Modeling Turbulent Combustion

Transcription

1 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.

2 Course Overview Part II: Turbulent Combustion Turbulence Turbulent Premixed Combustion Turbulent Non-Premixed Combustion Modeling Turbulent Combustion Applications 2 Moment Methods for reactive scalars Simple Models in Fluent: EBU,EDM, FRCM, EDM/FRCM Introduction in Statistical Methods: PDF, CDF, Transported PDF Model Modeling Turbulent Premixed Combustion BML-Model Level Set Approach/G-equation Modeling Turbulent Non-Premixed Combustion Conserved Scalar Based Models for Non-Premixed Turbulent Combustion Flamelet-Model Application: RIF, steady flamelet model

3 Moment Methods for Reactive Scalars Balance Equation for Reactive Scalars The term reactive scalar Mass fraction Y α of all components α = 1, N Temperature T Balance equation for D i : mass diffusivity, thermal diffusivity S i : mass/temperature source term 3

4 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

5 Moment Methods for Reactive Scalars Assumption: heat release expressed by B: includes frequency factor und heat of reaction T b : adiabatic flame temperature E: activation energy Approach for modeling the chemical source term Proven method decomposition into mean and fluctuation 5

6 Moment Methods for Reactive Scalars Taylor expansion at (for ) of terms Pre-exponential term Exponential term Leads to 6

7 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

8 Example: Non-Premixed Combustion in Isotropic Turbulence Favre averaged transport equation Gradient transport model One step global reaction Decaying isotropic turbulence 8

9 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

10 Course Overview Part II: Turbulent Combustion Turbulence Turbulent Premixed Combustion Turbulent Non-Premixed Combustion Modeling Turbulent Combustion Applications 10 Moment Methods for reactive scalars Simple Models in Fluent: EBU,EDM, FRCM, EDM/FRCM Introduction in Statistical Methods: PDF, CDF, Transported PDF Model Modeling Turbulent Premixed Combustion BML-Model Level Set Approach/G-equation Modeling Turbulent Non-Premixed Combustion Conserved Scalar Based Models for Non-Premixed Turbulent Combustion Flamelet-Model Application: RIF, steady flamelet model

11 Simple Models for Turbulent Combustion Example: standard models in Fluent Very simple models, e.g. based on very fast chemistry no consideration of turbulence Quelle: Fluent 12 user s guide 11

12 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

13 1. Eddy-Break-Up-Modell Averaged turbulent reaction rate for the products : variance of mass fraction of the product C EBU : 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-Dissipation- Model (EDM) 13

14 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 Y E, Y P : mass fraction of reactant/product A, B: Model parameter (determined by experiment) 14

15 2. Eddy-Dissipation-Model Example: diffusion flame, one step reaction Y F > Y F,st, therefore Y O < Y F Y E = Y O Y F < Y F,st Y E = Y F 15

16 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

17 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 k f,α, k b,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

18 Résumé FRCM Chemistry-controled Appropriate for t chemistry > t mixng (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

19 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

20 5. Eddy-Dissipation-Concept (EDC) Extension of EDM Considers detailed reaction kinetics Assumption: Reactions on small scales ( * : fine scale) Fluent: C ξ = 2,1377 Volume of small scales: Reaction rates are determined by Arrhenius expression (cf. FRCM) Time scale of the reactions Fluent: C τ = 0,

21 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 Mass fraction on small scales of species i after reaction time τ* 21

22 Résumé: Simple Combustion Models Solely calculation by Arrhenius equation turbulence is not considered Quelle: Fluent 12 user s guide Calculation of Arrhenius reaction rate and mixing rate; selection of the smaller one local choice: laminar/turbulent Solely calculation of mixing rate Chemical kinetic is not considered Modeling of turbulence/chemistry interaction; detailed chemistry 22

23 Course Overview Part II: Turbulent Combustion Turbulence Turbulent Premixed Combustion Turbulent Non-Premixed Combustion Modeling Turbulent Combustion Applications 23 Moment Methods for reactive scalars Simple Models in Fluent: EBU,EDM, FRCM, EDM/FRCM Introduction in Statistical Methods: PDF, CDF, Transported PDF Model Modeling Turbulent Premixed Combustion BML-Model Level Set Approach/G-equation Modeling Turbulent Non-Premixed Combustion Conserved Scalar Based Models for Non-Premixed Turbulent Combustion Flamelet-Model Application: RIF, steady flamelet model

