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Modeling Turbulent Combustion CEFRC Combustion Summer School 2014 Prof. Dr.-Ing. Heinz Pitsch Copyright ©2014 by Heinz Pitsch. This material is not to be sold, reproduced or distributed without prior written permission of the owner, Heinz Pitsch. Course Overview Part II: Turbulent Combustion • Moment Methods for reactive scalars • Simple Models in Fluent: EBU,EDM, FRCM, EDM/FRCM • Turbulence • Introduction in Statistical Methods: PDF, CDF,… • Turbulent Premixed Combustion • Transported PDF Model • Turbulent Non-Premixed • Modeling Turbulent Premixed Combustion Combustion • Modeling Turbulent Combustion • Applications • BML-Model • Level Set Approach/G-equation • Modeling Turbulent Non-Premixed Combustion • Conserved Scalar Based Models for Non-Premixed Turbulent Combustion • Flamelet-Model 2 • Application: RIF, steady flamelet model Moment Methods for Reactive Scalars Balance Equation for Reactive Scalars • The term „reactive scalar“ Mass fraction Yα of all components α = 1, … N Temperature T • Balance equation for Di: mass diffusivity, thermal diffusivity Si: mass/temperature source term 3 Balance Equation for Reactive Scalars • Neglecting the molecular transport (assumption: Re↑) • Gradient transport assumption for the turbulent transport → Averaged transport equation not closed → idea: approach similar to (simple) turbulence models: expression as a function of mean values 4 Moment Methods for Reactive Scalars • Assumption: heat release expressed by B: includes frequency factor und heat of reaction Tb: adiabatic flame temperature E: activation energy • Approach for modeling the chemical source term • Proven method decomposition into mean and fluctuation 5 Moment Methods for Reactive Scalars • Taylor expansion at Pre-exponential term Exponential term • Leads to 6 (for ) of terms Moment Methods for Reactive Scalars • As a function of Favre-mean at yields • Typical values in the reaction zone of a flame • Intense fluctuations of the chemical source term around the mean value • Moment method for reactive scalars inappropriate due to strong non-linear effect of the chemical source term 7 Example: Non-Premixed Combustion in Isotropic Turbulence • Favre averaged transport equation • Gradient transport model • One step global reaction • Decaying isotropic turbulence 8 Example: Non-Premixed Combustion in Isotropic Turbulence Product Mass Fraction Flamelet Closure Assumption DNS data Evaluation of chemical source term with mean quantities Closure by mean values does not work! 9 Course Overview Part II: Turbulent Combustion • Moment Methods for reactive scalars • Simple Models in Fluent: EBU,EDM, FRCM, EDM/FRCM • Turbulence • Introduction in Statistical Methods: PDF, CDF,… • Turbulent Premixed Combustion • Transported PDF Model • Turbulent Non-Premixed • Modeling Turbulent Premixed Combustion Combustion • Modeling Turbulent Combustion • Applications • BML-Model • Level Set Approach/G-equation • Modeling Turbulent Non-Premixed Combustion • Conserved Scalar Based Models for Non-Premixed Turbulent Combustion • Flamelet-Model 10 • Application: RIF, steady flamelet model Simple Models for Turbulent Combustion • Example: standard models in Fluent • Very simple models, e.g. based on 3. 4. 2. 5. very fast chemistry no consideration of turbulence Quelle: Fluent 12 user‘s guide 11 1. Eddy-Break-Up-Model First approach for closing the chemical source term was made by Spalding (1971) in premixed combustion unburnt mixture hot burnt gas flow • Assumption: very fast chemistry (after pre-heating) • Combustion process Breakup of eddies from the unburnt mixture smaller eddies → Large surface area (with hot burnt gas) → Duration of this breakup determines the pace → Eddy-Break-Up-Model (EBU) 12 1. Eddy-Break-Up-Modell • Averaged turbulent reaction rate for the products : variance of mass fraction of the product CEBU: Eddy-Break-Up constant • EBU-modell turbulent mixing sufficiently describes the combustion process chemical reaction rate is negligible • Problems with EGR, lean/rich combustion → further development by Magnussen & Hjertager (1977): Eddy-DissipationModel (EDM)… 13 2. Eddy-Dissipation-Model • EDM: typical model for eddy breakup Assumption: very fast