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Conditional moment closure for methane oxidation using two conditional variables and stochastic processes 2004

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Conditional Moment Closure for Methane Oxidation Using Two Conditional Variables and Stochastic Processes by Jorge R. Lozada-Ramirez B.A.Sc, Universidad de las Americas-Puebla, Mexico, 1998 A THESIS S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D SCIENCES in The Faculty of Graduate Studies (Department of Mechanical Engineering) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A June 16, 2004 © Jorge R. Lozada-Ramirez, 2004 Library Authorization In presenting this thesis in partial fulfi l lment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Name of Author (please print) Date (dd/mm/yyyy) Title of Thesis: C W . ' - r V i w f / W a ^ ^ - r - ClcS ore f ^ \)<k C\ck.M<?J OtorX STT>cU«xj-Uy (?, Degree: fi/[A Sr. Y e a r : Department of M<Lc U**.lr<J £ ruuJrtee.r/r>« . The University of British Columbia i) " Vancouver, BC Canada 20 OH Abstract i i Abstract 'Conditional Moment Closure for Methane Oxidation Using Two Conditional Variables and Stochastic Processes' by Jorge R . Lozada-Rajiurez The condi t ional moment closure method using two condi t ioning scalar variables is applied to derive the t ransport equation o f species mass f ract ion, temperature, and scalar dissipa- t i o n in a decaying, isotropic, homogeneous turbulent methane-air flow. T h e strain tensor in the t ranspor t equation of scalar dissipation of the condi t ioning variables is simulated using stochastic processes. T h e results o f this model are then compared to DNS and condi t ional moment closure w i t h one variable for the same test case. Contents iii Contents Abstract ii Contents i i i List of Figures v i Nomenclature vii Acknowledgements x 1 Introduction 1 1.1 Objective 1 1.2 Doubly Condi t ional Moment Closure ( D C M C ) 1 1.3 Stochastic Processes 2 1.4 Thesis Out l ine 2 2 Combustion, Turbulence, and Stochastic Processes 4 2.1 Premixed Combust ion 4 2.2 Non-premixed Combustion 5 2.3 Transport Equations 5 2.3.1 Transport o f Mass, Momentum, and Energy 5 2.3.2 Transport o f M ix tu re Fract ion 6 2.3.3 Transport o f Species Mass Fract ion 6 2.4 Chemical Kinet ics and Reaction Mechanisms 7 2.5 Scales o f Turbulence and Kolmogorov Scales 9 2.6 Cascade o f Turbulent K inet ic Energy 10 2.7 Turbulence Simulat ion and Model l ing 10 Contents i v 2.7.1 Direct Numerical Simulat ion (DNS) 11 2.7.2 Large-Eddy Simulat ion (LES) 11 2.7.3 T ime or Ensemble Averaging (Reynolds Decomposit ion) 11 2.7.4 Turbulence Models 12 2.8 Turbulent Combust ion 13 2.9 Turbulent Combust ion Simulat ion and Model l ing 13 2.9.1 DNS 14 2.9.2 Favre Averaging 14 2.9.3 LES 14 2.9.4 Laminar Flamelets 15 2.9.5 Probabi l i ty Density Funct ion (PDF) Model l ing 16 2.9.6 Condi t ional Moment Closure 16 2.10 Stochastic Processes and Monte Car lo Methods 17 2.10.1 Monte Car lo Simulations of Turbulence 17 2.10.2 Monte Car lo Simulations of Turbulent React ing Flows 19 2.11 Summary 21 3 The D C M C Method with Stochastic Processes 22 3.1 Condi t ional Moment Closure w i t h T w o Condi t ional Variables ( D C M C ) . . . . 24 3.2 Der ivat ion of the D C M C Equations 24 3.3 Model l ing Scalar Dissipat ion 26 3.4 Scalar Transport Closure Hypotheses 29 3.5 D C M C Closure Hypotheses 30 3.6 Mathemat ica l solut ion o f the Z-gradient squared te rm 31 3.7 Models for the Transport Equat ion o f Scalar Dissipat ion 32 3.7.1 Periodic Forcing w i t h Random Phase Shi f t ing (PF) 33 3.7.2 Coupled Map Latt ices ( C M L ) 34 3.8 Summary 36 4 Simulation of Methane Oxidation with D C M C - S P 37 4.1 The Reference DNS and C M C Databases 37 4.1.1 Flow F ie ld 37 Contents v 4.1.2 Chemical Kinet ics 38 4.1.3 I n i t i a l Condit ions 39 4.2 Simulat ion of CH4 Ox idat ion Using D C M C - S P . . 40 4.2.1 Simulat ion o f the Stra in Field 42 4.2.2 Predict ions o f Reactive Scalars 49 4.3 Summary 55 5 Conclusions and Recommendations 56 Bibliography 58 List of Figures v i List of Figures 2.1 Turbulent kinet ic energy spectrum, f r o m Peters [1] 10 3.1 Condi t ional average of temperature 23 3.2 DNS calculat ion of temperature, f rom Bushe [8] 23 4.1 DNS computat ional domain 38 4.2 DNS in i t i a l velocities 38 4.3 In i t ia l Condit ions of the Scalar Fields 40 4.4 D C M C - S P code structure 42 4.5 Strain fields simulated w i t h P F 44 4.6 u-velocity fields simulated w i t h C M L 46 4.7 Velocity fields simulated w i t h C M L 47 4.8 Strain fields simulated w i t h C M L 48 4.9 DNS, C M C , and D C M C Favre averages, of species mass fractions 50 4.10 Condi t ional averages at t = 30.0 51 4.11 a-variations in the intermediates field at t = 30.0 52 4.12 a-variations in the N O field at t = 30.0 53 4.13 a-variations in the temperature field at t = 30.0 54 Nomenclature vii Nomenclature A Deterministic coefficient, amplitude of periodic force a Conditioning variable a B Deterministic coefficient C Constant c Speed of sound, concentration Cp Constant pressure specific heat D Molecular diffusivity, dimension dW Wiener process E Activation energy F Fuel, a stochastic process 9 Gravitational force h Specific enthalpy I Intermediates J Diffusion of energy, molecular diffusion k Arrhenius constant, reaction rate coefficient, energy level in coupled map lattice L Length I Length Le Lewis number m Mass O Oxidiser P Products P Pressure Pr Prandtl number R° Ideal gas constant Re Reynolds number Nomenclature vi i i s Surface stretch rate Sc Schmidt number Sij Strain tensor t Time u Velocity V Volume V Velocity W Molecular mass X Mole fraction, stochastic variable X Coordinate axis Y Mass fraction, stochastic variable Z Mixture fraction Nomenclature i x Greek Letters a Value of the condi t ioning variable a r Di f fusion 7 Parameter i n the coupled map lat t ice, rat io o f specific heats Sij Kronecker de l ta e Dissipat ion o f turbulent kinetic energy £ Value of the condi t ioning variable Z T) Characterist ic length microscale 0 Product of the gradients of scalar dissipation 0 Random phase shift K Turbulent k inet ic energy A Parameter i n the coupled map lat t ice H Viscosity v K inemat ic viscosity, stoichiometric coefficients I I Product p Density S Summat ion T Characterist ic t ime microscale, parameter i n the coupled map lat t ice Tij Viscous tensor v Characterist ic velocity microscale $ Gradient squared of the condi t ioning variables X Scalar dissipation A scalar field, periodic force w i t h random phase shi f t ing CJ Chemical source term w Per iod of force Acknowledgements x Acknowledgements The work presented i n this thesis was made possible thanks to the support and encouragement of many people. I would l ike to express m y utmost appreciat ion to my supervisor, Dr . Kenda l Bushe, for his patience and t imely, expert advice. Thanks are also due to Dr . Andrea Frisque for her constant encouragement and for those long discussions regarding stochastic processes. The financial suppor t of Mexico's Nat ional Counci l of Science and Tecnology (CONACyT) and o f Westport Innovations, Inc. are also grateful ly acknowledged. Thanks are also given to Col in B la i r , Andrea, and Ray Grout for proof-reading preel imi- nary drafts of th is thesis and for their always insight ful discussions on combust ion, turbulence, model l ing, programming, and poli t ics. Thanks also go to Co l in and Jen Bruce for their un- condi t ional suppor t and for becoming m y fami ly away f rom home. Special thanks are due to M i n a Ha lpern for her continuous support , encouragement, and mot iva t ion in reaching this goal. Thank you, M ina , for al l those smiles i n the times o f hardship. Final ly , I wou ld like to dedicate th is thesis t o my parents Jorge and Tere, and to m y siblings Daniel and Raquel for always bel ieving i n me. Papa, Mama, Dany y Raque; muchas gracias por su apoyo y por sus oraciones. Este documento esta dedicado a ustedes como u n pequeno t r i bu to a su confianza y amor. Gracias por creer en m i . Jorge R. Lozada-J^imirez June, 2004 Chapter 1 i Introduction T h e complexi ty of the t ranspor t equations tha t describe turbulent combustion makes necessary the use o f averaging procedures so that pract ical problems can be analyzed numerical ly in an efficient, t ime ly fashion. Th is averaging process creates a closure prob lem by generating terms wh ich are not expl ici t funct ions of the averaged variables [2]. Generally speaking, closure problems are solved by model l ing the unclosed terms th rough the use of theoret ical or empir ical hypotheses or experimental results. I n the fol lowing chapters of this thesis, a closure method for the t ransport equations o f turbulent combustion w i l l be formulated. The results obta ined from the computat ional implementat ion of the resul t ing model on a decaying, isotropic, homegeneous, non-premixed turbulent flow of methane and air w i l l be shown and compared to a reference DNS database of a similar f low. 1.1 Objective T h e main objective o f th is research project is to explore the appl icat ion of stochastic processes to the model l ing of the strain tensor i n the scalar dissipation t ranspor t equat ion t h a t is de- r ived using the Doub ly Condi t ional Moment Closure ( D C M C ) method. T h e resul t ing system o f equations is used to simulate the oxidat ion of methane by air i n decaying, homogeneous, isotropic turbulence. I t is expected that the predictions of species mass fractions and tem- perature w i l l compare well w i t h those obtained from a direct numerical s imulat ion database and show an improvement over the results provided by a singly condi t ional moment closure s imulat ion o f the same test case. 1.2 Doubly Conditional Moment Closure (DCMC) T h e Condi t ional Moment Closure ( C M C ) o f Kl imenko [3] and Bi lger [4] was developed w i t h the purpose o f f inding closure for the chemical source te rm of the species mass f ract ion t ranspor t Chapter 1. Introduction 2 equat ion. This method is based on the assumption that the fluctuations of scalars, such as the chemical species mass f ract ion and temperature, are related to variat ions of other scalars. I n th is thesis, the C M C method has been expanded by using two different condi t ioning scalar variables. One o f those variables, m ix tu re fract ion, is used to express the degree o f mixedness of two or more elements i n a non-premixed, reacting flow. A second non-reactive scalar variable named a has been defined to capture the effects o f fluctuations due to st ra in along isosurfaces of m i x t u r e f ract ion. Th rough the use o f the doub ly condi t ional moment closure ( D C M C ) methodology, macro-mix ing phenomena can be decoupled from the chemistry, wh i le conserving the effects of micro-mix ing v ia the scalar dissipation term. 1.3 Stochastic Processes The st rong coupl ing between fluid dynamics and chemistry i n turbu lent react ing flows is frequent ly modelled using the scalar dissipation t e r m for wh ich a number of different models have been proposed i n the past. I n th is thesis, a t ransport equat ion is used to calculate scalar dissipat ion. Th is t ransport equation includes the strain tensor which is often model led i n turbulence research. Recognizing the chaotic behaviour of turbulence, the use of two stochastic processes to model the st ra in tensor i n the t ransport equation of scalar dissipat ion te rm is proposed and analyzed. Stochastic processes are events tha t depend on stochastic variables that exh ib i t an unpredictable behaviour. I n the first model , a periodic force w i t h random phase shif t (PF) is used to emulate the chaotic behaviour o f the s t ra in tensor. T h e second model uses the Coupled M a p Lat t ice (CML) o f Beck [5] and Hilgers and Beck [6], [7] t h a t generates realisations of the fluctuating velocity increments as a funct ion o f t ime for a fu l ly - developed turbulent flow. T h e strain tensor is then calculated using those velocity increments. The results from b o t h codes are then compared to the Direct Numerical S imulat ion database of Bushe et al [8] and the C M C results o f Bushe and Bilger [9]. 1.4 Thesis Outline I n th is chapter, the fundamental grounds and mot ivat ion for th is research project have been discussed. I n Chapter 2, the fundamentals o f combustion and turbulence, as wel l as turbu lent com- Chapter 1. Introduction 3 bust ion, axe addressed. The t ranspor t equations that describe turbu lent combust ion, as well as current turbu lent combust ion models, are also described. I n Chapter 3, the D C M C equations are derived. T h e closure assumptions and model l ing o f the strain tensor and o f scalar dissipation are also discussed i n th is chapter. Chapter 4 pertains t o the implementat ion o f the equations and models discussed i n Chap- ter 3 to a decaying, homogeneous, isotropic turbulent react ing flow. The turbulence model and chemical k inet ic mechanism used for this project are addressed and compared w i t h a direct numerical s imulat ion. Final ly, i n Chapter 5, the conclusions about the research are ar t icu lated and recommen- dat ions made for future research. 