24 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 Pope, Turbulent Flows 24

25 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

26 Probability Probability of the event Probability p impossible event sure event 26

27 Cumulative Distribution Function (CDF) Probability of any event can be determined from cumulative distribution function (CDF) Event A Event B 27

28 Cumulative Distribution Function (CDF) Three basic properties of a CDF 1. Occuring of event is impossible 2. Occuring of event is sure 3. F is a non-decreasing function as CDF of Gaussian distributed random variable 28

29 Probability Density Function (PDF) Derivative of the CDF probability density function Three basic properties of a PDF 1. CDF non-decreasing PDF 2. Satisfies the normalization condition 3. For infinite sample space variable PDF of Gaussian distributed random variable 29

30 Probability Density Function (PDF) Examining the particular interval V a U < V b Interval V b - V a 0: 30

31 Example for CDF/PDF Uniform distribution Source: Pope, Turbulent Flows 31

32 Example for CDF/PDF Exponential distribution Source: Pope, Turbulent Flows 32

33 Example for CDF/PDF Normal distribution Source: Pope, Turbulent Flows 33

34 Example for CDF/PDF Delta-function distribution or Source: Pope, Turbulent Flows 34

35 Moments of a PDF PDF of U is known n-th moment Q: random function, with Q = Q(U) Example: first moment (n = 1): mean of U 35

36 Central Moments n-th central moment Example: second central moment (n = 2): variance of U 36

37 Joint Cumulative Density Function Joint CDF (jcdf) of random variables U 1, U 2 (in general U i, i = 1,2, ) Source: Pope, Turbulent Flows 37

38 Joint Cumulative Density Function Basic properties of a jcdf Non-decreasing function Since is impossible Since is certain equally marginal CDF 38

39 Joint Probability Density Function Joint PDF (jpdf) Fundamental property: Source: Pope, Turbulent Flows 39

40 Joint Probability Density Function Basic properties of a jpdf Non-negative: Satisfies the normalization condition Marginal PDF 40

41 Joint Statistics For a function Q(U 1,U 2, ) Example: i = 1, 2; n = 1;, covariance of U 1 and U 2 Scatterplot of two velocitycomponents U 1 and U 2 Covariance shows the correlation of two variables 41

42 Conditional PDF PDF of U 2 conditioned on U 1 = V 1 Bayes-Theorem jpdf f 1,2 (V 1,V 2 ) scaled so that it satisfies the normalization condition Conditional mean of a function Q(U 1,U 2 ) 42

43 Statistical Independence If U 1 and U 2 are statistically independent, conditioning has no effect Bayes-Theorem Therefore: Independent variables uncorrelated In general the converse is not true 43

44 Course Overview Part II: Turbulent Combustion Turbulence Turbulent Premixed Combustion Turbulent Non-Premixed Combustion Modeling Turbulent Combustion Applications 44 Moment Methods for reactive scalars Simple Models in Fluent: EBU,EDM, FRCM, EDM/FRCM Introduction in Statistical Methods: PDF, CDF, Transported PDF Model Modeling Turbulent Premixed Combustion BML-Model Level Set Approach/G-equation Modeling Turbulent Non-Premixed Combustion Conserved Scalar Based Models for Non-Premixed Turbulent Combustion Flamelet-Model Application: RIF, steady flamelet model

45 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

46 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

47 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

48 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

49 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

50 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 50 This term represents molecular mixing

51 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

52 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

53 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 N 1/2 - Slow convergence 53

54 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

55 Application TPDF Model in LES of Turbulent Jet Flames Flame D: Temperature Flame E: Temperature Flame E: Dissipation Rate 55

