Download Lecture 10 - Modeling Turbulent Combustion

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=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
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