4 Chapter 2 Combustion, Turbulence, and Stochastic Processes Combust ion can be denned as the self-sustaining process th rough wh ich two or more com- pounds, fuels and oxidizer, are modif ied through a chemical reaction w i t h an associated release of energy. The study of combustion phenomena implies a combinat ion o f thermodynamics, fluid mechanics/dynamics, chemical kinetics, and transport phenomena. Combust ion pro- cesses can be classified according to the mix ing behavior and the flow velocity regime of the system. Premixed systems and non-premixed systems can exist i n b o t h the laminar and tur - bulent regimes of fluid flow. Th is research util izes mathemat ical model l ing and computer s imulat ion of combustion i n a deacying, isotropic, homogeneous, non-premixed turbu lent flow of air and methane. The Doub ly Condi t ional Moment Closure ( D C M C ) method has been used to solve the transport equations of mass fract ion o f species. I t is proposed to use a stochastic process to model the strain tensor i n the transport equation o f scalar dissipation responsible for the micro-mix ing o f species. Different options for the stochastic process are discussed. 2.1 Premixed Combustion Premixed combustion occurs i f the fuel and oxidizer are completely mixed before the chemical reaction takes place. Examples o f premixed combustion are Bunsen burners and spark- igni t ion engines. Premixed combust ion is often characterized by the burn ing velocity, wh ich is a funct ion o f the chemical composit ion, i n i t i a l temperature, and pressure o f the premixed flow. Chapter 2. Combustion, Turbulence, and Stochastic Processes 5 2.2 Non-premixed Combustion I n non-premixed combustion, the fuel and oxidizer are fed into a combustion chamber or reactor f rom different sources. Pract ical examples of non-premixed combust ion include diesel engines, gas turbines and gas stoves. Unlike premixed combustion, where m i x i n g has already taken place before the chemical reaction occurs, and the fue l /a i r ra t io is homogeneous i n the domain , i n non-premixed combustion fuel and oxidizer have to mix i n order for the react ion to proceed. The state of mixedness o f the reactants is characterized by mix tu re f ract ion, wh ich is defined as: Here, Yp and Yo axe defined as the mass fractions of the fuel and oxidizer, respectively. M i x t u r e f ract ion for the fuel stream is, Z(xj,t) — 1, and Z(xi,t) = 0 for the oxidizer stream. 2.3 Transport Equations I n general, the t ransport equations used in combustion express the coupl ing between the fluid mechanical and chemical reaction phenomena tak ing place in the combustion event. W h e n deal ing w i t h reactive mixtures of gases, for example, the t ransport equations are usual ly derived making use of the kinetic theory of gases in order to capture the specific behavior of the system under study. Unless noted otherwise, the fol lowing equations axe derived for the i r appl icat ion to problems involv ing ideal gases. 2.3.1 Transport of Mass, Momentum, and Energy T h e t ranspor t o f mass balances the amount o f mass entering and leaving any given contro l volume. dp dpuj _Q dt dx{ T h e equation of t ranspor t of momentum indicates the change i n momentum caused by the external forces (e.g. iner t ia l , pressure, viscous, and body forces) act ing on the fluid as, dt dxj dxj dxj Chapter 2. Combustion, Turbulence, and Stochastic Processes 6 where the viscous tensor for a Newtonian fluid, using the molecular viscosity, pi, and the Kronecker del ta 8%j, is described by: (dui duj\ 2 / duk\ "3 The equat ion o f t ransport o f energy balances the thermal , chemical, and kinetic energy of the system. Fol lowing Veynante and Vervisch [10], the t ranspor t equation of energy can be u2 expressed i n terms o f the to ta l specific enthalpy, ht = h + -_-, o f the system as follows: dpht dpuiht dp d opuiht op d ( rh \ + ~dx~ =di + dx-iVi+ UjTij) + UiP9i dt where the diffusion of energy, i n this case of specific enthalpy, is described by the Fourier Law, as: N Ji ~ pr dh ^fPr \ dYj dxi _—' \Sci J dxi Here: 1. The P rand t l number, Pr = ^j^-, is the non-dimensional group tha t measures the rat io of t ranspor t o f momentum due to diffusion to the temperature. Cp is the specific heat at constant pressure and A is the thermal conductivi ty. 2 . Scj, the Sdun id t number of the Ph species, is defined as Sc j = where Dj is the di f fus iv i ty of the Ph species relative t o a major or reference species. mi m 3. Yj is defined as the mass fract ion o f the P h species, as 2.3.2 Transport of Mixture Fraction M i x t u r e f ract ion is a conserved scalar. I t is only used to keep track of the mix ing process as i t evolves i n t ime and i n space. dpZ | udpZ d f p D z d Z \ _ o ( 2 i ) dt 1 dxi dxi \ dxij 2.3.3 Transport of Species Mass Fraction T h e t ranspor t equation of the mass fractions for J = 1...N chemical species consists o f accu- mula t ion , advection, diffusion, and product ion terms: dpYj doYj dJ\ ^ + Ui-dx- = -dx-+0}I- M Chapter 2. Combustion, Turbulence, and Stochastic Processes 7 Using Fick's Law of dif fusion [2], the molecular diffusion of species is defined as: J! = - A D , ^ : (2.3, 2.4 Chemical Kinetics and Reaction Mechanisms The chemical source te rm, CJJ, in Eq. (2.2) describes the net rate of p roduct ion o f species due to chemical reactions. The chemical source t e r m for M reactions is defined as [11], M N / Y K * / = w £ K * - 4 , * ) * n ( ^ ) ' » (2-4) K=l J = l V ' where Wj is defined as the molecular mass of the Ith species. i/T K a n d v" K are the stoichio- metr ic coefficients in the Kth chemical reaction, N N 7=1 7=1 w i t h Aj being the chemical symbol o f the Ith species i n the chemical reaction, k is the Arrhenius type constant o f the form, k = BTaexp(-^y (2.5) Here, J3/fTQfir corresponds to the frequency factor for the Kth react ion step, OLK is used to define the temperature dependence o f the frequency factor for the KTH reaction, while EK represents the act ivat ion energy required by the Kth react ion to proceed, and R° being the universal gas constant. F inal ly , the mole fract ion o f the Ph species Xj is defined as, w i t h the ideal-gas law, and, x,= f'W m pV = mR°T, (2.7) N m = p^(yr/Wi). (2.8) 7=1 Subst i tu t ing Eq. (2.8) in to Eq . (2.7) leads to, N PV = R0TpY,(Yj/Wr). (2.9) i=i Chapter 2. Combustion, Turbulence, and Stochastic Processes 8 Using Eqs. (2.5), (2.6), and (2.9), Eq . (2.4) can now be wr i t ten as */ = W, £ ) K « - BKT«« exp ( - J | ) JJ (^y^ • (2-10) The most basic analyses of combustion often rely on the assumption tha t the chemical reactions occur at faster rates t han those associated to the t ranspor t phenomena occurr ing i n the flow. This assumption is often referred to as 'fast chemistry ' and usually denotes equ i l ib r ium states for chemical systems. B y invoking fast chemistry, systems can be analyzed by using the basic laws of thermodynamics, which greatly simplif ies the problem i n hand. However, some chemical reactions occur at rates simi lar to those o f t ransport processes; most transient phenomena like ign i t ion, ext inct ion, and re- igni t ion cannot be predicted w i t h the use of fast chemistry. K inet ic mechanisms express the rates at which species are created or consumed dur ing a chemical reaction. For a system of reactions r = 1...R and species s = 1...S, s=l s=l where kT is the reaction rate coefficient, Eq . (2.5) and the stoichiometric coefficients o f reac- tants, frl\ and products uf). T h e rate of format ion for the Ith species as a funct ion o f the concentrat ion of S species, cs, as given by Warnatz [2]: *"=(#) =»{#-#)n<#- V / chem,r s = \ Reaction rates are strongly non-linear functions of temperature v ia the react ion rate coefficient k. Pressure is also known to affect the reaction rates by modi fy ing the concentrat ion o f species. Complex reaction mechanisms, e.g. for the ox idat ion of methane, can consist o f many different global reactions. Researchers have proposed methods to reduce the number o f re- actions to include only those tha t are deemed most relevant for the study o f the phenomena o f interest. Most reduct ion processes are based on assumptions of par t ia l equ ihbr ium and quasi-steady state, wh ich implies tha t reduced mechanisms are applicable on ly under certain l i m i t i n g condit ions. Peters [12], identif ied the reduct ion in computat ion t ime for the numer i - cal s imulat ion of combustion events, as well as the appl icabi l i ty o f asymptot ic methods to the analysis o f flame structures as the two most impor tan t applications of reduced kinet ic mech- anisms. As an example, Bilger et al [13] proposed a method to derive a four-step mechanism Chapter 2. Combustion, Turbulence, and Stochastic Processes 9 for a methane-air flame wh ich is in good agreement w i t h the more complex mechanisms f r o m which i t or iginated for a l im i ted class o f combustion problems. The reduced chemical k inet ic mechanism used i n this research project is described i n Chapter 4. 2.5 Scales of Turbulence and Kolmogorov Scales Turbu lent flows can be found in different scales o f magni tude ranging f r o m large scales such as those found i n oceans or atmospherical ly relevant scales, t o the very small scales such as those studied i n combust ion. Turbulence is a high-energy phenomenon. This k inet ic energy is later dissipated, mainly th rough viscosity. The effects of molecular viscosity, however, are relevant only at very small scales. Th is implies tha t , i n order for the energy to be dissipated at those scales, th is energy must be transferred f rom the largest scales down to the smaller scales. The vortices, also known as eddies, present i n a turbu lent flow vary in scale and they can be correlated to the development o f velocity gradients. I t has been found tha t the largest vortices are responsible for most of the transport o f momentum. However, viscosity act ing at molecular scales is responsible for the dissipation of energy through a conversion of k inet ic energy in to heat at the smal l scales of mot ion [14]. Since chemical reactions typ ica l ly take place at this same molecular level, i t is impor tant to consider these smal l turbulence scales. T h e K41 theory, proposed by Kolmogorov [15], states tha t , at the smallest scales of t u r b u - lence, the fluid mo t ion is self-similar and isotropic, giving or ig in to the concept o f universal i ty of turbulence at such small scales. Fol lowing Kolmogorov, the use of the kinematic viscosity o f the fluid, v, and the energy dissipation rate normalised per un i t mass, e, permits the def in i t ion of the fol lowing microscales, also known as the Kolmogorov scales. • Characterist ic length microscale, • Characterist ic t ime microscale, • Characteristic velocity microscale, Chapter 2. Combustion, Turbulence, and Stochastic Processes 10 A t these micro-scales, the result ing local Kolmogorov Reynolds number is equal t o unity, wh ich means that there is no turbulence at smaller scales. 2.6 Cascade of Turbulent Kinetic Energy Based on Kolmogorov's ideas, researchers have developed the concept of the energy cascade, wh ich states tha t tu rbu len t k inet ic energy transfers f rom large to small scales v i a vortex stretching and once the energy reaches the smaller scales of turbulence, viscous dissipation occurs. Tennekes and Lumley [14] provide a complete discussion on the vortex-stretching mechanism. F ig . 2.1 ( f rom Peters [1]) shows the turbu lent k inet ic energy spectrum as a funct ion o f wave number k, where k is the inverse of the vortex or eddy size. F rom this figure, i t can be seen that the kinet ic energy exhibi ts a monotonic decay i n the logar i thmic p lo t , w i t h a slope of —5/3 along the inert ia l subrange. Th is is followed by an abrupt decay i n the viscous subrange, where viscous dissipation affects the smallest scales of turbulence. log E(k) large energy inertial viscous scales containing subrange subrange integral I 1 1 0 r 1 7,-' log k Figure 2 .1 : Turbulent kinetic energy spectrum, from Peters [1] 2.7 Turbulence Simulation and Modelling T h e Navier-Stokes equations that describe turbulence can, i n pr inciple, be solved di rect ly using numerical techniques. I n reality, the strong coupling and non-l ineari ty of the equations, along w i t h the wide range of length and t ime scales involved i n the solut ion and the chaotic nature o f turbulence, make such direct calculations possible on ly in very l im i ted cases. T i m e and mass averaging techniques have been developed to circumvent this problem. T h e in t roduct ion o f such techniques, however, produces terms for which the solut ion is either not k n o w n or Chapter 2. Combustion, Turbulence, and Stochastic Processes 11 cannot be expressed as an explicit function of the averaged variables. Oftentimes, these terms have to be modelled. The following sections describe the simulation and modelling techniques most widely used in turbulence. 