56 Application TPDF Model in LES of Turbulent Jet Flames 56

57 Course Overview Part II: Turbulent Combustion Turbulence Turbulent Premixed Combustion Turbulent Non-Premixed Combustion Modeling Turbulent Combustion Applications 57 Moment Methods for reactive scalars Simple Models in Fluent: EBU,EDM, FRCM, EDM/FRCM Introduction in Statistical Methods: PDF, CDF, Transported PDF Model Modeling Turbulent Premixed Combustion BML-Model Level Set Approach/G-equation Modeling Turbulent Non-Premixed Combustion Conserved Scalar Based Models for Non-Premixed Turbulent Combustion Flamelet-Model Application: RIF, steady flamelet model

58 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

59 Bray-Moss-Libby-Model Assumption: very fast chemistry, flame size l F << η << l t 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

60 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

61 Bray-Moss-Libby-Model instantaneous flame front mean flame front unburnt burnt 61

62 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 delta function 62

63 BML-closure of Turbulent Transport With follows 63

64 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 Within the flame zone c conflict Gradient transport assumption would be u u u b ρ u u u = ρ b u b Conflict: countergradient diffusion 64

65 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) I 0 : strain factor local increase of burning velocity by strain Flame-surface-density Σ e.g. algebraic model: Or transport equation for Σ Flame crossing length 65

66 BML-closure of Chemical Source Term Transport equation for Σ local change convectiv change turbulent transport production due to stretching of the flame flameannihilation 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 s L 66

67 Course Overview Part II: Turbulent Combustion Turbulence Turbulent Premixed Combustion Turbulent Non-Premixed Combustion Modeling Turbulent Combustion Applications 67 Moment Methods for reactive scalars Simple Models in Fluent: EBU,EDM, FRCM, EDM/FRCM Introduction in Statistical Methods: PDF, CDF, Transported PDF Model Modeling Turbulent Premixed Combustion BML-Model Level Set Approach/G-equation Modeling Turbulent Non-Premixed Combustion Conserved Scalar Based Models for Non-Premixed Turbulent Combustion Flamelet-Model Application: RIF, steady flamelet model

68 Level-Set-Approach Kinematics of the flame front by examining the movement of single flame front- particles Movement influenced by Local flow velocity u i, i = 1,2,3 Burning velocity s L instantaneous flame front particle normal vector 68

69 G-Equation Instead of observing a lot of particles examination of a scalar field G Iso-surface G 0 is defined as the flame front Substantial derivative of G (on the flame front) unburnt burnt 69

70 Example: Level-Set-Method 70

71 G-Equation for Premixed Combustion Kinematics normal vector and lead to unburnt burnt G-Equation for premixed combustion 71

72 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 >> l F >> l δ ) unburnt burnt 72

73 G-Equation in the Regime of Corrugated Flamelets Kinematic equation f(ρ) Valid for flame position: G = G 0 (= 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 burnt 73

74 G-Equation in the Regime of Corrugated Flamelets Influence of chemistry by s L s L not necessarily constant, influenced by strain S curvature κ Lewis number effect Modified laminar burning velocity unburnt burnt 74

75 Laminar Burning Velocity: Curvature influence of curvature uncorrected laminar burning velocity unburnt G < 0 burnt G > 0 75

76 Laminar Burning Velocity: Markstein Length uncorrected laminar burning velocity Markstein length Determined by experiment Or by asymptotic analysis density ratio Zeldovich number Lewis number 76

77 Extended G-Equation influence of curvature uncorrected laminar burning velocity influence of strain Markstein length Extended G-Equation 77

78 G-Equation: Corrugated Flamelets/Thin Reaction Zones Previous examinations limited to the regime of corrugated flamelets Thin flame structures (η >> l F >> l δ ) Laminar burning velocity well-defined Regime of thin reaction zones Small scale eddies penetrate the preheating zone Transient flow Burning velocity not well-defined no longer valid Problem: Level-Set-Approach valid in the regime of thin reaction zones? 78

79 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, l δ << η Described by T(x i,t) = T 0 Temperature equation Iso temperature surface T(x i,t) = T 0 79

80 G-Equation: Regime of Thin Reaction Zones Equation of motion of the iso temperature surface T(x i,t) = T 0 With the displacement speed s d Normal vector 80

81 G-Equation: Regime of Thin Reaction Zones With G 0 = T 0 Diffusion term normal diffusion (~s n ) and curvature term (~κ) G-equation for the regime of thin reaction zones 81