chemistry Turbulent mixing time is the dominant time scale • Chemical source term YE, YP: mass fraction of reactant/product A, B: Model parameter (determined by experiment) 14 2. Eddy-Dissipation-Model Example: diffusion flame, one step reaction • YF > YF,st , therefore YO < YF YE = YO • YF < YF,st YE = YF 15 Résumé EDM • • • • Controlled by mixing Very fast chemistry Application: turbulent premixed and nonpremixed combustion Connects turbulent mixing with chemical reaction rich or lean? → full or partial conversion • Advantage: simple and robust model • Disadvantage No effects of chemical non-equilibrium (formation of NO, local extinction) Areas of finite-rate chemistry: • Fuel consumption is overestimated • Locally too high temperatures 16 3. Finite-Rate-Chemistry-Model (FRCM) • Chemical conversion with finite-rate • Capable of reverse reactions • Chemical source term for species i in a reaction α kf,α , kb,α: reaction rates(determined by Arrhenius kinetic expressions models the influence of third bodies • Linearization of the source term centered on the operating point Integration into equations for species, larger Δt realizable • Typical approach for detailed computation of homogeneous systems 17 ) Résumé FRCM • Chemistry-controled • Appropriate for tchemistry > tmixng (laminar/laminar-turbulent) • Application Laminar-turbulent Non-premixed • Source term: Arrhenius ansatz Mean values for temperature in Arrhenius expression → Effects of turbulent fluctuations are ignored → Temperature locally too low • Consideration of non-equilibrium effects 18 4. Combination EDM/FRCM • Turbulent flow Areas with high turbulence and intense mixing Laminar structures • Concept: Combination of EDM and FRCM For each cell: computation of both reaction rates and The smaller one is picked (determines the reaction rate) → Choses locally between chemistry- and mixing-controlled • Advantage: Meant for large range of applicability • Disadvantage: no turbulence/chemistry interaction 19 5. Eddy-Dissipation-Concept (EDC) • Extension of EDM Considers detailed reaction kinetics • Assumption: Reactions on small scales („*“: fine scale) Fluent: Cξ = 2,1377 • Volume of small scales: • Reaction rates are determined by Arrhenius expression (cf. FRCM) • Time scale of the reactions Fluent: Cτ = 0,4082 20 5. Eddy-Dissipation-Concept (EDC) • Boundary/initial conditions for reactions (on small scales) Assumption: pressure p = const. Initial condition: temperature and species concentration in a cell Reactions on time scale Numerical integration (e.g. ISAT-Algorithm) • Model for source term • Problem: Requires a lot of processing power Stiff differential equation 21 Mass fraction on small scales of species i after reaction time τ* Résumé: Simple Combustion Models Solely calculation by Arrhenius equation turbulence is not considered Calculation of Arrhenius reaction rate and mixing rate; selection of the smaller one local choice: laminar/turbulent 3. 4. 2. 5. Solely calculation of mixing rate Chemical kinetic is not considered Quelle: Fluent 12 user‘s guide Modeling of turbulence/chemistry interaction; detailed chemistry 22 Course Overview Part II: Turbulent Combustion • Moment Methods for reactive scalars • Simple Models in Fluent: EBU,EDM, FRCM, EDM/FRCM • Turbulence • Introduction in Statistical Methods: PDF, CDF,… • Turbulent Premixed Combustion • Transported PDF Model • Turbulent Non-Premixed • Modeling Turbulent Premixed Combustion Combustion • Modeling Turbulent Combustion • Applications • BML-Model • Level Set Approach/G-equation • Modeling Turbulent Non-Premixed Combustion • Conserved Scalar Based Models for Non-Premixed Turbulent Combustion • Flamelet-Model 23 • Application: RIF, steady flamelet model Introduction to Statistical Methods • Introduction to statistical methods Sample space Probability Cumulative distribution function(CDF) Probability density function(PDF) Examples for CDFs/PDFs Moments of a PDF Joint statistics Conditional statistics 24 Pope, „Turbulent Flows“ Sample Space • Probability of events in sample space • Sample space: set of all possible events Random variable U Sample space variable V (independent variable) • Event A • Event B 25 Probability • Probability of the event • Probability p impossible event 26 sure event Cumulative Distribution Function (CDF) • Probability of any event can be determined from cumulative distribution function (CDF) • Event A • Event B 27 Cumulative Distribution Function (CDF) • Three basic properties of a CDF 1. 