2.7.1 Direct Numerical Simulation (DNS) In direct numerical simulation (DNS), the Navier-Stokes equations are solved numerically. This requires very fine spatial and temporal discretisations so that the transport equations can be solved from the large scales down to the Kolmogorov microscales. In general, it is estimated that the computation time for a DNS simulation increases approximately at a rate of Re* [2]. Consequently, at present only low Reynolds number flows can be solved directly. This limitation prevents the use of DNS for most practical applications. Currently, most research using D N S is aimed at the analysis of small turbulent structures, and the development and validation of closure models. 2.7.2 Large-Eddy Simulation (LES) With Large-Eddy Simulation (LES), the large scales of turbulence are calculated directly in the same way as in DNS. The smaller scales are modelled using turbulence models and through processes of numerical filtering. L E S provides high-quality results at a much lower computational expense than DNS. 2.7.3 Time or Ensemble Averaging (Reynolds Decomposition) The Reynolds decomposition [14], is frequently used to obtain statistical information from a turbulent flow. The time-average velocity of the flow can be defined as, so that instantaneous fluctuations of velocity around the time-average mean can be expressed which can also be expressed as: Ui{x,t) N u'iixjt) = Ui{x,t) — Ui(x). Chapter 2. Combustion, Turbulence, and Stochastic Processes 12 A n impor tant , and widely used parameter i n turbulence characterization is the turbu lent k inet ic energy K , which is obtained f r o m tak ing the root mean square and t ime-averaging the veloci ty fluctuations as follows [16], - 1 — 2 wh ich , for a three-dimensional flow field becomes, '2 . *2 i '2 _ _ Uj + + M3 K ~ 2 • I f no addi t ional energy is transferred to the system, the kinet ic energy dissipates i n t ime at a rate given by, DK e=m- 2.7A Turbulence Models F r o m probabi l i ty theory, i t can be shown tha t the ensemble average of the velocity fluctua- t ions around the mean value is equal to zero [14], or i n mathemat ical fo rm (u[) = 0. The second moment (u^u'^j t ha t represents the stat ist ical correlat ion between u,- and Uj becomes an ex t ra unknown in the system of equations for which an equation as an expl ic i t func t ion of the averaged variables is not available. This is commonly known as a closure prob lem and the unclosed terms are often modelled using physical or mathemat ical tools. Launder and Spalding [17], and Pope [18] summarise different mathemat ica l turbulence models. Those models range f rom one-equation to mult i -equat ion mathemat ica l expressions to characterise turbulent phenomena and were derived as an al ternat ive to previous models such as the mix ing- length model. The numerical designation for the models is based on the number of par t ia l differential equations which are needed to calculate the model led parameters. Gatsk i and Rumsey [19], for example, identi f ied variat ions of previous models tha t have or ig inated 'zero-equation' models, wh ich are algebraic relations to define the vort ical or eddy viscosity, and 'hal f-equat ion' models where an ordinary differential equation is solved to calculate the eddy viscosity. Results obtained f r o m zero- t o mul t i -equat ion models may vary considerably, due to the different degrees of complexi ty and detai l that they may include i n their analyses. Most models are derived for specific applications, e.g. flow around a cyl inder, boundary- layer flows, and so f o r t h and they are usual ly tuned to provide acceptable to good approximat ions to experimental results. Chapter 2. Combustion, Turbulence, and Stochastic Processes 13 2.8 Turbulent Combustion Most practical applications involving combustion occur in turbulent regimes. Directly or in- directly, al l of the characteristics of turbulence have a major effect on the evolution of a com- bustion event. Vorticity, for example, increases the interface area between fuel and oxidiser in non-premixed combustion, or between reactants, intermediates, and products in premixed flames, which results in an increased rate of reaction. In premixed combustion, turbulence has several important effects. It increases the area of the reactant-intermediate-products interface, which in turn increases the rate of reaction. This increase in area is due to the appearance of 'wrinkles' in the flame front which are produced by the action of turbulent vortices. As a consequence of this stretching, the reaction zone be- comes thinner, promoting higher flame-propagation velocities. If the accelerated mixing rate produced by the stretch effect reaches a point where mixing takes place faster than reaction, extinction of the flame occurs [20]. In non-premixed combustion, the most significant role of turbulence is to enhance mixing. This is achieved through the increase in the surface area of the fuel-oxidiser interface due to the strain and stretch created by vorticity and the high diffusion velocity present in turbulent flows which is a result of an increase in gradients. Since non-premixed combustion is a primarily mixing-limited phenomenon, the result of these effects is of paramount importance. 2.9 Turbulent Combustion Simulation and Modelling There are a number of different techniques and approaches to solve the equations that describe the evolution of a turbulent reacting flow. Well established methods have been used for the study and analysis of turbulence. These methods have been modified and adapted to account for the interactions between fluid mechanics/dynamics and chemistry. Basically, most of the research work is dedicated to closure methods for the chemical source term. The works by Mel l et al [21], Swaminathan and Bilger [22], Bilger [23], and Peters [1], amongst others, offer a variety of options for the modelling of the chemical source term. A brief description of the most widely used techniques and their applications is given below. Chapter 2. Combustion, Turbulence, and Stochastic Processes 14 2.9.1 DNS Echekki et al [24] report the use of D N S , coupled w i t h a mult i -step chemical kinetics mechanism for the s imulat ion of a 2-D premixed turbulent reacting flow. Papers by Bushe et a l [8] and Swaminathan and Bi lger [22] describe the use of D N S w i t h reduced chemical k inet ic mechanisms in the study o f non-premixed combustion. The DNS database of the oxidat ion o f methane in a turbu lent flow of Bushe et al [8] is used as the baseline for the comparison w i t h the results of the model developed i n th is thesis. 2.9.2 Favre Averaging Combust ion processes are characterized by considerable density fluctuations, wh ich has p rompted the development o f the density-weighted, or Favre averaging. The Favre average o f a property, such as mass f ract ion for example, can be defined as y = ^ - (2.11) P T h e instantaneous local value o f mass fract ion can be defined i n terms of i ts average and fluctuation as Y{x,t)=Y{x,t)+Y'{x,t) (2.12) Similar ly, the instantaneous local value of mass f ract ion is defined in terms of i ts Favre average and fluctuation as Y(x,t)=Y(x,t)+Y"(x,t) (2.13) Combin ing the def in i t ion o f Eq. (2.11) w i t h Eq . (2.12) leads to an expression for the Favre average of mass f ract ion i n terms of the average mass f ract ion as ~ (p + d) (7 + Y') — pfY1 Y = \PTl>)\ —I = Y + P^_ , 2 M ) P P The t e r m p'Y' i n Eq. (2.14) denotes the correlation between the fluctuations o f density a n d mass f ract ion. Th i s is an unclosed t e r m which can be computed using a t ranspor t equation, or model led using theoretical or empir ical relations. 2.9.3 LES Chemical reactions occur at small scales where the flow is modelled i n LES, not resolved; th is implies the need for models for the chemical source term i n the t ranspor t equations of mass Chapter 2. Combustion, Turbulence, and Stochastic Processes 15 f ract ion o f species [16]. LES results have been used as benchmarks for a priori test ing of o ther simulations. T h e pi loted flame of Pi tsch and Steiner [25], as well as t h a t of Steiner and Bushe [26], have been used in comparative studies o f computer simulat ions. I n the case of the work by Steiner and Bushe, LES was incorporated w i t h the newly-developed condi t ional source-term est imat ion. Laurence [27] discussed applications of LES in indus t r ia l settings, and ident i f ied the ongoing work on numerical methods for finite volumes in unst ructured grids as one of the m a i n activit ies aimed at provid ing more opportuni t ies to such applications of the method. 2.9.4 Laminar Flamelets F low visualisation of reacting flows w i th low-intensity turbulence shows t h a t curved flame fronts propagate outwards from the source of ign i t ion to the boundaries o f the system. I n th is fashion, the turbulent flame can be conceptualised as an aggregation o f such flame fronts. T h i s is also known as the flamelet regime. Based on th is concept, the Laminar Flamelets, as described by Peters [28], [1], are t h i n (their w i d t h being smaller t han the Kolmogorov length- scale), reactive sheets tha t react and diffuse w i t h i n a non-reactive field. Th i s method assumes t h a t once ign i t ion has occurred, the chemical t ime scale is great ly accelerated, reducing the react ion zone in to a t h i n sheet. Th i s concept was fur ther developed into assuming tha t i f the t h i n reaction zone is smaller t h a n the Kolmogorov length-scale, then the flow in tha t zone is laminar , due to the fact that there exists no turbulence below the Kolmogorov microscales. Much research has been dedicated to the development of the flamelet model . Peters [1] provided a complete descript ion o f the model , as well as its formal appl icat ion to premixed a n d non-premixed turbulent combustion. Len t in i [29] explored the appl icat ion o f the stretched laminar flamelet method combined w i t h the K — e model for turbulence i n a tu rbu lent non- premixed flow and reported good agreement w i t h results obtained f rom experiments performed at the flamelet regime. Cook et al [30] presented the large-eddy laminar flamelet mode l in wh ich the chemistry at the smaller scales was modelled using the flamelet theory. The flamelet mode l offers good predictions of the major chemical species tha t originate i n combust ion of hydrocarbon-based fuels at what is known as the 'flamelet regime'. However, the flamelet the- o ry applies under very specific condit ions of the flow, e.g. the flamelet thickness is smaller than Chapter 2. Combustion, Turbulence, and Stochastic Processes 16 the smaller length-scales of turbulence, which limits the effective application of the flamelet method. Swaminathan and Bilger [22] pointed out that the flamelet model falls short of pro- viding good predictions of the minor species in combustion of hydrocarbon fuels. This point was also discussed by Mell et al [21] when comparing the flamelet and the conditional moment closure (CMC) models with a DNS database for different flow configurations to study the effects of variations of scalar dissipation on flamelet and C M C results. A n improvement over the laminar flamelet model is the unsteady laminar flamelet theory, which introduces a Langrangian time frame that records the time dependence in the flame structure. Mauss et al [31] presented results that seem to indicate that the transient variations of the flamelet structure of a steady turbulent non-premixed flame are responsible for the scattering of experimental data points obtained with one-point Raman measurements. Pitsch [32] obtained good agreement between experiments and the results of the unsteady flamelet model for a turbulent non-premixed CH^ffyf^-air flame. 2.9.5 Probability Density Function (PDF) Modelling In the P D F method, a transport equation for the joint P D F of relevant scalars is solved to model the evolution of a combustion event. These scalars can include mass fraction of chemical species, flow velocities, temperature, mixture fraction, and so forth. A n important advantage of this method is that the chemical source term is solved for directly, without the need of a model. Modelling is necessary, however, for the molecular mixing terms in scalar space, as well as for the turbulent transport terms in physical space [33]. Due to the high-dimensionality inherent to this method, traditional computing techniques are insufficient. This has prompted researchers to apply numerical methods capable of dealing with problems with a large number of variables and dimensions, such as Monte Carlo methods. These applications wil l be reviewed with more detail in the following sections of this thesis. 