82 Common Level Set Equation for Both Regimes Normalize G-equation with Kolmogorov scales (η, τ η, u η ) leads to 82

83 Order of Magnitude Analysis Non dimensional Derivatives, u i *, κ* O(1) O(Ka -1/2 ) O(1) Typical flame Sc = ν/d 1 D/ν = O(1) Parameter: s L /u η Ka = u η2 /s L 2 s L /u η = Ka -1/2 s L,s s L 83

84 G-Equation for both Regimes O(Ka -1/2 ) O(1) Thin reaction zones: Ka >> 1 curvature term is dominant Corrugated flamelets: Ka << 1 s L term is dominant Leading order equation in both regimes Assumption: const. const. 84

85 Statistical Description of Turbulent Flame Front Probability density function of finding G(x i,t) = G 0 = 0 85 Experimental determination in weak swirl burner

86 Statistical Description of Turbulent Flame Front Consider steady one-dimensional premixed turbulent mean flame at position x f Define flame brush thickness l f from f(x) If G is distance function then 86

87 Favre-Mean- and Variance-Equation Equation for Favre-mean instantaneous flame front Equation for variance averaged flame front can be interpreted as the area ratio of the flame A T /A Variance describes the average size of the flame averaged temperature profile instantaneous temperature profile 87

88 Modeling of the Variance Equation Sink terms in the variance equation Kinematic restoration Scalar dissipation are modeled by 88

89 G-Equation for Turbulent Flows Introducing turbulent burning velocity Equation for Favre mean Equation for variance 89

90 G-Equation for Turbulent Flows Modeling of turbulent burning velocity by Damköhler theory 90

91 G-Equation for Turbulent Flows Favre mean of G instantaneous flame front Favre-PDF averaged flame front Mean temperature (or other scalar) averaged temperature profile T(G)=T(x) taken from laminar premixed flame without strain instantaneous temperature profile 91

92 Example: Presumed Shape PDF Approach (RANS) experiment computed numerically 92

93 Example: LES of a Premixed Turbulent Bunsen Flame Premixed methan/air flame Re = Broad, low velocity pilot flame heat losses to burner Dilution by air co-flow temperature axial velocity 93

94 Time-Averaged Temperature and Axial Velocity at position x/d = 2.5 temperature axial velocity 94

95 Time-Averaged Temperature and Axial Velocity at position x/d = 6.5 temperature axial velocity 95

96 Turbulent Kinetic Energie at Position x/d = 2.5 and 6.5 x/d = 2,5 x/d = 6,5 96

97 LES Regime Diagram for Premixed Turbulent Combustion 97

98 Course Overview Part II: Turbulent Combustion Turbulence Turbulent Premixed Combustion Turbulent Non-Premixed Combustion Modeling Turbulent Combustion Applications 98 Moment Methods for reactive scalars Simple Models in Fluent: EBU,EDM, FRCM, EDM/FRCM Introduction in Statistical Methods: PDF, CDF, Transported PDF Model Modeling Turbulent Premixed Combustion BML-Model Level Set Approach/G-equation Modeling Turbulent Non-Premixed Combustion Conserved Scalar Based Models for Non-Premixed Turbulent Combustion Flamelet-Model Application: RIF, steady flamelet model

99 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: m 2 /m B Mass of air per fuel mass at stoichiometric conditions 99

100 Mixture Fraction Z Stoichiometric Conditions (λ=1) Stoichiometric mixture fraction Relation between Z and λ Z: normalized local λ λ = 0 Z = 1 λ = 1 Z = Z st λ = Z = 0 100

101 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: Y i (Z) and T(Z) 101

102 Presumed PDF Approach Equation for the mean and the variance of Z are known and closed 102

103 Presumed PDF Approach b-function pdf for mixture fraction Z With 103

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) 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