2. 3. Occuring of event is impossible Occuring of event is sure F is a non-decreasing function as CDF of Gaussian distributed random variable 28 Probability Density Function (PDF) • Derivative of the CDF probability density function • Three basic properties of a PDF 1. 2. CDF non-decreasing PDF Satisfies the normalization condition 3. For infinite sample space variable PDF of Gaussian distributed random variable 29 Probability Density Function (PDF) • Examining the particular interval Va ≤ U < Vb • Interval Vb - Va 0: 30 Example for CDF/PDF Uniform distribution Source: Pope, „Turbulent Flows“ 31 Example for CDF/PDF Exponential distribution Source: Pope, „Turbulent Flows“ 32 Example for CDF/PDF Normal distribution Source: Pope, „Turbulent Flows“ 33 Example for CDF/PDF Delta-function distribution or Source: Pope, „Turbulent Flows“ 34 Moments of a PDF • PDF of U is known n-th moment • Example: first moment (n = 1): mean of U 35 Q: random function, with Q = Q(U) Central Moments • n-th central moment • Example: second central moment (n = 2): variance of U 36 Joint Cumulative Density Function • Joint CDF (jCDF) of random variables U1, U2 (in general Ui, i = 1,2,…) Source: Pope, „Turbulent Flows“ 37 Joint Cumulative Density Function • Basic properties of a jCDF Non-decreasing function Since is impossible Since is certain equally marginal CDF 38 Joint Probability Density Function • Joint PDF (jPDF) • Fundamental property: Source: Pope, „Turbulent Flows“ 39 Joint Probability Density Function • Basic properties of a jPDF Non-negative: Satisfies the normalization condition Marginal PDF 40 Joint Statistics • For a function Q(U1,U2,…) • Example: i = 1, 2; n = 1; , covariance of U1 and U2 Scatterplot of two velocitycomponents U1 and U2 • Covariance shows the correlation of two variables 41 Conditional PDF • PDF of U2 conditioned on U1 = V1 Bayes-Theorem • jPDF f1,2(V1,V2) scaled so that it satisfies the normalization condition • Conditional mean of a function Q(U1,U2) 42 Statistical Independence • If U1 and U2 are statistically independent, conditioning has no effect • Bayes-Theorem • Therefore: • Independent variables uncorrelated • In general the converse is not true 43 Course Overview Part II: Turbulent Combustion • Moment Methods for reactive scalars • Simple Models in Fluent: EBU,EDM, FRCM, EDM/FRCM • Turbulence • Introduction in Statistical Methods: PDF, CDF,… • Turbulent Premixed Combustion • Transported PDF Model • Turbulent Non-Premixed • Modeling Turbulent Premixed Combustion Combustion • Modeling Turbulent Combustion • Applications • BML-Model • Level Set Approach/G-equation • Modeling Turbulent Non-Premixed Combustion • Conserved Scalar Based Models for Non-Premixed Turbulent Combustion • Flamelet-Model 44 • Application: RIF, steady flamelet model The PDF Transport Equation Model • Similar to moment methods, models based on a pdf transport equation for the velocity and the reactive scalars are usually formulated for one-point statistics • Within that framework, however, they represent a general statistical description of turbulent reacting flows, in principle, independent of the combustion regime • A joint pdf transport equation for the velocity and the reactive scalars can be derived, Pope (1990) 45 The PDF Transport Equation Model • There are several ways to derive a transport equation for the joint probability density function P(v, ψ ; x, t) of velocity v and the vector of reactive scalars ψ (cf. O'Brien, 1980) • We refer here to the presentation in Pope (1985, 2000), but write the convective terms in conservative form where is gradient with respect to velocity components, angular brackets are conditional means, and the same symbol is used for random and sample space variables 46 PDF Transport Equation: Closure Problem • The first two terms on the l.h.s. of are the local change and convection of the probability density function in physical space • The third term represents transport in velocity space by gravity and the mean pressure gradient • The last term on the l.h.s. contains the chemical source terms • All these terms are in closed form, since they are local in physical space 47 PDF Transport Equation: Closure Problem • Note that the mean pressure gradient does not present a closure problem, since the pressure is calculated independently of the pdf equation using the mean velocity field • For chemically reacting flows, it is of particular interest that the chemical source terms can be treated exactly • It has often been argued that in this respect the transported pdf formulation has a considerable advantage compared to other formulations 48 PDF Transport Equation: Closure Problem • However, on the r.h.s. of the transport equation there are two terms that contain gradients of quantities conditioned on the values of velocity and composition • Therefore, if gradients are not included as sample space variables in the pdf equation, these terms occur in unclosed form and have to be modeled 49 PDF Transport Equation: Closure Problem • The first unclosed term on the r.h.s. describes transport of the probability density function in velocity space induced by the viscous stresses and the fluctuating pressure gradient • The second term represents transport in reactive scalar space by molecular fluxes This term represents molecular mixing 50 PDF Transport Equation: Closure Problem • When chemistry is fast, mixing and reaction take place in thin layers where molecular transport and the chemical source term balance each other • Therefore, the closed chemical source term and the unclosed molecular mixing term, being leading order terms in a asymptotic description of the flame structure, are closely linked to each other • Pope and Anand (1984) have illustrated this for the case of premixed turbulent combustion by comparing a standard pdf closure for the molecular mixing term with a formulation, where the molecular diffusion term was combined with the chemical source term to define a modified reaction rate • They call the former distributed combustion and the latter flamelet combustion and find considerable differences in the Damköhler number dependence of the turbulent burning velocity normalized with the turbulent intensity 51 PDF Transport Equation: Solution • From a numerical point of view, the most apparent property of the pdf transport equation is its high dimensionality • Finite-volume and finite-difference techniques are not very attractive for this type of problem, as memory requirements increase roughly exponentially with dimensionality • Therefore, virtually all numerical implementations of pdf methods for turbulent reactive flows employ Monte-Carlo simulation techniques (cf. Pope, 1981, 1985) • The advantage of Monte-Carlo methods is that their memory requirements depend only linearly on the dimensionality of the problem 52 PDF Transport Equation: Solution • Monte-Carlo methods employ a large number, N, of so called notional particles (Pope, 1985) • Particles should be considered different realizations of the turbulent reactive flow problem under investigation • Particles should not be confused with real fluid elements, which behave similarly in a number of respects • Statistical error decreases with N1/2 - Slow convergence 53 Application TPDF Model in LES of Turbulent Jet Flames • LES/FDF of Sandia flames D and E (Raman & Pitsch, 2007) Joint scalar pdf Density computed through filtered enthalpy equation for improved numerical stability Detailed chemical mechanism • Modeled particle stochastic differential equations 54 Application TPDF Model in LES of Turbulent Jet Flames Flame D: Temperature 55 Flame E: Temperature Flame E: Dissipation Rate Application TPDF Model in LES of Turbulent Jet Flames 56 Course Overview Part II: Turbulent Combustion • Moment Methods for reactive scalars • Simple Models in Fluent: EBU,EDM, FRCM, EDM/FRCM • Turbulence • Introduction in Statistical Methods: PDF, CDF,… • Turbulent Premixed Combustion • Transported PDF Model • Turbulent Non-Premixed • Modeling Turbulent Premixed Combustion Combustion • Modeling Turbulent Combustion • Applications • BML-Model • Level Set Approach/G-equation • Modeling Turbulent Non-Premixed Combustion • Conserved Scalar Based Models for Non-Premixed Turbulent Combustion • Flamelet-Model 57 • Application: RIF, steady flamelet model Bray-Moss-Libby-Model • Flamelet concept for premixed turbulent combustion: Bray-Moss-Libby-Modell (BML) • Premixed combustion: progress variable c, e.g. or • Favre averaged transport equation (neglecting the molecular transport) not closed • Closure for turbulent transport and chemical source term by BML-Model 58 Bray-Moss-Libby-Model • Assumption: very fast chemistry, flame size lF << η << lt burnt burnt unburnt • Fuel conversion only in the area of thin flame front → in the flow field Burnt mixture or Unburnt mixture, Intermediate states are very unlikely 59 Bray-Moss-Libby-Model • Assumption: progress variable is expected solely to be c = 0 (unburnt) or c = 1 (burnt) • Probability densitiy function : probabilities, to encounter burnt or unburnt mixture in the flow field No intermediate states δ: Delta function 60 Bray-Moss-Libby-Model instantaneous flame front „unburnt“ 61 mean flame front „burnt“ BML-closure of Turbulent Transport • For a Favre average • Therefore the the unclosed correlation joint PDF for u and c (Bayes-Theorem) Introducing the BML approach for f(c) leads to conditional PDF 62 delta function BML-closure of Turbulent Transport • With follows 63 Bray-Moss-Libby-Model: „countergradient diffusion“ • Because of ρu = const. through flame front: u↑ just as much as ρ↓ • Because of c ≥ 0 Flame front c • Within the flame zone conflict • Gradient transport assumption would be uu ub ρuuu = ρbub • Conflict: „countergradient diffusion“ 64 BML-closure of Chemical Source Term • Closure by BML-model f(c) leads to • Closure of the chemical source term, e.g. by flame-surface-density-model local mass conversion per area Flächen-Dichte (flamen area per volume) I0: strain factor local increase of burning velocity by strain • Flame-surface-density Σ e.g. algebraic model: Or transport equation for Σ 65 Flame crossing length BML-closure of Chemical Source Term • Transport equation for Σ local change convectiv change turbulent transport production due to stretching of the flame • No chemical time scale Turbulent time (τ = k/ε) is the determining time scale Limit of infinitely fast chemistry By using transport equations model for chemical source term independent of sL 66 flameannihilation Course Overview Part II: Turbulent Combustion • Moment Methods for reactive scalars • Simple Models in Fluent: EBU,EDM, FRCM, EDM/FRCM • Turbulence • Introduction in Statistical Methods: PDF, CDF,… • Turbulent Premixed Combustion • Transported PDF Model • Turbulent Non-Premixed • Modeling Turbulent Premixed Combustion Combustion • Modeling Turbulent Combustion • Applications • BML-Model • Level Set Approach/G-equation • Modeling Turbulent Non-Premixed Combustion • Conserved Scalar Based Models for Non-Premixed Turbulent Combustion • Flamelet-Model 67 • Application: RIF, steady flamelet model Level-Set-Approach • Kinematics of the flame front by examining the movement of single flame front-„particles“ • Movement influenced by instantaneous flame front Local flow velocity ui, i = 1,2,3 Burning velocity sL particle normal vector 68 G-Equation • Instead of observing a lot of particles examination of a scalar field G • Iso-surface G0 is defined as the flame front • Substantial derivative of G (on the flame front) unburnt 69 burnt Example: Level-Set-Method 70 G-Equation for Premixed Combustion • Kinematics normal vector and lead to unburnt → G-Equation for premixed combustion 71 burnt G-Equation in the Regime of Corrugated Flamelets local change convective change progress of flame front by burning velocity • No diffusive term • Can be applied for Thin flames Well-defined burning velocity → Regime of corrugated flamelets (η >> lF >> lδ) 72 unburnt burnt G-Equation in the Regime of Corrugated Flamelets • Kinematic equation ≠ f(ρ) • Valid for flame position: G = G0 (= 0) For solving the field equation, G needs to be defined in the entire field Different possibilities to define G, e.g. signed distance function unburnt 73 burnt G-Equation in the Regime of Corrugated Flamelets • Influence of chemistry by sL • sL not necessarily constant, influenced by strain S curvature κ Lewis number effect • Modified laminar burning velocity unburnt 74 burnt Laminar Burning