2.9.6 Conditional Moment Closure The Conditional Moment Closure ( C M C ) of Klimenko [3] and Bilger [4] determines closure for the chemical source term of the transport equation of the mass fraction of species. This method is based on the assumption that the fluctuations of scalars, such as mass fraction Chapter 2. Combustion, Turbulence, and Stochastic Processes 17 of chemical species and temperature, are associated with variations of other scalars. The advantage of using the C M C method is that macromixing phenomena can be decoupled from the chemical kinetics, while conserving the effects of micromixing via the scalar dissipation term. The method is described and its equations derived in Chapter 3. 2.10 Stochastic Processes and Monte Carlo Methods The theory behind stochastic processes originated from the observations of the trajectories of pollen particles moving on a water surface made by Brown in 1827 and later independently developed into an elegant mathematical theory by Einstein [34], [35], Smoluchowski [36], and Langevin [37]. The experiments by Perrin [38], [39], [40], [41], [42] confirmed the previous theoretical analyses and his work was awarded the Nobel Prize of Physics in 1926. The ini- t ial applications of the method concentrated on physics, mathematics and chemistry. Today, stochastic processes are extensively applied in fields such as biology, finance, economics, de- mography, atmospheric and ocean sciences, amongst many others. Different numerical meth- ods, such as the Monte Carlo (MC) method, have been developed as tools for the study of stochastic processes. The M C method is intended to solve multi-dimensional problems with a large number of variables. This can be a very inefficient method when used to solve simple problems given the large number of computations required. 2.10.1 Monte Carlo Simulations of Turbulence Kramer [43] reviewed the application of the M C method in generic turbulence problems. In this work, a thorough description of the two different (Eulerian and Lagrangian) types of M C simulation is presented. The M C method is remarkably useful for the simulation of turbulent flows given its ability to treat multi-dimensional problems with a large number of degrees of freedom. Another attractive feature of this method is its ability to generate chaotic parti- cle trajectories and velocity field structures. In the Eulerian approach, a stochastic velocity field model replaces the turbulent velocity field to be simulated. This field is then solved in a spatial domain, which is computationally less demanding when compared to the solution of the Navier-Stokes partial differential equations. The stochastic field is tailored to the spe- cific application and is frequently used to simulate trajectories of particles suspended in a flow. Chapter 2. Combustion, Turbulence, and Stochastic Processes 18 A number of methods exist to numerically solve the equations derived using the Eulerian approach. The Fourier method is based on a spectral formulation for a homogeneous Gaussian stochastic field. A major drawback of this method is the generation of a spatial period that has detrimental effects on turbulent diffusion simulation. Using a randomization method, the stochastic Fourier integral is discretised by stochastic wave numbers. Despite the fact that this method provides useful data to simulate velocity fields with strong long-range correla- tions, those fields have non-Gaussian statistics. According to Elliot et al [44] and Sabelfeld [45] this limits the applications of this method for ideal mathematical or physical turbulence models. More physically meaningful models have also been developed. Among them are the moving-average and the wavelet methods. McCoy [46] introduced a Gaussian stochastic field represented in physical space using white noise convolved against a kernel. Long-range corre- lations in the velocity field can be appropriately simulated using the wavelet method. There exists a combination of Fourier-wavelet methods that present improvements over its precur- sors. This method has been reported to have shown excellent results when simulating velocity fields with the condition of self-similarity scaling of their structures. A Lagrangian M C simulation utilises particles evolving in a stochastic model. A l l the effects of turbulence are incorporated within the properties of each particle. Despite the fact that the Lagrangian M C simulation does not make reference to a fluid velocity field, Minier and Pozorsky [47] and Welton [48] have used it to simulate the motion of a turbulent flow by analysing the stochastic evolution of a group of particles. The average properties of this group of particles (and of the ones in the vicinity of the target field) define the properties of the fluid, and of the flow, at that specific location. The methods developed to deal with M C simulations using the Lagrangian approach include the simulation of fluid motion and particles, the simulation of immersed bodies and hybrid L E S / M C schemes. In the simulation of fluid motion and particles, a set of fluid particles is followed accord- ing to a stochastic formulation. Physical, or pseudo-physical properties are defined for the particles. The evolution of the dynamics of the flow is assumed to be the average of the simulated particles. This method offers flexibility in the way that it can provide some in- sight into the mixing process of the unresolved scales of the turbulent flow by the use of a Chapter 2. Combustion, Turbulence, and Stochastic Processes 19 simple Langevin formulation. If more details are desirable, the stochastic formulation must be changed. Stochastic differential equations in the form of Markov processes are often used with this purpose in mind [49]. The simulation of immersed bodies follows a similar proce- dure. Pozorsky and Minier [50] reported an approach in which two sets of stochastic particles are defined. The first set represents the immersed particles, while the second is intended to simulate the flow. The physical and pseudo-physical properties of both sets of particles are different from one another. A major drawback of these methods is that they can only simulate homogeneous turbulent flows. In the hybrid L E S / M C schemes the flow field is resolved using a L E S formulation, while the mixing (and chemical reaction, if reacting flow is being considered) simulations are resolved using a M C formulation. This scheme calls for the continuous feedback from one formulation to the other. Jaberi et al [51] and Obliego et al [52] have reported an increase in accuracy in their L E S calculations after the addition of the M C component. The apparent reason for this is that the M C simulation provides some information on the effects of the small scales otherwise neglected by the LES model. This model has also been used successfully in non- homogeneous turbulent flows, which extends its potential for applicability to more realistic problems. Le Maitre et al [53] proposed a method in which a stochastic spectral finite element method undertakes the formalism of a M C simulation and offered numerous results claiming improvements in computational expenses, amongst other advantages over the M C method. Unfortunately, this research group did not provide a comparison between the results of the two formulations. 2.10.2 Monte Carlo Simulations of Turbulent Reacting Flows Most applications of the M C method in turbulent reacting flows follow one of the Lagrangian formulations described in the preceding section. Pope [54] described what appears to be the first application of the M C simulation to a turbulent reacting flow. This method has shown great usefulness given its capabilities of dealing with problems with a large number of variables and dimensions. Its first and foremost application is intended to numerically solve the P D F transport equations derived for a trubulent reacting flow. Chapter 2. Combustion, Turbulence, and Stochastic Processes 20 The advantages o f the method are numerous. As described i n different papers [54], [55], the P D F method provides closure to the chemical source te rm i n the equat ion w i t h o u t the need of any model l ing and can be applied to either premixed or non-premixed flows. I t is also known t h a t this method incurs large computat ional demands, has no abi l i ty to model wal l - type boundaries, and is di f f icul t to apply i n complex flows. According to Pope [54], the computa- t ional requirements o f this method increase linearly w i t h the number of variables used ( in this case dimensions, as wel l ) , compared to the exponential increase of computat ions required by t rad i t iona l finite-difference formulations. Th is fact provides a way of dealing w i t h problems involv ing many species. As discussed i n the previous section, the M C method can deal w i t h the s imulat ion of walls and complex, inhomogeneous flows, hence solving the more relevant drawbacks of the P D F method. As described by Pope [54], stochastic particles were used to mimic the chemical and thermodynamic properties of the flow. These particles, however, were defined as a numerical too l and have no physical meaning. A plug-flow reactor problem w i t h imperfect mix ing was simulated using the M C method, using a single scalar to model the concentrat ion of reaction products. Pope's numerical results were in good agreement to those obtained i n experiments performed by B a t t [56]. Val ino [57] proposed an Euler ian M C formulat ion for the s imulat ion of the P D F o f a single scalar i n a turbulent flow. His proposal was to use stochastic part icles ' j ump ing ' f r o m node to node in the spatial domain according to specific rules (a stochastic equation). Hulek and L ind- stedt [58] offered a comparison between da ta obtained from D N S and f r o m M C simulations. The i r formulat ion used the M C method to solve the terms model led w i t h the P D F method i n the flamelet equation. This method, called flamelet acceleration, was also tested and i t ap- peared to be an improvement when compared w i t h the standard flamelet solut ion. Hulek and L indstedt [59] later modelled premixed turbulent flames using P D F and M C methods. The M C simulat ion was used to solve the mass density funct ion evolut ion equation. Stochastic par- ticles evolved fol lowing determinist ic processes representing the closed terms of the equation and stochastic processes designed to model the effect o f the unclosed terms. They showed the comparison between several different numerical formulat ions and experimental results, where i t can be seen that the P D F - M C simulations were in good agreement w i t h experiments. Chapter 2. Combustion, Turbulence, and Stochastic Processes 21 The special ability of the M C method to mimic turbulent mixing was explored by Kawan- abe et al [60]. The methodology was applied to a plug-flow flame in unsteady turbulent combustion. The turbulence model used was a standard K — e and the mixture is forced to avoid homogeneity. Cannon et al [61] and Kraft and Fey [62] have explored stochastic reactor models using the P D F and M C methods. Cannon's group offered some practical results, while Kraft and Fey mainly developed a purely analytical solution to the stochastic reactor model. Cannon's research was aimed at the prediction of CO and N O in premixed combustion of methane, using a partially stirred reactor (PaSR). Conceptually, the PaSR could be thought of as a M C process itself, since it also randomly selects particles that are then forced to interact amongst themselves. This method is an improvement over the perfectly stirred reactor (PSR) since finite-rate mixing effects can be included in the solution. The mixing term was modelled using a deterministic method rather than a M C method. The good agreement between the 9-step simulation and the detailed kinetic solution demonstrated the robustness and accuracy of this particular model. 2.11 Summary In this Chapter, the fundamentals of turbulent combustion, have been reviewed. Turbulence, scales of turbulence, and turbulence simulation and modelling were also treated, along with the interactions between turbulence and combustion. Descriptions of the regimes and equa- tions that define turbulent combustion, as well as some of the most important analytical tools available for the solution of those equations were also presented in this chapter. A detailed description of the applications of the Monte Carlo simulations in turbulence and turbulent combustion was provided. It has been observed that there exist a number of different options in which the M C method can be used to solve numerically demanding problems in turbulent combustion. Future possible applications include particulate matter and pollutant formation mechanisms and mass transport in permeable membranes. Of particular interest to this research is the capability of the method to be included to solve problems expressed in terms of different modelling theories such as conditional moment closure, L E S , or flamelet models. A n increasing number of research opportunities exist in this field, with promises of continuous and prolific growth in the near future. 