pro edure to he k for a grid-indepent solution will not work in this ase. Infa t, by redu ing the ell size in absurdum, par els will move independently of the othersand never ollide, sin e the ells are too small to hold more than one droplet.Investigating the results from the mesh independent formulation, it is obvious that theresults are truly mesh independent. Although this method is more expensive, it is stillattra tive as it o�ers a solution to an otherwise unsolvable problem. In the simulationsbelow the onstant, C2 = 0, in Equation (61), whi h means that all par els rossing ea hothers path, within the time step, will ollide. This overpredi ts the ollision frequen yand was used here only for demonstration purposes.5.4 Mi ro Mixing TimeTo investigate the in uen e of di�erent mi ro mixing times, �mix. Constant inje tion witha liquid temperature of 300K into a gas with 50bar, 800K was set up. The ' old' liquidwas hosen to make the ame lifto� longer, hen e enhan ing any di�eren e in �mix. The42

5 RESULTS AND DISCUSSION

(a) 20x50, no model (b) 20x50, kiva model ( ) 20x50, new model

(d) 21x50, no model (e) 21x50, kiva model (f) 21x50, new modelFigure 17: Comparison of ollision models for two di�erent grids, with 'smaller' ells�gures in Figure 19 are arranged in in reasing order, where the shortest expression for�mix is at the top and the longest at the bottom. It is immediately lear that by in reasingthe mixing time, the ame lift o� is in reased., whi h is in omplian e with Equations(87) and (88).

43

5 RESULTS AND DISCUSSION

(a) 10x50, no model (b) 10x50, kiva model ( ) 10x50, new model

(d) 11x50, no model (e) 11x50, kiva model (f) 11x50, new modelFigure 18: Comparison of ollision models for two di�erent grids, with 'larger' ells

44

5 RESULTS AND DISCUSSION

(a) �mix = 0(b) �mix = 0:2p�k�t( ) �mix = 0:02�t(d) �mix = 0:2�tFigure 19: The temperature �eld for ontinous spray inje tion using di�erent mixingtimes.

45

6 CONCLUSIONS6 Con lusionsI do not feel obliged to believe that the same God who has endowed us with sense, reason, andintelle t has intended us to forgo their use.- Galileo GalileiThis thesis presents the appli ation of CFD to spray ombustion analysis. Improvedsub-models for turbulen e/ hemistry intera tion using a detailed hemistry approa his presented. The omputer model was applied to a onstant volume apparatus withn-heptane and DME as a fuel and reasonable agreement with experimental data wasa hieved, although spray simulations are sensitive to the ell size.When omparing numeri al spray ombustion simulations with experimental data, itis important to know the initial onditions of the turbulen e hara teristi s and to use a orre t inje tion rate s hedule of the liquid fuel, sin e the simulation is espe ially sensitiveto these parameters. However, it is not ertain that the best result would be obtained ifthe experimental results of these parameters were obtained and used in the simulation,as this might depend on the hoi e of turbulen e model, as well. In this thesis the k-",either standard or RNG, has been used throughout and no spe ial investigation of thisstatements has been performed. However, the k-" model is 'tuned' only for stationarysituations, while spray ombustion is a highly transient phenomenon.Constant volume al ulations are attra tive, sin e they o�er the possibility to study thein uen e of ea h parameter. In an engine al ulation, in whi h the ell size and turbulentparameters are more diÆ ult to ontrol, it is even more important that experimentalresults are available to validate the al ulations. Still, when the problem setup has beentuned, good agreement an be obtained.In ontrast to single-phase ows where it is almost always possible to obtain meshindependent results, whi h are dependent only on the models, Lagrangian spray al u-lations do not obey this 'law,' be ause the smaller the ell size does not always yield abetter result. If a omplete solution to this problem ould be dis overed, it would be amajor breakthrough in the modeling.

