Modeling Turbulent Combustion

Modeling Turbulent Combustion

CEFRC Combustion Summer School

Prof. Dr.-Ing. Heinz Pitsch

2014

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

2

• Turbulence

• Turbulent Premixed Combustion

• Turbulent Non-Premixed

Combustion

• Modeling Turbulent Combustion

• Applications

• 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

Part II: Turbulent Combustion

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

→ idea: approach similar to (simple) turbulence models: expression as a function of mean values

4

not closed


• 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


• Taylor expansion at (for ) of terms

Pre-exponential term

Exponential term

• Leads to

6


• 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

9

Product Mass Fraction

DNS data

Flamelet Closure Assumption

Evaluation of chemical source term with mean quantities

Closure by mean values does not work!

Course Overview

10

• Turbulence



Combustion


• Applications



EDM/FRCM


CDF,…



• BML-Model



Combustion



• Flamelet-Model



Simple Models for Turbulent Combustion

• Example: standard models in Fluent

• Very simple models, e.g. based on

very fast chemistry

no consideration of turbulence

11

Quelle: Fluent 12 user‘s guide

3. 4. 2. 5.

1. Eddy-Break-Up-Model

First approach for closing the chemical source term was made by Spalding (1971) in premixed combustion

• 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

flow

hot burnt gas unburnt mixture

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-Dissipation-Model (EDM)…

13

2. Eddy-Dissipation-Model

14

• 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)

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)

• Volume of small scales:

• Reaction rates are determined by Arrhenius expression (cf. FRCM)

• Time scale of the reactions

20

Fluent: Cξ = 2,1377

Fluent: Cτ = 0,4082

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

22

Quelle: Fluent 12 user‘s guide

3. 4. 2. 5.

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

Solely calculation of mixing rate Chemical kinetic is not considered

Modeling of turbulence/chemistry interaction; detailed chemistry

Course Overview

23

• Turbulence



Combustion


• Applications



EDM/FRCM


CDF,…



• BML-Model



Combustion



• 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“

• 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

Sample Space

25

Probability

• Probability of the event

• Probability p

26

impossible event 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. Occuring of event is impossible

2. Occuring of event is sure

3. F is a non-decreasing function as

28

CDF of Gaussian distributed random variable

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

29

PDF of Gaussian distributed random variable

Probability Density Function (PDF)

• Examining the particular interval Va ≤ U < Vb

• Interval Vb - Va 0:

30

Example for CDF/PDF

Uniform distribution

31

Source: Pope, „Turbulent Flows“

Example for CDF/PDF

32


Exponential distribution

Example for CDF/PDF

33


Normal distribution

Example for CDF/PDF

34


or

Delta-function distribution

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,…)

37


Joint Cumulative Density Function

• Basic properties of a jCDF

Non-decreasing function

Since is impossible

Since is certain equally

38

marginal CDF

Joint Probability Density Function

• Joint PDF (jPDF)

• Fundamental property:

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

• Covariance shows the correlation of two variables

41

Scatterplot of two velocity- components U1 and U2

Conditional PDF

• PDF of U2 conditioned on U1 = V1

• jPDF f1,2(V1,V2) scaled so that it satisfies the normalization condition

• Conditional mean of a function Q(U1,U2)

42

Bayes-Theorem

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

44

• Turbulence



Combustion


• Applications



EDM/FRCM


CDF,…



• BML-Model



Combustion



• Flamelet-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)

The PDF Transport Equation Model

45

• 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

The PDF Transport Equation Model

46

• 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

PDF Transport Equation: Closure Problem

47

• 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

• 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

• 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

• 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

• 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

PDF Transport Equation: Solution

52

• 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

PDF Transport Equation: Solution

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


55

Flame D: Temperature

Flame E: Temperature

Flame E: Dissipation Rate


56

Course Overview

57

• Turbulence



Combustion


• Applications



EDM/FRCM


CDF,…



• BML-Model



Combustion



• Flamelet-Model



Bray-Moss-Libby-Model

• Flamelet concept for premixed turbulent combustion: Bray-Moss-Libby-Modell (BML)

• Premixed combustion: progress variable c, e.g.

• Favre averaged transport equation (neglecting the molecular transport)

• Closure for turbulent transport and chemical source term by BML-Model

58

not closed

or


• Assumption: very fast chemistry, flame size lF << η << lt

• 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

burnt burnt

unburnt

• 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

instantaneous flame front

mean flame front

„unburnt“ „burnt“

BML-closure of Turbulent Transport

• For a Favre average

• Therefore the the unclosed correlation

joint PDF for u and c

Introducing the BML approach for f(c) leads to

62

conditional PDF delta function

(Bayes-Theorem)

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

• Within the flame zone

• Gradient transport assumption would be

• Conflict: „countergradient diffusion“

64

Flame front

uu ub

ρuuu = ρbub

c

conflict

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

I0: strain factor local increase of burning velocity by strain

• Flame-surface-density Σ

e.g. algebraic model:

Or transport equation for Σ

65

local mass conversion per area

Flächen-Dichte (flamen area per volume)

Flame crossing length

BML-closure of Chemical Source Term

• Transport equation for Σ

• 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

local change

convectiv change

turbulent transport

production due to stretching of the flame

flame- annihilation

Course Overview

67

• Turbulence



Combustion


• Applications



EDM/FRCM


CDF,…



• BML-Model



Combustion



• Flamelet-Model



Level-Set-Approach

• Kinematics of the flame front by examining the movement of single flame front-„particles“