22 Chapter 3 The D C M C Method with Stochastic Processes Condi t ional moments are averages which are calculated for those members or realisations that satisfy some predefined condi t ion. The condit ional moment closure method ( C M C ) , inde- pendently developed by K l imenko [3] and Bi lger [4], provides closure for the chemical source t e r m in the t ranspor t equation of species and enthalpy or temperature by assuming tha t the var iabi l i ty i n temperature and the mass fractions of species can be l inked to the fluctuations o f a given scalar variable. The reaction progress variable and mix tu re f ract ion are common choices of scalar variables for premixed and non-premixed combustion analysis, respectively. I f the mass fractions o f species are expressed as functions of not only space and t ime, but also as funct ions of the condi t ioning variable, their condit ional averages can then be used to approximate the condi t ional average of the source t e r m as d\Z = £ « u \T\Z = £, Yj\Z = £ j . T h e overbar i n this nota t ion indicates that the mean value has been calculated f rom averaging over n = 1...N realisations. The vert ical bar means 'given' , and i n the case o f Yj denotes the value of the mass f ract ion given tha t m ix tu re fract ion has a value of £. As an i l lus t ra t ion o f the theoretical basis of the C M C method, F ig. 3.1 shows a plot o f ind iv idua l n = 1..JV realisations of a simulat ion o f temperature (blue fines). Where the uncondi t ional average o f temperature, T = YliLiTi/N, is represented by the hor izontal line at around T « 500, and the red curve is the calculated condit ional average o f temperature, T = (T\Z = Q, where the angle brackets indicate t h a t only those realisations that comply w i t h the condi t ion of Z = £ are considered i n the averaging process. I t is clear from F ig . 3.1 t h a t the in format ion obtained from condit ional averages is more representative of the actual relat ionship between T and Z t han that obtained from uncondi t ional averages. Chapter 3. The DCMC Method with Stochastic Processes 23 Z Figure 3.1: Conditional average of temperature; blue(.) DNS, red(-) (T\Z), black(-) T Researchers have previously explored the application of the C M C method and have found that using only one conditional variable offers good results in terms of species mass fraction and temperature predictions. However, the presence of triple flames and prediction of igni- tion, extinction, and re-ignition phenomena are not well reproduced by just one conditioning variable. Figure 3.2, from Bushe et al [8], shows a slice of a 3-dimensional DNS calculation of temperature. The contour lines represent isopleths of mixture fraction, which were formed by the intersection of the slicing plane and the corresponding isosurfaces of mixture fraction. The white isopleth corresponds to the stoichiometric mixture fraction. Figure 3.2: DNS calculation of temperature, from Bushe [8] It is clear that this temperature field exhibits strong fluctuations in regions of constant mixture fraction. This is evident in the 'cold pockets' shown in regions A and B in the figure. These cold zones are typical of extinguishing flames. From this figure, it can also be Chapter 3. The DCMC Method with Stochastic Processes 24 observed that there is a transition between hot zones, located to the right of region B. This transition could correspond to re-ignition or bifurcation of the flame. It has been proposed that using a second conditional variable would solve those issues [9], [16], [63]. This result is of paramount importance, given the close relationship between these phenomena and the formation of pollutants. In this thesis, the use of two conditioning variables has been explored. One variable is mixture fraction Z, which characterises the state of mixedness of the flow for a given instant in time and space. The other conditional variable is named a and is introduced to capture fluctuations along an isosurface of mixture fraction. This is achieved through the definition of the scalar a as a conserved scalar with a spatial gradient perpendicular to that of mixture fraction. 3.1 Conditional Moment Closure with Two Conditional Variables ( D C M C ) The use of a second conditional variable has been proposed as a way to solve the shortcomings of the original C M C method. Bushe [16] used a second conditioning variable that filtered between regions of a surface defined by a constant mixture fraction, hence detecting differences in the stretch rate history of the flow. The results of a simulation of auto-ignition of n-Heptane showed an improvement over the single-conditional method in detecting autoignition and double flames when compared to DNS databases. However, the same results failed to show the presence of triple flames present in the reference data. Cha et al [63] used the scalar dissipation rate as the second conditioning variable for the simulation of a methane-air turbulent reacting flow. After comparing their results with those obtained from an experimental jet and a DNS simulation, it was found that there exists good agreement in the prediction of extinction phenomena, but they obtained an early prediction of the onset of re-ignition. High fluctuations around the conditional means, specifically for low values of the scalar dissipation rate, were identified as a possible source for the discrepancies with the reference data. 3.2 Derivation of the D C M C Equations During the development of this project, two conditioning conditional variables have been used. • Mixture fraction Z — Z (x{,t) , Chapter 3. The DCMC Method with Stochastic Processes 25 • and a = a (xi,t). These two random variables are related to the non-random variables £ and a using: w |(Z = C,o = a) * co (r|(Z = C,« = a) ,Y/ |(Z = C,a = a)) , (3.1) such that the result is a function of what values are chosen for the conditioning variables £ and a. The definition of the conditional average of the mass fraction of the Ph species is Y[ = Yi{t,a;Xi,t) = (Yi(xi,t)\Z{xi,t) = C,a(xi,t)=a), so that the instantaneous value of species mass fraction using conditional average and fluctu- ations around that mean can be expressed as Yi(xi,t;n) = 17U(Xj)t;n))a(xi,t;n) +YI- E q . (2.3) is substituted in Eq. (2.2) to obtain a new version of the transport equation of species mass fractions as dpYi + dpuiYj = _d_ ( Di9YI\ + ^ 2 ) dt dxi dxi \ dxi J Using the experimental results from Vargaftik [64], it has been proposed [16] that pD\ = constant, and that Dz = Di, implying that Le = 1. Using these assumptions, Eq. (3.2) can be written as dYr | udYr D b^Yj = Co! dt 1 dxi dxidxi p To express E q . (3.3) in terms of conditional averages, it is required to derive partial derivatives, which were obtained by applying the chain rule of differential calculus. • The partial derivative of species mass fraction with respect to time is dt dt dC dt da dt dt ' ( ' ' • The first partial derivative of species mass fraction with respect to space is m = w + mM + m ^ + dY[ dx{ dx{ dC, dxi da dxi dxi Chapter 3. The DCMC Method with Stochastic Processes 26 The second partial derivative of species mass fraction with respect to space is &Yi dWjdZdZ c ^ l ? d a d a d2Yj dZ da dxidxi dC,2 dxi dxi da2 dxi dxt dC,da dxi dxi 3 2 Y} dZ n d2Yj da dYi d2Z +2 1-2 1- 'dxidC,dx{ ' ~dxidadxi ' d( dxtdxi + dYLJ^+F*_+&YL ( 3 6 ) da dxidxi dxidxi dxidxi Equations (3.4), (3.5), and (3.6) were substituted in Eq. (3.3) producing, *' *• +dY*r + d T l F n&Y'MOz ^Yj da da Yj dC, da ° dC,2 dx\ dxi da2 dxi dxi dC,da dxi dx{ dx^dC, dxi dxida dxi 1 where the following definitions are used: • The transport equation of the conditional average of the species mass fraction is E =  dyi ! dy? D d2Y Y l dt dxi dxidxi • The transport equation of mixture fraction, E q . (2.1), is written here as Z dt * dx{ dxidxi ^ ^ • The transport equation of the scalar a is „ da da ^ d2a n ,„ „„. E a = m + U i d ^ - D d ^ d x - = 0- ( 3 - 1 0 ) • The transport equation of the fluctuations around the conditional average of species mass fraction is Y* dt * dxi dxidxi ^ ^ 3.3 Modelling Scalar Dissipation Scalar dissipation, Xz = M7!^' ^ used to represent the coupling that exists between fluid dynamics and the chemical reactions taking place in a turbulent reacting flow [23]. Previous research [1] indicated that large values of scalar dissipation rates lead to extinction of flames. Conversely, ignition phenomena are associated to small scalar dissipation rates. Chapter 3. The DCMC Method with Stochastic Processes 27 Peters [1] proposed a model in which the scalar dissipation rate diminished as a function of mixture fraction with time in a one-dimensional mixing layer. This model is frequently used in the flamelet formulation discussed in section 2.6.4. However, the model is not capable of cap- turing extinction or ignition. A n improvement to better reproduce extinction phenomena was proposed by Peters [1] in which a conditional stoichiometric scalar dissipation rate, assumed to have a lognormal P D F , was used to calculate the right proportion of burning flamelets during an extinction event. Using C M C and the stretched diffusion concept, Bushe [16] proposed a model for the con- ditional scalar dissipation rate, which was found to provide good approximations of values of the scalar dissipation rate when compared to DNS of a non-reactive flow of Eswaran and Pope [65]. This same model, however, did not provide good estimations of ignition due to long decreasing times of the scalar dissipation [16]. A n improvement of this model was to in- corporate a second conditioning variable to the system of transport equations of species mass fractions, which was also a passive scalar. This model was found to provide good predictions of ignition and double-flame formation of an n-Heptane non-premixed flame when compared to data obtained by other researchers [66], [67], [68]. However, this formulation did not predict the presence of triple flames found in other research papers [67]. In this thesis, the D C M C method has been applied to define conditionally averaged trans- port equations for the scalar dissipation of Z and a. The conditional average of scalar dissi- pation is defined as Xl = Xl(t,(x;xi,t) = (xz{xi,t)\Z(xi,t) = t,a{xi,t) = a) leading to an instantaneous value of Xz{%k,t) = Xz\c=Z(xi,t),a=a(xi,t) + x'z- (3.12) Ruetsch and Maxey [69] proposed a transport equation for the scalar dissipation, with the terms on the rhs of the equation representing production, dissipation, and diffusion: DXz Dt OXi OXj d fdZ\ dxj \dxi J + D ______ dxjdxj (3.13) The Sjj term in Eq . (3.13) corresponds to the strain tensor that denotes the production of scalar dissipation through vortex stretching mechanisms. In premixed, turbulent reacting Chapter 3. The DCMC Method with Stochastic Processes 28 flows, strain is known to have a strong effect over the stretch history of the flame, which consequently affects the flame structure [11]. The definition of Sij [14] is, (3-14) In turbulent, non-premixed flames, high levels of strain imply high levels of scalar dissipa- tion, which in turn generate local extinction of the flame, since the chemical reactions cannot take place given the extremely high rate at which reactants are diffused into and through the reaction zone. Williams [11] provided a flamelet-based analysis of this phenomenon. Thevenin and Candel [67] studied the effect of variable strain on the ignition of a non-premixed flow of air and hydrogen through the use of computer simulations. While their research predicted self-ignition and ignition times in agreement with the experimental results of Cheng et al [70], the triple-flame phenomenon was observed only for the case where there was no strain present. Using the definition of a material derivative, Eq. (3.13) can be expanded to obtain dt z dxz = ~Ui-1 dxj OXjOXj -2D d2Z dxjdxi J \dxjdxij (3.15) uj dxi dxj In order to express Eq. (3.15) in terms of the conditioning variables, it is required to obtain derivatives analogues to Eqs. (3.4), (3.5), and (3.6). The definitions of Eqs. (3.9) and (3.10) are also used to produce dxi at HiZ ^ &n AC da TP X z dxi 2 ° l ™ S i + D - ^ dxidxj lJ dxidxi ^ x l d Z dZ d2xl da da ^^d2^ dZ da d2Xz dZ dC2 dxi dxi da2 dxi dxj dCda dxi dxi dxidC dxi ( d2z \ f d2z \ \ dxj dxij \ dxj dxi J (3.16) dxida dxi where the transport equation for the conditionally averaged fluctuation of scalar dissipation is Ezr = dx'z , dx + Ui- D d2x!z (3.17) X z dt dxi dxidxi The conditionally averaged transport equation of the scalar dissipation Xa follows analogously to Eqs. (3.16) and (3.17). Chapter 3. The DCMC Method with Stochastic Processes 29 3.4 Scalar Transport Closure Hypotheses Equations (3.7) and (3.16) contain unclosed terms for which closure hypotheses are formulated in this thesis. W i t h the following scalar transport hypotheses, some of the terms in those equations are closed: • Z and a are conserved scalars, hence its lack of a source term as shown in Eqs. (2.1) and (3.10); • The spatial gradient of a is perpendicular to the spatial gradient of Z to better capture the effects of fluctuations along isosurfaces of mixture fraction. Using these hypotheses, Eq . (3.7) is reduced to — = & Y, D&Yi dZ dZ ^Yj da da dC2 dxi dxi da2 dx{ dxi 2D d 2 T l d Z 2D d W l d Q 1 E , dxidC, dxi dxida dxi Yj Eq. (3.16) is also reduced to dx~z dt _E dxz x* * dx{ 2™ B*S„+ ' dxi dxj %3 dxidxi dC2 dxi dxi da2 dxi dxi dxidC, dxi +2DS^^-2D ( &z \ f d2z \ \dxjdxiJ \dxjdxi J dxida dxi Multiplying Eq.