46

7 FUTURE WORK7 Future Work I like work; it fas inates me. I an sit and look at it for hours- Jerome K. JeromeIn order to a urately predi t the emissions in spray ombustion, a minimum require-ment is, at least, that the initial fuel vapor distribution is orre t. Sin e this is suppliedby the spray models, it is important to have as a urate and reliable behavior of the sprayas possible. Future work will fo us on improving the Lagrangian treatment of the sprayin regions with a high liquid volume fra tion.It is still an open question whether or not this e�e t is important enough to in lude.A bug-free implementation of the above mentioned phenomena in a CFD ode built onfortran77 is an extremely tedious and time onsuming task. However, by using obje t-oriented programming te hniques it is hoped that a proper investigation of this an beturned into a reality. Improving the PaSR model is also, as always, a question for futureresear h.

47

8 SUMMARY OF PAPERS8 Summary of PapersNumeri al Evaluation of n-Heptane Spray Combustion at Diesel-like ConditionsThis paper fo uses on 2D axi-symmetri spray ombustion simulations in a onstantvolume with n-heptane as a fuel. A detailed hemi al me hanism was implemented in theKIVA ode with the purpose of verifying the hemistry model and �nding an optimalgrid size for the behavior of the spray. For this purpose, lo al measurements, using Ramanspe tros opy, of mixture on entration was used together with ignition delay and liquidspray penetration measurements, obtained at Aa hen [21℄.Numeri al Evaluation of Dual Oxygenated Fuel Setup for DIDiesel Appli ationUsing methanol as a diesel fuel is promising due to the redu tion of parti ulate and NOxemissions. However, the poor ignition quality of methanol ne essitates the use of anignition improver. In this study, Dimethy Ether (DME) was sele ted, sin e gaseous DME an be manufa tured from methanol on board the vehi le through a atalyti system, andthe ignition quality of DME is ex ellent. When adding DME to the intake-air, are shouldbe taken as to how mu h DME is to be added. Adding the wrong amount an prove tohave the opposite e�e t of that whi h is intended. To study this, a hemi al me hanismwas developed and implemented in the KIVA ode and the optimal addition amount ofDME was numeri ally evaluated.Computer Evaluation of DI Diesel Engine Fueled with NeatDimethyl EtherDimethyl ether has, due to Haldor Tops�e A/S (see below), re eived an in reasing amountof interest as a potential diesel fuel. However, there is a la k of fundamental data in thekineti s of DME oxidation at high pressures and temperatures. This paper fo uses on de-veloping a detailed hemi al me hanism of methanol+DME/Air oxidation and omparingthe ignition delay times obtained in sho k tube experiments. The hemi al me hanismwas implemented into the KIVA ode and applied to a onstant volume appli ation anda 2D axi-symmetri Volvo DI Diesel engine with methanol as a fuel and DME in a purelygaseous state.Neat Dimethyl Ether: Is It Really Real Diesel Fuel of Promise?This paper is a ontinuation of the work arried out in the paper des ribed above, Com-puter Evaluation of DI Diesel Engine Fueled with Neat Dimethyl Ether.48

8 SUMMARY OF PAPERSHaldor Tops�e A/S stated in 1995 that dimethyl ether has the potential of being a realdiesel fuel in the future. It has low parti ulate emissions and ex ellent ignition qualitiesand it does not require any spe ial treatment of the fuel inje tion equipment, sin e it isnon- orrosive to metals.This paper is a numeri al evaluation of how valid that statement was. For this purposea detailed hemi al me hanism was developed and implemented in the KIVA ode. The hemi al me hanism was ompared with experimental data for ignition delay times withgood agreement. The la k of spray ombustion experiments with DME as a fuel preventedany su h omparisons and the model was tuned with n-heptane as a fuel.Evaluation of Ignition Quality of DME at Diesel Engine Condi-tionsPreviously la king onstant volume experimental data for DME, this paper is a ontinu-ation of the paper above and presents a omparison with the experiments performed atthe University of Hiroshima. Good agreement is obtained and the simulations on�rmthe high ignition quality of DME.3-D Diesel Spray Simulations Using a New Detailed ChemistryTurbulent Combustion ModelThis is a presentation of the latest models for turbulen e/ hemistry intera tion togetherwith a new hemi al me hanism, to whi h a soot model has been added and the physi alproperties of the n-heptane have been repla ed with the properties of a real Diesel #2grade fuel. The Volve engines AH10A245 and D12C were simulated using a 3D se tormesh, and the simulations were ompared with experimental data for the pressure tra e.Fair agreement was obtained.