• Movement influenced by

Local flow velocity ui, i = 1,2,3

Burning velocity sL

68

normal vector


particle

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)

69

unburnt burnt

Example: Level-Set-Method

70

G-Equation for Premixed Combustion

• Kinematics and lead to

→ G-Equation for premixed combustion

71

normal vector

unburnt burnt

G-Equation in the Regime of Corrugated Flamelets

• No diffusive term

• Can be applied for

Thin flames

Well-defined burning velocity

→ Regime of corrugated flamelets (η >> lF >> lδ)

72

local change

convective change

progress of flame front by burning velocity

unburnt burnt


• 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

73

unburnt burnt


• Influence of chemistry by sL

• sL not necessarily constant, influenced by strain S

curvature κ

Lewis number effect

• Modified laminar burning velocity

74

unburnt burnt

Laminar Burning Velocity: Curvature

75

influence of curvature

uncorrected laminar burning velocity

unburnt G < 0

burnt G > 0

Laminar Burning Velocity: Markstein Length

• Markstein length

Determined by experiment

Or by asymptotic analysis

76

density ratio

Zeldovich number Lewis number


Extended G-Equation

→ Extended G-Equation

77

influence of strain

influence of curvature

Markstein length


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

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

no longer valid

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


• Equation of motion of the iso temperature surface T(xi,t) = T0

With the displacement speed sd

Normal vector

80


• 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

• Non dimensional

→ Derivatives, ui*, κ* ≈ O(1)

• Typical flame

→ Sc = ν/D ≈ 1 D/ν = O(1)

• Parameter: sL/uη

Ka = uη2/sL

2 sL/uη = Ka-1/2

sL,s ≈ sL

83

O(Ka-1/2) O(1)

G-Equation for both Regimes

• Thin reaction zones: Ka >> 1 curvature term is dominant

• Corrugated flamelets: Ka << 1 sL term is dominant

• Leading order equation in both regimes

84

O(Ka-1/2) O(1)

Assumption: const. const.

Statistical Description of Turbulent Flame Front

• Probability density function of finding G(xi,t) = G0 = 0

85

Experimental determination in weak swirl burner

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

• Equation for variance

• can be interpreted as the area ratio of the flame AT/A

• Variance describes the average size of the flame

87


averaged flame front

averaged temperature profile

instantaneous temperature profile

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


• Modeling of turbulent burning velocity by Damköhler theory

90


• Favre mean of G

• Favre-PDF

• Mean temperature (or other scalar)

91

T(G)=T(x) taken from laminar premixed flame without strain


averaged flame front

averaged temperature profile

instantaneous temperature profile

Example: Presumed Shape PDF Approach (RANS)

92

experiment

computed numerically

• Premixed methan/air flame

• Re = 23486

• Broad, low velocity pilot flame heat losses to burner

• Dilution by air co-flow

93

Example: LES of a Premixed Turbulent Bunsen Flame

temperature

axial velocity

Time-Averaged Temperature and Axial Velocity at position x/D = 2.5

94

temperature axial velocity

Time-Averaged Temperature and Axial Velocity at position x/D = 6.5

95

temperature axial velocity

Turbulent Kinetic Energie at Position x/D = 2.5 and 6.5

96

x/D = 2,5 x/D = 6,5

LES Regime Diagram for Premixed Turbulent Combustion

97

Course Overview

98

• Turbulence



Combustion


• Applications



EDM/FRCM


CDF,…



• BML-Model



Combustion



• Flamelet-Model



Conserved Scalar Based Models for Non-Premixed Combustion

Mixture fraction Z

• Definition of Z

• With the dimensionless quantity

99

Mass of air per fuel mass: m2/mB

Mass of air per fuel mass at stoichiometric conditions

1

Coupling function:

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 = 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

107

• Turbulence



Combustion


• Applications



EDM/FRCM


CDF,…



• BML-Model



Combustion



• 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

x1, Z

x2, Z2

x3, Z3

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

111

0 0 1

0 0 0

0 0

Analogous for x3

1

Flamelet Equations

• Temperature equation Transformed temperature equation:

112

• 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

Flamelet Equations

113

small

Local change Describes mixing

Source term

Example

• Example from DNS of Non-Premixed Combustion in Isotropic Turbulence

114

• Temperature (color)

• Stoichiometric mixture fraction (line)

Flamelet Equations

• Same procedure for the mass fraction…

• Flamelet structure is to leading order described by the one-dimensional time-dependent 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


• RIF-Temperature

125

0.00.2

0.40.6

0.81.0

Mischungsbruch -10

0

10

20

Kurbelwinkel [˚nOT]

300

600

900

1200

1500

1800

2100

2400

2700

T [K]

300 600 900 1200 1500 1800 2100 2400 2700

Temperatur [K]


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Mischungsbruch- verteilung

Schadstoff- bildung


• Comparison with Magnussen-/ Hiroyasu-Model

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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

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• 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

Example: LES of a Bluff-Body Stabilized Flame

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Exp. by Masri et al.


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Temperature CO Mass Fraction


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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

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Exp. by Barlow et al.


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Flame D: Temperature

Flame E: Temperature


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Flame E

Flame D

Flame E

Flame D

Summary

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• Turbulence



Combustion


• Applications



EDM/FRCM


CDF,…



• BML-Model



Combustion



• Flamelet-Model



Modeling Turbulent Combustion

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