(3.18) by p and taking its conditional average yields dWt LOI = pEyT + pEyJ DC" 0 „ <?Y, dZ dxidC, dxj Conditionally-averaging Eq . (3.19) yields dZ_dZ_ dxi dxi d2Yj &Yj Dda da da2 dxi dxi -pD— dxida dxi dt dxi dZ dZ d2x = -E^r-ui^-2——Sij + D X dxi dxi dxj dxidxi d2Tz dZ dZ d2Tzda da d2Xz~ dZ + J J r,>o ~—z= h U - » -—— V IU-dC2 dxi dxi da2 dxi dx{ dxidC, dxi dxida oxi / d2Z \ ( d2Z \ \ dxj dxi J \ dxjdxi J (3.18) (3.19) (3.20) (3.21) Chapter 3. The DCMC Method with Stochastic Processes 30 3.5 D C M C Closure Hypotheses Equations (3.20) and (3.21) were reduced to a conditionally-averaged form in the previous section. Whilst closure was determined for some unclosed terms, both equations still contain terms that can be further closed using the D C M C closure hypotheses described in the following modelling assumptions. • D C M C Modelling Assumption 1: the fluctuations around the conditional mean of species mass fraction and around the conditional mean of scalar dissipation are only functions of the scalar variables Z and a, independent of time and space. This produces tW1 BY7 cPY' ^ + m ^ - D ^ - = 0. (3.23) ut OX, OXiOXi • D C M C Modelling Assumption 2: assuming that the flow is homogeneous and isotropic, the ensemble of conditional averages of species mass fractions and scalar dissipation are equal at each point in space for the same instant in time, and the spatial gradient of the conditional mean of the species mass fraction is equal to zero [16]. This yields _ 3 * i ^dx- = 0 (3.24) ~D d W l dxidxi = 0 (3.25) d2Yj DdZ dxidC, dxi = 0 (3.26) cPYj p D da dxjda dxi = 0 (3.27) dxz OXi = 0 (3.28) D d2T> dxidxi = 0 (3.29) D d2Xl dZ dxidC, dxi = 0 (3.30) d 2x~z da dxida dxi - 0. (3.31) Substituting Eqs. (3.22) and (3.24) - (3.27) in Eq. (3.20) results in T _dYj PTi ndZdZ d2Yi nda da " I = p-d7- WP dx^dx- - L^^dx-dx-- ( 3 - 3 2 ) Chapter 3. The DCMC Method with Stochastic Processes 31 Here, the term on the lhs represents the condit ional average o f chemical source t e r m , whi le the second and t h i r d terms on the rhs represent the condit ional average of di f fusion of the species mass fract ions in the scalar f ield. Subst i tu t ing Eqs. (3.17) and (3.28) - (3.31) i n Eq. (3.21) results i n dxTz = - 2 dZ dZ oc2 Sij + D—^xl + D da2 Xa 2D- 82Z d2Z (3.33) dt ~dxidxj"lJ ' dC2 ' 2 /v" dxjdxidxjdxi Here, the first t e r m on the rhs represents the condit ional average of produc t ion o f scalar dissipat ion, the second and t h i r d terms represent the condit ional average of di f fusion of scalar dissipat ion, and the th i rd t e r m represents the condit ional average o f dissipat ion of scalar dissipation. 3.6 Mathematical solution of the Z-gradient squared term We define 9 = Expand ing 6 yields, d_ dx* dXz dxz dxj dxj (MdZ\ " \dxi dxi J , d2Z dZ dxjdxi dxi d2z d2z dxjdxi dxjdxi dXz dxz dxj dxj (dxj) d (OZ^dZY dxj \dxi dxi J „ d2Z dZ dxjdxi dxi Compar ing Eq. (3.36) to Eq. (3.34) i t can be seen that 0 = 4$;vvs>and 1 4x. 0. (3.34) (3.35) \dxjdxidxjdxi) \dxidxi) (3.37) Chapter 3. The DCMC Method with Stochastic Processes 32 Using Eq . (3.12) and tak ing par t ia l derivatives of Xz w i t h respect to space, Eq. (3.35) can be expressed as \ dxj J \ dC 9XJ ) \da dxj ) \ dxj ) .^dx^dx^dZ | dxidxi da | ^dxl dZ dxi da dxj dC, dxj ' " dxi da dxj ' " d£ dxj da dxj x ^dXzdx'z ^ 2dx~z dZ dx!z | jTXz da dX'z ( 3 3 g ) dxj dxj c?C dxj dxj da dxj dxj Using the scalar t ranspor t closure hypotheses and the model l ing assumptions, Eq . (3.38) reduces to Subst i tu t ing Eq. (3.39) i n Eq. (3.37) and condit ionally-averaging, yields 4 V d( J A\da) I t is fur ther assumed that / Xa \ Xa \Xz) xl' (3.41) Using th is assumption i n Eq. (3.40) and subst i tu t ing th is result i n Eq. (3.33) produces the final version o f the conditionally-averaged t ransport equation of scalar dissipation o f m ix tu re f rac t ion and o f the scalar a dxi = -2-^L^S -^(^V-^/^Vxi dt dxi dxj 13 2 V dC j 2\da)x~z +D^Xz + D ^ x a (3.42) dxZ = r>da da _ P_ (dx^\ _ & (d)g\ Xz dt dxi dxj i3 2 V dC J 2 V da J xi +D^Xz + D ^ T a . (3-43) T h e first t e r m on the rhs of these transport equations describes the condi t ional average of p roduc t ion o f scalar dissipation. Th is te rm represents the mapping of the st ra in tensor Sij onto the corresponding diffusing scalar's surface. The second and t h i r d terms represent the condi t ional average of dissipation o f scalar dissipation, whi le the last two terms represent the condi t ional average o f diffusion of scalar dissipation. A l l the terms i n these equations are closed, w i t h the exception of the te rm containing the strain tensor Sij, which has t o be model led. Chapter 3. The DCMC Method with Stochastic Processes 33 3.7 Models for the Transport Equation of Scalar Dissipation Two different models using stochastic processes are proposed to model the strain tensor Sij. The first model uses a periodic force with random phase shifting to simulate the chaotic be- haviour of the strain tensor. A second model uses the Coupled Map Lattices (CML) developed by Beck and Hilgers [5], [6], [7] to simulate velocity increments in an isotropic, homogeneous, fully-developed turbulent flow, from which the corresponding strain tensor is calculated. Pitsch and Fedotov [71] simulated the fluctuations of the scalar dissipation rate in non- premixed combustion. They used this approach in combination with a flamelet formulation to simulate a turbulent, non-premixed reactive flow of methane and air. They reported that the extinction process occured at smaller time scales than those relevant to the overall phe- nomenon, like turbulent or chemical time scales. This seems to indicate that the deviations are due to Gaussian white noise, resulting in a Wiener process. This research group also found de- limiting regimes that drive the burning and extinguishing states and that even small-amplitude scalar dissipation fluctuations have considerable effects on the state of the flame, taking it from the burning to the extinguished state. This finding in particular, further reinforced the need to study the influence of the scalar dissipation rate on a turbulent, non-premixed reacting flow, which is one objective of this thesis. 3.7.1 Periodic Forcing with Random Phase Shifting (PF) The use of forcing schemes involving harmonic functions has been explored in DNS simulations of isotropic, homogeneous turbulence [72], [73]. The general idea behind this method is to generate an influx of energy at low Fourier wavenumbers that compensates for the dissipative effects at the small scales of turbulence. Eswaran and Pope [73], used a stochastic forcing scheme with an Ornstein-Uhlenbeck for- mulation to study the validity and capabilities of the method in an isotropic, homogeneous turbulent flow. Hilgers et al [74] explored the use of stochastic resonance to the study of motion of an overdamped particle in a bistable potential, subject to a external periodic per- turbations and coloured noise. Chapter 3. The DCMC Method with Stochastic Processes 34 Similar to the work by Eswaran and Pope [73], the first model proposed in this thesis uses a combination of harmonic signals to mimic the chaotic influence of the turbulent flow field on the strain tensor by using the product of the signals in the x and y directions: = An sin (wnxx + 6nx) sin (u>nyy + 0ny). (3-44) Here, • The magnitude of the amplitudes is An+i = An/2, and is scaled by the Reynolds number of the flow. • The period of the signals is w n +i = 2w n and is used as an emulator of the viscosity of the flow. Larger periodicity implies a higher number of transitions, which are interpreted as lower viscosity. • The random phase shift 0n is produced with Gaussian random numbers. This shifting is used to produce random fluctuations in the strain field. • n = 1...N is the number of signals that are then combined to obtain the assumed strain tensor field in the x — y space: n Stf(x,y) = _T*n. (3.45) i=l The coupling between the strain field and the flow is obtained by forcing proportionality between the amplitudes An of the signals and the magnitude of the characteristic Reynolds number of the flow. This implies that this scheme can only simulate strain fields for one energy level at a time and must be calibrated for each energy level independently. For decay- ing, isotropic, homogeneous, turbulence, characteristic parameters can be used to scale the magnitude of the strain field accordingly with turbulence theory [14] as, I S y l = ^ (3-46) The structure of the strain fields simulated with the P F model can be modified by varying the number of signals used in the calculation of Eq. (3.45), by varying the scaling of the amplitudes between signals, or by varying the periodicity of Eq. (3.44). The P F scheme was used in this thesis to simulate strain fields with fluctuations and magnitudes relevant to the combustion process studied. The magnitudes were compared to Chapter 3. The DCMC Method with Stochastic Processes 35 those obtained from the DNS of Bushe [8] and the simulated fields in the x — y space were then transformed to the a — £ space using a simple mapping convention. The mapping is performed by assuming a linear correspondence between a and x, and £ = erf(y). The second model that has been implemented uses the turbulent flow simulation proposed in the works by Beck [5] and Hilgers and Beck [6], [7]. This model is a coupled map lattice that mimics the energy cascade in a turbulent flow. The system of equations is as follows In this model, • F(x) is a map that models the energy input from the large scales and is defined as F(x) = 1 - 2a:2, where x is the position of stochastic particles in D spatial dimensions. • The parameter k corresponds to the energy level in the cascade. High-values of A; imply higher energy-dissipation processes, which leads to a lower energy level. • The coupling parameter g is associated with, and should be scaled to, the molecular viscosity, p, of the fluid. • The damping parameter A is defined as A = exp(—jrk), where the product JT is pro- portional to the inverse Reynolds number. • The product C£ is used to determine the random fraction of the driving momentum that each of the 'daughter' eddies receive from the previous higher scale. The calculated temporal velocity difference u is later used to calculate the strain rate tensor using Eq. (3.46), or, under the assumption of decaying, isotropic, homogeneous turbulence, it can be further demonstrated [14] that, 3.7.2 Coupled Map Lattices (CML) Xd,n{i) F(xd,n-i{i)) +<«-1i)(0(i->»-i)«iJr-}(o. (3.47) (3.48) Chapter 3. The DCMC Method with Stochastic Processes 36 Using the C M L model, the strain fields were simulated in the x — y plane and later mapped onto the a — £ plane using the same mapping convention as used in the P F model: (a, C) —> (z,erf(y)). It is clear that while this model is capable of simulating strain fields for a wide range of energy levels, the main focus of this research is on the simulation of strain fields at the energy levels relevant to the combustion process studied. This means that the C M L had to be capable of simulating velocity increment fields at relatively small scales. This calibration was made possible by using the variable k that denotes the energy level at which the velocity fields are calculated. The target value of k was obtained by running a number of simulations and comparing the results with statistics from the DNS of Bushe et al [8]. A consequence of this is that all the previous energy levels still have to be calculated, which inevitably increased the computation time. 3 . 8 Summary The D C M C equations that describe turbulent combustion events using two conditioning vari- ables have been derived in this chapter, resulting in the following system of equations ZJ - ~DYL d 2 * / DdZ dZ d 2 Y 7 Dda da W l ^ dt dC,2 ̂  dxi dxi da2 ^ dx% dxi dt dxi dxj 13 2 V #C / 2 \da ) x~z dC,2 X z da2 X a dXa~ dt daaa _ D fdxA2 _ D fdxZ\2 xl dxi dxj 13 2 V d£ J 2 V da J x~a +D^Xz+D^x-a. (3.49) The properties and characteristics of the second conditional variable a have also been described and applied during the derivation of the equations. In this chapter, closure for the D C M C equations has been proposed using scalar transport and C M C closure hypotheses. The resulting equation contains a strain rate tensor term that has been modelled using two different stochastic processes. The results obtained from the use of these models are presented in the following chapter. 37 Chapter 4 Simulation of Methane Oxidation with DCMC-SP The system of equations developed in the previous chapter, Eq . (3.49), is used to simulate the combustion of methane and air. The present chapter describes the results obtained from this computational simulation. The chemical kinetic mechanism, as well as the initial and boundary conditions developed for the generation of both the DNS and C M C databases have been used in this research. The DNS database has also been used as a source of information for the evolution of the turbulent flow. This means that the results presented in this thesis have been produced in an a priori sense. A comparison between the D C M C - S P results of this thesis and the C M C [9] and DNS [8] for a similar test case shows that the D C M C - S P implementation offers an improvement over the singly-conditional moment closure method. 