105 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

106 Burke-Schumann Solution 106

107 Course Overview Part II: Turbulent Combustion Turbulence Turbulent Premixed Combustion Turbulent Non-Premixed Combustion Modeling Turbulent Combustion Applications 107 Moment Methods for reactive scalars Simple Models in Fluent: EBU,EDM, FRCM, EDM/FRCM Introduction in Statistical Methods: PDF, CDF, Transported PDF Model Modeling Turbulent Premixed Combustion BML-Model Level Set Approach/G-equation Modeling Turbulent Non-Premixed Combustion Conserved Scalar Based Models for Non-Premixed Turbulent Combustion Flamelet-Model Application: RIF, steady flamelet model

108 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

109 Flamelet Model for Non-Premixed Turbulent Combustion Balance equations for temperature, species and mixture fraction With it follows 109

110 Flamelet Equations Consider surface of stoichiometric mixture Reaction zone confined to thin layer around this surface Transformation to surface attached coordinate system x 1, x 2, x 3, t Z(x 1, x 2, x 3, t), Z 2, Z 3, τ x 1, Z x 3, Z 3 x 2, Z 2 110

111 Transformation rules Transformation: x 1, x 2, x 3, t Z(x 1, x 2, x 3, t), Z 2, Z 3, τ (where Z 2 = x 2, Z 3 = x 3, τ = t) Example: Temperature T Analogous for x 3 111

112 Flamelet Equations Temperature equation Transformed temperature equation: 112

113 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

114 Example Example from DNS of Non-Premixed Combustion in Isotropic Turbulence Temperature (color) Stoichiometric mixture fraction (line) 114

115 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 [χ st ] = 1/s: may be interpreted as the inverse of a characteristic diffusion time 115

116 Temperature profiles for methane-air flames Temperature profiles for methane-air flames 116

117 Flamelet Equations Asymptotic analysis by Seshadri (1988) Based on four-step model Close correspondence between layers identified in premixed diffusion flames 117

118 Flamelet Equations The calculations agree well with numerical and experimental data They also show the vertical slope of T 0 versus χ st which corresponds to extinction 118

119 Flamelet Equations Steady state flamelet equations provide ψ i = f(z,χ st ) If joint pdf is known Favre mean of ψ i : If the unsteady term in the flamelet equation must be retained, joint statistics of Z and χ st 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

120 Flamelet Equations Modeling the conditional Favre mean scalar dissipation rate Flamelet equations Favre mean 120

121 Flamelet Equations Model for conditional scalar dissipation rate One relates the conditional scalar dissipation rate to that at a fixed value Z st by With 121

122 n-heptane Air Ignition The initial air temperature is 1100 K and the initial fuel temperature is 400 K. 122

123 Representative-Interactive-Flamelet-Modell (RIF) 123

124 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 No x as function of EGR 124

125 Example: Diesel engine simulation RIF-Temperature T [K] Mischungsbruch Kurbelwinkel [ not] Temperatur [K] 125

126 Example: Diesel engine simulation Mischungsbruchverteilung Schadstoffbildung 126

127 Example: Diesel engine simulation Comparison with Magnussen-/ Hiroyasu-Model 127

128 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 c st Tabulate in terms of c st or progress variable C, e.g. C = Y CO2 + Y HO2 + Y CO + Y H2 Presumed PDF, typically beta function for Z, delta function for dissipation rate or reaction progress parameter 128

129 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 129 Exp. by Masri et al.

130 Example: LES of a Bluff-Body Stabilized Flame 130

131 Example: LES of a Bluff-Body Stabilized Flame Temperature CO Mass Fraction 131

132 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 132 Exp. by Barlow et al.

133 Flamelet Model Application to Sandia Jet Flames Flame D: Temperature Flame E: Temperature 133

134 Flamelet Model Application to Sandia Jet Flames Flame D Flame E Flame E 134 Flame D

135 Summary Part II: Turbulent Combustion Turbulence Turbulent Premixed Combustion Turbulent Non-Premixed Combustion Modeling Turbulent Combustion Applications 135 Moment Methods for reactive scalars Simple Models in Fluent: EBU,EDM, FRCM, EDM/FRCM Introduction in Statistical Methods: PDF, CDF, Transported PDF Model Modeling Turbulent Premixed Combustion BML-Model Level Set Approach/G-equation Modeling Turbulent Non-Premixed Combustion Conserved Scalar Based Models for Non-Premixed Turbulent Combustion Flamelet-Model Application: RIF, steady flamelet model