49

A ROBUST FACE-TO-FACE TRACKINGA Robust Fa e-To-Fa e Tra kingTo tra k par els through any mesh in a reliable and robust way is only possible when the ells are tetrahedral. This rather strong on lusion will be shown below. Sin e many CFD odes use hexahedral ells this requires the hex- ells to be split into tets. An operationquite ostly if performed every time it is required. Thus, the optimal way is to performthis de omposition in the beginning, or every time the mesh has hanged, and then storethis information.Be ause of the fa e to fa e addressing, a par el is not allowed to travel through avertex whi h in a dis rete world, where the representation of a real number is �nite, hasa 100 % han e of happening. Looking at Figure 20, this means that a par el going from ell 1 to ell 4 is not allowed to travel along path II, going dire tly from ell 1 to ell 4.Path I, however, is perfe tly valid. This is due to the fa t that a fa e is only shared bytwo ells, and when a par el hits a fa e it must go into the the other ell sharing this fa e.Thus, if the par el travels along path II, it an either go into ell 2 or ell 3, but not ell4. Hen e, the verti es, and edges as well, are holes, through whi h the par els an es ape,and this is one way to lose them.3

1 2

4

I

II

Figure 20: Two theoreti ally possible ways for a par el to go from ell 1 to ell 4The other way to lose par els is fa e-warpage, whi h happens when a fa e is des ribedby more than three points. Thus, for a hexahedral ell, in whi h a fa e has four points, itis possible for one point, or more, to not lie on the fa e plane. Be ause a fa e is des ribedby a fa e normal and a fa e point only, a situation like in Figure 21, where fa e � iswarped, may o ur. Cells 1 and 2 only share two verti es. The other two verti es for fa e� may thus be arranged so that one of them is higher, and the other lower, than the 2shared verti es. Resulting in a situation depi ted in Figure 21.A par el going from ell 1 hits fa e i. Be ause only ells 1 and 2 share fa e i, thepar el is now believed to travel into ell 2. Hen e, the par el moves to ell 2 and has nowbeen 'lost'. This will not be dis overed unless a he k is performed every time the par el hanges ells, whi h slows down the al ulations and makes the whole point of the F2Falgorithm useless. 50

B THE TURBULENCE/SPRAY INTERACTION CONSTANT CS1 2

i

αj

Figure 21: Tra king a par el on a warped mesh.To remedy fa e warpage it is ne essary to de ompose the ells into tetrahedrals usingthe verti es. This ensures that the par el enters the orre t ell as it rosses a fa e, anda situation like Figure 21 be omes impossible.B The turbulen e/spray intera tion onstant CsThe value of Cs = 1:5 (in front of the spray/turbulen e intera tion term in the "-equation)is hosen on basis that this onserves the turbulent length s ale, lt. It is, however, shownbelow that Cs = 1:5 is the worst possible hoi e.By assuming no gradients and onstant density, and without negle ting ", in the k-"equations, they are redu ed to��k�t = ��"+ _W s; ��"�t = "k ��C"2�"+ Cs _W s� (124)After some manipulation, the equation for the turbulent length s ale an be written as"k��1 ��lt�t = (�� Cs) _W s + (C"2 � �)�" (125)where the turbulent length s ale is lt = k�=" and � = 1.5.It is not possible to draw any de�nite onlusion about the value on Cs, and its in uen eon the length s ale sin e the magnitude of " is unknown.The on lusion that Cs = 1:5 onserves the length s ale is therefore erroneous.In fa t, by setting Cs = � = 1:5 Equation (125) is redu ed to"k��1 ��lt�t = (C"2 � �)�" > 0: (126)Thus, if Cs = 1:5, the only on lusion that an be drawn, is that the turbulent lengths ale in reases, when it should de rease.51

B THE TURBULENCE/SPRAY INTERACTION CONSTANT CSCommentIt is generally agreed that it is not possible to re onstru t a CFD ode using the do umen-tation only, meaning that there are undo umented features and tri ks, never mentionedin text, that make the ode work.It is the authors experien e that this also applies to model des riptions, and this isnot dis overed until one tries to implement the model. It is therefore the author's wishthat all details of the models des ribed herein and how they have been implemented aremade perfe tly lear, and for those using the KIVA3-V ode all models are therefore avail-able at the KIVA JumpStation, http://www.tfd. halmers.se/~nordin/KJS/, for anyone todownload.