4.1 The Reference D N S and C M C Databases The DNS database of Bushe et al [8] described the results of a number of independent runs of a shear-free, decaying, turbulent mixing layer. A C M C simulation of the same test case [9] provided good predictions of temperature and mass fractions of the major species while falling short of providing good predictions of NO and intermediate mass fractions. 4.1.1 Flow Field The flow field simulated in the DNS database is a three-dimensional shear-free, temporal mixing layer, discretised by a 240 x 120 x 120 grid, in which the governing equations for incompressible flow were resolved. The domain is periodic in two directions, while allowing outflow in the third direction. The domain is shown in Fig. 4.1. Here, region A denotes the approximate spatial domain where the mixing between fuel and oxidiser occurs. Chapter 4. Simulation of Methane Oxidation with DCMC-SP 38 q Oxidiser Fuel Figure 4.1: DNS computational domain The initial turbulent flow field was forced losing the pseudo-spectral scheme of Ruetsch and Maxey [69] that simulates a statistically stationary, incompressible turbulent flow. The initial Taylor Reynolds number was 59 and the Prandtl and Schmidt numbers used were both set to a constant value of 0.75. 4.1.2 Chemical Kinetics The chemical kinetic mechanism developed for the DNS database [8] has been used in this research project to determine the effectiveness of the model proposed in Chapter 3. This mechanism involves three steps and is a modification of the two-step reduced mechanism originally proposed by Williams [75], and later modified by Swaminathan and Bilger [76] for the oxidation of methane, with the addition of a third step for NO formation with the Zel'dovich mechanism. The resulting three-step mechanism is written here as F + Oxi-> I + P I-\-Oxi -> 2P N2 + Oxi -> 2NO where F is CH4, Oxi is O2, and / - (\* + lco) P = \^H20+\c02 A n important feature of this mechanism is the presence of a steady-state approximation for the mass fraction of the Hydrogen radical. This expression has been designed to emulate the Chapter 4. Simulation of Methane Oxidation with DCMC-SP 39 reduct ion in the concentration of the Hydrogen radical at low temperatures. The mechanism was fur ther simpli f ied by Bushe et al [8] by assuming peak temperatures in the 1200-2230 K range, so that the computat ional expense is reduced, while keeping acceptable levels of accuracy. The mechanism was non-dimensionalised by assuming that the constant pressure specific heat Cp, the rat io of specific heats 7, and the speed of sound c remain constant. I t was also found that the region where the chemical reaction takes place needed to be broadened to avoid the computat ional expense associated w i t h the calculat ion of sharp gradients. 4.1.3 I n i t i a l C o n d i t i o n s Since the chemical kinetic mechanism used in the DNS is not capable of auto- igni t ion, the in i t ia l condit ions of the scalars were taken f rom a planar, laminar flame, which also served as the in i t ia l conditions for the C M C simulations. The in i t ia l conditions as a funct ion of m ix tu re f ract ion are shown in F ig . 4.2. This laminar flame was located in the plane formed by points pqrs i n F ig. 4 .1 . 0.35 0.3 0.25 0.15 0.1 0.05 0, CH 4 INT - * - PROD N O x 1 0 7 I P \ y 4 $ \ 0.2 0.8 (a) Initial Conditions of Species Mass Fractions (b) Initial Conditions of Temperature Figure 4.2: In i t i a l Condit ions of the Scalar Fields 4.2 Simulation of CH4 Oxidation Using D C M C - S P Using the chemical kinet ic mechanism described in the previous section, Eq. (3.49) is expanded to include the species present i n the chemical kinetics mechanism, as well as the condi t ioning Chapter 4. Simulation of Methane Oxidation with DCMC-SP 40 scalar variables, resulting i n the following system of equations z _dY02 -- , d2Yp2 n _ , d2Yp2 _d¥F -- d2YF n _ 2 2 F F P-^- = "F + -^rpDXz + -^rpDXa _dYj -- d2Yj n _ _ 9 2 F / P~W = w ' + ^ ^ + ^ r ^ * ° _d¥P d2YP n - M d 2 Y P p~dT = »p + ^ P D x > + -fo?-PDx* _3YNO -r- , d2YNO , d2YNO P—QJ- = UNO + —^-pDXz+ d a 2 pDXa _&T -. d 2T d 2T „ p~dt = "T+dcJ^X'+da*^ dXa = 2 da da ? _ __ _ __ __ d< ftr* ft^ V 2 V aC 7 2 V da ) Xa~ ___ = _2dZdZ_- _P_(9xi\2 __V___T\2__ dt dxi dxj tJ 2 V d£ J 2\da) xl +D^Xz + D^x-a. (4.1) This system of equations was solved for a 50 x 50, a — £ grid, with the initial calculation of the strain fields in the x — y space and later mapped onto the a — £ space using the simple mapping convention described in Sections 3.7.1 and 3.7.2. The boundary conditions for the strain fields were set to be periodic for both directions. The boundary conditions for al l the reacting scalars were set to be periodic in the direction of a and constant in the direction of C The initial conditions of the reacting scalars and of scalar dissipation of mixture fraction were taken directly from the laminar flame described in Section 4.1.3. The results obtained from the numerical implementation of the system in Eq . (4.1) are presented in this section. Once the strain field was simulated using either of the stochastic processes, the O D E solver D V O D E [77] was used to solve Eq. (4.1). Fig. 4.3 serves as an illustration of the structure of the code. Chapter 4. Simulation of Methane Oxidation with DCMC-SP 41 Initial conditions Calculate strain tensor with stochastic process ODE solver dt df _ ~dt~ <?X . dt ' Write output fields. Compare to DNS and CMC data. DNS database |Sy| = (2u ( t ) /E( t ) ) 1/2 Figure 4.3: D C M C - S P code structure 4.2.1 Simulation of the Strain Field As discussed in Chapter 3, the strain field has been simulated using two different stochastic processes. This section summarises the results obtained for both cases. Unless expressed otherwise, the figures in this section show results obtained after a calculation time of t — 5 time units. In this computational implementation, the following model has been used: ndZ dZ „ n — -2-—-—Sij = -2xzniUjbij -2xzniUjSij. dxi dxj Where re, and rij axe unity vectors in the direction of the Xi and Xj directions of the gradients of Z. The value of niUj is then [—1 + 1]. Bushe and Cant [78] showed that positive values of niTij are statistically more frequent; as a result, it is assumed in this research that niUj is negative i f the strain Sij is negative and positive otherwise. This model applies analogously to the corresponding term in the transport equation of Xa- Chapter 4. Simulation of Methane Oxidation with DCMC-SP 42 Periodic Forcing with Random Phase Shifting (PF) This model was validated using the stochastic variables X(x, y; t) and Y(x, y; t) in the solution of Eqs. (3.44) and (3.45). Using statistics from the DNS database of Bushe et al [8], it was found that the appropriate values to simulate a strain field with similar characteristics using the P F model were, • Ai = 1.585 this parameter is proportional to the Taylor Reynolds number of the DNS database • &ix = uiy = 2n which indicates the periodicity of the signals. This parameter was set empirically by comparing the structure of the strain fields from the DNS database and the strain fields obtained using the P F model. The magnitude of the strain tensor was calculated using the values of the parameters e and v from the DNS database. In all cases, the strain field was first calculated in the x — y plane and later transformed to the a — C plane following the mapping rules described in Section 3.6. Figure 4.4(a) shows the results of the calculation of the strain field in the x — y plane using ^ = 3 periodic signals. Figure 4.4(b) shows the same strain field after being transformed to the a — £ plane. Comparing Figs. 4.4(a) and 4.4(b) it can be seen that the main effect of the coordinate transformation is an apparent reduction of the area of influence of the structures close to mixture fraction values of 0 and 1, followed by an elongation of the area of influence of those structures between the aforementioned regions of Z. These two figures also show that the overall magnitudes of the strain field are maintained, while the reduction - elongation - reduction effect makes some of the structures disappear, es- pecially for regions in the vicinity of Z = 0. This finding is of relevance since the areas where the elongation effect is prevalent, is where values of mixture fraction indicate a more vigorous mixing process, where chemical processes are taking place with more frequency. As expected, the reduction - elongation - reduction effect does not take place along the a direction. The strain fields are also shown to be periodic in the x, y, a, and £ directions. Chapter 4. Simulation of Methane Oxidation with DCMC-SP 43 Figures 4.4(c) and 4.4(d), show the s train fields i n the a — ( space for \I> = 4 and \& = 5 periodic signals, respectively. F r o m Figs . 4.4(b), 4.4(c), and 4.4(d), i t can be seen that the number of signals used has a significant effect on the structure and propagation of the calculated field. (c) * = 4, Q - C (d) * = 5, a - C Figure 4.4: S t ra in fields simulated wi th P F T h e strain field calculated wi th three periodic signals shows well-defined regions of positive and negative values of strain, which have the physical meaning of stretch and compression regions i n the flow. These structures tend to be very large for this part icular case and do not show much intermittency. T h e structures can be considered to be eddies, wh ich implies that large structures correspond to the energy-containing subrange or the iner t ia l subrange Chapter 4. Simulation of Methane Oxidation with DCMC-SP 44 in the cascade model. Consequently, smaller, finer structures would represent the behaviour of a flow at the dissipation subrange, at which combustion is assumed to take place. Figures 4.4(c) and 4.4(d) show a much finer structure with more intermittency than Fig. 4.4(b) while yielding similar strain magnitudes to those obtained from the DNS database. C o u p l e d M a p L a t t i c e ( C M L ) In Chapter 3, the C M L equation of Hilgers and Beck [6], [7] that was used to calculate the strain field in this thesis was described. Opposite to the P F model, the C M L model simulates velocities from which the strain field has to be calculated. Two independent velocity fields were simulated and assigned arbitrary directions u and v, which are orthogonal to each other. The simulations were run in the x — y space, with parameters 7T = 0.02, g = 0.0194, and C = 1.411 in Eq. (3.47) and transformed to the a — C, space using the mapping rules described in Section 3.6. Figure 4.5(a) shows the the u-velocity field simulated using the C M L model at a level of fc = 3, which is assumed to be in the integral sub-range of the energy cascade of F ig . 2.1. Figures 4.5(b) and 4.5(c) show the results of the simulation of velocity for levels of fc = 9 and fc = 15, respectively. The structures in these figures do not correspond to eddies, since eddies represent a vortical motion and these figures are only a two-dimensional representation of one component of velocity. A comparison between Figs. 4.5(a), 4.5(b) and 4.5(c) shows that the velocity magnitude decreases about 4 orders of magnitude from levels fc = 3 to fc = 15. This proves the capabilities of the model to represent the dissipation of energy from one level to the next. It can also be seen that the structure of the velocity field is more uniform as the fc level is increased. Figures 4.6(a), 4.6(b), and 4.6(c) show the vector form of the two-dimensional velocity fields at different levels of energy for the area delimited by 0.2 < a < 0.6 and 0.2 < £ < 0.6. The fields were calculated using the results of the simulations of velocities in two orthogonal directions for levels of fc = 3, fc = 9, and fc = 15, respectively. Chapter 4. Simulation of Methane Oxidation with DCMC-SP 45 Compar ing these figures, i t can be seen that some of the structures present i n Fig. 4.6(a) have been completely dissipated, or have been dissipated into smaller structures i n Fig. 4.6(c), g iv ing rise to a more orderly field. 0.2 0.4 0.6 0.8 a 0.2 0.4 0.6 a (a) k = 3 0.8 0.6 3 0.4 0 , (b) it = 9 l o . 0 2 0.02 • H ___-0.04 0.2 0.4 0.6 0.8 a (c) k = 15 Figure 4.5: u-velocity fields simulated w i t h C M L Figs. 4.7(a), 4.7(b), and 4.7(c) show the strain fields obtained using the C M L model and the def in i t ion of Eq. (3.48) for energy levels of k = 3, k = 9, and k = 15, respectively. The effects of the decaying energy on the strain fields are evident in these figures. I t can be seen that the magnitude of the strain field decays considerably as k advances fur ther in to the viscous subrange of the cascade model. This leads to a strain field w i t h fewer peaks and more neutral, or zero-strain areas as shown in Fig. 4.7(c). Using the values of the turbulence Chapter 4. Simulation of Methane Oxidation with DCMC-SP 46 parameters e and v from the reference DNS database, it was determined that the level of energy in the C M L model that would give the appropriate values corresponds to k = 9, which is shown in Fig. 4.7(b). 