Velocity: Curvature influence of curvature uncorrected laminar burning velocity unburnt G<0 burnt G>0 75 Laminar Burning Velocity: Markstein Length uncorrected laminar burning velocity • Markstein length Determined by experiment Or by asymptotic analysis density ratio 76 Zeldovich number Lewis number Extended G-Equation influence of curvature uncorrected laminar burning velocity Markstein length → Extended G-Equation 77 influence of strain G-Equation: Corrugated Flamelets/Thin Reaction Zones • Previous examinations limited to the regime of corrugated flamelets Thin flame structures (η >> lF >> lδ) Laminar burning velocity well-defined • Regime of thin reaction zones no longer valid Small scale eddies penetrate the preheating zone Transient flow Burning velocity not well-defined → Problem: Level-Set-Approach valid in the regime of thin reaction zones? 78 G-Equation: Regime of Thin Reaction Zones • Assumption: „G=0“ surface is represented by inner reaction zone • Inner reaction zone Thin compared to small scale eddies, lδ << η Described by T(xi,t) = T0 • Temperature equation • Iso temperature surface T(xi,t) = T0 79 G-Equation: Regime of Thin Reaction Zones • Equation of motion of the iso temperature surface T(xi,t) = T0 With the displacement speed sd Normal vector 80 G-Equation: Regime of Thin Reaction Zones • With G0 = T0 Diffusion term normal diffusion (~sn) and curvature term (~κ) → G-equation for the regime of thin reaction zones 81 Common Level Set Equation for Both Regimes • Normalize G-equation with Kolmogorov scales (η, τη, uη) leads to 82 Order of Magnitude Analysis O(Ka-1/2) • Non dimensional → Derivatives, ui*, κ* ≈ O(1) • Typical flame → Sc = ν/D ≈ 1 D/ν = O(1) • Parameter: sL/uη Ka = uη2/sL2 sL/uη = Ka-1/2 sL,s ≈ sL 83 O(1) G-Equation for both Regimes O(Ka-1/2) O(1) • Thin reaction zones: Ka >> 1 curvature term is dominant • Corrugated flamelets: Ka << 1 sL term is dominant • Leading order equation in both regimes Assumption: 84 const. const. Statistical Description of Turbulent Flame Front • Probability density function of finding G(xi,t) = G0 = 0 Experimental determination in weak swirl burner 85 Statistical Description of Turbulent Flame Front • Consider steady one-dimensional premixed turbulent mean flame at position xf • Define flame brush thickness lf from f(x) • If G is distance function then 86 Favre-Mean- and Variance-Equation • Equation for Favre-mean instantaneous flame front • Equation for variance • can be interpreted as the area ratio of the flame AT/A • Variance describes the average size of the flame averaged flame front averaged temperature profile instantaneous temperature profile 87 Modeling of the Variance Equation • Sink terms in the variance equation Kinematic restoration Scalar dissipation are modeled by 88 G-Equation for Turbulent Flows • Introducing turbulent burning velocity → Equation for Favre mean → Equation for variance 89 G-Equation for Turbulent Flows • Modeling of turbulent burning velocity by Damköhler theory 90 G-Equation for Turbulent Flows • Favre mean of G instantaneous flame front averaged flame front • Favre-PDF • Mean temperature (or other scalar) averaged temperature profile T(G)=T(x) taken from laminar premixed flame without strain 91 instantaneous temperature profile Example: Presumed Shape PDF Approach (RANS) experiment computed numerically 92 Example: LES of a Premixed Turbulent Bunsen Flame • • • • Premixed methan/air flame Re = 23486 Broad, low velocity pilot flame heat losses to burner Dilution by air co-flow temperature axial velocity 93 Time-Averaged Temperature and Axial Velocity at position x/D = 2.5 temperature 94 axial velocity Time-Averaged Temperature and Axial Velocity at position x/D = 6.5 temperature 95 axial velocity Turbulent Kinetic Energie at Position x/D = 2.5 and 6.5 x/D = 2,5 96 x/D = 6,5 LES Regime Diagram for Premixed Turbulent Combustion 97 Course Overview Part II: Turbulent Combustion • Moment Methods for reactive scalars • Simple Models in Fluent: EBU,EDM, FRCM, EDM/FRCM • Turbulence • Introduction in Statistical Methods: PDF, CDF,… • Turbulent Premixed Combustion • Transported PDF Model • Turbulent Non-Premixed • Modeling Turbulent Premixed Combustion Combustion • Modeling Turbulent Combustion • Applications • BML-Model • Level Set Approach/G-equation • Modeling Turbulent Non-Premixed Combustion • Conserved Scalar Based Models for Non-Premixed Turbulent Combustion • Flamelet-Model 98 • Application: RIF, steady flamelet model Conserved Scalar Based Models for Non-Premixed Combustion Mixture fraction Z • Definition of Z Coupling function: 1 • With the dimensionless quantity Mass of air per fuel mass: m2/mB Mass of air per fuel mass at stoichiometric conditions 99 Mixture Fraction Z • Stoichiometric Conditions (λ=1) Stoichiometric mixture fraction → • Relation between Z and λ • Z: normalized local λ λ=0Z=1 λ = 1 Z = Zst λ=∞Z=0 100 Transport Equation for Z • Transport equation Advantage: L(Z) = 0 No Chemical Source Term BC: Z = 0 in Oxidator, Z = 1 in Fuel • If species and temperature function of mixture fraction, then • Needed: Local statistics of Z (expressed by PDF) Species/temperature as function of Z: Yi(Z) and T(Z) 101 Presumed PDF Approach • Equation for the mean and the variance of Z are known and closed 102 Presumed PDF Approach • b-function pdf for mixture fraction Z • With 103 Conserved Scalar Based Models for Non-Premixed Turbulent Combustion • Infinitely fast irreversible chemistry Burke-Schumann solution Solution = f(Z) • Infinitely fast reversible chemistry Chemical equilibrium Solution = f(Z) • Flamelet model for non-premixed combustion Chemistry fast, but not infinitely fast Solution = f(Z, χ) • Conditional Moment Closure (CMC) Similar to flamelet model Solution = f(Z,< χ|Z>) 104 Conserved Scalar Based Models for Non-Premixed Turbulent Combustion • Infinitely fast irreversible chemistry Burke-Schumann solution Solution = f(Z) • Infinitely fast reversible chemistry Chemical equilibrium Solution = f(Z) 105 Burke-Schumann Solution 106 Course Overview Part II: Turbulent Combustion • Moment Methods for reactive scalars • Simple Models in Fluent: EBU,EDM, FRCM, EDM/FRCM • Turbulence • Introduction in Statistical Methods: PDF, CDF,… • Turbulent Premixed Combustion • Transported PDF Model • Turbulent Non-Premixed • Modeling Turbulent Premixed Combustion Combustion • Modeling Turbulent Combustion • Applications • BML-Model • Level Set Approach/G-equation • Modeling Turbulent Non-Premixed Combustion • Conserved Scalar Based Models for Non-Premixed Turbulent Combustion • Flamelet-Model 107 • Application: RIF, steady flamelet model Flamelet Model for Non-Premixed Turbulent Combustion • Basic idea: Scale separation • Assume fast, but not infinitely fast chemistry: 1 << Da << ∞ • Reaction zone is thin compared to small scales of turbulence and hence retains laminar structure • Transformation and asymptotic approximation leads to flamelet equations 108 Flamelet Model for Non-Premixed Turbulent Combustion • Balance equations for temperature, species and mixture fraction • With it follows 109 Flamelet Equations • Consider surface of stoichiometric mixture • Reaction zone confined to thin layer around this surface • Transformation to surface attached coordinate system • x1, x2, x3, t → Z(x1, x2, x3, t), Z2, Z3, τ 110 x3, Z3 x1, Z x2, Z2 Transformation rules • Transformation: x1, x2, x3, t → Z(x1, x2, x3, t), Z2, Z3, τ (where Z2 = x2 , Z3 = x3, τ = t) • Example: Temperature T 0 1 0 0 0 1 0 0 Analogous for x3 111 0 Flamelet Equations • Temperature equation Transformed temperature equation: 112 Flamelet Equations small Local change Describes mixing Source term • If the flamelet is thin in the Z direction, an order-of-magnitude analysis similar to that for a boundary layer shows that is the dominating term of the spatial derivatives • Equivalent to the assumption that temperature derivatives normal to the flame surface are much larger than those in tangential direction • ∂T/∂τ is important if very rapid changes, such as extinction, occur 113 Example • Example from DNS of Non-Premixed Combustion in Isotropic Turbulence • Temperature (color) • Stoichiometric mixture fraction (line) 114 Flamelet Equations • Same procedure for the mass fraction… • Flamelet structure is to leading order described by the one-dimensional timedependent equations → • Instantaneous scalar dissipation rate at stoichiometric conditions → [χst] = 1/s: may be interpreted as the inverse of a characteristic diffusion time 115 