52

Referen es[1℄ A.A. Amsden. KIVA-3: A KIVA Program with Blo k-Stru tured Mesh for ComplexGeometries. LA-12503-MS, Mar h, 1993.[2℄ A.A. Amsden. KIVA-3V: A Blo k-Stru tured KIVA Program for Engines with Ver-ti al or Canted Valves. LA-13313-MS, July, 1997.[3℄ A.A. Amsden, P.J. O'Rourke, and T.D. Butler. KIVA-II: A omputer Program forChemi ally Rea tive Flows with Sprays. LA-11560-MS, May, 1989.[4℄ C Bergstr�om. Numeri al Modeling of Fuel Sprays. PhD thesis, Lund Institute ofTe hnology, 1999.[5℄ A. Berlemont, F. Benoist, and G. Gousbet. In uen e of Collisions on Parti le Flu -tuating Velo ities Using a Lagrangian Approa h. Numeri al Methods in MultiphaseFlows, ASME, Vol. 185:23{28, 1994.[6℄ R. Borghi. Turbulent Combustion Modelling. Prog in Energy and Combustion S i-en e, Vol 14:245{292, 1988.[7℄ A. Brink, C. Mueller, P. Kilpinen, and M. Hupa. Brief Communi ation: Possibilitiesand Limitations of the Eddy Break-Up Model. Combustion and Flame, Vol. 123:275{279, 2000.[8℄ G.H. Chiang, M.S. Raju, and W.A. Sirignano. Numeri al Analysis of Conve ting,Vaporizing Fuel Droplets with Variable Properties. International Journal of Heatand Mass Transfer, 1991.[9℄ J. Chomiak and A. Karlsson. Flame Lifto� in Diesel Sprays. Twenty-Sixth Symposium(International) on Combustion/The Combustion Institute, pages 2557{2564, 1996.[10℄ C. Crowe, M. Sommerfeld, and Y. Tsuji. Multiphase Flows with Droplets and Parti- les. CRC Press LLC, 1998.[11℄ A.D. Gosman and D. Clerides. Diesel Spray Modelling: A Review. ILASS-EuropeAnnual Meeting, Floren e, Italy 9-11 July, 1997.[12℄ A.D. Gosman and E. Ioannides. Aspe ts of Computer Simulation of Liquid-FueledCombustors. J. of Energy, Vol 3(6):482{490, 1983.[13℄ G. Gousbet and A. Berlemont. Eulerian and Lagrangian approa hes for predi ting thebehaviour of dis rete parti les in turbulent ows. Progress in Energy and CombustionS ien e, Vol 25:133{159, 1999. 53