0.6 0.5 0.3 i \ \ / 1 \ f t \ ' t « N \ I * \ — - - • * * / ** * • . - y • » - *• f * • • * - •#* f * — -* • - / * - •» m .* r » - ^ ^ / -» m » + / / v m * + / / *• to • •*? f / - P • - • ' - > 0.3 0.4 a 0.5 i r I V 0.6 0.6i , „ 5 0.5 0.4 0.3 ft., ft., * / # _ * * / _ » » / - . ! / - . t / , \ » \ . / - . _ 0.3 0.4 a 0.5 0.6 (a) k (b) k = 9 0.6 r 0.5h 0.4 h 0.3K » * - M °-8s 0.3 0.4 a 0.5 0.6 (c) k = 15 Figure 4.6: Velocity fields simulated with C M L Chapter 4. Simulation of Methane Oxidation with DCMC-SP 47 0.2 0.4 _ 0.6 0.8 (c) k = 15 Figure 4.7: Stra in fields simulated w i t h C M L Compar ing Fig. 4.7(b) w i t h F ig. 4.4(d), i t can be seen that there are impor tant differences and similarit ies between the structure of the fields produced by the P F and the C M L models. F i rs t , the P F model is based on spectral methods which are inherently smooth due to the filtering that occurs as a consequence of Fourier-like transformations. Th is is proven by the fact that the structures produced by the P F are larger than those simulated w i t h the C M L model. I t can also be observed that , i n general, the fields produced by the P F behave very much as an ampl i f ied por t ion o f a selected area of the CML-produced field. Th is discrepancy could be el iminated by using a larger number of signals i n the P F model that would induce more var iabi l i ty along the domain. One of the similarit ies between the models is their capabi l i ty to Chapter 4. Simulation of Methane Oxidation with DCMC-SP 48 provide the right orders of magnitude for the field when compared to the calculated value of the strain tensor using v and e parameters from the DNS database. A t t = 5 time units, the DNS data yields Sij = y^e/2i/ = ±0.5656, while the both the C M L and P F models provide values in the range —0.8 < Sij < 0.8 for the case of k = 9, and ^ = 5, respectively. 4.2.2 Predictions of Reactive Scalars Two different sets of calculations using the P F and C M L models proposed in this thesis were performed and compared to the DNS [8] and C M C [9] reference databases. In addition to these, a third set of computations was performed using a generator of Gaussian random numbers that was used in place of the P F and C M L models to mimic the chaotic variations of strain. This was done with the objective of determine whether or not a simple set of random numbers would provide similar results to those obtained with the periodic forcing and coupled map lattice models. The results obtained with this scheme are identified as D C M C - R A N in the following figures. Figure 4.8 expresses results in terms of Favre averages which, for the case of species mass fractions, were calculated in a similar fashion as in the C M C formulation as f ^ lopWKpjQdc: fiP\CP(Q<K where P(Q is the P D F of mixture fraction which is represented by the /3-PDF taken from the DNS reference database [8]. In Fig . 4.8(a) it can be seen that the D C M C - P F and D C M C - C M L models provide predic- tions of the Favre average of oxidiser mass fraction in excellent agreement with the DNS results with a noticeable improvement over the C M C results, while the D C M C - R A N model provides only a marginal improvement over the C M C predictions. The Favre-averaged methane mass fraction is also well predicted by both the D C M C - P F and D C M C - C M L models, as shown in Fig. 4.8(b), with a tendency to under-predict, especially after 35 time units. Figures 4.8(c) and 4.9(a) show that the D C M C - P F and D C M C - C M L models provide good predictions of the Favre average and conditionally-averaged intermediate mass fraction after 30.0 time units, re- spectively. This result has important implications given that the intermediate includes CO. It is expected that i f the intermediate mass fraction is properly simulated, the ability to capture its effect in the overall generation of pollutants wi l l be positively affected. Chapter 4. Simulation of Methane Oxidation with DCMC-SP 49 0.125 0.12 0.115 0.11 0.105 0.1 D N S -a- C M C + D C M C - P F 0 D C M C - C M L — D C M C - R A N 10 20 30 Time 40 50 0.073 0.072 'fuel 0.071 0.07 0.069 D N S -e- C M C + D C M C - P F 0 D C M C - C M L • D C M C - R A N 10 20 30 Time 40 50 (a) Oxidiser (b) Fuel , x 10 ' int 2 0.045 0.04 0.035 'prod 0.03 0.025 0.02 DNS -e- C M C + D C M C - P F 0 D C M C - C M L - D C M C - R A N 4 ^ " * 10 20 30 40 50 Time (c) Intermediates (d) Products 10 20 30 Time 40 50 650i 600 f 550 500 450 20 30 Time (e) NO (f) Temperature Figure 4.8: DNS, C M C , and D C M C Favre averages of species mass fractions Chapter 4. Simulation of Methane Oxidation with DCMC-SP 50 The Favre-averaged mass fract ion of products is over-predicted for al l times by bo th DCMC-based simulations as shown i n Fig. 4.8(d). Th is over-prediction, along w i t h the under- predict ion of the Favre average of fuel mass fract ion, is merely a result of the overpredict ion of the reaction rates. (c) (T\Z) Figure 4.9: Condi t ional averages at t = 30.0 Probably the most significant achievement of this project is the good agreement between the DNS and D C M C - P F and D C M C - C M L predictions of NO. Figure 4.8(e) shows an i m - por tant improvement i n the predictions o f the Favre average of NO mass f ract ion over the C M C results. Th is improvement is also evident in the conditionally-averaged NO mass frac- t i on predictions shown after 30.0 t ime uni ts i n Fig. 4.9(b). The good predictions of NO Chapter 4. Simulation of Methane Oxidation with DCMC-SP 51 mass f ract ion are closely related to the qual i ty of the predictions of temperature, since the Zel 'dovich NO mechanism used to calculate the NO mass fract ion is strongly dependent on temperature. Figure 4.8(f) shows the Favre average of temperature, where i t can be seen that the D C M C - P F and D C M C - C M L results show a considerably improvement i n the predicted results over the C M C results. This result is fur ther confirmed by Fig. 4.9(c). _x1CT 6 __„x10^ 0.2 0.4 0.6 0.8 (c) n i n t , D C M C - R A N Figure 4.10: a-variations in the intermediates field at t = 30.0 Figures 4.10, 4.11, and 4.12 show comparisons between the variations around the condi- t iona l mean i n the direct ion o f a for different scalar fields at 30.0 t ime uni ts calculated w i t h the different models proposed in this research. Here, Hi = Y} (a, £) — (Y / |a ) . Chapter 4. Simulation of Methane Oxidation with DCMC-SP 52 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 (a) IIJVO, DCMC-PF (b) UNO, D C M C - C M L 0.2 0.4 0.6 0.8 (c) I W , D C M C - R A N Figure 4.11: a-variations in the N O field at t = 30.0 From these figures, it can be seen that the variations simulated by the P F and C M L models have similar structures and magnitudes for all cases shown, while the D C M C - R A N implemen- tation simulates variations one order of magnitude smaller in the same fields. For all cases, it is clear that the maximum and minimum variations occur in the vicinity of ( « 0.35, which is where the chemical reaction is assumed to occur with more intensity. While the P F and C M L show consistent results with this hypothesis, the R A N implementation shows large variations in the temperature field at a very different location, as shown in Fig. 4.12(c). This is due to the lack of structure in the tensor field calculated with the D C M C - R A N model. Chapter 4. Simulation of Methane Oxidation with DCMC-SP 53 The variations shown in F ig . 4.12(a) and 4.12(b) seem to indicate the presence of ex- t inc t ion and re- igni t ion phenomena, characterised by 'cold ' regions between 'hot ' regions. I n the case o f the figures shown, these variations are rather small, bu t large enough to produce changes i n the simulated values of N O and intermediates, as shown i n F ig . 4.11(a), 4.11(b), 4.10(a), and 4.10(b). B o t h the P F and R A N simulations took approximately 2.7 hours to complete i n a P4 Xeon 2.4 GHz cluster using a single processor, while the C M L simulations took approximately 30.0 hours to complete using the same hardware. xKf 3 xlO"3 (a) n T , D C M C - P F (b) n T , D C M C - C M L xlCf4 0.2 0.4 „ 0.6 0.8 (c) UT, D C M C - R A N Figure 4.12: a-variations i n the temperature field at t = 30.0 Chapter 4. Simulation of Methane Oxidation with DCMC-SP 54 4.3 Summary T h e results obtained from the numerical implementat ion of the models proposed i n this thesis have been presented i n this chapter. The most important observations are summarised as follows: • T h e stochastic processes used to simulate the s t ra in tensor proved to offer good results i n terms of magnitude and structure of the strain fields. T h e P F mode l requires at least 5 signals to offer consistent results, whi le the C M L model provides the structure and magni tude required for the test combustion process at a dissipation level of fc = 9 • It was observed that by us ing a second condit ional variable addi t ional variations can be simulated w i t h the D C M C model . These variations would have been ignored us ing the single condi t ional moment closure. • In general, the D C M C - P F and D C M C - C M L models provided improved predictions of species mass fractions and temperature when compared w i t h the C M C model . T h i s is especially evident in the Favre-averaged predictions of intermediates and NO mass fractions and temperature. • W h i l e the computat ional expense of the D C M C - R A N is comparable to that o f the D C M C - P F model , i t was observed that the lack of structure inherent to the R A N i m - plementation results i n inconsistent simulations of the scalar fields. Chapter 5 55 Conclusions and Recommendations T h e appl ica t ion of the D C M C method w i t h stochastic processes to simulate turbulent com- bust ion has been explored throughout the development of this thesis. In this chapter the findings produced w i t h this research are summarised and recommendations for future work are art iculated. T h e appl ica t ion of stochastic processes and Monte Car lo methods i n combust ion were de- scribed i n Chapter 2, along w i t h the general fundamentals of turbulent combustion. Examples of relevant applications to this research were also discussed i n Chapter 2. T h e derivat ion of the mathematical models proposed i n this thesis took place i n Chapter 3. D C M C transport equations of species mass fractions, temperature and scalar dissipat ion were formulated and the unclosed terms were modelled using two tools: scalar t ransport and condi t ional moment closure hypotheses. T w o different stochastic processes were also proposed i n Chapter 3 of this thesis to model the strain tensor i n the transport equation o f scalar dis- sipat ion. T h e numerical implementat ion of the models developed i n Chapter 3 showed encourag- ing results i n terms of simulations of the strain tensor, as wel l as predictions of temperature and intermediates and NO mass fraction. The addi t ion of the second scalar variable induced variations that cannot be simulated by the single condi t ional moment closure method . These results are discussed and summarised i n Chapter 4. A n example of the appl icabi l i ty and advantages of the D C M C method w i t h stochastic processes i n s imulat ion of turbulent combustion has been demonstrated w i t h the results ob- tained from this research. A s w i t h many other projects, however, areas o f improvement have Chapter 5. Conclusions and Recommendations 56 also been identified that would make the model more robust and applicable to a more general range of problems. It is then proposed to: • Explore ways in which this or a similar model could work linked to a C F D program. Wi th this model it is possible to use the fluid flow data provided by a C F D program as a real-time input. The code would then calculate the strain field and the solution for Eq. 4.1, returning updated values of density, temperature, viscosity, etc. to the C F D code. • Investigate the effects of increased variations around the conditional mean of a. These variations could be induced by modifying the initial conditions and coupling between the scalar dissipation of a and Z. • Investigate the use of the models presented in this thesis including variations in space. This implies a different approach for the closure of Eqs. (3.20) and (3.21) to that presented in Chapter 3. It is possible that the differences between the results presented in this research and those found in the DNS reference database could be due to the lack of a spatial variable. The use of such as a conditional variable will provide the model with a capability to discriminate among isosurfaces of mixture fraction and o. • Test a different case. 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