Temperature profiles for methane-air flames • Temperature profiles for methane-air flames 116 Flamelet Equations • Asymptotic analysis by Seshadri (1988) Based on four-step model Close correspondence between layers identified in premixed diffusion flames 117 Flamelet Equations • The calculations agree well with numerical and experimental data • They also show the vertical slope of T0 versus χst which corresponds to extinction 118 Flamelet Equations • Steady state flamelet equations provide ψi = f(Z,χst) • If joint pdf is known → Favre mean of ψi: • If the unsteady term in the flamelet equation must be retained, joint statistics of Z and χst become impractical • Then, in order to reduce the dimension of the statistics, it is useful to introduce multiple flamelets, each representing a different range of the χ-distribution • Such multiple flamelets are used in the Eulerian Particle Flamelet Model (EPFM) by Barths et al. (1998) • Then the scalar dissipation rate can be formulated as function of the mixture fraction 119 Flamelet Equations • Modeling the conditional Favre mean scalar dissipation rate • Flamelet equations • Favre mean 120 Flamelet Equations • Model for conditional scalar dissipation rate • One relates the conditional scalar dissipation rate to that at a fixed value Zst by With 121 n-Heptane Air Ignition • The initial air temperature is 1100 K and the initial fuel temperature is 400 K. 122 Representative-Interactive-Flamelet-Modell (RIF) 123 Example: Diesel engine simulation • VW 1,9 l DI-Diesel engine (Fuel: n-Heptan) • Simulation: KIVA-Code RIF-Model n-Heptan detailed chemistry Soot and Nox as function of EGR 124 Example: Diesel engine simulation • RIF-Temperature 2700 2400 2100 1800 1500 T [K] 1200 900 600 300 20 0.0 10 0.2 0.4 0.6 Mischungsbruch Kurbelwinkel [˚nOT] 0 -10 0.8 1.0 300 125 600 900 1200 1500 1800 2100 2400 2700 Temperatur [K] Example: Diesel engine simulation Mischungsbruchverteilung Schadstoffbildung 126 Example: Diesel engine simulation • Comparison with Magnussen-/ Hiroyasu-Model 127 Steady Laminar Flamelet Model • Assumption that flame structure is in steady state • Assumption often good, except slow chemical and physical processes, such as Pollutant formation Radiation Extinction/re-ignition • Model formulation Solve steady flamelet equations with varying cst Tabulate in terms of cst or progress variable C, e.g. C = YCO2 + YHO2 + YCO + YH2 Presumed PDF, typically beta function for Z, delta function for dissipation rate or reaction progress parameter 128 Example: LES of a Bluff-Body Stabilized Flame • • • • Bluff-body stablized methane/air flame Fuel issues through center of bluff body Flame stabilization by complex recirculating flow RANS models where unsuccessful in predicting experimental data • Here, LES using simple steady flamelet model • New recursive filter refinement method • Accurate models for scalar variance and scalar dissipation rate Exp. by Masri et al. 129 Example: LES of a Bluff-Body Stabilized Flame 130 Example: LES of a Bluff-Body Stabilized Flame Temperature 131 CO Mass Fraction Flamelet Model Application to Sandia Jet Flames Flamelet model application to jet flame with extinction and reignition • Flamelet/progress variable model (Ihme & Pitsch, 2008) • Definition of reaction progress parameter Based on progress variable C Defined to be independent of Z • Joint pdf of Z and l Z and l independent Beta function for Z Statistically most likely distribution for l Exp. by Barlow et al. 132 Flamelet Model Application to Sandia Jet Flames Flame D: Temperature 133 Flame E: Temperature Flamelet Model Application to Sandia Jet Flames Flame D Flame E Flame E Flame D 134 Summary Part II: Turbulent Combustion • Moment Methods for reactive scalars • Simple Models in Fluent: EBU,EDM, FRCM, EDM/FRCM • Turbulence • Introduction in Statistical Methods: PDF, CDF,… • Turbulent Premixed Combustion • Transported PDF Model • Turbulent Non-Premixed • Modeling Turbulent Premixed Combustion Combustion • Modeling Turbulent Combustion • Applications • BML-Model • Level Set Approach/G-equation • Modeling Turbulent Non-Premixed Combustion • Conserved Scalar Based Models for Non-Premixed Turbulent Combustion • Flamelet-Model 135 • Application: RIF, steady flamelet model