REFERENCES[14℄ I.R. Gran. Mathemati al Modeling and Numeri al Simulation of Chemi al Kineti sin Turbulent Combustion. PhD thesis, University of Trondheim, The NorwegianInstitute of Te hnology, Trondheim, 1994.[15℄ Z. Han and R.D. Reitz. Turbulen e Modeling of Internal Combustion Engines UsingRNG k-" Models. Combustion S ien e and Te hnology, Vol 106(4-6):267{295, 1995.[16℄ A.M. Kanury. Introdu tion to ombustion phenomena. Gordon and Brea h S ien ePublishers, 1975.[17℄ J.A.J. Karlsson. Modeling Auto-Ignition, Flame Propagation and Combustion inNon-stationary Turbulent Sprays. PhD thesis, Chalmers University of Te hnology,G�oteborg, 1995.[18℄ J.A.J. Karlsson and J. Chomiak. Physi al and Chemi al E�e ts in Diesel SprayIgnition. 21st Congress of CIMAC, Interlaken Switzerland, May 15-18, 1995.[19℄ L. Kj�aldman, A. Brink, and M. Hupa. Mi ro mixing time in the eddy dissipation on ept. Combustion S ien e and Te hnology, Vol. 154:207{227, 2000.[20℄ A.Y. Klimenko and R.W. Bilger. Conditional Moment Closure for Turbulent Com-bustion. Progress in Energy and Combustion S ien e, 25:595{687, 1996.[21℄ H.J. Koss, D. Br�uggeman, A. Wiartalla, H. B�a ker, and A. Breuer. Results fromFuel/Air Ratio Measurements in a n-Heptane Inje tion Spray. IDEA EFFECT,Subprogramme A.2, Revised Additional Report, 1992.[22℄ C. Kralj. Numeri al Simulation of Diesel Spray Pro esses. PhD thesis, ImperialCollege of S ien e Te hnology and Medi ine, 1995.[23℄ C. Ljus. On Parti le Transport and Turbulen e Modi� ation in Air-Parti le Flows.PhD thesis, Chalmers University of Te hnology, 2000.[24℄ B.F. Magnussen and B.H. Hjertager. On Mathemati al Modeling of Turbulent Com-bustion with Spe ial Emphasis on Soot Formation and Combustion. The 16th Sym-posium (International) on Combustion, the Combustion Institute, page 719, 1976.[25℄ C. Marooney, Computational Dynami s, 2000. Personal Communi ation.[26℄ P.J. O'Rourke. Colle tive Drop E�e ts on Vaporizing Liquid Sprays. Te hni alReport LA-9069-T, Los Alamos National Laboratory, 1981.[27℄ P.J. O'Rourke. oral presentation at the 23:rd Dire t Inje tion Strati�ed ChargeWorking Group Meeting, Los Alamos National Laboratory, Mar h, 1986.54

REFERENCES[28℄ P.J. O'Rourke. Statisti al Properties and Numeri al Implementation of a Modelfor Droplet Dispersion in a Turbulent Gas. Journal of Computational Physi s, Vol.83:345{360, 1989.[29℄ N. Peters. Laminar di�usion amelet models in non premixed turbulent ombustion.Progress in Energy and Combustion S ien e, Vol 10:319{339, 1984.[30℄ S.B Pope. PDF Methods For Turbulent Rea tive Flows. Progress in Energy andCombustion S ien e, Vol. 11:119{192, 1985.[31℄ R.D. Reitz. Modeling Atomization Pro esses in High-Pressure Vaporizing Sprays.Atomization and Spray Te hnology, Vol. 3:309{337, 1987.[32℄ M. Renksizbulut and M.C. Yuen. Experimental Study of Droplet Evaporation in aHigh-Temperature Air Stream. ASME Journal of Heat Transfer, Vol. 105:384{388,May 1983.[33℄ L.M. Ri art, J. Xin, G.R. Bower, and R.D. Reitz. In-Cylinder Measurement andModeling of Liquid Fuel Spray Penetration in a Heavy-Duty Diesel Engine. SAEPaper 971591, 1997.[34℄ T.F. Su, M. Patterson, R.D. Reitz, and P.V. Farrell. Experimental and Numeri alStudies of High Pressure Multiple Inje tion Sprays. SAE Paper 960861, 1996.[35℄ M.M. el Wakill, O.A. Uyehara, and P.S. Myers. Theoreti al investigation of heatingup period of inje ted fuel droplets vaporizing in air. NACA, TN 3179, 1954.[36℄ H.G. Weller, G. Tabor, H. Jasak, and C. Fureby. A Tensorial Approa h to Computa-tional Continuum Me hani s using Obje t Orientated Te hniques. Comp. in Phys.,12(6):620{630, 1998.[37℄ M.C. Yuen and L.W. Chen. On Drag of Evaporating Liquid Droplets. CombustionS ien e and Te hnology, Vol. 14:147{154, 1976.

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